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+# REWARD DESIGN WITH LANGUAGE MODELS
+
+Minae Kwon, Sang Michael Xie, Kalesha Bullard†, Dorsa Sadigh Stanford University, DeepMind† {minae, xie, dorsa}@cs.stanford.edu, ksbullard@deepmind.com†
+
+# ABSTRACT
+
+Reward design in reinforcement learning (RL) is challenging since specifying human notions of desired behavior may be difficult via reward functions or require many expert demonstrations. Can we instead cheaply design rewards using a natural language interface? This paper explores how to simplify reward design by prompting a large language model (LLM) such as GPT-3 as a proxy reward function, where the user provides a textual prompt containing a few examples (few-shot) or a description (zero-shot) of the desired behavior. Our approach leverages this proxy reward function in an RL framework. Specifically, users specify a prompt once at the beginning of training. During training, the LLM evaluates an RL agent’s behavior against the desired behavior described by the prompt and outputs a corresponding reward signal. The RL agent then uses this reward to update its behavior. We evaluate whether our approach can train agents aligned with user objectives in the Ultimatum Game, matrix games, and the DEALORNODEAL negotiation task. In all three tasks, we show that RL agents trained with our framework are well-aligned with the user’s objectives and outperform RL agents trained with reward functions learned via supervised learning. Code and prompts can be found here.
+
+# 1 INTRODUCTION
+
+Autonomous agents are becoming increasingly capable with the rise of compute and data. This underscores the importance for human users to be able to control what policies the agents learn and ensure the policies are aligned with their objectives. For instance, imagine training an agent to represent users in a salary negotiation. A working mother fighting for a livable wage may want their agent to be stubborn whereas a new hire looking to develop a good relationship with the company may want their agent to be more versatile.
+
+Currently, users specify desired behaviors by 1) designing reward functions or 2) providing large amounts of labeled data. Both approaches are challenging and impractical for different reasons. Designing reward functions is not an intuitive way to specify preferences. For instance, it isn’t straightforward how to write a reward function for a “versatile” negotiator. Furthermore, designing reward functions that balance between different objectives — also known as the “reward design problem” — is notoriously difficult because agents are susceptible to reward hacking (Amodei et al., 2016; Hadfield-Menell et al., 2017). On the other hand, one can learn a reward function from labeled examples. However, that is not possible with a single example; we need large amounts of labeled data to capture the nuances of different users’ preferences and objectives, which has shown to be costly (Zhang et al., 2016). Additionally, both approaches do not generalize well to new users who have different objectives — we would have to re-design our reward functions or re-collect data.
+
+Our aim is to create an easier way for users to communicate their preferences, where the interface is more intuitive than crafting a reward function and where they can cheaply specify their preferences with no more than a few examples. To do this, we leverage large language models (LLMs) that are trained on internet-scale text data and have shown an impressive ability to learn in-context from few or zero examples (Brown et al., 2020). Our key insight is that
+
+The scale of data that LLMs have been trained on make them great in-context learners and also allows them to capture meaningful commonsense priors about human behavior. Given a few examples or a description demonstrating the user’s objective, an LLM should be able to provide an accurate instantiation of reward values on a new test example, allowing for easier generalization to new objectives.
+
+To this end, we explore how to prompt an LLM as a proxy reward function to train RL agents from user inputs. In our approach, the user specifies an objective with a natural language prompt. Objectives can be specified with a few examples when they are difficult to define (such as “versatility”) or as a single phrase when they are well-known concepts (such as “Pareto-optimality”). We use the prompt and the LLM to define a reward function for training an RL agent. The LLM takes the user prompt and a trajectory from an RL episode as input and outputs a score (e.g., “No” or $\ " { } 0 \ " { }$ ) for whether the trajectory satisfies the user’s objective, which we parse as an integer reward for the RL agent (Figure 1).
+
+![](images/4d357745dfacf87e4d77f974ecc34842b5bd5df46781e597ec75dd1e0114449e.jpg)  
+Figure 1: Depiction of our framework on the DEALORNODEAL negotiation task. A user provides an example and explanation of desired negotiating behavior (e.g., versatility) before training. During training, (1) we provide the LLM with a task description, a user��s description of their objective, an outcome of an episode that is converted to a string, and a question asking if the outcome episode satisfies the user objective. (2-3) We then parse the LLM’s response back into a string and use that as the reward signal for the Alice the RL agent. (4) Alice updates their weights and rolls out a new episode. (5) We parse the episode outcome int a string and continue training. During evaluation, we sample a trajectory from Alice and evaluate whether it is aligned with the user’s objective.
+
+There are two advantages to prompting LLMs as a proxy reward function: (1) we can leverage LLM’s in-context learning abilities and prior knowledge on human behavior so that users only need to provide a handful of example desirable behaviors and (2) users can specify their preferences intuitively using language. On the other hand, a potential disadvantage is that it is unclear how much prompt design will be required for the LLM to reliably infer user intent (see Sec. 5 for a discussion). The goal of this paper is to explore how well LLMs can train objective-aligned agents by providing reward signals, and empirically examine whether we can do so with no more than a few examples. Our contributions are as follows:
+
+• We introduce the idea of using LLMs as a proxy reward function.   
+• We propose a general RL training framework that leverages this proxy reward and is agnostic to the RL algorithm used.   
+• We show that an LLM can more accurately train objective-aligned RL agents by an average of $3 5 \%$ compared the baseline. We use few-shot prompting for the Ultimatum Game and DEALORNODEAL negotiation task as well as zero-shot prompting in Matrix Games.   
+• We conduct a pilot study with 10 human users. Users rate our agent to be significantly more aligned with their objective than an agent trained with a different one, $p { < } 0 . 0 0 1$ .   
+• We provide further analysis quantifying the amount of user data required for our approach as well as the effect varying prompts has on the LLM’s reward signal accuracy.
+
+# 2 RELATED WORK
+
+Using Language for Reward Shaping. Recent Reinforcement Learning from Human Feedback (RLHF) works Ouyang et al. (2022); Bai et al. (2022) use LLMs as rewards by fine-tuning them on large amounts of user data. Our work does not fine-tune LLMs but uses in-context learning from only a handful of user data.
+
+Several works Goyal et al. (2019); Carta et al. (2022); Mirchandani et al. (2021) shape rewards by training an RL agent to learn and complete intermediate tasks guided by language. In contrast, our framework does not focus on generating subtasks but leverages the in-context learning abilities of an LLM to determine whether an agent’s policy satisfies the higher-level task.
+
+RL and Foundation Models. We leverage large language models such as GPT-3 (Brown et al., 2020) to learn a proxy reward function while avoiding the need for many expert demonstrations. Ahn et al. (2022); Huang et al. (2022) use an LLM to provide a plan which guides a robot with reasonable/feasible actions towards a human goal (e.g., with enumerating subtasks). In contrast, our work is different in that we are using an LLM to identify if a behavior satisfies “hard-to-specify” properties of a human’s objective and also offers users more control over how they want their policy to be executed. In the vision domain, Parisi et al. (2022) used pre-trained vision models as a feature extractor for the learned policy, but not to design a reward signal. In a similar spirit of leveraging self-supervised pre-training to design a flexible reward function, Chen et al. (2021) use a broad dataset of human videos and a small dataset of robot videos to train a reward function, which improves generalization to new environments and tasks. The interface for the desired task is a video of the task to be completed, instead of text in our framework, and the domain is restricted to robot tasks. More related works can be found in Sec. A.2.
+
+# 3 USING LLMS AS A REWARD SIGNAL
+
+Our goal is to use an LLM as a proxy reward function to train objective-aligned RL agents from user inputs. We formalize the task using a Markov Decision Process $\mathcal { M } { = } \langle S , \mathcal { A } , p , \mathcal { R } , \gamma \rangle$ , where $s$ is the state space (e.g., in DEALORNODEAL, the space of representations of utterances in the negotiation so far), $\mathcal { A }$ is the action space (e.g., set of all possible utterances), $p { : } S \times \mathcal { A } \times \mathcal { S }  [ 0 , 1 ]$ is the transition probability, and $\gamma$ is the discount factor. Traditionally the reward function maps states and actions to a real number $\mathcal { R } : \mathcal { S } \times \mathcal { A }  \mathbb { R }$ In our work, we use an LLM as a proxy reward function that takes in a text prompt and outputs a string. We define $A ^ { * }$ to be the set of all strings, $\rho \in A ^ { * }$ as our text prompt (input to the LLM) and the LLM as a function $L L M { \mathrel { : } } A ^ { * } { \mathrel { \to } } A ^ { * }$ . As illustrated in Fig. 1, the prompt $\rho$ is a concatenation of four components including a string to describe the task $\rho _ { 1 } \in A ^ { * }$ and a user-specified string that describes their objectives using examples or a description, $\rho _ { 2 } \in A ^ { * }$ . Additionally, we include a textual description of states and actions from an RL episode, $\rho _ { 3 }$ , using a parser $f : { \cal S } \times { \cal A }  { \cal A } ^ { * }$ . $\rho _ { 3 }$ can describe the final state, final action, a trajectory, or any other representation of the episode. Finally, we include a question $\rho _ { 4 } \in A ^ { * }$ that asks whether the RL agent’s behavior, $\rho _ { 3 }$ , satisfies the user’s objective, $\rho _ { 2 }$ . We define an additional parser $g \colon A ^ { * } \to \{ 0 , 1 \}$ that maps the textual output of $L L M$ to a binary value. We use this as the reward signal. Our framework replaces the traditional reward function with a proxy reward, $L L M$ , and can be used with any RL training algorithm.
+
+Our framework is depicted in Fig. 1. Before training, a user specifies $\rho _ { 2 }$ which can be $N$ examples describing their objective or a description of their objective using natural language. In Fig. 1 a user provides an example of their objective: versatile negotiating behavior. During training, we construct a prompt $\rho$ by concatenating a description of the task, the user-specified examples/description, an episode’s outcome, and a question asking if the outcome satisfies the objective. We (1) feed the prompt to the LLM, (2) take its output, and (3) parse it into an integer using function $g$ ; we use a handcrafted, task-specific parser. We use the integer as the reward signal. (4) The RL agent then updates its weights and rolls out an episode. (5) We parse the episode outcome into a string using $f$ and continue training; we also instantiate $f$ as handcrafted, task-specific parser. To evaluate our framework, we sample a trajectory (e.g., a negotiation) from the agent and evaluate whether the trajectory is aligned with the user’s objective (e.g., whether Alice demonstrated versatile negotiating behavior).
+
+# 4 EXPERIMENTS
+
+In this section we investigate three questions to determine the feasibility and efficacy of our approach: (Q1) Can LLMs produce reward signals that are consistent with user objectives from a few examples (few-shot prompting)? (Q2) When objectives are well-known, can LLMs produce objective-consistent reward signals without any examples (zero-shot prompting)? (Q3) Can LLMs provide objective-aligned reward signals from examples (few-shot prompting) in more complex, longer-horizon domains? We evaluate our approach on three tasks: the Ultimatum Game, 2-player Matrix Games, and the DEALORNODEAL negotiation task (Lewis et al., 2017). We address (Q1) using the Ultimatum Game. We use Matrix Games to address (Q2) because it has well-known solution concepts such as Pareto-optimality. The
+
+DEALORNODEAL negotiation task is a longer-horizon domain where the LLM rewards agents for negotiating in a user-specified style; we address (Q3) in this task.
+
+In practice, we do not have access to ground truth user reward functions — this is the function that we are trying to approximate. However, for most of our experiments, we assume access to the true reward by constructing user reward functions that humans have been shown to have inspired by prior work. We use the true rewards only to evaluate our framework’s performance. Finally, we include a pilot user study where we evaluate agent performance when we do not have access to the ground truth reward. We use the ‘text-davinci-002’ GPT-3 model with temperature 0 as our LLM and our results are reported across 3 random seeds. Details on how we trained RL agents for each task are in A.4. s
+
+Evaluation Metrics. We evaluate our approach using the following metrics across our tasks (task-specific metrics are described within each subsection):
+
+Labeling Accuracy. We construct ground-truth reward functions for each domain. We report the mean accuracy of predictions of the reward value during RL training with respect to the ground-truth reward functions. This assesses how effectively the LLM can produce reward signals that are consistent with the user’s objective.
+
+RL Agent Accuracy. After RL training, we evaluate the learned policy with respect to the ground truth reward functions. We report the mean accuracy of RL agents.
+
+# Baselines.
+
+SL (Few-shot baseline). A supervised learning (SL) model trained to predict reward signals using the same examples given to the LLM in our framework. Examples are represented using structured non-text inputs, making it an easier problem for the SL model. This baseline only applies to tasks where we use few-shot prompting (Ultimatum Game, DEALORNODEAL). See A.5 for details on training and model architecture for each task.
+
+No Objective (Zero-shot baseline). In our zero-shot task, Matrix Games, we do not use any examples so we do not use SL as a baseline. Instead, we use a No Objective baseline where we prompt the LLM without using the user’s description of their objective to isolate the effect the description has on the LLM’s response.
+
+RL trained with Ground Truth Reward Functions. RL agents trained with ground truth reward functions.   
+We use this as an oracle.
+
+4.1 ULTIMATUM GAME: TRAINING OBJECTIVE-ALIGNED AGENTS WITH FEW-SHOT PROMPTING
+
+When defining a precise objective is difficult, we can instead give a few examples of desired behavior. For instance, in a resource division game like the Ultimatum Game, it may be difficult for a user to specify the exact percentage (such as $3 2 . 4 \%$ ) of resources they would be happy with receiving. Instead it could be easier for a user to give examples of splits that they would be happy with. We explore whether LLMs can produce reward signals that are consistent with user objectives from a few examples in the Ultimatum Game.
+
+Task Description. The Ultimatum Game consists of two players, a Proposer and a Responder. A sum of money is endowed to the Proposer and they must propose a way to split the endowment with the Responder. The Responder can accept or reject the proposed split. If the Responder accepts, players receive money as per the split; if the Responder rejects, then both players get nothing. We train an RL agent to play the Responder. The agent learns to reject proposals according to a user’s preferences. The game consists of a single timestep and our RL agents are trained using DQN for 1e4 steps.
+
+Ground Truth User Objectives. A rational Responder would accept any proposal, even if it is unfair because getting something is better than getting nothing (in fact, this is a Nash Equilibrium of the game). However, prior work in behavioral economics shows that humans are willing to “punish” the Proposer by rejecting unfair proposals (Vavra et al., 2018). For instance, a student may reject an unfair proposal only if she receives less than $30 \%$ of the endowment whereas a wealthier person may reject if they receive less than $60 \%$ . We experiment with the following preferences:
+
+• Low vs High Percentages. Users will reject proposals if they receive less than $( 3 0 \%$ , $6 0 \% \}$ of the endowment.   
+• Low vs High Payoffs. Users will reject unfair proposals if they receive less than $\{ \$ 10,4100 \}$ . They accept unfair proposals otherwise.
+
+![](images/5554a8306018ca0c9c21790adcde59541b3517880405b9f852f19fbc9b381a9b.jpg)  
+Figure 2: Ultimatum Game, Few-shot. (Top) Accuracy of reward signals provided by LLM and SL during RL training when prompted with/trained on 10 vs 1 example. (Bottom) Corresponding accuracy of RL agents after training. LLM is able to maintain a high accuracy when prompted with a single example followed by an explanation. We do not provide figures of Inequity Aversion because both LLM and SL trivially achieve perfect labeling and RL agent accuracy.
+
+• Inequity Aversion (Fehr & Schmidt (2010)). Users will reject proposals if they do not receive exactly $5 0 \%$ of the endowment.
+
+Prompt Design. We describe a user’s objective using 10 examples of the Ultimatum Game. An example consists of the proposed split, the Responder’s action, and a “yes/no” label of whether the Responder’s action was desirable or undesirable. These examples do not have explanations and resemble a traditional dataset used for supervised learning. We also experiment with using a single example followed by a short explanation. Importantly, we do not explicitly mention the user’s ground truth objective in the prompt. See Fig. 10 in the Appendix for an example of both types of prompts.
+
+Design Procedure. We randomly generated 10 proposed splits used for our prompt and sampled one proposal from the set for our single-example case. For Low vs High Percentages and Low vs High Payoffs, we used the same set of proposals across variants (i.e., same proposals for $30 \%$ and $60 \%$ ) and $( \$ 10$ , $\$ 100)$ ). We also randomly generated 50 proposals used to evaluate the LLM. Due to limited resources when querying GPT-3, we query the model’s responses to the 50 evaluation splits in a batched manner and save them. We then use those responses as the reward signal.
+
+# 4.1.1 RESULTS
+
+Labeling Accuracy. We evaluated our approach on our test set of 50 proposals over 3 seeds; results are shown in Fig. $\cdot$ . When prompted with 10 examples without explanations, the LLM and SL perform similarly well (see Fig. 2, top row). This result is not surprising, given that the decision boundary for the binary decision tasks is relatively simple to learn with 10 training examples.
+
+Instead, if we prompt the LLM with a single example followed by an explanation, it maintains a high accuracy whereas SL trained on the same, single example drops in accuracy. We did not use the explanation as part of input when training SL because it only takes non-textual inputs. This result highlights the advantage of using an LLM over a supervised learning model: they require far fewer examples because they can learn from explanations (Lampinen et al., 2022). We find that explanations are critical, as removing explanations when prompting the LLM with a single example results in a drop in LLM labeling accuracy (avg. drop of $3 1 . 6 7 \%$ ) and a drop in RL agent accuracy (avg. drop of $2 8 . 8 \%$ ).
+
+RL Agent Accuracy. The accuracy of the trained RL agents mirror the labeling accuracy.
+
+Summary. LLMs are efficient in-context learners. They are able to provide reward signals that are consistent with a user’s objectives from examples — even a single example with an explanation will suffice.
+
+4.2 MATRIX GAMES: TRAINING OBJECTIVE-ALIGNED AGENTS WITH ZERO-SHOT PROMPTING
+
+When objectives are well-known concepts such as Pareto-optimality, can we prompt the LLM without giving any examples? We hypothesize that well-known objectives are likely to be in-distribution for LLMs, and thus LLMs may be able to produce objective-aligned reward signals from zero-shot prompting. Since we do not use examples, we do not use a SL baseline. Instead we use a baseline No Objective where we do not mention any objectives and ask the LLM for a reward signal (see example in Fig. 11 in the Appendix). This baseline evaluates whether the LLM can successfully apply its knowledge of each objective.
+
+![](images/6f2f78c08eb7a4cd75c7b78807517333e6a2477ab0ce57631d8946cca7d5e2af.jpg)  
+Figure 3: Matrix Games, Zero-shot. (Top) Accuracy of reward signals provided by LLM and a No Objective baseline during RL training. We report results for both regular and scrambled versions of matrix games. (Bottom) Accuracy of RL agents after training.
+
+Task Description. We consider two-player normal-form matrix games: Battle of the Sexes, Stag Hunt, Chicken, and Prisoner’s Dilemma. Each matrix game has four joint outcomes (i.e., a tuple of joint actions and rewards) and we address pure strategies in this task. The game consists of a single timestep and our RL agents are trained using DQN for 500 steps.
+
+Ground Truth User Objectives. Although (mixed) Nash Equilibria are traditional solution concepts for normal form matrix games, users may prefer a solution for other properties. For instance, in Prisoner’s Dilemma, users may prefer both agents to cooperate because they will maximize total welfare even though it is not a Pure Nash Equilibrium. We experiment with four well-known solution concepts (or objectives):
+
+• Total Welfare. Outcomes that achieve the greatest sum of player rewards.   
+• Equality. Outcomes that result in equal rewards between players.   
+• Rawlsian Fairness. Outcomes that maximize the minimum reward any player receives.   
+• Pareto-optimality. Outcomes where the one of the corresponding rewards cannot be improved without lowering the other.
+
+Prompt Design. Prompts for each solution concept are shown in Fig. 11 in the Appendix. Due to limited resources with querying GPT-3, we queried GPT-3 in a batched manner and saved the corresponding labels to train our RL agents. Our prompt enumerates the outcomes of a matrix game and then asks the LLM for the outcome(s) that satisfy a solution concept. We do not mention the name of the matrix game in the prompt. As in Kojima et al. (2022), we elicit intermediate reasoning steps by asking the LLM to “think step-by-step” and provide a definition of the solution concept. To prevent any bias the LLM may have towards the order in which the outcomes of a matrix game are presented, we also randomly scramble associations between joint actions and rewards (example shown in Fig. 12 in the Appendix).
+
+Design Procedure. We tuned the wording of our prompt (e.g., how to describe the matrix game, whether or not to use chain-of-thought prompting) on the Battle of the Sexes matrix game to find a prompt that gave us accurate results. During evaluation, we kept the structure of our prompt the same for all of the matrix games.
+
+# 4.2.1 RESULTS
+
+Labeling Accuracy. Given that each game can have many outcomes that satisfy a solution concept, we report the LLM’s accuracy if its response does not include any incorrect outcomes. If the LLM identifies any incorrect outcome, we report a score of 0. The LLM produces more objective-aligned reward signals with zero-shot prompting by applying its knowledge of well-known objectives, improving the labeling accuracy over having no objective by $48 \%$ on average with a regular ordering of matrix game outcomes and $36 \%$ with a scrambled order. Scrambling the order of matrix game outcomes in the prompt lowers accuracy for most solution concepts. We suspect that this is because the matrix games are well-known and likely to have been in the LLM’s training set, where each joint action is usually associated with particular payoffs. Scrambling the associations between joint actions and payoffs could make the matrix game more out-of-distribution for the LLM, and thus lower accuracy.
+
+RL Agent Accuracy. Compared to labeling accuracy, it is easier for the resulting RL agents to be accurate because they only need to learn one correct outcome, not all of them. Thus, LLMs that only identify one out of two correct outcomes can still train objective-aligned RL agents. Results are shown on the bottom row of Fig. 3. RL agents trained using rewards from the LLM receive perfect accuracy for Total Welfare and Equality and $7 5 \%$ accuracy for the other two objectives. The baseline receives lower accuracy for all objectives.
+
+Does the LLM Correctly Identify Each Objective? As part of our prompt, we ask the LLM to provide a definition of the objective before reasoning about whether an outcome satisfies the objective (see Fig. 11 in the Appendix). Table 2 in the Appendix shows how the LLM defines each objective zero-shot. The LLM is able to successfully recall the definitions for each objective except for Rawlsian Fairness – it gets it partially correct. However, the LLM varies in its ability to reason whether an outcome of a game satisfies the objective, which explains why the LLM does not receive perfect labeling accuracy for all objectives.
+
+Summary. An LLM is able to identify well-known objectives and provide objective-aligned reward signals in a zero-shot setting.
+
+# 4.3 DEALORNODEAL: TRAINING OBJECTIVE-ALIGNED AGENTS IN MULTI-TIMESTEP TASKS
+
+We have shown that an LLM can provide objective-aligned reward signals in single-timestep tasks. In longer horizon tasks we must give trajectories instead of states as examples in our prompts. Longer prompts can be challenging because it is less likely for an LLM to have seen them during training. LLMs also have a recency bias which makes it harder for them to remember context introduced earlier on (Zhao et al., 2021). Can an LLM provide objective-aligned signals in longer horizon tasks? We investigate this question in the DEALORNODEAL negotiation task (Lewis et al., 2017).
+
+Task Description. DEALORNODEAL is a long-horizon task with a maximum length of 100 timesteps. An agent Alice must come to an agreement with her partner Bob on the allocation of a set of objects (books, hats, and balls). Agents are shown a context, which includes the counts of each item and their private utilities for each item. In the original task, agents get rewarded based on the agreed upon split and their utilities. If Alice and Bob reach a disagreement, both agents get nothing. We train Alice using on-policy RL by negotiating against a fixed partner model, which we refer to as Bob. See Sec. A.4 for more details on the domain and training.
+
+Ground Truth User Objectives. We train Alice to negotiate in different styles. For this experiment, we assume we have access to precise definitions in order to evaluate our models. Importantly, we do not give definitions of each style to the LLM, only examples. We experiment with the following negotiation styles inspired by previous literature Sycara et al. (1997); Caputo et al. (2019):
+
+• Versatile. Alice does not suggest the same proposal more than once.   
+• Push-Over. Alice gets less points than Bob.   
+• Competitive. Alice gets more points than Bob.   
+• Stubborn. Alice repeatedly suggests the same proposal.
+
+Prompt Design. We describe user objectives using three examples. Each example contains a negotiation between Alice and Bob, a question asking whether Alice negotiated in a particular style, and a yes or no answer followed by a short explanation. For an example, see Fig. 13 in the Appendix.
+
+Design Procedure. To create example negotiations, we randomly sampled three negotiation contexts for each objective and trained an RL agent (using the original task reward, without the LLM) to negotiate against $B o b$ in these contexts. We then sampled negotiations from the trained model. We also made sure all three sampled negotiations did not have the same ground truth label. We use a separate set of contexts when training RL agents with an LLM in the loop.
+
+# 4.3.1 RESULTS
+
+Labeling Accuracy. The top row of Fig. 4 shows that the LLM labels more accurately than SL except for Versatile. For Versatile, both models perform similarly because SL learns to overwhelmingly predict a negative label (avg of $9 6 \%$ negative predictions) and the RL agent showed more negative examples of Versatile behavior (avg. of $7 0 \%$ ground truth negative labels). However, the large portion of negative examples prevents the agent from learning correct behavior as shown in the Versatile plot on the bottom of Fig. 4); here, we get a larger performance gap between the LLM and SL.
+
+![](images/85056cb312e34a15f0e98388facbd4a8fb45226608d6ad7fdb3e5e03152829ba.jpg)  
+Figure 4: DEALORNODEAL, Few-shot. (Top) Accuracy of reward signals provided by LLM and SL during RL training. (Bottom) Accuracy of RL agents after training. (Right) Pilot study results. Agents trained with the user’s preferred style were rated as significantly more aligned than an agent trained with the opposite style $p { < } 0 . 0 0 1$ .
+
+Table 1: Qualitative results describing negotiations produced by agents trained with LLM.   
+
+<table><tr><td></td><td>Advantage</td><td>Diversity</td><td>Agreement Rate</td></tr><tr><td>Versatile</td><td>0.17±0.91</td><td>0.99±0.01</td><td>0.98±1.89</td></tr><tr><td>Push-Over</td><td>-2.95±0.64</td><td>0.82±0.26</td><td>1.0±0.0</td></tr><tr><td>Competitive</td><td>2.92±0.64</td><td>0.74±0.25</td><td>0.88±6.5</td></tr><tr><td>Stubborn</td><td>1.36±2.24</td><td>0.52±0.1</td><td>0.82±12.35</td></tr></table>
+
+RL Agent Accuracy. Results are shown on bottom row of of Fig. 4. LLM improves RL agent accuracy over SL by $4 6 \%$ on average. Our method approaches the performance of using the true reward; we under perform by an average of $4 \%$ . We remind readers that it is possible to outperform an agent trained with the true reward — especially when the LLM’s labeling accuracy is near-perfect as is the case for Competitive and Stubborn — due to reward hacking or stochasticity during training. For instance, agents trained with the true reward for Competitive end in more disagreements, leading to 0 reward for both agents and a lower accuracy.
+
+Is There a Qualitative Difference in Styles? Example negotiations of agents trained with the LLM for each style are shown in Fig. 9 in the Appendix. To measure qualitative differences among styles, we looked at average advantage (Alice’s original task reward - Bob’s original task reward), diversity (percentage of Alice’s utterances that are unique in a negotiation), and agreement rate (percentage of negotiations that end in agreement). The results in Table 1 demonstrate qualitative differences in styles.
+
+# 4.3.2 PILOT USER STUDY
+
+We conduct a within-subjects pilot user study to determine whether our trained agents can meaningfully align themselves with different user objectives when we do not have access to ground truth objectives and users evaluate agent performance.
+
+Method We asked $N { = } 1 0$ users to select a style in which they wanted their agent to negotiate in. We gave them an option to choose from our existing styles (Versatile, Push-Over, Competitive, Stubborn), or come up with their own. Importantly, users did not know how we defined these styles. We then showed users a list of 10 example negotiations generated via selfplay using an RL agent trained with a greedy objective (no particular style). We asked users to select 3 examples (1 positive, 1 negative, and 1 positive or negative) where Alice displayed positive or negative behavior of the user’s chosen style. For each chosen example, we asked users whether Alice demonstrated their chosen style and asked them to provide a ”Yes/No” answer as well as a short explanation. These examples corresponded to unknown, user-specific ground truth reward functions that we did not have access to. We then trained a negotiation agent by incorporating the user-provided examples in the prompt as described in Sec. 3. We also trained an agent to negotiate in the opposite style by flipping the ”Yes/No” labels in the user-provided explanations. We hypothesize that users should perceive a significant difference between these two agents. To evaluate our trained agents, we had both agents negotiate on a set of 10 test negotiations. For each test negotiation, we asked users to rate how well each agent aligned with their chosen style on a scale from 1 (least aligned) to 5 (most aligned).
+
+Results Agents trained with the correct style were significantly more aligned (avg. $3 . 7 2 { \pm } 1 . 2 )$ than agents trained with the opposite style (avg. $1 . 5 6 \pm 1 . 0 5 )$ , $p < 0 . 0 0 1$ , see Fig. 4 (right). Users varied in which styles they preferred: 4 users chose styles such as Polite, Push-Over, Considerate and Compromising, 2 users chose Versatile, and 4 users chose styles such as Stubborn, Competitive and Ambitious. These results demonstrate that our framework can produce agents aligned with differently specified objectives by changing the examples in the prompt. Furthermore, these results suggest that our framework can be used when rewards are difficult to define and results are agents with humans.
+
+Summary. Our framework can train objective-aligned agents when ground truth rewards are not present in complex, longer-horizon tasks. Agents are able to align the style in which they complete a task as evaluated by automated metrics as well as human users.
+
+# 5 ANALYSIS OF DATA EFFICIENCY & PROMPT DESIGN
+
+We have shown that we can use an LLM as a proxy reward function to successfully train objective-aligned agents across different tasks. This is a promising result because it represents an important step in enabling human-compatible and value-aligned AI systems. In this section, 1) we further quantify how data efficient our method is and 2) also analyze how robust LLM is to variations of prompt design.
+
+1) How Data-efficient is Our Method? We quantify how much more user data a supervised learning baseline would need in order to achieve the same labeling accuracy as the LLM. Results in the Ultimatum Game demonstrate that 10 labeled examples is enough for an SL model to reach comparable performance, whereas a a single labeled example is not sufficient. In this section, we quantify the amount of data needed for DEALORNODEAL because it is an example of a task where the decision boundary is not as easy to learn as the Ultimatum Game. We train SL by adding additional class-balanced, labeled examples to the original three examples it was trained on. We plot the average labeling accuracy SL achieves when trained on increasingly larger amounts of examples as well as the accuracy LLM achieves with three examples, shown in Fig. 6 in the Appendix. Results show that SL requires on the order of hundreds of more labeled examples in order to be comparably accurate as LLM.
+
+2) How Much Does LLM’s Labeling Accuracy Change When We Vary the Prompt? As with many approaches that use LLMs, a limitation of our approach is that it requires prompt design. We attempt to quantify the effort required for designing prompts as well as determine the feasibility of using nonengineered prompts from humans. We analyze the effect of prompt variation on labeling accuracy in DEALORNODEAL for the Stubborn objective. We vary different parts of the user-specified prompt at a time: the keyword (i.e., replacing “Stubborn” with its synonyms), the example negotiations, and the explanations associated with each example. Fig. 7 provides a summary of our results. See Sec. A.7 for the full results. Results illustrate that an LLM can be quite robust to different prompts — they all outperform SL. Furthermore, the quality of the explanation seems to be the most important in determining LLM accuracy.
+
+# 6 LIMITATIONS & FUTURE WORK
+
+User Studies. This work takes a first step in determining whether we can use LLMs as proxy rewards.   
+Given promising results, we plan on evaluating our approach with a larger user study.
+
+Multimodal Foundation Models. Beyond language models, multimodal foundation models such as Flamingo (Alayrac et al., 2022) can enable us to provide more complex environment states to the foundation model through images or other modalities while preserving an intuitive language interface for specifying the user objective.
+
+Non-Binary Rewards. Another limitation of our framework is that the LLM only specifies binary rewards. We plan on exploring how we can incorporate the likelihoods that LLMs produce for each word as a non-binary reward signal.
+
+# ACKNOWLEDGMENTS
+
+This work was supported by NSF Award 1941722, 2125511, 2006388, AFOSR, DARPA YFA Award, ONR, and JP Morgan Faculty Award. We would also like to thank Karl Tuyls, Ian Gemp, Albert Gu, Siddharth Karamcheti, Kanishk Gandhi, and other reviewers for this paper.
+
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+
+# A APPENDIX0 Tabl
+
+# A.1 SUMMARY
+
+OF RESULTS: IS IT IS POSSIBLE TO USE LLM AS A PROXY REWARD IN RL TRAINING?   
+Figure 5: Average Labeling and RL Agent Accuracy across the different objectives for each task across 3 seeds.   
+
+<table><tr><td colspan="5">Avg.Labeling Accuracy</td><td colspan="3">Avg.RL Agent Accuracy</td></tr><tr><td></td><td>Zero-shot Baseline (No Obj.)</td><td>Few-shot Baseline (SL)</td><td>Ours</td><td>Zero-shot Baseline (No Obj.)</td><td>Few-shot Baseline (SL)</td><td>Ours</td><td>True Reward</td></tr><tr><td>Ultimatum Game</td><td>-</td><td>0.67 ±0.34</td><td>0.91±0.27</td><td>-</td><td>0.67 ±0.34</td><td>0.9±0.26</td><td>1.0±0.02</td></tr><tr><td>Matrix Games</td><td>0.19 ±0.29</td><td>-</td><td>0.61±0.41</td><td>0.54±0.29</td><td>-</td><td>0.88±0.</td><td>1.0±0.</td></tr><tr><td>DEALORNODEAL</td><td></td><td>0.5± 0.47</td><td>0.9±0.22</td><td></td><td>0.33± 0.42</td><td>0.8±0.33</td><td>0.84 ±0.27</td></tr></table>
+
+We provide a summary of results in Fig. 5. The figure depicts the average Labeling and RL Agent Accuracy computed across the different user objectives for each task, across 3 seeds. Overall, our approach is able to produce more objective-aligned reward signals than our baselines. Our approach is also able to produce objective-aligned policies that are close in accuracy to policies trained with the true reward.
+
+# A.2 MORE RELATED WORKS
+
+Reward Design. Our framework addresses reward design—how to engineer rewards so that they align with our objectives (Amodei et al., 2016). This is challenging because tasks often have conflicting objectives that a human must trade off (Pan et al., 2022). Misspecifying reward functions can lead to reward hacking, or the gaming of specified rewards. Reward hacking has appeared in various domains such as autonomous driving (Knox et al., 2021) and game-playing (Ibarz et al., 2018). We hope to address these challenges by leveraging LLMs and making it easier for humans to specify their objectives.
+
+Imitation Learning & Preference-based Learning. Another method of specifying user objectives is to learn them from expert demonstrations (Ross et al., 2011) or preferences Sadigh et al. (2017). These techniques either assume access to large datasets (Christiano et al., 2017) or place restrictive assumptions (such as linearity) about the reward function (Sadigh et al., 2017). Recent work attempts to learn reward functions from language instructions using pragmatic reasoning (Lin et al., 2022). In contrast, our work relies on an LLM’s in-context learning abilities to provide a reward.
+
+# A.3 LLM DEFINITION OF OBJECTIVES IN THE MATRIX GAME
+
+<table><tr><td rowspan=1 colspan=1>LLMDefns.of ObjectivesTotal welfare is the sum of the rewards of both players.(√)</td></tr><tr><td rowspan=1 colspan=1>Equality of rewards is only possible if both players receive the same reward. (√)</td></tr><tr><td rowspan=1 colspan=1>Rawlsian fairness is defined as the maxmin value of the game, which is theminimum reward that the player could get assuming that the other player ismaximizing their reward. (X)</td></tr><tr><td rowspan=1 colspan=1>An outcome is Pareto-optimal if there is no other outcome that would make one player beter off without making the other player worse off.(√)</td></tr></table>
+
+Table 2: Completion of each sentence given by LLM in pink. LLM provides correct definitions for objectives except for Rawlsian Fairness, which is partially correct.
+
+# A.4 DETAILS ON RL ENVIRONMENTS AND TRAINING
+
+Ultimatum Game. The environment is a single horizon with discrete actions (i.e., accept or reject) and continuous observations (i.e., a proposed split). We train DQN agents using the Stable Baselines3 implementation for 1e4 timesteps with a learning rate of 1e-4 across 3 seeds (Raffin et al., 2021). We instantiate our policy as a MLP with the default parameters used in Stable Baselines3.
+
+Parser $g$ . The parser $g$ that transforms the LLM’s output into an integer reward signal is defined using a handcrafted parser. When prompting the LLM, we structure the labels for each example to be in “Yes/No” form which enables the LLM to also reply using the same format. We are then able search for the “Yes” or “No” strings and parse them into a 1 or 0 respectively. In the rare occasion that the LLM does not respond in this form, we skip the episode during RL training and omit the example from our evaluation.
+
+Further Analysis on Performance of No Objective Baseline. The average LLM accuracy of a random baseline across the four matrix games are Welfare: 0.125, Equality: 0.125, Rawlsian Fairness: 0.078, Pareto-optimality: 0.172. The No Objective baseline’s performance is close to random as can be verified by comparing the random baseline results with Figure 3 (top row).\* Behaviorally, we observe that No Objective acts like a random baseline: the LLM often hallucinates matrix game rewards and also displays incoherent reasoning when selecting answers.
+
+Matrix Game.. The environment is a single horizon with discrete actions (i.e., one of the four joint actions) and no observations. We train DQN agents using the Stable Baselines3 implementation for 500 timesteps with a learning rate of 1e-4 across 3 seeds (Raffin et al., 2021). We instantiate our policy as a MLP with the default parameters used in Stable Baselines3.
+
+Parser $g$ . We parse the LLM’s response by hand, since LLM output can be variable in zero-shot settings.
+
+DEALORNODEAL. We use a version of the DEALORNODEAL environment used in Kwon et al. (2021). In the environment, the goal is for an agent $A$ to come to an agreement with a partner $B$ on the allocation of a set of objects (books, hats, and balls). During each negotiation, agents receive a context, $c _ { A } = [ i ; u _ { A } ] , c _ { B } =$ $[ i ; u _ { B } ]$ , detailing the count of each item $i$ as well as their private utilities, $u _ { A } , u _ { B }$ . Item counts and utilities are represented as vectors $i \in \{ 1 , . . . , 4 \} ^ { 3 }$ and $u _ { A } , u _ { B } \in \{ 0 , \bar { . . . } , 1 0 \} ^ { 3 }$ and are sampled uniformly.
+
+After receiving contexts $c _ { A } , c _ { B }$ , an agent is randomly selected to begin the negotiation. Agents negotiate for $T$ time steps by exchanging coarse dialogue acts $x _ { t }$ at each time step $1 \leq t \leq T$ (He et al., 2018). Rather than negotiate directly in natural language, where the generation problem is hard and can result in degenerate dialogues (He et al., 2018), we use these dialogue acts instead to focus on learning diverse and interpretable strategies.
+
+A dialogue act $x _ { t }$ is one of five actions: propose, insist, agree, disagree, or end. The propose and insist acts take allocations of items as arguments $o \ = \ [ o _ { A } ; o _ { B } ]$ where $o _ { A } , o _ { B } \in \{ 1 , . . . , 4 \} ^ { 3 }$ (e.g., propose: books $^ { = 1 }$ , hat $S { = } 2$ , $\mathtt { b a l l s } = 1$ ). When an agent selects end, the conversation terminates and each agent is asked to make their final selection.
+
+If agents agree on the final allocation of items, i.e., $o _ { A } + o _ { B } = i$ , agents are awarded points based on their private utilities, $r _ { A } = u _ { A } \cdot o _ { A } , r _ { B } = u _ { B } \cdot o _ { B }$ . If agents do not agree, they receive 0 points. Each agent’s context is constrained so that the agent can receive a maximum of 10 points.
+
+Agents are first trained using supervised learning on a dataset of human-human negotiations provided by (Lewis et al., 2017) to predict the next token. We use a learning rate of 1.0 and batch size of 16. We then fine-tune these agents using RL where they optimize the expected reward of each dialogue act using REINFORCE (Williams, 1992). Agents are trained on 250 contexts for 1 epoch with a learning rate of 0.1. We instantiate our policy with four GRUs (Chung et al., 2014). We closely follow the implementation outlined in (Kwon et al., 2021; Lewis et al., 2017), please refer to those papers for more training details.
+
+Parser $g$ . The parser $g$ that transforms the LLM’s output into an integer reward signal is defined using a handcrafted parser. When prompting the LLM, we structure the labels for each example to be in “Yes/No” form which enables the LLM to also reply using the same format. We are then able search for the “Yes” or “No” strings and parse them into a 1 or 0 respectively. In the rare occasion that the LLM does not respond in this form, we skip the episode during RL training and omit the example from our evaluation.
+
+# A.5 SL MODEL ARCHITECTURE AND TRAINING
+
+Ultimatum Game. SL is trained to predict binary labels for a batch of proposed splits. We implemented SL as a multi-layer perceptron (MLP) network that consists of a single hidden layer with depth 32. We also use ReLU activations after our input and hidden layers. We trained the model on the same 10 examples we gave to LLM for 5 epochs with the Adam optimizer. We evaluate the model on the 50 heldout test examples and save the model with the best test accuracy. We show the training and test accuracy for each user objective below:
+
+Table 3: Training accuracy for SL on the Ultimatum Game.   
+
+<table><tr><td></td><td>30%</td><td>60%</td><td>$10</td><td>$100</td><td>Ineq. Aversion</td></tr><tr><td>Train Acc. (10 examples)</td><td>1.0</td><td>1.0</td><td>0.9</td><td>1.0</td><td>1.0</td></tr><tr><td>Train Acc. (1 example)</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1.0</td><td>1.0</td></tr></table>
+
+DEALORNODEAL. SL is trained to predict binary labels given a negotiation as input. We closely follow the implementation of a SL model found in (Kwon et al., 2021; Lewis et al., 2017). A negotiation consists of a context, coarse dialogue acts exchanged between Alice and Bob, and the outcome of the negotiation (the final split of items and whether agents agreed or disagreed). Please refer to Sec. A.4 for more details on the environment. We implement SL using a MLP context encoder, MLP outcome encoder, and a GRU ((Chung et al., 2014)) to process the coarse dialogue acts. The MLP encoders consist of an embedding layer followed by a linear layer with a Tanh activation function; we use a hidden size of 64 for the context encoder’s linear layer. We similarly embed each coarse dialogue act before feeding it into the GRU. We use a hidden size of 128 for the GRU. We train SL on the same 3 examples we use in our prompt for LLM. We train for a maximum of 50 epochs using the Adam optimizer. SL received a training accuracy of $1 0 0 \%$ for all of our objectives.
+
+# A.6 HOW DATA-EFFICIENT IS OUR METHOD?
+
+![](images/4f6b808338f90d977425b0fd7632117383cc36d585ae233c2534f392778dd388.jpg)  
+SL Labeling Accuracy When Trained with # Additional Labeled Examples   
+Figure 6: SL requires on the order of hundreds of more labeled examples in order to be comparably accurate to the LLM.
+
+A.7 HOW MUCH DOES LLM’S LABELING ACCURACY CHANGE WHEN WE VARY THE PROMPT?   
+Effect of Prompt Variation on Labeling Accuracy for Stubborn (N=3, 3 seeds)   
+Figure 7: On average, varying the prompt does not have a large impact on the accuracy of the LLM.   
+
+<table><tr><td>SL</td><td>Ours</td><td>Vary Keyword</td><td>Vary Example Negotiations</td><td>Vary Explanations</td></tr><tr><td>0.58 ± 0.48</td><td>0.97 ± 0.16</td><td>0.91±0.28</td><td>0.93 ±0.28</td><td>0.79 ±0.33</td></tr></table>
+
+We analyze the effect of varying prompts on the LLM’s labeling accuracy for the Stubborn negotiating style in DEALORNODEAL. We vary prompts in three ways: we vary the keyword (i.e., replacing “Stubborn” with its synonyms), the example negotiations, and the explanations associated with each example.
+
+When varying the keyword, we use the synonyms: “Headstrong”, “Obstinate”, and a less commonly used word, “Froward”. To vary example negotiations, we randomly sample three new negotiations to have counterbalanced labels, all positive labels, or all negative labels. We vary the explanations by coming up with two plausible sets of explanations a user might have given for each example. We also experiment with the scenario where we give no explanations. Results are shown in Fig. 8. Overall, varying prompts do not have a large impact on labeling accuracy — they all outperform the baseline. However, the quality of explanation seems to have the largest impact on labeling accuracy.
+
+# A.8 WHAT IS THE IMPORTANCE OF INCLUDING THE TASK DESCRIPTION, $\rho _ { 1 }$ IN THE PROMPT?
+
+We experiment with removing $\rho _ { 1 }$ , the task description in the Ultimatum Game with a single example followed by an explanation. Performance increases slightly in LLM labeling accuracy (avg. of $8 \%$ ) and RL agent accuracy (avg. of $9 \%$ ). We run the same experiment in the Ultimatum game in the case of 10 examples with no explanation. Performance drops slightly in LLM labeling accuracy (avg. $4 . 4 \%$ ) and RL agent accuracy (avg. $5 . 3 \%$ ). We conclude that $\rho _ { 1 }$ is not conclusively influential in improving performance in few-shot settings.
+
+![](images/4b26b32a6fb585ccb6d6dedd5628c442fa267f1c69fedf71cf174a571a5c57ce.jpg)  
+Figure 8: Varying prompts do not have a large impact on labeling accuracy.   
+Figure 9: Example negotiations after Alice is trained with reward signals from LLM in DEALORNODEAL. We illustrate qualitative differences in how Alice negotiates for the same context. $B o b$ is an agent that is trained with supervised learning.
+
+# A.9 EXPERIMENTING WITH SMALLER LLM SIZES
+
+We experiment with GPT-2, a 1.5B parameter model. We We find that GPT-2 underperforms GPT-3 in both labeling (avg. $1 5 \%$ ) and RL agent accuracy (avg. $4 9 \%$ ). GPT-2 outperforms the SL baseline in labeling accuracy (avg. $2 4 \%$ ), and slightly underperforms the SL baseline for RL agent accuracy (avg. $2 . 7 \%$ ). Results are averaged across styles and seeds. GPT-2 (1.5B) is several orders smaller than GPT-3 (175B), and we expect models larger than GPT-2 to close the gap with GPT-3’s performance.
+
+A.10 EXAMPLE NEGOTIATIONS
+
+# Context
+
+Alice : book=(count:1 value:0) hat=(count:1 value:7) ball=(count:3 value:1) Bob : book=(count:1 value:3) hat=(count:1 value:7) ball=(count:3 value:0)
+
+<table><tr><td rowspan=1 colspan=1>Versatile</td></tr><tr><td rowspan=1 colspan=1>Alice : propose: item0=1 item1=1 item2=0Bob :propose: item0=0 item1=1 item2=0Alice : propose: item0=1 item1=0 item2=2Bob :agree</td></tr><tr><td rowspan=1 colspan=1>Agreement!Alice : 2 pointsBob :7 pointsCompetitive</td></tr><tr><td rowspan=1 colspan=1>Alice : propose: item0=1 item1=0 item2=3Bob : propose: item0=1 item1=1 item2=0Alice : insist: item0=0 item1=1 item2=3Bob :agree</td></tr><tr><td rowspan=1 colspan=1>Agreement!Alice : 10 pointsBob :3 points</td></tr></table>
+
+<table><tr><td>Push-over</td></tr><tr><td>Alice : propose: item0=1 item1=0 item2=3 Bob :propose: itemO=1 item1=1 item2=0 Alice :agree Agreement! Alice : 3 points</td></tr><tr><td>Bob :10 points Stubborn Alice : propose: item0=0 item1=1 item2=1</td></tr><tr><td>Bob :propose: item0=0 item1=1 item2=0 Alice : propose: item0=0 item1=1 item2=1 Bob : propose: item0=0 item1=1 item2=0 Alice : propose: item0=0 item1=1 item2=1 Bob :propose: item0=0 item1=1 item2=0 Alice : propose: item0=0 item1=1 item2=1 Bob:propose: item0=0 item1=1 item2=0 Alice : propose: item0=0 item1=1 item2=1</td></tr></table>
+
+Disagreement?! Alice : 0 (potential 0) Bob : 0 (potential 0)
+
+# A.11 EXAMPLE OF PROMPTS USED IN OUR EXPERIMENTS
+
+# A.11.1 ULTIMATUM GAME
+
+Further Explanation of Our Prompt Selection Process. When constructing our explanations, we encourage the LLM to produce intermediate reasoning steps by using the “Let’s think step by step” template usedUltimatum prompt in Kojima et al. (2022) which has been shown to improve performance.
+
+![](images/29d0ebc7ae13f4da3f5e7cac6aafe08e7307f316a879182340e85c2fc8204951.jpg)  
+Figure 10: An example of few-shot prompts used for the Ultimatum Game. We highlight the four parts of each pro
+
+# A.11.2 MATRIX GAMES
+
+Further Explanation of Our Prompt Selection Process. We found that structuring the outcomes of the game as a multiple choice question improved performance. We also encouraged the LLM to produce intermediate reasoning steps by using the “Let’s think step by step” template used in Kojima et al. (2022) which has been shown to improve performance.Matrix prompt
+
+![](images/29401b78cc06de5d45f4c3a39611eed45b44fbd7d818516e389fa9b1f09d0051.jpg)  
+Figure 11: Examples of a zero-shot prompts used for each objective, including the no-objective baseline, in the Matrix Games. Due to limited resources when querying GPT-3, we queried GPT-3 in a batched manner and saved the corresponding labels to train our RL agents. Consequently, we do not have an Episode outcome in our prompts.
+
+# A.11.3 DEALORNODEAL
+
+Further Explanation of Our Prompt Selection Process. We chose 3 counterbalanced examples from a training set of sample negotiations. This training set was generated via selfplay using an RL agent trained
+
+Figure 12: Examples of regular and scrambled outcomes for the Chicken Game (Matrix Game experiments). Scrambling creates new associations between joint actions and joint rewards. We scramble the outcomes in order to remove any bias LLM may have towards the order in which outcomes are normally presented.   
+
+<table><tr><td>Chicken Game (Regular Order)</td><td>Chicken Game (Scrambled Order)</td></tr><tr><td>We have a two-player game where P1 and P2 can choose one of these options.</td><td>We have a two-player game where P1 and P2 can choose one of these options.</td></tr><tr><td>Options: A.if action1(P1) and action1(P2)=&gt;P1 gets reward of 2, P2 gets reward of 2.</td><td>Options: A. if actionl(P1) and actionl(P2) =&gt;P1 gets reward of 3,P2 gets reward of 1.</td></tr><tr><td>B. if action1(P1) and action2(P2)=&gt;P1 gets reward of 1,P2 gets reward of 3.</td><td>B.if action1(P1) and action2(P2)=&gt;P1 gets reward of 2,P2 gets reward of 2.</td></tr><tr><td>C.if action2(P1) and action1(P2)=&gt;P1 gets reward of 3,P2 gets reward of 1.</td><td>C.if action2(P1) and action1(P2)=&gt;P1 gets reward of 1, P2 gets reward of 3.</td></tr><tr><td>D.if action2(P1) and action2(P2)=&gt;P1 gets reward of 0,P2 gets reward of 0.</td><td>D.if action2(P1)and action2(P2)=&gt;P1 gets reward of 0,P2 gets reward of 0.</td></tr><tr><td></td><td></td></tr><tr><td>Which option(s) are Pareto-optimal? Let&#x27;s think step by step:</td><td>Which option(s) are Pareto-optimal? Let&#x27;s think step by step:</td></tr><tr><td>An outcome is Pareto-optimal if</td><td>An outcome is Pareto-optimal if</td></tr></table>
+
+with a greedy reward (no particular style). We chose to complement our examples with simple and succinct explanations.
+
+![](images/46283f58af7f6fc1357e46c8a1d3b0735f3e69152e432c8c5133651396f848ce.jpg)  
+Figure 13: Example of a prompt used for DEALORNODEAL.

parse/dev/1KaSx3GrBBm/1KaSx3GrBBm.md

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+# MOVING BEYOND HANDCRAFTED ARCHITECTURES IN SELF-SUPERVISED LEARNING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+The current literature on self-supervised learning (SSL) focuses on developing learning objectives to train neural networks more effectively on unlabeled data. The typical development process involves taking well-established architectures, e.g., ResNet or ViT demonstrated on ImageNet, and using them to evaluate newly developed objectives on downstream scenarios. While convenient, this neglects the role of architectures which has been shown to be crucial in the supervised learning literature. In this work, we establish extensive empirical evidence showing that a network architecture plays a significant role in contrastive SSL. We conduct a large-scale study with over 100 variants of ResNet and MobileNet architectures and evaluate them across 11 downstream scenarios in the contrastive SSL setting. We show that there is no one network that performs consistently well across the scenarios. Based on this, we propose to learn not only network weights but also architecture topologies in the SSL regime. We show that “self-supervised architectures” outperform popular handcrafted architectures (ResNet18 and MobileNetV2) while performing competitively with the larger and computationally heavy ResNet50 on major image classification benchmarks (ImageNet-1K, iNat2021, and more). Our results suggest that it is time to consider moving beyond handcrafted architectures in contrastive SSL and start thinking about incorporating architecture search into self-supervised learning objectives.
+
+# 1 INTRODUCTION
+
+Self-supervised learning (SSL) achieves impressive results on challenging tasks involving image, video, audio, and text. Models pretrained on large unlabeled data perform nearly as good and sometimes even better than their supervised counterparts (Caron et al., 2020; Chen & He, 2021). So far, the focus has been on designing effective learning objectives – e.g., pretext tasks (Gidaris et al., 2018; Caron et al., 2018), contrastive (Oord et al., 2018; Chen et al., 2020a) and noncontrastive (Grill et al., 2020) tasks – together with empirical (Cole et al., 2021; Feichtenhofer et al., 2021) and theoretical (Arora et al., 2019; Poole et al., 2019) studies providing key insights and underpinnings. Recent works propose new objectives with a different class of network architectures such as vision transformers (ViT) (Bao et al., 2021) and masked autoencoders (He et al., 2022).
+
+However, there has been little focus on the role of architectures in SSL. Currently, the de facto protocol in SSL is to take architectures that perform well on established benchmarks in the supervised setting and to adapt them to the self-supervised setting by plugging in different learning objectives. For example, several existing work on contrastive learning use ResNet (He et al., 2016) as the backbone (Chen et al., 2020a; He et al., 2020). This is partly for convenience. Evaluating different architectures in SSL is computationally expensive; selecting an architecture in advance and fixing it throughout makes it easy to evaluate different learning objectives. This also stems from strong empirical success of those architectures in transfer learning, e.g., CNNs trained on large labeled data provide “unreasonable effectiveness” (Sun et al., 2017; Zhang et al., 2018; Sejnowski, 2020) in a variety of downstream cases. Nonetheless, one implicit assumption is that an architecture that works well in the supervised learning scenario will continue to be effective in the SSL regime.
+
+We argue that this assumption is incorrect and dangerous. It is valid only to a limited extent and the performance starts deteriorating significantly when SSL is conducted on data whose distribution deviates much from the original distribution the architecture was trained on. This is counter to the promise of SSL, where one can learn optimal representation for a wide range of tasks. One main reason for performance degradation is that different data distributions benefit from different inductive biases: An architecture with specific layer types and the wiring between them naturally encodes inductive biases, which may be optimal only for a certain data distribution (e.g., objectcentric imagery such as ImageNet) and not for others (e.g., medical and satellite imagery). In fact, numerous studies have shown that standard “recipes” for architecture design do not translate well across different data distributions (Tuggener et al., 2021; Dey et al., 2021; Kolesnikov et al., 2019).
+
+The main objective of this work is to show that the choice of network architecture crucially matters in SSL, and that it is not easy to handcraft architectures that are effective across different SSL scenarios. To see this, recall that the goal of SSL is to learn data representations capturing important features and attributes that generalize well across various downstream tasks. There has been extensive literature on the expressivity of neural networks (Raghu et al. (2017); Zhang et al. (2021a) and references therein); one of the important conclusions is that the network topology plays a significant role in determining the expressivity, i.e., the kinds of functions a network can approximate is bounded by the network capacity and available sample size in the finite sample regime. This implies that, in practice, SSL with a fixed architecture learns representations only within the scope of function space induced by the pre-selected architecture topology, and therefore, the ultimate success of SSL can be achieved when it finds optimal architecture from certain search space in conjunction with its weights for specific data distributions.
+
+In this paper, we establish extensive empirical evidence showing that architecture matters in selfsupervised learning. We do this in two sets of large-scale studies. First, we sample 116 variants of ResNet (He et al., 2016) and MobileNet (Sandler et al., 2018) architectures with different topologies and evaluate them on 11 downstream tasks in the SSL setting. We pretrain all models under the same setting, optimizing the SimCLR objective (Chen et al., 2020a) on ImageNet (Deng et al., 2009), and investigate if there exist any correlation between these models in downstream performance on different datasets. We observe no strong correlation, except for tasks highly similar to ImageNet. We further show that ImageNet downstream performance, the gold standard benchmark in the SSL literature, is not indicative of performance on other downstream tasks. This implies that we need to be careful in choosing an architecture for evaluating any newly developed SSL objectives, as one might get different conclusions based on different network architectures.
+
+This subsequently raises the question: Can we improve SSL by learning not only network weights but also architectures directly optimized for the given dataset? It removes the burden of manually searching for effective architectures in SSL, and if we succeed, it can substantially improve performance of SSL. To test this hypothesis, as the second set of our study, we apply a well-established NAS algorithm (Cai et al., 2018) to the SSL setting. Unlike the typical NAS setting that optimize on a labeled target dataset, we search for optimal architectures directly on an unlabeled pretraining dataset via contrastive learning (Chen et al., 2020a). We evaluate our $\mathrm { ^ { 6 6 } N A S } + \mathrm { S S L } ^ { \mathrm { 3 } }$ framework on datasets with different distributions, ImageNet-1K and iNat 2021 (Van Horn et al., 2021), and show that self-supervised architectures consistently outperform handcrafted ones in the same parameter range (MobileNetV2 and ResNet18) across 11 downstream tasks. This provides strong evidence suggesting the importance of learning architecture topologies in addition to their weights in SSL.
+
+Our work focuses on studying the role of architectures in contrastive SSL with the SimCLR framework for CNN-based architectures such as ResNets and MobileNets. As a first step in this direction, we provide an in-depth analysis through large-scale experiments in this specific (yet limited) setting. Extending our study to different SSL approaches (He et al., 2020; Grill et al., 2020; He et al., 2022) and architectures such as ViTs would be an interesting direction but beyond the scope of this paper.
+
+In summary, our main contributions are: 1) We establish extensive evidence showing that there isn’t one network architecture performing consistently well across different downstream scenarios in the SimCLR setting. We show this using 116 variants of ResNet and MobileNet architectures pretrained on ImageNet and evaluated on 11 downstream datasets. 2) We show that ImageNet performance (the gold standard in SSL benchmark) is not always indicative of downstream performance. This means that findings about SSL objectives shown only on ImageNet do not generalize across other data distributions. 3) We propose to self-supervise a CNN architecture topology and its network weights on unlabeled data. We show that self-supervised architectures outperform handcrafted ones in a similar parameter range for the SimCLR setting across different downstream datasets.
+
+# 2 RELATED WORK
+
+Role of architectures in SSL. There has been significant progress in learning representations via SSL (Gidaris et al., 2018; Chen et al., 2020a;c; Caron et al., 2020). Ericsson et al. (2021) provide an overview of different SSL setups and their performance on downstream tasks. Most works focus on improving self-supervised objectives to develop better representations while keeping architectures fixed. Our focus is orthogonal to this line of work. We study the role of architectures in SSL and investigate the benefits of self-supervising architecture topologies along with network weights.
+
+Similar to ours, Kornblith et al. (2019) analyze transfer performance of different architectures pretrained on ImageNet across several downstream datasets. However, they focused on supervised learning, which need not translate to self-supervised setups (Kolesnikov et al., 2019). Caron et al. (2021) propose a self-distillation based SSL objective and compare the performance of ResNets with ViTs. In contrast, we focuse on contrastive SSL and the importance of architecture topologies for the class of CNNs. Kolesnikov et al. (2019) is the most related to ours but are limited to pretext-task based SSL and show results only for a few variants of VGG and ResNet. In comparison, we conduct a study on a much larger scale in a contrastive learning setup. Also, we provide insights into generalization performance of the networks across different datasets. Crucially, unlike all previous work in SSL, we propose to learn both architecture topologies and their network weights.
+
+NAS and SSL. The literature on NAS has rapidly progressed in the past years; we refer the reader to the NAS survey by Elsken et al. (2019). Here we focus on the most directly relevant work to SSL. Mellor et al. (2021) search for networks without any training and verify their effectiveness of supervised benchmarks. Liu et al. (2020) show that highly performant architectures can be found without using any supervised labels during the search itself, and propose using DARTS (Liu et al., 2019) with self-supervised proxy tasks to search for architectures which will perform well on supervised datasets. In a similar vein, Zhang et al. (2021b) use random labels during the search phase of NAS, and Yan et al. (2020) learn representations of architectures via SSL and use them to improve NAS. Li et al. (2021) introduce a new self-supervised training scheme to search for architectures in a new hybrid search space. One common theme in this line of work is that they aim to harness SSL in aid of NAS. In contrast, we aim to utilize NAS to hunt for architectures which will perform well for self-supervised learning, thereby harnessing NAS in aid of SSL.
+
+# 3 DOES ONE NETWORK RULE THEM ALL IN SSL?
+
+We conduct a large scale study to investigate whether a particular architecture topology can be consistently effective across a wide range of SSL scenarios. To this end, we sample 116 models with different architecture topologies and analyze their performance on 11 downstream tasks.
+
+To maximize generality while taming complexity of our study, we choose two most representative CNN architectures: ResNets and MobileNets. The former is the de facto backbone for numerous modern visual models (Kirillov et al., 2019; Wu et al., 2019) and SSL approaches (Caron et al., 2020; Chen & He, 2021), while the latter is used in low-resource setups (Cheng et al., 2017). We create 69 ResNet-like and 47 MobileNet-like architectures for our evaluation, varying the number of blocks at each of the 4 stages of a ResNet, the block structure (BasicBlock and Bottleneck), width, and number of groups. For MobileNet, we vary the width multiplier parameter and the number of blocks in each of the 6 stages. We choose the SimCLR objective (Chen et al., 2020a) for our experiments and pretrain each of the 116 architectures on ImageNet-1K. Owing to the large scale nature of the experiments and computational constraints, we limit the pretraining to 100 epochs and a batch size of 512. Each of these jobs takes roughly 1 day to finish on a system with $8 \times \mathrm { ~ V 1 0 0 ~ }$ (32GB) GPUs. We report linear evaluation results in all cases.
+
+ImageNet performance is not indicative of downstream performance for SSL. To examine the correlation between ImageNet vs. downstream performance, we compute the Spearman’s rank correlation coefficient $\rho$ on top-1 validation accuracy between every dataset pair, shown in Fig. 1. We also show scatter plots in Fig. 2 revealing the relationship between ImageNet vs. downstream performance on the most representative cases; the complete set of scatter plots are in the appendix.
+
+For ResNet-like architectures (Fig. 2 top row), we see strong correlation between ImageNet and CIFAR-100 ( $\rho = . 8 )$ ) as these datasets contain similar categories; about 90 classes in CIFAR-100 are the same class or a superclass in ImageNet. A similar observation is made (in Fig. 1) for Stanford Dogs $( \rho = . 7 7 )$ due to the 120 dog categories in ImageNet. However, we observe high variance in transfer performance for out-of-domain datasets, e.g., Flowers $( \rho = . 3 5 )$ , represented by only two ImageNet categories (daisy and yellow lady slipper). Correlation becomes negative on Stanford Cars $( \rho = - . 2 9 )$ and FGVC Aircraft $( \rho = - . 2 4 )$ , likely because ImageNet contains only a few categories of cars (10 classes) and aircraft (4 classes). Low correlation means network ranks are inconsistent and models performing well on ImageNet do not keep their precedence in other tasks.
+
+![](images/d365bf971a98922758f9bba2584f03a8e3a1f78f6ce0f45f6babac180c901c8e.jpg)  
+Figure 2: ImageNet performance is not indicative of downstream performance in SSL. We show linear evaluation top-1 accuracy of ImageNet (x-axis) vs. downstream datasets (y-axis) obtained from variations of ResNet (top) and MobileNet (bottom); all models are pretrained on ImageNet-1K using SimCLR under the same protocol. The solid lines and shaded areas indicate fitted regression models and their confidence intervals.
+
+For MobileNet-like architectures (Fig. 2 bottom row), we see overall much lower correlation with ImageNet performance; the ResNet space showed correlation at least for in-distribution datasets. For e.g., we see correlation between ImageNet and CIFAR-100 drops from $\rho = . 8$ (ResNet) to $\rho \ = \ . 2 3$ (MobileNet). For other datasets like MIT67, correlation is higher $( \rho ~ = ~ . 5 2 )$ but still less meaningful due to high variability in performance (notice the cluster around $42 \%$ accuracy). These results indicate that MobileNets are even less tuned towards ImageNet than ResNets and any handcrafted architectures in this space is likely to be suboptimal on ImageNet and other downstream datasets.
+
+![](images/978f0c664da19142b0881594c45d900b24f08e3c1aad569ad28356448d151fe7.jpg)  
+Figure 1: ImageNet performance isn’t indicative of downstream performance in SSL. We show rank correlation between each pair of top-1 accuracy on 11 downstream tasks obtained from ImageNet-pretrained ResNets. We see no strong correlation except for ones similar to ImageNet, e.g., CIFAR-10/100 and Dogs120.
+
+Our results highlight that the same architecture recipes in terms of ImageNet accuracy, which is frequently used as a predictor for various selfsupervised tasks, do not work well for different datasets in the SimCLR setting.
+
+Larger networks do not always perform better in contrastive SSL. In general, large-parameter models yield better performance on ImageNet, both in supervised (Kornblith et al., 2019) and SSL setups (Chen et al., 2020a). We examine whether this trend holds for downstream datasets some of which are widely different from ImageNet. We follow the same setup as above and compute the
+
+![](images/cafca3409464ab3c90dd7c490f15db8e5376b27cf6d4909facd721ca5a0d87e0.jpg)  
+Figure 3: Larger models do not always perform better in SSL. We show the model size in terms of parameter counts ( $\mathbf { \widetilde { x } }$ -axis) vs. top-1 accuracy on different datasets obtained from variations of ResNet (top) and MobileNet (bottom) architectures; all models are pretrained on ImageNet-1K using SimCLR under the same protocol.
+
+correlation between number of model parameters and top-1 validation accuracy on different datasets.   
+Fig. 3 shows scatter plots of the most representative results; full results are in the appendix.
+
+For ResNet-like architectures (Fig. 3 top row), we do not see strong trends indicating larger models always perform better. In fact, on some downstream tasks we see negative correlation (Aircraft, $\rho =$ $- . 3 7 )$ with lighter networks being favored for better performance, or even non-linear relationship, e.g., notice the slight “U” pattern on Stanford Cars $\zeta = - . 1 4 )$ , indicating the behavior of ResNets on these datasets are wildly unexpected. Unsurprisingly, there is strong correlation with ImageNet $( \rho = . 8 5 )$ and CIFAR-100 $\langle \rho = . 8 7 \rangle$ , likely because ResNets are heavily hand-tuned on the kinds of images observed in these datasets. These results clearly suggest that the same architecture recipes which work well for ImageNet (increasing parameters through depth and width) do not hold for other downstream scenarios.
+
+For MobileNet-like architectures (Fig. 3 bottom row), we see that there exists almost no interpretable trend. The fitted regression models (solid lines) and their confidence intervals (shaded area) show that the relationships are highly non-linear and non-monotonic; we shouldn’t read too much into the correlation coefficients (reported for completeness). These results indicate that MobileNets do not favor any one architecture and it is heavily reliant on the dataset it is trained on.
+
+Our results suggest that larger models are not always better in SSL; lighter models can outperform heavier models on tasks like FGVC Aircraft. Previous work (Chen et al., 2020b) showed that increasing network depth/width improves downstream performance. However, they vary depth at a coarser level with 50/100/150 layers and width with $1 \times / 2 \times$ , leading to a large swing in network parameters (24-795M); here we show the same is not true at a finer level. Our results provide evidence that ResNet architectures are tuned to scale well on ImageNet but not the others; the trend is even weaker for MobileNet-like architectures, suggesting they are optimized for computational efficiency and not for achieving high accuracy on any particular dataset.
+
+There is no winner in the battle of top vs. bottom heavy networks in SSL. Raghu et al. (2017) showed that CNNs are more sensitive to lower (initial) layer weights, suggesting that “not all weights are created equal” across layers. This raises the question: If CNNs are more sensitive to lower layers, will increasing the parameter count for lower layers yield better performance in SSL? To get insights into this, we split a network into two halves: “top” (layers closer to output) and “bottom” (closer to input). We use the ratio of top to bottom parameters as a measure of networks being “top-heavy” (high ratio) or “bottom-heavy” (low ratio).
+
+Fig. 4 shows the downstream accuracy for ImageNet, Stanford Cars and Sports against this ratio. From the top-left subplot, we see that ResNet-ImageNet accuracy tend to increase with high top:bottom ratio, showing top-heavy networks generally perform better than the bottom-heavy counterparts. However, we no longer observe such trend in other datasets, and with MobileNets (Fig. 4 bottom row) we do not see such trend even for ImageNet.
+
+An important point to realize: It is necessary to allocate the right portion of parameters to different layers of a given network topology, instead of allocating parameters in the top-heavy or bottom-heavy fashion assuming one will generally lead to better performance. ResNets and many other handcrafted CNNs (Simonyan & Zisserman, 2014; Szegedy et al., 2016; Huang et al., 2017) are usually top-heavy because of GPU memory limits; bottom-heavy networks occupy more memory in terms of activation maps. MobileNets alleviate this to some extent with lighter convolutions, allowing it to have more parameters in early layers. In object detection, Liang et al. (2019) also show that allocation of computational resources in the backbone is important for improved performance. While this architectural difference provides explanations about the wildly different trends we observe above, the key message here is that one recipe (top vs. bottom heavy) does not apply equally to different architectures, bolstering our claim that one network doesn’t rule them all in SSL.
+
+Key takeaway: We need to move beyond handcrafted architectures in SSL. The three main observations above imply that finding an optimal architecture could be an important missing piece for selfsupervised learning. Our results show that the current practice in designing SSL objectives – i.e., optimizing for ImageNet performance based on ResNet backbones – could lead to misleading conclusions which do not generalize to other downstream scenarios. Also, the general belief that “the larger the better” in model size do not really hold in SSL, e.g., smaller
+
+![](images/05d93e00c55127665a07af3b122624e5289290a2942c4b8f7f95506fb063641e.jpg)  
+Figure 4: There is no winner in the top vs. bottom battle in SSL. Except for ResNet-ImageNet (top-left), we see no strong trend that suggests either top-heavy or bottom-heavy networks perform better.
+
+ResNets can outperform larger ones even if they are pretrained following the same protocol. Let’s say, based on these observations, one is compelled to hunt for a new architecture geared specifically towards SSL. Our top vs. bottom analysis suggests that it can be extremely tricky to find the right architecture topology with optimal parameter allocation across different layers. All this suggests that it is time to consider moving beyond handcrafted architectures in SSL and start thinking about searching for optimal architectures as part of self-supervised learning objectives.
+
+# 4 NEURAL ARCHITECTURE SEARCH FOR SSL
+
+We turn to the idea of learning both the architecture topology and its network weights in an SSL framework, using NAS to improve SSL (rather than using SSL to improve NAS). It is important to draw a clear distinction between our idea and prior work that used SSL to improve NAS (Li et al., 2021; Liu et al., 2020; 2019; Yan et al., 2020; Zhang et al., 2021b) as well as work that used NAS to improve supervised learning (Elsken et al., 2019); our goal here is to show the benefit of harnessing NAS in aid of SSL and not for comparison with more recent SOTA NAS approaches.
+
+While examining this idea appears to be straightforward, it requires careful design of experiments. The biggest hurdle is that both NAS and SSL require heavy compute resources; the former needs a search space large enough to cover a comprehensive range of architecture topologies, while the latter requires large datasets and batch sizes to be effective. This calls for an efficient framework to conduct our study. Furthermore, we need datasets large enough to pretrain the models on, and different enough to investigate the importance of data-dependent architectures in SSL. To meet our desiderata, we choose ProxylessNAS (Cai et al., 2018) as our NAS algorithm, MobileNet as our search space, and ImageNet-1K and iNat2021 (Van Horn et al., 2021) as our pretraining datasets.
+
+![](images/f884c9b3cec2a6d4dfcf115fe09d50c96657df2bb253448807cdf02bb24737c0.jpg)  
+Figure 5: Samples from datasets used in our study. We choose these datasets because of the apparent domain shift across them. ImageNet contains general yet coarsely categorized images compared to iNat2021, which contains an order of magnitude higher number of fine-grained categories; although some images look similar to each other, every image shown belongs to a different category highlighting the fine-grained nature of this dataset. The downstream datasets, except for CIFAR, are similarly fine-grained but on different domains.
+
+# 4.1 EXPERIMENTAL SETUP
+
+SSL objective. We use SimCLR (Chen et al., 2020a), one of the most well-established contrastive SSL framework. We follow the same augmentation methods and hyperparameter settings as in Chen et al. (2020a). While more recent SSL works exist (Caron et al., 2020; Chen & He, 2021), they are similar to SimCLR by utilizing a contrastive learning based objective. We adopt SimCLR for its simplicity and leave analysis of other SSL approaches for future work.
+
+NAS algorithm. We choose ProxylessNAS for two reasons: efficiency and flexibility. One-shot NAS algorithms produce an architecture topology in a two-step process. They first find an optimal cell structure by solving a proxy task over a small dataset (e.g., CIFAR-10) and a smaller architecture (e.g., 8 cells), and then stack/repeat the best found cell topology for the target task (e.g., ImageNet with 20 cells). This reduces complexity at the cost of flexibility and introduces an optimization gap (Chen et al., 2021), requiring strong correlation between proxy and the actual target datasets. In contrast, ProxylessNAS produces an architecture by directly optimizing on a target task, as it can significantly ameliorate memory requirements of one-shot NAS methods. To ensure flexibility, it uses a “supernet” with a broad range of candidate operations orchestrating depth (via zero operations), width (via wider convolutions), and block structure (by allowing for operations to differ by level). At each training step, it optimizes one “subnet” on the target task as a surrogate, which greatly reduces compute and memory requirements.
+
+Datasets. We use ImageNet-1K and iNat2021 to pretrain our models and evaluate them on their validation sets as well as on 10 downstream datasets used in Section 3. We deliberately choose the two pretraining datasets as they exhibit widely different characteristics, i.e., ImageNet-1K contains a variety of objects and scenes, while iNat2021 contains fine-grained species covering the tree of life. The former contains many inorganic object categories not present in the latter. This creates a domain gap, which allows us to investigate the importance of data-dependent architectures in SSL.
+
+iNat2021 contains 2.7 million images (twice the size of ImageNet) representing 10K species (ten times more than ImageNet). To investigate the effect of dataset size during architecture search and pretraining, we use both the full and the mini versions of iNat2021 – the latter contains 500K images representing the same 10K classes. This gives us pretraining datasets at three different scales: 500K (iNat2021-mini), 1.2M (ImageNet), 2.7M (iNat2021).
+
+Implementation details. For ProxylessNAS, we replace the original supervised classification loss with the contrastive loss of SimCLR and remove the latency loss as we currently do not consider hardware-constrained scenarios. We follow the original training schedule, i.e., a warmup phase for 40 epochs, which optimizes only the network weights and not the NAS parameters, followed by a search phase for 120 epochs. We use the SGD optimizer for network weights and the ADAM optimizer for NAS parameters, using initial learning rates of 0.25 and 0.1, respectively, and use the cosine decay schedule for both.
+
+To make our experiments tractable, we use the MobileNetV2 search space which typically yields 3 to 18 million parameters; the ResNet search space is larger, yielding 20 to 60 million parameters. Working with smaller models means we can use large batch sizes, which is important for contrastive learning to work effectively; we use the batch size 640 given our computational budget. The candidate set of NAS operations consists of mobile inverted bottleneck convolution (MBConv) with kernel sizes $\{ 3 , 5 , 7 \}$ , expansion ratios $\{ 3 , 6 \}$ and zero operations. A higher expansion ratio enables a wider network with more channels for convolutions, while zero operations allow for choosing to remove operations, thereby learning the optimal depth. Once the search is done, we take the architecture and discard the learned weights; we train it again from scratch using the SimCLR objective on different datasets. This allows us to compare different architectures on fair ground. After pretraining, we conduct linear evaluation on all downstream datasets.
+
+# 4.2 RESULTS AND DISCUSSION
+
+Self-supervised architectures outperform handcrafted architectures in SSL. Table 1 compares our self-supervised architectures to MobileNetV2 and ResNet18/50, by searching, pretraining, and evaluating on ImageNet-1K, iNat2021 and iNat2021-mini. We report linear evaluation results on validation splits. The results show that our selfsupervised architectures outperform MobileNetV2 by a large margin, even with similar parameters (about 3M).
+
+Table 1: Searched architectures vs. handcrafted architecture results. We search, pretrain, and evaluate ours on each of the three datasets in the last three columns. †SOTA results (in gray) from Chen et al. (2020a) for ImageNet and Cole et al. (2021) for iNat21 require larger batch sizes and longer training.   
+
+<table><tr><td>Model</td><td>Params</td><td>Batch</td><td>Epochs</td><td>ImageNet</td><td>iNat21</td><td>iNat21-mini</td></tr><tr><td>MobileV2</td><td>3.5M</td><td>640</td><td>100</td><td>41.9</td><td>30.2</td><td>13.6</td></tr><tr><td>Ours</td><td>3.3M</td><td>640</td><td>100</td><td>55.3</td><td>40.3</td><td>14.7</td></tr><tr><td>ResNet18</td><td>11M</td><td>640</td><td>100</td><td>49.8</td><td>30.3</td><td>20.1</td></tr><tr><td>ResNet50</td><td>23.5M</td><td>640</td><td>100</td><td>58.9</td><td>41.3</td><td>23.4</td></tr><tr><td>Ours</td><td>12-18M</td><td>640</td><td>100</td><td>59.1</td><td>43.8</td><td>25.1</td></tr><tr><td>ResNet50</td><td>23.5M</td><td>4096</td><td>1000</td><td>69.3t</td><td>50.6t</td><td>-</td></tr></table>
+
+The self-supervised architectures also beat ResNet18 and ResNet50, with even smaller model size than ResNet50. The superior downstream performance of our approach should not be attributed solely to NAS, as the architecture search was performed without ever solving the downstream tasks. It is rather the incorporation of NAS into SSL that improved the quality of representations, leading to downstream performance boost. This shows the effectiveness of learning both the architecture topology and its weights in SSL.
+
+Do self-supervised architectures generalize well to different data distributions? We take the three architectures searched on each dataset, discard their learned weights, and pretrain them on each dataset, yielding 9 pretrained models. We then evaluate the performance directly on validation splits of the respective datasets. Table 2 shows that transferring an architecture from $\mathrm { i N a t } 2 0 2 1$ to ImageNet leads to a marginal
+
+Table 2: Self-supervised architecture transfer results. We evaluate architectures in the cross-dataset setting, pretraining and evaluating the searched architectures across three datasets (last three columns).   
+
+<table><tr><td>Searched on</td><td>Params</td><td>ImageNet</td><td>iNat21</td><td>iNat21-mini</td></tr><tr><td>ImageNet</td><td>12-18M</td><td>59.1</td><td>21.5</td><td>23.9</td></tr><tr><td>iNat21</td><td>12-18M</td><td>58.3</td><td>43.8</td><td>27.9</td></tr><tr><td>iNat21-mini</td><td>12-18M</td><td>58.0</td><td>22.4</td><td>25.1</td></tr></table>
+
+performance drop compared to an architecture optimized directly on ImageNet $( 5 9 . 1 \%$ to $5 8 . 3 \%$ ); both these architectures still outperform handcrafted ResNet18 $( 4 9 . 8 \% )$ with a comparable model size. However, transferring an architecture from ImageNet to iNat2021 deteriorates performance significantly $( 4 3 . 8 \%$ to $2 1 . { \bar { 5 } } \%$ ). This implies an interesting finding, i.e., iNat21-searched architectures seem to be more resilient to domain shift than ImageNet-searched architectures. This could be due to the difference in dataset size (iNat21 has twice as many images as ImageNet), or due to the fine-grained nature of iNat21 resulting in an overall more difficult instance discrimination task (Chen et al., 2020a) that leads to more discriminative representations. The effect of dataset size on architecture search is also shown on iNat21-mini results. While transferring an architecture from ImageNet to iNat21-mini shows an expected drop in accuracy $( 2 5 . 1 \%$ to $2 3 . 9 \%$ ), transferring from the larger iNat21 improves performance $2 5 . 1 \%$ to $2 7 . 9 \%$ ). As both datasets are in the same domain and only differ in number of samples per class, higher search dataset size is the driving factor behind the gains in accuracy while pretraining on a smaller version of the dataset.
+
+Downstream transfer experiments. The results above show that self-supervised architectures are superior to handcrafted architectures when tested in in-distribution settings, which might reflect a practical use case of self-supervised pretraining in the real-world setting (e.g., one has access to only small labeled but large unlabeled data from the same distribution). We now evaluate our approach on a downstream transfer scenario with possible domain shift and with much smaller datasets. To this end, we again use the 10 downstream tasks used in Section 3, which contain datasets coming from both in-distributions and out-of-distributions relative to the pretraining datasets.
+
+Table 3: Downstream transfer results. We categorize downstream datasets as in-distribution (green) and outof-distribution (red) relative to the pretraining dataset based on class overlap; best viewed in color. We see that self-supervised architectures generally perform better on in-distribution downstream scenarios.   
+
+<table><tr><td>Pretrain Dataset</td><td>Arch.</td><td>Params</td><td>Pretrain Val. Set</td><td>CUB</td><td>NABirds</td><td>CIFAR10</td><td>Oxford Flowers</td><td>Stanford Dogs</td><td>Food101</td><td>Sport</td><td>Stanford Cars</td><td>MIT67</td><td>FGVC Aircraft</td></tr><tr><td rowspan="4">ImNet</td><td>MobileV2</td><td>3.5M</td><td>41.9</td><td>20.1</td><td>14.2</td><td>75.8</td><td>85.4</td><td>35.7</td><td>51.6</td><td>94.0</td><td>26.4</td><td>57.5</td><td>35.8</td></tr><tr><td>Ours</td><td>3.3M</td><td>55.3</td><td>31.8</td><td>24.4</td><td>78.8</td><td>91.7</td><td>49.2</td><td>61.7</td><td>94.3</td><td>30.4</td><td>62.5</td><td>38.1</td></tr><tr><td>ResNet18</td><td>11M</td><td>49.8</td><td>27.0</td><td>19.0</td><td>79.9</td><td>89.9</td><td>44.1</td><td>55.8</td><td>94.6</td><td>27.8</td><td>62.1</td><td>36.7</td></tr><tr><td>ResNet50</td><td>23.5M</td><td>58.9</td><td>31.4</td><td>24.6</td><td>85.9</td><td>92.8</td><td>52.3</td><td>65.7</td><td>94.3</td><td>35.1</td><td>69.6</td><td>42.0</td></tr><tr><td rowspan="3">iNat21</td><td>Ours</td><td>3-18M</td><td>59.1</td><td>34.3</td><td>26.1</td><td>81.8</td><td>92.2</td><td>51.0</td><td>64.7</td><td>94.7</td><td>33.2</td><td>66.1</td><td>39.4</td></tr><tr><td>ResNet18</td><td>11M</td><td>30.3</td><td>26.1</td><td>19.0</td><td>73.2</td><td>92.8</td><td>31.3</td><td>55.3</td><td>92.1</td><td>18.9</td><td>49.9</td><td>32.7</td></tr><tr><td>ResNet50</td><td>23.5M</td><td>41.3</td><td>31.2</td><td>23.2</td><td>75.1</td><td>95.1</td><td>39.7</td><td>65.1</td><td>94.3</td><td>22.3</td><td>55.0</td><td>37.6</td></tr><tr><td></td><td>Ours</td><td>3-18M</td><td>43.8</td><td>32.7</td><td>24.1</td><td>76.1</td><td>94.7</td><td>39.0</td><td>63.1</td><td>93.1</td><td>20.1</td><td>49.0</td><td>34.9</td></tr></table>
+
+Table 3 summarizes the results (we color code in/out-of-distribution datasets based on our crude categorization; see appendix for our justification). We first compare our self-supervised architectures to MobileNetV2 in the same parameter range (3.5M vs $3 . 3 \mathbf { M }$ ; top two rows). We notice that self-supervised architectures significantly outperform MobileNetV2 in all datasets regardless of distributional shift. This is encouraging (i.e., self-supervised architectures can learn generalizable representations) but at the same time not totally surprising (i.e., MobileNet is optimized for efficiency and not for accuracy). Next, we compare ours to ResNet18 that has a similar parameter range although belonging to a class of architectures much different from our search space. Ours outperforms ResNet18 on all pretraining and evaluation datasets by a considerable margin. This shows that our approach is generalizable and can outperform architectures in the ResNet18 search space even though they are generally more computationally expensive than the MobileNet search space.
+
+Finally, we compare ours to ResNet50 which is computationally heavier compared to ResNet18. We preface our analysis with a caveat that our self-supervised architectures are almost half the capacity of ResNet50, limiting their representational power. Keeping this in mind, we see that our approach starts to fail in some of the in-distribution and all of the out-of-distribution scenarios (red shaded cells). This is somewhat disappointing but perhaps expected: self-supervised architectures naturally encode inductive biases specific to the dataset they were optimized on. When a distributional shift happens, their performance can start deteriorating because out-of-domain data might require a different set of inductive biases. The strong performance by ResNet50 imply that the model might be striking the right balance across those datasets in terms of inductive biases, but our results in Table 1 and 2 show that ResNet50 can be less effective on newly developed datasets such as iNat2021.
+
+# 5 CONCLUSION
+
+This work lays the ground for moving beyond handcrafted architectures in SSL. By conducting large-scale experiments with 116 architectures and 11 downstream tasks, we established extensive empirical evidence showing that there isn’t one architecture that performs consistently well across different downstream scenarios in SSL. Motivated by this, we proposed to move beyond handcrafted architectures and learn both an architecture topology and its network weights in SSL. We provided convincing results demonstrating that the self-supervised architectures significantly outperform handcrafted MobileNetV2 and ResNet18 architectures on 11 downstream tasks, and competitively with ResNet50 even with almost half the model size. We re-emphasize that improvements are not solely due to NAS, as the architecture search was performed by solving SSL and not by optimizing directly on downstream tasks as in the typical NAS setting.
+
+Our work barely scratches the surface and opens up many doors for future directions. Will our findings hold for different architectures such as Transformers and different modalities such as video and text? How can we make architecture search more effective for SSL? Can ideas from domain generalization improve the transferability of self-supervised architectures in the out-of-distribution setting? Or is it even the right idea to expect learned architectures to generalize to widely different domains? We hope the readers are as excited as us to investigate these challenging questions.
+
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+
+# APPENDIX
+
+Fig. 6 provides an overview of our work. We show that no single handcrafted architecture performs consistently well across different tasks. It is therefore imperative to optimize for architecture topologies along with network weights for a specific task. Through extensive empirical results we show that such self-supervised architectures outperform their handcrafted counterparts in the same search space on the respective tasks.
+
+![](images/accf1356d2a88bdcd1df4d10e4829e92d0fd35757f21f6038b46032d63d2158b.jpg)  
+Figure 6: Conventional SSL frameworks learn network weights for a fixed handcrafted architecture (left). We show that learning architecture topologies along with their weights can improve performance in SSL (right).
+
+# A SUPERVISED TRAINING PERFORMANCE OF SELF-SUPERVISED ARCHITECTURES
+
+In addition to evaluating the performance of the searched architectures for SSL, we analyze their supervised training performance. We use the searched architectures and directly train them, from scratch without any pretraining, on downstream datasets using the supervised labels. Results are summarized in Table 4. We include architectures searched on ImageNet and iNat21, and MobileNetV2 for reference. It shows the searched architectures perform well even in the supervised setting, outperforming the handcrafted MobileNetV2 on most of the downstream datasets. However, a performance degradation is observed in out-of-distribution datasets like Stanford Cars and FGVC Aircraft. This is in line with the discussion in Section 4.2 of the main paper where the searched architecture performances deteriorate with distributional shift. Nevertheless, for the more in-distribution datasets, we obtain higher accuracies showing that the searched architectures are suitable for supervised training as well.
+
+Table 4: Supervised performance of searched architectures.   
+
+<table><tr><td></td><td>CUB</td><td>CIFAR10</td><td>CIFAR100</td><td>Food</td><td>Flowers</td><td>Sport</td><td>Cars</td><td>Aircraft</td></tr><tr><td>MobileNetV2</td><td>58.2</td><td>93.7</td><td>71.9</td><td>80.8</td><td>90.0</td><td>94.2</td><td>88.6</td><td>81.4</td></tr><tr><td>Ours (ImNet)</td><td>57.9</td><td>93.1</td><td>74.7</td><td>78.1</td><td>96.7</td><td>95.0</td><td>71.0</td><td>75.5</td></tr><tr><td>Ours (iNat21)</td><td>64.5</td><td>93.9</td><td>75.8</td><td>81.6</td><td>98.1</td><td>96.3</td><td>72.3</td><td>72.9</td></tr></table>
+
+# B CLASS MAPPING FROM IMAGENET TO DOWNSTREAM DATASETS
+
+We provide a justification for characterizing downstream datasets as in-distribution/out-ofdistribution with respect to ImageNet as shown in Table 3 of the main paper. We provide a rough class mapping between ImageNet and the 10 downstream datasets. Note that obtaining an exact class mapping is difficult due to only an approximate mapping existing between any 2 datasets. In addition, there can be classes which contribute to improved features for another class while still being semantically different. For example, zebra (n02391049) can contribute to improved features for horses (sorrel-n02389026) due to similar shapes. We now list datasets with corresponding ImageNet classes/superclasses. While some superclasses can contain additional subclasses in the WordNet hierarchy, we restrict to only those classes in the ImageNet-1k dataset. Numbers in bracket denote the total number of classes roughly overlapping.
+
+• CIFAR-10 (270): vehicle (n4524313), bird (n1503061), feline (n2120997), frog (n1639765), dog (n2084071), sorrel (n2389026)   
+• Stanford Dogs (120): dog (n2084071)   
+• CUB (60): bird (n1503061)   
+• NABirds (60): bird (n1503061)   
+• Food101 (20): nutriment (n7570720), beverage (n7881800), foodstuff (n7566340), sandwich (n7695965), bagel (n7693725), guacamole (n7583066), chocolate sauce (n7836838), carbonara (n7831146), french loaf (n7684084), pretzel (n7695742)   
+• Stanford Cars (10): car (n2958343)   
+• FGVC Aircraft (3): airliner (n2690373), warplane (n4552348), airship (n2692877)   
+• Oxford Flowers (2): yellow lady’s slipper (n12057211), daisy (n11939491)   
+• MIT67 (0): -   
+• Sports(0): -
+
+Due to the inductive biases encoded during the search process specific to the dataset it is searched on, the self-supervised architecture performs well on more in-distribution datasets like CUB or NABirds. However, we see that for datasets like Stanford Cars and subsequent ones, there is little direct class overlap with ImageNet classes. This leads to lesser images being available for self-supervised pretraining which are in-distribution for these datasets. Consequently, due to the relatively out-of-distribution nature of these datasets we see in Table 3 of the main paper, our selfsupervised architectures are outperformed by the ResNet-50 baseline.
+
+# C ADDITIONAL IMPLEMENTATION DETAILS
+
+We sample ResNet architectures by varying the number of blocks at each of the 4 stages choosing from the set of $2 , 3 , 4$ blocks and choose the ones in the parameter range shown in Fig. 3 while also fitting in GPU memory. For MobileNets, we have 7 sequences (stages) and a higher variation of the number of blocks from [2-6] while also choosing the width parameter from the set $1 . 0 , 1 . 2 , 1 . 4 , 1 . 6 , 1 . 8 , 2 . 0$ and choose the ones in the 2M-7M parameter range and fitting in GPU memory. Note that a high number of blocks in the earlier stages take significantly more GPU memory due to larger feature map sizes.
+
+For the architecture search phase, we use the optimizer hyperparameters as explained in Sec. 4.1 of the main paper. We use a weight decay of $4 e ^ { - 5 }$ for the weight parameters excluding batch normalization parameters. The initial convolution is a $3 { \tt X } 3$ convolution with stride 2. The network consists of 6 stages with 4 cells in the first 5 stages and 1 cell in the last stage. By default, the number of channels at each stage is 24, 40, 80, 96, 192, 320, which is multiplied by a constant width multiplier. We downsample it by a factor of 2 at the beginning of the first, second, third and fifth stage. Other architecture details are the default ones used in Cai et al. (2018). We use the same projection head as used normally for SimCLR Chen et al. (2020a) on top of the backbone network, which is a 2048 dimensional hidden layer and 128 dimensional output layer. For evaluation, we remove the projection head and use the output of the network backbone as the feature extractor. Augmentations are the same as in SimCLR with random resize scaling and cropping, flipping and color jitter. A temperature value of $\tau = 0 . 1$ is set for the contrastive loss.
+
+![](images/fd128bd086da0bc459cdfc02749b7abd25faae3e845591e86215b11e95baf798.jpg)  
+Figure 7: Self-supervised architectures for different pretraining datasets. MB3 and MB6 are the mobile inverted convolutions with expansion ratio of 3 and 6 respectively. We see that the majority of the preferred convolutions is MB6 $7 \times 7$ suggesting that the network prefers convolutions with more parameters for the self-supervised regime due to lots of data. For smaller datasets like iNat21Mini, MB6 convolutions are not as strongly preferred.
+
+# D VISUALIZING SELF-SUPERVISED ARCHITECTURES FOR DIFFERENT PRETRAINING DATASETS
+
+We visualize the types of convolutions searched at a $1 . 7 5 \mathrm { x }$ width multiplier for the 3 different pretraining datasets: ImageNet, iNat21 and iNat21Mini. Results are shown in Fig. 7. MB3 and MB6 are the mobile inverted convolutions with expansion ratio of 3 and 6 respectively. The grey lines denote the downsampling of image due to strided convolutions. All architectures are followed by a pooling layer to reduce the image size to $1 \times 1$ . In contrast to standard handcrafted architectures, larger $7 \times 7$ convolutions are preferred even in the later stages of the network. We also see that the majority of the preferred convolutions is MB6 $7 \times 7$ suggesting that the network prefers convolutions with more parameters for the self-supervised regime due to lots of data. This is less preferred in smaller datasets like iNat21Mini where MB3 convolutions are common especially in earlier stages of the network. It is difficult to draw conclusions on the type of network preferred between ImageNet and iNat21 showing that it is imperative to search for an optimal architecture rather than handcraft them.
+
+# E DOWNSTREAM DATASET PERFORMANCE CORRELATION WITH IMAGENET
+
+We show the downstream dataset correlation for all 10 downstream datasets in addition to ImageNet1K Deng et al. (2009): CIFAR10/100 Krizhevsky et al. (2009), Stanford Cars Krause et al. (2013) and Dogs Khosla et al. (2011), CUB-200 Welinder et al. (2010), MIT-67 Quattoni & Torralba (2009), SVHN Netzer et al. (2011), Flowers-102 Nilsback & Zisserman (2008), FGVC-Aircraft Maji et al. (2013), Sports8 Li & Fei-Fei (2007). These are shown for both ResNets (Fig. 8) and MobileNets (Fig. 9). We see similar results for the 5 datasets in addition to those shown in Fig. 3 of main paper. High correlation exists for datasets which are visually similar to ImageNet while it is less correlated for datasets which are out of domain. For MobileNets this correlation is even less pronounced with high variance in performance at higher ImageNet accuracies.
+
+![](images/0bba7b900089965a52388ce1f902c3c9b6cc53e5c1884cf0ca84918f10797416.jpg)  
+Figure 8: ImageNet performance correlation with 10 different downstream datasets for various ResNets. We show linear evaluation top-1 accuracy of ImageNet $\mathbf { \dot { x } }$ -axis) vs. different downstream datasets (y-axis) obtained from variations of ResNet; all models are pretrained on ImageNet-1K Deng et al. (2009) using SimCLR Chen et al. (2020a) under the same protocol. The solid lines and shaded areas indicate fitted regression models and their confidence intervals.
+
+![](images/70824b783eee5ebb816e486c629b8b2f0fb15cd94462f425e2695756953b96ee.jpg)  
+Figure 9: ImageNet performance correlation with 10 different downstream datasets for various MobileNets. We show linear evaluation top-1 accuracy of ImageNet $\mathbf { \widetilde { x } }$ -axis) vs. different downstream datasets (y-axis) obtained from variations of MobileNet; all models are pretrained on ImageNet-1K Deng et al. (2009) using SimCLR Chen et al. (2020a) under the same protocol. The solid lines and shaded areas indicate fitted regression models and their confidence intervals.
+
+![](images/f6a7746ca10efe2bcf8f6ff1e9ca1737ac46bc489713dce258d039d4835980b9.jpg)  
+Figure 10: Dataset performance correlation with 10 different datasets for various ResNets. We show the model size in terms of parameter counts $\mathbf { \widetilde { x } }$ -axis) vs. top-1 accuracy on different datasets obtained from variations of ResNet-like architectures; all models are pretrained on ImageNet-1K Deng et al. (2009) using SimCLR Chen et al. (2020a) under the same protocol. The solid lines and shaded areas indicate fitted regression models and their confidence intervals.
+
+![](images/43fda8482ba7f3742017ae263db37ba673a381a4bd5bda452e730fa8592579af.jpg)  
+Figure 11: Dataset performance correlation with 10 different datasets for various MobileNets. We show the model size in terms of parameter counts $\mathbf { \widetilde { x } }$ -axis) vs. top-1 accuracy on different datasets obtained from variations of MobileNet-like architectures; all models are pretrained on ImageNet-1K Deng et al. (2009) using SimCLR Chen et al. (2020a) under the same protocol. The solid lines and shaded areas indicate fitted regression models and their confidence intervals.
+
+![](images/7370ad50b35571a20834b016d378412186e599a7e683874e685da7feb210fd98.jpg)  
+Figure 12: Linear and rank correlation of ResNets between different pairs of 11 datasets We show correlation between every pair of top-1 accuracy on 11 downstream tasks obtained from ImageNet-pretrained ResNets. We see no strong correlation except for ones highly similar to the data the models were originally pretrained on, e.g., CIFAR-10/100 and Dogs120.
+
+![](images/efa643a0719af7cf0d7b6b0edc681bbc4514829f52ef93a07103374d1a8f41a3.jpg)
+
+![](images/7d32535940ca31ff4448230673c02a97883e7edfd1ef783b7da494664bd4913c.jpg)  
+Figure 13: Linear and rank correlation of MobileNets between different pairs of 11 datasets We show correlation between every pair of top-1 accuracy on 11 downstream tasks obtained from ImageNet-pretrained MobileNets. The correlation
+
+![](images/7b54e0d8565d1af9449e55fb75fc8fc58b76a0d58d19907da021698701e96d2a.jpg)
+
+# F DATASET PERFORMANCE AS A FUNCTION OF NUMBER OF PARAMETERS
+
+We show the dataset correlation with respect to number of parameters for 5 more datasets in addition to that shown in Fig. 4 of main paper. Fig. 10 summarizes the results for ResNets while Fig. 11 shows results for MobileNets. We see that similar results hold for the additional 5 datasets where more parameters, and consequently larger networks, does not always lead to better downstream performance.
+
+# G LINEAR AND RANK CORRELATION FOR RESNETS/MOBILENETS
+
+We show the summary of the correlation across different datasets for both ResNets (Fig. 12) and MobileNets (Fig. 13). In addition to Spearman’s rank correlation coefficient, we also show Pearson’s linear correlation coefficient. While Pearson’s linear coefficient is higher in the case of MobileNets, the linear fit still exhibits high variance for higher ImageNet accuracies, as seen in Fig. 9.
+
+![](images/e7eab649c5fd42ef4d4f2ac0653c9c6227eb0ae9a13f15160c00b95f22820cec.jpg)  
+Figure 14: ImageNet performance correlation with 10 different downstream datasets for various ResNets, MobileNets and searched architectures. The searched architectures outperform MobileNets while being comparable with ResNets at fewer parameters.
+
+# H DATASET LICENSES
+
+Table 5 lists some datasets we used and their licenses.
+
+Table 5: Licenses of datasets.   
+
+<table><tr><td>Dataset</td><td>License</td></tr><tr><td>CIFAR-10 Krizhevsky et al. (2009)</td><td>MIT</td></tr><tr><td>CIFAR-100 Krizhevsky et al. (2009)</td><td>MIT</td></tr><tr><td>ImageNet Deng et al. (2009)</td><td>BSD 3-Clause</td></tr><tr><td>Sport8 Li &amp; Fei-Fei (2007)</td><td>CCO:Public Domain</td></tr><tr><td>Stanford Dogs Khosla et al. (2011)</td><td>BSD3-Clause</td></tr><tr><td>Stanford Cars Krause et al. (2013)</td><td>BSD 3-Clause</td></tr><tr><td>CUB-200 Welinder et al. (2010)</td><td>Data files @ Original Authors</td></tr><tr><td>MIT-67 Quattoni &amp; Torralba a (2009)</td><td>MIT</td></tr><tr><td>SVHN Netzer et al. (2011)</td><td>CCO:Public Domain</td></tr><tr><td>Flowers-102 Nilsback &amp; Zisserman (2008)</td><td>GNU General Public License,version 2</td></tr></table>

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+    {
+        "type": "text",
+        "text": "MOVING BEYOND HANDCRAFTED ARCHITECTURES IN SELF-SUPERVISED LEARNING ",
+        "text_level": 1,
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+        "text": "Anonymous authors Paper under double-blind review ",
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+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+        ],
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+    },
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+        "text": "The current literature on self-supervised learning (SSL) focuses on developing learning objectives to train neural networks more effectively on unlabeled data. The typical development process involves taking well-established architectures, e.g., ResNet or ViT demonstrated on ImageNet, and using them to evaluate newly developed objectives on downstream scenarios. While convenient, this neglects the role of architectures which has been shown to be crucial in the supervised learning literature. In this work, we establish extensive empirical evidence showing that a network architecture plays a significant role in contrastive SSL. We conduct a large-scale study with over 100 variants of ResNet and MobileNet architectures and evaluate them across 11 downstream scenarios in the contrastive SSL setting. We show that there is no one network that performs consistently well across the scenarios. Based on this, we propose to learn not only network weights but also architecture topologies in the SSL regime. We show that “self-supervised architectures” outperform popular handcrafted architectures (ResNet18 and MobileNetV2) while performing competitively with the larger and computationally heavy ResNet50 on major image classification benchmarks (ImageNet-1K, iNat2021, and more). Our results suggest that it is time to consider moving beyond handcrafted architectures in contrastive SSL and start thinking about incorporating architecture search into self-supervised learning objectives. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
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+        "text": "Self-supervised learning (SSL) achieves impressive results on challenging tasks involving image, video, audio, and text. Models pretrained on large unlabeled data perform nearly as good and sometimes even better than their supervised counterparts (Caron et al., 2020; Chen & He, 2021). So far, the focus has been on designing effective learning objectives – e.g., pretext tasks (Gidaris et al., 2018; Caron et al., 2018), contrastive (Oord et al., 2018; Chen et al., 2020a) and noncontrastive (Grill et al., 2020) tasks – together with empirical (Cole et al., 2021; Feichtenhofer et al., 2021) and theoretical (Arora et al., 2019; Poole et al., 2019) studies providing key insights and underpinnings. Recent works propose new objectives with a different class of network architectures such as vision transformers (ViT) (Bao et al., 2021) and masked autoencoders (He et al., 2022). ",
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+        "text": "However, there has been little focus on the role of architectures in SSL. Currently, the de facto protocol in SSL is to take architectures that perform well on established benchmarks in the supervised setting and to adapt them to the self-supervised setting by plugging in different learning objectives. For example, several existing work on contrastive learning use ResNet (He et al., 2016) as the backbone (Chen et al., 2020a; He et al., 2020). This is partly for convenience. Evaluating different architectures in SSL is computationally expensive; selecting an architecture in advance and fixing it throughout makes it easy to evaluate different learning objectives. This also stems from strong empirical success of those architectures in transfer learning, e.g., CNNs trained on large labeled data provide “unreasonable effectiveness” (Sun et al., 2017; Zhang et al., 2018; Sejnowski, 2020) in a variety of downstream cases. Nonetheless, one implicit assumption is that an architecture that works well in the supervised learning scenario will continue to be effective in the SSL regime. ",
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+        "text": "We argue that this assumption is incorrect and dangerous. It is valid only to a limited extent and the performance starts deteriorating significantly when SSL is conducted on data whose distribution deviates much from the original distribution the architecture was trained on. This is counter to the promise of SSL, where one can learn optimal representation for a wide range of tasks. One main reason for performance degradation is that different data distributions benefit from different inductive biases: An architecture with specific layer types and the wiring between them naturally encodes inductive biases, which may be optimal only for a certain data distribution (e.g., objectcentric imagery such as ImageNet) and not for others (e.g., medical and satellite imagery). In fact, numerous studies have shown that standard “recipes” for architecture design do not translate well across different data distributions (Tuggener et al., 2021; Dey et al., 2021; Kolesnikov et al., 2019). ",
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+        "text": "The main objective of this work is to show that the choice of network architecture crucially matters in SSL, and that it is not easy to handcraft architectures that are effective across different SSL scenarios. To see this, recall that the goal of SSL is to learn data representations capturing important features and attributes that generalize well across various downstream tasks. There has been extensive literature on the expressivity of neural networks (Raghu et al. (2017); Zhang et al. (2021a) and references therein); one of the important conclusions is that the network topology plays a significant role in determining the expressivity, i.e., the kinds of functions a network can approximate is bounded by the network capacity and available sample size in the finite sample regime. This implies that, in practice, SSL with a fixed architecture learns representations only within the scope of function space induced by the pre-selected architecture topology, and therefore, the ultimate success of SSL can be achieved when it finds optimal architecture from certain search space in conjunction with its weights for specific data distributions. ",
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+        "text": "In this paper, we establish extensive empirical evidence showing that architecture matters in selfsupervised learning. We do this in two sets of large-scale studies. First, we sample 116 variants of ResNet (He et al., 2016) and MobileNet (Sandler et al., 2018) architectures with different topologies and evaluate them on 11 downstream tasks in the SSL setting. We pretrain all models under the same setting, optimizing the SimCLR objective (Chen et al., 2020a) on ImageNet (Deng et al., 2009), and investigate if there exist any correlation between these models in downstream performance on different datasets. We observe no strong correlation, except for tasks highly similar to ImageNet. We further show that ImageNet downstream performance, the gold standard benchmark in the SSL literature, is not indicative of performance on other downstream tasks. This implies that we need to be careful in choosing an architecture for evaluating any newly developed SSL objectives, as one might get different conclusions based on different network architectures. ",
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+        "text": "This subsequently raises the question: Can we improve SSL by learning not only network weights but also architectures directly optimized for the given dataset? It removes the burden of manually searching for effective architectures in SSL, and if we succeed, it can substantially improve performance of SSL. To test this hypothesis, as the second set of our study, we apply a well-established NAS algorithm (Cai et al., 2018) to the SSL setting. Unlike the typical NAS setting that optimize on a labeled target dataset, we search for optimal architectures directly on an unlabeled pretraining dataset via contrastive learning (Chen et al., 2020a). We evaluate our $\\mathrm { ^ { 6 6 } N A S } + \\mathrm { S S L } ^ { \\mathrm { 3 } }$ framework on datasets with different distributions, ImageNet-1K and iNat 2021 (Van Horn et al., 2021), and show that self-supervised architectures consistently outperform handcrafted ones in the same parameter range (MobileNetV2 and ResNet18) across 11 downstream tasks. This provides strong evidence suggesting the importance of learning architecture topologies in addition to their weights in SSL. ",
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+        "text": "Our work focuses on studying the role of architectures in contrastive SSL with the SimCLR framework for CNN-based architectures such as ResNets and MobileNets. As a first step in this direction, we provide an in-depth analysis through large-scale experiments in this specific (yet limited) setting. Extending our study to different SSL approaches (He et al., 2020; Grill et al., 2020; He et al., 2022) and architectures such as ViTs would be an interesting direction but beyond the scope of this paper. ",
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+        "text": "In summary, our main contributions are: 1) We establish extensive evidence showing that there isn’t one network architecture performing consistently well across different downstream scenarios in the SimCLR setting. We show this using 116 variants of ResNet and MobileNet architectures pretrained on ImageNet and evaluated on 11 downstream datasets. 2) We show that ImageNet performance (the gold standard in SSL benchmark) is not always indicative of downstream performance. This means that findings about SSL objectives shown only on ImageNet do not generalize across other data distributions. 3) We propose to self-supervise a CNN architecture topology and its network weights on unlabeled data. We show that self-supervised architectures outperform handcrafted ones in a similar parameter range for the SimCLR setting across different downstream datasets. ",
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+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+        "text": "Role of architectures in SSL. There has been significant progress in learning representations via SSL (Gidaris et al., 2018; Chen et al., 2020a;c; Caron et al., 2020). Ericsson et al. (2021) provide an overview of different SSL setups and their performance on downstream tasks. Most works focus on improving self-supervised objectives to develop better representations while keeping architectures fixed. Our focus is orthogonal to this line of work. We study the role of architectures in SSL and investigate the benefits of self-supervising architecture topologies along with network weights. ",
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+        "text": "Similar to ours, Kornblith et al. (2019) analyze transfer performance of different architectures pretrained on ImageNet across several downstream datasets. However, they focused on supervised learning, which need not translate to self-supervised setups (Kolesnikov et al., 2019). Caron et al. (2021) propose a self-distillation based SSL objective and compare the performance of ResNets with ViTs. In contrast, we focuse on contrastive SSL and the importance of architecture topologies for the class of CNNs. Kolesnikov et al. (2019) is the most related to ours but are limited to pretext-task based SSL and show results only for a few variants of VGG and ResNet. In comparison, we conduct a study on a much larger scale in a contrastive learning setup. Also, we provide insights into generalization performance of the networks across different datasets. Crucially, unlike all previous work in SSL, we propose to learn both architecture topologies and their network weights. ",
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+        "text": "NAS and SSL. The literature on NAS has rapidly progressed in the past years; we refer the reader to the NAS survey by Elsken et al. (2019). Here we focus on the most directly relevant work to SSL. Mellor et al. (2021) search for networks without any training and verify their effectiveness of supervised benchmarks. Liu et al. (2020) show that highly performant architectures can be found without using any supervised labels during the search itself, and propose using DARTS (Liu et al., 2019) with self-supervised proxy tasks to search for architectures which will perform well on supervised datasets. In a similar vein, Zhang et al. (2021b) use random labels during the search phase of NAS, and Yan et al. (2020) learn representations of architectures via SSL and use them to improve NAS. Li et al. (2021) introduce a new self-supervised training scheme to search for architectures in a new hybrid search space. One common theme in this line of work is that they aim to harness SSL in aid of NAS. In contrast, we aim to utilize NAS to hunt for architectures which will perform well for self-supervised learning, thereby harnessing NAS in aid of SSL. ",
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+        "type": "text",
+        "text": "3 DOES ONE NETWORK RULE THEM ALL IN SSL?",
+        "text_level": 1,
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+        "text": "We conduct a large scale study to investigate whether a particular architecture topology can be consistently effective across a wide range of SSL scenarios. To this end, we sample 116 models with different architecture topologies and analyze their performance on 11 downstream tasks. ",
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+        "text": "To maximize generality while taming complexity of our study, we choose two most representative CNN architectures: ResNets and MobileNets. The former is the de facto backbone for numerous modern visual models (Kirillov et al., 2019; Wu et al., 2019) and SSL approaches (Caron et al., 2020; Chen & He, 2021), while the latter is used in low-resource setups (Cheng et al., 2017). We create 69 ResNet-like and 47 MobileNet-like architectures for our evaluation, varying the number of blocks at each of the 4 stages of a ResNet, the block structure (BasicBlock and Bottleneck), width, and number of groups. For MobileNet, we vary the width multiplier parameter and the number of blocks in each of the 6 stages. We choose the SimCLR objective (Chen et al., 2020a) for our experiments and pretrain each of the 116 architectures on ImageNet-1K. Owing to the large scale nature of the experiments and computational constraints, we limit the pretraining to 100 epochs and a batch size of 512. Each of these jobs takes roughly 1 day to finish on a system with $8 \\times \\mathrm { ~ V 1 0 0 ~ }$ (32GB) GPUs. We report linear evaluation results in all cases. ",
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+        "text": "ImageNet performance is not indicative of downstream performance for SSL. To examine the correlation between ImageNet vs. downstream performance, we compute the Spearman’s rank correlation coefficient $\\rho$ on top-1 validation accuracy between every dataset pair, shown in Fig. 1. We also show scatter plots in Fig. 2 revealing the relationship between ImageNet vs. downstream performance on the most representative cases; the complete set of scatter plots are in the appendix. ",
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+        "text": "For ResNet-like architectures (Fig. 2 top row), we see strong correlation between ImageNet and CIFAR-100 ( $\\rho = . 8 )$ ) as these datasets contain similar categories; about 90 classes in CIFAR-100 are the same class or a superclass in ImageNet. A similar observation is made (in Fig. 1) for Stanford Dogs $( \\rho = . 7 7 )$ due to the 120 dog categories in ImageNet. However, we observe high variance in transfer performance for out-of-domain datasets, e.g., Flowers $( \\rho = . 3 5 )$ , represented by only two ImageNet categories (daisy and yellow lady slipper). Correlation becomes negative on Stanford Cars $( \\rho = - . 2 9 )$ and FGVC Aircraft $( \\rho = - . 2 4 )$ , likely because ImageNet contains only a few categories of cars (10 classes) and aircraft (4 classes). Low correlation means network ranks are inconsistent and models performing well on ImageNet do not keep their precedence in other tasks. ",
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+        "image_caption": [
+            "Figure 2: ImageNet performance is not indicative of downstream performance in SSL. We show linear evaluation top-1 accuracy of ImageNet (x-axis) vs. downstream datasets (y-axis) obtained from variations of ResNet (top) and MobileNet (bottom); all models are pretrained on ImageNet-1K using SimCLR under the same protocol. The solid lines and shaded areas indicate fitted regression models and their confidence intervals. "
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+        "text": "For MobileNet-like architectures (Fig. 2 bottom row), we see overall much lower correlation with ImageNet performance; the ResNet space showed correlation at least for in-distribution datasets. For e.g., we see correlation between ImageNet and CIFAR-100 drops from $\\rho = . 8$ (ResNet) to $\\rho \\ = \\ . 2 3$ (MobileNet). For other datasets like MIT67, correlation is higher $( \\rho ~ = ~ . 5 2 )$ but still less meaningful due to high variability in performance (notice the cluster around $42 \\%$ accuracy). These results indicate that MobileNets are even less tuned towards ImageNet than ResNets and any handcrafted architectures in this space is likely to be suboptimal on ImageNet and other downstream datasets. ",
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+        "image_caption": [
+            "Figure 1: ImageNet performance isn’t indicative of downstream performance in SSL. We show rank correlation between each pair of top-1 accuracy on 11 downstream tasks obtained from ImageNet-pretrained ResNets. We see no strong correlation except for ones similar to ImageNet, e.g., CIFAR-10/100 and Dogs120. "
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+        "text": "Our results highlight that the same architecture recipes in terms of ImageNet accuracy, which is frequently used as a predictor for various selfsupervised tasks, do not work well for different datasets in the SimCLR setting. ",
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+        "text": "Larger networks do not always perform better in contrastive SSL. In general, large-parameter models yield better performance on ImageNet, both in supervised (Kornblith et al., 2019) and SSL setups (Chen et al., 2020a). We examine whether this trend holds for downstream datasets some of which are widely different from ImageNet. We follow the same setup as above and compute the ",
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+        "image_caption": [
+            "Figure 3: Larger models do not always perform better in SSL. We show the model size in terms of parameter counts ( $\\mathbf { \\widetilde { x } }$ -axis) vs. top-1 accuracy on different datasets obtained from variations of ResNet (top) and MobileNet (bottom) architectures; all models are pretrained on ImageNet-1K using SimCLR under the same protocol. "
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+        "type": "text",
+        "text": "correlation between number of model parameters and top-1 validation accuracy on different datasets.   \nFig. 3 shows scatter plots of the most representative results; full results are in the appendix. ",
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+        "type": "text",
+        "text": "For ResNet-like architectures (Fig. 3 top row), we do not see strong trends indicating larger models always perform better. In fact, on some downstream tasks we see negative correlation (Aircraft, $\\rho =$ $- . 3 7 )$ with lighter networks being favored for better performance, or even non-linear relationship, e.g., notice the slight “U” pattern on Stanford Cars $\\zeta = - . 1 4 )$ , indicating the behavior of ResNets on these datasets are wildly unexpected. Unsurprisingly, there is strong correlation with ImageNet $( \\rho = . 8 5 )$ and CIFAR-100 $\\langle \\rho = . 8 7 \\rangle$ , likely because ResNets are heavily hand-tuned on the kinds of images observed in these datasets. These results clearly suggest that the same architecture recipes which work well for ImageNet (increasing parameters through depth and width) do not hold for other downstream scenarios. ",
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+        "type": "text",
+        "text": "For MobileNet-like architectures (Fig. 3 bottom row), we see that there exists almost no interpretable trend. The fitted regression models (solid lines) and their confidence intervals (shaded area) show that the relationships are highly non-linear and non-monotonic; we shouldn’t read too much into the correlation coefficients (reported for completeness). These results indicate that MobileNets do not favor any one architecture and it is heavily reliant on the dataset it is trained on. ",
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+        "text": "Our results suggest that larger models are not always better in SSL; lighter models can outperform heavier models on tasks like FGVC Aircraft. Previous work (Chen et al., 2020b) showed that increasing network depth/width improves downstream performance. However, they vary depth at a coarser level with 50/100/150 layers and width with $1 \\times / 2 \\times$ , leading to a large swing in network parameters (24-795M); here we show the same is not true at a finer level. Our results provide evidence that ResNet architectures are tuned to scale well on ImageNet but not the others; the trend is even weaker for MobileNet-like architectures, suggesting they are optimized for computational efficiency and not for achieving high accuracy on any particular dataset. ",
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+        "text": "There is no winner in the battle of top vs. bottom heavy networks in SSL. Raghu et al. (2017) showed that CNNs are more sensitive to lower (initial) layer weights, suggesting that “not all weights are created equal” across layers. This raises the question: If CNNs are more sensitive to lower layers, will increasing the parameter count for lower layers yield better performance in SSL? To get insights into this, we split a network into two halves: “top” (layers closer to output) and “bottom” (closer to input). We use the ratio of top to bottom parameters as a measure of networks being “top-heavy” (high ratio) or “bottom-heavy” (low ratio). ",
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+        "type": "text",
+        "text": "Fig. 4 shows the downstream accuracy for ImageNet, Stanford Cars and Sports against this ratio. From the top-left subplot, we see that ResNet-ImageNet accuracy tend to increase with high top:bottom ratio, showing top-heavy networks generally perform better than the bottom-heavy counterparts. However, we no longer observe such trend in other datasets, and with MobileNets (Fig. 4 bottom row) we do not see such trend even for ImageNet. ",
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+        "text": "An important point to realize: It is necessary to allocate the right portion of parameters to different layers of a given network topology, instead of allocating parameters in the top-heavy or bottom-heavy fashion assuming one will generally lead to better performance. ResNets and many other handcrafted CNNs (Simonyan & Zisserman, 2014; Szegedy et al., 2016; Huang et al., 2017) are usually top-heavy because of GPU memory limits; bottom-heavy networks occupy more memory in terms of activation maps. MobileNets alleviate this to some extent with lighter convolutions, allowing it to have more parameters in early layers. In object detection, Liang et al. (2019) also show that allocation of computational resources in the backbone is important for improved performance. While this architectural difference provides explanations about the wildly different trends we observe above, the key message here is that one recipe (top vs. bottom heavy) does not apply equally to different architectures, bolstering our claim that one network doesn’t rule them all in SSL. ",
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+        "text": "Key takeaway: We need to move beyond handcrafted architectures in SSL. The three main observations above imply that finding an optimal architecture could be an important missing piece for selfsupervised learning. Our results show that the current practice in designing SSL objectives – i.e., optimizing for ImageNet performance based on ResNet backbones – could lead to misleading conclusions which do not generalize to other downstream scenarios. Also, the general belief that “the larger the better” in model size do not really hold in SSL, e.g., smaller ",
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+        "image_caption": [
+            "Figure 4: There is no winner in the top vs. bottom battle in SSL. Except for ResNet-ImageNet (top-left), we see no strong trend that suggests either top-heavy or bottom-heavy networks perform better. "
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+        "type": "text",
+        "text": "ResNets can outperform larger ones even if they are pretrained following the same protocol. Let’s say, based on these observations, one is compelled to hunt for a new architecture geared specifically towards SSL. Our top vs. bottom analysis suggests that it can be extremely tricky to find the right architecture topology with optimal parameter allocation across different layers. All this suggests that it is time to consider moving beyond handcrafted architectures in SSL and start thinking about searching for optimal architectures as part of self-supervised learning objectives. ",
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+        "type": "text",
+        "text": "4 NEURAL ARCHITECTURE SEARCH FOR SSL ",
+        "text_level": 1,
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+        "text": "We turn to the idea of learning both the architecture topology and its network weights in an SSL framework, using NAS to improve SSL (rather than using SSL to improve NAS). It is important to draw a clear distinction between our idea and prior work that used SSL to improve NAS (Li et al., 2021; Liu et al., 2020; 2019; Yan et al., 2020; Zhang et al., 2021b) as well as work that used NAS to improve supervised learning (Elsken et al., 2019); our goal here is to show the benefit of harnessing NAS in aid of SSL and not for comparison with more recent SOTA NAS approaches. ",
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+        "type": "text",
+        "text": "While examining this idea appears to be straightforward, it requires careful design of experiments. The biggest hurdle is that both NAS and SSL require heavy compute resources; the former needs a search space large enough to cover a comprehensive range of architecture topologies, while the latter requires large datasets and batch sizes to be effective. This calls for an efficient framework to conduct our study. Furthermore, we need datasets large enough to pretrain the models on, and different enough to investigate the importance of data-dependent architectures in SSL. To meet our desiderata, we choose ProxylessNAS (Cai et al., 2018) as our NAS algorithm, MobileNet as our search space, and ImageNet-1K and iNat2021 (Van Horn et al., 2021) as our pretraining datasets. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/f884c9b3cec2a6d4dfcf115fe09d50c96657df2bb253448807cdf02bb24737c0.jpg",
+        "image_caption": [
+            "Figure 5: Samples from datasets used in our study. We choose these datasets because of the apparent domain shift across them. ImageNet contains general yet coarsely categorized images compared to iNat2021, which contains an order of magnitude higher number of fine-grained categories; although some images look similar to each other, every image shown belongs to a different category highlighting the fine-grained nature of this dataset. The downstream datasets, except for CIFAR, are similarly fine-grained but on different domains. "
+        ],
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+    {
+        "type": "text",
+        "text": "4.1 EXPERIMENTAL SETUP ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "SSL objective. We use SimCLR (Chen et al., 2020a), one of the most well-established contrastive SSL framework. We follow the same augmentation methods and hyperparameter settings as in Chen et al. (2020a). While more recent SSL works exist (Caron et al., 2020; Chen & He, 2021), they are similar to SimCLR by utilizing a contrastive learning based objective. We adopt SimCLR for its simplicity and leave analysis of other SSL approaches for future work. ",
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+    {
+        "type": "text",
+        "text": "NAS algorithm. We choose ProxylessNAS for two reasons: efficiency and flexibility. One-shot NAS algorithms produce an architecture topology in a two-step process. They first find an optimal cell structure by solving a proxy task over a small dataset (e.g., CIFAR-10) and a smaller architecture (e.g., 8 cells), and then stack/repeat the best found cell topology for the target task (e.g., ImageNet with 20 cells). This reduces complexity at the cost of flexibility and introduces an optimization gap (Chen et al., 2021), requiring strong correlation between proxy and the actual target datasets. In contrast, ProxylessNAS produces an architecture by directly optimizing on a target task, as it can significantly ameliorate memory requirements of one-shot NAS methods. To ensure flexibility, it uses a “supernet” with a broad range of candidate operations orchestrating depth (via zero operations), width (via wider convolutions), and block structure (by allowing for operations to differ by level). At each training step, it optimizes one “subnet” on the target task as a surrogate, which greatly reduces compute and memory requirements. ",
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+        "type": "text",
+        "text": "Datasets. We use ImageNet-1K and iNat2021 to pretrain our models and evaluate them on their validation sets as well as on 10 downstream datasets used in Section 3. We deliberately choose the two pretraining datasets as they exhibit widely different characteristics, i.e., ImageNet-1K contains a variety of objects and scenes, while iNat2021 contains fine-grained species covering the tree of life. The former contains many inorganic object categories not present in the latter. This creates a domain gap, which allows us to investigate the importance of data-dependent architectures in SSL. ",
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+    {
+        "type": "text",
+        "text": "iNat2021 contains 2.7 million images (twice the size of ImageNet) representing 10K species (ten times more than ImageNet). To investigate the effect of dataset size during architecture search and pretraining, we use both the full and the mini versions of iNat2021 – the latter contains 500K images representing the same 10K classes. This gives us pretraining datasets at three different scales: 500K (iNat2021-mini), 1.2M (ImageNet), 2.7M (iNat2021). ",
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+    {
+        "type": "text",
+        "text": "Implementation details. For ProxylessNAS, we replace the original supervised classification loss with the contrastive loss of SimCLR and remove the latency loss as we currently do not consider hardware-constrained scenarios. We follow the original training schedule, i.e., a warmup phase for 40 epochs, which optimizes only the network weights and not the NAS parameters, followed by a search phase for 120 epochs. We use the SGD optimizer for network weights and the ADAM optimizer for NAS parameters, using initial learning rates of 0.25 and 0.1, respectively, and use the cosine decay schedule for both. ",
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+    {
+        "type": "text",
+        "text": "To make our experiments tractable, we use the MobileNetV2 search space which typically yields 3 to 18 million parameters; the ResNet search space is larger, yielding 20 to 60 million parameters. Working with smaller models means we can use large batch sizes, which is important for contrastive learning to work effectively; we use the batch size 640 given our computational budget. The candidate set of NAS operations consists of mobile inverted bottleneck convolution (MBConv) with kernel sizes $\\{ 3 , 5 , 7 \\}$ , expansion ratios $\\{ 3 , 6 \\}$ and zero operations. A higher expansion ratio enables a wider network with more channels for convolutions, while zero operations allow for choosing to remove operations, thereby learning the optimal depth. Once the search is done, we take the architecture and discard the learned weights; we train it again from scratch using the SimCLR objective on different datasets. This allows us to compare different architectures on fair ground. After pretraining, we conduct linear evaluation on all downstream datasets. ",
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+    {
+        "type": "text",
+        "text": "4.2 RESULTS AND DISCUSSION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Self-supervised architectures outperform handcrafted architectures in SSL. Table 1 compares our self-supervised architectures to MobileNetV2 and ResNet18/50, by searching, pretraining, and evaluating on ImageNet-1K, iNat2021 and iNat2021-mini. We report linear evaluation results on validation splits. The results show that our selfsupervised architectures outperform MobileNetV2 by a large margin, even with similar parameters (about 3M). ",
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+    {
+        "type": "table",
+        "img_path": "images/938163a754c962a7f4d6be4497431cddf7d645e746efb5244f6d37d3881655c3.jpg",
+        "table_caption": [
+            "Table 1: Searched architectures vs. handcrafted architecture results. We search, pretrain, and evaluate ours on each of the three datasets in the last three columns. †SOTA results (in gray) from Chen et al. (2020a) for ImageNet and Cole et al. (2021) for iNat21 require larger batch sizes and longer training. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Params</td><td>Batch</td><td>Epochs</td><td>ImageNet</td><td>iNat21</td><td>iNat21-mini</td></tr><tr><td>MobileV2</td><td>3.5M</td><td>640</td><td>100</td><td>41.9</td><td>30.2</td><td>13.6</td></tr><tr><td>Ours</td><td>3.3M</td><td>640</td><td>100</td><td>55.3</td><td>40.3</td><td>14.7</td></tr><tr><td>ResNet18</td><td>11M</td><td>640</td><td>100</td><td>49.8</td><td>30.3</td><td>20.1</td></tr><tr><td>ResNet50</td><td>23.5M</td><td>640</td><td>100</td><td>58.9</td><td>41.3</td><td>23.4</td></tr><tr><td>Ours</td><td>12-18M</td><td>640</td><td>100</td><td>59.1</td><td>43.8</td><td>25.1</td></tr><tr><td>ResNet50</td><td>23.5M</td><td>4096</td><td>1000</td><td>69.3t</td><td>50.6t</td><td>-</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "The self-supervised architectures also beat ResNet18 and ResNet50, with even smaller model size than ResNet50. The superior downstream performance of our approach should not be attributed solely to NAS, as the architecture search was performed without ever solving the downstream tasks. It is rather the incorporation of NAS into SSL that improved the quality of representations, leading to downstream performance boost. This shows the effectiveness of learning both the architecture topology and its weights in SSL. ",
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+    {
+        "type": "text",
+        "text": "Do self-supervised architectures generalize well to different data distributions? We take the three architectures searched on each dataset, discard their learned weights, and pretrain them on each dataset, yielding 9 pretrained models. We then evaluate the performance directly on validation splits of the respective datasets. Table 2 shows that transferring an architecture from $\\mathrm { i N a t } 2 0 2 1$ to ImageNet leads to a marginal ",
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+        "img_path": "images/404693e87872fe92f91e8bd92a5e0ce727a4874fcfcd2a42b36b15ac6edfd306.jpg",
+        "table_caption": [
+            "Table 2: Self-supervised architecture transfer results. We evaluate architectures in the cross-dataset setting, pretraining and evaluating the searched architectures across three datasets (last three columns). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Searched on</td><td>Params</td><td>ImageNet</td><td>iNat21</td><td>iNat21-mini</td></tr><tr><td>ImageNet</td><td>12-18M</td><td>59.1</td><td>21.5</td><td>23.9</td></tr><tr><td>iNat21</td><td>12-18M</td><td>58.3</td><td>43.8</td><td>27.9</td></tr><tr><td>iNat21-mini</td><td>12-18M</td><td>58.0</td><td>22.4</td><td>25.1</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "performance drop compared to an architecture optimized directly on ImageNet $( 5 9 . 1 \\%$ to $5 8 . 3 \\%$ ); both these architectures still outperform handcrafted ResNet18 $( 4 9 . 8 \\% )$ with a comparable model size. However, transferring an architecture from ImageNet to iNat2021 deteriorates performance significantly $( 4 3 . 8 \\%$ to $2 1 . { \\bar { 5 } } \\%$ ). This implies an interesting finding, i.e., iNat21-searched architectures seem to be more resilient to domain shift than ImageNet-searched architectures. This could be due to the difference in dataset size (iNat21 has twice as many images as ImageNet), or due to the fine-grained nature of iNat21 resulting in an overall more difficult instance discrimination task (Chen et al., 2020a) that leads to more discriminative representations. The effect of dataset size on architecture search is also shown on iNat21-mini results. While transferring an architecture from ImageNet to iNat21-mini shows an expected drop in accuracy $( 2 5 . 1 \\%$ to $2 3 . 9 \\%$ ), transferring from the larger iNat21 improves performance $2 5 . 1 \\%$ to $2 7 . 9 \\%$ ). As both datasets are in the same domain and only differ in number of samples per class, higher search dataset size is the driving factor behind the gains in accuracy while pretraining on a smaller version of the dataset. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Downstream transfer experiments. The results above show that self-supervised architectures are superior to handcrafted architectures when tested in in-distribution settings, which might reflect a practical use case of self-supervised pretraining in the real-world setting (e.g., one has access to only small labeled but large unlabeled data from the same distribution). We now evaluate our approach on a downstream transfer scenario with possible domain shift and with much smaller datasets. To this end, we again use the 10 downstream tasks used in Section 3, which contain datasets coming from both in-distributions and out-of-distributions relative to the pretraining datasets. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/28791a22845f26d263e0c46c70519b62ea681b1e0efc91c53065a751bfe1f7fa.jpg",
+        "table_caption": [
+            "Table 3: Downstream transfer results. We categorize downstream datasets as in-distribution (green) and outof-distribution (red) relative to the pretraining dataset based on class overlap; best viewed in color. We see that self-supervised architectures generally perform better on in-distribution downstream scenarios. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Pretrain Dataset</td><td>Arch.</td><td>Params</td><td>Pretrain Val. Set</td><td>CUB</td><td>NABirds</td><td>CIFAR10</td><td>Oxford Flowers</td><td>Stanford Dogs</td><td>Food101</td><td>Sport</td><td>Stanford Cars</td><td>MIT67</td><td>FGVC Aircraft</td></tr><tr><td rowspan=\"4\">ImNet</td><td>MobileV2</td><td>3.5M</td><td>41.9</td><td>20.1</td><td>14.2</td><td>75.8</td><td>85.4</td><td>35.7</td><td>51.6</td><td>94.0</td><td>26.4</td><td>57.5</td><td>35.8</td></tr><tr><td>Ours</td><td>3.3M</td><td>55.3</td><td>31.8</td><td>24.4</td><td>78.8</td><td>91.7</td><td>49.2</td><td>61.7</td><td>94.3</td><td>30.4</td><td>62.5</td><td>38.1</td></tr><tr><td>ResNet18</td><td>11M</td><td>49.8</td><td>27.0</td><td>19.0</td><td>79.9</td><td>89.9</td><td>44.1</td><td>55.8</td><td>94.6</td><td>27.8</td><td>62.1</td><td>36.7</td></tr><tr><td>ResNet50</td><td>23.5M</td><td>58.9</td><td>31.4</td><td>24.6</td><td>85.9</td><td>92.8</td><td>52.3</td><td>65.7</td><td>94.3</td><td>35.1</td><td>69.6</td><td>42.0</td></tr><tr><td rowspan=\"3\">iNat21</td><td>Ours</td><td>3-18M</td><td>59.1</td><td>34.3</td><td>26.1</td><td>81.8</td><td>92.2</td><td>51.0</td><td>64.7</td><td>94.7</td><td>33.2</td><td>66.1</td><td>39.4</td></tr><tr><td>ResNet18</td><td>11M</td><td>30.3</td><td>26.1</td><td>19.0</td><td>73.2</td><td>92.8</td><td>31.3</td><td>55.3</td><td>92.1</td><td>18.9</td><td>49.9</td><td>32.7</td></tr><tr><td>ResNet50</td><td>23.5M</td><td>41.3</td><td>31.2</td><td>23.2</td><td>75.1</td><td>95.1</td><td>39.7</td><td>65.1</td><td>94.3</td><td>22.3</td><td>55.0</td><td>37.6</td></tr><tr><td></td><td>Ours</td><td>3-18M</td><td>43.8</td><td>32.7</td><td>24.1</td><td>76.1</td><td>94.7</td><td>39.0</td><td>63.1</td><td>93.1</td><td>20.1</td><td>49.0</td><td>34.9</td></tr></table>",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Table 3 summarizes the results (we color code in/out-of-distribution datasets based on our crude categorization; see appendix for our justification). We first compare our self-supervised architectures to MobileNetV2 in the same parameter range (3.5M vs $3 . 3 \\mathbf { M }$ ; top two rows). We notice that self-supervised architectures significantly outperform MobileNetV2 in all datasets regardless of distributional shift. This is encouraging (i.e., self-supervised architectures can learn generalizable representations) but at the same time not totally surprising (i.e., MobileNet is optimized for efficiency and not for accuracy). Next, we compare ours to ResNet18 that has a similar parameter range although belonging to a class of architectures much different from our search space. Ours outperforms ResNet18 on all pretraining and evaluation datasets by a considerable margin. This shows that our approach is generalizable and can outperform architectures in the ResNet18 search space even though they are generally more computationally expensive than the MobileNet search space. ",
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+    {
+        "type": "text",
+        "text": "Finally, we compare ours to ResNet50 which is computationally heavier compared to ResNet18. We preface our analysis with a caveat that our self-supervised architectures are almost half the capacity of ResNet50, limiting their representational power. Keeping this in mind, we see that our approach starts to fail in some of the in-distribution and all of the out-of-distribution scenarios (red shaded cells). This is somewhat disappointing but perhaps expected: self-supervised architectures naturally encode inductive biases specific to the dataset they were optimized on. When a distributional shift happens, their performance can start deteriorating because out-of-domain data might require a different set of inductive biases. The strong performance by ResNet50 imply that the model might be striking the right balance across those datasets in terms of inductive biases, but our results in Table 1 and 2 show that ResNet50 can be less effective on newly developed datasets such as iNat2021. ",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "This work lays the ground for moving beyond handcrafted architectures in SSL. By conducting large-scale experiments with 116 architectures and 11 downstream tasks, we established extensive empirical evidence showing that there isn’t one architecture that performs consistently well across different downstream scenarios in SSL. Motivated by this, we proposed to move beyond handcrafted architectures and learn both an architecture topology and its network weights in SSL. We provided convincing results demonstrating that the self-supervised architectures significantly outperform handcrafted MobileNetV2 and ResNet18 architectures on 11 downstream tasks, and competitively with ResNet50 even with almost half the model size. We re-emphasize that improvements are not solely due to NAS, as the architecture search was performed by solving SSL and not by optimizing directly on downstream tasks as in the typical NAS setting. ",
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+    {
+        "type": "text",
+        "text": "Our work barely scratches the surface and opens up many doors for future directions. Will our findings hold for different architectures such as Transformers and different modalities such as video and text? How can we make architecture search more effective for SSL? Can ideas from domain generalization improve the transferability of self-supervised architectures in the out-of-distribution setting? Or is it even the right idea to expect learned architectures to generalize to widely different domains? We hope the readers are as excited as us to investigate these challenging questions. ",
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+    {
+        "type": "text",
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+        "text": "APPENDIX ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Fig. 6 provides an overview of our work. We show that no single handcrafted architecture performs consistently well across different tasks. It is therefore imperative to optimize for architecture topologies along with network weights for a specific task. Through extensive empirical results we show that such self-supervised architectures outperform their handcrafted counterparts in the same search space on the respective tasks. ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/accf1356d2a88bdcd1df4d10e4829e92d0fd35757f21f6038b46032d63d2158b.jpg",
+        "image_caption": [
+            "Figure 6: Conventional SSL frameworks learn network weights for a fixed handcrafted architecture (left). We show that learning architecture topologies along with their weights can improve performance in SSL (right). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "A SUPERVISED TRAINING PERFORMANCE OF SELF-SUPERVISED ARCHITECTURES ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In addition to evaluating the performance of the searched architectures for SSL, we analyze their supervised training performance. We use the searched architectures and directly train them, from scratch without any pretraining, on downstream datasets using the supervised labels. Results are summarized in Table 4. We include architectures searched on ImageNet and iNat21, and MobileNetV2 for reference. It shows the searched architectures perform well even in the supervised setting, outperforming the handcrafted MobileNetV2 on most of the downstream datasets. However, a performance degradation is observed in out-of-distribution datasets like Stanford Cars and FGVC Aircraft. This is in line with the discussion in Section 4.2 of the main paper where the searched architecture performances deteriorate with distributional shift. Nevertheless, for the more in-distribution datasets, we obtain higher accuracies showing that the searched architectures are suitable for supervised training as well. ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/884caaa801d66d7e0f5db3eac488794bc42cc3ff4ac831f806cc59dac3533d44.jpg",
+        "table_caption": [
+            "Table 4: Supervised performance of searched architectures. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>CUB</td><td>CIFAR10</td><td>CIFAR100</td><td>Food</td><td>Flowers</td><td>Sport</td><td>Cars</td><td>Aircraft</td></tr><tr><td>MobileNetV2</td><td>58.2</td><td>93.7</td><td>71.9</td><td>80.8</td><td>90.0</td><td>94.2</td><td>88.6</td><td>81.4</td></tr><tr><td>Ours (ImNet)</td><td>57.9</td><td>93.1</td><td>74.7</td><td>78.1</td><td>96.7</td><td>95.0</td><td>71.0</td><td>75.5</td></tr><tr><td>Ours (iNat21)</td><td>64.5</td><td>93.9</td><td>75.8</td><td>81.6</td><td>98.1</td><td>96.3</td><td>72.3</td><td>72.9</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "B CLASS MAPPING FROM IMAGENET TO DOWNSTREAM DATASETS",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
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+        ],
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+    {
+        "type": "text",
+        "text": "We provide a justification for characterizing downstream datasets as in-distribution/out-ofdistribution with respect to ImageNet as shown in Table 3 of the main paper. We provide a rough class mapping between ImageNet and the 10 downstream datasets. Note that obtaining an exact class mapping is difficult due to only an approximate mapping existing between any 2 datasets. In addition, there can be classes which contribute to improved features for another class while still being semantically different. For example, zebra (n02391049) can contribute to improved features for horses (sorrel-n02389026) due to similar shapes. We now list datasets with corresponding ImageNet classes/superclasses. While some superclasses can contain additional subclasses in the WordNet hierarchy, we restrict to only those classes in the ImageNet-1k dataset. Numbers in bracket denote the total number of classes roughly overlapping. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "• CIFAR-10 (270): vehicle (n4524313), bird (n1503061), feline (n2120997), frog (n1639765), dog (n2084071), sorrel (n2389026)   \n• Stanford Dogs (120): dog (n2084071)   \n• CUB (60): bird (n1503061)   \n• NABirds (60): bird (n1503061)   \n• Food101 (20): nutriment (n7570720), beverage (n7881800), foodstuff (n7566340), sandwich (n7695965), bagel (n7693725), guacamole (n7583066), chocolate sauce (n7836838), carbonara (n7831146), french loaf (n7684084), pretzel (n7695742)   \n• Stanford Cars (10): car (n2958343)   \n• FGVC Aircraft (3): airliner (n2690373), warplane (n4552348), airship (n2692877)   \n• Oxford Flowers (2): yellow lady’s slipper (n12057211), daisy (n11939491)   \n• MIT67 (0): -   \n• Sports(0): - ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Due to the inductive biases encoded during the search process specific to the dataset it is searched on, the self-supervised architecture performs well on more in-distribution datasets like CUB or NABirds. However, we see that for datasets like Stanford Cars and subsequent ones, there is little direct class overlap with ImageNet classes. This leads to lesser images being available for self-supervised pretraining which are in-distribution for these datasets. Consequently, due to the relatively out-of-distribution nature of these datasets we see in Table 3 of the main paper, our selfsupervised architectures are outperformed by the ResNet-50 baseline. ",
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "C ADDITIONAL IMPLEMENTATION DETAILS ",
+        "text_level": 1,
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+        "page_idx": 14
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+    {
+        "type": "text",
+        "text": "We sample ResNet architectures by varying the number of blocks at each of the 4 stages choosing from the set of $2 , 3 , 4$ blocks and choose the ones in the parameter range shown in Fig. 3 while also fitting in GPU memory. For MobileNets, we have 7 sequences (stages) and a higher variation of the number of blocks from [2-6] while also choosing the width parameter from the set $1 . 0 , 1 . 2 , 1 . 4 , 1 . 6 , 1 . 8 , 2 . 0$ and choose the ones in the 2M-7M parameter range and fitting in GPU memory. Note that a high number of blocks in the earlier stages take significantly more GPU memory due to larger feature map sizes. ",
+        "bbox": [
+            174,
+            652,
+            825,
+            750
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "For the architecture search phase, we use the optimizer hyperparameters as explained in Sec. 4.1 of the main paper. We use a weight decay of $4 e ^ { - 5 }$ for the weight parameters excluding batch normalization parameters. The initial convolution is a $3 { \\tt X } 3$ convolution with stride 2. The network consists of 6 stages with 4 cells in the first 5 stages and 1 cell in the last stage. By default, the number of channels at each stage is 24, 40, 80, 96, 192, 320, which is multiplied by a constant width multiplier. We downsample it by a factor of 2 at the beginning of the first, second, third and fifth stage. Other architecture details are the default ones used in Cai et al. (2018). We use the same projection head as used normally for SimCLR Chen et al. (2020a) on top of the backbone network, which is a 2048 dimensional hidden layer and 128 dimensional output layer. For evaluation, we remove the projection head and use the output of the network backbone as the feature extractor. Augmentations are the same as in SimCLR with random resize scaling and cropping, flipping and color jitter. A temperature value of $\\tau = 0 . 1$ is set for the contrastive loss. ",
+        "bbox": [
+            174,
+            757,
+            825,
+            922
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/fd128bd086da0bc459cdfc02749b7abd25faae3e845591e86215b11e95baf798.jpg",
+        "image_caption": [
+            "Figure 7: Self-supervised architectures for different pretraining datasets. MB3 and MB6 are the mobile inverted convolutions with expansion ratio of 3 and 6 respectively. We see that the majority of the preferred convolutions is MB6 $7 \\times 7$ suggesting that the network prefers convolutions with more parameters for the self-supervised regime due to lots of data. For smaller datasets like iNat21Mini, MB6 convolutions are not as strongly preferred. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            99,
+            816,
+            412
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "D VISUALIZING SELF-SUPERVISED ARCHITECTURES FOR DIFFERENT PRETRAINING DATASETS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            518,
+            767,
+            553
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "We visualize the types of convolutions searched at a $1 . 7 5 \\mathrm { x }$ width multiplier for the 3 different pretraining datasets: ImageNet, iNat21 and iNat21Mini. Results are shown in Fig. 7. MB3 and MB6 are the mobile inverted convolutions with expansion ratio of 3 and 6 respectively. The grey lines denote the downsampling of image due to strided convolutions. All architectures are followed by a pooling layer to reduce the image size to $1 \\times 1$ . In contrast to standard handcrafted architectures, larger $7 \\times 7$ convolutions are preferred even in the later stages of the network. We also see that the majority of the preferred convolutions is MB6 $7 \\times 7$ suggesting that the network prefers convolutions with more parameters for the self-supervised regime due to lots of data. This is less preferred in smaller datasets like iNat21Mini where MB3 convolutions are common especially in earlier stages of the network. It is difficult to draw conclusions on the type of network preferred between ImageNet and iNat21 showing that it is imperative to search for an optimal architecture rather than handcraft them. ",
+        "bbox": [
+            173,
+            571,
+            825,
+            738
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "E DOWNSTREAM DATASET PERFORMANCE CORRELATION WITH IMAGENET ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            763,
+            820,
+            780
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "We show the downstream dataset correlation for all 10 downstream datasets in addition to ImageNet1K Deng et al. (2009): CIFAR10/100 Krizhevsky et al. (2009), Stanford Cars Krause et al. (2013) and Dogs Khosla et al. (2011), CUB-200 Welinder et al. (2010), MIT-67 Quattoni & Torralba (2009), SVHN Netzer et al. (2011), Flowers-102 Nilsback & Zisserman (2008), FGVC-Aircraft Maji et al. (2013), Sports8 Li & Fei-Fei (2007). These are shown for both ResNets (Fig. 8) and MobileNets (Fig. 9). We see similar results for the 5 datasets in addition to those shown in Fig. 3 of main paper. High correlation exists for datasets which are visually similar to ImageNet while it is less correlated for datasets which are out of domain. For MobileNets this correlation is even less pronounced with high variance in performance at higher ImageNet accuracies. ",
+        "bbox": [
+            173,
+            797,
+            825,
+            924
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/0bba7b900089965a52388ce1f902c3c9b6cc53e5c1884cf0ca84918f10797416.jpg",
+        "image_caption": [
+            "Figure 8: ImageNet performance correlation with 10 different downstream datasets for various ResNets. We show linear evaluation top-1 accuracy of ImageNet $\\mathbf { \\dot { x } }$ -axis) vs. different downstream datasets (y-axis) obtained from variations of ResNet; all models are pretrained on ImageNet-1K Deng et al. (2009) using SimCLR Chen et al. (2020a) under the same protocol. The solid lines and shaded areas indicate fitted regression models and their confidence intervals. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            192,
+            167,
+            808,
+            359
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/70824b783eee5ebb816e486c629b8b2f0fb15cd94462f425e2695756953b96ee.jpg",
+        "image_caption": [
+            "Figure 9: ImageNet performance correlation with 10 different downstream datasets for various MobileNets. We show linear evaluation top-1 accuracy of ImageNet $\\mathbf { \\widetilde { x } }$ -axis) vs. different downstream datasets (y-axis) obtained from variations of MobileNet; all models are pretrained on ImageNet-1K Deng et al. (2009) using SimCLR Chen et al. (2020a) under the same protocol. The solid lines and shaded areas indicate fitted regression models and their confidence intervals. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            191,
+            579,
+            808,
+            775
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/f6a7746ca10efe2bcf8f6ff1e9ca1737ac46bc489713dce258d039d4835980b9.jpg",
+        "image_caption": [
+            "Figure 10: Dataset performance correlation with 10 different datasets for various ResNets. We show the model size in terms of parameter counts $\\mathbf { \\widetilde { x } }$ -axis) vs. top-1 accuracy on different datasets obtained from variations of ResNet-like architectures; all models are pretrained on ImageNet-1K Deng et al. (2009) using SimCLR Chen et al. (2020a) under the same protocol. The solid lines and shaded areas indicate fitted regression models and their confidence intervals. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            162,
+            816,
+            363
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/43fda8482ba7f3742017ae263db37ba673a381a4bd5bda452e730fa8592579af.jpg",
+        "image_caption": [
+            "Figure 11: Dataset performance correlation with 10 different datasets for various MobileNets. We show the model size in terms of parameter counts $\\mathbf { \\widetilde { x } }$ -axis) vs. top-1 accuracy on different datasets obtained from variations of MobileNet-like architectures; all models are pretrained on ImageNet-1K Deng et al. (2009) using SimCLR Chen et al. (2020a) under the same protocol. The solid lines and shaded areas indicate fitted regression models and their confidence intervals. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            577,
+            816,
+            779
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/7370ad50b35571a20834b016d378412186e599a7e683874e685da7feb210fd98.jpg",
+        "image_caption": [
+            "Figure 12: Linear and rank correlation of ResNets between different pairs of 11 datasets We show correlation between every pair of top-1 accuracy on 11 downstream tasks obtained from ImageNet-pretrained ResNets. We see no strong correlation except for ones highly similar to the data the models were originally pretrained on, e.g., CIFAR-10/100 and Dogs120. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            102,
+            508,
+            316
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/efa643a0719af7cf0d7b6b0edc681bbc4514829f52ef93a07103374d1a8f41a3.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            529,
+            103,
+            849,
+            316
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/7d32535940ca31ff4448230673c02a97883e7edfd1ef783b7da494664bd4913c.jpg",
+        "image_caption": [
+            "Figure 13: Linear and rank correlation of MobileNets between different pairs of 11 datasets We show correlation between every pair of top-1 accuracy on 11 downstream tasks obtained from ImageNet-pretrained MobileNets. The correlation "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            400,
+            509,
+            617
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/7b54e0d8565d1af9449e55fb75fc8fc58b76a0d58d19907da021698701e96d2a.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            527,
+            401,
+            851,
+            617
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "F DATASET PERFORMANCE AS A FUNCTION OF NUMBER OF PARAMETERS ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            698,
+            805,
+            715
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "We show the dataset correlation with respect to number of parameters for 5 more datasets in addition to that shown in Fig. 4 of main paper. Fig. 10 summarizes the results for ResNets while Fig. 11 shows results for MobileNets. We see that similar results hold for the additional 5 datasets where more parameters, and consequently larger networks, does not always lead to better downstream performance. ",
+        "bbox": [
+            174,
+            729,
+            825,
+            800
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "G LINEAR AND RANK CORRELATION FOR RESNETS/MOBILENETS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            820,
+            741,
+            837
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "We show the summary of the correlation across different datasets for both ResNets (Fig. 12) and MobileNets (Fig. 13). In addition to Spearman’s rank correlation coefficient, we also show Pearson’s linear correlation coefficient. While Pearson’s linear coefficient is higher in the case of MobileNets, the linear fit still exhibits high variance for higher ImageNet accuracies, as seen in Fig. 9. ",
+        "bbox": [
+            174,
+            852,
+            825,
+            909
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/e7eab649c5fd42ef4d4f2ac0653c9c6227eb0ae9a13f15160c00b95f22820cec.jpg",
+        "image_caption": [
+            "Figure 14: ImageNet performance correlation with 10 different downstream datasets for various ResNets, MobileNets and searched architectures. The searched architectures outperform MobileNets while being comparable with ResNets at fewer parameters. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            192,
+            98,
+            807,
+            304
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "H DATASET LICENSES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            382,
+            375,
+            398
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "Table 5 lists some datasets we used and their licenses. ",
+        "bbox": [
+            174,
+            414,
+            526,
+            428
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "table",
+        "img_path": "images/e53d333c426c871ea086c0b388fb5f77f01feccbdf12b7f6bcb7f9326b0cf486.jpg",
+        "table_caption": [
+            "Table 5: Licenses of datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>License</td></tr><tr><td>CIFAR-10 Krizhevsky et al. (2009)</td><td>MIT</td></tr><tr><td>CIFAR-100 Krizhevsky et al. (2009)</td><td>MIT</td></tr><tr><td>ImageNet Deng et al. (2009)</td><td>BSD 3-Clause</td></tr><tr><td>Sport8 Li &amp; Fei-Fei (2007)</td><td>CCO:Public Domain</td></tr><tr><td>Stanford Dogs Khosla et al. (2011)</td><td>BSD3-Clause</td></tr><tr><td>Stanford Cars Krause et al. (2013)</td><td>BSD 3-Clause</td></tr><tr><td>CUB-200 Welinder et al. (2010)</td><td>Data files @ Original Authors</td></tr><tr><td>MIT-67 Quattoni &amp; Torralba a (2009)</td><td>MIT</td></tr><tr><td>SVHN Netzer et al. (2011)</td><td>CCO:Public Domain</td></tr><tr><td>Flowers-102 Nilsback &amp; Zisserman (2008)</td><td>GNU General Public License,version 2</td></tr></table>",
+        "bbox": [
+            212,
+            463,
+            784,
+            651
+        ],
+        "page_idx": 19
+    }
+]

parse/dev/3lge0p5o-M-/3lge0p5o-M-.md

@@ -0,0 +1,456 @@
+# DIFFEDIT: DIFFUSION-BASED SEMANTIC IMAGE EDITING WITH MASK GUIDANCE
+
+Guillaume Couairon, Jakob Verbeek, Holger Schwenk Meta AI {gcouairon,jjverbeek, schwenk}@meta.com
+
+Matthieu Cord Sorbonne Universite, Valeo.ai´ matthieu.cord@ sorbonne-universite.fr
+
+# ABSTRACT
+
+Image generation has recently seen tremendous advances, with diffusion models allowing to synthesize convincing images for a large variety of text prompts. In this article, we propose DIFFEDIT, a method to take advantage of text-conditioned diffusion models for the task of semantic image editing, where the goal is to edit an image based on a text query. Semantic image editing is an extension of image generation, with the additional constraint that the generated image should be as similar as possible to a given input image. Current editing methods based on diffusion models usually require to provide a mask, making the task much easier by treating it as a conditional inpainting task. In contrast, our main contribution is able to automatically generate a mask highlighting regions of the input image that need to be edited, by contrasting predictions of a diffusion model conditioned on different text prompts. Moreover, we rely on latent inference to preserve content in those regions of interest and show excellent synergies with mask-based diffusion. DIFFEDIT achieves state-of-the-art editing performance on ImageNet. In addition, we evaluate semantic image editing in more challenging settings, using imagesdit from the COCO dataset as well as text-based generated images.
+
+![](images/6df455b4a7fe4ab8f50834af47c859c5a1e062e2fc186e5e9d6d08a9a97b86e9.jpg)  
+Figure 1: In semantic image editing the goal is to modify an input image based on a textual query, while otherwise leaving the image as close as possible to the original. In our DIFFEDIT approach, a mask generation module determines which part of the image should be edited, and an encoder infers the latents, to provide inputs to a text-conditional diffusion model which produces the image edit.
+
+# 1 INTRODUCTION
+
+Mask The task of semantic image editing consists in modifying an input image in accordance with a Generation  Masked textual transformation query. For instance, given an image of a bowl of fruits and the query “fruits” $ ~ \mathrm { ^ { * } } p e a r s ^ { \prime \prime }$ Module  Diffusion , the aim is to produce a novel image where the fruits have been changed into pears, while keeping the bowl and the background as similar as possible to the input image. The text query Encode can also be a more elaborate description like “A basket of fruits”. See the example edits obtained with DIFFEDIT in Figure 1. Semantic image editing bears strong similarities with image generation Text Query A basket of fruits and can be viewed as extending text-conditional image generation with an additional constraint: the Mask generated image should be as close as possible to a given input image.
+
+Input Image  Module  Masked Text-conditional image generation is currently undergoing a revolution, with DALL-E (Ramesh et al., 2021), Cogview (Ding et al., 2021), Make-a-scene (Gafni et al., 2022), Latent Diffusion Models (Rombach et al., 2022), DALL-E 2 (Ramesh et al., 2022) and Imagen (Saharia et al., 2022b), vastly improving state of the art in modelling wide distributions of images and allowing for unprecedented compositionality of concepts in image generation. Scaling these models is a key to their success. State-of-the art models are now trained on vast amounts of data, which requires large computational resources. Similarly to language models pretrained on web-scale data and adapted in downstreams tasks with prompt engineering, the generative power of these big generative models can be harnessed to solve semantic image editing, avoiding to train specialized architectures (Li et al., 2020a; Wang et al., 2022a), or to use costly instance-based optimization (Crowson et al., 2022; Couairon et al., 2022; Patashnik et al., 2021).
+
+Diffusion models are an especially interesting class of model for image editing because of their iterative denoising process starting from random Gaussian noise. This process can be guided through a variety of techniques, like CLIP guidance (Nichol et al., 2021; Avrahami et al., 2022; Crowson, 2021), and inpainting by copy-pasting pixel values outside a user-given mask (Lugmayr et al., 2022). These previous works, however, lack two crucial properties for semantic image editing: (i) inpainting discards information about the input image that should be used in image editing (e.g. changing a dog into a cat should not modify the animal’s color and pose); (ii) a mask must be provided as input to tell the diffusion model what parts of the image should be edited. We believe that while drawing masks is common on image editing tools like Photoshop, language-guided editing offers a more intuitive interface to modify images that requires less effort from users.
+
+Conditioning a diffusion model on an input image can also be done without a mask, e.g. by considering the distance to input image as a loss function (Crowson, 2021; Choi et al., 2021), or by using a noised version of the input image as a starting point for the denoising process as in SDEdit (Meng et al., 2021). However, these editing methods tend to modify the entire image, whereas we aim for localized edits. Furthermore, adding noise to the input image discards important information, both inside the region that should be edited and outside.
+
+To leverage the best of both worlds, we propose DIFFEDIT, an algorithm that leverages a pretrained text-conditional diffusion model for zero-shot semantic image editing, without expensive editingspecific training. DIFFEDIT makes it possible by automatically finding what regions of an input image should be edited given a text query, by contrasting the predictions of a conditional and unconditional diffusion model. We also show how using a reference text describing the input image and similar to the query, can help obtain better masks. Moreover, we demonstrate that using a reverse denoising model, to encode the input image in latent space, rather than simply adding noise to it, allows to better integrate the edited region into the background and produces more subtle and natural edits. See Figure 1 for illustrations. We quantitatively evaluate our approach and compare to prior work using images of the ImageNet and COCO dataset, as well as a set of generated images.
+
+# 2 RELATED WORK
+
+Semantic image editing. The field of image editing encompasses many different tasks, from photo colorization and retouching (Shi et al., 2020), to style transfer (Jing et al., 2019), inserting objects in images (Gafni & Wolf, 2020; Brown et al., 2022), image-to-image translation (Zhu et al., 2017; Saharia et al., 2022a), inpainting (Yu et al., 2018), scene graph manipulation (Dhamo et al., 2020), and placing subjects in novel contexts (Ruiz et al., 2022). We focus on semantic image editing, where the instruction to modify an image is given in natural language. Some approaches involve training an end-to-end architecture with a proxy objective before being adapted to editing at inference time, based on GANs (Li et al., 2020b;a; Ma et al., 2018; Alami Mejjati et al., 2018; Mo et al., 2018) or transformers (Wang et al., 2022a; Brown et al., 2022; Issenhuth et al., 2021). Others (Crowson et al., 2022; Couairon et al., 2022; Patashnik et al., 2021; Bar-Tal et al., 2022) rely on optimization of the image itself, or a latent representation of it, to modify an image based on a high-level multimodal objective in an embedding space, typically using CLIP (Radford et al., 2021). These approaches are quite computationnaly intensive, and work best when the optimization is coupled with a powerful generative network. Given a pre-trained generative model such as a GAN, it has also been explored to find directions in the latent space that corresponds to specific semantic edits (Hark ¨ onen ¨ et al., 2020; Collins et al., 2020; Shen et al., 2020; Shoshan et al., 2021), which then requires GAN inversion to edit real images (Wang et al., 2022c; Zhu et al., 2020; Grechka et al., 2021).
+
+Image editing with diffusion models. Because diffusion models iteratively refine an image starting from random noise, they are easily adapted for inpainting when a mask is given as input. Song et al.
+
+(2021) proposed to condition the generation process by copy-pasting pixel values from the reference image at each denoising step. Nichol et al. (2021) use a similar technique by copy-pasting pixels in the estimated final version of the image. Wang et al. (2022b) use DDIM encoding of the input image, and then decode on edited sketches or semantic segmentation maps. The gradient of a CLIP score can also be used to match a given text query inside a mask, as in Paint by Word (Bau et al., 2021), local CLIP-guided diffusion (Crowson, 2021), or blended diffusion (Avrahami et al., 2022). Lugmayr et al. (2022) apply a sequence of noise-denoise operations to better inpaint a specific region. There are also a number of methods that do not require an editing mask. In DiffusionCLIP (Kim & Ye, 2021), the weights of the diffusion model themselves are updated via gradient descent from a CLIP loss with a target text. The high computational cost of fine-tuning a diffusion model for each input image, however, makes it impractical as an interactive image editing tool. In SDEdit (Meng et al., 2021) the image is corrupted with Gaussian noise, and then the diffusion network is used to denoise it. While this method is originally designed to transform sketches to real images and to make pixel-based collages more realistic, we adapt it by denoising the image conditionally to the text query. In ILVR (Choi et al., 2021), the decoding process of diffusion model is guided with the constraint that downsampled versions of the input image and decoded image should stay close. Finally, in recent work concurrent to ours, Hertz et al. (2022) propose to edit images by modifying attention maps during the diffusion process.
+
+# 3 DIFFEDIT FRAMEWORK
+
+In this section, we first give an overview of diffusion models. We then describe our DIFFEDIT approach in detail, and provide a theoretical analysis comparing DIFFEDIT with SDEdit.
+
+# 3.1 BACKGROUND: DIFFUSION MODELS, DDIM AND ENCODING
+
+Denoising diffusion probabilistic models (Ho et al., 2020) is a class of generative models that are trained to invert a diffusion process. For a number of timesteps $T$ , the diffusion process gradually adds noise to the input data, until the resulting distribution is (almost) Gaussian. A neural network is then trained to reverse that process, by minimizing the denoising objective
+
+$$
+\begin{array} { r } { \mathcal { L } = \mathbb { E } _ { \mathbf { x } _ { 0 } , t , \epsilon } \Vert \epsilon - \epsilon _ { \theta } ( \mathbf { x } _ { t } , t ) \Vert _ { 2 } ^ { 2 } , } \end{array}
+$$
+
+where $\epsilon _ { \theta }$ is the noise estimator which aims to find the noise $\mathbf { \epsilon } \gets \mathcal { N } ( \mathbf { 0 } , \mathbf { I } )$ that is mixed with an input image $\mathbf { x } _ { \mathrm { 0 } }$ to yield $\mathbf { x } _ { t } = \sqrt { \alpha _ { t } } \mathbf { x } _ { 0 } + \sqrt { 1 - \alpha _ { t } } \epsilon$ . The coefficient $\alpha _ { t }$ defines the level of noise and is a decreasing function of the timestep $t$ , with $\alpha _ { 0 } = 1$ (no noise) and $\alpha _ { T } \approx 0$ (almost pure noise).
+
+Song et al. (2021) propose to use $\epsilon _ { \theta }$ to generate new images with the $D D I M$ algorithm: starting from $\mathbf { x } _ { T } \sim \mathcal { N } ( \mathbf { 0 } , \mathbf { I } )$ , the following update rule is applied iteratively until step 0:
+
+$$
+\mathbf { x } _ { t - 1 } = \sqrt { \alpha _ { t - 1 } } \Bigg ( \frac { \mathbf { x } _ { t } - \sqrt { 1 - \alpha _ { t } } \epsilon _ { \theta } ( \mathbf { x } _ { t } , t ) } { \sqrt { \alpha _ { t } } } \Bigg ) + \sqrt { 1 - \alpha _ { t - 1 } } \epsilon _ { \theta } ( \mathbf { x } _ { t } , t ) .
+$$
+
+The variable $\mathbf { x }$ is updated by taking small steps in the direction of $\epsilon _ { \theta }$ . Equation 2 can be written as the neural ODE , taking $\mathbf { u } = \mathbf { x } / \sqrt { \alpha }$ and $\tau = \sqrt { 1 / \alpha - 1 }$ :
+
+$$
+d \mathbf { u } = \epsilon _ { \theta } ( \frac { \mathbf { u } } { \sqrt { 1 + \tau ^ { 2 } } } , t ) d \tau .
+$$
+
+This allows to view DDIM sampling as an Euler scheme for solving Equation 3 with initial condition $\mathbf u ( t = T ) \sim \mathcal { N } ( \mathbf 0 , \alpha _ { T } \mathbf I )$ . This illustrates that we can use fewer sampling steps during inference than the value of $T$ chosen during training, by using a coarser discretization of the ODE. In the remainder of the paper, we parameterize the timestep $t$ to be between 0 and 1, so that $t = 1$ corresponds to $T$ steps of diffusion in the original formulation. As proposed by Song et al. (2021), we can also use this ODE to encode an image $\mathbf { x } _ { \mathrm { 0 } }$ onto a latent variable ${ \bf x } _ { r }$ for a timestep $r \leq 1$ , by using the boundary condition $\mathbf { u } ( t = 0 ) = \mathbf { x } _ { 0 }$ instead of $\mathbf { u } ( t = 1 )$ , and applying an Euler scheme until timestep $r$ . In the remainder of the paper, we refer to this encoding process as DDIM encoding, we denote the corresponding function that maps $\mathbf { x } _ { \mathrm { 0 } }$ to ${ \bf x } _ { r }$ as $E _ { r }$ , and refer to the variable $r$ as the encoding ratio. Similarly, we note $D _ { r }$ the inverse function that maps ${ \bf x } _ { r }$ to $\mathbf { x } _ { \mathrm { 0 } }$ , which corresponds to regular DDIM decoding. With sufficiently small steps in the Euler scheme, decoding ${ \bf x } _ { r }$ approximately recovers the original image $\mathbf { x } _ { \mathrm { 0 } }$ . This property is particularly interesting in the context of image editing: all the information of the input image $\mathbf { x } _ { \mathrm { 0 } }$ is encoded in ${ \bf x } _ { r }$ , and can be accessed via DDIM sampling.
+
+![](images/5053df65c16305af893dfa5604f8e00fd29a6471e9a5f85b878435f4c75c2a56.jpg)  
+Step 1: Compute Mask   
+Figure 2: The three steps of DIFFEDIT. Step 1: we add noise to the input image, and denoise it: once conditioned on the query text, and once conditioned on a reference text (or unconditionally). We derive a mask based on the difference in the denoising results. Step 2: we encode the input image with DDIM, to estimate the latents corresponding to the input image. Step 3: we perform DDIM decoding conditioned on the text query, using the inferred mask to replace the background with pixel values coming from the encoding process at the corresponding timestep.
+
+# 3.2 SEMANTIC IMAGE EDITING WITH DIFFEDIT
+
+In many cases, semantic image edits can be restricted to only a part of the image, leaving other parts unchanged. However, the input text query does not explicitly identify this region, and a naive method could allow for edits all over the image, risking to modify the input in areas where it is not needed. To circumvent this, we propose DIFFEDIT, a method to leverage a text-conditioned diffusion model to infer a mask of the region that needs to be edited. Starting from a DDIM encoding of the input image, DIFFEDIT uses the inferred mask to guide the denoising process, minimizing edits outside the region of interest. Figure 2 illustrates the three steps of our approach, which we detail below.
+
+Step 1: Computing editing mask. When the denoising an image, a text-conditioned diffusion model will yield different noise estimates given different text conditionings. We can consider where the estimates are different, which gives information about what image regions are concerned by the change in conditioning text. For instance, in Figure 2, the noise estimates conditioned to the query zebra and reference text horse1 are different on the body of the animal, where they will tend to decode different colors and textures depending on the conditioning. For the background, on the other hand, there is little change in the noise estimates. The difference between the noise estimates can thus be used to infer a mask that identifies what parts on the image need to be changed to match the query. In our algorithm, we use a Gaussian noise with strength $50 \%$ (see analysis in Appendix A.1), remove extreme values in noise predictions and stabilize the effect by averaging spatial differences over a set of $n$ input noises, with $n { = } 1 0$ in our default configuration. The result is then rescaled to the range $[ 0 , 1 ]$ , and binarized with a threshold, which we set to 0.5 by default. The masks generally somewhat overshoot the region that requires editing, this is beneficial as it allows it to be smoothly embedded in it’s context, see examples in Section 4 and Section A.5.
+
+Step 2: Encoding. We encode the input image $\mathbf { x } _ { \mathrm { 0 } }$ in the implicit latent space at timestep $r$ with the DDIM encoding function $E _ { r }$ . This is done with the unconditional model, i.e. using conditioning text $\varnothing$ , so no text input is used for this step.
+
+Step 3: Decoding with mask guidance. After obtaining the latent ${ \bf x } _ { r }$ , we decode it with our diffusion model conditioned on the editing text query $Q$ , e.g. zebra in the example of Figure 2. We use our mask $M$ to guide this diffusion process. Outside the mask $M$ , the edited image should in principle be the same as the input image. We guide the diffusion model by replacing pixel values outside the mask with the latents $\mathbf { x } _ { t }$ inferred with DDIM encoding, which will naturally map back to the original pixels through decoding, unlike when using a noised version of $\mathbf { x } _ { \mathrm { 0 } }$ as typically done (Meng et al., 2021; Song et al., 2021). The mask-guided DDIM update can be written as $\tilde { \mathbf { y } } _ { t } = M \mathbf { y } _ { t } + ( 1 - M ) \mathbf { x } _ { t }$ , where $\mathbf { y } _ { t }$ is computed from $\mathbf { y } _ { t - d t }$ with Eq. 2, and $\mathbf { x } _ { t }$ is the corresponding DDIM encoded latent.
+
+The encoding ratio $r$ determines the strength of the edit: larger values of $r$ allow for stronger edits that allow to better match the text query, at the cost of more deviation from the input image which might not be needed. We evaluate the impact of this parameter in our experiments. We illustrate the effect of the encoding ratio in Appendix A.5.
+
+# 3.3 THEORETICAL ANALYSIS
+
+In DIFFEDIT, we use DDIM encoding to encode images before doing the actual editing step. In this section, we give theoretical insight on why this component yields better editing results than adding random noise as in SDEdit (Meng et al., 2021). With ${ \bf x } _ { r }$ being the encoded version of $\mathbf { x } _ { \mathrm { 0 } }$ , using DDIM decoding on ${ \bf x } _ { r }$ unconditionally would give back the original image $\mathbf { x } _ { \mathrm { 0 } }$ . In DIFFEDIT, we use DDIM decoding conditioned on the text query $Q$ , but there is still a strong bias to stay close to the original image. This is because the unconditional and conditional noise estimator networks $\epsilon _ { \theta }$ and $\epsilon _ { \theta } ( \cdot , Q )$ often produce similar estimates, yielding similar decoding behavior when initialized with the same starting point ${ \bf x } _ { r }$ . This means that the edited image will have a small distance w.r.t. the input image, a property critical in the context of image editing. We capture this phenomenon with the proposition below, where we compare the DDIM encoder √ $E _ { r } ( \mathbf { x } _ { 0 } )$ to the SDEdit encoder $G _ { r } ( \mathbf { x } _ { 0 } , \epsilon \bar { ) : = } \sqrt { \alpha _ { r } } \mathbf { x } _ { 0 } + \sqrt { 1 - \alpha _ { r } } \epsilon$ , which simply adds noise to the image $\mathbf { x } _ { \mathrm { 0 } }$ .
+
+Proposition 1. Let $\boldsymbol { \mathcal { X } } ~ = ~ \mathbb { R } ^ { d }$ be the space of input images, $p _ { D }$ be the data distribution of couples $( \mathbf { x } _ { 0 } , Q )$ where $\mathbf { x } _ { 0 } \in \mathcal { X }$ and $Q$ a textual query to edit that image. Suppose that $\| \epsilon _ { \theta } ( \mathbf { x } _ { t } , Q , t ) \| _ { 2 } \leq C$ for all $x \in \mathcal { X }$ , $t \in [ 0 , 1 ]$ , that $\epsilon _ { \theta } ( \cdot , \varnothing , t )$ is $K _ { 1 }$ -Lipschitz for all $t _ { : }$ , and let $\begin{array} { r } { K _ { 2 } = \mathbb { E } _ { ( \mathbf { x } _ { 0 } , Q ) \in p _ { D } } \operatorname* { m a x } _ { t \in [ 0 , 1 ] } | | \epsilon _ { \theta } ( \mathbf { x } , Q , t ) - \epsilon _ { \theta } ( \mathbf { x } , \theta , t ) | | } \end{array}$ . Then, for all encoding ratios $0 \leq r \leq 1$ , we have the two following bounds:
+
+$$
+\begin{array} { r l } & { \underset { ( \mathbf { x } _ { 0 } , Q ) \sim p _ { D } } { \mathbb { E } } \| \mathbf { x } _ { 0 } - D _ { r } ( G _ { r } ( \mathbf { x } _ { 0 } , \epsilon ) , Q ) \| _ { 2 } \leq ( C + 1 ) \tau , } \\ & { \underset { ( \mathbf { x } _ { 0 } , Q ) \sim p _ { D } } { \mathbb { E } } } \\ & { } \\ & { \underset { ( \mathbf { x } _ { 0 } , Q ) \sim p _ { D } } { \mathbb { E } } \| \mathbf { x } _ { 0 } - D _ { r } ( E _ { r } ( \mathbf { x } _ { 0 } ) , Q ) \| _ { 2 } \leq \frac { K _ { 2 } \tau } { \sqrt { \tau ^ { 2 } + 1 } } \Big ( \tau + \sqrt { \tau ^ { 2 } + 1 } \Big ) ^ { K _ { 1 } } , } \end{array}
+$$
+
+where $\tau = \sqrt { 1 / \alpha _ { r } - 1 }$ increases with the encoding ratio $r$ : $\tau ( r = 0 ) = 0$ and $\operatorname* { l i m } _ { r \to 1 } \tau = + \infty$ .
+
+We provide the proof in Appendix B. The first bound is associated with SDEdit, and is an extension of a bound proven in the original paper. The second bound we contribute is associated with DIFFEDIT. It is tighter than the first bound below a certain encoding ratio, see Figure 3. We empirically estimated the parameters $K _ { 1 } , K _ { 2 }$ and $C$ with the diffusion models that we are using. While the asymptotic behavior of the second bound is worse than the first with $K _ { 1 } > 1$ , it is the very small value of $K _ { 2 }$ that gives a tighter bound.
+
+This supports our argument from above: because the unconditional and text-conditional noise estimates generally give close results — $K _ { 2 }$ being a measure of the average difference— the Euler scheme with $\epsilon _ { \theta } ( \cdot , Q , \cdot )$ gives a sequence of intermediate latents $\mathbf { y } _ { r } , . . . , \mathbf { y } _ { 0 }$ that stays close to the trajectory $x _ { r } , \ldots , D _ { r } ( x _ { r } ) \approx \mathbf { x } _ { 0 }$ mapping back $\mathbf { x } _ { r }$ to $\mathbf { x } _ { \mathrm { 0 } }$ . While these upper bounds do not guarantee that DDIM encoding yields smaller edits than SDEdit, experimentally we find that it is indeed the case.
+
+![](images/5c2ca421a89ac85605b05044c4f62df4cbbb91f0fdbbdaa8ce02e1695cb99b66.jpg)  
+Figure 3: Illustration of the bounds from Proposition 1, with estimated parameters $C = 1 , K _ { 2 } = 0 . 0 2$ , and $K _ { 1 } = 3$ .
+
+# 4 EXPERIMENTS
+
+In this section, we describe our experimental setup, followed by qualitative and quantitative results.
+
+# 4.1 EXPERIMENTAL SETUP
+
+Datasets. We perform experiments on three datasets. First, on ImageNet (Deng et al., 2009) we follow the evaluation protocol of FlexIT (Couairon et al., 2022). Given an image belonging to one class, the goal is to edit it so that it will depict an object of another class as indicated by the query. Given the nature of the ImageNet dataset, edits often concern the main object in the scene. Second, we consider editing images generated by Imagen (Saharia et al., 2022b) based on structured text prompts, in order to evaluate edits that involve changing the background, replacing secondary objects, or changing object properties. Third, we consider edits based on images and queries from the $C O C O$ (Lin et al., 2014) dataset to evaluate edits based on more complex text prompts.
+
+Diffusion models. In our experiments we use latent diffusion models (Rombach et al., 2022). We use the class-conditional model trained on ImageNet at resolution $2 5 6 \times 2 5 6$ , as well as the 890M parameter text-conditional model trained on LAION-5B (Schuhmann et al., 2021), known as Stable Diffusion, at $5 1 2 \times 5 1 2$ resolution.2 Since these models operate in a VQGAN latent spaces (Esser et al., 2021), the resolution of our masks is $3 2 \times 3 2$ (ImageNet) or $6 4 \times 6 4$ (Imagen and COCO). We use 50 steps in DDIM sampling with a fixed schedule, and the encoding ratio parameter further decreases the number of updates used for our edits. This allows to edit images in ${ \sim } 1 0$ seconds on a single Quadro GP100 GPU. We also use classifier-free guidance (Ho & Salimans, 2022) with the recommended values: 5 on ImageNet, 7.5 for Stable Diffusion. For more details see Section A.2.
+
+Comparison to other methods. We use SDEdit (Meng et al., 2021) as our main point of comparison, since we can use the same diffusion model as for DIFFEDIT. We also compare to FlexIT (Couairon et al., 2022), a mask-free, optimization-based editing method based on VQGAN and CLIP. On ImageNet, we evaluate ILVR (Choi et al., 2021) which uses another diffusion model trained on ImageNet (Dhariwal & Nichol, 2021). Finally, on COCO and Imagen images, we compare to the concurrent work of Hertz et al. (2022). 3
+
+Evaluation. In semantic image editing, we have to satisfy the two contradictory objectives of (i) matching the text query and (ii) staying close to the input image. For a given editing method, better matching the text query comes at the cost of increased distance to the input image. Different editing methods often have a parameter that allows to control the editing strength: varying its value allows to get different operating points, forming a trade-off curve between the two objectives aforementioned. Therefore, we evaluate editing methods by comparing their trade-off curves. For diffusion-based methods, we use the encoding ratio to control the trade-off.
+
+# 4.2 EXPERIMENTS ON IMAGENET
+
+On ImageNet, we follow the evaluation protocol of Couairon et al. (2022), with the associated metrics: the LPIPS perceptual distance (Zhang et al., 2018) measures the distance with the input image, and the CSFID, which is a class-conditional FID metric (Heusel et al., 2017) measuring both image realism and consistency w.r.t. the transformation prompt. For both metrics, lower values indicate better edits. For more details see Couairon et al. (2022).
+
+We compare DIFFEDIT to other semantic editing methods from the literature in terms of CSFID-LPIPS tradeoff. Stronger edits improve (lower) the CSFID score as the edited images better adhere to the text query, but the resulting images tend to deviate more from the input image, leading to worse (increased) LPIPS distances.
+
+![](images/81da9fb5f22efa064f9fab04beb6895717d0558f98ba64004a495ab754af39da.jpg)  
+Figure 4: Comparison on ImageNet data of DIFFEDIT with other Image Editing methods. For DIFFEDIT we annotate the different operating points with the corresponding encoding ratios.
+
+![](images/9c9aefa8f89f5b928d13eedda0e4704bbef08a3cf7984fa9004c1a8a9cc2df0a.jpg)  
+Figure 5: Edits obtained on ImageNet with DIFFEDIT and ablated models. Encode-Decode is DIFFEDIT without masking, and SDEdit is obtained when not using masking nor encoding. When not using masking (SDEdit and Encode-Decode) we observe undesired edits to the background, see e.g. the sky in the second column. When not using DDIM encoding (SDEdit and DiffEdit w/o Encode), appearance information from the input —such as pose— is lost, see last two columns.
+
+The results in Figure 4 indicate that DIFFEDIT obtains the best trade-offs among the different methods. For fair comparison with previous methods, here we do not leverage the label of the input image and use the empty text as reference when inferring the editing mask. The Copy and Retrieve baselines are two opposite cases where we have best possible LPIPS distance —zero, by copying the input image— and best possible transformation score by discarding the input image and replacing it with a real image from the target class from the ImageNet dataset. DIFFEDIT, as well as the diffusion-based SDEdit and ILVR, are able to obtain CSFID values comparable to that of the retrieval baseline. Among the diffusion-based methods, our DIFFEDIT obtains comparable CSFID values at significantly better LPIPS scores. For FlexIT, the CSFID best value is significantly worse, indicating it is not able to produce both strong and realistic edits. Using more optimization steps does not solve this issue, as the distance to the input image is part of the loss it minimizes.
+
+Ablation experiments. We ablate the two core components of DIFFEDIT, mask inference and DDIM encoding, to measure their relative contributions in terms of CSFID-LPIPS trade-off. If we do not use either of these components our method reverts to SDEdit (Meng et al., 2021). The results in Figure 6, left panel, show that adding DDIM encoding (Encode-Decode) and the masking (DiffEdit w/o Encode) separately both improve the trade-off and reduce the average editing distance w.r.t. the input image compared to SDEdit. Moreover, combining these two elements into DIFFEDIT gives an even better trade-off, showing their complementarity: masking better preserves the background, while DDIM encoding retains visual appearance of the input inside the mask. See Figure 5 for qualitative examples of these ablations, along with the inferred masks.
+
+The right panel of Figure 6 shows DIFFEDIT with different mask binarization thresholds. Compared to our default value of 0.5, a lower threshold of 0.25 results in larger masks (more image modifications) and worse CSFID-LPIPS tradeoff. A higher threshold of 0.75 results in masks that are too restrictive: the CSFID score stagnates around 40, even at large encoding ratios.
+
+![](images/bd394577caeec14ce29c629ae45ab0ae5086faddc7bc6ae3177733fe31472844.jpg)  
+Figure 6: Ablations on ImageNet. Left: effect of masking and encoding component. Right: DIFFEDIT with different mask thresholds; with 0.5 our default setting.
+
+Finally, our mask guidance operator $\tilde { \mathbf { y } } _ { t } = M \mathbf { y } _ { t } + ( 1 - M ) \mathbf { x } _ { t }$ provides a better trade-off than the reference text operator used in GLIDE (Nichol et al., 2021), which interpolates $\mathbf { y } _ { t }$ with a mask-corrected version of the predicted denoised image $\hat { \mathbf { y } } _ { 0 }$ . With encoding ratio $80 \%$ , both operators produce edits with a LPIPS score of 30.5, but the GLIDE version yields a CSFID of 26.4 compared to 23.6 for ours.w/o reference text
+
+# 4.3 EXPERIMENTS ON IMAGES GENERATED BY IMAGEN
+
+In our second set of experiments we evaluate edits that involve changes in background, replacingw/ reference text “A bowl of fruits” secondary objects, and editing object properties. We find that images generated by Imagen (Saharia et al., 2022b) offer a well suited testbed for this purpose. Indeed, the authors tested the compositional abilities of Imagen with templated prompts of the form: “{A photo of a | An oil painting ofw/o reference text $^  a \}$ {fuzzy panda | British shorthair cat | Persian cat | Shiba Inu dog | raccoon} {wearing a cowboy hat and | wearing sunglasses and} {red shirt | black jacket} {playing a guitar | riding a bike | skateboarding} {in a garden | on a beach | on top of a mountain}”, resulting in 300 prompts.
+
+We use the generated images as input and ask to change the prompt to another prompt for which one of these elements is changed. Since we cannot use the CSFID metric as for ImageNet, as images do not carry a single class label, we use FID to measure image realism, and CLIPScore (Hessel et al., 2021) to measure the alignment of the query and A cup of output image. These two scores have become the standard in evaluating text-conditional image generation (Saharia et al., 2022b).
+
+Figure 7 displays the CLIP-LPIPS and FID-CLIP trade-offs. DIFFEDIT provides more accurate edits than SDEdit, FlexIT, and Cross Attention Control, by combining inferred masks with DDIM encoding. Two versions of DiffEdit are shown, which differ by how the mask is computed: they correspond to (i) using the original caption as reference text (labelled w/ ref. text) or (ii) using the empty text $\varnothing$ (labelled w/o ref. text).
+
+Computing the mask with the original caption as reference text yields the best overall trade-off. Leveraging the original caption yields better CLIP and FID scores. Figure 8 illustrates the difference in the masks obtained with and without reference text for two examples. The reference text allows to ignore parts of the image that are described both by the query and reference text (e.g. the fruits), because in both cases the network uses the common text on the corresponding image region to estimate the noise. On the contrary, parts where the query and reference text disagree, e.g. “bowl” vs. “basket”, will have different noise estimates.
+
+![](images/8bc62fea0c1cfe6b80c8348821bdd365ab6aba3a0859bf187dd92c5f07adbcaf.jpg)  
+Figure 7: Editing trade-offs on Imagen images.
+
+![](images/8c115111817cfa4530bd1beb3c4f7a43c220e79c8a3a12c32e2a436fc5a9674c.jpg)  
+Figure 8: Masks and edits obtained with and without reference text in the mask computation algorithm.
+
+![](images/a52dbaa4154524b092e23925ae0e8691fb6033c00d465c0e1b2e20f228d3e67a.jpg)  
+Figure 9: Edits on Imagen dataset. We use encoding ratio of $90 \%$ for DiffEdit and $70 \%$ for SDEdit for fair comparison: both methods have similar CLIPScore, for larger encoding ratios SDEdit drastically change the input.
+
+Qualitative transformation examples are shown in Figure 9, where the masks are inferred by contrasting the caption and query texts.
+
+# 4.4 EXPERIMENTS ON COCO
+
+To evaluate semantic image editing with more complex prompts, we use images and captions from the COCO dataset Lin et al. (2014). To this end, we leverage the annotations provided by Hu et al. (2019), which associate images from the COCO validation set with other COCO captions that are similar to the original ones, but in contradiction with the given image. This makes these annotations particularly interesting as queries for semantic image editing, as they can often be satisfied by editing only a part of the input image, see Figure 15 in the supplementary material for examples. Similar to our evaluation for Imagen images, here we evaluate edits in terms of CLIPScore, FID and LPIPS.
+
+The results in Figure 10 show that the CLIP-LPIPS trade-off of DIFFEDIT is the best, but that it reaches lower maximum CLIP score than SDEdit. The FID scores are similar to SDEdit, but significantly improves upon the Encode-Decode ablation, which does not use a mask.
+
+Moreover, in contrast to results on the Imagen data, leveraging the original image caption does not change the CLIP-LPIPS and FID-CLIP tradeoffs. We find that the caption often describes the input image differently compared to the query text, making it more difficult to identify which part of the image needs to be edited. We verify this hypothesis in Section A.3 by filtering the dataset according to the edit distance between the caption and edit query. When the caption and edit query are similar, leveraging the image caption boosts CLIP scores by 0.25 points, a similar improvement as seen on the Imagen data.
+
+Qualitative examples are shown in Figure 11. The first column illustrates the benefit of DDIM encoding: we are able to correctly maintain properties of the object inside the mask, such as clothes’ color. The three last columns illustrate how contrasting different pairs of reference and query text allows to select different objects in the input image to perform different edits. See Section A.5 for more examples.
+
+![](images/82a3728328ed1ad8bbb36cc6af19d1b25ae3a7477d2eaea9f2926e245f9cf7e2.jpg)  
+Figure 10: Quantitative evaluation on COCO.
+
+![](images/0a3a019573e15f7cc607d61aa9c49bfddaa017877dffe35637f99f62b067a970.jpg)  
+Figure 11: Examples edits on COCO images.
+
+# 5 CONCLUSION
+
+We introduced DIFFEDIT, a novel algorithm for semantic image editing based on diffusion models. Given a textual query, using the diffusion model, DIFFEDIT infers the relevant regions to be edited rather than requiring a user generated mask. Furthermore, in contrast to other diffusion-based methods, we initialize the generation process with a DDIM encoding of the input.We provide theoretical analysis that motivates this choice, and show experimentally that this approach conserves more appearance information from the input image, leading to lighter edits. Quantitative and qualitative evaluations on ImageNet, COCO, and images generated by Imagen, show that our approach leads excellent edits, improving over previous approaches. Although DIFFEDIT works better with a reference text describing the input image, we believe this additional information can be inferred from input image and target caption, which we leave for future work.
+
+# 6 ETHICS STATEMENT
+
+Image editing raises several ethical challenges that we wish to discuss here. First, as image editing is closely related to image generation, it inherits known concerns. Open-source diffusion models are trained on large amounts of web-scraped data like LAION, and inherit their biases. In particular, it was shown that LAION contains inappropriate content (violence, hate, pornography), along with racist and sexist stereotypes. Furthermore it was found that diffusion models trained on LAION, such as Imagen, can exhibit social and cultural bias. Therefore, the use of such models can raise ethical issues, whether the text prompt is intentionnally harmful or not. Because image editing is usually performed on real images, there are additionnal ethical challenges, such as potential skin tone change when editing a person or re-inforcing harmful social stereotypes. We believe that open-sourcing editing algorithms in a research context contributes to a better understanding of such problems, and can aid the community in efforts to mitigate them in the future. Furthermore, image editing tools could be used with harmful intent such as harrassement or propagating fake news. This use, known as deep fakes, has been largely discussed in previous work, e.g. in Etienne (2021). To mitigate potential misuse, the Stable Diffusion model is released under a license focused on ethical and legal use, stating explicitly that users “must not distribute harmful, offensive, dehumanizing content or otherwise harmful representations of people or their environments, cultures, religions, etc. produced with the model weights”.
+
+Our editing benchmark based on the COCO dataset also has some limitations. COCO has a predominant western cultural bias, and we are therefore evaluating transformations on a small subset of images mostly associated with western culture. Finding relevant transformation prompts for an image is challenging: while we found it relevant to leverage existing annotations based on COCO, we believe that evaluating image editing models on a less culturally biased dataset is needed.
+
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+# DIFFEDIT: Diffusion-based semantic image editing with mask generation Supplementary Material
+
+In this supplementary material we provide more details on the experiments and methods presented in the main paper. In Section A we provide additional experimental results, including assessment of the impact of the strength of classifier-free guidance, the impact of using reference texts describing the input image, an illustration of the effect of the encoding ratio, more qualitative editing examples, as well as a number of example images and associated reference and query texts on COCO. In Section B we provide proofs that support Proposition 1 in the main paper.
+
+# A ADDITIONAL EXPERIMENTAL RESULTS
+
+# A.1 ANALYSIS OF NOISE USED TO COMPUTE THE MASK
+
+In step one our method an editing mask is inferred by contrasting noise estimations on a noised version of the input image, see Section 3.2. In this section, we study the impact of the level of noise added to the input image, by varying its value between 0.1 and 0.8, where 0 corresponds to using the initial image as input, and 1 to replacing the input image with random Gaussian noise. We evaluate the obtained operating points on ImageNet with the CSFID and LPIPS metrics when using a encoding ratios of 0.7 and 0.8 for DDIM encoding and masked-guided denoising in steps two and three of DIFFEDIT. From the results in Figure 12, we find that best results are obtained for moderate values of noise addition of 0.6 and below. Indeed, with too much noise added to the
+
+![](images/b4f7638e52a2495478a3751fffd13fc96b6b3bc408e5414cd7c8ad0e0ca0dbb7.jpg)  
+Figure 12: Impact of the noise added to input image when computing the mask, for a encoding ratios of 0.7 or 0.8 on ImageNet.
+
+input image, it is difficult to correctly identify visual elements in the input image. We use a value of 0.5 in all our experiments.
+
+# A.2 CLASSIFIER-FREE GUIDANCE
+
+In diffusion models, the noise estimator $\epsilon _ { \theta }$ can be conditioned on text describing the image, which provides a signal to guide the noise estimation process (Nichol et al., 2021; Saharia et al., 2022b). Ho & Salimans (2022) introduced classifier-free guidance, a technique to greatly improve generation quality and imagetext alignment in text-conditional diffusion models. It consists in training both a conditional and unconditional model by dropping the conditioning text at train time with fixed probability, e.g. $10 \%$ . Then, after training, at each step $t$ during decoding, the noise estimation $\epsilon _ { \theta } ( \mathbf { x } _ { t } , Q , t )$ is extrapolated by using the unconditional noise estimation $\epsilon _ { \theta } ( \mathbf { x } _ { t } , \emptyset , \bar { t } )$ as origin. Formally, the noise that is used is
+
+$$
+\begin{array} { r } { \epsilon = \epsilon _ { \theta } ( \mathbf { x } _ { t } , \theta , t ) + \lambda ( \epsilon _ { \theta } ( \mathbf { x } _ { t } , Q , t ) - \epsilon _ { \theta } ( \mathbf { x } _ { t } , \theta , t ) ) , } \end{array}
+$$
+
+where $\lambda$ is the classifier-free guidance parame
+
+![](images/0a64148ed96aada078fbea7a59fbba8884b9f07e1968081881ebe91c0b2dad74.jpg)  
+Figure 13: Ablation on the value of the classifierfree guidance parameter. On ImageNet, we find that a value of at least 3 must be used to get good results. We chose to use 5, which is the default value recommended for generation.
+
+ter. We study the influence of this parameter on DIFFEDIT in Figure 13, finding that similarly to generation, a value above 3 yields the best results. Without classifier-free guidance, the obtained trade-off is not competitive. For our experiments we use a default guidance value of $\lambda = 5$ .
+
+Here, we investigate why there is little difference between using or not the reference text to compute the mask on our COCO queries. In Figure 15 we show several editing queries on the COCO dataset taken from the BISON dataset (Hu et al., 2019). Generally, the text query describes a scene similar to the one in the input image, and it is possible to match the text query by editing only a fraction of the input image.
+
+![](images/1f2d52295d3873752b14dbd03495d4a4c21274ed5eb5e9345e93b33551629d4f.jpg)  
+Figure 14: Results on COCO: unfiltered (left) and filtered (right). While having a small impact overall, for the filtered set using the reference text is beneficial, especially at high encoding ratios, e.g. $90 \%$ .
+
+However, we find that while queries have been built to be close to a caption of the input image, most of the time the query is not well aligned with the caption. We create a filtered version of this dataset, for which queries are structurally similar to the caption, i.e. where only a few words are changed, but the grammatical structure stays the same. We use the filtering criterion that the total number of words inserted/deleted/replaced must not exceed $2 5 \%$ of the total number of words in the original caption, resulting in a total of 272 queries out of $5 0 \mathrm { k }$ original queries. In Figure 14 we compare results with and without filtering, and observe that for the images with small caption edits the gain of DIFFEDIT (w/ ref. text) compared to Encode-Decode is somewhat larger than on the unfiltered dataset. Moreover, using the original caption as reference text to compute the mask gives higher CLIPScore, especially at high encoding ratio. This illustrates that a well chosen reference text helps to generate better editing masks.
+
+![](images/488f6a20b73e3d82f3e869fcb234a05356e8a7c901a0c2b27ba1d0537d357d97.jpg)  
+Figure 15: Editing queries on the COCO dataset.
+
+# A.4 VISUALISATION OF THE IMPACT OF ENCODING RATIO
+
+We show visual results for ablations of our two main components, mask inference and DDIM encoding, in Figure 16. The resulting methods are SDEdit, Encode-Decode, DIFFEDIT w/o Encoding, and DIFFEDIT. We demonstrate the qualitative behavior of these different methods, at varying encoding ratios between $3 0 \%$ and $8 0 \%$ . Compared to SDEdit, Encode-Decode allows to better match the query with less modifications of the main object and the background, especially at $6 0 \% - 7 0 \%$ .
+
+Mask inference allows to maintain exactly the background. Using DDIM inference on top of maskbased decoding allows to better retain of the content inside the mask, especially at $70 \%$ and $80 \%$ , c.f. row 3 vs. 4.
+
+![](images/98719c7ee5f1dbdd5d84668164eddd692f5076bf64a8a6a1f502daf850302b04.jpg)  
+Figure 16: Qualitative ablations of the mask and encoding components, using different encoding ratios from $30 \%$ to $80 \%$ .
+
+A.5 ADDITIONAL VISUALIZATIONS AND QUALITATIVE RESULTS
+
+Figure 17 illustrates editing examples on Imagen, in comparison with other mask-free editing methods. DIFFEDIT generally performs more targeted and accurate edits, leaving more of the original image in tact where possible. Consider for example the first column of Figure 17, where DIFFEDIT leaves the guitar as it, while other methods make unnecessary and unrealistic changes to the guitar.
+
+Additional qualitative examples on COCO images are shown in Figure 19.
+
+Figure 20 shows several failure cases of semantic image editing with DIFFEDIT. Some failure modes are inherited from the generative model itself: models trained on web-scrapped image-text data are known to struggle with understanding spatial positions in images, spatial reasoning, and counting (Ramesh et al., 2021). Others are specific to our mask-based method, like the difficulty to insert objects, because the mask often seeks an “anchor” visual element to insert an object, see first column.
+
+# A.6 DETAILS ON COMPARISONS WITH OTHER METHODS
+
+On the COCO and Imagen datasets, we do not compare with ILVR, since it cannot be used within the latent diffusion framework: the method needs image downsampling and upsampling, which does not work well with the latent spaces used in latent diffusion. Even adapted with a diffusion model without latent spaces like Imagen, we do not expect the CLIP-LPIPS trade-off to be favorable for this method, given the high editing distance obtained on ImageNet. Instead, we compare against CrossAttention Control Hertz et al. (2022), a recent method for text-driven image editing based on the unreleased Imagen diffusion model. The method is very recent and has been adapted to use with Stable Diffusion at https://github.com/bloc97/CrossAttentionControl/. We have performed lightweight hyperparameter search to optimize the CLIP-LPIPS trade-off on a subset of Imagen images. We generally find that this re-implementation, while producing edited images structurally similar to the input, changes local features more than SDEdit, leading to generally high LPIPS distances, resulting in a CLIP-LPIPS trade-off not competitive with other methods on our COCO and Imagen benchmark. In particular, LPIPS distance are high on the COCO dataset where text query and reference text have a high edit distance on average, whereas Cross-Attention Control was designed to perform well for prompt-to-prompt editing, i.e. the input and target text should almost exactly match. Given that our results are based on the unofficial re-implementation, we caution that they are temporary and we will update them when the official code (or official adaptation for Stable Diffusion) is released.
+
+![](images/78bc302f08d16abf3a851266266d94346544fd5ce0416324636d42d3f1df7b6f.jpg)  
+Figure 17: Example edits from Imagen, in comparison with other mask-free editing methods.
+
+![](images/75da0df749c4ad6b61b37d67664e96a9e20b15b7bf18f6dd1e207fc31ae9d8fe.jpg)  
+Figure 18: More qualitative examples on COCO. Baseline methods are shown for comparison. The mask is sometimes bigger or smaller that one could expect: In column 3, it is larger, but there are few edits outside the requested bread burger transformation (except for the wine bottle label), which is not the case without DDIM encoding. In column 4, the mask does not cover the interior of the truck, but this does not affect the edit quality.
+
+![](images/0d9ae0edcececceda8e7defeb40c065f1d7d0469bfa6a73d1336df98cabc0a22.jpg)  
+Figure 19: More qualitative examples on COCO. In the first column, the color of the objects to be edited is maintained, which would not be the case with regular inpainting methods. Contrasting similar text query and reference text allows to select the object to be edited.
+
+![](images/a7741a1f96ca0868b98f8a26ed65c06c87b95e062dd68424b0fb1acc9a0ae11c.jpg)  
+Figure 20: Illustration of failure modes. In the first two columns show difficulty to insert an object in a smooth region of the image. In column three the mask fails to identify a region where to add the zebra. Columns 4 and 5 show mask identification errors, where multiple similar objects are included in the mask, whereas matching the text query only requires to edit a single object. In both cases this results in over-editing. Col. 6 shows the failure to change a spatial relation in the image.
+
+# B THEORETICAL RESULTS
+
+Here, we prove the bounds given in the main paper. We reused notations from Proposition 1 in the main paper. We also discuss links to optimal transport.
+
+# B.1 PROOF OF SDEDIT BOUND
+
+Proposition 2. Suppose that $\| \epsilon _ { \theta } ( \mathbf { x } , Q , t ) \| _ { 2 } \leq C$ for all $x \in \mathcal { X }$ , $t \in [ 0 , 1 ]$ . Then
+
+$$
+\begin{array} { r l } & { \quad \underset { ( \mathbf { x } _ { 0 } , Q ) \sim p _ { D } } { \mathbb { E } } \| \mathbf { x } _ { 0 } - D _ { r } ( G _ { r } ( \mathbf { x } _ { 0 } , \epsilon ) , Q ) \| _ { 2 } \leq ( C + 1 ) \tau } \\ & { \quad \epsilon \sim { \cal N } ( 0 , 1 ) } \end{array}
+$$
+
+Proof. Let $T$ $\mathbf { \xi } , \mathbf { x } _ { r } = G _ { r } ( \mathbf { x } _ { 0 } , \epsilon ) , \mathbf { y } _ { r } = \mathbf { x } _ { r }$ and ${ \bf y } _ { 0 } = D _ { r } ( { \bf y } _ { r } , Q )$ . Then
+
+$$
+\| \frac { { \bf x } _ { r } } { \sqrt { \alpha _ { r } } } - { \bf y } _ { 0 } \| = \| \frac { { \bf y } _ { r } } { \sqrt { \alpha _ { r } } } - \frac { { \bf y } _ { 0 } } { \sqrt { \alpha _ { 0 } } } \| = \| \int _ { \tau } ^ { 0 } \epsilon _ { \theta } ( x _ { t } , Q , t ) d \tau \| \le C \tau .
+$$
+
+Since the pr $\begin{array} { r } { \frac { \mathbf { x } _ { r } } { \sqrt { \alpha _ { r } } } = \mathbf { x } _ { 0 } + \boldsymbol { \tau } \boldsymbol { \epsilon } } \end{array}$ , we have $\begin{array} { r } { \| \mathbf { x } _ { 0 } - \mathbf { y } _ { 0 } \| \leq \| \mathbf { x } _ { 0 } + \tau \epsilon - \mathbf { y } _ { 0 } \| + \| \tau \epsilon \| \leq C \tau + \tau } \end{array}$ which concludes
+
+In the SDEdit paper (Meng et al., 2021), a proof similar to what we state is given, with three main differences: (i) the proof is given in the case of variance-exploding Stochastic Differential Equation (VE-SDE), which needs adaption for our setting which uses variance-preserving SDE; (ii) the bound is derived in the case of a stochastic differential equation, whereas we use a deterministic DDIM process; (iii) the bound is given by controlling the probability tail, whereas we only consider the expectancy of edit distance. However, despite these differences, the spirit of the proof is the same as here.
+
+# B.2 PROOF OF PROPOSITION 2
+
+Proposition 3. Suppose that $\epsilon _ { \theta } ( \cdot , Q , t )$ is $K _ { 1 }$ -lipschitz and $\kappa _ { 2 }$ defined as
+
+$$
+\kappa _ { 2 } ( \mathbf { x } _ { 0 } ) = \operatorname* { m a x } _ { t \in [ 0 , 1 ] } \| \epsilon _ { \theta } ( E _ { t } ( \mathbf { x } _ { 0 } ) , Q , t ) - \epsilon _ { \theta } ( E _ { t } ( \mathbf { x } _ { 0 } ) , \emptyset , t ) \|
+$$
+
+Let $K _ { 2 } = \mathbb { E } _ { \mathbf { x } _ { 0 } } \kappa _ { 2 } ( \mathbf { x } _ { 0 } )$ . Then for all encoding ratio $r$ , with $\tau = \sqrt { \alpha _ { r } ^ { - 1 } - 1 }$ ,
+
+$$
+\mathbb { E } _ { \mathbf { x } _ { 0 } } \| \mathbf { x } _ { 0 } - D _ { r } ( E _ { r } ( \mathbf { x } _ { 0 } ) , Q ) \| \leq \frac { K _ { 2 } \tau } { \sqrt { \tau ^ { 2 } + 1 } } \Big ( \tau + \sqrt { \tau ^ { 2 } + 1 } \Big ) ^ { K _ { 1 } }
+$$
+
+Proof. Let $\sigma$ be a time-dependent variable defined as $\sigma ( t ) ~ = ~ \sqrt { \alpha _ { t } ^ { - 1 } - 1 }$ . Let $\textbf { u } = \textbf { x } / \sqrt { \alpha } =$ $\mathbf { x } \sqrt { 1 + \sigma ^ { 2 } }$ and $\mathbf { v } = \mathbf { y } { \sqrt { 1 + \sigma ^ { 2 } } }$ . u and $\mathbf { v }$ are solutions of the following differential system:
+
+$$
+\begin{array} { r l } & { { d { \mathbf { u } } } | _ { t } = \epsilon _ { \theta } ( { \mathbf { u } } / \sqrt { 1 + \sigma ^ { 2 } } , \varnothing , t ) { d \sigma } , } \\ & { { d { \mathbf { v } } } | _ { t } = \epsilon _ { \theta } ( { \mathbf { v } } / \sqrt { 1 + \sigma ^ { 2 } } , Q , t ) { d \sigma } , } \\ & { { \mathbf { u } } ( r ) = { \mathbf { v } } ( r ) = E _ { r } ( \mathbf { x } _ { 0 } ) \sqrt { 1 + \sigma ^ { 2 } } . } \end{array}
+$$
+
+Let $\mathbf { w } = \| \mathbf { u } - \mathbf { v } \|$ , then $\mathbf { w } | _ { t = r } = 0$ and
+
+$$
+\begin{array} { r l } & { \| \mathbf { w } | _ { t } \leq \| d \mathbf { u } | _ { t } - d \mathbf { v } | _ { t } \| = \| ( { \epsilon } _ { \theta } ( \mathbf { x } , { \theta } , t ) - { \epsilon } _ { \theta } ( \mathbf { y } , Q , t ) ) d \sigma \| } \\ & { \qquad \leq \| ( { \epsilon } _ { \theta } ( \mathbf { x } , { \theta } , t ) - { \epsilon } _ { \theta } ( \mathbf { x } , Q , t ) \| d \sigma + \| ( { \epsilon } _ { \theta } ( \mathbf { x } , Q , t ) - { \epsilon } _ { \theta } ( \mathbf { y } , Q , t ) \| d \sigma } \\ & { \qquad \leq { \kappa } _ { 2 } ( \mathbf { x } _ { 0 } ) d \sigma + K _ { 1 } \| \mathbf { x } - \mathbf { y } \| d \sigma } \\ & { \qquad \leq \Big ( { \kappa } _ { 2 } ( \mathbf { x } _ { 0 } ) + \frac { K _ { 1 } } { \sqrt { 1 + { \sigma } ^ { 2 } } } w \Big ) d \sigma . } \end{array}
+$$
+
+By integration we get
+
+$$
+{ \pmb w } ( t ) \le \kappa _ { 2 } ( { \bf x } _ { 0 } ) * ( \tau - t ) + \int _ { t } ^ { \tau } \frac { K _ { 1 } } { \sqrt { 1 + \sigma ^ { 2 } } } { \pmb w } ( \sigma ) d \sigma .
+$$
+
+From here we can apply Gronwall’s inequality: ¨
+
+$$
+\begin{array} { l } { w ( 0 ) \leq \kappa _ { 2 } ( \mathbf { x } _ { 0 } ) \tau \exp \Big ( \displaystyle \int _ { 0 } ^ { \tau } \frac { K _ { 1 } } { \sqrt { 1 + s ^ { 2 } } } d s \Big ) } \\ { \leq \kappa _ { 2 } ( \mathbf { x } _ { 0 } ) \tau \exp \Big ( K _ { 1 } \log ( \tau + \sqrt { \tau ^ { 2 } + 1 } ) \Big ) } \\ { \leq \kappa _ { 2 } ( \mathbf { x } _ { 0 } ) \tau \Big ( \tau + \sqrt { \tau ^ { 2 } + 1 } \Big ) ^ { K _ { 1 } } . } \end{array}
+$$
+
+Which finally gives
+
+$$
+\| \mathbf { x } _ { 0 } - \mathbf { y } _ { 0 } \| \leq \frac { \kappa _ { 2 } ( \mathbf { x } _ { 0 } ) \tau } { \sqrt { \tau ^ { 2 } + 1 } } \Big ( \tau + \sqrt { \tau ^ { 2 } + 1 } \Big ) ^ { K _ { 1 } } .
+$$
+
+Taking the expectation w.r.t. the input image $\mathbf { x } _ { \mathrm { 0 } }$ gives the final result:
+
+$$
+\mathbb { E } _ { \mathbf { x } _ { 0 } } \left\| \mathbf { x } _ { 0 } - D _ { T } ( E _ { T } ( \mathbf { x } _ { 0 } ) , Q ) \right\| \leq \frac { K _ { 2 } \tau } { \sqrt { \tau ^ { 2 } + 1 } } \Big ( \tau + \sqrt { \tau ^ { 2 } + 1 } \Big ) ^ { K _ { 1 } }
+$$
+
+which concludes the proof.
+
+# B.3 LINKS TO OPTIMAL TRANSPORT THEORY
+
+The reverse DDIM encoder $E _ { r }$ maps the distribution of images $p _ { 0 } = p _ { D }$ to the distribution $p _ { r }$ of images noised at timestep $r$ . Khrulkov & Oseledets (2022) suggested that $E _ { r }$ could be an optimal transport map between $p _ { 0 }$ and $p _ { r }$ , minimizing the transport cost $\mathbb { E } _ { \mathbf { x } _ { 0 } } \| \mathbf { x } _ { 0 } - E _ { r } ( \mathbf { x } _ { 0 } ) \| _ { 2 } ^ { 2 }$ . This means that the encoded images are, on average, as close as possible to the input images, while following the correct distribution $p _ { r }$ . It would entail that the unconditional decoder $\bar { D _ { r } } = E _ { r } ^ { - 1 }$ would be an optimal transport map between $p _ { r }$ and $p _ { 0 }$ , and moreover that the conditional decoder $D _ { r } ( \cdot , Q )$ would be an optimal transport map between the distributions $p _ { r } ( \cdot | Q )$ and $p _ { 0 } ( \cdot | Q )$ conditioned by text description $Q$ . Under the hypothesis that $p _ { r }$ is very close to $p _ { r } ( \cdot | Q )$ , then the Encode-Decode algorithm would be the combination of two optimal transport maps $E _ { r }$ and $D _ { r } ( \cdot , Q )$ , mapping $p _ { 0 }$ to $p _ { r }$ and then $p _ { r } \simeq p _ { r } ( \cdot | Q )$ to $p _ { 0 } ( \cdot | Q )$ . This is a very interesting property and we make the connection with the desired properties of semantic image editing, which can be expressed as an optimal transport problem. Given two distribution of images $p _ { 1 } , p _ { 2 }$ (lets say cats and dogs), the aim is to find the function $f$ that performs the expected edit (changing images of cats into images of dogs) while minimally editing the image, which can be expressed mathematically as:
+
+$$
+f = \underset { f } { \arg \operatorname* { m i n } } \mathbb { E } _ { \mathbf { x } } \| \mathbf { x } - f ( \mathbf { x } ) \| \quad \mathrm { s . t . } \quad p _ { 2 } = f _ { \# } p _ { 1 } ,
+$$
+
+where $f _ { \# }$ is the push-forward measure. The function $D _ { r } ( \cdot , Q ) \circ E _ { r }$ is not a solution of this optimal transport problem, because (i) it was proven that the reverse DDIM encoder is not the optimal transport map for some distributions (Lavenant & Santambrogio, 2022), and (ii) the composition of two optimal transport maps is not necessarily an optimal transport map. However, experiments and numerical simulations suggest that $E _ { r }$ is very close from an optimal transport map. It would be interesting to study the “optimality defect” of $E _ { r }$ and of the editing function $D _ { r } ( \cdot , Q ) \circ E _ { r }$ . We leave this for future work.

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@@ -0,0 +1,307 @@
+# VideoMAE: Masked Autoencoders are Data-Efficient Learners for Self-Supervised Video Pre-Training
+
+Zhan Tong 1,2∗ Yibing Song 2 Jue Wang 2 Limin Wang 1,3† 1State Key Laboratory for Novel Software Technology, Nanjing University 2Tencent AI Lab 3Shanghai AI Lab tongzhan@smail.nju.edu.cn {yibingsong.cv, arphid}@gmail.com lmwang@nju.edu.cn
+
+# Abstract
+
+Pre-training video transformers on extra large-scale datasets is generally required to achieve premier performance on relatively small datasets. In this paper, we show that video masked autoencoders (VideoMAE) are data-efficient learners for self-supervised video pre-training (SSVP). We are inspired by the recent ImageMAE [30] and propose customized video tube masking with an extremely high ratio. This simple design makes video reconstruction a more challenging and meaningful self-supervision task, thus encouraging extracting more effective video representations during the pre-training process. We obtain three important findings with VideoMAE: (1) An extremely high proportion of masking ratio (i.e., $9 0 \%$ to $9 5 \%$ ) still yields favorable performance for VideoMAE. The temporally redundant video content enables higher masking ratio than that of images. (2) VideoMAE achieves impressive results on very small datasets (i.e., around $3 \mathrm { k } { - } 4 \mathrm { k }$ videos) without using any extra data. This is partially ascribed to the challenging task of video reconstruction to enforce high-level structure learning. (3) VideoMAE shows that data quality is more important than data quantity for SSVP. Domain shift between pre-training and target datasets is an important factor. Notably, our VideoMAE with the vanilla ViT backbone can achieve $8 7 . 4 \%$ on Kinects-400, $7 5 . 4 \%$ on SomethingSomething V2, $9 1 . 3 \%$ on UCF101, and $6 2 . 6 \%$ on HMDB51, without using any extra data. Code is available at https://github.com/MCG-NJU/VideoMAE.
+
+# 1 Introduction
+
+Transformer [70] has brought significant progress in natural language processing [17, 7, 54]. The vision transformer [20] also improves a series of computer vision tasks including image classification [66, 88], object detection [8, 37], semantic segmentation [80], object tracking [13, 16], and video recognition [6, 3]. The multi-head self-attention upon linearly projected image/video tokens is capable of modeling global dependency among visual content either spatially or temporally. The inductive bias is effectively reduced via this flexible attention mechanism.
+
+Training effective vision transformers (ViTs) typically necessitates large-scale supervised datasets. Initially, the pre-trained ViTs achieve favorable performance by using hundreds of millions of labeled images [20]. For video transformers [3, 6], they are usually derived from image-based transformers and heavily depend on the pre-trained models from large-scale image data (e.g., ImageNet [57]). Previous trials [3, 6] on training video transformers from scratch yield unsatisfied results (except for MViT [21] with a strong inductive bias). Therefore, the learned video transformers are naturally biased by image-based models, and it still remains a challenge that how to effectively and efficiently train a vanilla vision transformer on the video dataset itself without using any pre-trained model or extra image data. Moreover, the existing video datasets are relatively small compared with image datasets, which further increases the difficulty of training video transformers from scratch. Meanwhile, self-supervised learning has shown remarkable performance by using large-scale image datasets [14, 9]. The learned representations have outperformed the ones via supervised learning when being transferred to downstream tasks. It is expected that this self-supervised learning paradigm can provide a promising solution to address the challenge of training video transformers.
+
+![](images/7b105b3d14f137e30020c4f998b46547045fcb84c9d865cb040b66ae6e581a13.jpg)  
+Figure 1: VideoMAE performs the task of masking random cubes and reconstructing the missing ones with an asymmetric encoder-decoder architecture. Due to high redundancy and temporal correlation in videos, we present the customized design of tube masking with an extremely high ratio $90 \%$ to $9 5 \%$ ). This simple design enables us to create a more challenging and meaningful self-supervised task to make the learned representations capture more useful spatiotemporal structures.
+
+Following the success of masked autoencoding in NLP [17] and images [30, 4], we present a new selfsupervised video pre-training (SSVP) method, termed as Video Masked Autoencoder (VideoMAE). Our VideoMAE inherits the simple pipeline of masking random cubes and reconstructing the missing ones. However, the extra time dimension of videos makes them different from images in this masked modeling. First, video frames are often densely captured, and their semantics varies slowly in time [87]. This temporal redundancy would increase the risk of recovering missing pixels from the spatiotemporal neighborhood with little high-level understanding. Furthermore, video could be viewed as the temporal evolution of static appearance, and there exists a correspondence between frames. This temporal correlation could lead to information leakage (i.e., masked spatiotemporal content re-occurrence) during reconstruction unless a specific masking strategy is considered. In this sense, for each masked cube, it is easy to find a corresponding and unmasked copy in adjacent frames. This property would make the learned models identify some “shortcut” features that are hard to generalize to new scenarios.
+
+To make video masked modeling more effective, in this paper, we present a customized design of tube masking with an extremely high ratio in our VideoMAE. First, due to temporal redundancy, we use an extremely high masking ratio to drop the cubes from the downsampled clips. This simple strategy not only effectively increases the pre-training performance but also greatly reduces the computational cost due to the asymmetric encoder-decoder architecture. Second, to consider temporal correlation, we devise a simple yet effective tube masking strategy, which turns out to be helpful in relieving the risk of information leakage for cubes with no or negligible motion during reconstruction. With this simple yet effective design in our VideoMAE, we are able to successfully train vanilla ViT backbones on the relatively small-scale video datasets such as Something-Something [25], UCF101 [60], and HMDB51 [34], which significantly outperform the previous state of the art under the setting without extra data. In summary, the main contribution of this paper is threefold:
+
+• We present a simple but effective video masked autoencoder that unleashes the potential of vanilla vision transformer for video recognition. To the best of our knowledge, this is the first masked video pre-training framework of simply using plain ViT backbones. To relieve the information leakage issue in masked video modeling, we present the tube masking with an extremely high ratio, which brings the performance improvement to the VideoMAE. • Aligned with the results in NLP and Images on masked modeling, our VideoMAE demonstrates that this simple masking and reconstruction strategy provides a good solution to self-supervised video pre-training. The models pre-trained with our VideoMAE significantly outperform those trained from scratch or pre-trained with contrastive learning methods. • We obtain extra important findings on masked modeling that might be ignored in previous research in NLP and Images. (1) We demonstrate that VideoMAE is a data-efficient learner that could be successfully trained with only $3 . 5 \mathrm { k }$ videos. (2) Data quality is more important than quantity for SSVP when a domain shift exists between the source and target dataset.
+
+# 2 Related Work
+
+Video representation learning. Learning good video representations has been heavily investigated in the literature. The supervised learning methods [58, 75, 69, 10, 6] usually depend on the image backbones. The video encoder backbones are first pre-trained with image data in a supervised form. Then, these backbones are fine-tuned on the video dataset for classifying human actions. Meanwhile, some methods [67, 22, 21] directly train video backbones from videos in a supervised manner. Besides supervised learning, semi-supervised video representation learning has also been studied [59]. The representations of labeled training samples are utilized to generate supervision signals for unlabeled ones. Supervised or semi-supervised representation learning mainly uses a top-down training paradigm, which is not effective in exploring the inherent video data structure itself. Meanwhile, some multimodal contrastive learning methods [36, 42, 62] have been developed to learn video representation from noisy text supervision.
+
+For self-supervised learning, the prior knowledge of temporal information has been widely exploited to design pretext tasks [78, 44, 82, 5] for SSVP. Recently, contrastive learning [28, 45, 29, 52, 24, 27] is popular to learn better visual representation. However these methods heavily rely on strong data augmentation and large batch size [23]. Predicting the video clip with autoencoders in pixel space has been explored for representation learning by using CNN or LSTM backbones [48, 61], or conducting video generation with autoregressive GPT [83]. Instead, our VideoMAE aims to use the simple masked autoencoder with recent ViT backbones to perform data-efficient SSVP.
+
+Masked visual modeling. Masked visual modeling has been proposed to learn effective visual representations based on the simple pipeline of masking and reconstruction. These works mainly focus on the image domain. The early work [72] treated the masking as a noise type in denoised autoencoders [71] or inpainted missing regions with context [47] by using convolutions. iGPT [11] followed the success of GPT [7, 55] in NLP and operated a sequence of pixels for prediction. The original ViT [20] investigated the masked token prediction for self-supervised pre-training. More recently, the success of vision transformer has led to investigation of Transformer-based architectures for masked visual modeling [4, 19, 30, 79, 81, 89]. BEiT [4], BEVT [76] and VIMPAC [64] followed BERT [17] and proposed to learn visual representations from images and videos by predicting the discrete tokens [56]. MAE [30] introduced an asymmetric encoder-decoder architecture for masked image modeling. MaskFeat [79] proposed to reconstruct the HOG features of masked tokens to perform self-supervised pre-training in videos. VideoMAE is inspired by the ImageMAE and introduces specific design in implementation for SSVP. In particular, compared with previous masked video modeling [30, 76, 64], we present a simpler yet more effective video masked autoencoder by directly reconstructing the pixels. Our VideoMAE is the first masked video pre-training framework of simply using plain ViT backbones.
+
+# 3 Proposed Method
+
+In this section, we first revisit ImageMAE [30]. Then we analyze the characteristics of video data.   
+Finally, we show how we explore MAE in the video data by presenting our VideoMAE.
+
+# 3.1 Revisiting Image Masked Autoencoders
+
+ImageMAE [30] performs the masking and reconstruction task with an asymmetric encoder-decoder architecture. The input image $I \in \mathcal { R } ^ { \mathbf { \breve { 3 } } \times H \times W }$ is first divided into regular non-overlapping patches of size $1 6 \times 1 6$ , and each patch is represented with token embedding. Then a subset of tokens are randomly masked with a high masking ratio $( 7 5 \% )$ , and only the remaining ones are fed into the transformer encoder $\Phi _ { \mathrm { e n c } }$ . Finally, a shallow decoder $\Phi _ { \mathrm { d e c } }$ is placed on top of the visible tokens from the encoder and learnable mask tokens to reconstruct the image. The loss function is mean squared error (MSE) loss between the normalized masked tokens and reconstructed ones in the pixel space:
+
+$$
+\mathcal { L } = \frac { 1 } { \Omega } \sum _ { p \in \Omega } | I ( p ) - \hat { I } ( p ) | ^ { 2 } ,
+$$
+
+where $p$ is the token index, $\Omega$ is the set of masked tokens, $I$ is the input image, and $\hat { I }$ is the reconstructed one.
+
+![](images/ae9479b1875af3b188fce7f930b56324b71788c64cb850c7133617d6ff4dbf9e.jpg)  
+Figure 2: Slowness is a general prior in (a) video data [87]. This leads to two important characteristics in time: temporal redundancy and temporal correlation. Temporal redundancy makes it possible to recover pixels under an extremely high masking ratio. Temporal correlation leads to easily reconstruct the missing pixels by finding those corresponding patches in adjacent frames under plain (b) frame masking or (c) random masking. To avoid this simple task and encourage learning representative representation, we propose a (d) tube masking, where the masking map is the same for all frames.
+
+# 3.2 Characteristics of Video Data
+
+Compared with static images, video data contain temporal relations. We show the motivation of our VideoMAE by analyzing video characteristics.
+
+Temporal redundancy. There are frequently captured frames in a video. The semantics vary slowly in the temporal dimension [87]. We observe that consecutive frames are highly redundant, as shown in Figure 2. This property leads to two critical issues in masked video autoencoding. First, it would be less efficient to keep the original temporal frame rate for pre-training. This would draw us to focus more on static or slow motions in our masked modeling. Second, temporal redundancy greatly dilutes motion representations. This would make the task of reconstructing missing pixels not difficult under the normal masking ratio (e.g., $50 \%$ to $7 5 \%$ ). The encoder backbone is not effective in capturing motion representations.
+
+Temporal correlation. Videos could be viewed as the temporal extension of static appearance, and therefore there exists an inherent correspondence between adjacent frames. This temporal correlation could increase the risk of information leakage in the masking and reconstruction pipeline. In this sense, as shown in Figure 2, we can reconstruct the masked patches by finding the spatiotemporal corresponding unmasked patches in the adjacent frames under plain random masking or frame masking. In this case, it might guide the VideoMAE to learn low-level temporal correspondence rather than high-level information such as spatiotemporal reasoning over the content. To alleviate this behavior, we need to propose a new masking strategy to make the reconstruction more challenging and encourage effective learning of spatiotemporal structure representations.
+
+# 3.3 VideoMAE
+
+To relieve the above issues in video masked modeling, we make the customized design in our VideoMAE, and the overall pipeline is shown in Figure 1. Our VideoMAE takes the downsampled frames as inputs and uses the cube embedding to obtain video tokens. Then, we propose a simple design of tube masking with high ratio to perform MAE pre-training with an asymmetric encoderdecoder architecture. Our backbone uses the vanilla ViT with joint space-time attention.
+
+Temporal downsampling. According to the above analysis on temporal redundancy over consecutive frames, we propose to use the strided temporal sampling strategy to perform more efficient video pre-training. Formally, one video clip consisting of $t$ consecutive frames is first randomly sampled from the original video $V$ . We then use temporal sampling to compress the clip to $T$ frames, each of which contains $H \times W \times 3$ pixels. In experiments, the stride $\tau$ is set to 4 and 2 on Kinetics and Something-Something, respectively.
+
+Cube embedding. We adopt the joint space-time cube embedding [3, 21, 38] in our VideoMAE, where we treat each cube of size $2 \times 1 6 \times 1 6$ as one token embedding. Thus, the cube embedding layer obtains T2 × H16 $\begin{array} { r } { \frac { T } { 2 } \times \frac { H } { 1 6 } \times \frac { W } { 1 6 } \ 3 } \end{array}$ D tokens and maps each token to the channel dimension $D$ . This design can decrease the spatial and temporal dimension of input, which helps to alleviate the spatiotemporal redundancy in videos.
+
+Tube masking with extremely high ratios. First, temporal redundancy is a factor affecting VideoMAE design. We find that VideoMAE is in favor of extremely high masking ratios (e.g. $90 \%$ to $9 5 \%$ ) compared with the ImageMAE. Video information density is much lower than images, and we expect a high ratio to increase the reconstruction difficulty. This high masking ratio is helpful to mitigate the information leakage during masked modeling and make masked video reconstruction a meaningful self-supervised pre-training task.
+
+Second, temporal correlation is another factor in our VideoMAE design. We find even under the extremely high masking ratio, we can still improve the masking efficiency by proposing the temporal tube masking mechanism. Temporal tube masking enforces a mask to expand over the whole temporal axis, namely, different frames sharing the same masking map. Mathematically, the tube mask mechanism can be expressed as $\mathbb { I } [ p _ { x , y , \cdot } \in \Omega ] \sim \mathrm { B e r n o u l l i } ( \bar { \rho _ { \mathrm { m a s k } } } )$ and different time $t$ shares the same value. With this mechanism, temporal neighbors of masked cubes are always masked. So for some cubes with no or small motion (e.g., finger cube in 4th row of Figure 2 (d)), we can not find the spatiotemporal corresponding content in all frames. In this way, it would encourage our VideoMAE to reason over high-level semantics to recover these totally missing cubes. This simple strategy can alleviate the information leakage for cubes with no or negligible motion, and turns out to be effective in practice for masked video pre-training.
+
+Backbone: joint space-time attention. Due to the high proportion of masking ratio mentioned above, only a few tokens are left as the input for the encoder. To better capture high-level spatio-temporal information in the remaining tokens, we use the vanilla ViT backbone [20] and adopt the joint space-time attention [3, 38]. Thus, all pair tokens could interact with each other in the multi-head self-attention layer [70]. The specific architecture design for the encoder and decoder is shown in supplementary materials. The quadratic complexity of the joint space-time attention mechanism is a computational bottleneck, while our design of an extremely high masking ratio alleviates this issue by only putting the unmasked tokens (e.g., $10 \%$ ) into the encoder during the pre-training phase.
+
+# 4 Experiments
+
+# 4.1 Datasets
+
+We evaluate our VideoMAE on five common video datasets: Kinetics-400 [33], Something-Something V2 [25], UCF101 [60], HMDB51 [34], and AVA [26]. The Kinetics-400 contains around 240k training videos and 20k validation videos of 10s from 400 classes. The Something-Something V2 is another large-scale video dataset, having around 169k videos for training and 20k videos for validation. In contrast to Kinetics-400, this dataset contains 174 motion-centric action classes. These two large-scale video datasets focus on different visual cues for action recognition. UCF101 and HMDB51 are two relatively small video datasets, which contain around $9 . 5 \mathrm { k } / 3 . 5 \mathrm { k }$ train/val videos and $3 . 5 \mathrm { k } / 1 . 5 \mathrm { k }$ train/val videos, respectively. Compared with those large-scale video datasets, these two small datasets are more suitable for verifying the effectiveness of VideoMAE, as training large ViT models is more challenging on small datasets. Moreover, we also transfer the learned ViT models by VideoMAE to downstream action detection task. We work on AVA, a dataset for spatiotemporal localization of human actions with 211k training and $5 7 \mathrm { k }$ validation video segments. In experiments of downstream tasks, we fine-tune the pre-trained VideoMAE models on the training set and report the results on the validation set. The implementation details are described in Appendix $\ S \mathrm { ~ B ~ }$ .
+
+# 4.2 Ablation Studies
+
+In this subsection, we perform in-depth ablation studies on VideoMAE design with the default backbone of 16-frame ViT-B on Something-Something V2 (SSV2) and Kinetics-400 (K400). The specific architectures for the encoder and decoder are shown in Appendix $\ S \mathrm { ~ A ~ }$ . For fine-tuning, we perform TSN [75] uniform sampling on SSV2 and dense sampling [77, 22] on K400. All models share the same inference protocol, i.e., $2 \mathrm { c l i p s } \times 3$ crops on SSV2 and 5 clips $\times 3$ crops on K400.
+
+Decoder design. The lightweight decoder is one key component of our VideoMAE. We conduct experiments with the different depths in Table 1a. Unlike in ImageMAE, a deep decoder here is important for better performance, while a shallow decoder could reduce the GPU memory consumption. We take 4 blocks for the decoder by default. The decoder width is set to half channel of the encoder (e.g., 384-d for ViT-B), following the design in the image domain.
+
+<table><tr><td rowspan=1 colspan=3>blocks SSV2K400 GPU mem.</td></tr><tr><td rowspan=1 colspan=3>1   68.579.0  7.9G</td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>69.279.2</td><td rowspan=1 colspan=1>10.2G</td></tr><tr><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>69.680.0</td><td rowspan=1 colspan=1>14.7G</td></tr><tr><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>69.379.7</td><td rowspan=1 colspan=1>23.7G</td></tr></table>
+
+(a) Decoder depth. 4 blocks of decoder achieve the best tradeoff. “GPU mem.” is GPU memory during pre-training, benchmarked in one GPU with a batch size of 16.
+
+<table><tr><td>case</td><td>ratio</td><td>SSV2</td><td>K400</td></tr><tr><td>tube</td><td>75</td><td>68.0</td><td>79.8</td></tr><tr><td>tube</td><td>90</td><td>69.6</td><td>80.0</td></tr><tr><td>random</td><td>90</td><td>68.3</td><td>79.5</td></tr><tr><td>frame</td><td>87.5*</td><td>61.5</td><td>76.5</td></tr></table>
+
+(b) Mask sampling. We compare different masking strategies. Our proposed tube masking with an extremely high ratio works the best. $* ^ { * } 8 7 . 5 ^ { * }$ means masking 14/16 frames.
+
+<table><tr><td>input</td><td>target</td><td>SSV2</td><td>K400</td></tr><tr><td>T×T</td><td>center</td><td>63.0</td><td>79.3</td></tr><tr><td>TX</td><td>Tx</td><td>68.9</td><td>79.8</td></tr><tr><td>T×T</td><td>T×T</td><td>69.6</td><td>80.0</td></tr><tr><td>T×T</td><td>2T×</td><td>69.2</td><td>80.1</td></tr></table>
+
+(c) Reconstruction target. $T \times$ $\tau$ denotes “frames $\times$ stride”. center denotes the center frame of the input clip. $T$ is set to 16 as default. $\tau$ is set to 2 and 4 on SSV2 and K400, respectively.
+
+<table><tr><td rowspan=1 colspan=2>case              SSV2K400fromscratch      32.668.8</td></tr><tr><td rowspan=1 colspan=2>fromscratch      32.668.8</td></tr><tr><td rowspan=1 colspan=2>ImageNet-21k sup.61.8 78.9</td></tr><tr><td></td><td rowspan=2 colspan=1>65.2   -</td></tr><tr><td rowspan=1 colspan=1>IN-21k+K400 sup.</td></tr><tr><td rowspan=1 colspan=1>VideoMAE</td><td rowspan=1 colspan=1>69.680.0</td></tr></table>
+
+(d) Pre-training strategy. Our VideoMAE works the best without using any extra data. “sup.” is supervised training.
+
+<table><tr><td>dataset</td><td>method SSV2</td><td>K400</td></tr><tr><td>IN-1K</td><td>ImageMAE 64.8</td><td>78.7</td></tr><tr><td>K400</td><td>VideoMAE 68.5</td><td>80.0</td></tr><tr><td>SSV2</td><td>VideoMAE 69.6</td><td>79.6</td></tr></table>
+
+(e) Pre-training dataset. Our VideoMAE works the best when directly pre-training the models on the source datasets.
+
+<table><tr><td>case</td><td>SSV2</td><td>K400</td></tr><tr><td>L1 loss</td><td>69.1</td><td>79.7</td></tr><tr><td>MSE loss</td><td>69.6</td><td>80.0</td></tr><tr><td>Smooth L1 loss</td><td>68.9</td><td>79.6</td></tr></table>
+
+(f) Loss function. MSE loss works the best for the masking and reconstruction task in VideoMAE.
+
+Table 1: Ablation experiments on Something-Something V2 and Kinetics-400. Our backbone is 16-frame vanilla ViT-B and all models are pre-trained with mask ratio $\rho { = } 9 0 \%$ for 800 epochs, and finetuned for evaluation. We perform TSN [75] uniform sampling on SSV2 and dense sampling [77, 22] on K400. All models share the same inference protocol, i.e., 2 clips $\times 3$ crops on SSV2 and 5 clips $\times 3$ crops on K400. The default choice for our model is colored in gray .
+
+Masking strategy. We compare different masking strategies in Table 1b. When increasing the masking ratio from $7 5 \%$ to $90 \%$ for tube masking, the performance on SSV2 boosts from $6 8 . 0 \%$ to $6 9 . 6 \%$ . Then, with an extremely high ratio, we find tube masking also achieves better performance than plain random masking and frame masking. We attribute these interesting observations to the redundancy and temporal correlation in videos. The conclusion on K400 is in accord with one on SSV2. One may note that the performance gap on K400 is lower than one on SSV2. We argue that the Kinetics videos are mostly stationary and scene-related. The effect of temporal modeling is not obvious. Overall, we argue that our default designs enforce the networks to capture more useful spatiotemporal structures and therefore make VideoMAE a more challenging task, which a good self-supervised learner hunger for.
+
+Reconstruction target. First, if we only employ the center frame as the target, the results would decrease greatly as shown in Table 1c. The sampling stride is also sensitive. The result of small sampling strid $\begin{array} { c } { { \vdots \frac { \tau } { 2 } } } \\ { { 2 T } } \end{array}$ is lower than default sampling stride $\tau$ $6 8 . 9 \%$ vs. $6 9 . 6 \%$ on SSV2). We also try to reconstruct frames from the downsampled $T$ frames, but it obtains slightly worse results on SSV2. For simplicity, we use the input downsampled clip as our default reconstruction target.
+
+Pre-training strategy. We compare different pre-training strategies in Table 1d. Similar to previous trials [3, 6], training video transformers from scratch yields unsatisfied results on video datasets. When pre-trained on the large-scale ImageNet-21K dataset, the video transformer obtains better accuracy from $3 2 . 6 \%$ to $6 1 . 8 \%$ on SSV2 and $6 8 . 8 \%$ to $78 . 9 \%$ on K400. Using the models pre-trained on both ImageNet-21K and Kinetics further increases accuracy to $6 5 . 2 \%$ on SSV2. Our VideoMAE can effectively train a video transformer on the video dataset itself without using any extra data and achieve the best performance $6 9 . 6 \%$ on SSV2 and $8 0 . 0 \%$ on K400).
+
+Pre-training dataset. First, we pre-train the ViT-B on ImageNet-1K for 1600 epochs, following the recipes in [30]. Then we inflate the 2D patch embedding layer to our cube embedding layer following [10] and fine-tune the model on the target video datasets. The results surpass the model trained from scratch as shown in Table 1e. We also compare the ImageMAE pre-trained model with VideoMAE models pre-trained on video datasets. We see that our VideoMAE models can achieve better performance than ImageMAE. However, when we try to transfer the pre-trained VideoMAE models to the other video datasets (e.g. from Kinetics to Something-Something), the results are slightly worse than their counterpart, which is directly pre-trained on its own target video datasets. We argue that domain shift between pre-training and target datasets could be an important issue.
+
+dataset training data from scratch MoCo v3 VideoMAE   
+Table 2: Comparisons with the results of previous self-supvised pre-training methods on different datasets. We take 16-frame ViT-B as the default backbone. Notably, here MoCo v3 and VideoMAE all only use the unlabelled data in the training set of each dataset for pre-training and are all fine-tuned for evaluation.   
+
+<table><tr><td>K400</td><td>240k</td><td>68.8</td><td>74.2</td><td>80.0</td></tr><tr><td>Sth-Sth V2</td><td>169k</td><td>32.6</td><td>54.2</td><td>69.6</td></tr><tr><td>UCF101</td><td>9.5k</td><td>51.4</td><td>81.7</td><td>91.3</td></tr><tr><td>HMDB51</td><td>3.5k</td><td>18.0</td><td>39.2</td><td>62.6</td></tr></table>
+
+Table 3: Comparisons with the efficiency and effectiveness on Something-Something V2. We report the fine-tuning (ft) and linear probing (lin) accuracy $( \% )$ . The wall-clock time of pre-training is benchmarked in 64 Tesla V100 GPUs with PyTorch.   
+
+<table><tr><td>method</td><td>epoch</td><td>ft. acc.</td><td>lin. acc.</td><td>hours</td><td>speedup</td></tr><tr><td>MoCov3</td><td>300</td><td>54.2</td><td>33.7</td><td>61.7</td><td>-</td></tr><tr><td>VideoMAE</td><td>800</td><td>69.6</td><td>38.9</td><td>19.5</td><td>3.2×</td></tr></table>
+
+Table 4: Comparisons with the feature transferability on smaller datasets. We take 16-frame ViT-B as the default backbone. Notably, here MoCo v3 and VideoMAE are all pre-trained on Kinetics-400 with unlabelled data in the training set. Then the pre-trained model is fine-tuned on target datasets for evaluation.   
+
+<table><tr><td>method</td><td>K400 → SSV2 K400 →UCF K400 →HMDB</td><td></td><td></td></tr><tr><td>MoCo v3</td><td>62.4</td><td>93.2</td><td>67.9</td></tr><tr><td>VideoMAE</td><td>68.5</td><td>96.1</td><td>73.3</td></tr></table>
+
+Loss function. Table 1f contains an ablation study of loss function. We find that the MSE loss could achieve a higher result compared with the L1 loss and smooth L1 loss. Therefore, we employ the MSE loss by default.
+
+# 4.3 Main Results and Analysis
+
+VideoMAE: data-efficient learner. The self-supervised video pre-training (SSVP) has been extensively studied in previous works, but they mainly use the CNN-based backbones. Few works have investigated transformer-based backbone in SSVP. Therefore, to demonstrate the effectiveness of VideoMAE for transformer-based SSVP, we compare two methods implemented by ourselves: (1) training from scratch and (2) pre-training with contrastive learning (MoCo v3 [14]). For training from scratch, we carefully tune these hyper-parameters to successfully pre-train ViT-Base from the training set of the dataset. For pre-training with MoCo v3, we strictly follow the training practice in its image counterpart and carefully avoid the collapse issue.
+
+The recognition accuracy is reported in Table 2. We see that our VideoMAE significantly outperforms other two training settings. For instance, on the largest dataset of Kinetics-400, our VideoMAE outperforms training from scratch by around $10 \%$ and MoCo v3 pre-training by around $5 \%$ . This superior performance demonstrates that masked autoencoder provides an effective pre-training mechanism for video transformers. We also see that the performance gap between our VideoMAE and the other two methods becomes larger as the training set becomes smaller. Notably, even with only $3 . 5 \mathrm { k }$ training clips on HMDB51, our VideoMAE pre-training can still obtain a satisfying accuracy (around $6 1 \%$ ). This new result demonstrates that VideoMAE is a more data-efficient learner for SSVP. This property is particularly important for scenarios with limited data available and different with contrastive learning methods.
+
+We compare the efficiency of VideoMAE pre-training and MoCo v3 pre-training in Table 3. The task of masked autoencoding with a high ratio is more challenging and thereby requires more training epochs (800 vs. 300). Thanks to the asymmetric encoder-decoder in our VideoMAE and extremely high masking ratio, our pre-training time is much shorter than MoCo v3 (19.5 vs. 61.7 hours).
+
+High masking ratio. In VideoMAE, one core design is the extremely high masking ratio. We perform an investigation of this design on the Kinetics-400 and Something-Something V2 datasets. The results are shown in Figure 3. We see that the best masking ratio is extremely high, and even $9 5 \%$ can achieve good performance for both datasets. This result is difference from BERT [17] in NLP and MAE [30] in images. We analyze the temporal redundancy and correlation in videos makes it possible for our VideoMAE to learn plausible outputs with such a high masking ratio.
+
+![](images/b75fa7e78a9ca4d039292fdaded0567cbb3ebfb7d0b845e469cbbc73517cc6c0.jpg)
+
+![](images/7fbbad8d22712458885f5f102f2b7248e1babda4d1f8cc723955cf62c79a6cde.jpg)  
+Figure 3: The effect of masking ratio Figure 4: Data efficiency of VideoMAE representations. on (a) Something-Something V2 and (b) Our default backbone is 16-frame vanilla ViT-B. • denotes Kinetics-400. We take 16-frame vanilla that all models are trained for the same 132k iterations, ViT-B as default. The results show that and $0$ denotes that all models are trained for the same 800 an extremely high masking ratio $( 9 0 \% )$ epochs. Note that it takes 132k iterations to pre-train the achieves the best efficiency and effective- model for 800 epochs on the full training set of Somethingness trade-off on both video datasets. Something V2.
+
+We also visualize the reconstructed examples in Appendix $\ S  { s e }$ . We see that even under an extremely high masking ratio, VideoMAE can produce satisfying reconstructed results. This implies VideoMAE is able to learn useful representations that capture the holistic spatiotemporal structure in videos.
+
+Transfer learning: quality vs. quantity. To further investigate the generalization ability of VideoMAE in representation learning, we transfer the learned VideoMAE from Kinetics-400 to Something-Something V2, UCF101, and HMDB51. The results are shown in Table 4, and we compare them with MoCo v3 pre-training. The models pre-trained by VideoMAE are better than those pre-trained by MoCo v3, demonstrating that our VideoMAE learns more transferable representations.
+
+Comparing Table 2 and Table 4, the transferred representation outperforms the original VideoMAE models trained from its own dataset on UCF101 and HMDB51. In contrast, the transferred representation is worse on Something-Something V2. To figure out whether this inconsistent result is caused by the large scale of Something-Something V2, we further perform a detailed investigation by decreasing the pre-training video numbers. In this study, we run two experiments: (1) pre-training with the same epochs and (2) pre-training with the same time budget. The result is shown in Figure 4. We see that more training iterations could contribute to better performance when we decrease the size of the pre-training set. Surprisingly, even with only $4 2 \mathrm { k }$ pre-training videos, we can still obtain better accuracy than the Kinetics pre-trained models with 240k videos $6 8 . 7 \%$ vs. $6 8 . 5 \%$ ). This result implies that domain shift is another important factor, and data quality is more important than data quantity in SSVP when there exists a difference between pre-training and target datasets. It also demonstrates that VideoMAE is a data-efficient learner for SSVP.
+
+Transfer learning: downstream action detection. We also transfer the learned VideoMAE on Kinetics-400 to downstream action detection dataset AVA. Following the standard setting [26], we evaluate on top 60 common classes with mean Average Precision (mAP) as the metric under IoU threshold of 0.5. The results are shown in the Table 5. After self-supervised pre-training on Kinetics400, our VideoMAE with the vanilla ViT-B can achieve $2 6 . 7 \mathrm { m A P }$ on AVA, which demonstrates the strong transferability of our VideoMAE. If the pre-trained ViT-B is additionally fine-tuned on Kinetics-400 with labels, the transfer learning performance can further increase about $5 \mathrm { m A P }$ (from 26.7 to 31.8). More remarkably, when we scale up the pre-training configurations with larger video datasets (e.g. Kinetics-700) or more powerful backbones (e.g. ViT-Large and ViT-Huge), VideoMAE can finally obtain better performance. For example, our ViT-L VideoMAE pre-trained on Kinetics-700 achieves $3 9 . 3 \mathrm { m A P }$ and ViT-H VideoMAE pre-trained on Kinetics-400 has $3 9 . 5 \mathrm { m A P } . $ . These results demonstrate that the self-supervised pre-trained models transfer well not only on action classification task but on more complex action detection task.
+
+Table 5: Comparison with the state-of-the-art methods on AVA v2.2. All models are pre-trained and fine-tuned at image size $2 2 4 ^ { 2 }$ . We report the mean Average Precision (mAP) on validation set. “Ex. labels $\pmb { \chi } ^ { , }$ means only unlabelled data is used during the pre-training phase and the pre-trained models are directly transferred to AVA. “Ex. labels $\curvearrowleft$ means pre-trained models are additionally fine-tuned on the pre-training dataset with labels before transferred to AVA. $T \times \tau$ refers to frame number and corresponding sample rate.   
+
+<table><tr><td>Method</td><td>Backbone</td><td>Pre-train Dataset Extra Labels|T×7</td><td></td><td></td><td>GFLOPs</td><td>Param</td><td>mAP</td></tr><tr><td>supervised [22]</td><td>SlowFast-R101</td><td>Kinetics-400</td><td>√</td><td>8×8</td><td>138</td><td>53</td><td>23.8</td></tr><tr><td>CVRL [53]</td><td>SlowOnly-R50</td><td>Kinetics-400</td><td>X</td><td>32×2</td><td>42</td><td>32</td><td>16.3</td></tr><tr><td>pBYOLp=3 [23]</td><td>SlowOnly-R50 SlowOnly-R50</td><td>Kinetics-400</td><td>X</td><td>8×8</td><td>42</td><td>32</td><td>23.4</td></tr><tr><td>pMoCop=3 [23]</td><td>MViT-L</td><td>Kinetics-400 Kinetics-400</td><td>X</td><td>8×8 40×3</td><td>42</td><td>32</td><td>20.3</td></tr><tr><td>MaskFeat↑312 [79] MaskFeat↑312 [79]</td><td>MViT-L</td><td>Kinetics-600</td><td>√ √</td><td>40×3</td><td>2828 2828</td><td>218 218</td><td>37.5</td></tr><tr><td>VideoMAE</td><td>ViT-S</td><td>Kinetics-400</td><td>X</td><td>16×4</td><td>57</td><td>22</td><td>38.8 22.5</td></tr><tr><td>VideoMAE</td><td>ViT-S</td><td>Kinetics-400</td><td>√</td><td>16×4</td><td>57</td><td>22</td><td>28.4</td></tr><tr><td>VideoMAE</td><td>ViT-B</td><td>Kinetics-400</td><td>X</td><td>16×4</td><td>180</td><td>87</td><td>26.7</td></tr><tr><td>VideoMAE</td><td>ViT-B</td><td>Kinetics-400</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>ViT-L</td><td>Kinetics-400</td><td>√</td><td>16×4</td><td>180</td><td>87</td><td>31.8</td></tr><tr><td>VideoMAE</td><td>ViT-L</td><td>Kinetics-400</td><td>X</td><td>16×4 16×4</td><td>597</td><td>305</td><td>34.3</td></tr><tr><td>VideoMAE VideoMAE</td><td>ViT-H</td><td>Kinetics-400</td><td>√</td><td>16×4</td><td>597</td><td>305</td><td>37.0</td></tr><tr><td>VideoMAE</td><td>ViT-H</td><td>Kinetics-400</td><td>X √</td><td>16×4</td><td>1192 1192</td><td>633 633</td><td>36.5</td></tr><tr><td>VideoMAE</td><td>ViT-L</td><td>Kinetics-700</td><td>X</td><td>16×4</td><td>597</td><td>305</td><td>39.5 36.1</td></tr><tr><td>VideoMAE</td><td>ViT-L</td><td>Kinetics-700</td><td>√</td><td>16×4</td><td>597</td><td>305</td><td>39.3</td></tr></table>
+
+<table><tr><td>Method</td><td>Backbone</td><td>Extra data</td><td>Ex. labels</td><td>Frames</td><td>GFLOPs</td><td>Param</td><td>Top-1</td><td>Top-5</td></tr><tr><td>TEINetEn [39]</td><td>ResNet50x2</td><td rowspan="3">ImageNet-1K</td><td>√</td><td>8+16</td><td>99×10x3</td><td>50</td><td>66.5</td><td>N/A</td></tr><tr><td>TANetEn [40]</td><td>ResNet50×2</td><td>√</td><td>8+16</td><td>99×2×3</td><td>51</td><td>66.0</td><td>90.1</td></tr><tr><td>TDNEn [74]</td><td>ResNet101×2</td><td>√</td><td>8+16</td><td>198×1×3</td><td>88</td><td>69.6</td><td>92.2</td></tr><tr><td>SlowFast [22] MViTv1[21]</td><td>ResNet101 MViTv1-B</td><td rowspan="2">Kinetics-400</td><td>√ √</td><td>8+32 64</td><td>106×1×3 455×1×3</td><td>53 37</td><td>63.1 67.7</td><td>87.6</td></tr><tr><td>TimeSformer [6]</td><td>ViT-B</td><td></td><td>8</td><td>196×1×3</td><td>121</td><td>59.5</td><td>90.9 N/A</td></tr><tr><td>TimeSformer [6]</td><td>ViT-L</td><td rowspan="2">ImageNet-21K</td><td>V</td><td>64</td><td>5549×1×3</td><td>430</td><td>62.4</td><td>N/A</td></tr><tr><td>ViViT FE [3]</td><td>ViT-L</td><td>√</td><td>32</td><td>995×4×3</td><td>N/A</td><td>65.9</td><td>89.9</td></tr><tr><td>Motionformer [50]</td><td>ViT-B</td><td rowspan="4">IN-21K+K400</td><td>√</td><td>16</td><td>370×1×3</td><td>109</td><td>66.5</td><td>90.1</td></tr><tr><td>Motionformer [50]</td><td>ViT-L</td><td>√</td><td>32</td><td>1185×1×3</td><td>382</td><td>68.1</td><td>91.2</td></tr><tr><td>Video Swin [38]</td><td>Swin-B</td><td>√</td><td>32</td><td>321×1×3</td><td>88</td><td>69.6</td><td>92.7</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>VIMPAC [64]</td><td>ViT-L</td><td>HowTo100M+DALLE IN-1K+K400+DALLE</td><td>X</td><td>10</td><td>N/A×10×3</td><td>307</td><td>68.1</td><td>N/A</td></tr><tr><td>BEVT[76]</td><td>Swin-B</td><td rowspan="2">Kinetics-600</td><td>×</td><td>32</td><td>321×1×3</td><td>88</td><td>70.6</td><td>N/A</td></tr><tr><td>MaskFeat↑312 [79]</td><td>MViT-L</td><td>√</td><td>40</td><td>2828×1×3</td><td>218</td><td>75.0</td><td>95.0</td></tr><tr><td>VideoMAE</td><td>ViT-B</td><td>Kinetics-400 Kinetics-400</td><td>X ×</td><td>16</td><td>180×2×3</td><td>87</td><td>69.7</td><td>92.3</td></tr><tr><td>VideoMAE</td><td>ViT-L</td><td rowspan="4"> no external data</td><td>X</td><td>16</td><td>597×2×3</td><td>305</td><td>74.0</td><td>94.6</td></tr><tr><td>VideoMAE</td><td>ViT-S</td><td>X</td><td>16</td><td>57×2×3</td><td>22</td><td>66.8</td><td>90.3</td></tr><tr><td>VideoMAE</td><td>ViT-B</td><td></td><td>16</td><td>180×2×3</td><td>87</td><td>70.8</td><td>92.4</td></tr><tr><td>VideoMAE VideoMAE</td><td>ViT-L ViT-L</td><td>X X</td><td>16 32</td><td>597×2×3 1436×1×3</td><td>305 305</td><td>74.3 75.4</td><td>94.6 95.2</td></tr></table>
+
+Table 6: Comparison with the state-of-the-art methods on Something-Something V2. Our VideoMAE reconstructs normalized cube pixels and is pre-trained with a masking ratio of $90 \%$ for 2400 epochs. “Ex. labels $\pmb { \chi } ^ { , }$ means only unlabelled data is used during the pre-training phase. “N/A” indicates the numbers are not available for us.
+
+# 4.4 Comparison with the state of the art
+
+We compare with the previous state-of-the-art performance on the Kinetics-400 and SomethingSomething V2 datasets. The results are reported in Table 6 and Table 7. Our VideoMAE can easily scale up with more powerful backbones (e.g. ViT-Large and ViT-Huge) and more frames (e.g. 32). Our VideoMAE achieves the top-1 accuracy of $7 5 . 4 \%$ on Something-Something V2 and $8 7 . 4 \%$ on Kinetics-400 without using any extra data. We see that the existing state-of-the-art methods all depend on the external data for pre-training on the Something-Something V2 dataset. On the contrary, our VideoMAE without any external data significantly outperforms previous methods with the same input resolution by around $5 \%$ . Our ViT-H VideoMAE also achieves very competitive performance on the Kinetics-400 dataset without using any extra data, which is even better than ViViT-H with on
+
+Table 7: Comparison with the state-of-the-art methods on Kinetics-400. Our VideoMAE reconstructs normalized cube pixels. Here models are self-supervised pre-trained with a masking ratio of $90 \%$ for 1600 epochs on Kinetics-400. VideoMAE↑320 is initialized from its $2 2 4 ^ { 2 }$ resolution counterpart and then fine-tuned for evaluation. “Ex. labels $\pmb { \chi } ^ { , }$ means only unlabelled data is used during the pre-training phase. “N/A” indicates the numbers are not available for us.   
+
+<table><tr><td>Method</td><td>Backbone</td><td>Extra data</td><td>Ex.labels|1</td><td>Frames</td><td>GFLOPs</td><td>Param</td><td>Top-1</td><td>Top-5</td></tr><tr><td>NL I3D [77]</td><td>ResNet101</td><td rowspan="3">ImageNet-1K</td><td>√</td><td>128</td><td>359×10×3</td><td>62</td><td>77.3</td><td>93.3</td></tr><tr><td>TANet [40]</td><td>ResNet152</td><td></td><td>16</td><td>242×4×3</td><td>59</td><td>79.3</td><td>94.1</td></tr><tr><td>TDNEn [74]</td><td>ResNet101</td><td></td><td>8+16</td><td>198×10×3</td><td>88</td><td>79.4</td><td>94.4</td></tr><tr><td>TimeSformer [6]</td><td>ViT-L</td><td rowspan="4">ImageNet-21K</td><td></td><td>96</td><td>8353×1×3</td><td>430</td><td>80.7</td><td>94.7</td></tr><tr><td>ViViT FE [3]</td><td>ViT-L</td><td></td><td>128</td><td>3980×1×3</td><td>N/A</td><td>81.7</td><td>93.8</td></tr><tr><td>Motionformer [50]</td><td>ViT-L</td><td></td><td>32</td><td>1185×10×3</td><td>382</td><td>80.2</td><td>94.8</td></tr><tr><td>Video Swin [38]</td><td>Swin-L</td><td></td><td>32</td><td>604×4×3</td><td>197</td><td>83.1</td><td>95.9</td></tr><tr><td>ViViT FE [3]</td><td>ViT-L</td><td>JFT-300M</td><td>√</td><td>128</td><td>3980×1×3</td><td>N/A</td><td>83.5</td><td>94.3</td></tr><tr><td>ViViT[3]</td><td>ViT-H</td><td>JFT-300M</td><td></td><td>32</td><td>3981×4×3</td><td>N/A</td><td>84.9</td><td>95.8</td></tr><tr><td>VIMPAC [64]</td><td>ViT-L</td><td>HowTo100M+DALLE</td><td>X</td><td>10</td><td>N/A×10×3</td><td>307</td><td>77.4</td><td>N/A</td></tr><tr><td>BEVT[76]</td><td>Swin-B</td><td>IN-1K+DALLE</td><td>X</td><td>32</td><td>282×4×3</td><td>88</td><td>80.6</td><td>N/A</td></tr><tr><td>MaskFeat↑352 [79]</td><td>MViT-L</td><td>Kinetics-600</td><td>X</td><td>40</td><td>3790×4×3</td><td>218</td><td>87.0</td><td>97.4</td></tr><tr><td>ip-CSN [68]</td><td>ResNet152</td><td rowspan="4"> no external data</td><td>X</td><td>32</td><td>109×10×3</td><td>33</td><td>77.8</td><td>92.8</td></tr><tr><td>SlowFast [22]</td><td>R101+NL</td><td>X</td><td>16+64</td><td>234×10×3</td><td>60</td><td>79.8</td><td>93.9</td></tr><tr><td>MViTv1[21]</td><td>MViTv1-B</td><td>X</td><td>32</td><td>170×5×1</td><td>37</td><td>80.2</td><td>94.4</td></tr><tr><td>MaskFeat [79]</td><td>MViT-L</td><td>×</td><td>16</td><td>377×10×1</td><td>218</td><td>84.3</td><td>96.3</td></tr><tr><td>VideoMAE</td><td>ViT-S</td><td rowspan="4">no external data</td><td>X</td><td>16</td><td>57×5×3</td><td>22</td><td>79.0</td><td>93.8</td></tr><tr><td>VideoMAE</td><td>ViT-B</td><td>X</td><td>16</td><td>180×5×3</td><td>87</td><td>81.5</td><td>95.1</td></tr><tr><td>VideoMAE</td><td>ViT-L</td><td>×</td><td>16</td><td>597×5×3</td><td>305</td><td>85.2</td><td>96.8</td></tr><tr><td>VideoMAE</td><td>ViT-H</td><td>×</td><td>16</td><td>1192×5×3</td><td>633</td><td>86.6</td><td>97.1</td></tr><tr><td>VideoMAE↑320</td><td>ViT-L</td><td rowspan="2">no external data</td><td>X</td><td>32</td><td>3958×4×3</td><td>305</td><td>86.1</td><td></td></tr><tr><td>VideoMAE↑320</td><td>ViT-H</td><td>×</td><td>32</td><td>7397×4×3</td><td>633</td><td>87.4</td><td>97.3 97.6</td></tr></table>
+
+JFT-300M pre-training $8 6 . 6 \%$ v.s. $8 4 . 9 \%$ ). When fine-tuned with larger spatial resolutions and input video frames, the performance of our ViT-H VideoMAE can further boost from $8 6 . 6 \%$ to $8 7 . 4 \%$ .
+
+# 5 Conclusion
+
+In this paper, we have presented a simple and data-efficient self-supervised learning method (VideoMAE) for video transformer pre-training. Our VideoMAE introduces two critical designs of extremely high masking ratio and tube masking strategy to make the video reconstruction task more challenging. This harder task would encourage VideoMAE to learn more representative features and relieve the information leakage issue. Empirical results demonstrate this simple algorithm works well for video datasets of different scales. In particular, we are able to learn effective VideoMAE only with thousands of video clips, which has significant practical value for scenarios with limited data available.
+
+Future work VideoMAE could be further improved by using larger webly datasets, larger models (e.g., ViT-G) and larger spatial resolutions of input video (e.g., $3 8 \hat { 4 ^ { 2 } }$ ). VideoMAE only leverages the RGB video stream without using additional audio or text stream. We expect that audio and text from the video data can provide more information for self-supervised pre-training.
+
+Broader impact Potential negative societal impacts of VideoMAE are mainly concerned with energy consumption. The pre-training phase may lead to a large amount of carbon emission. Though the pre-training is energy-consuming, we only need to pre-train the model once. Different downstream tasks can then share the same pre-trained model via additional fine-tuning. Our VideoMAE unleashes the great potential of vanilla vision transformer for video analysis, which could increase the risk of video understanding model or its outputs being used incorrectly, such as for unauthorized surveillance.
+
+Acknowledgements and disclosure of funding Thanks to Ziteng Gao, Lei Chen and Chongjian Ge for their help. This work is supported by National Natural Science Foundation of China (No. 62076119, No. 61921006), the Fundamental Research Funds for the Central Universities (No. 020214380091), Tencent AI Lab Rhino-Bird Focused Research Program (No. JR202125), and Collaborative Innovation Center of Novel Software Technology and Industrialization.
+
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+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] Shown in Section 5.   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] Shown in Section 5.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] Shown in Section 4.1 and Appendix $\ S \ B$ .   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] Shown in Appendix $\ S \ O $ .   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] Because of the computation costs, we did not run the experiments multiple times.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] Please see Table 3 and Appendix $\ S \mathrm { ~ B ~ }$ .
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes]   
+(b) Did you mention the license of the assets? [Yes] Shown in Appendix $\ S ~ ^ { \intercal }$ .   
+(c) Did you include any new assets either in the supplemental material or as a URL? [No]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes] We used publicly available datasets whose licenses allow research usage.   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No] To the best of our knowledge, the data we used contains no personally identifiable information or offensive content.
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

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+        "text": "VideoMAE: Masked Autoencoders are Data-Efficient Learners for Self-Supervised Video Pre-Training ",
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+        "text": "Zhan Tong 1,2∗ Yibing Song 2 Jue Wang 2 Limin Wang 1,3† 1State Key Laboratory for Novel Software Technology, Nanjing University 2Tencent AI Lab 3Shanghai AI Lab tongzhan@smail.nju.edu.cn {yibingsong.cv, arphid}@gmail.com lmwang@nju.edu.cn ",
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+        "text": "Abstract ",
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+        "text": "Pre-training video transformers on extra large-scale datasets is generally required to achieve premier performance on relatively small datasets. In this paper, we show that video masked autoencoders (VideoMAE) are data-efficient learners for self-supervised video pre-training (SSVP). We are inspired by the recent ImageMAE [30] and propose customized video tube masking with an extremely high ratio. This simple design makes video reconstruction a more challenging and meaningful self-supervision task, thus encouraging extracting more effective video representations during the pre-training process. We obtain three important findings with VideoMAE: (1) An extremely high proportion of masking ratio (i.e., $9 0 \\%$ to $9 5 \\%$ ) still yields favorable performance for VideoMAE. The temporally redundant video content enables higher masking ratio than that of images. (2) VideoMAE achieves impressive results on very small datasets (i.e., around $3 \\mathrm { k } { - } 4 \\mathrm { k }$ videos) without using any extra data. This is partially ascribed to the challenging task of video reconstruction to enforce high-level structure learning. (3) VideoMAE shows that data quality is more important than data quantity for SSVP. Domain shift between pre-training and target datasets is an important factor. Notably, our VideoMAE with the vanilla ViT backbone can achieve $8 7 . 4 \\%$ on Kinects-400, $7 5 . 4 \\%$ on SomethingSomething V2, $9 1 . 3 \\%$ on UCF101, and $6 2 . 6 \\%$ on HMDB51, without using any extra data. Code is available at https://github.com/MCG-NJU/VideoMAE. ",
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+        "text": "1 Introduction ",
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+        "text": "Transformer [70] has brought significant progress in natural language processing [17, 7, 54]. The vision transformer [20] also improves a series of computer vision tasks including image classification [66, 88], object detection [8, 37], semantic segmentation [80], object tracking [13, 16], and video recognition [6, 3]. The multi-head self-attention upon linearly projected image/video tokens is capable of modeling global dependency among visual content either spatially or temporally. The inductive bias is effectively reduced via this flexible attention mechanism. ",
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+        "text": "Training effective vision transformers (ViTs) typically necessitates large-scale supervised datasets. Initially, the pre-trained ViTs achieve favorable performance by using hundreds of millions of labeled images [20]. For video transformers [3, 6], they are usually derived from image-based transformers and heavily depend on the pre-trained models from large-scale image data (e.g., ImageNet [57]). Previous trials [3, 6] on training video transformers from scratch yield unsatisfied results (except for MViT [21] with a strong inductive bias). Therefore, the learned video transformers are naturally biased by image-based models, and it still remains a challenge that how to effectively and efficiently train a vanilla vision transformer on the video dataset itself without using any pre-trained model or extra image data. Moreover, the existing video datasets are relatively small compared with image datasets, which further increases the difficulty of training video transformers from scratch. Meanwhile, self-supervised learning has shown remarkable performance by using large-scale image datasets [14, 9]. The learned representations have outperformed the ones via supervised learning when being transferred to downstream tasks. It is expected that this self-supervised learning paradigm can provide a promising solution to address the challenge of training video transformers. ",
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+        "image_caption": [
+            "Figure 1: VideoMAE performs the task of masking random cubes and reconstructing the missing ones with an asymmetric encoder-decoder architecture. Due to high redundancy and temporal correlation in videos, we present the customized design of tube masking with an extremely high ratio $90 \\%$ to $9 5 \\%$ ). This simple design enables us to create a more challenging and meaningful self-supervised task to make the learned representations capture more useful spatiotemporal structures. "
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+        "text": "Following the success of masked autoencoding in NLP [17] and images [30, 4], we present a new selfsupervised video pre-training (SSVP) method, termed as Video Masked Autoencoder (VideoMAE). Our VideoMAE inherits the simple pipeline of masking random cubes and reconstructing the missing ones. However, the extra time dimension of videos makes them different from images in this masked modeling. First, video frames are often densely captured, and their semantics varies slowly in time [87]. This temporal redundancy would increase the risk of recovering missing pixels from the spatiotemporal neighborhood with little high-level understanding. Furthermore, video could be viewed as the temporal evolution of static appearance, and there exists a correspondence between frames. This temporal correlation could lead to information leakage (i.e., masked spatiotemporal content re-occurrence) during reconstruction unless a specific masking strategy is considered. In this sense, for each masked cube, it is easy to find a corresponding and unmasked copy in adjacent frames. This property would make the learned models identify some “shortcut” features that are hard to generalize to new scenarios. ",
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+        "type": "text",
+        "text": "To make video masked modeling more effective, in this paper, we present a customized design of tube masking with an extremely high ratio in our VideoMAE. First, due to temporal redundancy, we use an extremely high masking ratio to drop the cubes from the downsampled clips. This simple strategy not only effectively increases the pre-training performance but also greatly reduces the computational cost due to the asymmetric encoder-decoder architecture. Second, to consider temporal correlation, we devise a simple yet effective tube masking strategy, which turns out to be helpful in relieving the risk of information leakage for cubes with no or negligible motion during reconstruction. With this simple yet effective design in our VideoMAE, we are able to successfully train vanilla ViT backbones on the relatively small-scale video datasets such as Something-Something [25], UCF101 [60], and HMDB51 [34], which significantly outperform the previous state of the art under the setting without extra data. In summary, the main contribution of this paper is threefold: ",
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+        "text": "• We present a simple but effective video masked autoencoder that unleashes the potential of vanilla vision transformer for video recognition. To the best of our knowledge, this is the first masked video pre-training framework of simply using plain ViT backbones. To relieve the information leakage issue in masked video modeling, we present the tube masking with an extremely high ratio, which brings the performance improvement to the VideoMAE. • Aligned with the results in NLP and Images on masked modeling, our VideoMAE demonstrates that this simple masking and reconstruction strategy provides a good solution to self-supervised video pre-training. The models pre-trained with our VideoMAE significantly outperform those trained from scratch or pre-trained with contrastive learning methods. • We obtain extra important findings on masked modeling that might be ignored in previous research in NLP and Images. (1) We demonstrate that VideoMAE is a data-efficient learner that could be successfully trained with only $3 . 5 \\mathrm { k }$ videos. (2) Data quality is more important than quantity for SSVP when a domain shift exists between the source and target dataset. ",
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+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Video representation learning. Learning good video representations has been heavily investigated in the literature. The supervised learning methods [58, 75, 69, 10, 6] usually depend on the image backbones. The video encoder backbones are first pre-trained with image data in a supervised form. Then, these backbones are fine-tuned on the video dataset for classifying human actions. Meanwhile, some methods [67, 22, 21] directly train video backbones from videos in a supervised manner. Besides supervised learning, semi-supervised video representation learning has also been studied [59]. The representations of labeled training samples are utilized to generate supervision signals for unlabeled ones. Supervised or semi-supervised representation learning mainly uses a top-down training paradigm, which is not effective in exploring the inherent video data structure itself. Meanwhile, some multimodal contrastive learning methods [36, 42, 62] have been developed to learn video representation from noisy text supervision. ",
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+    {
+        "type": "text",
+        "text": "For self-supervised learning, the prior knowledge of temporal information has been widely exploited to design pretext tasks [78, 44, 82, 5] for SSVP. Recently, contrastive learning [28, 45, 29, 52, 24, 27] is popular to learn better visual representation. However these methods heavily rely on strong data augmentation and large batch size [23]. Predicting the video clip with autoencoders in pixel space has been explored for representation learning by using CNN or LSTM backbones [48, 61], or conducting video generation with autoregressive GPT [83]. Instead, our VideoMAE aims to use the simple masked autoencoder with recent ViT backbones to perform data-efficient SSVP. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Masked visual modeling. Masked visual modeling has been proposed to learn effective visual representations based on the simple pipeline of masking and reconstruction. These works mainly focus on the image domain. The early work [72] treated the masking as a noise type in denoised autoencoders [71] or inpainted missing regions with context [47] by using convolutions. iGPT [11] followed the success of GPT [7, 55] in NLP and operated a sequence of pixels for prediction. The original ViT [20] investigated the masked token prediction for self-supervised pre-training. More recently, the success of vision transformer has led to investigation of Transformer-based architectures for masked visual modeling [4, 19, 30, 79, 81, 89]. BEiT [4], BEVT [76] and VIMPAC [64] followed BERT [17] and proposed to learn visual representations from images and videos by predicting the discrete tokens [56]. MAE [30] introduced an asymmetric encoder-decoder architecture for masked image modeling. MaskFeat [79] proposed to reconstruct the HOG features of masked tokens to perform self-supervised pre-training in videos. VideoMAE is inspired by the ImageMAE and introduces specific design in implementation for SSVP. In particular, compared with previous masked video modeling [30, 76, 64], we present a simpler yet more effective video masked autoencoder by directly reconstructing the pixels. Our VideoMAE is the first masked video pre-training framework of simply using plain ViT backbones. ",
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+    {
+        "type": "text",
+        "text": "3 Proposed Method ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we first revisit ImageMAE [30]. Then we analyze the characteristics of video data.   \nFinally, we show how we explore MAE in the video data by presenting our VideoMAE. ",
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+    {
+        "type": "text",
+        "text": "3.1 Revisiting Image Masked Autoencoders ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "ImageMAE [30] performs the masking and reconstruction task with an asymmetric encoder-decoder architecture. The input image $I \\in \\mathcal { R } ^ { \\mathbf { \\breve { 3 } } \\times H \\times W }$ is first divided into regular non-overlapping patches of size $1 6 \\times 1 6$ , and each patch is represented with token embedding. Then a subset of tokens are randomly masked with a high masking ratio $( 7 5 \\% )$ , and only the remaining ones are fed into the transformer encoder $\\Phi _ { \\mathrm { e n c } }$ . Finally, a shallow decoder $\\Phi _ { \\mathrm { d e c } }$ is placed on top of the visible tokens from the encoder and learnable mask tokens to reconstruct the image. The loss function is mean squared error (MSE) loss between the normalized masked tokens and reconstructed ones in the pixel space: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/6024c3fa071d56d1e41057c77900bf1277368825b3b8ba1c9e55f8bd35803215.jpg",
+        "text": "$$\n\\mathcal { L } = \\frac { 1 } { \\Omega } \\sum _ { p \\in \\Omega } | I ( p ) - \\hat { I } ( p ) | ^ { 2 } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $p$ is the token index, $\\Omega$ is the set of masked tokens, $I$ is the input image, and $\\hat { I }$ is the reconstructed one. ",
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+    {
+        "type": "image",
+        "img_path": "images/ae9479b1875af3b188fce7f930b56324b71788c64cb850c7133617d6ff4dbf9e.jpg",
+        "image_caption": [
+            "Figure 2: Slowness is a general prior in (a) video data [87]. This leads to two important characteristics in time: temporal redundancy and temporal correlation. Temporal redundancy makes it possible to recover pixels under an extremely high masking ratio. Temporal correlation leads to easily reconstruct the missing pixels by finding those corresponding patches in adjacent frames under plain (b) frame masking or (c) random masking. To avoid this simple task and encourage learning representative representation, we propose a (d) tube masking, where the masking map is the same for all frames. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "3.2 Characteristics of Video Data ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Compared with static images, video data contain temporal relations. We show the motivation of our VideoMAE by analyzing video characteristics. ",
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+    {
+        "type": "text",
+        "text": "Temporal redundancy. There are frequently captured frames in a video. The semantics vary slowly in the temporal dimension [87]. We observe that consecutive frames are highly redundant, as shown in Figure 2. This property leads to two critical issues in masked video autoencoding. First, it would be less efficient to keep the original temporal frame rate for pre-training. This would draw us to focus more on static or slow motions in our masked modeling. Second, temporal redundancy greatly dilutes motion representations. This would make the task of reconstructing missing pixels not difficult under the normal masking ratio (e.g., $50 \\%$ to $7 5 \\%$ ). The encoder backbone is not effective in capturing motion representations. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Temporal correlation. Videos could be viewed as the temporal extension of static appearance, and therefore there exists an inherent correspondence between adjacent frames. This temporal correlation could increase the risk of information leakage in the masking and reconstruction pipeline. In this sense, as shown in Figure 2, we can reconstruct the masked patches by finding the spatiotemporal corresponding unmasked patches in the adjacent frames under plain random masking or frame masking. In this case, it might guide the VideoMAE to learn low-level temporal correspondence rather than high-level information such as spatiotemporal reasoning over the content. To alleviate this behavior, we need to propose a new masking strategy to make the reconstruction more challenging and encourage effective learning of spatiotemporal structure representations. ",
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+    {
+        "type": "text",
+        "text": "3.3 VideoMAE ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To relieve the above issues in video masked modeling, we make the customized design in our VideoMAE, and the overall pipeline is shown in Figure 1. Our VideoMAE takes the downsampled frames as inputs and uses the cube embedding to obtain video tokens. Then, we propose a simple design of tube masking with high ratio to perform MAE pre-training with an asymmetric encoderdecoder architecture. Our backbone uses the vanilla ViT with joint space-time attention. ",
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+    {
+        "type": "text",
+        "text": "Temporal downsampling. According to the above analysis on temporal redundancy over consecutive frames, we propose to use the strided temporal sampling strategy to perform more efficient video pre-training. Formally, one video clip consisting of $t$ consecutive frames is first randomly sampled from the original video $V$ . We then use temporal sampling to compress the clip to $T$ frames, each of which contains $H \\times W \\times 3$ pixels. In experiments, the stride $\\tau$ is set to 4 and 2 on Kinetics and Something-Something, respectively. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Cube embedding. We adopt the joint space-time cube embedding [3, 21, 38] in our VideoMAE, where we treat each cube of size $2 \\times 1 6 \\times 1 6$ as one token embedding. Thus, the cube embedding layer obtains T2 × H16 $\\begin{array} { r } { \\frac { T } { 2 } \\times \\frac { H } { 1 6 } \\times \\frac { W } { 1 6 } \\ 3 } \\end{array}$ D tokens and maps each token to the channel dimension $D$ . This design can decrease the spatial and temporal dimension of input, which helps to alleviate the spatiotemporal redundancy in videos. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Tube masking with extremely high ratios. First, temporal redundancy is a factor affecting VideoMAE design. We find that VideoMAE is in favor of extremely high masking ratios (e.g. $90 \\%$ to $9 5 \\%$ ) compared with the ImageMAE. Video information density is much lower than images, and we expect a high ratio to increase the reconstruction difficulty. This high masking ratio is helpful to mitigate the information leakage during masked modeling and make masked video reconstruction a meaningful self-supervised pre-training task. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Second, temporal correlation is another factor in our VideoMAE design. We find even under the extremely high masking ratio, we can still improve the masking efficiency by proposing the temporal tube masking mechanism. Temporal tube masking enforces a mask to expand over the whole temporal axis, namely, different frames sharing the same masking map. Mathematically, the tube mask mechanism can be expressed as $\\mathbb { I } [ p _ { x , y , \\cdot } \\in \\Omega ] \\sim \\mathrm { B e r n o u l l i } ( \\bar { \\rho _ { \\mathrm { m a s k } } } )$ and different time $t$ shares the same value. With this mechanism, temporal neighbors of masked cubes are always masked. So for some cubes with no or small motion (e.g., finger cube in 4th row of Figure 2 (d)), we can not find the spatiotemporal corresponding content in all frames. In this way, it would encourage our VideoMAE to reason over high-level semantics to recover these totally missing cubes. This simple strategy can alleviate the information leakage for cubes with no or negligible motion, and turns out to be effective in practice for masked video pre-training. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Backbone: joint space-time attention. Due to the high proportion of masking ratio mentioned above, only a few tokens are left as the input for the encoder. To better capture high-level spatio-temporal information in the remaining tokens, we use the vanilla ViT backbone [20] and adopt the joint space-time attention [3, 38]. Thus, all pair tokens could interact with each other in the multi-head self-attention layer [70]. The specific architecture design for the encoder and decoder is shown in supplementary materials. The quadratic complexity of the joint space-time attention mechanism is a computational bottleneck, while our design of an extremely high masking ratio alleviates this issue by only putting the unmasked tokens (e.g., $10 \\%$ ) into the encoder during the pre-training phase. ",
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+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "4.1 Datasets ",
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+        "type": "text",
+        "text": "We evaluate our VideoMAE on five common video datasets: Kinetics-400 [33], Something-Something V2 [25], UCF101 [60], HMDB51 [34], and AVA [26]. The Kinetics-400 contains around 240k training videos and 20k validation videos of 10s from 400 classes. The Something-Something V2 is another large-scale video dataset, having around 169k videos for training and 20k videos for validation. In contrast to Kinetics-400, this dataset contains 174 motion-centric action classes. These two large-scale video datasets focus on different visual cues for action recognition. UCF101 and HMDB51 are two relatively small video datasets, which contain around $9 . 5 \\mathrm { k } / 3 . 5 \\mathrm { k }$ train/val videos and $3 . 5 \\mathrm { k } / 1 . 5 \\mathrm { k }$ train/val videos, respectively. Compared with those large-scale video datasets, these two small datasets are more suitable for verifying the effectiveness of VideoMAE, as training large ViT models is more challenging on small datasets. Moreover, we also transfer the learned ViT models by VideoMAE to downstream action detection task. We work on AVA, a dataset for spatiotemporal localization of human actions with 211k training and $5 7 \\mathrm { k }$ validation video segments. In experiments of downstream tasks, we fine-tune the pre-trained VideoMAE models on the training set and report the results on the validation set. The implementation details are described in Appendix $\\ S \\mathrm { ~ B ~ }$ . ",
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+    {
+        "type": "text",
+        "text": "4.2 Ablation Studies ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this subsection, we perform in-depth ablation studies on VideoMAE design with the default backbone of 16-frame ViT-B on Something-Something V2 (SSV2) and Kinetics-400 (K400). The specific architectures for the encoder and decoder are shown in Appendix $\\ S \\mathrm { ~ A ~ }$ . For fine-tuning, we perform TSN [75] uniform sampling on SSV2 and dense sampling [77, 22] on K400. All models share the same inference protocol, i.e., $2 \\mathrm { c l i p s } \\times 3$ crops on SSV2 and 5 clips $\\times 3$ crops on K400. ",
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+    {
+        "type": "text",
+        "text": "Decoder design. The lightweight decoder is one key component of our VideoMAE. We conduct experiments with the different depths in Table 1a. Unlike in ImageMAE, a deep decoder here is important for better performance, while a shallow decoder could reduce the GPU memory consumption. We take 4 blocks for the decoder by default. The decoder width is set to half channel of the encoder (e.g., 384-d for ViT-B), following the design in the image domain. ",
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+    {
+        "type": "table",
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+        "table_caption": [],
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+        "table_body": "<table><tr><td rowspan=1 colspan=3>blocks SSV2K400 GPU mem.</td></tr><tr><td rowspan=1 colspan=3>1   68.579.0  7.9G</td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>69.279.2</td><td rowspan=1 colspan=1>10.2G</td></tr><tr><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>69.680.0</td><td rowspan=1 colspan=1>14.7G</td></tr><tr><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>69.379.7</td><td rowspan=1 colspan=1>23.7G</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "(a) Decoder depth. 4 blocks of decoder achieve the best tradeoff. “GPU mem.” is GPU memory during pre-training, benchmarked in one GPU with a batch size of 16. ",
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+    {
+        "type": "table",
+        "img_path": "images/85c2dcbea2782ef2307058aada2a4ed8c51e93e43a82cbc1da1d57fc699f708c.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>case</td><td>ratio</td><td>SSV2</td><td>K400</td></tr><tr><td>tube</td><td>75</td><td>68.0</td><td>79.8</td></tr><tr><td>tube</td><td>90</td><td>69.6</td><td>80.0</td></tr><tr><td>random</td><td>90</td><td>68.3</td><td>79.5</td></tr><tr><td>frame</td><td>87.5*</td><td>61.5</td><td>76.5</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "(b) Mask sampling. We compare different masking strategies. Our proposed tube masking with an extremely high ratio works the best. $* ^ { * } 8 7 . 5 ^ { * }$ means masking 14/16 frames. ",
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+    {
+        "type": "table",
+        "img_path": "images/5c7330a1ed4e4f0bc8ea37fd1a291664bd9b1c03825c7b29ee15d62eedb95f4e.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>input</td><td>target</td><td>SSV2</td><td>K400</td></tr><tr><td>T×T</td><td>center</td><td>63.0</td><td>79.3</td></tr><tr><td>TX</td><td>Tx</td><td>68.9</td><td>79.8</td></tr><tr><td>T×T</td><td>T×T</td><td>69.6</td><td>80.0</td></tr><tr><td>T×T</td><td>2T×</td><td>69.2</td><td>80.1</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "(c) Reconstruction target. $T \\times$ $\\tau$ denotes “frames $\\times$ stride”. center denotes the center frame of the input clip. $T$ is set to 16 as default. $\\tau$ is set to 2 and 4 on SSV2 and K400, respectively. ",
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+    {
+        "type": "table",
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+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=2>case              SSV2K400fromscratch      32.668.8</td></tr><tr><td rowspan=1 colspan=2>fromscratch      32.668.8</td></tr><tr><td rowspan=1 colspan=2>ImageNet-21k sup.61.8 78.9</td></tr><tr><td></td><td rowspan=2 colspan=1>65.2   -</td></tr><tr><td rowspan=1 colspan=1>IN-21k+K400 sup.</td></tr><tr><td rowspan=1 colspan=1>VideoMAE</td><td rowspan=1 colspan=1>69.680.0</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "(d) Pre-training strategy. Our VideoMAE works the best without using any extra data. “sup.” is supervised training. ",
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+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>dataset</td><td>method SSV2</td><td>K400</td></tr><tr><td>IN-1K</td><td>ImageMAE 64.8</td><td>78.7</td></tr><tr><td>K400</td><td>VideoMAE 68.5</td><td>80.0</td></tr><tr><td>SSV2</td><td>VideoMAE 69.6</td><td>79.6</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "(e) Pre-training dataset. Our VideoMAE works the best when directly pre-training the models on the source datasets. ",
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+        "type": "table",
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+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>case</td><td>SSV2</td><td>K400</td></tr><tr><td>L1 loss</td><td>69.1</td><td>79.7</td></tr><tr><td>MSE loss</td><td>69.6</td><td>80.0</td></tr><tr><td>Smooth L1 loss</td><td>68.9</td><td>79.6</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "(f) Loss function. MSE loss works the best for the masking and reconstruction task in VideoMAE. ",
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+    },
+    {
+        "type": "text",
+        "text": "Table 1: Ablation experiments on Something-Something V2 and Kinetics-400. Our backbone is 16-frame vanilla ViT-B and all models are pre-trained with mask ratio $\\rho { = } 9 0 \\%$ for 800 epochs, and finetuned for evaluation. We perform TSN [75] uniform sampling on SSV2 and dense sampling [77, 22] on K400. All models share the same inference protocol, i.e., 2 clips $\\times 3$ crops on SSV2 and 5 clips $\\times 3$ crops on K400. The default choice for our model is colored in gray . ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Masking strategy. We compare different masking strategies in Table 1b. When increasing the masking ratio from $7 5 \\%$ to $90 \\%$ for tube masking, the performance on SSV2 boosts from $6 8 . 0 \\%$ to $6 9 . 6 \\%$ . Then, with an extremely high ratio, we find tube masking also achieves better performance than plain random masking and frame masking. We attribute these interesting observations to the redundancy and temporal correlation in videos. The conclusion on K400 is in accord with one on SSV2. One may note that the performance gap on K400 is lower than one on SSV2. We argue that the Kinetics videos are mostly stationary and scene-related. The effect of temporal modeling is not obvious. Overall, we argue that our default designs enforce the networks to capture more useful spatiotemporal structures and therefore make VideoMAE a more challenging task, which a good self-supervised learner hunger for. ",
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+    },
+    {
+        "type": "text",
+        "text": "Reconstruction target. First, if we only employ the center frame as the target, the results would decrease greatly as shown in Table 1c. The sampling stride is also sensitive. The result of small sampling strid $\\begin{array} { c } { { \\vdots \\frac { \\tau } { 2 } } } \\\\ { { 2 T } } \\end{array}$ is lower than default sampling stride $\\tau$ $6 8 . 9 \\%$ vs. $6 9 . 6 \\%$ on SSV2). We also try to reconstruct frames from the downsampled $T$ frames, but it obtains slightly worse results on SSV2. For simplicity, we use the input downsampled clip as our default reconstruction target. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Pre-training strategy. We compare different pre-training strategies in Table 1d. Similar to previous trials [3, 6], training video transformers from scratch yields unsatisfied results on video datasets. When pre-trained on the large-scale ImageNet-21K dataset, the video transformer obtains better accuracy from $3 2 . 6 \\%$ to $6 1 . 8 \\%$ on SSV2 and $6 8 . 8 \\%$ to $78 . 9 \\%$ on K400. Using the models pre-trained on both ImageNet-21K and Kinetics further increases accuracy to $6 5 . 2 \\%$ on SSV2. Our VideoMAE can effectively train a video transformer on the video dataset itself without using any extra data and achieve the best performance $6 9 . 6 \\%$ on SSV2 and $8 0 . 0 \\%$ on K400). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Pre-training dataset. First, we pre-train the ViT-B on ImageNet-1K for 1600 epochs, following the recipes in [30]. Then we inflate the 2D patch embedding layer to our cube embedding layer following [10] and fine-tune the model on the target video datasets. The results surpass the model trained from scratch as shown in Table 1e. We also compare the ImageMAE pre-trained model with VideoMAE models pre-trained on video datasets. We see that our VideoMAE models can achieve better performance than ImageMAE. However, when we try to transfer the pre-trained VideoMAE models to the other video datasets (e.g. from Kinetics to Something-Something), the results are slightly worse than their counterpart, which is directly pre-trained on its own target video datasets. We argue that domain shift between pre-training and target datasets could be an important issue. ",
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+    {
+        "type": "table",
+        "img_path": "images/12c65c199af023af0cea89a7d75f02e26d83901381f5cf1dfd55d88c919f1fdc.jpg",
+        "table_caption": [
+            "dataset training data from scratch MoCo v3 VideoMAE ",
+            "Table 2: Comparisons with the results of previous self-supvised pre-training methods on different datasets. We take 16-frame ViT-B as the default backbone. Notably, here MoCo v3 and VideoMAE all only use the unlabelled data in the training set of each dataset for pre-training and are all fine-tuned for evaluation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>K400</td><td>240k</td><td>68.8</td><td>74.2</td><td>80.0</td></tr><tr><td>Sth-Sth V2</td><td>169k</td><td>32.6</td><td>54.2</td><td>69.6</td></tr><tr><td>UCF101</td><td>9.5k</td><td>51.4</td><td>81.7</td><td>91.3</td></tr><tr><td>HMDB51</td><td>3.5k</td><td>18.0</td><td>39.2</td><td>62.6</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/4984d80bb5931373d14a561e52f3030d30e0b8e911a65487ada3c3c03b1affcd.jpg",
+        "table_caption": [
+            "Table 3: Comparisons with the efficiency and effectiveness on Something-Something V2. We report the fine-tuning (ft) and linear probing (lin) accuracy $( \\% )$ . The wall-clock time of pre-training is benchmarked in 64 Tesla V100 GPUs with PyTorch. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>method</td><td>epoch</td><td>ft. acc.</td><td>lin. acc.</td><td>hours</td><td>speedup</td></tr><tr><td>MoCov3</td><td>300</td><td>54.2</td><td>33.7</td><td>61.7</td><td>-</td></tr><tr><td>VideoMAE</td><td>800</td><td>69.6</td><td>38.9</td><td>19.5</td><td>3.2×</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/8235a1ecef5b6377cd778faac984f55c895569c4b9d58e7a010b28300b9c0475.jpg",
+        "table_caption": [
+            "Table 4: Comparisons with the feature transferability on smaller datasets. We take 16-frame ViT-B as the default backbone. Notably, here MoCo v3 and VideoMAE are all pre-trained on Kinetics-400 with unlabelled data in the training set. Then the pre-trained model is fine-tuned on target datasets for evaluation. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>method</td><td>K400 → SSV2 K400 →UCF K400 →HMDB</td><td></td><td></td></tr><tr><td>MoCo v3</td><td>62.4</td><td>93.2</td><td>67.9</td></tr><tr><td>VideoMAE</td><td>68.5</td><td>96.1</td><td>73.3</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Loss function. Table 1f contains an ablation study of loss function. We find that the MSE loss could achieve a higher result compared with the L1 loss and smooth L1 loss. Therefore, we employ the MSE loss by default. ",
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+    },
+    {
+        "type": "text",
+        "text": "4.3 Main Results and Analysis ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "VideoMAE: data-efficient learner. The self-supervised video pre-training (SSVP) has been extensively studied in previous works, but they mainly use the CNN-based backbones. Few works have investigated transformer-based backbone in SSVP. Therefore, to demonstrate the effectiveness of VideoMAE for transformer-based SSVP, we compare two methods implemented by ourselves: (1) training from scratch and (2) pre-training with contrastive learning (MoCo v3 [14]). For training from scratch, we carefully tune these hyper-parameters to successfully pre-train ViT-Base from the training set of the dataset. For pre-training with MoCo v3, we strictly follow the training practice in its image counterpart and carefully avoid the collapse issue. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The recognition accuracy is reported in Table 2. We see that our VideoMAE significantly outperforms other two training settings. For instance, on the largest dataset of Kinetics-400, our VideoMAE outperforms training from scratch by around $10 \\%$ and MoCo v3 pre-training by around $5 \\%$ . This superior performance demonstrates that masked autoencoder provides an effective pre-training mechanism for video transformers. We also see that the performance gap between our VideoMAE and the other two methods becomes larger as the training set becomes smaller. Notably, even with only $3 . 5 \\mathrm { k }$ training clips on HMDB51, our VideoMAE pre-training can still obtain a satisfying accuracy (around $6 1 \\%$ ). This new result demonstrates that VideoMAE is a more data-efficient learner for SSVP. This property is particularly important for scenarios with limited data available and different with contrastive learning methods. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We compare the efficiency of VideoMAE pre-training and MoCo v3 pre-training in Table 3. The task of masked autoencoding with a high ratio is more challenging and thereby requires more training epochs (800 vs. 300). Thanks to the asymmetric encoder-decoder in our VideoMAE and extremely high masking ratio, our pre-training time is much shorter than MoCo v3 (19.5 vs. 61.7 hours). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "High masking ratio. In VideoMAE, one core design is the extremely high masking ratio. We perform an investigation of this design on the Kinetics-400 and Something-Something V2 datasets. The results are shown in Figure 3. We see that the best masking ratio is extremely high, and even $9 5 \\%$ can achieve good performance for both datasets. This result is difference from BERT [17] in NLP and MAE [30] in images. We analyze the temporal redundancy and correlation in videos makes it possible for our VideoMAE to learn plausible outputs with such a high masking ratio. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/b75fa7e78a9ca4d039292fdaded0567cbb3ebfb7d0b845e469cbbc73517cc6c0.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/7fbbad8d22712458885f5f102f2b7248e1babda4d1f8cc723955cf62c79a6cde.jpg",
+        "image_caption": [
+            "Figure 3: The effect of masking ratio Figure 4: Data efficiency of VideoMAE representations. on (a) Something-Something V2 and (b) Our default backbone is 16-frame vanilla ViT-B. • denotes Kinetics-400. We take 16-frame vanilla that all models are trained for the same 132k iterations, ViT-B as default. The results show that and $0$ denotes that all models are trained for the same 800 an extremely high masking ratio $( 9 0 \\% )$ epochs. Note that it takes 132k iterations to pre-train the achieves the best efficiency and effective- model for 800 epochs on the full training set of Somethingness trade-off on both video datasets. Something V2. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We also visualize the reconstructed examples in Appendix $\\ S  { s e }$ . We see that even under an extremely high masking ratio, VideoMAE can produce satisfying reconstructed results. This implies VideoMAE is able to learn useful representations that capture the holistic spatiotemporal structure in videos. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Transfer learning: quality vs. quantity. To further investigate the generalization ability of VideoMAE in representation learning, we transfer the learned VideoMAE from Kinetics-400 to Something-Something V2, UCF101, and HMDB51. The results are shown in Table 4, and we compare them with MoCo v3 pre-training. The models pre-trained by VideoMAE are better than those pre-trained by MoCo v3, demonstrating that our VideoMAE learns more transferable representations. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Comparing Table 2 and Table 4, the transferred representation outperforms the original VideoMAE models trained from its own dataset on UCF101 and HMDB51. In contrast, the transferred representation is worse on Something-Something V2. To figure out whether this inconsistent result is caused by the large scale of Something-Something V2, we further perform a detailed investigation by decreasing the pre-training video numbers. In this study, we run two experiments: (1) pre-training with the same epochs and (2) pre-training with the same time budget. The result is shown in Figure 4. We see that more training iterations could contribute to better performance when we decrease the size of the pre-training set. Surprisingly, even with only $4 2 \\mathrm { k }$ pre-training videos, we can still obtain better accuracy than the Kinetics pre-trained models with 240k videos $6 8 . 7 \\%$ vs. $6 8 . 5 \\%$ ). This result implies that domain shift is another important factor, and data quality is more important than data quantity in SSVP when there exists a difference between pre-training and target datasets. It also demonstrates that VideoMAE is a data-efficient learner for SSVP. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Transfer learning: downstream action detection. We also transfer the learned VideoMAE on Kinetics-400 to downstream action detection dataset AVA. Following the standard setting [26], we evaluate on top 60 common classes with mean Average Precision (mAP) as the metric under IoU threshold of 0.5. The results are shown in the Table 5. After self-supervised pre-training on Kinetics400, our VideoMAE with the vanilla ViT-B can achieve $2 6 . 7 \\mathrm { m A P }$ on AVA, which demonstrates the strong transferability of our VideoMAE. If the pre-trained ViT-B is additionally fine-tuned on Kinetics-400 with labels, the transfer learning performance can further increase about $5 \\mathrm { m A P }$ (from 26.7 to 31.8). More remarkably, when we scale up the pre-training configurations with larger video datasets (e.g. Kinetics-700) or more powerful backbones (e.g. ViT-Large and ViT-Huge), VideoMAE can finally obtain better performance. For example, our ViT-L VideoMAE pre-trained on Kinetics-700 achieves $3 9 . 3 \\mathrm { m A P }$ and ViT-H VideoMAE pre-trained on Kinetics-400 has $3 9 . 5 \\mathrm { m A P } . $ . These results demonstrate that the self-supervised pre-trained models transfer well not only on action classification task but on more complex action detection task. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/0dd820b66ca8a61b2d3147dc5da302a16fb9dfa2c7e7adc15b5b05b34b9f7aa1.jpg",
+        "table_caption": [
+            "Table 5: Comparison with the state-of-the-art methods on AVA v2.2. All models are pre-trained and fine-tuned at image size $2 2 4 ^ { 2 }$ . We report the mean Average Precision (mAP) on validation set. “Ex. labels $\\pmb { \\chi } ^ { , }$ means only unlabelled data is used during the pre-training phase and the pre-trained models are directly transferred to AVA. “Ex. labels $\\curvearrowleft$ means pre-trained models are additionally fine-tuned on the pre-training dataset with labels before transferred to AVA. $T \\times \\tau$ refers to frame number and corresponding sample rate. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Backbone</td><td>Pre-train Dataset Extra Labels|T×7</td><td></td><td></td><td>GFLOPs</td><td>Param</td><td>mAP</td></tr><tr><td>supervised [22]</td><td>SlowFast-R101</td><td>Kinetics-400</td><td>√</td><td>8×8</td><td>138</td><td>53</td><td>23.8</td></tr><tr><td>CVRL [53]</td><td>SlowOnly-R50</td><td>Kinetics-400</td><td>X</td><td>32×2</td><td>42</td><td>32</td><td>16.3</td></tr><tr><td>pBYOLp=3 [23]</td><td>SlowOnly-R50 SlowOnly-R50</td><td>Kinetics-400</td><td>X</td><td>8×8</td><td>42</td><td>32</td><td>23.4</td></tr><tr><td>pMoCop=3 [23]</td><td>MViT-L</td><td>Kinetics-400 Kinetics-400</td><td>X</td><td>8×8 40×3</td><td>42</td><td>32</td><td>20.3</td></tr><tr><td>MaskFeat↑312 [79] MaskFeat↑312 [79]</td><td>MViT-L</td><td>Kinetics-600</td><td>√ √</td><td>40×3</td><td>2828 2828</td><td>218 218</td><td>37.5</td></tr><tr><td>VideoMAE</td><td>ViT-S</td><td>Kinetics-400</td><td>X</td><td>16×4</td><td>57</td><td>22</td><td>38.8 22.5</td></tr><tr><td>VideoMAE</td><td>ViT-S</td><td>Kinetics-400</td><td>√</td><td>16×4</td><td>57</td><td>22</td><td>28.4</td></tr><tr><td>VideoMAE</td><td>ViT-B</td><td>Kinetics-400</td><td>X</td><td>16×4</td><td>180</td><td>87</td><td>26.7</td></tr><tr><td>VideoMAE</td><td>ViT-B</td><td>Kinetics-400</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>ViT-L</td><td>Kinetics-400</td><td>√</td><td>16×4</td><td>180</td><td>87</td><td>31.8</td></tr><tr><td>VideoMAE</td><td>ViT-L</td><td>Kinetics-400</td><td>X</td><td>16×4 16×4</td><td>597</td><td>305</td><td>34.3</td></tr><tr><td>VideoMAE VideoMAE</td><td>ViT-H</td><td>Kinetics-400</td><td>√</td><td>16×4</td><td>597</td><td>305</td><td>37.0</td></tr><tr><td>VideoMAE</td><td>ViT-H</td><td>Kinetics-400</td><td>X √</td><td>16×4</td><td>1192 1192</td><td>633 633</td><td>36.5</td></tr><tr><td>VideoMAE</td><td>ViT-L</td><td>Kinetics-700</td><td>X</td><td>16×4</td><td>597</td><td>305</td><td>39.5 36.1</td></tr><tr><td>VideoMAE</td><td>ViT-L</td><td>Kinetics-700</td><td>√</td><td>16×4</td><td>597</td><td>305</td><td>39.3</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/14299b0b16efb4597e3b38c0c1c492f49ae4e4738377e25b001909eedf380a15.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 6: Comparison with the state-of-the-art methods on Something-Something V2. Our VideoMAE reconstructs normalized cube pixels and is pre-trained with a masking ratio of $90 \\%$ for 2400 epochs. “Ex. labels $\\pmb { \\chi } ^ { , }$ means only unlabelled data is used during the pre-training phase. “N/A” indicates the numbers are not available for us. "
+        ],
+        "table_body": "<table><tr><td>Method</td><td>Backbone</td><td>Extra data</td><td>Ex. labels</td><td>Frames</td><td>GFLOPs</td><td>Param</td><td>Top-1</td><td>Top-5</td></tr><tr><td>TEINetEn [39]</td><td>ResNet50x2</td><td rowspan=\"3\">ImageNet-1K</td><td>√</td><td>8+16</td><td>99×10x3</td><td>50</td><td>66.5</td><td>N/A</td></tr><tr><td>TANetEn [40]</td><td>ResNet50×2</td><td>√</td><td>8+16</td><td>99×2×3</td><td>51</td><td>66.0</td><td>90.1</td></tr><tr><td>TDNEn [74]</td><td>ResNet101×2</td><td>√</td><td>8+16</td><td>198×1×3</td><td>88</td><td>69.6</td><td>92.2</td></tr><tr><td>SlowFast [22] MViTv1[21]</td><td>ResNet101 MViTv1-B</td><td rowspan=\"2\">Kinetics-400</td><td>√ √</td><td>8+32 64</td><td>106×1×3 455×1×3</td><td>53 37</td><td>63.1 67.7</td><td>87.6</td></tr><tr><td>TimeSformer [6]</td><td>ViT-B</td><td></td><td>8</td><td>196×1×3</td><td>121</td><td>59.5</td><td>90.9 N/A</td></tr><tr><td>TimeSformer [6]</td><td>ViT-L</td><td rowspan=\"2\">ImageNet-21K</td><td>V</td><td>64</td><td>5549×1×3</td><td>430</td><td>62.4</td><td>N/A</td></tr><tr><td>ViViT FE [3]</td><td>ViT-L</td><td>√</td><td>32</td><td>995×4×3</td><td>N/A</td><td>65.9</td><td>89.9</td></tr><tr><td>Motionformer [50]</td><td>ViT-B</td><td rowspan=\"4\">IN-21K+K400</td><td>√</td><td>16</td><td>370×1×3</td><td>109</td><td>66.5</td><td>90.1</td></tr><tr><td>Motionformer [50]</td><td>ViT-L</td><td>√</td><td>32</td><td>1185×1×3</td><td>382</td><td>68.1</td><td>91.2</td></tr><tr><td>Video Swin [38]</td><td>Swin-B</td><td>√</td><td>32</td><td>321×1×3</td><td>88</td><td>69.6</td><td>92.7</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>VIMPAC [64]</td><td>ViT-L</td><td>HowTo100M+DALLE IN-1K+K400+DALLE</td><td>X</td><td>10</td><td>N/A×10×3</td><td>307</td><td>68.1</td><td>N/A</td></tr><tr><td>BEVT[76]</td><td>Swin-B</td><td rowspan=\"2\">Kinetics-600</td><td>×</td><td>32</td><td>321×1×3</td><td>88</td><td>70.6</td><td>N/A</td></tr><tr><td>MaskFeat↑312 [79]</td><td>MViT-L</td><td>√</td><td>40</td><td>2828×1×3</td><td>218</td><td>75.0</td><td>95.0</td></tr><tr><td>VideoMAE</td><td>ViT-B</td><td>Kinetics-400 Kinetics-400</td><td>X ×</td><td>16</td><td>180×2×3</td><td>87</td><td>69.7</td><td>92.3</td></tr><tr><td>VideoMAE</td><td>ViT-L</td><td rowspan=\"4\"> no external data</td><td>X</td><td>16</td><td>597×2×3</td><td>305</td><td>74.0</td><td>94.6</td></tr><tr><td>VideoMAE</td><td>ViT-S</td><td>X</td><td>16</td><td>57×2×3</td><td>22</td><td>66.8</td><td>90.3</td></tr><tr><td>VideoMAE</td><td>ViT-B</td><td></td><td>16</td><td>180×2×3</td><td>87</td><td>70.8</td><td>92.4</td></tr><tr><td>VideoMAE VideoMAE</td><td>ViT-L ViT-L</td><td>X X</td><td>16 32</td><td>597×2×3 1436×1×3</td><td>305 305</td><td>74.3 75.4</td><td>94.6 95.2</td></tr></table>",
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+        "type": "text",
+        "text": "4.4 Comparison with the state of the art ",
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+        "text": "We compare with the previous state-of-the-art performance on the Kinetics-400 and SomethingSomething V2 datasets. The results are reported in Table 6 and Table 7. Our VideoMAE can easily scale up with more powerful backbones (e.g. ViT-Large and ViT-Huge) and more frames (e.g. 32). Our VideoMAE achieves the top-1 accuracy of $7 5 . 4 \\%$ on Something-Something V2 and $8 7 . 4 \\%$ on Kinetics-400 without using any extra data. We see that the existing state-of-the-art methods all depend on the external data for pre-training on the Something-Something V2 dataset. On the contrary, our VideoMAE without any external data significantly outperforms previous methods with the same input resolution by around $5 \\%$ . Our ViT-H VideoMAE also achieves very competitive performance on the Kinetics-400 dataset without using any extra data, which is even better than ViViT-H with on ",
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+        "table_caption": [
+            "Table 7: Comparison with the state-of-the-art methods on Kinetics-400. Our VideoMAE reconstructs normalized cube pixels. Here models are self-supervised pre-trained with a masking ratio of $90 \\%$ for 1600 epochs on Kinetics-400. VideoMAE↑320 is initialized from its $2 2 4 ^ { 2 }$ resolution counterpart and then fine-tuned for evaluation. “Ex. labels $\\pmb { \\chi } ^ { , }$ means only unlabelled data is used during the pre-training phase. “N/A” indicates the numbers are not available for us. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Backbone</td><td>Extra data</td><td>Ex.labels|1</td><td>Frames</td><td>GFLOPs</td><td>Param</td><td>Top-1</td><td>Top-5</td></tr><tr><td>NL I3D [77]</td><td>ResNet101</td><td rowspan=\"3\">ImageNet-1K</td><td>√</td><td>128</td><td>359×10×3</td><td>62</td><td>77.3</td><td>93.3</td></tr><tr><td>TANet [40]</td><td>ResNet152</td><td></td><td>16</td><td>242×4×3</td><td>59</td><td>79.3</td><td>94.1</td></tr><tr><td>TDNEn [74]</td><td>ResNet101</td><td></td><td>8+16</td><td>198×10×3</td><td>88</td><td>79.4</td><td>94.4</td></tr><tr><td>TimeSformer [6]</td><td>ViT-L</td><td rowspan=\"4\">ImageNet-21K</td><td></td><td>96</td><td>8353×1×3</td><td>430</td><td>80.7</td><td>94.7</td></tr><tr><td>ViViT FE [3]</td><td>ViT-L</td><td></td><td>128</td><td>3980×1×3</td><td>N/A</td><td>81.7</td><td>93.8</td></tr><tr><td>Motionformer [50]</td><td>ViT-L</td><td></td><td>32</td><td>1185×10×3</td><td>382</td><td>80.2</td><td>94.8</td></tr><tr><td>Video Swin [38]</td><td>Swin-L</td><td></td><td>32</td><td>604×4×3</td><td>197</td><td>83.1</td><td>95.9</td></tr><tr><td>ViViT FE [3]</td><td>ViT-L</td><td>JFT-300M</td><td>√</td><td>128</td><td>3980×1×3</td><td>N/A</td><td>83.5</td><td>94.3</td></tr><tr><td>ViViT[3]</td><td>ViT-H</td><td>JFT-300M</td><td></td><td>32</td><td>3981×4×3</td><td>N/A</td><td>84.9</td><td>95.8</td></tr><tr><td>VIMPAC [64]</td><td>ViT-L</td><td>HowTo100M+DALLE</td><td>X</td><td>10</td><td>N/A×10×3</td><td>307</td><td>77.4</td><td>N/A</td></tr><tr><td>BEVT[76]</td><td>Swin-B</td><td>IN-1K+DALLE</td><td>X</td><td>32</td><td>282×4×3</td><td>88</td><td>80.6</td><td>N/A</td></tr><tr><td>MaskFeat↑352 [79]</td><td>MViT-L</td><td>Kinetics-600</td><td>X</td><td>40</td><td>3790×4×3</td><td>218</td><td>87.0</td><td>97.4</td></tr><tr><td>ip-CSN [68]</td><td>ResNet152</td><td rowspan=\"4\"> no external data</td><td>X</td><td>32</td><td>109×10×3</td><td>33</td><td>77.8</td><td>92.8</td></tr><tr><td>SlowFast [22]</td><td>R101+NL</td><td>X</td><td>16+64</td><td>234×10×3</td><td>60</td><td>79.8</td><td>93.9</td></tr><tr><td>MViTv1[21]</td><td>MViTv1-B</td><td>X</td><td>32</td><td>170×5×1</td><td>37</td><td>80.2</td><td>94.4</td></tr><tr><td>MaskFeat [79]</td><td>MViT-L</td><td>×</td><td>16</td><td>377×10×1</td><td>218</td><td>84.3</td><td>96.3</td></tr><tr><td>VideoMAE</td><td>ViT-S</td><td rowspan=\"4\">no external data</td><td>X</td><td>16</td><td>57×5×3</td><td>22</td><td>79.0</td><td>93.8</td></tr><tr><td>VideoMAE</td><td>ViT-B</td><td>X</td><td>16</td><td>180×5×3</td><td>87</td><td>81.5</td><td>95.1</td></tr><tr><td>VideoMAE</td><td>ViT-L</td><td>×</td><td>16</td><td>597×5×3</td><td>305</td><td>85.2</td><td>96.8</td></tr><tr><td>VideoMAE</td><td>ViT-H</td><td>×</td><td>16</td><td>1192×5×3</td><td>633</td><td>86.6</td><td>97.1</td></tr><tr><td>VideoMAE↑320</td><td>ViT-L</td><td rowspan=\"2\">no external data</td><td>X</td><td>32</td><td>3958×4×3</td><td>305</td><td>86.1</td><td></td></tr><tr><td>VideoMAE↑320</td><td>ViT-H</td><td>×</td><td>32</td><td>7397×4×3</td><td>633</td><td>87.4</td><td>97.3 97.6</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "JFT-300M pre-training $8 6 . 6 \\%$ v.s. $8 4 . 9 \\%$ ). When fine-tuned with larger spatial resolutions and input video frames, the performance of our ViT-H VideoMAE can further boost from $8 6 . 6 \\%$ to $8 7 . 4 \\%$ . ",
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+    {
+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this paper, we have presented a simple and data-efficient self-supervised learning method (VideoMAE) for video transformer pre-training. Our VideoMAE introduces two critical designs of extremely high masking ratio and tube masking strategy to make the video reconstruction task more challenging. This harder task would encourage VideoMAE to learn more representative features and relieve the information leakage issue. Empirical results demonstrate this simple algorithm works well for video datasets of different scales. In particular, we are able to learn effective VideoMAE only with thousands of video clips, which has significant practical value for scenarios with limited data available. ",
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+    {
+        "type": "text",
+        "text": "Future work VideoMAE could be further improved by using larger webly datasets, larger models (e.g., ViT-G) and larger spatial resolutions of input video (e.g., $3 8 \\hat { 4 ^ { 2 } }$ ). VideoMAE only leverages the RGB video stream without using additional audio or text stream. We expect that audio and text from the video data can provide more information for self-supervised pre-training. ",
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+    {
+        "type": "text",
+        "text": "Broader impact Potential negative societal impacts of VideoMAE are mainly concerned with energy consumption. The pre-training phase may lead to a large amount of carbon emission. Though the pre-training is energy-consuming, we only need to pre-train the model once. Different downstream tasks can then share the same pre-trained model via additional fine-tuning. Our VideoMAE unleashes the great potential of vanilla vision transformer for video analysis, which could increase the risk of video understanding model or its outputs being used incorrectly, such as for unauthorized surveillance. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgements and disclosure of funding Thanks to Ziteng Gao, Lei Chen and Chongjian Ge for their help. This work is supported by National Natural Science Foundation of China (No. 62076119, No. 61921006), the Fundamental Research Funds for the Central Universities (No. 020214380091), Tencent AI Lab Rhino-Bird Focused Research Program (No. JR202125), and Collaborative Innovation Center of Novel Software Technology and Industrialization. ",
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+        "type": "text",
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parse/dev/EXnIyMVTL8s/EXnIyMVTL8s.md

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+# TOWARDS UNDERSTANDING AND MITIGATING DIMENSIONAL COLLAPSE IN HETEROGENEOUS FEDERATED LEARNING
+
+Yujun $\mathbf { S h i } ^ { 1 \mathrm { ~ * ~ } }$ Jian Liang3 Wenqing Zhang2 Vincent Y. F. Tan1 Song Bai2 1National University of Singapore 2ByteDance Inc. 3Institute of Automation, CAS shi.yujun@u.nus.edu vtan@nus.edu.sg songbai.site@gmail.com
+
+# ABSTRACT
+
+Federated learning aims to train models collaboratively across different clients without sharing data for privacy considerations. However, one major challenge for this learning paradigm is the data heterogeneity problem, which refers to the discrepancies between the local data distributions among various clients. To tackle this problem, we first study how data heterogeneity affects the representations of the globally aggregated models. Interestingly, we find that heterogeneous data results in the global model suffering from severe dimensional collapse, in which representations tend to reside in a lower-dimensional space instead of the ambient space. Moreover, we observe a similar phenomenon on models locally trained on each client and deduce that the dimensional collapse on the global model is inherited from local models. In addition, we theoretically analyze the gradient flow dynamics to shed light on how data heterogeneity result in dimensional collapse for local models. To remedy this problem caused by the data heterogeneity, we propose FEDDECORR, a novel method that can effectively mitigate dimensional collapse in federated learning. Specifically, FEDDECORR applies a regularization term during local training that encourages different dimensions of representations to be uncorrelated. FEDDECORR, which is implementation-friendly and computationally-efficient, yields consistent improvements over baselines on standard benchmark datasets. Code: https://github.com/bytedance/FedDecorr.
+
+# 1 INTRODUCTION
+
+With the rapid development deep learning and the availability of large amounts of data, concerns regarding data privacy have been attracting increasingly more attention from industry and academia. To address this concern, McMahan et al. (2017) propose Federated Learning—a decentralized training paradigm enabling collaborative training across different clients without sharing data.
+
+One major challenge in federated learning is the potential discrepancies in the distributions of local training data among clients, which is known as the data heterogeneity problem. In particular, this paper focuses on the heterogeneity of label distributions (see Fig. 1(a) for an example). Such discrepancies can result in drastic disagreements between the local optima of the clients and the desired global optimum, which may lead to severe performance degradation of the global model. Previous works attempting to tackle this challenge mainly focus on the model parameters, either during local training (Li et al., 2020; Karimireddy et al., 2020) or global aggregation (Wang et al., 2020b). However, these methods usually result in an excessive computation burden or high communication costs (Li et al., 2021a) because deep neural networks are typically heavily over-parameterized. In contrast, in this work, we focus on the representation space of the model and study the impact of data heterogeneity.
+
+To commence, we study how heterogeneous data affects the global model in federated learning in Sec. 3.1. Specifically, we compare representations produced by global models trained under different degrees of data heterogeneity. Since the singular values of the covariance matrix provide a comprehensive characterization of the distribution of high-dimensional embeddings, we use it to study the representations output by each global model. Interestingly, we find that as the degree of data heterogeneity increases, more singular values tend to evolve towards zero. This observation suggests that stronger data heterogeneity causes the trained global model to suffer from more severe dimensional collapse, whereby representations are biased towards residing in a lower-dimensional space (or manifold). A graphical illustration of how heterogeneous training data affect output representations is shown in Fig. 1(b-c). Our observations suggest that dimensional collapse might be one of the key reasons why federated learning methods struggle under data heterogeneity. Essentially, dimensional collapse is a form of oversimplification in terms of the model, where the representation space is not being fully utilized to discriminate diverse data of different classes.
+
+![](images/6bfb5b75681a9a3482c26de3896181a5d389c20e76c6c73da39d13fbe5d64381.jpg)  
+Figure 1: (a) illustrates data heterogeneity in terms of number of samples per class. (b), (c), (d) show representations (normalized to the unit sphere) of global models trained under homogeneous data, heterogeneous data, and heterogeneous data with FEDDECORR, respectively. Only (c) suffers dimensional collapse. (b), (c), (d) are produced with ResNet20 on CIFAR10. Best viewed in color.
+
+Given the observations made on the global model, we conjecture that the dimensional collapse of the global model is inherited from models locally trained on various clients. This is because the global model is a result of the aggregation of local models. To validate our conjecture, we further visualize the local models in terms of the singular values of representation covariance matrices in Sec. 3.2. Similar to the visualization on the global model, we observe dimensional collapse on representations produced by local models. With this observation, we establish the connection between dimensional collapse of the global model and local models. To further understand the dimensional collapse on local models, we analyze the gradient flow dynamics of local training in Sec. 3.3. Interestingly, we show theoretically that heterogeneous data drive the weight matrices of the local models to be biased to being low-rank, which further results in representation dimensional collapse.
+
+Inspired by the observations that dimensional collapse of the global model stems from local models, we consider mitigating dimensional collapse during local training in Sec. 4. In particular, we propose a novel federated learning method termed FEDDECORR. FEDDECORR adds a regularization term during local training to encourage the Frobenius norm of the correlation matrix of representations to be small. We show theoretically and empirically that this proposed regularization term can effectively mitigate dimensional collapse (see Fig. 1(d) for example). Next, in Sec. 5, through extensive experiments on standard benchmark datasets including CIFAR10, CIFAR100, and TinyImageNet, we show that FEDDECORR consistently improves over baseline federated learning methods. In addition, we find that FEDDECORR yields more dramatic improvements in more challenging federated learning setups such as stronger heterogeneity or more number of clients. Lastly, FEDDECORR has extremely low computation overhead and can be built on top of any existing federated learning baseline methods, which makes it widely applicable.
+
+Our contributions are summarized as follows. First, we discover through experiments that stronger data heterogeneity in federated learning leads to greater dimensional collapse for global and local models. Second, we develop a theoretical understanding of the dynamics behind our empirical discovery that connects data heterogeneity and dimensional collapse. Third, based on the motivation of mitigating dimensional collapse, we propose a novel method called FEDDECORR, which yields consistent improvements while being implementation-friendly and computationally-efficient.
+
+# 2 RELATED WORKS
+
+Federated Learning. McMahan et al. (2017) proposed FedAvg, which adopts a simple averaging scheme to aggregate local models into the global model. However, under data heterogeneity, FedAvg suffers from unstable and slow convergence, resulting in performance degradation. To tackle this challenge, previous works either improve local training (Li et al., 2021b; 2020; Karimireddy et al., 2020; Acar et al., 2021; Al-Shedivat et al., 2020; Wang et al., 2021) or global aggregation (Wang et al., 2020b; Hsu et al., 2019; Luo et al., 2021; Wang et al., 2020a; Lin et al., 2020; Reddi et al., 2020; Wang et al., 2020a). Most of these methods focus on the model parameter space, which may result in high computation or communication cost due to deep neural networks being overparameterized. Li et al. (2021b) focuses on model representations and uses a contrastive loss to maximize agreements between representations of local models and the global model. However, one drawback of Li et al. (2021b) is that it requires additional forward passes during training, which almost doubles the training cost. In this work, based on our study of how data heterogeneity affects model representations, we propose an effective yet highly efficient method to handle heterogeneous data. Another research trend is in personalized federated learning (Arivazhagan et al., 2019; Li et al., 2021c; Fallah et al., 2020; T Dinh et al., 2020; Hanzely et al., 2020; Huang et al., 2021; Zhang et al., 2020), which aims to train personalized local models for each client. In this work, however, we focus on the typical setting that aims to train one global model for all clients.
+
+Dimensional Collapse. Dimensional collapse of representations has been studied in metric learning (Roth et al., 2020), self-supervised learning (Jing et al., 2021), and class incremental learning (Shi et al., 2022). In this work, we focus on federated learning and discover that stronger data heterogeneity causes a higher degree of dimensional collapse for locally trained models. To the best of our knowledge, this work is the first to discover and analyze dimensional collapse of representations in federated learning.
+
+Gradient Flow Dynamics. Arora et al. (2018; 2019) introduce the gradient flow dynamics framework to analyze the dynamics of multi-layer linear neural networks under the $\ell _ { 2 }$ -loss and find deeper neural networks biasing towards low-rank solution during optimization. Following their works, Jing et al. (2021) finds two factors that cause dimensional collapse in self-supervised learning, namely strong data augmentation and implicit regularization from depth. Differently, we focus on federated learning with the cross-entropy loss. More importantly, our analysis focuses on dimensional collapse caused by data heterogeneity in federated learning instead of depth of neural networks.
+
+Feature Decorrelation. Feature decorrelation had been used for different purposes, such as preventing mode collapse in self-supervised learning (Bardes et al., 2021; Zbontar et al., 2021; Hua et al., 2021), boosting generalization (Cogswell et al., 2015; Huang et al., 2018; Xiong et al., 2016), and improving class incremental learning (Shi et al., 2022). We instead apply feature decorrelation to counter the undesired dimensional collapse caused by data heterogeneity in federated learning.
+
+# 3 DIMENSIONAL COLLAPSE CAUSED BY DATA HETEROGENEITY
+
+In this section, we first empirically visualize and compare representations of global models trained under different degrees of data heterogeneity in Sec. 3.1. Next, to better understand the observations on global models, we analyze representations of local models in Sec. 3.2. Finally, to theoretically understand our observations, we analyze the gradient flow dynamics of local training in Sec. 3.3.
+
+# 3.1 EMPIRICAL OBSERVATIONS ON THE GLOBAL MODEL
+
+We first empirically demonstrate that stronger data heterogeneity causes more severe dimensional collapse on the global model. Specifically, we first separate the training samples of CIFAR100 into 10 splits, each corresponding to the local data of one client. To simulate data heterogeneity among clients as in previous works (Yurochkin et al., 2019; Wang et al., 2020a; Li et al., 2021b), we sample a probability vector $\mathbf { p } _ { c } = ( p _ { c , 1 } , p _ { c , 2 } , \hdots , p _ { c , K } ) \sim { \mathrm { D i r } } _ { K } ( \alpha )$ and allocate a $p _ { c , k }$ proportion of instances of class $c \in [ C ] = \{ 1 , 2 , \ldots , C \}$ to client $k \in [ K ]$ , where $\operatorname { D i r } _ { K } ( \alpha )$ is the Dirichlet distribution with $K$ categories and $\alpha$ is the concentration parameter. A smaller $\alpha$ implies stronger data heterogeneity $\alpha = \infty$ corresponds to the homogeneous setting). We let $\alpha \in \{ 0 . 0 1 , 0 . 0 5 , 0 . 2 5 , \infty \}$ .
+
+For each of the settings generated by different $\alpha$ ’s, we apply FedAvg (McMahan et al., 2017) to train a MobileNetV2 (Sandler et al., 2018) with CIFAR100 (observations on other federated learning methods, model architectures, or datasets are similar and are provided in Appendix D). Next, for each of the four trained global models, we compute the covariance matrix $\begin{array} { r } { \mathbf { \tilde { \Sigma } } = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } ( \mathbf { z } _ { i } - } \end{array}$ $\bar { \mathbf { z } } ) ( \mathbf { z } _ { i } - \bar { \mathbf { z } } ) ^ { \top }$ of the representations over the $N$ test data points in CIFAR100. Here $\mathbf { z } _ { i }$ is the $i$ -th test data point and $\begin{array} { r } { \bar { \bf z } = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } { \bf z } _ { i } } \end{array}$ is their average.
+
+![](images/be2ad82d7c357ba4f28d2b26cf038606163450610b73a1ea913424c0cd048c69.jpg)  
+Figure 2: Data heterogeneity causes dimensional collapse on (a) global models and (b) local models. We plot the singular values of the covariance matrix of representations in descending order. The $x$ -axis $( k )$ is the index of singular values and the $y$ -axis is the logarithm of the singular values.
+
+Finally, we apply the singular value decomposition (SVD) on each of the covariance matrices and visualize the top 100 singular values in Fig. 2(a). If we define a small value $\tau$ as the threshold for a singular value to be significant (e.g., $\log \tau = - 2$ ), we observe that for the homogeneous setting, almost all the singular values are significant, i.e., they surpass $\tau$ . However, as $\alpha$ decreases, the number of singular values exceeding $\tau$ monotonically decreases. This implies that with stronger heterogeneity among local training data, the representation vectors produced by the trained global model tend to reside in a lower-dimensional space, corresponding to more severe dimensional collapse.
+
+# 3.2 EMPIRICAL OBSERVATIONS ON LOCAL MODELS
+
+Since the global model is obtained by aggregating locally trained models on each client, we conjecture that the dimensional collapse observed on the global model stems from the dimensional collapse of local models. To further validate our conjecture, we continue to study whether increasing data heterogeneity will also lead to more severe dimensional collapse on locally trained models.
+
+Specifically, for different $\alpha$ ’s, we visualize the locally trained model of one client (visualizations on local models of other clients are similar and are provided in Appendix E). Following the same procedure as in Sec. 3.1, we plot the singular values of covariance matrices of representations produced by the local models. We observe from Fig. 2(b) that locally trained models demonstrate the same trend as the global models—namely, that the presence of stronger data heterogeneity causes more severe dimensional collapse. These experiments corroborate that the global model inherit the adverse dimensional collapse phenomenon from the local models.
+
+# 3.3 A THEORETICAL EXPLANATION FOR DIMENSIONAL COLLAPSE
+
+Based on the empirical observations in Sec. 3.1 and Sec. 3.2, we now develop a theoretical understanding to explain why heterogeneous training data causes dimensional collapse for the learned representations.
+
+Since we have established that the dimensional collapse of global model stems from local models, we focus on studying local models in this section. Without loss of generality, we study local training of one arbitrary client. Specifically, we first analyze the gradient flow dynamics of the model weights during the local training. This analysis shows how heterogeneous local training data drives the model weights towards being low-rank, which leads to dimensional collapse for the representations.
+
+# 3.3.1 SETUPS AND NOTATIONS
+
+We denote the number of training samples as $N$ , the dimension of input data as $d _ { \mathrm { i n } }$ , and total number of classes as $C$ . The $i$ -th sample is denoted as $X _ { i } \in \mathbb { R } ^ { d _ { \mathrm { i n } } }$ , and its corresponding one-hot encoded label is $\mathbf { y } _ { i } \in \mathbb { R } ^ { C }$ . The collection of all $N$ training samples is denoted as $\bar { X } = [ \bar { X } _ { 1 } , X _ { 2 } \ldots , X _ { N } ] \in$ $\mathbb { R } ^ { d _ { \mathrm { i n } } \times N }$ , and the $N$ one-hot encoded training labels are denoted as $\mathbf { y } = [ \mathbf { y } _ { 1 } , \mathbf { y } _ { 2 } , \ldots , \mathbf { y } _ { N } ] \in \mathbb { R } ^ { C \times N }$ .
+
+For simplicity in exposition, we follow Arora et al. (2018; 2019) and Jing et al. (2021) and analyze linear neural networks (without nonlinear activation layers). We consider an $( L + 1 )$ -layer (where $L \geq 1 \AA$ ) linear neural network trained using the cross entropy loss under gradient flow (i.e., gradient descent with an infinitesimally small learning rate). The weight matrix of the $i$ -th layer $( i \in [ L + 1 ] )$ ) at the optimization time step $t$ is denoted as $W _ { i } ( t )$ . The dynamics can be expressed as
+
+$$
+\dot { W } _ { i } ( t ) = - \frac { \partial } { \partial W _ { i } } \ell ( W _ { 1 } ( t ) , \ldots , W _ { L + 1 } ( t ) ) ,
+$$
+
+where $\ell$ denotes the cross-entropy loss.
+
+In addition, at the optimization time step $t$ and given the input data $X _ { i }$ , we denote $\mathbf { z } _ { i } ( t ) \in \mathbb { R } ^ { d }$ as the output representation vector $d$ being the dimension of the representations) and $\gamma _ { i } ( t ) \in \mathbb { R } ^ { C }$ as the output softmax probability vector. We have
+
+$$
+\gamma _ { i } ( t ) = \mathrm { s o f t m a x } ( W _ { L + 1 } ( t ) { \mathbf z } _ { i } ( t ) ) = \mathrm { s o f t m a x } ( W _ { L + 1 } ( t ) W _ { L } ( t ) \ldots W _ { 1 } ( t ) X _ { i } ) .
+$$
+
+e define -dimens $\begin{array} { r } { \mu _ { c } = \frac { N _ { c } } { N } } \end{array}$ , where -hot vec $N _ { c }$ is number of d where only the a samples belonging to class -th entry is 1 (and the others $c$ . We denote re 0). In ad $\mathbf { e } _ { c }$ as theon, let $C$ $c$   
+$\begin{array} { r } { \bar { \gamma } _ { c } ( t ) = \frac { 1 } { N _ { c } } \sum _ { i = 1 } ^ { N } \gamma _ { i } ( t ) \mathbb { 1 } \{ \mathbf { y } _ { i } = \mathbf { e } _ { c } \} } \end{array}$ and $\begin{array} { r } { \bar { X } _ { c } = \frac { 1 } { N _ { c } } \sum _ { i = 1 } ^ { N } X _ { i } \mathbb { { l } } \{ \mathbf { y } _ { i } = \mathbf { e } _ { c } \} } \end{array}$ .
+
+# 3.3.2 ANALYSIS ON GRADIENT FLOW DYNAMICS
+
+Since our goal is to analyze model representations ${ \bf z } _ { i } ( t )$ , we focus on weight matrices that directly produce representations (i.e., the first $L$ layers). We denote the product of the weight matrices of the first $L$ layers as $\Pi ( t ) = W _ { L } ( t ) W _ { L - 1 } ( \dot { t } ) \dots W _ { 1 } ( t )$ and analyze the behavior of $\Pi ( t )$ under the gradient flow dynamics. In particular, we derive the following result for the singular values of $\Pi ( t )$ .
+
+Theorem 1 (Informal). Assuming that the mild conditions as stated in Appendix $A . 3$ hold. Let $\sigma _ { k } ( t )$ for $k \in [ d ]$ be the $k$ -th largest singular value of $\Pi ( t )$ . Then,
+
+$$
+\begin{array} { r } { \dot { \sigma } _ { k } ( t ) = N L \left( \sigma _ { k } ( t ) \right) ^ { 2 - \frac { 2 } { L } } \sqrt { ( \sigma _ { k } ( t ) ) ^ { \frac { 2 } { L } } + M } \left( \mathbf { u } _ { L + 1 , k } ( t ) \right) ^ { \top } G ( t ) \mathbf { v } _ { k } ( t ) , } \end{array}
+$$
+
+where ${ \mathbf { u } } _ { L + 1 , k } ( t )$ is the $k$ -th left singular vector of $W _ { L + 1 } ( t )$ , $\mathbf { v } _ { k } ( t )$ is the $k$ -th right singular vector of $\Pi ( t )$ , $M$ is a constant, and $G ( t )$ is defined as
+
+$$
+G ( t ) = \sum _ { c = 1 } ^ { C } \mu _ { c } ( \mathbf { e } _ { c } - \bar { \gamma } _ { c } ( t ) ) \bar { X } _ { c } ^ { \top } ,
+$$
+
+where $\mu _ { c }$ $\iota _ { c } , \mathbf { e } _ { c } , \bar { \gamma } _ { c } ( t )$ , $\bar { X } _ { c }$ are defined after Eqn. (2).
+
+The proof of the precise version of Theorem 1 is provided in Appendix A.
+
+Based on Theorem 1, we are able to explain why greater data heterogeneity causes $\Pi ( t )$ to be biased to become lower-rank. Note that strong data heterogeneity causes local training data of one client being highly imbalanced in terms of the number of data samples per class (recall Fig. 1(a)). This implies that $\mu _ { c }$ , which is the proportion of the class $c$ data, will be close to 0 for some classes.
+
+Next, based on the definition of $G ( t )$ in Eqn. (4), more $\mu _ { c }$ ’s being close to 0 leads to $G ( t )$ being biased towards a low-rank matrix. If this is so, the term $( \mathbf { \bar { u } } _ { L + 1 , k } ( \mathbf { \bar { \Psi } } _ { k } ( t ) ) ^ { \top } G ( t ) \mathbf { v } _ { k } ( t )$ in Eqn. (3) will only be significant (large in magnitude) for fewer values of $k$ . This is because ${ \bf u } _ { L + 1 , k } ( t )$ and $\mathbf { v } _ { k } ( t )$ are both singular vectors, which are orthogonal among different $k$ ’s. This further leads to $\dot { \sigma } _ { k } ( t )$ on the left-hand side of Eqn. (3), which is the evolving rate of $\sigma _ { k }$ , being small for most of the $k$ ’s throughout training. These observations imply that only relatively few singular values of $\Pi ( t )$ will increase significantly after training.
+
+Furthermore, $\Pi ( t )$ being biased towards being low-rank will directly lead to dimensional collapse for the representations. To see this, we simply write the covariance matrix of the representations in terms of ${ \bar { \Pi } } ( t )$ as
+
+$$
+\Sigma ( t ) = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } ( { \bf z } _ { i } ( t ) - \bar { \bf z } ( t ) ) ( { \bf z } _ { i } ( t ) - \bar { \bf z } ( t ) ) ^ { \top } = \Pi ( t ) \biggl ( \frac { 1 } { N } \sum _ { i = 1 } ^ { N } ( X _ { i } - \bar { X } ) ( X _ { i } - \bar { X } ) ^ { \top } \biggr ) \Pi ( t ) ^ { \top } .
+$$
+
+From Eqn. (5), we observe that if $\Pi ( t )$ evolves to being a lower-rank matrix, $\Sigma ( t )$ will also tend to be lower-rank, which corresponds to the stronger dimensional collapse observed in Fig. 2(b).
+
+![](images/cd76f8c0de5a83b2cdcf682823c303da52d014edf04f43ca5e75cb73f2f48aec.jpg)  
+Figure 3: FEDDECORR effectively mitigates dimensional collapse for (a-b) local models and (c-d) global models. For each heterogeneity parameter $\alpha \in \{ 0 . 0 1 , 0 . { \overset { \cdot } { 0 } } 5 \}$ , we apply FEDDECORR and plot the singular values of the representation covariance matrix. The $x$ -axis $( k )$ is the index of singular values. With FEDDECORR, the tail singular values are prevented from dropping to 0 too rapidly.
+
+# 4 MITIGATING DIMENSIONAL COLLAPSE WITH FEDDECORR
+
+Motivated by the above observations and analyses on dimensional collapse caused by data heterogeneity in federated learning, we explore how to mitigate excessive dimensional collapse.
+
+Since dimensional collapse on the global model is inherited from local models, we propose to alleviate the problem during local training. One natural way to achieve this is to add the following regularization term on the representations during training
+
+$$
+L _ { \mathrm { s i n g u l a r } } ( w , X ) = { \frac { 1 } { d } } \sum _ { i = 1 } ^ { d } { \bigg ( } \lambda _ { i } - { \frac { 1 } { d } } \sum _ { j = 1 } ^ { d } \lambda _ { j } { \bigg ) } ^ { 2 } ,
+$$
+
+where $\lambda _ { i }$ is the $i$ -th singular value of the covariance matrix of the representations. Essentially, $L _ { \mathrm { s i n g u l a r } }$ penalizes the variance among the singular values, thus discouraging the tail singular values from collapsing to 0, mitigating dimensional collapse. However, this regularization term is not practical as it requires calculating all the singular values, which is computationally expensive.
+
+Therefore, to derive a computationally-cheap training objective, we first apply the $\mathbf { Z }$ -score normalization on all the representation vectors $\mathbf { z } _ { i }$ as follows: $\hat { \mathbf { z } } _ { i } = ( \mathbf { z } _ { i } - \bar { \mathbf { z } } ) / \sqrt { \mathrm { V a r } ( \mathbf { z } ) }$ . This results in the covariance matrix of $\hat { \mathbf { z } } _ { i }$ being equal to its correlation matrix (i.e., the matrix of correlation coefficients). The following proposition suggests a more convenient cost function to regularize.
+
+Proposition 1. For a $d$ -by- $d$ correlation matrix $K$ with singular values $( \lambda _ { 1 } , \ldots , \lambda _ { d } )$ , we have:
+
+$$
+\sum _ { i = 1 } ^ { d } \left( \lambda _ { i } - \frac { 1 } { d } \sum _ { j = 1 } ^ { d } \lambda _ { j } \right) ^ { 2 } = \| K \| _ { \mathrm { F } } ^ { 2 } - d .
+$$
+
+The proof of Proposition 1 can be found in Appendix B. This proposition suggests that regularizing the Frobenius norm of the correlation matrix $\| K \| _ { \mathrm { F } }$ achieves the same effect as minimizing $L _ { \mathrm { s i n g u l a r } }$ . In contrast to the singular values, $\| K \| _ { \mathrm { F } }$ can be computed efficiently.
+
+To leverage this proposition, we propose a novel method, FEDDECORR, which regularizes the Frobenius norm of the correlation matrix of the representation vectors during local training on each client. Formally, the proposed regularization term is defined as:
+
+$$
+L _ { \mathrm { F e d D e c o r r } } ( w , X ) = \frac { 1 } { d ^ { 2 } } \| K \| _ { \mathrm { F } } ^ { 2 } ,
+$$
+
+where $w$ is the model parameters, $K$ is the correlation matrix of the representations. The overall objective of each local client is
+
+$$
+\operatorname* { m i n } _ { w } \ell ( w , X , \mathbf { y } ) + \beta L _ { \mathrm { F e d D e c o r r } } ( w , X ) ,
+$$
+
+where $\ell$ is the cross entropy loss, and $\beta$ is the regularization coefficient of FEDDECORR. The pseudocode of our method is provided in Appendix $\mathbf { G }$ .
+
+To visualize the effectiveness of $L _ { \mathrm { F e d D e c o r r } }$ in mitigating dimensional collapse, we now revisit the experiments of Fig. 2 and apply $L _ { \mathrm { F e d D e c o r r } }$ under the heterogeneous setting where $\alpha \in \{ 0 . 0 1 , 0 . 0 5 \}$ . We plot our results in Fig. 3 for both local and global models. Figs. 3(a-b) show that for local models, FEDDECORR encourages the tail singular values to not collapse to 0, thus effectively mitigating dimensional collapse. Moreover, as illustrated in Figs. 3(c-d), this desirable effect introduced by FEDDECORR on local models can also be inherited by the global models.
+
+<table><tr><td rowspan="2">Method</td><td colspan="4">CIFAR10</td><td colspan="4">CIFAR100</td></tr><tr><td>α= 0.05</td><td>0.1</td><td>0.5</td><td>8</td><td>0.05</td><td>0.1</td><td>0.5</td><td>8</td></tr><tr><td>FedAvg</td><td>64.85±2.01 76.28±1.22 89.84±0.13 92.39±0.26 59.87±0.25 66.46±0.16 71.69±0.15 74.54±0.15</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>FedProx</td><td>+ FEDDEC0RR 73.06±0.81 80.60±0.91 89.84±0.05 92.19±0.10 61.53±0.11 67.12±0.09 71.91±0.04 73.87±0.18</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+ FEDDEC0RR 71.38±0.81 81.74±0.34 89.96±0.26 92.14±0.20 61.33±0.19 67.00±0.46 71.64±0.10 74.15±0.06</td><td></td><td></td><td></td><td></td><td></td><td></td><td>64.11±0.84 76.10±0.40 89.57±0.04 92.38±0.09 60.02±0.46 66.41±0.27 71.78±0.19 74.34±0.03</td><td></td></tr><tr><td>FedAvgM</td><td></td><td></td><td></td><td></td><td>71.34±0.71 77.51±0.58 88.39±0.17 91.35±0.15 59.64±0.20 66.36±0.14 71.17±0.22 74.20±0.16</td><td></td><td></td><td></td></tr><tr><td>MOON</td><td>+ FEDDEC0RR 73.60±0.82 79.21±0.15 88.70±0.26 91.33±0.13 61.48±0.27 66.60±0.1171.26±0.21 73.86±0.25</td><td></td><td></td><td></td><td>68.79±0.69 78.70±0.66 90.08±0.10 92.62±0.17 56.79±0.17 65.48±0.29 71.81±0.14 74.30±0.12</td><td></td><td></td><td></td></tr><tr><td></td><td>+ FEDDEC0RR 73.46±0.84 81.63±0.5590.61±0.05 92.63±0.19 59.43±0.34 66.12±0.20 71.68±0.05 73.70±0.25</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+Table 1: CIFAR10/100 Experiments. We run experiments under various degrees of heterogeneity $( \alpha \in \{ 0 . 0 5 , 0 . 1 , 0 . 5 , \infty \} )$ and report the test accuracy $( \% )$ . All results are (re)produced by us and are averaged over 3 runs (mean $\pm$ std). Bold font highlights the highest accuracy in each column.
+
+# 5 EXPERIMENTS
+
+# 5.1 EXPERIMENTAL SETUPS
+
+Datasets: We adopt three popular benchmark datasets, namely CIFAR10, CIFAR100, and TinyImageNet. CIFAR10 and CIFAR100 both have 50, 000 training samples and 10, 000 test samples, and the size of each image is $3 2 \times 3 2$ . TinyImageNet contains 200 classes, with 100, 000 training samples and 10, 000 testing samples, and each image is $6 4 \times 6 4$ . The method generating local data for each client was introduced in Sec. 3.1.
+
+Implementation Details: Our code is based on the code of Li et al. (2021b). For all experiments, we use MobileNetV2 (Sandler et al., 2018). We run 100 communication rounds for all experiments on the CIFAR10/100
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">TinyImageNet</td></tr><tr><td>α = 0.050.1</td><td>0.5</td><td>8</td></tr><tr><td>FedAvg</td><td colspan="3">35.02±0.46 39.30±0.23 46.92±0.25 49.33±0.19</td></tr><tr><td></td><td colspan="3">+ FEDDEC0RR 40.29±0.18 43.86±0.50 50.01±0.27 52.63±0.26</td></tr><tr><td>FedProx</td><td colspan="3">35.20±0.30 39.66±0.43 47.16±0.0749.76±0.36 + FEDDEC0RR 40.63±0.05 44.19±0.14 50.26±0.27 52.37±0.36</td></tr><tr><td>FedAvgM</td><td colspan="3">34.81±0.09 39.72±0.11 47.11±0.04 49.67±0.25</td></tr><tr><td></td><td colspan="3">+ FEDDEC0RR 39.97±0.23 43.95±0.26 50.14±0.11 52.05±0.37</td></tr><tr><td>MOON</td><td colspan="3">35.23±0.26 40.53±0.28 47.25±0.6650.48±0.57 + FEDDEC0RR 40.40±0.24 44.20±0.22 50.81±0.51 53.01±0.45</td></tr></table>
+
+Table 2: TinyImageNet Experiments. We run with $\alpha \in$ $\{ 0 . 0 5 , 0 . 1 , 0 . 5 , \infty \} )$ and report the test accuracy $( \% )$ . All results are (re)produced by us and are averaged over 3 runs (mean $\pm$ std is reported). Bold font highlights the highest accuracy in each column.
+
+datasets and 50 communication rounds on the TinyImageNet dataset. We conduct local training for 10 epochs in each communication round using SGD optimizer with a learning rate of 0.01, a SGD momentum of 0.9, and a batch size of 64. The weight decay is set to $1 0 ^ { - 5 }$ for CIFAR10 and $1 0 ^ { - 4 }$ for CIFAR100 and TinyImageNet. We apply the data augmentation of Cubuk et al. (2018) in all CIFAR100 and TinyImageNet experiments. The $\beta$ of FEDDECORR (i.e., $\beta$ in Eqn. (9)) is tuned to be 0.1. The details of tuning hyper-parameters for other federated learning methods are described in Appendix F.
+
+# 5.2 FEDDECORR SIGNIFICANTLY IMPROVES BASELINE METHODS
+
+To validate the effectiveness of our method, we apply FEDDECORR to four baselines, namely FedAvg (McMahan et al., 2017), FedAvgM (Hsu et al., 2019), FedProx (Li et al., 2020), and MOON (Li et al., 2021b). We partition the three benchmark datasets (CIFAR10, CIFAR100, and TinyImageNet) into 10 clients with $\alpha \in \{ 0 . 0 5 , 0 . 1 , 0 . 5 , \infty \}$ . Since $\alpha = \infty$ is the homogeneous setting where local models should be free from the pitfall of excessive dimensional collapse, we only expect FEDDECORR to perform on par with the baselines in this setting.
+
+We display the CIFAR10/100 results in Tab. 1 and the TinyImageNet results in Tab. 2. We observe that for all of the heterogeneous settings on all datasets, the highest accuracies are achieved by adding FEDDECORR on top of a certain baseline method. In particular, in the strongly heterogeneous settings where $\alpha \in \{ 0 . 0 5 , 0 . 1 \}$ , adding FEDDECORR yields significant improvements of around $2 \% \sim \mathbf { \bar { g } } \%$ over baseline methods on all datasets. On the other hand, for the less heterogeneous setting of $\alpha = 0 . 5$ , the problem of dimensional collapse is less pronounced as discussed in Sec 3, leading to smaller improvements from FEDDECORR. Such decrease in improvements is a general trend and is also observed on FedProx, FedAvgM, and MOON. In addition, surprisingly, in the homogeneous setting of $\alpha = \infty$ , FEDDECORR still produces around $2 \%$ of improvements on the TinyImageNet dataset. We conjecture that this is because TinyImageNet is much more complicated than the CIFAR datasets, and some other factors besides heterogeneity of label may cause undesirable dimensional collapse in the federated learning setup. Therefore, federated learning on TinyImageNet can benefit from FEDDECORR even in the homogeneous setting.
+
+![](images/c9dde8fad2285892716733d22074999cb3445a8c763bebb4960ae6fc2a4747ea.jpg)  
+Figure 4: Test accuracy $( \% )$ at each communication round. Results are averaged over 3 runs. Shaded areas denote one standard deviation above and below the mean.
+
+To further demonstrate the advantages of FEDDECORR, we apply it on FedAvg and plot how the test accuracy of the global model evolves throughout the federated learning in Fig. 4. In this figure, if we set a certain value of the testing accuracy as a threshold, we see that adding FEDDECORR significantly reduces the number of communication rounds needed to achieve the given threshold. This further shows that FEDDECORR not only improves the final performance, but also greatly boosts the communication efficiency in federated learning.
+
+# 5.3 ABLATION STUDY ON THE NUMBER OF CLIENTS
+
+Table 3: Ablation study on the number of clients. Based on TinyImageNet, we run experiments with different number of clients and different amounts of data heterogeneity.   
+
+<table><tr><td># clients</td><td>Method</td><td>α = 0.05 0.1</td><td>0.5</td></tr><tr><td rowspan="2">10</td><td>FedAvg</td><td>35.02 39.30</td><td>46.92</td></tr><tr><td>+ FEDDECORR</td><td>40.29 43.86</td><td>50.01</td></tr><tr><td rowspan="2">20</td><td>FedAvg</td><td>31.21 35.30</td><td>43.64</td></tr><tr><td>+ FEDDECORR</td><td>39.41 41.27</td><td>46.17</td></tr><tr><td rowspan="2">30</td><td>FedAvg</td><td>26.20 30.88</td><td>37.22</td></tr><tr><td>+ FEDDECORR</td><td>36.50 39.02</td><td>44.38</td></tr><tr><td rowspan="2">50</td><td>FedAvg</td><td>25.70 28.88</td><td>34.89</td></tr><tr><td>+ FEDDECORR</td><td>34.50 36.674</td><td>42.34</td></tr><tr><td rowspan="2">100</td><td>FedAvg</td><td>21.53 </td><td>24.693 30.21</td></tr><tr><td> + FEDDECORR</td><td>30.55</td><td>33.85 38.65</td></tr></table>
+
+Next, we study whether the improvements brought by FEDDECORR are preserved as number of clients increases. We partition the TinyImageNet dataset into 10, 20, 30, 50, and 100 clients according to different $\alpha$ ’s, and then run FedAvg with and without FEDDECORR. For
+
+the experiments with 10, 20 and 30 clients, we run 50 communication rounds. For the experiments with 50 and 100 clients, we randomly select $2 0 \%$ of the total clients to participate the federated learning in each round and run 100 communication rounds. Results are shown in Tab. 3. From this table, we see that the performance improvements resulting from FEDDECORR increase from around ${ \mathbf 3 \% } \sim { \mathbf 5 \% }$ to around $\mathbf { 7 \% } \sim \mathbf { 1 0 \% }$ with the growth in the number of clients. Therefore, interestingly, we show through experiments that the improvements brought by FEDDECORR can be even more pronounced under the more challenging settings with more clients. Moreover, our experimental results under random client participation show that the improvements from FEDDECORR are robust to such uncertainties. These experiments demonstrate the potential of FEDDECORR to be applied to real world federated learning settings with massive numbers of clients and random client participation.
+
+![](images/50b0f82d753e33a03d0b8e9ffe4f7047490a6ddbcac4d877abf0313c35fdad02.jpg)  
+Figure 5: Ablation study on $\beta$ . We apply FEDDECORR with different choices of $\beta$ on FedAvg.
+
+5.4 ABLATION STUDY ON THE REGULARIZATION COEFFICIENT $\beta$
+
+Next, we study FEDDECORR’s robustness to the $\beta$ in Eqn. (9) by varying it in the set $\{ 0 . 0 1 , 0 . 0 5 , 0 . 1 , 0 . 2 , 0 . 3 \}$ . We partition the CIFAR10 and TinyImageNet datasets into 10 clients with $\alpha$ equals to 0.05 and 0.1 to simulate the heterogeneous setting. Results are shown in Fig. 5. We observe that, in general, when $\beta$ increases, the performance of FEDDECORR first increases, then plateaus, and finally decreases slightly. These results show that FEDDECORR is relatively insensitive to the choice of $\beta$ , which implies FEDDECORR is an easy-to-tune federated learning method. In addition, among all experimental setups, setting $\beta$ to be 0.1 consistently produces (almost) the best results. Therefore, we recommend $\beta = 0 . 1$ when having no prior information about the dataset.
+
+# 5.5 ABLATION STUDY ON THE NUMBER OF LOCAL EPOCHS
+
+Lastly, we ablate on the number of local epochs per communication round. We set the number of local epochs $E$ to be in the set $\{ 1 , 5 , 1 0 , 2 0 \}$ . We run experiments with and without FEDDECORR, and we use the CIFAR100 and TinyImageNet datasets with $\alpha$ being 0.05 and 0.1 for this ablation study. Results are shown in Tab. 4, in which one observes that with increasing $E$ , FEDAVG performance first increases and then decreases. This is because when $E$ is too small, the local training cannot converge properly in each communication round. On the other hand, when $E$ is too large, the model parameters of local clients might be driven to be too far from the global optimum. Nevertheless,
+
+Table 4: Ablation study on local epochs. Experiments with different number of local epochs $E$ .   
+
+<table><tr><td rowspan="2">E</td><td rowspan="2">Method</td><td>CIFAR100</td><td>TinyImageNet</td></tr><tr><td>α = 0.05 0.1</td><td>0.05 0.1</td></tr><tr><td rowspan="2">1</td><td>FedAvg</td><td>50.67 55.98</td><td>32.31 34.88</td></tr><tr><td>+ FEDDECORR</td><td>53.18 57.02</td><td>36.49 38.99</td></tr><tr><td rowspan="2">5</td><td>FedAvg</td><td>59.57</td><td>65.0236.02 40.75</td></tr><tr><td> + FEDDECORR</td><td>61.42 65.98</td><td>41.68 44.77</td></tr><tr><td rowspan="2">10</td><td>FedAvg</td><td>59.87 66.46</td><td>535.02 39.30</td></tr><tr><td> + FEDDECORR</td><td>61.53</td><td>67.12 40.29 43.86</td></tr><tr><td rowspan="2">20</td><td>FedAvg</td><td>58.50 </td><td>66.3731.23 37.23</td></tr><tr><td>+ FEDDECORR</td><td>60.65</td><td>66.86 35.44 42.04</td></tr></table>
+
+FEDDECORR consistently improves over the baselines across different choices of local epochs $E$ .
+
+# 5.6 ADDITIONAL EMPIRICAL ANALYSES
+
+We present more empirical analyses in Appendix C. These include comparing FEDDECORR with other baselines (Appendix C.4) and other decorrelation methods (Appendix C.2), experiments on other model architectures (Appendix C.3) and another type of data heterogeneity (Appendix C.5), and discussing the computational advantage of FEDDECORR (Appendix C.1).
+
+# 6 CONCLUSION
+
+In this work, we study representations of trained models under federated learning in which the data held by clients are heterogeneous. Through extensive empirical observations and theoretical analyses, we show that stronger data heterogeneity results in more severe dimensional collapse for both global and local representations. Motivated by this, we propose FEDDECORR, a novel method to mitigate dimensional collapse during local training, thus improving federated learning under the heterogeneous data setting. Extensive experiments on benchmark datasets show that FEDDECORR yields consistent improvements over existing baseline methods.
+
+# ACKNOWLEDGEMENTS
+
+The authors would like to thank anonymous reviewers for the constructive feedback. Yujun Shi and Vincent Tan are supported by Singapore Ministry of Education Tier 1 grants (Grant Number: A-0009042-01-00, A-8000189-01-00, A-8000980-00-00) and a Singapore National Research Foundation (NRF) Fellowship (Grant Number: A-0005077-01-00). Jian Liang is supported by National Natural Science Foundation of China (Grant No. 62276256) and Beijing Nova Program under Grant Z211100002121108.
+
+# REPRODUCIBILITY STATEMENT
+
+All source code has been released at https://github.com/bytedance/FedDecorr. Pseudo-code of FEDDECORR is provided in Appendix G. We introduced all the implementation details of baselines and our method in Sec. 5.1. In addition, the proofs of Theorem 1 and Proposition 1 are provided in Appendix A and Appendix B, respectively. All assumptions are stated and discussed in the proof.
+
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+
+# A PROOF OF THEOREM 1 IN MAIN PAPER
+
+# A.1 NOTATIONS REVISITED
+
+Here, for the reader’s convenience, we summarize the notations used in both the main text and this appendix.
+
+<table><tr><td>Notation N</td><td>Explanation Number of training data points.</td></tr><tr><td>C X y Y Wi(t) I(t) 01,k ul,k Vl,k 0k Uk Vk Nc μc ec</td><td>Total number of classes. The collection of the N training samples, X ∈ Rdin × N. The collection of one hot labels of the N training samples,y ∈ RC×N. The collection of model output softmax vectors givenall N input data,y ∈ RCN The i-th layer weight matrix at the t-th optimization step. The product of the weight matrices of the first L layers: II(t) = WL(t) ...Wi(t). The k-th singular value of Wt . The k-th left singular vector of Wt. The k-th right singular vector of Wt.</td></tr></table>
+
+# A.2 TWO LEMMAS
+
+Here, we elaborate two useful lemmas from Arora et al. (2019; 2018).
+
+The first lemma is adopted from Arora et al. (2019):
+
+Lemma 1. Assuming the weight matrix $W$ evolves under gradient descent dynamics with infinitesimally small learning rate, the $k$ -th singular value of this matrix (denoted as $\sigma _ { k }$ ) evolves as
+
+$$
+\begin{array} { r } { \dot { { \boldsymbol \sigma } } _ { k } ( t ) = ( { \mathbf u } _ { k } ( t ) ) ^ { \top } \dot { W } ( t ) { \mathbf v } _ { k } ( t ) , } \end{array}
+$$
+
+where ${ \bf u } _ { k } ( t )$ and $\mathbf { v } _ { k } ( t )$ are the $k$ -th left and right singular vectors of $W ( t )$ , respectively.
+
+Proof. By performing an SVD on $W ( t )$ , we have $W ( t ) = U ( t ) S ( t ) V ( t ) ^ { \top }$ . Therefore, by the chain rule in differention, we have:
+
+$$
+\dot { W } ( t ) = \dot { U } ( t ) S ( t ) V ( t ) ^ { \top } + U ( t ) \dot { S } ( t ) V ( t ) ^ { \top } + U ( t ) S ( t ) \dot { V } ( t ) ^ { \top } .
+$$
+
+Next, for both sides of the above equation, we left multiply $U ( t ) ^ { \top }$ and right multiply $V ( t )$
+
+$$
+U ( t ) ^ { \top } \dot { W } ( t ) V ( t ) = U ( t ) ^ { \top } \dot { U } ( t ) S ( t ) + \dot { S } ( t ) + S ( t ) ( \dot { V } ( t ) ) ^ { \top } V ( t ) .
+$$
+
+ince $S ( t )$ is a diagonal matrix, we consider the $k$ -th diagonal entry of $S ( t )$ , namely $\sigma _ { k } ( t )$
+
+$$
+\begin{array} { r } { ( \mathbf { u } _ { k } ( t ) ) ^ { \top } \dot { W } ( t ) \mathbf { v } _ { k } ( t ) = ( \mathbf { u } _ { k } ( t ) ) ^ { \top } \dot { \mathbf { u } } _ { k } ( t ) \sigma _ { k } ( t ) + \dot { \sigma } _ { k } ( t ) + \sigma _ { k } ( t ) ( \mathbf { v } _ { k } ( t ) ) ^ { \top } \dot { \mathbf { v } } _ { k } ( t ) . } \end{array}
+$$
+
+Since ${ \bf u } _ { k } ( t )$ and $\mathbf { v } _ { k } ( t )$ are unit vectors and are evolving in time with infinitesimal rate, we have $( { \mathbf { u } } _ { k } ( t ) ) ^ { \top } \dot { { \mathbf { u } } } _ { k } ( t ) = 0$ and $( { \bf v } _ { k } ( t ) ) ^ { \top } \dot { \bf v } _ { k } ( t ) = 0$ . Next, Eqn. (13) can be simplified as
+
+$$
+\begin{array} { r } { \dot { { \boldsymbol \sigma } } _ { k } ( t ) = ( { \mathbf u } _ { k } ( t ) ) ^ { \top } \dot { W } ( t ) { \mathbf v } _ { k } ( t ) . } \end{array}
+$$
+
+The proof is thus complete.
+
+The second lemma is adopted from Arora et al. (2018).
+
+Lemma 2. Given $L$ consecutive linear layers in a neural network characterized by weight matrices $W _ { 1 } , W _ { 2 } , \ldots , W _ { L }$ . We denote $\Pi = W _ { L } W _ { L - 1 } \ldots W _ { 1 }$ . We further denote $W _ { j } ( t )$ as weight matrix $W _ { j }$ after the $t$ -th gradient descent optimization step. Correspondingly, the initialization of $W _ { j }$ is $W _ { j } ( 0 )$ . Assuming we have $W _ { j } ( 0 ) ( W _ { j } ( 0 ) ) ^ { \top } = ( W _ { j + 1 } ( 0 ) ) ^ { \top } W _ { j + 1 } ( 0 )$ for any $j \in [ L - 1 ]$ at initialization. Then, under the gradient descent dynamics, $\Pi ( t )$ satisfies
+
+$$
+\dot { \Pi } ( t ) = - \sum _ { j = 1 } ^ { L } \left[ \Pi ( t ) \Pi ( t ) ^ { \top } \right] ^ { \frac { L - j } { L } } \frac { \partial \ell ( \Pi ( t ) ) } { \partial \Pi } \left[ \Pi ( t ) ^ { \top } \Pi ( t ) \right] ^ { \frac { j - 1 } { L } } ,
+$$
+
+where $[ \cdot ] ^ { \frac { L - j } { L } }$ and $[ \cdot ] ^ { \frac { j - 1 } { L } }$ are fractional power operators defined over positive semi-definite matrices.
+
+Proof. Here, we first define some additional notation. Given any square matrices (or possibly scalar) $A _ { 1 } , A _ { 2 } , \ldots , A _ { m }$ , we denote $\mathrm { d i a g } ( A _ { 1 } , A _ { 2 } , \ldots , A _ { m } )$ to be the block diagonal matrix
+
+$$
+\mathrm { d i a g } ( A _ { 1 } , A _ { 2 } , \ldots , A _ { m } ) = \left[ \begin{array} { c c c c } { { A _ { 1 } } } & { { 0 } } & { { 0 } } & { { 0 } } \\ { { 0 } } & { { A _ { 2 } } } & { { 0 } } & { { 0 } } \\ { { 0 } } & { { 0 } } & { { \ddots } } & { { 0 } } \\ { { 0 } } & { { 0 } } & { { 0 } } & { { A _ { m } } } \end{array} \right] .
+$$
+
+Here, we first consider dynamics of an arbitrary $W _ { j }$ where $j \in [ L - 1 ]$ . By the chain rule, we have
+
+$$
+\begin{array} { l } { \dot { W } _ { j } ( t ) = - \frac { \partial \ell ( W _ { 1 } ( t ) , \dots , W _ { L + 1 } ( t ) ) } { \partial W _ { j } ( t ) } } \\ { \quad \qquad = - ( W _ { j + 1 } ( t ) ^ { \top } \dots W _ { L } ( t ) ^ { \top } ) \frac { \partial \ell ( \Pi ( t ) ) } { \partial \Pi } ( W _ { 1 } ( t ) ^ { \top } \dots W _ { j - 1 } ( t ) ^ { \top } ) . } \end{array}
+$$
+
+Given Eqn. (16), we right multiply $\dot { W } _ { j } ( t )$ by $( W _ { j } ( t ) ) ^ { \top }$ and we left multiply $\dot { W } _ { j + 1 } ( t )$ by $( W _ { j + 1 } ( t ) ) ^ { \top }$ , which yields
+
+$$
+\dot { W } _ { j } ( t ) ( W _ { j } ( t ) ) ^ { \top } = ( W _ { j + 1 } ( t ) ) ^ { \top } \dot { W } _ { j + 1 } ( t ) .
+$$
+
+Applying the same trick on $W _ { j } ( t ) ^ { \top }$ and $W _ { j + 1 } ( t ) ^ { \top }$ yields
+
+$$
+W _ { j } ( t ) ( \dot { W } _ { j } ( t ) ) ^ { \top } = ( \dot { W } _ { j + 1 } ( t ) ) ^ { \top } W _ { j + 1 } ( t ) .
+$$
+
+Adding Eqns. (17) and (18) on both sides yields
+
+$$
+\dot { W } _ { j } ( t ) ( W _ { j } ( t ) ) ^ { \top } + W _ { j } ( t ) ( \dot { W } _ { j } ( t ) ) ^ { \top } = W _ { j + 1 } ( t ) ^ { \top } \dot { W } _ { j + 1 } ( t ) + ( \dot { W } _ { j + 1 } ( t ) ) ^ { \top } W _ { j + 1 } ( t ) .
+$$
+
+Next, by the chain rule for differentiation, Eqn. (19) directly implies that
+
+$$
+\frac { \mathrm { d } ( W _ { j } ( t ) W _ { j } ( t ) ^ { \top } ) } { \mathrm { d } t } = \frac { \mathrm { d } ( W _ { j + 1 } ( t ) ^ { \top } W _ { j + 1 } ( t ) ) } { \mathrm { d } t } .
+$$
+
+Since we have assumed that $W _ { j } ( 0 ) W _ { j } ( 0 ) ^ { \top } = W _ { j + 1 } ( 0 ) ^ { \top } W _ { j + 1 } ( 0 )$ , we can conclude that
+
+$$
+W _ { j } ( t ) W _ { j } ( t ) ^ { \top } = W _ { j + 1 } ( t ) ^ { \top } W _ { j + 1 } ( t ) .
+$$
+
+Next, we apply an SVD on $W _ { j } ( t )$ and $W _ { j + 1 } ( t )$ in Eqn. (21). This yields
+
+$$
+U _ { j } ( t ) S _ { j } ( t ) S _ { j } ^ { \top } ( t ) U _ { j } ^ { \top } ( t ) = V _ { j + 1 } ( t ) S _ { j + 1 } ^ { \top } ( t ) S _ { j + 1 } ( t ) V _ { j + 1 } ^ { \top } ( t ) .
+$$
+
+Based on Eqn. (22) and given the uniqueness property of SVD, we know:
+
+$$
+S _ { j } ( t ) S _ { j } ( t ) ^ { \top } = S _ { j + 1 } ^ { \top } ( t ) S _ { j + 1 } ( t ) = \mathrm { d i a g } ( \rho _ { 1 } I _ { d _ { 1 } } , \rho _ { 2 } I _ { d _ { 2 } } , \dots , \rho _ { m } I _ { d _ { m } } ) ,
+$$
+
+where ${ \sqrt { \rho _ { 1 } } } , \ldots , { \sqrt { \rho _ { m } } }$ represent the $m$ distinct singular values satisfying $\rho _ { 1 } > \rho _ { 2 } > . . . > \rho _ { m } \geq 0$ , and $I _ { d _ { r } }$ for any $r \in [ m ]$ are identity matrix of size $d _ { r } \times d _ { r }$ . Since Eqn. (23) holds for any $j$ , we know by induction that the set of values of $\rho$ ’s is the same across all layers $j \in [ L ]$ . In addition, there exist orthogonal matrices $O _ { j , r } \in \mathbb { R } ^ { d _ { r } \times \dot { d } _ { r } }$ for any $r \in [ m ]$ such that
+
+$$
+U _ { j } ( t ) = V _ { j + 1 } ( t ) \mathrm { d i a g } ( O _ { j , 1 } , O _ { j , 2 } , \dots , O _ { j , m } ) .
+$$
+
+Given Eqns. (24), next, we study $W _ { j + 1 } ( t ) W _ { j } ( t ) W _ { j } ^ { \top } ( t ) W _ { j + 1 } ^ { \top } ( t )$ for any $j \in [ N - 1 ]$ :
+
+$$
+\begin{array} { r l } & { W _ { j + 1 } ( t ) W _ { j } ( t ) W _ { j } ^ { \top } ( t ) W _ { j + 1 } ^ { \top } ( t ) } \\ & { = U _ { j + 1 } S _ { j + 1 } V _ { j + 1 } ^ { \top } U _ { j } S _ { j } S _ { j } ^ { \top } U _ { j } ^ { \top } V _ { j + 1 } S _ { j + 1 } ^ { \top } U _ { j + 1 } ^ { \top } } \\ & { = U _ { j + 1 } S _ { j + 1 } \mathrm { d i a g } ( O _ { j , 1 } , O _ { j , 2 } , \ldots , O _ { j , m } ) S _ { j } S _ { j } ^ { \top } \mathrm { d i a g } ( O _ { j , 1 } ^ { \top } , O _ { j , 2 } ^ { \top } , \ldots , O _ { j , m } ^ { \top } ) S _ { j + 1 } ^ { \top } U _ { j + 1 } ^ { \top } } \\ & { \qquad \quad \mathrm { ( p l u g g i n g } } \\ & { = U _ { j + 1 } S _ { j + 1 } S _ { j } S _ { j } ^ { \top } S _ { j + 1 } ^ { \top } U _ { j + 1 } ^ { \top } \qquad ( S _ { j } \mathrm { c o m m u t e s ~ w i t h ~ d i a g } ( O _ { j , 1 } , O _ { j , 2 } , \ldots , O _ { j , m } ) ) } \\ & { = U _ { j + 1 } \mathrm { d i a g } ( \rho _ { 1 } ^ { 2 } I _ { d _ { 1 } } , \rho _ { 2 } ^ { 2 } I _ { d _ { 2 } } , \ldots , \rho _ { m } ^ { 2 } I _ { d _ { m } } ) U _ { j + 1 } ^ { \top } . } \end{array}
+$$
+
+Similarly, it holds that
+
+$$
+\begin{array} { r } { W _ { j } ^ { \top } ( t ) W _ { j + 1 } ^ { \top } ( t ) W _ { j + 1 } ( t ) W _ { j } ( t ) = V _ { j } \mathrm { d i a g } ( \rho _ { 1 } ^ { 2 } I _ { d _ { 1 } } , \rho _ { 2 } ^ { 2 } I _ { d _ { 2 } } , \dots , \rho _ { m } ^ { 2 } I _ { d _ { m } } ) V _ { j } ^ { \top } . } \end{array}
+$$
+
+Next, by induction and Eqns. (25),
+
+$$
+\begin{array} { r l } & { W _ { L } ( t ) \ldots W _ { j } ( t ) W _ { j } ( t ) ^ { \top } \ldots W _ { L } ( t ) ^ { \top } } \\ & { \qquad = U _ { L } \mathrm { d i a g } ( \rho _ { 1 } ^ { L - j + 1 } I _ { d _ { 1 } } , \rho _ { 2 } ^ { L - j + 1 } I _ { d _ { 2 } } , \ldots , \rho _ { m } ^ { L - j + 1 } I _ { d _ { m } } ) U _ { L } ^ { \top } , } \end{array}
+$$
+
+by induction and Eqns. (26), it holds that
+
+$$
+\begin{array} { r } { W _ { 1 } ^ { \top } ( t ) \ldots W _ { j } ^ { \top } ( t ) W _ { j } ( t ) \ldots W _ { 1 } ( t ) = V _ { 1 } \mathrm { d i a g } ( \rho _ { 1 } ^ { j } I _ { d _ { 1 } } , \rho _ { 2 } ^ { j } I _ { d _ { 2 } } , \ldots , \rho _ { m } ^ { j } I _ { d _ { m } } ) V _ { 1 } ^ { \top } . } \end{array}
+$$
+
+From Eqns. (27), we know that for any $j \in [ L - 1 ]$ ,
+
+$$
+\begin{array} { r l } & { \Pi ( t ) \Pi ( t ) ^ { \top } = W _ { L } ( t ) \ldots W _ { 1 } ( t ) W _ { 1 } ( t ) ^ { \top } \ldots W _ { L } ( t ) ^ { \top } } \\ & { \qquad = U _ { L } \mathrm { d i a g } ( \rho _ { 1 } ^ { L } I _ { d _ { 1 } } , \rho _ { 2 } ^ { L } I _ { d _ { 2 } } , \ldots , \rho _ { m } ^ { L } I _ { d _ { m } } ) U _ { L } ^ { \top } } \\ & { \qquad = \left[ U _ { L } \mathrm { d i a g } ( \rho _ { 1 } ^ { L - j } I _ { d _ { 1 } } , \rho _ { 2 } ^ { L - j } I _ { d _ { 2 } } , \ldots , \rho _ { m } ^ { L - j } I _ { d _ { m } } ) U _ { L } ^ { \top } \right] ^ { \frac { L } { L - j } } } \\ & { \qquad = \left[ W _ { L } ( t ) \ldots W _ { j + 1 } ( t ) W _ { j + 1 } ( t ) ^ { \top } \ldots W _ { L } ( t ) ^ { \top } \right] ^ { \frac { L } { L - j } } . } \end{array}
+$$
+
+Similarly, from Eqn. (28), we know that for any $2 \leq j \leq L - 1$ ,
+
+$$
+\begin{array} { r l } & { \Pi ( t ) ^ { \top } \Pi ( t ) = W _ { 1 } ( t ) ^ { \top } \cdot \cdot \cdot W _ { L } ( t ) ^ { \top } W _ { L } ( t ) \cdot \cdot \cdot W _ { 1 } ( t ) } \\ & { \qquad = V _ { 1 } \mathrm { d i a g } ( \rho _ { 1 } ^ { L } I _ { d _ { 1 } } , \rho _ { 2 } ^ { L } I _ { d _ { 2 } } , \cdot \cdot \cdot , \rho _ { m } ^ { L } I _ { d _ { m } } ) V _ { 1 } ^ { \top } } \\ & { \qquad = \Big [ V _ { 1 } \mathrm { d i a g } ( \rho _ { 1 } ^ { j - 1 } I _ { d _ { 1 } } , \rho _ { 2 } ^ { j - 1 } I _ { d _ { 2 } } , \dots , \rho _ { m } ^ { j - 1 } I _ { d _ { m } } ) V _ { 1 } ^ { \top } \Big ] ^ { \frac { L } { j - 1 } } } \\ & { \qquad = \big [ W _ { 1 } ^ { \top } \cdot \dots W _ { j - 1 } ^ { \top } W _ { j - 1 } ( t ) \cdot \dots W _ { 1 } ( t ) \big ] ^ { \frac { L } { j - 1 } } . } \end{array}
+$$
+
+With everything derived above, we now study the dynamics of $\Pi ( t )$ as follows
+
+$$
+\begin{array} { l } { { \displaystyle \dot { \mathrm { I I } } ( t ) = \sum _ { j = 1 } ^ { L } \left[ W _ { L } ( t ) \dots W _ { j + 1 } ( t ) \right] ( \dot { W } _ { j } ( t ) ) \left[ W _ { j - 1 } ( t ) \dots W _ { 1 } ( t ) \right] \quad \mathrm { ( d i f f e r e n t i a l ~ c h a i n ~ r u l e ) } } } \\ { { \displaystyle = - \sum _ { j = 1 } ^ { L } \left[ W _ { L } ( t ) \dots W _ { j + 1 } ( t ) W _ { j + 1 } ( t ) ^ { \top } \dots W _ { L } ( t ) ^ { \top } \right] } } \\ { { \displaystyle \qquad \times \frac { \partial \ell ( [ \mathbf { I I } ( t ) ) } { \partial \Pi } \left[ W _ { 1 } ^ { \top } ( t ) \dots W _ { j - 1 } ^ { \top } ( t ) W _ { j - 1 } ( t ) \dots W _ { 1 } ( t ) \right] \quad \mathrm { ( p l u g g i n g - i n ~ ( 1 6 ) ) } } } \\ { { \displaystyle = - \sum _ { j = 1 } ^ { L } \left[ \Pi ( t ) \Pi ( t ) ^ { \top } \right] ^ { \frac { L - j } { \mathrm { x } _ { \mathrm { x } } } } \frac { \partial \ell ( [ \mathbf { I } ( t ) ) } { \partial \Pi } \left[ \Pi ( t ) ^ { \top } \Pi ( t ) \right] ^ { \frac { j - 1 } { \mathrm { x } _ { \mathrm { x } } } } \qquad \mathrm { ( p l u g g g i n g - i n ~ ( 2 9 ) ~ a n d ~ ( 2 6 ) ~ } } } \end{array}
+$$
+
+This completes the proof.
+
+![](images/323acb0638f29f5b7d6c06a6bb5198d719953d1062a80c427e5170fed35bb967.jpg)  
+Figure 6: Alignment effects between the singular spaces of $W _ { L + 1 } ( t )$ and $\Pi ( t )$ . We train a 3-layer linear neural network on the MNIST dataset and visualize the models at 3, 5, 7, 9 training epochs, respectively. In each figure, the $k ^ { \prime }$ -th row and $k$ -th column pixel is the value of $\vert { \mathbf u } _ { k } ( t ) ^ { \top } { \mathbf v } _ { L + 1 , k ^ { \prime } } ( t ) \vert$ . Darker colors denote values close to 1 while lighter colors denote values close to 0. From the figures, we empirically observe that $| \mathbf { u } _ { k } ( t ) ^ { \top } \mathbf { v } _ { L + 1 , k ^ { \prime } } ( \bar { t } ) | = \mathbb { 1 } \{ k = k ^ { \prime } \}$ approximately holds.
+
+# A.3 ASSUMPTIONS
+
+Assumption 1. We assume that the initial values of the weight matrices satisfy $W _ { i + 1 } ^ { \top } ( 0 ) W _ { i + 1 } ( 0 ) =$ $W _ { i } ( 0 ) W _ { i } ^ { \top } ( 0 )$ for any $i \in [ L - 1 ]$ .
+
+Assumption 2. We assume $| \mathbf { u } _ { k } ( t ) ^ { \top } \mathbf { v } _ { L + 1 , k ^ { \prime } } ( t ) | = \mathbb { 1 } \{ k = k ^ { \prime } \}$ holds for all $t ,$ , where ${ \bf u } _ { k } ( t )$ is the $k$ -th left singular vector of $\Pi ( t )$ and $\mathbf { v } _ { L + 1 , k ^ { \prime } } ( t )$ is the $k ^ { \prime }$ -th right singular vector of $W _ { L + 1 } ( t )$ .
+
+Remark: For Assumption 1, it can be achieved in practice by proper random initialization. For Assumption 2, Ji & Telgarsky (2018) proved that under some assumptions, gradient descent optimization will drive consecutive layers of linear networks to satisfy it. We also provide empirical evidence in Fig. 6 to corroborate that this assumption approximately holds.
+
+# A.4 PROOF OF THEOREM 1 IN THE MAIN TEXT
+
+Theorem 1 (formally stated). Let $\sigma _ { k } ( t )$ for $k \in [ d ]$ be the $k$ -th largest singular value of $\Pi ( t )$ . Then, under Assumptions $^ { l }$ and 2, we have
+
+$$
+\begin{array} { r } { \dot { \sigma } _ { k } ( t ) = N L \left( \sigma _ { k } ( t ) \right) ^ { 2 - \frac { 2 } { L } } \sqrt { \left( \sigma _ { k } ( t ) \right) ^ { \frac { 2 } { L } } + M } \left( \mathbf { u } _ { L + 1 , k } ( t ) \right) ^ { \top } G ( t ) \mathbf { v } _ { k } ( t ) , } \end{array}
+$$
+
+where ${ \mathbf { u } } _ { L + 1 , k } ( t )$ is the $k$ -th left singular vector of $W _ { L + 1 } ( t )$ , $\mathbf { v } _ { k } ( t )$ is the $k$ -th right singular vector of $\Pi ( t )$ , $M$ is a constant, and $G ( t )$ is defined as
+
+$$
+G ( t ) = \sum _ { c = 1 } ^ { C } \mu _ { c } ( \mathbf { e } _ { c } - \bar { \gamma } _ { c } ( t ) ) \bar { X } _ { c } ^ { \top } .
+$$
+
+Proof. Recall that for $( L + 1 )$ -layer linear neural networks, given the $i$ -th training sample $X _ { i } \in \mathbb { R } ^ { d }$ , we have
+
+$$
+\gamma _ { i } ( t ) = \mathrm { s o f t m a x } ( W _ { L + 1 } ( t ) { \mathbf z } _ { i } ( t ) ) = \mathrm { s o f t m a x } ( W _ { L + 1 } ( t ) \Pi ( t ) X _ { i } ) ,
+$$
+
+and the loss is the standard cross-entropy loss defined as follows
+
+$$
+\ell ( \Pi ( t ) , W _ { L + 1 } ( t ) ) = \sum _ { i = 1 } ^ { N } - \mathbf { y } _ { i } ^ { \top } \log \gamma _ { i } ( t ) .
+$$
+
+By the chain rule, we can derive gradient of $\ell$ with respect to $W _ { L + 1 }$ and $\Pi$ , which are respectively,
+
+$$
+\frac { \partial \ell ( \Pi ( t ) , W _ { L + 1 } ( t ) ) } { \partial W _ { L + 1 } } = - ( \mathbf { y } - \boldsymbol { \gamma } ( t ) ) \boldsymbol { X } ^ { \top } \Pi ( t ) ^ { \top } ,
+$$
+
+and
+
+$$
+\frac { \partial \ell ( \Pi ( t ) , W _ { L + 1 } ( t ) ) } { \partial \Pi } = - W _ { L + 1 } ( t ) ^ { \top } ( \mathbf { y } - \boldsymbol { \gamma } ( t ) ) \boldsymbol { X } ^ { \top } .
+$$
+
+Next, under the gradient descent dynamics, the dynamics on $W _ { L + 1 }$ satisfies
+
+$$
+\dot { W } _ { L + 1 } ( t ) = - \frac { \partial \ell ( \Pi ( t ) , W _ { L + 1 } ( t ) ) } { \partial W _ { L + 1 } } = ( \mathbf { y } - \boldsymbol { \gamma } ( t ) ) \boldsymbol { X } ^ { \top } \Pi ( t ) ^ { \top } ,
+$$
+
+while the dynamics on $\Pi$ requires invoking Lemma 2, which allows us to write
+
+$$
+\begin{array} { r l } & { \dot { \Pi } ( t ) = - \displaystyle \sum _ { j = 1 } ^ { L } [ \Pi ( t ) \Pi ( t ) ^ { \top } ] ^ { \frac { L - j } { L } } \frac { \partial \ell ( \Pi ( t ) ) } { \partial \Pi } [ \Pi ( t ) ^ { \top } \Pi ( t ) ] ^ { \frac { j - 1 } { L } } } \\ & { \quad \quad = \displaystyle \sum _ { j = 1 } ^ { L } [ \Pi ( t ) \Pi ( t ) ^ { \top } ] ^ { \frac { L - j } { L } } W _ { L + 1 } ( t ) ^ { \top } ( \mathbf { y } - \boldsymbol { \gamma } ( t ) ) \boldsymbol { X } ^ { \top } [ \Pi ( t ) ^ { \top } \Pi ( t ) ] ^ { \frac { j - 1 } { L } } . } \end{array}
+$$
+
+Next, we invoke Lemma 1 on Eqn. (39) and Eqn. (38), respectively, yielding:
+
+$$
+\begin{array} { r l } { \dot { \sigma } _ { k } ( t ) = } & { ( \mathbf { u } _ { k } ( t ) ) ^ { \top } \dot { \Pi } ( t ) ( \mathbf { v } _ { k } ( t ) ) } \\ & { = \displaystyle \sum _ { j = 1 } ^ { L } \mathbf { u } _ { k } ( t ) ^ { \top } [ \Pi ( t ) \Pi ( t ) ^ { \top } ] ^ { \frac { L - i } { L } } W _ { L + 1 } ( t ) ^ { \top } ( \mathbf { y } - \gamma ( t ) ) X ^ { \top } [ \Pi ( t ) ^ { \top } \Pi ( t ) ] ^ { \frac { i - 1 } { L } } \mathbf { v } _ { k } ( t ) } \\ & { = L ( \sigma _ { k } ( t ) ) ^ { 2 - \frac { 2 } { L } } \mathbf { u } _ { k } ( t ) ^ { \top } W _ { L + 1 } ( t ) ^ { \top } ( \mathbf { y } - \gamma ( t ) ) X ^ { \top } \mathbf { v } _ { k } ( t ) } \\ & { \qquad \mathrm { ( S V D ~ o n ~ I ~ } ] } \\ & { = L ( \sigma _ { k } ( t ) ) ^ { 2 - \frac { 2 } { L } } \displaystyle \sum _ { k ^ { \prime } } \sigma _ { L + 1 , k ^ { \prime } } \mathbf { u } _ { k } ( t ) ^ { \top } \mathbf { v } _ { L + 1 , k ^ { \prime } } ( t ) ( \mathbf { u } _ { L + 1 , k ^ { \prime } } ( t ) ) ^ { \top } ( \mathbf { y } - \gamma ( t ) ) X ^ { \top } \mathbf { v } _ { k } ( t ) } \\ & { \qquad \mathrm { ( S V D ~ o n ~ } b } \\ & { = L ( \sigma _ { k } ( t ) ) ^ { 2 - \frac { 2 } { L } } \sigma _ { L + 1 , k } ( \mathbf { u } _ { L + 1 , k } ( t ) ) ^ { \top } ( \mathbf { y } - \gamma ( t ) ) X ^ { \top } \mathbf { v } _ { k } ( t ) \qquad \mathrm { ( A s s u m p i o n 2 ) } . } \end{array}
+$$
+
+and
+
+$$
+\begin{array} { r l } & { \dot { \boldsymbol \sigma } _ { L + 1 , k } ( t ) = \mathbf { u } _ { L + 1 , k } ( t ) ^ { \top } ( \mathbf { y } - \boldsymbol \gamma ( t ) ) \boldsymbol X ^ { \top } \boldsymbol \Pi ( t ) ^ { \top } { \mathbf { v } } _ { L + 1 , k } ( t ) } \\ & { \qquad = \displaystyle \sum _ { k ^ { \prime } } \sigma _ { k ^ { \prime } } \mathbf { u } _ { L + 1 , k } ( t ) ^ { \top } ( \mathbf { y } - \boldsymbol \gamma ( t ) ) \boldsymbol X ^ { \top } { \mathbf { v } } _ { k ^ { \prime } } ( t ) \mathbf { u } _ { k ^ { \prime } } ^ { \top } { \mathbf { v } } _ { L + 1 , k } ( t ) } \\ & { \qquad = \sigma _ { k } \mathbf { u } _ { L + 1 , k } ( t ) ^ { \top } ( \mathbf { y } - \boldsymbol \gamma ( t ) ) \boldsymbol X ^ { \top } { \mathbf { v } } _ { k } ( t ) \qquad \mathrm { ( A s s u m p t i o n ~ 2 ) } . } \end{array}
+$$
+
+Combining Eqns. (40) and (41), we have:
+
+$$
+\frac { 1 } { L } ( \dot { \sigma } _ { k } ( t ) ) ( \sigma _ { k } ( t ) ) ^ { \frac { 2 } { L } - 1 } = \sigma _ { L + 1 , k } ( t ) ( \dot { \sigma } _ { L + 1 , k } ( t ) ) .
+$$
+
+Next, apply integration on both sides, which yields
+
+$$
+( \sigma _ { L + 1 , k } ( t ) ) ^ { 2 } = ( \sigma _ { k } ( t ) ) ^ { \frac { 2 } { L } } + M ,
+$$
+
+where $M$ a constant.
+
+By Eqn. (43), Eqn. (40) can be rewritten as
+
+$$
+\begin{array} { r } { \dot { \sigma } _ { k } ( t ) = L ( \sigma _ { k } ( t ) ) ^ { 2 - \frac { 2 } { L } } \sqrt { ( \sigma _ { k } ( t ) ) ^ { \frac { 2 } { L } } + M } \left( \mathbf { u } _ { L + 1 , k } ( t ) \right) ^ { \top } ( \mathbf { y } - \boldsymbol { \gamma } ( t ) ) { X } ^ { \top } \mathbf { v } _ { k } ( t ) . } \end{array}
+$$
+
+Finally, notice that $( \mathbf y - \gamma ( t ) ) X ^ { \top }$ can be rewritten as
+
+$$
+( \mathbf { y } - \boldsymbol { \gamma } ( t ) ) \boldsymbol { X } ^ { \top } = \sum _ { i = 1 } ^ { N } ( \mathbf { y } - \boldsymbol { \gamma } _ { i } ( t ) ) \boldsymbol { X } _ { i } ^ { \top } = N \sum _ { c = 1 } ^ { C } \mu _ { c } ( \mathbf { e } _ { c } - \bar { \boldsymbol { \gamma } } _ { c } ( t ) ) \bar { X } _ { c } ^ { \top } .
+$$
+
+We further substitute Eqn. (45) into Eqn. (44) and obtain
+
+$$
+\begin{array} { r } { \dot { \sigma } _ { k } ( t ) = N L ( \sigma _ { k } ( t ) ) ^ { 2 - \frac { 2 } { L } } \sqrt { ( \sigma _ { k } ( t ) ) ^ { \frac { 2 } { L } } + M } \left( \mathbf { u } _ { L + 1 , k } ( t ) \right) ^ { \top } G ( t ) \mathbf { v } _ { k } ( t ) , } \end{array}
+$$
+
+where $G ( t )$ is defined as
+
+$$
+G ( t ) = \sum _ { c = 1 } ^ { C } \mu _ { c } ( \mathbf { e } _ { c } - \bar { \gamma } _ { c } ( t ) ) \bar { X } _ { c } ^ { \top } .
+$$
+
+This completes the proof.
+
+# B PROOF OF PROPOSITION 1 IN THE MAIN PAPER
+
+Proposition 1 (restated). For a $d$ -by- $d$ correlation matrix $K$ with singular values $( \lambda _ { 1 } , \ldots , \lambda _ { d } )$ , we have:
+
+$$
+\sum _ { i = 1 } ^ { d } \left( \lambda _ { i } - \frac { 1 } { d } \sum _ { j = 1 } ^ { d } \lambda _ { j } \right) ^ { 2 } = \| K \| _ { \mathrm { F } } ^ { 2 } - d .
+$$
+
+Proof. Given a $d$ -by- $d$ correlation matrix $K$ , since the diagonal entries of $K$ are all 1, we have
+
+$$
+\sum _ { i = 1 } ^ { d } \lambda _ { i } = \operatorname { t r } ( K ) = d .
+$$
+
+This is because for any symmetric positive definite matrix, the sum of all singular values equals the trace of the matrix.
+
+Next, for the left-hand side of Eqn. (7), we have:
+
+$$
+\begin{array} { l l } { \displaystyle \sum _ { i = 1 } ^ { d } \left( \lambda _ { i } - \frac { 1 } { d } \sum _ { j = 1 } ^ { d } \lambda _ { j } \right) ^ { 2 } = \sum _ { i = 1 } ^ { d } ( \lambda _ { i } - 1 ) ^ { 2 } } & { ( \mathrm { P l u g - i n ~ E q n . ~ } ( 4 9 ) ) } \\ { \displaystyle } & { = \sum _ { i = 1 } ^ { d } \lambda _ { i } ^ { 2 } - 2 \sum _ { i = 1 } ^ { d } \lambda _ { i } + d } \\ { \displaystyle } & { = \sum _ { i = 1 } ^ { d } \lambda _ { i } ^ { 2 } - d } & { ( \mathrm { P l u g - i n ~ E q n . ~ } ( 4 9 ) ) . } \end{array}
+$$
+
+Next, for the right-hand side of Eqn. (7), we have:
+
+$$
+\begin{array} { r l } & { \| K \| _ { \mathrm { F } } ^ { 2 } - d = \mathrm { t r } ( K ^ { \top } K ) - d } \\ & { \qquad = \mathrm { t r } ( U S V ^ { \top } V S ^ { \top } U ^ { \top } ) - d } \\ & { \qquad = \mathrm { t r } ( U S S ^ { \top } U ^ { \top } ) - d } \\ & { \qquad = \displaystyle \sum _ { i = 1 } ^ { n } \lambda _ { i } ^ { 2 } - d . } \end{array}
+$$
+
+Therefore, we have shown that the left-hand side of Eqn. (7) equals its right-hand side.
+
+# C ADDITIONAL EMPIRICAL ANALYSES
+
+# C.1 COMPUTATIONAL EFFICIENCY
+
+We demonstrate FEDDECORR’s advantage vis- $\grave { \mathbf { a } }$ -vis some of its competitors in terms of its computational efficiency. We compare FEDDECORR with some other methods that also apply additional regularization terms during local training such as FedProx and MOON. We partition CIFAR10, CIFAR100 and TinyImageNet into 10 clients with $\alpha = 0 . 5$ and report the total computation times required for one round of training for FedAvg, FedProx, MOON, and FEDDECORR . Results are shown in Tab. 5. All results are produced with a NVIDIA Tesla V100 GPU. We see that FEDDECORR incurs a negligible computation overhead on top of the na¨ıve FedAvg, while FedProx and MOON introduce about $0 . 5 \sim 1 \times$ additional computation cost. The advantage of FEDDECORR in terms of efficiency is mainly because it only involves calculating the Frobenius norm of a matrix which is extremely cheap. Indeed this regularization operates on the output representation vectors of the model, without requiring computing parameter-wise regularization like FedProx nor extra forward passes like MOON.
+
+<table><tr><td></td><td>CIFAR10</td><td>CIFAR100</td><td>TinyImageNet</td></tr><tr><td>FedAvg</td><td>6.7</td><td>6.9</td><td>25.4</td></tr><tr><td>FedProx</td><td>12.1</td><td>12.3</td><td>33.2</td></tr><tr><td>MOON</td><td>12.2</td><td>12.7</td><td>38.1</td></tr><tr><td>FEDDECORR</td><td>6.9</td><td>7.1</td><td>25.7</td></tr></table>
+
+Table 5: Comparison of computation times. We report the total computation times (in minutes) for one round of training on the three datasets for FedAvg, FedProx, MOON, and FEDDECORR. Here, FEDDECORR stands for applying FEDDECORR to FedAvg.
+
+# C.2 COMPARISON WITH OTHER DECORRELATION METHODS
+
+Some decorrelation regularizations such as DeCov (Cogswell et al., 2015) and StructuredDeCov (Xiong et al., 2016) were proposed to improve the generalization capabilities in standard classification tasks. Both these methods operate directly on the covariance matrix of the representations instead of the correlation matrix like our proposed method—FEDDECORR. To compare our FEDDECORR with the existing decorrelation methods, we follow the same procedure as in FEDDECORR and apply DeCov and Structured-DeCov during local training. Our experiments are based on TinyImageNet and FedAvg. TinyImageNet is partitioned into 10 clients according to various $\alpha$ ’s. Results are shown in Tab. 6. Surprisingly, we see that unlike our FEDDECORR which steadily improves the baseline, adding DeCov or Structured-DeCov both degrade the performance in federated learning. We conjecture that this is because directly regularizing the covariance matrix is highly unstable, leading to undesired modification on the representations. This experiment shows that our design of regularization of the correlation matrix instead of the covariance matrix is of paramount importance.
+
+Table 6: Comparison with other decorrelation methods. Based on FedAvg and the TinyImageNet dataset, we use different decorrelation regularizers in local training.   
+
+<table><tr><td></td><td>|FedAvg</td><td>DeCov</td><td>St.-Decov</td><td>FEDDECORR</td></tr><tr><td>a= 0.05</td><td>35.02</td><td>32.88</td><td>32.04</td><td>40.29</td></tr><tr><td>α = 0.1</td><td>39.30</td><td>37.29</td><td>37.74</td><td>43.86</td></tr><tr><td>α = 0.5</td><td>46.92</td><td>46.29</td><td>45.85</td><td>50.01</td></tr></table>
+
+# C.3 EXPERIMENTS ON OTHER MODEL ARCHITECTURES
+
+In this section, we demonstrate the effectiveness of our method across different model architectures. Here, besides the MobileNetV2 used in the main paper, we also experiment on ResNet18 and ResNet32. Note that ResNet18 is the wider ResNet whose representation dimension is 512 and ResNet32 is the narrower ResNet whose representation dimension is 64. The coefficient of the FedDecorr objective is set to be 0.1 as suggested to be a good universal value of $\beta$ in the paper. The heterogeneity parameter $\alpha$ is set to be 0.05 and we use the CIFAR10 dataset. Our results are shown in Tab. 7. As can be seen, FedDecorr yields consistent improvements across different neural network architectures. One interesting phenomenon is that the improvements brought about by FedDecorr are much larger on wider networks (e.g., MobileNetV2, ResNet18) than on narrower ones (e.g. ResNet32). We conjecture this is because the dimension of the ambient space of wider networks are clearly higher than that of shallower networks. Therefore, relatively speaking, the dimensional collapse caused by data heterogeneity will be more severe for wider networks.
+
+Table 7: Effectiveness of FEDDECORR on other model architectures.   
+
+<table><tr><td></td><td colspan="3">|MobileNetV2 ResNet18 ResNet32</td></tr><tr><td>FedAvg</td><td>64.85</td><td>71.51</td><td>65.76</td></tr><tr><td>+ FEDDECORR</td><td>73.06</td><td>76.54</td><td>67.21</td></tr></table>
+
+Table 8: CIFAR10/100 Experiments. We run experiments under various degrees of heterogeneity $( \alpha \in \{ 0 . 0 5 , 0 . 1 , 0 . 5 , \infty \} )$ and report the test accuracy $( \% )$ . All results are (re)produced by us and are averaged over 3 runs (mean $\pm$ std). Bold font highlights the highest accuracy in each column. We add results of Scaffold and FedNova comparing to Tab. 1 in the main paper.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="4">CIFAR10</td><td colspan="4">CIFAR100</td></tr><tr><td></td><td>α=0.050.1</td><td>0.5</td><td>8</td><td>0.05</td><td>0.1</td><td>0.5</td><td>8</td></tr><tr><td>Scaffold</td><td></td><td></td><td></td><td></td><td>51.99±2.54 74.36±3.10 87.05±0.3989.77±0.24 54.51±0.26 61.42±0.54 68.37±0.44 70.97±0.04</td><td></td><td></td><td></td></tr><tr><td>FedNova</td><td></td><td></td><td></td><td></td><td>63.07±1.59 79.98±1.56 90.23±0.41 92.39±0.18 60.22±0.33 66.43±0.26 71.79±0.17 74.47±0.13</td><td></td><td></td><td></td></tr><tr><td>FedAvg</td><td></td><td></td><td></td><td></td><td>64.85±2.01 76.28±1.22 89.84±0.13 92.39±0.26 59.87±0.25 66.46±0.16 71.69±0.15 74.54±0.15</td><td></td><td></td><td></td></tr><tr><td>FedProx</td><td></td><td></td><td></td><td></td><td>+ FEDDEC0RR 73.06±0.81 80.60±0.91 89.84±0.05 92.19±0.10 61.53±0.11 67.12±0.09 71.91±0.04 73.87±0.18</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>64.11±0.84 76.10±0.40 89.57±0.04 92.38±0.09 60.02±0.46 66.41±0.27 71.78±0.19 74.34±0.03</td><td>+ FEDDEC0RR 71.38±0.81 81.74±0.34 89.96±0.26 92.14±0.20 61.33±0.19 67.00±0.46 71.64±0.10 74.15±0.06</td><td></td><td></td></tr><tr><td>FedAvgM</td><td></td><td></td><td></td><td></td><td>71.34±0.71 77.51±0.58 88.39±0.1791.35±0.15 59.64±0.20 66.36±0.1471.17±0.2 74.20±0.16</td><td></td><td></td><td></td></tr><tr><td>MOON</td><td>+ FEDDEC0RR 73.60±0.82 79.21±0.15 88.70±0.26 91.33±0.13 61.48±0.27 66.60±0.11 71.26±0.21 73.86±0.25</td><td></td><td></td><td></td><td></td><td>68.79±0.69 78.70±0.6690.08±0.10 92.62±0.17 56.79±0.17 65.48±0.29 71.81±0.14 74.30±0.12</td><td></td><td></td></tr></table>
+
+<table><tr><td rowspan="2">Method</td><td>TinyImageNet</td></tr><tr><td>α =0.050.1 0.5</td></tr><tr><td>Scaffold FedNova</td><td>8 35.16±0.77 37.87±0.78 44.24±0.14 44.88±0.29 35.28±0.04 39.73±0.07 47.05±0.42 49.57±0.09</td></tr><tr><td>FedAvg</td><td>35.02±0.46 39.30±0.23 46.92±0.25 49.33±0.19</td></tr><tr><td></td><td>+ FEDDEC0RR 40.29±0.18 43.86±0.50 50.01±0.27 52.63±0.26</td></tr><tr><td>FedProx</td><td>35.20±0.30 39.66±0.4347.16±0.07 49.76±0.36</td></tr><tr><td></td><td>+ FEDDEC0RR 40.63±0.05 44.19±0.14 50.26±0.27 52.37±0.36</td></tr><tr><td>FedAvgM</td><td>34.81±0.09 39.72±0.1147.11±0.04 49.67±0.25</td></tr><tr><td></td><td>+ FEDDEC0RR 39.97±0.23 43.95±0.26 50.14±0.11 52.05 ±0.37</td></tr><tr><td>MOON</td><td></td></tr><tr><td></td><td>35.23±0.26 40.53±0.28 47.25±0.66 50.48±0.57</td></tr><tr><td></td><td>+ FEDDEC0RR 40.40±0.24 44.20±0.22 50.81±0.51 53.01±0.45</td></tr></table>
+
+Table 9: TinyImageNet Experiments. We run with $\alpha \in \{ 0 . 0 5 , 0 . 1 , 0 . 5 , \infty \}$ and report the test accuracies $( \% )$ . All results are (re)produced by us and are averaged over 3 runs (mean $\pm$ std is reported). Bold font highlights the highest accuracy in each column. We add results of Scaffold and FedNova comparing to Tab. 2 in the main paper.
+
+# C.4 COMPARISON WITH OTHER FEDERATED LEARNING BASELINES
+
+In this section, we compare FEDDECORR with two other baselines, namely Scaffold (Karimireddy et al., 2020) and FedNova (Wang et al., 2020b). We use the same experimental setups as in the main paper to implement these two baselines. Results on CIFAR10/100 and TinyImageNet are shown in Tab. 8 and Tab. 9, respectively. As shown in the tables, across various datasets and degrees of heterogeneity, adding FEDDECORR on top of a baseline method can outperform the baselines when there is some heterogeneity across the agents, i.e., $\alpha < \infty$ .
+
+# C.5 EXPERIMENTS ON ANOTHER TYPE OF HETEROGENEITY
+
+In this section, we run experiments under another type of data heterogeneity. Specifically, we follow McMahan et al. (2017) and split the CIFAR10 dataset across different clients such that each client only has a fixed number of classes $C$ (e.g., $C = 2$ indicates each client only has data of two classes). We split the data across 10 clients and choose $C$ to be 2 and 3. Results are shown in Tab. 10. As can
+
+![](images/bfd5cdd999ac6f66e406cbcd381a7e83905bcce32499590652ea7972fe33f585.jpg)  
+Figure 7: Data heterogeneity causes similar dimensional collapse on other federated learning methods such as FedAvgM (Hsu et al., 2019), FedProx (Li et al., 2020), and MOON (Li et al., 2021b). The $\mathbf { X }$ -axis $( k )$ is the index of singular values.
+
+![](images/1e3a1f14e624f31497abb41b5bc9f7e7aea1230bc8d90c14592c9053c3efef84.jpg)  
+Figure 8: Data heterogeneity causes similar dimensional collapse on other model architectures during federated learning. The $\mathbf { X }$ -axis $( k )$ is the index of singular values.
+
+be observed, under this different heterogeneity scenario, FEDDECORR also yields noticeable and consistent improvements.   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>|C=2 C=3</td></tr><tr><td rowspan=1 colspan=1>FedAvg</td><td rowspan=1 colspan=1>45.61 67.53</td></tr><tr><td rowspan=1 colspan=1>+ FEDDECORR</td><td rowspan=1 colspan=1>47.6374.51</td></tr></table>
+
+Table 10: FEDDECORR yields noticeable and consistent improvements under another type of data heterogeneity.
+
+# D ADDITIONAL VISUALIZATIONS ON GLOBAL MODELS
+
+In this section, we provide additional visualizations on global models with different federated learning methods, model architectures, and datasets. Through our extensive experimental results, we demonstrate that dimensional collapse is a general problem under heterogeneous data in federated learning.
+
+# D.1 VISUALIZATION ON GLOBAL MODELS OF OTHER FEDERATED LEARNING METHODS
+
+In the main text, we have shown that global models produced by FedAvg (McMahan et al., 2017) suffer stronger dimensional collapse with increasing data heterogeneity. To further show such dimensional collapse phenomenon is a general problem in federated learning, we visualized global models produced by other federated learning methods such as FedAvg with server momentum (Hsu et al., 2019), FedProx (Li et al., 2020), and MOON (Li et al., 2021b). Specifically, we follow the same procedure as in the main text and plot the singular values of covariance matrices of representations. Results are shown in Fig. 7. From the figure, one can see that all these three other methods also demonstrated the similar hazard of dimensional collapse as in FedAvg.
+
+# D.2 VISUALIZATION ON GLOBAL MODELS OF OTHER MODEL ARCHITECTURES
+
+In the main text, we have shown the dimensional collapse on global models caused by data heterogeneity with MobileNetV2. In this section, we perform the similar visualization based on other model architectures such as ResNet32 and ResNet18. Note that ResNet32 is a narrower ResNet whose representation dimension is 64 and ResNet18 is a wider ResNet whose representation dimension is 64. We visualize the top 50 singular values for ResNet32 and the top 100 singular values for ResNet18. Results are shown in Fig. 8. From the figure, one can observe that heterogeneous data also lead to dimensional collapse on ResNet32 and ResNet18.
+
+![](images/68663f7a210a4ed41865bc862bacaa1913c33f17e48a3a3ce3035b45ee4379cb.jpg)  
+Figure 9: Data heterogeneity causes similar dimensional collapse on other datasets during federated learning. The $\mathbf { X }$ -axis $( k )$ is the index of singular values.
+
+# D.3 VISUALIZATION ON GLOBAL MODELS OF OTHER DATASETS
+
+In the main text, we use the CIFAR100 dataset for our visualizations. In this section, we perform similar visualizations with other datasets such as CIFAR10 and TinyImageNet. Results are shown in Fig. 9. From the figure, one can also observe that dimensional collapse results from data heterogeneity.
+
+# E VISUALIZATION ON OTHER LOCAL CLIENTS
+
+In the main text Fig. 2(b), under the four different degrees of data heterogeneity (i.e., $\alpha \in$ $\{ 0 . 0 1 , 0 . 0 5 , 0 . 2 5 , \infty \} )$ ), we compare representations of local models of client 1 and empirically show how data heterogeneity affects representations produced by the local models. In this section, to further corroborate our conclusion, we follow the same procedure and visualize singular values of the covariance matrix of representations produced by local models trained on the rest of the 9 clients under the same $\alpha$ ’s. Results are shown in Fig. 10. From the results, we can obtain the similar observations as in Fig. 2(b) of the main text, namely that stronger data heterogeneity causes more severe dimensional collapse for local models.
+
+# F HYPERPARAMETERS OF OTHER FEDERATED LEARNING METHODS
+
+The regularization coefficient of FedProx (Li et al., 2020) $\mu$ is tuned across $\{ 1 0 ^ { - 4 } , 1 \overset { \smile } { 0 } ^ { - 3 } , 1 0 ^ { - 2 } , 1 0 ^ { - 1 } \}$ and is selected to be $\begin{array} { r } { \dot { \mu } { } ~ = ~ 1 0 ^ { - 3 } } \end{array}$ ; the regularization coefficient of MOON (Li et al., 2021b) $\mu$ is tuned across $\{ 0 . 1 , 1 . 0 , 5 . 0 , 1 0 . 0 \}$ and is selected to be $\mu = 1 . 0$ ; the server momentum of FedAvgM (Hsu et al., 2019) $\rho$ is tuned across $\{ 0 . 1 , 0 . 5 , 0 . 9 \}$ and is selected to be $\rho = 0 . 5$ .
+
+# G PSEUDO-CODE OF FEDDECORR
+
+Here, we provide a pytorch-style pseudo-code for FEDDECORR in Alg. 1. All FEDDECORR-specific components are highlight in blue. As indicated in the pseudocode, the only additional operation of FEDDECORR is in adding a regularization term $L _ { \mathrm { F e d D e c o r r } } ( w , X )$ defined in Eqn. (8). This shows that FEDDECORR is an extremely convenient plug-and-play federated learning method.
+
+# H STABILITY OF FEDDECORR REGULARIZATION LOSS
+
+In this section, we first split CIFAR10 into 10 clients with $\alpha = 0 . 5$ . Then, we plot how FedDecorr loss evolve within 10 local epochs for all the 10 clients in Fig. 11. All training configurations are the same as in the main paper. From the results, one can observe that the optimization process of FedDecorr loss is stable.
+
+![](images/6f1ba99f72524a916c45b65cd2cb2d9231656632895b065b90e6a6e48bfdac68.jpg)  
+Figure 10: Heterogeneous local training data cause dimensional collapse. For each of the clients, given the four models trained under different degrees of heterogeneity, we plot the singular values of covariance matrix of representations in descending orders (the results of client 1 are shown in main text Fig. 2(b)). Representations are computed over the CIFAR100 test set. The $x$ -axis $( k )$ is the index of singular values and the $y$ -axis is the logarithm of the singular values.
+
+![](images/1a3240b5479ee3d199cd9088170457ca430ceb2639b46676fb93756a2b56466c.jpg)  
+Figure 11: How FedDecorr loss evolve within 10 local epochs.
+
+# Algorithm 1 PyTorch-style Pseudocode for FEDDECORR. (Blue highlights FEDDECORR-specific code)
+
+def FedDecorrLoss(z): # N: batch size # d: representation dimension # z: a batch of representation, with shape (N, d) N,d = z.shape # z-score normalization z = (z - z.mean(0)) / z.std(0) # estimate correlation matrix corr mat $\equiv$ 1/N\*torch.matmul(z.t(), z) # calculate FedDecorr loss loss fed decorr $=$ (corr mat.pow(2)).mean() return loss fed decorr
+
+def LocalTraining(train_loader, local_epochs, beta): for e in range(local_epochs): for data, targets in train_loader: # forward propagation. # given the batch of data, compute batch representations z and loss loss, z = . loss $+ =$ beta\*FedDecorr(z) # back propagation and update local model parameters ...
+
+# def GlobalAggregation():
+
+# receiving models from each clients # aggregating local models with certain schemes # sending aggregated models back to clients
+
+# def main():
+
+# n_comm_round: number of communication rounds.   
+# train_loader: data loader of training data.   
+# n_local_epochs: number of local trainig epochs on each client.   
+# beta: coefficient of the FedDecorr regularization.   
+for comm in range(n_comm_round): LocalTraining(train_loader, n_local_epochs, beta) GlobalAggregation()

parse/dev/LV8OmADmoOe/LV8OmADmoOe.md

@@ -0,0 +1,223 @@
+# IMPROVING THE TRANSFERABILITY OF ADVERSARIAL ATTACKS THROUGH EXPERIENCED PRECISE NESTEROV MOMENTUM
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Deep Neural Networks are vulnerable to adversarial attacks, which makes adversarial attacks serve as a method to evaluate the robustness of DNNs. However, adversarial attacks have high white-box attack success rates but poor transferability, making black-box attacks impracticable in the real world. Momentumbased attacks were proposed to accelerate optimization to improve transferability. Nevertheless, conventional momentum-based attacks accelerate optimization inefficiently during early iterations since the initial value of momentum is zero, which leads to unsatisfactory transferability. Therefore, we propose Experienced Momentum (EM), which is the pre-trained momentum. Initializing the momentum to EM can help accelerate optimization during the early iterations. Moreover, the pre-update of conventional Nesterov momentum based attacks is rough, prompting us to propose Precise Nesterov momentum (PN). PN refines the preupdate by considering the gradient of the current data point. Finally, we integrate EM with PN as Experienced Precise Nesterov momentum (EPN) to further improve transferability. Extensive experiments against normally trained and defense models demonstrate that our EPN is more effective than conventional momentum in the improvement of transferability. Specifically, the attack success rates of our EPN-based attacks are ${ \sim } 1 1 . 9 \%$ and ${ \sim } 1 3 . 1 \%$ higher than conventional momentum-based attacks on average against normally trained and defense models, respectively.
+
+# 1 INTRODUCTION
+
+Deep neural networks (DNNs) (Krizhevsky et al., 2012; Szegedy et al., 2015; He et al., 2016; Ioffe & Szegedy, 2015) have been widely applied in computer vision, e.g., autonomous driving (Franchi et al., 2022; Hao et al., 2019; Cococcioni et al., 2018), facial recognition (Chrysos et al., 2020; Ghenescu et al., 2018), and medical image analysis (Akselrod-Ballin et al., 2016; Ding et al., 2017; Liu et al., 2019). However, Szegedy et al. (2013) found that applying certain imperceptible perturbations to images can make DNNs misclassify, and they refer to such perturbed images as adversarial examples $( A E s )$ . Adversarial examples pose a huge threat to the security of DNNs, which attaches extensive attention from researchers.
+
+Adversarial attacks can be categorized into white-box attacks and black-box attacks. Typically, iterative gradient-based (Kurakin et al., 2016; Madry et al., 2017) and optimization-based attacks (Carlini & Wagner, 2017) have high white-box but low black-box attack success rates, which means that such two attacks are impracticable in the real world. Transferability, which means adversarial examples crafted on the source model remain effective on other models, makes black-box attacks feasible. Furthermore, iterative gradient-based attacks have the advantages of low computational cost and fast generation speed, thus improving the transferability of iterative gradient-based attacks has become a hotspot in the field of adversarial attacks.
+
+Many methods have been proposed to improve the transferability of iterative gradient-based attacks. These methods can be classified into three branches: improving optimization algorithms, input transformations, and disrupting feature space. For example, MI-FGSM (Dong et al., 2018), NI-FGSM (Lin et al., 2019), and VM(N)I-FGSM (Wang & He, 2021) improve gradient ascent (or descent) algorithm to escape from saddle points and poor local extrema to improve transferability; DIM (Xie et al., 2019), TIM (Dong et al., 2019), and SIM (Lin et al., 2019) craft adversarial examples on a set of models derived by input transformations to prevent overfitting and improve transferability; NRDM (Naseer et al., 2018), FDA (Ganeshan et al., 2019), and FIA (Wang et al., 2021) disrupt deep features of DNNs to craft highly transferable adversarial examples.
+
+![](images/43ff633c9b654c39becc3a37d6d52763e1946d4a2c40cf69095e67ae5381f60e.jpg)  
+Figure 1: Comparison of conventional Polyak momentum and experienced Polyak momentum. Adversarial examples are crafted on the source model (Inception-v3) and used to attack the target model (VGG16). Our Experienced MI-FGSM (EMI-FGSM), which integrates EM into MI-FGSM, causes misclassification with higher loss and confidence than MI-FGSM, thus EMI-FGSM can mislead the attention of the target model better than MI-FGSM.
+
+Those mentioned above adversarial attacks mostly adopt the momentum (Polyak, 1964; Nesterov, 1983) to accelerate optimization. However, such momentum-based adversarial attacks (e.g., M(N)IFGSM, VM(N)I-FGSM, and FIA) have the problem of initializing the momentum to zero, resulting in inefficient acceleration due to momentum accumulating few gradients during the first few iterations. Therefore, we propose Experienced Momentum (EM), which is the pre-trained momentum. Before the iterations, the momentum is initialized to EM instead of zero, leading to better acceleration in the first few iterations. The comparison of conventional Polyak momentum (Polyak, 1964) and experienced Polyak momentum is shown in Fig. 1. To prevent overfitting on the source model, we train EM on a set of models derived by Random Channels Swapping (RCS). EM and RCS are detailed in Sec. 3.1.
+
+Furthermore, adversarial attacks (e.g., NI-FGSM and VNI-FGSM) based on Nesterov momentum (i.e., Nesterov Accelerated Gradient, NAG (Nesterov, 1983)) have the disadvantage that the preupdate is rough. Specifically, during each iteration, the parameters are first pre-updated along the momentum to obtain the pre-update point, which is an estimation of the next position. Then the preupdate is modified by the gradient of the pre-update point. Such looking-ahead property of Nesterov momentum makes parameters escape from saddle points and poor local extrema easier and faster, resulting in improving transferability. However, pre-updating only along the momentum is rough, and the estimation of the next position of the parameters is imprecise. Therefore, we propose Precise Nesterov momentum (PN), which not only retains the looking-ahead property but also refines the pre-update by adopting the gradient of the current data point. To improve transferability further, we integrate EM with PN as Experienced Precise Nesterov momentum (EPN). PN and EPN are detailed in Sec. 3.2.
+
+Overall, we make the following contributions:
+
+• We propose Experienced Momentum (EM), which is trained on a set of models derived by Random Channels Swapping (RCS). Initializing the momentum to EM can accelerate optimization effectively during the early iterations to improve transferability.
+
+• We propose Precise Nesterov momentum (PN), which adopts the gradient of the current data point to refine the pre-update to escape from saddle points and poor local extrema easier and faster. We also integrate EM with PN as Experienced Precise Nesterov momentum (EPN) to improve transferability further. • Extensive experiments on normally trained and defense models demonstrate that our EPN is more effective than conventional momentum for improving transferability.
+
+# 2 RELATED WORK
+
+# 2.1 TRANSFERABLE ADVERSARIAL ATTACKS
+
+Since adversarial examples were discovered by Szegedy et al. (2013), many methods (Goodfellow et al., 2014; Kurakin et al., 2016; Carlini & Wagner, 2017) have been proposed to craft adversarial examples to demonstrate the vulnerability of DNNs. We focus on the transferability of iterative gradient-based attacks and review related works from three branches: improving optimization algorithms, input transformations, and disrupting feature space.
+
+Improving optimization algorithms. Dong et al. (2018) integrated Polyak momentum (Polyak, 1964) into I-FGSM (Kurakin et al., 2016) to accelerate gradient ascent (or descent) to improve transferability. Inspired by the fact that Nesterov momentum (Nesterov, 1983) is superior to Polyak momentum, Lin et al. (2019) integrated Nesterov momentum into I-FGSM to improve transferability further. Wang & He (2021) used the gradient variance of the previous iteration to tune the current gradient to stabilize the update direction and escape from saddle points and poor local extrema.
+
+Input transformations. The nature of input transformations is crafting adversarial examples on a set of derived models to prevent overfitting. Xie et al. (2019) performed random resizing and padding with probability $p$ to derive models. Dong et al. (2019) convolved the gradient to approximate translating input. Lin et al. (2019) scaled the input with the scale factor $1 / 2 ^ { i }$ to derive a set of models.
+
+Disrupting feature space. Naseer et al. (2018) created maximum distortions in the feature space to craft adversarial examples, based on the intuition that features of DNNs are highly generalizable. Ganeshan et al. (2019) highly corrupted deep features by disrupting features at each layer of DNNs to improve transferability. Wang et al. (2021) described feature importance with the aggregate gradient and disrupted important object-aware features to achieve stronger transferability.
+
+# 2.2 ADVERSARIAL TRAINING
+
+Adversarial training as a common defense measure can validate transferability further. Adversarial training increases robustness by adding adversarial examples to the training data. Goodfellow et al. (2014) showed that adversarially trained models are more robust. However, Kurakin et al. (2016) pointed out that adversarial training is not robust to iterative attacks. Moreover, Tramer et al. (2017) \` showed that adversarially trained models are still vulnerable to simple white-box and black-box attacks. Therefore, they proposed ensemble adversarial training adding adversarial examples crafted from other models to the training data.
+
+# 3 METHODOLOGY
+
+Given a target model $f ^ { \prime } ( x ; { \pmb \theta } ^ { \prime } )$ , where $_ { \textbf { \em x } }$ is an input, and $\pmb { \theta } ^ { \prime }$ is the parameters of $f ^ { \prime }$ . Let $J ( \cdot , y )$ be a loss function, where $y$ is the ground-truth label of the input $_ { \textbf { \em x } }$ . A non-targeted adversarial example $\pmb { x } ^ { a d v }$ satisfies $f ^ { \prime } ( { \pmb x } ; { \hat { \pmb \theta } } ^ { \prime } ) \neq { \bar { f ^ { \prime } } } ( { \pmb x } ^ { a d v } ; { \pmb \theta } ^ { \prime } )$ under the constraint of $| | { \pmb x } ^ { a d v } \bar { - } { \pmb x } | | _ { p } \le \epsilon .$ , where $| | \cdot | | _ { p }$ denotes the $L ^ { p }$ norm, and $p$ is generally $0 , 1 , 2 , \infty$ . In this paper, we focus on $p = \infty$ . Note that our methods can be generalized to $p = 0 , 1 , 2$ easily. Crafting non-targeted adversarial examples can be described as solving the following optimization problem:
+
+$$
+\underset { \pmb { x } ^ { a d v } } { \arg \operatorname* { m a x } } J ( f ^ { \prime } ( \pmb { x } ^ { a d v } ; \pmb { \theta } ^ { \prime } ) , y ) , \quad \mathrm { s . t . } | | \pmb { x } ^ { a d v } - \pmb { x } | | _ { p } \leq \epsilon .
+$$
+
+In this paper, we focus on non-targeted attacks. Our proposed methods can be easily transformed into targeted attacks by replacing the above objective function with $- J ( f ^ { \prime } ( x ^ { a d v } ; \pmb { \theta } ^ { \prime } ) , \mathbf { \bar { \psi } } ^ { * } )$ , where $y ^ { * }$ denotes the target label.
+
+![](images/87cfd090bbc4c049cc375cc349efb5a739a1a34309be3d1c013402ce1a4aa858.jpg)  
+Figure 2: Illustration of training EM during each iteration.
+
+Many gradient-based methods have been proposed to solve Eq. 1, e.g., FGSM (Goodfellow et al., 2014), I-FGSM (Kurakin et al., 2016), and PGD (Madry et al., 2017). However, the parameters $\pmb { \theta } ^ { \prime }$ of the target model $f ^ { \prime }$ is inaccessible for black-box attacks, resulting in the inability to solve Eq. 1 directly. Therefore, the target model $f ^ { \prime }$ is usually replaced with a model $f$ (i.e., the source model) with accessible parameters $\pmb \theta$ , and then adversarial examples are crafted on the source model $f$ to attack the target model $f ^ { \prime }$ . To achieve effective black-box attacks, adversarial examples crafted on the source model $f$ are required to have high transferability. Therefore, we propose Experienced Momentum (EM, detailed in Sec. 3.1) and Precise Nesterov momentum (PN, detailed in Sec. 3.2) to improve transferability. EM and PN can be naturally combined as Experienced Precise Nesterov momentum (EPN, detailed in Sec. 3.2) to further improve transferability.
+
+# 3.1 EXPERIENCED MOMENTUM
+
+Momentum-based attacks initialize momentum to zero, resulting in inefficient acceleration during the first few iterations. Therefore, we propose Experienced Momentum (EM), which is the pretrained momentum. Setting the initial momentum to EM can accelerate the optimization during the early iterations. To prevent overfitting of EM and improve transferability further, we train EM on a set of models derived by Random Channels Swapping (RCS). RCS derives models by randomly swapping the channels of the input image, which is equivalent to randomly swapping the “block” dimensions of the original model, leading to various decision boundaries of derived models. Therefore, training EM on derived models can prevent overfitting. The specific procedure for training EM is as follows.
+
+First of all, we perform RCS on the input image $_ { \textbf { \em x } }$ . Specifically, we denote the input image $_ { \textbf { \em x } }$ as an RGB triplet $( R , G , B )$ , and then the input image $_ { \textbf { \em x } }$ through RCS can be denoted as $\bar { S } ( { \pmb x } )$ , where $S ( { \pmb x } ) \in \{ ( R , G , B ) , ( R , B , G ) , ( G , R , B ) , ( G , B , R ) , ( B , R , G ) , ( B , G , R ) \} .$ , $S ( \cdot )$ denotes RCS. Secondly, $S ( { \pmb x } )$ is fed into the source model $f$ to derive $f ( S ( \cdot ) , y )$ . Thirdly, we pre-perturb the input image $_ { \textbf { \em x } }$ on the derived model $f ( S ( \cdot ) , y )$ by iterative gradient-based attacks to prevent overfitting. As shown in Fig. 2, we accumulate gradients to training EM during each iteration. Finally, we follow the above procedure repeatedly to make the EM more generalizable. After training EM, we set the initial value of momentum to EM to accelerate the early iterations.
+
+# 3.2 PRECISE NESTEROV MOMENTUM
+
+Nesterov momentum based Attack (e.g., NI-FGSM (Lin et al., 2019) and VNI-FGSM (Wang & He, 2021)) only pre-update along the momentum roughly, resulting in the imprecision of the pre-update point that is the estimate of the next iterative position. Against this disadvantage, we propose Precise Nesterov momentum (PN), which considers the gradient of the current data point in the pre-update to make the pre-update precise. Specifically, during each iteration, the pre-update is performed along the gradient of the current data point and momentum successively to obtain the pre-update point, and then we use the gradient of the pre-update point to modify the pre-update. We integrate PN into I-FGSM as PNI-FGSM. The $t$ -th iteration of PNI-FGSM can be formalized as follows:
+
+$$
+\widetilde { \pmb x } _ { t } ^ { a d v } = \pmb x _ { t } ^ { a d v } + \alpha \cdot \left[ \frac { \nabla _ { \pmb x _ { t } ^ { a d v } } J ( f ( \pmb x _ { t } ^ { a d v } ; \pmb \theta ) , y ) } { | | \nabla _ { \pmb x _ { t } ^ { a d v } } J ( f ( \pmb x _ { t } ^ { a d v } ; \pmb \theta ) , y ) | | _ { 1 } } + \mu \cdot { \pmb g } _ { t - 1 } \right] ,
+$$
+
+Input : A source model $f$ with parameters $\pmb \theta$ and a loss function $J$ . An original image $_ { \textbf { \em x } }$ with ground-truth label $y$ . Input : The maximum perturbation $\epsilon$ , the number of iterations $T$ , and the decay factor $\mu$ . Input : The epochs of pretraining epochs. Output: An adversarial example $\bar { \boldsymbol { x } } ^ { a \bar { d } v }$ . 1 $\alpha  \epsilon / T$ ; $\pmb { g } ^ { e x p }  \mathbf { 0 }$ ; 2 for $n \gets 1$ to epochs do 3 $\hat { \pmb { x } } _ { 1 } ^ { a d v }  { \pmb { x } }$ ; 4 for $t \gets 1$ to $T$ do 5 $\begin{array} { r l } & { \widetilde { x } _ { t } ^ { a d v } \gets \hat { x } _ { t } ^ { a d v } + \alpha \cdot \left[ \frac { \nabla _ { \hat { x } _ { t } ^ { a d v } } J ( f ( S ( \hat { x } _ { t } ^ { a d v } ) ; \pmb \theta ) , y ) } { | | \nabla _ { \hat { x } _ { t } ^ { a d v } } J ( f ( S ( \hat { x } _ { t } ^ { a d v } ) ; \pmb \theta ) , y ) | | _ { 1 } } + \mu \cdot g ^ { e x p } \right] ; } \\ & { g ^ { e x p } \gets \frac { \nabla _ { \hat { x } _ { t } ^ { a d v } } J ( f ( S ( \hat { x } _ { t } ^ { a d v } ) ; \pmb \theta ) , y ) } { | | \nabla _ { \hat { x } _ { t } ^ { a d v } } J ( f ( S ( \hat { x } _ { t } ^ { a d v } ) ; \pmb \theta ) , y ) | | _ { 1 } } + \mu \cdot g ^ { e x p } + \frac { \nabla _ { \widetilde { x } _ { t } ^ { a d v } } J ( f ( \widetilde { x } _ { t } ^ { a d v } ; \pmb \theta ) , y ) } { | | \nabla _ { \widetilde { x } _ { t } ^ { a d v } } J ( f ( \widetilde { x } _ { t } ^ { a d v } ; \pmb \theta ) , y ) | | _ { 1 } } ; } \\ & { \hat { x } _ { t + 1 } ^ { a d v } \gets \mathrm { C l i p } _ { ( \pmb { x } , \epsilon ) } \left\{ \hat { x } _ { t } ^ { a d v } + \alpha \cdot \mathrm { s i g n } ( g ^ { e x p } ) \right\} ; } \end{array}$ 6 7 8 end 9 end 10 ${ \pmb x } _ { 1 } ^ { a d v }  { \pmb x } ; { \pmb g } _ { 0 }  { \pmb g } ^ { e x p }$ ; 11 for $t \gets 1$ to $T$ do 12 Update $\mathbf { \nabla } _ { \mathbf { \boldsymbol { g } } _ { t } }$ and $\pmb { x } _ { t + 1 } ^ { a d v }$ by Eq. 2, 3, 4; 13 end 14 return ${ \pmb x } ^ { a d v }  { \pmb x } _ { T + 1 } ^ { a d v }$ .
+
+Algorithm 1: Experienced Precise Nesterov momentum I-FGSM (EPNI-FGSM)   
+Table 1: The abbreviations used in the paper.   
+
+<table><tr><td>Abbreviation</td><td>Explanation</td></tr><tr><td></td><td></td></tr><tr><td>D(T)I-MI-FGSM</td><td>the combination of D(T)IM and MI-FGSM</td></tr><tr><td>SI-NI-FGSM</td><td>the combination of SIMand NI-FGSM</td></tr><tr><td>D(T,S)I-EPNI-FGSM</td><td>the combination of D(T,S)IM and EPNI-FGSM</td></tr><tr><td>VT-M(N)I-FGSM</td><td>i.e., VM(N)I-FGSM</td></tr><tr><td>VT-EPNI-FGSM</td><td>the combination of Variance Tuning (VT)(Wang &amp; He,2021) and EPNI-FGSM</td></tr><tr><td>FI-MI-FGSM FI-EPNI-FGSM</td><td>i.e., FIA</td></tr><tr><td></td><td>the combination of Feature Importance-aware (FI) (Wang et al.,2021) and EPNI-FGSM</td></tr></table>
+
+$$
+g _ { t } = \frac { \nabla _ { x _ { t } ^ { a d v } } J ( f ( x _ { t } ^ { a d v } ; \pmb { \theta } ) , y ) } { | | \nabla _ { x _ { t } ^ { a d v } } J ( f ( x _ { t } ^ { a d v } ; \pmb { \theta } ) , y ) | | _ { 1 } } + \mu \cdot g _ { t - 1 } + \frac { \nabla _ { \widetilde { x } _ { t } ^ { a d v } } J ( f ( \widetilde { x } _ { t } ^ { a d v } ; \pmb { \theta } ) , y ) } { | | \nabla _ { \widetilde { x } _ { t } ^ { a d v } } J ( f ( \widetilde { x } _ { t } ^ { a d v } ; \pmb { \theta } ) , y ) | | _ { 1 } } ,
+$$
+
+$$
+\begin{array} { r } { \pmb { x } _ { t + 1 } ^ { a d v } = \mathrm { C l i p } _ { ( \pmb { x } , \epsilon ) } \left\{ \pmb { x } _ { t } ^ { a d v } + \alpha \cdot \mathrm { s i g n } ( \pmb { g } _ { t } ) \right\} , } \end{array}
+$$
+
+where $\mathbf { \nabla } _ { \mathbf { \boldsymbol { g } } _ { t } }$ denotes the momentum, ${ \bf { \mathit { g } } } _ { 0 } = { \bf { 0 } }$ , and $\mu$ denotes the decay factor.
+
+We combine EM and PN as Experienced Precise Nesterov momentum (EPN) to further improve transferability. The algorithm of EPNI-FGSM, which integrates EPN into I-FGSM, is summarized in Algorithm 1. Particularly, if $\frac { \nabla _ { \widetilde { \pmb { x } } _ { t } ^ { a d v } } J ( f ( \widetilde { \pmb { x } } _ { t } ^ { a d v } ; \pmb { \theta } ) , y ) } { | | \nabla _ { \widetilde { \pmb { x } } _ { t } ^ { a d v } } J ( f ( \widetilde { \pmb { x } } _ { t } ^ { a d v } ; \pmb { \theta } ) , y ) | | _ { 1 } } = \mathbf { 0 }$ , EPNI-FGSM degrades to Experienced MI-FGSM (EMI-FGSM). If $\frac { \nabla _ { \hat { \pmb { x } } _ { t } ^ { a d v } } J ( f ( S ( \hat { \pmb { x } } _ { t } ^ { a d v } ) ; \pmb { \theta } ) , y ) } { | | \nabla _ { \hat { \pmb { x } } _ { t } ^ { a d v } } J ( f ( S ( \hat { \pmb { x } } _ { t } ^ { a d v } ) ; \pmb { \theta } ) , y ) | | _ { 1 } } = \mathbf { 0 }$ , EPNI-FGSM degrades to Experienced NI-FGSM (ENI-FGSM). If $e p o c h s = 0$ , EPNI-FGSM degrades to PNI-FGSM.
+
+# 4 EXPERIMENTS
+
+We conduct extensive experiments on normally trained and defense models to validate that our EPN is more efficient than conventional momentum. We first present the experimental settings in Sec. 4.1. Then, we report the results for attacking normally trained and defense models in Sec. 4.2 and
+
+Table 2: The attack success rates $( \% )$ of adversarial examples crafted on source models against normally trained target models. “\*” indicates the model being white-box attacked. “Avg” means the average attack success rate.   
+
+<table><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="6">Iv3</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>69.2 56.7</td><td>67.9</td><td>68.2</td><td>66.2</td><td>63.0</td><td>61.4</td><td>61.2</td><td></td></tr><tr><td>TI-MI-FGSM SI-NI-FGSM DI-EPNI-FGSM(Ours) TI-EPNI-FGSM(Ours)</td><td>41.8 77.4 88.7 70.6 90.2</td><td>99.7* 100.0* 100.0* 100.0* 100.0*</td><td>37.9 72.7 87.3 68.8 87.1</td><td>31.5 69.8 86.0 59.6</td><td>46.4 67.6 78.5 69.3</td><td>47.6 67.3 81.3 69.8</td><td>44.9 68.0 83.4 68.9</td><td>41.9 61.5 80.2 65.5 76.4</td><td>38.6 62.1 80.5 63.3</td><td>54.9 70.0 80.4 73.5</td><td>72.8 83.8 76.5</td><td>56.2 73.8 83.8 78.5</td><td>58.3 71.9 84.2 77.9</td><td>47.9 70.7 86.1 74.7</td><td>46.5 68.4 84.9 72.9</td><td>43.5 64.8 82.9 70.8</td><td>42.8 65.9 83.4 71.5</td><td>66.0 49.2 70.9 84.4 72.5</td></tr><tr><td>SI-EPNI-FGSM(Ours) VT-MI-FGSM VT-NI-FGSM</td><td>70.5 76.2</td><td>100.0* 100.0*</td><td>69.7 76.2</td><td>85.1 64.9 70.5</td><td>80.1 61.7 65.6</td><td>81.2 63.6 68.4</td><td>81.7 65.7 71.2</td><td>60.0 65.2</td><td>78.2 58.6 64.1</td><td>80.6 65.0 70.5</td><td>83.1 69.2 73.4</td><td>84.4 68.7 73.4</td><td>82.6 66.7 73.2</td><td>86.7 65.8 71.2</td><td>83.8 63.8 71.3</td><td>80.9 65.0 69.2</td><td>81.7 62.6 68.2</td><td>83.8 67.1 72.2</td></tr><tr><td>VT-EPNI-FGSM(Ours) FI-MI-FGSM FI-EPNI-FGSM(Ours)</td><td>84.8 85.8 90.0</td><td>100.0* 97.1* 97.4*</td><td>83.3 85.4 88.0</td><td>78.1 81.8 84.8</td><td>77.0 79.4 81.0</td><td>78.8 80.1 82.4</td><td>79.4 79.5 84.3</td><td>75.8 76.3 79.6</td><td>73.3 74.5 78.5</td><td>80.2 81.0 85.5</td><td>83.3 82.7 87.0</td><td>82.0 82.7 85.9</td><td>81.5 83.2 86.3</td><td>79.7 81.3 84.0</td><td>80.1 78.8 83.5</td><td>78.3 77.2 80.5</td><td>78.5 77.8 81.7</td><td>80.8 81.4</td></tr><tr><td>MI-FGSM NI-FGSM EPNI-FGSM(Ours)</td><td>46.4 49.0 68.1</td><td>44.0 45.6 65.3</td><td>45.4 48.5 68.4</td><td>99.6* 100.0* 100.0*</td><td>40.0 40.6 55.2</td><td>39.2 38.7 53.8</td><td>41.3 42.1 55.5</td><td>33.9 34.9 48.4</td><td>32.7 32.9 47.9</td><td>49.9 53.5 63.8</td><td>53.2 54.3 67.8</td><td>51.4 54.6 66.6</td><td>50.6 54.3 65.3</td><td>38.6 37.5 54.7</td><td>35.3 37.5 50.7</td><td>35.0 33.8 49.0</td><td>32.7 33.2 48.1</td><td>84.7 45.2 46.5 60.5</td></tr><tr><td>DI-MI-FGSM TI-MI-FGSM SI-NI-FGSM DI-EPNI-FGSM(Ours) TI-EPNI-FGSM(Ours) SI-EPNI-FGSM (Ours) VT-MI-FGSM</td><td>63.2 41.5 74.7 82.8 65.1</td><td>62.3 44.5 81.7 83.2 68.2</td><td>65.4 42.0 75.1 84.1 67.6</td><td>98.4* 97.9* 99.3* 100.0* 99.9* 100.0*</td><td>54.2 47.1 65.4 70.7 66.6</td><td>54.2 47.6 65.8 72.3 69.5</td><td>55.7 47.3 69.7 74.4 68.9</td><td>48.6 42.9 65.4 67.5 65.2</td><td>51.0 41.8 64.7 71.1 63.2</td><td>59.3 55.9 68.9 74.6 73.8</td><td>64.4 56.7 71.6 79.5 75.0</td><td>62.8 55.5 71.8 78.2 73.6</td><td>64.7 53.9 73.0 79.6 74.2</td><td>57.0 49.2 72.1 75.0 72.9</td><td>52.8 45.3 68.7 73.0 68.4</td><td>51.8 46.9 67.3 69.6 69.1</td><td>52.6 41.9 68.0 70.9 65.4</td><td>59.9 50.5 72.0 76.9 71.0</td></tr><tr><td></td><td>VT-NI-FGSM VT-EPNI-FGSM(Ours)</td><td>90.9 63.2 66.3 81.0</td><td>93.1 66.6 71.0 82.1</td><td>88.6 68.8 99.6* 73.1 99.8*</td><td>79.1 57.8 59.3</td><td>81.3 57.8 59.3</td><td>85.5 60.7 62.3 75.4</td><td>79.6 53.6 56.9 70.0</td><td>82.9 54.9 56.9 70.6</td><td>83.2 61.7 66.8 77.3</td><td>85.5 63.5 68.2 80.4</td><td>86.0 64.4 68.2 79.9</td><td>86.1 64.0 68.1</td><td>86.3 59.8 60.9</td><td>83.6 56.4 58.0 73.4</td><td>81.5 54.2 57.2</td><td>84.6 56.1 58.5</td><td>85.8 62.5 65.3</td></tr><tr><td></td><td>FI-MI-FGSM FI-EPNI-FGSM (Ours)</td><td>76.3 78.0</td><td>76.0 77.3</td><td>83.1 76.3 89.7* 78.0 91.0*</td><td>100.0* 71.7 67.5 69.8</td><td>72.4 68.0 69.9</td><td>69.7 72.2</td><td>66.8 68.9</td><td>65.5 67.4</td><td>72.5 73.5</td><td>73.7 74.0</td><td>73.4 74.5</td><td>78.7 73.5 73.6</td><td>77.3 70.7 71.7</td><td>68.3 69.0</td><td>72.0 66.8 66.9</td><td>71.9 67.0 67.2</td><td>77.5 71.9 73.1</td></tr><tr><td rowspan="5">R152</td><td>MI-FGSM NI-FGSM EPNI-FGSM(Ours)</td><td>68.9 75.7 91.5</td><td>59.4 64.1 85.8</td><td>53.8 58.0 79.2 74.2</td><td>49.4 81.0 51.7 86.0</td><td>83.3 87.7 94.8 96.2</td><td>92.1 94.7 98.7</td><td>94.5 96.9 99.4</td><td></td><td>100.0* 100.0* 100.0*</td><td>72.7 74.1 76.1 77.4 88.7 89.8</td><td>72.5 77.0 89.5</td><td></td><td>72.5 76.2 89.6</td><td>86.5 87.4 96.5</td><td>83.0 85.1 96.5</td><td>82.5 84.0 85.5 86.7 97.6 96.7</td><td></td><td>77.1 80.4 92.0</td></tr><tr><td>DI-MI-FGSM TI-MI-FGSM SI-NI-FGSM DI-EPNI-FGSM(Ours) TI-EPNI-FGSM(Ours) SI-EPNI-FGSM(Ours)</td><td>85.6 57.1 85.1 97.6 80.3</td><td>82.9 51.3 77.2 96.9 78.2</td><td>75.5 50.1 71.1 94.2 75.7</td><td>72.1 41.4 66.5 92.1</td><td>91.8 70.9 89.6 98.0 89.3</td><td>93.8 75.0 92.0 98.9 91.4</td><td>96.0 80.5 95.3 99.4 94.0</td><td>96.8 84.5 97.8 99.6 95.5</td><td>100.0* 100.0* 100.0* 100.0* 100.0*</td><td>84.0 65.4 82.1 94.3 81.7</td><td>84.4 65.0 83.0 94.6 81.2</td><td>85.5 64.1 81.9 94.7 80.8</td><td>84.6 64.0 81.9 95.2 81.5</td><td>93.7 75.1 93.6 99.4 91.8</td><td>94.0 71.5 91.8 99.3 89.2</td><td>93.7 70.7 92.4 99.6 91.0</td><td>94.4 68.1 91.8 99.1 89.7</td><td>88.8 67.9 86.7 97.2 85.8</td></tr><tr><td>VT-MI-FGSM VT-NI-FGSM VT-EPNI-FGSM(Ours)</td><td>95.3 83.8 87.7 95.4</td><td>91.4 79.6 81.9 92.1</td><td>88.2 74.1 78.9</td><td>66.8 84.8 70.9 74.4</td><td>96.7 92.2 93.5</td><td>97.6 93.7 94.9</td><td>98.6 96.4 98.2</td><td>99.0 97.5 98.7</td><td>100.0* 100.0* 100.0*</td><td>90.3 83.6 85.3</td><td>91.2 82.8 87.2</td><td>92.2 84.4 88.2</td><td>90.5 83.8 86.9</td><td>99.0 94.2 96.1</td><td>98.2 92.4 94.9</td><td>98.1 93.1 95.1</td><td>98.2 93.6 95.7</td><td>94.7 88.0 90.4</td></tr><tr><td>FI-MI-FGSM FI-EPNI-FGSM(Ours) MI-FGSM</td><td>93.3 95.4 78.0</td><td>88.6 93.2</td><td>89.7 88.7 93.1</td><td>86.9 85.1 90.1</td><td>97.9 95.5 96.8</td><td>98.8 96.8 97.9</td><td>99.5 97.5 98.3</td><td>99.8 98.9 98.9</td><td>100.0* 99.9* 100.0*</td><td>93.6 92.5 94.1</td><td>93.5 92.0 94.8</td><td>93.7 93.0 95.0</td><td>94.4 93.8 95.1</td><td>99.4 97.0 97.6</td><td>98.7 95.7 97.4</td><td>99.4 96.6 97.3</td><td>98.8 96.9 97.6</td><td>96.0 94.2 96.0</td></tr><tr><td>NI-FGSM EPNI-FGSM(Ours) DI-MI-FGSM TI-MI-FGSM SI-NI-FGSM V16</td><td>78.8 93.5 88.4 60.6</td><td>59.2 61.6 82.3 75.6 51.1</td><td>63.8 68.8 87.0 77.5</td><td>43.8 47.3 68.1 59.5</td><td>80.0 82.3 91.5 87.8</td><td>73.3 76.1 90.0 83.9</td><td>75.0 78.0 90.0 86.2</td><td>63.7 67.0 82.9 73.8 53.6</td><td>58.7 60.4 77.6 69.7 48.4</td><td>94.7 96.8 99.0 98.3</td><td>98.3 99.2 100.0 98.8</td><td>99.8* 99.9* 100.0* 100.0*</td><td>99.2 99.1 99.9 99.4</td><td>77.0 79.1 91.2 88.0</td><td>69.5 72.1 87.3 81.5</td><td>65.3 66.4 85.8 78.2</td><td>64.9 66.0 87.0 77.5</td><td>74.4 76.4</td><td>89.0 83.8 64.6</td></tr><tr><td>VT-MI-FGSM VT-NI-FGSM</td><td>DI-EPNI-FGSM(Ours) TI-EPNI-FGSM(Ours) SI-EPNI-FGSM(Ours)</td><td>89.8 96.2 82.5 96.1 87.5</td><td>77.5 90.6 74.6 89.1 75.2</td><td>48.5 80.7 62.5 92.6 81.9 77.4 58.1 91.8 80.0 77.9 61.9</td><td>33.4</td><td>71.2 89.1 95.9 88.4 93.6 89.9</td><td>65.4 84.8 94.1 84.2 91.2 86.1</td><td>60.3 85.9 93.5 82.7 92.5 86.8</td><td>77.1 88.6 75.9 86.4 78.3</td><td>73.0 85.5 70.4 84.8 74.1</td><td>88.1 98.5 99.8 96.8 99.4 97.7</td><td>92.9 99.6 99.9 98.2 100.0 98.9</td><td>99.8* 100.0* 100.0* 100.0* 100.0* 99.9*</td><td>94.4 100.0 100.0 98.1 100.0 99.4</td><td>64.9 87.8 95.6 86.4 95.6 87.8</td><td>58.0 81.9 93.1 80.7 92.6 82.2</td><td>54.8 78.9 93.0 78.8 91.0 81.3</td><td>52.4 80.2 91.4 77.4 92.2 80.6</td><td>85.1 93.6 83.0 92.7 85.0</td></tr><tr><td>VT-EPNI-FGSM(Ours) FI-MI-FGSM</td><td></td><td>89.8 95.6 95.9</td><td>76.9 88.4 89.1</td><td>80.7 65.6 92.4 81.1 93.1 79.7</td><td>92.1 96.1 95.6</td><td>87.5 94.4 94.9</td><td>89.3 94.0 93.4</td><td>81.1 90.4 90.4</td><td>76.1 88.8 87.1</td><td></td><td>98.7 99.4 99.6</td><td>99.5 99.9 99.8</td><td>99.9* 99.9* 100.0*</td><td>99.4 99.9 99.8</td><td>88.7 95.6 94.3</td><td>85.4 93.7 91.5</td><td>82.5 93.2 90.2</td><td>82.9 92.2 88.6</td><td>86.8 93.8 93.1</td></tr><tr><td>FI-EPNI-FGSM(Ours)</td><td></td><td>96.7</td><td>90.4</td><td>94.1 84.3</td><td>96.7</td><td>95.9</td><td></td><td>96.0</td><td>92.6</td><td>90.1</td><td>99.8</td><td>99.8</td><td>100.0*</td><td>99.9</td><td>96.4</td><td>94.3</td><td>93.3</td><td>93.0</td><td>94.9</td></tr></table>
+
+Sec. 4.3, respectively. Finally, we provide ablation studies in Sec. 4.4. Table 1 introduces the abbreviations used in the paper.
+
+# 4.1 EXPERIMENTAL SETTINGS
+
+Dataset. We follow the previous works (Dong et al., 2019; Wang et al., 2021) to use the DEV dataset from the NIPS17 Adversarial Attacks and Defenses Competition. This dataset contains 1000 images with size $2 9 9 \times 2 9 9$ .
+
+Target Models. Seventeen normally trained models, i.e., GoogLeNet (Iv1) (Szegedy et al., 2015), Inception-v3 (Iv3) (Szegedy et al., 2016), Inception-v4 (Iv4), Inception-ResNet-v2 (IRv2) (Szegedy et al., 2017), ResNet-18 (R18), ResNet-34 (R34), ResNet-50 (R50), ResNet-101 (R101), ResNet152 (R152) (He et al., 2016), VGG11 (V11), VGG13 (V13), VGG16 (V16), VGG19 (V19) (Simonyan & Zisserman, 2014), DenseNet-121 (D121), DenseNet-169 (D169), DenseNet-201 (D201), and DenseNet-161 (D161) (Huang et al., 2017). Ten defense models (i.e., adversarially trained models), i.e., Adv-Inception-v3 $( \mathrm { I v } 3 _ { \mathrm { a d v } } )$ , Ens-Inception-Resnet-v2 $( \mathrm { I R } \mathrm { v } 2 _ { \mathrm { e n s } }$ ) Tramer et al. (2017), \` Adv-EfficientNet-b0 $( \mathrm { E b 0 _ { a d v } ) }$ to Adv-EfficientNet-b7 $( \mathrm { E b } 7 _ { \mathrm { a d v } } ,$ ).
+
+Baselines. For fair comparison of our EPN and conventional momentum, we replace conventional momentum with our EPN in momentum-based attacks, i.e., MI-FGSM (Dong et al., 2018), NI
+
+Table 3: The attack success rates $( \% )$ of adversarial examples crafted on source models against defense models. “Avg” means the average attack success rate.   
+
+<table><tr><td>Models</td><td>Attacks</td><td>Iv3adv</td><td>IRv2ens</td><td>Eb0adv</td><td>Ebladv</td><td>Eb2adv</td><td>Eb3adv</td><td>Eb4adv</td><td>Eb5adv</td><td>Eb6adv</td><td>Eb7adv</td><td>Avg</td></tr><tr><td rowspan="9">Iv3</td><td>MI-FGSM NI-FGSM EPNI-FGSM(Ours) DI-MI-FGSM</td><td>25.4 25.4 32.4 33.0</td><td>11.5 11.6 14.3 19.3</td><td>29.7 34.3 49.0 46.6</td><td>26.5 31.0 45.0 42.9</td><td>28.1 31.2 45.4 42.4</td><td>18.8 22.2 31.1 31.3</td><td>16.9 18.6 23.9 27.4</td><td>17.5 18.6 25.7 26.7</td><td>14.8 16.0 22.4</td><td>15.2 16.5 22.3 24.4</td><td>20.4 22.5 31.2 31.7</td></tr><tr><td>TI-MI-FGSM SI-NI-FGSM DI-EPNI-FGSM(Ours) TI-EPNI-FGSM(Ours)</td><td>29.9 37.4 41.5 54.1</td><td>21.5 23.2 25.2 41.2</td><td>35.2 50.2 65.0 60.7</td><td>32.6 46.1 65.5 58.2</td><td>34.9 46.3 64.7 57.0</td><td>27.8 33.3 45.5 46.6</td><td>29.3 30.9 39.1 46.7</td><td>28.2 29.6 41.6 45.3</td><td>22.9 24.5 27.6 36.7 43.4</td><td>26.6 25.8 35.6 44.2</td><td>29.1 35.0 46.0 49.7</td></tr><tr><td>SI-EPNI-FGSM(Ours) VT-MI-FGSM VT-NI-FGSM</td><td>47.4 36.8 39.1</td><td>28.4 25.1 26.4</td><td>66.3 50.5 54.0</td><td>64.7 46.3 50.6</td><td>63.1 44.9 49.5</td><td>46.7 33.5 36.6</td><td>41.8 29.4 31.0</td><td>42.1 29.4 32.7</td><td>37.9 26.8 30.3</td><td>37.3 26.2 28.5</td><td>47.6 34.9 37.9</td></tr><tr><td>VT-EPNI-FGSM(Ours) FI-MI-FGSM FI-EPNI-FGSM(Ours)</td><td>43.1 55.3 58.9</td><td>26.5 38.0 39.9</td><td>63.2 68.2 76.2</td><td>61.8 67.8 73.3</td><td>60.6 64.9 72.7</td><td>46.9 53.6 61.1</td><td>41.1 50.0 56.0</td><td>39.7 49.3 55.6</td><td>38.2 46.6 52.9</td><td>36.8 45.1 50.5</td><td>45.8 53.9 59.7</td></tr><tr><td>MI-FGSM NI-FGSM EPNI-FGSM(Ours) DI-MI-FGSM</td><td>27.3 25.9 33.5 33.1</td><td>15.5 14.9 15.9 24.7</td><td>25.2 24.8 36.3</td><td>22.8 23.1 32.4</td><td>24.3 23.6 34.9</td><td>17.0 17.8 23.9 27.1</td><td>13.8 14.1 20.7 24.6</td><td>14.6 14.9 20.4 23.2</td><td>12.1 12.8 17.6 22.0</td><td>13.2 13.7 17.8 20.8</td><td>18.6 18.6 25.3 28.7</td></tr><tr><td>TI-MI-FGSM SI-NI-FGSM DI-EPNI-FGSM(Ours) TI-EPNI-FGSM(Ours) SI-EPNI-FGSM(Ours)</td><td>38.1 37.0 39.4 58.3 45.6</td><td>32.2 33.0 29.7 52.9 41.5</td><td>38.1 39.0 50.8 57.4 60.5 68.9</td><td>35.8 36.7 48.2 52.4 58.3</td><td>37.9 39.4 46.9 53.4 56.6 66.8</td><td>30.2 35.7 41.0 49.7 50.2</td><td>33.8 32.0 33.6 50.9 42.6</td><td>31.4 28.7 33.3 47.0 43.5</td><td>29.2 26.9 30.2 48.0 39.7</td><td>29.7 27.5 30.6 46.5 40.1</td><td>34.0 36.7 40.1 52.9 50.3</td></tr><tr><td>VT-MI-FGSM VT-NI-FGSM VT-EPNI-FGSM(Ours) FI-MI-FGSM</td><td>38.8 40.5 48.2</td><td>36.4 34.8 36.6</td><td>42.3 44.6 59.5</td><td>64.3 38.9 41.1 55.3</td><td>41.2 43.4 54.1</td><td>31.2 33.1 43.0</td><td>28.4 28.2 38.8</td><td>26.4 26.6 38.0</td><td>25.4 25.8 36.9</td><td>25.4 26.2 35.3</td><td>33.4 34.4 44.6 46.7</td></tr><tr><td>FI-EPNI-FGSM(Ours) MI-FGSM NI-FGSM EPNI-FGSM(Ours)</td><td>54.5 53.3 36.5 40.1</td><td>40.1 47.1 27.8 29.4</td><td>57.4 59.6 46.6 49.8</td><td>56.7 56.8 43.6 47.3</td><td>56.3 57.9 47.3 47.8</td><td>45.7 47.0 30.9 33.2</td><td>40.9 43.3 27.3 29.1</td><td>39.7 42.1 26.5 27.7</td><td>38.2 40.6 22.0 25.2</td><td>37.4 38.6 24.8 25.4</td><td>48.6 33.3 35.5</td></tr><tr><td>DI-MI-FGSM TI-MI-FGSM SI-NI-FGSM R152</td><td>53.1 57.6 46.8 51.7 DI-EPNI-FGSM(Ours) 78.3</td><td>43.1 51.7 41.1 43.4 73.1</td><td>71.9 72.6 50.7 62.8 91.6</td><td>68.1 68.3 46.4 57.3 89.9</td><td>70.5 71.2 50.1 61.2 90.6</td><td>50.2 54.7 41.8 43.4 75.7</td><td>41.0 47.1 43.2 39.3 67.3</td><td></td><td>41.0 37.5 44.7 44.2 41.0 36.2 36.0 35.4 64.9 62.6</td><td>40.2 44.3 38.7 33.8 64.3</td><td>51.7 55.6 43.6 46.4 75.8</td></tr><tr><td>VT-MI-FGSM VT-NI-FGSM FI-MI-FGSM</td><td>TI-EPNI-FGSM(Ours) SI-EPNI-FGSM(Ours) VT-EPNI-FGSM(Ours)</td><td>72.2 67.4 67.5 59.3 59.1 52.6 61.6 55.5 72.6 71.6</td><td>74.7 79.4 69.6 73.6 86.2</td><td>70.1 78.5 64.9 68.2 85.3</td><td></td><td>74.2 81.1 69.3 72.1 88.4</td><td>62.9 61.8 55.1 56.1 74.1</td><td>62.7 53.7 48.8 50.6 66.0</td><td>60.7 51.5 46.1 48.7 64.8</td><td>59.3 58.3 48.8 50.3 43.6 44.6 45.5 47.9 62.2 65.1</td><td></td><td>66.3 63.2 55.4 58.0 73.6</td></tr><tr><td>TI-MI-FGSM V16</td><td>FI-EPNI-FGSM(Ours) MI-FGSM NI-FGSM EPNI-FGSM(Ours) DI-MI-FGSM</td><td>81.0 33.0 33.0 46.7 44.9</td><td>73.3 20.3 22.5 29.0 31.6</td><td>88.5 48.8 49.5 71.2 62.4</td><td>87.0 41.8 43.6 65.7 55.9</td><td>82.6 88.6 41.3 44.3 63.5 56.9</td><td>76.3 25.1 27.7 41.5 37.5</td><td>70.9 21.5 22.2 31.3 32.0</td><td>70.1 21.7 22.7 31.5 30.9</td><td>66.4 19.0 18.9 28.5 27.8</td><td>66.4 21.8 20.1 28.7 28.1</td><td>76.9 29.4 30.5 43.8 40.8</td></tr><tr><td></td><td>SI-NI-FGSM DI-EPNI-FGSM(Ours) TI-EPNI-FGSM(Ours) SI-EPNI-FGSM(Ours) VT-MI-FGSM VT-NI-FGSM</td><td>38.9 50.0 61.2 61.8 62.8 49.4 49.3</td><td>29.4 33.5 44.7 49.9 42.6 35.4 37.5</td><td>44.3 61.4 80.3 68.6 78.6 65.3</td><td>38.0 56.4 77.9 65.2 75.4 58.7</td><td>41.4 54.4 78.2 63.7 74.3 58.6</td><td>31.1 36.5 52.2 49.2 48.1 39.7</td><td>32.0 29.9 43.5 50.3 40.6 34.3</td><td>31.5 30.0 42.6 47.7 41.9 33.6</td><td>28.5 27.2 40.2 47.6 37.9 29.2</td><td>28.2 26.4 38.8 46.1 35.5 32.2</td><td>34.3 40.6 56.0 55.0 53.8</td></tr></table>
+
+FGSM (Lin et al., 2019), DI-MI-FGSM (Xie et al., 2019), TI-MI-FGSM (Dong et al., 2019), SI-NIFGSM (Lin et al., 2019), VT-MI-FGSM (Wang & He, 2021), VT-NI-FGSM (Wang & He, 2021) and FI-MI-FGSM (Wang et al., 2021). Then we compare the transferability of conventional momentumbased attacks and our EPN-based attacks.
+
+Hyperparameters. In all experiments, we follow the official default settings for hyperparameters. Specifically, the maximum perturbation $\epsilon = 1 6$ , the number of iterations $T = 1 0$ , the step size $\alpha = \epsilon / T = 1 . 6$ , and the decay factor $\mu = 1 . 0$ . For DIM (Xie et al., 2019), the probability $p$ is set to 0.5. For TIM (Dong et al., 2019), the size of the Gaussian kernel is set to $1 5 { \times } 1 5$ . For SIM (Lin et al., 2019), the number of scale copies $m$ is set to 5. For VT-MI-FGSM (Wang & He, 2021) and VT-NIFGSM (Wang & He, 2021), the number of sampled examples $N$ is set to 20, and the parameter $\beta$ for the upper bound of the neighborhood is set to 1.5. For FI-MI-FGSM (Wang et al., 2021), the drop probability $p _ { d }$ is set to 0.3 when attacking normally trained models and 0.1 when attacking defense models, the ensemble number $N$ is set to 30 in aggregate gradient, and the intermediate layer is set to Mixed ${ } _ { 5 b }$ for Iv3, Conv 4a for IRv2, Conv3 3 for V16 as well as the last layer of the second block for R152. For our EM-based attacks, epochs is set to 5.
+
+![](images/fafd2e428be0acdf7ede725979fa5a3c2e99054d62d3d5a67513a93fbcd1c074.jpg)  
+Figure 3: The average attack success rates $( \% )$ of the adversarial examples crafted on source models against normally trained models (except the source model) and defense models for various $\mu$ .
+
+![](images/13699bec5c115b8c6b25669e76ccd9babb24eaeb2ee8d0e2bc3561395399d59a.jpg)  
+Figure 4: The average attack success rates $( \% )$ of the adversarial examples crafted on source models against normally trained models (except the source model) and defense models for various epochs.
+
+# 4.2 ATTACK NORMALLY TRAINED MODELS
+
+To validate that EPN-based attacks have higher transferability than conventional momentum-based attacks, we choose Iv3, IRv2, R152, and V16 as the source model, respectively, and attack normally trained target models via our EPN-based methods and baseline methods. The attack success rates are shown in Table 2. The results show that the attack success rates of our EPN-based methods are ${ \sim } 1 1 . 9 \%$ higher than baseline methods on average, In particular, our EPN-based attacks have the best transferability against normally trained target models when the source model is R152. Specifically, the attack success rates of EPNI-FGSM, DI-EPNI-FGSM, VT-EPNI-FGSM, and FI-EPNI-FGSM are $9 2 . 0 \%$ , $9 7 . 2 \%$ , $9 6 . 0 \%$ , and $9 6 . 0 \%$ , respectively, on average. Therefore, the experiments demonstrate that our EPN improves transferability more effectively than conventional momentum against normally trained models.
+
+# 4.3 ATTACK DEFENSE MODELS
+
+To further compare the transferability, we also use defense models as the target models, and the source models are still Iv3, IRv2, R152, and V16. We craft adversarial examples on the source model via our EPN-based methods and baseline methods to attack defense models. The attack success rates are shown in Table 3. The results show that the attack success rates of our EPNbased methods are ${ \sim } 1 3 . 1 \%$ higher than baseline methods on average. Adversarial examples crafted on R152 still show the best transferability against defense models. Specifically, the attack success rates of EPNI-FGSM, DI-EPNI-FGSM, VT-EPNI-FGSM, and FI-EPNI-FGSM are $5 1 . 7 \%$ , $7 5 . 8 \%$ , $7 3 . 6 \%$ , and $7 6 . 9 \%$ , respectively, on average. The results of experiments indicate that our EPN is still more effective than conventional momentum against defense models.
+
+# 4.4 ABLATION STUDY
+
+We conduct ablation studies for EPNI-FGSM. We investigate the impacts of two hyperparameters (i.e., the decay factor $\mu$ and the epochs of pretraining epochs) on the transferability of EPNI-FGSM in Sec. 4.4.1. We further study the impacts of EM and PN on transferability in Sec. 4.4.2.
+
+# 4.4.1 IMPACTS OF $\mu$ AND epochs
+
+The source models are set to Iv3, IRv2, R152, and V16. We use EPNI-FGSM to craft adversarial examples to attack normally trained models and defense models, respectively. We investigate the impacts of $\mu$ and epochs on the transferability of EPNI-FGSM by counting the average attack success rates against normally trained models (except the source model) and defense models.
+
+The decay factor $\mu .$ . The decay factor $\mu$ plays a vital role for momentum. If $\mu = 0$ , the momentumbased attacks degrade to vanilla iterative gradient-based attacks. If $0 < \mu < 1$ , the previous gradients accumulated in the momentum decay exponentially. If $\mu = 1$ , the momentum simply adds up all previous gradients. If $\mu > 1$ , the previous gradients accumulated in the momentum grow exponentially. We pre-set epoch $s = 5$ and set $\mu$ from 0.0 to 2.0 with a step size of 0.1. The average attack success rates are shown in Fig. 3. When $\mu \leq 1 . 0$ , the average attack success rates show an upward trend, and when $\mu \geq 1 . 0$ , the average attack success rates show a downward trend. Therefore, we set $\mu = 1 . 0$ for EPNI-FGSM to achieve the best transferability.
+
+![](images/e2f2ae241e3bdb3bc56d7fb243e0dfe1fdab5ecc0bb0f52c3961f4f1620f6522.jpg)  
+Figure 5: The average attack success rates $( \% )$ of the adversarial examples crafted on source models against normally trained models and defense models via NI-FGSM, ENI-FGSM, PNI-FGSM, and EPNI-FGSM.
+
+The epochs of pretraining epochs. epochs affects the amount of gradient accumulated in EM. We pre-set $\mu = 1 . 0$ and set epochs from 0 to 10 with a step size of 1. The average attack success rates are shown in Fig. 4. As epochs increases, the average attack success rates increase and gradually converge. Since the larger epochs, the higher the computational cost, we set epochs $= 5$ for EPNIFGSM to strike a balance between computational cost and transferability.
+
+In summary, we set the decay factor $\mu = 1 . 0$ and the epochs of pretraining epochs $= 5$ for EPNIFGSM. Similarly, such two hyperparameters of other EPN-based attacks have the same settings as EPNI-FGSM.
+
+# 4.4.2 IMPACTS OF EM AND PN
+
+The source models are the same as in Sec 4.4.1. To investigate the impacts of EM and PN, we craft adversarial examples on source models via ENI-FGSM (only with EM), PNI-FGSM (only with PN), and EPNI-FGSM (with EM and PN), respectively. In addition, we also use MI-FGSM and NI-FGSM (without EM and PN) as baselines. For ENI-FGSM, the epochs of pretraining epochs is set to 5. For ENI-FGSM and PNI-FGSM, the decay factor $\mu$ is set to 1.0. The average attack success rates of the adversarial examples against normally trained models and defense models are shown in Fig. 5. The average attack success rates of ENI-FGSM are higher than MI-FGSM and NIFGSM, demonstrating that EM improves transferability more than conventional momentum. The same is true for PN. Besides, the average attack success rates of EPNI-FGSM are higher than that of ENI-FGSM and PNI-FGSM, demonstrating that the combination of EM and PN can further improve transferability.
+
+# 5 CONCLUSION
+
+In this work, we proposed Experienced Momentum (EM) and Precise Nesterov momentum (PN) to boost transferability. Specifically, EM is trained on a set of derived models by Random Channels Swapping (RCS), and then conventional momentum is initialized to EM, which can accelerate optimization to escape from saddle points and poor local extrema during early iterations to improve transferability. Additionally, we adopted the current gradient to refine the pre-update of conventional Nesterov momentum, called PN. Then, we naturally combined EM and PN as EPN to improve transferability further. Extensive experiments demonstrate that EPN-based attacks have higher transferability than conventional momentum-based attacks. However, our methods still adopt a fixed learning rate or step size that is crucial for the optimizer. Therefore, we will explore the impact of learning rate or step size on transferability in future work.
+
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+        "type": "text",
+        "text": "IMPROVING THE TRANSFERABILITY OF ADVERSARIAL ATTACKS THROUGH EXPERIENCED PRECISE NESTEROV MOMENTUM ",
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+            176,
+            98,
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+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+        "text": "Deep Neural Networks are vulnerable to adversarial attacks, which makes adversarial attacks serve as a method to evaluate the robustness of DNNs. However, adversarial attacks have high white-box attack success rates but poor transferability, making black-box attacks impracticable in the real world. Momentumbased attacks were proposed to accelerate optimization to improve transferability. Nevertheless, conventional momentum-based attacks accelerate optimization inefficiently during early iterations since the initial value of momentum is zero, which leads to unsatisfactory transferability. Therefore, we propose Experienced Momentum (EM), which is the pre-trained momentum. Initializing the momentum to EM can help accelerate optimization during the early iterations. Moreover, the pre-update of conventional Nesterov momentum based attacks is rough, prompting us to propose Precise Nesterov momentum (PN). PN refines the preupdate by considering the gradient of the current data point. Finally, we integrate EM with PN as Experienced Precise Nesterov momentum (EPN) to further improve transferability. Extensive experiments against normally trained and defense models demonstrate that our EPN is more effective than conventional momentum in the improvement of transferability. Specifically, the attack success rates of our EPN-based attacks are ${ \\sim } 1 1 . 9 \\%$ and ${ \\sim } 1 3 . 1 \\%$ higher than conventional momentum-based attacks on average against normally trained and defense models, respectively. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+    },
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+        "type": "text",
+        "text": "Deep neural networks (DNNs) (Krizhevsky et al., 2012; Szegedy et al., 2015; He et al., 2016; Ioffe & Szegedy, 2015) have been widely applied in computer vision, e.g., autonomous driving (Franchi et al., 2022; Hao et al., 2019; Cococcioni et al., 2018), facial recognition (Chrysos et al., 2020; Ghenescu et al., 2018), and medical image analysis (Akselrod-Ballin et al., 2016; Ding et al., 2017; Liu et al., 2019). However, Szegedy et al. (2013) found that applying certain imperceptible perturbations to images can make DNNs misclassify, and they refer to such perturbed images as adversarial examples $( A E s )$ . Adversarial examples pose a huge threat to the security of DNNs, which attaches extensive attention from researchers. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Adversarial attacks can be categorized into white-box attacks and black-box attacks. Typically, iterative gradient-based (Kurakin et al., 2016; Madry et al., 2017) and optimization-based attacks (Carlini & Wagner, 2017) have high white-box but low black-box attack success rates, which means that such two attacks are impracticable in the real world. Transferability, which means adversarial examples crafted on the source model remain effective on other models, makes black-box attacks feasible. Furthermore, iterative gradient-based attacks have the advantages of low computational cost and fast generation speed, thus improving the transferability of iterative gradient-based attacks has become a hotspot in the field of adversarial attacks. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Many methods have been proposed to improve the transferability of iterative gradient-based attacks. These methods can be classified into three branches: improving optimization algorithms, input transformations, and disrupting feature space. For example, MI-FGSM (Dong et al., 2018), NI-FGSM (Lin et al., 2019), and VM(N)I-FGSM (Wang & He, 2021) improve gradient ascent (or descent) algorithm to escape from saddle points and poor local extrema to improve transferability; DIM (Xie et al., 2019), TIM (Dong et al., 2019), and SIM (Lin et al., 2019) craft adversarial examples on a set of models derived by input transformations to prevent overfitting and improve transferability; NRDM (Naseer et al., 2018), FDA (Ganeshan et al., 2019), and FIA (Wang et al., 2021) disrupt deep features of DNNs to craft highly transferable adversarial examples. ",
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+    {
+        "type": "image",
+        "img_path": "images/43ff633c9b654c39becc3a37d6d52763e1946d4a2c40cf69095e67ae5381f60e.jpg",
+        "image_caption": [
+            "Figure 1: Comparison of conventional Polyak momentum and experienced Polyak momentum. Adversarial examples are crafted on the source model (Inception-v3) and used to attack the target model (VGG16). Our Experienced MI-FGSM (EMI-FGSM), which integrates EM into MI-FGSM, causes misclassification with higher loss and confidence than MI-FGSM, thus EMI-FGSM can mislead the attention of the target model better than MI-FGSM. "
+        ],
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+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Those mentioned above adversarial attacks mostly adopt the momentum (Polyak, 1964; Nesterov, 1983) to accelerate optimization. However, such momentum-based adversarial attacks (e.g., M(N)IFGSM, VM(N)I-FGSM, and FIA) have the problem of initializing the momentum to zero, resulting in inefficient acceleration due to momentum accumulating few gradients during the first few iterations. Therefore, we propose Experienced Momentum (EM), which is the pre-trained momentum. Before the iterations, the momentum is initialized to EM instead of zero, leading to better acceleration in the first few iterations. The comparison of conventional Polyak momentum (Polyak, 1964) and experienced Polyak momentum is shown in Fig. 1. To prevent overfitting on the source model, we train EM on a set of models derived by Random Channels Swapping (RCS). EM and RCS are detailed in Sec. 3.1. ",
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+    {
+        "type": "text",
+        "text": "Furthermore, adversarial attacks (e.g., NI-FGSM and VNI-FGSM) based on Nesterov momentum (i.e., Nesterov Accelerated Gradient, NAG (Nesterov, 1983)) have the disadvantage that the preupdate is rough. Specifically, during each iteration, the parameters are first pre-updated along the momentum to obtain the pre-update point, which is an estimation of the next position. Then the preupdate is modified by the gradient of the pre-update point. Such looking-ahead property of Nesterov momentum makes parameters escape from saddle points and poor local extrema easier and faster, resulting in improving transferability. However, pre-updating only along the momentum is rough, and the estimation of the next position of the parameters is imprecise. Therefore, we propose Precise Nesterov momentum (PN), which not only retains the looking-ahead property but also refines the pre-update by adopting the gradient of the current data point. To improve transferability further, we integrate EM with PN as Experienced Precise Nesterov momentum (EPN). PN and EPN are detailed in Sec. 3.2. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Overall, we make the following contributions: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "• We propose Experienced Momentum (EM), which is trained on a set of models derived by Random Channels Swapping (RCS). Initializing the momentum to EM can accelerate optimization effectively during the early iterations to improve transferability. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "• We propose Precise Nesterov momentum (PN), which adopts the gradient of the current data point to refine the pre-update to escape from saddle points and poor local extrema easier and faster. We also integrate EM with PN as Experienced Precise Nesterov momentum (EPN) to improve transferability further. • Extensive experiments on normally trained and defense models demonstrate that our EPN is more effective than conventional momentum for improving transferability. ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "2.1 TRANSFERABLE ADVERSARIAL ATTACKS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Since adversarial examples were discovered by Szegedy et al. (2013), many methods (Goodfellow et al., 2014; Kurakin et al., 2016; Carlini & Wagner, 2017) have been proposed to craft adversarial examples to demonstrate the vulnerability of DNNs. We focus on the transferability of iterative gradient-based attacks and review related works from three branches: improving optimization algorithms, input transformations, and disrupting feature space. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Improving optimization algorithms. Dong et al. (2018) integrated Polyak momentum (Polyak, 1964) into I-FGSM (Kurakin et al., 2016) to accelerate gradient ascent (or descent) to improve transferability. Inspired by the fact that Nesterov momentum (Nesterov, 1983) is superior to Polyak momentum, Lin et al. (2019) integrated Nesterov momentum into I-FGSM to improve transferability further. Wang & He (2021) used the gradient variance of the previous iteration to tune the current gradient to stabilize the update direction and escape from saddle points and poor local extrema. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Input transformations. The nature of input transformations is crafting adversarial examples on a set of derived models to prevent overfitting. Xie et al. (2019) performed random resizing and padding with probability $p$ to derive models. Dong et al. (2019) convolved the gradient to approximate translating input. Lin et al. (2019) scaled the input with the scale factor $1 / 2 ^ { i }$ to derive a set of models. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Disrupting feature space. Naseer et al. (2018) created maximum distortions in the feature space to craft adversarial examples, based on the intuition that features of DNNs are highly generalizable. Ganeshan et al. (2019) highly corrupted deep features by disrupting features at each layer of DNNs to improve transferability. Wang et al. (2021) described feature importance with the aggregate gradient and disrupted important object-aware features to achieve stronger transferability. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 ADVERSARIAL TRAINING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Adversarial training as a common defense measure can validate transferability further. Adversarial training increases robustness by adding adversarial examples to the training data. Goodfellow et al. (2014) showed that adversarially trained models are more robust. However, Kurakin et al. (2016) pointed out that adversarial training is not robust to iterative attacks. Moreover, Tramer et al. (2017) \\` showed that adversarially trained models are still vulnerable to simple white-box and black-box attacks. Therefore, they proposed ensemble adversarial training adding adversarial examples crafted from other models to the training data. ",
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+    {
+        "type": "text",
+        "text": "3 METHODOLOGY ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Given a target model $f ^ { \\prime } ( x ; { \\pmb \\theta } ^ { \\prime } )$ , where $_ { \\textbf { \\em x } }$ is an input, and $\\pmb { \\theta } ^ { \\prime }$ is the parameters of $f ^ { \\prime }$ . Let $J ( \\cdot , y )$ be a loss function, where $y$ is the ground-truth label of the input $_ { \\textbf { \\em x } }$ . A non-targeted adversarial example $\\pmb { x } ^ { a d v }$ satisfies $f ^ { \\prime } ( { \\pmb x } ; { \\hat { \\pmb \\theta } } ^ { \\prime } ) \\neq { \\bar { f ^ { \\prime } } } ( { \\pmb x } ^ { a d v } ; { \\pmb \\theta } ^ { \\prime } )$ under the constraint of $| | { \\pmb x } ^ { a d v } \\bar { - } { \\pmb x } | | _ { p } \\le \\epsilon .$ , where $| | \\cdot | | _ { p }$ denotes the $L ^ { p }$ norm, and $p$ is generally $0 , 1 , 2 , \\infty$ . In this paper, we focus on $p = \\infty$ . Note that our methods can be generalized to $p = 0 , 1 , 2$ easily. Crafting non-targeted adversarial examples can be described as solving the following optimization problem: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cc94b61db3a9da4bf4fe2a37bfd7d7aeee25432a5c2b6813899fb9c851273ea6.jpg",
+        "text": "$$\n\\underset { \\pmb { x } ^ { a d v } } { \\arg \\operatorname* { m a x } } J ( f ^ { \\prime } ( \\pmb { x } ^ { a d v } ; \\pmb { \\theta } ^ { \\prime } ) , y ) , \\quad \\mathrm { s . t . } | | \\pmb { x } ^ { a d v } - \\pmb { x } | | _ { p } \\leq \\epsilon .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "In this paper, we focus on non-targeted attacks. Our proposed methods can be easily transformed into targeted attacks by replacing the above objective function with $- J ( f ^ { \\prime } ( x ^ { a d v } ; \\pmb { \\theta } ^ { \\prime } ) , \\mathbf { \\bar { \\psi } } ^ { * } )$ , where $y ^ { * }$ denotes the target label. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/87cfd090bbc4c049cc375cc349efb5a739a1a34309be3d1c013402ce1a4aa858.jpg",
+        "image_caption": [
+            "Figure 2: Illustration of training EM during each iteration. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Many gradient-based methods have been proposed to solve Eq. 1, e.g., FGSM (Goodfellow et al., 2014), I-FGSM (Kurakin et al., 2016), and PGD (Madry et al., 2017). However, the parameters $\\pmb { \\theta } ^ { \\prime }$ of the target model $f ^ { \\prime }$ is inaccessible for black-box attacks, resulting in the inability to solve Eq. 1 directly. Therefore, the target model $f ^ { \\prime }$ is usually replaced with a model $f$ (i.e., the source model) with accessible parameters $\\pmb \\theta$ , and then adversarial examples are crafted on the source model $f$ to attack the target model $f ^ { \\prime }$ . To achieve effective black-box attacks, adversarial examples crafted on the source model $f$ are required to have high transferability. Therefore, we propose Experienced Momentum (EM, detailed in Sec. 3.1) and Precise Nesterov momentum (PN, detailed in Sec. 3.2) to improve transferability. EM and PN can be naturally combined as Experienced Precise Nesterov momentum (EPN, detailed in Sec. 3.2) to further improve transferability. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 EXPERIENCED MOMENTUM ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Momentum-based attacks initialize momentum to zero, resulting in inefficient acceleration during the first few iterations. Therefore, we propose Experienced Momentum (EM), which is the pretrained momentum. Setting the initial momentum to EM can accelerate the optimization during the early iterations. To prevent overfitting of EM and improve transferability further, we train EM on a set of models derived by Random Channels Swapping (RCS). RCS derives models by randomly swapping the channels of the input image, which is equivalent to randomly swapping the “block” dimensions of the original model, leading to various decision boundaries of derived models. Therefore, training EM on derived models can prevent overfitting. The specific procedure for training EM is as follows. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "First of all, we perform RCS on the input image $_ { \\textbf { \\em x } }$ . Specifically, we denote the input image $_ { \\textbf { \\em x } }$ as an RGB triplet $( R , G , B )$ , and then the input image $_ { \\textbf { \\em x } }$ through RCS can be denoted as $\\bar { S } ( { \\pmb x } )$ , where $S ( { \\pmb x } ) \\in \\{ ( R , G , B ) , ( R , B , G ) , ( G , R , B ) , ( G , B , R ) , ( B , R , G ) , ( B , G , R ) \\} .$ , $S ( \\cdot )$ denotes RCS. Secondly, $S ( { \\pmb x } )$ is fed into the source model $f$ to derive $f ( S ( \\cdot ) , y )$ . Thirdly, we pre-perturb the input image $_ { \\textbf { \\em x } }$ on the derived model $f ( S ( \\cdot ) , y )$ by iterative gradient-based attacks to prevent overfitting. As shown in Fig. 2, we accumulate gradients to training EM during each iteration. Finally, we follow the above procedure repeatedly to make the EM more generalizable. After training EM, we set the initial value of momentum to EM to accelerate the early iterations. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 PRECISE NESTEROV MOMENTUM ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Nesterov momentum based Attack (e.g., NI-FGSM (Lin et al., 2019) and VNI-FGSM (Wang & He, 2021)) only pre-update along the momentum roughly, resulting in the imprecision of the pre-update point that is the estimate of the next iterative position. Against this disadvantage, we propose Precise Nesterov momentum (PN), which considers the gradient of the current data point in the pre-update to make the pre-update precise. Specifically, during each iteration, the pre-update is performed along the gradient of the current data point and momentum successively to obtain the pre-update point, and then we use the gradient of the pre-update point to modify the pre-update. We integrate PN into I-FGSM as PNI-FGSM. The $t$ -th iteration of PNI-FGSM can be formalized as follows: ",
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+    {
+        "type": "equation",
+        "img_path": "images/bc34467f464d519ac6f8687411051e4810b902b979b67545313ad46e73a5b6c2.jpg",
+        "text": "$$\n\\widetilde { \\pmb x } _ { t } ^ { a d v } = \\pmb x _ { t } ^ { a d v } + \\alpha \\cdot \\left[ \\frac { \\nabla _ { \\pmb x _ { t } ^ { a d v } } J ( f ( \\pmb x _ { t } ^ { a d v } ; \\pmb \\theta ) , y ) } { | | \\nabla _ { \\pmb x _ { t } ^ { a d v } } J ( f ( \\pmb x _ { t } ^ { a d v } ; \\pmb \\theta ) , y ) | | _ { 1 } } + \\mu \\cdot { \\pmb g } _ { t - 1 } \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Input : A source model $f$ with parameters $\\pmb \\theta$ and a loss function $J$ . An original image $_ { \\textbf { \\em x } }$ with ground-truth label $y$ . Input : The maximum perturbation $\\epsilon$ , the number of iterations $T$ , and the decay factor $\\mu$ . Input : The epochs of pretraining epochs. Output: An adversarial example $\\bar { \\boldsymbol { x } } ^ { a \\bar { d } v }$ . 1 $\\alpha  \\epsilon / T$ ; $\\pmb { g } ^ { e x p }  \\mathbf { 0 }$ ; 2 for $n \\gets 1$ to epochs do 3 $\\hat { \\pmb { x } } _ { 1 } ^ { a d v }  { \\pmb { x } }$ ; 4 for $t \\gets 1$ to $T$ do 5 $\\begin{array} { r l } & { \\widetilde { x } _ { t } ^ { a d v } \\gets \\hat { x } _ { t } ^ { a d v } + \\alpha \\cdot \\left[ \\frac { \\nabla _ { \\hat { x } _ { t } ^ { a d v } } J ( f ( S ( \\hat { x } _ { t } ^ { a d v } ) ; \\pmb \\theta ) , y ) } { | | \\nabla _ { \\hat { x } _ { t } ^ { a d v } } J ( f ( S ( \\hat { x } _ { t } ^ { a d v } ) ; \\pmb \\theta ) , y ) | | _ { 1 } } + \\mu \\cdot g ^ { e x p } \\right] ; } \\\\ & { g ^ { e x p } \\gets \\frac { \\nabla _ { \\hat { x } _ { t } ^ { a d v } } J ( f ( S ( \\hat { x } _ { t } ^ { a d v } ) ; \\pmb \\theta ) , y ) } { | | \\nabla _ { \\hat { x } _ { t } ^ { a d v } } J ( f ( S ( \\hat { x } _ { t } ^ { a d v } ) ; \\pmb \\theta ) , y ) | | _ { 1 } } + \\mu \\cdot g ^ { e x p } + \\frac { \\nabla _ { \\widetilde { x } _ { t } ^ { a d v } } J ( f ( \\widetilde { x } _ { t } ^ { a d v } ; \\pmb \\theta ) , y ) } { | | \\nabla _ { \\widetilde { x } _ { t } ^ { a d v } } J ( f ( \\widetilde { x } _ { t } ^ { a d v } ; \\pmb \\theta ) , y ) | | _ { 1 } } ; } \\\\ & { \\hat { x } _ { t + 1 } ^ { a d v } \\gets \\mathrm { C l i p } _ { ( \\pmb { x } , \\epsilon ) } \\left\\{ \\hat { x } _ { t } ^ { a d v } + \\alpha \\cdot \\mathrm { s i g n } ( g ^ { e x p } ) \\right\\} ; } \\end{array}$ 6 7 8 end 9 end 10 ${ \\pmb x } _ { 1 } ^ { a d v }  { \\pmb x } ; { \\pmb g } _ { 0 }  { \\pmb g } ^ { e x p }$ ; 11 for $t \\gets 1$ to $T$ do 12 Update $\\mathbf { \\nabla } _ { \\mathbf { \\boldsymbol { g } } _ { t } }$ and $\\pmb { x } _ { t + 1 } ^ { a d v }$ by Eq. 2, 3, 4; 13 end 14 return ${ \\pmb x } ^ { a d v }  { \\pmb x } _ { T + 1 } ^ { a d v }$ . ",
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+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/32b9f28d4027958a04c50a5fb1ee427db1722078d20df053ea46e2aaa82cc641.jpg",
+        "table_caption": [
+            "Algorithm 1: Experienced Precise Nesterov momentum I-FGSM (EPNI-FGSM) ",
+            "Table 1: The abbreviations used in the paper. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Abbreviation</td><td>Explanation</td></tr><tr><td></td><td></td></tr><tr><td>D(T)I-MI-FGSM</td><td>the combination of D(T)IM and MI-FGSM</td></tr><tr><td>SI-NI-FGSM</td><td>the combination of SIMand NI-FGSM</td></tr><tr><td>D(T,S)I-EPNI-FGSM</td><td>the combination of D(T,S)IM and EPNI-FGSM</td></tr><tr><td>VT-M(N)I-FGSM</td><td>i.e., VM(N)I-FGSM</td></tr><tr><td>VT-EPNI-FGSM</td><td>the combination of Variance Tuning (VT)(Wang &amp; He,2021) and EPNI-FGSM</td></tr><tr><td>FI-MI-FGSM FI-EPNI-FGSM</td><td>i.e., FIA</td></tr><tr><td></td><td>the combination of Feature Importance-aware (FI) (Wang et al.,2021) and EPNI-FGSM</td></tr></table>",
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+        "type": "equation",
+        "img_path": "images/8dd749b920db925cee00f283cff9af8d380e6a843b7221a6e2d3687702e4bfc6.jpg",
+        "text": "$$\ng _ { t } = \\frac { \\nabla _ { x _ { t } ^ { a d v } } J ( f ( x _ { t } ^ { a d v } ; \\pmb { \\theta } ) , y ) } { | | \\nabla _ { x _ { t } ^ { a d v } } J ( f ( x _ { t } ^ { a d v } ; \\pmb { \\theta } ) , y ) | | _ { 1 } } + \\mu \\cdot g _ { t - 1 } + \\frac { \\nabla _ { \\widetilde { x } _ { t } ^ { a d v } } J ( f ( \\widetilde { x } _ { t } ^ { a d v } ; \\pmb { \\theta } ) , y ) } { | | \\nabla _ { \\widetilde { x } _ { t } ^ { a d v } } J ( f ( \\widetilde { x } _ { t } ^ { a d v } ; \\pmb { \\theta } ) , y ) | | _ { 1 } } ,\n$$",
+        "text_format": "latex",
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+        "type": "equation",
+        "img_path": "images/33838553fa2bb80a284ecae401f7852a5fe62f09189224b91fefa46a7d728fd2.jpg",
+        "text": "$$\n\\begin{array} { r } { \\pmb { x } _ { t + 1 } ^ { a d v } = \\mathrm { C l i p } _ { ( \\pmb { x } , \\epsilon ) } \\left\\{ \\pmb { x } _ { t } ^ { a d v } + \\alpha \\cdot \\mathrm { s i g n } ( \\pmb { g } _ { t } ) \\right\\} , } \\end{array}\n$$",
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+    {
+        "type": "text",
+        "text": "where $\\mathbf { \\nabla } _ { \\mathbf { \\boldsymbol { g } } _ { t } }$ denotes the momentum, ${ \\bf { \\mathit { g } } } _ { 0 } = { \\bf { 0 } }$ , and $\\mu$ denotes the decay factor. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We combine EM and PN as Experienced Precise Nesterov momentum (EPN) to further improve transferability. The algorithm of EPNI-FGSM, which integrates EPN into I-FGSM, is summarized in Algorithm 1. Particularly, if $\\frac { \\nabla _ { \\widetilde { \\pmb { x } } _ { t } ^ { a d v } } J ( f ( \\widetilde { \\pmb { x } } _ { t } ^ { a d v } ; \\pmb { \\theta } ) , y ) } { | | \\nabla _ { \\widetilde { \\pmb { x } } _ { t } ^ { a d v } } J ( f ( \\widetilde { \\pmb { x } } _ { t } ^ { a d v } ; \\pmb { \\theta } ) , y ) | | _ { 1 } } = \\mathbf { 0 }$ , EPNI-FGSM degrades to Experienced MI-FGSM (EMI-FGSM). If $\\frac { \\nabla _ { \\hat { \\pmb { x } } _ { t } ^ { a d v } } J ( f ( S ( \\hat { \\pmb { x } } _ { t } ^ { a d v } ) ; \\pmb { \\theta } ) , y ) } { | | \\nabla _ { \\hat { \\pmb { x } } _ { t } ^ { a d v } } J ( f ( S ( \\hat { \\pmb { x } } _ { t } ^ { a d v } ) ; \\pmb { \\theta } ) , y ) | | _ { 1 } } = \\mathbf { 0 }$ , EPNI-FGSM degrades to Experienced NI-FGSM (ENI-FGSM). If $e p o c h s = 0$ , EPNI-FGSM degrades to PNI-FGSM. ",
+        "bbox": [
+            173,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            849,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We conduct extensive experiments on normally trained and defense models to validate that our EPN is more efficient than conventional momentum. We first present the experimental settings in Sec. 4.1. Then, we report the results for attacking normally trained and defense models in Sec. 4.2 and ",
+        "bbox": [
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+            881,
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+        ],
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+    },
+    {
+        "type": "table",
+        "img_path": "images/95f94de990d614b9f271a99dcbf24c84578b37b989b360a791135bb430333780.jpg",
+        "table_caption": [
+            "Table 2: The attack success rates $( \\% )$ of adversarial examples crafted on source models against normally trained target models. “\\*” indicates the model being white-box attacked. “Avg” means the average attack success rate. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=\"6\">Iv3</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>69.2 56.7</td><td>67.9</td><td>68.2</td><td>66.2</td><td>63.0</td><td>61.4</td><td>61.2</td><td></td></tr><tr><td>TI-MI-FGSM SI-NI-FGSM DI-EPNI-FGSM(Ours) TI-EPNI-FGSM(Ours)</td><td>41.8 77.4 88.7 70.6 90.2</td><td>99.7* 100.0* 100.0* 100.0* 100.0*</td><td>37.9 72.7 87.3 68.8 87.1</td><td>31.5 69.8 86.0 59.6</td><td>46.4 67.6 78.5 69.3</td><td>47.6 67.3 81.3 69.8</td><td>44.9 68.0 83.4 68.9</td><td>41.9 61.5 80.2 65.5 76.4</td><td>38.6 62.1 80.5 63.3</td><td>54.9 70.0 80.4 73.5</td><td>72.8 83.8 76.5</td><td>56.2 73.8 83.8 78.5</td><td>58.3 71.9 84.2 77.9</td><td>47.9 70.7 86.1 74.7</td><td>46.5 68.4 84.9 72.9</td><td>43.5 64.8 82.9 70.8</td><td>42.8 65.9 83.4 71.5</td><td>66.0 49.2 70.9 84.4 72.5</td></tr><tr><td>SI-EPNI-FGSM(Ours) VT-MI-FGSM VT-NI-FGSM</td><td>70.5 76.2</td><td>100.0* 100.0*</td><td>69.7 76.2</td><td>85.1 64.9 70.5</td><td>80.1 61.7 65.6</td><td>81.2 63.6 68.4</td><td>81.7 65.7 71.2</td><td>60.0 65.2</td><td>78.2 58.6 64.1</td><td>80.6 65.0 70.5</td><td>83.1 69.2 73.4</td><td>84.4 68.7 73.4</td><td>82.6 66.7 73.2</td><td>86.7 65.8 71.2</td><td>83.8 63.8 71.3</td><td>80.9 65.0 69.2</td><td>81.7 62.6 68.2</td><td>83.8 67.1 72.2</td></tr><tr><td>VT-EPNI-FGSM(Ours) FI-MI-FGSM FI-EPNI-FGSM(Ours)</td><td>84.8 85.8 90.0</td><td>100.0* 97.1* 97.4*</td><td>83.3 85.4 88.0</td><td>78.1 81.8 84.8</td><td>77.0 79.4 81.0</td><td>78.8 80.1 82.4</td><td>79.4 79.5 84.3</td><td>75.8 76.3 79.6</td><td>73.3 74.5 78.5</td><td>80.2 81.0 85.5</td><td>83.3 82.7 87.0</td><td>82.0 82.7 85.9</td><td>81.5 83.2 86.3</td><td>79.7 81.3 84.0</td><td>80.1 78.8 83.5</td><td>78.3 77.2 80.5</td><td>78.5 77.8 81.7</td><td>80.8 81.4</td></tr><tr><td>MI-FGSM NI-FGSM EPNI-FGSM(Ours)</td><td>46.4 49.0 68.1</td><td>44.0 45.6 65.3</td><td>45.4 48.5 68.4</td><td>99.6* 100.0* 100.0*</td><td>40.0 40.6 55.2</td><td>39.2 38.7 53.8</td><td>41.3 42.1 55.5</td><td>33.9 34.9 48.4</td><td>32.7 32.9 47.9</td><td>49.9 53.5 63.8</td><td>53.2 54.3 67.8</td><td>51.4 54.6 66.6</td><td>50.6 54.3 65.3</td><td>38.6 37.5 54.7</td><td>35.3 37.5 50.7</td><td>35.0 33.8 49.0</td><td>32.7 33.2 48.1</td><td>84.7 45.2 46.5 60.5</td></tr><tr><td>DI-MI-FGSM TI-MI-FGSM SI-NI-FGSM DI-EPNI-FGSM(Ours) TI-EPNI-FGSM(Ours) SI-EPNI-FGSM (Ours) VT-MI-FGSM</td><td>63.2 41.5 74.7 82.8 65.1</td><td>62.3 44.5 81.7 83.2 68.2</td><td>65.4 42.0 75.1 84.1 67.6</td><td>98.4* 97.9* 99.3* 100.0* 99.9* 100.0*</td><td>54.2 47.1 65.4 70.7 66.6</td><td>54.2 47.6 65.8 72.3 69.5</td><td>55.7 47.3 69.7 74.4 68.9</td><td>48.6 42.9 65.4 67.5 65.2</td><td>51.0 41.8 64.7 71.1 63.2</td><td>59.3 55.9 68.9 74.6 73.8</td><td>64.4 56.7 71.6 79.5 75.0</td><td>62.8 55.5 71.8 78.2 73.6</td><td>64.7 53.9 73.0 79.6 74.2</td><td>57.0 49.2 72.1 75.0 72.9</td><td>52.8 45.3 68.7 73.0 68.4</td><td>51.8 46.9 67.3 69.6 69.1</td><td>52.6 41.9 68.0 70.9 65.4</td><td>59.9 50.5 72.0 76.9 71.0</td></tr><tr><td></td><td>VT-NI-FGSM VT-EPNI-FGSM(Ours)</td><td>90.9 63.2 66.3 81.0</td><td>93.1 66.6 71.0 82.1</td><td>88.6 68.8 99.6* 73.1 99.8*</td><td>79.1 57.8 59.3</td><td>81.3 57.8 59.3</td><td>85.5 60.7 62.3 75.4</td><td>79.6 53.6 56.9 70.0</td><td>82.9 54.9 56.9 70.6</td><td>83.2 61.7 66.8 77.3</td><td>85.5 63.5 68.2 80.4</td><td>86.0 64.4 68.2 79.9</td><td>86.1 64.0 68.1</td><td>86.3 59.8 60.9</td><td>83.6 56.4 58.0 73.4</td><td>81.5 54.2 57.2</td><td>84.6 56.1 58.5</td><td>85.8 62.5 65.3</td></tr><tr><td></td><td>FI-MI-FGSM FI-EPNI-FGSM (Ours)</td><td>76.3 78.0</td><td>76.0 77.3</td><td>83.1 76.3 89.7* 78.0 91.0*</td><td>100.0* 71.7 67.5 69.8</td><td>72.4 68.0 69.9</td><td>69.7 72.2</td><td>66.8 68.9</td><td>65.5 67.4</td><td>72.5 73.5</td><td>73.7 74.0</td><td>73.4 74.5</td><td>78.7 73.5 73.6</td><td>77.3 70.7 71.7</td><td>68.3 69.0</td><td>72.0 66.8 66.9</td><td>71.9 67.0 67.2</td><td>77.5 71.9 73.1</td></tr><tr><td rowspan=\"5\">R152</td><td>MI-FGSM NI-FGSM EPNI-FGSM(Ours)</td><td>68.9 75.7 91.5</td><td>59.4 64.1 85.8</td><td>53.8 58.0 79.2 74.2</td><td>49.4 81.0 51.7 86.0</td><td>83.3 87.7 94.8 96.2</td><td>92.1 94.7 98.7</td><td>94.5 96.9 99.4</td><td></td><td>100.0* 100.0* 100.0*</td><td>72.7 74.1 76.1 77.4 88.7 89.8</td><td>72.5 77.0 89.5</td><td></td><td>72.5 76.2 89.6</td><td>86.5 87.4 96.5</td><td>83.0 85.1 96.5</td><td>82.5 84.0 85.5 86.7 97.6 96.7</td><td></td><td>77.1 80.4 92.0</td></tr><tr><td>DI-MI-FGSM TI-MI-FGSM SI-NI-FGSM DI-EPNI-FGSM(Ours) TI-EPNI-FGSM(Ours) SI-EPNI-FGSM(Ours)</td><td>85.6 57.1 85.1 97.6 80.3</td><td>82.9 51.3 77.2 96.9 78.2</td><td>75.5 50.1 71.1 94.2 75.7</td><td>72.1 41.4 66.5 92.1</td><td>91.8 70.9 89.6 98.0 89.3</td><td>93.8 75.0 92.0 98.9 91.4</td><td>96.0 80.5 95.3 99.4 94.0</td><td>96.8 84.5 97.8 99.6 95.5</td><td>100.0* 100.0* 100.0* 100.0* 100.0*</td><td>84.0 65.4 82.1 94.3 81.7</td><td>84.4 65.0 83.0 94.6 81.2</td><td>85.5 64.1 81.9 94.7 80.8</td><td>84.6 64.0 81.9 95.2 81.5</td><td>93.7 75.1 93.6 99.4 91.8</td><td>94.0 71.5 91.8 99.3 89.2</td><td>93.7 70.7 92.4 99.6 91.0</td><td>94.4 68.1 91.8 99.1 89.7</td><td>88.8 67.9 86.7 97.2 85.8</td></tr><tr><td>VT-MI-FGSM VT-NI-FGSM VT-EPNI-FGSM(Ours)</td><td>95.3 83.8 87.7 95.4</td><td>91.4 79.6 81.9 92.1</td><td>88.2 74.1 78.9</td><td>66.8 84.8 70.9 74.4</td><td>96.7 92.2 93.5</td><td>97.6 93.7 94.9</td><td>98.6 96.4 98.2</td><td>99.0 97.5 98.7</td><td>100.0* 100.0* 100.0*</td><td>90.3 83.6 85.3</td><td>91.2 82.8 87.2</td><td>92.2 84.4 88.2</td><td>90.5 83.8 86.9</td><td>99.0 94.2 96.1</td><td>98.2 92.4 94.9</td><td>98.1 93.1 95.1</td><td>98.2 93.6 95.7</td><td>94.7 88.0 90.4</td></tr><tr><td>FI-MI-FGSM FI-EPNI-FGSM(Ours) MI-FGSM</td><td>93.3 95.4 78.0</td><td>88.6 93.2</td><td>89.7 88.7 93.1</td><td>86.9 85.1 90.1</td><td>97.9 95.5 96.8</td><td>98.8 96.8 97.9</td><td>99.5 97.5 98.3</td><td>99.8 98.9 98.9</td><td>100.0* 99.9* 100.0*</td><td>93.6 92.5 94.1</td><td>93.5 92.0 94.8</td><td>93.7 93.0 95.0</td><td>94.4 93.8 95.1</td><td>99.4 97.0 97.6</td><td>98.7 95.7 97.4</td><td>99.4 96.6 97.3</td><td>98.8 96.9 97.6</td><td>96.0 94.2 96.0</td></tr><tr><td>NI-FGSM EPNI-FGSM(Ours) DI-MI-FGSM TI-MI-FGSM SI-NI-FGSM V16</td><td>78.8 93.5 88.4 60.6</td><td>59.2 61.6 82.3 75.6 51.1</td><td>63.8 68.8 87.0 77.5</td><td>43.8 47.3 68.1 59.5</td><td>80.0 82.3 91.5 87.8</td><td>73.3 76.1 90.0 83.9</td><td>75.0 78.0 90.0 86.2</td><td>63.7 67.0 82.9 73.8 53.6</td><td>58.7 60.4 77.6 69.7 48.4</td><td>94.7 96.8 99.0 98.3</td><td>98.3 99.2 100.0 98.8</td><td>99.8* 99.9* 100.0* 100.0*</td><td>99.2 99.1 99.9 99.4</td><td>77.0 79.1 91.2 88.0</td><td>69.5 72.1 87.3 81.5</td><td>65.3 66.4 85.8 78.2</td><td>64.9 66.0 87.0 77.5</td><td>74.4 76.4</td><td>89.0 83.8 64.6</td></tr><tr><td>VT-MI-FGSM VT-NI-FGSM</td><td>DI-EPNI-FGSM(Ours) TI-EPNI-FGSM(Ours) SI-EPNI-FGSM(Ours)</td><td>89.8 96.2 82.5 96.1 87.5</td><td>77.5 90.6 74.6 89.1 75.2</td><td>48.5 80.7 62.5 92.6 81.9 77.4 58.1 91.8 80.0 77.9 61.9</td><td>33.4</td><td>71.2 89.1 95.9 88.4 93.6 89.9</td><td>65.4 84.8 94.1 84.2 91.2 86.1</td><td>60.3 85.9 93.5 82.7 92.5 86.8</td><td>77.1 88.6 75.9 86.4 78.3</td><td>73.0 85.5 70.4 84.8 74.1</td><td>88.1 98.5 99.8 96.8 99.4 97.7</td><td>92.9 99.6 99.9 98.2 100.0 98.9</td><td>99.8* 100.0* 100.0* 100.0* 100.0* 99.9*</td><td>94.4 100.0 100.0 98.1 100.0 99.4</td><td>64.9 87.8 95.6 86.4 95.6 87.8</td><td>58.0 81.9 93.1 80.7 92.6 82.2</td><td>54.8 78.9 93.0 78.8 91.0 81.3</td><td>52.4 80.2 91.4 77.4 92.2 80.6</td><td>85.1 93.6 83.0 92.7 85.0</td></tr><tr><td>VT-EPNI-FGSM(Ours) FI-MI-FGSM</td><td></td><td>89.8 95.6 95.9</td><td>76.9 88.4 89.1</td><td>80.7 65.6 92.4 81.1 93.1 79.7</td><td>92.1 96.1 95.6</td><td>87.5 94.4 94.9</td><td>89.3 94.0 93.4</td><td>81.1 90.4 90.4</td><td>76.1 88.8 87.1</td><td></td><td>98.7 99.4 99.6</td><td>99.5 99.9 99.8</td><td>99.9* 99.9* 100.0*</td><td>99.4 99.9 99.8</td><td>88.7 95.6 94.3</td><td>85.4 93.7 91.5</td><td>82.5 93.2 90.2</td><td>82.9 92.2 88.6</td><td>86.8 93.8 93.1</td></tr><tr><td>FI-EPNI-FGSM(Ours)</td><td></td><td>96.7</td><td>90.4</td><td>94.1 84.3</td><td>96.7</td><td>95.9</td><td></td><td>96.0</td><td>92.6</td><td>90.1</td><td>99.8</td><td>99.8</td><td>100.0*</td><td>99.9</td><td>96.4</td><td>94.3</td><td>93.3</td><td>93.0</td><td>94.9</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Sec. 4.3, respectively. Finally, we provide ablation studies in Sec. 4.4. Table 1 introduces the abbreviations used in the paper. ",
+        "bbox": [
+            176,
+            650,
+            823,
+            679
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1 EXPERIMENTAL SETTINGS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            700,
+            398,
+            714
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Dataset. We follow the previous works (Dong et al., 2019; Wang et al., 2021) to use the DEV dataset from the NIPS17 Adversarial Attacks and Defenses Competition. This dataset contains 1000 images with size $2 9 9 \\times 2 9 9$ . ",
+        "bbox": [
+            176,
+            728,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Target Models. Seventeen normally trained models, i.e., GoogLeNet (Iv1) (Szegedy et al., 2015), Inception-v3 (Iv3) (Szegedy et al., 2016), Inception-v4 (Iv4), Inception-ResNet-v2 (IRv2) (Szegedy et al., 2017), ResNet-18 (R18), ResNet-34 (R34), ResNet-50 (R50), ResNet-101 (R101), ResNet152 (R152) (He et al., 2016), VGG11 (V11), VGG13 (V13), VGG16 (V16), VGG19 (V19) (Simonyan & Zisserman, 2014), DenseNet-121 (D121), DenseNet-169 (D169), DenseNet-201 (D201), and DenseNet-161 (D161) (Huang et al., 2017). Ten defense models (i.e., adversarially trained models), i.e., Adv-Inception-v3 $( \\mathrm { I v } 3 _ { \\mathrm { a d v } } )$ , Ens-Inception-Resnet-v2 $( \\mathrm { I R } \\mathrm { v } 2 _ { \\mathrm { e n s } }$ ) Tramer et al. (2017), \\` Adv-EfficientNet-b0 $( \\mathrm { E b 0 _ { a d v } ) }$ to Adv-EfficientNet-b7 $( \\mathrm { E b } 7 _ { \\mathrm { a d v } } ,$ ). ",
+        "bbox": [
+            173,
+            776,
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+            888
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Baselines. For fair comparison of our EPN and conventional momentum, we replace conventional momentum with our EPN in momentum-based attacks, i.e., MI-FGSM (Dong et al., 2018), NI",
+        "bbox": [
+            173,
+            895,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/a957ef153e01230375406a247adb7e50ffabb5ed67bc7d3946922fab3b5fad65.jpg",
+        "table_caption": [
+            "Table 3: The attack success rates $( \\% )$ of adversarial examples crafted on source models against defense models. “Avg” means the average attack success rate. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Models</td><td>Attacks</td><td>Iv3adv</td><td>IRv2ens</td><td>Eb0adv</td><td>Ebladv</td><td>Eb2adv</td><td>Eb3adv</td><td>Eb4adv</td><td>Eb5adv</td><td>Eb6adv</td><td>Eb7adv</td><td>Avg</td></tr><tr><td rowspan=\"9\">Iv3</td><td>MI-FGSM NI-FGSM EPNI-FGSM(Ours) DI-MI-FGSM</td><td>25.4 25.4 32.4 33.0</td><td>11.5 11.6 14.3 19.3</td><td>29.7 34.3 49.0 46.6</td><td>26.5 31.0 45.0 42.9</td><td>28.1 31.2 45.4 42.4</td><td>18.8 22.2 31.1 31.3</td><td>16.9 18.6 23.9 27.4</td><td>17.5 18.6 25.7 26.7</td><td>14.8 16.0 22.4</td><td>15.2 16.5 22.3 24.4</td><td>20.4 22.5 31.2 31.7</td></tr><tr><td>TI-MI-FGSM SI-NI-FGSM DI-EPNI-FGSM(Ours) TI-EPNI-FGSM(Ours)</td><td>29.9 37.4 41.5 54.1</td><td>21.5 23.2 25.2 41.2</td><td>35.2 50.2 65.0 60.7</td><td>32.6 46.1 65.5 58.2</td><td>34.9 46.3 64.7 57.0</td><td>27.8 33.3 45.5 46.6</td><td>29.3 30.9 39.1 46.7</td><td>28.2 29.6 41.6 45.3</td><td>22.9 24.5 27.6 36.7 43.4</td><td>26.6 25.8 35.6 44.2</td><td>29.1 35.0 46.0 49.7</td></tr><tr><td>SI-EPNI-FGSM(Ours) VT-MI-FGSM VT-NI-FGSM</td><td>47.4 36.8 39.1</td><td>28.4 25.1 26.4</td><td>66.3 50.5 54.0</td><td>64.7 46.3 50.6</td><td>63.1 44.9 49.5</td><td>46.7 33.5 36.6</td><td>41.8 29.4 31.0</td><td>42.1 29.4 32.7</td><td>37.9 26.8 30.3</td><td>37.3 26.2 28.5</td><td>47.6 34.9 37.9</td></tr><tr><td>VT-EPNI-FGSM(Ours) FI-MI-FGSM FI-EPNI-FGSM(Ours)</td><td>43.1 55.3 58.9</td><td>26.5 38.0 39.9</td><td>63.2 68.2 76.2</td><td>61.8 67.8 73.3</td><td>60.6 64.9 72.7</td><td>46.9 53.6 61.1</td><td>41.1 50.0 56.0</td><td>39.7 49.3 55.6</td><td>38.2 46.6 52.9</td><td>36.8 45.1 50.5</td><td>45.8 53.9 59.7</td></tr><tr><td>MI-FGSM NI-FGSM EPNI-FGSM(Ours) DI-MI-FGSM</td><td>27.3 25.9 33.5 33.1</td><td>15.5 14.9 15.9 24.7</td><td>25.2 24.8 36.3</td><td>22.8 23.1 32.4</td><td>24.3 23.6 34.9</td><td>17.0 17.8 23.9 27.1</td><td>13.8 14.1 20.7 24.6</td><td>14.6 14.9 20.4 23.2</td><td>12.1 12.8 17.6 22.0</td><td>13.2 13.7 17.8 20.8</td><td>18.6 18.6 25.3 28.7</td></tr><tr><td>TI-MI-FGSM SI-NI-FGSM DI-EPNI-FGSM(Ours) TI-EPNI-FGSM(Ours) SI-EPNI-FGSM(Ours)</td><td>38.1 37.0 39.4 58.3 45.6</td><td>32.2 33.0 29.7 52.9 41.5</td><td>38.1 39.0 50.8 57.4 60.5 68.9</td><td>35.8 36.7 48.2 52.4 58.3</td><td>37.9 39.4 46.9 53.4 56.6 66.8</td><td>30.2 35.7 41.0 49.7 50.2</td><td>33.8 32.0 33.6 50.9 42.6</td><td>31.4 28.7 33.3 47.0 43.5</td><td>29.2 26.9 30.2 48.0 39.7</td><td>29.7 27.5 30.6 46.5 40.1</td><td>34.0 36.7 40.1 52.9 50.3</td></tr><tr><td>VT-MI-FGSM VT-NI-FGSM VT-EPNI-FGSM(Ours) FI-MI-FGSM</td><td>38.8 40.5 48.2</td><td>36.4 34.8 36.6</td><td>42.3 44.6 59.5</td><td>64.3 38.9 41.1 55.3</td><td>41.2 43.4 54.1</td><td>31.2 33.1 43.0</td><td>28.4 28.2 38.8</td><td>26.4 26.6 38.0</td><td>25.4 25.8 36.9</td><td>25.4 26.2 35.3</td><td>33.4 34.4 44.6 46.7</td></tr><tr><td>FI-EPNI-FGSM(Ours) MI-FGSM NI-FGSM EPNI-FGSM(Ours)</td><td>54.5 53.3 36.5 40.1</td><td>40.1 47.1 27.8 29.4</td><td>57.4 59.6 46.6 49.8</td><td>56.7 56.8 43.6 47.3</td><td>56.3 57.9 47.3 47.8</td><td>45.7 47.0 30.9 33.2</td><td>40.9 43.3 27.3 29.1</td><td>39.7 42.1 26.5 27.7</td><td>38.2 40.6 22.0 25.2</td><td>37.4 38.6 24.8 25.4</td><td>48.6 33.3 35.5</td></tr><tr><td>DI-MI-FGSM TI-MI-FGSM SI-NI-FGSM R152</td><td>53.1 57.6 46.8 51.7 DI-EPNI-FGSM(Ours) 78.3</td><td>43.1 51.7 41.1 43.4 73.1</td><td>71.9 72.6 50.7 62.8 91.6</td><td>68.1 68.3 46.4 57.3 89.9</td><td>70.5 71.2 50.1 61.2 90.6</td><td>50.2 54.7 41.8 43.4 75.7</td><td>41.0 47.1 43.2 39.3 67.3</td><td></td><td>41.0 37.5 44.7 44.2 41.0 36.2 36.0 35.4 64.9 62.6</td><td>40.2 44.3 38.7 33.8 64.3</td><td>51.7 55.6 43.6 46.4 75.8</td></tr><tr><td>VT-MI-FGSM VT-NI-FGSM FI-MI-FGSM</td><td>TI-EPNI-FGSM(Ours) SI-EPNI-FGSM(Ours) VT-EPNI-FGSM(Ours)</td><td>72.2 67.4 67.5 59.3 59.1 52.6 61.6 55.5 72.6 71.6</td><td>74.7 79.4 69.6 73.6 86.2</td><td>70.1 78.5 64.9 68.2 85.3</td><td></td><td>74.2 81.1 69.3 72.1 88.4</td><td>62.9 61.8 55.1 56.1 74.1</td><td>62.7 53.7 48.8 50.6 66.0</td><td>60.7 51.5 46.1 48.7 64.8</td><td>59.3 58.3 48.8 50.3 43.6 44.6 45.5 47.9 62.2 65.1</td><td></td><td>66.3 63.2 55.4 58.0 73.6</td></tr><tr><td>TI-MI-FGSM V16</td><td>FI-EPNI-FGSM(Ours) MI-FGSM NI-FGSM EPNI-FGSM(Ours) DI-MI-FGSM</td><td>81.0 33.0 33.0 46.7 44.9</td><td>73.3 20.3 22.5 29.0 31.6</td><td>88.5 48.8 49.5 71.2 62.4</td><td>87.0 41.8 43.6 65.7 55.9</td><td>82.6 88.6 41.3 44.3 63.5 56.9</td><td>76.3 25.1 27.7 41.5 37.5</td><td>70.9 21.5 22.2 31.3 32.0</td><td>70.1 21.7 22.7 31.5 30.9</td><td>66.4 19.0 18.9 28.5 27.8</td><td>66.4 21.8 20.1 28.7 28.1</td><td>76.9 29.4 30.5 43.8 40.8</td></tr><tr><td></td><td>SI-NI-FGSM DI-EPNI-FGSM(Ours) TI-EPNI-FGSM(Ours) SI-EPNI-FGSM(Ours) VT-MI-FGSM VT-NI-FGSM</td><td>38.9 50.0 61.2 61.8 62.8 49.4 49.3</td><td>29.4 33.5 44.7 49.9 42.6 35.4 37.5</td><td>44.3 61.4 80.3 68.6 78.6 65.3</td><td>38.0 56.4 77.9 65.2 75.4 58.7</td><td>41.4 54.4 78.2 63.7 74.3 58.6</td><td>31.1 36.5 52.2 49.2 48.1 39.7</td><td>32.0 29.9 43.5 50.3 40.6 34.3</td><td>31.5 30.0 42.6 47.7 41.9 33.6</td><td>28.5 27.2 40.2 47.6 37.9 29.2</td><td>28.2 26.4 38.8 46.1 35.5 32.2</td><td>34.3 40.6 56.0 55.0 53.8</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "FGSM (Lin et al., 2019), DI-MI-FGSM (Xie et al., 2019), TI-MI-FGSM (Dong et al., 2019), SI-NIFGSM (Lin et al., 2019), VT-MI-FGSM (Wang & He, 2021), VT-NI-FGSM (Wang & He, 2021) and FI-MI-FGSM (Wang et al., 2021). Then we compare the transferability of conventional momentumbased attacks and our EPN-based attacks. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Hyperparameters. In all experiments, we follow the official default settings for hyperparameters. Specifically, the maximum perturbation $\\epsilon = 1 6$ , the number of iterations $T = 1 0$ , the step size $\\alpha = \\epsilon / T = 1 . 6$ , and the decay factor $\\mu = 1 . 0$ . For DIM (Xie et al., 2019), the probability $p$ is set to 0.5. For TIM (Dong et al., 2019), the size of the Gaussian kernel is set to $1 5 { \\times } 1 5$ . For SIM (Lin et al., 2019), the number of scale copies $m$ is set to 5. For VT-MI-FGSM (Wang & He, 2021) and VT-NIFGSM (Wang & He, 2021), the number of sampled examples $N$ is set to 20, and the parameter $\\beta$ for the upper bound of the neighborhood is set to 1.5. For FI-MI-FGSM (Wang et al., 2021), the drop probability $p _ { d }$ is set to 0.3 when attacking normally trained models and 0.1 when attacking defense models, the ensemble number $N$ is set to 30 in aggregate gradient, and the intermediate layer is set to Mixed ${ } _ { 5 b }$ for Iv3, Conv 4a for IRv2, Conv3 3 for V16 as well as the last layer of the second block for R152. For our EM-based attacks, epochs is set to 5. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/fafd2e428be0acdf7ede725979fa5a3c2e99054d62d3d5a67513a93fbcd1c074.jpg",
+        "image_caption": [
+            "Figure 3: The average attack success rates $( \\% )$ of the adversarial examples crafted on source models against normally trained models (except the source model) and defense models for various $\\mu$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/13699bec5c115b8c6b25669e76ccd9babb24eaeb2ee8d0e2bc3561395399d59a.jpg",
+        "image_caption": [
+            "Figure 4: The average attack success rates $( \\% )$ of the adversarial examples crafted on source models against normally trained models (except the source model) and defense models for various epochs. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "4.2 ATTACK NORMALLY TRAINED MODELS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To validate that EPN-based attacks have higher transferability than conventional momentum-based attacks, we choose Iv3, IRv2, R152, and V16 as the source model, respectively, and attack normally trained target models via our EPN-based methods and baseline methods. The attack success rates are shown in Table 2. The results show that the attack success rates of our EPN-based methods are ${ \\sim } 1 1 . 9 \\%$ higher than baseline methods on average, In particular, our EPN-based attacks have the best transferability against normally trained target models when the source model is R152. Specifically, the attack success rates of EPNI-FGSM, DI-EPNI-FGSM, VT-EPNI-FGSM, and FI-EPNI-FGSM are $9 2 . 0 \\%$ , $9 7 . 2 \\%$ , $9 6 . 0 \\%$ , and $9 6 . 0 \\%$ , respectively, on average. Therefore, the experiments demonstrate that our EPN improves transferability more effectively than conventional momentum against normally trained models. ",
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+    },
+    {
+        "type": "text",
+        "text": "4.3 ATTACK DEFENSE MODELS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To further compare the transferability, we also use defense models as the target models, and the source models are still Iv3, IRv2, R152, and V16. We craft adversarial examples on the source model via our EPN-based methods and baseline methods to attack defense models. The attack success rates are shown in Table 3. The results show that the attack success rates of our EPNbased methods are ${ \\sim } 1 3 . 1 \\%$ higher than baseline methods on average. Adversarial examples crafted on R152 still show the best transferability against defense models. Specifically, the attack success rates of EPNI-FGSM, DI-EPNI-FGSM, VT-EPNI-FGSM, and FI-EPNI-FGSM are $5 1 . 7 \\%$ , $7 5 . 8 \\%$ , $7 3 . 6 \\%$ , and $7 6 . 9 \\%$ , respectively, on average. The results of experiments indicate that our EPN is still more effective than conventional momentum against defense models. ",
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+    {
+        "type": "text",
+        "text": "4.4 ABLATION STUDY ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We conduct ablation studies for EPNI-FGSM. We investigate the impacts of two hyperparameters (i.e., the decay factor $\\mu$ and the epochs of pretraining epochs) on the transferability of EPNI-FGSM in Sec. 4.4.1. We further study the impacts of EM and PN on transferability in Sec. 4.4.2. ",
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+    {
+        "type": "text",
+        "text": "4.4.1 IMPACTS OF $\\mu$ AND epochs ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The source models are set to Iv3, IRv2, R152, and V16. We use EPNI-FGSM to craft adversarial examples to attack normally trained models and defense models, respectively. We investigate the impacts of $\\mu$ and epochs on the transferability of EPNI-FGSM by counting the average attack success rates against normally trained models (except the source model) and defense models. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The decay factor $\\mu .$ . The decay factor $\\mu$ plays a vital role for momentum. If $\\mu = 0$ , the momentumbased attacks degrade to vanilla iterative gradient-based attacks. If $0 < \\mu < 1$ , the previous gradients accumulated in the momentum decay exponentially. If $\\mu = 1$ , the momentum simply adds up all previous gradients. If $\\mu > 1$ , the previous gradients accumulated in the momentum grow exponentially. We pre-set epoch $s = 5$ and set $\\mu$ from 0.0 to 2.0 with a step size of 0.1. The average attack success rates are shown in Fig. 3. When $\\mu \\leq 1 . 0$ , the average attack success rates show an upward trend, and when $\\mu \\geq 1 . 0$ , the average attack success rates show a downward trend. Therefore, we set $\\mu = 1 . 0$ for EPNI-FGSM to achieve the best transferability. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/e2f2ae241e3bdb3bc56d7fb243e0dfe1fdab5ecc0bb0f52c3961f4f1620f6522.jpg",
+        "image_caption": [
+            "Figure 5: The average attack success rates $( \\% )$ of the adversarial examples crafted on source models against normally trained models and defense models via NI-FGSM, ENI-FGSM, PNI-FGSM, and EPNI-FGSM. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The epochs of pretraining epochs. epochs affects the amount of gradient accumulated in EM. We pre-set $\\mu = 1 . 0$ and set epochs from 0 to 10 with a step size of 1. The average attack success rates are shown in Fig. 4. As epochs increases, the average attack success rates increase and gradually converge. Since the larger epochs, the higher the computational cost, we set epochs $= 5$ for EPNIFGSM to strike a balance between computational cost and transferability. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In summary, we set the decay factor $\\mu = 1 . 0$ and the epochs of pretraining epochs $= 5$ for EPNIFGSM. Similarly, such two hyperparameters of other EPN-based attacks have the same settings as EPNI-FGSM. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.4.2 IMPACTS OF EM AND PN ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The source models are the same as in Sec 4.4.1. To investigate the impacts of EM and PN, we craft adversarial examples on source models via ENI-FGSM (only with EM), PNI-FGSM (only with PN), and EPNI-FGSM (with EM and PN), respectively. In addition, we also use MI-FGSM and NI-FGSM (without EM and PN) as baselines. For ENI-FGSM, the epochs of pretraining epochs is set to 5. For ENI-FGSM and PNI-FGSM, the decay factor $\\mu$ is set to 1.0. The average attack success rates of the adversarial examples against normally trained models and defense models are shown in Fig. 5. The average attack success rates of ENI-FGSM are higher than MI-FGSM and NIFGSM, demonstrating that EM improves transferability more than conventional momentum. The same is true for PN. Besides, the average attack success rates of EPNI-FGSM are higher than that of ENI-FGSM and PNI-FGSM, demonstrating that the combination of EM and PN can further improve transferability. ",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this work, we proposed Experienced Momentum (EM) and Precise Nesterov momentum (PN) to boost transferability. Specifically, EM is trained on a set of derived models by Random Channels Swapping (RCS), and then conventional momentum is initialized to EM, which can accelerate optimization to escape from saddle points and poor local extrema during early iterations to improve transferability. Additionally, we adopted the current gradient to refine the pre-update of conventional Nesterov momentum, called PN. Then, we naturally combined EM and PN as EPN to improve transferability further. Extensive experiments demonstrate that EPN-based attacks have higher transferability than conventional momentum-based attacks. However, our methods still adopt a fixed learning rate or step size that is crucial for the optimizer. Therefore, we will explore the impact of learning rate or step size on transferability in future work. ",
+        "bbox": [
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+    {
+        "type": "text",
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parse/dev/Sxk8Bse3RKO/Sxk8Bse3RKO.md

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+# Reconstructing Training Data from Trained Neural Networks
+
+Niv Haim∗ Weizmann Institute of Science niv.haim@weizmann.ac.il
+
+Gal Vardi∗† TTI-Chicago and Hebrew University galvardi@ttic.edu
+
+Gilad Yehudai∗ Weizmann Institute of Science gilad.yehudai@weizmann.ac.il
+
+Ohad Shamir Weizmann Institute of Science ohad.shamir@weizmann.ac.il
+
+Michal Irani Weizmann Institute of Science michal.irani@weizmann.ac.il
+
+Project page: https://giladude1.github.io/reconstruction
+
+# Abstract
+
+Understanding to what extent neural networks memorize training data is an intriguing question with practical and theoretical implications. In this paper we show that in some cases a significant fraction of the training data can in fact be reconstructed from the parameters of a trained neural network classifier. We propose a novel reconstruction scheme that stems from recent theoretical results about the implicit bias in training neural networks with gradient-based methods. To the best of our knowledge, our results are the first to show that reconstructing a large portion of the actual training samples from a trained neural network classifier is generally possible. This has negative implications on privacy, as it can be used as an attack for revealing sensitive training data. We demonstrate our method for binary MLP classifiers on a few standard computer vision datasets.
+
+# 1 Introduction
+
+It is commonly believed that neural networks memorize the training data, even when they are able to generalize well to unseen test data (e.g., [Zhang et al., 2021, Feldman, 2020]). Exploring this memorization phenomenon is of great importance both practically and theoretically. Indeed, it has implications on our understanding of generalization in deep learning, on the hidden representations learnt by neural networks, and on the extent to which they are vulnerable to privacy attacks.
+
+A fundamental question for understanding memorization is:
+
+Are the specific training samples encoded in the parameters of a trained classifier? Can they be recovered from the network parameters?
+
+In this work, we study this question, and devise a novel scheme which allows us to reconstruct a significant portion of the training data from the parameters of a trained neural network alone, without having any additional information on the data. Thus, we provide a proof-of-concept that the learning (a) Top 24 images reconstructed from a binary classifier trained on 50 CIFAR10 images (b) Their corresponding nearest neighbours from the training-set of the model process can sometimes be reversed: That is, instead of learning a model given a training dataset, it is possible to find the training data given a trained model. In Figure 1 we show how our approach reconstructs images from the CIFAR10 dataset, given a simple trained binary classifier.
+
+![](images/ed282a7049c35ade42ab306a6ef56700e2d645bdbbdfcd5644417349b411ddc9.jpg)
+
+![](images/7cf960fa2f82434a9efbbc4e9e6c61ec64f0bfc24a1869967b39feb8256f2cee.jpg)  
+Figure 1: Reconstruction of training images from a pretrained binary classifier, trained on 50 CIFAR10 images. The two classes are “animals” and “vehicles”. We calculate the nearest neighbor using the SSIM metric.
+
+Many works try to “crack” neural networks by analyzing and visualizing either their learnt parameters or representations [Erhan et al., 2009, Mahendran and Vedaldi, 2015, Olah et al., 2017, 2020]. This is usually done by “inverting” the model, namely finding inputs that are strongly correlated with the model’s activations [Mordvintsev et al., 2015, Yin et al., 2020, Fredrikson et al., 2015]. Unsurprisingly, the results are semantically correlated with the training dataset. However, one rarely sees an exact version of a training sample.
+
+Our results have potential negative implications on privacy in deep learning. Our scheme can be viewed as a training-data reconstruction attack, since an adversary might recover sensitive training data. For example, if a medical device includes a model trained on sensitive medical records, an adversary might reconstruct this data and thus violate the privacy of the patients. Privacy attacks in deep learning have been widely studied in recent years (cf. Liu et al. [2021]), but as far as we are aware, the known attacks cannot reconstruct portions of the training data from a trained model.
+
+Our approach relies on theoretical results about the implicit bias in training neural networks with gradient-based methods. The implicit bias has been studied extensively in recent years with the motivation of explaining generalization in deep learning (see Section 2). We use results by Lyu and Li [2019], Ji and Telgarsky [2020], which establish that, under some technical assumptions, if we train a neural network with the binary cross entropy loss, its parameters will converge to a stationary point of a certain margin-maximization problem. This result implies that the parameters of the trained network satisfy a set of equations w.r.t. the training dataset. In our approach, given a trained network, we find a dataset that solves this set of equations w.r.t. the trained parameters.
+
+Our Contributions We show that large portions of the training samples are encoded in the parameters of a trained classifier. We also provide a practical scheme to decode the training samples, without any assumptions on the data. As far as we know, this is the first work that shows that reconstruction of actual training samples from a trained neural network classifier is possible.
+
+# 2 Related Work
+
+Understanding and Visualizing what is learnt by Neural Networks. The most common approach for analysing what is learnt by a neural network is by searching inputs that maximize the class output or the activations of neurons in intermediate layers [Erhan et al., 2009, Olah et al., 2020]. Oftentimes this is done via optimization with respect to the model input. Optimizing without any prior on the input usually results in noise inputs. Therefore, most approaches incorporate priors such as smoothness regularization or the use of pre-trained image generators [Mahendran and Vedaldi, 2015, Yosinski et al., 2015, Mordvintsev et al., 2015, Nguyen et al., 2016a,b, 2017] (see Olah et al. [2017] for a comprehensive summary). Optimization w.r.t. the input may also result in adversarial examples [Szegedy et al., 2013, Goodfellow et al., 2014]. Recently, [Tsipras et al., 2018, Engstrom et al., 2019] showed that classifiers trained to be robust to adversarial examples tend to learn representations that are more aligned with human vision. This was later utilized by [Santurkar et al., 2019, Mejia et al., 2019] to generate class-conditional images from a trained classifier. While all those approaches indicate that, unsurprisingly, the learnt representations are strongly correlated with the datasets on which the model was trained, none of them demonstrate the reconstruction of exact training samples from the trained models.
+
+Privacy Attacks in Deep Learning. Many methods deal with extracting sensitive information from trained models. Perhaps the closest to our approach is model-inversion that aims to reconstruct class representatives from the training data of a trained model [Fredrikson et al., 2015, He et al., 2019, Yang et al., 2019, Yin et al., 2020]. It is important to note that the reconstructed images, albeit semantically similar to some input images, are still not actual samples from the training set. Carlini et al. [2021, 2019] demonstrated reconstruction of training data from generative language models. By completing sentences, they reveal sensitive information from the training data. We note that this approach is specific to generative language models, while our approach considers classifiers and is less data specific. Membership-inference attacks [Shokri et al., 2017] aim to determine whether a given data point was used to train the model or not. For these methods to work, the adversary must be able to guess a specific input, whereas our approach does not assume such ability. Lastly, avoiding leakage of sensitive information on the training dataset is the motivation behind differential privacy in machine learning, which has been extensively studied [Abadi et al., 2016, Dwork et al., 2006, Chaudhuri et al., 2011]. For an elaborated discussion on the relation of these approaches to ours see Appendix A.
+
+Implicit Bias. In overparameterized neural networks one might expect overfitting to occur, but it seems that gradient-based methods are biased towards networks that generalize well [Zhang et al., 2021, Neyshabur et al., 2017]. Mathematically characterizing this implicit bias is a major problem in the theory of deep learning. Our approach is based on a characterization of the implicit bias of gradient flow in homogeneous neural networks due to Lyu and Li [2019] and Ji and Telgarsky [2020] (see Section 3 for details). The implicit bias of gradient-based methods in neural networks was extensively studied in recent years both for classification tasks (e.g., Soudry et al. [2018], Gunasekar et al. [2018c], Ji and Telgarsky [2018], Nacson et al. [2019], Vardi et al. [2021], Chizat and Bach [2020], Gunasekar et al. [2018a], Moroshko et al. [2020]) and regression tasks (e.g., Gunasekar et al. [2018b], Arora et al. [2019], Azulay et al. [2021], Yun et al. [2020], Woodworth et al. [2020], Razin and Cohen [2020], Li et al. [2020], Vardi and Shamir [2021], Timor et al. [2022]). See Vardi [2022] for a survey.
+
+# 3 Background and Reconstruction Scheme
+
+In this section we present our training data reconstruction scheme, as well as provide a brief overview on the theoretical results about implicit bias, which motivate our approach.
+
+# 3.1 On the Implicit Bias of Neural Networks
+
+Let $S = \{ ( \mathbf { x } _ { i } , y _ { i } ) \} _ { i = 1 } ^ { n } \subseteq \mathbb { R } ^ { d } \times \{ - 1 , 1 \}$ be a binary classification training dataset. Let $\Phi ( \pmb \theta ; \cdot ) :$ $\mathbb { R } ^ { d } \to \mathbb { R }$ be a neural network parameterized by $\pmb { \theta } \in \mathbb { R } ^ { p }$ . For a loss function $\ell : \mathbb { R } \to \mathbb { R }$ the empirical loss of $\Phi ( \theta ; \cdot )$ on the dataset $S$ is $\begin{array} { r } { \mathcal { L } ( \pmb { \theta } ) : = \sum _ { i = 1 } ^ { n } \ell \big ( y _ { i } \Phi ( \pmb { \theta } ; \mathbf { x } _ { i } ) \big ) } \end{array}$ . We focus on the logistic loss (a.k.a. binary cross entropy), namely, $\ell ( q ) = \log ( 1 + e ^ { - q } )$ .
+
+Our approach is based on Theorem 3.1 below, which holds for gradient flow (i.e., gradient descent with an infinitesimally small step size). Before stating the theorem, we need the following definitions: (1) We say that gradient flow converges in direction to $\tilde { \pmb { \theta } }$ if $\begin{array} { r } { \operatorname* { l i m } _ { t  \infty } \frac { \pmb { \theta } ( t ) } { \lVert \pmb { \theta } ( t ) \rVert } = \frac { \tilde { \pmb { \theta } } } { \lVert \tilde { \pmb { \theta } } \rVert } } \end{array}$ , where $\pmb \theta ( t )$ is the parameter vector at time $t$ ; (2) We say that a network $\Phi$ is homogeneous w.r.t. the parameters $\pmb \theta$ if there exists $L > 0$ such that for every $\alpha > 0$ and $\theta , \mathbf { x }$ we have $\bar { \Phi ( \alpha \pmb { \theta } ; \mathbf { x } ) } = \alpha ^ { L } \Phi ( \bar { \pmb { \theta } } ; \mathbf { x } )$ . Thus, scaling the parameters by any factor $\alpha > 0$ scales the outputs by $\alpha ^ { L }$ . We note that essentially any fully-connected or convolutional neural network with ReLU activations is homogeneous w.r.t. the parameters $\pmb \theta$ if it does not have any skip-connections (i.e., residual connections) or bias terms, except possibly for the first layer.
+
+Theorem 3.1 (Paraphrased from Lyu and Li [2019], Ji and Telgarsky [2020]) Let $\Phi ( \theta ; \cdot )$ be $a$ homogeneous ReLU neural network. Consider minimizing the logistic loss over a binary classification dataset $\{ ( \mathbf { x } _ { i } , y _ { i } ) \} _ { i = 1 } ^ { n }$ using gradient flow. Assume that there exists time $t _ { 0 }$ such that $\mathcal { L } ( \pmb { \theta } ( t _ { 0 } ) ) < 1 ^ { \ddagger }$ Then, gradient flow converges in direction to a first order stationary point (KKT point) of the following maximum-margin problem:
+
+$$
+\operatorname* { m i n } _ { \pmb { \theta } ^ { \prime } } \frac { 1 } { 2 } \left\| \pmb { \theta } ^ { \prime } \right\| ^ { 2 } s . t . \forall i \in [ n ] \ y _ { i } \Phi ( \pmb { \theta } ^ { \prime } ; \mathbf { x } _ { i } ) \geq 1 .
+$$
+
+Moreover, $\mathcal { L } ( \pmb \theta ( t ) )  0$ as $t \to \infty$ .
+
+The above theorem guarantees directional convergence to a first order stationary point (of the optimization problem (1)), which is also called Karush–Kuhn–Tucker point, or KKT point for short. The KKT approach allows inequality constraints, and is a generalization of the method of Lagrange multipliers, which allows only equality constraints.
+
+The great virtue of Theorem 3.1 is that it characterizes the implicit bias of gradient flow with the logistic loss for homogeneous networks. Namely, even though there are many possible directions of $\frac { \bar { \pmb { \theta } } } { \| \pmb { \theta } \| }$ that classify the dataset correctly, gradient flow converges only to directions that are KKT points of Problem (1). In particular, if the trajectory $\pmb \theta ( t )$ of gradient flow under the regime of Theorem 3.1 converges in direction to a KKT point $\tilde { \pmb { \theta } }$ , then we have the following: There exist $\lambda _ { 1 } , \ldots , \lambda _ { n } \in \mathbb { R }$ such that
+
+$$
+\begin{array} { l } { \displaystyle \tilde { \theta } = \sum _ { i = 1 } ^ { n } \lambda _ { i } y _ { i } \nabla _ { \theta } \Phi ( \tilde { \theta } ; { \bf x } _ { i } ) } \\ { \displaystyle \forall i \in [ n ] , ~ y _ { i } \Phi ( \tilde { \theta } ; { \bf x } _ { i } ) \geq 1 } \\ { \displaystyle \lambda _ { 1 } , \ldots , \lambda _ { n } \geq 0 } \\ { \displaystyle \forall i \in [ n ] , ~ \lambda _ { i } = 0 \mathrm { i f } y _ { i } \Phi ( \tilde { \theta } ; { \bf x } _ { i } ) \neq 1 } \end{array}
+$$
+
+Our main insight is based on Eq. (2), which implies that the parameters $\tilde { \pmb { \theta } }$ are a linear combinations of the derivatives of the network at the training data points. We say that a data point $\mathbf { x } _ { i }$ is on the margin if $y _ { i } \Phi ( \tilde { \pmb { \theta } } ; { \mathbf x } _ { i } ) = 1$ (i.e. $| \Phi ( \tilde { \pmb \theta } ; { \mathbf x } _ { i } ) | = 1 )$ . Note that Eq. (5) implies that only samples which are on the margin affect Eq. (2), since samples not on the margin have a coefficient $\lambda _ { i } = 0$ .
+
+# 3.2 Dataset Reconstruction
+
+Suppose we are given a trained neural network with parameters $\pmb \theta$ , and our goal is to reconstruct the dataset that the network was trained on. Although Theorem 3.1 holds asymptotically as the time $t$ tends to infinity, it suggests that also after training for a finite number of iterations the parameters of the network might approximately satisfy Eq. (2), and the coefficients $\lambda _ { i }$ satisfy Eq. (4). Since $n$ is unknown (and so is the number of samples on the margin) we set $m \geq 2 n$ which represents the number of samples we want to reconstruct (thus, we only need to upper bound $n$ ), and fix $y _ { i } = 1$ for $i = 1 , \ldots , m / 2$ and $y _ { i } = - 1$ for $i = m / 2 + 1 , \dots , m$ . We define the following losses:
+
+$$
+\begin{array} { r l } & { L _ { \mathrm { s t a t i o n a r y } } ( \mathbf { x } _ { 1 } , \ldots , \mathbf { x } _ { m } , \boldsymbol { \lambda } _ { 1 } , \ldots , \boldsymbol { \lambda } _ { m } ) = \displaystyle \left\| \pmb { \theta } - \sum _ { i = 1 } ^ { m } \lambda _ { i } y _ { i } \nabla _ { \pmb { \theta } } \Phi ( \pmb { \theta } ; \mathbf { x } _ { i } ) \right\| _ { 2 } ^ { 2 } } \\ & { L _ { \lambda } ( \lambda _ { 1 } , \ldots , \lambda _ { m } ) = \displaystyle \sum _ { i = 1 } ^ { m } \operatorname* { m a x } \{ - \lambda _ { i } , 0 \} } \end{array}
+$$
+
+Note that the unknown parameters are the $\mathbf { x } _ { i }$ ’s and $\lambda _ { i }$ ’s, and that $\pmb { \theta }$ and the $y _ { i }$ ’s are given. The loss $L _ { \mathrm { s t a t i o n a r y } }$ represents the stationarity condition that the parameters of the network satisfy, and $L _ { \lambda }$ represents the dual feasibility condition. We additionally define $L _ { \mathrm { p r i o r } }$ which represents some prior knowledge we might have about the dataset. For example, if we know that the dataset contains images, prior knowledge would be that each input coordinate (i.e. each pixel) is between 0 and 1. Given no prior knowledge on the data, we can define $L _ { \mathrm { p r i o r } } \equiv 0$ . Finally, we define the reconstruction loss as:
+
+$$
+L _ { \mathrm { r e c o n s t r u c t } } ( \{ \mathbf { x } _ { i } \} _ { i = 1 } ^ { m } , \{ \lambda _ { i } \} _ { i = 1 } ^ { m } ) = \alpha _ { 1 } L _ { \mathrm { s t a t i o n a r y } } + \alpha _ { 2 } L _ { \lambda } + \alpha _ { 3 } L _ { \mathrm { p r i o r } }
+$$
+
+where $\alpha _ { 1 } , \alpha _ { 2 } , \alpha _ { 3 } \in \mathbb { R }$ are tunable hyperparameters of the different losses. To reconstruct the dataset, we can use any nonconvex optimization method (e.g. SGD) to find the $\mathbf { x } _ { 1 } , \hdots , \mathbf { x } _ { m } , \lambda _ { 1 } , \hdots , \lambda _ { m }$ which minimize Eq. (8). We note that the $\lambda _ { i }$ ’s are not part of the training data, but finding them is necessary in order to solve this optimization problem. Finally, we emphasize that there are many other possible options to formulate the KKT conditions Eq. (2)-(5) as an unconstrained optimization problem. However, this simple choice seemed to work quite well in practice.
+
+We note that if there exist $\{ { \bf x } _ { i } \} _ { i = 1 } ^ { n }$ and $\{ \lambda _ { i } \} _ { i = 1 } ^ { n }$ which satisfy the KKT conditions, then there are $\{ { \bf x } _ { i } \} _ { i = 1 } ^ { m }$ and $\{ \lambda _ { i } \} _ { i = 1 } ^ { m }$ which achieve zero loss in Eq. (8). Indeed, such a solution can be obtained by adding to $\{ { \bf x } _ { i } \} _ { i = 1 } ^ { n }$ additional points $\mathbf { x } _ { j }$ with $\lambda _ { j } = 0$ , or by duplicating some points in $\{ { \bf x } _ { i } \} _ { i = 1 } ^ { n }$ and modifying the $\lambda$ ’s accordingly. Also, note that since we choose $m \geq 2 n$ , then we set at least $n$ labels $y _ { i }$ to 1 and at least $n$ labels to $- 1$ . Hence, there is a solution to Eq. (8) even though we do not know the real distribution of labels in the actual training data.
+
+We cannot simply use Eq. (3) and (5) in our reconstruction scheme, because they contain the constant "1" which corresponds to the margin (i.e., $\operatorname* { m i n } _ { i } | \Phi ( \tilde { \pmb { \theta } } ; { \mathbf { x } } _ { i } ) | )$ . Namely, we only converge in direction to a point $\tilde { \pmb { \theta } }$ that attains margin 1, but in practice we approach some point $\pmb \theta$ which attains an unknown margin $\gamma$ (i.e., $\mathrm { m i n } _ { i } \left| \Phi ( \pmb { \theta } ; \mathbf { x } _ { i } ) \right| = \gamma )$ , and we do not know in advance how to normalize it to attain a margin of exactly 1. On the other hand, Eq. (2) and (4) hold not only for $\tilde { \pmb { \theta } }$ but also for any $\pmb { \theta }$ that points at the direction of $\tilde { \pmb { \theta } }$ , and therefore in our loss in Eq. (8) we rely only on these conditions.
+
+Intuitively, a reason to believe that there is enough information in Eq. (2) to reconstruct the data, is the following observation: Eq. (2) represents a set of $p$ equations with $O ( n d )$ unknown variables, where $p$ is the number of parameters in the network. In practice, neural networks are often highly overparameterized (i.e., $p > n d ,$ , suggesting more equations than variables.
+
+Finally, since by Eq. (5) we have $\lambda _ { i } = 0$ for every $\mathbf { x } _ { i }$ that is not on the margin, then Eq. (2) implies that $\tilde { \pmb { \theta } }$ is determined only by the gradients w.r.t. the data points that are on the margin. Hence, we can only expect to reconstruct training samples that are on the margin (see also Subsection 5.3).
+
+# 4 A Simple Experiment in Two Dimensions
+
+![](images/4695b7b15f87a2c4a89697189e1241eb252753b4423bcb1c728119f24ad7122f.jpg)  
+Figure 2: Exemplifying our reconstruction scheme on a simple 2D dataset (see text for explanation).
+
+In this section we exemplify our dataset reconstruction scheme on a toy example of 2-dimensional data, i.e. we consider $( \dot { \bf x } , y ) \in \mathbb { R } ^ { 2 } \times \{ \pm 1 \}$ . We set $n = 2 0$ training samples on the unit circle, with alternating labels. For a visualization of the dataset see Figure 2a, blue and red " $" \times "$ represent the two classes. We trained a 3-layer model with 1000 neurons in each layer on this dataset. The model learns to correctly classify the training set. In Figure 2b, we visualize the output of the model as a function of its input. Blue and red regions correspond to smaller and larger outputs of the model, respectively.
+
+We now demonstrate our reconstruction scheme. We first randomly initialize $m = 1 0 0$ points in $\mathbb { R } ^ { 2 }$ , and assign 50 points to each class. This is depicted in Figure $2 \mathrm { c }$ , where green points correspond to the blue class, and magenta points correspond to the red class. Next, we optimize the loss in Eq. (8), with $L _ { \mathrm { p r i o r } } \equiv 0$ . The results of our reconstruction scheme are in Figure 2d. Note that our approach reconstructed all the input samples, up to some noise.
+
+To further improve our reconstruction results, we remove some of the extra points which did not converge to a training sample. In Figure 2e we removed points $\mathbf { x } _ { i }$ with corresponding $\lambda _ { i } ~ < ~ 5$ . According to Eq. (2), points with $\lambda _ { i } = 0$ should not affect the parameters, hence their corresponding $\mathbf { x } _ { i }$ can take any value. In practice, it is sufficient to remove points with a small enough corresponding $\lambda _ { i }$ . Finally, to remove duplicates, we greedily remove points which are very close to other points. That is, we randomly order the points, and iteratively remove points that are at distance $< 0 . 0 3$ from another point. The final reconstruction result is depicted in Figure 2f.
+
+# 5 Results
+
+![](images/9ba1d585e03e450dfc4f5ad8ac520e3a120d423feabaea0e9bf57b6db37b2f8e.jpg)  
+Top 45 images reconstructed from a model trained on CIFAR10 (rows 1, 3, 5), and their corresponding nearest-neighbors from the training-set of the model (rows 2, 4, 6)
+
+![](images/0fad3d3c0032f4f7495de8818c7a26d4e9430c33bef1601d39dc5dbb7c22559c.jpg)  
+Top 45 images reconstructed from a model trained on MNIST (rows 1, 3, 5), and their corresponding nearest-neighbors from the training-set of the model (rows 2, 4, 6)   
+Figure 3: Reconstructing training samples from two binary classifiers – one trained on 500 images with labels animals/vehicles (CIFAR), and the other trained on 500 odd/even digit images (MNIST). Train errors are zero, test accuracies are $8 8 . 0 \% / 7 7 . 6 \%$ for MNIST/CIFAR
+
+# 5.1 Experimental Setup
+
+Datasets. We conduct experiments on binary classification tasks where images are taken from the MNIST [LeCun et al., 2010] and CIFAR10 [Krizhevsky et al., 2009] datasets and the labels are set to odd vs. even digits (MNIST), and vehicles vs. animals§ (CIFAR10). We make sure that the class distribution in the training and test sets is balanced, and normalize the train and test sets by reducing the mean of the training set from both.
+
+Training. We consider MLP architectures. Unless stated otherwise, our models comprise of three fully-connected layers with dimensions $d$ -1000-1000-1 (where $d$ is the dimension of the input) with ReLU activations. Biases are set to zero except for the first layer, to line up with the theoretical assumption of homogeneous models in Section 3. The parameters are initialized using standard Kaiming He initialization [He et al., 2015] except for the weights of the first layer that are initialized to a Gaussian distribution with standard deviation $1 0 ^ { - 4 }$ (see discussion in Subsection 5.2). We train our models using full batch gradient descent for $1 0 ^ { 6 }$ epochs with a learning rate of 0.01. All models achieve zero training error (i.e., all the train samples are labeled correctly), and a training loss $< 1 0 ^ { - 6 }$ To compute the test accuracy, we use the original test sets of MNIST/CIFAR10 with 10000/8000 images respectively, and labeled accordingly.
+
+# 5.2 Training Set Reconstruction
+
+We minimize the loss defined in Eq. (8) with $\alpha _ { 1 } = 1$ , $\alpha _ { 2 } = 5$ , $\alpha _ { 3 } = 1$ . We initialize $\mathbf { x } _ { i } \sim \mathcal { N } ( 0 , \sigma _ { x } I )$ , where $\sigma _ { x }$ is a hyperparameter, and $\lambda _ { i } \sim \mathcal { U } [ 0 , 1 ]$ . We set the number of reconstructed samples to $m =$ $2 n$ (where $n$ is the size of the original training set). Note that our loss contains the derivative of ReLU Eq. (6). This derivative is a step function, containing only flat regions which are hard to optimize. We replace the derivative of the ReLU layer (backward function) with a sigmoid, which is the derivative of softplus (a smooth version of ReLU). We use the fact that our inputs are images to penalize values outside the range $[ - 1 , 1 ]$ . To this end we set $L _ { \mathrm { p r i o r } } ( z ) = \operatorname* { m a x } \{ z - 1 , 0 \} + \operatorname* { m a x } \{ - z - 1 , 0 \}$ for each pixel $z$ , and average over all dimensions (pixels) in $\mathbf { x } _ { i }$ . We optimize our loss for 100, 000 iterations using an SGD optimizer with momentum 0.9. We conduct a total of 100 runs using a random grid search on the hyperparameters (e.g. learning rate, $\sigma _ { x }$ . See Appendix $\mathbf { B }$ for full details). This results in $1 0 0 m$ “reconstructed” inputs.
+
+While some $\mathbf { x } _ { i }$ end up converging to a training sample, some end as noise (similar phenomenon can be observed in 2D in Figure 2d). To identify the reconstructions that are most similar to a training image we use the SSIM metric [Wang et al., 2004].
+
+In Figure 3 we show the best reconstruction results (in terms of SSIM) for models trained on $n { = } 5 0 0$ samples from MNIST/CIFAR10 datasets (with test accuracy $8 8 . 0 \% / 7 7 . 6 \%$ resp.). Note that the reconstructed images are very similar to the real input data, although a bit noisy. The source of this noise is not entirely clear. Possible reasons may be the complexity of the optimization problem, or the possibility that the trained model has not fully converged to the KKT point of Problem (1).
+
+We observed that small initializations significantly improve the quality of the reconstructed samples. We conjecture that small initialization causes faster convergence to the direction of the KKT point. This is also theoretically implied in Moroshko et al. [2020] (for certain linear models). Similarly, training for more epochs also improves the quality of the reconstruction. In Appendix C we show results for reconstructions from networks trained with standard initialization or trained for much fewer epochs. During the training phase, we used full batch gradient descent, to remain as much aligned to the theoretical setting. In Appendix C we show that our approach can reconstruct training data also from models trained with mini-batch SGD.
+
+# 5.3 Practice vs. Theory
+
+In this section we analyze some relations between our experimental results to the theory laid down in Section 3. Given a trained model and its reconstructed samples, we match each training sample to its best reconstruction (in terms of SSIM score). We then plot this SSIM score against $\bar { \Phi } ( \pmb \theta ; \mathbf x )$ (the value of the model’s output on this training sample) – for all training samples. In Figure 4 each cell shows such plot for a given model. The top row shows models trained on the same architecture with
+
+Models with the same architecture (1000-1000) trained on different number of training samples $( n )$
+
+![](images/acf9a903284c286521927ed5267fbe8b20e8a007d4ef72864bb8eb94f4607bdc.jpg)
+
+![](images/5f8fefd7b02bbd13b61929e28694bbeafe49126e63ee1df602eedf813bf6d4eb.jpg)  
+Models trained on $n = 5 0 0$ samples with different architectures   
+Figure 4: Each point represents a training sample. The y-axis is the highest SSIM score achieved by a reconstruction of this sample, the $\mathbf { X }$ -axis is the output of the model. Top: The effect of training the same model on different number of training samples $( n )$ . Bottom: The effect of training models with different architectures (on $n = 5 0 0$ training samples). The right-most plot shows a 3-layer non-homogeneous MLP (with bias terms in all hidden layers). See discussion in Section 5.3.
+
+different number of training samples $( n )$ , where in the bottom row we show the results for models trained on $n = 5 0 0$ training samples, with different architectures (all results are on CIFAR10).
+
+Recall that we do not expect to reconstruct samples that are far from the margin (Subsection 3.2). It is evident from Figure 4 that good reconstructions (e.g., $\mathrm { S S I M } > 0 . 4 \AA ,$ ) are obtained for samples that lie on the margin, as expected from theory. The plots indicate that increasing training size makes reconstruction more difficult. Lastly, as seen from the rightmost plot in the bottom row, we manage to get high-quality reconstructions from a non-homogeneous model (trained with biases in all hidden layers). This indicates that our approach may work beyond the theoretical limitations of Theorem 3.1.
+
+# 5.4 Comparison to other Reconstruction Schemes
+
+Model Inversion. Given a trained model $\Phi ( \theta ; \cdot )$ , we search for $\mathbf { x }$ which maximizes or minimizes $\Phi ( \pmb \theta ; \mathbf x )$ , corresponding to positive or negative labels. We initialize $\mathbf { x } \sim \mathcal { N } ( 0 , \sigma I )$ for several values of $\sigma$ and optimize w.r.t. the model output (see Appendix B for the choice of hyperparameters). In Figure 5a (left) it is apparent that in our two-dimensional experiment, model inversion successfully reconstructed 7 training samples, which indeed lie on a local minimum or maximum. However, note that our scheme reconstructs all 20 samples (Figure 2). In high dimensions, namely, in MNIST and CIFAR, while our scheme can reconstruct a large portion of the training set (Figure 3a), model inversion converges to noisy/blurry class representatives that correspond to high/low output values (Figure 5a, right). Such results are typical with model inversion since not all class members from the training set are visually similar (see discussions in Shokri et al. [2017], Melis et al. [2019]).
+
+Weights Visualization. The weights of the first fully-connected layer have the same dimension as that of the input. One may wonder whether training samples are directly encoded there. In Figure 5b we show the weights that are most similar (SSIM) to a training sample, or all of them in the 2D case. As seen in the 2D case, most weights are in the general direction of a training sample, however the scale is unknown without prior knowledge on the data. For images (MNIST/CIFAR10), not more than 3 or 4 of the weights have resemblance to training samples, while our scheme manages to reconstruct dozens of samples. See Appendix B and C for details and all 1000 weights of the models.
+
+# 6 Discussion and Conclusion
+
+Even though our results are shown for relatively small-scale models, they are the first to show that the parameters of trained networks may contain enough information to fully reconstruct training samples,
+
+# (a) Model Inversion
+
+![](images/2a1e1da7792eaea32ec559b3a37f2a5fe38b259f2bcb1d5335d7c7dd321138da.jpg)
+
+![](images/6f832c5d79b42554e21e18bfc5782279a9167f3d917b448259e4b1db5c1ae8e3.jpg)  
+(b) Weights of the first Fully-Connected Layer   
+Figure 5: Comparison to other reconstruction schemes. Top: Model inversion on the 2D experiment (left), on CIFAR10 (top right) and MNIST (bottom right). The CIFAR and MNIST images are ordered by the value of their output from left (smallest) to right (largest). Bottom: Weights of the first (fully-connected) layer for the 2D experiment (left), CIFAR10 (top right) and MNIST (bottom right). The weights for the 2D experiment are the small purple dots. For the CIFAR and MNIST experiments we show the 10 weights with the highest SSIM score.
+
+and the first to reconstruct a substantial amount of training samples. Moreover, the theoretical basis of the implicit bias in neural networks provides an analytic explanation to this phenomenon.
+
+Solving our optimization problem for convolutional neural networks turned out to be more challenging and is therefore a subject of future research. We note that the theoretical results that we rely on (i.e., Theorem 3.1) also covers convolutional neural networks. We believe that the homogeneity restriction might be relaxed, and showed reconstructions also from a non-homogeneous model (Figure 4, bottomrightmost). We also believe that our method may be extended to multi-class classifiers using an extension of Theorem 3.1. Finally, showing reconstructions on larger models and datasets, or on tabular or textual data are interesting future directions.
+
+On the theoretical side, it is not entirely clear why our optimization problem in Eq. (8) converges to actual training samples, even though there is no guarantee that the solution is unique, especially when using no prior (other than simple bounding to $[ - 1 , 1 ] ,$ . As a final note, our work brings up the question: are samples on margin the only ones that can be recovered from a trained classifier? or there exist better reconstruction schemes to reconstruct even more training samples from a trained neural network.
+
+# Acknowledgements
+
+This project received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 788535), and ERC grant 754705, and from the D. Dan and Betty Kahn Foundation, and was supported by the Carolito Stiftung.
+
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+F. Tramèr, F. Zhang, A. Juels, M. K. Reiter, and T. Ristenpart. Stealing machine learning models via prediction {APIs}. In 25th USENIX security symposium (USENIX Security 16), pages 601–618, 2016.   
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+G. Vardi. On the implicit bias in deep-learning algorithms. arXiv preprint arXiv:2208.12591, 2022.   
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+Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli. Image quality assessment: from error visibility to structural similarity. IEEE transactions on image processing, 13(4):600–612, 2004.   
+B. Woodworth, S. Gunasekar, J. D. Lee, E. Moroshko, P. Savarese, I. Golan, D. Soudry, and N. Srebro. Kernel and rich regimes in overparametrized models. In Conference on Learning Theory, pages 3635–3673. PMLR, 2020.   
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+L. Zhu, Z. Liu, and S. Han. Deep leakage from gradients. Advances in Neural Information Processing Systems, 32, 2019.
+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes]   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes]   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [N/A] We rely on known theoretical results, so proofs are not required.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes]   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [N/A]   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes]
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes]   
+(b) Did you mention the license of the assets? [Yes]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [N/A] We do not have new assets.   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A] We used only publicly available assets.   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] We used only publicly available data.
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/dev/Sxk8Bse3RKO/Sxk8Bse3RKO_content_list.json

@@ -0,0 +1,1205 @@
+[
+    {
+        "type": "text",
+        "text": "Reconstructing Training Data from Trained Neural Networks ",
+        "text_level": 1,
+        "bbox": [
+            233,
+            122,
+            766,
+            171
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Niv Haim∗ Weizmann Institute of Science niv.haim@weizmann.ac.il ",
+        "bbox": [
+            240,
+            222,
+            442,
+            263
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Gal Vardi∗† TTI-Chicago and Hebrew University galvardi@ttic.edu ",
+        "bbox": [
+            514,
+            220,
+            758,
+            263
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Gilad Yehudai∗ Weizmann Institute of Science gilad.yehudai@weizmann.ac.il ",
+        "bbox": [
+            233,
+            285,
+            473,
+            327
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ohad Shamir Weizmann Institute of Science ohad.shamir@weizmann.ac.il ",
+        "bbox": [
+            539,
+            285,
+            764,
+            327
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Michal Irani Weizmann Institute of Science michal.irani@weizmann.ac.il ",
+        "bbox": [
+            382,
+            348,
+            614,
+            388
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Project page: https://giladude1.github.io/reconstruction ",
+        "bbox": [
+            232,
+            401,
+            683,
+            415
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
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+            460
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Understanding to what extent neural networks memorize training data is an intriguing question with practical and theoretical implications. In this paper we show that in some cases a significant fraction of the training data can in fact be reconstructed from the parameters of a trained neural network classifier. We propose a novel reconstruction scheme that stems from recent theoretical results about the implicit bias in training neural networks with gradient-based methods. To the best of our knowledge, our results are the first to show that reconstructing a large portion of the actual training samples from a trained neural network classifier is generally possible. This has negative implications on privacy, as it can be used as an attack for revealing sensitive training data. We demonstrate our method for binary MLP classifiers on a few standard computer vision datasets. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            648,
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+            666
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "It is commonly believed that neural networks memorize the training data, even when they are able to generalize well to unseen test data (e.g., [Zhang et al., 2021, Feldman, 2020]). Exploring this memorization phenomenon is of great importance both practically and theoretically. Indeed, it has implications on our understanding of generalization in deep learning, on the hidden representations learnt by neural networks, and on the extent to which they are vulnerable to privacy attacks. ",
+        "bbox": [
+            174,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A fundamental question for understanding memorization is: ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Are the specific training samples encoded in the parameters of a trained classifier? Can they be recovered from the network parameters? ",
+        "bbox": [
+            230,
+            781,
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+            809
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this work, we study this question, and devise a novel scheme which allows us to reconstruct a significant portion of the training data from the parameters of a trained neural network alone, without having any additional information on the data. Thus, we provide a proof-of-concept that the learning (a) Top 24 images reconstructed from a binary classifier trained on 50 CIFAR10 images (b) Their corresponding nearest neighbours from the training-set of the model process can sometimes be reversed: That is, instead of learning a model given a training dataset, it is possible to find the training data given a trained model. In Figure 1 we show how our approach reconstructs images from the CIFAR10 dataset, given a simple trained binary classifier. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/ed282a7049c35ade42ab306a6ef56700e2d645bdbbdfcd5644417349b411ddc9.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/7cf960fa2f82434a9efbbc4e9e6c61ec64f0bfc24a1869967b39feb8256f2cee.jpg",
+        "image_caption": [
+            "Figure 1: Reconstruction of training images from a pretrained binary classifier, trained on 50 CIFAR10 images. The two classes are “animals” and “vehicles”. We calculate the nearest neighbor using the SSIM metric. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
+            210,
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+            294
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            357,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Many works try to “crack” neural networks by analyzing and visualizing either their learnt parameters or representations [Erhan et al., 2009, Mahendran and Vedaldi, 2015, Olah et al., 2017, 2020]. This is usually done by “inverting” the model, namely finding inputs that are strongly correlated with the model’s activations [Mordvintsev et al., 2015, Yin et al., 2020, Fredrikson et al., 2015]. Unsurprisingly, the results are semantically correlated with the training dataset. However, one rarely sees an exact version of a training sample. ",
+        "bbox": [
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+            405,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our results have potential negative implications on privacy in deep learning. Our scheme can be viewed as a training-data reconstruction attack, since an adversary might recover sensitive training data. For example, if a medical device includes a model trained on sensitive medical records, an adversary might reconstruct this data and thus violate the privacy of the patients. Privacy attacks in deep learning have been widely studied in recent years (cf. Liu et al. [2021]), but as far as we are aware, the known attacks cannot reconstruct portions of the training data from a trained model. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our approach relies on theoretical results about the implicit bias in training neural networks with gradient-based methods. The implicit bias has been studied extensively in recent years with the motivation of explaining generalization in deep learning (see Section 2). We use results by Lyu and Li [2019], Ji and Telgarsky [2020], which establish that, under some technical assumptions, if we train a neural network with the binary cross entropy loss, its parameters will converge to a stationary point of a certain margin-maximization problem. This result implies that the parameters of the trained network satisfy a set of equations w.r.t. the training dataset. In our approach, given a trained network, we find a dataset that solves this set of equations w.r.t. the trained parameters. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our Contributions We show that large portions of the training samples are encoded in the parameters of a trained classifier. We also provide a practical scheme to decode the training samples, without any assumptions on the data. As far as we know, this is the first work that shows that reconstruction of actual training samples from a trained neural network classifier is possible. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            768,
+            321,
+            786
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Understanding and Visualizing what is learnt by Neural Networks. The most common approach for analysing what is learnt by a neural network is by searching inputs that maximize the class output or the activations of neurons in intermediate layers [Erhan et al., 2009, Olah et al., 2020]. Oftentimes this is done via optimization with respect to the model input. Optimizing without any prior on the input usually results in noise inputs. Therefore, most approaches incorporate priors such as smoothness regularization or the use of pre-trained image generators [Mahendran and Vedaldi, 2015, Yosinski et al., 2015, Mordvintsev et al., 2015, Nguyen et al., 2016a,b, 2017] (see Olah et al. [2017] for a comprehensive summary). Optimization w.r.t. the input may also result in adversarial examples [Szegedy et al., 2013, Goodfellow et al., 2014]. Recently, [Tsipras et al., 2018, Engstrom et al., 2019] showed that classifiers trained to be robust to adversarial examples tend to learn representations that are more aligned with human vision. This was later utilized by [Santurkar et al., 2019, Mejia et al., 2019] to generate class-conditional images from a trained classifier. While all those approaches indicate that, unsurprisingly, the learnt representations are strongly correlated with the datasets on which the model was trained, none of them demonstrate the reconstruction of exact training samples from the trained models. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Privacy Attacks in Deep Learning. Many methods deal with extracting sensitive information from trained models. Perhaps the closest to our approach is model-inversion that aims to reconstruct class representatives from the training data of a trained model [Fredrikson et al., 2015, He et al., 2019, Yang et al., 2019, Yin et al., 2020]. It is important to note that the reconstructed images, albeit semantically similar to some input images, are still not actual samples from the training set. Carlini et al. [2021, 2019] demonstrated reconstruction of training data from generative language models. By completing sentences, they reveal sensitive information from the training data. We note that this approach is specific to generative language models, while our approach considers classifiers and is less data specific. Membership-inference attacks [Shokri et al., 2017] aim to determine whether a given data point was used to train the model or not. For these methods to work, the adversary must be able to guess a specific input, whereas our approach does not assume such ability. Lastly, avoiding leakage of sensitive information on the training dataset is the motivation behind differential privacy in machine learning, which has been extensively studied [Abadi et al., 2016, Dwork et al., 2006, Chaudhuri et al., 2011]. For an elaborated discussion on the relation of these approaches to ours see Appendix A. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Implicit Bias. In overparameterized neural networks one might expect overfitting to occur, but it seems that gradient-based methods are biased towards networks that generalize well [Zhang et al., 2021, Neyshabur et al., 2017]. Mathematically characterizing this implicit bias is a major problem in the theory of deep learning. Our approach is based on a characterization of the implicit bias of gradient flow in homogeneous neural networks due to Lyu and Li [2019] and Ji and Telgarsky [2020] (see Section 3 for details). The implicit bias of gradient-based methods in neural networks was extensively studied in recent years both for classification tasks (e.g., Soudry et al. [2018], Gunasekar et al. [2018c], Ji and Telgarsky [2018], Nacson et al. [2019], Vardi et al. [2021], Chizat and Bach [2020], Gunasekar et al. [2018a], Moroshko et al. [2020]) and regression tasks (e.g., Gunasekar et al. [2018b], Arora et al. [2019], Azulay et al. [2021], Yun et al. [2020], Woodworth et al. [2020], Razin and Cohen [2020], Li et al. [2020], Vardi and Shamir [2021], Timor et al. [2022]). See Vardi [2022] for a survey. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Background and Reconstruction Scheme ",
+        "text_level": 1,
+        "bbox": [
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+            609,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section we present our training data reconstruction scheme, as well as provide a brief overview on the theoretical results about implicit bias, which motivate our approach. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 On the Implicit Bias of Neural Networks ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            686,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Let $S = \\{ ( \\mathbf { x } _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { n } \\subseteq \\mathbb { R } ^ { d } \\times \\{ - 1 , 1 \\}$ be a binary classification training dataset. Let $\\Phi ( \\pmb \\theta ; \\cdot ) :$ $\\mathbb { R } ^ { d } \\to \\mathbb { R }$ be a neural network parameterized by $\\pmb { \\theta } \\in \\mathbb { R } ^ { p }$ . For a loss function $\\ell : \\mathbb { R } \\to \\mathbb { R }$ the empirical loss of $\\Phi ( \\theta ; \\cdot )$ on the dataset $S$ is $\\begin{array} { r } { \\mathcal { L } ( \\pmb { \\theta } ) : = \\sum _ { i = 1 } ^ { n } \\ell \\big ( y _ { i } \\Phi ( \\pmb { \\theta } ; \\mathbf { x } _ { i } ) \\big ) } \\end{array}$ . We focus on the logistic loss (a.k.a. binary cross entropy), namely, $\\ell ( q ) = \\log ( 1 + e ^ { - q } )$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our approach is based on Theorem 3.1 below, which holds for gradient flow (i.e., gradient descent with an infinitesimally small step size). Before stating the theorem, we need the following definitions: (1) We say that gradient flow converges in direction to $\\tilde { \\pmb { \\theta } }$ if $\\begin{array} { r } { \\operatorname* { l i m } _ { t  \\infty } \\frac { \\pmb { \\theta } ( t ) } { \\lVert \\pmb { \\theta } ( t ) \\rVert } = \\frac { \\tilde { \\pmb { \\theta } } } { \\lVert \\tilde { \\pmb { \\theta } } \\rVert } } \\end{array}$ , where $\\pmb \\theta ( t )$ is the parameter vector at time $t$ ; (2) We say that a network $\\Phi$ is homogeneous w.r.t. the parameters $\\pmb \\theta$ if there exists $L > 0$ such that for every $\\alpha > 0$ and $\\theta , \\mathbf { x }$ we have $\\bar { \\Phi ( \\alpha \\pmb { \\theta } ; \\mathbf { x } ) } = \\alpha ^ { L } \\Phi ( \\bar { \\pmb { \\theta } } ; \\mathbf { x } )$ . Thus, scaling the parameters by any factor $\\alpha > 0$ scales the outputs by $\\alpha ^ { L }$ . We note that essentially any fully-connected or convolutional neural network with ReLU activations is homogeneous w.r.t. the parameters $\\pmb \\theta$ if it does not have any skip-connections (i.e., residual connections) or bias terms, except possibly for the first layer. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.1 (Paraphrased from Lyu and Li [2019], Ji and Telgarsky [2020]) Let $\\Phi ( \\theta ; \\cdot )$ be $a$ homogeneous ReLU neural network. Consider minimizing the logistic loss over a binary classification dataset $\\{ ( \\mathbf { x } _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { n }$ using gradient flow. Assume that there exists time $t _ { 0 }$ such that $\\mathcal { L } ( \\pmb { \\theta } ( t _ { 0 } ) ) < 1 ^ { \\ddagger }$ Then, gradient flow converges in direction to a first order stationary point (KKT point) of the following maximum-margin problem: ",
+        "bbox": [
+            173,
+            90,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e1b2c8ac454534a70e287b922fc4cdd2f0681650953e0e1a7c532b5c7fbc4c25.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\pmb { \\theta } ^ { \\prime } } \\frac { 1 } { 2 } \\left\\| \\pmb { \\theta } ^ { \\prime } \\right\\| ^ { 2 } s . t . \\forall i \\in [ n ] \\ y _ { i } \\Phi ( \\pmb { \\theta } ^ { \\prime } ; \\mathbf { x } _ { i } ) \\geq 1 .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Moreover, $\\mathcal { L } ( \\pmb \\theta ( t ) )  0$ as $t \\to \\infty$ . ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The above theorem guarantees directional convergence to a first order stationary point (of the optimization problem (1)), which is also called Karush–Kuhn–Tucker point, or KKT point for short. The KKT approach allows inequality constraints, and is a generalization of the method of Lagrange multipliers, which allows only equality constraints. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The great virtue of Theorem 3.1 is that it characterizes the implicit bias of gradient flow with the logistic loss for homogeneous networks. Namely, even though there are many possible directions of $\\frac { \\bar { \\pmb { \\theta } } } { \\| \\pmb { \\theta } \\| }$ that classify the dataset correctly, gradient flow converges only to directions that are KKT points of Problem (1). In particular, if the trajectory $\\pmb \\theta ( t )$ of gradient flow under the regime of Theorem 3.1 converges in direction to a KKT point $\\tilde { \\pmb { \\theta } }$ , then we have the following: There exist $\\lambda _ { 1 } , \\ldots , \\lambda _ { n } \\in \\mathbb { R }$ such that ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/10a5b80a8bbdde1d43bd6cc8f9d151d5719fd201ad176f6f6bef7032058b8c92.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\tilde { \\theta } = \\sum _ { i = 1 } ^ { n } \\lambda _ { i } y _ { i } \\nabla _ { \\theta } \\Phi ( \\tilde { \\theta } ; { \\bf x } _ { i } ) } \\\\ { \\displaystyle \\forall i \\in [ n ] , ~ y _ { i } \\Phi ( \\tilde { \\theta } ; { \\bf x } _ { i } ) \\geq 1 } \\\\ { \\displaystyle \\lambda _ { 1 } , \\ldots , \\lambda _ { n } \\geq 0 } \\\\ { \\displaystyle \\forall i \\in [ n ] , ~ \\lambda _ { i } = 0 \\mathrm { i f } y _ { i } \\Phi ( \\tilde { \\theta } ; { \\bf x } _ { i } ) \\neq 1 } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            251,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our main insight is based on Eq. (2), which implies that the parameters $\\tilde { \\pmb { \\theta } }$ are a linear combinations of the derivatives of the network at the training data points. We say that a data point $\\mathbf { x } _ { i }$ is on the margin if $y _ { i } \\Phi ( \\tilde { \\pmb { \\theta } } ; { \\mathbf x } _ { i } ) = 1$ (i.e. $| \\Phi ( \\tilde { \\pmb \\theta } ; { \\mathbf x } _ { i } ) | = 1 )$ . Note that Eq. (5) implies that only samples which are on the margin affect Eq. (2), since samples not on the margin have a coefficient $\\lambda _ { i } = 0$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 Dataset Reconstruction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            559,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Suppose we are given a trained neural network with parameters $\\pmb \\theta$ , and our goal is to reconstruct the dataset that the network was trained on. Although Theorem 3.1 holds asymptotically as the time $t$ tends to infinity, it suggests that also after training for a finite number of iterations the parameters of the network might approximately satisfy Eq. (2), and the coefficients $\\lambda _ { i }$ satisfy Eq. (4). Since $n$ is unknown (and so is the number of samples on the margin) we set $m \\geq 2 n$ which represents the number of samples we want to reconstruct (thus, we only need to upper bound $n$ ), and fix $y _ { i } = 1$ for $i = 1 , \\ldots , m / 2$ and $y _ { i } = - 1$ for $i = m / 2 + 1 , \\dots , m$ . We define the following losses: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/e0c44a360b2594a8f67677a4d1fdf0f04c302705f6ce1d82dfeded590f0dde30.jpg",
+        "text": "$$\n\\begin{array} { r l } & { L _ { \\mathrm { s t a t i o n a r y } } ( \\mathbf { x } _ { 1 } , \\ldots , \\mathbf { x } _ { m } , \\boldsymbol { \\lambda } _ { 1 } , \\ldots , \\boldsymbol { \\lambda } _ { m } ) = \\displaystyle \\left\\| \\pmb { \\theta } - \\sum _ { i = 1 } ^ { m } \\lambda _ { i } y _ { i } \\nabla _ { \\pmb { \\theta } } \\Phi ( \\pmb { \\theta } ; \\mathbf { x } _ { i } ) \\right\\| _ { 2 } ^ { 2 } } \\\\ & { L _ { \\lambda } ( \\lambda _ { 1 } , \\ldots , \\lambda _ { m } ) = \\displaystyle \\sum _ { i = 1 } ^ { m } \\operatorname* { m a x } \\{ - \\lambda _ { i } , 0 \\} } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            277,
+            685,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Note that the unknown parameters are the $\\mathbf { x } _ { i }$ ’s and $\\lambda _ { i }$ ’s, and that $\\pmb { \\theta }$ and the $y _ { i }$ ’s are given. The loss $L _ { \\mathrm { s t a t i o n a r y } }$ represents the stationarity condition that the parameters of the network satisfy, and $L _ { \\lambda }$ represents the dual feasibility condition. We additionally define $L _ { \\mathrm { p r i o r } }$ which represents some prior knowledge we might have about the dataset. For example, if we know that the dataset contains images, prior knowledge would be that each input coordinate (i.e. each pixel) is between 0 and 1. Given no prior knowledge on the data, we can define $L _ { \\mathrm { p r i o r } } \\equiv 0$ . Finally, we define the reconstruction loss as: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bd9175db7eeae1bc308df6a9019b2c5dbd058575839ca746e18c10ab59593088.jpg",
+        "text": "$$\nL _ { \\mathrm { r e c o n s t r u c t } } ( \\{ \\mathbf { x } _ { i } \\} _ { i = 1 } ^ { m } , \\{ \\lambda _ { i } \\} _ { i = 1 } ^ { m } ) = \\alpha _ { 1 } L _ { \\mathrm { s t a t i o n a r y } } + \\alpha _ { 2 } L _ { \\lambda } + \\alpha _ { 3 } L _ { \\mathrm { p r i o r } }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            871,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\alpha _ { 1 } , \\alpha _ { 2 } , \\alpha _ { 3 } \\in \\mathbb { R }$ are tunable hyperparameters of the different losses. To reconstruct the dataset, we can use any nonconvex optimization method (e.g. SGD) to find the $\\mathbf { x } _ { 1 } , \\hdots , \\mathbf { x } _ { m } , \\lambda _ { 1 } , \\hdots , \\lambda _ { m }$ which minimize Eq. (8). We note that the $\\lambda _ { i }$ ’s are not part of the training data, but finding them is necessary in order to solve this optimization problem. Finally, we emphasize that there are many other possible options to formulate the KKT conditions Eq. (2)-(5) as an unconstrained optimization problem. However, this simple choice seemed to work quite well in practice. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We note that if there exist $\\{ { \\bf x } _ { i } \\} _ { i = 1 } ^ { n }$ and $\\{ \\lambda _ { i } \\} _ { i = 1 } ^ { n }$ which satisfy the KKT conditions, then there are $\\{ { \\bf x } _ { i } \\} _ { i = 1 } ^ { m }$ and $\\{ \\lambda _ { i } \\} _ { i = 1 } ^ { m }$ which achieve zero loss in Eq. (8). Indeed, such a solution can be obtained by adding to $\\{ { \\bf x } _ { i } \\} _ { i = 1 } ^ { n }$ additional points $\\mathbf { x } _ { j }$ with $\\lambda _ { j } = 0$ , or by duplicating some points in $\\{ { \\bf x } _ { i } \\} _ { i = 1 } ^ { n }$ and modifying the $\\lambda$ ’s accordingly. Also, note that since we choose $m \\geq 2 n$ , then we set at least $n$ labels $y _ { i }$ to 1 and at least $n$ labels to $- 1$ . Hence, there is a solution to Eq. (8) even though we do not know the real distribution of labels in the actual training data. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We cannot simply use Eq. (3) and (5) in our reconstruction scheme, because they contain the constant \"1\" which corresponds to the margin (i.e., $\\operatorname* { m i n } _ { i } | \\Phi ( \\tilde { \\pmb { \\theta } } ; { \\mathbf { x } } _ { i } ) | )$ . Namely, we only converge in direction to a point $\\tilde { \\pmb { \\theta } }$ that attains margin 1, but in practice we approach some point $\\pmb \\theta$ which attains an unknown margin $\\gamma$ (i.e., $\\mathrm { m i n } _ { i } \\left| \\Phi ( \\pmb { \\theta } ; \\mathbf { x } _ { i } ) \\right| = \\gamma )$ , and we do not know in advance how to normalize it to attain a margin of exactly 1. On the other hand, Eq. (2) and (4) hold not only for $\\tilde { \\pmb { \\theta } }$ but also for any $\\pmb { \\theta }$ that points at the direction of $\\tilde { \\pmb { \\theta } }$ , and therefore in our loss in Eq. (8) we rely only on these conditions. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Intuitively, a reason to believe that there is enough information in Eq. (2) to reconstruct the data, is the following observation: Eq. (2) represents a set of $p$ equations with $O ( n d )$ unknown variables, where $p$ is the number of parameters in the network. In practice, neural networks are often highly overparameterized (i.e., $p > n d ,$ , suggesting more equations than variables. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Finally, since by Eq. (5) we have $\\lambda _ { i } = 0$ for every $\\mathbf { x } _ { i }$ that is not on the margin, then Eq. (2) implies that $\\tilde { \\pmb { \\theta } }$ is determined only by the gradients w.r.t. the data points that are on the margin. Hence, we can only expect to reconstruct training samples that are on the margin (see also Subsection 5.3). ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 A Simple Experiment in Two Dimensions ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/4695b7b15f87a2c4a89697189e1241eb252753b4423bcb1c728119f24ad7122f.jpg",
+        "image_caption": [
+            "Figure 2: Exemplifying our reconstruction scheme on a simple 2D dataset (see text for explanation). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section we exemplify our dataset reconstruction scheme on a toy example of 2-dimensional data, i.e. we consider $( \\dot { \\bf x } , y ) \\in \\mathbb { R } ^ { 2 } \\times \\{ \\pm 1 \\}$ . We set $n = 2 0$ training samples on the unit circle, with alternating labels. For a visualization of the dataset see Figure 2a, blue and red \" $\" \\times \"$ represent the two classes. We trained a 3-layer model with 1000 neurons in each layer on this dataset. The model learns to correctly classify the training set. In Figure 2b, we visualize the output of the model as a function of its input. Blue and red regions correspond to smaller and larger outputs of the model, respectively. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We now demonstrate our reconstruction scheme. We first randomly initialize $m = 1 0 0$ points in $\\mathbb { R } ^ { 2 }$ , and assign 50 points to each class. This is depicted in Figure $2 \\mathrm { c }$ , where green points correspond to the blue class, and magenta points correspond to the red class. Next, we optimize the loss in Eq. (8), with $L _ { \\mathrm { p r i o r } } \\equiv 0$ . The results of our reconstruction scheme are in Figure 2d. Note that our approach reconstructed all the input samples, up to some noise. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To further improve our reconstruction results, we remove some of the extra points which did not converge to a training sample. In Figure 2e we removed points $\\mathbf { x } _ { i }$ with corresponding $\\lambda _ { i } ~ < ~ 5$ . According to Eq. (2), points with $\\lambda _ { i } = 0$ should not affect the parameters, hence their corresponding $\\mathbf { x } _ { i }$ can take any value. In practice, it is sufficient to remove points with a small enough corresponding $\\lambda _ { i }$ . Finally, to remove duplicates, we greedily remove points which are very close to other points. That is, we randomly order the points, and iteratively remove points that are at distance $< 0 . 0 3$ from another point. The final reconstruction result is depicted in Figure 2f. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5 Results ",
+        "text_level": 1,
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+    },
+    {
+        "type": "image",
+        "img_path": "images/9ba1d585e03e450dfc4f5ad8ac520e3a120d423feabaea0e9bf57b6db37b2f8e.jpg",
+        "image_caption": [
+            "Top 45 images reconstructed from a model trained on CIFAR10 (rows 1, 3, 5), and their corresponding nearest-neighbors from the training-set of the model (rows 2, 4, 6) "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/0fad3d3c0032f4f7495de8818c7a26d4e9430c33bef1601d39dc5dbb7c22559c.jpg",
+        "image_caption": [
+            "Top 45 images reconstructed from a model trained on MNIST (rows 1, 3, 5), and their corresponding nearest-neighbors from the training-set of the model (rows 2, 4, 6) ",
+            "Figure 3: Reconstructing training samples from two binary classifiers – one trained on 500 images with labels animals/vehicles (CIFAR), and the other trained on 500 odd/even digit images (MNIST). Train errors are zero, test accuracies are $8 8 . 0 \\% / 7 7 . 6 \\%$ for MNIST/CIFAR "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.1 Experimental Setup ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Datasets. We conduct experiments on binary classification tasks where images are taken from the MNIST [LeCun et al., 2010] and CIFAR10 [Krizhevsky et al., 2009] datasets and the labels are set to odd vs. even digits (MNIST), and vehicles vs. animals§ (CIFAR10). We make sure that the class distribution in the training and test sets is balanced, and normalize the train and test sets by reducing the mean of the training set from both. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Training. We consider MLP architectures. Unless stated otherwise, our models comprise of three fully-connected layers with dimensions $d$ -1000-1000-1 (where $d$ is the dimension of the input) with ReLU activations. Biases are set to zero except for the first layer, to line up with the theoretical assumption of homogeneous models in Section 3. The parameters are initialized using standard Kaiming He initialization [He et al., 2015] except for the weights of the first layer that are initialized to a Gaussian distribution with standard deviation $1 0 ^ { - 4 }$ (see discussion in Subsection 5.2). We train our models using full batch gradient descent for $1 0 ^ { 6 }$ epochs with a learning rate of 0.01. All models achieve zero training error (i.e., all the train samples are labeled correctly), and a training loss $< 1 0 ^ { - 6 }$ To compute the test accuracy, we use the original test sets of MNIST/CIFAR10 with 10000/8000 images respectively, and labeled accordingly. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.2 Training Set Reconstruction ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            356,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We minimize the loss defined in Eq. (8) with $\\alpha _ { 1 } = 1$ , $\\alpha _ { 2 } = 5$ , $\\alpha _ { 3 } = 1$ . We initialize $\\mathbf { x } _ { i } \\sim \\mathcal { N } ( 0 , \\sigma _ { x } I )$ , where $\\sigma _ { x }$ is a hyperparameter, and $\\lambda _ { i } \\sim \\mathcal { U } [ 0 , 1 ]$ . We set the number of reconstructed samples to $m =$ $2 n$ (where $n$ is the size of the original training set). Note that our loss contains the derivative of ReLU Eq. (6). This derivative is a step function, containing only flat regions which are hard to optimize. We replace the derivative of the ReLU layer (backward function) with a sigmoid, which is the derivative of softplus (a smooth version of ReLU). We use the fact that our inputs are images to penalize values outside the range $[ - 1 , 1 ]$ . To this end we set $L _ { \\mathrm { p r i o r } } ( z ) = \\operatorname* { m a x } \\{ z - 1 , 0 \\} + \\operatorname* { m a x } \\{ - z - 1 , 0 \\}$ for each pixel $z$ , and average over all dimensions (pixels) in $\\mathbf { x } _ { i }$ . We optimize our loss for 100, 000 iterations using an SGD optimizer with momentum 0.9. We conduct a total of 100 runs using a random grid search on the hyperparameters (e.g. learning rate, $\\sigma _ { x }$ . See Appendix $\\mathbf { B }$ for full details). This results in $1 0 0 m$ “reconstructed” inputs. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "While some $\\mathbf { x } _ { i }$ end up converging to a training sample, some end as noise (similar phenomenon can be observed in 2D in Figure 2d). To identify the reconstructions that are most similar to a training image we use the SSIM metric [Wang et al., 2004]. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In Figure 3 we show the best reconstruction results (in terms of SSIM) for models trained on $n { = } 5 0 0$ samples from MNIST/CIFAR10 datasets (with test accuracy $8 8 . 0 \\% / 7 7 . 6 \\%$ resp.). Note that the reconstructed images are very similar to the real input data, although a bit noisy. The source of this noise is not entirely clear. Possible reasons may be the complexity of the optimization problem, or the possibility that the trained model has not fully converged to the KKT point of Problem (1). ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We observed that small initializations significantly improve the quality of the reconstructed samples. We conjecture that small initialization causes faster convergence to the direction of the KKT point. This is also theoretically implied in Moroshko et al. [2020] (for certain linear models). Similarly, training for more epochs also improves the quality of the reconstruction. In Appendix C we show results for reconstructions from networks trained with standard initialization or trained for much fewer epochs. During the training phase, we used full batch gradient descent, to remain as much aligned to the theoretical setting. In Appendix C we show that our approach can reconstruct training data also from models trained with mini-batch SGD. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.3 Practice vs. Theory ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section we analyze some relations between our experimental results to the theory laid down in Section 3. Given a trained model and its reconstructed samples, we match each training sample to its best reconstruction (in terms of SSIM score). We then plot this SSIM score against $\\bar { \\Phi } ( \\pmb \\theta ; \\mathbf x )$ (the value of the model’s output on this training sample) – for all training samples. In Figure 4 each cell shows such plot for a given model. The top row shows models trained on the same architecture with ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Models with the same architecture (1000-1000) trained on different number of training samples $( n )$ ",
+        "bbox": [
+            156,
+            88,
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+            103
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/acf9a903284c286521927ed5267fbe8b20e8a007d4ef72864bb8eb94f4607bdc.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/5f8fefd7b02bbd13b61929e28694bbeafe49126e63ee1df602eedf813bf6d4eb.jpg",
+        "image_caption": [
+            "Models trained on $n = 5 0 0$ samples with different architectures ",
+            "Figure 4: Each point represents a training sample. The y-axis is the highest SSIM score achieved by a reconstruction of this sample, the $\\mathbf { X }$ -axis is the output of the model. Top: The effect of training the same model on different number of training samples $( n )$ . Bottom: The effect of training models with different architectures (on $n = 5 0 0$ training samples). The right-most plot shows a 3-layer non-homogeneous MLP (with bias terms in all hidden layers). See discussion in Section 5.3. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            156,
+            218,
+            807,
+            306
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "different number of training samples $( n )$ , where in the bottom row we show the results for models trained on $n = 5 0 0$ training samples, with different architectures (all results are on CIFAR10). ",
+        "bbox": [
+            171,
+            414,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Recall that we do not expect to reconstruct samples that are far from the margin (Subsection 3.2). It is evident from Figure 4 that good reconstructions (e.g., $\\mathrm { S S I M } > 0 . 4 \\AA ,$ ) are obtained for samples that lie on the margin, as expected from theory. The plots indicate that increasing training size makes reconstruction more difficult. Lastly, as seen from the rightmost plot in the bottom row, we manage to get high-quality reconstructions from a non-homogeneous model (trained with biases in all hidden layers). This indicates that our approach may work beyond the theoretical limitations of Theorem 3.1. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.4 Comparison to other Reconstruction Schemes ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Model Inversion. Given a trained model $\\Phi ( \\theta ; \\cdot )$ , we search for $\\mathbf { x }$ which maximizes or minimizes $\\Phi ( \\pmb \\theta ; \\mathbf x )$ , corresponding to positive or negative labels. We initialize $\\mathbf { x } \\sim \\mathcal { N } ( 0 , \\sigma I )$ for several values of $\\sigma$ and optimize w.r.t. the model output (see Appendix B for the choice of hyperparameters). In Figure 5a (left) it is apparent that in our two-dimensional experiment, model inversion successfully reconstructed 7 training samples, which indeed lie on a local minimum or maximum. However, note that our scheme reconstructs all 20 samples (Figure 2). In high dimensions, namely, in MNIST and CIFAR, while our scheme can reconstruct a large portion of the training set (Figure 3a), model inversion converges to noisy/blurry class representatives that correspond to high/low output values (Figure 5a, right). Such results are typical with model inversion since not all class members from the training set are visually similar (see discussions in Shokri et al. [2017], Melis et al. [2019]). ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Weights Visualization. The weights of the first fully-connected layer have the same dimension as that of the input. One may wonder whether training samples are directly encoded there. In Figure 5b we show the weights that are most similar (SSIM) to a training sample, or all of them in the 2D case. As seen in the 2D case, most weights are in the general direction of a training sample, however the scale is unknown without prior knowledge on the data. For images (MNIST/CIFAR10), not more than 3 or 4 of the weights have resemblance to training samples, while our scheme manages to reconstruct dozens of samples. See Appendix B and C for details and all 1000 weights of the models. ",
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+        "text": "6 Discussion and Conclusion ",
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+        "text": "Even though our results are shown for relatively small-scale models, they are the first to show that the parameters of trained networks may contain enough information to fully reconstruct training samples, ",
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+        "text": "(a) Model Inversion ",
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+            "(b) Weights of the first Fully-Connected Layer ",
+            "Figure 5: Comparison to other reconstruction schemes. Top: Model inversion on the 2D experiment (left), on CIFAR10 (top right) and MNIST (bottom right). The CIFAR and MNIST images are ordered by the value of their output from left (smallest) to right (largest). Bottom: Weights of the first (fully-connected) layer for the 2D experiment (left), CIFAR10 (top right) and MNIST (bottom right). The weights for the 2D experiment are the small purple dots. For the CIFAR and MNIST experiments we show the 10 weights with the highest SSIM score. "
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+    {
+        "type": "text",
+        "text": "and the first to reconstruct a substantial amount of training samples. Moreover, the theoretical basis of the implicit bias in neural networks provides an analytic explanation to this phenomenon. ",
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+    {
+        "type": "text",
+        "text": "Solving our optimization problem for convolutional neural networks turned out to be more challenging and is therefore a subject of future research. We note that the theoretical results that we rely on (i.e., Theorem 3.1) also covers convolutional neural networks. We believe that the homogeneity restriction might be relaxed, and showed reconstructions also from a non-homogeneous model (Figure 4, bottomrightmost). We also believe that our method may be extended to multi-class classifiers using an extension of Theorem 3.1. Finally, showing reconstructions on larger models and datasets, or on tabular or textual data are interesting future directions. ",
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+    {
+        "type": "text",
+        "text": "On the theoretical side, it is not entirely clear why our optimization problem in Eq. (8) converges to actual training samples, even though there is no guarantee that the solution is unique, especially when using no prior (other than simple bounding to $[ - 1 , 1 ] ,$ . As a final note, our work brings up the question: are samples on margin the only ones that can be recovered from a trained classifier? or there exist better reconstruction schemes to reconstruct even more training samples from a trained neural network. ",
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+        "text": "Acknowledgements ",
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+        "text": "This project received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 788535), and ERC grant 754705, and from the D. Dan and Betty Kahn Foundation, and was supported by the Carolito Stiftung. ",
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+        "type": "text",
+        "text": "References ",
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+    },
+    {
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parse/dev/TatRHT_1cK/TatRHT_1cK.md

@@ -0,0 +1,314 @@
+# QUANTIFYING MEMORIZATION ACROSS NEURAL LANGUAGE MODELS
+
+Nicholas Carlini∗1 Daphne Ippolito1,2 Matthew Jagielski1   
+Katherine Lee1,3 Florian Tramèr1 Chiyuan Zhang1
+
+1Google Research 2University of Pennsylvania 3Cornell University
+
+# ABSTRACT
+
+Large language models (LMs) have been shown to memorize parts of their training data, and when prompted appropriately, they will emit the memorized training data verbatim. This is undesirable because memorization violates privacy (exposing user data), degrades utility (repeated easy-to-memorize text is often low quality), and hurts fairness (some texts are memorized over others).
+
+We describe three log-linear relationships that quantify the degree to which LMs emit memorized training data. Memorization significantly grows as we increase (1) the capacity of a model, (2) the number of times an example has been duplicated, and (3) the number of tokens of context used to prompt the model. Surprisingly, we find the situation becomes more complicated when generalizing these results across model families. On the whole, we find that memorization in LMs is more prevalent than previously believed and will likely get worse as models continues to scale, at least without active mitigations.
+
+# 1 INTRODUCTION
+
+The performance of neural language models has continuously improved as these models have grown from millions to trillions of parameters (Fedus et al., 2021), with their training sets similarly growing from millions to trillions of tokens. In anticipation of future, even larger models trained on minimally curated datasets, it is important to quantify factors that lead to increased memorization of a model’s training set. Indeed, recent work has shown that training data extraction attacks are a practical threat for current language models (Carlini et al., 2020); an adversary interacting with a pretrained model can extract individual sequences that were used to train the model.
+
+While current attacks are effective, they only represent a lower bound on how much memorization occurs in existing models. For example, by querying the GPT-2 language model, Carlini et al. (2020) (manually) identified just 600 memorized training examples out of a 40GB training dataset. This attack establishes a (loose) lower bound that at least $0 . 0 0 0 0 0 0 0 1 5 \%$ of the dataset is memorized. In contrast, we are able to show that the 6 billion parameter GPT-J model (Black et al., 2021; Wang and Komatsuzaki, 2021) memorizes at least $1 \%$ of its training dataset: The Pile (Gao et al., 2020).
+
+In addition to prior work’s loose estimates of models’ memorization capabilities, there is a limited understanding of how memorization varies across different neural language models and datasets of different scales. Prior studies of memorization in language models either focus on models or datasets of a fixed size (Carlini et al., 2019; Zhang et al., 2021; Thakkar et al., 2020) or identify a narrow memorization-versus-scale relationship (Carlini et al., 2020; Lee et al., 2021). While McCoy et al. (2021) broadly study the extent to which language models memorize, their focus is on how to avoid the problem and ensure novelty of model outputs, rather than on studying model risk through identifying the maximal amount of data memorization.
+
+This paper addresses both of the above open questions by comprehensively quantifying memorization across three families of neural language models and their associated datasets. We leverage access to each model’s original training set to provide order-of-magnitude more precise bounds on the amount of extractable data that an adversary could recover than in prior works.
+
+We first construct a set of prompts from the model’s training set. By feeding prefixes of these prompts into the trained model, we check whether the model has the ability to complete the rest of the example verbatim. This allows us to measure memorization across models, datasets, and prompts of varying sizes. We identify three properties that significantly impact memorization:
+
+1. Model scale: Within a model family, larger models memorize $2 { - } 5 \times$ more than smaller models.   
+2. Data duplication: Examples repeated more often are more likely to be extractable.   
+3. Context: It is orders of magnitude easier to extract sequences when given a longer context.
+
+Our analysis suggests that future research on neural language modeling will need to take steps to prevent future (larger) models from memorizing their training datasets.
+
+# 2 RELATED WORK
+
+There is extensive prior work that qualitatively studies memorization in neural language models. Prior work has demonstrated extraction attacks that recover memorized data including URLs, phone numbers, and other personal information (Carlini et al., 2020; Ziegler, 2021)—or synthetically injected “canaries” (Carlini et al., 2019; Henderson et al., 2018; Thakkar et al., 2020; Thomas et al., 2020). However most of these works are qualitative and aim to demonstrate the existence of extractable data, rather than precisely quantifying how much models memorize. For example, the unprompted memorization evaluation of Carlini et al. (2020) found just 600 examples of memorization in GPT-2. Our paper aims to establish tighter bounds on the fraction of a dataset that is memorized.
+
+Our analysis is relevant to the broad literature on privacy attacks on machine learning. For example, membership inference attacks (Shokri et al., 2017; Yeom et al., 2018) let an adversary detect the presence of a given example in a model’s training set; other forms of data leakage let an adversary learn dataset properties (Ganju et al., 2018; Fredrikson et al., 2015). We focus on extraction attacks due to their relevance for language modeling—extraction implies significant leakage from a model, and grows with data duplication (Lee et al., 2021), a common feature of large-scale text datasets.
+
+Various definitions of memorization in deep neural networks have been studied in prior work (Carlini et al., 2019; 2020; Feldman and Zhang, 2020; Zhang et al., 2021). A detailed comparison with those existing formulations is presented in Section 3.1. One leading general memorization definition is differential privacy (Dwork et al., 2006), which formalizes the idea that removing any one example from the training set should not change the trained model. However, while differential privacy protects a single user’s private information, it is ineffective for preventing memorization of highly duplicated data, and does not capture the complexity of social, linguistic data (Brown et al., 2022). Also, differentially private learning algorithms (Abadi et al., 2016) generally suffer from expensive computation, slow convergence, and poor model utility, despite recent advances (Anil et al., 2021).
+
+In concurrent work, Kandpal et al. (2022) study how often models emit memorized data as a function of data duplication. Their analysis focuses on evaluating why training data extraction attacks succeed. In contrast, we explicitly prompt models with training data prefixes in order to measure memorization in the worst case, something that a practical attack cannot necessarily do.
+
+Prior scaling hypotheses. Our motivation to study scaling phenomena stems from anecdotal evidence in prior work that memorization ability relates to various aspects of scale. In particular, our analysis on model scale is informed by preliminary experiments in (Zhang et al., 2017; Carlini et al., 2020), our data duplication experiments follow in the line of Lee et al. (2021), and our context length experiments build on hypotheses by Carlini et al. (2020); Ziegler (2021).
+
+# 3 METHODOLOGY
+
+# 3.1 DEFINITION OF MEMORIZATION
+
+To begin, we first select a precise definition for memorization:
+
+Definition 3.1. A string $s$ is extractable with $k$ tokens of context from a model $f$ if there exists a (length- $k$ ) string $p$ , such that the concatenation $\left[ p \mid \mid s \right]$ is contained in the training data for $f$ , and $f$ produces $s$ when prompted with $p$ using greedy decoding.
+
+For example, if a model’s training dataset contains the sequence “My phone number is 555-6789”, and given the length $k = 4$ prefix “My phone number is”, the most likely output is “555-6789”, then this sequence is extractable (with 4 words of context). We focus on greedy sampling in this paper, and verify in Section 4.1 that our choice of decoding strategy does not significantly impact our results.
+
+While prior work proposed other definitions, we prefer ours in this paper as it is more actionable. Some memorization definitions, including lower-bounds on differential privacy (Dwork et al., 2006; Jagielski et al., 2020; Nasr et al., 2021) or counterfactual memorization (Feldman and Zhang, 2020; Zhang et al., 2021), require training hundreds or thousands of models, which is impractical for large language models. Alternatively, computing exposure (Carlini et al., 2019) requires thousands of generations per sequence, and is only designed for carefully crafted training examples.Finally, $k$ -eidetic memorization (Carlini et al., 2020), is a useful definition for unprompted memorization, but less useful for tightly bounding memorization by prompting with training data (as we will do). Future work might explore how our three scaling observations apply to other definitions of memorization.
+
+# 3.2 SELECTION OF EVALUATION DATA
+
+Having chosen a definition, we next describe our evaluation procedure. Ideally, we would consider every sequence $x = [ p \mid \mid s ]$ in the model’s training dataset (where $x$ has been split into a length- $k$ prefix $p$ and a suffix $s$ ). For each sequence, we would report if the model exactly reproduces $s$ when prompted with $p$ , following Definition 3.1. Unfortunately, performing this test on every sequence in the training data would be prohibitively expensive. For example, the largest 6 billion parameter GPT-Neo model has a throughput of roughly one 100-token generation per second on a V100 GPU. Extrapolating to the 800GB training dataset, this would require over 30 GPU-years of compute.
+
+Instead, we query on a smaller subset of the training data, that still produces statistically confident estimates. In this paper we randomly choose subsets of roughly 50,000 sequences, allowing us to efficiently run inference in just a few hours. The primary criteria when choosing a subset of the training data is to obtain a representative sample that allows us to draw meaningful conclusions from the data. We consider two approaches to constructing a subset of the data.
+
+Our first subset is a uniformly random sample of 50,000 sequences, drawn from the training dataset without repetition. While a uniform sample is useful to estimate the absolute amount of memorization in a model, it is poorly suited for studying how memorization scales with data properties that are not uniformly represented in the training set. For example, prior work has identified that data duplication (i.e., how often the same sequence is repeated either exactly or approximately) is an important factor for memorization. Yet, because the frequency of training data duplication decays extremely quickly (Lee et al., 2021), a uniformly random sample of 50,000 sequences (accounting for $\leq 0 . 0 \dot { 2 } \%$ of the dataset) is unlikely to contain any signal that would allow us to accurately measure the tail of this repeated data distribution. A similar concern arises for measuring how memorization scales with prompt length, since very long sentences account for only a small fraction of the training set.
+
+Therefore, our second subset is a random sample normalized by both sequence lengths and duplication counts, which allows us to accurately measure memorization of large language models in the worst-case, on highly duplicated data with long prompts. For each sequence length $\ell \in \{ 5 0 , 1 0 0 , 1 5 0 , \ldots , 5 0 0 \}$ , and integer $n$ , we select 1,000 sequences of length $\ell$ that are contained in the training dataset between $2 ^ { n / 4 }$ and $2 ^ { ( n + 1 ) / 4 }$ times. We do this until we reach an $n$ for which 1,000 sequences are not available. This gives us 1,000 sequences that repeat between 6 and 8 times $( \approx 2 ^ { 1 1 / 4 }$ and $\approx 2 ^ { 1 2 / 4 }$ ) and also 1,000 sequences that repeat between 724 and 861 times $( \approx 2 ^ { 3 8 / 4 }$ and $\approx 2 ^ { 3 9 / 4 }$ ). This biased sampling allows us to more accurately measure memorization as a function of a sample’s duplication factor and prompt length, without querying the entire dataset. Note that constructing this duplicate-normalized data subset requires some work, as efficiently identifying duplicate substrings in an 800GB training dataset is computationally challenging. We make use of the suffix array construction from Lee et al. (2021) (see Appendix).
+
+For each length from 50 to 500 tokens, we collect 50,000 examples duplicated varying numbers of times, totaling roughly 500,000 sequences. For each sequence of length $\ell$ , we prompt the model with the first $\ell - 5 0$ tokens and report the sequence as “extractable” if the model exactly emits the next 50 token suffix of this sequence. Fifty tokens corresponds to an average of 127 characters or 25 wordsin the GPT-Neo training set, well over the length of a typical English sentence. Finally, we compute the average probability that a sequence is extractable by averaging over all lengths $\ell$ .
+
+![](images/58cdd75dde2fb8cf29242afb07a5e17dbb8a0c99e5ec6d53633791dd78c9bd34.jpg)  
+Figure 1: We prompt various sizes of GPT-Neo models (green) with data sampled from their training set—The Pile, and normalized by sequence lengths and duplication counts. As a baseline (yellow), we also prompt the GPT-2 family of models with the same Pile-derived prompts, even though these models were trained on WebText, a different training dataset. (a) Larger models memorize a larger fraction of their training dataset, following a log-linear relationship. This is not just a result of better generalization, as shown by the lack of growth for the GPT-2 baseline models. (b) Examples that are repeated more often in the training set are more likely to be extractable, again following a log-linear trend (baseline is GPT-2 XL). (c) As the number of tokens of context available increases, so does our ability to extract memorized text (baseine is GPT-2 XL).
+
+# 4 EXPERIMENTS
+
+We primarily study the GPT-Neo model family (Black et al., 2021; Wang and Komatsuzaki, 2021) trained on the Pile dataset (Gao et al., 2020). The GPT-Neo models are causal language models trained with the objective of predicting the next token in a sequence given the previous ones. They come in four sizes: 125 million, 1.3 billion, 2.7 billion and 6 billion parameters.1 The Pile is a dataset of 825GB of text collected from various sources (e.g., books, Web scrapes, open source code). Prior to the recent release of OPT (Zhang et al., 2022), the GPT-Neo models were the largest language models available for public download, and The Pile is the largest public text dataset available.
+
+# 4.1 BIGGER MODELS MEMORIZE MORE
+
+We begin by considering the impact of model size on memorization, expanding on prior studies which qualitatively established a relationship between the size of GPT-2 models and their ability to memorize $< 3 0$ URLs (Carlini et al., 2020). In contrast, we study a million model generations in order to describe how model scale relates to memorization.
+
+Results. We first study our biased random data sample normalized by duplication count and sequence lengths. The results of this experiment are given in Figure 1a. The y-axis reports the fraction of generations which exactly reproduce the true suffix for their prompt, averaged over all prompt and sequence lengths in our evaluation set. Because our biased sampling over-represents duplicated strings, the absolute degree of memorization in Figure 1a is not particularly important here—rather, we are interested in how memorization varies with scale.2 We find that larger models memorize significantly more than smaller models do, with a near-perfect log-linear fit ( $R ^ { 2 }$ of $9 9 . 8 \%$ ): a ten fold increase in model size corresponds to an increase in memorization of 19 percentage points.
+
+To confirm that larger models are indeed memorizing more data, and not simply generalizing better, we repeat the analysis with the GPT-2 model family as a baseline. The GPT-2 models are similarly sized, and also trained on Internet-scraped data. If our “larger models memorize more” result was due to the predictive strength of larger models, and not the memorization of specific training data, we would expect a similar relationship between comparably sized GPT-2 models trained on similar data. Put differently, this baseline allows to establish what fraction of the training data is sufficiently “easy” that any language model can correctly predict the 50-token suffix, even if the example has not been seen during training. For example, a language model trained on multiple examples of number sequences can likely correctly complete some other unseen number sequences. We find that GPT-2 correctly completes approximately $6 \%$ of the examples in our evaluation set, compared to $4 0 \%$ for the similarly sized 1.3B parameter GPT-Neo model. A qualitative analysis (see examples in Appendix Figure 15) suggests that examples “memorized” by GPT-2 are largely uninteresting sequences (e.g., number sequences, repetitions of the same few tokens, or common phrases). Therefore, we conclude that larger models have a higher fraction of extractable training data because they have actually memorized the data; it is not simply that the larger models are more accurate.
+
+# 4.2 REPEATED STRINGS ARE MEMORIZED MORE
+
+Prior work provides preliminary evidence that memorization in language models increases with the number of times sequences are repeated in the training set (Carlini et al., 2020; Lee et al., 2021). We expand on this observation and quantitatively measure the effect of data duplication on memorization. Using our duplication-normalized data sample, we measure the fraction of sequences which are extractable, for buckets of sequences duplicated between 2 and 900 times. Each bucket consists of 1,000 distinct sentences, and we compute the average amount of memorization for each bucket.
+
+Results. Figure 1b shows our results, aggregated over all sequence lengths. We observe a clear log-linear trend in memorization. While models rarely regurgitate strings that are repeated only a few times, this probability increases severely for highly duplicated strings. The small memorization values at low numbers of repetitions corroborates the positive impact of training dataset deduplication on memorization observed by Lee et al. (2021). However, we find that memorization does still happen, even with just a few duplicates—thus, deduplication will not perfectly prevent leakage. While this relationship is perhaps obvious, and has been corroborated for specific training examples in prior work (Carlini et al., 2019; 2020), our results show that it holds across the entire training set.
+
+# 4.3 LONGER CONTEXT DISCOVERS MORE MEMORIZATION
+
+The previous two questions evaluated how data collection and model training decisions impact the leakage of a model’s training data when it is provided a fixed number of tokens from a sequence as context. As a result, those experiments suggest particular actions that could be taken to mitigate memorization (by reducing model size, or limiting the number of duplicate examples).
+
+However, even when the model is fixed, it is possible to vary the amount of extractable training data by controlling the length of the prefix passed to the model. By studying how the number of tokens of context impacts extractability, we demonstrate the difficulty of discovering memorization—language models may only exhibit their memorization under favorable conditions.
+
+Results. In Figure 1c, we observe that the fraction of extractable sequences increases log-linearly with the number of tokens of context. For example, $33 \%$ of training sequences in our evaluation set are extractable from the 6B model at 50 tokens of context, compared to $65 \%$ with 450 tokens of context. We call this the discoverability phenomenon: some memorization only becomes apparent under certain conditions, such as when the model is prompted with a sufficiently long context.
+
+The discoverability phenomenon may seem natural: conditioning a model on 100 tokens of context is more specific than conditioning the model on 50 tokens of context, and it is natural that the model would estimate the probability of the training data as higher in this situation. However, the result is that some strings are “hidden” in the model and require more knowledge than others to be extractable.
+
+From one point of view, it is good that some memorization is difficult to discover. This makes it harder for attackers to perform training data extraction attacks (Carlini et al., 2020), or otherwise exploit memorization. Indeed, if an exact 100 token prompt is required to make the model output a given string, then, in practice, an adversary will likely be unable to perform the attack. The difficulty in discovering memorization also reduces the likelihood of non-adversarial training data regurgitation. For example, the GitHub Copilot model (Chen et al., 2021) reportedly rarely emits memorized code in benign situations, and most memorization occurs only when the model has been prompted with long code excerpts that are very similar to the training data (Ziegler, 2021). Practitioners building language generation APIs could (until stronger attacks are developed) significantly reduce extraction risk by restricting the maximum prompt length available to users.
+
+![](images/0bedac6815f44affbcc628726f723f2ee181583068aac9bec5893bf523140593.jpg)  
+Figure 2: (a) Fraction of sequences extracted as a function of model scale where we sample uniformly from the training set. (b) Fraction of sequences extracted as we vary the length of the prompt. For each sequence length $n$ , $n { - } 5 0$ tokens are used as the prefix, and we check for extraction of the remaining 50 tokens. (c-left) Using beam search with $\scriptstyle b = 1 0 0$ slightly increases the data extracted. (c-right) We observe considerably more memorization when checking whether the generated sequence occurs anywhere in the entire training set (Section C). However, this approach is very computationally expensive so we do not use it for our other experiments.
+
+Viewed differently, however, the difficulty of discovering memorization can also harm our ability to audit privacy in machine learning models. Because provably-correct approaches for privacypreserving training of machine learning models are applied only rarely in practice (Abadi et al., 2016; Thakkar et al., 2020; Ramaswamy et al., 2020), it is common to attempt post-hoc privacy auditing (Jayaraman and Evans, 2019; Jagielski et al., 2020; Nasr et al., 2021). Our results suggest that correctly auditing large language models likely requires prompting the model with training data, as there are no known techniques to identify the tail of memorized data without conditioning the model with a large context. Improving upon this limitation is an interesting problem for future work.
+
+# 4.4 ALTERNATE EXPERIMENTAL SETTINGS
+
+In this section, we briefly review other strategies that we could have used to quantify memorization.
+
+Random dataset sampling. The majority of this paper uses subsets of the training data that were explicitly sampled according to training data duplication frequency. Now, we consider how our results would differ if we chose a truly random subset of the training data, where each sequence is sampled uniformly, instead of sampling a duplicate-normalized dataset. Specifically, we randomly sample 100,000 sequences of varying lengths from The Pile dataset, then prompt the model and test for memorization as before (more details in Appendix C).
+
+Figure 2a and Figure 2b present the results. We observe similar qualitative trends with model scale and context length as in Figure 1. Larger models memorize more training examples than smaller models—and much more than the GPT-2 models that were not trained on The Pile. Similarly, providing more context to a model increases the likelihood we discover memorization. We can extract the last 50 tokens of a length-1000 sequence with $7 \%$ probability for the largest GPT-J 6B model, compared to $4 \%$ probability for the smallest 125M GPT-Neo model. (And both of these are much larger than the $2 \%$ probability of extraction for the 1.5B parameter GPT2-XL model.) These results, taken together, allow us to estimate a lower bound that there is at least $1 \%$ of The Pile dataset that is extractable by the 6B GPT-J model, but not by GPT-2 XL.
+
+Alternate decoding strategies. We have defined memorization as a model’s ability to generate the true continuation when choosing the most likely token at every step of decoding. Yet, this greedy decoding strategy does not produce the overall most likely sequence. Many language model applications use other decoding strategies, such as beam search to find the generation with highest likelihood. To understand how our choice of decoding strategy affects the amount of memorization we measure, we compare greedy decoding with beam search in Figure 2(c). We find that using beam search with 100 beams results in marginally more extracted memorization. The difference in extractable memorization is just under 2 percentage points on average, with a maximum of $5 . 6 \%$ Interestingly, beam search and greedy decoding generated the same output $45 \%$ of the time.
+
+Figure 3: Text examples that are memorized by the 6B model, but not by smaller models. Green highlighted text matches the ground truth continuation, while red text indicates incorrect generation.   
+
+<table><tr><td rowspan=1 colspan=1>Prompt</td><td rowspan=1 colspan=1>Continuation (== 6B)</td><td rowspan=1 colspan=2>2.7B</td><td rowspan=1 colspan=1>1.3B</td><td rowspan=1 colspan=1>125M</td></tr><tr><td rowspan=1 colspan=1>Gallery &quot;Though defensive violence will</td><td rowspan=1 colspan=1>condemned as ridiculous,and</td><td rowspan=1 colspan=2>condemned as ridiculous,and then</td><td rowspan=1 colspan=1>tried inthe mind of achild</td><td rowspan=1 colspan=1>and foremostaman&#x27;s dutyto</td></tr><tr><td rowspan=3 colspan=1>always be&#x27;a sad necessity&#x27;in the eyesof men of principle, it would be stillmore unfortunate if wrongdoers shoulddominate just men.&quot;- St.Augustine &quot;Anew idea is first</td><td rowspan=2 colspan=1>thendismissedastrivial,utilfinally it becomes what everybodyknows.&quot;-William James &quot;This istherealtaskbefore us:toreassert</td><td rowspan=2 colspan=2>dismissedastivial,bforengacceptedas the mostimportant thingin the world.George Bernard</td><td rowspan=3 colspan=1>and then it is tried in themind of a man.&quot;- ThomasPaine &quot;Theonly thingnecessary for the triumphof evil is for good mento donothing.&quot;- Edmund Burke&quot;</td><td rowspan=3 colspan=1>his fellow man.It isa duty tobea manof principleandto beaman ofprinciple,andtobeaman ofprinciple,andtobeaman ofprinciple,and tobea</td></tr><tr><td rowspan=1 colspan=1>acceptedas the mostimportant thingin the world.George Bernard</td></tr><tr><td rowspan=1 colspan=1>therealtaskbefore us:toreassertourcommitmentasanation toalaw higher than our own,</td><td rowspan=1 colspan=2>Shaw&quot;The ultimate measure of a manis not where he standsin moments ofcomfort and convenience,but where</td><td rowspan=1 colspan=1>aman</td></tr><tr><td rowspan=2 colspan=1>_GPL(crypto_unregister_alg); intcrypto_register_template(structcrypto_template *tmpl){structcrypto_template *q; interr=-EIS;</td><td rowspan=2 colspan=1>down_write(&amp;crypto_alg_sem);list_for_each_entry(q,&amp;cryptotemplatelist,istif==tmpl)</td><td rowspan=1 colspan=2>list_for_each_entry(q,&amp;cryptoag_list,list){if(tmp-</td><td rowspan=2 colspan=1>q = kzalloc(sizeof(*q),GFP_KERNEL); if(q)goto out; q-&gt;alg=tmpl-&gt;alg;q-&gt;base</td><td rowspan=2 colspan=1>stuctcrypto_template*tmpl=crypto_template_new(tmpl);if (err)returemp-&gt;tmpl = q;tmpl-&gt;tmpl-&gt;tm</td></tr><tr><td rowspan=1 colspan=2>&gt;name&amp;&amp; tmpl-&gt;name!= q-&gt;alg.cra_name)</td></tr></table>
+
+The most common decoding strategy employed by modern LMs is random sampling, where the next token is selected at random according to a probability distribution derived from the model’s predictions. McCoy et al. (2021) found that random sampling resulted in generated text with a greater number of novel $n$ -grams. Since the goal of our study is to maximize discoverability—an antithetical goal to maximizing linguistic novelty—we do not present experiments that use random sampling.
+
+Alternate definition of extractability. Our main experiments report a sequence as “extractable” if the model’s generation is identical to the true suffix of the considered training example. However it is possible this suffix is still present (elsewhere) in the dataset. We now consider a loose lower bound on memorization that considers a sequence memorized if the generation $[ p | | f ( p ) ]$ from a prompt $p$ is contained anywhere in the training dataset. Searching within the entire dataset finds more memorized content than comparing with the ground truth (Figure 2c). For examples at 100 repetitions, $3 2 . 6 \%$ of outputs are contained somewhere in the dataset but just $1 5 . 8 \%$ match the ground truth continuation.
+
+# 4.5 QUALITATIVE EXAMPLES OF MEMORIZATION
+
+In Figure 3, we present qualitative examples that are only memorized by the largest (6B) model, but not the smaller ones. We highlight some interesting patterns in these sequences: while the generations from the smaller models do not match the training data, they are generally thematically-relevant and locally consistent. However, a closer inspection reveals that those generations are only syntactically sound, but semantically incorrect. Appendix Figure 8 shows further examples of sequences that are memorized by all the models. We found most of these universally-memorized sequences to be “unconventional” texts such as code snippets or highly duplicated texts such as open source licenses. Figure 13 shows sequences which are memorized by the 6B parameter model despite being infrequent in the training set. These tend to be easily completed text– Figure 14 shows sequences which are repeated thousands of times but are surprisingly not memorized by the 6B parameter model. Many of these are mostly correctly completed, only differing on semantically unimportant characters.
+
+# 5 REPLICATION STUDY
+
+The above analysis provides evidence that memorization scales log-linearly with model size, data duplicates, and context length. We now replicate this analysis for other language models trained with different datasets and training objectives, namely: (1) the T5 family of models trained on the C4 dataset (Raffel et al., 2020), (2) models from Lee et al. (2021), trained on a deduplicated version of C4, and (3) the OPT family of models (Zhang et al., 2022), also trained on the Pile. We expected our results to cleanly generalize across settings, and this is indeed true for model scale. Yet, the situation is more complicated when considering data duplication, due to training set idiosyncrasies.
+
+![](images/90276b71ff45e18c305de9a4befba9abd4ccc032b0e6fcc2437cae7d08e47802.jpg)  
+Figure 4: (a) Masked language model objective: Larger models have a higher fraction of sequences extractable on T5. (b) Masked language model objective: Relationship between number of repetitions and extractable tokens on T5. (c) Causal language model objective: Relationship between number of repetitions and memorization on language models trained with deduplicated data.
+
+# 5.1 T5 MASKED LANGUAGE MODELING
+
+Model and dataset. The T5 v1.1 models are masked encoder-decoder models trained to reproduce randomly deleted spans from an input sequence. The models vary in size from 77M to 11B billion parameters, and are trained on C4—a 806 GB curated version of English web pages from the Common Crawl. The largest T5 model (11B parameters) is the largest publicly available masked language model. T5 models are thus good candidates for studying how memorization scales with model size.
+
+We must first define what is meant by “extractable data” for the masked language modeling task. T5 models are trained by removing a random $1 5 \%$ of tokens from each training sequence (i.i.d), and the model must then “fill in the blanks” to restore the tokens that were dropped from the input. As a result of this different training objective, Definition 3.1 is not directly applicable: the model does not operate on a prefix and output a suffix. We instead call a sequence memorized if the model perfectly solves the masked language modeling task on that sequence. For example, we call a 200-token sequence memorized if the model can use the 170 $( = 2 0 0 \cdot 0 . 8 5 )$ ) tokens of context to perfectly predict the remaining 30 tokens $( = 2 0 0 \cdot 0 . 1 5 )$ . Because this token-dropping procedure is stochastic, it is possible that one set of dropped tokens might yield an output of “memorized” and another might not. For simplicity, we inspect only one set of masked tokens per sequence; because we are already averaging over 50,000 sequences this additional randomness does not harm the results of our analysis.
+
+Results. In Figure 4a, we reproduce the model scaling effect (from Figure 1a) for T5 models. Larger models similarly have an increased ability to perfectly solve the masked prediction task. Surprisingly, while a scaling trend does hold here as well, the absolute memorization in masked models is an order of magnitude lower than for comparably sized causal language models. For example, the 3B parameter T5-XL model memorizes $3 . { \bar { 5 } } \%$ of sequences repeated 100 times, whereas the GPT-Neo 2.7B model memorizes $5 3 . 6 \%$ of sequences repeated 100 times (with 150 tokens of context).
+
+Next, we turn to reproducing the analysis of how memorization scales with data duplication. The situation here becomes significantly less clear. As shown in Figure 4b, sequences duplicated more often tend to be easier to memorize, but there is no monotonic scaling relationship. Compared to the case of the GPT-Neo models trained on The Pile, the relation between data duplication counts and memorization for T5 models trained on C4 exhibits large variance. This variance is statistically significant: sequences repeated 159 to 196 times are memorized with probability less than $5 . 1 \%$ with $9 9 . 7 \%$ confidence (three standard deviations from the mean), however sequences repeated 138 to 158 times (that is, less often) are memorized with probability at least $6 . 2 \%$ (also with $9 9 . 7 \%$ confidence). That is, for some reason, sequences that occur ${ \sim } 1 4 0$ times are more likely to be memorized, despite occurring less often, even if we assume a three-sigma error in both measurements simultaneously.
+
+In order to explain this counter-intuitive phenomenon, we qualitatively study each of these two buckets of examples to understand this difference. We find that most of the duplicate examples repeated 138-158 times consist mainly of whitespace tokens. These sequences are thus much easier to predict correctly than other sequences, even if they are repeated more often. This effect, to a lesser extent, can be found in other buckets which contain many approximately near duplicates.
+
+# 5.2 LANGUAGE MODELS TRAINED ON DEDUPLICATED DATA
+
+Model and dataset. The models used in Lee et al. (2021) are 1.5B parameter causal language models. This model family consists of one model trained on C4 (the same dataset as T5), one model trained on a version of C4 that was deduplicated by removing all documents which were near-duplicates of other documents, and one model trained on a version of C4 that was deduplicated by deleting any string of length-50 tokens that occurred more than once. Lee et al. (2021) found that both types of deduplication reduced the likelihood of memorization.
+
+Results. We were most interested in whether models trained on deduplicated data would still exhibit increased memorization of examples which were repeated frequently in the original, non-deduplicated C4 dataset (e.g., because the deduplication missed some near-duplicates). Figure 4c plots the fraction of sequences memorized by these three models. We draw two interesting conclusions from this data.
+
+First, we confirm that models trained on deduplicated datasets memorize less data than models trained without deduplication. For example, for sequences repeated below 35 times, the exact deduplicated model memorizes an average of $1 . 2 \%$ of sequences, compared to $3 . 6 \%$ without deduplication, a statistically significant $\left( p < 1 0 ^ { - 1 5 } \right.$ ) decrease by a factor of $3 \times$ . Second, while deduplication does help for sequences repeated up to ${ \sim } 1 0 0$ times, it does not help for sequences repeated more often! The extractability of examples repeated at least 408 times is statistically significantly higher than any other number of repeats before this. We hypothesize that this is due to the fact that any deduplication strategy is necessarily imperfect in order to efficiently scale to hundreds of gigabytes of training data. Thus, while it may be possible to remove most instances of duplicate data, different and valid definitions of duplicates can mean deduplication is not exhaustive.
+
+# 5.3 LANGUAGE MODELS TRAINED ON A MODIFIED VERSION OF THE PILE
+
+Model and dataset. We finally study the OPT family of models (Zhang et al., 2022), that vary from 125 million to 175 billion parameters.3 These models were trained on a 800GB dataset that overlaps with The Pile but is not identical and contains data from many new sources, while also removing some data from the Pile. This dataset was also deduplicated prior to training, and so we do not expect to see duplicate sequences memorized (much) more than sequences repeated only a few times.
+
+Results. Overall, we find that while there are nearly identical scaling trends to those we found for GPT-Neo’s model family, the effect size is orders-of-magnitude smaller (figure 7). Even the 66 billion parameter model memorizes a smaller fraction of The Pile than the smallest 125 million parameter GPT Neo model. This suggests two possible conclusions: (a) careful data curation and training can mitigate memorization, or (b) even slight shifts in data distribution can significantly alter what content gets memorized. Without direct access to the original training dataset, we can not distinguish between these two conclusions and hope future work will be able to resolve this question.
+
+# 6 CONCLUSION
+
+Our paper presents the first comprehensive quantitative analysis of memorization in large language models, by re-processing the training set to find memorized data. Our work has two broad conclusions.
+
+For the study of generalization, we have shown that while current LMs do accurately model the statistics of their training data, this need not imply that they faithfully model the desired underlying data distribution. In particular, when the training data distribution is skewed (e.g., by containing many duplicates of some sequences) larger models are likely to learn these unintended dataset peculiarities. It is therefore important to carefully analyze the datasets used to train ever larger models, as future (larger) models are likely to remember even more training details than current (smaller) models.
+
+For the study of privacy, our work indicates that current large language models memorize a significant fraction of their training datasets. Memorization scales log-linear with model size—by doubling the number of parameters in a model we can extract a significantly larger fraction of the dataset. Given that current state-of-the-art models contain more than $2 0 0 \times$ as many parameters as the largest 6B parameter model we analyze, it is likely that these even larger models memorize many sequences that are repeated just a handful of times. At the same time, we have shown that this memorization is often hard to discover, and for an attack to actually extract this data it will be necessary to develop qualitatively new attack strategies. Fortunately, it appears that (for the comparatively small models we study) training data inserted just once is rarely memorized, and so deduplicating training datasets (Lee et al., 2021) is likely a practical technique to mitigate the harms of memorization.
+
+# REFERENCES
+
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+
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+
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+Leo Gao, Stella Biderman, Sid Black, Laurence Golding, Travis Hoppe, Charles Foster, Jason Phang, Horace He, Anish Thite, Noa Nabeshima, et al. The Pile: An 800GB dataset of diverse text for language modeling. arXiv preprint arXiv:2101.00027, 2020.
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+Bargav Jayaraman and David Evans. Evaluating differentially private machine learning in practice. In 28th {USENIX} Security Symposium ({USENIX} Security 19), pages 1895–1912, 2019.
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+Nikhil Kandpal, Eric Wallace, and Colin Raffel. Deduplicating training data mitigates privacy risks in language models. arXiv preprint arXiv:2202.06539, 2022.
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+Katherine Lee, Daphne Ippolito, Andrew Nystrom, Chiyuan Zhang, Douglas Eck, Chris CallisonBurch, and Nicholas Carlini. Deduplicating training data makes language models better. CoRR, abs/2107.06499, 2021. URL https://arxiv.org/abs/2107.06499.
+
+R. Thomas McCoy, Paul Smolensky, Tal Linzen, Jianfeng Gao, and Asli Celikyilmaz. How much do language models copy from their training data? Evaluating linguistic novelty in text generation using RAVEN. CoRR, abs/2111.09509, 2021. URL https://arxiv.org/abs/2111.09509.
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+Milad Nasr, Shuang Song, Abhradeep Thakurta, Nicolas Papernot, and Nicholas Carlini. Adversary instantiation: Lower bounds for differentially private machine learning. arXiv preprint arXiv:2101.04535, 2021.
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+Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J. Liu. Exploring the limits of transfer learning with a unified text-to-text transformer. Journal of Machine Learning Research, 21(140):1–67, 2020. URL http://jmlr.org/papers/v21/20-074.html.
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+Reza Shokri, Marco Stronati, Congzheng Song, and Vitaly Shmatikov. Membership inference attacks against machine learning models. In 2017 IEEE Symposium on Security and Privacy (SP), pages 3–18. IEEE, 2017.
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+Aleena Thomas, David Ifeoluwa Adelani, Ali Davody, Aditya Mogadala, and Dietrich Klakow. Investigating the impact of pre-trained word embeddings on memorization in neural networks. In International Conference on Text, Speech, and Dialogue, pages 273–281. Springer, 2020.
+
+Ben Wang and Aran Komatsuzaki. GPT-J-6B: A 6 Billion Parameter Autoregressive Language Model. https://github.com/kingoflolz/mesh-transformer-jax, May 2021.
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+
+Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning requires rethinking generalization. ICLR, 2017.
+
+Chiyuan Zhang, Daphne Ippolito, Katherine Lee, Matthew Jagielski, Florian Tramèr, and Nicholas Carlini. Counterfactual memorization in neural language models. arXiv preprint arXiv:2112.12938, 2021.
+
+Susan Zhang, Stephen Roller, Naman Goyal, Mikel Artetxe, Moya Chen, Shuohui Chen, Christopher Dewan, Mona Diab, Xian Li, Xi Victoria Lin, et al. Opt: Open pre-trained transformer language models. arXiv preprint arXiv:2205.01068, 2022.
+
+Albert Ziegler. GitHub Copilot: Parrot or crow? https://docs.github.com/en/github/copilot/researchrecitation, 2021.
+
+# A IMPLEMENTATION DETAILS FOR DATASET CREATION
+
+Intuitively speaking, it is straightforward to construct a dataset containing specifiable proportions of documents at various frequencies. We need only enumerate all sequences repeated various numbers of times, and then sample uniformly at random from each of these subsets. However in practice this is difficult to do, given the scale of these datasets: even asking the question “how many times is this sequence present in the training dataset” requires linear work for each query, and so repeating this thousands of times for an 800GB dataset would be infeasible.
+
+To do this efficiently, we build on the work of Lee et al. (2021) and construct a suffix array over the training dataset. Such a data structure allows efficient queries to enumerate all sequences of length $k$ that are repeated between $N$ and $M$ times for any $N , M$ . This can be accomplished by a linear scan of the suffix array. As notation, write $i$ as the pointer into the dataset at a certain position $j$ of the suffix array (i.e., $A [ j ] = i$ ), $i ^ { \prime }$ as the index at position $j + N$ (so that $A [ j + N ] = i ^ { \prime } )$ , and $i ^ { \prime \prime }$ as the index at position $j + M$ (so that $A [ j + M ] ^ { - } = i ^ { \prime \prime }$ . Then, if $D [ i : i + k ] = D [ i ^ { \prime } : i ^ { \prime } + k ]$ but $D [ i : i + k ] \neq D [ i ^ { \prime \prime } : i ^ { \prime \prime } + k ]$ , the sequence $D [ k : i + k ]$ is guaranteed to appear between $N$ and $M$ times in the dataset. As a result, we can scan linearly through the suffix array and enumerate all values of j $j$ to efficiently find all potential sequences repeated between $\mathbf { N }$ and $\mathbf { M }$ times. From here, we then randomly sample 1,000 indices within these buckets to construct all of our sequences.
+
+# B LONGER DOCUMENTS ARE NOT EASIER TO MEMORIZE THAN SHORTER DOCUMENTS
+
+![](images/1f1fd78d3e6a830529d1a41ffd177190e2b301675e49b12fd2e25fbd4c9cbae1.jpg)  
+Figure 5: Longer sequences are not easier to extract. We compute the probability that an adversary can extract a sequence as a function of the number of tokens of context available, when varying the length of the sequences. All sequences are repeated the same number of times, and evaluated with the same 6B parameter model. Each line represents the fraction extractable in sequences of increasing lengths. Because all lines nearly perfectly overlap, longer sequences are not fundamentally “easier” to extract than shorter sequences.
+
+Intuitively, one might think that longer sequences are more likely in the tail of the distribution, and if the model is trained to a low perplexity, then the tail of the distribution may be more likely to be memorized. This could lead our context length results to be exaggerated (as it would be difficult to untangle the tail effect of memorization from the context length effect). To check if sequence length plays a role in the amount of memorization we can extract with this method, we generated the next 50 tokens after the prompt for various sequence lengths and various prompt lengths. Figure 5 shows the fraction of extractable tokens in the next 50 tokens after the prompt. Each line on the figure represents a set of sequences with sequence lengths between 100 and 500 tokens. For each sequence length, we looked at prompt lengths from 50 tokens to (sequence length − 50) tokens. We do not see significant differences between the fraction of extractable tokens with varying prompt lengths across various sequence lengths.
+
+Figure 6: Text examples that are memorized by the 6B model, but not by smaller models. Text highlighted in green matches the ground truth continuation, while text in red indicates incorrect (novel) generation.   
+
+<table><tr><td rowspan=1 colspan=1>Prompt</td><td rowspan=1 colspan=1>Continuation (= 6B)</td><td rowspan=1 colspan=1>2.7B</td><td rowspan=1 colspan=1>1.3B</td><td rowspan=1 colspan=1>125M</td></tr><tr><td rowspan=3 colspan=1>Gallery &quot;Though defensive violencewill always be&#x27;a sad necessity&#x27; inthe eyes of men of principle,itwould be still more unfortunate ifwrongdoers should dominate justmen.&quot;- St. Augustine &quot;A new idea isfirst</td><td rowspan=1 colspan=1>condemned as ridiculous,and thendismissedastrivial,untilfinallyit</td><td rowspan=3 colspan=1>condemned as ridiculous,and thendismissedasiialoegaccepted asthe most important thingin the world.&quot;- George BernardShaw&quot;The ultimate measure of a manis not where he stands in moments ofcomfort and convenience,but where</td><td rowspan=2 colspan=1>tried in the mindof achild,and thenit istried in themind ofaman.&quot;- ThomasPaine &quot;The only thingnecessary for the triumphof evil is for goodmen todo</td><td rowspan=3 colspan=1>and foremostaman&#x27;sdutytohis fellow man.It isa duty tobe a man of principle,andto bea man ofprincipleand tobeaman ofprinciple,andtobeamanofprinciple,and tobe</td></tr><tr><td rowspan=2 colspan=1>becomeswhat everybody knows.&quot;-William James &quot;This is the real taskbeforeus:toreassertourcommitmentasanationtoalawhigherthanourown,</td></tr><tr><td rowspan=1 colspan=1>nothing.&quot;- Edmund Burke &quot;</td></tr><tr><td rowspan=1 colspan=1>_GPL(crypto_unregister_alg);intcrypto_register_template(structcrypto_template *tmpl){structcrypto_template *q; int err =-EEXIST;</td><td rowspan=1 colspan=1>down_write(&amp;cryptoalg_sem);list_for_each_entry(q,&amp;cryptotemateistst{tmpl)</td><td rowspan=1 colspan=1>list_for_each_entry(q,&amp;cryptoaglist,ist{i&gt;name &amp;&amp;tmpl-&gt;name!= q-&gt;alg.cra_name)</td><td rowspan=1 colspan=1>q = kzaloc(sizeof(*q),GFP_KERNEL); if(!q)goto out; q-&gt;alg=tmpl-&gt;alg; q-&gt;base</td><td rowspan=1 colspan=1>structcrypto_template*tmpl=crypto_templatenew(tmpl);if(err)returnerr;tmpl-&gt;tmpl = q;tmpl-&gt;tmpl-&gt;tm</td></tr><tr><td rowspan=1 colspan=1>ions:before {content: &quot;\f5eb&quot;;}.fa-discord:before {content:&quot;\f392&quot;;}.fa-discourse:before{ content: &quot;\f393</td><td rowspan=1 colspan=1>&quot;;}.fa-divide:before{content:&quot;\f529&quot;;}.fa-dizzy:before{content:&quot;\f567&quot;;}.fa-dna:before</td><td rowspan=1 colspan=1>&quot;;}.fa-digg:before{content:&quot;\f391&quot;;}.fdochubbeore{ot：&quot;\f394&quot;;}.fockeboe</td><td rowspan=1 colspan=1>;}.fa-dggbeore{ot：&quot;\f96c&quot;;}.fa-dollar-sign:before{content:&quot;f155&quot;;}.fa-digniter</td><td rowspan=1 colspan=1>&quot;;}.fa-discus:before{content:&quot;\f394&quot;;}.f-drupal:before{content:&quot;\f395&quot;;}.f-drupaldiscod</td></tr><tr><td rowspan=1 colspan=1>new users as an exploration tourand getting started guide, withexercises at the end of eachchapter. For more advanced traineesit can be a desktop reference,and acollection of the base knowledgeneeded to proceed with system andnetwork administration. This</td><td rowspan=1 colspan=1>book contains many real life examplesderivedfromtheauthor&#x27;sexperienceasa Linux systemand networkadministrator,trainerand consultant.They hope these examples will helpyoutogeta better understanding ofthe Linux systemand that you feelencouraged to tryoutthingson</td><td rowspan=1 colspan=1>bookisdesigned to give thereaderafirm understanding of thetechnologies needed to install andmanage Linux systems, using thevarous available tools and techniquesfor the task.The book begins with arapid-fire introduction to the basic principles of the Linux operating</td><td rowspan=1 colspan=1>is a good place to start foranew user. A: I wouldrecommendthe book&amp;quot;LinuxNetworking&quot; by David S.Itisavery goodbook forbeginners.A: Iwouldrecommend</td><td rowspan=1 colspan=1>isa great way to get startedwitha new project.A:Iwould suggest youto usethefollowing:Createanew project Create a newuser CreateanewuserCreateanew user Create</td></tr></table>
+
+# C ALTERNATE EXPERIMENTAL SETTINGS
+
+In this section, we study other strategies that we could have used to quantify memorization.
+
+Random dataset sampling. In Section 4.4, we explored what would happen if we instead chose a truly random subset of the training data, where each sequence is sampled uniformly. Specifically, we randomly sample 100,000 sequences from The Pile dataset of length 100, 200, 500, and 1,000; prompt the model with the first $N - 5 0$ tokens; and then test for memorization by verifying if the model can emit the remaining 50 tokens perfectly. In our analysis in Figures 2a and 2b, we vary the size of the trained model and the context length we provide it to understand how these factors impact memorization—but this time through prompting the models with randomly sampled training sequences. As expected, the absolute probability of memorization is much lower than in Figure 1 where we prompted models with training data from the sampled duplication-normalized subset.
+
+We observe similar trends with model scale and context length as in our other results. Larger models memorize more training examples than smaller models—and much more than the baseline GPT-2 model that was not trained on The Pile. Similarly, providing more context to a model increases the likelihood we can discover memorization. In Figure 2b, we prompt models with: prompt length $=$ sequence length − 50. We see that the longer prompts are easier to predict correctly than shorter prompts. The baseline GPT-2 model is nearly twice as accurate on sequences of length 1,000 (prompt length $= 9 5 0$ ) compared to sequences of length 100 (prompt length $= 5 0$ ).
+
+Alternate definition of extractability. Our main experiments report a sequence as “extractable” if the model’s generated continuation is identical to the true suffix within that training example. This method is a loose lower bound on memorization. Consider two sequences $x _ { 1 }$ , $x _ { 2 }$ both contained in the training dataset. Suppose these two sequences share the same prefix, and differ only in the final suffix; that is, $x _ { 1 } = [ p | | s _ { 1 } ]$ and $x _ { 2 } = [ p | | s _ { 2 } ]$ . When we select $x _ { 1 }$ and prompt the model on the prefix $p$ , we will report “success” only if the output equals $s _ { 1 }$ , but not if the output is $s _ { 2 }$ , even though this is also a form of memorization.
+
+We now consider how our results would change if we instead checked that the generation $[ p | | f ( p ) ]$ from a prompt $p$ was contained anywhere in the training dataset. This gives a strictly larger measurement of memorization. By comparing these two methods (checking for memorization within the ground truth continuation, and within the entire dataset), we can understand how the choice of measurement affects the results in our experiments.
+
+Searching within the entire dataset finds more memorized content than comparing with the ground truth (Figure 2c). For examples at 100 repetitions $3 2 . 6 \%$ of outputs are contained somewhere in the dataset but just $1 5 . 8 \%$ match the ground truth continuation. This difference becomes more pronounced as the number of repetitions increases. The maximum difference between these approaches is $2 8 . 4 \%$ , at 2,200 repetitions.
+
+We refrain from using this approach for our main experiments, because this definition requires vastly larger computation resources; it requires querying whether hundreds of thousands of sequences are contained in an 800GB training dataset. Therefore, to promote reproducability, the remainder of this paper continues with testing the generated suffix against the single expected training suffix.
+
+# D TEXT MEMORIZED BY ONLY SOME MODELS
+
+Table 1: The number of sequences memorized by one model, and not memorized by another.   
+
+<table><tr><td></td><td></td><td colspan="4">Not Memorized By</td></tr><tr><td>Model</td><td>Memorized</td><td>125M</td><td>1.3B</td><td>2.7B</td><td>6B</td></tr><tr><td>125M</td><td>4,812</td><td>=</td><td>328</td><td>295</td><td>293</td></tr><tr><td>1.3B</td><td>10,391</td><td>5,907</td><td>=</td><td>1,205</td><td>1,001</td></tr><tr><td>2.7B</td><td>12,148</td><td>7,631</td><td>2.962</td><td></td><td>1,426</td></tr><tr><td>6B</td><td>14,792</td><td>10,273</td><td>5,402</td><td>4,070</td><td>-</td></tr></table>
+
+Table 1 shows the total number of sequences that are memorized by one model but not another. Larger models have more uniquely memorized sequences, although every model has some memorization not shared by any other model. (Even the 125M model memorizes a few sequences that the 6B model does not.)
+
+# E MEMORIZATION IN OPT MODELS
+
+![](images/b4258f34d08fcde4ad2d3433830b3b37edccf6aae45ffb70995755120e86a1e5.jpg)  
+Figure 7: We prompt OPT models with data sampled from their training set. We use a prompt length of 100 here. (a) Fraction of sequences extracted as a function of model scale. (b) Fraction of sequences extracted as the number of repetitions of that sequence in the training set increases.
+
+# F EXAMPLES OF MEMORIZED TEXTS
+
+We show examples of texts that are memorized by different models. We consider the case of 50-token prompts and 50-token generation. We sample texts with various number of repetitions in the training data. It is impossible to inspect all the generated examples, so we random sample examples satisfying a certain criterion and show a few interesting ones in the paper. Figure 8 lists examples that are memorized by models of all sizes, in the sense that the 50-token generations match the groundtruth continuations of the prompts.
+
+![](images/4b2fa768251cd5dd0d4a0b299a6d2144bc7e70f30cea7dc7a96b9fb636b408dd.jpg)  
+Figure 8: Text examples that are memorized by all the models: given 50-token prompts on the left, the next 50 tokens generated by all the models match the groundtruth continuation.
+
+Figure 9 lists examples that are memorized by the 6B model but not by smaller ones. Specifically, the 50-token generations of the 6B model match the groundtruth continuations exactly, but the generations from the smaller models match neither the groundtruth continuations of the prompted examples nor any other training examples with the same prompts. We find that when smaller models do not get the groundtruth continuation right, they are generally still able to stick to similar topics. However, in many cases, the texts generated by the smaller models are only syntactically sound, but semantically incorrect. Figure 10 and Figure 11 show more examples.
+
+In Figure 12 we show examples that are only memorized by the smallest model, using similar criterion as when we filter examples that are only memorized by the largest model. There are significantly fewer number of examples that are only memorized by the smallest model (35) than that of the largest model (2860). One of those examples (the first row of Figure 12) is particularly interesting: the groundtruth continuation contains a typo due to formatting cutoff. While the smallest model memorized the typo, larger models try to fix the typo.
+
+In Figure 13 and Figure 14 we show examples that are memorized but not heavily duplicated in the training set, and examples that are heavily duplicated but not memorized, respectively. Finally, we show examples that are memorized by GPT2-XL in Figure 15.
+
+![](images/af75c63b4b84f24044fb3de4fe439cf1a2cade35280e1f033b47d5a3a4a25b5b.jpg)  
+Figure 9: Text examples that are memorized by the 6B model (according to true-continuation match), but not memorized by smaller models (the generated texts do not match the true continuation, nor any other training examples). The first column shows the prompt. The second column shows the prediction from the 6B model, which matches the groundtruth continuation exactly. The remaining columns shows predictions from smaller models.
+
+![](images/706495db9f05416273dce96673059dfa28b2bfaa34aaab2a13c82452f13a4945.jpg)  
+Figure 10: Continuation of Figure 9.
+
+![](images/79d84d735b3294a769d41ff4777c5b4fa00e7bf1841434ea4f9f079758fd7425.jpg)  
+Figure 11: Continuation of Figure 9.
+
+![](images/164ea4bad0bf0873b77d8491a0a471744bcb879d8db2d165fe58d4ce0df81fe6.jpg)  
+Figure 12: Text examples that are memorized by the $1 2 5 \mathbf { M }$ model (according to true-continuation match), but not memorized by larger models (the generated texts do not match the true continuation, nor any other training examples). The first column shows the prompt. The last column shows the prediction from the 125M model, which matches the groundtruth continuation exactly.
+
+Figure 13: Text examples that are memorized but are not heavily duplicated in the training set. Many of these have a simple sequential structure (the middle three), may be boilerplate code (the first), or starts out with unique text, and completes with frequently repeated text (the last example). Overall, these are easily completed sequences.   
+
+<table><tr><td rowspan=1 colspan=1>Frequency</td><td rowspan=1 colspan=1> Prompt</td><td rowspan=1 colspan=1>Continuation ( == 6B)</td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>L_LONG_LONG*/__STL_TEMPLATE_NULL struCt_type_traits&lt;float&gt;{typedef_true_typehas_trivial_default_</td><td rowspan=1 colspan=1>constructor; typedef__true_type has_trivial_copy_constructor;typedef__true_type has_trivial_assignment_operator;</td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>&quot;groupby4_map&quot;,&quot;groupby4_map_skew&quot;,&quot;groupby4_noskew&quot;,&quot;groupby5&quot;,</td><td rowspan=1 colspan=1>&quot;groupby5_map&quot;,&quot;groupby5_map_skew&quot;,&quot;groupby5_noskew&quot;,&quot;groupby6&quot;,</td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>pair&lt;K12&gt;(_12),fusion:make_pair&lt;K13&gt;(13),fusion:make_pair&lt;K14&gt;(14),fusion:makepair&lt;K15&gt;(5));</td><td rowspan=1 colspan=1>J namespace result_of{template&lt;typename K0,typename K1,typenameK</td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>_GLSL_400))list += QLatin1String(&quot;dmat2&quot;); if (variant&amp;(Lexer:Variant_GLSL_40O)) ist+= QLatin</td><td rowspan=1 colspan=1>1String(&quot;dmat3&quot;); f (variant &amp; (Lexer:Variant_GLSL_40o))list +=QLatin1String(&quot;dmat4&quot;); if (variant&amp;(</td></tr><tr><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>disabled.BCG.com will work better for you if you enableJavaScript or switch to a JavaScript supported browser.Boston Consulting Group is an Equal Opportunity Employer.All qualified applicants will receive consideration foremployment without regard to race,color,</td><td rowspan=1 colspan=1>age,religion,sex,xualentationgederdiy/expressionioalginoectedeatothercharacteristic protected under federal,stateor local law,whereapplicable,and those with criminal histories will beconsidered ina manner consistent with</td></tr></table>
+
+Figure 14: Text examples that are heavily replicated in the training set but not memorized. We find many examples which have slight differences with no semantic (English) meaning. This includes comment characters in code, non-English characters, template values, error messages, and meaningless symbols. We also surprisingly find a large number of slightly different but heavily repeated documents about dumpsters.   
+
+<table><tr><td rowspan=1 colspan=1>Frequency</td><td rowspan=1 colspan=1>Prompt</td><td rowspan=1 colspan=1>Continuation</td><td rowspan=1 colspan=1>6B</td></tr><tr><td rowspan=1 colspan=1>5526</td><td rowspan=1 colspan=1>contributors.All rights reserved./// Licensedunder the Apache License,Version 2.0 (the&quot;License&quot;);//you may not usethisfile exceptin compliance with the License.// You mayobtain a copy of the</td><td rowspan=1 colspan=1>License at //http://www.apache.org/licenses/LICENSE-2.0/// Unless requiredby applicable law or agreed to in writing,software // distributed under the License isdistributed on an &quot;AS</td><td rowspan=1 colspan=1>Licenseat////http://www.apache.org/licenses/LICENSE-2.0///Unlessrequiredbyapplicablelaworagreedto inwriting,software//distributedunderthe License</td></tr><tr><td rowspan=1 colspan=1>5652</td><td rowspan=1 colspan=1>ators are Standing By Our Dumpster Specialistsare waiting by the phone to help you getstarted. However, there is some importantinformation we need before we can give you aprice quote and delivery time.Some things weneed to know</td><td rowspan=1 colspan=1>: What size dumpster you will need Whatyou plan on puting in your dumpster Whenyou need us to drop your roll off containeroff How long you plan on keeping your roll-off City &amp;</td><td rowspan=1 colspan=1>include:What size dumpster do you need?What type of waste do you have?How much wastedo you have?What is the delivery time?What isthepickup&#x27;time?What</td></tr><tr><td rowspan=1 colspan=1>4323</td><td rowspan=1 colspan=1>a copy ofthe License at ## http:/lwww.apache.org/licenses/LICENSE-2.0 ##Unless required by applicable law</td><td rowspan=1 colspan=1>or agreed to in writing,software #distributed under the License is distributedon an &quot;AS IS&quot; BASIS,WITHOUT #WARRANTIES OR CONDITIONS OF ANYKIND,either express or implied. See</td><td rowspan=1 colspan=1>oragreed to in writing,software #distributed underthe License is distributed onan &quot;AS IS&quot;BASIS,#VWITHOUT WARRANTIES OR CONDITIONS OFANY KIND,either express or implied.</td></tr><tr><td rowspan=1 colspan=1>3556</td><td rowspan=1 colspan=1>date_default_timezone_set() function.In caseyou used any of those methods and you arestill geting this warning,you most likelymisspelled the timezone identifier. We selectedthe timezone &#x27;UTC&#x27; for now,but please</td><td rowspan=1 colspan=1>setdate.timezone to select your timezone.in/home/erlypro/public_html/natural/datas/persobanner_center.php on line 17Deprecated: Function eregi_replace</td><td rowspan=1 colspan=1>setdate.timezone toselect your timezone.in/home/notes/public_html/wp-includes/functions.phponline3812Warning:date()[functiondatel:Itisnotsafe to rely</td></tr><tr><td rowspan=2 colspan=1>3920</td><td rowspan=2 colspan=1>|&amp;nbsp;&amp;nbsp;[手把手翻墙教程]https://github.com/gfw-breaker/guides/wiki) &amp;nbsp;&amp;</td><td rowspan=1 colspan=1>nbsp;l&amp;nbsp;&amp;nbsp;[禁闻聚合安卓版](https://github.com/gfw-breaker/bn-android)</td><td rowspan=2 colspan=1>nbsp;[网门安卓版](https://github.com/gfw-breaker/webpack-interpreter)&amp;nbsp;&amp;nbsp;[神州正道</td></tr><tr><td rowspan=1 colspan=1>&amp;nbsp;&amp;n</td></tr><tr><td rowspan=1 colspan=1>3675</td><td rowspan=1 colspan=1>*/_II\\)11*</td><td rowspan=1 colspan=1>=(_-_-_△</td><td rowspan=1 colspan=1>I(_ILI1_11*</td></tr><tr><td rowspan=1 colspan=1>3030</td><td rowspan=1 colspan=1>1775,&quot;date&quot;: &quot;1775\u20131851&quot;,&quot;displayOrder&quot;:1,&quot;fc&quot;: &quot;Joseph Mallord William</td><td rowspan=1 colspan=1>Turner&quot;,&quot;gender&quot;:&quot;Male&quot;,&quot;id&quot;:558,&quot;mda&quot;:&quot;Turner,Joseph Mallord William&quot;,</td><td rowspan=1 colspan=1>Turner&quot;,&quot;gender&quot;:&quot;Male&quot;,&quot;id&quot;:604,&quot;mda&quot;: &quot;Turner,Joseph MallordWilliam&quot;,</td></tr></table>
+
+Figure 15: Text examples that are from The Pile and memorized by GPT2-XL. The first two examples have a natural sequential structure, while the others appear to represent an overlap in GPT2-XL’s training set and The Pile.   
+
+<table><tr><td rowspan=1 colspan=1> Prompt</td><td rowspan=1 colspan=1>Continuation (== GPT2-XL)</td></tr><tr><td rowspan=1 colspan=1>165.00 $175.00 $180.00 $185.00 $190.00 $195.00 $200.00$210.00 $220.00 $230.00 $240.00 $250.00 $260.</td><td rowspan=1 colspan=1>00 $270.00 $280.00 $290.00 $300.00 $310.00 $320.00$330.00 $340.00 $350.00 $360.00 $370.00 $380.00 $</td></tr><tr><td rowspan=1 colspan=1>ISRC3DEC2,ARIZONA_ISRC3DEC2MIX_INPUT_1_SOURCE);ARIZONA_MUX_ENUMS(ISRC3DEC3,ARIZONA_</td><td rowspan=1 colspan=1>ISRC3DEC3MIX_INPUT_1_SOURCE);ARIZONA_MUX_ENUMS(ISRC3DEC4,ARIZONA_ISRC3DEC4MIX_INPUT</td></tr><tr><td rowspan=1 colspan=1>the councils of government,we must guard against theacquisition of unwarranted influence,whether sought orunsought,by the militaryindustrial complex.The potentialfor the disastrous rise of misplaced power exists and willpersist.We must never let the weight</td><td rowspan=1 colspan=1>of this combination endanger our liberties or democraticprocesses.Weshould take nothing forgranted.Onlyanalertand knowledgeable citizenrycan compel the propermeshing of the huge industrial and military machinery ofdefense with our peaceful methods and goals,so thatsecurityand</td></tr><tr><td rowspan=1 colspan=1>full spreadsheet NEW! Buy our book·Facts are Sacred: the power of data (on Kindle) More open data Data journalismand data visualisations from the Guardian World government data ·Search the</td><td rowspan=1 colspan=1>world&#x27;s government data with our gateway Developmentand aid data·Search the world&#x27;s global development datawith our gateway Can you do something with this data?·Flickr Please post your visualisations and mash-ups on</td></tr><tr><td rowspan=1 colspan=1>Original press release Get ahead of the crowd by signing upfor 420 Investor,the largest&amp; most comprehensivepremium subscription service for cannabis traders andinvestors since 2013.Published by NCV Newswire The NCVNewswire</td><td rowspan=2 colspan=1>by New Cannabis Venturesaims to curate high qualitycontent and information about leading cannabis companiesto help our readers flterout the noiseand to stay on top ofthe most important cannabis business news.The NCVNewswire is hand-curated byas long the source and copyright are acknowledgedtogetherwithahyperlinktotheoriginalGlobalResearcharticle.For publication of Global Research articles in printor other forms including commercial internet sites,contact:[email protected]www.globalresearch.ca</td></tr><tr><td rowspan=1 colspan=1>of sole responsibility of the author(s).The Centre forResearch on Globalization will not be responsible for any inaccurate or incorrect statement in this article.The Centreof Research on Globalization grants permission to cross- post Global Research articles on community internet sites</td></tr></table>

parse/dev/TntbHxxGd6j/TntbHxxGd6j_content_list.json

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+    {
+        "type": "text",
+        "text": "OUTPUT DISTRIBUTION OVER THE ENTIRE INPUT SPACE: A NOVEL PERSPECTIVE TO UNDERSTAND NEURAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            184,
+            195,
+            400,
+            223
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            261,
+            544,
+            275
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Understanding the input-output mapping relationship in the entire input space contributes a novel perspective to a comprehensive understanding of deep neural networks. In this paper, we focus on binary neural classifiers and propose to first uncover the histogram about the number of inputs that are mapped to certain output values and then scrutinize the representative inputs from a certain output range of interest, such as the positive-logit region that corresponds to one of the classes. A straightforward solution is uniform sampling (or exhaustive enumeration) in the entire input space but when the inputs are high dimensional, it can take almost forever to converge. We connect the output histogram to the density of states in physics by making an analogy between the energy of a system and the neural network output. Inspired by the Wang-Landau algorithm designed for sampling the density of states, we propose an efficient sampler that is driven to explore the under-explored output values through a gradient-based proposal. Compared with the random proposal in Wang-Landau algorithm, our gradientbased proposal converges faster as it can propose the inputs corresponding to the under-explored output values. Extensive experiments have verified the accuracy of the histogram generated by our sampler and also demonstrated interesting findings. For example, the models map many human unrecognizable images to very negative logit values. These properties of a neural model are revealed for the first time through our sampled statistics. We believe that our approach opens a new gate for neural model evaluation and shall be further explored in future works. ",
+        "bbox": [
+            233,
+            296,
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+            588
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            627,
+            336,
+            643
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Understanding the input-output mapping relationship in the entire input space contributes a novel perspective to a comprehensive understanding of deep neural networks. Existing methods approximate such mapping relations through the evaluation on a certain subset of the entire input space, such as measuring the accuracy on in-distribution test sets Dosovitskiy et al. (2021); Tolstikhin et al. (2021); Steiner et al. (2021); Chen et al. (2021); Zhuang et al. (2022); He et al. (2015), out-ofdistribution (OOD) test sets (Liu et al., 2020; Hendrycks & Gimpel, 2016; Hendrycks et al., 2019; Hsu et al., 2020; Lee et al., 2017; 2018), and adversarial test sets Szegedy et al. (2013); Rozsa et al. (2016); Miyato et al. (2018); Kurakin et al. (2016). However, none of the existing evaluations can offer a comprehensive understanding that covers the entire input space, including all kinds of inputs mentioned above and even those human unrecognizable inputs as shown in Fig 1a. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "As a pilot study, we focus on binary classification — given a trained binary classifier, we aim to uncover a histogram that counts how many samples in the entire input space are mapped to certain logit values, i.e., the distribution of the output values, as shown in Fig 1b. A straightforward solution is uniform sampling (or exhaustive enumeration) in the entire input space but when the inputs are high dimensional, it can take almost forever to converge. Therefore, it calls for a novel efficient sampling method over a neural model’s output space. Note that, as a side product of the sampling procedure, one can expect that this histogram also offers fine-grained information such as some representative input samples corresponding to a certain range of output values. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/29ec5d1242f09bc8849324efbf60a3490ac355a746485b26678407d230c3dadb.jpg",
+        "image_caption": [
+            "Figure 1: Input types and the example output histogram when the task is binary classification between digits 0 and 1. The entire input space covers all possible gray-scale images of the same shape. y is the output (logit) of input $\\mathbf { X }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We connect the output histogram problem to the density of states (DOS) problem in physics by making an analogy between the system energy and neural network output, as shown in the right figure. If one follows the physics language to describe our problem, the input $\\mathbf { x }$ to the neural network can be viewed as the configuration $\\mathbf { x }$ of the system; the neural network output (e.g., logit values in binary classifier) $y ( \\mathbf { x } )$ corresponds to the energy function $E ( \\mathbf { x } )$ ; the desired output histogram can be obtained through the DOS (a.k.a., the entropy, $S ( E ( \\mathbf { x } ) { \\bar { ) } }$ , the log scale of DOS), which is the count of the configurations given the energy value. Note that the density of states by definition is over the entire input space, which aligns perfectly with our objective. ",
+        "bbox": [
+            174,
+            332,
+            678,
+            470
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/ad5a7aeb42a570ca2f9b842f8a6e68d599dbba1e4e3359f8c99a0d5fe36c7fcc.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            694,
+            335,
+            816,
+            468
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Inspired by the Wang-Landau algorithm (Wang & Landau, 2001) designed for density of states, we propose an efficient sampler that is driven to explore the under-explored output values through a gradient-based proposal. If the binary classifier is well-trained, it is reasonable to believe that in-distribution inputs (images) which usually have certain semantic structures are concentrated in certain output ranges. If one follows the random proposal in the Wang-Landau algorithm, it is difficult to propose the inputs with meaningful structure (or even in-distribution inputs), thus possibly preventing the sampler from exploring the corresponding output values. Thus, we propose to apply a gradient-based proposal called Gibbs-with-Gradients (GWG) (Grathwohl et al., 2021) which proves to be efficient to propose in-distribution inputs for a trained model. ",
+        "bbox": [
+            174,
+            478,
+            825,
+            603
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "With the help of this new sampler, we can reveal some new understanding of the models for the entire input space. First, in our experiments on a real-world dataset, the dominant output values are very negative and correspond to the human-unrecognizable inputs. This indicates the models may map an overwhelmingly large number of unrecognizable images to the overconfident prediction probabilities. Second, we can derive the relative difference between the dominant peak of output values and the other output values, especially those where the in-distribution inputs correspond to. The output values where the in-distribution inputs correspond to are also dominated by the humanunrecognizable inputs. This result presents significant challenges to the OOD detection problems. Third, we observe a clear trend of the representative samples in a CNN model and speculate it simply utilizes the background to predict the labels of the digits. ",
+        "bbox": [
+            174,
+            611,
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+            750
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our contributions are summarized as follows. ",
+        "bbox": [
+            174,
+            756,
+            472,
+            770
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We work on the challenging problem to uncover the output distribution over the entire input space. Such output distributions offer a novel perspective to understand deep neural networks. • We connect this output distribution problem to the density of states problem in physic and successfully tailor Wang-Landau algorithm using a gradient-based proposal, which is a must-have component to sample the entire output space as much as possible, improving the efficiency. • We conduct extensive experiments on toy and real-world datasets based on CNN and ResNet-18 to confirm the correctness of our proposed sampler and discover novel and interesting findings. ",
+        "bbox": [
+            174,
+            777,
+            825,
+            875
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We believe that our approach opens a new gate for neural model evaluation and shall be further explored in future works. For example, one can can utilize our sampler to estimate the intrinsic ratio of in-distribution samples given a range of interest with human evaluation as shown in Sec. 5.3. ",
+        "bbox": [
+            174,
+            882,
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+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 PROBLEM DEFINITION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            390,
+            118
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In the traditional setting, binary neural classifiers model the class distribution through logit $z$ . A neural classifier parameterized by $\\theta$ learns $p _ { \\theta } ( z | \\mathbf { x } ) = \\delta ( z - y _ { \\theta } ( \\mathbf { x } ) )$ through a function $y _ { \\theta } : \\mathbf { x }  z \\in$ $\\mathbb { R }$ , where $\\mathbf { x } \\in \\Omega$ , $\\Omega \\subseteq \\{ 0 , . . . , N \\} ^ { D }$ for images, and $\\delta$ is the Dirac delta function. $\\Omega$ is aligned with Gibbs-With-Gradient’s setting to be discrete. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "What the above model does not define is the distribution of the data $\\mathbf { x }$ . This paper aims to obtain the output value distribution of binary classifiers in the entire input space: $\\Omega \\doteq \\dot { \\{ 0 , . . . , N \\} } ^ { D }$ . Here we assume that the data distribution $p ( \\mathbf { x } )$ follows the uniform distribution over the domain $\\Omega$ of $\\mathbf { x }$ and denote its measure by $\\mu$ . We define the joint distribution ",
+        "bbox": [
+            173,
+            195,
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+            252
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8cda377b4319d2958baf932e0d698f57240311b130c7671ed50e4ec4447ac5c9.jpg",
+        "text": "$$\np _ { \\theta } ( z , \\mathbf { x } ) = p _ { \\theta } ( z | \\mathbf { x } ) \\mu ( \\mathbf { x } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            416,
+            257,
+            580,
+            275
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our goal is only the logit (output) distribution. We marginalize the above joint distribution to define the density given the logit $z$ : ",
+        "bbox": [
+            174,
+            286,
+            823,
+            315
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4d97f2c2f096b6f20e6d971cf1201b2fa8857671e277c8bc34829d1f66a95a8c.jpg",
+        "text": "$$\np _ { \\theta } ( z ) = \\sum _ { \\Omega } p _ { \\theta } ( z | \\mathbf { x } ) \\mu ( \\mathbf { x } ) = \\sum _ { \\mathbf { x } \\in \\Omega } \\delta ( z - y _ { \\theta } ( \\mathbf { x } ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            343,
+            320,
+            653,
+            354
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To sample from the distribution $p _ { \\theta } ( z )$ , we can first sample $\\mathbf { x } _ { i } \\sim \\mathrm { U n i f o r m } ( \\Omega )$ , then condition on the sampled $\\mathbf { x } _ { i }$ , sample $z _ { i } \\sim p _ { \\theta } ( z | \\mathbf { x } _ { i } )$ . While uniform sampler in principle can resolve our problem, it takes almost forever to converge. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 METHOD ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            422,
+            282,
+            438
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section, we first discuss the connection between our problem to density of states (DOS), introduce both Wang-Landau algorithm and Gibbs-with-Gradient as background, and present our new sampler Gradient-Wang-Landau algorithm. ",
+        "bbox": [
+            174,
+            453,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 CONNECTION TO DENSITY OF STATES (DOS) IN PHYSICS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            512,
+            614,
+            527
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In statistical physics, given the energy function $E : \\mathbf { x }  \\mathcal { E } \\in \\mathbb { R }$ , the DOS $\\rho ( \\mathcal { E } )$ is defined as ",
+        "bbox": [
+            168,
+            537,
+            777,
+            554
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/546328d62bee1475970760535c3aa9834edeed7e45686bd3d701a8c370a6fe68.jpg",
+        "text": "$$\n\\rho ( \\mathcal { E } ) = \\sum _ { \\mathbf { x } \\in \\Omega } \\delta ( \\mathcal { E } - E ( \\mathbf { x } ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            413,
+            558,
+            584,
+            593
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\delta$ is the Dirac delta function and $\\Omega$ is the domain of $\\mathbf { x }$ where $\\mathbf { x }$ is valid. The DOS is treated as a probability distribution in the energy space, whose log-probability is defined as the entropy $S$ : ",
+        "bbox": [
+            173,
+            597,
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+            626
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7751010ef8ce91f63b057ece1e72e8ddbcd2a2b7533cbd43043ab3e970c287e9.jpg",
+        "text": "$$\n\\rho ( \\mathcal { E } ) = \\exp ( S ( \\mathcal { E } ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            434,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Boltzmann constant is assumed to be 1 in our setting. DOS is meaningful because many physical quantities depend on energy or its integration but not the specific input $\\mathbf { x }$ . ",
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+        "type": "text",
+        "text": "We connect the output histogram to DOS in physics by making an analogy between the system energy $\\mathcal { E } = E ( \\mathbf { x } )$ and neural network output $z = y ( \\mathbf { x } )$ . This connection is based on the observation that the energy function in physics maps an input configuration to a scalar-valued energy; similarly, a binary neural classifier maps an image to a logit. Both the logit and energy are treated as the direct output of the mapping. Other quantities, such as the loss, are derived from the output. The desired output histogram can be obtained similarly through sampling the DOS (a.k.a., the entropy $S ( E ( \\mathbf { x } ) )$ or $\\bar { S } ( y ( { \\bf x } ) )$ in the log scale) which is the count of the configurations given the energy value. The output histogram and DOS are defined in the entire input space. ",
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+        "type": "text",
+        "text": "3.2 WANG-LANDAU ALGORITHM AND GIBBS-WITH-GRADIENT ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Wang-Landau algorithm is a Markov chain Monte Carlo sampler that samples DOS. Since the true distribution $\\rho ( \\mathcal { E } )$ is what we are interested in sampling but its formula/model is unknown, we need to approximate it. The Wang-Landau algorithm uses a histogram to store the current estimation $\\tilde { S }$ . It improves the sampling efficiency by sampling the inverted distribution: ",
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+    {
+        "type": "equation",
+        "img_path": "images/115dcc95d101a8d7496740de4da467a508b3eb0e197d17daef448a8be69d2916.jpg",
+        "text": "$$\np ( \\mathbf { x } ) \\propto \\exp ( - \\tilde { S } ( E ( \\mathbf { x } ) ) )\n$$",
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+        "type": "text",
+        "text": "By sampling $p ( \\mathbf { x } )$ , we can get an ensemble of $\\mathcal { E }$ via $E ( \\cdot )$ whose probability distribution is: ",
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+    {
+        "type": "equation",
+        "img_path": "images/2926a6d93cecd4732fede8730e48ee4967de16040ee9e8d63f94df0554166535.jpg",
+        "text": "$$\n\\pi ( \\mathcal { E } ) = \\sum _ { \\mathbf { x } \\sim p ( \\mathbf { x } ) } \\delta ( \\mathcal { E } - E ( \\mathbf { x } ) )\n$$",
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+        "type": "text",
+        "text": "When $\\tilde { S } ( \\mathcal { E } )$ approaches $S ( \\mathcal { E } )$ , the energy distribution $\\pi ( \\mathcal { E } )$ approaches to $\\mathcal { E }$ -independent constant for all the accessible energy $\\mathcal { E }$ . ",
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+        "type": "text",
+        "text": "Gibbs-With-Gradient (GWG) is used for energy-based models (EBM) by sampling ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/49901da0b3bf7607159dfd7b5e8d7ce036c83431dc9e1743c2cf58fc1c2036dc.jpg",
+        "text": "$$\n\\log p ( \\mathbf { x } ) = f ( \\mathbf { x } ) - \\log Z ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $f ( \\mathbf { x } )$ is the unnormalized log-probability, $Z$ is the partition function, and $\\mathbf { x }$ is discrete. Typical Gibbs sampler iterates every dimension $x _ { i }$ of $\\mathbf { x }$ , computes the conditional probability $p ( x _ { i } | x _ { 1 } , . . . x _ { i - 1 } , x _ { i + 1 } , . . . , x _ { D } )$ , and samples according to this conditional probability. ",
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+    {
+        "type": "text",
+        "text": "When the training data $\\mathbf { x }$ are natural images and the EBM learns $\\mathbf { x }$ decently well, the traditional Gibbs sampler wastes much of the computation. For example, most pixel-by-pixel iterations over $x _ { i }$ in MNIST dataset will be on the background which should stay black. GWG proposes a smart proposal that picks the pixel $x _ { i }$ that is more likely to change, such as the pixels around the edge between the bright and dark region of the digits. ",
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+    {
+        "type": "text",
+        "text": "3.3 WANG-LANDAU WITH GRADIENT PROPOSAL ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Directly applying Wang-Landau algorithm is not enough as it uses random proposal, because a trained neural model learns preferred mapping through the loss function. For example, a binary classifier should map the training inputs to either the sufficiently positive or negative logit values which ideally should correspond to the extremely rare but semantically meaningful inputs. After the sampler explores and generates the peak centered at 0 where most random samples correspond to as shown in Fig. 1b, it is almost impossible for the sampler with a random proposal to propose the inputs with meaningful structure (or even in-distribution inputs) so that the other possible output values are explored. Of course, whether those output values correspond to in-distribution inputs is only confirmable after sampling. In summary, it is extremely difficult for the random proposal in Wang-Landau algorithm to explore (almost all) the possible output values. ",
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+    {
+        "type": "text",
+        "text": "We propose to use the framework of Wang-Landau algorithm but replace the proposal distribution with the Gibbs-With-Gradients (GWG) sampler which has a gradient proposal, since the gradient proposal takes the advantage of model’s learned weights to propose inputs. In order to sample the distribution of the output prediction through GWG, we define log-probability $f ( x )$ as: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/3724791a0308e376d5f26a18fc822591aff3ec25adaa5e7979121308143b2582.jpg",
+        "text": "$$\nf ( x ) = S ( y ( \\mathbf { x } ) )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $S$ is the count for the bin corresponding to $y ( \\mathbf x )$ . The fixed $f ( \\cdot )$ in the original GWG is changing in our sampling process given the input $\\mathbf { x }$ , since the formula for $S$ is unknown and we can only estimate the output distribution as we did in Wang-Landau algorithm. Moreover, the GWG requires the gradient of $f$ , but the $S$ is not differentiable since it is approximated through discrete bins. We adopt a first-order differentiable interpolation for the discrete histogram of entropy. ",
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+        "type": "text",
+        "text": "In summary, similar to the original Wang-Landau algorithm, we first initialize two histograms with all of their bins to 0. One of these histograms is for entropy $S$ , and the other histogram, $H$ , is a counter of how many times the sampler workers visited a specific bin and $H$ is also for the flatness check. We first preset the number of iterations. When $H$ passes the flatness check, it enters the next iteration loop with the step counter reset to 0. Every step in the while loop until the flatness check passes, we interpolate the entropy in the histogram to get a differentiable interpolation and take the derivative of the negation of the entropy with respect to the output $z$ and the inputs $\\mathbf { x }$ through the chain rule. GWG uses this gradient to propose the next input that is likely to have lower entropy and be accepted by the sampler. his procedure drives the sampler to visit rare samples whose logit values correspond to the lower entropy until $S$ converges. This proposal also goes through a Monte-Carlo accept-reject procedure in the GWG. Once the flatness is met, the bins of $H$ are reset to 0 and the step size is halved before a new iteration. Our proposed algorithm is in Alg. 1 in Appendix. ",
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+    {
+        "type": "text",
+        "text": "4 RELATED WORKS AND DISCUSSIONS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Performance Characterization has long been explored even before the era of deep learning (Haralick, 1992; Klette et al., 2000; Thacker et al., 2008). The input-output relationship has been explored for simple functions (Hammitt & Bartlett, 1995) and mathematical morphological operators (Gao et al., 2002; Kanungo & Haralick, 1990). Compared to existing performance characterization approaches (Ramesh et al., 1997; Bowyer & Phillips, 1998; Aghdasi, 1994; Ramesh & Haralick, 1992; 1994), our work focuses on the output distribution (Greiffenhagen et al., 2001) of a neural network over the entire input space (i.e., not task specific) following the blackbox approach (Courtney et al., 1997; Cho et al., 1997) where the system transfer function from input to output is unknown. Our setting shall be viewed as the most general forward uncertainty quantification case (Lee & Chen, 2009) where the model performance is characterized when the inputs are perturbed (Roberts et al., 2021). To our best knowledge, we demonstrate for the first time that the challenging task of sampling the entire input space for modern neural networks is feasible and efficient by drawing the connection between neural network and physics models. Our proposed method can offer samples to be further integrated with the performance characterization methods mentioned above. ",
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+    {
+        "type": "text",
+        "text": "Density Estimation and Energy Landscape Mapping Previous works in density estimation focus on data density Tabak & Turner (2013); Liu et al. (2021), where class samples are given and the goal is to estimate the density of samples. Here we are not interested in the density of the given dataset, but the density of all the valid samples in the pixel space for a trained model. Hill et al. (2019); Barbu & Zhu (2020) have done the pioneering work in sampling the energy landscape for energy-based models. Their methods specifically focus on the local minimum and barriers of the energy landscape. We can relax the requirement and generalize the mapping on the “output” space where either sufficiently positive or sufficiently negative output (logit) values are meaningful in binary classifiers and other models. ",
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+    {
+        "type": "text",
+        "text": "Open-world Model Evaluation Though many neural models have achieved the SOTA performance, most of them are only on in-distribution test sets (Dosovitskiy et al., 2021; Tolstikhin et al., 2021; Steiner et al., 2021; Chen et al., 2021; Zhuang et al., 2022; He et al., 2015; Simonyan & Zisserman, 2014; Szegedy et al., 2015; Huang et al., 2017; Zagoruyko & Komodakis, 2016). Openworld settings where the test set distribution differs from the in-distribution training set create special challenges for the model. While the models have to detect the OOD samples from in-distribution samples (Liu et al., 2020; Hendrycks & Gimpel, 2016; Hendrycks et al., 2019; Hsu et al., 2020; Lee et al., 2017; 2018; Liang et al., 2018; Mohseni et al., 2020; Ren et al., 2019), we also expect sometimes the model could generalize what it learns to OOD datasets (Cao et al., 2022; Sun & Li, 2022). It has been discovered that models have over-confident predictions for some OOD samples that obviously do not align with human judgments (Nguyen et al., 2015). The OOD generalization becomes more challenging because of this discovery, because the models may not be as reliable as we thought they were. Adversarial test sets Szegedy et al. (2013); Rozsa et al. (2016); Miyato et al. (2018); Kurakin et al. (2016); Xie et al. (2019); Madry et al. (2017) also present special challenges as models decisions are different from those of humans. Having a full view of input-output relation with all the above different kinds of test sets under consideration is important. ",
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+    },
+    {
+        "type": "text",
+        "text": "Samplers MCMC samplers (Chen et al., 2014; Welling & Teh, 2011; Li et al., 2016; Xu et al., 2018) are developed to scale to big datasets and sample efficient with gradients. Recently, GibbsWith-Gradients (GWG) (Grathwohl et al., 2021) is proposed to pick the promising pixel(s) as the proposal. To further improve sampling efficiency, CSGLD (Deng et al., 2020) drives the sampler to explore the under-explored energy using similar idea as Wang-Landau algorithm (Wang & Landau, 2001). The important difference between our problem setting and the previous ones solved by other MCMC samplers is the function or model as distribution to be sampled from is unknown. Wang-Landau algorithm utilizes previous approximation of the distribution to drive the sampler to explore the under-explored energy regions. This algorithm can be more efficient through parallelization (Vogel et al., 2013; Cunha-Netto et al., 2008), bin-free (Junghans et al., 2014; Li & Eisenbach, 2017) and extended to multi-dimensional outputs (Zhou et al., 2006). While the previous samplers can be applied to high dimensional inputs, the energy functions written by physicists are relative simple and symmetric. However, modern neural networks are complex and hard to characterize performance (Roberts et al., 2021). We assume agnostic of the output properties of the model and thus apply the Wang-Landau algorithm to sample the entropy as a function of energy but with the gradient proposal in GWG to make the sampler more efficient. Similar to GWG, our sampler can propose the inputs corresponding to the under-explored regions of outputs. Improvements of efficiency can benefit from a patch of pixel changes. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we apply our proposed Gradient Wang-Landau sampler to inspect a few neural network models and present the discovered output histogram together with representative samples. The dataset and model training details are introduced in Sec. 5.1. We first empirically confirm our sampler performance through a toy example in Sec. 5.2. We then discuss results for modern binary classifiers in Sec. 5.3 and Sec. 5.4. Hyperparameters of the samplers tested in are Appendix C. ",
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+    {
+        "type": "text",
+        "text": "5.1 DATASETS, MODELS, AND OTHER EXPERIMENT SETTINGS ",
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+        "type": "text",
+        "text": "Datasets As aforementioned, we focus on binary classification. Therefore, we derive two datasets from the MNIST datasets by only including samples with labels $\\{ 0 , 1 \\}$ . The training and test splits are the same as those in the original MNIST dataset. ",
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+        "type": "text",
+        "text": "• Toy is a simple dataset with $5 \\times 5$ binary input images we construct. It is designed to make feasible the bruteforce enumeration over the entire input space (only $2 ^ { 5 \\times 5 }$ different samples). We center crop the MNIST samples from $\\{ 0 , 1 \\}$ classes and resize them to $5 \\times 5$ images. We compute the average of the pixel values and use the average as the threshold to binarize the images — the pixel value lower than this threshold becomes 0; otherwise, it becomes 1. The duplicates are not removed for accuracy after resizing since PyTorch does not find duplicate row indices. • MNIST-0/1 is an MNIST dataset whose samples only have the 0,1 labels. To align with the GWG setting, the inputs are discrete and not $\\mathrm { _ { Z } }$ -normalized. Therefore, in this dataset, the input $\\mathbf { x }$ is $2 8 \\times 2 8$ dimensional with discrete pixel values from $\\{ 0 , . . . , 2 5 5 \\}$ . ",
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+    {
+        "type": "text",
+        "text": "Neural Network Models for Evaluation Since the focus of this paper is not to compare different neural architectures, given the relatively small datasets we have, we train two types of models, a simple CNN and ResNet-18 (He et al., 2015). Each pixel of the inputs is first transformed to the one-hot encoding and passed to a 3-by-3 convolution layer with 3 channel output. The CNN model contains 2 convolution layers with 3-by-3 filter size. The output channels are 32 and 128. The final features are average-pooled and passed to a fully-connected layer for the binary classification. ",
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+    {
+        "type": "text",
+        "text": "Please keep in mind that our goal in this experiment section is to showcase that our proposed sampler can uncover some novel interesting empirical insights for neural network models. Models with different architectures, weights due to different initialization, optimization, and/or datasets will lead to different results. Therefore, our results and discussions are all model-specific. Specifically, we train a simple CNN model to classify the $5 \\times 5$ binary images in the Toy dataset (CNN-Toy). The test accuracy of this CNN-Toy model reaches $9 9 . 7 \\%$ , which is almost perfect. We train a simple CNN model to classify the $2 8 \\times 2 8$ grey-scale images in the MNIST-0/1 dataset (CNN-MNIST-0/1). The test accuracy of CNN-MNIST-0/1 model is $9 7 . 8 \\%$ . We train a ResNet-18 model to classify the $2 8 \\times 2 8$ grey-scale images in the MNIST-0/1 dataset (ResNet-18-MNIST-0/1). The test accuracy of ResNet-18-MNIST-0/1 model is $1 0 0 \\%$ . ",
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+    {
+        "type": "text",
+        "text": "Sampling Methods for Comparison We compare several different sampling methods (including our proposed method) to obtain the output histogram over the entire input space. ",
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+        "type": "text",
+        "text": "• Enumeration generates the histogram by enumerating all the possible pixel values as inputs. This is a rather slow but the most accurate method.   \n• In-dist Test Samples generates the histogram of the inputs based on the fixed test set.This is commonly used in machine learning evaluation. It is based on a very small and potentially biased subset of the entire input space.   \n• Wang-Landau algorithm (WL) generates the histogram the Wang-Landau algorithm with the random proposal. Specifically, we randomly pick one pixel at a time and change it to any valid (discrete) value as in this implementation 1   \n• Gradient Wang-Landau (GWL) generates the histogram by our proposed sampler of WangLandau algorithm with gradient proposal. ",
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+    {
+        "type": "image",
+        "img_path": "images/0c66faa4a0eb7a34a59abefd6c5d10be5d13bb2f8cef0d5ec07bef93354745d8.jpg",
+        "image_caption": [
+            "Figure 2: Output histograms of CNN-Toy obtained by different sampling methods. The indistribution samples are only a very small portion in the output histogram. We also present the representative samples obtained by GWL given different logit values. "
+        ],
+        "image_footnote": [],
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+        "image_caption": [
+            "Figure 3: Output histograms of CNN-MNIST-0/1 obtained by different sampling methods. The blue scale is for GWL and the red scale is for In-distribution Test Samples. We also present the representative samples obtained by GWL given different logit values (more in Fig. 6 in Appendix) "
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+    {
+        "type": "text",
+        "text": "5.2 RESULTS OF CNN-TOY ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Given the CNN-Toy model, we apply Enumeration, GWL, and In-dist Test Samples to obtain the output histograms, as shown in Fig. 2. Note that our GWL method samples the relative entropy of different energy values as duplicate x may be proposed. After normalization with the maximum entropy, the GWL histogram almost exactly matches the Enumeration histogram which is the ground truth histogram. This confirms the accuracy of our GWL sampler and we can apply it further to more complicated models with confidence. ",
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+    {
+        "type": "text",
+        "text": "Remarkably, this histogram is quite different from the expectation we presented in Fig. 1b — this histogram is even not centered at 0 or has the expected subdominant peaks on both the positive and negative sides. Instead, the dominant peak is so wide that it covers almost the entire spectrum of the possible output values. From a coarse-grained overview, most of the samples are mapped to the center of logit $- 5$ with a decay from $- 5$ to both sides in the CNN-Toy model. This shows the CNN-Toy model is biased to predict more samples to the negative logit values. ",
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+    {
+        "type": "text",
+        "text": "In Fig. 2, we also present the representative samples obtained by GWL given different logit values in the CNN-Toy model. The visualization results suggest that the CNN-Toy model probably learns the digit “1” for positive logit values as the center pixels of the representative samples are white (see the three representative samples with logit values from 0 to 20) and $\\ \" 0 \\ \"$ for the very negative logit values as the center pixels of the representative samples are black (see two representative samples with logit values from -20 to -30). From this example, one can see that the output histogram over the entire input space can offer a comprehensive understanding of the neural network models, helping researchers better understand critical questions such as the distribution of the outputs, where the model maps the samples to, and what the representative samples with high likelihood are. ",
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+    {
+        "type": "text",
+        "text": "5.3 RESULTS OF CNN-MNIST-0/1 ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Histogram Results by GWL The trial of applying GWL on the CNN-Toy model is encouraging and we now apply GWL to the CNN-MNIST-0/1 that is trained on a real-world dataset. The results are shown in Fig. 3. As our GWL reveals, the output histogram of CNN-MNIST-0/1, similar to CNN-Toy’s histogram, does not have the subdominant peaks. It is also different from the presumed case in Fig. 1b. Compared with the output histogram of the CNN-Toy model (i.e., Fig. 2), this time, the peak is on the negative boundary and the histogram is skewed towards the negative logit values. $S$ almost linearly decays to the positive logit values. While the in-distribution samples have logit values between $- 2 0$ and 15 as we expect, these samples are exponentially (i.e., $e ^ { 1 3 0 0 }$ at logit value -20 to $e ^ { 3 1 0 0 }$ at logit value 13, thousands in log scale) less often found than the majority samples whose logit values are around $- 5 5$ . From a fine-grained view, the CNN-MNIST-0/1 model tends to map the human-unrecognizable samples to the very negative logit values. While previous work (Nguyen et al., 2015) showed the existence of the overconfident prediction samples, our result shows a rough but quantitative performance of this CNN which can serve as a baseline for further improvements. One may notice that in Fig. 3, there is still some output values (e.g., the rightmost positive logit region) that are not yet covered by our GWL sampler. We believe that this calls for more future work to follow on more advanced efficient samplers. ",
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+        "img_path": "images/e9ff3076d4a132b973f8030a8b19784ad990e1a23965a3fd9c23de67d84f3d1b.jpg",
+        "image_caption": [
+            "Figure 4: Intermediate output histogram $S$ per iteration. (a) GWL gradually explores the logit values in the first iteration. (b) GWL discovers the output histogram well within 2 iterations. (c) The original WL explores the output distribution much slower. "
+        ],
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+    {
+        "type": "text",
+        "text": "GWL is much more efficient than WL Since WL takes a much longer time to converge, we are not able to obtain the converged results from WL. For the comparison purpose, we inspect the intermediate $S$ results of the GWL and WL samplers, as shown in Fig. 4. As one can see from Fig. 4a, in the first iteration, GWL has already been able to gradually explore the logit values efficiently from the most dominant output value around $- 5 5$ to the positive logit values. Within only two iterations, as shown in Fig. 4b, GWL can discover the output histogram covering the value range from $- 5 5$ to 13. On the other hand, as presented in Fig. 4c, in the first two iterations, the original WL can only explore the output ranges from around $- 5 5$ to $- 4 5$ ; in the 3rd iteration, WL converges significantly slower and never ends in a reasonable time. This result indicates that the GWL converges much faster than the original WL and is able to explore a much more diverse range of output values. ",
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+    {
+        "type": "text",
+        "text": "Manual inspection on more representative samples As show in Fig. 3, for the CNN-MNIST$_ { 0 / 1 }$ model, GWL can effectively sample input images from logit values ranging from -55 to 13. We further group these logit values per 100 bins (100 bins correspond to a difference of 10 in logit value) in $S$ , resulting in about 7 groups. For every group, we sample 200 representative input images. To make sure they are not correlated, we sample every 1000 steps. For demonstration purposes, we randomly pick 10-out-of-200 samples from every group in Fig. 6 in Appendix. We manually inspect the sufficiently positive group (e.g., the last column in Fig. 6) and the sufficiently negative groups (e.g., the first five columns in Fig. 6) , and there are no human recognizable samples of digits. We also observe an interesting pattern that as the logit value increases, more and more representative samples have black background. This result suggests that the CNN-MNIST-0/1 model may heavily rely on the background to classify the images (Xiao et al., 2020). We conjecture that is because the samples in the most dominant peak are closer to class 0 samples than class 1 samples (Appendix. D). In summary, although CNN-MNIST-0/1 holds a very high in-distribution test accuracy, it is far from a robust model because it does not truly understand the semantic structure of the digits. ",
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+    {
+        "type": "text",
+        "text": "Discussion Fig. 3 presents challenges to the OOD detection methods that may be more modeldependent than we thought before. If the model cannot map most of the human unrecognizable samples with high uncertainty, the likelihood-based OOD detection methods (Liu et al., 2020; Hendrycks & Gimpel, 2016) cannot perform well for samples in the entire input space. Fig. 6 shows the inputs with the in-distribution output values (output logits of the red plot) of the CNN model may not uniquely correspond to in-distribution samples. More rigorous experiments to a definite conclusion are yet required as future work. ",
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+    {
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+        "img_path": "images/12832be2d3026b42c4b661d4a5448d60b6a476c48ec9c0a5dde3fb23f35a2113.jpg",
+        "image_caption": [
+            "(a) Results with random re-initialization. "
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+        "img_path": "images/092109fc749bf3dd1476c50096882b69ff635bf9f62d94622d943b3cda11f427.jpg",
+        "image_caption": [
+            "(b) Results with test set re-initialization. ",
+            "Figure 5: Output histograms of ResNet-18-MNIST-0/1 obtained by different sampling methods. There may be a sharp local minima in the output landscape causing a cliff around the logit value of -33 and making GWL “trapped”. We have tried two variants to address the “trapped” issue via (a) random re-initialization and (b) test set re-initialization. The blue scale is for GWL and the red scale is for In-distribution Test Samples. We also present the representative samples obtained by GWL given different logit values. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "5.4 RESULTS OF RESNET-18-MNIST-0/1 ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "When applying our GWL samplers to the ResNet-18-MNIST-0/1 model, we observe that the sampler can easily get trapped in some output regions. This is a fairly common phenomenon for WangLandau-based samplers, as reported in (Vogel et al., 2018). We follow the common practice to re-initialize the sampler to the random samples every time it gets trapped. We let those workers run 1000 steps (10,000 pixels selected with replacement) before counting to $S$ again. As shown in Fig. 5a, the smallest logit values in ResNet-18-MNIST-0/1 are around -220, much lower than those of CNN-MNIST-0/1. A wide range of negative logit values corresponds to human unrecognizable inputs and there is no obvious pattern observed in contrast to CNN-MNIST-0/1’s results. ",
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+    {
+        "type": "text",
+        "text": "Interestingly, we observe a cliff around the logit value of -33. We try another variant for reinitialization: we re-initialize using the test set samples every time it gets trapped in certain output value. This time, as shown in Figure 5b, it can explore the output values larger than -33. We believe there may be a sharp local minima in the output landscape, similar to the case discussed before Vogel et al. (2018). ",
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+    {
+        "type": "text",
+        "text": "Because of the “trapping” issue and the complexity of ResNet-18 over CNN, we have to relax the flatness check a bit to let GWL converge for the first iteration. Because of the less rigorous flatness check, we do not draw conclusions about ResNet-18-MNIST-0/1 evaluation of the relative entropy differences. Compared with the CNN-MNIST-0/1 model, ResNet-18-MNIST-0/1 has more interesting phenomena and further exploration is needed to understand these phenomena. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We aim to get a full picture of the input-output relationship of a model through the inputs valid in the pixel space. We propose to use a histogram to better understand the input-output distribution. When the inputs are high-dimensional, enumeration or uniform sampling is either impossible or takes too long to converge. We connect the density of states in physics to this histogram sampling problem. We propose to use an efficient sampler to achieve this goal. We confirm empirically this can be achieved and uncover some new aspects of neural networks. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For future work, it is interesting to develop a new and more efficient sampler that has theoretical guarantees to acquire this input-output relationship in order to sample with more pixels, such as the ImageNet (Deng et al., 2009). Most importantly, with this new sampler, we can develop new insights into network architectures developed in the last decade for open-world applications. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "7 REPRODUCIBILITY ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We provide fairly amount of information to re-implement our sampler. The data processing is in the Sec. 5.1 and algorithm is in Appendix B. The hyperparameters and sampling details are also listed in the Sec. 5. We also provide different time stamp of steps for our samplers to indicate what to expect during the sampling procedure in Fig. 4. Of course, the Wang-Landau algorithm we adopted is the prototypical one and it subjects to some issues reported in its follow-up works, such as the discontinuity of the boundaries between bins and trapping in one of the bins. These problems lead to some issues in our experiments and we discussed them in Sec. 5.4. More advanced algorithms have been developed to resolve these issues. ",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "8 ETHICS STATEMENT ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Our method aims to provide a comprehensitve understanding of the neural models. This work will be applicable to many applications, such as those in the safety and trusty-worthy machine learning. As a pilor study, we do not anticipate the negative aspects of our work. ",
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+    {
+        "type": "text",
+        "text": "A APPENDIX A: REPRESENTATIVE INPUTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            103,
+            540,
+            118
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Here we list more representative samples of the CNN-MNIST-0/1 scenario. The samples are bounded by a black box of boundaries. ",
+        "bbox": [
+            174,
+            132,
+            823,
+            161
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/bf1948b09fba60548ba41be433ac25ba3ecc7334f0121c861a922a4307083b04.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>-50.6 国</td><td>-43.2 -43.9</td><td>-31.5 国</td><td>-18.3 国 -18.5</td><td>-13.8 国</td><td>-0.5 □</td><td></td><td>10.3 ■</td></tr><tr><td>-54.2 国</td><td>国</td><td>-28.9 国</td><td></td><td>国</td><td>-13.9 国</td><td>0.6 国</td><td>8.8 □</td></tr><tr><td>-46.7 国</td><td>-44.2 国</td><td>-32.8 √</td><td>-17.4 √</td><td>-13.9 国</td><td></td><td>-3.5 国</td><td>11.2 ■</td></tr><tr><td>-49.3 国</td><td>-39.0 国</td><td>-26.0 国</td><td>-20.2 国</td><td>-9.8 国</td><td></td><td>0.7 国</td><td>7.9 □</td></tr><tr><td>-49.1 图</td><td>-39.6 国</td><td>-33.4 国</td><td>-23.4 国</td><td>-11.0 □</td><td></td><td>0.8 ■</td><td>6.3 1</td></tr><tr><td>-47.9 国</td><td>-35.7 園</td><td>-32.2 国</td><td>-18.6 国</td><td>-7.6 √</td><td></td><td>-0.7</td><td>9.3</td></tr><tr><td>-47.2</td><td>-43.1</td><td>-33.7</td><td>-16.2</td><td>-14.9</td><td></td><td>国 3.4</td><td>□ 10.4</td></tr><tr><td>国</td><td>国 -38.1</td><td>国 -28.8</td><td>□ -23.4</td><td>国</td><td></td><td>□</td><td>□</td></tr><tr><td>-45.1 国</td><td>国</td><td>国</td><td>国</td><td>-12.2 ?</td><td></td><td>3.9 □</td><td>14.8 ■</td></tr><tr><td>-53.1</td><td>-36.7</td><td>-33.5</td><td>-22.5</td><td>-5.7</td><td></td><td>-1.9</td><td>8.8</td></tr><tr><td>国</td><td>国</td><td>国</td><td>国</td><td>国</td><td></td><td>国</td><td>□</td></tr><tr><td>-48.7</td><td>-43.8</td><td>-31.7</td><td>-22.2</td><td>-8.0</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td>0.2</td><td></td></tr><tr><td>国</td><td>国</td><td>国</td><td>国</td><td></td><td>国</td><td>国</td><td>5.8 □</td></tr></table>",
+        "bbox": [
+            197,
+            175,
+            805,
+            549
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Figure 6: More representative samples of the CNN-MNIST-0/1 model obtained by GWL at different logit values, grouped by logit values. We further group these logit values per 100 bins (100 bins correspond to a difference of 10 in logit value) in $S$ , resulting in about 10 groups. The output values in the first column are within the range [-55,-45) and the second is from [-45,-35], etc. ",
+        "bbox": [
+            173,
+            565,
+            826,
+            622
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "B APPENDIX B: GRADIENT WANG-LANDAU ALGORITHM ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            666,
+            118
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Here we provide the algorithms of the GWL algorithm. The input and output are listed. The hyperparameters are determined mostly by the toy-example. ",
+        "bbox": [
+            176,
+            133,
+            820,
+            161
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/5621d2b0fd7f21d9204f4c768b3ebd0c68ceb94b25527cb1e676172069f87584.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Algorithm 1 Our proposed Gradient Wang-Landau (GWL)</td></tr><tr><td>lation function g(z, S) for t=1,2,...,Tdo X~Dte while His not flat do</td><td>Require: pretrained model y: X → z, flat histogram H = O,entropy histogram S = 0, increment/step-size lnf, number of iterations T,test set Dte, GWG sampler GWG(z,S), interpo- &gt; Get the continuous interpolation entropy Sin at output z</td></tr><tr><td colspan=\"2\">z =y(x) Sin=g(z,S) X ~ GWG(z,-Sin) = round(z) Round z to the nearest z&#x27; that corresponds to one of the bins S[]←S[]+lnf H[]←H[]+1 end while lnf ←lnf/2</td></tr></table>",
+        "bbox": [
+            171,
+            174,
+            826,
+            438
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "C HYPER-PARAMETERS AND IMPLEMENTATION DETAILS FOR GWL AND WL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            467,
+            797,
+            497
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "The hyper-parameters for GWL and WL are extremely similar, if not identical, as the only major difference between GWL and WL is the gradient proposal versus the random proposal. We first preset a large enough range of output values for the sampler to explore the trained neural network models. In our experiments, we found that the output (logit) values of the binary classifiers typically fall in the range of -300 to 100 (based on ResNet). Therefore, we use this range for all experiments. For flatness histogram $H$ , the bin window size is set to be 1, resulting in 400 bins. The histogram $H$ is considered flat if the difference between maximum bin value and minimum bin value is smaller than the average bin value. For output histogram $S$ , we set the bin window size to be 0.1, resulting in 4000 bins. Instead of updating one bin at a time for $S$ , we update the neighbor bins with exponential decay. We use the linear interpolation to approximate the bins for continuous queries. We iterate 5 times with test set initialization. Every step the GWG tries to at most update 10 pixels. ",
+        "bbox": [
+            173,
+            515,
+            825,
+            667
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "D SAMPLES SIMILARITY ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            688,
+            393,
+            704
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "The samples in the most dominant peak may be closer to class 0 than to class 1. We compute the L2 pixel-wise distance from the uniform noise image to the samples of class 1 and 0 respectively. The mean L2 distance from uniform noise to 0 is around 0.3121 and that from uniform noise to 1 is around 0.3236. The distance between 1 and 0 samples is 0.1652. This result shows the samples in the most dominant peak are closer to class 0 samples than class 1 samples. More rigorous experiments to a definite conclusion is yet required as future work. ",
+        "bbox": [
+            174,
+            719,
+            825,
+            804
+        ],
+        "page_idx": 14
+    }
+]