Datasets:

zai-org
/

RPC-Bench

+# EFFICIENT EXPLORATION VIA STATE MARGINAL MATCHING
+Anonymous authors Paper under double-blind review
+# ABSTRACT
+Reinforcement learning agents need to explore their unknown environments to solve the tasks given to them. The Bayes optimal solution to exploration is intractable for complex environments, and while several exploration methods have been proposed as approximations, it remains unclear what underlying objective is being optimized by existing exploration methods, or how they can be altered to incorporate prior knowledge about the task. Moreover, it is unclear how to acquire a single exploration strategy that will be useful for solving multiple downstream tasks. We address these shortcomings by learning a single exploration policy that can quickly solve a suite of downstream tasks in a multi-task setting, amortizing the cost of learning to explore. We recast exploration as a problem of State Marginal Matching (SMM), where we aim to learn a policy for which the state marginal distribution matches a given target state distribution, which can incorporate prior knowledge about the task. We optimize the objective by reducing it to a two-player, zero-sum game between a state density model and a parametric policy. Our theoretical analysis of this approach suggests that prior exploration methods do not learn a policy that does distribution matching, but acquire a replay buffer that performs distribution matching, an observation that potentially explains prior methods’ success in single-task settings. On both simulated and real-world tasks, we demonstrate that our algorithm explores faster and adapts more quickly than prior methods.1
+# 1 INTRODUCTION
+Reinforcement learning (RL) algorithms must be equipped with exploration mechanisms to effectively solve tasks with limited reward signals. These tasks arise in many real-world applications where providing human supervision is expensive. The inability of current RL algorithms to adequately explore limits their applicability to long-horizon control tasks.
+A wealth of prior work has studied exploration for RL. While, in theory, the Bayes-optimal exploration strategy is optimal, it is intractable to compute exactly, motivating work on tractable heuristics for exploration. Exploration methods based on random actions have limited ability to cover a wide range of states. More sophisticated techniques, such as intrinsic motivation, accelerate learning in the single-task setting. However, these methods have two limitations. First, they do not explicitly define an objective to quantify “good exploration,” but rather argue that exploration arises implicitly through some iterative procedure. Lacking a well-defined optimization objective, it remains challenging to understand what these methods are doing and why they work. Similarly, the lack of a metric to quantify exploration, even if only for evaluation, makes it challenging to compare exploration methods and assess progress in this area. The second limitation is that these methods target the single-task setting. Because these methods aim to converge to the optimal policy for a particular task, it is challenging to repurpose these methods to solve multiple tasks.
+We address these shortcomings by considering a multi-task setting, where many different reward functions can be provided for the same set of states and dynamics. Rather than exploring from scratch for each task, we aim to learn a single, task-agnostic exploration policy that can be adapted to many possible downstream reward functions, amortizing the cost of learning to explore. This exploration policy can be viewed as a prior on the policy for solving downstream tasks. Learning will consist of two phases: during training, we acquire this task-agnostic exploration policy; during testing, we use this exploration policy to quickly explore and maximize the task reward.
+Learning a single exploration policy is considerably more difficult than doing exploration throughout the course of learning a single task. The latter is done by intrinsic motivation (Pathak et al., 2017; Tang et al., 2017; Oudeyer et al., 2007) and count-based exploration methods (Bellemare et al., 2016), which can effectively explore to find states with high reward, at which point the agent can decrease exploration and increase exploitation of those high-reward states. While these methods perform efficient exploration for learning a single task, the policy at any particular iteration is not a good exploration policy. For example, the final policy at convergence would only visit the high-reward states discovered for the current task.
+What objective should be optimized to obtain a good exploration policy? We recast exploration as a problem of State Marginal Matching: given a desired state distribution, we learn a mixture of policies for which the state marginal distribution matches this desired distribution. Without any prior information, this objective reduces to maximizing the marginal state entropy $\mathcal { H } [ s ]$ , which encourages the policy to visit as many states as possible. The distribution matching objective also provides a convenient mechanism to incorporate prior knowledge about the task, whether in the form of safety constraints that the agent should obey; preferences for some states over other states; reward shaping; or the relative importance of each state dimension for a particular task.
+We also propose an algorithm to optimize the State Marginal Matching (SMM) objective. First, we reduce the problem of SMM to a two-player, zero-sum game between a policy player and a density player. We find a Nash Equilibrium for this game using fictitious play (Brown, 1951), a classic procedure from game theory. Our resulting algorithm iteratively fits a state density model and then updates the policy to visit states with low density under this model. Our analysis of this approach sheds light on prior work on exploration. In particular, while the policy learned by existing exploration algorithms does not perform distribution matching, the replay buffer does, an observation that potentially explains the success of prior methods. On both simulated and real-world tasks, we demonstrate that our algorithm explores more effectively and adapts more quickly to new tasks than state-of-the-art baselines.
+# 2 RELATED WORK
+Most prior work on exploration has looked at exploration bonuses and intrinsic motivation. One class of exploration methods uses prediction error of some auxiliary task as an exploration bonus, which provides high (intrinsic) reward in states where the predictive model performs poorly (Pathak et al., 2017; Oudeyer et al., 2007; Schmidhuber, 1991; Houthooft et al., 2016; Burda et al., 2018). Another set of approaches (Tang et al., 2017; Bellemare et al., 2016; Schmidhuber, 2010) directly encourage the agent to visit novel states. While all methods effectively explore during the course of solving a single task (Taïga et al., 2019), the policy obtained at convergence is often not a good exploration policy (see Section 4). In contrast, our method converges to a highly-exploratory policy by maximizing state entropy in the training objective (Eq. 2).
+Many exploration algorithms can be classified by whether they explore in the space of actions, policy parameters, goals, or states. Common exploration strategies including $\epsilon { \cdot }$ -greedy and Ornstein–Uhlenbeck noise (Lillicrap et al., 2015), as well as standard MaxEnt RL algorithms (Ziebart, 2010; Haarnoja et al., 2018), explore in the action space. Recent work (Fortunato et al., 2017; Plappert et al., 2017) shows that adding noise to the parameters of the policy can result in good exploration. Most closely related to our work are methods that perform exploration in the space of states or goals (Colas et al., 2018; Held et al., 2017; Nair et al., 2018; Pong et al., 2019; Hazan et al., 2018). In fact, Hazan et al. (2018) consider the same State Marginal Matching objective that we examine and propose a similar algorithm. In relation to Hazan et al. (2018), our main contributions are (1) empirically showing that exploration based on state-entropy is competitive with existing state-of-the-art exploration methods, and (2) explaining how existing exploration methods based on prediction error are implicitly maximizing this state-entropy objective. In Appendix C.1, we also discuss how goal-conditioned RL (Kaelbling, 1993; Schaul et al., 2015) can be viewed as a special case of State Marginal Matching when the goal-sampling distribution is learned jointly with the policy.
+The problems of exploration and meta-reinforcement learning are tightly coupled. Meta-reinforcement learning algorithms (Duan et al., 2016; Finn et al., 2017; Rakelly et al., 2019; Mishra et al., 2017) must perform effective exploration if they hope to solve a downstream task. Some prior work has explicitly looked at the problem of learning to explore (Gupta et al., 2018; Xu et al., 2018). Our problem statement is similar to meta-learning, in that we also aim to learn a policy as a prior for solving downstream tasks. However, whereas meta-RL requires a distribution of task reward functions, our method will require only a single target state marginal distribution. Due to the simpler problem assumptions and training procedure, our method may be easier to apply in real-world domains.
+Related to our approach are standard maximum action entropy algorithms (Haarnoja et al., 2018; Kappen et al., 2012; Rawlik et al., 2013; Ziebart et al., 2008; Theodorou & Todorov, 2012). While these algorithms are referred to as MaxEnt RL, they are maximizing entropy over actions, not states. These algorithms can be viewed as performing inference on a graphical model where the likelihood of a trajectory is given by its exponentiated reward (Toussaint & Storkey, 2006; Levine, 2018; Abdolmaleki et al., 2018). While distributions over trajectories induce distributions over states, computing the exact relationship requires integrating over all possible trajectories, an intractable problem for most MDPs. A related but distinct class of relative entropy methods use a similar entropy-based objective to limit the size of policy updates (Peters et al., 2010; Schulman et al., 2015).
+Finally, the idea of distribution matching has been employed successfully in imitation learning (Ziebart et al., 2008; Ho & Ermon, 2016; Finn et al., 2016; Fu et al., 2017). Similar to some inverse RL algorithms (Ho & Ermon, 2016; Fu et al., 2018), our method iterates between learning a policy and learning a reward function, though our reward function is obtained via a density model instead of a discriminator. While inverse RL algorithms assume access to expert trajectories, we instead assume access to the density of the target state marginal distribution. In many realistic settings, such as robotic control with many degrees of freedom, providing fully-specified trajectories may be much more challenging than defining a target state marginal distribution. The latter only requires some aggregate statistics about expert behavior, and does not even need to be realizable by any policy.
+In summary, our work unifies prior exploration methods as performing approximate distribution matching, and explains how state distribution matching can be performed properly. This perspective provides a clearer picture of exploration, and this observation is useful particularly because many of the underlying ingredients, such as adversarial games and density estimation, have seen recent progress and therefore might be adopted to improve exploration methods.
+# 3 STATE MARGINAL MATCHING
+In this section, we propose the State Marginal Matching problem as a principled objective for learning to explore, and offer an algorithm for optimizing it. We consider a parametric policy $\pi _ { \theta } \in \Pi \triangleq \{ \pi _ { \theta } \mid \theta \in \Theta \}$ that chooses actions $a \in { \mathcal { A } }$ in a Markov Decision Process (MDP) $\mathcal { M }$ with fixed episode lengths $T$ , dynamics distribution $p ( s _ { t + 1 } \mid s _ { t } , a _ { t } )$ , and initial state distribution $p _ { 0 } ( s )$ . The MDP $\mathcal { M }$ together with the policy $\pi _ { \theta }$ form an implicit generative model over states. We define the state marginal distribution $\rho _ { \pi } ( s )$ as the probability that the policy visits state $s$ :
+$$
+\rho _ { \pi } ( s ) \triangleq \mathbb { E } _ { \begin{array} { c } { s _ { 1 } \sim p _ { 0 } ( S ) , } \\ { a _ { t } \sim \pi _ { \theta } ( A \vert s _ { t } ) } \end{array} } \left[ \frac { 1 } { T } \sum _ { t = 1 } ^ { T } \mathbb { 1 } ( s _ { t } = s ) \right]
+$$
+We emphasize that $\rho _ { \pi } ( s )$ is not a distribution over trajectories, and is not the stationary distribution of the policy after infinitely many steps, but rather the distribution over states visited in a finite-length episode.2 We also note that any trajectory distribution matching problem can be reduced to a state marginal matching problem by augmenting the current state to include all previous states.
+We assume that we are given a target distribution $p ^ { * } ( s )$ over states $s \in S$ that encodes our belief about the tasks we may be given at test-time. For example, a roboticist might assign small values of $p ^ { * } ( s )$ to states that are dangerous, regardless of the desired task. Alternatively, we might also learn $p ^ { * } ( s )$ from data about human preferences (Christiano et al., 2017). For goal-reaching tasks, we can analytically derive the optimal target distribution (Appendix C). Given $p ^ { * } ( s )$ , our goal is to find a parametric policy that is “closest” to this target distribution, where we measure discrepancy using the Kullback-Leibler (KL) divergence:
+![](images/5824c542baced011f2fba4616e707b9e68fb67136fe4ee0c9fffa185c6d5856c.jpg)
+Figure 1: State Marginal Matching: (Left) Our goal is to learn a policy whose distribution over states (blue histogram) matches some target density (black line). Our algorithm iteratively increases the reward on states visited too infrequently (green arrow) and decreases the reward on states visited too frequently (red arrow). (Center) At convergence, these two distributions are equal. (Right) For complex target distributions, we use a mixture of policies $\begin{array} { r } { { \dot { \rho } } _ { \pi } ( s ) = \int \rho _ { \pi _ { z } } ( s ) p ( z ) d z } \end{array}$ . (See Appendix B.)
+$$
+\begin{array} { r l } { \displaystyle \underset { \pi \in \Pi } { \operatorname* { m a x } } \mathcal { F } ( \rho _ { \pi } ( s ) , p ^ { * } ( s ) ) \triangleq \underset { \pi \in \Pi } { \operatorname* { m a x } } - D _ { \mathrm { K L } } ( \rho _ { \pi } ( s ) \parallel p ^ { * } ( s ) ) } & { } \\ { = \underset { \pi \in \Pi } { \operatorname* { m a x } } \mathbb { E } _ { s \sim \rho _ { \pi } ( s ) } \left[ \log p ^ { * } ( s ) - \log \rho _ { \pi } ( s ) \right] } & { } \\ { = \underset { \pi \in \Pi } { \operatorname* { m a x } } \mathbb { E } _ { s \sim \rho _ { \pi } ( s ) } [ \log p ^ { * } ( s ) ] + \mathcal { H } _ { \pi } [ s ] } \end{array}
+$$
+This is the same objective as in Hazan et al. (2018). Note that we use the reverse-KL (Bishop, 2006), which is mode-seeking (i.e., exploratory). We show in Appendix C that the policies obtained via State Marginal Matching provide an optimal exploration strategy for a particular distribution over reward functions. To gain intuition for the State Marginal Matching objective, we decomposed it in two ways. In Equation 2, we see that State Marginal Matching is equivalent to maximizing the reward function $r ( s ) \triangleq \log p ^ { * } ( s )$ while simultaneously maximizing the entropy of states. Note that, unlike traditional MaxEnt RL algorithms (Ziebart et al., 2008; Haarnoja et al., 2018), we regularize the entropy of the state distribution, not the conditional distribution of actions given states, which results in exploration in the space of states rather than in actions. Moreover, Equation 1 suggests that State Marginal Matching maximizes a pseudo-reward $r ( s ) \triangleq \log p ^ { * } ( s ) - \log \rho _ { \pi } ( s )$ , which assigns positive utility to states that the agent visits too infrequently and negative utility to states visited too frequently (see Figure 1). We emphasize that maximizing this pseudo-reward is not a RL problem because the pseudo-reward depends on the policy.
+# 3.1 OPTIMIZING THE STATE MARGINAL MATCHING OBJECTIVE
+Optimizing Equation 1 to obtain a single exploration policy is more challenging than standard RL because the reward function itself depends on the policy. To break this cyclic dependency, we introduce a parametric state density model $q _ { \psi } ( s ) \in Q \triangleq \{ q _ { \psi } \mid \psi \in \Psi \}$ to approximate the policy’s state marginal distribution, $\rho _ { \pi } ( s )$ . We assume that the class of density models $Q$ is sufficiently expressive to represent every policy:
+Assumption 1. For every policy $\pi \in \Pi$ , there exists $q \in Q$ such that $D _ { K L } ( \rho _ { \pi } ( s ) \parallel q ( s ) ) = 0 .$ .
+Under this assumption, optimizing the policy w.r.t. this approximate distribution $q ( s )$ will yield the same solution as Equation 1 (see Appendix A for the proof):
+Proposition 3.1. Let policies $\Pi$ and density models $Q$ satisfying Assumption $^ { l }$ be given. For any target distribution $p ^ { * }$ , the following optimization problems are equivalent:
+$$
+\operatorname* { m a x } _ { \pi } \mathbb { E } _ { \rho _ { \pi } ( s ) } [ \log p ^ { * } ( s ) - \log \rho _ { \pi } ( s ) ] = \operatorname* { m a x } _ { \pi } \operatorname* { m i n } _ { q } \mathbb { E } _ { \rho _ { \pi } ( s ) } [ \log p ^ { * } ( s ) - \log q ( s ) ]
+$$
+Solving the new max-min optimization problem is equivalent to finding the Nash equilibrium of a two-player, zero-sum game: a policy player chooses the policy $\pi$ while the density player chooses the density model $q$ . To avoid confusion, we use actions to refer to controls $a \in { \mathcal { A } }$ output by the policy $\pi$ in the traditional RL problem and strategies to refer to the decisions $\pi \in \Pi$ of the policy player and decisions $q \in Q$ of the density player. The Nash existence theorem (Nash, 1951) proves that such a stationary point always exists for such a two-player, zero-sum game.
+One common approach to saddle point games is to alternate between updating player A w.r.t. player B, and updating player B w.r.t. player A. However, games such as Rock-Paper-Scissors illustrate that such a greedy approach is not guaranteed to converge to a stationary point. A slight variant,
+# Algorithm 1 Learning to Explore via Fictitious Play
+<table><tr><td>Input: Target distribution p*(s) Initialize policy π(a | s),density model q(s),and replay buffer B.</td></tr><tr><td>while not converged do</td></tr><tr><td>q(m） ← arg maxqEs~B(m-1)[logq(s)]</td></tr><tr><td>π(m) ← arg maxπ Es~pπ(s)[r(s)] wherer(s)=logp*(s)-logq(m)(s)</td></tr><tr><td>B(m)← B(m-1) U{(st,at,St+1)}T=1 with new transitions{(st,at,St+1)}T=1 sampled from π(m)</td></tr><tr><td>return historical policies {(1),.,π(m)}</td></tr></table>
+Algorithm 1: An algorithm for optimizing the State Marginal Matching objective (Equation 1). The algorithm iterates between (1) fitting a density model ${ \boldsymbol q } ^ { ( m ) }$ and (2) training the policy $\bar { \pi } ^ { ( m ) }$ with a RL objective to optimize the expected return w.r.t. the updated reward function $r ( s )$ . The algorithm returns the collection of policies from each iteration, which do distribution matching in aggregate.
+fictitious play (Brown, 1951) does converge to a Nash equilibrium in finite time (Robinson, 1951; Daskalakis & Pan, 2014). At each iteration, each player chooses their best strategy in response to the historical average of the opponent’s strategies. In our setting, fictitious play alternates between fitting the density model to the historical average of policies (Equation 4), and updating the policy with RL to minimize the log-density of the state, using a historical average of the density models (Equation 5):
+$$
+\begin{array} { l l } { { q _ { m + 1 }  \displaystyle \operatorname * { a r g m a x } _ { q } \mathbb E _ { s \sim \bar { \rho } _ { m } ( s ) } [ \log q ( s ) ] } } & { { \quad \mathrm { w h e r e } \quad \bar { \rho } _ { \pi } ^ { ( m ) } ( s ) \triangleq \displaystyle \frac { 1 } { m } \sum _ { i = 1 } ^ { m } \rho _ { \pi _ { i } } ( s ) } } \\ { { \displaystyle \pi _ { m + 1 }  \operatorname * { a r g m a x } _ { \pi } \mathbb E _ { s \sim \rho _ { \pi } ( s ) } \mathbb { I o g } p ^ { * } ( s ) - \log \bar { q } _ { m } ( s ) ] } } & { { \quad \mathrm { ~ w h e r e } \quad \bar { q } _ { m } ( s ) \triangleq \displaystyle \frac { 1 } { m } \sum _ { i = 1 } ^ { m } q _ { i } ( s ) } } \end{array}
+$$
+Crucially, the exploration policy is not the last policy, $\pi _ { m + 1 }$ , but rather the historical average policy:
+Definition 3.1. A historical average policy $\bar { \pi } ( \boldsymbol { a } \ \mid \ \boldsymbol { s } )$ , parametrized by a collection of policies $\pi _ { 1 } , \cdots , \pi _ { m }$ , is a policy that randomly samples one of the policy iterates $\pi _ { i } \sim \operatorname { U n i f } [ \pi _ { 1 } , \cdot \cdot \cdot , \pi _ { m } ]$ at the start of each episode and takes actions according to that policy for each step in the episode. A new policy is sampled for the next episode.
+We summarize the resulting algorithm in Algorithm 1. In practice, we can efficiently implement Equation 4 and avoid storing the policy parameters from every iteration by instead storing sampled states from each iteration.3 We cannot perform the same trick for Equation 5, and instead resort to approximating the historical average of density models with the most recent iterate. Algorithm 1 looks similar to prior exploration methods based on prediction-error, suggesting that we might use SMM to understand how these prior methods work.
+# 4 WHY DOES PREDICTION-ERROR EXPLORATION WORK?
+Exploration methods based on prediction error (Burda et al., 2018; Stadie et al., 2015; Pathak et al., 2017; Schmidhuber, 1991; Chentanez et al., 2005) do not converge to an exploratory policy, even in the absence of extrinsic reward. For example, consider the asymptotic behavior of ICM (Pathak et al., 2017) in a deterministic MDP, such as the Atari games where it was evaluated. At convergence, the predictive model will have zero error in all states, so the exploration bonus is zero – the ICM objective has no effect on the policy at convergence. Similarly, consider the exploration bonus in Pseudocounts (Bellemare et al., 2016): $1 / \hat { n } ( s )$ , where $\hat { n } ( s )$ is the (estimated) number of times that state $s$ has been visited. In the infinite limit, each state has been visited infinitely many times, so the Pseudocount exploration bonus also goes to zero — Pseudocounts has no effect at convergence. Similar reasoning can be applied to other methods based on prediction error (Burda et al., 2018; Stadie et al., 2015). More broadly, we can extend this analysis to stochastic MDPs, where we consider an abstract exploration algorithm that alternates between computing some intrinsic reward and performing RL (to convergence) on that intrinsic reward. Existing prediction-error exploration methods are all special cases. At each iteration, the RL step solves a fully-observed MDP, which always admits a deterministic policy as a solution (Puterman, 2014). Thus, any exploration algorithm in this class cannot converge to a single, exploratory policy.
+![](images/9c0339dc7796b3aaecebd42f912eab1db1565a55e5d7a2143b6ec2512e7530d6.jpg)
+Figure 2: Exploration in State Space (SMM) vs. Action Space (SAC) for Navigation: (a): A point-mass agent is spawned at the center of $m$ long hallways that extend radially outward, and the target state distribution places uniform probability mass $\frac { 1 } { m }$ at the end of each hallway. We can vary the length of the hallway and the number of hallways to control the task difficulty. (b) A heatmap showing states visited by SAC and SMM during training illustrates that SMM explores a wider range of states. (c) SMM reaches more goals than the MaxEnt baseline. SM4 is an extension of SMM that incorporates mixture modelling with $n > 1$ skills (see Appendix B), and further improves exploration of SMM.
+Despite these observations, prior methods do excel at solving hard exploration tasks. We draw an analogy to fictitious play to explain their success. While these methods never acquire an exploratory policy, over the course of training they will eventually visit all states. In other words, the historical average over policies will visit a wide range of states. Since the replay buffer exactly corresponds to this historical average over states, these methods will obtain a replay buffer with a diverse range of experience, possibly explaining why they succeed at solving hard exploration tasks. Moreover, this analysis suggests a surprisingly simple method for obtaining an exploration from these prior methods: use a mixture of the policy iterates throughout training. The following section will not only compare SMM against prior exploration methods, but also show that this historical averaging trick can be used to improve existing exploration methods.
+# 5 SIMULATED EXPERIMENTS
+We used simulated control tasks to determine if SMM learns an exploratory policy, to compare SMM to prior exploration methods, and to study the effect of historical averaging. More details can be found in Appendix D, and code will be released upon publication.
+Baselines and Implementation Details: We compare to a state-of-the-art off-policy MaxEnt RL algorithm, Soft Actor-Critic (SAC) (Haarnoja et al., 2018); an inverse RL algorithm, Generative Adversarial Imitation Learning (GAIL) (Ho & Ermon, 2016); and three exploration methods:
+• Count-based Exploration (C), which discretizes states and uses − $\log \hat { \pi } ( s )$ as an exploration bonus • Pseudo-counts (PC) (Bellemare et al., 2016), which uses the recoding probability as a bonus. • Intrinsic Curiosity Module (ICM) (Pathak et al., 2017), which uses prediction error as a bonus.
+We used SAC as the base RL algorithm for all exploration methods (SMM, C, PC, ICM). To implement SMM, we define the target distribution in terms of the extrinsic environment reward: $p ^ { * } \bar { ( } s ) \propto \exp ( r _ { \mathrm { e n v } } ( s ) )$ . We use a variational autoencoder (VAE) to model the density $q ( s )$ for both SMM and Pseudocounts (PC). For the GAIL baseline, we generated synthetic expert data by sampling expert states from the target distribution $p ^ { * } ( s )$ (see Appendix D.2 for details). Results for all experiments are averaged over 4-5 random seeds.
+We start with a sanity check: Is exploration in state space (as done by SMM) better than exploration in action space (as done by MaxEnt RL, e.g., SAC)? To study this question, we implemented a 2D Navigation environment, shown in Figure 2a. To evaluate each method, we counted the number of hallways that the agent fully explored (i.e., reached the end) during training. Figure 2b shows the state visitations for the three hallway environment, illustrating that SAC only explores one hallway, whereas SMM explores all three. Figure 2c also shows that SMM consistently explores $60 \%$ of hallways, whereas SAC rarely visits more than $20 \%$ of hallways.
+The remaining simulated experiments used the Manipulation environment (Plappert et al., 2018), shown in Figure 3a. Our first experiment evaluates whether the exploration policy acquired by SMM allows us to solve downstream tasks more quickly. We defined the target distribution to be uniform over the entire state space (joint $^ +$ block configuration), with the constraints that we put low probability mass on states where the block has fallen off the table; that actions should be small; and that the arm should be close to the object. As shown in Figure 3b, SMM adapts substantially more quickly than other exploration methods, achieving a success rate $20 \%$ higher than the next best method, and reaching the same level of performance of the next baseline (ICM) in $4 \mathbf { x }$ fewer episodes. SMM without historical averaging attains similar performance as the next best baseline (ICM), suggesting that historical averaging is the key ingredient, while the particular choice of prediction error or VAE is less important. We provide further ablation studies of SMM in Appendix B.2.
+![](images/1e10def1d2b3ed892870abe9c349a7f43577052b4da01db66a212264993cbe9a.jpg)
+Figure 3: Exploration for Manipulation. (a) Task: The robot agent controls a single gripper arm to move a block object to a goal location on the table surface. The goal is not observed by the robot, thus requiring the robot to explore by moving the block to different locations on the table. (b) Test-Time Exploration: At test-time, we sample goal locations uniformly on the table, and plot the percentage of goals found within $N$ episodes. SMM and its mixture-model variant SM4 (Algorithm 2) both explore faster than the baselines, allowing it to successfully find the goal in fewer episodes. (c) State Entropy: After training, we rollout the policy for 1e3 epochs, and record the entropy of the object and gripper positions. SMM achieves higher state entropy than the other methods. Historical averaging also improves the exploration of prior methods. (d) Non-Uniform Exploration: We measure the discrepancy between the state marginal distribution, $\rho _ { \pi } ( s )$ , and a non-uniform target distribution. SMM matches the target distribution better than SAC and is on par with Count. Error bars show std. dev. across 4 random seeds.
+While historical averaging is necessary to guarantee convergence $( \ S \ 3 . 1 )$ , most prior exploration methods do not employ historical averaging, raising the question of whether it is necessary in practice. To answer this question, we compare SMM to three exploration methods. In Figure 3c, we compare the policy obtained at convergence with the historical average of policy iterates over training for each method. We measure how well each method explores by computing the marginal state entropy, which we compute by discretizing the state space.4 The results show that SMM maximizes state entropy at least as effectively as prior methods, if not better. While this comparison is somewhat unfair, as we measure exploration using the objective that SMM maximizes, none of the methods we compare against propose metrics for exploration that we could use instead. Furthermore, we see that historical averaging is not only beneficial to SMM, but also improves the exploration of prior methods.
+In our final simulated experiment, we check whether prior knowledge injected via the target distribution is reflected in the policy obtained from State Marginal Matching. Using the same Manipulation environment as above, we modified the target distribution to assign larger probability to states where the block was on the left half of the table than on the right half. In Figure 3d, we measure whether SMM is able to achieve the target distribution by measuring the discrepancy between the block’s horizontal coordinate and the target distribution. Compared to the SAC baseline, SMM and the Count baseline are half the distance to the target distribution. No method achieves zero discrepancy, suggesting that future methods could be better at matching state marginals.
+# 6 REAL-WORLD EXPERIMENTS
+While almost all research on exploration focus on simulated domains, attributes of the real world such as partial observability, nonstationarity, and stochasticity may make the exploration more challenging. The aim of this section is to see if SMM explores effectively on a real-world robotic control task. We used the $D$ ’Claw (Ahn et al., 2019) robotic manipulator, which is a 3-fingered hand positioned vertically above a handle that it can turn. For all experiments on the $D$ ’Claw, we used a target distribution that places uniform mass over all object angles $\left[ - 1 8 0 ^ { \circ } , 1 8 0 ^ { \circ } \right]$ .
+![](images/59b349d6d5df5d28a9bad6be1af0a12fe93bf184830bea590f2f684e9cbe3682.jpg)
+Figure 4: Real-World Exploration: (a) D’Claw is a 9-DoF robotic hand (Ahn et al., 2019) that is trained to turn a valve object. (b) Sim2Real: We trained each algorithm in simulation, and then measured how far the trained policy rotated the knob on the hardware robot. We also measured the maximum angle that the agent turned the knob in the clockwise and counter-clockwise directions within one episode. (c) Training on Hardware: We trained SAC and SMM on the real robot for 1e5 environment steps (about 9 hours in real time), and measured the maximum angle turned throughout training. We see that SMM moves the knob more and visits a wider range of states than SAC. All results are averaged over 4-5 seeds.
+In a first experiment, we trained SMM and other baselines in simulation, and then evaluated the acquired exploration policy on the real robot using two metrics: the total number of rotations (in either direction), and the maximum radians turned (in both directions). For each method, we computed the average metric across 100 evaluation episodes. We repeated this process for 5 independent training runs. Figure 4b shows that SMM turns the knob more than the baselines, and it turns the knob to a wider range of angles. To test for statistical significance, we used a 1-sided Student’s t-test to test the hypothesis that SMM turned the knob more and to a wider range of angles than SAC. The p-values were all less than 0.05: $p = 0 . 0 4 6$ for number of rotations, $p = 0 . 0 1 9$ for maximum clockwise angle, and $p = 0 . 0 0 1$ for maximum counter-clockwise angle.
+In our second experiment, we investigated whether it was possible to learn an exploration policy directly in the real world, without the need for a simulator. Learning to explore in the real world is quite important, as building faithful simulators of complex systems is challenging. The physical constraints of the real robot make data efficiency paramount, suggesting that learning to explore will require an effective exploration strategy. In Figure $_ { 4 \mathrm { c } }$ , we plot the range of angles that the policy explores throughout training. Not only does SMM explore a wider range of angles than SAC, but its ability to explore increases throughout training, suggesting that the SMM objective is correlated with real-world metrics of exploration.
+In summary, the results in this section suggests that exploration techniques may actually be useful in the real world, which may encourage future work to study exploration methods on real-world tasks.
+# 7 DISCUSSION
+In this paper, we introduced a formal objective for exploration. While it is often unclear what existing exploration algorithms will converge to, our State Marginal Matching objective has a clear solution: at convergence, the policy should visit states in proportion to their density under a target distribution. Not only does this objective encourage exploration, it also provides human users with a flexible mechanism to bias exploration towards states they prefer and away from dangerous states. Upon convergence, the resulting policy can thereafter be used as a prior in a multi-task setting, amortizing exploration and enabling faster adaptation to new, potentially sparse, reward functions. The algorithm we proposed looks quite similar to previous exploration methods based on prediction error, suggesting that those methods are also performing some form of distribution matching. However, by deriving our method from first principles, we note that these prior methods omit a crucial historical averaging step, without which the algorithm is not guaranteed to converge. Experiments on both simulated and real-world tasks demonstrated how SMM learns to explore, enabling an agent to efficiently explore in new tasks provided at test time.
+In future work, we aim to study connections between inverse RL, MaxEnt RL and state marginal matching, all of which perform some form of distribution matching. Empirically, we aim to scale to more complex tasks by parallelizing the training of all mixture components simultaneously. Broadly, we expect the state distribution matching problem formulation to enable the development of more effective and principled RL methods that reason about distributions rather than individual states.
+# REFERENCES
+Abbas Abdolmaleki, Jost Tobias Springenberg, Yuval Tassa, Remi Munos, Nicolas Heess, and Martin Riedmiller. Maximum a posteriori policy optimisation. arXiv preprint arXiv:1806.06920, 2018.
+Joshua Achiam, Harrison Edwards, Dario Amodei, and Pieter Abbeel. Variational option discovery algorithms. arXiv preprint arXiv:1807.10299, 2018.
+David Barber Felix Agakov. The im algorithm: a variational approach to information maximization. Advances in Neural Information Processing Systems, 16:201, 2004.
+Michael Ahn, Henry Zhu, Kristian Hartikainen, Hugo Ponte, Abhishek Gupta, Sergey Levine, and VIKASH KUMAR. ROBEL: RObotics BEnchmarks for Learning with low-cost robots. In Conference on Robot Learning (CoRL), 2019.
+Marc Bellemare, Sriram Srinivasan, Georg Ostrovski, Tom Schaul, David Saxton, and Remi Munos. Unifying count-based exploration and intrinsic motivation. In Advances in Neural Information Processing Systems, pp. 1471–1479, 2016.
+Christopher M Bishop. Pattern recognition and machine learning. springer, 2006.
+G. Brown. Iterative solution of games by fictitious play. Activity Analysis of Production and Allocation, 1951.
+Yuri Burda, Harrison Edwards, Amos Storkey, and Oleg Klimov. Exploration by random network distillation. arXiv preprint arXiv:1810.12894, 2018.
+Nuttapong Chentanez, Andrew G Barto, and Satinder P Singh. Intrinsically motivated reinforcement learning. In Advances in neural information processing systems, pp. 1281–1288, 2005.
+Paul F Christiano, Jan Leike, Tom Brown, Miljan Martic, Shane Legg, and Dario Amodei. Deep reinforcement learning from human preferences. In Advances in Neural Information Processing Systems, pp. 4299–4307, 2017.
+John D Co-Reyes, YuXuan Liu, Abhishek Gupta, Benjamin Eysenbach, Pieter Abbeel, and Sergey Levine. Self-consistent trajectory autoencoder: Hierarchical reinforcement learning with trajectory embeddings. arXiv preprint arXiv:1806.02813, 2018.
+Cédric Colas, Olivier Sigaud, and Pierre-Yves Oudeyer. Curious: Intrinsically motivated multi-task, multi-goal reinforcement learning. arXiv preprint arXiv:1810.06284, 2018.
+Constantinos Daskalakis and Qinxuan Pan. A counter-example to karlin’s strong conjecture for fictitious play. In 2014 IEEE 55th Annual Symposium on Foundations of Computer Science, pp. 11–20. IEEE, 2014.
+Maria Dimakopoulou and Benjamin Van Roy. Coordinated exploration in concurrent reinforcement learning. arXiv preprint arXiv:1802.01282, 2018.
+Yan Duan, John Schulman, Xi Chen, Peter L Bartlett, Ilya Sutskever, and Pieter Abbeel. Rl2: Fast reinforcement ˆ learning via slow reinforcement learning. arXiv preprint arXiv:1611.02779, 2016.
+Benjamin Eysenbach, Abhishek Gupta, Julian Ibarz, and Sergey Levine. Diversity is all you need: Learning skills without a reward function. arXiv preprint arXiv:1802.06070, 2018.
+Chelsea Finn, Sergey Levine, and Pieter Abbeel. Guided cost learning: Deep inverse optimal control via policy optimization. In International Conference on Machine Learning, pp. 49–58, 2016.
+Chelsea Finn, Pieter Abbeel, and Sergey Levine. Model-agnostic meta-learning for fast adaptation of deep networks. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pp. 1126– 1135. JMLR. org, 2017.
+Meire Fortunato, Mohammad Gheshlaghi Azar, Bilal Piot, Jacob Menick, Ian Osband, Alex Graves, Vlad Mnih, Remi Munos, Demis Hassabis, Olivier Pietquin, et al. Noisy networks for exploration. arXiv preprint arXiv:1706.10295, 2017.
+Justin Fu, Katie Luo, and Sergey Levine. Learning robust rewards with adversarial inverse reinforcement learning. arXiv preprint arXiv:1710.11248, 2017.
+Justin Fu, Katie Luo, and Sergey Levine. Learning robust rewards with adverserial inverse reinforcement learning. In International Conference on Learning Representations, 2018. URL https://openreview. net/forum?id $_ { \cdot } =$ rkHywl-A-.
+Abhishek Gupta, Russell Mendonca, YuXuan Liu, Pieter Abbeel, and Sergey Levine. Meta-reinforcement learning of structured exploration strategies. In Advances in Neural Information Processing Systems, pp. 5302–5311, 2018.
+Tuomas Haarnoja, Aurick Zhou, Pieter Abbeel, and Sergey Levine. Soft actor-critic: Off-policy maximum entropy deep reinforcement learning with a stochastic actor. arXiv preprint arXiv:1801.01290, 2018.
+Elad Hazan, Sham M Kakade, Karan Singh, and Abby Van Soest. Provably efficient maximum entropy exploration. arXiv preprint arXiv:1812.02690, 2018.
+David Held, Xinyang Geng, Carlos Florensa, and Pieter Abbeel. Automatic goal generation for reinforcement learning agents. arXiv preprint arXiv:1705.06366, 2017.
+Jonathan Ho and Stefano Ermon. Generative adversarial imitation learning. In Advances in Neural Information Processing Systems, pp. 4565–4573, 2016.
+Rein Houthooft, Xi Chen, Yan Duan, John Schulman, Filip De Turck, and Pieter Abbeel. Vime: Variational information maximizing exploration. In Advances in Neural Information Processing Systems, pp. 1109–1117, 2016.
+Leslie Pack Kaelbling. Learning to achieve goals. In IJCAI, pp. 1094–1099. Citeseer, 1993.
+Hilbert J Kappen, Vicenç Gómez, and Manfred Opper. Optimal control as a graphical model inference problem. Machine learning, 87(2):159–182, 2012.
+Sergey Levine. Reinforcement learning and control as probabilistic inference: Tutorial and review. arXiv preprint arXiv:1805.00909, 2018.
+Timothy P Lillicrap, Jonathan J Hunt, Alexander Pritzel, Nicolas Heess, Tom Erez, Yuval Tassa, David Silver, and Daan Wierstra. Continuous control with deep reinforcement learning. arXiv preprint arXiv:1509.02971, 2015.
+Nancy A Lynch. Distributed algorithms. Elsevier, 1996.
+Nikhil Mishra, Mostafa Rohaninejad, Xi Chen, and Pieter Abbeel. A simple neural attentive meta-learner. arXiv preprint arXiv:1707.03141, 2017.
+Ashvin V Nair, Vitchyr Pong, Murtaza Dalal, Shikhar Bahl, Steven Lin, and Sergey Levine. Visual reinforcement learning with imagined goals. In Advances in Neural Information Processing Systems, pp. 9191–9200, 2018.
+John Nash. Non-cooperative games. Annals of mathematics, pp. 286–295, 1951.
+Pierre-Yves Oudeyer, Frdric Kaplan, and Verena V Hafner. Intrinsic motivation systems for autonomous mental development. IEEE transactions on evolutionary computation, 11(2):265–286, 2007.
+Emilio Parisotto, Soham Ghosh, Sai Bhargav Yalamanchi, Varsha Chinnaobireddy, Yuhuai Wu, and Ruslan Salakhutdinov. Concurrent meta reinforcement learning. arXiv preprint arXiv:1903.02710, 2019.
+Deepak Pathak, Pulkit Agrawal, Alexei A Efros, and Trevor Darrell. Curiosity-driven exploration by selfsupervised prediction. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops, pp. 16–17, 2017.
+Jan Peters, Katharina Mulling, and Yasemin Altun. Relative entropy policy search. In Twenty-Fourth AAAI Conference on Artificial Intelligence, 2010.
+Matthias Plappert, Rein Houthooft, Prafulla Dhariwal, Szymon Sidor, Richard Y Chen, Xi Chen, Tamim Asfour, Pieter Abbeel, and Marcin Andrychowicz. Parameter space noise for exploration. arXiv preprint arXiv:1706.01905, 2017.
+Matthias Plappert, Marcin Andrychowicz, Alex Ray, Bob McGrew, Bowen Baker, Glenn Powell, Jonas Schneider, Josh Tobin, Maciek Chociej, Peter Welinder, et al. Multi-goal reinforcement learning: Challenging robotics environments and request for research. arXiv preprint arXiv:1802.09464, 2018.
+Vitchyr H Pong, Murtaza Dalal, Steven Lin, Ashvin Nair, Shikhar Bahl, and Sergey Levine. Skew-fit: Statecovering self-supervised reinforcement learning. arXiv preprint arXiv:1903.03698, 2019.
+Martin L Puterman. Markov Decision Processes.: Discrete Stochastic Dynamic Programming. John Wiley & Sons, 2014.
+Kate Rakelly, Aurick Zhou, Deirdre Quillen, Chelsea Finn, and Sergey Levine. Efficient off-policy metareinforcement learning via probabilistic context variables. arXiv preprint arXiv:1903.08254, 2019.
+Konrad Rawlik, Marc Toussaint, and Sethu Vijayakumar. On stochastic optimal control and reinforcement learning by approximate inference. In Twenty-Third International Joint Conference on Artificial Intelligence, 2013.
+Julia Robinson. An iterative method of solving a game. Annals of mathematics, pp. 296–301, 1951.
+Tom Schaul, Daniel Horgan, Karol Gregor, and David Silver. Universal value function approximators. In International conference on machine learning, pp. 1312–1320, 2015.
+Jürgen Schmidhuber. A possibility for implementing curiosity and boredom in model-building neural controllers. In Proc. of the international conference on simulation of adaptive behavior: From animals to animats, pp. 222–227, 1991.
+Jürgen Schmidhuber. Formal theory of creativity, fun, and intrinsic motivation (1990–2010). IEEE Transactions on Autonomous Mental Development, 2(3):230–247, 2010.
+John Schulman, Sergey Levine, Pieter Abbeel, Michael Jordan, and Philipp Moritz. Trust region policy optimization. In International conference on machine learning, pp. 1889–1897, 2015.
+Bradly C Stadie, Sergey Levine, and Pieter Abbeel. Incentivizing exploration in reinforcement learning with deep predictive models. arXiv preprint arXiv:1507.00814, 2015.
+Adrien Ali Taïga, William Fedus, Marlos C Machado, Aaron Courville, and Marc G Bellemare. Benchmarking bonus-based exploration methods on the arcade learning environment. arXiv preprint arXiv:1908.02388, 2019.
+Haoran Tang, Rein Houthooft, Davis Foote, Adam Stooke, OpenAI Xi Chen, Yan Duan, John Schulman, Filip DeTurck, and Pieter Abbeel. # exploration: A study of count-based exploration for deep reinforcement learning. In Advances in neural information processing systems, pp. 2753–2762, 2017.
+Evangelos A Theodorou and Emanuel Todorov. Relative entropy and free energy dualities: Connections to path integral and kl control. In 2012 IEEE 51st IEEE Conference on Decision and Control (CDC), pp. 1466–1473. IEEE, 2012.
+Emanuel Todorov, Tom Erez, and Yuval Tassa. Mujoco: A physics engine for model-based control. In 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 5026–5033. IEEE, 2012.
+Marc Toussaint and Amos Storkey. Probabilistic inference for solving discrete and continuous state markov decision processes. In Proceedings of the 23rd international conference on Machine learning, pp. 945–952. ACM, 2006.
+Dan Xie, Sinisa Todorovic, and Song-Chun Zhu. Inferring "dark matter" and "dark energy" from videos. In The IEEE International Conference on Computer Vision (ICCV), December 2013.
+Tianbing Xu, Qiang Liu, Liang Zhao, and Jian Peng. Learning to explore with meta-policy gradient. arXiv preprint arXiv:1803.05044, 2018.
+Henry Zhu, Abhishek Gupta, Aravind Rajeswaran, Sergey Levine, and Vikash Kumar. Dexterous manipulation with deep reinforcement learning: Efficient, general, and low-cost. In International Conference on Robotics and Automation, ICRA 2019, Montreal, QC, Canada, May 20-24, 2019, pp. 3651–3657, 2019. doi: 10.1109/ ICRA.2019.8794102. URL https://doi.org/10.1109/ICRA.2019.8794102.
+Brian D Ziebart. Modeling purposeful adaptive behavior with the principle of maximum causal entropy. 2010.
+Brian D Ziebart, Andrew L Maas, J Andrew Bagnell, and Anind K Dey. Maximum entropy inverse reinforcement learning. In Aaai, volume 8, pp. 1433–1438. Chicago, IL, USA, 2008.
+Brian D Ziebart, Nathan Ratliff, Garratt Gallagher, Christoph Mertz, Kevin Peterson, J Andrew Bagnell, Martial Hebert, Anind K Dey, and Siddhartha Srinivasa. Planning-based prediction for pedestrians. In 2009 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 3931–3936. IEEE, 2009.
+# A PROOFS
+Proof of Proposition 3.1. Note that
+$$
+\mathbb { E } _ { \rho _ { \pi } ( s ) } [ \log p ^ { * } ( s ) - \log q ( s ) ] = \mathbb { E } _ { \rho _ { \pi } ( s ) } [ \log p ^ { * } ( s ) - \log \rho _ { \pi } ( s ) ] + D _ { \mathrm { K L } } ( \rho _ { \pi } ( s ) \parallel q ( s ) ) .
+$$
+By Assumption 1, $D _ { \mathrm { K L } } ( \rho _ { \pi } ( s ) \parallel q ( s ) ) = 0$ for some $q \in Q$ , so we obtain the desired result:
+$$
+\begin{array} { l } { \displaystyle \operatorname* { m a x } _ { \boldsymbol \pi } \bigg ( \operatorname* { m i n } _ { \boldsymbol q } \mathbb { E } _ { \rho _ { \boldsymbol \pi } ( s ) } \big [ \log p ^ { * } ( s ) - \log q ( s ) \big ] \bigg ) = \operatorname* { m a x } _ { \boldsymbol \pi } \bigg ( \mathbb { E } _ { \rho _ { \boldsymbol \pi } ( s ) } \big [ \log p ^ { * } ( s ) - \log \rho _ { \boldsymbol \pi } ( s ) \big ] + \operatorname* { m i n } _ { \boldsymbol q } D _ { \mathrm { K L } } \big ( \rho _ { \boldsymbol \pi } ( s ) \ \big \| } \\ { \displaystyle \qquad = \operatorname* { m a x } _ { \boldsymbol \pi } \mathbb { E } _ { \rho _ { \boldsymbol \pi } ( s ) } \big [ \log p ^ { * } ( s ) - \log \rho _ { \boldsymbol \pi } ( s ) \big ] . \qquad \quad \qquad \quad \big | } \end{array}
+$$
+# B BETTER MARGINAL MATCHING WITH MIXTURES OF MIXTURES
+This section introduces an extension of SMM, SM4, that incorporates mixure modelling. Given the challenging problem of exploration in large state spaces, it is natural to wonder whether we can accelerate exploration by automatically decomposing the potentially-multimodal target distribution into a mixture of “easier-to-learn” distributions and learn a corresponding set of policies to do distribution matching for each component. Note that the mixture model we introduce here is orthogonal to the historical averaging step discussed before. Using $\rho _ { \pi _ { z } } ( s )$ to denote the state distribution of the policy conditioned on the latent variable $z \in { \mathcal { Z } }$ , the state marginal distribution of the mixture of policies is
+$$
+\rho _ { \pi } ( s ) = \int _ { \mathcal Z } \rho _ { \pi _ { z } } ( s ) p ( z ) d z = \mathbb { E } _ { z \sim p ( z ) } \left[ \rho _ { \pi _ { z } } ( s ) \right] ,
+$$
+where $p ( z )$ is a latent prior. As before, we will minimize the $\mathrm { K L }$ divergence between this mixture distribution and the target distribution. Using Bayes’ rule to re-write $\rho _ { \pi } ( s )$ in terms of conditional probabilities, we obtain the following optimization problem:
+$$
+\operatorname* { m a x } _ { \pi _ { z } ) _ { z \in z } } \mathbb { E } _ { \underset { s \sim p _ { \pi _ { z } } ( s ) } { z } } \left[ \log \frac { p ^ { * } ( s ) } { \frac { \rho _ { \pi _ { z } } ( s ) p ( z ) } { p ( z | s ) } } \right] = \mathbb { E } _ { \underset { s \sim \rho _ { \pi _ { z } } ( s ) } { z \sim p ( z ) } } \left[ \underbrace { \log p ^ { * } ( s ) } _ { ( a ) } - \underbrace { \log \rho _ { \pi _ { z } } ( s ) } _ { ( b ) } + \underbrace { \log p ( z \mid s ) } _ { ( c ) } - \underbrace { \log p ( z ) } _ { ( d ) } \right]
+$$
+Intuitively, this says that the agent should go to states (a) with high density under the target state distribution, (b) where this agent has not been before, and (c) where this agent is clearly distinguishable from the other agents. The last term (d) says to explore in the space of mixture components $z$ . This decomposition bears a resemblance to the mutual-information objectives in recent work (Achiam et al., 2018; Eysenbach et al., 2018; Co-Reyes et al., 2018). Thus, one interpretation of our work is as explaining that mutual information objectives almost perform distribution matching. The caveat is that prior work omits the state entropy term $- \log \rho _ { \pi _ { z } } ( s )$ which provides high reward for visiting novel states, possibly explaining why these previous works have failed to scale to complex tasks.
+# B.1 ALGORITHMIC SUMMARY
+We summarize the resulting procedure in Algorithm 2, which we refer to as SM4 (State Marginal Matching with Mixtures of Mixtures). The algorithm (1) fits a density model $q _ { z } ^ { ( m ) } ( s )$ to approximate the state marginal distribution for each policy $\pi _ { z }$ ; (2) learns a discriminator $d ^ { ( m ) } ( z \mid s )$ to predict which policy $\pi _ { z }$ will visit state $s$ ; and (3) uses RL to update each policy $\pi _ { z }$ to maximize the expected return of its corresponding reward function $r _ { z } ( s ) \triangleq \log p ^ { * } ( s ) - \log \rho _ { \pi _ { z } } ( s ) + \log p ( z \mid s ) - \log p ( z )$ derived in Equation 8.
+The only difference from Algorithm 1 is that we learn a discriminator $d ( z \mid s )$ , in addition to updating the density models $q _ { z } ( s )$ and the policies $\pi _ { z } ( a \mid s )$ . Jensen’s inequality tells us that maximizing the log-density of the learned discriminator will maximize a lower bound on the true density (see Agakov (2004)):
+$$
+\mathbb { E } _ { s \sim \rho _ { \pi _ { z } } ( s ) , z \sim p ( z ) } [ \log d ( z \mid s ) ] \le \mathbb { E } _ { s \sim \rho _ { \pi _ { z } } ( s ) , z \sim p ( z ) } [ \log p ( z \mid s ) ]
+$$
+# Algorithm 2 State Marginal Matching with Mixtures of Mixtures (SM4)
+<table><tr><td>Input: Target distribution p*(s) Initialize policy πz(a | s),density model qz(s), discriminator d(z | s),and replay buffer B. while not converged do &gt;(1) Update density model for each policy πz.</td></tr><tr><td>for z = 1,.. ,n do</td></tr><tr><td>d(m) ← arg maxdE(z,s)~B(m-1)[logd(z | s)] &gt; (2) Update discriminator.</td></tr><tr><td>for z = 1,.. ,n do r(m)(s)=logp*（s)-logm)(s)+lgd(m)(z|s)-logp(z)</td></tr><tr><td>π(m）← argmaxx Ep(s) [(m)(s)] (3) Update each policy Tz.</td></tr><tr><td>Sample latent skill z(m） ~ p(z)</td></tr><tr><td>Sample transitions {(Sst,at,St+1)}T=1 with πm (m）(a|s) return {π(1),</td></tr></table>
+Algorithm 2. An algorithm for learning a mixture of policies $\pi _ { 1 } , \pi _ { 2 } , \cdots , \pi _ { n }$ that do state marginal matching in aggregate. The algorithm (1) fits a density model $\overset { \cdot } { q } _ { z } ^ { ( m ) } ( s )$ to approximate the state marginal distribution for each policy $\pi _ { z }$ ; (2) learns a discriminator $d ^ { ( m ) } ( z \mid s )$ to predict which policy $\pi _ { z }$ will visit state $s$ ; and (3) uses RL to update each policy $\pi _ { z }$ to maximize the expected return of its corresponding reward function derived in Equation 8: $r _ { z } ( s ) \triangleq \log p ^ { * } ( s ) - \log \rho _ { \pi _ { z } } ( s ) + \log p ( z \mid s ) - \log p ( z )$ . In our implementation, the density model $q _ { z } \left( s \right)$ is a VAE that inputs the concatenated vector $\{ s , z \}$ of the state $s$ and the latent skill $z$ used to obtain this sample $s$ ; and the discriminator is a feedforward MLP. The algorithm returns the historical average of mixtures of policies (a total of $n \cdot m$ policies).
+In our implementation, the density model $q _ { z } ( s )$ is a VAE that inputs the concatenated vector $\{ s , z \}$ of the state $s$ and the latent skill $z$ used to obtain this sample $s$ ; and the discriminator is a feedforward MLP. The algorithm returns the historical average of mixtures of policies (a total of $n \cdot m$ policies). Our implementation uses a uniform categorical distribution for the prior $p ( z )$ , and does not implement the update for $p ( z )$ .
+Note that the updates for each $z$ can be conducted in parallel. A distributed implementation would emulate broadcast-collect algorithms (Lynch, 1996), with each worker updating the policy independently, and periodically aggregating results to update the discriminator $\bar { d } ( z \mid s )$ . Such a distributed implementation has the appealing property that each compute node would explore a different part of the state space. While there has been some work on multi-agent coordinated exploration (Parisotto et al., 2019) and concurrent exploration (Dimakopoulou & Van Roy, 2018), it remains a fairly unexplored area (pun intended) and we believe that SMM with Mixtures of Mixtures offers a simple approach to this problem.
+# B.2 ABLATION STUDY
+To understand the relative contribution of each component in the SM4 objective (Equation 8), we compare SM4 to baselines that lack conditional state entropy $\mathcal { H } _ { \pi _ { z } } [ s ] = - \log \rho _ { \pi _ { z } } ( s )$ , latent conditional action entropy $\log p ( z \mid s )$ , or both (i.e, SAC). In Figure 5a, we plot the training time performance on the Navigation task with 3 halls of length 50. We see that SM4 relies heavily on both key differences from SAC.
+In Figure 5b, we study the effect of mixture modelling on test-time exploration in the Manipulation environment. After running SMM/SM4 with a uniform distribution, we count the number of episodes required to find an (unknown) goal state. We run each method for the same number of environment transitions; a mixture of three policies does not get to take three times more transitions. We find that increasing the number of mixture components increases the agents success. However, the effect was smaller when using historical averaging. Taken together, this result suggests that efficient exploration requires either historical averaging or mixture modelling, but might not need both.
+![](images/5778a92ab21e18081c8e75ddd5d2322cb42d3442ed9b139c560edc26d76fbbe3.jpg)
+Figure 5: Ablation Analysis of State Marginal Matching with Mixtures of Mixtures (SM4). (a): On the Navigation task, we compare SM4 (with three mixture components) against ablation baselines that lack conditional state entropy, latent conditional action entropy, or both (i.e., SAC) in the SM4 objective (Equation 8). We see that both terms contribute heavily to the exploration ability of SM4, but the state entropy term is especially critical. (b): We compare SMM/SM4 with different numbers of mixtures, and with vs. without historical averaging. We found that increasing the number of latent mixture components $n \in \{ 1 , 2 , 4 \}$ accelerates exploration, as does historical averaging. Error bars show std. dev. across 4 random seeds.
+# C CHOOSING $p ^ { * } ( s )$ FOR GOAL-REACHING TASKS
+In general, the choice of the target distribution $p ^ { * } ( s )$ will depend on the distribution of test-time tasks. In this section, we consider the special case where the test-time tasks correspond to goal-reaching derive the optimal target distribution $p ^ { * } ( s )$ . We consider the setting where goals $g \sim p _ { g } ( g )$ are sampled from some known distribution. Our goal is to minimize the number of episodes required to reach that goal state. We define reaching the goal state as visiting a state that lies within an $\epsilon$ ball of the goal, where both $\epsilon > 0$ and the distance metric are known.
+We start with a simple lemma that shows that the probability that we reach the goal at any state in a trajectory is at least the probability that we reach the goal at a randomly chosen state in that same trajectory. Defining the binary random variable $z _ { t } \triangleq \mathbb { 1 } ( \left. s _ { t } - g \right. \leq \epsilon )$ as the event that the state at time $t$ reaches the goal state, we can formally state the claim as follows:
+Lemma C.1.
+$$
+p \left( \sum _ { t = 1 } ^ { T } z _ { t } > 0 \right) \geq p ( z _ { \mathbf { t } } ) \qquad w h e r e \quad \mathbf { t } \sim U n i f [ 1 , \cdots , H ]
+$$
+Proof. We start by noting the following implication:
+$$
+z _ { \mathbf { t } } = 1 \implies \sum _ { t = 1 } ^ { T } z _ { t } > 0
+$$
+Thus, the probability of the event on the RHS must be at least as large as the probability of the event on the LHS:
+$$
+p ( z _ { \mathbf { t } } ) \leq p \left( \sum _ { t = 1 } ^ { T } z _ { t } > 0 \right)
+$$
+Next, we look at the expected number of episodes to reach the goal state. Since each episode is independent, the expected hitting time is simply
+$$
+\mathrm { H I T T I N G T I M E } ( s ) = { \frac { 1 } { p ( { \mathrm { s o m e ~ s t a t e ~ r e a c h e s ~ } } s ) } } = { \frac { 1 } { p \left( \sum _ { t = 1 } ^ { T } z _ { t } > 0 \right) } } \leq { \frac { 1 } { p ( z _ { \mathbf { t } } ) } }
+$$
+Note that we have upper-bounded the hitting time using Lemma C.1. Since the goal $g$ is a random variable, we take an expectation over $g$ :
+$$
+\mathbb { E } _ { s \sim p _ { g } ( s ) } \left[ \mathrm { H I T T I N G T I M E } ( s ) \right] \le \mathbb { E } _ { s \sim p ( s ) } \left[ \frac { 1 } { p ( z _ { \mathbf { t } } ) } \right]
+$$
+We can rewrite the RHS using $p ^ { * } ( s )$ to denote the target state marginal distribution:
+$$
+\mathbb { E } _ { s \sim p ^ { * } ( s ) } \left[ \mathrm { H r T T I N G T I M E } ( s ) \right] \le \mathbb { E } _ { s \sim p _ { g } ( s ) } \left[ \frac { 1 } { \int p ^ { * } ( \tilde { s } ) \mathbb { 1 } ( \| s - \tilde { s } \| \le \epsilon ) d \tilde { s } } \right] \triangleq \mathcal { F } ( p ^ { * } )
+$$
+We will minimize $\mathcal { F }$ , an upper bound on the expected hitting time.
+Lemma C.2. The state marginal distribution $\begin{array} { r l r } { p ^ { * } ( s ) } & { { } \propto } & { \sqrt { \tilde { p } ( s ) } } \end{array}$ minimizes $\mathcal { F } ( p ^ { * } )$ , where $\begin{array} { r } { \tilde { p } ( s ) \triangleq \int p _ { g } ( \tilde { s } ) \mathbb { 1 } ( \lVert s - \tilde { s } \rVert \leq \epsilon ) d \tilde { s } } \end{array}$ is a smoothed version of the target density.
+Before presenting the proof, we provide a bit of intuition. In the case where $\epsilon \to 0$ , the optimal target distribution is $p ^ { * } ( s ) \propto \sqrt { p _ { g } ( s ) }$ . For non-zero $\epsilon$ , the policy in Lemma C.2 is equivalent to convolving $p _ { g } ( s )$ with a box filter before taking the square root. In both cases, we see that the optimal policy does distribution matching to some function of the goal distribution. Note that $\tilde { p } ( \cdot )$ may not sum to one and therefore is not a proper probability distribution.
+Proof. We start by forming the Lagrangian:
+$$
+\mathcal { L } ( p ^ { * } ) \triangleq \int \frac { p _ { g } ( s ) } { \int p ^ { * } ( \tilde { s } ) \mathbb { 1 } ( \lVert s - \tilde { s } \rVert \leq \epsilon ) d \tilde { s } } d s + \lambda \left( \int p ^ { * } ( \tilde { s } ) d \tilde { s } - 1 \right)
+$$
+The first derivative is
+$$
+\frac { d \mathcal { L } } { d p ^ { * } ( \tilde { s } ) } = \int \frac { - p _ { g } ( s ) \mathbb { 1 } ( \lVert s - \tilde { s } \rVert \leq \epsilon ) } { { p ^ { * } } ^ { 2 } ( \tilde { s } ) } d s + \lambda = 0
+$$
+Note that the second derivative is positive, indicating that this Lagrangian is convex, so all stationary points must be global minima:
+$$
+\frac { d ^ { 2 } \mathcal { L } } { d p ^ { \ast } ( \tilde { s } ) ^ { 2 } } = \int \frac { 2 p _ { g } ( s ) \mathbb { 1 } ( \| s - \tilde { s } \| \leq \epsilon ) } { { p ^ { \ast } } ^ { 3 } ( \tilde { s } ) } d s > 0
+$$
+Setting the first derivative equal to zero and rearranging terms, we obtain
+$$
+\pi ( { \tilde { s } } ) \propto \sqrt { \int p _ { g } ( s ) \mathbb { 1 } ( \| s - { \tilde { s } } \| \leq \epsilon ) d s }
+$$
+Renaming ${ \tilde { s } }  s$ , we obtain the desired result.
+# C.1 CONNECTIONS TO GOAL-CONDITIONED RL
+Goal-Conditioned RL (Kaelbling, 1993; Nair et al., 2018; Held et al., 2017) can be viewed as a special case of State Marginal Matching when the goal-sampling distribution is learned jointly with the policy. In particular, consider the State Marginal Matching with a mixture policy (Algorithm 2), where the mixture component $z$ maps bijectively to goal states $g$ . In this case, we learn goal-conditioned policies of the form $\pi ( \boldsymbol { a } \mid s , g )$ . We start by swapping $g$ for $z$ in the SMM objective with Mixtures of Policies (Equation 8):
+$$
+D _ { \mathrm { K L } } ( \rho _ { \pi } ( s ) \parallel p ^ { * } ( s ) ) = \mathbb { E } _ { \underset { s \sim \pi ( s | g ) } { \mathbb { E } _ { g \sim \pi ( g ) } } } \left[ \log p ^ { * } ( s ) + \log p ( g \mid s ) - \log \rho _ { \pi } ( s \mid g ) - \log \pi ( g ) \right]
+$$
+The second term $p ( g \mid s )$ is an estimate of which goal the agent is trying to reach, similar to objectives in intent inference (Ziebart et al., 2009; Xie et al., 2013). The third term $\pi ( s \mid g )$ is the distribution over states visited by the policy when attempting to reach goal $g$ . For an optimal goal-conditioned policy in an infinite-horizon MDP, both of these terms are Dirac functions:
+$$
+\pi ( g \mid s ) = \rho _ { \pi } ( s \mid g ) = \mathbb { 1 } ( s = g )
+$$
+In this setting, the State Marginal Matching objective simply says to sample goals $g \sim \pi ( g )$ with probability equal to the density of that goal under the target distribution.
+$$
+D _ { \mathrm { K L } } ( \rho _ { \pi } ( s ) \parallel p ^ { * } ( s ) ) = \mathbb { E } _ { \underset { s \sim \pi ( s \mid g ) } { \mathbb { E } } } \left[ \log p ^ { * } ( s ) - \log \pi ( g ) \right]
+$$
+Whether goal-conditioned RL is the preferable way to do distribution matching depends on (1) the difficulty of sampling goals and (2) the supervision that will be provided at test time. It is natural to use goal-conditioned RL in settings where it is easy to sample goals, such as when the space of goals is small and finite or otherwise low-dimensional. If a large collection of goals is available apriori, we could use importance sampling to generate goals to train the goal-conditioned policy (Pong et al., 2019). However, many real-world settings have high-dimensional goals, which can be challenging to sample. While goal-conditioned RL is likely the right approach when we will be given a test-time task, a latent-conditioned policy may explore better in settings where the goal-state is not provided at test-time.
+# D ADDITIONAL EXPERIMENTS & EXPERIMENTAL DETAILS
+# D.1 ENVIRONMENT DETAILS
+We summarize the environment parameters for Navigation (Figures 2, 5a), Manipulation (Figures 3, 5b, 7, 8, 9, 10), and $D$ ’Claw (Figure 4) in Table 1.
+Navigation: Episodes have a maximum time horizon of 100 steps. The environment reward is
+$$
+r _ { \mathrm { e n v } } ( s ) = { \left\{ \begin{array} { l l } { p _ { i } } & { { \mathrm { i f ~ } } \| s _ { \mathrm { r o b o t } } - g _ { i } \| _ { 2 } ^ { 2 } < \epsilon { \mathrm { ~ f o r ~ a n y ~ } } i \in [ n ] } \\ { 0 } & { { \mathrm { o t h e r w i s e } } } \end{array} \right. }
+$$
+where $s _ { x y }$ is the xy-position of the agent. We used a uniform target distribution over the end of all $m$ halls, so the environment reward at training time is $\textstyle r _ { \mathrm { e n v } } ( s ) = { \frac { 1 } { m } }$ if the robot is close enough to the end of any of the halls.
+We used a fixed hall length of 10 in Figures 2b and 2c, and length 50 in Figure 5a. All experiments used $m = 3$ halls, except in Figure 2c where we varied the number of halls $\{ 3 , 5 , 7 \}$ .
+Manipulation. We used the simulated Fetch Robotics $\mathrm { a r m } ^ { 5 }$ implemented by Plappert et al. (2018) using the MuJoCo simulator Todorov et al. (2012). The state vector $s \in \mathbb { R } ^ { 2 8 }$ includes the xyzcoordinates $s _ { \mathrm { o b j } }$ , $s _ { \mathrm { r o b o t } } \in \mathbb { R } ^ { 3 }$ of the block and the robot gripper respectively, as well as their velocities, orientations, and relative position $s _ { \mathrm { o b j } } - s _ { \mathrm { r o b o t } }$ . At the beginning of each episode, we spawn the object at the center of the table, and the robot gripper above the initial block position. We terminate each episode after 50 environment steps, or if the block falls off the table.
+We considered two target state marginal distributions. In Manipulation-Uniform, the target density is given by
+$$
+p ^ { * } ( s ) \propto \exp \left( \alpha _ { 1 } r _ { \mathrm { g o a l } } ( s ) + \alpha _ { 2 } r _ { \mathrm { r o b o t } } ( s ) + \alpha _ { 3 } r _ { \mathrm { a c t i o n } } ( s ) \right)
+$$
+where $\alpha _ { 1 } , \alpha _ { 2 } , \alpha _ { 3 } > 0$ are fixed weights, and the rewards
+$$
+\begin{array} { r l } & { r _ { \mathrm { g o a l } } ( s ) : = 1 - \mathbb { 1 } \left( s _ { \mathrm { o b j } } \mathrm { ~ i s ~ o n ~ t h e ~ t a b l e } \right. } \\ & { r _ { \mathrm { r o b o t } } ( s ) : = \mathbb { 1 } \left( \| s _ { \mathrm { o b j } } - s _ { \mathrm { r o b o t } } \| _ { 2 } ^ { 2 } < 0 . 1 \right) } \\ & { r _ { \mathrm { a c t i o n } } ( s ) : = - \| a \| _ { 2 } ^ { 2 } } \end{array}
+$$
+correspond to (1) a uniform distribution of the block position over the table surface (the agent receives $+ 0$ reward while the block is on the table), (2) an indicator reward for moving the robot gripper close to the block, and (3) action penalty, respectively. The environment reward is a weighted sum of the three reward terms: $r _ { \mathrm { e n v } } ( s ) \triangleq 2 0 r _ { \mathrm { g o a l } } ( s ) + r _ { \mathrm { r o b o t } } ( s ) + 0 . 1 r _ { \mathrm { a c t i o n } } ( s )$ . At test-time, we sample a goal block location $g \in \mathbb { R } ^ { 3 }$ uniformly on the table surface, and the goal is not observed by the agent.
+In Manipulation-Half, the target state density places higher probability mass to states where the block is on the left-side of the table. This is implemented by replacing $r _ { \mathrm { g o a l } } ( s )$ with a reward function that gives a slightly higher reward $+ 0 . 1$ for states where the block is on the left-side of the table.
+D’Claw. The $D$ ’Claw robot (Ahn et al., 2019; Zhu et al., $2 0 1 9 ) ^ { 6 }$ controls three claws to rotate a valve object. The environment consists of a 9-dimensional action space (three joints per claw) and a 12-dimensional observation space that encodes the joint angles and object orientation. We fixed each episode at 50 timesteps, which is about 5 seconds on the real robot. In the hardware experiments, each algorithm was trained on the same four $D$ ’Claw robots to ensure consistency.
+Table 1: Environment parameters specifying the observation space dimension $| S |$ ; action space dimension $| { \cal A } |$ ; max episode length $T$ ; the environment reward, related to the target distribution by $\exp \{ r _ { \mathrm { e n v } } ( s ) \} \propto p ^ { * } ( s )$ , and other environment parameters.
+<table><tr><td rowspan=1 colspan=1>Environment</td><td rowspan=1 colspan=1>[S</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>A</td><td rowspan=1 colspan=1>T</td><td rowspan=1 colspan=1>Env Reward (log p*(s))</td><td rowspan=1 colspan=1>Other Parameters</td><td rowspan=1 colspan=1>Figure</td></tr><tr><td rowspan=2 colspan=1>Navigation</td><td rowspan=2 colspan=2>2</td><td rowspan=2 colspan=1>2</td><td rowspan=2 colspan=1>100</td><td rowspan=1 colspan=1>Uniformoverallmhalls</td><td rowspan=1 colspan=1>#Halls: 3,5,7Hall length: 10</td><td rowspan=1 colspan=1>2</td></tr><tr><td rowspan=1 colspan=1>Uniformoverallmhalls</td><td rowspan=1 colspan=1># Halls: 3Hall length: 50</td><td rowspan=1 colspan=1>5a</td></tr><tr><td rowspan=2 colspan=1>Manipulation</td><td rowspan=2 colspan=2>25</td><td rowspan=2 colspan=1>4</td><td rowspan=2 colspan=1>50</td><td rowspan=1 colspan=1>Uniform block pos.over table surface</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>3b,3c,5b</td></tr><tr><td rowspan=1 colspan=1>More block pos. densityon left-half of table</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>3d</td></tr><tr><td rowspan=1 colspan=1>D&#x27;Claw</td><td rowspan=1 colspan=2>12</td><td rowspan=1 colspan=1>9</td><td rowspan=1 colspan=1>50</td><td rowspan=1 colspan=1>Uniform object angleover[-180°,180°]</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>4</td></tr></table>
+We defined the target state distribution to place uniform probability mass over all object angles in $\left[ - 1 8 0 ^ { \circ } , 1 8 0 ^ { \circ } \right]$ . It also incorporates reward shaping terms that place lower probability mass on states with high joint velocity and on states with joint positions that deviate far from the initial position (see (Zhu et al., 2019)).
+# D.2 GAIL
+GAIL assumes access to expert demonstrations, which SMM and the other exploration methods do not require. To compare GAIL with the exploration methods on a level footing, we sampled synthetic states from $p ^ { * } ( s )$ to train GAIL, and restricted the GAIL discriminator input to states only (no actions).
+For $D$ ’Claw (Fig. 4), we sampled the valve object angle uniformly in $\left[ - 1 8 0 ^ { \circ } , 1 8 0 ^ { \circ } \right]$ . For Manipulation-Uniform (Fig. 3c), we sampled object positions $\boldsymbol { s } _ { \mathrm { o b j e c t } }$ uniformly on the table surface, and tried two different sampling distributions for the gripper position $s _ { \mathrm { r o b o t } }$ (see Fig. 6). For both environments, all other state dimensions were sampled uniformly in $[ - 1 0 , 1 0 ]$ , and used 1e4 synthetic state samples to train GAIL.
+Since the state samples from $p ^ { * } ( s )$ may not be reachable from the initial state, the policy may not be able to fool the discriminator. To get around this problem, we also tried training GAIL with the discriminator input restricted to only the state dimensions corresponding to the object position or gripper position (Manipulation), or the object angle $( D ^ { \prime } C l a w )$ . We summarize these GAIL ablation experiments in Fig. 6. In our experiments, we used the best GAIL ablation model to compare against the exploration baselines in Figures 3c and 4.
+# D.3 VAE DENSITY MODEL
+In our SMM implementation, we estimated the density of data $x$ as $p ( x ) \approx \operatorname { d e c o d e r } ( { \hat { x } } = x | z =$ encoder $\left( x \right)$ ). That is, we encoded $x$ to $z$ , reconstruction $\hat { x }$ from $z$ , and then took the likelihood of the true data $x$ under a unit-variance Gaussian distribution centered at the reconstructed $\hat { x }$ . The log-likelihood is therefore given by the mean-squared error between the data $x$ and the reconstruction $\hat { x }$ , plus a constant that is independent of $x$ : $\begin{array} { r } { \log ^ { * } { q ( x ) } = \frac { 1 } { 2 } \| x - \hat { x } \| _ { 2 } ^ { 2 } + C } \end{array}$ .
+# D.4 COMPUTATIONAL COMPLEXITY
+We compare the wall-clock time of each exploration method in Table 2. The computational cost of our method is comparable with prior work.
+![](images/ba6a52177d16e7dde348c661a44f561b7c16a95a0361e5635b488461291ccf3d.jpg)
+Figure 6: GAIL Ablation Study: We studied the effect of restricting the GAIL discriminator input to fewer state dimensions. (a) Manipulation: We trained the GAIL discriminator on the entire state vector $s$ ; on the object and gripper positions $\{ s _ { \mathrm { o b j e c t } } , s _ { \mathrm { r o b o t } } \}$ only; or on the object position $\boldsymbol { s } _ { \mathrm { o b j e c t } }$ only. We also varied the sampling distribution for the gripper position, $p ^ { * } ( s _ { \mathrm { r o b o t } } )$ : we compare using a normal distribution, $\mathcal { N } ( s _ { \mathrm { o b j e c t } } , I _ { 3 } )$ , to sample gripper positions closer to the object, versus a uniform distribution, Uniform $\left[ - 1 0 , 1 0 \right]$ , for greater entropy of the sampled gripper positions. We observe that sampling gripper positions closer to the object position improves the entropy of the object position ${ \mathcal { H } } _ { \pi } [ s _ { \mathrm { o b j e c t } } ]$ , but hurts the entropy of the gripper position $\bar { \mathcal { H } } _ { \pi } [ s _ { \mathrm { r o b o t } } ]$ . (b) $\mathbf { \nabla } D ^ { \prime } C l a w$ : We restricted the discriminator to the entire state vector $s$ , or to the object angle and position $\boldsymbol { s } _ { \mathrm { o b j e c t } }$ . Analysis: In both domains, we observe that restricting the discriminator input to fewer state dimensions (e.g., to $s _ { \mathrm { o b j e c t } } )$ makes the discriminator less capable of distinguishing between expert and policy states (orange and green curves). On the other hand, training on the entire state vector $s$ causes the discriminator loss to approach 0 (i.e., perfect classification), partly because some of the “expert” states sampled from $p ^ { * } ( s )$ are not reachable from the initial state, and the policy is thus unable to fool the discriminator.
+Table 2: Per-epoch wall-clock time on the Manipulation environment. One epoch is 1e3 env. steps.
+<table><tr><td>SAC</td><td>ICM</td><td>Count</td><td>SMM (ours)</td><td>PseudoCounts</td></tr><tr><td>17.95s (+0%)</td><td>22.74s (+27%)</td><td>25.24s (+41%)</td><td>25.82s (+44%)</td><td>33.87s (+89%)</td></tr></table>
+# D.5 ALGORITHM HYPERPARAMETERS
+We summarize hyperparameter settings in Table 3. All algorithms were trained for 1e5 steps on Navigation, 1e6 steps on Manipulation, 1e6 steps on $D$ ’Claw Sim2Real, and 1e5 steps on $D$ ’Claw hardware.
+Loss Hyperparameters. For each exploration method, we tuned the weights of the different loss components. SAC reward scale controls the weight of the action entropy reward relative to the extrinsic reward. Count coeff controls the intrinsic count-based exploration reward w.r.t. the extrinsic reward and SAC action entropy reward. Similarly, Pseudocount coeff controls the intrinsic pseudocount exploration reward. SMM coeff for $\mathcal { H } [ s \mid z ]$ and $\mathcal { H } [ z \mid s ]$ control the weight of the different loss components (state entropy and latent conditional entropy) of the SMM objective in Equation 8.
+Historical Averaging. In the Manipulation experiments, we tried the following sampling strategies for historical averaging: (1) Uniform: Sample policies uniformly across training iterations. (2) Exponential: Sample policies, with recent policies sampled exponentially more than earlier ones. (3) Last: Sample the $N$ latest policies uniformly at random. We found that Uniform worked less well, possibly due to the policies at early iterations not being trained enough. We found negligible difference in the state entropy metric between Exponential vs. Last, and between sampling 5 vs. 10 historical policies, and we also note that it is unnecessary to keep checkpoints from every iteration.
+Network Hyperparameters. For all algorithms, we use a Gaussian policy with two hidden layers with Tanh activation and a final fully-connected layer. The Value function and Q-function each are a feedforward MLP with two hidden layers with ReLU activation and a final fully-connected layer. Each hidden layer is of size 300 (SMM, SAC, ICM, C, PC) or 256 (GAIL). The same network configuration is used for the SMM discriminator, $d ( z \mid s )$ , and the GAIL discriminator, but with different input and output sizes. The SMM density model, $q ( s )$ , is modeled by a VAE with encoder and decoder networks each consisting of two hidden layers of size (150, 150) with ReLU activation. The same VAE network configuration is used for Pseudocount.
+GAIL Hyperparameters: The replay buffer is filled with 1e4 random actions before training, for training stability. We perform one discriminator update per SAC update. For both Manipulation and $D ^ { \prime } C l a w$ , we used 1e4 states sampled from $p ^ { * } ( s )$ . Other hyperparameter settings, such as batch size for both discriminator and policy updates, are summarized in Table 3. We observed that GAIL training is more unstable compared to the exploration baselines. Thus, for GAIL, we did not take the final iterate (e.g., policy at convergence) but instead used early termination (e.g., take the best iterate according to the state entropy metric).
+# D.6 VISUALIZING THE MANIPULATION ENVIRONMENT
+We visualize where different methods push the block in the Manipulation environment. More precisely, we visualize the log state marginal $\log \rho _ { \pi _ { z } } ( s )$ over block XY-coordinates $\boldsymbol { s } = ( x , y )$ in Figures 7 and 8. In Figure 9, we plot goals sampled at test-time, colored by the number of episodes each method required to push the block to that goal location. Blue dots indicate that the agent found the goal quickly. We observe that SMM has the most blue dots, indicating that it succeeds in exploring a wide range of states at test-time.
+![](images/917294c588d1908dfebab5f53e9e879024cd86a34b328b890d54ed139a4d9aea.jpg)
+Figure 7: The log state marginal $\log \rho _ { \pi } ( s )$ over block XY-coordinates, averaged over 1e3 epochs.
+![](images/6975547d160be40dbd7b3ba3b9c4cb2f726af59354daa69558810c2a04865822.jpg)
+Figure 8: SM4 with Eight Mixture Components. The log state marginal $\log \rho _ { \pi _ { z } } ( s )$ over block XY-coordinates for each latent skill $z \in \{ 0 , \ldots , 7 \}$ , averaged over 1000 epochs.
+![](images/fbdc903631f297030358029798c84705eb7dbb2a521d86a1823a8e131fb01753.jpg)
+Figure 9: Goals sampled uniformly on the table surface, colored by the number of episodes until the policy finds the goal. Red (100 episodes) indicates failure. The block always starts at the center.
+![](images/44abb1684c2a3365097b763c5c692f3541b22c4f9df728c7a654fd7f1098a7d3.jpg)
+Figure 10: Train curves on Manipulation. One epoch is 1e3 steps. (a) . The environment reward is a weighted sum of three terms: $r _ { \mathrm { g o a l } } ( s )$ $_ { + 0 }$ if object is on table, $^ { - 1 }$ otherwise), $r _ { \mathrm { r o b o t } } ( s )$ $( + 1$ if robot gripper is close to block), and $r _ { \mathrm { a c t i o n } }$ (action penalty term), with weights -20, 1, 0.1 respectively (see Appendix D.1). The three exploration methods (ICM, Count, SMM) also optimize an auxilliary exploration loss, which makes the agent more likely to move around the block. Compared to SAC, this causes the exploration methods to get worse returns for $r _ { \mathrm { g o a l } } ( s )$ and $r _ { \mathrm { a c t i o n } } ( s )$ (due to the agent moving the block around), but also quickly learns to maximize the sparse reward $r _ { \mathrm { r o b o t } } ( s )$ (indicator reward for moving gripper within a threshold distance to the block). (b) The latent action entropy $\mathcal { H } [ \boldsymbol { z } \mid s ]$ (discriminator) and latent state entropy $\mathcal { H } [ s \ | \ z ]$ (density model) per epoch.
+Table 3: Hyperparameter Settings. Hyperparameters were chosen according to the following eval metrics: Manip.-Uniform: State entropy of the discretized gripper and block positions (bin size 0.05), after rolling out the trained policy for 50K env steps. Manip.-Half : $D _ { \mathrm { K L } } ( p ^ { * } ( s ) \parallel \rho _ { \pi } ( \bar { s } ) )$ and $\mathrm { T V } ( p ^ { * } ( s ) , \rho _ { \pi } ( s ) )$ of the discretized gripper and block positions (bin size 0.01), after rolling out the trained policy for 50K env steps. 2D Navigation: State entropy of the discretized XY-positions of the trained policy. $D ^ { \prime } C l a w$ : State entropy of the object angle.
+<table><tr><td rowspan=1 colspan=1>Environment</td><td rowspan=1 colspan=1>Algorithm</td><td rowspan=1 colspan=1>HyperparametersUsed</td><td rowspan=1 colspan=1>HyperparametersConsidered</td></tr><tr><td rowspan=2 colspan=1>All</td><td rowspan=1 colspan=1>SMM, SAC,ICM, Count,Pseudocount</td><td rowspan=1 colspan=1>Batch size: 128le6 env training stepsRL discount: 0.99Network size: 300Policy lr: 3e-4Q-function lr: 3e-4Value function lr: 3e-4</td><td rowspan=1 colspan=1> N/A (Default SAC hyperparameters)</td></tr><tr><td rowspan=1 colspan=1>GAIL</td><td rowspan=1 colspan=1>1e6 env training stepsPolicy lr: 1e-5Critic Ir: 1e-3# Random actionsbefore training: le4Network size: 256</td><td rowspan=1 colspan=1>N/A (Default GAIL hyperparameters)</td></tr><tr><td rowspan=2 colspan=1>Navigation(Fig. 2, 5a)</td><td rowspan=1 colspan=1>SMM, SAC</td><td rowspan=1 colspan=1>SAC reward scale: 25</td><td rowspan=1 colspan=1>SAC reward scale: 1e-2, 0.1,1,10,25,100</td></tr><tr><td rowspan=1 colspan=1>SMM</td><td rowspan=1 colspan=1>SMMH[sz]coeff: 1SMMHzscoeff:1</td><td rowspan=1 colspan=1>SMMH[s|z]coeff: 1e-3,1e-2,1e-1,1,10SMMH[z|s]coeff: 1e-3,1e-2,1e-1,1,10</td></tr><tr><td rowspan=6 colspan=1>Manip.-Uniform(Fig.3b,3c,5b)</td><td rowspan=1 colspan=1>SMM</td><td rowspan=1 colspan=1>Num skills: 4VAE Ir: 1e-2SMM H[s|z]coeff: 1SMM H[z|s]coeff: 1HA sampling: Exponential# HA policies: 10SMM Latent Prior Coeff: 1</td><td rowspan=1 colspan=1>Num skills: 1,2, 4,8, 16VAE lr: 1e-4,1e-3,1e-2HA sampling: Exponential, Uniform, Last#HA policies: 5,10SMMLatent Prior Coeff: 1, 4</td></tr><tr><td rowspan=1 colspan=1>SAC</td><td rowspan=1 colspan=1>SAC reward scale: 0.1</td><td rowspan=1 colspan=1>SAC reward scale: 0.1,1,10,100</td></tr><tr><td rowspan=1 colspan=1>Count</td><td rowspan=1 colspan=1>Count coeff: 10Histogram bin width: 0.05</td><td rowspan=1 colspan=1>Count coeff: O.1,1,10</td></tr><tr><td rowspan=1 colspan=1>Pseudocount</td><td rowspan=1 colspan=1>Pseudocount coeff: 1VAE lr: 1e-2</td><td rowspan=1 colspan=1>Pseudocount coeff: 0.1,1,10(Use same VAE lr as SMM)</td></tr><tr><td rowspan=1 colspan=1>ICM</td><td rowspan=1 colspan=1>Learning rate: le-3</td><td rowspan=1 colspan=1>Learning rate:le-4,le-3,1e-2</td></tr><tr><td rowspan=1 colspan=1>GAIL</td><td rowspan=1 colspan=1>Batch size: 512# SAC updates per step: 1Discriminator input: sTraining iterate: 1e6# State Samples: le4</td><td rowspan=1 colspan=1>Batch size:128,512,1024# SAC updates per step: 1, 4Discriminator input: s,Sobject,{Sobject, Srobot}Training iterate: 1e5,2e5,3e5,...,9e5,1e6# State Samples: le4</td></tr><tr><td rowspan=4 colspan=1>Manip.-Half(Fig.3d)</td><td rowspan=1 colspan=1>SMM, SAC,ICM, Count</td><td rowspan=1 colspan=1>SAC reward scale: 0.1</td><td rowspan=1 colspan=1>(Best reward scale for Manip.-Uniform)</td></tr><tr><td rowspan=1 colspan=1>SMM</td><td rowspan=1 colspan=1>Num skills: 4SMMH[s| z] coeff: 1SMMH[z|s] coeff:1</td><td rowspan=1 colspan=1>Num skills: 1, 2, 4, 8</td></tr><tr><td rowspan=1 colspan=1>Count</td><td rowspan=1 colspan=1>Count coeff: 10Histogram bin width: 0.05</td><td rowspan=1 colspan=1>Count coeff: 0.1,1,10</td></tr><tr><td rowspan=1 colspan=1>ICM</td><td rowspan=1 colspan=1>Learning rate: le-3</td><td rowspan=1 colspan=1>Learning rate:le-4,le-3,1e-2</td></tr><tr><td rowspan=6 colspan=1>D&#x27;Claw(Fig.4)</td><td rowspan=1 colspan=1>SMM, SAC</td><td rowspan=1 colspan=1>SAC reward scale: 5</td><td rowspan=1 colspan=1>SAC reward scale:1e-2,0.1,1,5,10, 100</td></tr><tr><td rowspan=1 colspan=1>SMM</td><td rowspan=1 colspan=1>SMMHs|zcoeff: 250</td><td rowspan=1 colspan=1>SMMH[s|z]c0eff: 1,10,100,250,500,1e3</td></tr><tr><td rowspan=1 colspan=1>Count</td><td rowspan=1 colspan=1>Count coeff: 1Histogram bin width: 0.05</td><td rowspan=1 colspan=1>Count coeff: 1,10Histogram bin width: 0.05, 0.1</td></tr><tr><td rowspan=1 colspan=1>Pseudocount</td><td rowspan=1 colspan=1>Pseudocount coeff: 1VAE lr: 1e-3</td><td rowspan=1 colspan=1>Pseudocount coeff: 1, 10VAE lr: le-1,1e-2, 1e-3</td></tr><tr><td rowspan=1 colspan=1>ICM</td><td rowspan=1 colspan=1>Learning rate: le-3VAE lr: 1e-1</td><td rowspan=1 colspan=1>Learning rate:1e-2,1e-3,1e-4VAE lr: 1e-1,1e-2, 1e-3</td></tr><tr><td rowspan=1 colspan=1>GAIL</td><td rowspan=1 colspan=1>Batch size: 512 # SAC updates per step: 4Discriminator input: SobjectTraining iterate: 1e5# State Samples: 1e4</td><td rowspan=1 colspan=1>Batch size:128,512,1024# SAC updates per step: 1, 4Discriminator input: s, SobjectTraining iterate: 1e5,2e5,3e5,...,9e5,1e6# State Samples: 1e4</td></tr></table>

parse/train/Hkla1eHFvS/Hkla1eHFvS_content_list.json ADDED Viewed

The diff for this file is too large to render. See raw diff

parse/train/Hkla1eHFvS/Hkla1eHFvS_middle.json ADDED Viewed

The diff for this file is too large to render. See raw diff

parse/train/Hkla1eHFvS/Hkla1eHFvS_model.json ADDED Viewed

The diff for this file is too large to render. See raw diff

parse/train/Hyewf3AqYX/Hyewf3AqYX.md ADDED Viewed

	@@ -0,0 +1,572 @@

+# A FRANK-WOLFE FRAMEWORK FOR EFFICIENT AND EFFECTIVE ADVERSARIAL ATTACKS
+Anonymous authors Paper under double-blind review
+# ABSTRACT
+Depending on how much information an adversary can access to, adversarial attacks can be classified as white-box attack and black-box attack. In both cases, optimization-based attack algorithms can achieve relatively low distortions and high attack success rates. However, they usually suffer from poor time and query complexities, thereby limiting their practical usefulness. In this work, we focus on the problem of developing efficient and effective optimization-based adversarial attack algorithms. In particular, we propose a novel adversarial attack framework for both white-box and black-box settings based on the non-convex Frank-Wolfe algorithm. We show in theory that the proposed attack algorithms are efficient with an $O ( 1 / \sqrt { T } )$ convergence rate. The empirical results of attacking Inception V3 model and ResNet V2 model on the ImageNet dataset also verify the efficiency and effectiveness of the proposed algorithms. More specific, our proposed algorithms attain the highest attack success rate in both white-box and black-box attacks among all baselines, and are more time and query efficient than the stateof-the-art.
+# 1 INTRODUCTION
+Deep Neural Networks (DNNs) have made many breakthroughs in different areas of artificial intelligence such as image classification (Krizhevsky et al., 2012; He et al., 2016a), object detection (Ren et al., 2015; Girshick, 2015), and speech recognition (Mohamed et al., 2012; Bahdanau et al., 2016). However, recent studies show that deep neural networks can be vulnerable to adversarial examples (Szegedy et al., 2013; Goodfellow et al., 2015) – a tiny perturbation on an image that is almost invisible to human eyes could mislead a well-trained image classifier towards misclassification. Soon later this is proved to be not a coincidence: similar phenomena have been observed in other problems such as speech recognition (Carlini et al., 2016), visual QA (Xu et al., 2017), image captioning (Chen et al., 2017a), machine translation (Cheng et al., 2018), reinforcement learning (Pattanaik et al., 2018), and even on systems that operate in the physical world (Kurakin et al., 2016).
+Depending on how much information an adversary can access to, adversarial attacks can be classified into two classes: white-box attack (Szegedy et al., 2013; Goodfellow et al., 2015) and black-box attack (Papernot et al., 2016a; Chen et al., 2017c). In the white-box setting, the adversary has full access to the target model, while in the black-box setting, the adversary can only access the input and output of the target model but not its internal configurations. Among the approaches proposed for white-box and black-box attacks, optimization-based methods (Carlini & Wagner, 2017; Chen et al., 2017b;c; Ilyas et al., 2018) are most effective: they usually achieve relatively low distortions and high attack success rates. However, these methods are far from efficient. In the white-box setting, they need to solve constrained optimization problems (Carlini & Wagner, 2017), and are usually significantly slower than Fast Gradient Sign Method (FGSM) (Goodfellow et al., 2015) or Iterative FGSM (I-FGM) (Kurakin et al., 2016). Applying those methods with one or two examples are fine, yet in the case of attacking hundreds of thousands examples, e.g. in adversarial training (Kurakin et al., 2016; Madry et al., 2018), this is far from satisfactory.
+In the black-box setting, it becomes even more severe since they need to make gradient estimations (Chen et al., 2017c). Therefore, a large number of queries are needed for them to perform a successful attack, especially when the data dimension is large. For example, attacking a $2 9 9 \times 2 9 9 \times 3$ Imagenet image may take them hundreds of thousands of queries. This significantly limits their practical usefulness since they can be easily defeated by limiting the number of queries that an adversary can make to the target model.
+In this study, we aim to examine the following questions in this study:
+Can we improve the efficiency of the optimization-based attack algorithms? In other words, can we use less time and queries to conduct adversarial attacks?
+In this work, we provide an affirmative answer to this question by proposing an efficient FrankWolfe optimization framework for both white-box and black-box attacks. In summary, we make the following main contributions:
+• We propose a novel Frank-Wolfe based adversarial attack framework. The white-box attack algorithm is an iterative first-order method which admits the fast gradient sign method (FGSM) as the one-step special case. And the corresponding black-box attack algorithm adopts zeroth-order optimization with two sensing vector options (either from the Euclidean unit sphere or from the standard Gaussian distribution) provided. We show that the proposed white-box and black-box attack algorithms enjoy an $O ( 1 / \sqrt { T } )$ convergence rate. Also we show that the query complexity of the proposed black-box attack algorithm is linear in data dimension $d$ . Our empirical results on attacking Inception V3 model with the ImageNet dataset show that (i) the proposed white-box attack algorithm is more efficient than all the baseline whitebox algorithms evaluated here, and (ii) the proposed black-box attack algorithm is highly efficient and is also the only one algorithm that achieves a $1 0 0 \%$ attack success rate.
+# 2 RELATED WORK
+There is a large body of work on adversarial attacks. In this section, we review the most relevant work in both white-box and black-box attack settings, as well as the non-convex Frank-Wolfe optimization.
+White-box Attacks: Szegedy et al. (2013) proposed to use box-constrained L-BFGS algorithm for conducting white-box attacks. Goodfellow et al. (2015) proposed the Fast Gradient Sign Method (FGSM) based on linearization of the network as a simple alternative to L-BFGS. Kurakin et al. (2016) proposed to iteratively perform one-step FGSM (Goodfellow et al., 2015) algorithm and clips the adversarial point back to the distortion limit after every iteration. It is called Basic Iterative Method (BIM) or I-FGM in the literature. Madry et al. (2018) showed that for the $L _ { \infty }$ norm case, BIM/I-FGM is equivalent to Projected Gradient Descent (PGD), which is a standard tool for constrained optimization. Papernot et al. (2016b) proposed JSMA to greedily attack the most significant pixel based on the Jacobian-based saliency map. Moosavi-Dezfooli et al. (2016) proposed attack methods by projecting the data to the closest separating hyperplane. Carlini & Wagner (2017) introduced the so-called CW attack by proposing multiple new loss functions for generating adversarial examples. Chen et al. (2017b) followed CW’s framework and use an Elastic Net term as the distortion penalty.
+Black-box Attacks: One popular family of black-box attacks (Hu & Tan, 2017; Papernot et al., 2016a; 2017) is based on the transferability of adversarial examples (Liu et al., 2018; Bhagoji et al., 2017), where an adversarial example generated for one DNN may be reused to attack other neural networks. This allows the adversary to construct a substitute model that mimics the targeted DNN, and then attack the constructed substitute model using white-box attack methods. However, this type of attack algorithms usually suffer from large distortions and relatively low success rates (Chen et al., 2017c). To address this issue, Chen et al. (2017c) proposed the Zeroth-Order Optimization (ZOO) algorithm that extends the CW attack to the black-box setting and uses a zeroth-order optimization approach to conduct the attack. Although ZOO achieves much higher attack success rates than the substitute model-based black-box attacks, it suffers from a poor query complexity since its naive implementation requires to estimate the gradients of all the coordinates (pixels) of the image. To improve its query complexity, several approaches have been proposed. For example, Tu et al. (2018) introduces an adaptive random gradient estimation algorithm and a well-trained Autoencoder to speed up the attack process. Ilyas et al. (2018) and Liu et al. (2018) improved ZOO’s query complexity by using Natural Evolutionary Strategies (NES) (Wierstra et al., 2014; Salimans et al., 2017) and active learning, respectively.
+Non-convex Frank-Wolfe Algorithms: The Frank-Wolfe algorithm (Frank & Wolfe, 1956), also known as the conditional gradient method, is an iterative optimization method for constrained optimization problem. Jaggi (2013) revisited Frank-Wolfe algorithm in 2013 and provided a stronger and more general convergence analysis in the convex setting. Yu et al. (2017) proved the first convergence rate for Frank-Wolfe type algorithm in the non-convex setting. Lacoste-Julien (2016) provided the convergence guarantee for Frank-Wolfe algorithm in the non-convex setting with adaptive step sizes. Reddi et al. (2016) further studied the convergence rate of non-convex stochastic Frank-Wolfe algorithm in the finite-sun optimization setting. Very recently, Staib & Jegelka (2017) proposed to use Frank-Wolfe for distributionally robust training (Sinha et al., 2018). Balasubramanian & Ghadimi (2018) proved the convergence rate for zeroth-order nonconvex Frank-Wolfe algorithm using one-side finite difference gradient estimator with standard Gaussian sensing vectors.
+# 3 METHODOLOGY
+# 3.1 NOTATIONS
+Throughout the paper, scalars are denoted by lower case letters, vectors by lower case bold face letters and sets by calligraphy upper cae letters. For a vector $\mathbf { x } \in \mathbb { R } ^ { d }$ , we denote the $L _ { p }$ norm of $\mathbf { x }$ by $\begin{array} { r } { \| \mathbf { x } \| _ { p } = ( \sum _ { i = 1 } ^ { d } x _ { i } ^ { p } ) ^ { 1 / p } } \end{array}$ . Specially, for $p = \infty$ , the $L _ { \infty }$ norm of $\mathbf { x }$ by $\| \mathbf { x } \| _ { \infty } = \operatorname* { m a x } _ { i = 1 } ^ { d } \left| \theta _ { i } \right|$ . We denote ${ \mathcal { P } } _ { \mathcal { X } } ( \mathbf { x } )$ as the projection operation of projecting vector $\mathbf { x }$ into the set $\mathcal { X }$ .
+# 3.2 PROBLEM FORMULATION
+According to the attack purposes, attacks can be divided into two categories: untargeted attack and targeted attack. In particular, untargeted attack aims to turn the prediction into any incorrect label, while the targeted attack, which is considerably harder, requires to mislead the classifier to a specific target class. In this work, we follow the literature (Carlini & Wagner, 2017; Ilyas et al., 2018) and focus on the strictly harder targeted attack setting. It is worth noting that our proposed algorithm can be extended to untargeted attack straightforwardly.
+Let us define $f ( \cdot )$ as the classification loss function of the targeted DNN. For targeted attacks, we aim to learn an adversarial example $\mathbf { x }$ that is close enough to the original input $\mathbf { x } _ { \mathrm { o r i } }$ and can be misclassified to the target class $y _ { \mathrm { t a r } }$ . The corresponding optimization problem 1 is defined as:
+$$
+\begin{array} { r l } { \operatorname* { m i n } _ { \mathbf { x } } ~ } & { f ( \mathbf { x } , \ y _ { \mathrm { t a r } } ) } \\ { \mathrm { s u b j e c t \ t o } ~ } & { \| \mathbf { x } - \mathbf { x } _ { \mathrm { o r i } } \| _ { p } \leq \epsilon . } \end{array}
+$$
+Evidently, the constraint set $\mathcal { X } : = \{ \mathbf { x } \mid \| \mathbf { x } - \mathbf { x } _ { \mathrm { o r i } } \| _ { p } \leq \epsilon \}$ is a bounded convex set when $p \geq 1$ . Normally, $p = 2$ and $p = \infty$ are used to measure the distortions $\left\| \mathbf { x } - \mathbf { x } _ { \mathrm { o r i } } \right\| _ { p }$ , resulting in $L _ { 2 }$ attack model and $L _ { \infty }$ attack model respectively. In this work, we study both attack models. In the sequel, since we mainly focus on the targeted attack case, we use $f ( \mathbf { x } )$ to denote $f ( \mathbf { x } , \ y _ { \mathrm { t a r } } )$ for simplicity.
+# 3.3 FRANK-WOLFE WHITE-BOX ATTACKS
+Frank-Wolfe algorithm (Frank & Wolfe, 1956), also known as the conditional gradient descent, is a popular optimization tool for constrained optimization. Different from PGD that first performs gradient descent followed by a projection step at each iteration, Frank-Wolfe algorithm calls a Linear Minimization Oracle (LMO) over the the constraint set $\mathcal { X }$ at each iteration, i.e.,
+$$
+\begin{array} { r } { \mathrm { L M O } \in \underset { \mathbf { v } \in \mathcal { X } } { \mathrm { a r g m i n } } \langle \mathbf { v } , \nabla f ( \mathbf { x } _ { t } ) \rangle . } \end{array}
+$$
+The LMO can be seen as the minimization of the first-order Taylor expansion of $f ( \cdot )$ at point $\mathbf { x } _ { t }$ :
+$$
+\operatorname* { m i n } _ { \mathbf { v } \in \mathcal { X } } f ( \mathbf { x } _ { t } ) + \langle \mathbf { v } - \mathbf { x } _ { t } , \nabla f ( \mathbf { x } _ { t } ) \rangle .
+$$
+By calling LMO, Frank Wolfe solves the linear problem in $\mathcal { X }$ and then perform weighted average with previous iterate to obtain the final update formula.
+We present our proposed Frank-Wolfe white-box attack algorithm in Algorithm 1, which is built upon the original Frank-Wolfe algorithm. The key difference between Algorithm 1 and the standard Frank-Wolfe algorithm is in Line 4, where the LMO is called over a slightly relaxed constraint set
+$\mathcal { X } _ { \lambda } : = \{ \mathbf { x } | \| \mathbf { x } - \mathbf { x } _ { \mathrm { o r i } } \| _ { p } \leq \lambda \epsilon \}$ with $\lambda \geq 1$ , instead of the original constraint set $\mathcal { X }$ . When $\lambda = 1$ , set $\mathcal { X } _ { \lambda }$ reduces to $\mathcal { X }$ , and Algorithm 1 reduces to standard Frank Wolfe. We argue that this modification makes our algorithm more general, and gives rise to better attack results.
+# Algorithm 1 Frank-Wolfe White-box Attack Algorithm
+1: input: number of iterations $T$ , step sizes $\{ \gamma _ { t } \}$ , $\lambda > 0$ , original image $\mathbf { x } _ { \mathrm { o r i } }$ ;
+2: x0 = xori
+3: for $t = 0 , \ldots , T - 1$ do
+4: $\mathbf { v } _ { t } = \operatorname { a r g m i n } _ { \mathbf { v } \in \mathcal { X } _ { \lambda } } \langle \mathbf { v } , \nabla f ( \mathbf { x } _ { t } ) \rangle ~ / / \operatorname { L M C }$
+5: ${ \bf d } _ { t } = { \bf v } _ { t } - { \bf x } _ { t }$
+6: $\mathbf x _ { t + 1 } = \mathbf x _ { t } + \boldsymbol \gamma _ { t } \mathbf d _ { t }$
+7: if $\lambda > 1$ then
+8: $\mathbf { x } _ { t + 1 } = \mathcal { P } _ { \mathcal { X } } ( \mathbf { x } _ { t + 1 } )$
+9: end if
+10: end for
+11: output: $\mathbf { x } _ { T }$
+The LMO solution itself can be expensive to obtain in general. Fortunately, applying Frank-Wolfe to solve (3.1) actually gives us a closed-form LMO solution. We provide the solutions of LMO (Line 4 in Algorithm 1) for $L _ { 2 }$ norm and $L _ { \infty }$ norm cases respectively:
+$$
+\begin{array} { r l } & { \mathbf { v } _ { t } = - \displaystyle \frac { \lambda \boldsymbol { \epsilon } \cdot \nabla f ( \mathbf { x } _ { t } ) } { \| \nabla f ( \mathbf { x } _ { t } ) \| _ { 2 } } + \mathbf { x } _ { \mathrm { o r i } } , } \\ & { \mathbf { v } _ { t } = - \lambda \boldsymbol { \epsilon } \cdot \mathrm { s i g n } ( \nabla f ( \mathbf { x } _ { t } ) ) + \mathbf { x } _ { \mathrm { o r i } } . } \end{array}
+$$
+The derivation can be found in the supplemental materials.
+Note that when $T = 1 , \lambda = 1$ , substituting the above LMO solutions into Algorithm 1 yields the final update of $x _ { 1 } = x _ { 0 } - \gamma _ { t } \epsilon \cdot \nabla f ( { \bf x } _ { t } )$ , which reduces to FGSM 2 when $\gamma _ { t } = 1$ . Similar derivation also applies to $L _ { 2 }$ norm case. Therefore, just like PGD, our proposed Frank-Wolfe white-box attack also includes FGSM (FGM) as a one-step special instance.
+# 3.4 FRANK-WOLFE BLACK-BOX ATTACKS
+Next we consider the black-box setting, where we cannot perform back-propagation to calculate the gradient of the loss function anymore. Instead, we can only query the DNN system’s outputs with specific inputs. To clarify, here the output refers to the logit layer’s output (confidence scores for classification), not the final prediction label. The label-only setting is doable under our framework, but will incur extra difficulty such as designing new loss functions. For simplicity, here we consider the confidence score output.
+We propose a zeroth-order Frank-Wolfe based algorithm to solve this problem. Algorithm 2 show our proposed Frank-Wolfe black-box attack algorithm. The key difference between our proposed black-box attack and white-box attack is one extra gradient estimation step, which is presented in Line 4 in Algorithm 2. Also note that for the final output, we provide two options. While option II is the common choice in practice, option I is also provided for the ease of theoretical analysis.
+As many other zeroth-order optimization algorithms (Shamir, 2017; Flaxman et al., 2005), Algorithm 3 uses symmetric finite differences to estimate the gradient and therefore, gets rid of the dependence on back-propagation in white-box setting. Different from Chen et al. (2017c), here we do not utilize natural basis as our sensing vectors, instead, we provide two options: one is to use vectors uniformly sampled from Euclidean unit sphere and the other is to use vectors uniformly sampled from standard multivarite Gaussian distribution. This will greatly improve the gradient estimation efficiency comparing to sensing with natural basis as such option will only be able to estimate one coordinate of the gradient vector per query. In practice, both options here provide us competitive experimental results. It is worth noting that NES method (Wierstra et al., 2014) with antithetic sampling (Salimans et al., 2017) used in Ilyas et al. (2018) yields similar formula as our Option II in Algorithm 3.
+# Algorithm 2 Frank-Wolfe Black-box Attack Algorithm
+1: input: number of iterations $T$ , step sizes $\{ \gamma _ { t } \}$ , $\lambda > 0$ , original image $\mathbf { x } _ { \mathrm { o r i } }$ , target label $y _ { \mathrm { t a r } }$ ;
+2: $\mathbf { x } _ { \mathrm { 0 } } = \mathbf { x } _ { \mathrm { o r i } }$
+3: for $t = 0 , \ldots , T - 1$ do
+4: $\mathbf { q } _ { t } = \mathrm { Z E R O \_ O R D \_ G R A D \_ E S T } ( \mathbf { x } _ { \mathrm { t } } )$ // Algorithm 3
+5: $\mathbf { v } _ { t } = \operatorname { a r g m i n } _ { \mathbf { v } \in \mathcal { X } _ { \lambda } } \langle \mathbf { v } , \mathbf { q } _ { t } \rangle$
+6: dt = vt − xt
+7: xt+1 = xt + γtdt
+8: if λ > 1 then
+9: $\mathbf { x } _ { t + 1 } = \mathcal { P } _ { \mathcal { X } } ( \mathbf { x } _ { t + 1 } )$
+10: end if
+11: end for
+12: Option I: $\mathbf { x } _ { a }$ is uniformly random chosen from $\{ { \bf x } _ { t } \} _ { t = 1 } ^ { T }$
+13: Option II: ${ \bf x } _ { a } = { \bf x } _ { T }$
+14: output: $\mathbf { x } _ { a }$
+# Algorithm 3 Zeroth-Order Gradient Estimation (ZERO ORD GRAD EST)
+1: parameters: number of gradient estimation samples $b$ , sampling parameter $\delta _ { t }$ ;
+2: $\mathbf q = \mathbf 0$
+3: for $i = 1 , \ldots , b$ do
+4: Option I: Sample $\mathbf { u } _ { i }$ uniformly from the Euclidean unit sphere with $\| \mathbf { u } _ { i } \| _ { 2 } = 1$ $\begin{array} { r } { \mathbf { q } = \mathbf { q } + \frac { d } { 2 \delta _ { t } b } \big ( f ( \mathbf { x } _ { t } + \delta _ { t } \mathbf { u } _ { i } ) - f ( \mathbf { x } _ { t } - \delta _ { t } \mathbf { u } _ { i } ) \big ) \mathbf { u } _ { i } } \end{array}$
+5: Option II: Sample $\mathbf { u } _ { i }$ uniformly from the standard Gaussian distribution $\mathcal { N } ( \mathbf { 0 } , \mathbf { I } )$ $\begin{array} { r } { \mathbf { q } = \mathbf { q } + \frac { 1 } { 2 \delta _ { t } b } \big ( f ( \mathbf { x } _ { t } + \delta _ { t } \mathbf { u } _ { i } ) - f ( \mathbf { x } _ { t } - \delta _ { t } \mathbf { u } _ { i } ) \big ) \mathbf { u } _ { i } } \end{array}$
+6: end for
+7: output: q
+# 4 MAIN THEORY
+In this section, we establish the convergence guarantees for our proposed Frank-Wolfe adversarial attack algorithms described in Section 3. First, we introduce the convergence criterion for our FrankWolfe adversarial attack framework.
+# 4.1 CONVERGENCE CRITERION
+The loss function for common DNN models are generally nonconvex. In addition, (3.1) is a constrained optimization. For such general nonconvex constrained optimization, we typically adopt the Frank-Wolfe gap as the convergence criterion (since gradient norm of $f$ is no longer a proper criterion for constrained optimization problems):
+$$
+g ( \mathbf { x } _ { t } ) = \operatorname* { m a x } _ { \mathbf { x } \in \mathcal { X } } \langle \mathbf { x } - \mathbf { x } _ { t } , - \nabla f ( \mathbf { x } _ { t } ) \rangle .
+$$
+Note that for the Frank-Wolfe gap, we always have $g ( \mathbf { x } _ { t } ) \geq 0$ and $\mathbf { x } _ { t }$ is a stationary point for the constrained optimization problem if and only if $g ( { \bf x } _ { t } ) = 0$ . Also the Frank-Wolfe gap is affine invariant and do not tie to any specific choice of norm, which makes itself a perfect convergence criterion for Frank-Wolfe based algorithms.
+# 4.2 CONVERGENCE GUARANTEE FOR FRANK-WOLFE WHITE-BOX ATTACK
+Before we are going to provide the convergence guarantee of Frank-Wolfe white-box attack (Algorithm 1), we introduce the following assumptions that are essential to the convergence analysis.
+Assumption 4.1. Function $f ( \cdot )$ is $L$ -smooth with respect to $\mathbf { x }$ , i.e., for any $\mathbf { x } , \mathbf { x } ^ { \prime }$ , it holds that
+$$
+f ( \mathbf { x } ^ { \prime } ) \leq f ( \mathbf { x } ) + \nabla f ( \mathbf { x } ) ^ { \top } ( \mathbf { x } ^ { \prime } - \mathbf { x } ) + \frac { L } { 2 } \| \mathbf { x } ^ { \prime } - \mathbf { x } \| _ { 2 } ^ { 2 } .
+$$
+Assumption 4.1 is a standard assumption in nonconvex optimization, and is also adopted in other Frank-Wolfe literature such as Lacoste-Julien (2016); Reddi et al. (2016). Note that even though the smoothness assumption does not hold for general DNN models, a recent study (Santurkar et al., 2018) shows that batch normalization that is used in many modern DNNs such as Inception V3 model, actually makes the optimization landscape significantly smoother 3. This justifies the validity of Assumption 4.1.
+Assumption 4.2. Set $\mathcal { X }$ is bounded with diameter $D$ , i.e., $\| \mathbf { x } - \mathbf { x } ^ { \prime } \| _ { 2 } \leq D$ for all $\mathbf { x } , \mathbf { x } ^ { \prime } \in \mathcal { X }$ .
+Assumption 4.2 implies that the input space is bounded. For common tasks such as image classification, given the fact that images have bounded pixel range and $\epsilon$ is a small constant, this assumption trivially holds.
+Now we present the theorem, which characterizes the convergence rate of our proposed Frank-Wolfe white-box adversarial attack algorithm presented in Algorithm 1.
+Theorem 4.3. Under Assumptions 4.1 and 4.2, let $\gamma _ { t } = \gamma = \sqrt { 2 ( f ( \mathbf { x } _ { 0 } ) - f ( \mathbf { x } ^ { * } ) ) / ( L D ^ { 2 } T ) } ,$ , denote $\widetilde { g } _ { T } = \mathrm { m i n } _ { 1 \leq k \leq T } g ( \mathbf { x } _ { k } )$ where $\{ { \bf x } _ { k } \} _ { k = 1 } ^ { T }$ are iterates in Algorithm 1 with $\lambda = 1$ , we have:
+$$
+\widetilde { g } _ { T } \leq \sqrt { \frac { L D ^ { 2 } ( f ( \mathbf { x } _ { 0 } ) - f ( \mathbf { x } ^ { * } ) ) } { 2 T } } ,
+$$
+where $\mathbf { x } ^ { * }$ is the optimal solution to (3.1).
+Remark 4.4. Theorem 4.3 suggests that our proposed Frank-Wolfe white-box attack algorithm√ achieves a $O ( 1 / \sqrt { T } )$ rate of convergence. Note that similar result has been proved in Lacoste-Julien (2016) under a different choice of step size.
+4.3 CONVERGENCE GUARANTEE FOR FRANK-WOLFE BLACK-BOX ATTACK
+Next we analyze the convergence of our proposed Frank-Wolfe black-box adversarial attack algorithm presented in Algorithm 2.
+In order to prove the convergence of our proposed Frank-Wolfe black-box attack algorithm, we need the following additional assumption that $\| { \bar { \nabla } } f ( \mathbf { 0 } ) \| _ { 2 }$ is bounded.
+Assumption 4.5. Gradient of $f ( \cdot )$ at zero point $\nabla f ( \mathbf { 0 } )$ satisfies $\operatorname* { m a x } _ { y } \| \nabla f ( \mathbf { 0 } ) \| _ { 2 } \leq C _ { g } .$
+Following the analysis in Shamir (2017), let $f _ { \delta } ( \mathbf { x } ) = \mathbb { E } _ { \mathbf { u } } [ f ( \mathbf { x } + \delta \mathbf { u } ) ]$ , which is the smoothed version of $f ( \mathbf { x } )$ . This smoothed function value plays a central role in our theoretical analysis, since it bridges the finite difference gradient approximation with the actual gradient. The following lemma shows this relationship.
+Lemma 4.6. For the gradient estimator $\mathbf { q } _ { t }$ in Algorithm 3, its expectation and variance satisfy
+$$
+\mathbb { E } [ \mathbf { q } _ { t } ] = \nabla f _ { \delta } ( \mathbf { x } _ { t } ) , \qquad \mathbb { E } \| \mathbf { q } _ { t } - \mathbb { E } [ \mathbf { q } _ { t } ] \| _ { 2 } ^ { 2 } \leq \frac { 1 } { b } \bigg ( 2 d ( C _ { g } + L D ) ^ { 2 } + \frac { 1 } { 2 } \delta _ { t } ^ { 2 } L ^ { 2 } d ^ { 2 } \bigg ) .
+$$
+Now we are going to present the theorem, which characterizes the convergence rate of Algorithm 2.
+Theorem 4.7. Under Assumptions 4.1, 4.2 and 4.5, let $\gamma _ { t } = \gamma = \sqrt { 2 ( f ( \mathbf { x } _ { 0 } ) - f ( \mathbf { x } ^ { * } ) ) / ( L D ^ { 2 } T ) }$ , $b = T d$ and $\delta _ { t } = \sqrt { 2 / T d ^ { 2 } }$ , suppose we use Option I in Algorithm 2 and option II for Algorithm 3, then the output $\mathbf { x } _ { a }$ from Algorithm 2 with $\lambda = 1$ satisfies:
+$$
+\mathbb { E } [ g ( \mathbf { x } _ { a } ) ] \leq \frac { D } { \sqrt { 2 T } } \Big ( \sqrt { L ( f ( \mathbf { x } _ { 0 } ) - f ( \mathbf { x } ^ { * } ) ) } + 2 ( L + C _ { g } + L D ) \Big ) ,
+$$
+where $\mathbf { x } ^ { * }$ is the optimal solution to (3.1).
+Remark 4.8. Theorem 4.7 suggests that Algorithm 2 also enjoys a $O ( 1 / \sqrt { T } )$ rate of convergence. In terms of query complexity, the total number of queries needed is $T b = T ^ { 2 } d$ , which is linear in the data dimension $d$ . In fact, in the experiment part, we observed that this number can be substantially smaller than $d$ , e.g., $b = 2 5$ , which is much lower than the theorem suggests. Note that although we only prove for option I in Algorithm 3, our result can be readily extended to Option II (the Gaussian sensing vector case).
+# 5 EXPERIMENTS
+In this section, we present the experimental results for our proposed Frank-Wolfe attack framework against other state-of-the-art adversarial attack algorithms in both white-box and black-box settings. All of our experiments are conducted on Amazon AWS ${ \tt p } 3 . 2$ xlarge servers which come with Intel Xeon E5 CPU and one NVIDIA Tesla V100 GPU (16G RAM). All experiments are implemented in Tensorflow platform version 1.10.0 within Python 3.6.4.
+# 5.1 EVALUATION SETUP AND METRICS
+We test the attack effectiveness of all algorithms by evaluating on a pre-trained Inception V3 model (Szegedy et al., 2016) and a ResNet V2 50 (He et al., 2016b) model that are trained on ImageNet dataset (Deng et al., 2009). The pre-trained Inception V3 model is reported to have a $7 8 . 0 \%$ top-1 accuracy and a $9 3 . 9 \%$ top-5 accuracy. The pre-trained ResNet V2 model is reported to have a $7 5 . 6 \%$ top-1 and a $9 2 . 8 \%$ top-5 accuracy. We randomly choose 500 images from the ImageNet validation set that are verified to be correctly classified by the pre-trained model and also randomly choose a target class for each image. Each image has a dimension of $2 9 9 \times 2 9 9 \times 3$ and we test all attack algorithms through the same randomly chosen data samples and target labels.
+We test for both $L _ { 2 }$ norm based and $L _ { \infty }$ norm based attacks. In the white-box setting, we perform binary search $/$ grid search for the best distortion parameter $\stackrel { \cdot } { \epsilon }$ in our formulation and $c$ in CW’s regularized formulation). In the black-box setting, for $L _ { 2 }$ norm based attack, we set $\epsilon = 5$ and for $L _ { \infty }$ based attack, we set $\epsilon = 0 . 0 5$ . For white-box attack, we restrict a maximum of 1, 000 iterations per attack for each method. And for black-box attack, we set a maximum query limit of 500, 000 per attack per image for each method.
+For all algorithms, we stop the algorithm when a successful attack is found. For our proposed blackbox attack, we use option II in Algorithm 2 and test both options in Algorithm 3. We set the number of gradient estimation samples $b = 2 5$ for Algorithm 2. More detailed description on parameter settings can be found in the supplemental materials.
+We evaluate the final performance through attack success rate where the success is defined as making the classifier output the exact target class label (not any incorrect labels). We also measure average attack time per image, average distortion (only on successful attacked samples) and average number of queries needed (only for black-box attack) per image. For a fair time comparison, even though some of the algorithms including ours can be written in batch form (attack multiple images at one time), all algorithms are set to attack one image at a time.
+Due to page limit, we leave all experimental results on ResNet V2 model in the supplemental materials.
+# 5.2 BASELINE METHODS
+We compare the proposed algorithms with several state-of-the-art baseline algorithms. Specifically, we compare the proposed white-box attack algorithm with 4 (i) PGD (Madry et al., 2018) (which is essentially I-FGM (Kurakin et al., 2016)), (ii) CW attack (Carlini & Wagner, 2017) and (iii) EAD attack (Chen et al., 2017b). We compare the proposed black-box attack algorithm with (i) ZOO attack (Chen et al., 2017c) and (ii) NES-PGD attack (Ilyas et al., 2018).
+# 5.3 WHITE-BOX ATTACK EXPERIMENTS
+In this subsection, we present the white-box attack experiments on Inception V3 model. Tables 1 and 2 present our experimental results for $L _ { 2 }$ norm and $L _ { \infty }$ norm based white-box attacks respectively. As we can observe from the tables, the attack success rate is $1 0 0 \%$ for every method. For the other baselines in the $L _ { 2 }$ norm case, CW method achieves the smallest average distortion, yet it comes with an expansive time cost. EAD method does not have either time advantage or distortion advantage in this experiment, probably due to its different motivation in attacking. PGD has moderate average distortion, yet it also costs quite some time to finish the attack. On the other hand, our proposed algorithm achieves the shortest attack time with moderate distortion. It significantly reduces the time complexity needed for attacking data with large dimensionality. For the $L _ { \infty }$ norm case, CW method takes significantly longer time and does not perform very well on average distortion either.
+Table 1: Comparison of $L _ { 2 }$ norm based white-box attacks on Inception V3 model with $\epsilon = 5$ . We report attack success rate, average time and average distortion.
+<table><tr><td>METHODS</td><td>SUCCESSRATE（%）</td><td>AVERAGE TIME (s)</td><td>）AVERAGEDISTORTION</td></tr><tr><td>PGD</td><td>100.0</td><td>143.2</td><td>0.74</td></tr><tr><td>CW</td><td>100.0</td><td>169.9</td><td>0.57</td></tr><tr><td>EAD</td><td>100.0</td><td>167.8</td><td>1.09</td></tr><tr><td>FW-White</td><td>100.0</td><td>50.6</td><td>0.85</td></tr></table>
+Table 2: Comparison of $L _ { \infty }$ norm based white-box attacks on Inception V3 model with $\epsilon = 0 . 0 5$ We report attack success rate, average time and average distortion.
+<table><tr><td>METHODS</td><td></td><td></td><td>SUCCESSRATE（%）AVERAGETIME(S）AVERAGEDISTORTION</td></tr><tr><td>PGD</td><td>100.0</td><td>39.1</td><td>0.0027</td></tr><tr><td>CW</td><td>100.0</td><td>745.2</td><td>0.0071</td></tr><tr><td>FW-White</td><td>100.0</td><td>13.7</td><td>0.0034</td></tr></table>
+This is largely due to the original CW was designed for $L _ { 2 }$ norm attack, and in order to apply it to $L _ { \infty }$ norm attack, special design is needed, which sacrifices its performance in terms of runtime. Again, our proposed white-box attack algorithm achieves the shortest average attack time and a moderate average distortion.
+In Figure 1, we also examine the effect of $\lambda$ in our proposed Frank-Wolfe white-box attack algorithm. We plot the objective loss function value of attacking one example against the number of iterations for both $L _ { 2 }$ and $L _ { \infty }$ based white-box attack on Inception V3 model. From the plot, we can see that larger $\lambda$ indeed leads to faster convergence.
+![](images/222c76ef0ea8abcc3ea126edf292fab6d6acac47b5236a7233e9a21a381b0142.jpg)
+Figure 1: Loss against the number of iterations plot for PGD and FW algorithms in both $L _ { 2 }$ norm and $L _ { \infty }$ norm based white-box attacks on Inception V3 model.
+# 5.4 BLACK-BOX ATTACK EXPERIMENTS
+In this subsection, we present the black-box attack experiments on Inception V3 model. For blackbox attacks, attack success rate, time and number of queries needed are more meaningful evaluation metrics than distortion distances. Therefore, we omit all the grid search / binary search steps that are used in the white-box setting since extra time $/$ queries are needed for finding parameters that can obtain better distortion distances.
+Tables 3 and 4 present our experimental results for $L _ { 2 }$ norm and $L _ { \infty }$ norm based black-box attacks respectively. For ZOO method, note that it only has the $L _ { 2 }$ norm version and it follows CW’s framework and thus uses different loss function and problem formulation (cannot exactly control the adversarial example to be within the distortion limit, we manage to keep the average distortion around $\epsilon$ for ZOO while other methods have average distortions very close to $\epsilon$ ). Furthermore, we can observe that ZOO is quite slow in this task. Attack on a single image can take up to 2 hours for ZOO and it is only able to achieve a $7 4 . 8 \%$ success rate (compared with the $8 8 . 9 \%$ success rate in the original paper, we think the main reason is the query limit here is only half of the query limit in the original paper). NES-PGD method, while greatly improving ZOO’s performance, still cannot achieve $1 0 0 \%$ success rate in both attack models and takes relatively more time and queries. In sharp contrast, our proposed Frank-Wolfe black-box attacks (both option I and option II) achieve the highest success rate in both $L _ { 2 }$ norm and $L _ { \infty }$ norm based black-box attacks and further largely improve the attack efficiency.
+![](images/6a7f50fecaff02fa57ee962370d5cdac5ab37708bafa9897544631eaee69cc0b.jpg)
+Figure 2: Attack success rate against the number of queries plot for different black-box attack algorithms in both $L _ { 2 }$ norm $L _ { \infty }$ norm cases on Inception V3 model.
+Table 3: Comparison of $L _ { 2 }$ norm based black-box attacks on Inception V3 model with $\epsilon = 5$ . We report attack success rate, average time and average number of queries needed per image. Opt I and Opt II refer to the two options in Algorithm 2.
+Table 4: Comparison of $L _ { \infty }$ norm based black-box attacks on Inception V3 model with $\epsilon = 0 . 0 5$ . We report attack success rate, average time and average number of queries needed per image. Opt I and Opt II refer to the two options in Algorithm 2.
+<table><tr><td>METHODS</td><td>SUCCESS RATE（%)</td><td>AVERAGE TIME (s)</td><td>AVERAGEQUERIES</td></tr><tr><td>ZOO</td><td>74.8</td><td>5692.6</td><td>296867.0</td></tr><tr><td>NES-PGD</td><td>96.7</td><td>133.0</td><td>58921.8</td></tr><tr><td>FW-Black (Opt I)</td><td>100.0</td><td>102.9</td><td>45994.5</td></tr><tr><td>FW-Black (Opt II)</td><td>100.0</td><td>100.9</td><td>45156.0</td></tr></table>
+<table><tr><td>METHODS</td><td>SUCCESSRATE(%)</td><td>AVERAGETIME (s)</td><td>AVERAGEQUERIES</td></tr><tr><td>NES-PGD</td><td>98.0</td><td>76.9</td><td>34062.2</td></tr><tr><td>FW-Black (Opt I)</td><td>100.0</td><td>50.4</td><td>22313.2</td></tr><tr><td>FW-Black (Opt II)</td><td>100.0</td><td>50.6</td><td>22424.1</td></tr></table>
+Figure 2 illustrates the attack success rate against the number of queries plot for different algorithms in both $L _ { 2 }$ norm and $L _ { \infty }$ norm based black-box attacks on Inception V3 model. As we can see from the plot, our proposed Frank-Wolfe black-box attack algorithm (both options) achieves the highest attack success rate and best efficiency (least queries needed for achieving the same success rate), especially in the $L _ { 2 }$ norm case.
+# 6 CONCLUSIONS
+In this work, we propose a Frank-Wolfe framework for efficient and effective adversarial attacks. Our proposed white-box and black-box attack algorithms enjoy an $O ( 1 / \sqrt { T } )$ rate of convergence, and the query complexity of the proposed black-box attack algorithm is linear in data dimension $d$ . Finally, our empirical study on attacking Inception V3 model with ImageNet dataset yields a $1 0 0 \%$ attack success rate for our proposed algorithms, even in the setting of black-box attack.
+# REFERENCES
+Dzmitry Bahdanau, Jan Chorowski, Dmitriy Serdyuk, Philemon Brakel, and Yoshua Bengio. Endto-end attention-based large vocabulary speech recognition. In Acoustics, Speech and Signal Processing (ICASSP), 2016 IEEE International Conference on, pp. 4945–4949. IEEE, 2016.
+Krishnakumar Balasubramanian and Saeed Ghadimi. Zeroth-order (non)-convex stochastic optimization via conditional gradient and gradient updates. arXiv preprint arXiv:1809.06474, 2018.
+Arjun Nitin Bhagoji, Warren He, Bo Li, and Dawn Song. Exploring the space of black-box attacks on deep neural networks. arXiv preprint arXiv:1712.09491, 2017.
+Nicholas Carlini and David Wagner. Towards evaluating the robustness of neural networks. In 2017 IEEE Symposium on Security and Privacy $( S P )$ , pp. 39–57. IEEE, 2017.
+Nicholas Carlini, Pratyush Mishra, Tavish Vaidya, Yuankai Zhang, Micah Sherr, Clay Shields, David Wagner, and Wenchao Zhou. Hidden voice commands. In USENIX Security Symposium, pp. 513– 530, 2016.
+Hongge Chen, Huan Zhang, Pin-Yu Chen, Jinfeng Yi, and Cho-Jui Hsieh. Show-and-fool: Crafting adversarial examples for neural image captioning. arXiv preprint arXiv:1712.02051, 2017a.
+Pin-Yu Chen, Yash Sharma, Huan Zhang, Jinfeng Yi, and Cho-Jui Hsieh. Ead: elastic-net attacks to deep neural networks via adversarial examples. arXiv preprint arXiv:1709.04114, 2017b.
+Pin-Yu Chen, Huan Zhang, Yash Sharma, Jinfeng Yi, and Cho-Jui Hsieh. Zoo: Zeroth order optimization based black-box attacks to deep neural networks without training substitute models. In Proceedings of the 10th ACM Workshop on Artificial Intelligence and Security, pp. 15–26. ACM, 2017c.
+Minhao Cheng, Jinfeng Yi, Huan Zhang, Pin-Yu Chen, and Cho-Jui Hsieh. Seq2sick: Evaluating the robustness of sequence-to-sequence models with adversarial examples. arXiv preprint arXiv:1803.01128, 2018.
+Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In Computer Vision and Pattern Recognition, 2009. CVPR 2009. IEEE Conference on, pp. 248–255. Ieee, 2009.
+Abraham D Flaxman, Adam Tauman Kalai, and H Brendan McMahan. Online convex optimization in the bandit setting: gradient descent without a gradient. In Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms, pp. 385–394. Society for Industrial and Applied Mathematics, 2005.
+Marguerite Frank and Philip Wolfe. An algorithm for quadratic programming. Naval research logistics quarterly, 3(1-2):95–110, 1956.
+Xiang Gao, Bo Jiang, and Shuzhong Zhang. On the information-adaptive variants of the admm: an iteration complexity perspective. Journal of Scientific Computing, 76(1):327–363, 2018.
+Ross Girshick. Fast r-cnn. In Proceedings of the IEEE international conference on computer vision, pp. 1440–1448, 2015.
+Ian J Goodfellow, Jonathon Shlens, and Christian Szegedy. Explaining and harnessing adversarial examples. ICLR, 2015.
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In CVPR, pp. 770–778, 2016a.
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Identity mappings in deep residual networks. In European Conference on Computer Vision, pp. 630–645. Springer, 2016b.
+Weiwei Hu and Ying Tan. Generating adversarial malware examples for black-box attacks based on gan. arXiv preprint arXiv:1702.05983, 2017.
+Andrew Ilyas, Logan Engstrom, Anish Athalye, Jessy Lin, Anish Athalye, Logan Engstrom, Andrew Ilyas, and Kevin Kwok. Black-box adversarial attacks with limited queries and information. In Proceedings of the 35th International Conference on Machine Learning, $\{ I C M L \}$ 2018, 2018.
+Martin Jaggi. Revisiting frank-wolfe: Projection-free sparse convex optimization. In ICML (1), pp. 427–435, 2013.
+Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In Advances in neural information processing systems, pp. 1097–1105, 2012.
+Alexey Kurakin, Ian Goodfellow, and Samy Bengio. Adversarial examples in the physical world. arXiv preprint arXiv:1607.02533, 2016.
+Simon Lacoste-Julien. Convergence rate of frank-wolfe for non-convex objectives. arXiv preprint arXiv:1607.00345, 2016.
+Yanpei Liu, Xinyun Chen, Chang Liu, and Dawn Song. Delving into transferable adversarial examples and black-box attacks. International Conference on Data Mining (ICDM), 2018.
+Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and Adrian Vladu. Towards deep learning models resistant to adversarial attacks. International Conference on Learning Representations, 2018.
+Abdel-rahman Mohamed, George E Dahl, Geoffrey Hinton, et al. Acoustic modeling using deep belief networks. IEEE Trans. Audio, Speech & Language Processing, 20(1):14–22, 2012.
+Seyed-Mohsen Moosavi-Dezfooli, Alhussein Fawzi, and Pascal Frossard. Deepfool: a simple and accurate method to fool deep neural networks. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2574–2582, 2016.
+Nicolas Papernot, Patrick McDaniel, and Ian Goodfellow. Transferability in machine learning: from phenomena to black-box attacks using adversarial samples. arXiv preprint arXiv:1605.07277, 2016a.
+Nicolas Papernot, Patrick McDaniel, Somesh Jha, Matt Fredrikson, Z Berkay Celik, and Ananthram Swami. The limitations of deep learning in adversarial settings. In Security and Privacy (EuroS&P), 2016 IEEE European Symposium on, pp. 372–387. IEEE, 2016b.
+Nicolas Papernot, Patrick McDaniel, Ian Goodfellow, Somesh Jha, Z Berkay Celik, and Ananthram Swami. Practical black-box attacks against machine learning. In Proceedings of the 2017 ACM on Asia Conference on Computer and Communications Security, pp. 506–519. ACM, 2017.
+Anay Pattanaik, Zhenyi Tang, Shuijing Liu, Gautham Bommannan, and Girish Chowdhary. Robust deep reinforcement learning with adversarial attacks. In Proceedings of the 17th International Conference on Autonomous Agents and MultiAgent Systems, pp. 2040–2042. International Foundation for Autonomous Agents and Multiagent Systems, 2018.
+Sashank J Reddi, Suvrit Sra, Barnabas P ´ oczos, and Alex Smola. Stochastic frank-wolfe methods ´ for nonconvex optimization. In Communication, Control, and Computing (Allerton), 2016 54th Annual Allerton Conference on, pp. 1244–1251. IEEE, 2016.
+Shaoqing Ren, Kaiming He, Ross Girshick, and Jian Sun. Faster r-cnn: Towards real-time object detection with region proposal networks. In Advances in neural information processing systems, pp. 91–99, 2015.
+Tim Salimans, Jonathan Ho, Xi Chen, Szymon Sidor, and Ilya Sutskever. Evolution strategies as a scalable alternative to reinforcement learning. arXiv preprint arXiv:1703.03864, 2017.
+Shibani Santurkar, Dimitris Tsipras, Andrew Ilyas, and Aleksander Madry. How does batch normalization help optimization?(no, it is not about internal covariate shift). arXiv preprint arXiv:1805.11604, 2018.
+Ohad Shamir. An optimal algorithm for bandit and zero-order convex optimization with two-point feedback. Journal of Machine Learning Research, 18(52):1–11, 2017.
+Aman Sinha, Hongseok Namkoong, and John Duchi. Certifying some distributional robustness with principled adversarial training. International Conference on Learning Representations, 2018.
+Matthew Staib and Stefanie Jegelka. Distributionally robust deep learning as a generalization of adversarial training. Machine Learning and Computer Security Workshop, 2017.
+Christian Szegedy, Wojciech Zaremba, Ilya Sutskever, Joan Bruna, Dumitru Erhan, Ian Goodfellow, and Rob Fergus. Intriguing properties of neural networks. arXiv preprint arXiv:1312.6199, 2013.
+Christian Szegedy, Vincent Vanhoucke, Sergey Ioffe, Jon Shlens, and Zbigniew Wojna. Rethinking the inception architecture for computer vision. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 2818–2826, 2016.
+Chun-Chen Tu, Pai-Shun Ting, Pin-Yu Chen, Sijia Liu, Huan Zhang, Jinfeng Yi, Cho-Jui Hsieh, and Shin-Ming Cheng. Autozoom: Autoencoder-based zeroth order optimization method for attacking black-box neural networks. CoRR, abs/1805.11770, 2018.
+Daan Wierstra, Tom Schaul, Tobias Glasmachers, Yi Sun, Jan Peters, and Jurgen Schmidhuber. ¨ Natural evolution strategies. The Journal of Machine Learning Research, 15(1):949–980, 2014.
+Xiaojun Xu, Xinyun Chen, Chang Liu, Anna Rohrbach, Trevor Darell, and Dawn Song. Can you fool ai with adversarial examples on a visual turing test? arXiv preprint arXiv:1709.08693, 2017.
+Yaoliang Yu, Xinhua Zhang, and Dale Schuurmans. Generalized conditional gradient for sparse estimation. The Journal of Machine Learning Research, 18(1):5279–5324, 2017.
+# A LINEAR MINIMIZATION ORACLE (LMO) SOLUTIONS
+Denote $\mathbf { u } = ( \mathbf { v } - \mathbf { x _ { \mathrm { o r i } } } ) / ( \lambda \epsilon )$ , the linear minimization problem can be written as
+$$
+\begin{array} { r l } & { \underset { | | \mathbf { v } - \mathbf { x } _ { \mathrm { o r i } } | | _ { p } \leq \lambda \epsilon } { \mathrm { m i n } } \langle \mathbf { v } , \nabla f ( \mathbf { x } _ { t } ) \rangle = \underset { | | \mathbf { u } | | _ { p } \leq 1 } { \mathrm { m i n } } \lambda \epsilon \cdot \langle \mathbf { u } , \nabla f ( \mathbf { x } _ { t } ) \rangle } \\ & { = \underset { | | \mathbf { u } | | _ { p } \leq 1 } { \mathrm { m a x } } \lambda \epsilon \cdot \langle \mathbf { u } , - \nabla f ( \mathbf { x } _ { t } ) \rangle } \\ & { = \lambda \epsilon \cdot \| \nabla f ( \mathbf { x } _ { t } ) \| _ { p * } , } \end{array}
+$$
+where $\| \cdot \| _ { p * }$ denotes the dual norm of $\| \cdot \| _ { p }$ . For $p = 2$ case, we have
+$$
+\langle ( \mathbf { v } - \mathbf { x } _ { \mathrm { o r i } } ) / ( \lambda \epsilon ) , - \nabla f ( \mathbf { x } _ { t } ) \rangle = \| \nabla f ( \mathbf { x } _ { t } ) \| _ { 2 } .
+$$
+It immediately implies that
+$$
+\mathbf { v } = - \frac { \lambda \boldsymbol { \epsilon } \cdot \nabla f ( \mathbf { x } _ { t } ) } { \| \nabla f ( \mathbf { x } _ { t } ) \| _ { 2 } } + \mathbf { x } _ { \mathrm { o r i } } .
+$$
+For $p = \infty$ case, we have
+$$
+\langle ( \mathbf { v } - \mathbf { x } _ { \mathrm { o r i } } ) / ( \lambda \epsilon ) , - \nabla f ( \mathbf { x } _ { t } ) \rangle = \| \nabla f ( \mathbf { x } _ { t } ) \| _ { 1 } .
+$$
+It immediately implies that
+$$
+\mathbf { v } = - \lambda \epsilon \cdot \mathrm { s i g n } ( \nabla f ( \mathbf { x } _ { t } ) ) + \mathbf { x } _ { \mathrm { o r i } } .
+$$
+For the ease of comparison, we show the full update formula (before final projection step) for our algorithm. In detail, for $p = \infty$ case, our algorithm takes the following update formulate:
+$$
+\begin{array} { r l } & { \mathbf { x } _ { t + 1 } = ( 1 - \gamma _ { t } ) \mathbf { x } _ { t } + \gamma _ { t } \mathbf { v } _ { t } } \\ & { \qquad = ( 1 - \gamma _ { t } ) \mathbf { x } _ { t } - \lambda \gamma _ { t } \boldsymbol { \epsilon } \cdot \mathrm { s i g n } ( \nabla f ( \mathbf { x } _ { t } ) ) + \gamma _ { t } \cdot \mathbf { x } _ { \mathrm { o r i } } } \\ & { \qquad = \mathbf { x } _ { t } - \lambda \gamma _ { t } \boldsymbol { \epsilon } \cdot \mathrm { s i g n } ( \nabla f ( \mathbf { x } _ { t } ) ) - \gamma _ { t } ( \mathbf { x } _ { t } - \mathbf { x } _ { \mathrm { o r i } } ) , } \end{array}
+$$
+and for $p = 2$ case, it takes
+$$
+\mathbf { x } _ { t + 1 } = \mathbf { x } _ { t } - \lambda \gamma _ { t } \epsilon \cdot \frac { \nabla f ( \mathbf { x } _ { t } ) } { \| \nabla f ( \mathbf { x } _ { t } ) \| _ { 2 } } - \gamma _ { t } ( \mathbf { x } _ { t } - \mathbf { x } _ { \mathrm { o r i } } ) .
+$$
+Compared with PGD, the full update (before final projection step) of Frank-Wolfe white-box attack includes an extra parameter $\lambda$ before the normalized gradient, as well as an extra term $\left( \mathbf { x } _ { t } - \mathbf { x } _ { \mathrm { o r i } } \right)$ . This difference makes the behavior of Frank-Wolfe based attacks different from that of PGD based attacks.
+# B PROOF OF THE MAIN THEORY IN SECTION 4
+B.1 PROOF OF THEOREM 4.3
+Proof. For simplicity, we denote $f ( \mathbf { x } _ { t } )$ by $f ( \mathbf { x } _ { t } )$ for the rest of the proof. First by Assumption 4.1, we have
+$$
+\begin{array} { r l } & { f ( \mathbf { x } _ { t + 1 } ) \leq f ( \mathbf { x } _ { t } ) + \nabla f ( \mathbf { x } _ { t } ) ^ { \top } ( \mathbf { x } _ { t + 1 } - \mathbf { x } _ { t } ) + \displaystyle \frac { L } { 2 } \| \mathbf { x } _ { t + 1 } - \mathbf { x } _ { t } \| _ { 2 } ^ { 2 } } \\ & { \qquad = f ( \mathbf { x } _ { t } ) + \gamma \nabla f ( \mathbf { x } _ { t } ) ^ { \top } ( \mathbf { v } _ { t } - \mathbf { x } _ { t } ) + \displaystyle \frac { L \gamma ^ { 2 } } { 2 } \| \mathbf { v } _ { t } - \mathbf { x } _ { t } \| _ { 2 } ^ { 2 } } \\ & { \qquad \leq f ( \mathbf { x } _ { t } ) + \gamma \nabla f ( \mathbf { x } _ { t } ) ^ { \top } ( \mathbf { v } _ { t } - \mathbf { x } _ { t } ) + \displaystyle \frac { L D ^ { 2 } \gamma ^ { 2 } } { 2 } , } \end{array}
+$$
+where the last inequality uses the bounded domain condition in Assumption 4.2. Note that by definition of the Frank-Wolfe gap, we have
+$$
+f ( \mathbf { x } _ { t + 1 } ) \leq f ( \mathbf { x } _ { t } ) - \gamma g ( \mathbf { x } _ { t } ) + \frac { L D ^ { 2 } \gamma ^ { 2 } } { 2 } .
+$$
+Summation over $t$ of the above inequality, we obtain
+$$
+\begin{array} { l } { f ( \mathbf { x } _ { T } ) \leq f ( \mathbf { x } _ { 0 } ) - { \displaystyle \sum _ { k = 0 } ^ { T - 1 } } \gamma g ( \mathbf { x } _ { k } ) + \frac { T L D ^ { 2 } \gamma ^ { 2 } } { 2 } } \\ { \leq f ( \mathbf { x } _ { 0 } ) - \gamma T \widetilde { g } _ { T } + \frac { T L D ^ { 2 } \gamma ^ { 2 } } { 2 } , } \end{array}
+$$
+where the second inequality follows from the definition of $\widetilde { g } _ { t }$ . Note that by optimality we easily have $f ( \mathbf { x } _ { t + 1 } ) \geq f ( \mathbf { x } ^ { * } )$ e. Rearrange the above inequality we have
+$$
+\begin{array} { r } { \widetilde { g } _ { T } \le \displaystyle \frac { f ( \mathbf { x } _ { 0 } ) - f ( \mathbf { x } ^ { * } ) } { T \gamma } + \displaystyle \frac { L D ^ { 2 } \gamma } { 2 } } \\ { \le \sqrt { \displaystyle \frac { L D ^ { 2 } ( f ( \mathbf { x } _ { 0 } ) - f ( \mathbf { x } ^ { * } ) ) } { 2 T } } , } \end{array}
+$$
+where the second inequality is achieved when $\gamma = \sqrt { 2 ( f ( \mathbf { x } _ { 0 } ) - f ( \mathbf { x } ^ { * } ) ) / ( L D ^ { 2 } T ) }$ .
+# B.2 PROOF OF LEMMA 4.6
+Proof. For simplicity we denote $f ( \cdot )$ by $f ( \cdot )$ for the rest of the proof. Let us denote $\psi _ { i } ~ =$ $\begin{array} { r } { \frac { d } { 2 \delta _ { t } b } \big ( f ( \mathbf { x } _ { t } + \delta _ { t } \mathbf { u } _ { i } ) - f ( \mathbf { x } _ { t } - \delta _ { t } \mathbf { u } _ { i } ) \big ) \mathbf { u } _ { i } } \end{array}$ . For the first part, we have
+$$
+\begin{array} { l } { \displaystyle \mathbb { E } [ \psi _ { i } ] = \mathbb { E } _ { \mathbf { u } } \bigg [ \frac { d } { 2 \delta _ { t } b } \big ( f ( \mathbf { x } _ { t } + \delta _ { t } \mathbf { u } _ { i } ) - f ( \mathbf { x } _ { t } - \delta _ { t } \mathbf { u } _ { i } ) \big ) \mathbf { u } _ { i } \bigg ] } \\ { = \mathbb { E } _ { \mathbf { u } } \bigg [ \frac { d } { 2 \delta _ { t } b } f ( \mathbf { x } _ { t } + \delta _ { t } \mathbf { u } _ { i } ) \mathbf { u } _ { i } \bigg ] + \mathbb { E } _ { \mathbf { u } } \bigg [ \frac { d } { 2 \delta _ { t } b } f ( \mathbf { x } _ { t } - \delta _ { t } \mathbf { u } _ { i } ) ( - \mathbf { u } _ { i } ) \bigg ] } \\ { = \mathbb { E } _ { \mathbf { u } } \bigg [ \frac { d } { \delta _ { t } b } f ( \mathbf { x } _ { t } + \delta _ { t } \mathbf { u } _ { i } ) \mathbf { u } _ { i } \bigg ] } \\ { = \frac { 1 } { b } \nabla f _ { \delta } ( \mathbf { x } _ { t } ) , } \end{array}
+$$
+where the third equality holds due to symmetric property of $\mathbf { u } _ { i }$ and the last equality follows from Lemma 4.1(a) in Gao et al. (2018). Therefore, we have
+$$
+\mathbb { E } [ \mathbf { q } _ { t } ] = \mathbb { E } \bigg [ \sum _ { i = 1 } ^ { b } \psi _ { i } \bigg ] = \nabla f _ { \delta } ( \mathbf { x } _ { t } ) .
+$$
+For second part, note that $\psi _ { i }$ ’s are independent from each other due to the independence of $\mathbf { u } _ { i }$ , we have
+$$
+\mathbb { E } \Vert \mathbf { q } _ { t } - \mathbb { E } [ \mathbf { q } _ { t } ] \Vert _ { 2 } ^ { 2 } = \mathbb { E } \bigg \Vert \sum _ { i = 1 } ^ { b } \Big [ \psi _ { i } - \mathbb { E } \psi _ { i } \Big ] \bigg \Vert _ { 2 } ^ { 2 } = \sum _ { i = 1 } ^ { b } \mathbb { E } \big \Vert \psi _ { i } - \mathbb { E } \psi _ { i } \big \Vert ^ { 2 } \leq \sum _ { i = 1 } ^ { b } \mathbb { E } \big \Vert \psi _ { i } \big \Vert ^ { 2 } .
+$$
+Now take a look at $\mathbb { E } { \left. { \psi _ { i } } \right. } ^ { 2 }$
+$$
+\begin{array} { r l } & { \mathbb { E } \Big \| \psi _ { i } \Big \| ^ { 2 } = \mathbb { E } _ { \mathbf { u } } \Bigg \| \frac { d } { 2 \delta _ { t } b } \big ( f ( \mathbf { x } _ { t } + \delta _ { t } \mathbf { u } _ { i } ) - f ( \mathbf { x } _ { t } ) + f ( \mathbf { x } _ { t } ) - f ( \mathbf { x } _ { t } - \delta _ { t } \mathbf { u } _ { i } ) \big ) \mathbf { u } _ { i } \Bigg \| _ { 2 } ^ { 2 } } \\ & { \qquad \leq \frac { 1 } { 2 b ^ { 2 } } \mathbb { E } _ { \mathbf { u } } \Bigg \| \frac { d } { \delta _ { t } } \big ( f ( \mathbf { x } _ { t } + \delta _ { t } \mathbf { u } _ { i } ) - f ( \mathbf { x } _ { t } ) \big ) \mathbf { u } _ { i } \Bigg \| _ { 2 } ^ { 2 } + \frac { 1 } { 2 b ^ { 2 } } \mathbb { E } _ { \mathbf { u } } \Bigg \| \frac { d } { \delta _ { t } } \big ( f ( \mathbf { x } _ { t } ) - f ( \mathbf { x } _ { t } - \delta _ { t } \mathbf { u } _ { i } ) \big ) \mathbf { u } _ { i } \Bigg \| _ { 2 } ^ { 2 } } \\ & { \qquad = \frac { 1 } { b ^ { 2 } } \mathbb { E } _ { \mathbf { u } } \Bigg \| \frac { d } { \delta _ { t } } \big ( f ( \mathbf { x } _ { t } + \delta _ { t } \mathbf { u } _ { i } ) - f ( \mathbf { x } _ { t } ) \big ) \mathbf { u } _ { i } \Bigg \| _ { 2 } ^ { 2 } } \\ & { \qquad \leq \frac { 1 } { b ^ { 2 } } \bigg ( 2 d \| \nabla f ( \mathbf { x } _ { t } ) \| _ { 2 } ^ { 2 } + \frac { 1 } { 2 } \delta _ { t } ^ { 2 } L ^ { 2 } d ^ { 2 } \bigg ) , } \end{array}
+$$
+where the first inequality is due to the fact that $( a + b ) ^ { 2 } \leq 2 a ^ { 2 } + 2 b ^ { 2 }$ , the second equality follows from the symmetric property of $\mathbf { u } _ { i }$ and the last inequality is by Lemma 4.1(b) in Gao et al. (2018). Also note that by Assumption 4.1 and 4.5 we have
+$$
+\| \nabla f ( \mathbf { x } _ { t } ) \| _ { 2 } ^ { 2 } \leq ( \| \nabla f ( \mathbf { 0 } ) ) \| _ { 2 } + L \| \mathbf { x } _ { t } \| _ { 2 } ) ^ { 2 } \leq ( C _ { g } + L D ) ^ { 2 } .
+$$
+Combine all above results, we obtain
+$$
+\mathbb { E } \| \mathbf { q } _ { t } - \mathbb { E } [ \mathbf { q } _ { t } ] \| _ { 2 } ^ { 2 } \leq \frac { 1 } { b } \bigg ( 2 d ( C _ { g } + L D ) ^ { 2 } + \frac { 1 } { 2 } \delta _ { t } ^ { 2 } L ^ { 2 } d ^ { 2 } \bigg ) .
+$$
+# B.3 PROOF OF THEOREM 4.7
+Proof. For simplicity we denote $f ( \mathbf { x } _ { t } )$ by $f ( \mathbf { x } _ { t } )$ for the rest of the proof. First by Assumption 4.1, we have
+$$
+\begin{array} { l } { f ( { \mathbf x } _ { t + 1 } ) \leq f ( { \mathbf x } _ { t } ) + \nabla f ( { \mathbf x } _ { t } ) ^ { \top } ( { \mathbf x } _ { t + 1 } - { \mathbf x } _ { t } ) + \displaystyle \frac { L } { 2 } \| { \mathbf x } _ { t + 1 } - { \mathbf x } _ { t } \| _ { 2 } ^ { 2 } } \\ { = f ( { \mathbf x } _ { t } ) + \gamma \nabla f ( { \mathbf x } _ { t } ) ^ { \top } ( { \mathbf v } _ { t } - { \mathbf x } _ { t } ) + \displaystyle \frac { L \gamma ^ { 2 } } { 2 } \| { \mathbf v } _ { t } - { \mathbf x } _ { t } \| _ { 2 } ^ { 2 } } \\ { \leq f ( { \mathbf x } _ { t } ) + \gamma \nabla f ( { \mathbf x } _ { t } ) ^ { \top } ( { \mathbf v } _ { t } - { \mathbf x } _ { t } ) + \displaystyle \frac { L D ^ { 2 } \gamma ^ { 2 } } { 2 } } \\ { = f ( { \mathbf x } _ { t } ) + \gamma { \mathbf q } _ { t } ^ { \top } ( { \mathbf v } _ { t } - { \mathbf x } _ { t } ) + \gamma ( \nabla f ( { \mathbf x } _ { t } ) - { \mathbf q } _ { t } ) ^ { \top } ( { \mathbf v } _ { t } - { \mathbf x } _ { t } ) + \displaystyle \frac { L D ^ { 2 } \gamma ^ { 2 } } { 2 } , } \end{array}
+$$
+where the second inequality uses the bounded domain condition in Assumption 4.2. Now define an auxiliary quantity:
+$$
+\widehat { \mathbf { v } } _ { t } = \underset { \mathbf { v } \in \mathcal { X } } { \mathrm { a r g m i n } } \langle \mathbf { v } , \nabla f ( \mathbf { x } _ { t } ) \rangle .
+$$
+According to the definition of $g ( \mathbf { x } _ { t } )$ , this immediately implies
+$$
+g ( \mathbf x _ { t } ) = \langle \widehat { \mathbf v } _ { t } , \nabla f ( \mathbf x _ { t } ) \rangle .
+$$
+Then we further have
+$$
+\begin{array} { r l } & { f ( { \mathbf x } _ { t + 1 } ) \leq f ( { \mathbf x } _ { t } ) + \gamma { \mathbf q } _ { t } ^ { \top } ( \widehat { \mathbf v } _ { t } - { \mathbf x } _ { t } ) + \gamma ( \nabla f ( { \mathbf x } _ { t } ) - { \mathbf q } _ { t } ) ^ { \top } ( { \mathbf v } _ { t } - { \mathbf x } _ { t } ) + \frac { L D ^ { 2 } \gamma ^ { 2 } } { 2 } } \\ & { \qquad = f ( { \mathbf x } _ { t } ) + \gamma \nabla f ( { \mathbf x } _ { t } ) ^ { \top } ( \widehat { \mathbf v } _ { t } - { \mathbf x } _ { t } ) + \gamma ( \nabla f ( { \mathbf x } _ { t } ) - { \mathbf q } _ { t } ) ^ { \top } ( { \mathbf v } _ { t } - \widehat { \mathbf v } _ { t } ) + \frac { L D ^ { 2 } \gamma ^ { 2 } } { 2 } } \\ & { \qquad = f ( { \mathbf x } _ { t } ) - \gamma g ( { \mathbf x } _ { t } ) + \gamma ( \nabla f ( { \mathbf x } _ { t } ) - { \mathbf q } _ { t } ) ^ { \top } ( { \mathbf v } _ { t } - \widehat { \mathbf v } _ { t } ) + \frac { L D ^ { 2 } \gamma ^ { 2 } } { 2 } } \\ & { \qquad \leq f ( { \mathbf x } _ { t } ) - \gamma g ( { \mathbf x } _ { t } ) + \gamma D \cdot \Vert \nabla f ( { \mathbf x } _ { t } ) - { \mathbf q } _ { t } \Vert _ { 2 } + \frac { L D ^ { 2 } \gamma ^ { 2 } } { 2 } , } \end{array}
+$$
+where the first inequality follows from the optimally of $\mathbf { v } _ { t }$ in Algorithm 2 and the last inequality holds due to Cauchy-Schwarz inequality. Take expectations for both sides of the above inequality, we have
+$$
+\begin{array} { r l } & { \mathbb { E } [ f ( \mathbf { x } _ { + 1 } ) ] } \\ & { \leq \mathbb { E } [ f ( \mathbf { x } _ { + 1 } ) ] - \gamma \mathbb { E } [ g ( \mathbf { x } _ { s } ) ] + \gamma D \cdot \mathbb { E } \| \nabla f ( \mathbf { x } _ { s } ) - \mathbf { q } _ { \mathrm { t } } \| _ { 2 } + \frac { L D ^ { 2 } \gamma ^ { 2 } } { 2 } } \\ & { \leq \mathbb { E } [ f ( \mathbf { x } _ { s } ) ] - \gamma \| \mathcal { D } [ g ( \mathbf { x } _ { s } ) ] + \gamma D \cdot \big ( \| \nabla f ( \mathbf { x } _ { s } ) - \mathbf { E } ( \mathbf { y } _ { \mathrm { t } } ) \| _ { 2 } + \mathbb { E } \| \mathbf { q } _ { s } - \mathbb { E } [ \mathbf { q } _ { \mathrm { t } } ] \| _ { 2 } \big ) + \frac { L D ^ { 2 } \gamma ^ { 2 } } { 2 } , } \\ & { \leq \mathbb { E } [ f ( \mathbf { x } _ { s } ) ] - \gamma \| \mathcal { D } [ g ( \mathbf { x } _ { s } ) ] + \gamma D \cdot \Big ( \| \nabla f ( \mathbf { x } _ { s } ) - \mathbf { B } [ \mathbf { q } _ { \mathrm { t } } ] \| _ { 2 } + \sqrt { \mathbb { E } \| \mathbf { q } _ { s } - \mathbb { B } [ \mathbf { q } _ { \mathrm { t } } ] \| _ { 2 } } \Big ) + \frac { L D ^ { 2 } \gamma ^ { 2 } } { 2 } } \\ & { \leq \mathbb { E } [ f ( \mathbf { x } _ { s } ) ] - \gamma \| \mathcal { D } [ g ( \mathbf { x } _ { s } ) ] + \gamma D \cdot \bigg ( \| \nabla f ( \mathbf { x } _ { s } ) - \mathbf { B } [ \mathbf { q } _ { \mathrm { t } } ] \| _ { 2 } + \sqrt { \mathbb { E } \| \mathbf { q } _ { s } - \mathbb { B } [ \mathbf { q } _ { \mathrm { t } } ] \| _ { 2 } ^ { 2 } } \Big ) + \frac { L D ^ { 2 } \gamma ^ { 2 } } { 2 } } \\ &  \leq \mathbb { E } [ f ( \mathbf { x } _ { s } ) ] - \gamma \mathbb { E } [ g ( \mathbf { x } _ { s } ) ] + \gamma D \cdot ( \| \nabla f ( \mathbf { x } _ { s } ) - \nabla f ( \mathbf { x } _ { s } ) \| _ { 2 } + \sqrt  \frac  4 d ( C _ { s } + L D ) ^ { 2 } + \ \end{array}
+$$
+where the second inequality follows from triangle inequality, the third inequality is due to Jenson’s inequality and the last inequality holds due to Lemma 4.6.
+Summation over $t$ of the above inequality, we obtain
+$$
+\begin{array} { r l } & { \mathbb { E } [ f ( \mathbf { x } _ { T } ) ] } \\ & { \leq f ( \mathbf { x } _ { 0 } ) - \displaystyle \sum _ { t = 0 } ^ { T - 1 } \gamma \mathbb { E } [ g ( \mathbf { x } _ { t } ) ] + \gamma D T \bigg ( \frac { \delta _ { t } L d } { 2 } + \frac { 2 \sqrt { d } ( C _ { g } + L D ) + \delta _ { t } L d } { \sqrt { 2 b } } \bigg ) + \frac { T L D ^ { 2 } \gamma ^ { 2 } } { 2 } } \\ & { \leq f ( \mathbf { x } _ { 0 } ) - \gamma T g _ { a } + \gamma D T \bigg ( \frac { \delta _ { t } L d } { 2 } + \frac { 2 \sqrt { d } ( C _ { g } + L D ) + \delta _ { t } L d } { \sqrt { 2 b } } \bigg ) + \frac { T L D ^ { 2 } \gamma ^ { 2 } } { 2 } , } \end{array}
+$$
+where the second inequality follows from the definition of $\widetilde { g } _ { a }$ . Note that by the zeroth-order optimality, we have $f ( \mathbf { x } _ { t + 1 } ) \geq f ( \mathbf { x } ^ { * } )$ e. Rearrange the above inequality we obtain
+$$
+\begin{array} { l } { \displaystyle \mathbb { E } [ g _ { a } ] \leq \frac { f ( \mathbf { x } _ { 0 } ) - f ( \mathbf { x } ^ { * } ) } { T \gamma } + \frac { L D ^ { 2 } \gamma } { 2 } + D \bigg ( \frac { \delta _ { t } L d } { 2 } + \frac { 2 \sqrt { d } ( C _ { g } + L D ) + \delta _ { t } L d } { \sqrt { 2 b } } \bigg ) } \\ { \displaystyle \qquad \leq \frac { D } { \sqrt { 2 T } } \Big ( \sqrt { L ( f ( \mathbf { x } _ { 0 } ) - f ( \mathbf { x } ^ { * } ) ) } + 2 ( L + C _ { g } + L D ) \Big ) , } \end{array}
+$$
+where the second inequality is achieved by setting $\gamma = \sqrt { 2 ( f ( \mathbf { x } _ { 0 } ) - f ( \mathbf { x } ^ { * } ) ) / ( L D ^ { 2 } T ) }$ , $b = T d$ and $\delta _ { t } = \sqrt { 2 / T d ^ { 2 } }$ .
+# C PARAMETERS SETTINGS FOR SECTION 5
+For Frank-Wolfe white-box attack algorithm, we list the parameters we use in Section 5 at Table 5.
+Table 5: Parameters used in Frank-Wolfe white-box attack.
+<table><tr><td>PARAMETER</td><td>L2 CASE</td><td>Linf CASE</td></tr><tr><td></td><td></td><td></td></tr><tr><td>T</td><td>1000</td><td>1000</td></tr><tr><td>{2t}</td><td>0.03</td><td>0.005</td></tr><tr><td>入</td><td>20</td><td>5</td></tr></table>
+Similarly, for Frank-Wolfe black-box attack algorithm, we also list the parameters we use in Section 5 at Table 6.
+Table 6: Parameters used in Frank-Wolfe black-box attack.
+<table><tr><td>PARAMETER</td><td>L2 CASE Linf</td><td>CASE</td></tr><tr><td></td><td></td><td></td></tr><tr><td>T</td><td>10000</td><td>10000</td></tr><tr><td>{2}</td><td>0.05/√t</td><td>0.03/√t</td></tr><tr><td>入</td><td>50</td><td>30</td></tr><tr><td>b</td><td>25</td><td>25</td></tr><tr><td>St</td><td>0.001</td><td>0.01</td></tr></table>
+We also list the hyperparameters we use for baseline algorithms. Specifically, for PGD, we set a step size of 0.05 for $L _ { 2 }$ case and 0.01 for $L _ { \infty }$ case. For CW, we set a step size of 0.002 for $L _ { 2 }$ case step size of 0.005 for $L _ { \infty }$ case. The confidence is set to 0 and we perform 10 times binary search for the constant starting from 0.01 $L _ { 2 }$ case) and 0.001 $L _ { \infty }$ case). For EAD, we use a step size of 0.01 and the same binary search strategy as CW and $\beta$ is set to be 0.001. In terms of black-box experiments, for ZOO, we set a step size of 0.01 and the initial constant is set to be 1 without binary search to achieve better query complexity. For NES-PGD, we set a step size of 0.3 for $L _ { 2 }$ case and 0.01 for $L _ { \infty }$ case.
+Table 7: Comparison of $L _ { 2 }$ norm based white-box attacks on ResNet V2 model with $\epsilon = 5$ . We report attack success rate, average time and average distortion.
+<table><tr><td>METHODS</td><td>SUCCESS RATE（%）</td><td>AVERAGETIME (s)</td><td>AVERAGEDISTORTION</td></tr><tr><td>PGD</td><td>99.8</td><td>168.2</td><td>0.88</td></tr><tr><td>CW</td><td>98.6</td><td>278.8</td><td>1.45</td></tr><tr><td>EAD</td><td>73.0</td><td>109.2</td><td>2.81</td></tr><tr><td>FW-White</td><td>100.0</td><td>47.1</td><td>0.93</td></tr></table>
+Table 8: Comparison of $L _ { \infty }$ norm based white-box attacks on ResNet V2 model with $\epsilon = 0 . 0 5$ . We report attack success rate, average time and average distortion.
+<table><tr><td>METHODS</td><td></td><td></td><td>SUCCESSRATE（%）AVERAGETIME(S）AVERAGEDISTORTION</td></tr><tr><td>PGD</td><td>100.0</td><td>26.8</td><td>0.0031</td></tr><tr><td>CW</td><td>100.0</td><td>538.9</td><td>0.0251</td></tr><tr><td>FW-White</td><td>100.0</td><td>14.9</td><td>0.0031</td></tr></table>
+# D ADDITIONAL EXPERIMENTS
+# D.1 RESNET V2 WHITE-BOX ATTACK RESULTS
+In this subsection, we present the white-box attack experiments on ResNet V2 model. Tables 7 and 8 present our experimental results for $L _ { 2 }$ norm and $L _ { \infty }$ norm based white-box attacks respectively. For the other baselines in the $L _ { 2 }$ norm case, surprisingly, CW method cannot achieve the best $L _ { 2 }$ distortion as it does in Inception V3 model. EAD method is relatively faster than CW in terms of attack time yet it has the largest distortion and a quite low success rate of $7 3 . 0 \%$ . PGD has the smallest average distortion in this setting, yet it also costs a lot of attack time. On the other hand, our proposed algorithm achieves the highest attack success rate within very short attack time with very small distortion. It significantly reduces the time complexity needed for effective attacking data with large dimensionality. For the $L _ { \infty }$ norm case, CW method takes significantly longer time and does not perform very well on average distortion either. Our proposed white-box attack algorithm, on the other hand, again achieves the shortest average attack time and $100 \%$ success rate.
+# D.2 RESNET V2 BLACK-BOX ATTACK RESULTS
+Table 9: Comparison of $L _ { 2 }$ norm based black-box attacks on ResNet V2 model with $\epsilon = 5$ . Query limit is set to be 50, 000. We report attack success rate, average time and average number of queries needed per image. Opt I and Opt II refer to the two options in Algorithm 2.
+<table><tr><td>METHODS</td><td>SUCCESSRATE(%)</td><td>AVERAGETIME (s)</td><td>AVERAGEQUERIES</td></tr><tr><td>ZOO</td><td>2.4</td><td>696.0</td><td>49495.2</td></tr><tr><td>NES-PGD</td><td>58.0</td><td>75.3</td><td>36748.1</td></tr><tr><td>FW-Black (Opt I)</td><td>57.4</td><td>75.0</td><td>34382.5</td></tr><tr><td>FW-Black (Opt II)</td><td>58.8</td><td>74.7</td><td>34362.8</td></tr></table>
+Table 10: Comparison of $L _ { \infty }$ norm based black-box attacks on Inception V3 model with $\epsilon = 0 . 0 5$ . Query limit is set to be 50, 000. We report attack success rate, average time and average number of queries needed per image. Opt I and Opt II refer to the two options in Algorithm 2.
+<table><tr><td>METHODS</td><td>SUCCESSRATE（%)</td><td>AVERAGETIME(s)</td><td>AVERAGEQUERIES</td></tr><tr><td>NES-PGD</td><td>90.4</td><td>44.7</td><td>20914.0</td></tr><tr><td>FW-Black (Opt I)</td><td>90.8</td><td>44.1</td><td>19934.6</td></tr><tr><td>FW-Black (Opt II)</td><td>91.6</td><td>44.0</td><td>20004.6</td></tr></table>
+In this subsection, we present the black-box experiments on ResNet V2 model. We again mainly focus on evaluating attack success rate, time and number of queries needed. In previous experiments on Inception V3 model, we show the performance of different black-box attack algorithms given enough number of queries (i.e., 500,000 per attack per image). And it shows that basically all algorithms can achieve very high attack success rate (almost $100 \%$ ). Now we examine a much harder case, where we reduce the the number of allowed queries per attack per image to only 50, 000. Tables 9 and 10 present our experimental results for $L _ { 2 }$ norm and $L _ { \infty }$ norm based black-box attacks respectively. We still set $\epsilon = 5$ for $L _ { 2 }$ case and $\epsilon = 0 . 0 5$ for $L _ { \infty }$ case.
+![](images/81f0ac93d395c052150eaa4241cf87f7470c153048e5de7271c5635ba7c5ed50.jpg)
+Figure 3: Attack success rate against the number of queries plot for different algorithms in both $L _ { 2 }$ norm and $L _ { \infty }$ norm based black-box attacks on ResNet V2 model.
+For the $L _ { 2 }$ norm case, ZOO method barely succeeds due to the strict query limit of 50, 00 while it typically requires over $1 0 ^ { 6 }$ queries to attack successfully. Our proposed Frank-Wolfe black-box attacks, on the other hand, achieve nearly $6 0 \%$ attack success rate under such a stringent query budget. For the $L _ { \infty }$ norm case, both NES-PGD method and ours achieve over $9 0 \%$ success rate. Even though they share similar average attack time and average number of queries needed, our Frank-Wolfe based methods still achieve the best in terms of all three evaluation metrics.
+Figure 3 illustrates the attack success rate against the number of queries plot for different algorithms in both $L _ { 2 }$ norm and $L _ { \infty }$ norm based black-box attack on ResNet V2 model. Note that here we have a query limit of $5 0 , 0 0 0$ , which is especially hard for the $L _ { 2 }$ norm case. As we can see from the Figure 3, our proposed Frank-Wolfe black-box attack algorithm (both options) achieves the best performance (highest attack success rate and smallest queries needed for achieving the same success rate).
+# D.3 VISUALIZATION EXAMPLES
+For the completeness, we also provide some visual illustrations on the adversarial examples generated by various algorithms. Figure 4 shows some adversarial examples generated through different $L _ { 2 }$ norm based white-box attacks. Figure 5 shows some adversarial examples generated through different $L _ { \infty }$ norm based black-box attacks.
+![](images/8a88e01a06bd25abfa5f31cbe474ee5a4b0b5274636ffa074782a7890d799d72.jpg)
+Figure 4: Sample adversarial examples generated through different $L _ { 2 }$ norm based white-box attacks. Left side labels denote the original class label and right side labels denote the target class label.
+![](images/fddedae3eb323873bfd9b835b7adf4ea6820b27cd3b93a7b844e40ffbd9a86cc.jpg)
+Figure 5: Sample adversarial examples generated through different $L _ { \infty }$ norm based black-box attacks. Left side labels denote the original class label and right side labels denote the target class label.

parse/train/Hyewf3AqYX/Hyewf3AqYX_content_list.json ADDED Viewed

The diff for this file is too large to render. See raw diff

parse/train/Hyewf3AqYX/Hyewf3AqYX_middle.json ADDED Viewed

The diff for this file is too large to render. See raw diff

parse/train/Hyewf3AqYX/Hyewf3AqYX_model.json ADDED Viewed

The diff for this file is too large to render. See raw diff

parse/train/KJSC_AsN14/KJSC_AsN14.md ADDED Viewed

	@@ -0,0 +1,248 @@

+# CONTRASTIVE LEARNING WITH STRONGER AUGMENTATIONS
+Anonymous authors Paper under double-blind review
+# ABSTRACT
+Representation learning has been greatly improved with the advance of contrastive learning methods with the performance being closer to their supervised learning counterparts. Those methods have greatly benefited from various data augmentations that are carefully designated to maintain their identities so that the images transformed from the same instance can still be retrieved. Although stronger augmentations could expose novel patterns of representations to improve their generalizability, directly using stronger augmentations in instance discrimination-based contrastive learning may even deteriorate the performance, because the distortions induced from the stronger augmentations could ridiculously change the image structures and thus the transformed images can not be viewed as the same as the original ones any more. Additional efforts are needed for us to explore the role of the stronger augmentations in further pushing the performance of unsupervised learning to the fully supervised upper bound. Instead of applying the stronger augmentations directly to minimize the contrastive loss, we propose to minimize the distribution divergence between the weakly and strongly augmented images over the representation bank to supervise the retrieval of strongly augmented queries from a pool of candidates. This avoids an overoptimistic assumption that could overfit the strongly augmented queries containing distorted visual structures into the positive targets in the representation bank, while still being able to distinguish them from the negative samples by leveraging the distributions of weakly augmented counterparts. The proposed method achieves top-1 accuracy of $7 6 . 2 \%$ on ImageNet with a standard ResNet-50 architecture with a single-layer classifier fine-tuned. This is almost the same as $7 6 . 5 \%$ of top-1 accuracy with a fully supervised ResNet-50. Moreover, it outperforms the previous self-supervised and supervised methods on both the transfer learning and object detection tasks.
+# 1 INTRODUCTION
+Deep neural network has shown its sweeping successes in learning from large-scale labeled datasets like ImageNet (Deng et al. (2009)). However, such successes hinge on the availability of a large amount of labeled examples that are expensive to collect. To address this challenge, unsupervised visual representation learning and self-supervised learning, have been studied to learn feature representations without labels. Among them is the contrastive learning (Hadsell et al. (2006); Misra & Maaten (2020); Chen et al. (2020b); He et al. (2020); Caron et al. (2020)), showing great potentials to close the performance gap with supervised methods.
+In contrastive learning Hadsell et al. (2006), each image is considered as an instance, and we wish to train the network so that the representations of different augmentations of the same instance are as close as possible to each other (He et al. (2020); Chen et al. (2020a); Wu et al. (2018); Hjelm et al. (2018); Oord et al. (2018); Bachman et al. (2019); Zhuang et al. (2019); Tian et al. (2019); Henaff et al. (2019)). Meanwhile, the representations of different instances can be also distinguished ´ between each other.
+It is worth noting that these methods usually rely on image augmentations that are carefully designated to maintain their instance identities so that the augmentation of an instance can be accurately retrieved from a dictionary of instances. On the other hand, we believe stronger augmentations could expose novel patterns which can further improve the generalizability of learned representations and eventually close the gap with the fully supervised models. However, directly using stronger augmentations in the contrastive learning could deteriorate the performance, because the induced distortions could ridiculously change the image structures and thus the transformed images cannot keep the identity of the original instances. Thus, additional efforts are needed for us to explore the role of the stronger augmentations to further boost the performance of self-supervised learning.
+Thus we propose the CLSA (Contrastive Learning with Stronger Augmentations) framework to address this challenge. Instead of applying strongly augmented views to the contrastive loss, we propose to minimize the distribution divergence between the weakly and strongly augmented images over a representation bank to supervise the retrieval of stronger queries. This avoids an overoptimistic assumption that could overfit the strongly augmented queries containing distorted visual structures into the positive targets, while still being able to distinguish them from the negative samples by leveraging the distributions of weakly augmented counterparts. The learned representation will not only explore the novel patterns exposed by the stronger augmentations, but also inherits the knowledge about the relative similarities to the negative samples.
+The experiments on various datasets demonstrate that the proposed framework can greatly boost the performance by learning from stronger augmentations. On the ImageNet linear evaluation protocol, we reach a record $7 6 . 2 \%$ top-1 accuracy with the standard ResNet-50 backbone, which is almost as high as $7 6 . 5 \%$ top-1 accuracy of the fully supervised model. Meanwhile, it also achieves the competitive performances on several downstream tasks. Among them is a top-1 accuracy of $9 3 . 6 \%$ on VOC07 with the linear classifier on the pretrained ResNet-50 compared to the previous record of $8 8 . 9 \%$ top-1 accuracy. For the COCO object detection, the $A P _ { S }$ for small object detection has been improved to $2 4 . 4 \%$ from the previous best $A P _ { S }$ of $2 0 . 8 \%$ . These results show that the CLSA can more effectively leverage stronger augmentations than the previous self-supervised methods on downstream tasks. We also conduct ablation study to show a naive application of stronger augmentations in the contrastive learning would degrade the performances.
+# 2 RELATED WORK
+Unsupervised and self-supervised learning methods have been widely studied to close the gap with supervised learning. These methods can be categorized into four different major aspects.
+Instance Discrimination and Contrastive Learning Each image is considered as an individual class in an instance discrimination setting (Bojanowski & Joulin (2017); Dosovitskiy et al. (2015); Wu et al. (2018); Chen et al. (2020a); He et al. (2020)). It can be further formulated as contrastive learning (Hadsell et al. (2006)). In particular, Wu et al. (2018) built a memory bank that stores pre-computed representations from which positive examples are retrieved given some queries. Following this work, He et al. (2020) used a momentum update mechanism to maintain a long queue of negative examples for contrastive learning. Chen et al. (2020a) proposed a rich family of data augmentations on cropped images which has significantly boosted the classification accuracy. However, these methods failed to further improve the performance by naively applying stronger augmentations to minimize the contrastive loss, and this motivated the proposed work.
+Generative Methods The generative methods typically adopt auto-encoders (Vincent et al. (2008); Kingma & Welling (2013)), and adversarial learning (Donahue et al. (2016); Donahue & Simonyan (2019)) to train an unsupervised representation. Usually, they focused on the pixel-wise information of images to distinguish images from different classes.
+Self-supervised Clustering Data clustering (Asano et al. (2019); Caron et al. (2018; 2019; 2020); Yan et al. (2020)) can also be used to learn visual representations by assigning pseudo cluster labels to individual samples. DeepCluster (Caron et al. (2018)) generalized k-means by alternating between assigning pseudo-labels and updating networks. Recently, the SWAV (Caron et al. (2020)) is proposed to learn a cluster of prototypes as the negative examples for the contrastive learning. Combined with the multi-crops of training examples, the SWAV has achieved the state-of-the-art performance on ImageNet.
+Pretext Tasks In addition to the contrastive learning, there exist many alternative methods using different pretext tasks (Agrawal et al. (2015); Qi et al. (2019); Doersch et al. (2015); Kim et al. (2018); Larsson et al. (2016); Zhang et al. (2019)) to train unsupervised deep networks. For example, Doersch et al. (2015) used the relative positions of two randomly sampled patches as the supervised signal. Agrawal et al. (2015); Zhang et al. (2019); Gidaris et al. (2018) adopted various geometric transformations on images and used the transformation parameters to train deep networks. For more details about these works, please refer to the survey by Jing & Tian (2020).
+![](images/fc9396798788c1e2df6331a2190dd5aab1b2d2e388e0702049ba14da1706f1c3.jpg)
+Figure 1: Contrastive instance learning framework
+# 3 THE PROPOSED METHOD
+In this section, we will first review the preliminary work on the contrastive learning, and discuss their limitations on applying stronger augmentations to explore the novel patterns of representations. Then we will present a new distributional divergence loss between weakly and strongly augmented images to self-train the representations over a representation bank consisting of both negative samples and positive targets. After that, the algorithm and the implementation details will also be explained.
+# 3.1 PRELIMINARY METHODS AND LIMITATIONS
+Contrastive learning (Hadsell et al. (2006)) is a popular self-supervised idea and made great success in recent years with the advance of computation and various image augmentations. In contrastive learning, each image is considered as an instance, so it’s also known as instance learning.
+Fig. 1 illustrates general framework of contrastive learning methods.For each image $x$ in batch $B$ , we apply two transformations $T$ and $T ^ { \prime }$ to obtain two different views $V$ and $V ^ { \prime }$ of the same instance $x$ . Then they go through a key encoder and a query encoder respectively, followed with MLP projection layers, resulting in two embedded representations $z$ and $z ^ { \prime }$ to calculate the constrastive loss. The key factor of contrastive learning is the quality and the number of negative examples. To deal with those issues, various methods have been proposed, such as Memory Bank (Wu et al. (2018)), momentum encoder (He et al. (2020)), and online learning with bigger batch (Chen et al. (2020a)).
+Specifically, the contrastive loss is developed to maximize the agreement of representations of different views of the same instance while minimizing the agreement with other negative samples. Hence, the contrastive loss for the different views of the same instance can be defined in Eq. (1).
+$$
+\mathcal { L } _ { C } = - \frac { 1 } { | B | } \sum _ { i \in B } \log \frac { \exp ( s i m ( z _ { i } ^ { \prime } , z _ { i } ) / \tau ) } { \exp ( s i m ( z _ { i } ^ { \prime } , z _ { i } ) / \tau ) + \sum _ { k = 0 } ^ { K } \exp ( s i m ( z _ { i } ^ { \prime } , z _ { k } ) / \tau ) }
+$$
+with the cosine similarity
+$$
+s i m ( z _ { i } , z _ { j } ) = z _ { i } ^ { T } z _ { j } / ( | | z _ { i } | | \cdot | | z _ { j } | | )
+$$
+where $\mathcal { L } _ { C }$ is the contrastive loss, $z _ { i }$ and $z _ { i } ^ { \prime }$ are the projected representations of the different augmentations of the same sample $x _ { i }$ , the summation is taken over the samples $x _ { i }$ in the current batch $B$ , and $\tau$ is the temperature parameter set to 0.2. Also, the negative pool $Q = \{ z _ { k } | k = 1 , \cdot \cdot \cdot , K \}$ shown in Fig. 1 is a queue of size $K$ storing the embedded features from the past batches in a FIFO fashion. This will keep the examples from the most recent batches in the queue while removing these obsoleted ones from it, which has been widely adopted in previous works (Chen et al. (2020b); He et al. (2020)).
+As illustrated in Fig. 1, previous contrastive learning works use two transformations $T$ and $T ^ { \prime }$ to generate two different views $V$ and $V ^ { \prime }$ , in which the transformations are carefully designated. Thus, the two views are not transformed aggressively so that they can still be viewed as the same instance. However, directly adopting stronger transformations (e.g., with larger rotation angles, more aggressive colorjittering and cutout) in contrastive learning fails to further improve the performance or even deteriorate it for downstream tasks, which is not surprising. Stronger transformations could distort image structures and their perceptual patterns in the learned representation so that strongly augmented samples from the same instance cannot be viewed as keeping the same instance identity for training the underlying network. However, stronger augmentations can expose useful clues to the novel patterns that cannot be revealed from moderately augmented images. In supervised learning (Cubuk et al. (2018); Lim et al. (2019); Hataya et al. (2019); Cubuk et al. (2020)), data augmentation search have been widely studied and greatly boost the performance with the novel pattern exposed by strongly augmented images. The findings in RandAugment (Cubuk et al. (2020)) have verified that strongly augmented views can provide more clues even without an explicit augmentation policy. We believe learning the representations from these novel patterns will pave the last mile to close the gap with the fully supervised representations. Indeed, in semi-supervised learning and supervised learning (Cubuk et al. (2020); Qi et al. (2019); Wang et al. (2019)), more aggressive augmentations have been adopted and achieved extraordinary performances. For example, AET Qi et al. (2019) has adopted the parameters of augmentations as supervised signal to self-supervise the training of networks. All of these findings have inspired us to explore novel ways to utilize stronger transformations in self-supervised learning while avoiding deteriorated performances by naively using them in a contrastive model (Chen et al. (2020a)). All of these have inspired us to explore novel ways to utilize stronger transformations in self-supervised learning.
+![](images/268a14d1dc57c7c205146a98e8e874c8e5fc108c8bbb61a6ea5de761f98921e4.jpg)
+Figure 2: Comparison of the strongly weakly augmented images. The left is the original image, the middle is the weakly augmented image, and the right is the strongly augmented one with overcontrastive details.
+However, it is not easy. As shown in Fig. 2, a strongly augmented image may look perceptually different from the original counterpart. Consequently, the representation of a strongly augmented image can be far apart from that of the weakly augmented one. Thus, naively using strongly augmented images in contrastive learning can be over-optimistic since the induced distortions could dramatically ruin their image structures. To this end, in Section 3.2, we instead proposed the Distributional Divergence Minimization (DDM) between weakly and strongly augmented images over a representation bank to avoid overfitting the representation of a strongly augmented image with that of the corresponding positive target.
+# 3.2 DISTRIBUTIONAL DIVERGENCE MINIMIZATION BETWEEN WEAKLY AND STRONGLY AUGMENTED IMAGES
+Due to the aforementioned limitation of stronger augmentations in contrastive learning, a direct retrieval of a strongly augmented query is infeasible to self-train deep networks. Fortunately, the distribution of relative similarities of a weakly augmented image from the same instance over the representation bank can provide useful information to bridge the gap. As explained below, it does not only avoid directly placing the representation of a strongly augmented image too closer to that of the positive target, but also allows it to explore the novel patterns of variations exposed by the strong augmentation.
+![](images/472bf4d8591573c6a5b08b9f46db3447cb762de241cd82f4bc5df537b81d7bb7.jpg)
+Figure 3: Diagram of distributional divergence minimization. Here the representation bank consists of $K$ stored features $z _ { k }$ in the negative pool and online features $z _ { i }$ from the key encoder. They will be used to calculate the conditional probability of current weakly and strongly augmented images.
+Formally, as shown in Fig. 3, for an original image $x _ { i }$ and the representation $z _ { i }$ of the corresponding positive target, we applied a weaker and a stronger augmentation to obtain two separate views $V _ { i } ^ { \prime }$ and $V _ { i } ^ { \prime \prime }$ , and their embeddings $z _ { i } ^ { \prime }$ (weak representation) and $z _ { i } ^ { \prime \prime }$ (strong representation). Given a pool $Q$ (shown in Fig. 1) of $K$ negative samples $\{ z _ { k } | k = 1 , \dot { \cdots } , K \}$ accumulated from the past iterations, we obtain a conditional distribution
+$$
+p ( z _ { k } | z _ { i } ^ { \prime } ) = \frac { e x p ( s i m ( z _ { i } ^ { \prime } , z _ { k } ) / \tau ) } { e x p ( s i m ( z _ { i } ^ { \prime } , z _ { i } ) / \tau ) + \sum _ { k = 0 } ^ { K } e x p ( s i m ( z _ { i } ^ { \prime } , z _ { k } ) / \tau ) }
+$$
+which encodes the likelihood of the weaker query $z _ { i } ^ { \prime }$ being assigned to $z _ { k }$ . In the similar way, we can define the likelihood $p ( z _ { i } | z _ { i } ^ { \prime } )$ of the query being assigned to the positive target $z _ { i }$ , as well as the likelihoods $p ( z _ { i } | z _ { i } ^ { \prime \prime } )$ and $p ( z _ { k } | z _ { i } ^ { \prime \prime } )$ for the stronger query $z _ { i } ^ { \prime \prime }$ . Here the representation bank consists of the negative pool $Q$ and the positive query targets $z _ { i }$ from the current batch $B$ .
+Then, we propose to minimize the following distributional divergence between the weak and the strong queries such that
+$$
+\mathcal { L } _ { D } = \frac { 1 } { | B | } \sum _ { i \in B } \left[ - \sum _ { k = 1 } ^ { K } p ( z _ { k } | z _ { i } ^ { \prime } ) \log ( p ( z _ { k } | z _ { i } ^ { \prime \prime } ) ) - p ( z _ { i } | z _ { i } ^ { \prime } ) \log ( p ( z _ { i } | z _ { i } ^ { \prime \prime } ) ) \right]
+$$
+By minimizing this divergence, we assume the learned representation $z _ { i } ^ { \prime \prime }$ of the strongly augmented query should inherit the representation $z _ { i } ^ { \prime }$ of the weakly augmented one regarding not only its belief of the query being assigned to the corresponding positive target $z _ { i }$ , but also its relations with the negative samples $z _ { k }$ in the representation bank through the conditional distribution $p ( z _ { k } | z _ { i } ^ { \prime } )$ .
+This will prevent a direct overfitting of the strong query representation to the positive target as well as improve the generalization of the learned representation with additional clues from the other examples in the representation pool. In a more general sense, this extends the idea of knowledge distillation (Hinton et al. (2015)). However, we did not use the predicted labels by a teacher model to supervise the training of a student model as in the knowledge distillation. Instead, we used the distribution of the likelihoods of a weak query to supervise the retrieval of a strong query from a pool of representations.
+# 3.3 IMPLEMENTATION DETAILS
+Algorithm 1 gives the pseudo code to implement the proposed CLSA method. In the following, we will discuss the details about the applied strong and weak augmentations for distributional divergence minimization.
+Stronger Augmentations $\pmb { S }$ As explored in the previous works (Cubuk et al. (2018); Wang et al. (2019); Qi et al. (2019)), strong augmentations usually have two types: geometric and non-geometric augmentations. Specifically, we considered 14 types of augmentations: ShearX/Y, TranslateX/Y,
+# Algorithm 1 Pseudo code for the proposed CLSA
+Input: $f _ { \theta } , f _ { \phi }$ : the query and the key encoder networks; $g _ { \boldsymbol { \theta } } , g _ { \boldsymbol { \phi } }$ : the MLP projection layers for the query and the key; $Q$ : a queue of representations of $K$ negative samples; $\alpha$ : momentum decay for the key network; $\tau$ : the temperature; $T$ and $T ^ { \prime }$ : weak augmentation; $S$ : strong augmentation; $\beta$ : the balancing coefficient.
+1: Initialize the network $f _ { \phi } = f _ { \theta }$ , $g _ { \phi } = g _ { \theta }$ and $Q$ ;
+. see Eq. (1) . see Eq. (4)
+Rotate, AutoContrast, Invert, Equalize, Solarize, Posterize, Contrast, Color, Brightness, Sharpness. The magnitude of each augmentation is significant enough to produce as strong augmentations as possible. More details are shown in Table 1. For example, the shear is drawn from a range of [-0.3,0.3], which results in aggressively transformed images that can be hard to retrieve given a counterpart target. In particular, to transform an image, we randomly select an augmentation from the above 14 types of transformations, and apply it to the image with a probability of 0.5. This process is repeated five times and that will strongly augment an image as the example shown in the right panel of Fig. 2.
+Table 1: Various augmentations we applied in experiments to strongly augment training images.
+<table><tr><td>Operation Mag Range</td><td>ShearX(Y) [-0.3,0.3]</td><td>TranslateX(Y) [-0.3,0.3]</td><td>Rotate [-30,30]</td><td>AutoContrast 0or1</td><td>Invert 0or 1</td><td>Equalize 0or1</td></tr><tr><td>Operation Mag Range</td><td>Solarize [0,256]</td><td>Posterize [4,8]</td><td>Contrast [0.05,0.95]</td><td>Color [0.05,0.95]</td><td>Brightness [0.05,0.95]</td><td>Sharpeness [0.05,0.95]</td></tr></table>
+10: end for Output: The representation encoder $f _ { \theta }$ .
+Weaker Augmentations $\mathbf { T }$ Weak augmentations are drawn by following most of existing contrastive learning methods in literature (Chen et al. (2020a;b); Caron et al. (2020); He et al. (2020)): an image is first cropped from an input image and resized to $2 2 4 \times 2 2 4$ pixels. Then random color jittering, Gaussian Blur, grayscale conversion, horizontal flip, channel-wise color normalization are sequentially applied to generate weakly augmented images with an example shown in the middle of Fig. 2.
+Technical Details Similar to the previous works (He et al. (2020); Chen et al. (2020a); Caron et al. (2020)), we used the ResNet-50 (He et al. (2016)) as our encoder backbones $f _ { \theta }$ and $f _ { \phi }$ and a 2- layer MLP (2048-d hidden layer with the ReLU) as the projection head $g _ { \boldsymbol { \theta } }$ and $g _ { \phi }$ . The projected representation $z$ is first L2-normalized (Wu et al. (2018)) before calculating the cosine similarity. The temperature $\tau$ is set to 0.2, with a momentum smoothing factor $\alpha$ of 0.999 and a fixed balancing coefficient $\beta$ of 1.0. We set the size $K$ of the queue $Q$ to 65536 to store the negative examples used to compute the conditional distribution of weakly and strongly augmented queries and minimize their divergence.
+# 4 EXPERIMENTS
+# 4.1 TRAINING DETAILS
+For the unsupervised pretraining on ImageNet with the CLSA, we used the SGD optimizer (Bottou (2010)) with an initial learning rate of 0.03, a weight decay of 0.0001 and a momentum of 0.9. We used cosine scheduler (Loshchilov & Hutter (2016)) to gradually decay the learning rate to 0. Usually, the batch size is set to 256. When multiple GPU cluster servers are used, the batch size will be multiplied by the same number of servers by convention. Typically, the experiment with a single strong augmentation for each training image takes roughly 70 hours to finish on 8 V100 GPUs.
+Table 2: Top-1 accuracy under the linear evaluation on ImageNet with the ResNet-50 backbone. The left table compared methods trained over 200 epochs, and the right table compared the methods with various numbers of epochs.
+<table><tr><td>Method</td><td>Top1</td><td>Method</td><td>Top1</td></tr><tr><td>InstDisc (Wu et al. (2018))</td><td>54.0</td><td>BigBiGAN (Donahue &amp; Simonyan (2019))</td><td>56.6</td></tr><tr><td>LocalAgg (Zhuang et al. (2019))</td><td>58.8</td><td>SeLa-400epochs (Asano et al. (2019))</td><td>61.5</td></tr><tr><td>MoCo (He et al. (2020))</td><td>60.8</td><td>PIRL-800epochs (Misra &amp; Maaten (2020))</td><td>63.6</td></tr><tr><td>SimCLR (Chen et al. (2020a))</td><td>61.9</td><td>CMC (Tian et al. (2019))</td><td>66.2</td></tr><tr><td>CPC v2 (Hénaff et al. (2019))</td><td>63.8</td><td>SimCLR-800epochs (Chen et al. (2020a))</td><td>70.0</td></tr><tr><td>PCL (Li et al. (2020))</td><td>65.9</td><td>MoCo v2-800epochs (Chen et al. (2020b))</td><td>71.1</td></tr><tr><td>MoCo v2 (Chen et al. (2020b))</td><td>67.5</td><td>InfoMin Aug-800epochs (Tian et al. (2020))</td><td>73.0</td></tr><tr><td>InfoMin Aug (Tian et al. (2020))</td><td>70.1</td><td>BYOL-1000epochs (Grill et al. (2020))</td><td>74.3</td></tr><tr><td>SWAV (Caron et al. (2020))</td><td>72.7</td><td>SWAV-800epochs (Caron et al. (2020))</td><td>75.3</td></tr><tr><td>CLSA-Single</td><td>69.4</td><td>CLSA-Single-800epochs</td><td>72.2</td></tr><tr><td>CLSA-Multi</td><td>73.3</td><td>CLSA-Multi-800epochs</td><td>76.2</td></tr><tr><td>Supervised</td><td>76.5</td><td>Supervised</td><td>76.5</td></tr></table>
+For the fine-tuning on ImageNet, we trained a linear classifier on top of the frozen feature vector (2048-D) upon the pretrained ResNet-50 with CLSA. This linear layer is trained for 100 epochs, with a learning rate of 10 without weight decay. We used the cosine learning rate decay and a batch size of 256.
+For the transfer learning on the VOC dataset, we trained a linear classifier upon the pretrained Resnet-50 in the similar way for ImageNet – we trained 100 epochs with the SGD optimizer and a learning rate of 0.05, a momentum of 0.9 and no weight decay. The batch size is 256 without learning rate scheduler.
+Finally, for object detection, we adopted the same protocol in (He et al. (2020)) to fine-tune the pretrained Resnet-50 backbone based on detectron2 (Wu et al. (2019)) for the sake of a fair and straight comparison with the other methods.
+# 4.2 LINEAR CLASSIFICATION ON IMAGENET
+For the linear evaluation on ImageNet, we trained the CLSA in two settings. In the first setting named CLSA-Single, we used a single stronger augmentation (see Table 1) that crops each training image to a smaller size of $9 6 \times 9 6$ , which does not incur too much computing overhead in processing these smaller augmented images. In the second setting named CLSA-Multi, we adopted five different stronger augmentations that crop each image into various sizes: $2 2 4 \times 2 2 4$ , $1 9 2 \times 1 9 2$ , $1 6 0 \times$ 160, $1 2 8 \times 1 2 8$ , and $9 6 \times 9 6$ . The DDM loss in Eq. (4) is the sum over these multiple stronger augmentations. The similar multi-crop strategy has been adopted in contrastive learning literature before. For example, the SWAV (Caron et al. (2020)) reached the state-of-the-art top-1 accuracy by applying such multi-crop augmentations. To ensure a fair comparison with the SWAV, we chose five stronger augmentations such that the self-training with CLSA-Multi consumed the same computing time (i.e., 166 hours with a cluster of 8 V100 GPUs for 200 epochs of pre-training with a batch size of 256).
+As shown in Table 2, we compared the performance with the other unsupervised methods. All the experiments are based on a pretrained ResNet-50 backbone that is fine-tuned with a linear classifier. The left table showed the performance of different methods pretrained over 200 epochs, and the right table reported models pretrained over more epochs.
+First, under the same contrastive protocol, the CLSA-Single has a higher $6 9 . 4 \%$ top-1 accuracy than both MoCo v2 $( 6 7 . 5 \% )$ and SimCLR $( 6 1 . 9 \% )$ with 200 epochs training. With multiple stronger augmentations, the CLSA-Multi outperforms the State-of-the-art SWAV model using multi-crops of training images over 200 epochs $7 3 . 3 \%$ vs. $7 2 . 7 \%$ . Moreover, as shown in the right table, the CLSA
+Single outperforms MoCo v2 and SimCLR with the same training epochs. It is also noteworthy that the CLSA-Multi achieves almost the same top-1 accuracy as that of the fully supervised network ( $7 6 . 2 \%$ vs. $7 6 . 5 \%$ ).
+# 4.3 TRANSFER LEARNING RESULTS ON DOWNSTREAM TASKS
+Table 3: Transfer learning results on various downstream tasks.
+<table><tr><td rowspan="3">Measurement</td><td>Classification</td><td colspan="3">Object Detection</td></tr><tr><td>VOC07</td><td>VOC07+12</td><td></td><td>COCO</td></tr><tr><td>Accuracy</td><td>AP50</td><td>AP</td><td>APs</td></tr><tr><td>RotNet (Gidaris et al. (2018))</td><td>64.6</td><td>=</td><td>=</td><td>=</td></tr><tr><td>NPID++ (Wu et al. (2018))</td><td>76.6</td><td>79.1</td><td></td><td></td></tr><tr><td>MoCo (He et al. (2020))</td><td>79.8</td><td>81.5</td><td>=</td><td></td></tr><tr><td>PIRL (Misra &amp; Maaten (2020))</td><td>81.1</td><td>80.7</td><td>=</td><td></td></tr><tr><td>PCL (Li et al. (2020))</td><td>84.0</td><td>1</td><td></td><td></td></tr><tr><td>BoWNet (Gidaris et al. (2020))</td><td>79.3</td><td>81.3</td><td></td><td></td></tr><tr><td>SimCLR (Chen et al. (2020a))</td><td>86.4</td><td>=</td><td>=</td><td>=</td></tr><tr><td>MoCov2 (Chen et al. (2020b))</td><td>87.1</td><td>82.5</td><td>42.0</td><td>20.8</td></tr><tr><td>SWAV (Caron et al. (2020))</td><td>88.9</td><td>82.6</td><td>42.1</td><td>19.7</td></tr><tr><td>CLSA</td><td>93.6</td><td>83.2</td><td>42.3</td><td>24.4</td></tr><tr><td>Supervised</td><td>87.5</td><td>81.3</td><td>40.8</td><td>20.1</td></tr></table>
+We test the generalizability of the ResNet-50 pre-trained on ImageNet to several downstream tasks. Specifically, we focused on two tasks: cross-dataset image classification and object detection. The pre-trained ResNet-50 was frozon, and we fine-tuned the linear classifier on the VOC07trainval (Everingham et al. (2010)) and tested it on the VOC07test. For object detection, we evaluated the pre-trained network on two datasets using the detectron2 (Wu et al. (2019)) used in the previous methods He et al. (2020); Chen et al. (2020a). On the VOC dataset (Everingham et al. (2010)), we trained the detection head with $\mathrm { \ V O C { 0 7 + 1 2 } }$ trainval dataset and tested on VOC07 test dataset. On the COCO dataset (Lin et al. (2014)), we fine-tuned the network on the train2017 set with $1 1 8 \mathrm { k }$ images and evaluate on the val2017. For the sake of a fair comparison, the object detection tasks are completed by detectron2 (He et al. (2017)) based on the pretrained ResNet-50.
+As shown in Table 3, the performances on both tasks are much better than the supervised model trained on ImageNet. This suggests that the proposed method has better generalization ability in downstream tasks. The pre-trained network on ImageNet by the CLSA outperformed the compared models after being fine-tuned on different datasets. Among them is a top-1 accuracy of $9 3 . 6 \%$ on the VOC07 with the linear classifier on the pretrained ResNet-50 in comparison with the previous record of $8 8 . 9 \%$ top-1 accuracy by the SWAV. On the COCO dataset, the $A P _ { S }$ for small object detection has been significantly improved to $2 4 . 4 \%$ from the previously best $A P _ { S }$ of $2 0 . 8 \%$ . As well known, it is much challenging to detect small objects on the COCO dataset. Thus, the better performance of the CLSA could be attributed to the ability of involving the stronger augmentations that result in many small objects to pretrain the network.
+# 4.4 ABLATION STUDY
+Table 4: Ablation study of the CLSA on ImageNet with 200 epochs of pre-training.
+<table><tr><td>Model</td><td>Top-1 Accuracy</td></tr><tr><td>MoCo V2</td><td>67.5</td></tr><tr><td>MoCo V2 with Strong query</td><td>67.7</td></tr><tr><td>MoCo V2 with Strong query &amp; Strong key</td><td>67.0</td></tr><tr><td>CLSA-Single with contrastive loss</td><td>68.0</td></tr><tr><td>CLSA-Single</td><td>69.4</td></tr></table>
+In the ablation study shown in Table 4, we studied the role of the proposed DDM loss in the CLSA. First, we naively used the stronger augmentation applied in the CLSA-Single as the query and/or the key in the MoCo V2. Both results (Strong query and Strong query & Strong key) showed the performance can not be improved or even degraded. Second, we replaced the DDM loss in the CLSA-Single with the contrastive loss, and we found it can only achieved a top-1 accuracy of $6 8 . 0 \%$ compared to that of $6 9 . 4 \%$ with the DDM loss. Both studies showed that the proposed CLSA and its DDM loss help us learn from stronger augmentations by avoiding the performance degeneration that would be incurred by the distortions of augmented images.
+# 5 CONCLUSION
+In this paper, we present CLSA, a novel method that can utilize the distributional divergence to learn the information from strongly augmented images. The proposed method outperforms the state-ofthe-art methods on all the datasets and achieved almost same performance compared to supervised ImageNet network. Meanwhile, it outperforms the previous supervised and self-supervised methods on downstream tasks, which suggests CLSA learned more reliable and fine-grained features that can contribute to the development of other areas.
+# REFERENCES
+Pulkit Agrawal, Joao Carreira, and Jitendra Malik. Learning to see by moving. In Proceedings of the IEEE international conference on computer vision, pp. 37–45, 2015.
+Yuki Markus Asano, Christian Rupprecht, and Andrea Vedaldi. Self-labelling via simultaneous clustering and representation learning. arXiv preprint arXiv:1911.05371, 2019.
+Philip Bachman, R Devon Hjelm, and William Buchwalter. Learning representations by maximizing mutual information across views. In Advances in Neural Information Processing Systems, pp. 15535–15545, 2019.
+Piotr Bojanowski and Armand Joulin. Unsupervised learning by predicting noise. arXiv preprint arXiv:1704.05310, 2017.
+Leon Bottou. Large-scale machine learning with stochastic gradient descent. In ´ Proceedings of COMPSTAT’2010, pp. 177–186. Springer, 2010.
+Mathilde Caron, Piotr Bojanowski, Armand Joulin, and Matthijs Douze. Deep clustering for unsupervised learning of visual features. In Proceedings of the European Conference on Computer Vision (ECCV), pp. 132–149, 2018.
+Mathilde Caron, Piotr Bojanowski, Julien Mairal, and Armand Joulin. Unsupervised pre-training of image features on non-curated data. In Proceedings of the IEEE International Conference on Computer Vision, pp. 2959–2968, 2019.
+Mathilde Caron, Ishan Misra, Julien Mairal, Priya Goyal, Piotr Bojanowski, and Armand Joulin. Unsupervised learning of visual features by contrasting cluster assignments. arXiv preprint arXiv:2006.09882, 2020.
+Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton. A simple framework for contrastive learning of visual representations. arXiv preprint arXiv:2002.05709, 2020a.
+Xinlei Chen, Haoqi Fan, Ross Girshick, and Kaiming He. Improved baselines with momentum contrastive learning. arXiv preprint arXiv:2003.04297, 2020b.
+Ekin D Cubuk, Barret Zoph, Dandelion Mane, Vijay Vasudevan, and Quoc V Le. Autoaugment: Learning augmentation policies from data. arXiv preprint arXiv:1805.09501, 2018.
+Ekin D Cubuk, Barret Zoph, Jonathon Shlens, and Quoc V Le. Randaugment: Practical automated data augmentation with a reduced search space. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops, pp. 702–703, 2020.
+Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In 2009 IEEE conference on computer vision and pattern recognition, pp. 248–255. Ieee, 2009.
+Carl Doersch, Abhinav Gupta, and Alexei A Efros. Unsupervised visual representation learning by context prediction. In Proceedings of the IEEE international conference on computer vision, pp. 1422–1430, 2015.
+Jeff Donahue and Karen Simonyan. Large scale adversarial representation learning. In Advances in Neural Information Processing Systems, pp. 10542–10552, 2019.
+Jeff Donahue, Philipp Krahenb ¨ uhl, and Trevor Darrell. Adversarial feature learning. ¨ arXiv preprint arXiv:1605.09782, 2016.
+Alexey Dosovitskiy, Philipp Fischer, Jost Tobias Springenberg, Martin Riedmiller, and Thomas Brox. Discriminative unsupervised feature learning with exemplar convolutional neural networks. IEEE transactions on pattern analysis and machine intelligence, 38(9):1734–1747, 2015.
+Mark Everingham, Luc Van Gool, Christopher KI Williams, John Winn, and Andrew Zisserman. The pascal visual object classes (voc) challenge. International journal of computer vision, 88(2): 303–338, 2010.
+Spyros Gidaris, Praveer Singh, and Nikos Komodakis. Unsupervised representation learning by predicting image rotations. arXiv preprint arXiv:1803.07728, 2018.
+Spyros Gidaris, Andrei Bursuc, Nikos Komodakis, Patrick Perez, and Matthieu Cord. Learning ´ representations by predicting bags of visual words. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 6928–6938, 2020.
+Jean-Bastien Grill, Florian Strub, Florent Altche, Corentin Tallec, Pierre H Richemond, Elena ´ Buchatskaya, Carl Doersch, Bernardo Avila Pires, Zhaohan Daniel Guo, Mohammad Gheshlaghi Azar, et al. Bootstrap your own latent: A new approach to self-supervised learning. arXiv preprint arXiv:2006.07733, 2020.
+Raia Hadsell, Sumit Chopra, and Yann LeCun. Dimensionality reduction by learning an invariant mapping. In 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’06), volume 2, pp. 1735–1742. IEEE, 2006.
+Ryuichiro Hataya, Jan Zdenek, Kazuki Yoshizoe, and Hideki Nakayama. Faster autoaugment: Learning augmentation strategies using backpropagation. arXiv preprint arXiv:1911.06987, 2019.
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770–778, 2016.
+Kaiming He, Georgia Gkioxari, Piotr Dollar, and Ross Girshick. Mask r-cnn. In ´ Proceedings of the IEEE international conference on computer vision, pp. 2961–2969, 2017.
+Kaiming He, Haoqi Fan, Yuxin Wu, Saining Xie, and Ross Girshick. Momentum contrast for unsupervised visual representation learning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 9729–9738, 2020.
+Olivier J Henaff, Aravind Srinivas, Jeffrey De Fauw, Ali Razavi, Carl Doersch, SM Eslami, and ´ Aaron van den Oord. Data-efficient image recognition with contrastive predictive coding. arXiv preprint arXiv:1905.09272, 2019.
+Geoffrey Hinton, Oriol Vinyals, and Jeff Dean. Distilling the knowledge in a neural network. arXiv preprint arXiv:1503.02531, 2015.
+R Devon Hjelm, Alex Fedorov, Samuel Lavoie-Marchildon, Karan Grewal, Phil Bachman, Adam Trischler, and Yoshua Bengio. Learning deep representations by mutual information estimation and maximization. arXiv preprint arXiv:1808.06670, 2018.
+Longlong Jing and Yingli Tian. Self-supervised visual feature learning with deep neural networks: A survey. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2020.
+Dahun Kim, Donghyeon Cho, Donggeun Yoo, and In So Kweon. Learning image representations by completing damaged jigsaw puzzles. In 2018 IEEE Winter Conference on Applications of Computer Vision (WACV), pp. 793–802. IEEE, 2018.
+Under review as a conference paper at ICLR 2021
+Diederik P Kingma and Max Welling. Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114, 2013.
+Gustav Larsson, Michael Maire, and Gregory Shakhnarovich. Learning representations for automatic colorization. In European conference on computer vision, pp. 577–593. Springer, 2016.
+Junnan Li, Pan Zhou, Caiming Xiong, Richard Socher, and Steven CH Hoi. Prototypical contrastive learning of unsupervised representations. arXiv preprint arXiv:2005.04966, 2020.
+Sungbin Lim, Ildoo Kim, Taesup Kim, Chiheon Kim, and Sungwoong Kim. Fast autoaugment. In Advances in Neural Information Processing Systems, pp. 6665–6675, 2019.
+Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dollar, and C Lawrence Zitnick. Microsoft coco: Common objects in context. In ´ European conference on computer vision, pp. 740–755. Springer, 2014.
+Ilya Loshchilov and Frank Hutter. Sgdr: Stochastic gradient descent with warm restarts. arXiv preprint arXiv:1608.03983, 2016.
+Ishan Misra and Laurens van der Maaten. Self-supervised learning of pretext-invariant representations. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 6707–6717, 2020.
+Aaron van den Oord, Yazhe Li, and Oriol Vinyals. Representation learning with contrastive predictive coding. arXiv preprint arXiv:1807.03748, 2018.
+Guo-Jun Qi, Liheng Zhang, and Xiao Wang. Learning generalized transformation equivariant representations via autoencoding transformations. arXiv preprint arXiv:1906.08628, 2019.
+Yonglong Tian, Dilip Krishnan, and Phillip Isola. Contrastive multiview coding. arXiv preprint arXiv:1906.05849, 2019.
+Yonglong Tian, Chen Sun, Ben Poole, Dilip Krishnan, Cordelia Schmid, and Phillip Isola. What makes for good views for contrastive learning. arXiv preprint arXiv:2005.10243, 2020.
+Pascal Vincent, Hugo Larochelle, Yoshua Bengio, and Pierre-Antoine Manzagol. Extracting and composing robust features with denoising autoencoders. In Proceedings of the 25th international conference on Machine learning, pp. 1096–1103, 2008.
+Xiao Wang, Daisuke Kihara, Jiebo Luo, and Guo-Jun Qi. Enaet: Self-trained ensemble autoencoding transformations for semi-supervised learning. arXiv preprint arXiv:1911.09265, 2019.
+Yuxin Wu, Alexander Kirillov, Francisco Massa, Wan-Yen Lo, and Ross Girshick. Detectron2. https://github.com/facebookresearch/detectron2, 2019.
+Zhirong Wu, Yuanjun Xiong, Stella X Yu, and Dahua Lin. Unsupervised feature learning via nonparametric instance discrimination. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 3733–3742, 2018.
+Xueting Yan, Ishan Misra, Abhinav Gupta, Deepti Ghadiyaram, and Dhruv Mahajan. Clusterfit: Improving generalization of visual representations. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 6509–6518, 2020.
+Liheng Zhang, Guo-Jun Qi, Liqiang Wang, and Jiebo Luo. Aet vs. aed: Unsupervised representation learning by auto-encoding transformations rather than data. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2547–2555, 2019.
+Chengxu Zhuang, Alex Lin Zhai, and Daniel Yamins. Local aggregation for unsupervised learning of visual embeddings. In Proceedings of the IEEE International Conference on Computer Vision, pp. 6002–6012, 2019.

parse/train/KJSC_AsN14/KJSC_AsN14_content_list.json ADDED Viewed

	@@ -0,0 +1,1270 @@

+[
+    {
+        "type": "text",
+        "text": "CONTRASTIVE LEARNING WITH STRONGER AUGMENTATIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            99,
+            821,
+            145
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            171,
+            398,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            236,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Representation learning has been greatly improved with the advance of contrastive learning methods with the performance being closer to their supervised learning counterparts. Those methods have greatly benefited from various data augmentations that are carefully designated to maintain their identities so that the images transformed from the same instance can still be retrieved. Although stronger augmentations could expose novel patterns of representations to improve their generalizability, directly using stronger augmentations in instance discrimination-based contrastive learning may even deteriorate the performance, because the distortions induced from the stronger augmentations could ridiculously change the image structures and thus the transformed images can not be viewed as the same as the original ones any more. Additional efforts are needed for us to explore the role of the stronger augmentations in further pushing the performance of unsupervised learning to the fully supervised upper bound. Instead of applying the stronger augmentations directly to minimize the contrastive loss, we propose to minimize the distribution divergence between the weakly and strongly augmented images over the representation bank to supervise the retrieval of strongly augmented queries from a pool of candidates. This avoids an overoptimistic assumption that could overfit the strongly augmented queries containing distorted visual structures into the positive targets in the representation bank, while still being able to distinguish them from the negative samples by leveraging the distributions of weakly augmented counterparts. The proposed method achieves top-1 accuracy of $7 6 . 2 \\%$ on ImageNet with a standard ResNet-50 architecture with a single-layer classifier fine-tuned. This is almost the same as $7 6 . 5 \\%$ of top-1 accuracy with a fully supervised ResNet-50. Moreover, it outperforms the previous self-supervised and supervised methods on both the transfer learning and object detection tasks. ",
+        "bbox": [
+            232,
+            266,
+            764,
+            611
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            628,
+            336,
+            643
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep neural network has shown its sweeping successes in learning from large-scale labeled datasets like ImageNet (Deng et al. (2009)). However, such successes hinge on the availability of a large amount of labeled examples that are expensive to collect. To address this challenge, unsupervised visual representation learning and self-supervised learning, have been studied to learn feature representations without labels. Among them is the contrastive learning (Hadsell et al. (2006); Misra & Maaten (2020); Chen et al. (2020b); He et al. (2020); Caron et al. (2020)), showing great potentials to close the performance gap with supervised methods. ",
+        "bbox": [
+            174,
+            659,
+            825,
+            756
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In contrastive learning Hadsell et al. (2006), each image is considered as an instance, and we wish to train the network so that the representations of different augmentations of the same instance are as close as possible to each other (He et al. (2020); Chen et al. (2020a); Wu et al. (2018); Hjelm et al. (2018); Oord et al. (2018); Bachman et al. (2019); Zhuang et al. (2019); Tian et al. (2019); Henaff et al. (2019)). Meanwhile, the representations of different instances can be also distinguished ´ between each other. ",
+        "bbox": [
+            174,
+            762,
+            825,
+            847
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "It is worth noting that these methods usually rely on image augmentations that are carefully designated to maintain their instance identities so that the augmentation of an instance can be accurately retrieved from a dictionary of instances. On the other hand, we believe stronger augmentations could expose novel patterns which can further improve the generalizability of learned representations and eventually close the gap with the fully supervised models. However, directly using stronger augmentations in the contrastive learning could deteriorate the performance, because the induced distortions could ridiculously change the image structures and thus the transformed images cannot keep the identity of the original instances. Thus, additional efforts are needed for us to explore the role of the stronger augmentations to further boost the performance of self-supervised learning. ",
+        "bbox": [
+            174,
+            854,
+            823,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            823,
+            159
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Thus we propose the CLSA (Contrastive Learning with Stronger Augmentations) framework to address this challenge. Instead of applying strongly augmented views to the contrastive loss, we propose to minimize the distribution divergence between the weakly and strongly augmented images over a representation bank to supervise the retrieval of stronger queries. This avoids an overoptimistic assumption that could overfit the strongly augmented queries containing distorted visual structures into the positive targets, while still being able to distinguish them from the negative samples by leveraging the distributions of weakly augmented counterparts. The learned representation will not only explore the novel patterns exposed by the stronger augmentations, but also inherits the knowledge about the relative similarities to the negative samples. ",
+        "bbox": [
+            174,
+            166,
+            825,
+            291
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The experiments on various datasets demonstrate that the proposed framework can greatly boost the performance by learning from stronger augmentations. On the ImageNet linear evaluation protocol, we reach a record $7 6 . 2 \\%$ top-1 accuracy with the standard ResNet-50 backbone, which is almost as high as $7 6 . 5 \\%$ top-1 accuracy of the fully supervised model. Meanwhile, it also achieves the competitive performances on several downstream tasks. Among them is a top-1 accuracy of $9 3 . 6 \\%$ on VOC07 with the linear classifier on the pretrained ResNet-50 compared to the previous record of $8 8 . 9 \\%$ top-1 accuracy. For the COCO object detection, the $A P _ { S }$ for small object detection has been improved to $2 4 . 4 \\%$ from the previous best $A P _ { S }$ of $2 0 . 8 \\%$ . These results show that the CLSA can more effectively leverage stronger augmentations than the previous self-supervised methods on downstream tasks. We also conduct ablation study to show a naive application of stronger augmentations in the contrastive learning would degrade the performances. ",
+        "bbox": [
+            174,
+            299,
+            825,
+            452
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            467,
+            344,
+            483
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Unsupervised and self-supervised learning methods have been widely studied to close the gap with supervised learning. These methods can be categorized into four different major aspects. ",
+        "bbox": [
+            173,
+            506,
+            823,
+            534
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Instance Discrimination and Contrastive Learning Each image is considered as an individual class in an instance discrimination setting (Bojanowski & Joulin (2017); Dosovitskiy et al. (2015); Wu et al. (2018); Chen et al. (2020a); He et al. (2020)). It can be further formulated as contrastive learning (Hadsell et al. (2006)). In particular, Wu et al. (2018) built a memory bank that stores pre-computed representations from which positive examples are retrieved given some queries. Following this work, He et al. (2020) used a momentum update mechanism to maintain a long queue of negative examples for contrastive learning. Chen et al. (2020a) proposed a rich family of data augmentations on cropped images which has significantly boosted the classification accuracy. However, these methods failed to further improve the performance by naively applying stronger augmentations to minimize the contrastive loss, and this motivated the proposed work. ",
+        "bbox": [
+            174,
+            540,
+            825,
+            680
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Generative Methods The generative methods typically adopt auto-encoders (Vincent et al. (2008); Kingma & Welling (2013)), and adversarial learning (Donahue et al. (2016); Donahue & Simonyan (2019)) to train an unsupervised representation. Usually, they focused on the pixel-wise information of images to distinguish images from different classes. ",
+        "bbox": [
+            174,
+            686,
+            825,
+            742
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Self-supervised Clustering Data clustering (Asano et al. (2019); Caron et al. (2018; 2019; 2020); Yan et al. (2020)) can also be used to learn visual representations by assigning pseudo cluster labels to individual samples. DeepCluster (Caron et al. (2018)) generalized k-means by alternating between assigning pseudo-labels and updating networks. Recently, the SWAV (Caron et al. (2020)) is proposed to learn a cluster of prototypes as the negative examples for the contrastive learning. Combined with the multi-crops of training examples, the SWAV has achieved the state-of-the-art performance on ImageNet. ",
+        "bbox": [
+            174,
+            750,
+            825,
+            847
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Pretext Tasks In addition to the contrastive learning, there exist many alternative methods using different pretext tasks (Agrawal et al. (2015); Qi et al. (2019); Doersch et al. (2015); Kim et al. (2018); Larsson et al. (2016); Zhang et al. (2019)) to train unsupervised deep networks. For example, Doersch et al. (2015) used the relative positions of two randomly sampled patches as the supervised signal. Agrawal et al. (2015); Zhang et al. (2019); Gidaris et al. (2018) adopted various geometric transformations on images and used the transformation parameters to train deep networks. For more details about these works, please refer to the survey by Jing & Tian (2020). ",
+        "bbox": [
+            174,
+            854,
+            823,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/fc9396798788c1e2df6331a2190dd5aab1b2d2e388e0702049ba14da1706f1c3.jpg",
+        "image_caption": [
+            "Figure 1: Contrastive instance learning framework "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            241,
+            87,
+            751,
+            284
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            314,
+            823,
+            343
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 THE PROPOSED METHOD ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            353,
+            416,
+            371
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section, we will first review the preliminary work on the contrastive learning, and discuss their limitations on applying stronger augmentations to explore the novel patterns of representations. Then we will present a new distributional divergence loss between weakly and strongly augmented images to self-train the representations over a representation bank consisting of both negative samples and positive targets. After that, the algorithm and the implementation details will also be explained. ",
+        "bbox": [
+            173,
+            385,
+            825,
+            469
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 PRELIMINARY METHODS AND LIMITATIONS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            486,
+            519,
+            501
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Contrastive learning (Hadsell et al. (2006)) is a popular self-supervised idea and made great success in recent years with the advance of computation and various image augmentations. In contrastive learning, each image is considered as an instance, so it’s also known as instance learning. ",
+        "bbox": [
+            174,
+            512,
+            825,
+            554
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Fig. 1 illustrates general framework of contrastive learning methods.For each image $x$ in batch $B$ , we apply two transformations $T$ and $T ^ { \\prime }$ to obtain two different views $V$ and $V ^ { \\prime }$ of the same instance $x$ . Then they go through a key encoder and a query encoder respectively, followed with MLP projection layers, resulting in two embedded representations $z$ and $z ^ { \\prime }$ to calculate the constrastive loss. The key factor of contrastive learning is the quality and the number of negative examples. To deal with those issues, various methods have been proposed, such as Memory Bank (Wu et al. (2018)), momentum encoder (He et al. (2020)), and online learning with bigger batch (Chen et al. (2020a)). ",
+        "bbox": [
+            173,
+            560,
+            825,
+            659
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Specifically, the contrastive loss is developed to maximize the agreement of representations of different views of the same instance while minimizing the agreement with other negative samples. Hence, the contrastive loss for the different views of the same instance can be defined in Eq. (1). ",
+        "bbox": [
+            174,
+            665,
+            823,
+            708
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/db4e895bc5ec54e5d74b2b92f9b4d21412c9896aa285755eecf67662012c3fd8.jpg",
+        "text": "$$\n\\mathcal { L } _ { C } = - \\frac { 1 } { | B | } \\sum _ { i \\in B } \\log \\frac { \\exp ( s i m ( z _ { i } ^ { \\prime } , z _ { i } ) / \\tau ) } { \\exp ( s i m ( z _ { i } ^ { \\prime } , z _ { i } ) / \\tau ) + \\sum _ { k = 0 } ^ { K } \\exp ( s i m ( z _ { i } ^ { \\prime } , z _ { k } ) / \\tau ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            264,
+            713,
+            732,
+            752
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "with the cosine similarity ",
+        "bbox": [
+            174,
+            757,
+            341,
+            772
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6d44b61b35e593811cec6f4320a4dde352b7583ab8f32ea297598e2c4894b552.jpg",
+        "text": "$$\ns i m ( z _ { i } , z _ { j } ) = z _ { i } ^ { T } z _ { j } / ( | | z _ { i } | | \\cdot | | z _ { j } | | )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            383,
+            770,
+            614,
+            789
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { L } _ { C }$ is the contrastive loss, $z _ { i }$ and $z _ { i } ^ { \\prime }$ are the projected representations of the different augmentations of the same sample $x _ { i }$ , the summation is taken over the samples $x _ { i }$ in the current batch $B$ , and $\\tau$ is the temperature parameter set to 0.2. Also, the negative pool $Q = \\{ z _ { k } | k = 1 , \\cdot \\cdot \\cdot , K \\}$ shown in Fig. 1 is a queue of size $K$ storing the embedded features from the past batches in a FIFO fashion. This will keep the examples from the most recent batches in the queue while removing these obsoleted ones from it, which has been widely adopted in previous works (Chen et al. (2020b); He et al. (2020)). ",
+        "bbox": [
+            173,
+            791,
+            825,
+            888
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As illustrated in Fig. 1, previous contrastive learning works use two transformations $T$ and $T ^ { \\prime }$ to generate two different views $V$ and $V ^ { \\prime }$ , in which the transformations are carefully designated. Thus, the two views are not transformed aggressively so that they can still be viewed as the same instance. However, directly adopting stronger transformations (e.g., with larger rotation angles, more aggressive colorjittering and cutout) in contrastive learning fails to further improve the performance or even deteriorate it for downstream tasks, which is not surprising. Stronger transformations could distort image structures and their perceptual patterns in the learned representation so that strongly augmented samples from the same instance cannot be viewed as keeping the same instance identity for training the underlying network. However, stronger augmentations can expose useful clues to the novel patterns that cannot be revealed from moderately augmented images. In supervised learning (Cubuk et al. (2018); Lim et al. (2019); Hataya et al. (2019); Cubuk et al. (2020)), data augmentation search have been widely studied and greatly boost the performance with the novel pattern exposed by strongly augmented images. The findings in RandAugment (Cubuk et al. (2020)) have verified that strongly augmented views can provide more clues even without an explicit augmentation policy. We believe learning the representations from these novel patterns will pave the last mile to close the gap with the fully supervised representations. Indeed, in semi-supervised learning and supervised learning (Cubuk et al. (2020); Qi et al. (2019); Wang et al. (2019)), more aggressive augmentations have been adopted and achieved extraordinary performances. For example, AET Qi et al. (2019) has adopted the parameters of augmentations as supervised signal to self-supervise the training of networks. All of these findings have inspired us to explore novel ways to utilize stronger transformations in self-supervised learning while avoiding deteriorated performances by naively using them in a contrastive model (Chen et al. (2020a)). All of these have inspired us to explore novel ways to utilize stronger transformations in self-supervised learning. ",
+        "bbox": [
+            173,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
+            825,
+            395
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/268a14d1dc57c7c205146a98e8e874c8e5fc108c8bbb61a6ea5de761f98921e4.jpg",
+        "image_caption": [
+            "Figure 2: Comparison of the strongly weakly augmented images. The left is the original image, the middle is the weakly augmented image, and the right is the strongly augmented one with overcontrastive details. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            410,
+            825,
+            577
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "However, it is not easy. As shown in Fig. 2, a strongly augmented image may look perceptually different from the original counterpart. Consequently, the representation of a strongly augmented image can be far apart from that of the weakly augmented one. Thus, naively using strongly augmented images in contrastive learning can be over-optimistic since the induced distortions could dramatically ruin their image structures. To this end, in Section 3.2, we instead proposed the Distributional Divergence Minimization (DDM) between weakly and strongly augmented images over a representation bank to avoid overfitting the representation of a strongly augmented image with that of the corresponding positive target. ",
+        "bbox": [
+            174,
+            646,
+            825,
+            757
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 DISTRIBUTIONAL DIVERGENCE MINIMIZATION BETWEEN WEAKLY AND STRONGLY AUGMENTED IMAGES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            784,
+            795,
+            810
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Due to the aforementioned limitation of stronger augmentations in contrastive learning, a direct retrieval of a strongly augmented query is infeasible to self-train deep networks. Fortunately, the distribution of relative similarities of a weakly augmented image from the same instance over the representation bank can provide useful information to bridge the gap. As explained below, it does not only avoid directly placing the representation of a strongly augmented image too closer to that of the positive target, but also allows it to explore the novel patterns of variations exposed by the strong augmentation. ",
+        "bbox": [
+            173,
+            827,
+            825,
+            924
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/472bf4d8591573c6a5b08b9f46db3447cb762de241cd82f4bc5df537b81d7bb7.jpg",
+        "image_caption": [
+            "Figure 3: Diagram of distributional divergence minimization. Here the representation bank consists of $K$ stored features $z _ { k }$ in the negative pool and online features $z _ { i }$ from the key encoder. They will be used to calculate the conditional probability of current weakly and strongly augmented images. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            106,
+            818,
+            292
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Formally, as shown in Fig. 3, for an original image $x _ { i }$ and the representation $z _ { i }$ of the corresponding positive target, we applied a weaker and a stronger augmentation to obtain two separate views $V _ { i } ^ { \\prime }$ and $V _ { i } ^ { \\prime \\prime }$ , and their embeddings $z _ { i } ^ { \\prime }$ (weak representation) and $z _ { i } ^ { \\prime \\prime }$ (strong representation). Given a pool $Q$ (shown in Fig. 1) of $K$ negative samples $\\{ z _ { k } | k = 1 , \\dot { \\cdots } , K \\}$ accumulated from the past iterations, we obtain a conditional distribution ",
+        "bbox": [
+            174,
+            362,
+            825,
+            433
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/67abee15978e0981aae5d2e3696b4b5c0866c1e0ae749e02f5b51f7e49407a62.jpg",
+        "text": "$$\np ( z _ { k } | z _ { i } ^ { \\prime } ) = \\frac { e x p ( s i m ( z _ { i } ^ { \\prime } , z _ { k } ) / \\tau ) } { e x p ( s i m ( z _ { i } ^ { \\prime } , z _ { i } ) / \\tau ) + \\sum _ { k = 0 } ^ { K } e x p ( s i m ( z _ { i } ^ { \\prime } , z _ { k } ) / \\tau ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            295,
+            438,
+            700,
+            476
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "which encodes the likelihood of the weaker query $z _ { i } ^ { \\prime }$ being assigned to $z _ { k }$ . In the similar way, we can define the likelihood $p ( z _ { i } | z _ { i } ^ { \\prime } )$ of the query being assigned to the positive target $z _ { i }$ , as well as the likelihoods $p ( z _ { i } | z _ { i } ^ { \\prime \\prime } )$ and $p ( z _ { k } | z _ { i } ^ { \\prime \\prime } )$ for the stronger query $z _ { i } ^ { \\prime \\prime }$ . Here the representation bank consists of the negative pool $Q$ and the positive query targets $z _ { i }$ from the current batch $B$ . ",
+        "bbox": [
+            174,
+            481,
+            825,
+            537
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Then, we propose to minimize the following distributional divergence between the weak and the strong queries such that ",
+        "bbox": [
+            173,
+            544,
+            823,
+            571
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/144232bca21264c0ddf109e0131768c1c7b579e1300d624345beb4367f003ecb.jpg",
+        "text": "$$\n\\mathcal { L } _ { D } = \\frac { 1 } { | B | } \\sum _ { i \\in B } \\left[ - \\sum _ { k = 1 } ^ { K } p ( z _ { k } | z _ { i } ^ { \\prime } ) \\log ( p ( z _ { k } | z _ { i } ^ { \\prime \\prime } ) ) - p ( z _ { i } | z _ { i } ^ { \\prime } ) \\log ( p ( z _ { i } | z _ { i } ^ { \\prime \\prime } ) ) \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            259,
+            579,
+            736,
+            622
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "By minimizing this divergence, we assume the learned representation $z _ { i } ^ { \\prime \\prime }$ of the strongly augmented query should inherit the representation $z _ { i } ^ { \\prime }$ of the weakly augmented one regarding not only its belief of the query being assigned to the corresponding positive target $z _ { i }$ , but also its relations with the negative samples $z _ { k }$ in the representation bank through the conditional distribution $p ( z _ { k } | z _ { i } ^ { \\prime } )$ . ",
+        "bbox": [
+            174,
+            628,
+            825,
+            685
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "This will prevent a direct overfitting of the strong query representation to the positive target as well as improve the generalization of the learned representation with additional clues from the other examples in the representation pool. In a more general sense, this extends the idea of knowledge distillation (Hinton et al. (2015)). However, we did not use the predicted labels by a teacher model to supervise the training of a student model as in the knowledge distillation. Instead, we used the distribution of the likelihoods of a weak query to supervise the retrieval of a strong query from a pool of representations. ",
+        "bbox": [
+            173,
+            691,
+            825,
+            790
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            806,
+            405,
+            820
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 gives the pseudo code to implement the proposed CLSA method. In the following, we will discuss the details about the applied strong and weak augmentations for distributional divergence minimization. ",
+        "bbox": [
+            174,
+            832,
+            825,
+            875
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Stronger Augmentations $\\pmb { S }$ As explored in the previous works (Cubuk et al. (2018); Wang et al. (2019); Qi et al. (2019)), strong augmentations usually have two types: geometric and non-geometric augmentations. Specifically, we considered 14 types of augmentations: ShearX/Y, TranslateX/Y, ",
+        "bbox": [
+            174,
+            881,
+            823,
+            924
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 Pseudo code for the proposed CLSA ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            503,
+            117
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Input: $f _ { \\theta } , f _ { \\phi }$ : the query and the key encoder networks; $g _ { \\boldsymbol { \\theta } } , g _ { \\boldsymbol { \\phi } }$ : the MLP projection layers for the query and the key; $Q$ : a queue of representations of $K$ negative samples; $\\alpha$ : momentum decay for the key network; $\\tau$ : the temperature; $T$ and $T ^ { \\prime }$ : weak augmentation; $S$ : strong augmentation; $\\beta$ : the balancing coefficient. ",
+        "bbox": [
+            178,
+            122,
+            825,
+            178
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "1: Initialize the network $f _ { \\phi } = f _ { \\theta }$ , $g _ { \\phi } = g _ { \\theta }$ and $Q$ ; ",
+        "bbox": [
+            183,
+            179,
+            511,
+            193
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": ". see Eq. (1) . see Eq. (4) ",
+        "bbox": [
+            736,
+            236,
+            823,
+            262
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Rotate, AutoContrast, Invert, Equalize, Solarize, Posterize, Contrast, Color, Brightness, Sharpness. The magnitude of each augmentation is significant enough to produce as strong augmentations as possible. More details are shown in Table 1. For example, the shear is drawn from a range of [-0.3,0.3], which results in aggressively transformed images that can be hard to retrieve given a counterpart target. In particular, to transform an image, we randomly select an augmentation from the above 14 types of transformations, and apply it to the image with a probability of 0.5. This process is repeated five times and that will strongly augment an image as the example shown in the right panel of Fig. 2. ",
+        "bbox": [
+            174,
+            362,
+            825,
+            474
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/0cda85b789c6d756347bd228af8d5098903d4d87b50af4077798761715452875.jpg",
+        "table_caption": [
+            "Table 1: Various augmentations we applied in experiments to strongly augment training images. "
+        ],
+        "table_footnote": [
+            "10: end for Output: The representation encoder $f _ { \\theta }$ . "
+        ],
+        "table_body": "<table><tr><td>Operation Mag Range</td><td>ShearX(Y) [-0.3,0.3]</td><td>TranslateX(Y) [-0.3,0.3]</td><td>Rotate [-30,30]</td><td>AutoContrast 0or1</td><td>Invert 0or 1</td><td>Equalize 0or1</td></tr><tr><td>Operation Mag Range</td><td>Solarize [0,256]</td><td>Posterize [4,8]</td><td>Contrast [0.05,0.95]</td><td>Color [0.05,0.95]</td><td>Brightness [0.05,0.95]</td><td>Sharpeness [0.05,0.95]</td></tr></table>",
+        "bbox": [
+            176,
+            513,
+            823,
+            578
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Weaker Augmentations $\\mathbf { T }$ Weak augmentations are drawn by following most of existing contrastive learning methods in literature (Chen et al. (2020a;b); Caron et al. (2020); He et al. (2020)): an image is first cropped from an input image and resized to $2 2 4 \\times 2 2 4$ pixels. Then random color jittering, Gaussian Blur, grayscale conversion, horizontal flip, channel-wise color normalization are sequentially applied to generate weakly augmented images with an example shown in the middle of Fig. 2. ",
+        "bbox": [
+            173,
+            587,
+            825,
+            671
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Technical Details Similar to the previous works (He et al. (2020); Chen et al. (2020a); Caron et al. (2020)), we used the ResNet-50 (He et al. (2016)) as our encoder backbones $f _ { \\theta }$ and $f _ { \\phi }$ and a 2- layer MLP (2048-d hidden layer with the ReLU) as the projection head $g _ { \\boldsymbol { \\theta } }$ and $g _ { \\phi }$ . The projected representation $z$ is first L2-normalized (Wu et al. (2018)) before calculating the cosine similarity. The temperature $\\tau$ is set to 0.2, with a momentum smoothing factor $\\alpha$ of 0.999 and a fixed balancing coefficient $\\beta$ of 1.0. We set the size $K$ of the queue $Q$ to 65536 to store the negative examples used to compute the conditional distribution of weakly and strongly augmented queries and minimize their divergence. ",
+        "bbox": [
+            173,
+            678,
+            825,
+            790
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            810,
+            326,
+            825
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1 TRAINING DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            842,
+            351,
+            856
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For the unsupervised pretraining on ImageNet with the CLSA, we used the SGD optimizer (Bottou (2010)) with an initial learning rate of 0.03, a weight decay of 0.0001 and a momentum of 0.9. We used cosine scheduler (Loshchilov & Hutter (2016)) to gradually decay the learning rate to 0. Usually, the batch size is set to 256. When multiple GPU cluster servers are used, the batch size will be multiplied by the same number of servers by convention. Typically, the experiment with a single strong augmentation for each training image takes roughly 70 hours to finish on 8 V100 GPUs. ",
+        "bbox": [
+            174,
+            867,
+            823,
+            924
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/29de55d0090b30b89597bd022e591d2f86e023f4bc773f4e743c95a2093debac.jpg",
+        "table_caption": [
+            "Table 2: Top-1 accuracy under the linear evaluation on ImageNet with the ResNet-50 backbone. The left table compared methods trained over 200 epochs, and the right table compared the methods with various numbers of epochs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Top1</td><td>Method</td><td>Top1</td></tr><tr><td>InstDisc (Wu et al. (2018))</td><td>54.0</td><td>BigBiGAN (Donahue &amp; Simonyan (2019))</td><td>56.6</td></tr><tr><td>LocalAgg (Zhuang et al. (2019))</td><td>58.8</td><td>SeLa-400epochs (Asano et al. (2019))</td><td>61.5</td></tr><tr><td>MoCo (He et al. (2020))</td><td>60.8</td><td>PIRL-800epochs (Misra &amp; Maaten (2020))</td><td>63.6</td></tr><tr><td>SimCLR (Chen et al. (2020a))</td><td>61.9</td><td>CMC (Tian et al. (2019))</td><td>66.2</td></tr><tr><td>CPC v2 (Hénaff et al. (2019))</td><td>63.8</td><td>SimCLR-800epochs (Chen et al. (2020a))</td><td>70.0</td></tr><tr><td>PCL (Li et al. (2020))</td><td>65.9</td><td>MoCo v2-800epochs (Chen et al. (2020b))</td><td>71.1</td></tr><tr><td>MoCo v2 (Chen et al. (2020b))</td><td>67.5</td><td>InfoMin Aug-800epochs (Tian et al. (2020))</td><td>73.0</td></tr><tr><td>InfoMin Aug (Tian et al. (2020))</td><td>70.1</td><td>BYOL-1000epochs (Grill et al. (2020))</td><td>74.3</td></tr><tr><td>SWAV (Caron et al. (2020))</td><td>72.7</td><td>SWAV-800epochs (Caron et al. (2020))</td><td>75.3</td></tr><tr><td>CLSA-Single</td><td>69.4</td><td>CLSA-Single-800epochs</td><td>72.2</td></tr><tr><td>CLSA-Multi</td><td>73.3</td><td>CLSA-Multi-800epochs</td><td>76.2</td></tr><tr><td>Supervised</td><td>76.5</td><td>Supervised</td><td>76.5</td></tr></table>",
+        "bbox": [
+            173,
+            155,
+            833,
+            359
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            377,
+            823,
+            406
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For the fine-tuning on ImageNet, we trained a linear classifier on top of the frozen feature vector (2048-D) upon the pretrained ResNet-50 with CLSA. This linear layer is trained for 100 epochs, with a learning rate of 10 without weight decay. We used the cosine learning rate decay and a batch size of 256. ",
+        "bbox": [
+            174,
+            412,
+            825,
+            468
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For the transfer learning on the VOC dataset, we trained a linear classifier upon the pretrained Resnet-50 in the similar way for ImageNet – we trained 100 epochs with the SGD optimizer and a learning rate of 0.05, a momentum of 0.9 and no weight decay. The batch size is 256 without learning rate scheduler. ",
+        "bbox": [
+            174,
+            476,
+            825,
+            531
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Finally, for object detection, we adopted the same protocol in (He et al. (2020)) to fine-tune the pretrained Resnet-50 backbone based on detectron2 (Wu et al. (2019)) for the sake of a fair and straight comparison with the other methods. ",
+        "bbox": [
+            174,
+            539,
+            825,
+            580
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2 LINEAR CLASSIFICATION ON IMAGENET ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            603,
+            495,
+            617
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For the linear evaluation on ImageNet, we trained the CLSA in two settings. In the first setting named CLSA-Single, we used a single stronger augmentation (see Table 1) that crops each training image to a smaller size of $9 6 \\times 9 6$ , which does not incur too much computing overhead in processing these smaller augmented images. In the second setting named CLSA-Multi, we adopted five different stronger augmentations that crop each image into various sizes: $2 2 4 \\times 2 2 4$ , $1 9 2 \\times 1 9 2$ , $1 6 0 \\times$ 160, $1 2 8 \\times 1 2 8$ , and $9 6 \\times 9 6$ . The DDM loss in Eq. (4) is the sum over these multiple stronger augmentations. The similar multi-crop strategy has been adopted in contrastive learning literature before. For example, the SWAV (Caron et al. (2020)) reached the state-of-the-art top-1 accuracy by applying such multi-crop augmentations. To ensure a fair comparison with the SWAV, we chose five stronger augmentations such that the self-training with CLSA-Multi consumed the same computing time (i.e., 166 hours with a cluster of 8 V100 GPUs for 200 epochs of pre-training with a batch size of 256). ",
+        "bbox": [
+            174,
+            631,
+            825,
+            797
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "As shown in Table 2, we compared the performance with the other unsupervised methods. All the experiments are based on a pretrained ResNet-50 backbone that is fine-tuned with a linear classifier. The left table showed the performance of different methods pretrained over 200 epochs, and the right table reported models pretrained over more epochs. ",
+        "bbox": [
+            174,
+            805,
+            823,
+            861
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "First, under the same contrastive protocol, the CLSA-Single has a higher $6 9 . 4 \\%$ top-1 accuracy than both MoCo v2 $( 6 7 . 5 \\% )$ and SimCLR $( 6 1 . 9 \\% )$ with 200 epochs training. With multiple stronger augmentations, the CLSA-Multi outperforms the State-of-the-art SWAV model using multi-crops of training images over 200 epochs $7 3 . 3 \\%$ vs. $7 2 . 7 \\%$ . Moreover, as shown in the right table, the CLSA",
+        "bbox": [
+            174,
+            867,
+            823,
+            924
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Single outperforms MoCo v2 and SimCLR with the same training epochs. It is also noteworthy that the CLSA-Multi achieves almost the same top-1 accuracy as that of the fully supervised network ( $7 6 . 2 \\%$ vs. $7 6 . 5 \\%$ ). ",
+        "bbox": [
+            174,
+            103,
+            823,
+            146
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.3 TRANSFER LEARNING RESULTS ON DOWNSTREAM TASKS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            161,
+            611,
+            176
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/fff186a31bced870242059f910409488891a14677cacab47df76b9c52b8fdbdf.jpg",
+        "table_caption": [
+            "Table 3: Transfer learning results on various downstream tasks. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"3\">Measurement</td><td>Classification</td><td colspan=\"3\">Object Detection</td></tr><tr><td>VOC07</td><td>VOC07+12</td><td></td><td>COCO</td></tr><tr><td>Accuracy</td><td>AP50</td><td>AP</td><td>APs</td></tr><tr><td>RotNet (Gidaris et al. (2018))</td><td>64.6</td><td>=</td><td>=</td><td>=</td></tr><tr><td>NPID++ (Wu et al. (2018))</td><td>76.6</td><td>79.1</td><td></td><td></td></tr><tr><td>MoCo (He et al. (2020))</td><td>79.8</td><td>81.5</td><td>=</td><td></td></tr><tr><td>PIRL (Misra &amp; Maaten (2020))</td><td>81.1</td><td>80.7</td><td>=</td><td></td></tr><tr><td>PCL (Li et al. (2020))</td><td>84.0</td><td>1</td><td></td><td></td></tr><tr><td>BoWNet (Gidaris et al. (2020))</td><td>79.3</td><td>81.3</td><td></td><td></td></tr><tr><td>SimCLR (Chen et al. (2020a))</td><td>86.4</td><td>=</td><td>=</td><td>=</td></tr><tr><td>MoCov2 (Chen et al. (2020b))</td><td>87.1</td><td>82.5</td><td>42.0</td><td>20.8</td></tr><tr><td>SWAV (Caron et al. (2020))</td><td>88.9</td><td>82.6</td><td>42.1</td><td>19.7</td></tr><tr><td>CLSA</td><td>93.6</td><td>83.2</td><td>42.3</td><td>24.4</td></tr><tr><td>Supervised</td><td>87.5</td><td>81.3</td><td>40.8</td><td>20.1</td></tr></table>",
+        "bbox": [
+            232,
+            215,
+            763,
+            434
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We test the generalizability of the ResNet-50 pre-trained on ImageNet to several downstream tasks. Specifically, we focused on two tasks: cross-dataset image classification and object detection. The pre-trained ResNet-50 was frozon, and we fine-tuned the linear classifier on the VOC07trainval (Everingham et al. (2010)) and tested it on the VOC07test. For object detection, we evaluated the pre-trained network on two datasets using the detectron2 (Wu et al. (2019)) used in the previous methods He et al. (2020); Chen et al. (2020a). On the VOC dataset (Everingham et al. (2010)), we trained the detection head with $\\mathrm { \\ V O C { 0 7 + 1 2 } }$ trainval dataset and tested on VOC07 test dataset. On the COCO dataset (Lin et al. (2014)), we fine-tuned the network on the train2017 set with $1 1 8 \\mathrm { k }$ images and evaluate on the val2017. For the sake of a fair comparison, the object detection tasks are completed by detectron2 (He et al. (2017)) based on the pretrained ResNet-50. ",
+        "bbox": [
+            173,
+            435,
+            825,
+            574
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "As shown in Table 3, the performances on both tasks are much better than the supervised model trained on ImageNet. This suggests that the proposed method has better generalization ability in downstream tasks. The pre-trained network on ImageNet by the CLSA outperformed the compared models after being fine-tuned on different datasets. Among them is a top-1 accuracy of $9 3 . 6 \\%$ on the VOC07 with the linear classifier on the pretrained ResNet-50 in comparison with the previous record of $8 8 . 9 \\%$ top-1 accuracy by the SWAV. On the COCO dataset, the $A P _ { S }$ for small object detection has been significantly improved to $2 4 . 4 \\%$ from the previously best $A P _ { S }$ of $2 0 . 8 \\%$ . As well known, it is much challenging to detect small objects on the COCO dataset. Thus, the better performance of the CLSA could be attributed to the ability of involving the stronger augmentations that result in many small objects to pretrain the network. ",
+        "bbox": [
+            173,
+            582,
+            825,
+            720
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.4 ABLATION STUDY ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            737,
+            341,
+            751
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/4fc4ffef4b3c0c0172da65274f1c56f8f4cd2ca439c52a4de8c9ccd4eaf045de.jpg",
+        "table_caption": [
+            "Table 4: Ablation study of the CLSA on ImageNet with 200 epochs of pre-training. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Top-1 Accuracy</td></tr><tr><td>MoCo V2</td><td>67.5</td></tr><tr><td>MoCo V2 with Strong query</td><td>67.7</td></tr><tr><td>MoCo V2 with Strong query &amp; Strong key</td><td>67.0</td></tr><tr><td>CLSA-Single with contrastive loss</td><td>68.0</td></tr><tr><td>CLSA-Single</td><td>69.4</td></tr></table>",
+        "bbox": [
+            282,
+            790,
+            712,
+            891
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In the ablation study shown in Table 4, we studied the role of the proposed DDM loss in the CLSA. First, we naively used the stronger augmentation applied in the CLSA-Single as the query and/or the key in the MoCo V2. Both results (Strong query and Strong query & Strong key) showed the performance can not be improved or even degraded. Second, we replaced the DDM loss in the CLSA-Single with the contrastive loss, and we found it can only achieved a top-1 accuracy of $6 8 . 0 \\%$ compared to that of $6 9 . 4 \\%$ with the DDM loss. Both studies showed that the proposed CLSA and its DDM loss help us learn from stronger augmentations by avoiding the performance degeneration that would be incurred by the distortions of augmented images. ",
+        "bbox": [
+            174,
+            895,
+            820,
+            924
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            825,
+            188
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            207,
+            318,
+            223
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we present CLSA, a novel method that can utilize the distributional divergence to learn the information from strongly augmented images. The proposed method outperforms the state-ofthe-art methods on all the datasets and achieved almost same performance compared to supervised ImageNet network. Meanwhile, it outperforms the previous supervised and self-supervised methods on downstream tasks, which suggests CLSA learned more reliable and fine-grained features that can contribute to the development of other areas. ",
+        "bbox": [
+            174,
+            238,
+            825,
+            321
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            343,
+            285,
+            358
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Pulkit Agrawal, Joao Carreira, and Jitendra Malik. Learning to see by moving. In Proceedings of the IEEE international conference on computer vision, pp. 37–45, 2015. ",
+        "bbox": [
+            173,
+            366,
+            825,
+            395
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Yuki Markus Asano, Christian Rupprecht, and Andrea Vedaldi. Self-labelling via simultaneous clustering and representation learning. arXiv preprint arXiv:1911.05371, 2019. ",
+        "bbox": [
+            173,
+            402,
+            823,
+            433
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Philip Bachman, R Devon Hjelm, and William Buchwalter. Learning representations by maximizing mutual information across views. In Advances in Neural Information Processing Systems, pp. 15535–15545, 2019. ",
+        "bbox": [
+            173,
+            439,
+            823,
+            483
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Piotr Bojanowski and Armand Joulin. Unsupervised learning by predicting noise. arXiv preprint arXiv:1704.05310, 2017. ",
+        "bbox": [
+            171,
+            491,
+            825,
+            520
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Leon Bottou. Large-scale machine learning with stochastic gradient descent. In ´ Proceedings of COMPSTAT’2010, pp. 177–186. Springer, 2010. ",
+        "bbox": [
+            169,
+            529,
+            825,
+            558
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Mathilde Caron, Piotr Bojanowski, Armand Joulin, and Matthijs Douze. Deep clustering for unsupervised learning of visual features. In Proceedings of the European Conference on Computer Vision (ECCV), pp. 132–149, 2018. ",
+        "bbox": [
+            173,
+            565,
+            825,
+            608
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Mathilde Caron, Piotr Bojanowski, Julien Mairal, and Armand Joulin. Unsupervised pre-training of image features on non-curated data. In Proceedings of the IEEE International Conference on Computer Vision, pp. 2959–2968, 2019. ",
+        "bbox": [
+            173,
+            616,
+            823,
+            660
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Mathilde Caron, Ishan Misra, Julien Mairal, Priya Goyal, Piotr Bojanowski, and Armand Joulin. Unsupervised learning of visual features by contrasting cluster assignments. arXiv preprint arXiv:2006.09882, 2020. ",
+        "bbox": [
+            173,
+            667,
+            823,
+            710
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton. A simple framework for contrastive learning of visual representations. arXiv preprint arXiv:2002.05709, 2020a. ",
+        "bbox": [
+            169,
+            718,
+            823,
+            747
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Xinlei Chen, Haoqi Fan, Ross Girshick, and Kaiming He. Improved baselines with momentum contrastive learning. arXiv preprint arXiv:2003.04297, 2020b. ",
+        "bbox": [
+            169,
+            756,
+            823,
+            785
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Ekin D Cubuk, Barret Zoph, Dandelion Mane, Vijay Vasudevan, and Quoc V Le. Autoaugment: Learning augmentation policies from data. arXiv preprint arXiv:1805.09501, 2018. ",
+        "bbox": [
+            171,
+            792,
+            823,
+            821
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Ekin D Cubuk, Barret Zoph, Jonathon Shlens, and Quoc V Le. Randaugment: Practical automated data augmentation with a reduced search space. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops, pp. 702–703, 2020. ",
+        "bbox": [
+            176,
+            829,
+            823,
+            873
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In 2009 IEEE conference on computer vision and pattern recognition, pp. 248–255. Ieee, 2009. ",
+        "bbox": [
+            174,
+            881,
+            823,
+            922
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Carl Doersch, Abhinav Gupta, and Alexei A Efros. Unsupervised visual representation learning by context prediction. In Proceedings of the IEEE international conference on computer vision, pp. 1422–1430, 2015. ",
+        "bbox": [
+            176,
+            103,
+            823,
+            146
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jeff Donahue and Karen Simonyan. Large scale adversarial representation learning. In Advances in Neural Information Processing Systems, pp. 10542–10552, 2019. ",
+        "bbox": [
+            171,
+            154,
+            823,
+            183
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jeff Donahue, Philipp Krahenb ¨ uhl, and Trevor Darrell. Adversarial feature learning. ¨ arXiv preprint arXiv:1605.09782, 2016. ",
+        "bbox": [
+            171,
+            190,
+            823,
+            219
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Alexey Dosovitskiy, Philipp Fischer, Jost Tobias Springenberg, Martin Riedmiller, and Thomas Brox. Discriminative unsupervised feature learning with exemplar convolutional neural networks. IEEE transactions on pattern analysis and machine intelligence, 38(9):1734–1747, 2015. ",
+        "bbox": [
+            174,
+            227,
+            823,
+            271
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Mark Everingham, Luc Van Gool, Christopher KI Williams, John Winn, and Andrew Zisserman. The pascal visual object classes (voc) challenge. International journal of computer vision, 88(2): 303–338, 2010. ",
+        "bbox": [
+            174,
+            277,
+            823,
+            320
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Spyros Gidaris, Praveer Singh, and Nikos Komodakis. Unsupervised representation learning by predicting image rotations. arXiv preprint arXiv:1803.07728, 2018. ",
+        "bbox": [
+            171,
+            328,
+            823,
+            358
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Spyros Gidaris, Andrei Bursuc, Nikos Komodakis, Patrick Perez, and Matthieu Cord. Learning ´ representations by predicting bags of visual words. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 6928–6938, 2020. ",
+        "bbox": [
+            174,
+            364,
+            823,
+            409
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jean-Bastien Grill, Florian Strub, Florent Altche, Corentin Tallec, Pierre H Richemond, Elena ´ Buchatskaya, Carl Doersch, Bernardo Avila Pires, Zhaohan Daniel Guo, Mohammad Gheshlaghi Azar, et al. Bootstrap your own latent: A new approach to self-supervised learning. arXiv preprint arXiv:2006.07733, 2020. ",
+        "bbox": [
+            173,
+            416,
+            823,
+            472
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Raia Hadsell, Sumit Chopra, and Yann LeCun. Dimensionality reduction by learning an invariant mapping. In 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’06), volume 2, pp. 1735–1742. IEEE, 2006. ",
+        "bbox": [
+            176,
+            481,
+            821,
+            523
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Ryuichiro Hataya, Jan Zdenek, Kazuki Yoshizoe, and Hideki Nakayama. Faster autoaugment: Learning augmentation strategies using backpropagation. arXiv preprint arXiv:1911.06987, 2019. ",
+        "bbox": [
+            176,
+            531,
+            821,
+            560
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770–778, 2016. ",
+        "bbox": [
+            174,
+            568,
+            823,
+            611
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Kaiming He, Georgia Gkioxari, Piotr Dollar, and Ross Girshick. Mask r-cnn. In ´ Proceedings of the IEEE international conference on computer vision, pp. 2961–2969, 2017. ",
+        "bbox": [
+            173,
+            618,
+            823,
+            648
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Kaiming He, Haoqi Fan, Yuxin Wu, Saining Xie, and Ross Girshick. Momentum contrast for unsupervised visual representation learning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 9729–9738, 2020. ",
+        "bbox": [
+            176,
+            655,
+            823,
+            699
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Olivier J Henaff, Aravind Srinivas, Jeffrey De Fauw, Ali Razavi, Carl Doersch, SM Eslami, and ´ Aaron van den Oord. Data-efficient image recognition with contrastive predictive coding. arXiv preprint arXiv:1905.09272, 2019. ",
+        "bbox": [
+            174,
+            707,
+            823,
+            750
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Geoffrey Hinton, Oriol Vinyals, and Jeff Dean. Distilling the knowledge in a neural network. arXiv preprint arXiv:1503.02531, 2015. ",
+        "bbox": [
+            173,
+            756,
+            823,
+            786
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "R Devon Hjelm, Alex Fedorov, Samuel Lavoie-Marchildon, Karan Grewal, Phil Bachman, Adam Trischler, and Yoshua Bengio. Learning deep representations by mutual information estimation and maximization. arXiv preprint arXiv:1808.06670, 2018. ",
+        "bbox": [
+            174,
+            792,
+            823,
+            837
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Longlong Jing and Yingli Tian. Self-supervised visual feature learning with deep neural networks: A survey. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2020. ",
+        "bbox": [
+            174,
+            844,
+            821,
+            873
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Dahun Kim, Donghyeon Cho, Donggeun Yoo, and In So Kweon. Learning image representations by completing damaged jigsaw puzzles. In 2018 IEEE Winter Conference on Applications of Computer Vision (WACV), pp. 793–802. IEEE, 2018. ",
+        "bbox": [
+            174,
+            882,
+            825,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Under review as a conference paper at ICLR 2021   \nDiederik P Kingma and Max Welling. Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114, 2013.   \nGustav Larsson, Michael Maire, and Gregory Shakhnarovich. Learning representations for automatic colorization. In European conference on computer vision, pp. 577–593. Springer, 2016.   \nJunnan Li, Pan Zhou, Caiming Xiong, Richard Socher, and Steven CH Hoi. Prototypical contrastive learning of unsupervised representations. arXiv preprint arXiv:2005.04966, 2020.   \nSungbin Lim, Ildoo Kim, Taesup Kim, Chiheon Kim, and Sungwoong Kim. Fast autoaugment. In Advances in Neural Information Processing Systems, pp. 6665–6675, 2019.   \nTsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dollar, and C Lawrence Zitnick. Microsoft coco: Common objects in context. In ´ European conference on computer vision, pp. 740–755. Springer, 2014.   \nIlya Loshchilov and Frank Hutter. Sgdr: Stochastic gradient descent with warm restarts. arXiv preprint arXiv:1608.03983, 2016.   \nIshan Misra and Laurens van der Maaten. Self-supervised learning of pretext-invariant representations. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 6707–6717, 2020.   \nAaron van den Oord, Yazhe Li, and Oriol Vinyals. Representation learning with contrastive predictive coding. arXiv preprint arXiv:1807.03748, 2018.   \nGuo-Jun Qi, Liheng Zhang, and Xiao Wang. Learning generalized transformation equivariant representations via autoencoding transformations. arXiv preprint arXiv:1906.08628, 2019.   \nYonglong Tian, Dilip Krishnan, and Phillip Isola. Contrastive multiview coding. arXiv preprint arXiv:1906.05849, 2019.   \nYonglong Tian, Chen Sun, Ben Poole, Dilip Krishnan, Cordelia Schmid, and Phillip Isola. What makes for good views for contrastive learning. arXiv preprint arXiv:2005.10243, 2020.   \nPascal Vincent, Hugo Larochelle, Yoshua Bengio, and Pierre-Antoine Manzagol. Extracting and composing robust features with denoising autoencoders. In Proceedings of the 25th international conference on Machine learning, pp. 1096–1103, 2008.   \nXiao Wang, Daisuke Kihara, Jiebo Luo, and Guo-Jun Qi. Enaet: Self-trained ensemble autoencoding transformations for semi-supervised learning. arXiv preprint arXiv:1911.09265, 2019.   \nYuxin Wu, Alexander Kirillov, Francisco Massa, Wan-Yen Lo, and Ross Girshick. Detectron2. https://github.com/facebookresearch/detectron2, 2019.   \nZhirong Wu, Yuanjun Xiong, Stella X Yu, and Dahua Lin. Unsupervised feature learning via nonparametric instance discrimination. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 3733–3742, 2018.   \nXueting Yan, Ishan Misra, Abhinav Gupta, Deepti Ghadiyaram, and Dhruv Mahajan. Clusterfit: Improving generalization of visual representations. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 6509–6518, 2020.   \nLiheng Zhang, Guo-Jun Qi, Liqiang Wang, and Jiebo Luo. Aet vs. aed: Unsupervised representation learning by auto-encoding transformations rather than data. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2547–2555, 2019.   \nChengxu Zhuang, Alex Lin Zhai, and Daniel Yamins. Local aggregation for unsupervised learning of visual embeddings. In Proceedings of the IEEE International Conference on Computer Vision, pp. 6002–6012, 2019. ",
+        "bbox": [
+            171,
+            27,
+            826,
+            882
+        ],
+        "page_idx": 10
+    }
+]

parse/train/KJSC_AsN14/KJSC_AsN14_middle.json ADDED Viewed

The diff for this file is too large to render. See raw diff

parse/train/KJSC_AsN14/KJSC_AsN14_model.json ADDED Viewed

The diff for this file is too large to render. See raw diff

parse/train/rqfq0CYIekd/rqfq0CYIekd.md ADDED Viewed

	@@ -0,0 +1,324 @@

+# Reliable Post hoc Explanations: Modeling Uncertainty in Explainability
+Dylan Slack
+UC Irvine
+dslack@uci.edu
+Sophie Hilgard Harvard University ash798@g.harvard.edu
+Sameer Singh UC Irvine sameer@uci.edu
+Himabindu Lakkaraju Harvard University hlakkaraju@hbs.edu
+# Abstract
+As black box explanations are increasingly being employed to establish model credibility in high stakes settings, it is important to ensure that these explanations are accurate and reliable. However, prior work demonstrates that explanations generated by state-of-the-art techniques are inconsistent, unstable, and provide very little insight into their correctness and reliability. In addition, these methods are also computationally inefficient, and require significant hyper-parameter tuning. In this paper, we address the aforementioned challenges by developing a novel Bayesian framework for generating local explanations along with their associated uncertainty. We instantiate this framework to obtain Bayesian versions of LIME and KernelSHAP which output credible intervals for the feature importances, capturing the associated uncertainty. The resulting explanations not only enable us to make concrete inferences about their quality (e.g., there is a $9 5 \%$ chance that the feature importance lies within the given range), but are also highly consistent and stable. We carry out a detailed theoretical analysis that leverages the aforementioned uncertainty to estimate how many perturbations to sample, and how to sample for faster convergence. This work makes the first attempt at addressing several critical issues with popular explanation methods in one shot, thereby generating consistent, stable, and reliable explanations with guarantees in a computationally efficient manner. Experimental evaluation with multiple real world datasets and user studies demonstrate that the efficacy of the proposed framework.1
+# 1 Introduction
+As machine learning (ML) models get increasingly deployed in domains such as healthcare and criminal justice, it is important to ensure that decision makers have a clear understanding of the behavior of these models. However, ML models that achieve state-of-the-art accuracy are typically complex black boxes that are hard to understand. As a consequence, there has been a surge in post hoc techniques for explaining black box models [1–10]. Most popular among these techniques are local explanation methods which explain complex black box models by constructing interpretable local approximations (e.g., LIME [2], SHAP [4], MAPLE [11], Anchors [1]). Due to their generality, these methods are being leveraged to explain a number of classifiers including deep neural networks and ensemble models in a variety of domains such as law, medicine, and finance [12, 13].
+Existing local explanation methods, however, suffer from several drawbacks. Explanations generated using these methods may be unstable [14–18], i.e., negligibly small perturbations to an instance can result in substantially different explanations. These methods are also inconsistent [19] i.e., multiple runs on the same input instance with the same parameter settings may result in vastly different explanations. There are also no reliable metrics to ascertain the quality of the explanations
+![](images/d47efc4990be75e69296fffac100fa38760e5883dfb996999588185ce3d0e586.jpg)
+(b) Explanation with 2000 perturbations
+![](images/3f716bc0fdf329c8651b34a0ee8151d109ef2249803f50f834c077a2687e7548.jpg)
+Figure 1: Example explanations on for an instance from the COMPAS dataset, where vertical lines indicate the feature importance by LIME (red is negative effect, green is positive) and the shaded region visualizes the uncertainty estimated by BayesLIME. While LIME produces very different and contradictory feature importance for different number of perturbations (1a and 1b), BayesLIME provides more context. The overlapping uncertainty intervals in the explanation computed with 100 perturbations (1a) indicate that it is unclear which feature is the most important. However, the tighter uncertainty intervals in the explanation computed with 2K perturbations (1b) clearly indicates that Female is the most important.
+(a) Explanation computed with 100 perturbations
+output by these methods. Commonly used metrics such as explanation fidelity rely heavily on the implementation details of the explanation method (e.g., the perturbation function used in LIME) and do not provide a true picture of the explanation quality [20]. Furthermore, there exists little to no guidance on determining the values of certain hyperparameters that are critical to the quality of the resulting local explanations (e.g., number of perturbations in case of LIME). Local explanation methods are also computationally inefficient i.e., they typically require a large number of black box model queries to construct local approximations [21]. This can be prohibitively slow especially in case of complex neural models.
+In this paper, we identify that modeling uncertainty in black box explanations is the key to addressing all the aforementioned challenges. To this end, we propose a novel Bayesian framework for generating local explanations along with their associated uncertainty. We instantiate this framework to obtain Bayesian versions of LIME and KernelSHAP, namely BayesLIME and BayesSHAP, that not only output point-wise estimates of feature importance but also their associated uncertainty in the form of credible intervals (See Figure 1). We derive closed form expressions for the posteriors of the explanations thereby eliminating the need for any additional computational complexity. The credible intervals produced by our framework not only allow us to make concrete inferences about the quality of the resulting explanations but also produce explanations that satisfy user specified levels of uncertainty (e.g., an end user may request for explanations that satisfy a certain $9 5 \%$ confidence level). In addition, the resulting explanations are also highly consistent and stable. To the best of our knowledge, this work makes the first attempt at addressing several critical challenges in popular explanation methods in one-shots, thereby generating consistent, stable, and reliable explanations with guarantees in a computationally efficient manner.
+We carry out theoretical analysis that leverages the measures of uncertainty (credible intervals) produced by our framework to estimate the values of critical hyperparameters. More specifically, we derive a closed form expression for the number of perturbations required to generate explanations that satisfy desired levels of confidence. We also propose a novel sampling technique called focused sampling that leverages uncertainty to determine how to sample perturbations for faster convergence, thereby enabling our framework to generate explanations in a computationally efficient manner.
+We evaluate the efficacy of the proposed framework on a variety of datasets including COMPAS, German Credit, ImageNet, and MNIST. Our results demonstrate that the explanations output by our framework are not only highly reliable, but also very consistent and stable $5 3 \%$ more stable than LIME/SHAP on an average). Our experimental results also confirm that we can accurately estimate the number of perturbations needed to generate explanations with a desired level of uncertainty, and that our uncertainty sampling technique speeds up the process of generating explanations by up to a factor of 2 relative to random sampling of perturbations. Lastly, we carry out a user study with 31 human subjects to evaluate the quality of the explanations generated by our framework, demonstrating that our explanations accurately capture the importance of the most influential features.
+# 2 Notation & Background
+Here we introduce notation and discuss two relevant prior approaches, LIME and KernelSHAP.
+Notation Let $f : \mathbb { R } ^ { d }  [ 0 , 1 ]$ denote a black box classifier that takes a data point $x$ with $d$ features, and returns the probability that $x$ belongs to a certain class. Our goal is to explain individual predictions of $f$ . Let $\phi \in \mathbb { R } ^ { d }$ denote the explanation in terms of feature importances for the prediction $f ( x )$ , i.e. coefficients $\phi$ are treated as the feature contributions to the black box prediction. Note that $\phi$ captures the coefficients of a linear model. Let $\mathcal { Z }$ be a set of $N$ randomly sampled instances (perturbations) around $x$ . The proximity between $x$ and any $z \in { \mathcal { Z } }$ is given by $\pi _ { x } ( z ) \in \mathbb { R }$ . We denote the vector of these distances over the $N$ perturbations in $\mathcal { Z }$ as $\Pi _ { x } ( \hat { \mathcal { Z } } ) \in \mathbb { R } ^ { \mathrm { \tilde { \cal N } } }$ . Let $Y \in [ 0 , 1 ]$ be the vector of the black box predictions $f ( z )$ corresponding to each of the $N$ instances in $\mathcal { Z }$ .
+LIME [2] and KernelSHAP [4] are popular model-agnostic local explanation approaches that explain predictions of a classifier $f$ by learning a linear model $\phi$ locally around each prediction (i.e. $y \overset { \cdot } { \sim } \phi ^ { T } \overset { \cdot } { z } ,$ ). The objective function for both LIME and KernelSHAP constructs an explanation that approximates the behavior of the black box accurately in the vicinity (neighborhood) of $x$ .
+$$
+\underset { \phi } { \arg \operatorname* { m i n } } \sum _ { z \in \mathcal { Z } } [ f ( z ) - \phi ^ { T } z ] ^ { 2 } \pi _ { x } ( z ) .
+$$
+The above objective function has the following closed form solution:
+$$
+\hat { \phi } = ( \mathcal { Z } ^ { T } \mathrm { d i a g } ( \Pi _ { x } ( \mathcal { Z } ) ) \mathcal { Z } + \mathbb { I } ) ^ { - 1 } ( \mathcal { Z } ^ { T } \mathrm { d i a g } ( \Pi _ { x } ( \mathcal { Z } ) ) Y )
+$$
+The main difference between LIME and KernelSHAP lies in how $\pi _ { x } ( z )$ is chosen. In LIME, it is chosen heuristically: $\pi _ { x } ( z )$ is computed as the cosine or $l _ { 2 }$ distance. KernelSHAP leverages game theoretic principles to compute $\pi _ { x } ( z )$ , guaranteeing that explanations satisfy certain properties.
+# 3 Our Framework: Bayesian Local Explanations
+In this section, we introduce our Bayesian framework which is designed to capture the uncertainty associated with local explanations of black box models. First, we discuss the generative process and inference procedure for the framework. Then, we highlight how our framework can be instantiated to obtain Bayesian versions of LIME and SHAP. Lastly, we present detailed theoretical analysis for estimating the values of critical hyperparameters, and discuss how to efficiently construct highly accurate explanations with uncertainty guarantees using our framework.
+# 3.1 Constructing Bayesian Local Explanations
+Our goal here is to explain the behavior of a given black box model $f$ in the vicinity of an instance $x$ while also capturing the uncertainty associated with the explanation. To this end, we propose a Bayesian framework for constructing local linear model based explanations and capturing their associated uncertainty. We model the black box prediction of each perturbation $z$ as a linear combination of the corresponding feature values $( \phi ^ { \dot { T } } z )$ plus an error term () as shown in Eqn (4). While the weights of the linear combination $\phi$ capture the feature importances and thereby constitute our explanation, $\epsilon$ captures the error that arises due to the mismatch between our explanation $\phi$ and the local decision surface of the black box model $f$ . Our complete generative process is shown below:
+$$
+\begin{array} { r l r } & { } & { y | z , \phi , \epsilon \sim \phi ^ { T } z + \epsilon \qquad \epsilon \sim \mathcal { N } ( 0 , \displaystyle \frac { \sigma ^ { 2 } } { \pi _ { x } ( z ) } ) } \\ & { } & { \phi | \sigma ^ { 2 } \sim \mathcal { N } ( 0 , \sigma ^ { 2 } \mathbb { I } ) \qquad \sigma ^ { 2 } \sim \mathrm { I n v } - \chi ^ { 2 } ( n _ { 0 } , \sigma _ { 0 } ^ { 2 } ) . } \end{array}
+$$
+The error term is modeled as a Gaussian whose variance relies on the proximity function $\pi _ { x } ( z )$ i.e., $\begin{array} { r } { \epsilon \sim \mathcal { N } ( 0 , \frac { \sigma ^ { 2 } } { \pi _ { x } ( z ) } ) } \end{array}$ . This proximity function ensures that perturbations closer to the data point $x$ are modeled accurately, while allowing more room for error in case of perturbations that are farther away. $\pi _ { x } ( z )$ can be computed using cosine or $l _ { 2 }$ distance or other game theoretic principles similar to that of LIME and KernelSHAP (see Section 2). The conjugate priors on $\phi$ and $\bar { \sigma } ^ { 2 }$ are shown in Eqn (4). Note that, the distributions on error $\epsilon$ and feature importance $\phi$ both consider the parameter $\sigma ^ { 2 }$ . The fact that the prior on the feature importances considers $\sigma ^ { 2 }$ has an intuitive interpretation: if we have prior knowledge that the error of the explanation is small, we expect to be more confident about the feature importances. Similarly, if we have prior knowledge the error is large, we expect to be less confident about the feature importances.
+Thus, our generative process corresponds to the Bayesian version of the weighted least squares formulation of LIME and KernelSHAP outlined in Eqn. (1), with additional terms to model uncertainty. As in Eqns. (4), the process captures two sources of uncertainty in local explanations: 1) feature importance uncertainty: the uncertainty associated with the feature importances $\phi$ , and (2) error uncertainty: the uncertainty associated with the error term $\epsilon$ which captures how well our explanation $\phi$ models the local decision surface of the underlying black box.
+Inference Our inference process involves estimating the values of two key parameters: $\phi$ and $\sigma ^ { 2 }$ . By doing so, we can compute the local explanation as well as the uncertainties associated with feature importances and the error term. Posterior distributions on $\phi$ and $\sigma ^ { 2 }$ are normal and scaled Inv- $\chi ^ { 2 }$ , respectively, due to the corresponding conjugate priors [22]:
+$$
+\begin{array} { r l } & { \sigma ^ { 2 } | \mathcal { Z } , Y \sim \mathrm { S c a l e d - I n v - } \chi ^ { 2 } \left( n _ { 0 } + N , \frac { n _ { 0 } \sigma _ { 0 } ^ { 2 } + N s ^ { 2 } } { n _ { 0 } + N } \right) } \\ & { \phi | \sigma ^ { 2 } , \mathcal { Z } , Y \sim \mathrm { N o r m a l } ( \hat { \phi } , V _ { \phi } \sigma ^ { 2 } ) } \end{array}
+$$
+Further, $\hat { \phi } , V _ { \phi }$ , and $s ^ { 2 }$ can be directly computed:
+$$
+\begin{array} { r l } & { \hat { \phi } = V _ { \phi } ( \mathcal { Z } ^ { T } \mathrm { d i a g } ( \Pi _ { x } ( \mathcal { Z } ) ) Y ) } \\ & { V _ { \phi } = \left( \mathcal { Z } ^ { T } \mathrm { d i a g } ( \Pi _ { x } ( \mathcal { Z } ) ) \mathcal { Z } + \mathbb { I } \right) ^ { - 1 } } \\ & { s ^ { 2 } = \displaystyle \frac { 1 } { N } \left[ ( Y - \mathcal { Z } \hat { \phi } ) ^ { T } \mathrm { d i a g } ( \Pi _ { x } ( \mathcal { Z } ) ) ( Y - \mathcal { Z } \hat { \phi } ) + \hat { \phi } ^ { T } \hat { \phi } \right] } \end{array}
+$$
+Details of the complete inference procedure including derivations of Eqns. (5-7) are provided in the Appendix A. Note that our estimate of the posterior mean feature importances $\hat { \phi }$ (Eqn. (6)) is the same as that of the feature importances computed in case of LIME and KernelSHAP (Eqn. (2)).
+Remark 3.1. If we use the same proximity function $\pi _ { x } ( z )$ in our framework as in LIME or KernelSHAP, the posterior mean of the feature importance $\hat { \phi }$ output by our framework $E q$ (6)) will be equivalent to the feature importances output by LIME or KernelSHAP, respectively.
+Feature Importance Uncertainty To obtain the local feature importances and their associated uncertainty, we first compute the posterior mean of the local feature importances $\hat { \phi }$ using the closed form expression in Eqn. (7). We then estimate the credible interval (measure of uncertainty) around the mean feature importances by repeatedly sampling from the posterior distribution of $\phi$ (Eq (5)).
+Error Uncertainty The error term $\epsilon$ can serve as a proxy for explanation quality because it captures the mismatch between the constructed explanation and the local decision surface of the underlying black box. We first calculate the marginal posterior distribution of $\epsilon$ by leveraging Eqn (4) and integrating out $\sigma ^ { 2 }$ . This results in a three parameter Student’s t distribution (derivation in appendix A):
+$$
+\epsilon | \mathcal { Z } , Y \sim t _ { ( \nu = n _ { 0 } + N ) } ( 0 , \frac { n _ { 0 } \sigma _ { 0 } ^ { 2 } + N s ^ { 2 } } { n _ { 0 } + N } ) .
+$$
+We then evaluate the probability density function (PDF) of the above posterior at 0, i.e., $P ( \epsilon = 0 )$ by substituting the value of $s ^ { 2 }$ computed using Eqn. (7) into the Student’s t distribution above (Eqn. (8)). The resulting expression gives us the probability density that the explanation output by our framework perfectly captures the local decision surface underlying the black box. This operation is performed in constant time, adding minimal overhead to non-Bayesian LIME and SHAP. We illustrate how these computed intervals capture the variance in the explanations in Figure 9.
+Proposition 3.2. As the number of perturbations around $x$ goes to $\infty$ i.e., $N \to \infty$ : $( l )$ the estimate of $\phi$ converges to the true feature importance scores, and its uncertainty to 0. (2) uncertainty of the error term  converges to the bias of the local linear model $\phi$ . [Details in Appendix B]
+BayesLIME and BayesSHAP Our framework can be instantiated to obtain the Bayesian version of LIME by setting the proximity function to $\pi _ { x } ( z ) = \exp ( - D ( x , z ) ^ { 2 } / \sigma ^ { 2 } )$ where $D$ is a distance metric
+(e.g. cosine or $l _ { 2 }$ distance), and $n _ { 0 }$ and $\sigma _ { 0 } ^ { 2 }$ to small values $( 1 0 ^ { - 6 } )$ so that the prior is uninformative.
+We compute feature importance uncertainty and error uncertainty for LIME’s feature importances.
+Our framework can also be instantiated to obtain the Bayesian version of KernelSHAP by setting uninformative prior on $\sigma ^ { 2 }$ and d−1(d choose |z|)|z|(d−|z|) where |z| denotes the number of the variables in the variable combination represented by the data point $z$ i.e., the number of non-zero valued features in the vector representation of $z$ . Note that the original SHAP method views the problem of constructing a local linear model as estimating the Shapley values corresponding to each of the features [4]. These Shapley values represent the contribution of each of the features to the black box prediction i.e., $f ( x ) = \phi _ { 0 } + \sum \phi _ { i }$ . Therefore, the measures of uncertainty output by our method BayesSHAP capture the reliability of the estimated variable contributions.
+To encourage BayesLIME and BayesSHAP explanations to be sparse, we can use dimensionality reduction or feature selection techniques as used by LIME and SHAP to obtain the top K features [2, 4, 23]. We can then construct our explanations using the data corresponding to these top K features.
+# 3.2 Estimating the Number of Perturbations
+One of the major drawbacks of approaches such as LIME and KernelSHAP is that they do not provide any guidance on how to choose the number of perturbations, a key factor in obtaining reliable explanations in an efficient manner. To address this, we leverage the uncertainty estimates output by our framework to compute perturbations-to-go $( G )$ , an estimate of how many more perturbations are required to obtain explanations that satisfy a desired level of certainty. This estimate thus predicts the computational cost of generating an explanation with a desired level of certainty and can help determine whether it is even worthwhile to do so. The user specifies the confidence level of the credible interval (denoted as $\alpha$ ) and the maximum width of the credible interval $( W )$ , e.g. “width of $9 5 \%$ credible interval should be less than $0 . 1 ^ { \mathfrak { s } }$ corresponds to $\alpha = 0 . 9 5$ and $W = 0 . 1$ . To estimate $G$ for the local explanation of a data point $x$ , we first generate $S$ perturbations around $x$ (where $S$ is small and chosen by the user) and fit a local linear model using our method2. This provides initial estimates of various parameters shown in Eqns (5)-(7) which can then be used to compute $G$ .
+Theorem 3.3. Given $S$ seed perturbations, the number of additional perturbations required $( G )$ to achieve a credible interval width $W$ of feature importance for a data point $x$ at user-specified confidence level $\alpha$ can be computed as:
+$$
+G ( W , \alpha , x ) = \frac { 4 s _ { S } ^ { 2 } } { \bar { \pi } _ { S } \times \left[ \frac { W } { \Phi ^ { - 1 } ( \alpha ) } \right] ^ { 2 } } - S
+$$
+where $\bar { \pi } _ { S }$ is the average proximity $\pi _ { x } ( z )$ for the $S$ perturbations, $s _ { S } ^ { 2 }$ is the empirical sum of squared errors (SSE) between the black box and local linear model predictions, weighted by $\pi _ { x } ( z )$ , as in (7), and $\Phi ^ { - 1 } ( \alpha )$ is the two-tailed inverse normal CDF at confidence level $\alpha$ .
+Proof (Sketch). To estimate $G$ , we first relate $W$ and $\alpha$ to $\operatorname { V a r } ( \phi _ { i } )$ , the marginal variance of the feature importance3 for any feature $i$ , obtained by integrating out $\sigma ^ { 2 }$ . Because Student’s t can be approximated by a Normal distribution for large degrees of freedom (here, $S$ should be large enough), we use the inverse normal CDF to calculate credible interval width at level $\alpha$ . We compute $V _ { \phi }$ from (6) using $\mathcal { Z }$ , treating its entries as Bernoulli distributed with probability 0.5. Due to the covariance structure of this sampling procedure, the resulting variance estimate after $N$ samples is the sample SSE $s _ { S } ^ { 2 }$ scaled by $\approx \frac { 4 } { \hat { \pi } _ { S } N }$ (derivation in appendix B). If we assume SSE scales linearly with $S$ , we can take this to be a reasonable estimate of $s _ { N } ^ { 2 }$ at any $N$ . We can then estimate $G$ as
+$$
+\left[ \frac { W } { \Phi ^ { - 1 } ( \alpha ) } \right] ^ { 2 } = \mathrm { V a r } ( \phi _ { i } ) = \frac { 4 s _ { S } ^ { 2 } } { \bar { \pi } _ { S } \times ( G + S ) } \Longrightarrow G = \frac { 4 s _ { S } ^ { 2 } } { \bar { \pi } _ { S } \times \left[ \frac { W } { \Phi ^ { - 1 } ( \alpha ) } \right] ^ { 2 } } - S .
+$$
+# 3.3 Focused Sampling of Perturbations
+Perturbations-to-go $( G )$ provides us with an estimate of how many samples are required to achieve reliable explanations. However, if $G$ is large, querying the black-box model for its predictions on a large number of perturbations can be computationally expensive for larger models [24, 25]. To reduce this cost, we develop an alternative sampling procedure called focused sampling which leverages uncertainty estimates to query the black box in a more targeted fashion (instead of querying randomly), thereby reducing the computational cost associated with generating reliable explanations. Inspired by active learning [26], focused sampling strategically prioritizes perturbations whose predictions the explanation is most uncertain about, when querying the black box. This enables the focused sampling procedure to query the black box only for the predictions of the most informative perturbations and thereby learn an accurate explanation with far fewer queries to the black box.
+To determine how uncertain our explanation $\phi$ is about the black box label for any given instance $z$ , we first compute the posterior predictive distribution for $z$ (derivation in Appendix A), given as $\boldsymbol { \hat { y } } ( z ) | \mathcal { Z } , \boldsymbol { Y } \sim t _ { ( \mathcal { V } = N ) } ( \boldsymbol { \hat { \phi } } ^ { T } \boldsymbol { z } , ( \boldsymbol { z } ^ { T } V _ { \phi } \boldsymbol { z } + 1 ) \boldsymbol { s } ^ { 2 } )$ . The variance of this three parameter student’s t distribution is,
+$$
+\mathrm { v a r } ( \hat { y } ( z ) ) = ( ( z ^ { T } V _ { \phi } z + 1 ) s ^ { 2 } ) ( N / ( N - 2 ) )
+$$
+We refer to this variance as the predictive variance $\mathrm { v a r } ( \hat { y } ( z ) )$ , and it captures how uncertain our explanation $\phi$ is about the black box prediction.
+The focus sampling procedure first fits the explanation with an initial $S$ perturbations (where $S$ is a small number). We then iterate the following procedure until the desired explanation certainty level is reached. We draw a batch of $A$ candidate perturbations, compute their predictive variance with the Bayesian explanation, and induce a distribution over the perturbations by running softmax on the variances with tempurature parameter $\tau$ . We draw a batch of $B$ perturbations from this distribution and query the black box model for their labels. Finally, we refit the Bayesian explanation on all the labeled perturbations collected so far. We provide pseudocode for the uncertainty sampling procedure in Algorithm 1.
+Algorithm 1 Focused sampling for local explanations
+<table><tr><td colspan="2">Require: Model f,Data instance x, Number of perturbations N,Number of seed perturbations S,</td></tr><tr><td colspan="2">Batch size B,Pool size A, tempurature T 1: function FOCUSED SAMPLE</td></tr><tr><td></td><td>Initialize Z with S seed perturbations.</td></tr><tr><td>2:</td><td></td></tr><tr><td>3:</td><td>Fit on Z Using Eqn (6)</td></tr><tr><td>4:</td><td>fori←1toN-Sinincrements ofBdo</td></tr><tr><td>5:</td><td>Q ←Generate Acandidate perturbations Using Eqn (11)</td></tr><tr><td>6:</td><td>Compute var(y(z)) on Q</td></tr><tr><td>7:</td><td>Define Qdist as X exp(var(g(z))/τ)</td></tr><tr><td>8:</td><td>Qnew ← Draw B samples from Qdist</td></tr><tr><td>9:</td><td>Z ← ZU Qnew; Fit on Z Using Eqn (6)</td></tr><tr><td>10: end for</td><td></td></tr><tr><td>11: return $</td><td></td></tr><tr><td>12: end function</td><td></td></tr></table>
+# 4 Experiments
+We evaluate the proposed framework by first analyzing the quality of our uncertainty estimates i.e., feature importance uncertainty and error uncertainty. We also assess our estimates of required perturbations $( G )$ , and evaluate the computational efficiency of focused sampling. Last, we describe a user study with 31 subjects to assess the informativeness of the explanations output by our framework.
+Setup We experiment with a variety of real world datasets spanning multiple applications (e.g., criminal justice, credit scoring) as well as modalities (e.g., structured data, images). Our first structured dataset is COMPAS [27], containing criminal history, jail and prison time, and demographic attributes of 6172 defendants, with class labels that represent whether each defendant was rearrested within 2 years of release. The second structured dataset is the German Credit dataset from the UCI repository [28] containing financial and demographic information (including account information, credit history, employment, gender) for 1000 loan applications, each labeled as a “good” or “bad” customer. We create 80/20 train/test splits for these two datasets, and train a random forest classifier (sklearn implementation with 100 estimators) as black box models for each (test accuracy of $8 2 . 8 \%$ and $7 2 . 5 \%$ , respectively). We also include popular image datasets–MNIST and Imagenet. For the MNIST [29] handwritten digits dataset, we train a 2-layer CNN to predict the digits (test accuracy of $9 9 . 2 \%$ ). For Imagenet [30], we use the off-the-shelf VGG16 model [31] as the black box. We select a sample of 100 images of the following classes French Bulldog, Scuba Diver, Corn, and Broccoli to use in the experiments. For generating explanations, we use standard implementations of the baselines LIME and KernelSHAP with default settings [2, 4]. For images, we construct super pixels as described in [2] and use them as features (number of super pixels is fixed to 20 per image). For our framework, the desired level of certainty is expressed as the width of the $9 5 \%$ credible interval.
+<table><tr><td></td><td>BayesLIME</td><td>BayesSHAP</td><td></td><td>BayesLIME</td><td>BayesSHAP</td></tr><tr><td>TABULAR DATASETS</td><td></td><td></td><td>MNIST</td><td></td><td></td></tr><tr><td>COMPAS</td><td>95.5</td><td>87.9</td><td>Digit 1</td><td>95.8</td><td>98.4</td></tr><tr><td>German Credit</td><td>96.9</td><td>89.6</td><td>Digit 2</td><td>95.8</td><td>97.4</td></tr><tr><td>IMAGENET</td><td></td><td></td><td>Digit 3</td><td>95.2</td><td>96.3</td></tr><tr><td>Corn</td><td>94.6</td><td>91.8</td><td>Digit 4</td><td>97.2</td><td>90.1</td></tr><tr><td>Broccoli</td><td>91.4</td><td>89.2</td><td>Digit 5</td><td>95.2</td><td>95.6</td></tr><tr><td>French Bulldog</td><td>94.8</td><td>89.9</td><td>Digit 6</td><td>96.7</td><td>96.8</td></tr><tr><td>Scuba Diver</td><td>92.4</td><td>94.6</td><td>Digit 7</td><td>95.7</td><td>95.3</td></tr></table>
+Table 1: Evaluating Credible Intervals. We report the $\%$ of time the $9 5 \%$ credible intervals with 100 perturbations include their true values (estimated on 10, 000 perturbations). Closer to 95.0 is better. Both BayesLIME and BayesSHAP are well calibrated.
+Quality of Uncertainty Estimates A critical component of our explanations is the feature importance uncertainty. To evaluate the correctness of these estimates, we compute how often true feature importances lie within the $9 5 \%$ credible intervals estimated by BayesLIME and BayesSHAP. Note, that by true feature importance, we refer to the best fit linear model output using either the LIME or SHAP kernels. We evaluate the quality of our credible interval estimates by running our methods with 100 perturbations to estimate feature importances and taking the corresponding $9 5 \%$ credible intervals for each test instance. We compute what fraction of the true feature importances fall within our $9 5 \%$ credible intervals. Note, because there are no methods to provide uncertainty estimates for LIME and SHAP, we do not provide further baselines. Since we do not have access to the true feature importances of the complex black box models, following Prop 3.2, we use feature importances computed using a large value of $N$ $( N = 1 0 , 0 0 0 )$ , and treat the resulting estimates as ground truth.
+Results for BayesLIME in Table 1 indicate that the true feature importances are close to ideal and indicate the estimates are well calibrated. While the estimates by BayesSHAP are somewhat less calibrated (true feature importances fall within our estimated $9 5 \%$ credible intervals about 89.2 to $9 8 . 4 \%$ of the time), they still are quite close to ideal. All in all, these results confirm that the credible intervals learned by our methods are well calibrated and therefore highly reliable in capturing the uncertainty of the feature importances. Lastly, though we set our priors to be uninformative in general, we also investigate how sensitive our uncertainty estimates are to hyperparameter choices in Figure 5 in the Appendix. We find that the explanation uncertainty becomes uncalibrated with strong priors. However, our explanations seem to be robust to hyperparameter choices in general.
+Correctness of Estimated Number of Perturbations We assess whether our estimate of perturbations-to-go $G$ ; Section 3.2) is an accurate estimate of the additional number of perturbations needed to reach a desired level of feature importance certainty. We carry out this experiment on MNIST data for the digit $" 4 > "$ (additional datasets explored in Appendix C) and use $S = 2 0 0$ as the initial number of perturbations to obtain a preliminary explanation and its associated uncertainty estimates. We then leverage these estimates to compute $G$ for 6 different certainty levels. First, we observe significant differences in $G$ estimates across instances (details in appendix C) i.e. number of perturbations needed to obtain a particular level of certainty varied significantly across instances– ranging from 200-5, 000 for the lowest level of certainty to 200-20, 000 for higher levels of certainty. Next, for each image and certainty level, we run our method for the estimated number of perturbations $( G )$ to determine if the observed estimates of uncertainty (observed credible interval width $W$ ) match the desired levels of uncertainty (desired credible interval width $W$ ). Results in Figure 2 show that the observed and desired levels of certainty are well calibrated, demonstrating that $G$ estimates are reliable approximations of the additional number of perturbations needed.
+![](images/a1ce895bb1fbdbf243ad90abc4bbd6376035f01308ad9e7c0365cb10fc50cc0d.jpg)
+Figure 2: Perturbations-to-go $( G )$ . We generate explanation with $G$ perturbations, where $G$ is computed using the desired credible interval width $\mathbf { \bar { X } }$ -axis), and compare desired levels to the observed credible interval width (y-axis) (blue line indicates ideal calibration). Results are averaged over 100 MNIST images of the digit $" 4 > "$ We see that $G$ provides a good approximation of the additional perturbations needed.
+Efficiency of Focused Sampling Focused sampling uses the predictive variance to strategically choose perturbations that will reduce uncertainty in order to be labeled by the black box (section 3.3). Here, we will evaluate the efficiency of the focused sampling procedure. First, we assess whether focused sampling converges (as measured by error uncertainty $P ( \epsilon = 0 ) )$ ) more efficiently than random sampling. To this end, we experiment with BayesLIME on Imagenet data for the “French bulldog” class to carry out this analysis. This setting replicates scenarios where LIME is applied to a computationally expensive black box model, making it highly desirable to limit the number of perturbations to reduce total running time. We run each sampling strategy for 2,000 perturbations and plot the number of model queries versus error uncertainty. During focused sampling, we set the batch size $B$ to 50. The results in Figure 3 show that focused sampling results in faster convergence to reliable and high quality explanations; focused sampling stabilizes within a couple hundred model queries while random sampling takes over 1,000. Note, as the inefficiency of querying the black box model increases, the advantages of focused sampling decreasing total running time of the explanations will only become more pronounced. These results clearly demonstrate that focused sampling can significantly speed up the process of generating high quality local explanations. Additionally, in Appendix C, we also check if focused sampling causes any bias (due to sampling based on uncertainty estimates) that results in convergence to a different/wrong explanation, however our results clearly indicate that this is not the case.
+Stability of BayesLIME & BayesSHAP Recall that LIME & SHAP are not stable: small changes to instances can produce substantially different explanations. We consider whether BayesLIME & BayesSHAP produce more stable explanations than their LIME & SHAP counterparts. To perform this analysis, we use the local Lipschitz metric for explanation stability [18]:
+$$
+\hat { L } ( x _ { i } ) = \operatorname * { a r g m a x } _ { x _ { j } \in N _ { \epsilon } ( x _ { i } ) } \frac { | | \phi _ { i } - \phi _ { j } | | _ { 2 } } { | | x _ { i } - x _ { j } | | _ { 2 } }
+$$
+where $x _ { i }$ refers to an instance, $N _ { \epsilon } ( x _ { i } )$ is the $\epsilon$ -ball centered at $x _ { i }$ , and $\phi _ { i }$ and $\phi _ { j }$ are the explanation parameters for $x _ { i }$ and $x _ { j }$ . Lower values indicate more stable explanations. We follow the setup outline by Alvarez-Melis and Jaakkola [18] and compute the local Lipschitz values, comparing both LIME & BayesLIME and SHAP & BayesSHAP across Compas, German Credit, MNIST digit $\cdot _ { 4 } \cdot \cdot$ , and Imagenet “French Bulldog.” We perform the comparison using the default number of perturbations in both LIME & SHAP, and use this same number in the respective Bayesian variants and set the batch size $B$ to half this value. We use focused sampling for BayesLIME and BayesSHAP, and report the $\%$ increase in stability of these approaches over LIME and SHAP for 40 test points. The results given in Figure 4 show a clear improvement (on average $5 3 \%$ ) in stability in all cases except German Credit for BayesSHAP. Further, we run a Wilcoxon signed-rank test and find our results are statistically significant in all cases $\mathrm { / e < 1 e { - } 2 ) }$ except for BayesSHAP for German Credit, where there is not a significant difference between the methods $\zeta _ { \rho } > 0 . 0 5 )$ . These results demonstrate BayesLIME and BayesSHAP are more stable than previous methods.
+![](images/dbdcf8d6b10a60e362cba9afcfbbaed135a2dd3c956ce32266eb747c42dae508.jpg)
+Figure 3: Efficiency of focused sampling for 100 Imagenet “French bulldog” images, with random sampling as a baseline. We provide mean and standard error. We assess the efficiency of focused sampling by comparing error uncertainty over model queries and show quicker convergence than random sampling.
+![](images/988605277650e44dda266f35f97a56551dc27c3e649264948462a5108a6c7e8f.jpg)
+Figure 4: Assessing the $\%$ increase in stability of BayesLIME and BayesSHAP over LIME and SHAP respectively. Our Bayesian methods are significant more stable $\rho \ < \ 1 { \mathrm { e } } { \mathrm { - } } 2$ according to Wilcoxon signed-rank test) except for BayesSHAP on German Credit, where there is not a significant difference between the methods $\zeta _ { \rho } > 0 . 0 5 )$ .
+User Study We perform a user study with 31 subjects to compare BayesLIME and LIME explanations on MNIST. We evaluate the following: are explanations with low levels of uncertainty (i.e., most confident explanations) more meaningful to humans? To answer this question, we follow prior work and mask the most important features selected by BayesLIME and LIME [32, 4]. We ask users to guess the digit of the masked images. The better the explanation, the more difficult it should be for the users to get it right. Further, the choice to mask the important features is motivated by its success in prior work. We randomly select 15 correctly predicted test images, generate explanations by sweeping over a range of perturbation amounts $[ 1 \dot { 0 } ^ { 5 } , . . . , 1 0 ^ { 3 . 5 } ]$ incremented by 0.5. We choose the top explanation for each image based on either fidelity (for LIME) or $P ( \epsilon = 0 )$ (for BayesLIME). We sent the user study out to students and researchers with background in computer science. A screen shot of the task is shown in Figure 7 in the Appendix. We find that the explanations output by our methods focus on more informative parts of the image, since hiding them makes it difficult for humans to guess the digit. Users had an error rate of $2 5 . 7 \%$ for LIME, while it was $3 0 . 7 \%$ for BayesLIME, both with standard error 0.003 $\mathrm { \Delta } \rho = 0 . 0 2 8$ through a one-tailed two sample t-test). This result indicates that our method BayesLIME and the associated measure of explanation uncertainty result in more high quality and reliable explanations compared to LIME and its associated fidelity metric.
+# 5 Related Work
+Interpretability Methods A variety of interpretability methods have been proposed. Some methods that are inherently interpretable include additive models [33, 34], decision lists and sets [35, 36], and instance-based explanations [37]. However, black-box models are often more flexible, accurate, and easier to use; thus, there has been a lot of interest in constructing post hoc explanations[38]. These include LIME [2] and SHAP [4, 39], which are among the most popular due to their broad applicability and code availability, but saliency maps [5–8], permutation feature importance [40], and partial dependency plots [41] also follow this paradigm. Other approaches to post hoc explanations focus on rule-based models [1, 3], counterfactuals [42, 43], and influence functions [9].
+Vulnerabilities of Post hoc Explanations Recent work has shed light on the downsides of post hoc explanation techniques. These methods are often highly sensitive to small changes in inputs [14], are susceptible to manipulation [15, 16, 44, 45], and are not faithful to the underlying black boxes [46]. Perturbation-based explanation methods such as LIME and SHAP are subject to additional criticisms: results vary between runs of the algorithms [18–20, 47, 21], and hyperparameters used to select the perturbations can greatly influence the resulting explanation [20]. Prior work has attempted to tackle the problem of instability in perturbation-based explanations by averaging over several explanations [48, 19], however, this is computationally expensive. Other works related to creating more trustworthy explanations include development of sanity checks for explainers [49, 17, 50]. These techniques represent an important step towards improved usability, given experimental evidence that humans are often too eager to accept inaccurate machine explanations [51–54]. Recent works theoretically analyze the sources of non-robustness in black box explanations [55–57].
+Logical and Formal Reasoning Additional related works have considered explaining classifiers through identifying a subset of features that are “sufficient” to explain a prediction [58–62]. Though these methods offer strong guarantees surrounding which features ensure a prediction is achieved, they are not model agnostic. Further, they do not define feature importances associated with the local explanations nor consider ways to improve locally weighted explanations, such as LIME and SHAP.
+Bayesian Methods in Explainable ML Few recent works have adopted Bayesian formulations to explain black box models [63–65]. Guo et al. [63] introduce a Bayesian non-parametric approach to fit a global surrogate model. Their formulation seeks to fit a mixture of generalizable explanations across instances. Zhao et al. [64] study whether incorporating informative priors improves the stability of the resulting explanations. However, neither of these works focus on modeling the uncertainty of local explanations. Further, these approaches also do not tackle the critical problems of estimating key hyperparameters or improving efficiency of computing explanations.
+# 6 Conclusion
+We developed a Bayesian framework for generating local explanations along with their associated uncertainty. We instantiated this framework to obtain Bayesian versions of LIME and SHAP that output pointwise estimates of feature importances as well as their associated credible intervals. These intervals enabled us to infer the quality of the explanations and output explanations that satisfied user specified levels of uncertainty. We carried out theoretical analysis that leverages these uncertainty measures (credible intervals) to estimate the values of critical hyperparameters (e.g., the number of perturbations). We also proposed a novel sampling technique called focused sampling that leverages uncertainty estimates to determine how to sample perturbations for faster convergence.
+While the Bayesian framework addresses several critical challenges (i.e., consistency, stability, modeling uncertainty) associated with LIME and SHAP, there are still certain aspects where it would exhibit the same shortcomings as LIME and SHAP [4, 66]. For instance, if the local decision surface of a given black box classifier is highly non-linear, our framework, which relies on local linear approximations, may not be able to capture this non-linear decision surface accurately. In addition, if the perturbation sampling procedures used in LIME and SHAP are used in BayesLIME and BayesSHAP, they will likely be vulnerable to the attacks proposed by Slack et al. [15]. In the future, it would be interesting to extend our framework to produce global explanations with uncertainty guarantees and explore how uncertainty quantification can help calibrate user trust in model explanations.
+# 7 Acknowledgments
+We would like to thank the anonymous reviewers for their insightful feedback. This work is supported in part by the NSF awards #IIS-2008461, #IIS-2008956, and #IIS-2040989, and research awards from the Harvard Data Science Institute, Amazon, Bayer, Google, and the HPI Research Center in Machine Learning and Data Science at UC Irvine. The views expressed are those of the authors and do not reflect the official policy or position of the funding agencies.
+References
+[1] Marco Tulio Ribeiro, Sameer Singh, and Carlos Guestrin. Anchors: High-precision modelagnostic explanations. In Thirty-Second AAAI Conference on Artificial Intelligence, 2018.
+[2] Marco Tulio Ribeiro, Sameer Singh, and Carlos Guestrin. Why Should I Trust You? explaining the predictions of any classifier. In Knowledge Discovery and Data mining (KDD), 2016.
+[3] Himabindu Lakkaraju, Ece Kamar, Rich Caruana, and Jure Leskovec. Faithful and customizable explanations of black box models. In Proceedings of the 2019 AAAI/ACM Conference on AI, Ethics, and Society, pages 131–138. ACM, 2019.
+[4] Scott M Lundberg and Su-In Lee. A unified approach to interpreting model predictions. In Advances in Neural Information Processing Systems, pages 4765–4774, 2017.
+[5] Karen Simonyan, Andrea Vedaldi, and Andrew Zisserman. Deep inside convolutional networks: Visualising image classification models and saliency maps. In Workshop at International Conference on Learning Representations, 2014.
+[6] Mukund Sundararajan, Ankur Taly, and Qiqi Yan. Axiomatic attribution for deep networks. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pages 3319–3328. JMLR. org, 2017.
+[7] Ramprasaath R Selvaraju, Michael Cogswell, Abhishek Das, Ramakrishna Vedantam, Devi Parikh, and Dhruv Batra. Grad-cam: Visual explanations from deep networks via gradient-based localization. In ICCV, 2017.
+[8] Daniel Smilkov, Nikhil Thorat, Been Kim, Fernanda Viégas, and Martin Wattenberg. Smoothgrad: removing noise by adding noise. Workshop on Visualization for Deep Learning, ICML, 2017.
+[9] Pang Wei Koh and Percy Liang. Understanding black-box predictions via influence functions. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pages 1885–1894. JMLR. org, 2017.
+[10] Osbert Bastani, Carolyn Kim, and Hamsa Bastani. Interpretability via model extraction. FAT/ML Workshop 2017, 2017.
+[11] Gregory Plumb, Denali Molitor, and Ameet S Talwalkar. Model agnostic supervised local explanations. In Neural Information Processing Systems, 2018.
+[12] Radwa Elshawi, Mouaz H Al-Mallah, and Sherif Sakr. On the interpretability of machine learning-based model for predicting hypertension. BMC medical informatics and decision making, 19(1):146, 2019.
+[13] Leanne S Whitmore, Anthe George, and Corey M Hudson. Mapping chemical performance on molecular structures using locally interpretable explanations. arXiv preprint arXiv:1611.07443, 2016.
+[14] Amirata Ghorbani, Abubakar Abid, and James Zou. Interpretation of neural networks is fragile. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pages 3681–3688, 2019.
+[15] Dylan Slack, Sophie Hilgard, Emily Jia, Sameer Singh, and Himabindu Lakkaraju. Fooling lime and shap: Adversarial attacks on post hoc explanation methods. Conference on Artificial Intelligence, Ethics, and Society (AIES), 2020.
+[16] Ann-Kathrin Dombrowski, Maximilian Alber, Christopher J Anders, Marcel Ackermann, KlausRobert Müller, and Pan Kessel. Explanations can be manipulated and geometry is to blame. arXiv preprint arXiv:1906.07983, 2019.
+[17] Julius Adebayo, Justin Gilmer, Michael Muelly, Ian Goodfellow, Moritz Hardt, and Been Kim. Sanity checks for saliency maps. In Advances in Neural Information Processing Systems, pages 9505–9515, 2018.
+[18] David Alvarez-Melis and Tommi S. Jaakkola. On the robustness of interpretability methods. ICML Workshop on Human Interpretability in Machine Learning, 2018.
+[19] Eunjin Lee, David Braines, Mitchell Stiffler, Adam Hudler, and Daniel Harborne. Developing the sensitivity of lime for better machine learning explanation. In Artificial Intelligence and Machine Learning for Multi-Domain Operations Applications, volume 11006, page 1100610. International Society for Optics and Photonics, 2019.
+[20] Hui Fen Tan, Kuangyan Song, Madeilene Udell, Yiming Sun, and Yujia Zhang. “why should you trust my explanation?” understanding uncertainty in lime explanations. In ICML Workshop on AI for Social Good, 2019.
+[21] Jianbo Chen, Le Song, Martin J. Wainwright, and Michael I. Jordan. L-shapley and c-shapley: Efficient model interpretation for structured data. In International Conference on Learning Representations, 2019.
+[22] Andrew Moore. Locally weighted bayesian regression, January 1995.
+[23] Kacper Sokol, Alexander Hepburn, Raúl Santos-Rodríguez, and Peter A. Flach. blimey: Surrogate prediction explanations beyond lime. NeurIPS HCML Workshop, 2019.
+[24] Emily L. Denton, Wojciech Zaremba, Joan Bruna, Yann LeCun, and Rob Fergus. Exploiting linear structure within convolutional networks for efficient evaluation. In NIPS, 2014.
+[25] Max Jaderberg, Andrea Vedaldi, and Andrew Zisserman. Speeding up convolutional neural networks with low rank expansions. BMVC 2014 - Proceedings of the British Machine Vision Conference 2014, 05 2014.
+[26] Burr Settles. Active learning literature survey. 2010.
+[27] Julia Angwin, Jeff Larson, Surya Mattu, and Lauren Kirchner. Machine bias. In ProPublica, 2016.
+[28] Dheeru Dua and Casey Graff. Uci machine learning repository, 2017. URL http://archive. ics.uci.edu/ml.
+[29] Yann LeCun, Corinna Cortes, and CJ Burges. Mnist handwritten digit database. ATT Labs [Online]. Available: http://yann.lecun.com/exdb/mnist, 2, 2010.
+[30] J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei. ImageNet: A Large-Scale Hierarchical Image Database. In CVPR09, 2009.
+[31] Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. In ICLR, 2015.
+[32] Patrick Schwab and Walter Karlen. Cxplain: Causal explanations for model interpretation under uncertainty. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-Buc, E. Fox, and R. Garnett, editors, Advances in Neural Information Processing Systems 32, pages 10220–10230. Curran Associates, Inc., 2019. URL http://papers.nips.cc/paper/ 9211-cxplain-causal-explanations-for-model-interpretation-under-uncertainty. pdf.
+[33] Yin Lou, Rich Caruana, Johannes Gehrke, and Giles Hooker. Accurate intelligible models with pairwise interactions. In KDD, 2013.
+[34] Berk Ustun, Stefano Traca, and Cynthia Rudin. Supersparse linear integer models for interpretable classification. arXiv preprint arXiv:1306.6677, 2013.
+[35] Himabindu Lakkaraju, Stephen H Bach, and Jure Leskovec. Interpretable decision sets: A joint framework for description and prediction. In Knowledge Discovery and Data mining (KDD), 2016.
+[36] Elaine Angelino, Nicholas Larus-Stone, Daniel Alabi, Margo Seltzer, and Cynthia Rudin. Learning certifiably optimal rule lists for categorical data. arXiv preprint arXiv:1704.01701, 2017.
+[37] Been Kim, Cynthia Rudin, and Julie Shah. The bayesian case model: A generative approach for case-based reasoning and prototype classification. arXiv preprint arXiv:1503.01161, 2015.
+[38] Marco Tulio Ribeiro, Sameer Singh, and Carlos Guestrin. Model-Agnostic Interpretability of Machine Learning. In ICML Workshop on Human Interpretability in Machine Learning (WHI), June 2016.
+[39] Ian Covert and Su-In Lee. Improving kernelshap: Practical shapley value estimation using linear regression. In Arindam Banerjee and Kenji Fukumizu, editors, Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, volume 130 of Proceedings of Machine Learning Research, pages 3457–3465. PMLR, 13–15 Apr 2021. URL https: //proceedings.mlr.press/v130/covert21a.html.
+[40] Leo Breiman. Random forests. Machine learning, 45(1):5–32, 2001.
+[41] Jerome H Friedman. Greedy function approximation: a gradient boosting machine. Annals of statistics, pages 1189–1232, 2001.
+[42] Berk Ustun, Alexander Spangher, and Yang Liu. Actionable recourse in linear classification. In Proceedings of the Conference on Fairness, Accountability, and Transparency, FAT\* ’19, pages 10–19, 2019. ISBN 978-1-4503-6125-5.
+[43] Sandra Wachter, Brent Mittelstadt, and Chris Russell. Counterfactual explanations without opening the black box: Automated decisions and the gdpr. Harv. JL & Tech., 31:841, 2017.
+[44] Juyeon Heo, Sunghwan Joo, and Taesup Moon. Fooling neural network interpretations via adversarial model manipulation. In Advances in Neural Information Processing Systems 32, pages 2921–2932. 2019.
+[45] Junlin Wang, Jens Tuyls, Eric Wallace, and Sameer Singh. Gradient-based Analysis of NLP Models is Manipulable. In Findings of the Association for Computational Linguistics: EMNLP (EMNLP Findings), page 247–258, 2020.
+[46] Cynthia Rudin. Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead. Nature Machine Intelligence, 1(5):206, 2019.
+[47] Muhammad Rehman Zafar and Naimul Mefraz Khan. Dlime: a deterministic local interpretable model-agnostic explanations approach for computer-aided diagnosis systems. arXiv preprint arXiv:1906.10263, 2019.
+[48] Chih-Kuan Yeh, Cheng-Yu Hsieh, Arun Suggala, David I Inouye, and Pradeep K Ravikumar. On the (in) fidelity and sensitivity of explanations. In Advances in Neural Information Processing Systems, pages 10965–10976, 2019.
+[49] Oana-Maria Camburu, Eleonora Giunchiglia, Jakob Foerster, Thomas Lukasiewicz, and Phil Blunsom. Can i trust the explainer? verifying post-hoc explanatory methods. arXiv preprint arXiv:1910.02065, 2019.
+[50] Mengjiao Yang and Been Kim. Bim: Towards quantitative evaluation of interpretability methods with ground truth. arXiv:1907.09701, 2019.
+[51] Harmanpreet Kaur, Harsha Nori, Samuel Jenkins, Rich Caruana, Hanna Wallach, and Jennifer Wortman Vaughan. Interpreting interpretability: Understanding data scientists’ use of interpretability tools for machine learning. In CHI, April 2020.
+[52] Fred Hohman, Andrew Head, Rich Caruana, Robert DeLine, and Steven M Drucker. Gamut: A design probe to understand how data scientists understand machine learning models. In Proceedings of the 2019 CHI Conference on Human Factors in Computing Systems, pages 1–13, 2019.
+[53] Forough Poursabzi-Sangdeh, Daniel G Goldstein, Jake M Hofman, Jennifer Wortman Vaughan, and Hanna Wallach. Manipulating and measuring model interpretability. arXiv preprint arXiv:1802.07810, 2018.
+[54] Himabindu Lakkaraju and Osbert Bastani. " how do i fool you?" manipulating user trust via misleading black box explanations. In Proceedings of the AAAI/ACM Conference on AI, Ethics, and Society, pages 79–85, 2020.
+[55] Damien Garreau and Ulrike von Luxburg. Looking deeper into lime. arXiv preprint arXiv:2008.11092, 2020.
+[56] Prasad Chalasani, Jiefeng Chen, Amrita Roy Chowdhury, Xi Wu, and Somesh Jha. Concise explanations of neural networks using adversarial training. In International Conference on Machine Learning, pages 1383–1391. PMLR, 2020.
+[57] Alexander Levine, Sahil Singla, and Soheil Feizi. Certifiably robust interpretation in deep learning. arXiv preprint arXiv:1905.12105, 2019.
+[58] Eric Wang, Pasha Khosravi, and Guy Van den Broeck. Probabilistic sufficient explanations. In IJCAI, 2021.
+[59] Andy Shih, Arthur Choi, and Adnan Darwiche. A symbolic approach to explaining bayesian network classifiers. In Proceedings of the 27th International Joint Conference on Artificial Intelligence, IJCAI’18, page 5103–5111. AAAI Press, 2018. ISBN 9780999241127.
+[60] Weijia Shi, Andy Shih, Adnan Darwiche, and Arthur Choi. On tractable representations of binary neural networks. In Diego Calvanese, Esra Erdem, and Michael Thielscher, editors, Proceedings of the 17th International Conference on Principles of Knowledge Representation and Reasoning, KR 2020, Rhodes, Greece, September 12-18, 2020, pages 882–892, 2020. doi: 10.24963/kr.2020/91. URL https://doi.org/10.24963/kr.2020/91.
+[61] Yacine Izza, Alexey Ignatiev, and Joao Marques-Silva. 2020.
+[62] Alexey Ignatiev, Nina Narodytska, and Joao Marques-Silva. Abduction-based explanations for machine learning models. Proceedings of the AAAI Conference on Artificial Intelligence, 33 (01):1511–1519, Jul. 2019. doi: 10.1609/aaai.v33i01.33011511. URL https://ojs.aaai. org/index.php/AAAI/article/view/3964.
+[63] Wenbo Guo, Sui Huang, Yunzhe Tao, Xinyu Xing, and Lin Lin. Explaining deep learning models – a bayesian non-parametric approach. In Neural Information Processing Systems (NeurIPS). 2018.
+[64] Xingyu Zhao, Xiaowei Huang, Valentin Robu, and David Flynn. Baylime: Bayesian local interpretable model-agnostic explanations. arXiv preprint arXiv:2012.03058, 2020.
+[65] Kirill Bykov, Marina Höhne, Klaus-Robert Müller, Shinichi Nakajima, and Marius Kloft. How much can i trust you? – quantifying uncertainties in explaining neural networks. arXiv, 06 2020.
+[66] Sushant Agarwal, Shahin Jabbari, Chirag Agarwal, Sohini Upadhyay, Steven Wu, and Himabindu Lakkaraju. Towards the unification and robustness of perturbation and gradient based explanations. In Marina Meila and Tong Zhang, editors, Proceedings of the 38th International Conference on Machine Learning, volume 139 of Proceedings of Machine Learning Research, pages 110–119. PMLR, 18–24 Jul 2021. URL https://proceedings.mlr. press/v139/agarwal21c.html.
+[67] Ludwig Fahrmeir, Thomas Kneib, and Stefan Lang. Regression. Statistik und ihre Anwendungen. Springer, 2007. ISBN 978-3-540-33932-8.
+[68] Christopher M. Bishop. Pattern Recognition and Machine Learning. Springer, 2006.

parse/train/rqfq0CYIekd/rqfq0CYIekd_content_list.json ADDED Viewed

	@@ -0,0 +1,1626 @@

+[
+    {
+        "type": "text",
+        "text": "Reliable Post hoc Explanations: Modeling Uncertainty in Explainability ",
+        "text_level": 1,
+        "bbox": [
+            259,
+            122,
+            736,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Dylan Slack   \nUC Irvine   \ndslack@uci.edu ",
+        "bbox": [
+            184,
+            227,
+            305,
+            267
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Sophie Hilgard Harvard University ash798@g.harvard.edu ",
+        "bbox": [
+            323,
+            226,
+            495,
+            267
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Sameer Singh UC Irvine sameer@uci.edu ",
+        "bbox": [
+            516,
+            227,
+            633,
+            267
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Himabindu Lakkaraju Harvard University hlakkaraju@hbs.edu ",
+        "bbox": [
+            651,
+            226,
+            813,
+            267
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            304,
+            535,
+            320
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "As black box explanations are increasingly being employed to establish model credibility in high stakes settings, it is important to ensure that these explanations are accurate and reliable. However, prior work demonstrates that explanations generated by state-of-the-art techniques are inconsistent, unstable, and provide very little insight into their correctness and reliability. In addition, these methods are also computationally inefficient, and require significant hyper-parameter tuning. In this paper, we address the aforementioned challenges by developing a novel Bayesian framework for generating local explanations along with their associated uncertainty. We instantiate this framework to obtain Bayesian versions of LIME and KernelSHAP which output credible intervals for the feature importances, capturing the associated uncertainty. The resulting explanations not only enable us to make concrete inferences about their quality (e.g., there is a $9 5 \\%$ chance that the feature importance lies within the given range), but are also highly consistent and stable. We carry out a detailed theoretical analysis that leverages the aforementioned uncertainty to estimate how many perturbations to sample, and how to sample for faster convergence. This work makes the first attempt at addressing several critical issues with popular explanation methods in one shot, thereby generating consistent, stable, and reliable explanations with guarantees in a computationally efficient manner. Experimental evaluation with multiple real world datasets and user studies demonstrate that the efficacy of the proposed framework.1 ",
+        "bbox": [
+            232,
+            335,
+            766,
+            612
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            640,
+            310,
+            657
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "As machine learning (ML) models get increasingly deployed in domains such as healthcare and criminal justice, it is important to ensure that decision makers have a clear understanding of the behavior of these models. However, ML models that achieve state-of-the-art accuracy are typically complex black boxes that are hard to understand. As a consequence, there has been a surge in post hoc techniques for explaining black box models [1–10]. Most popular among these techniques are local explanation methods which explain complex black box models by constructing interpretable local approximations (e.g., LIME [2], SHAP [4], MAPLE [11], Anchors [1]). Due to their generality, these methods are being leveraged to explain a number of classifiers including deep neural networks and ensemble models in a variety of domains such as law, medicine, and finance [12, 13]. ",
+        "bbox": [
+            174,
+            671,
+            825,
+            796
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Existing local explanation methods, however, suffer from several drawbacks. Explanations generated using these methods may be unstable [14–18], i.e., negligibly small perturbations to an instance can result in substantially different explanations. These methods are also inconsistent [19] i.e., multiple runs on the same input instance with the same parameter settings may result in vastly different explanations. There are also no reliable metrics to ascertain the quality of the explanations ",
+        "bbox": [
+            174,
+            803,
+            825,
+            872
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/d47efc4990be75e69296fffac100fa38760e5883dfb996999588185ce3d0e586.jpg",
+        "image_caption": [
+            "(b) Explanation with 2000 perturbations "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            511,
+            104,
+            810,
+            204
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/3f716bc0fdf329c8651b34a0ee8151d109ef2249803f50f834c077a2687e7548.jpg",
+        "image_caption": [
+            "Figure 1: Example explanations on for an instance from the COMPAS dataset, where vertical lines indicate the feature importance by LIME (red is negative effect, green is positive) and the shaded region visualizes the uncertainty estimated by BayesLIME. While LIME produces very different and contradictory feature importance for different number of perturbations (1a and 1b), BayesLIME provides more context. The overlapping uncertainty intervals in the explanation computed with 100 perturbations (1a) indicate that it is unclear which feature is the most important. However, the tighter uncertainty intervals in the explanation computed with 2K perturbations (1b) clearly indicates that Female is the most important. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            191,
+            104,
+            488,
+            205
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "(a) Explanation computed with 100 perturbations ",
+        "bbox": [
+            192,
+            218,
+            483,
+            232
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "output by these methods. Commonly used metrics such as explanation fidelity rely heavily on the implementation details of the explanation method (e.g., the perturbation function used in LIME) and do not provide a true picture of the explanation quality [20]. Furthermore, there exists little to no guidance on determining the values of certain hyperparameters that are critical to the quality of the resulting local explanations (e.g., number of perturbations in case of LIME). Local explanation methods are also computationally inefficient i.e., they typically require a large number of black box model queries to construct local approximations [21]. This can be prohibitively slow especially in case of complex neural models. ",
+        "bbox": [
+            174,
+            380,
+            825,
+            491
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we identify that modeling uncertainty in black box explanations is the key to addressing all the aforementioned challenges. To this end, we propose a novel Bayesian framework for generating local explanations along with their associated uncertainty. We instantiate this framework to obtain Bayesian versions of LIME and KernelSHAP, namely BayesLIME and BayesSHAP, that not only output point-wise estimates of feature importance but also their associated uncertainty in the form of credible intervals (See Figure 1). We derive closed form expressions for the posteriors of the explanations thereby eliminating the need for any additional computational complexity. The credible intervals produced by our framework not only allow us to make concrete inferences about the quality of the resulting explanations but also produce explanations that satisfy user specified levels of uncertainty (e.g., an end user may request for explanations that satisfy a certain $9 5 \\%$ confidence level). In addition, the resulting explanations are also highly consistent and stable. To the best of our knowledge, this work makes the first attempt at addressing several critical challenges in popular explanation methods in one-shots, thereby generating consistent, stable, and reliable explanations with guarantees in a computationally efficient manner. ",
+        "bbox": [
+            174,
+            497,
+            825,
+            690
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We carry out theoretical analysis that leverages the measures of uncertainty (credible intervals) produced by our framework to estimate the values of critical hyperparameters. More specifically, we derive a closed form expression for the number of perturbations required to generate explanations that satisfy desired levels of confidence. We also propose a novel sampling technique called focused sampling that leverages uncertainty to determine how to sample perturbations for faster convergence, thereby enabling our framework to generate explanations in a computationally efficient manner. ",
+        "bbox": [
+            174,
+            696,
+            825,
+            780
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We evaluate the efficacy of the proposed framework on a variety of datasets including COMPAS, German Credit, ImageNet, and MNIST. Our results demonstrate that the explanations output by our framework are not only highly reliable, but also very consistent and stable $5 3 \\%$ more stable than LIME/SHAP on an average). Our experimental results also confirm that we can accurately estimate the number of perturbations needed to generate explanations with a desired level of uncertainty, and that our uncertainty sampling technique speeds up the process of generating explanations by up to a factor of 2 relative to random sampling of perturbations. Lastly, we carry out a user study with 31 human subjects to evaluate the quality of the explanations generated by our framework, demonstrating that our explanations accurately capture the importance of the most influential features. ",
+        "bbox": [
+            174,
+            786,
+            825,
+            911
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Notation & Background ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            89,
+            406,
+            107
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Here we introduce notation and discuss two relevant prior approaches, LIME and KernelSHAP. ",
+        "bbox": [
+            171,
+            119,
+            797,
+            136
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Notation Let $f : \\mathbb { R } ^ { d }  [ 0 , 1 ]$ denote a black box classifier that takes a data point $x$ with $d$ features, and returns the probability that $x$ belongs to a certain class. Our goal is to explain individual predictions of $f$ . Let $\\phi \\in \\mathbb { R } ^ { d }$ denote the explanation in terms of feature importances for the prediction $f ( x )$ , i.e. coefficients $\\phi$ are treated as the feature contributions to the black box prediction. Note that $\\phi$ captures the coefficients of a linear model. Let $\\mathcal { Z }$ be a set of $N$ randomly sampled instances (perturbations) around $x$ . The proximity between $x$ and any $z \\in { \\mathcal { Z } }$ is given by $\\pi _ { x } ( z ) \\in \\mathbb { R }$ . We denote the vector of these distances over the $N$ perturbations in $\\mathcal { Z }$ as $\\Pi _ { x } ( \\hat { \\mathcal { Z } } ) \\in \\mathbb { R } ^ { \\mathrm { \\tilde { \\cal N } } }$ . Let $Y \\in [ 0 , 1 ]$ be the vector of the black box predictions $f ( z )$ corresponding to each of the $N$ instances in $\\mathcal { Z }$ . ",
+        "bbox": [
+            173,
+            140,
+            825,
+            253
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "LIME [2] and KernelSHAP [4] are popular model-agnostic local explanation approaches that explain predictions of a classifier $f$ by learning a linear model $\\phi$ locally around each prediction (i.e. $y \\overset { \\cdot } { \\sim } \\phi ^ { T } \\overset { \\cdot } { z } ,$ ). The objective function for both LIME and KernelSHAP constructs an explanation that approximates the behavior of the black box accurately in the vicinity (neighborhood) of $x$ . ",
+        "bbox": [
+            173,
+            257,
+            825,
+            314
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5b5d797713a6469acad5fb0bf6e7bc946bba80013ead1410fa91caa14bcccc0b.jpg",
+        "text": "$$\n\\underset { \\phi } { \\arg \\operatorname* { m i n } } \\sum _ { z \\in \\mathcal { Z } } [ f ( z ) - \\phi ^ { T } z ] ^ { 2 } \\pi _ { x } ( z ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            377,
+            320,
+            602,
+            356
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The above objective function has the following closed form solution: ",
+        "bbox": [
+            176,
+            361,
+            622,
+            377
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0afcf2cad957bb91d713ca1d977d180eabe894afd8197cdb69554ea594d79514.jpg",
+        "text": "$$\n\\hat { \\phi } = ( \\mathcal { Z } ^ { T } \\mathrm { d i a g } ( \\Pi _ { x } ( \\mathcal { Z } ) ) \\mathcal { Z } + \\mathbb { I } ) ^ { - 1 } ( \\mathcal { Z } ^ { T } \\mathrm { d i a g } ( \\Pi _ { x } ( \\mathcal { Z } ) ) Y )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            323,
+            383,
+            674,
+            404
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The main difference between LIME and KernelSHAP lies in how $\\pi _ { x } ( z )$ is chosen. In LIME, it is chosen heuristically: $\\pi _ { x } ( z )$ is computed as the cosine or $l _ { 2 }$ distance. KernelSHAP leverages game theoretic principles to compute $\\pi _ { x } ( z )$ , guaranteeing that explanations satisfy certain properties. ",
+        "bbox": [
+            173,
+            417,
+            825,
+            460
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Our Framework: Bayesian Local Explanations ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            478,
+            598,
+            497
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section, we introduce our Bayesian framework which is designed to capture the uncertainty associated with local explanations of black box models. First, we discuss the generative process and inference procedure for the framework. Then, we highlight how our framework can be instantiated to obtain Bayesian versions of LIME and SHAP. Lastly, we present detailed theoretical analysis for estimating the values of critical hyperparameters, and discuss how to efficiently construct highly accurate explanations with uncertainty guarantees using our framework. ",
+        "bbox": [
+            173,
+            510,
+            825,
+            594
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 Constructing Bayesian Local Explanations ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            609,
+            509,
+            626
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our goal here is to explain the behavior of a given black box model $f$ in the vicinity of an instance $x$ while also capturing the uncertainty associated with the explanation. To this end, we propose a Bayesian framework for constructing local linear model based explanations and capturing their associated uncertainty. We model the black box prediction of each perturbation $z$ as a linear combination of the corresponding feature values $( \\phi ^ { \\dot { T } } z )$ plus an error term (\u000f) as shown in Eqn (4). While the weights of the linear combination $\\phi$ capture the feature importances and thereby constitute our explanation, $\\epsilon$ captures the error that arises due to the mismatch between our explanation $\\phi$ and the local decision surface of the black box model $f$ . Our complete generative process is shown below: ",
+        "bbox": [
+            173,
+            635,
+            826,
+            747
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d6895be82c9552d896bdeba4d8aa98055324bd5c698e6bfedb43377deffa4bb8.jpg",
+        "text": "$$\n\\begin{array} { r l r } & { } & { y | z , \\phi , \\epsilon \\sim \\phi ^ { T } z + \\epsilon \\qquad \\epsilon \\sim \\mathcal { N } ( 0 , \\displaystyle \\frac { \\sigma ^ { 2 } } { \\pi _ { x } ( z ) } ) } \\\\ & { } & { \\phi | \\sigma ^ { 2 } \\sim \\mathcal { N } ( 0 , \\sigma ^ { 2 } \\mathbb { I } ) \\qquad \\sigma ^ { 2 } \\sim \\mathrm { I n v } - \\chi ^ { 2 } ( n _ { 0 } , \\sigma _ { 0 } ^ { 2 } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            351,
+            753,
+            647,
+            810
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The error term is modeled as a Gaussian whose variance relies on the proximity function $\\pi _ { x } ( z )$ i.e., $\\begin{array} { r } { \\epsilon \\sim \\mathcal { N } ( 0 , \\frac { \\sigma ^ { 2 } } { \\pi _ { x } ( z ) } ) } \\end{array}$ . This proximity function ensures that perturbations closer to the data point $x$ are modeled accurately, while allowing more room for error in case of perturbations that are farther away. $\\pi _ { x } ( z )$ can be computed using cosine or $l _ { 2 }$ distance or other game theoretic principles similar to that of LIME and KernelSHAP (see Section 2). The conjugate priors on $\\phi$ and $\\bar { \\sigma } ^ { 2 }$ are shown in Eqn (4). Note that, the distributions on error $\\epsilon$ and feature importance $\\phi$ both consider the parameter $\\sigma ^ { 2 }$ . The fact that the prior on the feature importances considers $\\sigma ^ { 2 }$ has an intuitive interpretation: if we have prior knowledge that the error of the explanation is small, we expect to be more confident about the feature importances. Similarly, if we have prior knowledge the error is large, we expect to be less confident about the feature importances. ",
+        "bbox": [
+            173,
+            820,
+            825,
+            912
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            90,
+            825,
+            147
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Thus, our generative process corresponds to the Bayesian version of the weighted least squares formulation of LIME and KernelSHAP outlined in Eqn. (1), with additional terms to model uncertainty. As in Eqns. (4), the process captures two sources of uncertainty in local explanations: 1) feature importance uncertainty: the uncertainty associated with the feature importances $\\phi$ , and (2) error uncertainty: the uncertainty associated with the error term $\\epsilon$ which captures how well our explanation $\\phi$ models the local decision surface of the underlying black box. ",
+        "bbox": [
+            173,
+            152,
+            825,
+            237
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Inference Our inference process involves estimating the values of two key parameters: $\\phi$ and $\\sigma ^ { 2 }$ . By doing so, we can compute the local explanation as well as the uncertainties associated with feature importances and the error term. Posterior distributions on $\\phi$ and $\\sigma ^ { 2 }$ are normal and scaled Inv- $\\chi ^ { 2 }$ , respectively, due to the corresponding conjugate priors [22]: ",
+        "bbox": [
+            174,
+            247,
+            825,
+            304
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ce99269200b8311cd63ef3c46c41140dd45c6a22a0c985b5bfc62b9be77124a6.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\sigma ^ { 2 } | \\mathcal { Z } , Y \\sim \\mathrm { S c a l e d - I n v - } \\chi ^ { 2 } \\left( n _ { 0 } + N , \\frac { n _ { 0 } \\sigma _ { 0 } ^ { 2 } + N s ^ { 2 } } { n _ { 0 } + N } \\right) } \\\\ & { \\phi | \\sigma ^ { 2 } , \\mathcal { Z } , Y \\sim \\mathrm { N o r m a l } ( \\hat { \\phi } , V _ { \\phi } \\sigma ^ { 2 } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
+            306,
+            671,
+            364
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Further, $\\hat { \\phi } , V _ { \\phi }$ , and $s ^ { 2 }$ can be directly computed: ",
+        "bbox": [
+            173,
+            367,
+            490,
+            383
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/756ed874aec151dfbac77feb2be03dd078941ebbec8dbb8c6401fcce987809c5.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\hat { \\phi } = V _ { \\phi } ( \\mathcal { Z } ^ { T } \\mathrm { d i a g } ( \\Pi _ { x } ( \\mathcal { Z } ) ) Y ) } \\\\ & { V _ { \\phi } = \\left( \\mathcal { Z } ^ { T } \\mathrm { d i a g } ( \\Pi _ { x } ( \\mathcal { Z } ) ) \\mathcal { Z } + \\mathbb { I } \\right) ^ { - 1 } } \\\\ & { s ^ { 2 } = \\displaystyle \\frac { 1 } { N } \\left[ ( Y - \\mathcal { Z } \\hat { \\phi } ) ^ { T } \\mathrm { d i a g } ( \\Pi _ { x } ( \\mathcal { Z } ) ) ( Y - \\mathcal { Z } \\hat { \\phi } ) + \\hat { \\phi } ^ { T } \\hat { \\phi } \\right] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            318,
+            386,
+            678,
+            462
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Details of the complete inference procedure including derivations of Eqns. (5-7) are provided in the Appendix A. Note that our estimate of the posterior mean feature importances $\\hat { \\phi }$ (Eqn. (6)) is the same as that of the feature importances computed in case of LIME and KernelSHAP (Eqn. (2)). ",
+        "bbox": [
+            176,
+            463,
+            821,
+            507
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Remark 3.1. If we use the same proximity function $\\pi _ { x } ( z )$ in our framework as in LIME or KernelSHAP, the posterior mean of the feature importance $\\hat { \\phi }$ output by our framework $E q$ (6)) will be equivalent to the feature importances output by LIME or KernelSHAP, respectively. ",
+        "bbox": [
+            174,
+            508,
+            825,
+            554
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Feature Importance Uncertainty To obtain the local feature importances and their associated uncertainty, we first compute the posterior mean of the local feature importances $\\hat { \\phi }$ using the closed form expression in Eqn. (7). We then estimate the credible interval (measure of uncertainty) around the mean feature importances by repeatedly sampling from the posterior distribution of $\\phi$ (Eq (5)). ",
+        "bbox": [
+            173,
+            568,
+            825,
+            627
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Error Uncertainty The error term $\\epsilon$ can serve as a proxy for explanation quality because it captures the mismatch between the constructed explanation and the local decision surface of the underlying black box. We first calculate the marginal posterior distribution of $\\epsilon$ by leveraging Eqn (4) and integrating out $\\sigma ^ { 2 }$ . This results in a three parameter Student’s t distribution (derivation in appendix A): ",
+        "bbox": [
+            173,
+            638,
+            825,
+            707
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/62d140f78c59e2efd62ba5c9efa76014ee9044c50f3c428d1da005a3d6e1aac5.jpg",
+        "text": "$$\n\\epsilon | \\mathcal { Z } , Y \\sim t _ { ( \\nu = n _ { 0 } + N ) } ( 0 , \\frac { n _ { 0 } \\sigma _ { 0 } ^ { 2 } + N s ^ { 2 } } { n _ { 0 } + N } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            367,
+            705,
+            630,
+            739
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We then evaluate the probability density function (PDF) of the above posterior at 0, i.e., $P ( \\epsilon = 0 )$ by substituting the value of $s ^ { 2 }$ computed using Eqn. (7) into the Student’s t distribution above (Eqn. (8)). The resulting expression gives us the probability density that the explanation output by our framework perfectly captures the local decision surface underlying the black box. This operation is performed in constant time, adding minimal overhead to non-Bayesian LIME and SHAP. We illustrate how these computed intervals capture the variance in the explanations in Figure 9. ",
+        "bbox": [
+            173,
+            739,
+            825,
+            824
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 3.2. As the number of perturbations around $x$ goes to $\\infty$ i.e., $N \\to \\infty$ : $( l )$ the estimate of $\\phi$ converges to the true feature importance scores, and its uncertainty to 0. (2) uncertainty of the error term \u000f converges to the bias of the local linear model $\\phi$ . [Details in Appendix B] ",
+        "bbox": [
+            174,
+            825,
+            825,
+            868
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "BayesLIME and BayesSHAP Our framework can be instantiated to obtain the Bayesian version of LIME by setting the proximity function to $\\pi _ { x } ( z ) = \\exp ( - D ( x , z ) ^ { 2 } / \\sigma ^ { 2 } )$ where $D$ is a distance metric ",
+        "bbox": [
+            173,
+            882,
+            823,
+            912
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "(e.g. cosine or $l _ { 2 }$ distance), and $n _ { 0 }$ and $\\sigma _ { 0 } ^ { 2 }$ to small values $( 1 0 ^ { - 6 } )$ so that the prior is uninformative.   \nWe compute feature importance uncertainty and error uncertainty for LIME’s feature importances. ",
+        "bbox": [
+            171,
+            90,
+            825,
+            119
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Our framework can also be instantiated to obtain the Bayesian version of KernelSHAP by setting uninformative prior on $\\sigma ^ { 2 }$ and d−1(d choose |z|)|z|(d−|z|) where |z| denotes the number of the variables in the variable combination represented by the data point $z$ i.e., the number of non-zero valued features in the vector representation of $z$ . Note that the original SHAP method views the problem of constructing a local linear model as estimating the Shapley values corresponding to each of the features [4]. These Shapley values represent the contribution of each of the features to the black box prediction i.e., $f ( x ) = \\phi _ { 0 } + \\sum \\phi _ { i }$ . Therefore, the measures of uncertainty output by our method BayesSHAP capture the reliability of the estimated variable contributions. ",
+        "bbox": [
+            173,
+            126,
+            825,
+            241
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To encourage BayesLIME and BayesSHAP explanations to be sparse, we can use dimensionality reduction or feature selection techniques as used by LIME and SHAP to obtain the top K features [2, 4, 23]. We can then construct our explanations using the data corresponding to these top K features. ",
+        "bbox": [
+            174,
+            247,
+            826,
+            289
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 Estimating the Number of Perturbations ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            304,
+            495,
+            319
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "One of the major drawbacks of approaches such as LIME and KernelSHAP is that they do not provide any guidance on how to choose the number of perturbations, a key factor in obtaining reliable explanations in an efficient manner. To address this, we leverage the uncertainty estimates output by our framework to compute perturbations-to-go $( G )$ , an estimate of how many more perturbations are required to obtain explanations that satisfy a desired level of certainty. This estimate thus predicts the computational cost of generating an explanation with a desired level of certainty and can help determine whether it is even worthwhile to do so. The user specifies the confidence level of the credible interval (denoted as $\\alpha$ ) and the maximum width of the credible interval $( W )$ , e.g. “width of $9 5 \\%$ credible interval should be less than $0 . 1 ^ { \\mathfrak { s } }$ corresponds to $\\alpha = 0 . 9 5$ and $W = 0 . 1$ . To estimate $G$ for the local explanation of a data point $x$ , we first generate $S$ perturbations around $x$ (where $S$ is small and chosen by the user) and fit a local linear model using our method2. This provides initial estimates of various parameters shown in Eqns (5)-(7) which can then be used to compute $G$ . ",
+        "bbox": [
+            173,
+            329,
+            825,
+            496
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.3. Given $S$ seed perturbations, the number of additional perturbations required $( G )$ to achieve a credible interval width $W$ of feature importance for a data point $x$ at user-specified confidence level $\\alpha$ can be computed as: ",
+        "bbox": [
+            173,
+            501,
+            825,
+            542
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/40186f46e1519fa4d25e8379f3069f26be6dc308e2e04c8e31c4467cc0aa5026.jpg",
+        "text": "$$\nG ( W , \\alpha , x ) = \\frac { 4 s _ { S } ^ { 2 } } { \\bar { \\pi } _ { S } \\times \\left[ \\frac { W } { \\Phi ^ { - 1 } ( \\alpha ) } \\right] ^ { 2 } } - S\n$$",
+        "text_format": "latex",
+        "bbox": [
+            375,
+            547,
+            622,
+            603
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\bar { \\pi } _ { S }$ is the average proximity $\\pi _ { x } ( z )$ for the $S$ perturbations, $s _ { S } ^ { 2 }$ is the empirical sum of squared errors (SSE) between the black box and local linear model predictions, weighted by $\\pi _ { x } ( z )$ , as in (7), and $\\Phi ^ { - 1 } ( \\alpha )$ is the two-tailed inverse normal CDF at confidence level $\\alpha$ . ",
+        "bbox": [
+            174,
+            608,
+            825,
+            652
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proof (Sketch). To estimate $G$ , we first relate $W$ and $\\alpha$ to $\\operatorname { V a r } ( \\phi _ { i } )$ , the marginal variance of the feature importance3 for any feature $i$ , obtained by integrating out $\\sigma ^ { 2 }$ . Because Student’s t can be approximated by a Normal distribution for large degrees of freedom (here, $S$ should be large enough), we use the inverse normal CDF to calculate credible interval width at level $\\alpha$ . We compute $V _ { \\phi }$ from (6) using $\\mathcal { Z }$ , treating its entries as Bernoulli distributed with probability 0.5. Due to the covariance structure of this sampling procedure, the resulting variance estimate after $N$ samples is the sample SSE $s _ { S } ^ { 2 }$ scaled by $\\approx \\frac { 4 } { \\hat { \\pi } _ { S } N }$ (derivation in appendix B). If we assume SSE scales linearly with $S$ , we can take this to be a reasonable estimate of $s _ { N } ^ { 2 }$ at any $N$ . We can then estimate $G$ as ",
+        "bbox": [
+            173,
+            665,
+            826,
+            782
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/86c08edb74635b95fbbf45f531e8136345e4c7137d6e42a24af5f7be0dd7f494.jpg",
+        "text": "$$\n\\left[ \\frac { W } { \\Phi ^ { - 1 } ( \\alpha ) } \\right] ^ { 2 } = \\mathrm { V a r } ( \\phi _ { i } ) = \\frac { 4 s _ { S } ^ { 2 } } { \\bar { \\pi } _ { S } \\times ( G + S ) } \\Longrightarrow G = \\frac { 4 s _ { S } ^ { 2 } } { \\bar { \\pi } _ { S } \\times \\left[ \\frac { W } { \\Phi ^ { - 1 } ( \\alpha ) } \\right] ^ { 2 } } - S .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            238,
+            787,
+            758,
+            845
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 Focused Sampling of Perturbations ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            90,
+            457,
+            106
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Perturbations-to-go $( G )$ provides us with an estimate of how many samples are required to achieve reliable explanations. However, if $G$ is large, querying the black-box model for its predictions on a large number of perturbations can be computationally expensive for larger models [24, 25]. To reduce this cost, we develop an alternative sampling procedure called focused sampling which leverages uncertainty estimates to query the black box in a more targeted fashion (instead of querying randomly), thereby reducing the computational cost associated with generating reliable explanations. Inspired by active learning [26], focused sampling strategically prioritizes perturbations whose predictions the explanation is most uncertain about, when querying the black box. This enables the focused sampling procedure to query the black box only for the predictions of the most informative perturbations and thereby learn an accurate explanation with far fewer queries to the black box. ",
+        "bbox": [
+            173,
+            116,
+            825,
+            256
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To determine how uncertain our explanation $\\phi$ is about the black box label for any given instance $z$ , we first compute the posterior predictive distribution for $z$ (derivation in Appendix A), given as $\\boldsymbol { \\hat { y } } ( z ) | \\mathcal { Z } , \\boldsymbol { Y } \\sim t _ { ( \\mathcal { V } = N ) } ( \\boldsymbol { \\hat { \\phi } } ^ { T } \\boldsymbol { z } , ( \\boldsymbol { z } ^ { T } V _ { \\phi } \\boldsymbol { z } + 1 ) \\boldsymbol { s } ^ { 2 } )$ . The variance of this three parameter student’s t distribution is, ",
+        "bbox": [
+            174,
+            261,
+            825,
+            319
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/924ee084bc57d7be88147e6536f4ee59feef084ad947ea7a4fcf28f3831f4841.jpg",
+        "text": "$$\n\\mathrm { v a r } ( \\hat { y } ( z ) ) = ( ( z ^ { T } V _ { \\phi } z + 1 ) s ^ { 2 } ) ( N / ( N - 2 ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            351,
+            316,
+            647,
+            335
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We refer to this variance as the predictive variance $\\mathrm { v a r } ( \\hat { y } ( z ) )$ , and it captures how uncertain our explanation $\\phi$ is about the black box prediction. ",
+        "bbox": [
+            171,
+            339,
+            821,
+            367
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The focus sampling procedure first fits the explanation with an initial $S$ perturbations (where $S$ is a small number). We then iterate the following procedure until the desired explanation certainty level is reached. We draw a batch of $A$ candidate perturbations, compute their predictive variance with the Bayesian explanation, and induce a distribution over the perturbations by running softmax on the variances with tempurature parameter $\\tau$ . We draw a batch of $B$ perturbations from this distribution and query the black box model for their labels. Finally, we refit the Bayesian explanation on all the labeled perturbations collected so far. We provide pseudocode for the uncertainty sampling procedure in Algorithm 1. ",
+        "bbox": [
+            173,
+            372,
+            825,
+            484
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/2aaf3ed471491c6419c0b3b2d1fe53312178ed130d8e0cc93c1ffde5ad05dc98.jpg",
+        "table_caption": [
+            "Algorithm 1 Focused sampling for local explanations "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Require: Model f,Data instance x, Number of perturbations N,Number of seed perturbations S,</td></tr><tr><td colspan=\"2\">Batch size B,Pool size A, tempurature T 1: function FOCUSED SAMPLE</td></tr><tr><td></td><td>Initialize Z with S seed perturbations.</td></tr><tr><td>2:</td><td></td></tr><tr><td>3:</td><td>Fit on Z Using Eqn (6)</td></tr><tr><td>4:</td><td>fori←1toN-Sinincrements ofBdo</td></tr><tr><td>5:</td><td>Q ←Generate Acandidate perturbations Using Eqn (11)</td></tr><tr><td>6:</td><td>Compute var(y(z)) on Q</td></tr><tr><td>7:</td><td>Define Qdist as X exp(var(g(z))/τ)</td></tr><tr><td>8:</td><td>Qnew ← Draw B samples from Qdist</td></tr><tr><td>9:</td><td>Z ← ZU Qnew; Fit on Z Using Eqn (6)</td></tr><tr><td>10: end for</td><td></td></tr><tr><td>11: return $</td><td></td></tr><tr><td>12: end function</td><td></td></tr></table>",
+        "bbox": [
+            176,
+            518,
+            826,
+            722
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            756,
+            312,
+            773
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We evaluate the proposed framework by first analyzing the quality of our uncertainty estimates i.e., feature importance uncertainty and error uncertainty. We also assess our estimates of required perturbations $( G )$ , and evaluate the computational efficiency of focused sampling. Last, we describe a user study with 31 subjects to assess the informativeness of the explanations output by our framework. ",
+        "bbox": [
+            174,
+            787,
+            825,
+            844
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Setup We experiment with a variety of real world datasets spanning multiple applications (e.g., criminal justice, credit scoring) as well as modalities (e.g., structured data, images). Our first structured dataset is COMPAS [27], containing criminal history, jail and prison time, and demographic attributes of 6172 defendants, with class labels that represent whether each defendant was rearrested within 2 years of release. The second structured dataset is the German Credit dataset from the UCI repository [28] containing financial and demographic information (including account information, credit history, employment, gender) for 1000 loan applications, each labeled as a “good” or “bad” customer. We create 80/20 train/test splits for these two datasets, and train a random forest classifier (sklearn implementation with 100 estimators) as black box models for each (test accuracy of $8 2 . 8 \\%$ and $7 2 . 5 \\%$ , respectively). We also include popular image datasets–MNIST and Imagenet. For the MNIST [29] handwritten digits dataset, we train a 2-layer CNN to predict the digits (test accuracy of $9 9 . 2 \\%$ ). For Imagenet [30], we use the off-the-shelf VGG16 model [31] as the black box. We select a sample of 100 images of the following classes French Bulldog, Scuba Diver, Corn, and Broccoli to use in the experiments. For generating explanations, we use standard implementations of the baselines LIME and KernelSHAP with default settings [2, 4]. For images, we construct super pixels as described in [2] and use them as features (number of super pixels is fixed to 20 per image). For our framework, the desired level of certainty is expressed as the width of the $9 5 \\%$ credible interval. ",
+        "bbox": [
+            174,
+            856,
+            825,
+            911
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/2efd1dd6a443caac7d848e203e6d424ec21eceafae245c5245181b231f109fe5.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 1: Evaluating Credible Intervals. We report the $\\%$ of time the $9 5 \\%$ credible intervals with 100 perturbations include their true values (estimated on 10, 000 perturbations). Closer to 95.0 is better. Both BayesLIME and BayesSHAP are well calibrated. "
+        ],
+        "table_body": "<table><tr><td></td><td>BayesLIME</td><td>BayesSHAP</td><td></td><td>BayesLIME</td><td>BayesSHAP</td></tr><tr><td>TABULAR DATASETS</td><td></td><td></td><td>MNIST</td><td></td><td></td></tr><tr><td>COMPAS</td><td>95.5</td><td>87.9</td><td>Digit 1</td><td>95.8</td><td>98.4</td></tr><tr><td>German Credit</td><td>96.9</td><td>89.6</td><td>Digit 2</td><td>95.8</td><td>97.4</td></tr><tr><td>IMAGENET</td><td></td><td></td><td>Digit 3</td><td>95.2</td><td>96.3</td></tr><tr><td>Corn</td><td>94.6</td><td>91.8</td><td>Digit 4</td><td>97.2</td><td>90.1</td></tr><tr><td>Broccoli</td><td>91.4</td><td>89.2</td><td>Digit 5</td><td>95.2</td><td>95.6</td></tr><tr><td>French Bulldog</td><td>94.8</td><td>89.9</td><td>Digit 6</td><td>96.7</td><td>96.8</td></tr><tr><td>Scuba Diver</td><td>92.4</td><td>94.6</td><td>Digit 7</td><td>95.7</td><td>95.3</td></tr></table>",
+        "bbox": [
+            187,
+            88,
+            803,
+            228
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            314,
+            825,
+            493
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Quality of Uncertainty Estimates A critical component of our explanations is the feature importance uncertainty. To evaluate the correctness of these estimates, we compute how often true feature importances lie within the $9 5 \\%$ credible intervals estimated by BayesLIME and BayesSHAP. Note, that by true feature importance, we refer to the best fit linear model output using either the LIME or SHAP kernels. We evaluate the quality of our credible interval estimates by running our methods with 100 perturbations to estimate feature importances and taking the corresponding $9 5 \\%$ credible intervals for each test instance. We compute what fraction of the true feature importances fall within our $9 5 \\%$ credible intervals. Note, because there are no methods to provide uncertainty estimates for LIME and SHAP, we do not provide further baselines. Since we do not have access to the true feature importances of the complex black box models, following Prop 3.2, we use feature importances computed using a large value of $N$ $( N = 1 0 , 0 0 0 )$ , and treat the resulting estimates as ground truth. ",
+        "bbox": [
+            174,
+            506,
+            825,
+            659
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Results for BayesLIME in Table 1 indicate that the true feature importances are close to ideal and indicate the estimates are well calibrated. While the estimates by BayesSHAP are somewhat less calibrated (true feature importances fall within our estimated $9 5 \\%$ credible intervals about 89.2 to $9 8 . 4 \\%$ of the time), they still are quite close to ideal. All in all, these results confirm that the credible intervals learned by our methods are well calibrated and therefore highly reliable in capturing the uncertainty of the feature importances. Lastly, though we set our priors to be uninformative in general, we also investigate how sensitive our uncertainty estimates are to hyperparameter choices in Figure 5 in the Appendix. We find that the explanation uncertainty becomes uncalibrated with strong priors. However, our explanations seem to be robust to hyperparameter choices in general. ",
+        "bbox": [
+            174,
+            664,
+            825,
+            790
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Correctness of Estimated Number of Perturbations We assess whether our estimate of perturbations-to-go $G$ ; Section 3.2) is an accurate estimate of the additional number of perturbations needed to reach a desired level of feature importance certainty. We carry out this experiment on MNIST data for the digit $\" 4 > \"$ (additional datasets explored in Appendix C) and use $S = 2 0 0$ as the initial number of perturbations to obtain a preliminary explanation and its associated uncertainty estimates. We then leverage these estimates to compute $G$ for 6 different certainty levels. First, we observe significant differences in $G$ estimates across instances (details in appendix C) i.e. number of perturbations needed to obtain a particular level of certainty varied significantly across instances– ranging from 200-5, 000 for the lowest level of certainty to 200-20, 000 for higher levels of certainty. Next, for each image and certainty level, we run our method for the estimated number of perturbations $( G )$ to determine if the observed estimates of uncertainty (observed credible interval width $W$ ) match the desired levels of uncertainty (desired credible interval width $W$ ). Results in Figure 2 show that the observed and desired levels of certainty are well calibrated, demonstrating that $G$ estimates are reliable approximations of the additional number of perturbations needed. ",
+        "bbox": [
+            174,
+            800,
+            825,
+            911
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/a1ce895bb1fbdbf243ad90abc4bbd6376035f01308ad9e7c0365cb10fc50cc0d.jpg",
+        "image_caption": [
+            "Figure 2: Perturbations-to-go $( G )$ . We generate explanation with $G$ perturbations, where $G$ is computed using the desired credible interval width $\\mathbf { \\bar { X } }$ -axis), and compare desired levels to the observed credible interval width (y-axis) (blue line indicates ideal calibration). Results are averaged over 100 MNIST images of the digit $\" 4 > \"$ We see that $G$ provides a good approximation of the additional perturbations needed. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            196,
+            98,
+            800,
+            265
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            378,
+            825,
+            462
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Efficiency of Focused Sampling Focused sampling uses the predictive variance to strategically choose perturbations that will reduce uncertainty in order to be labeled by the black box (section 3.3). Here, we will evaluate the efficiency of the focused sampling procedure. First, we assess whether focused sampling converges (as measured by error uncertainty $P ( \\epsilon = 0 ) )$ ) more efficiently than random sampling. To this end, we experiment with BayesLIME on Imagenet data for the “French bulldog” class to carry out this analysis. This setting replicates scenarios where LIME is applied to a computationally expensive black box model, making it highly desirable to limit the number of perturbations to reduce total running time. We run each sampling strategy for 2,000 perturbations and plot the number of model queries versus error uncertainty. During focused sampling, we set the batch size $B$ to 50. The results in Figure 3 show that focused sampling results in faster convergence to reliable and high quality explanations; focused sampling stabilizes within a couple hundred model queries while random sampling takes over 1,000. Note, as the inefficiency of querying the black box model increases, the advantages of focused sampling decreasing total running time of the explanations will only become more pronounced. These results clearly demonstrate that focused sampling can significantly speed up the process of generating high quality local explanations. Additionally, in Appendix C, we also check if focused sampling causes any bias (due to sampling based on uncertainty estimates) that results in convergence to a different/wrong explanation, however our results clearly indicate that this is not the case. ",
+        "bbox": [
+            173,
+            474,
+            825,
+            723
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Stability of BayesLIME & BayesSHAP Recall that LIME & SHAP are not stable: small changes to instances can produce substantially different explanations. We consider whether BayesLIME & BayesSHAP produce more stable explanations than their LIME & SHAP counterparts. To perform this analysis, we use the local Lipschitz metric for explanation stability [18]: ",
+        "bbox": [
+            174,
+            733,
+            825,
+            790
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/520ca044f1ba8cc0ca59bc40e01f2d02aa400e93eb6f6741a3ef7a14ee11a80c.jpg",
+        "text": "$$\n\\hat { L } ( x _ { i } ) = \\operatorname * { a r g m a x } _ { x _ { j } \\in N _ { \\epsilon } ( x _ { i } ) } \\frac { | | \\phi _ { i } - \\phi _ { j } | | _ { 2 } } { | | x _ { i } - x _ { j } | | _ { 2 } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            395,
+            796,
+            602,
+            833
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "where $x _ { i }$ refers to an instance, $N _ { \\epsilon } ( x _ { i } )$ is the $\\epsilon$ -ball centered at $x _ { i }$ , and $\\phi _ { i }$ and $\\phi _ { j }$ are the explanation parameters for $x _ { i }$ and $x _ { j }$ . Lower values indicate more stable explanations. We follow the setup outline by Alvarez-Melis and Jaakkola [18] and compute the local Lipschitz values, comparing both LIME & BayesLIME and SHAP & BayesSHAP across Compas, German Credit, MNIST digit $\\cdot _ { 4 } \\cdot \\cdot$ , and Imagenet “French Bulldog.” We perform the comparison using the default number of perturbations in both LIME & SHAP, and use this same number in the respective Bayesian variants and set the batch size $B$ to half this value. We use focused sampling for BayesLIME and BayesSHAP, and report the $\\%$ increase in stability of these approaches over LIME and SHAP for 40 test points. The results given in Figure 4 show a clear improvement (on average $5 3 \\%$ ) in stability in all cases except German Credit for BayesSHAP. Further, we run a Wilcoxon signed-rank test and find our results are statistically significant in all cases $\\mathrm { / e < 1 e { - } 2 ) }$ except for BayesSHAP for German Credit, where there is not a significant difference between the methods $\\zeta _ { \\rho } > 0 . 0 5 )$ . These results demonstrate BayesLIME and BayesSHAP are more stable than previous methods. ",
+        "bbox": [
+            174,
+            842,
+            823,
+            911
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/dbdcf8d6b10a60e362cba9afcfbbaed135a2dd3c956ce32266eb747c42dae508.jpg",
+        "image_caption": [
+            "Figure 3: Efficiency of focused sampling for 100 Imagenet “French bulldog” images, with random sampling as a baseline. We provide mean and standard error. We assess the efficiency of focused sampling by comparing error uncertainty over model queries and show quicker convergence than random sampling. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            97,
+            455,
+            255
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/988605277650e44dda266f35f97a56551dc27c3e649264948462a5108a6c7e8f.jpg",
+        "image_caption": [
+            "Figure 4: Assessing the $\\%$ increase in stability of BayesLIME and BayesSHAP over LIME and SHAP respectively. Our Bayesian methods are significant more stable $\\rho \\ < \\ 1 { \\mathrm { e } } { \\mathrm { - } } 2$ according to Wilcoxon signed-rank test) except for BayesSHAP on German Credit, where there is not a significant difference between the methods $\\zeta _ { \\rho } > 0 . 0 5 )$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            498,
+            94,
+            787,
+            256
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            385,
+            825,
+            496
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "User Study We perform a user study with 31 subjects to compare BayesLIME and LIME explanations on MNIST. We evaluate the following: are explanations with low levels of uncertainty (i.e., most confident explanations) more meaningful to humans? To answer this question, we follow prior work and mask the most important features selected by BayesLIME and LIME [32, 4]. We ask users to guess the digit of the masked images. The better the explanation, the more difficult it should be for the users to get it right. Further, the choice to mask the important features is motivated by its success in prior work. We randomly select 15 correctly predicted test images, generate explanations by sweeping over a range of perturbation amounts $[ 1 \\dot { 0 } ^ { 5 } , . . . , 1 0 ^ { 3 . 5 } ]$ incremented by 0.5. We choose the top explanation for each image based on either fidelity (for LIME) or $P ( \\epsilon = 0 )$ (for BayesLIME). We sent the user study out to students and researchers with background in computer science. A screen shot of the task is shown in Figure 7 in the Appendix. We find that the explanations output by our methods focus on more informative parts of the image, since hiding them makes it difficult for humans to guess the digit. Users had an error rate of $2 5 . 7 \\%$ for LIME, while it was $3 0 . 7 \\%$ for BayesLIME, both with standard error 0.003 $\\mathrm { \\Delta } \\rho = 0 . 0 2 8$ through a one-tailed two sample t-test). This result indicates that our method BayesLIME and the associated measure of explanation uncertainty result in more high quality and reliable explanations compared to LIME and its associated fidelity metric. ",
+        "bbox": [
+            174,
+            508,
+            825,
+            742
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 Related Work ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            762,
+            321,
+            780
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Interpretability Methods A variety of interpretability methods have been proposed. Some methods that are inherently interpretable include additive models [33, 34], decision lists and sets [35, 36], and instance-based explanations [37]. However, black-box models are often more flexible, accurate, and easier to use; thus, there has been a lot of interest in constructing post hoc explanations[38]. These include LIME [2] and SHAP [4, 39], which are among the most popular due to their broad applicability and code availability, but saliency maps [5–8], permutation feature importance [40], and partial dependency plots [41] also follow this paradigm. Other approaches to post hoc explanations focus on rule-based models [1, 3], counterfactuals [42, 43], and influence functions [9]. ",
+        "bbox": [
+            174,
+            800,
+            825,
+            911
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Vulnerabilities of Post hoc Explanations Recent work has shed light on the downsides of post hoc explanation techniques. These methods are often highly sensitive to small changes in inputs [14], are susceptible to manipulation [15, 16, 44, 45], and are not faithful to the underlying black boxes [46]. Perturbation-based explanation methods such as LIME and SHAP are subject to additional criticisms: results vary between runs of the algorithms [18–20, 47, 21], and hyperparameters used to select the perturbations can greatly influence the resulting explanation [20]. Prior work has attempted to tackle the problem of instability in perturbation-based explanations by averaging over several explanations [48, 19], however, this is computationally expensive. Other works related to creating more trustworthy explanations include development of sanity checks for explainers [49, 17, 50]. These techniques represent an important step towards improved usability, given experimental evidence that humans are often too eager to accept inaccurate machine explanations [51–54]. Recent works theoretically analyze the sources of non-robustness in black box explanations [55–57]. ",
+        "bbox": [
+            174,
+            90,
+            825,
+            257
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Logical and Formal Reasoning Additional related works have considered explaining classifiers through identifying a subset of features that are “sufficient” to explain a prediction [58–62]. Though these methods offer strong guarantees surrounding which features ensure a prediction is achieved, they are not model agnostic. Further, they do not define feature importances associated with the local explanations nor consider ways to improve locally weighted explanations, such as LIME and SHAP. ",
+        "bbox": [
+            174,
+            268,
+            825,
+            338
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Bayesian Methods in Explainable ML Few recent works have adopted Bayesian formulations to explain black box models [63–65]. Guo et al. [63] introduce a Bayesian non-parametric approach to fit a global surrogate model. Their formulation seeks to fit a mixture of generalizable explanations across instances. Zhao et al. [64] study whether incorporating informative priors improves the stability of the resulting explanations. However, neither of these works focus on modeling the uncertainty of local explanations. Further, these approaches also do not tackle the critical problems of estimating key hyperparameters or improving efficiency of computing explanations. ",
+        "bbox": [
+            174,
+            349,
+            825,
+            446
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6 Conclusion ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            467,
+            299,
+            483
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We developed a Bayesian framework for generating local explanations along with their associated uncertainty. We instantiated this framework to obtain Bayesian versions of LIME and SHAP that output pointwise estimates of feature importances as well as their associated credible intervals. These intervals enabled us to infer the quality of the explanations and output explanations that satisfied user specified levels of uncertainty. We carried out theoretical analysis that leverages these uncertainty measures (credible intervals) to estimate the values of critical hyperparameters (e.g., the number of perturbations). We also proposed a novel sampling technique called focused sampling that leverages uncertainty estimates to determine how to sample perturbations for faster convergence. ",
+        "bbox": [
+            174,
+            497,
+            825,
+            608
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "While the Bayesian framework addresses several critical challenges (i.e., consistency, stability, modeling uncertainty) associated with LIME and SHAP, there are still certain aspects where it would exhibit the same shortcomings as LIME and SHAP [4, 66]. For instance, if the local decision surface of a given black box classifier is highly non-linear, our framework, which relies on local linear approximations, may not be able to capture this non-linear decision surface accurately. In addition, if the perturbation sampling procedures used in LIME and SHAP are used in BayesLIME and BayesSHAP, they will likely be vulnerable to the attacks proposed by Slack et al. [15]. In the future, it would be interesting to extend our framework to produce global explanations with uncertainty guarantees and explore how uncertainty quantification can help calibrate user trust in model explanations. ",
+        "bbox": [
+            174,
+            614,
+            825,
+            753
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "7 Acknowledgments ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            772,
+            357,
+            790
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We would like to thank the anonymous reviewers for their insightful feedback. This work is supported in part by the NSF awards #IIS-2008461, #IIS-2008956, and #IIS-2040989, and research awards from the Harvard Data Science Institute, Amazon, Bayer, Google, and the HPI Research Center in Machine Learning and Data Science at UC Irvine. The views expressed are those of the authors and do not reflect the official policy or position of the funding agencies. ",
+        "bbox": [
+            174,
+            803,
+            825,
+            873
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "References   \n[1] Marco Tulio Ribeiro, Sameer Singh, and Carlos Guestrin. Anchors: High-precision modelagnostic explanations. In Thirty-Second AAAI Conference on Artificial Intelligence, 2018.   \n[2] Marco Tulio Ribeiro, Sameer Singh, and Carlos Guestrin. Why Should I Trust You? explaining the predictions of any classifier. In Knowledge Discovery and Data mining (KDD), 2016.   \n[3] Himabindu Lakkaraju, Ece Kamar, Rich Caruana, and Jure Leskovec. Faithful and customizable explanations of black box models. In Proceedings of the 2019 AAAI/ACM Conference on AI, Ethics, and Society, pages 131–138. ACM, 2019.   \n[4] Scott M Lundberg and Su-In Lee. A unified approach to interpreting model predictions. In Advances in Neural Information Processing Systems, pages 4765–4774, 2017.   \n[5] Karen Simonyan, Andrea Vedaldi, and Andrew Zisserman. Deep inside convolutional networks: Visualising image classification models and saliency maps. In Workshop at International Conference on Learning Representations, 2014.   \n[6] Mukund Sundararajan, Ankur Taly, and Qiqi Yan. Axiomatic attribution for deep networks. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pages 3319–3328. JMLR. org, 2017.   \n[7] Ramprasaath R Selvaraju, Michael Cogswell, Abhishek Das, Ramakrishna Vedantam, Devi Parikh, and Dhruv Batra. Grad-cam: Visual explanations from deep networks via gradient-based localization. In ICCV, 2017.   \n[8] Daniel Smilkov, Nikhil Thorat, Been Kim, Fernanda Viégas, and Martin Wattenberg. Smoothgrad: removing noise by adding noise. Workshop on Visualization for Deep Learning, ICML, 2017.   \n[9] Pang Wei Koh and Percy Liang. Understanding black-box predictions via influence functions. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pages 1885–1894. JMLR. org, 2017.   \n[10] Osbert Bastani, Carolyn Kim, and Hamsa Bastani. Interpretability via model extraction. FAT/ML Workshop 2017, 2017.   \n[11] Gregory Plumb, Denali Molitor, and Ameet S Talwalkar. Model agnostic supervised local explanations. In Neural Information Processing Systems, 2018.   \n[12] Radwa Elshawi, Mouaz H Al-Mallah, and Sherif Sakr. On the interpretability of machine learning-based model for predicting hypertension. BMC medical informatics and decision making, 19(1):146, 2019.   \n[13] Leanne S Whitmore, Anthe George, and Corey M Hudson. Mapping chemical performance on molecular structures using locally interpretable explanations. arXiv preprint arXiv:1611.07443, 2016.   \n[14] Amirata Ghorbani, Abubakar Abid, and James Zou. Interpretation of neural networks is fragile. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pages 3681–3688, 2019.   \n[15] Dylan Slack, Sophie Hilgard, Emily Jia, Sameer Singh, and Himabindu Lakkaraju. Fooling lime and shap: Adversarial attacks on post hoc explanation methods. Conference on Artificial Intelligence, Ethics, and Society (AIES), 2020.   \n[16] Ann-Kathrin Dombrowski, Maximilian Alber, Christopher J Anders, Marcel Ackermann, KlausRobert Müller, and Pan Kessel. Explanations can be manipulated and geometry is to blame. arXiv preprint arXiv:1906.07983, 2019.   \n[17] Julius Adebayo, Justin Gilmer, Michael Muelly, Ian Goodfellow, Moritz Hardt, and Been Kim. Sanity checks for saliency maps. In Advances in Neural Information Processing Systems, pages 9505–9515, 2018. ",
+        "bbox": [
+            171,
+            45,
+            826,
+            912
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "[18] David Alvarez-Melis and Tommi S. Jaakkola. On the robustness of interpretability methods. ICML Workshop on Human Interpretability in Machine Learning, 2018. ",
+        "bbox": [
+            169,
+            90,
+            825,
+            119
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[19] Eunjin Lee, David Braines, Mitchell Stiffler, Adam Hudler, and Daniel Harborne. Developing the sensitivity of lime for better machine learning explanation. In Artificial Intelligence and Machine Learning for Multi-Domain Operations Applications, volume 11006, page 1100610. International Society for Optics and Photonics, 2019. ",
+        "bbox": [
+            173,
+            128,
+            826,
+            184
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[20] Hui Fen Tan, Kuangyan Song, Madeilene Udell, Yiming Sun, and Yujia Zhang. “why should you trust my explanation?” understanding uncertainty in lime explanations. In ICML Workshop on AI for Social Good, 2019. ",
+        "bbox": [
+            173,
+            193,
+            823,
+            234
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[21] Jianbo Chen, Le Song, Martin J. Wainwright, and Michael I. Jordan. L-shapley and c-shapley: Efficient model interpretation for structured data. In International Conference on Learning Representations, 2019. ",
+        "bbox": [
+            174,
+            243,
+            825,
+            286
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[22] Andrew Moore. Locally weighted bayesian regression, January 1995. ",
+        "bbox": [
+            176,
+            294,
+            665,
+            310
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[23] Kacper Sokol, Alexander Hepburn, Raúl Santos-Rodríguez, and Peter A. Flach. blimey: Surrogate prediction explanations beyond lime. NeurIPS HCML Workshop, 2019. ",
+        "bbox": [
+            174,
+            318,
+            823,
+            347
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[24] Emily L. Denton, Wojciech Zaremba, Joan Bruna, Yann LeCun, and Rob Fergus. Exploiting linear structure within convolutional networks for efficient evaluation. In NIPS, 2014. ",
+        "bbox": [
+            176,
+            354,
+            823,
+            383
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[25] Max Jaderberg, Andrea Vedaldi, and Andrew Zisserman. Speeding up convolutional neural networks with low rank expansions. BMVC 2014 - Proceedings of the British Machine Vision Conference 2014, 05 2014. ",
+        "bbox": [
+            174,
+            392,
+            825,
+            434
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[26] Burr Settles. Active learning literature survey. 2010. ",
+        "bbox": [
+            173,
+            443,
+            555,
+            458
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[27] Julia Angwin, Jeff Larson, Surya Mattu, and Lauren Kirchner. Machine bias. In ProPublica, 2016. ",
+        "bbox": [
+            174,
+            465,
+            825,
+            496
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[28] Dheeru Dua and Casey Graff. Uci machine learning repository, 2017. URL http://archive. ics.uci.edu/ml. ",
+        "bbox": [
+            174,
+            502,
+            825,
+            532
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[29] Yann LeCun, Corinna Cortes, and CJ Burges. Mnist handwritten digit database. ATT Labs [Online]. Available: http://yann.lecun.com/exdb/mnist, 2, 2010. ",
+        "bbox": [
+            173,
+            540,
+            825,
+            570
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[30] J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei. ImageNet: A Large-Scale Hierarchical Image Database. In CVPR09, 2009. ",
+        "bbox": [
+            173,
+            577,
+            825,
+            606
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[31] Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. In ICLR, 2015. ",
+        "bbox": [
+            173,
+            614,
+            825,
+            643
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[32] Patrick Schwab and Walter Karlen. Cxplain: Causal explanations for model interpretation under uncertainty. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-Buc, E. Fox, and R. Garnett, editors, Advances in Neural Information Processing Systems 32, pages 10220–10230. Curran Associates, Inc., 2019. URL http://papers.nips.cc/paper/ 9211-cxplain-causal-explanations-for-model-interpretation-under-uncertainty. pdf. ",
+        "bbox": [
+            174,
+            651,
+            862,
+            736
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[33] Yin Lou, Rich Caruana, Johannes Gehrke, and Giles Hooker. Accurate intelligible models with pairwise interactions. In KDD, 2013. ",
+        "bbox": [
+            173,
+            743,
+            826,
+            773
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[34] Berk Ustun, Stefano Traca, and Cynthia Rudin. Supersparse linear integer models for interpretable classification. arXiv preprint arXiv:1306.6677, 2013. ",
+        "bbox": [
+            173,
+            781,
+            825,
+            810
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[35] Himabindu Lakkaraju, Stephen H Bach, and Jure Leskovec. Interpretable decision sets: A joint framework for description and prediction. In Knowledge Discovery and Data mining (KDD), 2016. ",
+        "bbox": [
+            171,
+            818,
+            826,
+            861
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[36] Elaine Angelino, Nicholas Larus-Stone, Daniel Alabi, Margo Seltzer, and Cynthia Rudin. Learning certifiably optimal rule lists for categorical data. arXiv preprint arXiv:1704.01701, 2017. ",
+        "bbox": [
+            174,
+            868,
+            826,
+            911
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "[37] Been Kim, Cynthia Rudin, and Julie Shah. The bayesian case model: A generative approach for case-based reasoning and prototype classification. arXiv preprint arXiv:1503.01161, 2015. ",
+        "bbox": [
+            171,
+            90,
+            825,
+            121
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "[38] Marco Tulio Ribeiro, Sameer Singh, and Carlos Guestrin. Model-Agnostic Interpretability of Machine Learning. In ICML Workshop on Human Interpretability in Machine Learning (WHI), June 2016. ",
+        "bbox": [
+            174,
+            128,
+            823,
+            171
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "[39] Ian Covert and Su-In Lee. Improving kernelshap: Practical shapley value estimation using linear regression. In Arindam Banerjee and Kenji Fukumizu, editors, Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, volume 130 of Proceedings of Machine Learning Research, pages 3457–3465. PMLR, 13–15 Apr 2021. URL https: //proceedings.mlr.press/v130/covert21a.html. ",
+        "bbox": [
+            174,
+            181,
+            826,
+            252
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "[40] Leo Breiman. Random forests. Machine learning, 45(1):5–32, 2001. ",
+        "bbox": [
+            173,
+            261,
+            661,
+            276
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "[41] Jerome H Friedman. Greedy function approximation: a gradient boosting machine. Annals of statistics, pages 1189–1232, 2001. ",
+        "bbox": [
+            174,
+            285,
+            825,
+            314
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "[42] Berk Ustun, Alexander Spangher, and Yang Liu. Actionable recourse in linear classification. In Proceedings of the Conference on Fairness, Accountability, and Transparency, FAT\\* ’19, pages 10–19, 2019. ISBN 978-1-4503-6125-5. ",
+        "bbox": [
+            173,
+            324,
+            826,
+            366
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "[43] Sandra Wachter, Brent Mittelstadt, and Chris Russell. Counterfactual explanations without opening the black box: Automated decisions and the gdpr. Harv. JL & Tech., 31:841, 2017. ",
+        "bbox": [
+            171,
+            376,
+            825,
+            405
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "[44] Juyeon Heo, Sunghwan Joo, and Taesup Moon. Fooling neural network interpretations via adversarial model manipulation. In Advances in Neural Information Processing Systems 32, pages 2921–2932. 2019. ",
+        "bbox": [
+            173,
+            414,
+            823,
+            457
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "[45] Junlin Wang, Jens Tuyls, Eric Wallace, and Sameer Singh. Gradient-based Analysis of NLP Models is Manipulable. In Findings of the Association for Computational Linguistics: EMNLP (EMNLP Findings), page 247–258, 2020. ",
+        "bbox": [
+            173,
+            465,
+            825,
+            508
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "[46] Cynthia Rudin. Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead. Nature Machine Intelligence, 1(5):206, 2019. ",
+        "bbox": [
+            169,
+            518,
+            823,
+            547
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "[47] Muhammad Rehman Zafar and Naimul Mefraz Khan. Dlime: a deterministic local interpretable model-agnostic explanations approach for computer-aided diagnosis systems. arXiv preprint arXiv:1906.10263, 2019. ",
+        "bbox": [
+            174,
+            556,
+            825,
+            599
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "[48] Chih-Kuan Yeh, Cheng-Yu Hsieh, Arun Suggala, David I Inouye, and Pradeep K Ravikumar. On the (in) fidelity and sensitivity of explanations. In Advances in Neural Information Processing Systems, pages 10965–10976, 2019. ",
+        "bbox": [
+            174,
+            608,
+            825,
+            651
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "[49] Oana-Maria Camburu, Eleonora Giunchiglia, Jakob Foerster, Thomas Lukasiewicz, and Phil Blunsom. Can i trust the explainer? verifying post-hoc explanatory methods. arXiv preprint arXiv:1910.02065, 2019. ",
+        "bbox": [
+            174,
+            660,
+            825,
+            703
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "[50] Mengjiao Yang and Been Kim. Bim: Towards quantitative evaluation of interpretability methods with ground truth. arXiv:1907.09701, 2019. ",
+        "bbox": [
+            171,
+            713,
+            823,
+            742
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "[51] Harmanpreet Kaur, Harsha Nori, Samuel Jenkins, Rich Caruana, Hanna Wallach, and Jennifer Wortman Vaughan. Interpreting interpretability: Understanding data scientists’ use of interpretability tools for machine learning. In CHI, April 2020. ",
+        "bbox": [
+            173,
+            751,
+            823,
+            794
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "[52] Fred Hohman, Andrew Head, Rich Caruana, Robert DeLine, and Steven M Drucker. Gamut: A design probe to understand how data scientists understand machine learning models. In Proceedings of the 2019 CHI Conference on Human Factors in Computing Systems, pages 1–13, 2019. ",
+        "bbox": [
+            174,
+            803,
+            826,
+            858
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "[53] Forough Poursabzi-Sangdeh, Daniel G Goldstein, Jake M Hofman, Jennifer Wortman Vaughan, and Hanna Wallach. Manipulating and measuring model interpretability. arXiv preprint arXiv:1802.07810, 2018. ",
+        "bbox": [
+            174,
+            869,
+            825,
+            911
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "[54] Himabindu Lakkaraju and Osbert Bastani. \" how do i fool you?\" manipulating user trust via misleading black box explanations. In Proceedings of the AAAI/ACM Conference on AI, Ethics, and Society, pages 79–85, 2020. ",
+        "bbox": [
+            171,
+            90,
+            823,
+            133
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "[55] Damien Garreau and Ulrike von Luxburg. Looking deeper into lime. arXiv preprint arXiv:2008.11092, 2020. ",
+        "bbox": [
+            171,
+            142,
+            825,
+            171
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "[56] Prasad Chalasani, Jiefeng Chen, Amrita Roy Chowdhury, Xi Wu, and Somesh Jha. Concise explanations of neural networks using adversarial training. In International Conference on Machine Learning, pages 1383–1391. PMLR, 2020. ",
+        "bbox": [
+            174,
+            179,
+            823,
+            223
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "[57] Alexander Levine, Sahil Singla, and Soheil Feizi. Certifiably robust interpretation in deep learning. arXiv preprint arXiv:1905.12105, 2019. ",
+        "bbox": [
+            173,
+            231,
+            825,
+            261
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "[58] Eric Wang, Pasha Khosravi, and Guy Van den Broeck. Probabilistic sufficient explanations. In IJCAI, 2021. ",
+        "bbox": [
+            171,
+            268,
+            825,
+            297
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "[59] Andy Shih, Arthur Choi, and Adnan Darwiche. A symbolic approach to explaining bayesian network classifiers. In Proceedings of the 27th International Joint Conference on Artificial Intelligence, IJCAI’18, page 5103–5111. AAAI Press, 2018. ISBN 9780999241127. ",
+        "bbox": [
+            174,
+            306,
+            825,
+            349
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "[60] Weijia Shi, Andy Shih, Adnan Darwiche, and Arthur Choi. On tractable representations of binary neural networks. In Diego Calvanese, Esra Erdem, and Michael Thielscher, editors, Proceedings of the 17th International Conference on Principles of Knowledge Representation and Reasoning, KR 2020, Rhodes, Greece, September 12-18, 2020, pages 882–892, 2020. doi: 10.24963/kr.2020/91. URL https://doi.org/10.24963/kr.2020/91. ",
+        "bbox": [
+            174,
+            358,
+            826,
+            428
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "[61] Yacine Izza, Alexey Ignatiev, and Joao Marques-Silva. 2020. ",
+        "bbox": [
+            173,
+            436,
+            609,
+            452
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "[62] Alexey Ignatiev, Nina Narodytska, and Joao Marques-Silva. Abduction-based explanations for machine learning models. Proceedings of the AAAI Conference on Artificial Intelligence, 33 (01):1511–1519, Jul. 2019. doi: 10.1609/aaai.v33i01.33011511. URL https://ojs.aaai. org/index.php/AAAI/article/view/3964. ",
+        "bbox": [
+            174,
+            460,
+            826,
+            517
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "[63] Wenbo Guo, Sui Huang, Yunzhe Tao, Xinyu Xing, and Lin Lin. Explaining deep learning models – a bayesian non-parametric approach. In Neural Information Processing Systems (NeurIPS). 2018. ",
+        "bbox": [
+            174,
+            526,
+            823,
+            568
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "[64] Xingyu Zhao, Xiaowei Huang, Valentin Robu, and David Flynn. Baylime: Bayesian local interpretable model-agnostic explanations. arXiv preprint arXiv:2012.03058, 2020. ",
+        "bbox": [
+            171,
+            577,
+            823,
+            607
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "[65] Kirill Bykov, Marina Höhne, Klaus-Robert Müller, Shinichi Nakajima, and Marius Kloft. How much can i trust you? – quantifying uncertainties in explaining neural networks. arXiv, 06 2020. ",
+        "bbox": [
+            169,
+            614,
+            825,
+            645
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "[66] Sushant Agarwal, Shahin Jabbari, Chirag Agarwal, Sohini Upadhyay, Steven Wu, and Himabindu Lakkaraju. Towards the unification and robustness of perturbation and gradient based explanations. In Marina Meila and Tong Zhang, editors, Proceedings of the 38th International Conference on Machine Learning, volume 139 of Proceedings of Machine Learning Research, pages 110–119. PMLR, 18–24 Jul 2021. URL https://proceedings.mlr. press/v139/agarwal21c.html. ",
+        "bbox": [
+            174,
+            652,
+            826,
+            737
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "[67] Ludwig Fahrmeir, Thomas Kneib, and Stefan Lang. Regression. Statistik und ihre Anwendungen. Springer, 2007. ISBN 978-3-540-33932-8. ",
+        "bbox": [
+            171,
+            746,
+            825,
+            773
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "[68] Christopher M. Bishop. Pattern Recognition and Machine Learning. Springer, 2006. ",
+        "bbox": [
+            168,
+            784,
+            764,
+            799
+        ],
+        "page_idx": 13
+    }
+]

parse/train/rqfq0CYIekd/rqfq0CYIekd_middle.json ADDED Viewed

The diff for this file is too large to render. See raw diff

parse/train/rqfq0CYIekd/rqfq0CYIekd_model.json ADDED Viewed

The diff for this file is too large to render. See raw diff