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+# WHAT WOULD $\pi ^ { * }$ DO?: IMITATION LEARNING VIA OFF-POLICY REINFORCEMENT LEARNING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Learning to imitate expert actions given demonstrations containing image observations is a difficult problem in robotic control. The key challenge is generalizing behavior to out-of-distribution states that differ from those in the demonstrations. State-of-the-art imitation learning algorithms perform well in environments with low-dimensional observations, but typically involve adversarial optimization procedures, which can be difficult to use with high-dimensional image observations. We propose a remarkably simple alternative based on off-policy soft Q-learning, which we call soft $Q$ imitation learning (SQIL, pronounced “skill”), that rewards the agent for matching demonstrated actions in demonstrated states. The key idea is initially filling the agent’s experience replay buffer with demonstrations, where rewards are set to a positive constant, and setting rewards to zero in all additional experiences. We derive SQIL from first principles as a method for performing approximate inference under the MaxCausalEnt model of expert behavior. The approximate inference objective trades off between a pure behavioral cloning loss and a regularization term that incorporates information about state transitions via the soft Bellman error. Our experiments show that SQIL matches the state of the art in low-dimensional environments, and significantly outperforms prior work in playing video games from high-dimensional images.
+
+# 1 INTRODUCTION
+
+Many real-world sequential decision-making problems can be tackled by imitation learning, where an expert demonstrates near-optimal behavior to an agent that then attempts to replicate that behavior in novel situations (Argall et al., 2009). This paper considers the problem of training an agent to imitate an expert policy, given expert action demonstrations and access to the environment. The agent does not get to observe a reward signal or query the expert, and does not know the state transition dynamics ex-ante.
+
+Standard approaches to this problem based on behavioral cloning seek to imitate the expert’s actions, but do not reason about the consequences of actions (Pomerleau, 1991). As a result, they suffer from state distribution shift, and fail to generalize to states that are very different from those seen in the demonstrations (Ross & Bagnell, 2010; Ross et al., 2011). Approaches based on inverse reinforcement learning (IRL) deal with this issue by fitting a reward function that represents preferences over trajectories rather than individual actions $\mathrm { N g }$ et al., 2000; Ziebart et al., 2008), and using the learned reward function to train the imitation agent through RL (Wulfmeier et al., 2015; Finn et al., 2016; Fu et al., 2017). This is the core idea behind generative adversarial imitation learning (GAIL), which implicitly combines IRL and RL using generative adversarial networks (GANs) (Ho & Ermon, 2016; Goodfellow et al., 2014). GAIL is state-of-the-art in environments with low-dimensional observations, but requires additional reward augmentation and feature engineering when applied to environments with high-dimensional image observations (Li et al., 2017), due to the difficulty of training GANs (Kurach et al., 2018). In short, imitation learning from raw pixel inputs without prior knowledge of the domain remains a challenge.
+
+The key insight in this paper is that instead of using a learned reward function or discriminator to provide a reward signal to the imitation policy, as is done in IRL-based imitation methods and GAIL, the imitation agent can simply be rewarded for matching demonstrated actions in demonstrated states using off-policy soft Q-learning. The agent’s experience replay buffer is initially filled with demonstrations where rewards are set to some positive constant, and gradually accumulates additional experiences with rewards set to zero. The agent has an incentive to imitate the expert in demonstrated states, and to take actions that lead it back to demonstrated states when it encounters new states that differ from the demonstrations. We call this algorithm soft $Q$ imitation learning (SQIL, pronounced “skill”).
+
+We motivate the SQIL algorithm by deriving it from first principles as a method for performing approximate inference under the MaxCausalEnt model of expert behavior (Ziebart et al., 2010). Our analysis shows that SQIL maximizes a lower bound on the log-posterior of the soft Q values given the demonstrations – a bound that trades off between a pure behavioral cloning loss and a regularization term that incorporates state transition information via the soft Bellman error.
+
+The main contribution of this paper is SQIL: a practical and general imitation learning algorithm that is effective in MDPs with high-dimensional, continuous observations – including raw pixel inputs – and has a natural theoretical interpretation as approximate inference. We run experiments in the image-based Car Racing and low-dimensional Lunar Lander game environments from OpenAI Gym, the Humanoid and HalfCheetah continuous control tasks from MuJoCo, and image-based Atari Pong, to compare SQIL to two prior methods: behavioral cloning and GAIL. The results show that SQIL matches GAIL in low-dimensional environments, and significantly outperforms GAIL on the image-based Car Racing and Pong tasks.
+
+# 2 PRELIMINARIES
+
+This work builds on the maximum causal entropy (MaxCausalEnt) model of expert behavior (Ziebart et al., 2010; Levine, 2018). In a finite-horizon Markov Decision Process (MDP) with a finite, discrete action space $\mathcal { A }$ ,1 the demonstrator is assumed to follow a policy $\pi$ that maximizes reward $R ( s , a )$ . To simplify exposition, we assume the state transition dynamics are deterministic: let $s ^ { \prime }$ denote the state that follows from taking action $a$ in state $s$ . The policy $\pi$ forms a Boltzmann distribution over actions,
+
+$$
+\pi ( a | s ) \triangleq \exp { \left( Q ( s , a ) - V ( s ) \right) } ,
+$$
+
+where $Q$ is the soft Q function, and $V$ is the soft value function,
+
+$$
+\begin{array} { r } { V ( s ) \triangleq \left\{ \begin{array} { l l } { 0 } & { \mathrm { ~ i f ~ } s \mathrm { ~ i s ~ a ~ t e r m i n a l ~ s t a t e } } \\ { \log \left( \sum _ { a \in A } \exp \left( Q ( s , a ) \right) \right) } & { \mathrm { ~ o t h e r w i s e } } \end{array} \right. } \end{array}
+$$
+
+In the standard MaxCausalEnt model, the soft Q values are a deterministic function of the rewards and dynamics, given by the soft Bellman equation,
+
+$$
+Q ( s , a ) \triangleq R ( s , a ) + \gamma V ( s ^ { \prime } ) .
+$$
+
+# 2.1 APPROXIMATE MAXCAUSALENT MODEL
+
+In the analysis in Section 3, the standard MaxCausalEnt model is approximated by a probabilistic model that treats the soft $\mathrm { Q }$ values as random variables,
+
+$$
+Q ( s , a ) \sim \mathcal { N } ( R ( s , a ) + \gamma V ( s ^ { \prime } ) , \sigma ^ { 2 } ) ,
+$$
+
+where $\sigma \in \mathbb { R }$ is a constant hyperparameter, and $R ( s , a )$ is a random variable with zero mean and unit variance. To simplify the analysis in Section 3, we assume that the dynamics of the environment satisfy the property that if state $s _ { i }$ is reachable from state $s _ { j }$ , then $s _ { j }$ is not reachable from $s _ { i }$ – this ensures that the probabilistic graphical model that captures the joint distribution over soft Q values is acyclic. The experiments in Section 5 show that even when this assumption is violated, SQIL performs well empirically.
+
+# 3 IMITATION LEARNING VIA APPROXIMATE MAXCAUSALENT INFERENCE
+
+We aim for an imitation learning algorithm that generalizes to new states, without resorting to adversarial optimization procedures that would be difficult to use with image observations. Our approach is to derive a lower bound on the log-posterior of the soft Q values given the demonstrations; in particular, a lower bound that is maximized by the SQIL algorithm.
+
+# 3.1 DERIVING THE APPROXIMATE INFERENCE OBJECTIVE
+
+Consider a demonstration rollout $\tau = ( s _ { 0 } , a _ { 0 } , s _ { 1 } , a _ { 1 } , . . . , s _ { T } )$ , where $s _ { T }$ is a terminal state. Standard MaxCausalEnt IRL searches for the reward function that maximizes the posterior probability of the rewards given the demonstration. Our goal, however, is not to infer the expert’s reward function; it is to infer the expert’s policy, which is represented by their soft $\mathrm { Q }$ function. We approach this problem by formalizing uncertainty about the soft Q values using the probabilistic model described in Section 2.1. We fit a soft Q function to optimize the posterior probability of the soft $\mathrm { Q }$ values $Q$ given the demonstration $\tau$ , where $Q \stackrel { \triangle } { = } \{ Q ( s , a ) \mid ( s , a ) \in \mathcal { X } \}$ , and $\mathcal { X }$ is a finite subset of $\boldsymbol { S } \times \boldsymbol { A }$ that includes the state-action pairs in $\tau$ . Since $Q ( s , a )$ may depend on $Q ( s ^ { \prime } , a ^ { \prime } )$ (see Equations 4 and 2), $\mathcal { X }$ is assumed to satisfy the closure property,
+
+$$
+\forall ( s , a ) \in \mathcal { X } . \big ( s ^ { \prime } \mathrm { ~ n o t ~ t e r m i n a l } \implies ( \forall a ^ { \prime } \in \mathcal { A } . ( s ^ { \prime } , a ^ { \prime } ) \in \mathcal { X } ) \big ) .
+$$
+
+The goal is to derive a lower bound on the log-posterior, $\log p ( Q | \tau )$ , that is maximized by the SQIL algorithm. The first step is to marginalize out the rewards, $\pmb { R } \triangleq \{ R ( s , a ) \mid ( s , a ) \in \chi \}$ , as follows.
+
+$$
+\begin{array} { r } { p ( Q | \tau ) \propto p ( \tau | Q ) p ( Q ) = \mathbb { E } _ { R } [ p ( \tau | Q , R ) p ( Q | R ) ] = \mathbb { E } _ { R } [ p ( \tau | Q ) p ( Q | R ) ] . } \end{array}
+$$
+
+First bound. Optimizing the log of the expectation in Equation 6 is hard, for a number of reasons: we haven’t assumed a particular prior distribution over rewards (only that it has zero mean and unit variance), and the soft Q values in $Q$ are not conditionally independent given $\pmb { R }$ . Optimizing the expectation of the log is easier. By Jensen’s inequality,
+
+$$
+\begin{array} { r } { \log \left( \mathbb E _ { R } [ p ( \tau | Q ) p ( Q | R ) ] \right) \geq \mathbb E _ { R } [ \log p ( \tau | Q ) + \log p ( Q | R ) ] . } \end{array}
+$$
+
+The conditional likelihood of the demonstration $\tau$ given the soft Q values $Q$ is given by Equation 1:
+
+$$
+p ( \tau | \mathbf { \Delta } Q ) = \exp \left( \sum _ { t = 0 } ^ { T - 1 } Q ( s _ { t } , a _ { t } ) - V ( s _ { t } ) \right) .
+$$
+
+The conditional likelihood of the soft Q values $Q$ given the rewards $\pmb { R }$ is given by Equation 4:
+
+$$
+p ( \pmb { Q } | \pmb { R } ) = \prod _ { ( s , a ) \in \mathcal { X } } \mathcal { N } ( Q ( s , a ) ; \pmb { R } ( s , a ) + \gamma V ( s ^ { \prime } ) , \sigma ^ { 2 } ) ,
+$$
+
+where the joint distribution over $Q$ can be factored due to the state re-entry assumption about the dynamics made in Section 2.1 and the closure assumption about $\mathcal { X }$ made in Equation 5. Simplifying the lower bound in Equation 7,
+
+$$
+\begin{array} { r l r } { \left. { ( 7 ) = \sum _ { t = 0 } ^ { T - 1 } Q ( s _ { t } , a _ { t } ) - V ( s _ { t } ) } } \\ & { } & { + \displaystyle \sum _ { ( s , a ) \in \mathcal { X } } - \frac { 1 } { 2 \sigma ^ { 2 } } \left( ( Q ( s , a ) - ( \mathbb { E } [ R ( s , a ) ] + \gamma V ( s ^ { \prime } ) ) \right) ^ { 2 } + \mathrm { V a r } [ R ( s , a ) ] \right) - \log \sqrt { 2 \pi \sigma ^ { 2 } } . } \end{array}
+$$
+
+Applying the assumption from Section 2.1 that the rewards have zero mean and unit variance,
+
+$$
+( 1 0 ) = \sum _ { t = 0 } ^ { T - 1 } Q ( s _ { t } , a _ { t } ) - V ( s _ { t } ) + \sum _ { ( s , a ) \in \mathcal { X } } - \frac { 1 } { 2 \sigma ^ { 2 } } ( ( Q ( s , a ) - \gamma V ( s ^ { \prime } ) ) ^ { 2 } + 1 ) - \log { \sqrt { 2 \pi \sigma ^ { 2 } } } .
+$$
+
+Second bound. Next, the bound is intentionally loosened, in order to arrive at an objective that is maximized by the SQIL algorithm.2 By the MaxCausalEnt assumptions in Section 2,
+
+$$
+V ( s _ { 0 } ) = \mathbb { E } _ { \tau \sim p ( \tau \mid \pi ) } \left[ \sum _ { t = 0 } ^ { T - 1 } \gamma ^ { t } \left( Q ( s _ { t } , a _ { t } ) - \gamma V ( s _ { t + 1 } ) + \mathcal { H } ( \pi ( \cdot | s _ { t } ) ) \right) \right] ,
+$$
+
+where $\mathcal { H }$ denotes entropy. Maximizing Equation 11 with respect to $Q$ yields a solution in which $Q ( s , a ) - \gamma V ( s ^ { \prime } ) \leq \bar { f } ( \bar { \sigma } ^ { 2 } )$ , where $f$ is an increasing function of $\sigma ^ { 2 }$ . Thus, $V ( s _ { 0 } ) \leq ( f ( \sigma ^ { 2 } ) +$ $\log | \mathcal { A } | ) / ( 1 - \gamma )$ , and
+
+$$
+( 1 1 ) \geq ( 1 1 ) + V ( s _ { 0 } ) - ( f ( \sigma ^ { 2 } ) + \log | A | ) / ( 1 - \gamma ) .
+$$
+
+The soft Q function that maximizes the lower bound in Equation 13 is
+
+$$
+Q ^ { * } \triangleq \operatorname * { a r g m a x } _ { Q } V ( s _ { 0 } ) + \sum _ { t = 0 } ^ { T - 1 } Q ( s _ { t } , a _ { t } ) - V ( s _ { t } ) - \frac { 1 } { 2 \sigma ^ { 2 } } \sum _ { ( s , a ) \in \mathcal X } ( Q ( s , a ) - \gamma V ( s ^ { \prime } ) ) ^ { 2 } .
+$$
+
+Practical implementation. For environments with a small, discrete state space $s$ and known dynamics, $\mathcal { X }$ can be set to $\boldsymbol { s } \times \boldsymbol { A }$ . For environments with a continuous state space $s$ and unknown dynamics, we set $\mathcal { X }$ to be the set of state-action pairs in the demonstration rollouts $\mathcal { D } _ { \mathrm { d e m o } }$ and rollouts $\mathcal { D } _ { \mathrm { s a m p } }$ of the imitation policy, which are periodically sampled during training.3 We fit a parameterized soft Q function $\bar { Q _ { \theta } }$ to minimize the loss,
+
+$$
+\ell ( \pmb \theta ) \triangleq \sum _ { \tau \in \mathcal { D } _ { \pmb { \mathscr { A } } _ { \mathrm { r m o } } } } \left( - V _ { \pmb \theta } ( s _ { 0 } ) + \sum _ { t = 0 } ^ { T - 1 } - ( Q _ { \theta } ( s _ { t } , a _ { t } ) - V _ { \pmb \theta } ( s _ { t } ) ) \right) + \lambda _ { \sf d e m o } B _ { \theta } ( \mathcal { D } _ { \sf d e m o } , 0 ) + \lambda _ { \sf s a m p } B _ { \theta } ( \mathcal { D } _ { \sf s a m p } , 0 ) ,
+$$
+
+where $\alpha , \lambda _ { \mathrm { d e m o } } , \lambda _ { \mathrm { s a m p } } \in \mathbb { R } _ { \ge 0 }$ are constant hyperparameters, $V _ { \pmb { \theta } }$ denotes the soft value function given by $Q _ { \theta }$ and Equation 2,4 and $B _ { \theta }$ denotes the sum of squared soft Bellman errors,
+
+$$
+B _ { \theta } ( \mathcal { D } , r ) \triangleq \sum _ { \tau \in \mathcal { D } } \sum _ { t = 0 } ^ { T - 1 } ( Q _ { \theta } ( s _ { t } , a _ { t } ) - ( r + \gamma V _ { \theta } ( s _ { t + 1 } ) ) ) ^ { 2 } ,
+$$
+
+where $r \in \mathbb { R }$ is a constant that does not depend on the state or action. The experiments in Section 5 use a convolutional neural network or multi-layer perceptron to model $Q _ { \theta }$ , where $\pmb { \theta }$ are the weights of the neural network (Schmidhuber, 2015).
+
+The loss in Equation 15 bounds the log-posterior of the soft $\mathrm { Q }$ values given the demonstrations. Searching for a soft Q function that optimizes this objective is equivalent to performing approximate inference under the MaxCausalEnt model of expert behavior. The objective combines a behavioral cloning loss, which does not reason about the consequences of actions, with a regularization term that incorporates information about state transitions. The regularization term performs soft Bellman backups on a reward of zero – a consequence of the prior distribution over rewards, which has an expected value of zero – and can be seen as imposing a prior distribution on the soft $\mathrm { Q }$ values that is independent of the expert demonstrations.
+
+# 3.2 IMPLEMENTING APPROXIMATE INFERENCE WITH OFF-POLICY SOFT Q-LEARNING
+
+Surprisingly, the gradient of the approximate inference objective in Equation 15 is equivalent to the gradient of the objective of a soft Q-learning algorithm (Haarnoja et al., 2017) that gives the agent a constant positive reward for matching the demonstrated action in a demonstrated state, and zero reward otherwise. Differentiating the objective in Equation 15,
+
+$$
+\begin{array} { r l } & { \nabla _ { \theta } \ell ( \theta ) = \alpha \underset { \tau \in \mathcal { D } _ { \mathrm { d a m } } } { \sum } \Bigg ( - \nabla V _ { \theta } ( s _ { 0 } ) + \underset { t = 0 } { \overset { T - 1 } { \sum } } - ( \nabla Q _ { \theta } ( s _ { t } , a _ { t } ) - \nabla V _ { \theta } ( s _ { t } ) ) \Bigg ) } \\ & { \quad \quad \quad + \lambda _ { \mathrm { d e m } } \underset { \tau \in \mathcal { D } _ { \mathrm { d a m } } } { \sum } \underset { t = 0 } { \overset { T - 1 } { \sum } } \nabla ( Q _ { \theta } ( s _ { t } , a _ { t } ) - \gamma V _ { \theta } ( s _ { t + 1 } ) ) ^ { 2 } + \lambda _ { \mathrm { s a m p } } \nabla B _ { \theta } ( \mathcal { D } _ { \mathrm { s a m p } } , 0 ) } \\ & { \quad \quad = \underset { \tau \in \mathcal { D } _ { \mathrm { d a m } } } { \sum } \Bigg ( - \alpha \nabla V _ { \theta } ( s _ { 0 } ) + \underset { t = 0 } { \overset { T - 1 } { \sum } } \alpha \nabla V _ { \theta } ( s _ { t } ) - \alpha \gamma \nabla V _ { \theta } ( s _ { t + 1 } ) \Bigg ) } \\ & { \quad \quad \quad + \lambda _ { \mathrm { d e m } } \nabla B _ { \theta } \left( \mathcal { D } _ { \mathrm { d a m } } , \frac { \alpha } { 2 \lambda _ { \mathrm { d a m o } } } \right) + \lambda _ { \mathrm { s a m p } } \nabla B _ { \theta } ( \mathcal { D } _ { \mathrm { s a m p } } , 0 ) . } \end{array}
+$$
+
+3The demonstrations alone may not satisfy the closure property in Equation 5. Augmenting the demonstrations with rollouts of the imitation policy improves our approximation of the objective in Equation 14.
+
+4Note that there is a Q network $Q _ { \theta }$ and a value function $V$ that is defined in terms of $Q _ { \theta }$ (see Equation 2), rather than a separate Q network and V network.
+
+# Algorithm 1 Soft Q Imitation Learning (SQIL, pronounced “skill”)
+
+<table><tr><td colspan="3">1:Requirer,入samp ∈R&gt;o,k∈N</td></tr><tr><td>2:InitializeDsamp←@ 3:fori=1,2,.do</td><td></td><td>&gt;See Equation 16</td></tr><tr><td>4: 5:</td><td>θ ←θ-nVe(Be(Ddemo,r)+入sampBe(Dsamp,O)) ifimodk=0then</td><td></td></tr><tr><td>6:</td><td>Sample rollout T =(so,ao,S1,a1,.,sT) with imitation policy πe(at|St)</td><td> See Equation 1</td></tr><tr><td>7:</td><td>Dsamp ←DsampU{𝑇}</td><td></td></tr><tr><td>8: end if</td><td></td><td></td></tr><tr><td>9: end for</td><td></td><td></td></tr></table>
+
+Setting $\gamma \triangleq 1$ turns the inner sum in the first term into a telescoping sum, yielding
+
+$$
+( 1 7 ) = \nabla \left( \lambda _ { \mathrm { d e m o } } B _ { \theta } \left( \mathcal { D } _ { \mathrm { d e m o } } , \frac { \alpha } { 2 \lambda _ { \mathrm { d e m o } } } \right) + \lambda _ { \mathrm { s a m p } } B _ { \theta } \big ( \mathcal { D } _ { \mathrm { s a m p } } , 0 \big ) \right) .
+$$
+
+Setting $\lambda _ { \mathrm { d e m o } } \triangleq 1$ and $\alpha \triangleq 2 \lambda _ { \mathrm { d e m o } } r$ yields the gradient of a soft Q-learning algorithm that gives the agent a reward of $r$ for taking the demonstrated action in a demonstrated state, and zero reward otherwise. The agent’s experience replay buffer consists of $\mathcal { D } _ { \mathrm { d e m o } } \cup \mathcal { D } _ { \mathrm { s a m p } }$ . We call this algorithm soft $Q$ imitation learning (SQIL, pronounced “skill”).
+
+# 3.3 SOFT Q IMITATION LEARNING
+
+SQIL is summarized in Algorithm 1. It performs soft Q-learning with two important modifications: (1) it initially fills the agent’s experience replay buffer with demonstrations, where the rewards are set to some positive constant (e.g., $r \triangleq 1$ ), and (2) as the agent interacts with the world and accumulates new experiences, it adds them to the replay buffer, and sets the rewards for these additional experiences to zero. The agent has an incentive to imitate the expert in states that are similar to those encountered in the demonstrations. During training, the agent may encounter out-of-distribution states that are very different from those in the demonstrations. In such states, the agent has an incentive to take actions that lead it back to demonstration states, where rewards are positive.
+
+Since the number of demonstrations is fixed and the number of additional experiences grows over time, we balance the number of demonstration experiences and additional experiences sampled for each gradient step in line 4 of Algorithm 1. As the imitation policy learns to behave more like the expert, the state-action distributions of $\mathcal { D } _ { \mathrm { s a m p } }$ and $\mathcal { D } _ { \mathrm { d e m o } }$ grow closer, which causes the reward for taking the expert action in an expert state to decay from $r$ to an expected value of $\frac { r } { 1 + \lambda _ { \mathrm { s a m p } } }$ This reward decay has the effect of making the imitation policy more stochastic over time, and could potentially be countered by decreasing $\lambda _ { \mathrm { s a m p } }$ to zero over time, or decreasing the temperature parameter of the stochastic policy, but neither of these fixes were necessary in our experiments.
+
+One potential issue with SQIL is that by setting the reward for taking demonstrated actions in demonstrated states to a positive constant, and implicitly setting rewards at terminal states to zero (see Equation 2), SQIL may encourage the agent to deviate from the expert trajectories in order to avoid terminating the episode, especially in environments with long horizons. This is also an issue with GAIL, and is addressed in Anonymous (2019) by explicitly learning the rewards at absorbing states. One potential way to fix this issue in SQIL is to set the rewards for demonstrated transitions into terminal states to be $\frac { r } { 1 - \gamma }$ instead of $r$ . This effectively augments the demonstration data with infinitely many rewarding transitions at absorbing states. This was not necessary in our experiments, since the horizons were relatively short (on the order of 1000 steps).
+
+# 4 RELATED WORK
+
+SQIL is inspired by the inverse soft Q-learning (ISQL) algorithm for internal dynamics estimation (Reddy et al., 2018), in that the objective in Equation 15 combines a behavioral cloning loss with soft Bellman error penalty terms, which is similar to the ISQL objective. The details of the two methods and their motivations are, however, completely different. ISQL is an internal dynamics estimation algorithm, while SQIL is meant for imitation learning. ISQL assumes that the demonstrations include observations of the expert’s reward signal, while SQIL does not.
+
+SQIL also resembles the Deep Q-learning from Demonstrations (DQfD) (Hester et al., 2017) and Normalized Actor-Critic (NAC) algorithms (Gao et al., 2018), in that all three algorithms fill the agent’s experience replay buffer with demonstrations and include an imitation loss in the agent’s objective. The key difference between SQIL and these prior methods is that DQfD and NAC are RL algorithms that assume access to a reward signal, while SQIL is an imitation learning algorithm that does not require an extrinsic reward signal from the environment. Instead, SQIL automatically constructs a reward signal from the demonstrations.5
+
+# 5 SIMULATION EXPERIMENTS
+
+The purpose of our experiments is three-fold. The first objective is to compare SQIL to prior work; in particular, on tasks with high-dimensional image observations, where prior methods are difficult to use. The second objective is to show that SQIL can be successfully deployed in MDPs with continuous action spaces. The third objective is to understand which components of SQIL contribute most to its performance. To accomplish the first objective, we use the image-based Car Racing and Atari Pong games, as well as the low-dimensional Lunar Lander environment from OpenAI Gym (Brockman et al., 2016), to benchmark SQIL against GAIL and behavioral cloning (BC). SQIL consistently outperforms BC, matches GAIL on low-dimensional Lunar Lander, and outperforms GAIL on the image-based Car Racing and Pong tasks. To accomplish the second objective, we use the Humanoid and HalfCheetah continuous control tasks from MuJoCo (Todorov et al., 2012) to show that SQIL performs comparably to GAIL on low-dimensional problems with continuous actions. To accomplish the third objective, we use the Lunar Lander game to conduct an ablation study that tests various hypotheses about which components of SQIL contribute to its performance.
+
+# 5.1 COMPARISON TO PRIOR METHODS ON LOW-DIMENSIONAL LUNAR LANDER
+
+The experiments in this section use the low-dimensional Lunar Lander game to verify that SQIL performs well on a relatively easy task. To emphasize the gap between imitation methods that take into account the consequences of actions, like SQIL and GAIL, and methods that do not, like BC, we exacerbate the state distribution shift that usually occurs due to compounding errors, by training the imitation agents in an environment with a different initial state distribution $S _ { 0 } ^ { \mathrm { t r a i n } }$ than that of the expert demonstrations $S _ { 0 } ^ { \mathrm { d e m o } }$ . This intervention explicitly tests the generalization capabilities of imitation learning, by placing the agent in start states $\bar { S } _ { 0 } ^ { \mathrm { t r a i n } }$ that are very different from those seen in the demonstrations.
+
+Experimental setup. The Lunar Lander game is a navigation task with deterministic dynamics in which the objective is to land on the ground, without crashing or flying out of bounds, using two lateral thrusters and a main engine. The action space consists of six discrete actions for steering and firing the main engine. Low-dimensional state observations $s \in \mathbb { R } ^ { 9 }$ encode position, velocity, orientation, and the location of the landing site. Demonstrations are collected from an expert trained using deep Q-learning (Mnih et al., 2015). The goal of this experiment is to study not only how well each method can mimic the expert demonstrations, but also how well they can acquire policies that generalize effectively to new states. After all, the main benefit of actually learning a policy, as opposed to simply playing back the expert’s actions, is to acquire a strategy that generalizes to new situations. Therefore, we manipulate the initial state distribution of the environment in which the imitation agents are trained and evaluated. The expert demonstrations are collected starting from the initial state distribution starting from a differen $S _ { 0 } ^ { \mathrm { d e m o } }$ . The imitation agl state distribution re then trai. To create the same environment, butin the Lunar Lander game, $S _ { 0 } ^ { \mathrm { t r a i n } }$ $S _ { 0 } ^ { \mathrm { t r a i n } }$   
+the agent is placed in a starting position that is rarely visited in the demonstrations. The agents are evaluated by training and testing with $S _ { 0 } ^ { \mathrm { d e m o } }$ , as well as training and testing with $S _ { 0 } ^ { \mathrm { t r a i n } }$ – both results are reported in Table 1 and Figure 1. We measure the agents’ success rate in landing at the target without crashing, flying out of bounds, or running out of time.
+
+![](images/6e039e2f9d4847a0bb5e8475f6b5b637724c54840af50bfb2cd53fa3e47a5a11.jpg)  
+Figure 1: Error regions show standard error across five random seeds. BC performance after training. $\mathrm { X }$ -axis represents amount of interaction with the environment (not expert demonstrations).
+
+<table><tr><td></td><td>Strain</td><td>Sdemo</td></tr><tr><td>Random</td><td>0.10 ± 0.30</td><td>0.04± 0.02</td></tr><tr><td>BC</td><td>0.07 ±0.03</td><td>0.93 ± 0.03</td></tr><tr><td>GAIL</td><td>0.98 ± 0.01</td><td>0.95 ± 0.02</td></tr><tr><td>SQIL-T</td><td>0.66 ± 0.02</td><td>0.89 ± 0.01</td></tr><tr><td>SQIL</td><td>0.89 ±0.02</td><td>0.88 ± 0.03</td></tr><tr><td>Expert</td><td>0.93 ±0.03</td><td>0.89± 0.31</td></tr></table>
+
+Table 1: Lunar Lander. Best performance on 100 consecutive episodes during training. Standard errors shown for five random seeds.
+
+Analysis. The results show that when there is no variation in the initial state, all methods perform equally well (see the $S _ { 0 } ^ { \mathrm { d e m o } }$ column in Table 1). The task is easy enough that even behavioral cloning achieves a high success rate. When the initial state is changed for the purpose of testing generalization, SQIL and GAIL both perform much better than BC (see Figure 1 and the $S _ { 0 } ^ { \mathrm { t r a i n } }$ column in Table 1). This is unsurprising, since BC suffers from state distribution shift, and GAIL tends to perform well on tasks with low-dimensional observations.
+
+GAIL achieves a higher success rate than SQIL, but requires many more interactions with the environment. This can most likely be attributed to the fact that GAIL uses TRPO (Schulman et al., 2015) for policy optimization, while SQIL uses Q-learning. Since this difference in the underlying RL algorithm makes it difficult to fairly compare SQIL to GAIL, the results are merely intended to illustrate that both SQIL and GAIL can outperform BC and perform on par with the expert. In the following two sections, we show that SQIL significantly outperforms GAIL on high-dimensional, image-based tasks.
+
+SQIL, which optimizes the bound in Equation 13, performs better than its cousin, SQIL-T, which optimizes the tighter bound in Equation 11. This is a somewhat surprising result, since optimizing a tighter bound on the log-posterior should lead to better performance. One possible explanation for this not being the case is that in practice, optimizing Equation 11 via gradient descent is harder than optimizing Equation 13.
+
+# 5.2 COMPARISON TO PRIOR METHODS ON IMAGE-BASED CAR RACING
+
+Section 5.1 showed that SQIL performs on par with GAIL and better than BC in environments with low-dimensional observations. This section shows that SQIL can significantly outperform GAIL and BC in environments with high-dimensional image observations. With images, the policy learning problem becomes substantially harder: now, the policy must learn to interpret raw image observations and determine the best course of action. This is notoriously difficult for methods based on adversarial learning, such as GAIL.
+
+Experimental setup. The Car Racing game is a navigation task with deterministic dynamics in which the objective is to visit as many road markers as possible, while avoiding the grass. The agent receives $6 4 \mathrm { x } 6 4 ~ \mathrm { R G B }$ images as observations, and outputs a discrete steering command (left, right, or straight). An episode lasts at most 1000 steps, and terminates early if the car deviates too far from the road. Demonstrations are collected from an expert trained using model-based RL (Ha & Schmidhuber, 2018). The initial state distribution is manipulated as in Section 5.1. To create $S _ { 0 } ^ { \mathrm { t r a i n } }$ in the Car Racing game, the car is rotated 90 degrees so that it begins perpendicular to the track, instead of parallel to the track as in $S _ { 0 } ^ { \mathrm { d e m o } }$ . This simple intervention presents a significant generalization challenge to the imitation learner, since the expert demonstrations do not contain any examples of states where the car is perpendicular to the road, or even significantly off the road axis. The agent must learn to make a tight turn to get back on the road, then stabilize its orientation so that it is parallel to the road, and only then proceed forward to mimic the expert demonstrations. We measure the reward collected by the agent, which counts the number of road markers it visits, and includes a -100 penalty for deviating too far from the road and causing the episode to terminate early.
+
+![](images/5ccda721a068728935469264a88f52cfb04b41153d1c4392255ca63c7b0bb91b.jpg)  
+Figure 2: Error regions show standard error across five random seeds. GAIL and BC performance after training.
+
+Table 2: Car Racing. Final performance on 20 episodes after training. Standard errors shown for five random seeds.   
+
+<table><tr><td></td><td>Strain</td><td>Sdemo</td></tr><tr><td>Random</td><td>-21± 56</td><td>-68±4</td></tr><tr><td>BC</td><td>-45±18</td><td>698 ±10</td></tr><tr><td>GAIL</td><td>17±0</td><td>7±1</td></tr><tr><td>SQIL-T</td><td>311 ± 10</td><td>693±7</td></tr><tr><td>SQIL</td><td>375 ±19</td><td>704±6</td></tr><tr><td>Expert</td><td>480 ± 11</td><td>704±79</td></tr></table>
+
+Analysis. Table 2 and Figure 2 show that SQIL performs much better than both BC and GAIL, when starting from $S _ { 0 } ^ { \mathrm { t r a i n } }$ . SQIL learns to make a tight turn that takes the car through the grass and back onto the road, then stabilizes the car’s orientation so that it is parallel to the track, and then proceeds forward like the expert does in the demonstrations. GAIL does not improve upon its initialized policy, which keeps turning in a loop, and does not move forward once the agent is parallel to the road. BC tends to drive straight ahead into the grass instead of turning back onto the road. One explanation of these results is that GAIL uses TRPO (Schulman et al., 2015) to fit the imitation policy, which inherits the challenges of RL with image observations. SQIL, on the other hand, inherits the ease of fitting an image-based policy from behavioral cloning via the analysis in Section 3, and only relies on soft Bellman backups for regularization and fine-tuning (vs. RL from scratch). SQIL outperforms SQIL-T, as in Section 5.1, though the performance gap is smaller here than in Section 5.1.
+
+# 5.3 COMPARISON TO PRIOR METHODS ON IMAGE-BASED ATARI PONG
+
+The experiments in the previous sections compare SQIL, which uses Q-learning, to GAIL, which uses TRPO. This can make it difficult to compare the performance and sample efficiency of SQIL and GAIL. The experiment in this section compares SQIL to a version of GAIL that uses Q-learning instead of TRPO, on the Pong Atari game with raw image observations. This allows for a head-to-head comparison of SQIL and GAIL: both algorithms use the same underlying RL algorithm (Q-learning) but provide the agent with different rewards – SQIL provides constant rewards, while GAIL provides learned rewards. The results in the figure to the right show that SQIL outperforms GAIL on this challenging image-based task.
+
+![](images/402459cacca9774bd62fc2c18999caeb251e9ad7805566fe7c0e3b5afce5fafc.jpg)
+
+Smoothed with a rolling window of 100 episodes. Standard error on three random seeds.
+
+The experiments in the previous sections evaluate SQIL in environments with discrete action spaces. The experiments in this section illustrate how SQIL can be adapted to continuous action spaces. We instantiate SQIL using soft actor-critic (SAC) – an offpolicy RL algorithm that can solve continuous con
+
+![](images/c5aaaee80e51069341b603bc966bd842d7177a233701c8bb328f3f791417af11.jpg)  
+5.4 COMPARISON TO PRIOR METHODS ON CONTINUOUS CONTROL TASKS
+
+SQIL: best performance on 10 consecutive training episodes. GAIL: results from Dhariwal et al. (2017).
+
+trol tasks (Haarnoja et al., 2018). In particular, SAC is modified in the following ways: (1) the agent’s experience replay buffer is initially filled with expert demonstrations, where rewards are set to a positive constant, (2) when taking gradient steps to fit the agent’s soft Q function, we sample a balanced number of demonstration experiences and new experiences from the replay buffer, and (3) the agent observes rewards of zero during their interactions with the environment, instead of an extrinsic reward signal that specifies the desired task. This instantiation of SQIL is compared to GAIL on the Humanoid (17 DoF) and HalfCheetah (6 DoF) tasks from MuJoCo (Todorov et al., 2012). The results in the figure to the right show that SQIL and GAIL perform comparably to each other on both Humanoid and HalfCheetah, demonstrating that SQIL can be successfully deployed on problems with continuous action spaces.
+
+# 5.5 ABLATION STUDY
+
+The experiments in Sections 5.1 and 5.2 show that SQIL can outperform prior methods, but they do not illuminate how the various terms in the SQIL gradient in line 4 of Algorithm 1 interact to yield these results. The experiments in this section use the Lunar Lander environment to conduct an ablation study to better understand the importance of each component of SQIL. We hypothesize that incorporating the right information about state transitions into the agent’s experience replay buffer is essential: the replay buffer needs to contain more than just the demonstrations, and these additional experiences need to be sampled on-policy (see line 6 of Algorithm 1).
+
+Experimental setup. We manipulate the parameters of SQIL to hobble it in various ways that test our hypotheses. In the first condition, $\lambda _ { \mathrm { s a m p } }$ is set to zero, to prevent SQIL from using additional samples drawn from the training environment. In the second condition, $\gamma$ is set to zero to prevent SQIL from accessing information about state transitions. In the third condition, a uniform random policy is used to sample additional rollouts instead of the imitation policy $\pi _ { \pmb { \theta } }$ in line 6 of Algorithm 1. We measure the agents’ success rate, as in Section 5.1.
+
+<table><tr><td>Strain</td><td>Sdemo</td></tr><tr><td>Random 0.10± 0.30</td><td>0.04±0.02</td></tr><tr><td>Xsamp = 0 0.12 ± 0.02</td><td>0.87 ± 0.02</td></tr><tr><td>γ=0 0.41 ± 0.02</td><td>0.84± 0.02</td></tr><tr><td>π=Unif 0.47 ±0.02</td><td>0.82 ± 0.02</td></tr><tr><td>SQIL-T</td><td>0.66 ±0.02 0.89 ± 0.01</td></tr><tr><td>SQIL 0.89 ± 0.02</td><td>0.88 ± 0.03</td></tr><tr><td>Expert</td><td>0.93± 0.03 0.89± 0.31</td></tr></table>
+
+![](images/fb48069d37826b185ea44410b0bd67b43fa4517950bb9d10e8f348fb3a966c77.jpg)  
+Figure 3: Error regions show standard error across five random seeds.
+
+Table 3: Lunar Lander. Best performance on 100 consecutive episodes during training. Standard errors shown for five random seeds.
+
+Analysis. The results in Figure 3 and Table 3 show that the original method performs significantly better than its hobbled counterparts when starting from $S _ { 0 } ^ { \mathrm { t r a i n } }$ , suggesting that SQIL relies on information about the environment dynamics encoded in state transitions (SQIL vs. $\gamma = 0$ ), and that SQIL requires on-policy sampling (SQIL vs. $\pi = \mathrm { U n i f }$ and $\lambda _ { \mathrm { s a m p } } = 0 \mathrm { , }$ ).
+
+# 6 DISCUSSION
+
+Summary. We contribute a practical and general algorithm for learning to imitate an expert given action demonstrations and access to the environment. Simulation experiments demonstrate the effectiveness of our method at recovering an imitation policy that performs well in comparison to GAIL and behavioral cloning on tasks with high-dimensional, continuous observations as well as continuous actions.
+
+Limitations. The analysis in Section 3, which proves that SQIL is equivalent to approximate inference of the expert policy under the MaxCausalEnt model of expert behavior, is limited in that does not show that SQIL exactly recovers the expert policy in the limit of having infinite demonstrations, or that the optimal imitation policy’s state-action occupancy measure converges to the expert’s. These are desirable properties for an imitation learning algorithm, and it is unclear if SQIL possesses them.
+
+Future work. The ability to robustly recover an imitation policy from image observations, even when the initial state distribution differs between demonstration time and training time, enables open-world robotics experiments in which using sensors other than a camera would be expensive, and consistently resetting the environment to the same set of start states is difficult.
+
+# REFERENCES
+
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+Greg Brockman, Vicki Cheung, Ludwig Pettersson, Jonas Schneider, John Schulman, Jie Tang, and Wojciech Zaremba. Openai gym, 2016.   
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+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.   
+Todd Hester, Matej Vecerik, Olivier Pietquin, Marc Lanctot, Tom Schaul, Bilal Piot, Dan Horgan, John Quan, Andrew Sendonaris, Gabriel Dulac-Arnold, et al. Deep q-learning from demonstrations. arXiv preprint arXiv:1704.03732, 2017.   
+Jonathan Ho and Stefano Ermon. Generative adversarial imitation learning. In Advances in Neural Information Processing Systems, pp. 4565–4573, 2016.   
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+Karol Kurach, Mario Lucic, Xiaohua Zhai, Marcin Michalski, and Sylvain Gelly. The gan landscape: Losses, architectures, regularization, and normalization. arXiv preprint arXiv:1807.04720, 2018.
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+Michael Laskey, Sam Staszak, Wesley Yu-Shu Hsieh, Jeffrey Mahler, Florian T Pokorny, Anca D Dragan, and Ken Goldberg. Shiv: Reducing supervisor burden in dagger using support vectors for efficient learning from demonstrations in high dimensional state spaces. In Robotics and Automation (ICRA), 2016 IEEE International Conference on, pp. 462–469. IEEE, 2016.
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+Sergey Levine. Reinforcement learning and control as probabilistic inference: Tutorial and review. arXiv preprint arXiv:1805.00909, 2018.
+
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+
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+
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+
+# A APPENDIX
+
+A.1 SQIL-T
+
+SQIL-T uses Adam (Kingma & Ba, 2014) to minimize the loss function,
+
+$$
+\ell _ { \mathrm { S O I L - T } } ( \theta ) \triangleq \alpha \sum _ { \tau \in \mathcal { D } _ { \mathrm { d e m } } } \sum _ { t = 0 } ^ { T - 1 } - ( Q _ { \theta } ( s _ { t } , a _ { t } ) - V _ { \theta } ( s _ { t } ) ) + \lambda _ { \mathrm { d e m o } } B _ { \theta } ( \mathcal { D } _ { \mathrm { d e m o } } , 0 ) + \lambda _ { \mathrm { s a m p } } B _ { \theta } ( \mathcal { D } _ { \mathrm { s a m p } } , 0 ) ,
+$$
+
+which is identical to the SQIL loss in Equation 15, except that the $V _ { \pmb { \theta } } ( s _ { 0 } )$ terms are not present in Equation 19.
+
+# A.2 IMPLEMENTATION DETAILS
+
+Code and data for the experiments in Section 5 are publicly available.6
+
+To ensure fair comparisons, the same network architectures were used to evaluate SQIL, GAIL, and BC. For Lunar Lander, we used a network architecture with two fully-connected layers containing 128 hidden units each to represent the Q network in SQIL, the policy and discriminator networks in GAIL, and the policy network in BC. For Car Racing, we used four convolutional layers (following Ha & Schmidhuber (2018)) and two fully-connected layers containing 256 hidden units each. For Humanoid and HalfCheetah, we used two fully-connected layers containing 256 hidden units each. For Pong, we used the convolutional neural network described in Mnih et al. (2015) to represent the Q network in SQIL, as well as the Q network and discriminator network in GAIL.
+
+To ensure fair comparisons, the same demonstration data was used to train SQIL, GAIL, and BC. For Lunar Lander, we collected 100 demonstration rollouts. For Car Racing, we collected 20 demonstration rollouts. For Pong, we collected 50 demonstration rollouts. Expert demonstrations were generated from scratch for Lunar Lander using DQN (Mnih et al., 2015), and collected from opensource pre-trained policies for Car Racing (Ha & Schmidhuber, 2018) as well as Humanoid and HalfCheetah (Dhariwal et al., 2017). The Humanoid demonstrations were generated by a stochastic expert policy, while the HalfCheetah demonstrations were generated by a deterministic expert policy; both experts were trained using TRPO.7 We used two open-source implementations of GAIL: Fu et al. (2017) for Lunar Lander and Car Racing, and Dhariwal et al. (2017) for MuJoCo. We adapted the OpenAI Baselines implementation of GAIL to use Q-learning for Pong. Expert demonstrations were generated from scratch for Pong using DQN.
+
+For Lunar Lander, we set $r = 1$ , $k = 2 0 0$ , and $\lambda _ { \mathrm { s a m p } } = 1 0 ^ { - 6 }$ . For Car Racing, we set $r = 1$ , $k = 5 0$ , and $\lambda _ { \mathrm { s a m p } } = 0 . 0 1$ . For Humanoid, we set $r = 5$ and $\lambda _ { \mathrm { s a m p } } = 1$ . For HalfCheetah, we set $r = 1 0$ and $\lambda _ { \operatorname* { s a m p } } = \mathrm { \hat { 1 } }$ . For Pong, we set $r = 1 0 0$ , $k = 1$ , and $\lambda _ { \operatorname* { s a m p } } \doteq 1$ .
+
+SQIL was not pre-trained in any of the experiments. GAIL was pre-trained using behavioral cloning for HalfCheetah, but was not pre-trained in any other experiments.
+
+In standard implementations of Q-learning and SAC, the agent’s experience replay buffer typically has a fixed size, and once the buffer is full, old experiences are deleted to make room for new experiences. In SQIL, we never delete demonstration experiences from the replay buffer, but otherwise follow the standard implementation.
+
+We use Adam (Kingma & Ba, 2014) to take the gradient step in line 4 of Algorithm 1.
+
+The GAIL performance metrics in Section 5.4 are taken from Dhariwal et al. (2017).8
+
+The GAIL and SQIL policies in Section 5.4 are set to be deterministic during the evaluation rollouts used to measure performance.
+
+![](images/d483066567d61ea3c53e78fcb064cb3a42fcadeacee800aa714121661be45cd4.jpg)  
+Figure 4: Standard error shown for five random seeds on the first $2 0 0 \mathrm { k }$ steps. No smoothing across training steps. The first four random seeds are from the experiment in Section 5.4, which only train for $2 0 0 \mathrm { k }$ steps. To measure the effect of training on more episodes, we ran another experiment with a fifth random seed and trained for $5 0 0 \mathrm { k }$ steps. Training for a much larger number of steps does not lead to significant deterioration in performance.
+
+# A.3 RELATED WORK
+
+Various approaches have been developed to address state distribution shift in behavioral cloning, without relying on IRL. Hand-engineering a domain-specific loss function and carefully designing the demonstration collection process have enabled researchers to train effective imitation policies for self-driving cars (Bojarski et al., 2016), autonomous drones (Giusti et al., 2016), and robotic manipulators (Zhang et al., 2017; Rahmatizadeh et al., 2017; 2016). DAgger-based methods query the expert for on-policy action labels (Ross et al., 2011; Laskey et al., 2016). These approaches either require domain knowledge or the ability to query the expert, while SQIL requires neither.

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+        "text": "WHAT WOULD $\\pi ^ { * }$ DO?: IMITATION LEARNING VIA OFF-POLICY REINFORCEMENT LEARNING ",
+        "text_level": 1,
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+        "text": "Anonymous authors Paper under double-blind review ",
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+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
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+        "text": "Learning to imitate expert actions given demonstrations containing image observations is a difficult problem in robotic control. The key challenge is generalizing behavior to out-of-distribution states that differ from those in the demonstrations. State-of-the-art imitation learning algorithms perform well in environments with low-dimensional observations, but typically involve adversarial optimization procedures, which can be difficult to use with high-dimensional image observations. We propose a remarkably simple alternative based on off-policy soft Q-learning, which we call soft $Q$ imitation learning (SQIL, pronounced “skill”), that rewards the agent for matching demonstrated actions in demonstrated states. The key idea is initially filling the agent’s experience replay buffer with demonstrations, where rewards are set to a positive constant, and setting rewards to zero in all additional experiences. We derive SQIL from first principles as a method for performing approximate inference under the MaxCausalEnt model of expert behavior. The approximate inference objective trades off between a pure behavioral cloning loss and a regularization term that incorporates information about state transitions via the soft Bellman error. Our experiments show that SQIL matches the state of the art in low-dimensional environments, and significantly outperforms prior work in playing video games from high-dimensional images. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+    },
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+        "text": "Many real-world sequential decision-making problems can be tackled by imitation learning, where an expert demonstrates near-optimal behavior to an agent that then attempts to replicate that behavior in novel situations (Argall et al., 2009). This paper considers the problem of training an agent to imitate an expert policy, given expert action demonstrations and access to the environment. The agent does not get to observe a reward signal or query the expert, and does not know the state transition dynamics ex-ante. ",
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+        "text": "Standard approaches to this problem based on behavioral cloning seek to imitate the expert’s actions, but do not reason about the consequences of actions (Pomerleau, 1991). As a result, they suffer from state distribution shift, and fail to generalize to states that are very different from those seen in the demonstrations (Ross & Bagnell, 2010; Ross et al., 2011). Approaches based on inverse reinforcement learning (IRL) deal with this issue by fitting a reward function that represents preferences over trajectories rather than individual actions $\\mathrm { N g }$ et al., 2000; Ziebart et al., 2008), and using the learned reward function to train the imitation agent through RL (Wulfmeier et al., 2015; Finn et al., 2016; Fu et al., 2017). This is the core idea behind generative adversarial imitation learning (GAIL), which implicitly combines IRL and RL using generative adversarial networks (GANs) (Ho & Ermon, 2016; Goodfellow et al., 2014). GAIL is state-of-the-art in environments with low-dimensional observations, but requires additional reward augmentation and feature engineering when applied to environments with high-dimensional image observations (Li et al., 2017), due to the difficulty of training GANs (Kurach et al., 2018). In short, imitation learning from raw pixel inputs without prior knowledge of the domain remains a challenge. ",
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+        "type": "text",
+        "text": "The key insight in this paper is that instead of using a learned reward function or discriminator to provide a reward signal to the imitation policy, as is done in IRL-based imitation methods and GAIL, the imitation agent can simply be rewarded for matching demonstrated actions in demonstrated states using off-policy soft Q-learning. The agent’s experience replay buffer is initially filled with demonstrations where rewards are set to some positive constant, and gradually accumulates additional experiences with rewards set to zero. The agent has an incentive to imitate the expert in demonstrated states, and to take actions that lead it back to demonstrated states when it encounters new states that differ from the demonstrations. We call this algorithm soft $Q$ imitation learning (SQIL, pronounced “skill”). ",
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+        "type": "text",
+        "text": "",
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+        "type": "text",
+        "text": "We motivate the SQIL algorithm by deriving it from first principles as a method for performing approximate inference under the MaxCausalEnt model of expert behavior (Ziebart et al., 2010). Our analysis shows that SQIL maximizes a lower bound on the log-posterior of the soft Q values given the demonstrations – a bound that trades off between a pure behavioral cloning loss and a regularization term that incorporates state transition information via the soft Bellman error. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The main contribution of this paper is SQIL: a practical and general imitation learning algorithm that is effective in MDPs with high-dimensional, continuous observations – including raw pixel inputs – and has a natural theoretical interpretation as approximate inference. We run experiments in the image-based Car Racing and low-dimensional Lunar Lander game environments from OpenAI Gym, the Humanoid and HalfCheetah continuous control tasks from MuJoCo, and image-based Atari Pong, to compare SQIL to two prior methods: behavioral cloning and GAIL. The results show that SQIL matches GAIL in low-dimensional environments, and significantly outperforms GAIL on the image-based Car Racing and Pong tasks. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 PRELIMINARIES ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This work builds on the maximum causal entropy (MaxCausalEnt) model of expert behavior (Ziebart et al., 2010; Levine, 2018). In a finite-horizon Markov Decision Process (MDP) with a finite, discrete action space $\\mathcal { A }$ ,1 the demonstrator is assumed to follow a policy $\\pi$ that maximizes reward $R ( s , a )$ . To simplify exposition, we assume the state transition dynamics are deterministic: let $s ^ { \\prime }$ denote the state that follows from taking action $a$ in state $s$ . The policy $\\pi$ forms a Boltzmann distribution over actions, ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1594f38899a5898ffdc021b7f3245ef2b18bf10650ca235008d82c2a7e834d42.jpg",
+        "text": "$$\n\\pi ( a | s ) \\triangleq \\exp { \\left( Q ( s , a ) - V ( s ) \\right) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            501,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $Q$ is the soft Q function, and $V$ is the soft value function, ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2a9dbd73914cf197746e398a525d832547f494f411bc591f1217a241c489698b.jpg",
+        "text": "$$\n\\begin{array} { r } { V ( s ) \\triangleq \\left\\{ \\begin{array} { l l } { 0 } & { \\mathrm { ~ i f ~ } s \\mathrm { ~ i s ~ a ~ t e r m i n a l ~ s t a t e } } \\\\ { \\log \\left( \\sum _ { a \\in A } \\exp \\left( Q ( s , a ) \\right) \\right) } & { \\mathrm { ~ o t h e r w i s e } } \\end{array} \\right. } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In the standard MaxCausalEnt model, the soft Q values are a deterministic function of the rewards and dynamics, given by the soft Bellman equation, ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/099f874ec21ffe5cb6b8bed01cf53939e4715728b3e3941e84abe95a08ab7b53.jpg",
+        "text": "$$\nQ ( s , a ) \\triangleq R ( s , a ) + \\gamma V ( s ^ { \\prime } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 APPROXIMATE MAXCAUSALENT MODEL ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In the analysis in Section 3, the standard MaxCausalEnt model is approximated by a probabilistic model that treats the soft $\\mathrm { Q }$ values as random variables, ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/88237b16218f2065e698935990746264d72a6fb32522da69a4f6a95fba967185.jpg",
+        "text": "$$\nQ ( s , a ) \\sim \\mathcal { N } ( R ( s , a ) + \\gamma V ( s ^ { \\prime } ) , \\sigma ^ { 2 } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $\\sigma \\in \\mathbb { R }$ is a constant hyperparameter, and $R ( s , a )$ is a random variable with zero mean and unit variance. To simplify the analysis in Section 3, we assume that the dynamics of the environment satisfy the property that if state $s _ { i }$ is reachable from state $s _ { j }$ , then $s _ { j }$ is not reachable from $s _ { i }$ – this ensures that the probabilistic graphical model that captures the joint distribution over soft Q values is acyclic. The experiments in Section 5 show that even when this assumption is violated, SQIL performs well empirically. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 IMITATION LEARNING VIA APPROXIMATE MAXCAUSALENT INFERENCE ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We aim for an imitation learning algorithm that generalizes to new states, without resorting to adversarial optimization procedures that would be difficult to use with image observations. Our approach is to derive a lower bound on the log-posterior of the soft Q values given the demonstrations; in particular, a lower bound that is maximized by the SQIL algorithm. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 DERIVING THE APPROXIMATE INFERENCE OBJECTIVE ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Consider a demonstration rollout $\\tau = ( s _ { 0 } , a _ { 0 } , s _ { 1 } , a _ { 1 } , . . . , s _ { T } )$ , where $s _ { T }$ is a terminal state. Standard MaxCausalEnt IRL searches for the reward function that maximizes the posterior probability of the rewards given the demonstration. Our goal, however, is not to infer the expert’s reward function; it is to infer the expert’s policy, which is represented by their soft $\\mathrm { Q }$ function. We approach this problem by formalizing uncertainty about the soft Q values using the probabilistic model described in Section 2.1. We fit a soft Q function to optimize the posterior probability of the soft $\\mathrm { Q }$ values $Q$ given the demonstration $\\tau$ , where $Q \\stackrel { \\triangle } { = } \\{ Q ( s , a ) \\mid ( s , a ) \\in \\mathcal { X } \\}$ , and $\\mathcal { X }$ is a finite subset of $\\boldsymbol { S } \\times \\boldsymbol { A }$ that includes the state-action pairs in $\\tau$ . Since $Q ( s , a )$ may depend on $Q ( s ^ { \\prime } , a ^ { \\prime } )$ (see Equations 4 and 2), $\\mathcal { X }$ is assumed to satisfy the closure property, ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f7e0f1ec9dcefdf40d71259603b830e9414934e5d7676780b37fb3390328485e.jpg",
+        "text": "$$\n\\forall ( s , a ) \\in \\mathcal { X } . \\big ( s ^ { \\prime } \\mathrm { ~ n o t ~ t e r m i n a l } \\implies ( \\forall a ^ { \\prime } \\in \\mathcal { A } . ( s ^ { \\prime } , a ^ { \\prime } ) \\in \\mathcal { X } ) \\big ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The goal is to derive a lower bound on the log-posterior, $\\log p ( Q | \\tau )$ , that is maximized by the SQIL algorithm. The first step is to marginalize out the rewards, $\\pmb { R } \\triangleq \\{ R ( s , a ) \\mid ( s , a ) \\in \\chi \\}$ , as follows. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0d8c6fb796b90b7add80048a5a0ef90e58c5dddadceee14535544c86fa1fc80f.jpg",
+        "text": "$$\n\\begin{array} { r } { p ( Q | \\tau ) \\propto p ( \\tau | Q ) p ( Q ) = \\mathbb { E } _ { R } [ p ( \\tau | Q , R ) p ( Q | R ) ] = \\mathbb { E } _ { R } [ p ( \\tau | Q ) p ( Q | R ) ] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "First bound. Optimizing the log of the expectation in Equation 6 is hard, for a number of reasons: we haven’t assumed a particular prior distribution over rewards (only that it has zero mean and unit variance), and the soft Q values in $Q$ are not conditionally independent given $\\pmb { R }$ . Optimizing the expectation of the log is easier. By Jensen’s inequality, ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/19eb269c8b8a7f0b39b2ac994aa0caa1b15aa2e06e14acedf8d518a98b546568.jpg",
+        "text": "$$\n\\begin{array} { r } { \\log \\left( \\mathbb E _ { R } [ p ( \\tau | Q ) p ( Q | R ) ] \\right) \\geq \\mathbb E _ { R } [ \\log p ( \\tau | Q ) + \\log p ( Q | R ) ] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The conditional likelihood of the demonstration $\\tau$ given the soft Q values $Q$ is given by Equation 1: ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fd815a2f19c2b23ffe0d64b4bbf8b63929fe8aee7a540429823f675054841370.jpg",
+        "text": "$$\np ( \\tau | \\mathbf { \\Delta } Q ) = \\exp \\left( \\sum _ { t = 0 } ^ { T - 1 } Q ( s _ { t } , a _ { t } ) - V ( s _ { t } ) \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The conditional likelihood of the soft Q values $Q$ given the rewards $\\pmb { R }$ is given by Equation 4: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bec061aecdc37e13078984e847da8efae9c6fa57a77f839c9eca8206ca8eb77d.jpg",
+        "text": "$$\np ( \\pmb { Q } | \\pmb { R } ) = \\prod _ { ( s , a ) \\in \\mathcal { X } } \\mathcal { N } ( Q ( s , a ) ; \\pmb { R } ( s , a ) + \\gamma V ( s ^ { \\prime } ) , \\sigma ^ { 2 } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where the joint distribution over $Q$ can be factored due to the state re-entry assumption about the dynamics made in Section 2.1 and the closure assumption about $\\mathcal { X }$ made in Equation 5. Simplifying the lower bound in Equation 7, ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/abb05fecc93ee2c1de7477f6363a52b11656317f81ef0ca42f0b75c9e4276d2a.jpg",
+        "text": "$$\n\\begin{array} { r l r } { \\left. { ( 7 ) = \\sum _ { t = 0 } ^ { T - 1 } Q ( s _ { t } , a _ { t } ) - V ( s _ { t } ) } } \\\\ & { } & { + \\displaystyle \\sum _ { ( s , a ) \\in \\mathcal { X } } - \\frac { 1 } { 2 \\sigma ^ { 2 } } \\left( ( Q ( s , a ) - ( \\mathbb { E } [ R ( s , a ) ] + \\gamma V ( s ^ { \\prime } ) ) \\right) ^ { 2 } + \\mathrm { V a r } [ R ( s , a ) ] \\right) - \\log \\sqrt { 2 \\pi \\sigma ^ { 2 } } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Applying the assumption from Section 2.1 that the rewards have zero mean and unit variance, ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/94e1acfd9425bf7260b9ca9bf9a229c942c2c67eb122821ed5882bf9957ee64c.jpg",
+        "text": "$$\n( 1 0 ) = \\sum _ { t = 0 } ^ { T - 1 } Q ( s _ { t } , a _ { t } ) - V ( s _ { t } ) + \\sum _ { ( s , a ) \\in \\mathcal { X } } - \\frac { 1 } { 2 \\sigma ^ { 2 } } ( ( Q ( s , a ) - \\gamma V ( s ^ { \\prime } ) ) ^ { 2 } + 1 ) - \\log { \\sqrt { 2 \\pi \\sigma ^ { 2 } } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Second bound. Next, the bound is intentionally loosened, in order to arrive at an objective that is maximized by the SQIL algorithm.2 By the MaxCausalEnt assumptions in Section 2, ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/195e3afca159cc710d9365af5d0018230b060cf40ef2deeb289f43c6760bf9b7.jpg",
+        "text": "$$\nV ( s _ { 0 } ) = \\mathbb { E } _ { \\tau \\sim p ( \\tau \\mid \\pi ) } \\left[ \\sum _ { t = 0 } ^ { T - 1 } \\gamma ^ { t } \\left( Q ( s _ { t } , a _ { t } ) - \\gamma V ( s _ { t + 1 } ) + \\mathcal { H } ( \\pi ( \\cdot | s _ { t } ) ) \\right) \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { H }$ denotes entropy. Maximizing Equation 11 with respect to $Q$ yields a solution in which $Q ( s , a ) - \\gamma V ( s ^ { \\prime } ) \\leq \\bar { f } ( \\bar { \\sigma } ^ { 2 } )$ , where $f$ is an increasing function of $\\sigma ^ { 2 }$ . Thus, $V ( s _ { 0 } ) \\leq ( f ( \\sigma ^ { 2 } ) +$ $\\log | \\mathcal { A } | ) / ( 1 - \\gamma )$ , and ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1c5c1a8263ad5b18c1645110b62b3cdd802c7535e02217e9f334c4f27b9f5d78.jpg",
+        "text": "$$\n( 1 1 ) \\geq ( 1 1 ) + V ( s _ { 0 } ) - ( f ( \\sigma ^ { 2 } ) + \\log | A | ) / ( 1 - \\gamma ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The soft Q function that maximizes the lower bound in Equation 13 is ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e4dffc6fd83cb67268719d1f65717b5e824806fdd0988494aab265eb1e38ed0c.jpg",
+        "text": "$$\nQ ^ { * } \\triangleq \\operatorname * { a r g m a x } _ { Q } V ( s _ { 0 } ) + \\sum _ { t = 0 } ^ { T - 1 } Q ( s _ { t } , a _ { t } ) - V ( s _ { t } ) - \\frac { 1 } { 2 \\sigma ^ { 2 } } \\sum _ { ( s , a ) \\in \\mathcal X } ( Q ( s , a ) - \\gamma V ( s ^ { \\prime } ) ) ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Practical implementation. For environments with a small, discrete state space $s$ and known dynamics, $\\mathcal { X }$ can be set to $\\boldsymbol { s } \\times \\boldsymbol { A }$ . For environments with a continuous state space $s$ and unknown dynamics, we set $\\mathcal { X }$ to be the set of state-action pairs in the demonstration rollouts $\\mathcal { D } _ { \\mathrm { d e m o } }$ and rollouts $\\mathcal { D } _ { \\mathrm { s a m p } }$ of the imitation policy, which are periodically sampled during training.3 We fit a parameterized soft Q function $\\bar { Q _ { \\theta } }$ to minimize the loss, ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/df5b56365013ce7d5993706408a221e33306f05e30816425321da024d59d5660.jpg",
+        "text": "$$\n\\ell ( \\pmb \\theta ) \\triangleq \\sum _ { \\tau \\in \\mathcal { D } _ { \\pmb { \\mathscr { A } } _ { \\mathrm { r m o } } } } \\left( - V _ { \\pmb \\theta } ( s _ { 0 } ) + \\sum _ { t = 0 } ^ { T - 1 } - ( Q _ { \\theta } ( s _ { t } , a _ { t } ) - V _ { \\pmb \\theta } ( s _ { t } ) ) \\right) + \\lambda _ { \\sf d e m o } B _ { \\theta } ( \\mathcal { D } _ { \\sf d e m o } , 0 ) + \\lambda _ { \\sf s a m p } B _ { \\theta } ( \\mathcal { D } _ { \\sf s a m p } , 0 ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\alpha , \\lambda _ { \\mathrm { d e m o } } , \\lambda _ { \\mathrm { s a m p } } \\in \\mathbb { R } _ { \\ge 0 }$ are constant hyperparameters, $V _ { \\pmb { \\theta } }$ denotes the soft value function given by $Q _ { \\theta }$ and Equation 2,4 and $B _ { \\theta }$ denotes the sum of squared soft Bellman errors, ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ec3d59cbba6e933040b522b9bb21dc19d91da9ebcf9dbf8a16940a5f940e69b9.jpg",
+        "text": "$$\nB _ { \\theta } ( \\mathcal { D } , r ) \\triangleq \\sum _ { \\tau \\in \\mathcal { D } } \\sum _ { t = 0 } ^ { T - 1 } ( Q _ { \\theta } ( s _ { t } , a _ { t } ) - ( r + \\gamma V _ { \\theta } ( s _ { t + 1 } ) ) ) ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $r \\in \\mathbb { R }$ is a constant that does not depend on the state or action. The experiments in Section 5 use a convolutional neural network or multi-layer perceptron to model $Q _ { \\theta }$ , where $\\pmb { \\theta }$ are the weights of the neural network (Schmidhuber, 2015). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The loss in Equation 15 bounds the log-posterior of the soft $\\mathrm { Q }$ values given the demonstrations. Searching for a soft Q function that optimizes this objective is equivalent to performing approximate inference under the MaxCausalEnt model of expert behavior. The objective combines a behavioral cloning loss, which does not reason about the consequences of actions, with a regularization term that incorporates information about state transitions. The regularization term performs soft Bellman backups on a reward of zero – a consequence of the prior distribution over rewards, which has an expected value of zero – and can be seen as imposing a prior distribution on the soft $\\mathrm { Q }$ values that is independent of the expert demonstrations. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 IMPLEMENTING APPROXIMATE INFERENCE WITH OFF-POLICY SOFT Q-LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            608,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Surprisingly, the gradient of the approximate inference objective in Equation 15 is equivalent to the gradient of the objective of a soft Q-learning algorithm (Haarnoja et al., 2017) that gives the agent a constant positive reward for matching the demonstrated action in a demonstrated state, and zero reward otherwise. Differentiating the objective in Equation 15, ",
+        "bbox": [
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+            633,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bb1c1628e47c71e08bb2c3271929bc6c446aa2fe19c88c1f4c6048ec2b4da556.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\nabla _ { \\theta } \\ell ( \\theta ) = \\alpha \\underset { \\tau \\in \\mathcal { D } _ { \\mathrm { d a m } } } { \\sum } \\Bigg ( - \\nabla V _ { \\theta } ( s _ { 0 } ) + \\underset { t = 0 } { \\overset { T - 1 } { \\sum } } - ( \\nabla Q _ { \\theta } ( s _ { t } , a _ { t } ) - \\nabla V _ { \\theta } ( s _ { t } ) ) \\Bigg ) } \\\\ & { \\quad \\quad \\quad + \\lambda _ { \\mathrm { d e m } } \\underset { \\tau \\in \\mathcal { D } _ { \\mathrm { d a m } } } { \\sum } \\underset { t = 0 } { \\overset { T - 1 } { \\sum } } \\nabla ( Q _ { \\theta } ( s _ { t } , a _ { t } ) - \\gamma V _ { \\theta } ( s _ { t + 1 } ) ) ^ { 2 } + \\lambda _ { \\mathrm { s a m p } } \\nabla B _ { \\theta } ( \\mathcal { D } _ { \\mathrm { s a m p } } , 0 ) } \\\\ & { \\quad \\quad = \\underset { \\tau \\in \\mathcal { D } _ { \\mathrm { d a m } } } { \\sum } \\Bigg ( - \\alpha \\nabla V _ { \\theta } ( s _ { 0 } ) + \\underset { t = 0 } { \\overset { T - 1 } { \\sum } } \\alpha \\nabla V _ { \\theta } ( s _ { t } ) - \\alpha \\gamma \\nabla V _ { \\theta } ( s _ { t + 1 } ) \\Bigg ) } \\\\ & { \\quad \\quad \\quad + \\lambda _ { \\mathrm { d e m } } \\nabla B _ { \\theta } \\left( \\mathcal { D } _ { \\mathrm { d a m } } , \\frac { \\alpha } { 2 \\lambda _ { \\mathrm { d a m o } } } \\right) + \\lambda _ { \\mathrm { s a m p } } \\nabla B _ { \\theta } ( \\mathcal { D } _ { \\mathrm { s a m p } } , 0 ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            220,
+            693,
+            777,
+            866
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3The demonstrations alone may not satisfy the closure property in Equation 5. Augmenting the demonstrations with rollouts of the imitation policy improves our approximation of the objective in Equation 14. ",
+        "bbox": [
+            173,
+            871,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4Note that there is a Q network $Q _ { \\theta }$ and a value function $V$ that is defined in terms of $Q _ { \\theta }$ (see Equation 2), rather than a separate Q network and V network. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 Soft Q Imitation Learning (SQIL, pronounced “skill”) ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            103,
+            617,
+            118
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/8d9fb390b0885e9b5f168bc11b14659e650cdb0f3ea8d67f1c8e29eb85455857.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"3\">1:Requirer,入samp ∈R&gt;o,k∈N</td></tr><tr><td>2:InitializeDsamp←@ 3:fori=1,2,.do</td><td></td><td>&gt;See Equation 16</td></tr><tr><td>4: 5:</td><td>θ ←θ-nVe(Be(Ddemo,r)+入sampBe(Dsamp,O)) ifimodk=0then</td><td></td></tr><tr><td>6:</td><td>Sample rollout T =(so,ao,S1,a1,.,sT) with imitation policy πe(at|St)</td><td> See Equation 1</td></tr><tr><td>7:</td><td>Dsamp ←DsampU{𝑇}</td><td></td></tr><tr><td>8: end if</td><td></td><td></td></tr><tr><td>9: end for</td><td></td><td></td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Setting $\\gamma \\triangleq 1$ turns the inner sum in the first term into a telescoping sum, yielding ",
+        "bbox": [
+            174,
+            261,
+            710,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/511460d1a40dd26f709c0267afd2918d50d5ebeb7adb2d94f0aa343023960a8d.jpg",
+        "text": "$$\n( 1 7 ) = \\nabla \\left( \\lambda _ { \\mathrm { d e m o } } B _ { \\theta } \\left( \\mathcal { D } _ { \\mathrm { d e m o } } , \\frac { \\alpha } { 2 \\lambda _ { \\mathrm { d e m o } } } \\right) + \\lambda _ { \\mathrm { s a m p } } B _ { \\theta } \\big ( \\mathcal { D } _ { \\mathrm { s a m p } } , 0 \\big ) \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            287,
+            280,
+            707,
+            315
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Setting $\\lambda _ { \\mathrm { d e m o } } \\triangleq 1$ and $\\alpha \\triangleq 2 \\lambda _ { \\mathrm { d e m o } } r$ yields the gradient of a soft Q-learning algorithm that gives the agent a reward of $r$ for taking the demonstrated action in a demonstrated state, and zero reward otherwise. The agent’s experience replay buffer consists of $\\mathcal { D } _ { \\mathrm { d e m o } } \\cup \\mathcal { D } _ { \\mathrm { s a m p } }$ . We call this algorithm soft $Q$ imitation learning (SQIL, pronounced “skill”). ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 SOFT Q IMITATION LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+            393,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "SQIL is summarized in Algorithm 1. It performs soft Q-learning with two important modifications: (1) it initially fills the agent’s experience replay buffer with demonstrations, where the rewards are set to some positive constant (e.g., $r \\triangleq 1$ ), and (2) as the agent interacts with the world and accumulates new experiences, it adds them to the replay buffer, and sets the rewards for these additional experiences to zero. The agent has an incentive to imitate the expert in states that are similar to those encountered in the demonstrations. During training, the agent may encounter out-of-distribution states that are very different from those in the demonstrations. In such states, the agent has an incentive to take actions that lead it back to demonstration states, where rewards are positive. ",
+        "bbox": [
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+            420,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Since the number of demonstrations is fixed and the number of additional experiences grows over time, we balance the number of demonstration experiences and additional experiences sampled for each gradient step in line 4 of Algorithm 1. As the imitation policy learns to behave more like the expert, the state-action distributions of $\\mathcal { D } _ { \\mathrm { s a m p } }$ and $\\mathcal { D } _ { \\mathrm { d e m o } }$ grow closer, which causes the reward for taking the expert action in an expert state to decay from $r$ to an expected value of $\\frac { r } { 1 + \\lambda _ { \\mathrm { s a m p } } }$ This reward decay has the effect of making the imitation policy more stochastic over time, and could potentially be countered by decreasing $\\lambda _ { \\mathrm { s a m p } }$ to zero over time, or decreasing the temperature parameter of the stochastic policy, but neither of these fixes were necessary in our experiments. ",
+        "bbox": [
+            173,
+            540,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "One potential issue with SQIL is that by setting the reward for taking demonstrated actions in demonstrated states to a positive constant, and implicitly setting rewards at terminal states to zero (see Equation 2), SQIL may encourage the agent to deviate from the expert trajectories in order to avoid terminating the episode, especially in environments with long horizons. This is also an issue with GAIL, and is addressed in Anonymous (2019) by explicitly learning the rewards at absorbing states. One potential way to fix this issue in SQIL is to set the rewards for demonstrated transitions into terminal states to be $\\frac { r } { 1 - \\gamma }$ instead of $r$ . This effectively augments the demonstration data with infinitely many rewarding transitions at absorbing states. This was not necessary in our experiments, since the horizons were relatively short (on the order of 1000 steps). ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            808,
+            344,
+            824
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "SQIL is inspired by the inverse soft Q-learning (ISQL) algorithm for internal dynamics estimation (Reddy et al., 2018), in that the objective in Equation 15 combines a behavioral cloning loss with soft Bellman error penalty terms, which is similar to the ISQL objective. The details of the two methods and their motivations are, however, completely different. ISQL is an internal dynamics estimation algorithm, while SQIL is meant for imitation learning. ISQL assumes that the demonstrations include observations of the expert’s reward signal, while SQIL does not. ",
+        "bbox": [
+            174,
+            839,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "SQIL also resembles the Deep Q-learning from Demonstrations (DQfD) (Hester et al., 2017) and Normalized Actor-Critic (NAC) algorithms (Gao et al., 2018), in that all three algorithms fill the agent’s experience replay buffer with demonstrations and include an imitation loss in the agent’s objective. The key difference between SQIL and these prior methods is that DQfD and NAC are RL algorithms that assume access to a reward signal, while SQIL is an imitation learning algorithm that does not require an extrinsic reward signal from the environment. Instead, SQIL automatically constructs a reward signal from the demonstrations.5 ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5 SIMULATION EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            222,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The purpose of our experiments is three-fold. The first objective is to compare SQIL to prior work; in particular, on tasks with high-dimensional image observations, where prior methods are difficult to use. The second objective is to show that SQIL can be successfully deployed in MDPs with continuous action spaces. The third objective is to understand which components of SQIL contribute most to its performance. To accomplish the first objective, we use the image-based Car Racing and Atari Pong games, as well as the low-dimensional Lunar Lander environment from OpenAI Gym (Brockman et al., 2016), to benchmark SQIL against GAIL and behavioral cloning (BC). SQIL consistently outperforms BC, matches GAIL on low-dimensional Lunar Lander, and outperforms GAIL on the image-based Car Racing and Pong tasks. To accomplish the second objective, we use the Humanoid and HalfCheetah continuous control tasks from MuJoCo (Todorov et al., 2012) to show that SQIL performs comparably to GAIL on low-dimensional problems with continuous actions. To accomplish the third objective, we use the Lunar Lander game to conduct an ablation study that tests various hypotheses about which components of SQIL contribute to its performance. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.1 COMPARISON TO PRIOR METHODS ON LOW-DIMENSIONAL LUNAR LANDER ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            450,
+            740,
+            464
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The experiments in this section use the low-dimensional Lunar Lander game to verify that SQIL performs well on a relatively easy task. To emphasize the gap between imitation methods that take into account the consequences of actions, like SQIL and GAIL, and methods that do not, like BC, we exacerbate the state distribution shift that usually occurs due to compounding errors, by training the imitation agents in an environment with a different initial state distribution $S _ { 0 } ^ { \\mathrm { t r a i n } }$ than that of the expert demonstrations $S _ { 0 } ^ { \\mathrm { d e m o } }$ . This intervention explicitly tests the generalization capabilities of imitation learning, by placing the agent in start states $\\bar { S } _ { 0 } ^ { \\mathrm { t r a i n } }$ that are very different from those seen in the demonstrations. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Experimental setup. The Lunar Lander game is a navigation task with deterministic dynamics in which the objective is to land on the ground, without crashing or flying out of bounds, using two lateral thrusters and a main engine. The action space consists of six discrete actions for steering and firing the main engine. Low-dimensional state observations $s \\in \\mathbb { R } ^ { 9 }$ encode position, velocity, orientation, and the location of the landing site. Demonstrations are collected from an expert trained using deep Q-learning (Mnih et al., 2015). The goal of this experiment is to study not only how well each method can mimic the expert demonstrations, but also how well they can acquire policies that generalize effectively to new states. After all, the main benefit of actually learning a policy, as opposed to simply playing back the expert’s actions, is to acquire a strategy that generalizes to new situations. Therefore, we manipulate the initial state distribution of the environment in which the imitation agents are trained and evaluated. The expert demonstrations are collected starting from the initial state distribution starting from a differen $S _ { 0 } ^ { \\mathrm { d e m o } }$ . The imitation agl state distribution re then trai. To create the same environment, butin the Lunar Lander game, $S _ { 0 } ^ { \\mathrm { t r a i n } }$ $S _ { 0 } ^ { \\mathrm { t r a i n } }$   \nthe agent is placed in a starting position that is rarely visited in the demonstrations. The agents are evaluated by training and testing with $S _ { 0 } ^ { \\mathrm { d e m o } }$ , as well as training and testing with $S _ { 0 } ^ { \\mathrm { t r a i n } }$ – both results are reported in Table 1 and Figure 1. We measure the agents’ success rate in landing at the target without crashing, flying out of bounds, or running out of time. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/6e039e2f9d4847a0bb5e8475f6b5b637724c54840af50bfb2cd53fa3e47a5a11.jpg",
+        "image_caption": [
+            "Figure 1: Error regions show standard error across five random seeds. BC performance after training. $\\mathrm { X }$ -axis represents amount of interaction with the environment (not expert demonstrations). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            186,
+            101,
+            488,
+            231
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/d66fa62df7d58d09dad6da6625a95a909d9444dddd259a2bd6c8aedf9ce4b17c.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 1: Lunar Lander. Best performance on 100 consecutive episodes during training. Standard errors shown for five random seeds. "
+        ],
+        "table_body": "<table><tr><td></td><td>Strain</td><td>Sdemo</td></tr><tr><td>Random</td><td>0.10 ± 0.30</td><td>0.04± 0.02</td></tr><tr><td>BC</td><td>0.07 ±0.03</td><td>0.93 ± 0.03</td></tr><tr><td>GAIL</td><td>0.98 ± 0.01</td><td>0.95 ± 0.02</td></tr><tr><td>SQIL-T</td><td>0.66 ± 0.02</td><td>0.89 ± 0.01</td></tr><tr><td>SQIL</td><td>0.89 ±0.02</td><td>0.88 ± 0.03</td></tr><tr><td>Expert</td><td>0.93 ±0.03</td><td>0.89± 0.31</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Analysis. The results show that when there is no variation in the initial state, all methods perform equally well (see the $S _ { 0 } ^ { \\mathrm { d e m o } }$ column in Table 1). The task is easy enough that even behavioral cloning achieves a high success rate. When the initial state is changed for the purpose of testing generalization, SQIL and GAIL both perform much better than BC (see Figure 1 and the $S _ { 0 } ^ { \\mathrm { t r a i n } }$ column in Table 1). This is unsurprising, since BC suffers from state distribution shift, and GAIL tends to perform well on tasks with low-dimensional observations. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "GAIL achieves a higher success rate than SQIL, but requires many more interactions with the environment. This can most likely be attributed to the fact that GAIL uses TRPO (Schulman et al., 2015) for policy optimization, while SQIL uses Q-learning. Since this difference in the underlying RL algorithm makes it difficult to fairly compare SQIL to GAIL, the results are merely intended to illustrate that both SQIL and GAIL can outperform BC and perform on par with the expert. In the following two sections, we show that SQIL significantly outperforms GAIL on high-dimensional, image-based tasks. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "SQIL, which optimizes the bound in Equation 13, performs better than its cousin, SQIL-T, which optimizes the tighter bound in Equation 11. This is a somewhat surprising result, since optimizing a tighter bound on the log-posterior should lead to better performance. One possible explanation for this not being the case is that in practice, optimizing Equation 11 via gradient descent is harder than optimizing Equation 13. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.2 COMPARISON TO PRIOR METHODS ON IMAGE-BASED CAR RACING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            608,
+            681,
+            623
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Section 5.1 showed that SQIL performs on par with GAIL and better than BC in environments with low-dimensional observations. This section shows that SQIL can significantly outperform GAIL and BC in environments with high-dimensional image observations. With images, the policy learning problem becomes substantially harder: now, the policy must learn to interpret raw image observations and determine the best course of action. This is notoriously difficult for methods based on adversarial learning, such as GAIL. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Experimental setup. The Car Racing game is a navigation task with deterministic dynamics in which the objective is to visit as many road markers as possible, while avoiding the grass. The agent receives $6 4 \\mathrm { x } 6 4 ~ \\mathrm { R G B }$ images as observations, and outputs a discrete steering command (left, right, or straight). An episode lasts at most 1000 steps, and terminates early if the car deviates too far from the road. Demonstrations are collected from an expert trained using model-based RL (Ha & Schmidhuber, 2018). The initial state distribution is manipulated as in Section 5.1. To create $S _ { 0 } ^ { \\mathrm { t r a i n } }$ in the Car Racing game, the car is rotated 90 degrees so that it begins perpendicular to the track, instead of parallel to the track as in $S _ { 0 } ^ { \\mathrm { d e m o } }$ . This simple intervention presents a significant generalization challenge to the imitation learner, since the expert demonstrations do not contain any examples of states where the car is perpendicular to the road, or even significantly off the road axis. The agent must learn to make a tight turn to get back on the road, then stabilize its orientation so that it is parallel to the road, and only then proceed forward to mimic the expert demonstrations. We measure the reward collected by the agent, which counts the number of road markers it visits, and includes a -100 penalty for deviating too far from the road and causing the episode to terminate early. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/5ccda721a068728935469264a88f52cfb04b41153d1c4392255ca63c7b0bb91b.jpg",
+        "image_caption": [
+            "Figure 2: Error regions show standard error across five random seeds. GAIL and BC performance after training. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            186,
+            138,
+            486,
+            267
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/d67ecb8f76196ae7184014819408bd6014deb864f70adda31ea9f4cee11d7f04.jpg",
+        "table_caption": [
+            "Table 2: Car Racing. Final performance on 20 episodes after training. Standard errors shown for five random seeds. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Strain</td><td>Sdemo</td></tr><tr><td>Random</td><td>-21± 56</td><td>-68±4</td></tr><tr><td>BC</td><td>-45±18</td><td>698 ±10</td></tr><tr><td>GAIL</td><td>17±0</td><td>7±1</td></tr><tr><td>SQIL-T</td><td>311 ± 10</td><td>693±7</td></tr><tr><td>SQIL</td><td>375 ±19</td><td>704±6</td></tr><tr><td>Expert</td><td>480 ± 11</td><td>704±79</td></tr></table>",
+        "bbox": [
+            531,
+            155,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Analysis. Table 2 and Figure 2 show that SQIL performs much better than both BC and GAIL, when starting from $S _ { 0 } ^ { \\mathrm { t r a i n } }$ . SQIL learns to make a tight turn that takes the car through the grass and back onto the road, then stabilizes the car’s orientation so that it is parallel to the track, and then proceeds forward like the expert does in the demonstrations. GAIL does not improve upon its initialized policy, which keeps turning in a loop, and does not move forward once the agent is parallel to the road. BC tends to drive straight ahead into the grass instead of turning back onto the road. One explanation of these results is that GAIL uses TRPO (Schulman et al., 2015) to fit the imitation policy, which inherits the challenges of RL with image observations. SQIL, on the other hand, inherits the ease of fitting an image-based policy from behavioral cloning via the analysis in Section 3, and only relies on soft Bellman backups for regularization and fine-tuning (vs. RL from scratch). SQIL outperforms SQIL-T, as in Section 5.1, though the performance gap is smaller here than in Section 5.1. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.3 COMPARISON TO PRIOR METHODS ON IMAGE-BASED ATARI PONG ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            520,
+            678,
+            534
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The experiments in the previous sections compare SQIL, which uses Q-learning, to GAIL, which uses TRPO. This can make it difficult to compare the performance and sample efficiency of SQIL and GAIL. The experiment in this section compares SQIL to a version of GAIL that uses Q-learning instead of TRPO, on the Pong Atari game with raw image observations. This allows for a head-to-head comparison of SQIL and GAIL: both algorithms use the same underlying RL algorithm (Q-learning) but provide the agent with different rewards – SQIL provides constant rewards, while GAIL provides learned rewards. The results in the figure to the right show that SQIL outperforms GAIL on this challenging image-based task. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/402459cacca9774bd62fc2c18999caeb251e9ad7805566fe7c0e3b5afce5fafc.jpg",
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+        "text": "Smoothed with a rolling window of 100 episodes. Standard error on three random seeds. ",
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+        "text": "The experiments in the previous sections evaluate SQIL in environments with discrete action spaces. The experiments in this section illustrate how SQIL can be adapted to continuous action spaces. We instantiate SQIL using soft actor-critic (SAC) – an offpolicy RL algorithm that can solve continuous con",
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+        "image_caption": [
+            "5.4 COMPARISON TO PRIOR METHODS ON CONTINUOUS CONTROL TASKS "
+        ],
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+    {
+        "type": "text",
+        "text": "SQIL: best performance on 10 consecutive training episodes. GAIL: results from Dhariwal et al. (2017). ",
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+        "type": "text",
+        "text": "trol tasks (Haarnoja et al., 2018). In particular, SAC is modified in the following ways: (1) the agent’s experience replay buffer is initially filled with expert demonstrations, where rewards are set to a positive constant, (2) when taking gradient steps to fit the agent’s soft Q function, we sample a balanced number of demonstration experiences and new experiences from the replay buffer, and (3) the agent observes rewards of zero during their interactions with the environment, instead of an extrinsic reward signal that specifies the desired task. This instantiation of SQIL is compared to GAIL on the Humanoid (17 DoF) and HalfCheetah (6 DoF) tasks from MuJoCo (Todorov et al., 2012). The results in the figure to the right show that SQIL and GAIL perform comparably to each other on both Humanoid and HalfCheetah, demonstrating that SQIL can be successfully deployed on problems with continuous action spaces. ",
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+    {
+        "type": "text",
+        "text": "5.5 ABLATION STUDY ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The experiments in Sections 5.1 and 5.2 show that SQIL can outperform prior methods, but they do not illuminate how the various terms in the SQIL gradient in line 4 of Algorithm 1 interact to yield these results. The experiments in this section use the Lunar Lander environment to conduct an ablation study to better understand the importance of each component of SQIL. We hypothesize that incorporating the right information about state transitions into the agent’s experience replay buffer is essential: the replay buffer needs to contain more than just the demonstrations, and these additional experiences need to be sampled on-policy (see line 6 of Algorithm 1). ",
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+    {
+        "type": "text",
+        "text": "Experimental setup. We manipulate the parameters of SQIL to hobble it in various ways that test our hypotheses. In the first condition, $\\lambda _ { \\mathrm { s a m p } }$ is set to zero, to prevent SQIL from using additional samples drawn from the training environment. In the second condition, $\\gamma$ is set to zero to prevent SQIL from accessing information about state transitions. In the third condition, a uniform random policy is used to sample additional rollouts instead of the imitation policy $\\pi _ { \\pmb { \\theta } }$ in line 6 of Algorithm 1. We measure the agents’ success rate, as in Section 5.1. ",
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+    {
+        "type": "table",
+        "img_path": "images/1f8ad29ae5dcb0cf47ea3dd79a5adad7ed064560785a80201f9fef5e07695f94.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Strain</td><td>Sdemo</td></tr><tr><td>Random 0.10± 0.30</td><td>0.04±0.02</td></tr><tr><td>Xsamp = 0 0.12 ± 0.02</td><td>0.87 ± 0.02</td></tr><tr><td>γ=0 0.41 ± 0.02</td><td>0.84± 0.02</td></tr><tr><td>π=Unif 0.47 ±0.02</td><td>0.82 ± 0.02</td></tr><tr><td>SQIL-T</td><td>0.66 ±0.02 0.89 ± 0.01</td></tr><tr><td>SQIL 0.89 ± 0.02</td><td>0.88 ± 0.03</td></tr><tr><td>Expert</td><td>0.93± 0.03 0.89± 0.31</td></tr></table>",
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+    },
+    {
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+        "img_path": "images/fb48069d37826b185ea44410b0bd67b43fa4517950bb9d10e8f348fb3a966c77.jpg",
+        "image_caption": [
+            "Figure 3: Error regions show standard error across five random seeds. "
+        ],
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+    {
+        "type": "text",
+        "text": "Table 3: Lunar Lander. Best performance on 100 consecutive episodes during training. Standard errors shown for five random seeds. ",
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+    {
+        "type": "text",
+        "text": "Analysis. The results in Figure 3 and Table 3 show that the original method performs significantly better than its hobbled counterparts when starting from $S _ { 0 } ^ { \\mathrm { t r a i n } }$ , suggesting that SQIL relies on information about the environment dynamics encoded in state transitions (SQIL vs. $\\gamma = 0$ ), and that SQIL requires on-policy sampling (SQIL vs. $\\pi = \\mathrm { U n i f }$ and $\\lambda _ { \\mathrm { s a m p } } = 0 \\mathrm { , }$ ). ",
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+    {
+        "type": "text",
+        "text": "6 DISCUSSION ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Summary. We contribute a practical and general algorithm for learning to imitate an expert given action demonstrations and access to the environment. Simulation experiments demonstrate the effectiveness of our method at recovering an imitation policy that performs well in comparison to GAIL and behavioral cloning on tasks with high-dimensional, continuous observations as well as continuous actions. ",
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+    {
+        "type": "text",
+        "text": "Limitations. The analysis in Section 3, which proves that SQIL is equivalent to approximate inference of the expert policy under the MaxCausalEnt model of expert behavior, is limited in that does not show that SQIL exactly recovers the expert policy in the limit of having infinite demonstrations, or that the optimal imitation policy’s state-action occupancy measure converges to the expert’s. These are desirable properties for an imitation learning algorithm, and it is unclear if SQIL possesses them. ",
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+    {
+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "Future work. The ability to robustly recover an imitation policy from image observations, even when the initial state distribution differs between demonstration time and training time, enables open-world robotics experiments in which using sensors other than a camera would be expensive, and consistently resetting the environment to the same set of start states is difficult. ",
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+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.1 SQIL-T ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "SQIL-T uses Adam (Kingma & Ba, 2014) to minimize the loss function, ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9962f6059d82f206b3ffb4833d6cf71638163f42fd17f05c450306928a35e72c.jpg",
+        "text": "$$\n\\ell _ { \\mathrm { S O I L - T } } ( \\theta ) \\triangleq \\alpha \\sum _ { \\tau \\in \\mathcal { D } _ { \\mathrm { d e m } } } \\sum _ { t = 0 } ^ { T - 1 } - ( Q _ { \\theta } ( s _ { t } , a _ { t } ) - V _ { \\theta } ( s _ { t } ) ) + \\lambda _ { \\mathrm { d e m o } } B _ { \\theta } ( \\mathcal { D } _ { \\mathrm { d e m o } } , 0 ) + \\lambda _ { \\mathrm { s a m p } } B _ { \\theta } ( \\mathcal { D } _ { \\mathrm { s a m p } } , 0 ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "which is identical to the SQIL loss in Equation 15, except that the $V _ { \\pmb { \\theta } } ( s _ { 0 } )$ terms are not present in Equation 19. ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.2 IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
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+            291,
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+            308
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Code and data for the experiments in Section 5 are publicly available.6 ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "To ensure fair comparisons, the same network architectures were used to evaluate SQIL, GAIL, and BC. For Lunar Lander, we used a network architecture with two fully-connected layers containing 128 hidden units each to represent the Q network in SQIL, the policy and discriminator networks in GAIL, and the policy network in BC. For Car Racing, we used four convolutional layers (following Ha & Schmidhuber (2018)) and two fully-connected layers containing 256 hidden units each. For Humanoid and HalfCheetah, we used two fully-connected layers containing 256 hidden units each. For Pong, we used the convolutional neural network described in Mnih et al. (2015) to represent the Q network in SQIL, as well as the Q network and discriminator network in GAIL. ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "To ensure fair comparisons, the same demonstration data was used to train SQIL, GAIL, and BC. For Lunar Lander, we collected 100 demonstration rollouts. For Car Racing, we collected 20 demonstration rollouts. For Pong, we collected 50 demonstration rollouts. Expert demonstrations were generated from scratch for Lunar Lander using DQN (Mnih et al., 2015), and collected from opensource pre-trained policies for Car Racing (Ha & Schmidhuber, 2018) as well as Humanoid and HalfCheetah (Dhariwal et al., 2017). The Humanoid demonstrations were generated by a stochastic expert policy, while the HalfCheetah demonstrations were generated by a deterministic expert policy; both experts were trained using TRPO.7 We used two open-source implementations of GAIL: Fu et al. (2017) for Lunar Lander and Car Racing, and Dhariwal et al. (2017) for MuJoCo. We adapted the OpenAI Baselines implementation of GAIL to use Q-learning for Pong. Expert demonstrations were generated from scratch for Pong using DQN. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "For Lunar Lander, we set $r = 1$ , $k = 2 0 0$ , and $\\lambda _ { \\mathrm { s a m p } } = 1 0 ^ { - 6 }$ . For Car Racing, we set $r = 1$ , $k = 5 0$ , and $\\lambda _ { \\mathrm { s a m p } } = 0 . 0 1$ . For Humanoid, we set $r = 5$ and $\\lambda _ { \\mathrm { s a m p } } = 1$ . For HalfCheetah, we set $r = 1 0$ and $\\lambda _ { \\operatorname* { s a m p } } = \\mathrm { \\hat { 1 } }$ . For Pong, we set $r = 1 0 0$ , $k = 1$ , and $\\lambda _ { \\operatorname* { s a m p } } \\doteq 1$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "SQIL was not pre-trained in any of the experiments. GAIL was pre-trained using behavioral cloning for HalfCheetah, but was not pre-trained in any other experiments. ",
+        "bbox": [
+            173,
+            667,
+            823,
+            695
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In standard implementations of Q-learning and SAC, the agent’s experience replay buffer typically has a fixed size, and once the buffer is full, old experiences are deleted to make room for new experiences. In SQIL, we never delete demonstration experiences from the replay buffer, but otherwise follow the standard implementation. ",
+        "bbox": [
+            174,
+            702,
+            825,
+            758
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We use Adam (Kingma & Ba, 2014) to take the gradient step in line 4 of Algorithm 1. ",
+        "bbox": [
+            174,
+            765,
+            736,
+            780
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The GAIL performance metrics in Section 5.4 are taken from Dhariwal et al. (2017).8 ",
+        "bbox": [
+            174,
+            786,
+            733,
+            801
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The GAIL and SQIL policies in Section 5.4 are set to be deterministic during the evaluation rollouts used to measure performance. ",
+        "bbox": [
+            174,
+            808,
+            823,
+            835
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/d483066567d61ea3c53e78fcb064cb3a42fcadeacee800aa714121661be45cd4.jpg",
+        "image_caption": [
+            "Figure 4: Standard error shown for five random seeds on the first $2 0 0 \\mathrm { k }$ steps. No smoothing across training steps. The first four random seeds are from the experiment in Section 5.4, which only train for $2 0 0 \\mathrm { k }$ steps. To measure the effect of training on more episodes, we ran another experiment with a fifth random seed and trained for $5 0 0 \\mathrm { k }$ steps. Training for a much larger number of steps does not lead to significant deterioration in performance. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            339,
+            103,
+            655,
+            279
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.3 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            390,
+            333,
+            405
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Various approaches have been developed to address state distribution shift in behavioral cloning, without relying on IRL. Hand-engineering a domain-specific loss function and carefully designing the demonstration collection process have enabled researchers to train effective imitation policies for self-driving cars (Bojarski et al., 2016), autonomous drones (Giusti et al., 2016), and robotic manipulators (Zhang et al., 2017; Rahmatizadeh et al., 2017; 2016). DAgger-based methods query the expert for on-policy action labels (Ross et al., 2011; Laskey et al., 2016). These approaches either require domain knowledge or the ability to query the expert, while SQIL requires neither. ",
+        "bbox": [
+            174,
+            416,
+            825,
+            515
+        ],
+        "page_idx": 12
+    }
+]

parse/train/BJDH5M-AW/BJDH5M-AW.md

@@ -0,0 +1,514 @@
+# SYNTHESIZING ROBUST ADVERSARIAL EXAMPLES
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Neural network-based classifiers parallel or exceed human-level accuracy on many common tasks and are used in practical systems. Yet, neural networks are susceptible to adversarial examples, carefully perturbed inputs that cause networks to misbehave in arbitrarily chosen ways. When generated with standard methods, these examples do not consistently fool a classifier in the physical world due to a combination of viewpoint shifts, camera noise, and other natural transformations. Adversarial examples generated using standard techniques require complete control over direct input to the classifier, which is impossible in many real-world systems.
+
+We introduce the first method for constructing real-world 3D objects that consistently fool a neural network across a wide distribution of angles and viewpoints. We present a general-purpose algorithm for generating adversarial examples that are robust across any chosen distribution of transformations. We demonstrate its application in two dimensions, producing adversarial images that are robust to noise, distortion, and affine transformation. Finally, we apply the algorithm to produce arbitrary physical 3D-printed adversarial objects, demonstrating that our approach works end-to-end in the real world. Our results show that adversarial examples are a practical concern for real-world systems.
+
+# 1 INTRODUCTION
+
+The existence of adversarial examples for neural networks has until now been largely a theoretical concern. While minute, carefully-crafted perturbations can cause targeted misclassification in a neural network, adversarial examples produced using standard techniques lose adversariality when directly translated to the physical world as they are captured over varying viewpoints and affected by natural phenomena such as lighting and camera noise. This suggests that practical systems may not be at risk because adversarial examples generated using standard techniques are not robust in the physical world.
+
+![](images/e7ff76dd7ad74bacab138890651367af1c995c337660f74ccd1530dd873a6cbc.jpg)  
+Figure 1: Randomly sampled poses of a single 3D-printed turtle adversarially perturbed to classify as a rifle at every viewpoint by an ImageNet classifier. An unperturbed model is classified correctly as a turtle $100 \%$ of the time. See https://youtu.be/YXy6oX1iNoA for a video where every frame is fed through the classifier: the turtle is consistently classified as a rifle.
+
+We show that neural network-based classifiers are vulnerable to physical-world adversarial examples. We introduce a new algorithm for reliably producing physical 3D objects that are adversarial over a distribution of viewpoints. Figure 1 shows an example of an adversarial object constructed using our approach, where a 3D-printed turtle is consistently classified as rifle by an ImageNet classifier. In this paper, we demonstrate the efficacy and generality of our method, demonstrating conclusively that adversarial examples are a concern in real-world systems.
+
+# 1.1 CHALLENGES
+
+Methods for transforming ordinary two-dimensional images into adversarial examples are wellknown; however, the extent to which these techniques affect specific real-world systems is still an open question. Prior work has shown that adversarial examples generated using standard techniques lose their adversarial nature once subjected to minor transformations (Luo et al., 2016; Lu et al., 2017). These results suggest that adversarial examples, while serving as interesting phenomena, have no hope of applying to physical-world systems, where transformations such as viewpoint changes and camera noise are inevitable.
+
+Prior techniques attempting to synthesize robust adversarial examples for the physical world have had limited success. While some progress has been made, current efforts have demonstrated a small number of datapoints on nonstandard classifiers, and only in the two-dimensional case, with no clear generalization to three dimensions (further discussed in Section 4).
+
+The entirety of prior work has only explored generating adversarial examples in the two-dimensional case, where “viewpoints” can be approximated by an affine transformations of an original image. Constructing adversarial examples for the physical world requires the ability to generate entire 3D adversarial objects, which must remain adversarial in the face of complex transformations not applicable to 2D objects, such as 3D rotations and perspective projection.
+
+# 1.2 CONTRIBUTIONS
+
+In this work, we definitively show that adversarial examples pose a real threat in the physical world. We propose a general-purpose algorithm for reliably constructing adversarial examples robust over a chosen distribution of transformations, and we demonstrate the efficacy of this algorithm in both the 2D and 3D case. We succeed in producing physical-world 3D adversarial objects that are robust over a large, realistic distribution of 3D viewpoints, proving that the algorithm produces adversarial three-dimensional objects that are adversarial in the physical world. Specifically, our contributions are as follows:
+
+• We develop Expectation Over Transformation (EOT), a novel algorithm that produces single adversarial examples that are simultaneously adversarial over an entire distribution of transformations   
+• We consider the problem of constructing 3D adversarial examples under the EOT framework, viewing the 3D rendering process as part of the transformation, and we show that the approach successfully synthesizes adversarial objects   
+• We fabricate adversarial objects and show that they remain adversarial, demonstrating that our approach works end-to-end in the physical world, showing that adversarial examples are of real concern in practical deep learning systems
+
+# 2 APPROACH
+
+First, we present the Expectation Over Transformation (EOT) algorithm, a general framework allowing for the construction of adversarial examples that remain adversarial over a chosen transformation distribution $T$ . We then describe our end-to-end approach for generating adversarial objects using a specialized application of EOT in conjunction with differentiating through the 3D rendering process.
+
+# 2.1 EXPECTATION OVER TRANSFORMATION
+
+When constructing adversarial examples in the white-box case (that is, with access to a classifier and its gradient), we know in advance a set of possible classes $Y$ and a space of valid inputs $X$ to the classifier; we have access to the function $P ( \boldsymbol { y } | \boldsymbol { x } )$ and its gradient $\bar { \nabla } P ( y | x )$ , for any class $y \in Y$ and input $x \in X$ . In the standard case, adversarial examples are produced by maximizing the log-likelihood of the target class over a $\epsilon$ -radius ball around the original image, that is:
+
+$$
+\hat { x } = \underset { x ^ { \prime } \in X } { \arg \operatorname* { m a x } } \log P ( y | x ^ { \prime } ) \qquad \mathrm { s . t . } \ | | x ^ { \prime } - x | | _ { p } < \epsilon
+$$
+
+This approach has been shown to be both feasible and effective at generating adversarial examples for any given classifier. However, prior work has shown adversarial examples’ inability to remain adversarial even under minor perturbations inevitable in any real-world observation (Luo et al., 2016; Lu et al., 2017).
+
+To address this issue, we introduce Expectation Over Transformation $( E O T )$ . The key insight behind EOT is to model such perturbations within the optimization procedure. Rather than optimizing the log-likelihood of a single example, EOT uses a chosen distribution $T$ of transformation functions $t$ taking an input $x ^ { \prime }$ supplied by the adversary to the “true” input $t ( x ^ { \prime } )$ perceived by the classifier. Furthermore, rather than simply taking the norm of $x ^ { \prime } - x$ to constrain the solution space, EOT instead aims to constrain the effective distance between the adversarial and original inputs, which we define as:
+
+$$
+\delta = \mathbb { E } _ { t \sim T } [ d ( t ( x ) - t ( x ^ { \prime } ) ) ]
+$$
+
+Intuitively, this is how different we expect the true input to the classifer will be, given our new input. Then, EOT solves the following optimization problem:
+
+$$
+\hat { x } = \underset { x ^ { \prime } } { \arg \operatorname* { m a x } } \mathbb { E } _ { t \sim T } [ \log P ( y | t ( x ^ { \prime } ) ) ] \qquad \mathrm { s . t . } \ \mathbb { E } _ { t \sim T } [ d ( t ( x ^ { \prime } ) , t ( x ) ) ] < \epsilon
+$$
+
+In practice, the distribution $T$ can model perceptual distortions such as random rotation, translation, or addition of noise. However, EOT generalizes beyond simple transformations. EOT finds examples robust under any perception distribution $Q ( \cdot ; x )$ parameterized by the generated example $x$ as long as ${ \frac { d } { d x } } Q ( \cdot ; x )$ is well-defined. The objective function can be optimized by stochastic gradient descent, approximating the derivative of the expected value through sampling transformations independently at each gradient descent step and differentiating through the transformation.
+
+# 2.2 SYSTEM OVERVIEW
+
+Given its ability to synthesize robust adversarial examples, we use the EOT framework for generating 2D examples, 3D models, and ultimately physical-world adversarial objects. Within the framework, however, there is a great deal of freedom in the actual method by which examples are generated, including choice of $T$ , distance metric, and optimization method.
+
+In the 2D case, we design $T$ to approximate a realistic space of possible distortions involved in printing out an image and taking a natural picture of it. This amounts to a set of random transformations of the form $t ( x ) = A x + \epsilon$ , which are more thoroughly described in Section 3.
+
+In the 3D case, we define the transformations $t$ from a texture to a perceived image by mapping the texture to a given 3D object, and simulating highly nonlinear functions such as rendering, rotation, translation, and perspective distortion of the object in addition to the perceptual mechanisms used in the 2D case.
+
+Finding adversarial examples across these nonlinear transformations is what allows for the transfer of adversarial examples to the physical world, but also introduces implementation complexities not found in the 2D case. EOT requires the ability to differentiate though the transformation distribution with respect to the input. In the 3D case, this implies differentiating through the 3D renderer with respect to the texture.
+
+To do this, we model the rendering process as a sparse matrix multiplication between the texture and a coordinate map, which is generated from a standard 3D renderer. Rather than attempting to differentiate through the complex renderer, we use the renderer to find a coordinate map for each rendering, which maps coordinates on the texture to coordinates in the classifier’s field of view. We can then simulate this specific rendering as a sparse matrix multiplication and differentiate through it; we differentiate through different matrix multiplications at each sampling step.
+
+Once EOT has been parameterized, i.e. once a distribution $T$ is chosen, the issue of actually optimizing the induced objective function remains. Rather than solving the constrained optimization problem given above, we use the Lagrangian-relaxed form of the problem, as Carlini $\&$ Wagner (2017) do in the conventional (non-EOT, single-viewpoint) case:
+
+$$
+\hat { x } = \underset { x ^ { \prime } } { \arg \operatorname* { m i n } } \mathbb { E } _ { t \sim T } [ - \log P ( y | t ( x ^ { \prime } ) ) ] + \lambda \mathbb { E } _ { t \sim T } [ d ( t ( x ^ { \prime } ) , t ( x ) ] )
+$$
+
+In order to encourage imperceptibility of the generated images, we set $d ( x ^ { \prime } , x )$ to be the $\ell _ { 2 }$ norm in the LAB color space, a perceptually uniform color space where Euclidean distance roughly corresponds with perceptual distance. Note that the $\mathbb { E } _ { t \sim T } [ | | L A B ( t ( x ) ) - L A B ( t ( \hat { x } ) ) | | _ { 2 } ^ { 2 } ]$ can be sampled and estimated in conjunction with $\mathbb { E } [ P ( y | t ( x ) ) ]$ ; in general, the Lagrangian formulation gives EOT the ability to intricately constrain the search space (in our case, using LAB distance) at insignificant computational cost (without computing a complex projection). Our optimization, then, is:
+
+$$
+\hat { x } = \underset { x ^ { \prime } } { \arg \operatorname* { m i n } } \mathbb { E } _ { t \sim T } [ - \log P ( y | t ( x ^ { \prime } ) ) + \lambda | | L A B ( t ( x ) ) - L A B ( t ( x ^ { \prime } ) ) | | _ { 2 } ^ { 2 } ]
+$$
+
+We use projected gradient descent to find the optimum, clipping to the set of valid inputs (e.g. $[ 0 , 1 ]$   
+for images).
+
+# 3 EVALUATION
+
+We show that we can reliably produce transformation-tolerant adversarial examples in both the 2D and 3D case. Furthermore, we show that we can synthesize and fabricate 3D adversarial objects, even those with complex shapes, in the physical world: these adversarial objects remain adversarial regardless of viewpoint, camera noise, and other similar real-world factors. Adversariality in this case refers to the propensity of the classifier to predict an attacker-chosen target class given an image or object.
+
+In our evaluation, given a source object $x$ and a set of correct classes $\{ y _ { 1 } , . . . y _ { n } \}$ , as well as an attacker-chosen target class $y _ { a d v }$ and a crafted adversarial example $x ^ { \prime }$ , we use the following terms in order to characterize the effectiveness of $x ^ { \prime }$ . We call a viewpoint “adversarial” if the top output of the classifier is $y _ { a d v }$ ; “correct” if the adversarial example fails and the classifier outputs one of $\{ y _ { 1 } , \ldots , y _ { n } \}$ , and “misclassified” if the classifier predicts a class unrelated to that of the source object (i.e. not in $\{ y _ { 1 } , . . . y _ { n } \} )$ but not the target class $y _ { a d v }$ . By randomly sampling a set of viewpoints, we can evaluate an adversarial example $x ^ { \prime }$ by examining the proportion of adversarial, correct, and misclassified viewpoints.
+
+We evaluate robust adversarial examples in the 2D and 3D case, and furthermore, we evaluate physical-world 3D adversarial objects. The two cases are fundamentally different. In the virtual case, we know that we want to construct adversarial examples robust over a certain distribution of transformations, and we can simply use EOT over that distribution to synthesize a robust adversarial example. In the case of the physical world, however, we cannot capture the exact distribution unless we perfectly model all physical phenomena. Therefore, we must approximate the distribution and perform EOT over the proxy distribution. This works well in practice for producing adversarial objects that remain adversarial under the “true” physical-world distribution, as we demonstrate. See Appendix A for the exact parameters of the distributions we use in the 2D, 3D simulation, and 3D physical-world cases. In all cases, the various transformation parameters are sampled as continuous random variables from a uniform distribution between the minimum and maximum values given, unless otherwise indicated in Appendix A (i.e. Gaussian noise).
+
+In our experiments, we use TensorFlow’s standard pre-trained InceptionV3 classifier (Szegedy et al., 2015) which has $7 8 . 0 \%$ top-1 accuracy on ImageNet. In all of our experiments, we use randomly chosen target classes, and we use EOT to synthesize adversarial examples over a chosen distribution. We measure the $\ell _ { 2 }$ distance per pixel between the original and adversarial example (in LAB space), and we also measure classification accuracy (percent of randomly sampled viewpoints classified as the true class) and adversariality (percent of randomly sampled viewpoints classified as the adversarial class) for both the original and adversarial example. When working in simulation, we evaluate over a large number of transformations sampled randomly from the distribution; in the physical world, we evaluate over a large number of manually-captured images of our adversarial objects taken over different viewpoints.
+
+Table 1: Evaluation of 1000 2D adversarial examples with random targets. We evaluate each example over 1000 randomly sampled transformations to calculate classification accuracy and adversariality (percent classified as the adversarial class).   
+
+<table><tr><td rowspan="2">Images</td><td colspan="2">Classification Accuracy</td><td colspan="2">Adversariality</td><td>l2</td></tr><tr><td>mean</td><td>stdev</td><td>mean</td><td>stdev</td><td>mean</td></tr><tr><td>Original</td><td>70.0%</td><td>36.4%</td><td>0.01%</td><td>0.3%</td><td>N/A</td></tr><tr><td>Adversarial</td><td>0.9%</td><td>2.0%</td><td>96.4%</td><td>4.4%</td><td>5.6 ×10-5</td></tr></table>
+
+![](images/8a1365085cfc8403c7fd2cf555137be9d735f49017aa2822c1b4193b33cc2c53.jpg)  
+Figure 2: A 2D adversarial example, showing classifier confidence in true and adversarial classes for original and corresponding adversarial image over randomly sampled poses.
+
+# 3.1 ROBUST 2D ADVERSARIAL EXAMPLES
+
+In the 2D case, we consider the distribution of transformations that includes rescaling, rotation, lightening or darkening by an additive factor, adding Gaussian noise, and any in-bounds translation of the image.
+
+We take the first 1000 images in the ImageNet validation set, randomly choose a target class for each image, and use EOT to synthesize an adversarial example that is robust over the chosen distribution. For each adversarial example, we evaluate over 1000 random transformations sampled from the distribution at evaluation time. Table 1 summarizes the results. On average, the adversarial examples have an adversariality of $9 6 . 4 \%$ , showing that our approach is highly effective in producing robust adversarial examples. Figure 2 shows an example of a synthesized adversarial example, along with the classification confidence in true and adversarial classes for original and corresponding adversarial images. See Appendix B for more examples.
+
+# 3.2 ROBUST 3D ADVERSARIAL EXAMPLES
+
+We produce 3D adversarial examples by modeling the 3D rendering as a transformation under EOT. Given a textured 3D object, we optimize over the texture such that the rendering is adversarial from any viewpoint. We consider a distribution that incorporates different camera distances, lateral translation, rotation of the object, and solid background colors. We approximate the expectation over transformation by taking the mean loss over batches of size 40; furthermore, due to the computational expense of computing a batch, we reuse up to $80 \%$ of the batch at each iteration, but enforce that each batch contain at least 8 new images. As previously mentioned, the parameters of the distribution we use is specified in Appendix A, sampled as independent continuous random variables (that are uniform except for Gaussian noise).
+
+Table 2: Evaluation of 200 3D adversarial examples with random targets. We evaluate each example over 100 randomly sampled poses to calculate classification accuracy and adversariality (percent classified as the adversarial class).   
+
+<table><tr><td rowspan="2">Images</td><td colspan="2">Classification Accuracy</td><td colspan="2">Adversariality</td><td>l</td></tr><tr><td>mean</td><td>stdev</td><td>mean</td><td>stdev</td><td>mean</td></tr><tr><td>Original</td><td>68.8%</td><td>31.2%</td><td>0.01%</td><td>0.1%</td><td>N/A</td></tr><tr><td>Adversarial</td><td>1.1%</td><td>3.1%</td><td>83.4%</td><td>21.7%</td><td>5.9 ×10-3</td></tr></table>
+
+![](images/1cf2978cb6d307e968db9ecc20ecdbb43bcd09fc113e3588f289f279b20e61d3.jpg)  
+Figure 3: A 3D adversarial example, showing classifier confidence in true and adversarial classes for original and corresponding adversarial object over randomly sampled poses.
+
+We consider 10 complex 3D models, choose 20 random target classes per 3D model, and use EOT to synthesize adversarial textures for the 3D models with minimal parameter search (four constant, pre-chosen $\lambda$ values were tested across each [3D model, target] pair). For each of the 200 adversarial examples, we sample 100 random transformations from the distribution at evaluation time. Table 2 summarizes results, and Figure 3 shows renderings of drawn samples with classification probabilities. See Appendix C for more examples.
+
+The simulated adversarial object have an average adversariality of $8 3 . 4 \%$ with a long left tail, showing that EOT usually produces highly adversarial objects. See Appendix C for a plot of the distribution.
+
+# 3.3 PHYSICAL ADVERSARIAL EXAMPLES
+
+In order to fabricate physical-world adversarial examples, beyond modeling the 3D rendering process, we need to model physical-world phenomena such as lighting effects and camera noise. Furthermore, we need to model the 3D printing process: in our case, we use commercially available full-color 3D printing. With the 3D printing technology we use, we find that color accuracy varies between prints, so we model printing errors as well. We approximate all of these phenomena by a distribution of transformations under EOT. In addition to the transformations considered for 3D in simulation, we consider camera noise, additive and multiplicative lighting, and per-channel color inaccuracies.
+
+![](images/0ce2a65d6a0c86019a0fd73535135727a2d3d4663220c6b525796904fe6f4d8d.jpg)  
+Figure 4: A sample of photos of unperturbed 3D prints. The unperturbed 3D-printed objects are consistently classified as the true class.
+
+<table><tr><td>Object</td><td>Adversarial</td><td>Misclassified</td><td>Correct</td></tr><tr><td>Turtle</td><td>82%</td><td>16%</td><td>2%</td></tr><tr><td>Baseball</td><td>59%</td><td>31%</td><td>10%</td></tr></table>
+
+Table 3: Quantitative analysis of the two adversarial objects, over 100 photos of each object over a wide distribution of viewpoints. Both objects are classified as the adversarial target class in the majority of viewpoints.
+
+We evaluate physical adversarial examples over two 3D-printed objets: one of a turtle (where we consider any of the 5 turtle classes in ImageNet as the “true” class), and one of a baseball. The unperturbed 3D-printed objects are correctly classified as the true class with $100 \%$ accuracy over a large number of samples. Figure 4 shows example photographs of unperturbed objects, along with their classifications.
+
+We choose target classes for each of the 3D models at random — “rifle” for the turtle, and “espresso” for the baseball — and we use EOT to synthesize adversarial examples. We evaluate the performance of our two 3D-printed adversarial objects by taking 100 photos of each object over a variety of viewpoints1. Figure 5 shows a random sample of these images, along with their classifications. Table 3 gives a quantitative analysis over all images, showing that our 3D-printed adversarial objects are strongly adversarial over a wide distribution of transformations. See Appendix D for more examples.
+
+# 3.4 DISCUSSION
+
+The results and quantative analysis in this section demonstrate the efficacy of EOT and confirm the existence of physical adversarial examples. Here, we perform a qualitative analysis of the results:
+
+Modeling Perception. The EOT algorithm as presented in Section 2 presents a general method to construct adversarial examples over a chosen perceptual distribution, but notably gives no guarantees for observations of the image outside of the chosen distribution. In constructing physical-world adversarial objects, we use a crude, high-variance approximation of the rendering and capture process, and this succeeds in ensuring robustness to a diverse set of environments; see, for example, Figure 6, which shows the same adversarial turtle in vastly different environments. In specialized domains, however, a domain expert may opt to model the perceptual distribution precisely in order to better constrain the search space.
+
+![](images/2d838726c5c8ee8f55c20ec69eb7940cf0838e9a4739614e4b6be8128e2a551c.jpg)  
+Figure 5: Random sample of photographs of the two 3D-printed adversarial objects. The 3Dprinted adversarial objects are strongly adversarial over a wide distribution of viewpoints.
+
+![](images/5868a039d1fd427be0e20aaccff39f6ed047d81f56294c207a550fa80938aaca.jpg)  
+Figure 6: Three pictures of the same adversarial turtle (all classified as “rifle”), demonstrating the need for a wide distribution, and the efficacy of EOT in finding examples robust across wide distributions of physical-world effects like lighting.
+
+![](images/7680637ccd673478b874fc51b0f957fc4bba607403fde58f404df0cfd76e6021.jpg)  
+Figure 7: A side-by-side comparison of a 3D-printed model (left) along with a printout of the corresponding texture, printed on a standard laser color printer (center) and the original digital texture (right), showing significant error in color accuracy in printing.
+
+Error in Printing. We also find significant error in the color accuracy of even state of the art commercially available color 3D printing; Figure 7 shows a comparison of a 3D-printed model along with a printout of the model’s texture, printed on a standard laser color printer. Still, EOT was able to overcome the problem and produce robust physical-world adversarial objects. We predict that we could have produced adversarial examples with smaller $\ell _ { 2 }$ perturbation with a higher-fidelity printing process.
+
+Semantically Relevant Misclassification. Interestingly, for the majority of viewpoints where the adversarial target class is not the top-1 predicted class, the classifier also fails to correctly predict the source class. Instead, we find that the classifier often classifies the object as an object that is semantically similar to the adversarial target; while generating the adversarial turtle to be classified as a rifle, for example, the second most popular class (after “rifle”) was “revolver,” followed by “holster” and then “assault rifle.” Similarly, when generating the baseball to be classified as an espresso, the example was often classified as “coffee” or “bakery”.
+
+# 4 RELATED WORK
+
+State of the art neural networks are vulnerable to adversarial examples (Szegedy et al., 2013). Researchers have proposed a number of methods for synthesizing adversarial examples in the whitebox scenario (with access to the gradient of the classifier), including L-BFGS (Szegedy et al., 2013), the Fast Gradient Sign Method (FGSM) (Goodfellow et al., 2015), and a Lagrangian relaxation formulation (Carlini & Wagner, 2017), all for the single-viewpoint case where the adversary directly controls the input to the neural network. Projected Gradient Descent (PGD) can be seen as a universal first-order adversary (Madry et al., 2017).
+
+Moosavi-Dezfooli et al. (2017) show the existence of universal (image-agnostic) adversarial perturbations, small perturbation vectors that can be applied to any image to induce misclassification. Their paper proposes an algorithm that finds perturbations that are universal over images; in our work, we give an algorithm that finds a perturbation to a single image or object that is universal over a chosen distribution of transformations. In preliminary experiments, we found that universal adversarial perturbations, like standard adversarial perturbations to single images, are not inherently robust to transformation.
+
+In the first work on 2D physical-world adversarial examples, Kurakin et al. (2016) demonstrate the transferability of FGSM-generated adversarial misclassification on a printed page. In their setup, a photo is taken of a printed image with QR code guides, and the resultant image is warped, cropped, and resized to become a square of the same size as the source image before classifying it. Their results show the existence of 2D physical-world adversarial examples for approximately axis-aligned views, demonstrating that adversarial perturbations produced using FGSM can translate to the physical world and are robust to camera noise, rescaling, and lighting effects. Kurakin et al. (2016) do not synthesize targeted physical-world adversarial examples, they do not evaluate other real-world 2D transformations such as rotation, skew, translation, or zoom, and their approach does not translate to the 3D case.
+
+Sharif et al. (2016) develop a real-world adversarial attack on a state-of-the-art face recognition algorithm, where adversarial eyeglass frames cause targeted misclassification in portrait photos. The algorithm produces robust perturbations through optimizing over a fixed set of inputs: the attacker collects a set of images and finds a perturbation that minimizes cross entropy loss over the set. The algorithm solves a different problem than we do in our work: it produces adversarial perturbations universal over portrait photos taken head-on from a single viewpoint, while EOT produces 2D/3D adversarial examples robust over transformations. The approach also includes a mechanism for enhancing perturbations’ printability using a color map to address the limited color gamut and color inaccuracy of the printer. Note that this differs from our approach in achieving printability: rather than creating a color map, we find an adversarial example that is robust to color inaccuracy. Our approach has the advantage of working in settings where color accuracy varies between prints, as was the case with our 3D-printer.
+
+Recently, Evtimov et al. (2017) proposed a method for generating robust physical-world adversarial examples in the 2D case by optimizing over a fixed set of manually-captured images. However, the approach is limited to the 2D case, with no clear translation to 3D, where there is no simple mapping between what the adversary controls (the texture) and the observed input to the classifier (an image). Furthermore, the approach requires the taking and preprocessing of a large number of photos in order to produce each adversarial example, which may be expensive or even infeasible for many objects.
+
+Lu et al. (2017) argued that adversarial examples may not be a practical concern in physical-world systems because adversarial examples generated for the single viewpoint case lose adversariality at viewpoints with differing scale and rotation. While this is the case with standard adversarial examples as evaluated in the paper and in Luo et al. (2016), we have shown that with EOT, it is in fact possible to construct physical-world adversarial images and objects that are classified as a chosen target class over a wide range of viewpoints.
+
+# 5 CONCLUSION
+
+Our work shows that adversarial examples pose a practical threat to systems using neural networkbased image classifiers. By introducing EOT, a general-purpose algorithm for creating robust adversarial examples under any chosen distribution, and modeling 3D rendering and printing within the framework of EOT, we succeed in fabricating three-dimensional adversarial objects. With access only to low-cost commercially available 3D printing technology, we successfully print physical adversarial objects that are strongly classified as a chosen target class over a variety of angles, viewpoints, and lighting conditions by a standard ImageNet classifier.
+
+# REFERENCES
+
+Nicholas Carlini and David Wagner. Towards evaluating the robustness of neural networks. In IEEE Symposium on Security & Privacy, 2017.
+
+Ivan Evtimov, Kevin Eykholt, Earlence Fernandes, Tadayoshi Kohno, Bo Li, Atul Prakash, Amir Rahmati, and Dawn Song. Robust Physical-World Attacks on Deep Learning Models. 2017. URL https://arxiv.org/abs/1707.08945.
+
+Ian J. Goodfellow, Jonathon Shlens, and Christian Szegedy. Explaining and harnessing adversarial examples. In Proceedings of the International Conference on Learning Representations (ICLR), 2015.
+
+Alexey Kurakin, Ian Goodfellow, and Samy Bengio. Adversarial examples in the physical world. 2016. URL https://arxiv.org/abs/1607.02533.
+
+Jiajun Lu, Hussein Sibai, Evan Fabry, and David Forsyth. No need to worry about adversarial examples in object detection in autonomous vehicles. 2017. URL https://arxiv.org/ abs/1707.03501.
+
+Yan Luo, Xavier Boix, Gemma Roig, Tomaso Poggio, and Qi Zhao. Foveation-based mechanisms alleviate adversarial examples. 2016. URL https://arxiv.org/abs/1511.06292.
+
+Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and Adrian Vladu. Towards deep learning models resistant to adversarial attacks. 2017. URL https://arxiv. org/abs/1706.06083.
+
+Seyed-Mohsen Moosavi-Dezfooli, Alhussein Fawzi, Omar Fawzi, and Pascal Frossard. Universal adversarial perturbations. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2017.
+
+Mahmood Sharif, Sruti Bhagavatula, Lujo Bauer, and Michael K. Reiter. Accessorize to a crime: Real and stealthy attacks on state-of-the-art face recognition. In Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security, CCS ’16, pp. 1528–1540, New York, NY, USA, 2016. ACM. ISBN 978-1-4503-4139-4. doi: 10.1145/2976749.2978392. URL http://doi.acm.org/10.1145/2976749.2978392.
+
+Christian Szegedy, Wojciech Zaremba, Ilya Sutskever, Joan Bruna, Dumitru Erhan, Ian Goodfellow, and Rob Fergus. Intriguing properties of neural networks. 2013. URL https://arxiv.org/ abs/1312.6199.
+
+Christian Szegedy, Vincent Vanhoucke, Sergey Ioffe, Jonathon Shlens, and Zbigniew Wojna. Rethinking the inception architecture for computer vision. 2015. URL https://arxiv.org/ abs/1512.00567.
+
+Table 4: Distribution of transformations for the 2D case, where each parameter is sampled uniformly at random from the specified range.   
+
+<table><tr><td>Transformation</td><td>Minimum</td><td>Maximum</td></tr><tr><td>Scale</td><td>0.9</td><td>1.4</td></tr><tr><td>Rotation</td><td>-22.5°</td><td>22.5°</td></tr><tr><td>Lighten /Darken</td><td>-0.05</td><td>0.05</td></tr><tr><td>Gaussian Noise (stdev)</td><td>0.0</td><td>0.1</td></tr><tr><td>Translation</td><td colspan="2">any in-bounds</td></tr></table>
+
+<table><tr><td>Transformation</td><td>Minimum</td><td>Maximum</td></tr><tr><td>Camera distance</td><td>2.5</td><td>3.0</td></tr><tr><td>X/Y translation</td><td>-0.05</td><td>0.05</td></tr><tr><td>Rotation</td><td>any</td><td></td></tr><tr><td>Background</td><td>(0.1, 0.1, 0.1)</td><td>(1.0, 1.0, 1.0)</td></tr></table>
+
+Table 5: Distribution of transformations for the 3D case when working in simulation, where each parameter is sampled uniformly at random from the specified range.
+
+# A DISTRIBUTIONS OF TRANSFORMATIONS
+
+Under the EOT framework, we must choose a distribution of transformations, and the optimization produces an adversarial example that is robust under the distribution of transformations. Here, we give the specific parameters we chose in the 2D (Table 4), 3D (Table 5), and physical-world case (Table 6).
+
+# B ROBUST 2D ADVERSARIAL EXAMPLES
+
+We give a random sample out of our 1000 2D adversarial examples in Figures 8 and 9.
+
+# C ROBUST 3D ADVERSARIAL EXAMPLES
+
+We give a random sample out of our 200 3D adversarial examples in Figures 10 and 11 and 12. We give a histogram of adversariality (percent classified as the adversarial class) over all 200 examples in Figure 13.
+
+<table><tr><td>Transformation</td><td>Minimum</td><td>Maximum</td></tr><tr><td>Camera distance</td><td>2.5</td><td>3.0</td></tr><tr><td>X/Y translation</td><td>-0.05</td><td>0.05</td></tr><tr><td>Rotation</td><td>any</td><td></td></tr><tr><td>Background</td><td>(0.1, 0.1, 0.1)</td><td>(1.0, 1.0, 1.0)</td></tr><tr><td>Lighten /Darken (additive)</td><td>-0.15</td><td>0.15</td></tr><tr><td>Lighten /Darken (multiplicative)</td><td>0.5</td><td>2.0</td></tr><tr><td>Per-channel (additive)</td><td>-0.15</td><td>0.15</td></tr><tr><td>Per-channel (multiplicative)</td><td>0.7</td><td>1.3</td></tr><tr><td>Gaussian Noise (stdev)</td><td>0.0</td><td>0.1</td></tr></table>
+
+Table 6: Distribution of transformations for the physical-world 3D case, approximating rendering, physical-world phenomena, and printing error.
+
+![](images/b083ea72bfac9f4868888f30c3c69dd4a49ec8ff2c8e79b652ea528b891770cb.jpg)  
+Figure 8: A random sample of 2D adversarial examples.
+
+![](images/f0f817fd8983b1e1d4054c974e670dd80c5166c6dac2cf13472fe01de10cd927.jpg)  
+Figure 9: A random sample of 2D adversarial examples.
+
+![](images/ed727815dd9fc97ab760d41d0cd4b6c21479351d5269e868470b0cd22de36447.jpg)
+
+![](images/c6f2063325c3f07b635032ee0c16b39697e906d77447ea490acbee04d5767b64.jpg)
+
+P (true): 96% P (adv): 0%
+
+P (true): 99% P (adv): 0%
+
+![](images/4cf3233d10102773051dc716a7368d5ab8850e906521a1b690ab87971dd83841.jpg)
+
+![](images/7515ed6a492794f537c1cc35e8dd4a29b2a68b15dabe753c0156aa14106740a7.jpg)
+
+P (true): 96% P (adv): 0%
+
+P (true): 97% P (adv): 0%
+
+Adversarial:   
+guillotine   
+\`2 = 3.1 × 10−5
+
+![](images/54725ecf42ea0dc7381f3cf3c3f859630c7958461ff2b8481e04ec046fcab82e.jpg)
+
+P (true): 1% P (adv): 10%
+
+![](images/8fd61dc806d9100aa12ab70aa0390d56a6b914ea4e50edcd9b099b652ae752cb.jpg)
+
+P (true): 0% P (adv): 95%
+
+![](images/daf5e022f68af61c84a6bccf67e1347bf62cd13906f887400733c8b202810466.jpg)
+
+P (true): 0% P (adv): 91%
+
+![](images/474cab26c2ad858b19ed19a9311ee4b9ffcbfd7636be1d0624df670ca2b536fe.jpg)
+
+P (true): 3% P (adv): 4%
+
+Original: baseball
+
+P (true): 100% P (adv): 0%
+
+![](images/446ebf8f295c43d48bb077f813088cceaa533e663bb7e9d921c7cbd3a4a523e8.jpg)
+
+P (true): 100% P (adv): 0%
+
+![](images/5801cd3b24e8979b4b93b46ce27a6964716fc3c454e51bf9ccf090205d092684.jpg)
+
+P (true): 100% P (adv): 0%
+
+![](images/c1a6f0835e8c8f8a5c5fc2ccf9378302270b489b6e4166a1ab5a5864fe7f68ae.jpg)
+
+P (true): 100% P (adv): 0%
+
+Adversarial:   
+green lizard   
+\`2 = 2.9 × 10−5
+
+![](images/b8cae5f69364c82edf35080f1259de8117c87c2ebc06e60fe98d2323ade5ad3f.jpg)
+
+![](images/cd986a37a5e612d93931ce84585be956cec7ce04e63438e588371e7e2c12a2ee.jpg)  
+Figure 10: A random sample of 3D adversarial examples.
+
+P (true): 0% P (adv): 66%
+
+![](images/a3a7f87be5edcfd09f8959b9e465d1b7734d5164f2d7f81f8941ba5aa235682e.jpg)
+
+P (true): 0% P (adv): 94%
+
+![](images/1a7be731f6e47133ef61ad9c4d7d80834dc5a12f838d2951ea6bdf17d3db6a9d.jpg)
+
+P (true): 0% P (adv): 87%
+
+![](images/c8c03105f7c59ea309bbb24efb5bbbd48af3a326bc24fb8ba8108beb2c9fdee4.jpg)
+
+P (true): 0% P (adv): 94%
+
+![](images/f5b8650edd0e56e908f26f6719451967643d51b73b5d96c42f3ec7114124cf94.jpg)
+
+![](images/47d685b13d576c782a76ea4310002346cc13e74471d60396a67bfb51531a72c1.jpg)
+
+P (true): 97% P (adv): 0%
+
+Adversarial:   
+Bouvier des   
+Flandres   
+$\ell _ { 2 } = 1 . 4 \times 1 0 ^ { - 4 }$
+
+Original: baseball
+
+![](images/249278e25d05f089dd06880dd49751146a6947e241fbe364d0ccdb463743f917.jpg)
+
+P (true): 100% $P ( a d v ) \colon 0 \%$
+
+![](images/66e03cb43930ef19e402f0f01e7787aadc51dc8239ed85223e3f8fbdc34dabd1.jpg)
+
+P (true): 100% P (adv): 0%
+
+![](images/6bb9dfef8cd85248a240d643c4b9500f131cac1df5328f3c08edc3f29e4fcbcf.jpg)
+
+P (true): 100% P (adv): 0%
+
+![](images/e2bbfdc90bb9e85b72a9335652ad544ee2642d4fdf0ec0b23d09248c608a7584.jpg)
+
+P (true): 100% P (adv): 0%
+
+Adversarial:   
+Airedale   
+\`2 = 3.4 × 10−5
+
+![](images/95900f3926ccab52c23ca303bc16c1eed02813c826f1675f2d45b02624161e6e.jpg)
+
+P (true): $0 \%$ $P ( a d v ) \colon 9 4 \%$
+
+![](images/a05c59801466e8c0cf5c36693564fe27616f7a486313b5569b8c4811ee21279b.jpg)
+
+P (true): 0% P (adv): 6%
+
+![](images/0d85b317da683e42fe93849c0db9cc79d3ea2e8124cdb3d10ed17f8df82cb1b4.jpg)
+
+P (true): 0% P (adv): 96%
+
+![](images/239fdb3245349dc1af9e16b14f520ec7678daa9356f3d0edf4bd37ee369a35b9.jpg)
+
+P (true): 0% P (adv): 18%
+
+Original: orange
+
+P (true): 73% $\dot { P } ( a d v ) \colon 0 \%$
+
+![](images/3b9c622a8360406355546717b32d8259f72994852b21e2ffe08b5133db3996a9.jpg)
+
+P (true): 29% P (adv): 0%
+
+![](images/6a4c9beddfe761c6bba3f18f1abfcc0ca223dfda977436934cd48abd07d7fd6a.jpg)
+
+P (true): 20% P (adv): 0%
+
+![](images/d5d1abe73147d2d10dbda4e5624b9e7dfb7a5e1b5ee787d4a091463cba747ab1.jpg)
+
+P (true): 85% P (adv): 0%
+
+Adversarial: power drill $\ell _ { 2 } = 4 . 3 \times 1 0 ^ { - 5 }$
+
+![](images/eb1b9d31583bd8c3c01ea86cabd1a3d1da86a0485b68f02f976c61a975132d00.jpg)
+
+P (true): 0% $P ( a d v ) \colon 8 9 \%$
+
+![](images/ceefc8ab855bea62d9afd910734084e0186d0b5f35c6b4da4978d71ef2e89ad4.jpg)
+
+![](images/3c3fb815e0066ce35a615779596beb636e43cd985cab46489e5d91c4606ba99c.jpg)  
+Figure 11: A random sample of 3D adversarial examples.
+
+P (true): 4% P (adv): 75%
+
+![](images/bcbc07463244d7b2b5fbe8bf013b56b2f5a4d250075fc2233ceb8817d2eeba77.jpg)
+
+P (true): 0% P (adv): 98%
+
+![](images/54f666abab66b32348d729e33d0dc873dc9ab5c0c9b70e37eab3f9f8e64229a6.jpg)
+
+P (true): 0% P (adv): 84%
+
+![](images/cb48e24288fd98194840193196c2f1c91cbe041c05552ab90195f16dc5eda6aa.jpg)
+
+![](images/0ad0ed609336cdf3edbeb9c8b639044946d9a6eadc326762b7b9144d498e8340.jpg)
+
+![](images/23c084bba5647d1f819303736d3f648ac671523d7199dd941ae192d5489896ea.jpg)
+
+![](images/f2cfff3e1d998a748bb6f9050dca68c368fbf8b5cf7c0ecd36f76daec2047b0d.jpg)
+
+P (true): 0% P (adv): 0%
+
+Adversarial:   
+bittern   
+$\ell _ { 2 } = 7 . 4 \times 1 0 ^ { - 5 }$
+
+P (true): 0% P (adv): 97%
+
+Original: teddybear
+
+![](images/6ad66586a045e8d66368d2977c1e998f6406417daf8d764b80c22f97afb000ea.jpg)
+
+P (true): 90% $P ( a d v ) \colon 0 \%$
+
+![](images/30755d556abc6422ec694c4576b70359e6018f4f9a724cabbdeb85c9bb4875a3.jpg)
+
+P (true): 1% P (adv): 0%
+
+![](images/5851e0b808bde9a1bf46c0794c0cf4b9232eb22a41c478dec454f86a98f6ab58.jpg)
+
+P (true): 98% P (adv): 0%
+
+![](images/1fe7cda29d006d909f7781f949ce04c9a7a60aaf3ecd6a7eac67e7fe96129995.jpg)
+
+P (true): 5% P (adv): 0%
+
+Adversarial: sock $\ell _ { 2 } = 3 . 3 \times 1 0 ^ { - 5 }$
+
+![](images/a9816a9b180b8f7bef67cbba62b49ec4d4cd3d317f357ede8d4343f8a39b0741.jpg)
+
+P (true): 0% P (adv): 99%
+
+![](images/c5fb2c2baa6b7cabfa3e65c92cc6bb38be0bd7020c3189bb36ee515c02bca7f6.jpg)
+
+P (true): 0% P (adv): 99%
+
+![](images/586093914075b318ca8bb6e5b7a6291befd2e976ad524a85192839ae0e695bed.jpg)
+
+P (true): 0% P (adv): 98%
+
+![](images/ebe47435f3be92f455951303cf89ec71f729d8c89a6e7f9af0411deb2620a93a.jpg)
+
+P (true): 0% P (adv): 99%
+
+Original: clownfish
+
+![](images/ac2dd4c42b02bc94fdb060c2e3d702c62369d6b0fbadc4d7f63e446d6b0a874c.jpg)
+
+P (true): 46% P (adv): 0%
+
+![](images/abd96785a2d7187f97cbb07657d511dcec4010242e79c96ce9edebceee8967bb.jpg)
+
+P (true): 14% P (adv): 0%
+
+![](images/347b879d28ef7690e64636129cc7eff717d83136b182259b8cb02ff40871203b.jpg)
+
+P (true): 2% P (adv): 0%
+
+![](images/65db67d6cbe560af1e9677e0329a5578b1b09ba62293dcd8f8654a46d51f6db1.jpg)
+
+P (true): 65% P (adv): 0%
+
+Adversarial:   
+panpipe   
+\`2 = 4.8 × 10−5
+
+![](images/e87b26949fe94ff2c31df5ba8cbd5a454d8160a7139bda171c60cb616dac6778.jpg)
+
+P (true): 0% P (adv): 100%
+
+![](images/bbb9c568b2e2c1cbbc549f938a340227515e212830866283b2ef9b4fb7fd50c4.jpg)
+
+P (true): 0% $P ( a d v ) \colon 1 \%$
+
+![](images/5d9935f54c75a0b8412dbf0ec7991dd36a9c4c4d61513020c87ffd4aca2658d6.jpg)
+
+P (true): 0% P (adv): 12%
+
+![](images/1f1bb7351508549b5ae2f69e4c29d4b3d9746bc865ffdd8c1868b987d04b8aef.jpg)
+
+P (true): 0% P (adv): 0%
+
+![](images/c29b80e8cfe88b28fff222bcbc5ed1d58a95ac9f5218e28749e09c9d7b97a622.jpg)
+
+![](images/0c9b9888052e6fae14934e6ffad213cbf504db46a4fba1429e7ad6e40c146804.jpg)
+
+![](images/0331f9e468f15008972fc47f1d99e1c16a46cf3d2510be331bec5a96f4b04fa8.jpg)
+
+![](images/9d96fa85337f06f1b358e57888978077a76cd1babff1c801deb58405babbf030.jpg)
+
+P (true): $70 \%$ P (adv): $0 \%$
+
+Adversarial:   
+sturgeon   
+\`2 = 1.3 × 10−5
+
+![](images/6514990055938b032cbe2d054224c99a04b627fe03e7037e27f07c7cd4cefd7c.jpg)  
+Figure 12: A random sample of 3D adversarial examples.
+
+![](images/8eaea64977a95f871993846895af7ee86e7ffd8785da694f9017c5519ff5fc86.jpg)  
+Figure 13: A histogram of adversariality (percent of 100 samples classified as the adversarial class) across the 200 3D adversarial examples.
+
+# D PHYSICAL ADVERSARIAL EXAMPLES
+
+Figure 14 gives all 100 photographs of our adversarial 3D-printed turtle, and Figure 15 gives all 100 photographs of our adversarial 3D-printed baseball.
+
+![](images/f9a72c737c6938de7d0ea9e2da871aa2b1b23446360d174129041b1c8e16bf96.jpg)  
+Figure 14: All 100 photographs of our physical-world 3D adversarial turtle.
+
+![](images/a3b9d7a1a22a885462f5210bd4bb0261e6148a4e194778f64bc986e7c7483fca.jpg)  
+Figure 15: All 100 photographs of our physical-world 3D adversarial baseball.

parse/train/BJDH5M-AW/BJDH5M-AW_content_list.json

@@ -0,0 +1,2837 @@
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+    {
+        "type": "text",
+        "text": "SYNTHESIZING ROBUST ADVERSARIAL EXAMPLES ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            98,
+            787,
+            121
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            145,
+            398,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            210,
+            544,
+            226
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Neural network-based classifiers parallel or exceed human-level accuracy on many common tasks and are used in practical systems. Yet, neural networks are susceptible to adversarial examples, carefully perturbed inputs that cause networks to misbehave in arbitrarily chosen ways. When generated with standard methods, these examples do not consistently fool a classifier in the physical world due to a combination of viewpoint shifts, camera noise, and other natural transformations. Adversarial examples generated using standard techniques require complete control over direct input to the classifier, which is impossible in many real-world systems. ",
+        "bbox": [
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+            242,
+            764,
+            368
+        ],
+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "We introduce the first method for constructing real-world 3D objects that consistently fool a neural network across a wide distribution of angles and viewpoints. We present a general-purpose algorithm for generating adversarial examples that are robust across any chosen distribution of transformations. We demonstrate its application in two dimensions, producing adversarial images that are robust to noise, distortion, and affine transformation. Finally, we apply the algorithm to produce arbitrary physical 3D-printed adversarial objects, demonstrating that our approach works end-to-end in the real world. Our results show that adversarial examples are a practical concern for real-world systems. ",
+        "bbox": [
+            233,
+            371,
+            764,
+            497
+        ],
+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            527,
+            336,
+            542
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The existence of adversarial examples for neural networks has until now been largely a theoretical concern. While minute, carefully-crafted perturbations can cause targeted misclassification in a neural network, adversarial examples produced using standard techniques lose adversariality when directly translated to the physical world as they are captured over varying viewpoints and affected by natural phenomena such as lighting and camera noise. This suggests that practical systems may not be at risk because adversarial examples generated using standard techniques are not robust in the physical world. ",
+        "bbox": [
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+        "img_path": "images/e7ff76dd7ad74bacab138890651367af1c995c337660f74ccd1530dd873a6cbc.jpg",
+        "image_caption": [
+            "Figure 1: Randomly sampled poses of a single 3D-printed turtle adversarially perturbed to classify as a rifle at every viewpoint by an ImageNet classifier. An unperturbed model is classified correctly as a turtle $100 \\%$ of the time. See https://youtu.be/YXy6oX1iNoA for a video where every frame is fed through the classifier: the turtle is consistently classified as a rifle. "
+        ],
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+    {
+        "type": "text",
+        "text": "We show that neural network-based classifiers are vulnerable to physical-world adversarial examples. We introduce a new algorithm for reliably producing physical 3D objects that are adversarial over a distribution of viewpoints. Figure 1 shows an example of an adversarial object constructed using our approach, where a 3D-printed turtle is consistently classified as rifle by an ImageNet classifier. In this paper, we demonstrate the efficacy and generality of our method, demonstrating conclusively that adversarial examples are a concern in real-world systems. ",
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+    {
+        "type": "text",
+        "text": "1.1 CHALLENGES ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Methods for transforming ordinary two-dimensional images into adversarial examples are wellknown; however, the extent to which these techniques affect specific real-world systems is still an open question. Prior work has shown that adversarial examples generated using standard techniques lose their adversarial nature once subjected to minor transformations (Luo et al., 2016; Lu et al., 2017). These results suggest that adversarial examples, while serving as interesting phenomena, have no hope of applying to physical-world systems, where transformations such as viewpoint changes and camera noise are inevitable. ",
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+    {
+        "type": "text",
+        "text": "Prior techniques attempting to synthesize robust adversarial examples for the physical world have had limited success. While some progress has been made, current efforts have demonstrated a small number of datapoints on nonstandard classifiers, and only in the two-dimensional case, with no clear generalization to three dimensions (further discussed in Section 4). ",
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+        "type": "text",
+        "text": "The entirety of prior work has only explored generating adversarial examples in the two-dimensional case, where “viewpoints” can be approximated by an affine transformations of an original image. Constructing adversarial examples for the physical world requires the ability to generate entire 3D adversarial objects, which must remain adversarial in the face of complex transformations not applicable to 2D objects, such as 3D rotations and perspective projection. ",
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+    {
+        "type": "text",
+        "text": "1.2 CONTRIBUTIONS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this work, we definitively show that adversarial examples pose a real threat in the physical world. We propose a general-purpose algorithm for reliably constructing adversarial examples robust over a chosen distribution of transformations, and we demonstrate the efficacy of this algorithm in both the 2D and 3D case. We succeed in producing physical-world 3D adversarial objects that are robust over a large, realistic distribution of 3D viewpoints, proving that the algorithm produces adversarial three-dimensional objects that are adversarial in the physical world. Specifically, our contributions are as follows: ",
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+        "type": "text",
+        "text": "• We develop Expectation Over Transformation (EOT), a novel algorithm that produces single adversarial examples that are simultaneously adversarial over an entire distribution of transformations   \n• We consider the problem of constructing 3D adversarial examples under the EOT framework, viewing the 3D rendering process as part of the transformation, and we show that the approach successfully synthesizes adversarial objects   \n• We fabricate adversarial objects and show that they remain adversarial, demonstrating that our approach works end-to-end in the physical world, showing that adversarial examples are of real concern in practical deep learning systems ",
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+    {
+        "type": "text",
+        "text": "2 APPROACH",
+        "text_level": 1,
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+        "type": "text",
+        "text": "First, we present the Expectation Over Transformation (EOT) algorithm, a general framework allowing for the construction of adversarial examples that remain adversarial over a chosen transformation distribution $T$ . We then describe our end-to-end approach for generating adversarial objects using a specialized application of EOT in conjunction with differentiating through the 3D rendering process. ",
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+    {
+        "type": "text",
+        "text": "2.1 EXPECTATION OVER TRANSFORMATION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "When constructing adversarial examples in the white-box case (that is, with access to a classifier and its gradient), we know in advance a set of possible classes $Y$ and a space of valid inputs $X$ to the classifier; we have access to the function $P ( \\boldsymbol { y } | \\boldsymbol { x } )$ and its gradient $\\bar { \\nabla } P ( y | x )$ , for any class $y \\in Y$ and input $x \\in X$ . In the standard case, adversarial examples are produced by maximizing the log-likelihood of the target class over a $\\epsilon$ -radius ball around the original image, that is: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/bf0ae510601bea8196f65266e719e3859019904e8b1485467d155211cfd039de.jpg",
+        "text": "$$\n\\hat { x } = \\underset { x ^ { \\prime } \\in X } { \\arg \\operatorname* { m a x } } \\log P ( y | x ^ { \\prime } ) \\qquad \\mathrm { s . t . } \\ | | x ^ { \\prime } - x | | _ { p } < \\epsilon\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "This approach has been shown to be both feasible and effective at generating adversarial examples for any given classifier. However, prior work has shown adversarial examples’ inability to remain adversarial even under minor perturbations inevitable in any real-world observation (Luo et al., 2016; Lu et al., 2017). ",
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+    {
+        "type": "text",
+        "text": "To address this issue, we introduce Expectation Over Transformation $( E O T )$ . The key insight behind EOT is to model such perturbations within the optimization procedure. Rather than optimizing the log-likelihood of a single example, EOT uses a chosen distribution $T$ of transformation functions $t$ taking an input $x ^ { \\prime }$ supplied by the adversary to the “true” input $t ( x ^ { \\prime } )$ perceived by the classifier. Furthermore, rather than simply taking the norm of $x ^ { \\prime } - x$ to constrain the solution space, EOT instead aims to constrain the effective distance between the adversarial and original inputs, which we define as: ",
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+    {
+        "type": "equation",
+        "img_path": "images/5671cc220eda2196123e05d263a279f79336535045c74d53cd49aca8e74a6e44.jpg",
+        "text": "$$\n\\delta = \\mathbb { E } _ { t \\sim T } [ d ( t ( x ) - t ( x ^ { \\prime } ) ) ]\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Intuitively, this is how different we expect the true input to the classifer will be, given our new input. Then, EOT solves the following optimization problem: ",
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+    {
+        "type": "equation",
+        "img_path": "images/e1649647c9731c4228bb619ed0d75abc00e677204427ba8a122c8e743382402f.jpg",
+        "text": "$$\n\\hat { x } = \\underset { x ^ { \\prime } } { \\arg \\operatorname* { m a x } } \\mathbb { E } _ { t \\sim T } [ \\log P ( y | t ( x ^ { \\prime } ) ) ] \\qquad \\mathrm { s . t . } \\ \\mathbb { E } _ { t \\sim T } [ d ( t ( x ^ { \\prime } ) , t ( x ) ) ] < \\epsilon\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "In practice, the distribution $T$ can model perceptual distortions such as random rotation, translation, or addition of noise. However, EOT generalizes beyond simple transformations. EOT finds examples robust under any perception distribution $Q ( \\cdot ; x )$ parameterized by the generated example $x$ as long as ${ \\frac { d } { d x } } Q ( \\cdot ; x )$ is well-defined. The objective function can be optimized by stochastic gradient descent, approximating the derivative of the expected value through sampling transformations independently at each gradient descent step and differentiating through the transformation. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "2.2 SYSTEM OVERVIEW ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Given its ability to synthesize robust adversarial examples, we use the EOT framework for generating 2D examples, 3D models, and ultimately physical-world adversarial objects. Within the framework, however, there is a great deal of freedom in the actual method by which examples are generated, including choice of $T$ , distance metric, and optimization method. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the 2D case, we design $T$ to approximate a realistic space of possible distortions involved in printing out an image and taking a natural picture of it. This amounts to a set of random transformations of the form $t ( x ) = A x + \\epsilon$ , which are more thoroughly described in Section 3. ",
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+    {
+        "type": "text",
+        "text": "In the 3D case, we define the transformations $t$ from a texture to a perceived image by mapping the texture to a given 3D object, and simulating highly nonlinear functions such as rendering, rotation, translation, and perspective distortion of the object in addition to the perceptual mechanisms used in the 2D case. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Finding adversarial examples across these nonlinear transformations is what allows for the transfer of adversarial examples to the physical world, but also introduces implementation complexities not found in the 2D case. EOT requires the ability to differentiate though the transformation distribution with respect to the input. In the 3D case, this implies differentiating through the 3D renderer with respect to the texture. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To do this, we model the rendering process as a sparse matrix multiplication between the texture and a coordinate map, which is generated from a standard 3D renderer. Rather than attempting to differentiate through the complex renderer, we use the renderer to find a coordinate map for each rendering, which maps coordinates on the texture to coordinates in the classifier’s field of view. We can then simulate this specific rendering as a sparse matrix multiplication and differentiate through it; we differentiate through different matrix multiplications at each sampling step. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Once EOT has been parameterized, i.e. once a distribution $T$ is chosen, the issue of actually optimizing the induced objective function remains. Rather than solving the constrained optimization problem given above, we use the Lagrangian-relaxed form of the problem, as Carlini $\\&$ Wagner (2017) do in the conventional (non-EOT, single-viewpoint) case: ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/f3d1c9a59fb0f6af5e56ae2c0a808908b6d994adebbf8fc55fc14b30ccca0125.jpg",
+        "text": "$$\n\\hat { x } = \\underset { x ^ { \\prime } } { \\arg \\operatorname* { m i n } } \\mathbb { E } _ { t \\sim T } [ - \\log P ( y | t ( x ^ { \\prime } ) ) ] + \\lambda \\mathbb { E } _ { t \\sim T } [ d ( t ( x ^ { \\prime } ) , t ( x ) ] )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "In order to encourage imperceptibility of the generated images, we set $d ( x ^ { \\prime } , x )$ to be the $\\ell _ { 2 }$ norm in the LAB color space, a perceptually uniform color space where Euclidean distance roughly corresponds with perceptual distance. Note that the $\\mathbb { E } _ { t \\sim T } [ | | L A B ( t ( x ) ) - L A B ( t ( \\hat { x } ) ) | | _ { 2 } ^ { 2 } ]$ can be sampled and estimated in conjunction with $\\mathbb { E } [ P ( y | t ( x ) ) ]$ ; in general, the Lagrangian formulation gives EOT the ability to intricately constrain the search space (in our case, using LAB distance) at insignificant computational cost (without computing a complex projection). Our optimization, then, is: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/45c796cf2f8208c892489f2a05251fe065d4f75ef9679e3288b83c4936e1c2a6.jpg",
+        "text": "$$\n\\hat { x } = \\underset { x ^ { \\prime } } { \\arg \\operatorname* { m i n } } \\mathbb { E } _ { t \\sim T } [ - \\log P ( y | t ( x ^ { \\prime } ) ) + \\lambda | | L A B ( t ( x ) ) - L A B ( t ( x ^ { \\prime } ) ) | | _ { 2 } ^ { 2 } ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We use projected gradient descent to find the optimum, clipping to the set of valid inputs (e.g. $[ 0 , 1 ]$   \nfor images). ",
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+    {
+        "type": "text",
+        "text": "3 EVALUATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We show that we can reliably produce transformation-tolerant adversarial examples in both the 2D and 3D case. Furthermore, we show that we can synthesize and fabricate 3D adversarial objects, even those with complex shapes, in the physical world: these adversarial objects remain adversarial regardless of viewpoint, camera noise, and other similar real-world factors. Adversariality in this case refers to the propensity of the classifier to predict an attacker-chosen target class given an image or object. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In our evaluation, given a source object $x$ and a set of correct classes $\\{ y _ { 1 } , . . . y _ { n } \\}$ , as well as an attacker-chosen target class $y _ { a d v }$ and a crafted adversarial example $x ^ { \\prime }$ , we use the following terms in order to characterize the effectiveness of $x ^ { \\prime }$ . We call a viewpoint “adversarial” if the top output of the classifier is $y _ { a d v }$ ; “correct” if the adversarial example fails and the classifier outputs one of $\\{ y _ { 1 } , \\ldots , y _ { n } \\}$ , and “misclassified” if the classifier predicts a class unrelated to that of the source object (i.e. not in $\\{ y _ { 1 } , . . . y _ { n } \\} )$ but not the target class $y _ { a d v }$ . By randomly sampling a set of viewpoints, we can evaluate an adversarial example $x ^ { \\prime }$ by examining the proportion of adversarial, correct, and misclassified viewpoints. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We evaluate robust adversarial examples in the 2D and 3D case, and furthermore, we evaluate physical-world 3D adversarial objects. The two cases are fundamentally different. In the virtual case, we know that we want to construct adversarial examples robust over a certain distribution of transformations, and we can simply use EOT over that distribution to synthesize a robust adversarial example. In the case of the physical world, however, we cannot capture the exact distribution unless we perfectly model all physical phenomena. Therefore, we must approximate the distribution and perform EOT over the proxy distribution. This works well in practice for producing adversarial objects that remain adversarial under the “true” physical-world distribution, as we demonstrate. See Appendix A for the exact parameters of the distributions we use in the 2D, 3D simulation, and 3D physical-world cases. In all cases, the various transformation parameters are sampled as continuous random variables from a uniform distribution between the minimum and maximum values given, unless otherwise indicated in Appendix A (i.e. Gaussian noise). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In our experiments, we use TensorFlow’s standard pre-trained InceptionV3 classifier (Szegedy et al., 2015) which has $7 8 . 0 \\%$ top-1 accuracy on ImageNet. In all of our experiments, we use randomly chosen target classes, and we use EOT to synthesize adversarial examples over a chosen distribution. We measure the $\\ell _ { 2 }$ distance per pixel between the original and adversarial example (in LAB space), and we also measure classification accuracy (percent of randomly sampled viewpoints classified as the true class) and adversariality (percent of randomly sampled viewpoints classified as the adversarial class) for both the original and adversarial example. When working in simulation, we evaluate over a large number of transformations sampled randomly from the distribution; in the physical world, we evaluate over a large number of manually-captured images of our adversarial objects taken over different viewpoints. ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/6ed4064bdc1d4deba14427032e88507f29852cc75e767101687b19a8324778fe.jpg",
+        "table_caption": [
+            "Table 1: Evaluation of 1000 2D adversarial examples with random targets. We evaluate each example over 1000 randomly sampled transformations to calculate classification accuracy and adversariality (percent classified as the adversarial class). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Images</td><td colspan=\"2\">Classification Accuracy</td><td colspan=\"2\">Adversariality</td><td>l2</td></tr><tr><td>mean</td><td>stdev</td><td>mean</td><td>stdev</td><td>mean</td></tr><tr><td>Original</td><td>70.0%</td><td>36.4%</td><td>0.01%</td><td>0.3%</td><td>N/A</td></tr><tr><td>Adversarial</td><td>0.9%</td><td>2.0%</td><td>96.4%</td><td>4.4%</td><td>5.6 ×10-5</td></tr></table>",
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/8a1365085cfc8403c7fd2cf555137be9d735f49017aa2822c1b4193b33cc2c53.jpg",
+        "image_caption": [
+            "Figure 2: A 2D adversarial example, showing classifier confidence in true and adversarial classes for original and corresponding adversarial image over randomly sampled poses. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "3.1 ROBUST 2D ADVERSARIAL EXAMPLES ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In the 2D case, we consider the distribution of transformations that includes rescaling, rotation, lightening or darkening by an additive factor, adding Gaussian noise, and any in-bounds translation of the image. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We take the first 1000 images in the ImageNet validation set, randomly choose a target class for each image, and use EOT to synthesize an adversarial example that is robust over the chosen distribution. For each adversarial example, we evaluate over 1000 random transformations sampled from the distribution at evaluation time. Table 1 summarizes the results. On average, the adversarial examples have an adversariality of $9 6 . 4 \\%$ , showing that our approach is highly effective in producing robust adversarial examples. Figure 2 shows an example of a synthesized adversarial example, along with the classification confidence in true and adversarial classes for original and corresponding adversarial images. See Appendix B for more examples. ",
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+    {
+        "type": "text",
+        "text": "3.2 ROBUST 3D ADVERSARIAL EXAMPLES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We produce 3D adversarial examples by modeling the 3D rendering as a transformation under EOT. Given a textured 3D object, we optimize over the texture such that the rendering is adversarial from any viewpoint. We consider a distribution that incorporates different camera distances, lateral translation, rotation of the object, and solid background colors. We approximate the expectation over transformation by taking the mean loss over batches of size 40; furthermore, due to the computational expense of computing a batch, we reuse up to $80 \\%$ of the batch at each iteration, but enforce that each batch contain at least 8 new images. As previously mentioned, the parameters of the distribution we use is specified in Appendix A, sampled as independent continuous random variables (that are uniform except for Gaussian noise). ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/688baa71546e46332901536953bcde0916ac2bde68e5e267bb8e9d19cdbd409c.jpg",
+        "table_caption": [
+            "Table 2: Evaluation of 200 3D adversarial examples with random targets. We evaluate each example over 100 randomly sampled poses to calculate classification accuracy and adversariality (percent classified as the adversarial class). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Images</td><td colspan=\"2\">Classification Accuracy</td><td colspan=\"2\">Adversariality</td><td>l</td></tr><tr><td>mean</td><td>stdev</td><td>mean</td><td>stdev</td><td>mean</td></tr><tr><td>Original</td><td>68.8%</td><td>31.2%</td><td>0.01%</td><td>0.1%</td><td>N/A</td></tr><tr><td>Adversarial</td><td>1.1%</td><td>3.1%</td><td>83.4%</td><td>21.7%</td><td>5.9 ×10-3</td></tr></table>",
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+        "type": "image",
+        "img_path": "images/1cf2978cb6d307e968db9ecc20ecdbb43bcd09fc113e3588f289f279b20e61d3.jpg",
+        "image_caption": [
+            "Figure 3: A 3D adversarial example, showing classifier confidence in true and adversarial classes for original and corresponding adversarial object over randomly sampled poses. "
+        ],
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "We consider 10 complex 3D models, choose 20 random target classes per 3D model, and use EOT to synthesize adversarial textures for the 3D models with minimal parameter search (four constant, pre-chosen $\\lambda$ values were tested across each [3D model, target] pair). For each of the 200 adversarial examples, we sample 100 random transformations from the distribution at evaluation time. Table 2 summarizes results, and Figure 3 shows renderings of drawn samples with classification probabilities. See Appendix C for more examples. ",
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+    {
+        "type": "text",
+        "text": "The simulated adversarial object have an average adversariality of $8 3 . 4 \\%$ with a long left tail, showing that EOT usually produces highly adversarial objects. See Appendix C for a plot of the distribution. ",
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+    {
+        "type": "text",
+        "text": "3.3 PHYSICAL ADVERSARIAL EXAMPLES ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In order to fabricate physical-world adversarial examples, beyond modeling the 3D rendering process, we need to model physical-world phenomena such as lighting effects and camera noise. Furthermore, we need to model the 3D printing process: in our case, we use commercially available full-color 3D printing. With the 3D printing technology we use, we find that color accuracy varies between prints, so we model printing errors as well. We approximate all of these phenomena by a distribution of transformations under EOT. In addition to the transformations considered for 3D in simulation, we consider camera noise, additive and multiplicative lighting, and per-channel color inaccuracies. ",
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+    {
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+        "img_path": "images/0ce2a65d6a0c86019a0fd73535135727a2d3d4663220c6b525796904fe6f4d8d.jpg",
+        "image_caption": [
+            "Figure 4: A sample of photos of unperturbed 3D prints. The unperturbed 3D-printed objects are consistently classified as the true class. "
+        ],
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+    {
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+        "table_body": "<table><tr><td>Object</td><td>Adversarial</td><td>Misclassified</td><td>Correct</td></tr><tr><td>Turtle</td><td>82%</td><td>16%</td><td>2%</td></tr><tr><td>Baseball</td><td>59%</td><td>31%</td><td>10%</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 3: Quantitative analysis of the two adversarial objects, over 100 photos of each object over a wide distribution of viewpoints. Both objects are classified as the adversarial target class in the majority of viewpoints. ",
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+    {
+        "type": "text",
+        "text": "We evaluate physical adversarial examples over two 3D-printed objets: one of a turtle (where we consider any of the 5 turtle classes in ImageNet as the “true” class), and one of a baseball. The unperturbed 3D-printed objects are correctly classified as the true class with $100 \\%$ accuracy over a large number of samples. Figure 4 shows example photographs of unperturbed objects, along with their classifications. ",
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+    {
+        "type": "text",
+        "text": "We choose target classes for each of the 3D models at random — “rifle” for the turtle, and “espresso” for the baseball — and we use EOT to synthesize adversarial examples. We evaluate the performance of our two 3D-printed adversarial objects by taking 100 photos of each object over a variety of viewpoints1. Figure 5 shows a random sample of these images, along with their classifications. Table 3 gives a quantitative analysis over all images, showing that our 3D-printed adversarial objects are strongly adversarial over a wide distribution of transformations. See Appendix D for more examples. ",
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+    {
+        "type": "text",
+        "text": "3.4 DISCUSSION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The results and quantative analysis in this section demonstrate the efficacy of EOT and confirm the existence of physical adversarial examples. Here, we perform a qualitative analysis of the results: ",
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+    {
+        "type": "text",
+        "text": "Modeling Perception. The EOT algorithm as presented in Section 2 presents a general method to construct adversarial examples over a chosen perceptual distribution, but notably gives no guarantees for observations of the image outside of the chosen distribution. In constructing physical-world adversarial objects, we use a crude, high-variance approximation of the rendering and capture process, and this succeeds in ensuring robustness to a diverse set of environments; see, for example, Figure 6, which shows the same adversarial turtle in vastly different environments. In specialized domains, however, a domain expert may opt to model the perceptual distribution precisely in order to better constrain the search space. ",
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+        "img_path": "images/2d838726c5c8ee8f55c20ec69eb7940cf0838e9a4739614e4b6be8128e2a551c.jpg",
+        "image_caption": [
+            "Figure 5: Random sample of photographs of the two 3D-printed adversarial objects. The 3Dprinted adversarial objects are strongly adversarial over a wide distribution of viewpoints. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
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+    {
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+        "img_path": "images/5868a039d1fd427be0e20aaccff39f6ed047d81f56294c207a550fa80938aaca.jpg",
+        "image_caption": [
+            "Figure 6: Three pictures of the same adversarial turtle (all classified as “rifle”), demonstrating the need for a wide distribution, and the efficacy of EOT in finding examples robust across wide distributions of physical-world effects like lighting. "
+        ],
+        "image_footnote": [],
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/7680637ccd673478b874fc51b0f957fc4bba607403fde58f404df0cfd76e6021.jpg",
+        "image_caption": [
+            "Figure 7: A side-by-side comparison of a 3D-printed model (left) along with a printout of the corresponding texture, printed on a standard laser color printer (center) and the original digital texture (right), showing significant error in color accuracy in printing. "
+        ],
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+        "bbox": [
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+    {
+        "type": "text",
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+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Error in Printing. We also find significant error in the color accuracy of even state of the art commercially available color 3D printing; Figure 7 shows a comparison of a 3D-printed model along with a printout of the model’s texture, printed on a standard laser color printer. Still, EOT was able to overcome the problem and produce robust physical-world adversarial objects. We predict that we could have produced adversarial examples with smaller $\\ell _ { 2 }$ perturbation with a higher-fidelity printing process. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Semantically Relevant Misclassification. Interestingly, for the majority of viewpoints where the adversarial target class is not the top-1 predicted class, the classifier also fails to correctly predict the source class. Instead, we find that the classifier often classifies the object as an object that is semantically similar to the adversarial target; while generating the adversarial turtle to be classified as a rifle, for example, the second most popular class (after “rifle”) was “revolver,” followed by “holster” and then “assault rifle.” Similarly, when generating the baseball to be classified as an espresso, the example was often classified as “coffee” or “bakery”. ",
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+    {
+        "type": "text",
+        "text": "4 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "State of the art neural networks are vulnerable to adversarial examples (Szegedy et al., 2013). Researchers have proposed a number of methods for synthesizing adversarial examples in the whitebox scenario (with access to the gradient of the classifier), including L-BFGS (Szegedy et al., 2013), the Fast Gradient Sign Method (FGSM) (Goodfellow et al., 2015), and a Lagrangian relaxation formulation (Carlini & Wagner, 2017), all for the single-viewpoint case where the adversary directly controls the input to the neural network. Projected Gradient Descent (PGD) can be seen as a universal first-order adversary (Madry et al., 2017). ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Moosavi-Dezfooli et al. (2017) show the existence of universal (image-agnostic) adversarial perturbations, small perturbation vectors that can be applied to any image to induce misclassification. Their paper proposes an algorithm that finds perturbations that are universal over images; in our work, we give an algorithm that finds a perturbation to a single image or object that is universal over a chosen distribution of transformations. In preliminary experiments, we found that universal adversarial perturbations, like standard adversarial perturbations to single images, are not inherently robust to transformation. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In the first work on 2D physical-world adversarial examples, Kurakin et al. (2016) demonstrate the transferability of FGSM-generated adversarial misclassification on a printed page. In their setup, a photo is taken of a printed image with QR code guides, and the resultant image is warped, cropped, and resized to become a square of the same size as the source image before classifying it. Their results show the existence of 2D physical-world adversarial examples for approximately axis-aligned views, demonstrating that adversarial perturbations produced using FGSM can translate to the physical world and are robust to camera noise, rescaling, and lighting effects. Kurakin et al. (2016) do not synthesize targeted physical-world adversarial examples, they do not evaluate other real-world 2D transformations such as rotation, skew, translation, or zoom, and their approach does not translate to the 3D case. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Sharif et al. (2016) develop a real-world adversarial attack on a state-of-the-art face recognition algorithm, where adversarial eyeglass frames cause targeted misclassification in portrait photos. The algorithm produces robust perturbations through optimizing over a fixed set of inputs: the attacker collects a set of images and finds a perturbation that minimizes cross entropy loss over the set. The algorithm solves a different problem than we do in our work: it produces adversarial perturbations universal over portrait photos taken head-on from a single viewpoint, while EOT produces 2D/3D adversarial examples robust over transformations. The approach also includes a mechanism for enhancing perturbations’ printability using a color map to address the limited color gamut and color inaccuracy of the printer. Note that this differs from our approach in achieving printability: rather than creating a color map, we find an adversarial example that is robust to color inaccuracy. Our approach has the advantage of working in settings where color accuracy varies between prints, as was the case with our 3D-printer. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Recently, Evtimov et al. (2017) proposed a method for generating robust physical-world adversarial examples in the 2D case by optimizing over a fixed set of manually-captured images. However, the approach is limited to the 2D case, with no clear translation to 3D, where there is no simple mapping between what the adversary controls (the texture) and the observed input to the classifier (an image). Furthermore, the approach requires the taking and preprocessing of a large number of photos in order to produce each adversarial example, which may be expensive or even infeasible for many objects. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Lu et al. (2017) argued that adversarial examples may not be a practical concern in physical-world systems because adversarial examples generated for the single viewpoint case lose adversariality at viewpoints with differing scale and rotation. While this is the case with standard adversarial examples as evaluated in the paper and in Luo et al. (2016), we have shown that with EOT, it is in fact possible to construct physical-world adversarial images and objects that are classified as a chosen target class over a wide range of viewpoints. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            667,
+            318,
+            683
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Our work shows that adversarial examples pose a practical threat to systems using neural networkbased image classifiers. By introducing EOT, a general-purpose algorithm for creating robust adversarial examples under any chosen distribution, and modeling 3D rendering and printing within the framework of EOT, we succeed in fabricating three-dimensional adversarial objects. With access only to low-cost commercially available 3D printing technology, we successfully print physical adversarial objects that are strongly classified as a chosen target class over a variety of angles, viewpoints, and lighting conditions by a standard ImageNet classifier. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            819,
+            285,
+            833
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
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+        "bbox": [
+            174,
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+    },
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+        "type": "text",
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+    },
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+    {
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+        "bbox": [
+            171,
+            516,
+            823,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/9af007c8eb8a825902b0157af886e7166f6386a70f020a0a457dc3ca4e09073e.jpg",
+        "table_caption": [
+            "Table 4: Distribution of transformations for the 2D case, where each parameter is sampled uniformly at random from the specified range. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Transformation</td><td>Minimum</td><td>Maximum</td></tr><tr><td>Scale</td><td>0.9</td><td>1.4</td></tr><tr><td>Rotation</td><td>-22.5°</td><td>22.5°</td></tr><tr><td>Lighten /Darken</td><td>-0.05</td><td>0.05</td></tr><tr><td>Gaussian Noise (stdev)</td><td>0.0</td><td>0.1</td></tr><tr><td>Translation</td><td colspan=\"2\">any in-bounds</td></tr></table>",
+        "bbox": [
+            320,
+            101,
+            676,
+            200
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/7e06c801799777460e530df13b67db30fa694732e4768c870b632a8f33debafd.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Transformation</td><td>Minimum</td><td>Maximum</td></tr><tr><td>Camera distance</td><td>2.5</td><td>3.0</td></tr><tr><td>X/Y translation</td><td>-0.05</td><td>0.05</td></tr><tr><td>Rotation</td><td>any</td><td></td></tr><tr><td>Background</td><td>(0.1, 0.1, 0.1)</td><td>(1.0, 1.0, 1.0)</td></tr></table>",
+        "bbox": [
+            321,
+            258,
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+            345
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Table 5: Distribution of transformations for the 3D case when working in simulation, where each parameter is sampled uniformly at random from the specified range. ",
+        "bbox": [
+            171,
+            356,
+            825,
+            385
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A DISTRIBUTIONS OF TRANSFORMATIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            416,
+            544,
+            433
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Under the EOT framework, we must choose a distribution of transformations, and the optimization produces an adversarial example that is robust under the distribution of transformations. Here, we give the specific parameters we chose in the 2D (Table 4), 3D (Table 5), and physical-world case (Table 6). ",
+        "bbox": [
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+            508
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B ROBUST 2D ADVERSARIAL EXAMPLES ",
+        "text_level": 1,
+        "bbox": [
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+            536,
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+            554
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We give a random sample out of our 1000 2D adversarial examples in Figures 8 and 9. ",
+        "bbox": [
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+            589
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "C ROBUST 3D ADVERSARIAL EXAMPLES ",
+        "text_level": 1,
+        "bbox": [
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+            633
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We give a random sample out of our 200 3D adversarial examples in Figures 10 and 11 and 12. We give a histogram of adversariality (percent classified as the adversarial class) over all 200 examples in Figure 13. ",
+        "bbox": [
+            176,
+            654,
+            825,
+            696
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/c8b0712f4c0d656a72060882cc4cc7afc26eec7182cc14a9e6282e2def3dba51.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 6: Distribution of transformations for the physical-world 3D case, approximating rendering, physical-world phenomena, and printing error. "
+        ],
+        "table_body": "<table><tr><td>Transformation</td><td>Minimum</td><td>Maximum</td></tr><tr><td>Camera distance</td><td>2.5</td><td>3.0</td></tr><tr><td>X/Y translation</td><td>-0.05</td><td>0.05</td></tr><tr><td>Rotation</td><td>any</td><td></td></tr><tr><td>Background</td><td>(0.1, 0.1, 0.1)</td><td>(1.0, 1.0, 1.0)</td></tr><tr><td>Lighten /Darken (additive)</td><td>-0.15</td><td>0.15</td></tr><tr><td>Lighten /Darken (multiplicative)</td><td>0.5</td><td>2.0</td></tr><tr><td>Per-channel (additive)</td><td>-0.15</td><td>0.15</td></tr><tr><td>Per-channel (multiplicative)</td><td>0.7</td><td>1.3</td></tr><tr><td>Gaussian Noise (stdev)</td><td>0.0</td><td>0.1</td></tr></table>",
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+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/b083ea72bfac9f4868888f30c3c69dd4a49ec8ff2c8e79b652ea528b891770cb.jpg",
+        "image_caption": [
+            "Figure 8: A random sample of 2D adversarial examples. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/f0f817fd8983b1e1d4054c974e670dd80c5166c6dac2cf13472fe01de10cd927.jpg",
+        "image_caption": [
+            "Figure 9: A random sample of 2D adversarial examples. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 14
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+    {
+        "type": "image",
+        "img_path": "images/c6f2063325c3f07b635032ee0c16b39697e906d77447ea490acbee04d5767b64.jpg",
+        "image_caption": [],
+        "image_footnote": [],
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "P (true): 96% P (adv): 0% ",
+        "bbox": [
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+            219,
+            429,
+            247
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "P (true): 99% P (adv): 0% ",
+        "bbox": [
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+            247
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/4cf3233d10102773051dc716a7368d5ab8850e906521a1b690ab87971dd83841.jpg",
+        "image_caption": [],
+        "image_footnote": [],
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/7515ed6a492794f537c1cc35e8dd4a29b2a68b15dabe753c0156aa14106740a7.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "P (true): 96% P (adv): 0% ",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "P (true): 97% P (adv): 0% ",
+        "bbox": [
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+            219,
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+            247
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Adversarial:   \nguillotine   \n\\`2 = 3.1 × 10−5 ",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/54725ecf42ea0dc7381f3cf3c3f859630c7958461ff2b8481e04ec046fcab82e.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "P (true): 1% P (adv): 10% ",
+        "bbox": [
+            334,
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+            426,
+            363
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/8fd61dc806d9100aa12ab70aa0390d56a6b914ea4e50edcd9b099b652ae752cb.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "P (true): 0% P (adv): 95% ",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/daf5e022f68af61c84a6bccf67e1347bf62cd13906f887400733c8b202810466.jpg",
+        "image_caption": [],
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+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "P (true): 0% P (adv): 91% ",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/474cab26c2ad858b19ed19a9311ee4b9ffcbfd7636be1d0624df670ca2b536fe.jpg",
+        "image_caption": [],
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+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "P (true): 3% P (adv): 4% ",
+        "bbox": [
+            697,
+            335,
+            787,
+            363
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Original: baseball ",
+        "bbox": [
+            222,
+            404,
+            282,
+            431
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "P (true): 100% P (adv): 0% ",
+        "bbox": [
+            338,
+            450,
+            423,
+            492
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/446ebf8f295c43d48bb077f813088cceaa533e663bb7e9d921c7cbd3a4a523e8.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+            552,
+            449
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "P (true): 100% P (adv): 0% ",
+        "bbox": [
+            459,
+            450,
+            544,
+            492
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/5801cd3b24e8979b4b93b46ce27a6964716fc3c454e51bf9ccf090205d092684.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "P (true): 100% P (adv): 0% ",
+        "bbox": [
+            580,
+            450,
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+            492
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/c1a6f0835e8c8f8a5c5fc2ccf9378302270b489b6e4166a1ab5a5864fe7f68ae.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "P (true): 100% P (adv): 0% ",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Adversarial:   \ngreen lizard   \n\\`2 = 2.9 × 10−5 ",
+        "bbox": [
+            196,
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+            308,
+            563
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/b8cae5f69364c82edf35080f1259de8117c87c2ebc06e60fe98d2323ade5ad3f.jpg",
+        "image_caption": [],
+        "image_footnote": [],
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/cd986a37a5e612d93931ce84585be956cec7ce04e63438e588371e7e2c12a2ee.jpg",
+        "image_caption": [
+            "Figure 10: A random sample of 3D adversarial examples. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            328,
+            371,
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+            449
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "P (true): 0% P (adv): 66% ",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/a3a7f87be5edcfd09f8959b9e465d1b7734d5164f2d7f81f8941ba5aa235682e.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+            500,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "P (true): 0% P (adv): 94% ",
+        "bbox": [
+            455,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/1a7be731f6e47133ef61ad9c4d7d80834dc5a12f838d2951ea6bdf17d3db6a9d.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "P (true): 0% P (adv): 87% ",
+        "bbox": [
+            575,
+            579,
+            668,
+            607
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/c8c03105f7c59ea309bbb24efb5bbbd48af3a326bc24fb8ba8108beb2c9fdee4.jpg",
+        "image_caption": [],
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "P (true): 0% P (adv): 94% ",
+        "bbox": [
+            696,
+            579,
+            789,
+            606
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/f5b8650edd0e56e908f26f6719451967643d51b73b5d96c42f3ec7114124cf94.jpg",
+        "image_caption": [],
+        "image_footnote": [],
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/47d685b13d576c782a76ea4310002346cc13e74471d60396a67bfb51531a72c1.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "P (true): 97% P (adv): 0% ",
+        "bbox": [
+            694,
+            694,
+            792,
+            722
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Adversarial:   \nBouvier des   \nFlandres   \n$\\ell _ { 2 } = 1 . 4 \\times 1 0 ^ { - 4 }$ ",
+        "bbox": [
+            196,
+            750,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Original: baseball ",
+        "bbox": [
+            222,
+            179,
+            282,
+            205
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/249278e25d05f089dd06880dd49751146a6947e241fbe364d0ccdb463743f917.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            328,
+            146,
+            431,
+            224
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "P (true): 100% $P ( a d v ) \\colon 0 \\%$ ",
+        "bbox": [
+            339,
+            226,
+            423,
+            266
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/66e03cb43930ef19e402f0f01e7787aadc51dc8239ed85223e3f8fbdc34dabd1.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+            223
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "P (true): 100% P (adv): 0% ",
+        "bbox": [
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+            226,
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+            267
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/6bb9dfef8cd85248a240d643c4b9500f131cac1df5328f3c08edc3f29e4fcbcf.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "P (true): 100% P (adv): 0% ",
+        "bbox": [
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+            226,
+            663,
+            267
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/e2bbfdc90bb9e85b72a9335652ad544ee2642d4fdf0ec0b23d09248c608a7584.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "P (true): 100% P (adv): 0% ",
+        "bbox": [
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+            226,
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Adversarial:   \nAiredale   \n\\`2 = 3.4 × 10−5 ",
+        "bbox": [
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+            337
+        ],
+        "page_idx": 15
+    },
+    {
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+        "img_path": "images/95900f3926ccab52c23ca303bc16c1eed02813c826f1675f2d45b02624161e6e.jpg",
+        "image_caption": [],
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "P (true): $0 \\%$ $P ( a d v ) \\colon 9 4 \\%$ ",
+        "bbox": [
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+            381
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/a05c59801466e8c0cf5c36693564fe27616f7a486313b5569b8c4811ee21279b.jpg",
+        "image_caption": [],
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+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "P (true): 0% P (adv): 6% ",
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
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+        "img_path": "images/0d85b317da683e42fe93849c0db9cc79d3ea2e8124cdb3d10ed17f8df82cb1b4.jpg",
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+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "P (true): 0% P (adv): 96% ",
+        "bbox": [
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+            381
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/239fdb3245349dc1af9e16b14f520ec7678daa9356f3d0edf4bd37ee369a35b9.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+            352
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "P (true): 0% P (adv): 18% ",
+        "bbox": [
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+            354,
+            789,
+            381
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Original: orange ",
+        "bbox": [
+            194,
+            424,
+            307,
+            438
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "P (true): 73% $\\dot { P } ( a d v ) \\colon 0 \\%$ ",
+        "bbox": [
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+            469,
+            429,
+            497
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/3b9c622a8360406355546717b32d8259f72994852b21e2ffe08b5133db3996a9.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+            390,
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+            467
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "P (true): 29% P (adv): 0% ",
+        "bbox": [
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+            469,
+            550,
+            497
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/6a4c9beddfe761c6bba3f18f1abfcc0ca223dfda977436934cd48abd07d7fd6a.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "P (true): 20% P (adv): 0% ",
+        "bbox": [
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+            469,
+            671,
+            497
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/d5d1abe73147d2d10dbda4e5624b9e7dfb7a5e1b5ee787d4a091463cba747ab1.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            691,
+            390,
+            794,
+            468
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "P (true): 85% P (adv): 0% ",
+        "bbox": [
+            694,
+            469,
+            790,
+            497
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Adversarial: power drill $\\ell _ { 2 } = 4 . 3 \\times 1 0 ^ { - 5 }$ ",
+        "bbox": [
+            196,
+            526,
+            308,
+            568
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/eb1b9d31583bd8c3c01ea86cabd1a3d1da86a0485b68f02f976c61a975132d00.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            330,
+            506,
+            429,
+            579
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "P (true): 0% $P ( a d v ) \\colon 8 9 \\%$ ",
+        "bbox": [
+            334,
+            584,
+            426,
+            612
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/ceefc8ab855bea62d9afd910734084e0186d0b5f35c6b4da4978d71ef2e89ad4.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+            582
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/3c3fb815e0066ce35a615779596beb636e43cd985cab46489e5d91c4606ba99c.jpg",
+        "image_caption": [
+            "Figure 11: A random sample of 3D adversarial examples. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            330,
+            390,
+            429,
+            463
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "P (true): 4% P (adv): 75% ",
+        "bbox": [
+            455,
+            584,
+            547,
+            612
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/bcbc07463244d7b2b5fbe8bf013b56b2f5a4d250075fc2233ceb8817d2eeba77.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            571,
+            505,
+            671,
+            582
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "P (true): 0% P (adv): 98% ",
+        "bbox": [
+            576,
+            584,
+            668,
+            612
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/54f666abab66b32348d729e33d0dc873dc9ab5c0c9b70e37eab3f9f8e64229a6.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            691,
+            505,
+            794,
+            583
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "P (true): 0% P (adv): 84% ",
+        "bbox": [
+            696,
+            584,
+            789,
+            612
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/cb48e24288fd98194840193196c2f1c91cbe041c05552ab90195f16dc5eda6aa.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+            685
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/0ad0ed609336cdf3edbeb9c8b639044946d9a6eadc326762b7b9144d498e8340.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+            621,
+            552,
+            698
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/23c084bba5647d1f819303736d3f648ac671523d7199dd941ae192d5489896ea.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/f2cfff3e1d998a748bb6f9050dca68c368fbf8b5cf7c0ecd36f76daec2047b0d.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+            621,
+            794,
+            698
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "P (true): 0% P (adv): 0% ",
+        "bbox": [
+            697,
+            699,
+            787,
+            727
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Adversarial:   \nbittern   \n$\\ell _ { 2 } = 7 . 4 \\times 1 0 ^ { - 5 }$ ",
+        "bbox": [
+            196,
+            757,
+            308,
+            797
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "P (true): 0% P (adv): 97% ",
+        "bbox": [
+            696,
+            815,
+            789,
+            842
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Original: teddybear ",
+        "bbox": [
+            217,
+            179,
+            284,
+            207
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/6ad66586a045e8d66368d2977c1e998f6406417daf8d764b80c22f97afb000ea.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            330,
+            154,
+            431,
+            231
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "P (true): 90% $P ( a d v ) \\colon 0 \\%$ ",
+        "bbox": [
+            331,
+            232,
+            429,
+            260
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/30755d556abc6422ec694c4576b70359e6018f4f9a724cabbdeb85c9bb4875a3.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            450,
+            154,
+            552,
+            231
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "P (true): 1% P (adv): 0% ",
+        "bbox": [
+            457,
+            232,
+            547,
+            260
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/5851e0b808bde9a1bf46c0794c0cf4b9232eb22a41c478dec454f86a98f6ab58.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            571,
+            154,
+            671,
+            229
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "P (true): 98% P (adv): 0% ",
+        "bbox": [
+            573,
+            232,
+            671,
+            260
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/1fe7cda29d006d909f7781f949ce04c9a7a60aaf3ecd6a7eac67e7fe96129995.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            692,
+            154,
+            794,
+            229
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "P (true): 5% P (adv): 0% ",
+        "bbox": [
+            697,
+            232,
+            789,
+            260
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Adversarial: sock $\\ell _ { 2 } = 3 . 3 \\times 1 0 ^ { - 5 }$ ",
+        "bbox": [
+            196,
+            289,
+            310,
+            330
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/a9816a9b180b8f7bef67cbba62b49ec4d4cd3d317f357ede8d4343f8a39b0741.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            330,
+            268,
+            429,
+            344
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "P (true): 0% P (adv): 99% ",
+        "bbox": [
+            333,
+            347,
+            426,
+            375
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/c5fb2c2baa6b7cabfa3e65c92cc6bb38be0bd7020c3189bb36ee515c02bca7f6.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            450,
+            268,
+            552,
+            345
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "P (true): 0% P (adv): 99% ",
+        "bbox": [
+            455,
+            347,
+            549,
+            375
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/586093914075b318ca8bb6e5b7a6291befd2e976ad524a85192839ae0e695bed.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            571,
+            268,
+            671,
+            344
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "P (true): 0% P (adv): 98% ",
+        "bbox": [
+            575,
+            347,
+            668,
+            375
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/ebe47435f3be92f455951303cf89ec71f729d8c89a6e7f9af0411deb2620a93a.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            692,
+            268,
+            794,
+            344
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "P (true): 0% P (adv): 99% ",
+        "bbox": [
+            696,
+            347,
+            789,
+            375
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Original: clownfish ",
+        "bbox": [
+            218,
+            409,
+            284,
+            436
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/ac2dd4c42b02bc94fdb060c2e3d702c62369d6b0fbadc4d7f63e446d6b0a874c.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            330,
+            383,
+            431,
+            462
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "P (true): 46% P (adv): 0% ",
+        "bbox": [
+            333,
+            463,
+            429,
+            489
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/abd96785a2d7187f97cbb07657d511dcec4010242e79c96ce9edebceee8967bb.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            450,
+            383,
+            552,
+            460
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "P (true): 14% P (adv): 0% ",
+        "bbox": [
+            452,
+            463,
+            550,
+            489
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/347b879d28ef7690e64636129cc7eff717d83136b182259b8cb02ff40871203b.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            571,
+            383,
+            671,
+            460
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "P (true): 2% P (adv): 0% ",
+        "bbox": [
+            578,
+            462,
+            666,
+            489
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/65db67d6cbe560af1e9677e0329a5578b1b09ba62293dcd8f8654a46d51f6db1.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            691,
+            383,
+            794,
+            462
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "P (true): 65% P (adv): 0% ",
+        "bbox": [
+            694,
+            462,
+            792,
+            489
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Adversarial:   \npanpipe   \n\\`2 = 4.8 × 10−5 ",
+        "bbox": [
+            196,
+            520,
+            310,
+            560
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/e87b26949fe94ff2c31df5ba8cbd5a454d8160a7139bda171c60cb616dac6778.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            330,
+            498,
+            431,
+            575
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "P (true): 0% P (adv): 100% ",
+        "bbox": [
+            331,
+            577,
+            429,
+            606
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/bbb9c568b2e2c1cbbc549f938a340227515e212830866283b2ef9b4fb7fd50c4.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            450,
+            498,
+            552,
+            575
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "P (true): 0% $P ( a d v ) \\colon 1 \\%$ ",
+        "bbox": [
+            455,
+            577,
+            545,
+            606
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/5d9935f54c75a0b8412dbf0ec7991dd36a9c4c4d61513020c87ffd4aca2658d6.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            571,
+            497,
+            671,
+            575
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "P (true): 0% P (adv): 12% ",
+        "bbox": [
+            575,
+            577,
+            668,
+            604
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/1f1bb7351508549b5ae2f69e4c29d4b3d9746bc865ffdd8c1868b987d04b8aef.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            691,
+            497,
+            794,
+            575
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "P (true): 0% P (adv): 0% ",
+        "bbox": [
+            697,
+            577,
+            789,
+            604
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/c29b80e8cfe88b28fff222bcbc5ed1d58a95ac9f5218e28749e09c9d7b97a622.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            330,
+            612,
+            431,
+            690
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/0c9b9888052e6fae14934e6ffad213cbf504db46a4fba1429e7ad6e40c146804.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            450,
+            613,
+            552,
+            690
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/0331f9e468f15008972fc47f1d99e1c16a46cf3d2510be331bec5a96f4b04fa8.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            570,
+            613,
+            671,
+            691
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/9d96fa85337f06f1b358e57888978077a76cd1babff1c801deb58405babbf030.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            691,
+            613,
+            794,
+            690
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "P (true): $70 \\%$ P (adv): $0 \\%$ ",
+        "bbox": [
+            694,
+            693,
+            792,
+            719
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Adversarial:   \nsturgeon   \n\\`2 = 1.3 × 10−5 ",
+        "bbox": [
+            196,
+            750,
+            308,
+            791
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/6514990055938b032cbe2d054224c99a04b627fe03e7037e27f07c7cd4cefd7c.jpg",
+        "image_caption": [
+            "Figure 12: A random sample of 3D adversarial examples. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            691,
+            728,
+            794,
+            805
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/8eaea64977a95f871993846895af7ee86e7ffd8785da694f9017c5519ff5fc86.jpg",
+        "image_caption": [
+            "Figure 13: A histogram of adversariality (percent of 100 samples classified as the adversarial class) across the 200 3D adversarial examples. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            245,
+            138,
+            735,
+            436
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "D PHYSICAL ADVERSARIAL EXAMPLES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            507,
+            522,
+            523
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Figure 14 gives all 100 photographs of our adversarial 3D-printed turtle, and Figure 15 gives all 100 photographs of our adversarial 3D-printed baseball. ",
+        "bbox": [
+            171,
+            539,
+            826,
+            568
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/f9a72c737c6938de7d0ea9e2da871aa2b1b23446360d174129041b1c8e16bf96.jpg",
+        "image_caption": [
+            "Figure 14: All 100 photographs of our physical-world 3D adversarial turtle. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            236,
+            820,
+            762
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/a3b9d7a1a22a885462f5210bd4bb0261e6148a4e194778f64bc986e7c7483fca.jpg",
+        "image_caption": [
+            "Figure 15: All 100 photographs of our physical-world 3D adversarial baseball. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            236,
+            820,
+            762
+        ],
+        "page_idx": 19
+    }
+]

parse/train/Bklzkh0qFm/Bklzkh0qFm.md

@@ -0,0 +1,382 @@
+# Relational Graph Attention Networks
+
+Anonymous authors Paper under double-blind review
+
+# Abstract
+
+We investigate Relational Graph Attention Networks, a class of models that extends non-relational graph attention mechanisms to incorporate relational information, opening up these methods to a wider variety of problems. A thorough evaluation of these models is performed, and comparisons are made against established benchmarks. To provide a meaningful comparison, we retrain Relational Graph Convolutional Networks, the spectral counterpart of Relational Graph Attention Networks, and evaluate them under the same conditions. We find that Relational Graph Attention Networks perform worse than anticipated, although some configurations are marginally beneficial for modelling molecular properties. We provide insights as to why this may be, and suggest both modifications to evaluation strategies, as well as directions to investigate for future work.
+
+# 1 Introduction
+
+Convolutional Neural Networks (CNNs) successfully solve a variety of tasks in Euclidean grid-like domains, such as image captioning (Donahue et al., 2017) and classifying videos (Karpathy et al., 2014). CNNs are successful because they assume the data is locally stationary and compositional (Defferrard et al., 2016; Henaff et al., 2015; Bruna et al., 2013).
+
+However, data often occurs in the form of graphs or manifolds, which are classic examples of non-Euclidean domains. Specific instances include knowledge bases, molecules, and point clouds captured by 3D data acquisition devices (Wang et al., 2018). The generalisation of Neural Networks (NNs) to non-Euclidean domains is termed Geometric Deep Learning (GDL), and may be roughly divided into spectral, spatial and hybrid approaches (Bronstein et al., 2017).
+
+Spectral approaches (Defferrard et al., 2016), most notably Graph Convolutional Networks (GCNs) (Kipf and Welling, 2016), are limited by their basis-dependence. A filter that is learned with respect to a basis on one domain is not guaranteed to behave similarly when applied to another basis and domain. Spatial approaches are limited by an absence of shift invariance and lack of coordinate system (Duvenaud et al., 2015; Atwood and Towsley, 2016; Monti et al., 2017). Hybrid approaches combine spectral and spatial approaches, trading their advantages and deficiencies against each-other (Bronstein et al., 2017; Rustamov and Guibas, 2013; Szlam et al., 2005; Gavish et al., 2010).
+
+A recent approach that began with Graph Attention Networks (GATs), applied attention mechanisms to graphs, and does not share these limitations (Veličković et al., 2017; Gong and Cheng, 2018; Zhang et al., 2018; Monti et al., 2018; Lee et al., 2018).
+
+An alternative direction has been to generalise Recurrent Neural Networks (RNNs) from sequential message passing on one-dimensional signals, to message passing on graphs (Sperduti and Starita, 1997; Frasconi et al., 1997; Gori et al., 2005). Incorporating gating mechanisms led to the development of Gated Graph Neural Networks (GGNNs) (Scarselli et al., 2009; Allamanis et al., 2017)1.
+
+Relational Graph Convolutional Networks (RGCNs) have been proposed as an extension of GCNs to the domain of relational graphs (Schlichtkrull et al., 2018). This model has achieved impressive performance on node classification and link prediction tasks, however, its mechanisms still resides within spectral methods and shares their deficiencies. The focus of this work investigate generalisations of RGCN away from its spectral origins.
+
+We take RGCN as a starting point, and investigate a class of models we term Relational Graph Attention Networks (RGATs), extending attention mechanisms to the relational graph domain. We consider two variants, Within-Relation Graph Attention (WIRGAT) and Across-Relation Graph Attention (ARGAT), each with either additive or multiplicative attention. We perform an extensive hyperparameter search, and evaluate these models on challenging transductive node classification and inductive graph classification tasks. These models are compared against established benchmarks, as well as a re-tuned RGCN model.
+
+We show that RGAT performs worse than expected, although some configurations produce marginal benefits on inductive graph classification tasks. In order to aid further investigation in this direction, we present the full Cumulative Distribution Functions (CDFs) for the hyperparameter searches in Appendix D, and statistical hypothesis tests in Appendix E. We also provide a vectorised, sparse, batched implementation of RGAT and RGCN in TensorFlow which is compatible with eager execution mode to open up research into these models to a wider audience2.
+
+# 2 RGAT architecture
+
+# 2.1 Relational graph attention layer
+
+We follow the construction of the GAT layer in Veličković et al. (2017), extending to the relational setting, using ideas from Schlichtkrull et al. (2018).
+
+Layer input and output The input to the layer is a graph with $R = | \mathcal R |$ relation types and $N$ nodes. The $i ^ { \mathrm { t h } }$ node is represented by a feature vector $\boldsymbol { h } _ { i } \in \mathbb { R } ^ { F }$ , and the features of all nodes are summarised in the feature matrix ${ \pmb H } = [ { \pmb h } _ { 1 } { \pmb h } _ { 2 } \dots { \pmb h } _ { N } ] \in \mathbb { R } ^ { N \times F }$ . The output of the layer is the transformed feature matrix $\pmb { H } ^ { \prime } = [ h _ { 1 } ^ { \prime } h _ { 2 } ^ { \prime } \dots h _ { N } ^ { \prime } ] \in \mathbb { R } ^ { N \times F ^ { \prime } }$ , where $\pmb { h } _ { i } ^ { \prime } \in \mathbb { R } ^ { F ^ { \prime } }$ is the transformed feature vector of the $i ^ { \mathrm { t h } }$ node.
+
+Intermediate representations Different relations convey distinct pieces of information. The update rule of Schlichtkrull et al. (2018) made this manifest by assigning each node $_ i$ a distinct intermediate representation $\pmb { g } _ { i } ^ { ( r ) } \in \mathbb { R } ^ { F ^ { \prime } }$ under relation $r$
+
+$$
+\pmb { G } ^ { ( r ) } = \pmb { H } \pmb { W } ^ { ( r ) } \in \mathbb { R } ^ { N \times F ^ { \prime } } ,
+$$
+
+where $\pmb { G } ^ { ( r ) } = \left[ \pmb { g } _ { 1 } ^ { ( r ) } \pmb { g } _ { 2 } ^ { ( r ) } \cdot \cdot \cdot \pmb { g } _ { N } ^ { ( r ) } \right]$ is the intermediate representation feature matrix under relation $r$ , and $W ^ { ( r ) } \in \mathbb { R } ^ { F \times F ^ { \prime } }$ are the learnable parameters of a shared linear transformation.
+
+Logits Following Veličković et al. (2017); Zhang et al. (2018), we assume the attention coefficient between two nodes is based only on the features of those nodes up to a neighborhoodlevel normalisation. To keep computational complexity linear in $R$ , we assume that, given linear transformations $W ^ { ( r ) }$ , the logits $E _ { i , j } ^ { ( r ) }$ of each relation $r$ are independent
+
+$$
+E _ { i , j } ^ { ( r ) } = a \left( \pmb { g } _ { i } ^ { ( r ) } , \pmb { g } _ { j } ^ { ( r ) } \right) ,
+$$
+
+and indicate the importance of node ’s intermediate representation to that of node $_ i$ under relation $r$ . The attention is masked so that, for node $i$ , coefficients $\alpha _ { i , j } ^ { ( r ) }$ exist only for $j \in n _ { i } ^ { ( r ) }$ , where $n _ { i } ^ { ( r ) }$ denotes the set of neighbor indices of node $_ i$ under relation $r \in \mathcal R$ .
+
+Queries, keys and values The logits are composed from queries and keys, and specify how the values, i.e. the intermediate representations $\mathbf { \Delta } _ { g _ { i } ^ { ( r ) } }$ , will combine to produce the updated
+
+![](images/9a4bf818da78a1f6aaa34d99c50e0d9b06bf898c39cb24788316934d42cf5262.jpg)
+
+Figure 1: WIRGAT. The intermediate representations for node $i$ (left red rectangle) are combined with the intermediate representations for nodes in its neighborhood (blue rectangles) under each relation $r$ , to form each logit $E _ { i , j } ^ { ( r ) }$ . A softmax is taken over each logit matrix for each relation type to form the attention coefficients $\alpha _ { i , j } ^ { ( r ) }$ . These attention coefficients construct a weighted sum over the nodes in the neighborhood for each relation (black rectangle). These are then aggregated and passed through a nonlinearity to produce the updated representation for node $_ i$ (right red rectangle).
+
+key kernel node representations $K ^ { ( r ) } \in \mathbb { R } ^ { F ^ { \prime } \times D }$ $ { \boldsymbol { h } } _ { i } ^ { \prime }$ (Vaswani et al., 2017). A separate query kernel project the intermediate representations $\mathbf { \Delta } _ { g _ { i } ^ { ( r ) } }$ , into query and key $Q ^ { ( r ) } \in \mathbb { R } ^ { F ^ { \prime } \times D }$ and representations of dimensionality
+
+$$
+\pmb { q } _ { i } ^ { ( r ) } = \pmb { g } _ { i } ^ { ( r ) } \pmb { Q } ^ { ( r ) } \in \mathbb { R } ^ { D } , \qquad \pmb { k } _ { i } ^ { ( r ) } = \pmb { g } _ { i } ^ { ( r ) } \pmb { K } ^ { ( r ) } \in \mathbb { R } ^ { D } .
+$$
+
+For convenience, the query and key kernels are combined to form the attention kernels $\pmb { A } ^ { ( r ) } = \pmb { Q } ^ { ( r ) } \oplus \pmb { K } ^ { ( r ) } \in \mathbb { R } ^ { 2 F ^ { \prime } \times D }$ . These query and key representations are the building blocks of the two specific realisations of $a$ in Equation (2) that we now consider.
+
+Additive attention logits The first realisation of $a$ we consider is the relational modification of the logit mechanism of Veličković et al. (2017)
+
+$$
+\begin{array} { r } { \boldsymbol { E } _ { i , j } ^ { ( r ) } = \mathrm { L e a k y R e L u } \left( \boldsymbol { q } _ { i } ^ { ( r ) } + \boldsymbol { k } _ { j } ^ { ( r ) } \right) , } \end{array}
+$$
+
+where the query and key dimensionality are both $D = 1$ , and $q _ { i } ^ { ( r ) }$ and $k _ { i } ^ { ( r ) }$ are scalar flattenings of their one-dimensional vector counterparts $\pmb { q } _ { i } ^ { ( r ) } , \pmb { k } _ { i } ^ { ( r ) } \in \mathbb { R } ^ { 1 }$ . We refer to any instance of RGAT using logits of the form in Equation (4) as additive RGAT.
+
+Multiplicative attention logits The second realisation we consider is the multiplicative mechanism of Vaswani et al. (2017); Zhang et al. (2018)3
+
+$$
+E _ { i , j } ^ { ( r ) } = \pmb { q } _ { i } ^ { ( r ) } \cdot \pmb { k } _ { j } ^ { ( r ) } ,
+$$
+
+where the query and key dimensionality $D$ can be any positive integer. We refer to any instance of RGAT using logits of the form in Equation (4) as multiplicative RGAT.
+
+It should be noted that there are many types of attention mechanisms beyond vanilla additive and multiplicative. These include mechanisms leveraging the structure of the dual graph Monti et al. (2018) as well as learned edge features Gong and Cheng (2018).
+
+The attention coefficients should be comparable across nodes. This can be achieved by applying softmax appropriately to any logits $E _ { i , j } ^ { ( r ) }$ . We investigate two candidates, each
+
+![](images/ef3930c93ca203f3d08151a1206bea28eef4dd567e160e21b8eb98831fd1cf88.jpg)  
+Figure 2: ARGAT. The logits are produced identically to those in Figure 1. A softmax is taken across all logits independent of relation type to form the attention coefficients α(r)i,j . The remaining weighting and aggregation steps are the same as those in Figure 1.
+
+WIRGAT The simplest way to take the softmax over the logits $E _ { i , j } ^ { ( r ) }$ of Equation (4) or
+
+$$
+\alpha _ { i , j } ^ { ( r ) } = \operatorname * { s o f t m a x } _ { j } \left( E _ { i , j } ^ { ( r ) } \right) = \frac { \exp \left( E _ { i , j } ^ { ( r ) } \right) } { \sum _ { k \in { \cal { n } } _ { i } ^ { ( r ) } } \exp \left( E _ { i , k } ^ { ( r ) } \right) } , \qquad \forall i , r : \sum _ { j \in { \cal { n } } _ { i } ^ { ( r ) } } \alpha _ { i , j } ^ { ( r ) } = 1 .
+$$
+
+We call the attention in Equation (6) Within-Relation Graph Attention (WIRGAT), and it is shown in Figure 1. This mechanism encodes the prior that relation importance is a purely global property of the graph by implementing an independent probability distribution over nodesnodes hborhood of yield compe $i$ for each relation ing attention coe $r$ . Explcients y, forand node with $i$ and relation izes dependin $r$ , $j , k \in n _ { i } ^ { ( r ) }$ $\alpha _ { i , j } ^ { ( r ) }$ α(r)i,k on their corresponding representations $\mathbf { \Delta } _ { g _ { j } ^ { ( r ) } }$ and $\mathbf { \Delta } _ { g _ { k } ^ { ( r ) } }$ . There is no competition between any attention coefficients $\alpha _ { i , j } ^ { ( r ) }$ and α(r )i,k for all nodes $i$ and nodes $j \in n _ { i } ^ { ( r ) } , j ^ { \prime } \in n ^ { ( r ^ { \prime } ) }$ where $\boldsymbol { r } ^ { \prime } \neq \boldsymbol { r }$ irrespective of node representations.
+
+ARGAT An alternative way to take the softmax over the logit $E _ { i , j } ^ { ( r ) }$ of Equation (4) or $r$
+
+$$
+\alpha _ { i , j } ^ { ( r ) } = \operatorname * { s o f } _ { j , r } \mathrm { a x } \left( E _ { i , j } ^ { ( r ) } \right) = \frac { \exp \left( E _ { i , j } ^ { ( r ) } \right) } { \sum _ { r ^ { \prime } \in \mathcal { R } } \sum _ { k \in { n _ { i } ^ { ( r ^ { \prime } ) } } } \exp \left( E _ { i , k } ^ { ( r ^ { \prime } ) } \right) } , \forall i : \sum _ { r \in \mathcal { R } } \sum _ { j \in { n _ { i } ^ { ( r ) } } } \alpha _ { i , j } ^ { ( r ) } = 1 .
+$$
+
+We call the attention in Equation (7) Equation (6) Across-Relation Graph Attention (ARGAT), and it is shown in Figure 2. This mechanism encodes the prior that relation importance is a local property of the graph by implementing a single probability distribution over the different representations $\mathbf { \Delta } _ { g _ { j } ^ { ( r ) } }$ for nodes $\mathrm { j }$ in the neighborhood of node $i$ . Explicitly, for any node $i$ and all $r , r ^ { \prime } \in \mathcal { R }$ , all nodes $j \in n _ { i } ^ { ( r ) }$ and $k \in \mathcal { n } _ { i } ^ { ( r ^ { \prime } ) }$ yield competing attention coefficients α(r)i,j and α(r  )i,k with sizes depending on their corresponding representations $\mathbf { \Delta } _ { g _ { j } ^ { ( r ) } }$ and g(rk $\pmb { g } _ { k } ^ { ( r ^ { \prime } ) }$
+
+Comparison to RGCN For comparison, the coefficients of RGCN are given by α(r)i,j $| n _ { i } ^ { ( r ) } | ^ { - 1 }$ . This encodes the prior that the intermediate representations of nodes $j \in n _ { i } ^ { ( r ) }$ to node $i$
+
+![](images/b51b37fc3f64eff99f9add08eebbdb9785410870141a6ca7965e0d2334b14044.jpg)  
+Figure 3: (a) The network architecture used for node classification on AIFB and MUTAG. This architecture is the same as in Schlichtkrull et al. (2018) except with RGCNs replaced with RGATs. (b) The network architecture used for multi-task graph classification on Tox21. This architecture is the same as the GCNs architecture in Altae-Tran et al. (2016) except with RGCNs replaces with GATs and we do not use graph pooling.
+
+Propagation rule Combining the attention mechanism of either Equation (6) or Equation (7) with the neighborhood aggregation step of Schlichtkrull et al. (2018) gives
+
+$$
+\pmb { h } _ { i } ^ { \prime } = \sigma \left( \sum _ { r \in \mathcal { R } } \sum _ { j \in { \cal n } _ { i } ^ { ( r ) } } { \alpha } _ { i , j } ^ { ( r ) } \pmb { g } _ { j } ^ { ( r ) } \right) \in \mathbb { R } ^ { N \times F ^ { \prime } } ,
+$$
+
+where $\sigma$ represents an optional nonlinearity. Similar to Vaswani et al. (2017); Veličković et al. (2017), we also find that using multiple heads in the attention mechanism can enhance performance
+
+$$
+\pmb { h } _ { i } ^ { \prime } = \bigoplus _ { k = 1 } ^ { K } \sigma \left( \sum _ { r \in \mathcal { R } } \sum _ { j \in { \cal n } _ { i } ^ { ( r ) } } \alpha _ { i , j } ^ { ( r , k ) } \pmb { g } _ { j } ^ { ( r , k ) } \right) \in \mathbb { R } ^ { N \times K F ^ { \prime } } ,
+$$
+
+where $\oplus$ denotes vector concatenation, $\alpha _ { i , j } ^ { ( r , k ) }$ are the normalised attention coefficients under relation $r$ computed by either WIRGAT or ARGAT, and $\pmb { g } _ { i } ^ { ( r , k ) } = \pmb { h } _ { i }$ $\left( \boldsymbol { W } ^ { ( r , k ) } \right) ^ { T }$ is the head specific intermediate representation of node $i$ under relation $r$ .
+
+It might be interesting to consider cases where there are a different number of heads for different relationship types, as well as when a mixture of ARGAT and WIRGAT produce the attention coefficients, however, we leave that subject for future investigation and will not consider it further.
+
+Basis decomposition The number of parameters in the RGAT layer increases linearly with the number of relations $R$ and heads $K$ , and can lead quickly to overparameterisation. In RGCNs it was found that decomposing the kernels was beneficial for generalisation, although it comes at the cost of increased model bias (Schlichtkrull et al., 2018). We follow this approach, decomposing both the kernels $W ^ { ( r , k ) }$ as well as the kernels of attention mechanism $A ^ { ( r , k ) }$ into $B _ { V }$ basis matrices $V ^ { ( b ) } \in \mathbb { R } ^ { F \times F ^ { \prime } }$ and $B _ { X }$ basis vectors $\pmb { X } ^ { ( b ) } \in \mathbb { R } ^ { 2 F ^ { \prime } \times D }$
+
+$$
+{ \pmb W } ^ { ( r , k ) } = \sum _ { b = 1 } ^ { B _ { W } } c _ { b } ^ { ( r , k ) } { \pmb V } ^ { ( b ) } , \qquad { \pmb A } ^ { ( r , k ) } = \sum _ { b = 1 } ^ { B _ { X } } d _ { b } ^ { ( r , k ) } { \pmb X } ^ { ( b ) } ,
+$$
+
+whe re c(r,k)b , d $c _ { b } ^ { ( r , k ) } , d _ { b } ^ { ( r , k ) } \in \mathbb { R }$ are basis coefficients. We consider models using full and decomposed $W$ and $\pmb { A }$ .
+
+# 2.2 Node classification
+
+For the transductive task of semi-supervised node classification, we employ a two-layer RGAT architecture shown in Figure 3a. We use a Rectified Linear Unit (ReLU) activation after the RGAT concat layer, and a node-wise softmax on the final layer to produce an estimate for the probability that the $i ^ { \mathrm { t h } }$ label is in the class $\alpha$
+
+$$
+P ( \mathrm { c l a s s } _ { i } = \alpha ) \approx \hat { y } _ { i , \alpha } = \mathrm { s o f t m a x } ( \pmb { h } _ { i } ^ { ( 2 ) } ) _ { \alpha } .
+$$
+
+Table 1: A summary of the datasets used in our experiments and how they are partitioned.   
+
+<table><tr><td>Datasets</td><td>AIFB</td><td>MUTAG</td><td>Tox21</td></tr><tr><td>Task</td><td>Transductive</td><td>Transductive</td><td>Inductive</td></tr><tr><td>Nodes</td><td>8,285 (1 graph)</td><td>23,644 (1 graph)</td><td>145,459 (8014 graphs)</td></tr><tr><td>Edges</td><td>29,043</td><td>74,227</td><td>151,095</td></tr><tr><td>Relations</td><td>45</td><td>23</td><td>4</td></tr><tr><td>Labelled</td><td>176</td><td>340</td><td>96,168 (12 per graph)</td></tr><tr><td>Classes</td><td>4</td><td>2</td><td>12 (multi-label)</td></tr><tr><td>Train nodes</td><td>112</td><td>218</td><td>(6411 graphs)</td></tr><tr><td>Validation nodes</td><td>28</td><td>54</td><td>(801 graphs)</td></tr><tr><td>Test nodes</td><td>28</td><td>54</td><td>(802 graphs)</td></tr></table>
+
+We then employ a masked cross-entropy loss $\mathcal { L }$ to constrain the network updates to the subset of nodes $y$ whose labels are known
+
+$$
+\mathcal { L } = - \sum _ { i \in \mathcal { Y } } \sum _ { \alpha = 1 } ^ { n _ { \mathrm { c l a s s e s } } } y _ { i , \alpha } \ln \left( \hat { y } _ { i , \alpha } \right) ,
+$$
+
+where ${ \bf { { y } } } _ { i }$ is the one-hot representation of the true label for node $i$ .
+
+# 2.3 Graph classification
+
+For inductive graph classification, we employ a two-layer RGAT followed by a graph gather and dense network architecture shown in Figure 3b. We use ReLU activations after each RGAT layer and the first dense layer. We use a tanh activation after the GraphGather : $\mathbb { R } ^ { N \times F } \to \mathbb { R } ^ { 2 F }$ , which is a vector concatenation of the mean of the node representations with the feature-wise max of the node representations
+
+$$
+H ^ { \prime } = \mathrm { G r a p h G a t h e r } ( H ) = \left( \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \pmb { h } _ { i } \right) \oplus \left[ \bigoplus _ { f = 1 } ^ { F } \operatorname* { m a x } _ { i } \pmb { h } _ { i , f } \right] .
+$$
+
+The final dense layer then produces logits of the size $n _ { \mathrm { c l a s s e s } } \times n _ { \mathrm { t a s k s } }$ , and we apply a task-wise softmax to its output to produce an estimate $\hat { y } _ { t , \alpha }$ for the probability that the graph is in class $\alpha$ for a given task $t$ , analogous to Equation (11). Weighted cross-entropy loss $\mathcal { L }$ is then used to form the learning objective
+
+$$
+\mathcal { L } ( w , y , \hat { y } ) = - \sum _ { t = 1 } ^ { n _ { \mathrm { t a s k s } } } \sum _ { \alpha = 1 } ^ { n _ { \mathrm { c l a s s e s } } } w _ { t , \alpha } y _ { t , \alpha } \ln \left( \hat { y } _ { t , \alpha } \right) ,
+$$
+
+where $w _ { t , \alpha }$ and $y _ { t , \alpha }$ are the weights and one-hot true labels for task $t$ and class $\alpha$ respectivel
+
+# 3 Evaluation
+
+# 3.1 Datasets
+
+We evaluate the models on transductive and inductive tasks. Following the experimental setup of Schlichtkrull et al. (2018) for the transductive tasks, we evaluate our model on the Resource Description Framework (RDF) datasets AIFB and MUTAG. We also evaluate our model for an inductive task on the molecular dataset, Tox21. Details of these data sets are given in Table 1. For further details on the transductive and inductive datasets, please see Ristoski and Paulheim (2016) and Wu et al. (2018) respectively.
+
+Transductive baselines We consider as a baseline the recent state-of-the-art results from Schlichtkrull et al. (2018) obtained with a two-layer RGCN model with 16 hidden units and basis function decomposition. We also include the same challenging baselines of FEAT (Paulheim and Fümkranz, 2012), WL (Shervashidze et al., 2011; de Vries and de Rooij, 2015) and RDF2Vec (Ristoski and Paulheim, 2016). In-depth details of these baselines are given by Ristoski and Paulheim (2016).
+
+Inductive baselines As baselines for Tox21, we compare against the most competitive methods on Tox21 reported in Wu et al. (2018). Specifically, we compare against deep multitask networks Ramsundar et al. (2015), deep bypass multitask networks Wu et al. (2018), Weave Kearnes et al. (2016), and a RGCN model whose relational structure is determined by the degree of the node to be updated Altae-Tran et al. (2016). Specifically, up to and including some maximum degree $D _ { \mathrm { m a x } }$ ,
+
+$$
+\pmb { h } _ { i } ^ { \prime } = \sigma \left[ ( W ^ { \mathrm { d e g ( i ) } } ) ^ { T } \pmb { h } _ { i } + \sum _ { j \in \mathcal { n } _ { i } } ( U ^ { \mathrm { d e g ( i ) } } ) ^ { T } \pmb { h } _ { j } + b ^ { \mathrm { d e g } ( i ) } \right] ,
+$$
+
+where $W ^ { \mathrm { d e g } ( i ) } ~ \in ~ \mathbb { R } ^ { F \times F ^ { \prime } }$ is a degree-specific linear transformation for self-connections, $U ^ { \mathrm { d e g ( i ) } } ~ \in ~ \mathbb { R } ^ { F \times F ^ { \prime } }$ is a degree-specific linear transformation for neighbours into their intermediate representations $g _ { i } \in \mathbb { R } ^ { F ^ { \prime } }$ , and $\boldsymbol { b } ^ { \mathrm { d e g } ( i ) }$ is a degree-specific bias. Any update for any degree $d ( i ) > D _ { \operatorname* { m a x } }$ gets assigned to the update for the maximum degree $D _ { \mathrm { m a x } }$ .
+
+# 3.2 Experimental setup
+
+Transductive learning For the transductive learning tasks, the architecture discussed in Section 2.2 was applied. Its hyperparameters were optimised for both AIFB and MUTAG on their respective training/validation sets defined in Ristoski and Paulheim (2016), using 5-fold cross validation. Using the found hyperparameters, we retrain on the full training set and report results on the test set across 200 seeds. We employ early stopping on the validation set during cross-validation to determine the number of epochs we will run on the final training set. Hyperparameter optimisation details are given in Table 4 of Appendix B.
+
+Inductive learning For the inductive learning tasks, the architecture discussed in Section 2.3 was applied. In order to optimise hyperparameters once, ensure no data leakage, but also provide comparable benchmarks to those presented in Wu et al. (2018), three benchmark splits were taken from the MolNet benchmarks4, and graphs belonging to any of the test sets were isolated. Using the remaining graphs we performed a hyperparameter search using 2 folds of 10-fold cross validation. Using the found hyperparameters, we then retrained on the three benchmark splits provided with 2 seeds each, giving an unbiased estimate of model performance. We employ early stopping during both the cross-validation and final run (the validation set of the inductive task is available for the final benchmark, in contrast to the transductive tasks) to determine the number of training epochs. Hyperparameter optimisation details are given in Table 5 of Appendix B.
+
+Constant attention In all experiments, we train with the attention mechanism turned on. At evaluation time, however, we report results with and without the attention mechanism to provide insight into whether the attention mechanism helps. ARGAT (WIRGAT) without the attention is called C-ARGAT (C-WIRGAT).
+
+# 3.3 Results
+
+# 3.3.1 Benchmarks and additional analyses
+
+Model means and standard deviations are presented in Table 2. To provide a picture of characteristic model behaviour, the CDFs for the hyperparameter sweep are presented in Figure 5 of Appendix D. To draw meaningful conclusions, we compare against our own implementation of RGCN rather than the results reported in Schlichtkrull et al. (2018); Wu et al. (2018).
+
+We will occasionally employ a one-sided hypothesis test in order to make concrete statements about model performance. The details and complete results of this test are presented in Appendix E. When we refer to significant results this corresponds to a test statistic supporting our hypothesis with a $p -$ value $p \leq 0 . 0 5$ .
+
+Table 2: (a) Entity classification results accuracy (mean and standard deviation over 10 seeds) for FEAT (Paulheim and Fümkranz, 2012), WL (Shervashidze et al., 2011; de Vries and de Rooij, 2015), RDF2Vec (Ristoski and Paulheim, 2016) and RGCN (Schlichtkrull et al., 2018), and (mean and standard deviation over 200 seeds) for our implementation of RGCN, as well as additive and multiplicative attention for (C-)WIRGAT and (C-)WIRGAT (this work). Test performance is reported on the splits provided in Ristoski and Paulheim (2016). (b) Graph classification mean Receiver Operating Characteristic (ROC) Area Under the Curve (AUC) across all 12 tasks (mean and standard deviation over 3 splits) for Multitask (Ramsundar et al., 2015), Bypass (Wu et al., 2018), Weave (Kearnes et al., 2016), RGCN (Altae-Tran et al., 2016), and (mean and standard deviation over 3 splits, 2 seeds per split) our implementation of RGCN, additive and multiplicative attention for (C-)WIRGAT and (C-)ARGAT (this work). Test performance is reported on the splits provided in Wu et al. (2018). Best performance in class in boldened, and best performance overall is underlined. For completeness, we present the training and validation mean ROC-AUCs alongside the test ROC-AUCs in Appendix A. For a graphical representation of these results, see Figure 4 in Appendix C.
+
+<table><tr><td>Model</td><td>AIFB</td><td>MUTAG</td></tr><tr><td>Feat</td><td>55.55 ± 0.00 80.55 ± 0.00</td><td>77.94 ± 0.00 80.88 ± 0.00</td></tr><tr><td>WL RDF2Vec</td><td>88.88 ±0.00</td><td>67.20 ± 1.24</td></tr><tr><td>RGCN</td><td>95.83 ± 0.62</td><td>73.23 ± 0.48</td></tr><tr><td>RGCN (ours)</td><td>94.64 ±2.75</td><td>74.15 ± 2.40</td></tr><tr><td colspan="3">Additive attention</td></tr><tr><td>C-WIRGAT WIRGAT</td><td>96.86 ± 0.94 96.83 ± 1.01</td><td>69.37 ± 2.75 69.83 ± 2.74</td></tr><tr><td>C-ARGAT</td><td>93.05 ± 3.05</td><td>63.69 ± 8.41</td></tr><tr><td>ARGAT</td><td>94.01 ± 2.76</td><td>65.54 ± 6.25</td></tr><tr><td colspan="3">Multiplicative attention</td></tr><tr><td>C-WIRGAT WIRGAT</td><td>93.71 ± 3.33 92.92 ± 3.75</td><td>69.57 ± 3.70</td></tr><tr><td>C-ARGAT</td><td>95.89 ± 1.93 96.19 ± 1.70</td><td>69.60 ± 3.75 74.38 ± 3.78 73.17 ± 3.41</td></tr></table>
+
+(a) Transductive
+
+<table><tr><td>Model</td><td>Tox21</td></tr><tr><td>Multitask</td><td>0.803 ± 0.012</td></tr><tr><td>Bypass</td><td>0.810 ± 0.013</td></tr><tr><td>Weave RGCN</td><td>0.820 ± 0.010 0.829 ±0.006</td></tr><tr><td>RGCN (ours)</td><td>0.835 士 0.008</td></tr><tr><td>Additive attention</td><td></td></tr><tr><td>C-WIRGAT</td><td>0.832 ± 0.009</td></tr><tr><td>WIRGAT</td><td>0.835 ± 0.006</td></tr><tr><td>C-ARGAT</td><td>0.829 ± 0.010</td></tr><tr><td>ARGAT</td><td>0.835 ±0.006</td></tr><tr><td>Multiplicative</td><td>attention</td></tr><tr><td>C-WIRGAT WIRGAT</td><td>0.811 ± 0.008 0.838 ± 0.007</td></tr><tr><td>C-ARGAT ARGAT</td><td>0.802 ± 0.007 0.837 ± 0.007</td></tr></table>
+
+(b) Inductive
+
+# 3.3.2 Transductive learning
+
+In Table 2a we evaluate RGAT on MUTAG and AIFB. With additive attention, WIRGAT outperforms ARGAT, consistent with Schlichtkrull et al. (2018). Interestingly, when employing multiplicative attention, the converse appears true. For node classification tasks on RDF data, this indicates that the importance of a particular relation type does not vary much (if at all) across the graph unless one employs a multiplicative comparison $^ 5$ between node representations.
+
+AIFB On AIFB, the best to worst performing models are: 1) additive WIRGAT 2) multiplicative ARGAT 3) RGCN 4) additive ARGAT, and 5) multiplicative WIRGAT, with each comparison being significant.
+
+When comparing against their constant attention counterparts, the significant differences observed were for additive and multiplicative ARGAT, where attention gives a relative mean performance improvements of $1 . 0 3 \%$ and $0 . 3 1 \%$ respectively, and multiplicative WIRGAT, where attention gives a relative mean performance drop of $0 . 8 4 \%$ .
+
+Although we present state-of-the art result on AIFB with additive WIRGAT, since its performance with and without attention are not significantly different, it is unlikely that this is due to the attention mechanism itself, at least at inference time. Over the hyperparameter space, additive WIRGAT and RGCN are comparable in performance (see Figure 5a in Appendix D), leading us to conclude that the result is more likely attributable to finding a better hyperparameter point for additive WIRGAT during the search.
+
+MUTAG On MUTAG, the best to worst performing models are: 1) RGCN 2) multiplicative ARGAT 3) additive WIRGAT tied with multiplicative WIRGAT, and 4) additive ARGAT, with each comparison being significant.
+
+When comparing against their constant attention counterparts, the significant differences observed were for additive WIRGAT and ARGAT, where attention gives relative mean performance improvements of $0 . 6 6 \%$ and $2 . 9 0 \%$ respectively, and multiplicative ARGAT, where attention gives a relative mean performance drop of $1 . 6 3 \%$ .
+
+We note that RGCN consistently outperforms RGAT on MUTAG, contrary to what might be expected (Schlichtkrull et al., 2018). The result is surprising given that RGCN lies within the parameter space of RGAT (where the attention kernel is zero), a configuration we check through evaluating C-WIRGAT. In our experiments we have observed that both RGCN and RGAT can memorise the MUTAG training set with $1 0 0 \%$ accuracy without difficulty (this is not the case for AIFB). The performance gap between RGCN and RGAT could then be explained by the following:
+
+- During training, the RGAT layer uses its attention mechanism to solve the learning objective. Once the objective is solved, the model is not encouraged by the loss function to seek a point in the parameter space that would also behave well when attention is set to a normalising constant within neighbourhoods (i.e. the parameter space point that would be found by RGCN).   
+The RDF tasks are transductive, meaning that a basis-dependent spectral approach   
+is sufficient to solve them. As RGCN already memorises the MUTAG training set, a model more complex $\cdot$ than RGCN, for example RGAT, that can also memorise the training set is unlikely to generalise as well, although this is a hotly debated topic - see e.g. Zhang et al. (2016).
+
+We employed a suite of regularisation techniques to get RGAT to generalise on MUTAG, including L2-norm penalties, dropout in multiple places, batch normalisation, parameter reduction and early stopping, however, no evaluated harshly regularised points for RGAT generalise well on MUTAG.
+
+Our final observation is that the attention mechanism presented in Section 2.1 relies on node features. The node features for the above tasks are learned from scratch (the input feature matrix is a one-hot node index) as part of the task. It is possible that in this semi-supervised setup, there is insufficient signal in the data to learn both the input node embeddings as well as a meaningful attention mechanism to act upon them.
+
+# 3.3.3 Inductive learning
+
+In Table 2b we evaluate RGAT on Tox21. The number of samples is lower for these evaluations than for the transductive tasks, and so fewer model comparisons will be accompanied with a reasonable significance, although there are still some conclusions we can draw.
+
+Through a thorough hyperparameter search, and incorporating various regularisation techniques, we obtained the relative mean performance of $0 . 7 2 \%$ for RGCN compared to the result reported in Wu et al. (2018), providing a much stronger baseline.
+
+Both additive attention models match the performance of RGCN, whereas multiplicative WIRGAT and ARGAT marginally outperform RGCN, although this is not significant $p = 0 . 2 4$ and $p = 0 . 4 1$ respectively).
+
+When comparing against their constant attention counterparts, significant differences observed were for multiplicative WIRGAT and ARGAT, where attention gives a relative mean performance improvements of $3 . 3 3 \%$ and $4 . 3 6 \%$ respectively. We do not observe any significant gains coming from additive attention when compared to their constant counterparts.
+
+# 4 Conclusion
+
+We have investigated a class of models we call Relational Graph Attention Networks (RGATs). These models act upon graph structures, inducing a masked self-attention that takes account of local relational structure as well as node features. This allows both nodes and their properties under specific relations to be dynamically assigned an importance for different nodes in the graph, and opens up graph attention mechanisms to a wider variety of problems.
+
+We evaluted two specific attention mechanisms, Within-Relation Graph Attention (WIRGAT) and Across-Relation Graph Attention (ARGAT), under both an additive and multiplicative logit construction, and compared them to their equivalently evaluated spectral counterpart Relational Graph Convolutional Networks (RGCNs).
+
+We find RGATs perform competitively or poorly on established baselines. This behavior appears strongly task-dependent. Specifically, relational inductive tasks such as graph classification benefit from multiplicative ARGAT, whereas transductive relational tasks, such as knowledge base completion, at least in the absence of node features, are better tackled using spectral methods like RGCNs or other graph feature extraction methods like Weisfeiler-Lehman (WL) graph kernels.
+
+In general we have found that WIRGAT should be paired with an additive logit mechanism, and fares marginally better than ARGAT on transductive tasks, whereas ARGAT should be paired with a multiplicative logit mechanism, and fares marginally better on inductive tasks.
+
+We have found no cases where choosing any variation of RGAT is guaranteed to significantly outperform RGCN, although we have found that in cases where RGCN can memorise the training set, we are confident that RGAT will not perform as well as RGCN. Consequently, we suggest that before attempting to train RGAT, a good first test is to inspect the training set performance of RGCN.
+
+Through our thorough evaluation and presentation of the behaviours and limitations of these models, insights can be derived that will enable the discovery of more powerful model architectures that act upon relational structures. Observing that model variance on all of the tasks presented here is high, any future work developing and expanding these methods must choose larger, more challenging datasets. In addition, a comparison between the generalisation of spectral methods, like those presented here, and generalisations of Recurrent Neural Networks (RNNs), like Gated Graph Sequence Networks, is a necessary ingredient for determining the most promising future direction for these models.
+
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+
+# A Tox21 Results
+
+For completeness, we present the training, validation and test set performance of our models in addition to those in Wu et al. (2018) in Table 3.
+
+Table 3: Graph classification mean Area Under the Curve (AUC) across all 12 tasks (mean and standard deviation over 3 splits) for Multitask (Ramsundar et al., 2015), Bypass (Wu et al., 2018), Weave (Kearnes et al., 2016), RGCN (Altae-Tran et al., 2016), our implementation of RGCN, additive and multiplicative attention versions of WIRGAT and ARGAT (this work). Training, validation and test performance is reported on the splits provided in Wu et al. (2018). Best performance in class in boldened, and best performance overall is underlined.
+
+<table><tr><td>Model</td><td>Training</td><td>Validation</td><td>Test</td></tr><tr><td>Multitask</td><td>0.884± 0.001</td><td>0.795 ± 0.017</td><td>0.803 ± 0.012</td></tr><tr><td>Bypass</td><td>0.938 ± 0.001</td><td>0.800 ±0.008</td><td>0.810 ± 0.013</td></tr><tr><td>Weave</td><td>0.875 ± 0.004</td><td>0.828 ±0.008</td><td>0.820 ± 0.010</td></tr><tr><td>RGCN</td><td>0.905 ± 0.004</td><td>0.825 ± 0.013</td><td>0.829 ± 0.006</td></tr><tr><td>RGCN (ours)</td><td>0.883 ± 0.010</td><td>0.845 ±0.003</td><td>0.835 ±0.008</td></tr><tr><td colspan="4">Additive attention</td></tr><tr><td>C-WIRGAT</td><td>0.897 ± 0.022</td><td>0.842 ± 0.004</td><td>0.832 ± 0.009</td></tr><tr><td>WIRGAT</td><td>0.902 ± 0.024</td><td>0.845 ± 0.005</td><td>0.835 ± 0.006</td></tr><tr><td>C-ARGAT</td><td>0.884± 0.012</td><td>0.848 ± 0.003</td><td>0.829 ± 0.010</td></tr><tr><td>ARGAT</td><td>0.896 ± 0.016</td><td>0.851 ± 0.004</td><td>0.835 ± 0.006</td></tr><tr><td colspan="4">Multiplicative attention</td></tr><tr><td>C-WIRGAT</td><td>0.859 ± 0.016</td><td>0.830 ± 0.007</td><td>0.811 ± 0.008</td></tr><tr><td>WIRGAT</td><td>0.904± 0.022</td><td>0.852 ± 0.002</td><td>0.838 ± 0.007</td></tr><tr><td>C-ARGAT</td><td>0.838 ± 0.007</td><td>0.816 ± 0.007</td><td>0.802 ± 0.007</td></tr><tr><td>ARGAT</td><td>0.802 ± 0.007</td><td>0.846 ± 0.003</td><td>0.837 ± 0.007</td></tr></table>
+
+# B Hyperparameters
+
+We perform hyperparameter optimisation using hyperopt Bergstra et al. (2013) with priors for the transductive tasks specified in Table 4 and priors for the inductive tasks specified in Table 5. In all experiments we use the Adam optimiser (Kingma and Ba, 2014).
+
+Table 4: Priors on the hyperparameter search space for the transductive tasks. When multihead attention is used, the number of units per head is appropriately reduced in order to keep the total number of output units of an RGAT layer independent of the number of heads.   
+
+<table><tr><td>Hyperparameter</td><td>Prior</td></tr><tr><td>Graph kernel units</td><td>MultiplesOfFour(4,20)</td></tr><tr><td>Heads</td><td>OneOf(1,2, 4)</td></tr><tr><td>Feature dropout rate</td><td>Uniform(0.0, 0.8)</td></tr><tr><td>Edge dropout</td><td>Uniform(0.0, 0.8)</td></tr><tr><td>W basis size Graph layer 1 W L2 coef</td><td>OneOf(Full, 5,10,20,30)</td></tr><tr><td></td><td>LogUniform(10-6,10-1)</td></tr><tr><td>Graph layer 2 W L2 coef</td><td>LogUniform(10-6,10-1)</td></tr><tr><td>A basis size</td><td>OneOf(Full, 5,10,20,30)</td></tr><tr><td>Graph layer 1 A L2 coef</td><td>LogUniform(10-6,10-1)</td></tr><tr><td>Graph layer 2 A L2 coef</td><td>LogUniform(10-6,10-1)</td></tr><tr><td>Learning rate</td><td>LogUniform(10-5,10-1)</td></tr><tr><td>Use bias</td><td></td></tr><tr><td></td><td>OneOf(Yes, No)</td></tr><tr><td>Use batch normalisation</td><td>OneOf(Yes, No)</td></tr></table>
+
+Table 5: Priors on the hyperparameter for the inductive task. The batch size was held at 64, and no bases decomposition is used. When multihead attention is used, the number of units per head is appropriately reduced in order to keep the total number of output units of an RGAT layer independent of the number of heads.   
+
+<table><tr><td>Hyperparameter</td><td>Prior</td></tr><tr><td>Graph kernel units</td><td>MultiplesOfEight(32,128)</td></tr><tr><td>Dense units</td><td>MultiplesOfEight(32,128)</td></tr><tr><td>Heads</td><td>OneOf(1,2,4,8)</td></tr><tr><td>Feature dropout</td><td>Uniform(0.0, 0.8)</td></tr><tr><td>Edge dropout</td><td>Uniform(0.0,0.8)</td></tr><tr><td>W L2 coef (1)</td><td>LogUniform(10-6,10-1)</td></tr><tr><td>W L2 coef (2)</td><td>LogUniform(10-6,10-1)</td></tr><tr><td>a L2 coef (1)</td><td>LogUniform(10-6,10-1)</td></tr><tr><td>α L2 coef (2)</td><td>LogUniform(10-6,10-1)</td></tr><tr><td>Learning rate</td><td>LogUniform(10-5,10-1)</td></tr><tr><td>Use bias</td><td>OneOf(Yes, No)</td></tr><tr><td>Use batch normalisation</td><td>OneOf(Yes, No)</td></tr></table>
+
+# C Charts
+
+To aid interpretability of the results presented in Table 2 we present a chart representation in Figure 4.
+
+![](images/de4678adc2e8472e364e37ce3447a0c4098cab4e69ba828e58f0badf8865fce2.jpg)  
+Figure 4: (a) and (b): Blue Baseline entity classification accuracy (mean and standard deviation over 10 seeds) for FEAT (Paulheim and Fümkranz, 2012), WL (Shervashidze et al., 2011; de Vries and de Rooij, 2015), RDF2Vec (Ristoski and Paulheim, 2016) and RGCN (Schlichtkrull et al., 2018), and (mean and standard deviation over 200 runs) for our implementation of RGCN. Yellow Entity classification accuracy (mean and standard deviation over 200 seeds) for additive attention (this work). Red Entity classification accuracy (mean and standard deviation over 200 seeds) for multiplicative attention (this work). Test performance is reported on the splits provided in Ristoski and Paulheim (2016). (c): Blue Baseline graph classification mean Receiver Operating Characteristic (ROC) AUC across all 12 tasks (mean and standard deviation over 3 splits) for Multitask (Ramsundar et al., 2015), Bypass (Wu et al., 2018), Weave (Kearnes et al., 2016), RGCN (Altae-Tran et al., 2016), and (mean and standard deviation over 3 splits, 2 seeds per split) our implementation of RGCN. Yellow Additive attention graph classification mean ROC-AUC (mean and standard deviation over 200 seeds) across all 12 tasks (this work). Red Multiplicative attention graph classification mean ROC-AUC (mean and standard deviation over 200 seeds) across all 12 tasks (this work). All raw values are given in Table 2.
+
+# D Cumulative distribution functions
+
+To aid further insight into our results, we present the Cumulative Distribution Functions (CDFs) for each model on each task in Figure 4. In this context, we treat the performance metric of interest during the hyperparameter search as the empirical distribution of some random variable $X$ . We then define its CDF $F _ { X } ( x )$ in the standard way
+
+$$
+F _ { X } ( x ) = P ( X \leq x ) ,
+$$
+
+where $P ( X \leq x )$ is the probability that $X$ takes on a value less than or equal to $x$ . The CDF allows one to gauge whether any given architecture typically performs better than another across the whole space, rather than comparison of the tuned hyperparameters, which in some cases may be outliers in terms of generic behavior for that architecture.
+
+![](images/e18a7451652d67e0875b57cde0428249e4526e12940f5831f6cac80e66daab95.jpg)  
+Figure 5: CDFs for all models on a) AIFB, b) MUTAG and c) TOX21. Green lines correspond to our implementation of RGCN, blue lines correspond to ARGAT, and orange lines correspond to WIRGAT. Solid lines correspond to additive attention (and RGCN), whereas dashed lines correspond to multiplicative attention. A lower CDF value is better in the sense that a greater proportion of models of achieve a higher value of that metric.
+
+AIFB Additive and multiplicative ARGAT perform poorly for most areas of the hyperparameter space, whereas RGCN and multiplicative WIRGAT perform comparably across the entire hyperparameter space.
+
+MUTAG Interestingly, the models that have a greater amount of hyperparameter space covering poor performance (i.e. RGCN, multiplicative and additive WIRGAT) are also the models which also have a greater amount of hyperparameter space covering good performance. In other words, on the MUTAG, the ARGAT prior resulted in a model whose test set performance was relatively insensitive to hyperparameter choice when compared against the other candidates. Given that the ARGAT model was the most flexible of the models evaluated, and that it was able to memorise the training set, this suggests that the task contained insufficient information for the model to learn its attention mechanism. Given that WIRGAT was able to at least partially learn to its attention mechanism suggests that WIRGAT is less data hungry than ARGAT.
+
+Tox21 The multiplicative attention models fare poorly on the majority of the hyperparameter space compared to the other models. There is a slice of the hyperparameter space where the multiplicative attention models outperform the other models, however, indicating that although they are difficult to train, it may be worth spending time hyperoptimising them if you need the best performing model on a relational inductive task. The additive attention models and RGCN perform comparably across the entirety of the hyperparameter space and generally perform better than the multiplicative methods except for the very small region of hyperparameter space mentioned above.
+
+# E Significance testing
+
+In order to determine if any of our model comparisons are significant, we employ the one-sided Mann-Whitney $U$ test Mann and Whitney (1947) as we are interested in the direction of movement (i.e. performance) and do not want to make any parametric assumptions about model response. For two populations $X$ and $Y$ :
+
+• The null hypothesis $H _ { 0 }$ is that the two populations are equal, and • The alternative hypothesis $H _ { 1 }$ is that the probability of an observation from population X exceeding an observation from population Y is larger than the probability of an observation from Y exceeding an observation from X; i.e., $H _ { 1 } : P ( X > Y ) > P ( Y > X )$ .
+
+We treat the empirical distributions of Model A as samples from population $X$ and the empirical distributions of Model B as samples from population $Y$ . This allows us a window into whether, given a task, whether which is the better model out of a pair of models. Results on AIFB, MUTAG and TOX21 are given in Figure 6, Figure 7 and Figure 8 respectively.
+
+![](images/d6b58d3b82577979f8ec66718246c6906d164643a0b8f52bdd1cffc5ea125905.jpg)  
+Figure 6: The $p$ -values for Mann-Whitney $U$ test with alternative hypothesis $H _ { 1 }$ of Model $A$ outperforming Model $B$ on AIFB.
+
+![](images/9f9346c252e13a119c74ac1fe44dbc25e2361ed2b3215116b217f2205f39108c.jpg)  
+Figure 7: The $p$ -values for Mann-Whitney $U$ test with alternative hypothesis $H _ { 1 }$ of Model $A$ outperforming Model $B$ on MUTAG.
+
+![](images/ce51e1157f7064678b8b1ea8ed556f9c0ae21e45c2183e579c2593a488d00115.jpg)  
+Figure 8: The $p$ -values for Mann-Whitney $U$ test with alternative hypothesis $H _ { 1 }$ of Model $A$ outperforming Model $B$ on TOX21.

parse/train/Bklzkh0qFm/Bklzkh0qFm_content_list.json

@@ -0,0 +1,1862 @@
+[
+    {
+        "type": "text",
+        "text": "Relational Graph Attention Networks ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            98,
+            745,
+            121
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            184,
+            145,
+            419,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            452,
+            209,
+            545,
+            226
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We investigate Relational Graph Attention Networks, a class of models that extends non-relational graph attention mechanisms to incorporate relational information, opening up these methods to a wider variety of problems. A thorough evaluation of these models is performed, and comparisons are made against established benchmarks. To provide a meaningful comparison, we retrain Relational Graph Convolutional Networks, the spectral counterpart of Relational Graph Attention Networks, and evaluate them under the same conditions. We find that Relational Graph Attention Networks perform worse than anticipated, although some configurations are marginally beneficial for modelling molecular properties. We provide insights as to why this may be, and suggest both modifications to evaluation strategies, as well as directions to investigate for future work. ",
+        "bbox": [
+            233,
+            239,
+            764,
+            406
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            431,
+            339,
+            446
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Convolutional Neural Networks (CNNs) successfully solve a variety of tasks in Euclidean grid-like domains, such as image captioning (Donahue et al., 2017) and classifying videos (Karpathy et al., 2014). CNNs are successful because they assume the data is locally stationary and compositional (Defferrard et al., 2016; Henaff et al., 2015; Bruna et al., 2013). ",
+        "bbox": [
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+            462,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "However, data often occurs in the form of graphs or manifolds, which are classic examples of non-Euclidean domains. Specific instances include knowledge bases, molecules, and point clouds captured by 3D data acquisition devices (Wang et al., 2018). The generalisation of Neural Networks (NNs) to non-Euclidean domains is termed Geometric Deep Learning (GDL), and may be roughly divided into spectral, spatial and hybrid approaches (Bronstein et al., 2017). ",
+        "bbox": [
+            174,
+            525,
+            823,
+            608
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Spectral approaches (Defferrard et al., 2016), most notably Graph Convolutional Networks (GCNs) (Kipf and Welling, 2016), are limited by their basis-dependence. A filter that is learned with respect to a basis on one domain is not guaranteed to behave similarly when applied to another basis and domain. Spatial approaches are limited by an absence of shift invariance and lack of coordinate system (Duvenaud et al., 2015; Atwood and Towsley, 2016; Monti et al., 2017). Hybrid approaches combine spectral and spatial approaches, trading their advantages and deficiencies against each-other (Bronstein et al., 2017; Rustamov and Guibas, 2013; Szlam et al., 2005; Gavish et al., 2010). ",
+        "bbox": [
+            173,
+            614,
+            825,
+            727
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A recent approach that began with Graph Attention Networks (GATs), applied attention mechanisms to graphs, and does not share these limitations (Veličković et al., 2017; Gong and Cheng, 2018; Zhang et al., 2018; Monti et al., 2018; Lee et al., 2018). ",
+        "bbox": [
+            174,
+            734,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "An alternative direction has been to generalise Recurrent Neural Networks (RNNs) from sequential message passing on one-dimensional signals, to message passing on graphs (Sperduti and Starita, 1997; Frasconi et al., 1997; Gori et al., 2005). Incorporating gating mechanisms led to the development of Gated Graph Neural Networks (GGNNs) (Scarselli et al., 2009; Allamanis et al., 2017)1. ",
+        "bbox": [
+            174,
+            782,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Relational Graph Convolutional Networks (RGCNs) have been proposed as an extension of GCNs to the domain of relational graphs (Schlichtkrull et al., 2018). This model has achieved impressive performance on node classification and link prediction tasks, however, its mechanisms still resides within spectral methods and shares their deficiencies. The focus of this work investigate generalisations of RGCN away from its spectral origins. ",
+        "bbox": [
+            174,
+            859,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            823,
+            146
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We take RGCN as a starting point, and investigate a class of models we term Relational Graph Attention Networks (RGATs), extending attention mechanisms to the relational graph domain. We consider two variants, Within-Relation Graph Attention (WIRGAT) and Across-Relation Graph Attention (ARGAT), each with either additive or multiplicative attention. We perform an extensive hyperparameter search, and evaluate these models on challenging transductive node classification and inductive graph classification tasks. These models are compared against established benchmarks, as well as a re-tuned RGCN model. ",
+        "bbox": [
+            173,
+            152,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We show that RGAT performs worse than expected, although some configurations produce marginal benefits on inductive graph classification tasks. In order to aid further investigation in this direction, we present the full Cumulative Distribution Functions (CDFs) for the hyperparameter searches in Appendix D, and statistical hypothesis tests in Appendix E. We also provide a vectorised, sparse, batched implementation of RGAT and RGCN in TensorFlow which is compatible with eager execution mode to open up research into these models to a wider audience2. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RGAT architecture ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            375,
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+            391
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 Relational graph attention layer ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            406,
+            504,
+            420
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We follow the construction of the GAT layer in Veličković et al. (2017), extending to the relational setting, using ideas from Schlichtkrull et al. (2018). ",
+        "bbox": [
+            176,
+            431,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Layer input and output The input to the layer is a graph with $R = | \\mathcal R |$ relation types and $N$ nodes. The $i ^ { \\mathrm { t h } }$ node is represented by a feature vector $\\boldsymbol { h } _ { i } \\in \\mathbb { R } ^ { F }$ , and the features of all nodes are summarised in the feature matrix ${ \\pmb H } = [ { \\pmb h } _ { 1 } { \\pmb h } _ { 2 } \\dots { \\pmb h } _ { N } ] \\in \\mathbb { R } ^ { N \\times F }$ . The output of the layer is the transformed feature matrix $\\pmb { H } ^ { \\prime } = [ h _ { 1 } ^ { \\prime } h _ { 2 } ^ { \\prime } \\dots h _ { N } ^ { \\prime } ] \\in \\mathbb { R } ^ { N \\times F ^ { \\prime } }$ , where $\\pmb { h } _ { i } ^ { \\prime } \\in \\mathbb { R } ^ { F ^ { \\prime } }$ is the transformed feature vector of the $i ^ { \\mathrm { t h } }$ node. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Intermediate representations Different relations convey distinct pieces of information. The update rule of Schlichtkrull et al. (2018) made this manifest by assigning each node $_ i$ a distinct intermediate representation $\\pmb { g } _ { i } ^ { ( r ) } \\in \\mathbb { R } ^ { F ^ { \\prime } }$ under relation $r$ ",
+        "bbox": [
+            174,
+            563,
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c0d60d8d843e165eb94f18a0f74f4b855c9a9c2bb3f49897b1d0cd914b50d2a7.jpg",
+        "text": "$$\n\\pmb { G } ^ { ( r ) } = \\pmb { H } \\pmb { W } ^ { ( r ) } \\in \\mathbb { R } ^ { N \\times F ^ { \\prime } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            405,
+            618,
+            591,
+            637
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $\\pmb { G } ^ { ( r ) } = \\left[ \\pmb { g } _ { 1 } ^ { ( r ) } \\pmb { g } _ { 2 } ^ { ( r ) } \\cdot \\cdot \\cdot \\pmb { g } _ { N } ^ { ( r ) } \\right]$ is the intermediate representation feature matrix under relation $r$ , and $W ^ { ( r ) } \\in \\mathbb { R } ^ { F \\times F ^ { \\prime } }$ are the learnable parameters of a shared linear transformation. ",
+        "bbox": [
+            174,
+            643,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Logits Following Veličković et al. (2017); Zhang et al. (2018), we assume the attention coefficient between two nodes is based only on the features of those nodes up to a neighborhoodlevel normalisation. To keep computational complexity linear in $R$ , we assume that, given linear transformations $W ^ { ( r ) }$ , the logits $E _ { i , j } ^ { ( r ) }$ of each relation $r$ are independent ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e8240fc2f8450c0676cf446ca809194bce81fdc9c7be857687cabffc7c140abc.jpg",
+        "text": "$$\nE _ { i , j } ^ { ( r ) } = a \\left( \\pmb { g } _ { i } ^ { ( r ) } , \\pmb { g } _ { j } ^ { ( r ) } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            768,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "and indicate the importance of node ’s intermediate representation to that of node $_ i$ under relation $r$ . The attention is masked so that, for node $i$ , coefficients $\\alpha _ { i , j } ^ { ( r ) }$ exist only for $j \\in n _ { i } ^ { ( r ) }$ , where $n _ { i } ^ { ( r ) }$ denotes the set of neighbor indices of node $_ i$ under relation $r \\in \\mathcal R$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Queries, keys and values The logits are composed from queries and keys, and specify how the values, i.e. the intermediate representations $\\mathbf { \\Delta } _ { g _ { i } ^ { ( r ) } }$ , will combine to produce the updated ",
+        "bbox": [
+            176,
+            867,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/9a4bf818da78a1f6aaa34d99c50e0d9b06bf898c39cb24788316934d42cf5262.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+            111,
+            684,
+            284
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Figure 1: WIRGAT. The intermediate representations for node $i$ (left red rectangle) are combined with the intermediate representations for nodes in its neighborhood (blue rectangles) under each relation $r$ , to form each logit $E _ { i , j } ^ { ( r ) }$ . A softmax is taken over each logit matrix for each relation type to form the attention coefficients $\\alpha _ { i , j } ^ { ( r ) }$ . These attention coefficients construct a weighted sum over the nodes in the neighborhood for each relation (black rectangle). These are then aggregated and passed through a nonlinearity to produce the updated representation for node $_ i$ (right red rectangle). ",
+        "bbox": [
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+            309,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "key kernel node representations $K ^ { ( r ) } \\in \\mathbb { R } ^ { F ^ { \\prime } \\times D }$ $ { \\boldsymbol { h } } _ { i } ^ { \\prime }$ (Vaswani et al., 2017). A separate query kernel project the intermediate representations $\\mathbf { \\Delta } _ { g _ { i } ^ { ( r ) } }$ , into query and key $Q ^ { ( r ) } \\in \\mathbb { R } ^ { F ^ { \\prime } \\times D }$ and representations of dimensionality ",
+        "bbox": [
+            173,
+            443,
+            825,
+            489
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3a841fc2464537719ea785cef19e8a70105025cfa7ea2112ed0358fa3ce9c2ba.jpg",
+        "text": "$$\n\\pmb { q } _ { i } ^ { ( r ) } = \\pmb { g } _ { i } ^ { ( r ) } \\pmb { Q } ^ { ( r ) } \\in \\mathbb { R } ^ { D } , \\qquad \\pmb { k } _ { i } ^ { ( r ) } = \\pmb { g } _ { i } ^ { ( r ) } \\pmb { K } ^ { ( r ) } \\in \\mathbb { R } ^ { D } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "For convenience, the query and key kernels are combined to form the attention kernels $\\pmb { A } ^ { ( r ) } = \\pmb { Q } ^ { ( r ) } \\oplus \\pmb { K } ^ { ( r ) } \\in \\mathbb { R } ^ { 2 F ^ { \\prime } \\times D }$ . These query and key representations are the building blocks of the two specific realisations of $a$ in Equation (2) that we now consider. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Additive attention logits The first realisation of $a$ we consider is the relational modification of the logit mechanism of Veličković et al. (2017) ",
+        "bbox": [
+            173,
+            574,
+            823,
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1c33c237c6d640072e9f31ac6f66d937f9faa71c6ae2ca6d0ef53d5426885312.jpg",
+        "text": "$$\n\\begin{array} { r } { \\boldsymbol { E } _ { i , j } ^ { ( r ) } = \\mathrm { L e a k y R e L u } \\left( \\boldsymbol { q } _ { i } ^ { ( r ) } + \\boldsymbol { k } _ { j } ^ { ( r ) } \\right) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            380,
+            607,
+            617,
+            633
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where the query and key dimensionality are both $D = 1$ , and $q _ { i } ^ { ( r ) }$ and $k _ { i } ^ { ( r ) }$ are scalar flattenings of their one-dimensional vector counterparts $\\pmb { q } _ { i } ^ { ( r ) } , \\pmb { k } _ { i } ^ { ( r ) } \\in \\mathbb { R } ^ { 1 }$ . We refer to any instance of RGAT using logits of the form in Equation (4) as additive RGAT. ",
+        "bbox": [
+            173,
+            638,
+            825,
+            688
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Multiplicative attention logits The second realisation we consider is the multiplicative mechanism of Vaswani et al. (2017); Zhang et al. (2018)3 ",
+        "bbox": [
+            174,
+            700,
+            823,
+            729
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2b0ab0586b2d10feaeb12c322e9a7a35202461e40f33e6a895ba2e08e0b3303c.jpg",
+        "text": "$$\nE _ { i , j } ^ { ( r ) } = \\pmb { q } _ { i } ^ { ( r ) } \\cdot \\pmb { k } _ { j } ^ { ( r ) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            436,
+            733,
+            560,
+            756
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where the query and key dimensionality $D$ can be any positive integer. We refer to any instance of RGAT using logits of the form in Equation (4) as multiplicative RGAT. ",
+        "bbox": [
+            173,
+            758,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "It should be noted that there are many types of attention mechanisms beyond vanilla additive and multiplicative. These include mechanisms leveraging the structure of the dual graph Monti et al. (2018) as well as learned edge features Gong and Cheng (2018). ",
+        "bbox": [
+            174,
+            792,
+            825,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The attention coefficients should be comparable across nodes. This can be achieved by applying softmax appropriately to any logits $E _ { i , j } ^ { ( r ) }$ . We investigate two candidates, each ",
+        "bbox": [
+            174,
+            842,
+            825,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/ef3930c93ca203f3d08151a1206bea28eef4dd567e160e21b8eb98831fd1cf88.jpg",
+        "image_caption": [
+            "Figure 2: ARGAT. The logits are produced identically to those in Figure 1. A softmax is taken across all logits independent of relation type to form the attention coefficients α(r)i,j . The remaining weighting and aggregation steps are the same as those in Figure 1. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            254,
+            109,
+            730,
+            285
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "WIRGAT The simplest way to take the softmax over the logits $E _ { i , j } ^ { ( r ) }$ of Equation (4) or ",
+        "bbox": [
+            173,
+            383,
+            825,
+            417
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0f147711a1f3e78f49ae6f8264a662e1df1e4b82efa9c744da6528d3cf3ba8fa.jpg",
+        "text": "$$\n\\alpha _ { i , j } ^ { ( r ) } = \\operatorname * { s o f t m a x } _ { j } \\left( E _ { i , j } ^ { ( r ) } \\right) = \\frac { \\exp \\left( E _ { i , j } ^ { ( r ) } \\right) } { \\sum _ { k \\in { \\cal { n } } _ { i } ^ { ( r ) } } \\exp \\left( E _ { i , k } ^ { ( r ) } \\right) } , \\qquad \\forall i , r : \\sum _ { j \\in { \\cal { n } } _ { i } ^ { ( r ) } } \\alpha _ { i , j } ^ { ( r ) } = 1 .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            232,
+            425,
+            767,
+            478
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We call the attention in Equation (6) Within-Relation Graph Attention (WIRGAT), and it is shown in Figure 1. This mechanism encodes the prior that relation importance is a purely global property of the graph by implementing an independent probability distribution over nodesnodes hborhood of yield compe $i$ for each relation ing attention coe $r$ . Explcients y, forand node with $i$ and relation izes dependin $r$ , $j , k \\in n _ { i } ^ { ( r ) }$ $\\alpha _ { i , j } ^ { ( r ) }$ α(r)i,k on their corresponding representations $\\mathbf { \\Delta } _ { g _ { j } ^ { ( r ) } }$ and $\\mathbf { \\Delta } _ { g _ { k } ^ { ( r ) } }$ . There is no competition between any attention coefficients $\\alpha _ { i , j } ^ { ( r ) }$ and α(r )i,k for all nodes $i$ and nodes $j \\in n _ { i } ^ { ( r ) } , j ^ { \\prime } \\in n ^ { ( r ^ { \\prime } ) }$ where $\\boldsymbol { r } ^ { \\prime } \\neq \\boldsymbol { r }$ irrespective of node representations. ",
+        "bbox": [
+            173,
+            486,
+            826,
+            617
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "ARGAT An alternative way to take the softmax over the logit $E _ { i , j } ^ { ( r ) }$ of Equation (4) or $r$ ",
+        "bbox": [
+            173,
+            632,
+            825,
+            666
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e2749d9449f4d39fc4d9aa78f092e3b65b25e2a8902afa719bc67723e50ef2ab.jpg",
+        "text": "$$\n\\alpha _ { i , j } ^ { ( r ) } = \\operatorname * { s o f } _ { j , r } \\mathrm { a x } \\left( E _ { i , j } ^ { ( r ) } \\right) = \\frac { \\exp \\left( E _ { i , j } ^ { ( r ) } \\right) } { \\sum _ { r ^ { \\prime } \\in \\mathcal { R } } \\sum _ { k \\in { n _ { i } ^ { ( r ^ { \\prime } ) } } } \\exp \\left( E _ { i , k } ^ { ( r ^ { \\prime } ) } \\right) } , \\forall i : \\sum _ { r \\in \\mathcal { R } } \\sum _ { j \\in { n _ { i } ^ { ( r ) } } } \\alpha _ { i , j } ^ { ( r ) } = 1 .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We call the attention in Equation (7) Equation (6) Across-Relation Graph Attention (ARGAT), and it is shown in Figure 2. This mechanism encodes the prior that relation importance is a local property of the graph by implementing a single probability distribution over the different representations $\\mathbf { \\Delta } _ { g _ { j } ^ { ( r ) } }$ for nodes $\\mathrm { j }$ in the neighborhood of node $i$ . Explicitly, for any node $i$ and all $r , r ^ { \\prime } \\in \\mathcal { R }$ , all nodes $j \\in n _ { i } ^ { ( r ) }$ and $k \\in \\mathcal { n } _ { i } ^ { ( r ^ { \\prime } ) }$ yield competing attention coefficients α(r)i,j and α(r  )i,k with sizes depending on their corresponding representations $\\mathbf { \\Delta } _ { g _ { j } ^ { ( r ) } }$ and g(rk $\\pmb { g } _ { k } ^ { ( r ^ { \\prime } ) }$ ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Comparison to RGCN For comparison, the coefficients of RGCN are given by α(r)i,j $| n _ { i } ^ { ( r ) } | ^ { - 1 }$ . This encodes the prior that the intermediate representations of nodes $j \\in n _ { i } ^ { ( r ) }$ to node $i$ ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/b51b37fc3f64eff99f9add08eebbdb9785410870141a6ca7965e0d2334b14044.jpg",
+        "image_caption": [
+            "Figure 3: (a) The network architecture used for node classification on AIFB and MUTAG. This architecture is the same as in Schlichtkrull et al. (2018) except with RGCNs replaced with RGATs. (b) The network architecture used for multi-task graph classification on Tox21. This architecture is the same as the GCNs architecture in Altae-Tran et al. (2016) except with RGCNs replaces with GATs and we do not use graph pooling. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Propagation rule Combining the attention mechanism of either Equation (6) or Equation (7) with the neighborhood aggregation step of Schlichtkrull et al. (2018) gives ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a871ee0006ab53936b1f7709b40dcd48dd4a3cf1027c738ac754af41edb02641.jpg",
+        "text": "$$\n\\pmb { h } _ { i } ^ { \\prime } = \\sigma \\left( \\sum _ { r \\in \\mathcal { R } } \\sum _ { j \\in { \\cal n } _ { i } ^ { ( r ) } } { \\alpha } _ { i , j } ^ { ( r ) } \\pmb { g } _ { j } ^ { ( r ) } \\right) \\in \\mathbb { R } ^ { N \\times F ^ { \\prime } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            357,
+            347,
+            640,
+            404
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\sigma$ represents an optional nonlinearity. Similar to Vaswani et al. (2017); Veličković et al. (2017), we also find that using multiple heads in the attention mechanism can enhance performance ",
+        "bbox": [
+            174,
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b4f9ce5d93b12987b37e36fda88b7a5319cd323836e24ba59b51a1a2d56d96e0.jpg",
+        "text": "$$\n\\pmb { h } _ { i } ^ { \\prime } = \\bigoplus _ { k = 1 } ^ { K } \\sigma \\left( \\sum _ { r \\in \\mathcal { R } } \\sum _ { j \\in { \\cal n } _ { i } ^ { ( r ) } } \\alpha _ { i , j } ^ { ( r , k ) } \\pmb { g } _ { j } ^ { ( r , k ) } \\right) \\in \\mathbb { R } ^ { N \\times K F ^ { \\prime } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            326,
+            446,
+            669,
+            503
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\oplus$ denotes vector concatenation, $\\alpha _ { i , j } ^ { ( r , k ) }$ are the normalised attention coefficients under relation $r$ computed by either WIRGAT or ARGAT, and $\\pmb { g } _ { i } ^ { ( r , k ) } = \\pmb { h } _ { i }$ $\\left( \\boldsymbol { W } ^ { ( r , k ) } \\right) ^ { T }$ is the head specific intermediate representation of node $i$ under relation $r$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "It might be interesting to consider cases where there are a different number of heads for different relationship types, as well as when a mixture of ARGAT and WIRGAT produce the attention coefficients, however, we leave that subject for future investigation and will not consider it further. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Basis decomposition The number of parameters in the RGAT layer increases linearly with the number of relations $R$ and heads $K$ , and can lead quickly to overparameterisation. In RGCNs it was found that decomposing the kernels was beneficial for generalisation, although it comes at the cost of increased model bias (Schlichtkrull et al., 2018). We follow this approach, decomposing both the kernels $W ^ { ( r , k ) }$ as well as the kernels of attention mechanism $A ^ { ( r , k ) }$ into $B _ { V }$ basis matrices $V ^ { ( b ) } \\in \\mathbb { R } ^ { F \\times F ^ { \\prime } }$ and $B _ { X }$ basis vectors $\\pmb { X } ^ { ( b ) } \\in \\mathbb { R } ^ { 2 F ^ { \\prime } \\times D }$ ",
+        "bbox": [
+            173,
+            632,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/09b8270c3c79a7a10dde5f8780b5cc0afac07699fc87a8b4d4e9abce614dc03b.jpg",
+        "text": "$$\n{ \\pmb W } ^ { ( r , k ) } = \\sum _ { b = 1 } ^ { B _ { W } } c _ { b } ^ { ( r , k ) } { \\pmb V } ^ { ( b ) } , \\qquad { \\pmb A } ^ { ( r , k ) } = \\sum _ { b = 1 } ^ { B _ { X } } d _ { b } ^ { ( r , k ) } { \\pmb X } ^ { ( b ) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            279,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "whe re c(r,k)b , d $c _ { b } ^ { ( r , k ) } , d _ { b } ^ { ( r , k ) } \\in \\mathbb { R }$ are basis coefficients. We consider models using full and decomposed $W$ and $\\pmb { A }$ . ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "2.2 Node classification ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For the transductive task of semi-supervised node classification, we employ a two-layer RGAT architecture shown in Figure 3a. We use a Rectified Linear Unit (ReLU) activation after the RGAT concat layer, and a node-wise softmax on the final layer to produce an estimate for the probability that the $i ^ { \\mathrm { t h } }$ label is in the class $\\alpha$ ",
+        "bbox": [
+            173,
+            844,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5b79b43b00ed5c667b90a58145dea257b8b4f85342dc50ae4f571235c62e5efa.jpg",
+        "text": "$$\nP ( \\mathrm { c l a s s } _ { i } = \\alpha ) \\approx \\hat { y } _ { i , \\alpha } = \\mathrm { s o f t m a x } ( \\pmb { h } _ { i } ^ { ( 2 ) } ) _ { \\alpha } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            359,
+            905,
+            637,
+            925
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/02f440a75975cc57780df2c00694cd5e2009b2b659dca8e4cd26ea76403ebc29.jpg",
+        "table_caption": [
+            "Table 1: A summary of the datasets used in our experiments and how they are partitioned. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Datasets</td><td>AIFB</td><td>MUTAG</td><td>Tox21</td></tr><tr><td>Task</td><td>Transductive</td><td>Transductive</td><td>Inductive</td></tr><tr><td>Nodes</td><td>8,285 (1 graph)</td><td>23,644 (1 graph)</td><td>145,459 (8014 graphs)</td></tr><tr><td>Edges</td><td>29,043</td><td>74,227</td><td>151,095</td></tr><tr><td>Relations</td><td>45</td><td>23</td><td>4</td></tr><tr><td>Labelled</td><td>176</td><td>340</td><td>96,168 (12 per graph)</td></tr><tr><td>Classes</td><td>4</td><td>2</td><td>12 (multi-label)</td></tr><tr><td>Train nodes</td><td>112</td><td>218</td><td>(6411 graphs)</td></tr><tr><td>Validation nodes</td><td>28</td><td>54</td><td>(801 graphs)</td></tr><tr><td>Test nodes</td><td>28</td><td>54</td><td>(802 graphs)</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We then employ a masked cross-entropy loss $\\mathcal { L }$ to constrain the network updates to the subset of nodes $y$ whose labels are known ",
+        "bbox": [
+            178,
+            303,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d76d83741bda7f830df6d87e154e0e9789431aec8a58d6e6633f519350c6f5df.jpg",
+        "text": "$$\n\\mathcal { L } = - \\sum _ { i \\in \\mathcal { Y } } \\sum _ { \\alpha = 1 } ^ { n _ { \\mathrm { c l a s s e s } } } y _ { i , \\alpha } \\ln \\left( \\hat { y } _ { i , \\alpha } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            392,
+            334,
+            604,
+            376
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where ${ \\bf { { y } } } _ { i }$ is the one-hot representation of the true label for node $i$ . ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "2.3 Graph classification ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            406,
+            390,
+            421
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For inductive graph classification, we employ a two-layer RGAT followed by a graph gather and dense network architecture shown in Figure 3b. We use ReLU activations after each RGAT layer and the first dense layer. We use a tanh activation after the GraphGather : $\\mathbb { R } ^ { N \\times F } \\to \\mathbb { R } ^ { 2 F }$ , which is a vector concatenation of the mean of the node representations with the feature-wise max of the node representations ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d26b74d490948f19e78b0e1e9257fb747412a7eb0bbb047c0f2cae11d4e66b61.jpg",
+        "text": "$$\nH ^ { \\prime } = \\mathrm { G r a p h G a t h e r } ( H ) = \\left( \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\pmb { h } _ { i } \\right) \\oplus \\left[ \\bigoplus _ { f = 1 } ^ { F } \\operatorname* { m a x } _ { i } \\pmb { h } _ { i , f } \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            290,
+            505,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The final dense layer then produces logits of the size $n _ { \\mathrm { c l a s s e s } } \\times n _ { \\mathrm { t a s k s } }$ , and we apply a task-wise softmax to its output to produce an estimate $\\hat { y } _ { t , \\alpha }$ for the probability that the graph is in class $\\alpha$ for a given task $t$ , analogous to Equation (11). Weighted cross-entropy loss $\\mathcal { L }$ is then used to form the learning objective ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/64b06b7f607749f9c0257d057ca7db0fb8cd3a02a3751462b025b7952a73d74d.jpg",
+        "text": "$$\n\\mathcal { L } ( w , y , \\hat { y } ) = - \\sum _ { t = 1 } ^ { n _ { \\mathrm { t a s k s } } } \\sum _ { \\alpha = 1 } ^ { n _ { \\mathrm { c l a s s e s } } } w _ { t , \\alpha } y _ { t , \\alpha } \\ln \\left( \\hat { y } _ { t , \\alpha } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            341,
+            612,
+            655,
+            652
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $w _ { t , \\alpha }$ and $y _ { t , \\alpha }$ are the weights and one-hot true labels for task $t$ and class $\\alpha$ respectivel ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3 Evaluation ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.1 Datasets ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We evaluate the models on transductive and inductive tasks. Following the experimental setup of Schlichtkrull et al. (2018) for the transductive tasks, we evaluate our model on the Resource Description Framework (RDF) datasets AIFB and MUTAG. We also evaluate our model for an inductive task on the molecular dataset, Tox21. Details of these data sets are given in Table 1. For further details on the transductive and inductive datasets, please see Ristoski and Paulheim (2016) and Wu et al. (2018) respectively. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Transductive baselines We consider as a baseline the recent state-of-the-art results from Schlichtkrull et al. (2018) obtained with a two-layer RGCN model with 16 hidden units and basis function decomposition. We also include the same challenging baselines of FEAT (Paulheim and Fümkranz, 2012), WL (Shervashidze et al., 2011; de Vries and de Rooij, 2015) and RDF2Vec (Ristoski and Paulheim, 2016). In-depth details of these baselines are given by Ristoski and Paulheim (2016). ",
+        "bbox": [
+            173,
+            839,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Inductive baselines As baselines for Tox21, we compare against the most competitive methods on Tox21 reported in Wu et al. (2018). Specifically, we compare against deep multitask networks Ramsundar et al. (2015), deep bypass multitask networks Wu et al. (2018), Weave Kearnes et al. (2016), and a RGCN model whose relational structure is determined by the degree of the node to be updated Altae-Tran et al. (2016). Specifically, up to and including some maximum degree $D _ { \\mathrm { m a x } }$ , ",
+        "bbox": [
+            174,
+            103,
+            825,
+            188
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5a84d77620531500cc4029f1c4bb90856aa7979ec746d6256274036ae82494b0.jpg",
+        "text": "$$\n\\pmb { h } _ { i } ^ { \\prime } = \\sigma \\left[ ( W ^ { \\mathrm { d e g ( i ) } } ) ^ { T } \\pmb { h } _ { i } + \\sum _ { j \\in \\mathcal { n } _ { i } } ( U ^ { \\mathrm { d e g ( i ) } } ) ^ { T } \\pmb { h } _ { j } + b ^ { \\mathrm { d e g } ( i ) } \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            310,
+            194,
+            687,
+            242
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where $W ^ { \\mathrm { d e g } ( i ) } ~ \\in ~ \\mathbb { R } ^ { F \\times F ^ { \\prime } }$ is a degree-specific linear transformation for self-connections, $U ^ { \\mathrm { d e g ( i ) } } ~ \\in ~ \\mathbb { R } ^ { F \\times F ^ { \\prime } }$ is a degree-specific linear transformation for neighbours into their intermediate representations $g _ { i } \\in \\mathbb { R } ^ { F ^ { \\prime } }$ , and $\\boldsymbol { b } ^ { \\mathrm { d e g } ( i ) }$ is a degree-specific bias. Any update for any degree $d ( i ) > D _ { \\operatorname* { m a x } }$ gets assigned to the update for the maximum degree $D _ { \\mathrm { m a x } }$ . ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "3.2 Experimental setup ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            327,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Transductive learning For the transductive learning tasks, the architecture discussed in Section 2.2 was applied. Its hyperparameters were optimised for both AIFB and MUTAG on their respective training/validation sets defined in Ristoski and Paulheim (2016), using 5-fold cross validation. Using the found hyperparameters, we retrain on the full training set and report results on the test set across 200 seeds. We employ early stopping on the validation set during cross-validation to determine the number of epochs we will run on the final training set. Hyperparameter optimisation details are given in Table 4 of Appendix B. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Inductive learning For the inductive learning tasks, the architecture discussed in Section 2.3 was applied. In order to optimise hyperparameters once, ensure no data leakage, but also provide comparable benchmarks to those presented in Wu et al. (2018), three benchmark splits were taken from the MolNet benchmarks4, and graphs belonging to any of the test sets were isolated. Using the remaining graphs we performed a hyperparameter search using 2 folds of 10-fold cross validation. Using the found hyperparameters, we then retrained on the three benchmark splits provided with 2 seeds each, giving an unbiased estimate of model performance. We employ early stopping during both the cross-validation and final run (the validation set of the inductive task is available for the final benchmark, in contrast to the transductive tasks) to determine the number of training epochs. Hyperparameter optimisation details are given in Table 5 of Appendix B. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Constant attention In all experiments, we train with the attention mechanism turned on. At evaluation time, however, we report results with and without the attention mechanism to provide insight into whether the attention mechanism helps. ARGAT (WIRGAT) without the attention is called C-ARGAT (C-WIRGAT). ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "3.3 Results ",
+        "text_level": 1,
+        "bbox": [
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+            705,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "3.3.1 Benchmarks and additional analyses ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            732,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Model means and standard deviations are presented in Table 2. To provide a picture of characteristic model behaviour, the CDFs for the hyperparameter sweep are presented in Figure 5 of Appendix D. To draw meaningful conclusions, we compare against our own implementation of RGCN rather than the results reported in Schlichtkrull et al. (2018); Wu et al. (2018). ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We will occasionally employ a one-sided hypothesis test in order to make concrete statements about model performance. The details and complete results of this test are presented in Appendix E. When we refer to significant results this corresponds to a test statistic supporting our hypothesis with a $p -$ value $p \\leq 0 . 0 5$ . ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 2: (a) Entity classification results accuracy (mean and standard deviation over 10 seeds) for FEAT (Paulheim and Fümkranz, 2012), WL (Shervashidze et al., 2011; de Vries and de Rooij, 2015), RDF2Vec (Ristoski and Paulheim, 2016) and RGCN (Schlichtkrull et al., 2018), and (mean and standard deviation over 200 seeds) for our implementation of RGCN, as well as additive and multiplicative attention for (C-)WIRGAT and (C-)WIRGAT (this work). Test performance is reported on the splits provided in Ristoski and Paulheim (2016). (b) Graph classification mean Receiver Operating Characteristic (ROC) Area Under the Curve (AUC) across all 12 tasks (mean and standard deviation over 3 splits) for Multitask (Ramsundar et al., 2015), Bypass (Wu et al., 2018), Weave (Kearnes et al., 2016), RGCN (Altae-Tran et al., 2016), and (mean and standard deviation over 3 splits, 2 seeds per split) our implementation of RGCN, additive and multiplicative attention for (C-)WIRGAT and (C-)ARGAT (this work). Test performance is reported on the splits provided in Wu et al. (2018). Best performance in class in boldened, and best performance overall is underlined. For completeness, we present the training and validation mean ROC-AUCs alongside the test ROC-AUCs in Appendix A. For a graphical representation of these results, see Figure 4 in Appendix C. ",
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+        "type": "table",
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+        "table_caption": [],
+        "table_footnote": [
+            "(a) Transductive "
+        ],
+        "table_body": "<table><tr><td>Model</td><td>AIFB</td><td>MUTAG</td></tr><tr><td>Feat</td><td>55.55 ± 0.00 80.55 ± 0.00</td><td>77.94 ± 0.00 80.88 ± 0.00</td></tr><tr><td>WL RDF2Vec</td><td>88.88 ±0.00</td><td>67.20 ± 1.24</td></tr><tr><td>RGCN</td><td>95.83 ± 0.62</td><td>73.23 ± 0.48</td></tr><tr><td>RGCN (ours)</td><td>94.64 ±2.75</td><td>74.15 ± 2.40</td></tr><tr><td colspan=\"3\">Additive attention</td></tr><tr><td>C-WIRGAT WIRGAT</td><td>96.86 ± 0.94 96.83 ± 1.01</td><td>69.37 ± 2.75 69.83 ± 2.74</td></tr><tr><td>C-ARGAT</td><td>93.05 ± 3.05</td><td>63.69 ± 8.41</td></tr><tr><td>ARGAT</td><td>94.01 ± 2.76</td><td>65.54 ± 6.25</td></tr><tr><td colspan=\"3\">Multiplicative attention</td></tr><tr><td>C-WIRGAT WIRGAT</td><td>93.71 ± 3.33 92.92 ± 3.75</td><td>69.57 ± 3.70</td></tr><tr><td>C-ARGAT</td><td>95.89 ± 1.93 96.19 ± 1.70</td><td>69.60 ± 3.75 74.38 ± 3.78 73.17 ± 3.41</td></tr></table>",
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+    {
+        "type": "table",
+        "img_path": "images/c4dfd5a451a9b1cfee5e88cd56d26c6ff22742dbedfd60b8b6789e2e9cfe993f.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "(b) Inductive "
+        ],
+        "table_body": "<table><tr><td>Model</td><td>Tox21</td></tr><tr><td>Multitask</td><td>0.803 ± 0.012</td></tr><tr><td>Bypass</td><td>0.810 ± 0.013</td></tr><tr><td>Weave RGCN</td><td>0.820 ± 0.010 0.829 ±0.006</td></tr><tr><td>RGCN (ours)</td><td>0.835 士 0.008</td></tr><tr><td>Additive attention</td><td></td></tr><tr><td>C-WIRGAT</td><td>0.832 ± 0.009</td></tr><tr><td>WIRGAT</td><td>0.835 ± 0.006</td></tr><tr><td>C-ARGAT</td><td>0.829 ± 0.010</td></tr><tr><td>ARGAT</td><td>0.835 ±0.006</td></tr><tr><td>Multiplicative</td><td>attention</td></tr><tr><td>C-WIRGAT WIRGAT</td><td>0.811 ± 0.008 0.838 ± 0.007</td></tr><tr><td>C-ARGAT ARGAT</td><td>0.802 ± 0.007 0.837 ± 0.007</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "3.3.2 Transductive learning ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In Table 2a we evaluate RGAT on MUTAG and AIFB. With additive attention, WIRGAT outperforms ARGAT, consistent with Schlichtkrull et al. (2018). Interestingly, when employing multiplicative attention, the converse appears true. For node classification tasks on RDF data, this indicates that the importance of a particular relation type does not vary much (if at all) across the graph unless one employs a multiplicative comparison $^ 5$ between node representations. ",
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+    {
+        "type": "text",
+        "text": "AIFB On AIFB, the best to worst performing models are: 1) additive WIRGAT 2) multiplicative ARGAT 3) RGCN 4) additive ARGAT, and 5) multiplicative WIRGAT, with each comparison being significant. ",
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+        "type": "text",
+        "text": "When comparing against their constant attention counterparts, the significant differences observed were for additive and multiplicative ARGAT, where attention gives a relative mean performance improvements of $1 . 0 3 \\%$ and $0 . 3 1 \\%$ respectively, and multiplicative WIRGAT, where attention gives a relative mean performance drop of $0 . 8 4 \\%$ . ",
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+    {
+        "type": "text",
+        "text": "Although we present state-of-the art result on AIFB with additive WIRGAT, since its performance with and without attention are not significantly different, it is unlikely that this is due to the attention mechanism itself, at least at inference time. Over the hyperparameter space, additive WIRGAT and RGCN are comparable in performance (see Figure 5a in Appendix D), leading us to conclude that the result is more likely attributable to finding a better hyperparameter point for additive WIRGAT during the search. ",
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+    {
+        "type": "text",
+        "text": "MUTAG On MUTAG, the best to worst performing models are: 1) RGCN 2) multiplicative ARGAT 3) additive WIRGAT tied with multiplicative WIRGAT, and 4) additive ARGAT, with each comparison being significant. ",
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+    {
+        "type": "text",
+        "text": "When comparing against their constant attention counterparts, the significant differences observed were for additive WIRGAT and ARGAT, where attention gives relative mean performance improvements of $0 . 6 6 \\%$ and $2 . 9 0 \\%$ respectively, and multiplicative ARGAT, where attention gives a relative mean performance drop of $1 . 6 3 \\%$ . ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We note that RGCN consistently outperforms RGAT on MUTAG, contrary to what might be expected (Schlichtkrull et al., 2018). The result is surprising given that RGCN lies within the parameter space of RGAT (where the attention kernel is zero), a configuration we check through evaluating C-WIRGAT. In our experiments we have observed that both RGCN and RGAT can memorise the MUTAG training set with $1 0 0 \\%$ accuracy without difficulty (this is not the case for AIFB). The performance gap between RGCN and RGAT could then be explained by the following: ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "- During training, the RGAT layer uses its attention mechanism to solve the learning objective. Once the objective is solved, the model is not encouraged by the loss function to seek a point in the parameter space that would also behave well when attention is set to a normalising constant within neighbourhoods (i.e. the parameter space point that would be found by RGCN).   \nThe RDF tasks are transductive, meaning that a basis-dependent spectral approach   \nis sufficient to solve them. As RGCN already memorises the MUTAG training set, a model more complex $\\cdot$ than RGCN, for example RGAT, that can also memorise the training set is unlikely to generalise as well, although this is a hotly debated topic - see e.g. Zhang et al. (2016). ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We employed a suite of regularisation techniques to get RGAT to generalise on MUTAG, including L2-norm penalties, dropout in multiple places, batch normalisation, parameter reduction and early stopping, however, no evaluated harshly regularised points for RGAT generalise well on MUTAG. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Our final observation is that the attention mechanism presented in Section 2.1 relies on node features. The node features for the above tasks are learned from scratch (the input feature matrix is a one-hot node index) as part of the task. It is possible that in this semi-supervised setup, there is insufficient signal in the data to learn both the input node embeddings as well as a meaningful attention mechanism to act upon them. ",
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+    {
+        "type": "text",
+        "text": "3.3.3 Inductive learning ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In Table 2b we evaluate RGAT on Tox21. The number of samples is lower for these evaluations than for the transductive tasks, and so fewer model comparisons will be accompanied with a reasonable significance, although there are still some conclusions we can draw. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Through a thorough hyperparameter search, and incorporating various regularisation techniques, we obtained the relative mean performance of $0 . 7 2 \\%$ for RGCN compared to the result reported in Wu et al. (2018), providing a much stronger baseline. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Both additive attention models match the performance of RGCN, whereas multiplicative WIRGAT and ARGAT marginally outperform RGCN, although this is not significant $p = 0 . 2 4$ and $p = 0 . 4 1$ respectively). ",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "When comparing against their constant attention counterparts, significant differences observed were for multiplicative WIRGAT and ARGAT, where attention gives a relative mean performance improvements of $3 . 3 3 \\%$ and $4 . 3 6 \\%$ respectively. We do not observe any significant gains coming from additive attention when compared to their constant counterparts. ",
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+        "page_idx": 9
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+    {
+        "type": "text",
+        "text": "4 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We have investigated a class of models we call Relational Graph Attention Networks (RGATs). These models act upon graph structures, inducing a masked self-attention that takes account of local relational structure as well as node features. This allows both nodes and their properties under specific relations to be dynamically assigned an importance for different nodes in the graph, and opens up graph attention mechanisms to a wider variety of problems. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We evaluted two specific attention mechanisms, Within-Relation Graph Attention (WIRGAT) and Across-Relation Graph Attention (ARGAT), under both an additive and multiplicative logit construction, and compared them to their equivalently evaluated spectral counterpart Relational Graph Convolutional Networks (RGCNs). ",
+        "bbox": [
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+        "page_idx": 9
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+    {
+        "type": "text",
+        "text": "We find RGATs perform competitively or poorly on established baselines. This behavior appears strongly task-dependent. Specifically, relational inductive tasks such as graph classification benefit from multiplicative ARGAT, whereas transductive relational tasks, such as knowledge base completion, at least in the absence of node features, are better tackled using spectral methods like RGCNs or other graph feature extraction methods like Weisfeiler-Lehman (WL) graph kernels. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In general we have found that WIRGAT should be paired with an additive logit mechanism, and fares marginally better than ARGAT on transductive tasks, whereas ARGAT should be paired with a multiplicative logit mechanism, and fares marginally better on inductive tasks. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We have found no cases where choosing any variation of RGAT is guaranteed to significantly outperform RGCN, although we have found that in cases where RGCN can memorise the training set, we are confident that RGAT will not perform as well as RGCN. Consequently, we suggest that before attempting to train RGAT, a good first test is to inspect the training set performance of RGCN. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Through our thorough evaluation and presentation of the behaviours and limitations of these models, insights can be derived that will enable the discovery of more powerful model architectures that act upon relational structures. Observing that model variance on all of the tasks presented here is high, any future work developing and expanding these methods must choose larger, more challenging datasets. In addition, a comparison between the generalisation of spectral methods, like those presented here, and generalisations of Recurrent Neural Networks (RNNs), like Gated Graph Sequence Networks, is a necessary ingredient for determining the most promising future direction for these models. ",
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
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+        "text": "Veličković, P., G. Cucurull, A. Casanova, A. Romero, P. Liò, and Y. Bengio 2017. Graph Attention Networks. ",
+        "bbox": [
+            176,
+            704,
+            717,
+            733
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Wang, Y., Y. Sun, Z. Liu, S. E. Sarma, M. M. Bronstein, and J. M. Solomon 2018. Dynamic Graph CNN for Learning on Point Clouds. Technical report. ",
+        "bbox": [
+            173,
+            741,
+            735,
+            770
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Wu, Z., B. Ramsundar, E. N. Feinberg, J. Gomes, C. Geniesse, A. S. Pappu, K. Leswing, and V. Pande 2018. MoleculeNet: A benchmark for molecular machine learning. Chemical Science, 9(2):513–530. ",
+        "bbox": [
+            173,
+            779,
+            828,
+            835
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Zhang, C., S. Bengio, M. Hardt, B. Recht, and O. Vinyals 2016. Understanding deep learning requires rethinking generalization. CoRR, abs/1611.03530. ",
+        "bbox": [
+            171,
+            843,
+            831,
+            886
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Zhang, J., X. Shi, J. Xie, H. Ma, I. King, and D.-Y. Yeung 2018. GaAN: Gated Attention Networks for Learning on Large and Spatiotemporal Graphs. ",
+        "bbox": [
+            174,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A Tox21 Results ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            356,
+            118
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For completeness, we present the training, validation and test set performance of our models in addition to those in Wu et al. (2018) in Table 3. ",
+        "bbox": [
+            174,
+            132,
+            821,
+            161
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Table 3: Graph classification mean Area Under the Curve (AUC) across all 12 tasks (mean and standard deviation over 3 splits) for Multitask (Ramsundar et al., 2015), Bypass (Wu et al., 2018), Weave (Kearnes et al., 2016), RGCN (Altae-Tran et al., 2016), our implementation of RGCN, additive and multiplicative attention versions of WIRGAT and ARGAT (this work). Training, validation and test performance is reported on the splits provided in Wu et al. (2018). Best performance in class in boldened, and best performance overall is underlined. ",
+        "bbox": [
+            173,
+            174,
+            825,
+            272
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/9f063f98d241dd9dd1f43327cd85840e87bbbe67ae95fccf10b22282edce9c40.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Training</td><td>Validation</td><td>Test</td></tr><tr><td>Multitask</td><td>0.884± 0.001</td><td>0.795 ± 0.017</td><td>0.803 ± 0.012</td></tr><tr><td>Bypass</td><td>0.938 ± 0.001</td><td>0.800 ±0.008</td><td>0.810 ± 0.013</td></tr><tr><td>Weave</td><td>0.875 ± 0.004</td><td>0.828 ±0.008</td><td>0.820 ± 0.010</td></tr><tr><td>RGCN</td><td>0.905 ± 0.004</td><td>0.825 ± 0.013</td><td>0.829 ± 0.006</td></tr><tr><td>RGCN (ours)</td><td>0.883 ± 0.010</td><td>0.845 ±0.003</td><td>0.835 ±0.008</td></tr><tr><td colspan=\"4\">Additive attention</td></tr><tr><td>C-WIRGAT</td><td>0.897 ± 0.022</td><td>0.842 ± 0.004</td><td>0.832 ± 0.009</td></tr><tr><td>WIRGAT</td><td>0.902 ± 0.024</td><td>0.845 ± 0.005</td><td>0.835 ± 0.006</td></tr><tr><td>C-ARGAT</td><td>0.884± 0.012</td><td>0.848 ± 0.003</td><td>0.829 ± 0.010</td></tr><tr><td>ARGAT</td><td>0.896 ± 0.016</td><td>0.851 ± 0.004</td><td>0.835 ± 0.006</td></tr><tr><td colspan=\"4\">Multiplicative attention</td></tr><tr><td>C-WIRGAT</td><td>0.859 ± 0.016</td><td>0.830 ± 0.007</td><td>0.811 ± 0.008</td></tr><tr><td>WIRGAT</td><td>0.904± 0.022</td><td>0.852 ± 0.002</td><td>0.838 ± 0.007</td></tr><tr><td>C-ARGAT</td><td>0.838 ± 0.007</td><td>0.816 ± 0.007</td><td>0.802 ± 0.007</td></tr><tr><td>ARGAT</td><td>0.802 ± 0.007</td><td>0.846 ± 0.003</td><td>0.837 ± 0.007</td></tr></table>",
+        "bbox": [
+            256,
+            286,
+            735,
+            563
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B Hyperparameters ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            388,
+            118
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We perform hyperparameter optimisation using hyperopt Bergstra et al. (2013) with priors for the transductive tasks specified in Table 4 and priors for the inductive tasks specified in Table 5. In all experiments we use the Adam optimiser (Kingma and Ba, 2014). ",
+        "bbox": [
+            174,
+            133,
+            825,
+            175
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/b8939f4d9684f2b69c92bf06e51c4da9a1eef565fde508ffc526f228e0c2a4db.jpg",
+        "table_caption": [
+            "Table 4: Priors on the hyperparameter search space for the transductive tasks. When multihead attention is used, the number of units per head is appropriately reduced in order to keep the total number of output units of an RGAT layer independent of the number of heads. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Hyperparameter</td><td>Prior</td></tr><tr><td>Graph kernel units</td><td>MultiplesOfFour(4,20)</td></tr><tr><td>Heads</td><td>OneOf(1,2, 4)</td></tr><tr><td>Feature dropout rate</td><td>Uniform(0.0, 0.8)</td></tr><tr><td>Edge dropout</td><td>Uniform(0.0, 0.8)</td></tr><tr><td>W basis size Graph layer 1 W L2 coef</td><td>OneOf(Full, 5,10,20,30)</td></tr><tr><td></td><td>LogUniform(10-6,10-1)</td></tr><tr><td>Graph layer 2 W L2 coef</td><td>LogUniform(10-6,10-1)</td></tr><tr><td>A basis size</td><td>OneOf(Full, 5,10,20,30)</td></tr><tr><td>Graph layer 1 A L2 coef</td><td>LogUniform(10-6,10-1)</td></tr><tr><td>Graph layer 2 A L2 coef</td><td>LogUniform(10-6,10-1)</td></tr><tr><td>Learning rate</td><td>LogUniform(10-5,10-1)</td></tr><tr><td>Use bias</td><td></td></tr><tr><td></td><td>OneOf(Yes, No)</td></tr><tr><td>Use batch normalisation</td><td>OneOf(Yes, No)</td></tr></table>",
+        "bbox": [
+            295,
+            261,
+            696,
+            473
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/5b3eba4a7e128871bd834410122d055342019a5e8faa378f5879598f5d3ef0c9.jpg",
+        "table_caption": [
+            "Table 5: Priors on the hyperparameter for the inductive task. The batch size was held at 64, and no bases decomposition is used. When multihead attention is used, the number of units per head is appropriately reduced in order to keep the total number of output units of an RGAT layer independent of the number of heads. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Hyperparameter</td><td>Prior</td></tr><tr><td>Graph kernel units</td><td>MultiplesOfEight(32,128)</td></tr><tr><td>Dense units</td><td>MultiplesOfEight(32,128)</td></tr><tr><td>Heads</td><td>OneOf(1,2,4,8)</td></tr><tr><td>Feature dropout</td><td>Uniform(0.0, 0.8)</td></tr><tr><td>Edge dropout</td><td>Uniform(0.0,0.8)</td></tr><tr><td>W L2 coef (1)</td><td>LogUniform(10-6,10-1)</td></tr><tr><td>W L2 coef (2)</td><td>LogUniform(10-6,10-1)</td></tr><tr><td>a L2 coef (1)</td><td>LogUniform(10-6,10-1)</td></tr><tr><td>α L2 coef (2)</td><td>LogUniform(10-6,10-1)</td></tr><tr><td>Learning rate</td><td>LogUniform(10-5,10-1)</td></tr><tr><td>Use bias</td><td>OneOf(Yes, No)</td></tr><tr><td>Use batch normalisation</td><td>OneOf(Yes, No)</td></tr></table>",
+        "bbox": [
+            295,
+            564,
+            694,
+            763
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C Charts ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            282,
+            118
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "To aid interpretability of the results presented in Table 2 we present a chart representation in Figure 4. ",
+        "bbox": [
+            174,
+            132,
+            823,
+            161
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/de4678adc2e8472e364e37ce3447a0c4098cab4e69ba828e58f0badf8865fce2.jpg",
+        "image_caption": [
+            "Figure 4: (a) and (b): Blue Baseline entity classification accuracy (mean and standard deviation over 10 seeds) for FEAT (Paulheim and Fümkranz, 2012), WL (Shervashidze et al., 2011; de Vries and de Rooij, 2015), RDF2Vec (Ristoski and Paulheim, 2016) and RGCN (Schlichtkrull et al., 2018), and (mean and standard deviation over 200 runs) for our implementation of RGCN. Yellow Entity classification accuracy (mean and standard deviation over 200 seeds) for additive attention (this work). Red Entity classification accuracy (mean and standard deviation over 200 seeds) for multiplicative attention (this work). Test performance is reported on the splits provided in Ristoski and Paulheim (2016). (c): Blue Baseline graph classification mean Receiver Operating Characteristic (ROC) AUC across all 12 tasks (mean and standard deviation over 3 splits) for Multitask (Ramsundar et al., 2015), Bypass (Wu et al., 2018), Weave (Kearnes et al., 2016), RGCN (Altae-Tran et al., 2016), and (mean and standard deviation over 3 splits, 2 seeds per split) our implementation of RGCN. Yellow Additive attention graph classification mean ROC-AUC (mean and standard deviation over 200 seeds) across all 12 tasks (this work). Red Multiplicative attention graph classification mean ROC-AUC (mean and standard deviation over 200 seeds) across all 12 tasks (this work). All raw values are given in Table 2. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            188,
+            826,
+            636
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "D Cumulative distribution functions ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            102,
+            563,
+            118
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "To aid further insight into our results, we present the Cumulative Distribution Functions (CDFs) for each model on each task in Figure 4. In this context, we treat the performance metric of interest during the hyperparameter search as the empirical distribution of some random variable $X$ . We then define its CDF $F _ { X } ( x )$ in the standard way ",
+        "bbox": [
+            173,
+            135,
+            825,
+            191
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/75f9c2ec65e75b8438f01dd99c3c9e3c1713128d9e13e07634f116ab343fe58a.jpg",
+        "text": "$$\nF _ { X } ( x ) = P ( X \\leq x ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            426,
+            203,
+            570,
+            218
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "where $P ( X \\leq x )$ is the probability that $X$ takes on a value less than or equal to $x$ . The CDF allows one to gauge whether any given architecture typically performs better than another across the whole space, rather than comparison of the tuned hyperparameters, which in some cases may be outliers in terms of generic behavior for that architecture. ",
+        "bbox": [
+            173,
+            229,
+            825,
+            285
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/e18a7451652d67e0875b57cde0428249e4526e12940f5831f6cac80e66daab95.jpg",
+        "image_caption": [
+            "Figure 5: CDFs for all models on a) AIFB, b) MUTAG and c) TOX21. Green lines correspond to our implementation of RGCN, blue lines correspond to ARGAT, and orange lines correspond to WIRGAT. Solid lines correspond to additive attention (and RGCN), whereas dashed lines correspond to multiplicative attention. A lower CDF value is better in the sense that a greater proportion of models of achieve a higher value of that metric. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            186,
+            313,
+            846,
+            763
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "AIFB Additive and multiplicative ARGAT perform poorly for most areas of the hyperparameter space, whereas RGCN and multiplicative WIRGAT perform comparably across the entire hyperparameter space. ",
+        "bbox": [
+            176,
+            881,
+            825,
+            924
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "MUTAG Interestingly, the models that have a greater amount of hyperparameter space covering poor performance (i.e. RGCN, multiplicative and additive WIRGAT) are also the models which also have a greater amount of hyperparameter space covering good performance. In other words, on the MUTAG, the ARGAT prior resulted in a model whose test set performance was relatively insensitive to hyperparameter choice when compared against the other candidates. Given that the ARGAT model was the most flexible of the models evaluated, and that it was able to memorise the training set, this suggests that the task contained insufficient information for the model to learn its attention mechanism. Given that WIRGAT was able to at least partially learn to its attention mechanism suggests that WIRGAT is less data hungry than ARGAT. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            242
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Tox21 The multiplicative attention models fare poorly on the majority of the hyperparameter space compared to the other models. There is a slice of the hyperparameter space where the multiplicative attention models outperform the other models, however, indicating that although they are difficult to train, it may be worth spending time hyperoptimising them if you need the best performing model on a relational inductive task. The additive attention models and RGCN perform comparably across the entirety of the hyperparameter space and generally perform better than the multiplicative methods except for the very small region of hyperparameter space mentioned above. ",
+        "bbox": [
+            174,
+            257,
+            825,
+            369
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "E Significance testing ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            415,
+            118
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "In order to determine if any of our model comparisons are significant, we employ the one-sided Mann-Whitney $U$ test Mann and Whitney (1947) as we are interested in the direction of movement (i.e. performance) and do not want to make any parametric assumptions about model response. For two populations $X$ and $Y$ : ",
+        "bbox": [
+            176,
+            132,
+            825,
+            189
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "• The null hypothesis $H _ { 0 }$ is that the two populations are equal, and • The alternative hypothesis $H _ { 1 }$ is that the probability of an observation from population X exceeding an observation from population Y is larger than the probability of an observation from Y exceeding an observation from X; i.e., $H _ { 1 } : P ( X > Y ) > P ( Y > X )$ . ",
+        "bbox": [
+            215,
+            200,
+            825,
+            276
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "We treat the empirical distributions of Model A as samples from population $X$ and the empirical distributions of Model B as samples from population $Y$ . This allows us a window into whether, given a task, whether which is the better model out of a pair of models. Results on AIFB, MUTAG and TOX21 are given in Figure 6, Figure 7 and Figure 8 respectively. ",
+        "bbox": [
+            176,
+            287,
+            825,
+            344
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/d6b58d3b82577979f8ec66718246c6906d164643a0b8f52bdd1cffc5ea125905.jpg",
+        "image_caption": [
+            "Figure 6: The $p$ -values for Mann-Whitney $U$ test with alternative hypothesis $H _ { 1 }$ of Model $A$ outperforming Model $B$ on AIFB. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            362,
+            794,
+            849
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/9f9346c252e13a119c74ac1fe44dbc25e2361ed2b3215116b217f2205f39108c.jpg",
+        "image_caption": [
+            "Figure 7: The $p$ -values for Mann-Whitney $U$ test with alternative hypothesis $H _ { 1 }$ of Model $A$ outperforming Model $B$ on MUTAG. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            242,
+            794,
+            734
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/ce51e1157f7064678b8b1ea8ed556f9c0ae21e45c2183e579c2593a488d00115.jpg",
+        "image_caption": [
+            "Figure 8: The $p$ -values for Mann-Whitney $U$ test with alternative hypothesis $H _ { 1 }$ of Model $A$ outperforming Model $B$ on TOX21. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            243,
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+            736
+        ],
+        "page_idx": 19
+    }
+]

parse/train/SJg7IsC5KQ/SJg7IsC5KQ.md

@@ -0,0 +1,889 @@
+# ON THE CONVERGENCE AND ROBUSTNESS OF BATCHNORMALIZATION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Despite its empirical success, the theoretical underpinnings of the stability, convergence and acceleration properties of batch normalization (BN) remain elusive. In this paper, we attack this problem from a modeling approach, where we perform a thorough theoretical analysis on BN applied to a simplified model: ordinary least squares (OLS). We discover that gradient descent on OLS with BN has interesting properties, including a scaling law, convergence for arbitrary learning rates for the weights, acceleration effects, as well as insensitivity to the choice of learning rates. We then demonstrate numerically that these findings are not specific to the OLS problem and hold qualitatively for more complex supervised learning problems. This points to a new direction towards uncovering the mathematical principles that underlies batch normalization.
+
+# 1 INTRODUCTION
+
+Batch normalization (Ioffe & Szegedy, 2015) (BN) is one of the most important techniques for training deep neural networks and has proven extremely effective in avoiding gradient blowups during back-propagation and speeding up convergence. In its original introduction (Ioffe & Szegedy, 2015), the desirable effects of BN are attributed to the so-called “reduction of covariate shift”. However, it is unclear what this statement means in precise mathematical terms. To date, there lacks a comprehensive theoretical analysis of the effect of batch normalization.
+
+In this paper, we study the convergence and stability of gradient descent with batch normalization (BNGD) via a modeling approach. More concretely, we consider a simplified supervised learning problem: ordinary least squares regression, and analyze precisely the effect of BNGD when applied to this problem. Much akin to the mathematical modeling of physical processes, the least-squares problem serves as an idealized “model” of the effect of BN for general supervised learning tasks. A key reason for this choice is that the dynamics of GD without BN (hereafter called GD for simplicity) in least-squares regression is completely understood, thus allowing us to isolate and contrast the additional effects of batch normalization.
+
+The modeling approach proceeds in the following steps. First, we derive precise mathematical results on the convergence and stability of BNGD applied to the least-squares problem. In particular, we show that BNGD converges for any constant learning rate $\varepsilon \in ( 0 , 1 ]$ , regardless of the conditioning of the regression problem. This is in stark contrast with GD, where the condition number of the problem adversely affect stability and convergence. Many insights can be distilled from the analysis of the OLS model. For instance, we may attribute the stability of BNGD to an interesting scaling law governing $\varepsilon$ and the initial condition; This scaling law is not present in GD. The preceding analysis also implies that if we are allowed to use different learning rates for the BN rescaling variables $( \varepsilon _ { a } )$ and the remaining trainable variables $( \varepsilon )$ , we may conclude that BNGD on our model converges for any $\varepsilon > 0$ as long as ${ \varepsilon } _ { a } \in ( 0 , 1 ]$ . Furthermore, we discover an acceleration effect of BNGD and moreover, there exist regions of $\varepsilon$ such that the performance of BNGD is insensitive to changes in $\varepsilon$ , which help to explain the robustness of BNGD to the choice of learning rates. We reiterate that contrary to many previous works, all the preceding statements are precise mathematical results that we derive for our simplified model.
+
+The last step in our modeling approach is also the most important: we need to demonstrate that these insights are not specific features of our idealized model. Indeed, they should be true characteristics, at least in an approximate sense, of BNGD for general supervised learning problems. We do this by numerically investigating the convergence, stability and scaling behaviors of BNGD on various datasets and model architectures. We find that the key insights derived from our idealized analysis indeed correspond to practical scenarios.
+
+# 1.1 RELATED WORK
+
+Batch normalization was originally introduced in (Ioffe & Szegedy, 2015) and subsequently studied in further detail in (Ioffe, 2017). Since its introduction, it has become an important practical tool to improve stability and efficiency of training deep neural networks (He et al., 2016; Bottou et al., 2018). Initial heuristic arguments attribute the desirable features of BN to concepts such as “covariate shift”, which lacks a concrete mathematical interpretation and alternative explanations have been given (Santurkar et al., 2018). Recent theoretical studies of BN includes (Ma & Klabjan, 2017), where the authors proposed a variant of BN, the diminishing batch normalization (DBN) algorithm and analyzed the convergence of the DBN algorithm, showing that it converges to a stationary point of the loss function. More recently, (Bjorck et al., 2018) demonstrated that the higher learning rates of batch normalization induce a regularizing effect. Another related work is (Kohler et al., 2018), where the authors also considered the convergence properties of BNGD on linear networks (similar to the least-squares problem), as well as other special problems, such as learning halfspaces and extensions. In the OLS case, the authors showed that for a particularly adaptive choice of dynamic learning rate schedule, which can be seen as a fixed effective step size in our terminology (see equation (11) and the discussion that immediately follows), BNGD converges linearly if $\lambda _ { m a x }$ is known. Moreover, the analysis also requires setting the rescaling parameter $a$ every step to satisfy a stationarity condition, instead of simply performing gradient descent on $a$ , as is done in the original BNGD.
+
+The present research differs from these previous analysis in an important way - we study the BNGD algorithm itself, and not a special variant. More specifically, we consider constant learning rates (without knowledge of properties of the OLS loss function) and we perform gradient descent on rescaling parameters. We prove that the convergence occurs for even in this case (and in fact, for arbitrarily large learning rates for $\varepsilon$ , as long as $0 < \varepsilon _ { a } \le 1 _ { , } ^ { }$ ). This poses more challenges in the analysis and contrasts our work with previous analysis on modified versions of BNGD. This is an important distinction; While a decaying or dynamic learning rate is sometimes used in practice, in the case of BN it is critical to analyze the non-asymptotic, constant learning rate case, precisely because one of the key practical advantages of BN is that a bigger learning rate can be used than that in GD. Hence, it is desirable, as in the results presented in this work, to perform our analysis in this regime.
+
+Finally, through the lens of the least-squares example, BN can be viewed as a type of overparameterization, where additional parameters, which do not increase model expressivity, are introduced to improve algorithm convergence and stability. In this sense, this is related in effect to the recent analysis of the implicit acceleration effects of over-parameterization on gradient descent (Arora et al., 2018).
+
+# 1.2 ORGANIZATION
+
+Our paper is organized as follows. In Section 2, we outline the ordinary least squares (OLS) problem and present GD and BNGD as alternative means to solve this problem. In Section 3, we demonstrate and analyze the convergence of the BNGD for the OLS model, and in particular contrast the results with the behavior of GD, which is completely known for this model. We also discuss the important insights to BNGD that these results provide us with. We then validate these findings on more general supervised learning problems in Section 4. Finally, we conclude in Section 5.
+
+# 2 BACKGROUND
+
+Consider the simple linear regression model where $x \in \mathbb { R } ^ { d }$ is a random input column vector and $y$ is the corresponding output variable. Since batch normalization is applied for each feature separately, in order to gain key insights it is sufficient to the case of $y \in \mathbb { R }$ . A noisy linear relationship is assumed between the dependent variable $y$ and the independent variables $x$ , i.e. $y = x ^ { T } w + n \dot { o } i s e$ where $w \in \mathbb { R } ^ { d }$ is the parameters. Denote the following moments:
+
+$$
+H : = E [ x x ^ { T } ] , \quad g : = E [ x y ] , \quad c : = E [ y ^ { 2 } ] .
+$$
+
+To simplify the analysis, we assume the covariance matrix $H$ of $x$ is positive definite and the mean $E [ x ]$ of $x$ is zero. The eigenvalues of $H$ are denoted as $\lambda _ { i } ( H ) , i = 1 , 2 , . . . d ,$ . Particularly, the maximum and minimum eigenvalue of $H$ is denoted by $\lambda _ { m a x }$ and $\lambda _ { m i n }$ respectively. The condition number of $H$ is defined as $\begin{array} { r } { \bar { { \kappa } } : = \frac { \lambda _ { m a x } } { \lambda _ { m i n } } } \end{array}$ . Note that the positive definiteness of $H$ allows us to define the vector norms $\left\| . \right\| _ { H }$ and $\lVert . \rVert _ { H ^ { - 1 } }$ by $\| x \| _ { H } ^ { 2 } = x ^ { T } H x$ and $\| x \| _ { H ^ { - 1 } } ^ { 2 } = x ^ { T } H ^ { - 1 } x$ respectively.
+
+# 2.1 ORDINARY LEAST SQUARES
+
+The ordinary least squares (OLS) method for estimating the unknown parameters $w$ leads to the following optimization problem
+
+$$
+\begin{array} { r } { \underset { w \in \mathbb { R } ^ { d } } { \operatorname* { m i n } } J _ { 0 } ( w ) : = \frac { 1 } { 2 } E _ { x , y } [ ( y - x ^ { T } w ) ^ { 2 } ] = \frac { c } { 2 } - g ^ { T } w + \frac { 1 } { 2 } w ^ { T } H w . } \end{array}
+$$
+
+The gradient of $J _ { 0 }$ with respect to $w$ is $\nabla _ { w } J _ { 0 } ( w ) = H w - g$ , and the unique minimizer is $w =$ $u : = H ^ { - 1 } g$ . The gradient descent (GD) method (with step size or learning rate $\varepsilon$ ) for solving the optimization problem (2) is given by the iterating sequence,
+
+$$
+w _ { k + 1 } = w _ { k } - \varepsilon \nabla _ { w } J _ { 0 } ( w _ { k } ) = ( I - \varepsilon H ) w _ { k } + \varepsilon g ,
+$$
+
+which converges if 0 < ε < 2λmax , and the convergence rate is determined by the spectral radius $\rho _ { \varepsilon } : = \rho ( I - \varepsilon H ) = \operatorname* { m a x } _ { i } \{ | 1 - \varepsilon \lambda _ { i } ( H ) | \}$ with
+
+$$
+\lVert u - w _ { k + 1 } \rVert \leq \rho _ { \varepsilon } \lVert u - w _ { k } \rVert .
+$$
+
+It is well known (for example see Chapter 4 of (Saad, 2003)) that the optimal learning rate is $\varepsilon _ { o p t } =$ $\frac { 2 } { \lambda _ { m a x } + \lambda _ { m i n } }$ , where the convergence estimate is related to the condition number $\kappa ( H )$ :
+
+$$
+\begin{array} { r } { \| u - w _ { k + 1 } \| \leq \frac { \kappa - 1 } { \kappa + 1 } \| u - w _ { k } \| . } \end{array}
+$$
+
+# 2.2 BATCH NORMALIZATION
+
+Batch normalization is a feature-wise normalization procedure typically applied to the output, which in this case is simply $z = x ^ { T } w$ . The normalization transform is defined as follows:
+
+$$
+\begin{array} { r } { N _ { B N } ( z ) : = \frac { z - E [ z ] } { \sqrt { \mathrm { V a r } [ z ] } } = \frac { x ^ { T } w } { \sigma } , } \end{array}
+$$
+
+where $\sigma : = \sqrt { w ^ { T } H w }$ . After this rescaling, $N _ { B N } ( z )$ will be order 1, and hence in order to reintroduce the scale (Ioffe & Szegedy, 2015), we multiply $N _ { B N } ( z )$ with a rescaling parameter $a$ (Note that the shift parameter can be set zero since $\mathbb { E } [ w ^ { \hat { T } } x ] w ] = 0 \mathrm { . }$ ). Hence, we get the BN version of the OLS problem (2):
+
+$$
+\begin{array} { r l } & { \underset { w \in \mathbb { R } ^ { d } , a \in \mathbb { R } } { \mathrm { m i n } } J ( a , w ) : = \frac { 1 } { 2 } E _ { x , y } \big [ \big ( y - a N _ { B N } ( x ^ { T } w ) \big ) ^ { 2 } \big ] = \frac { c } { 2 } - \frac { w ^ { T } g } { \sigma } a + \frac { 1 } { 2 } a ^ { 2 } . } \end{array}
+$$
+
+The objective function ${ J } ( a , w )$ is no longer convex. In fact, it has trivial critical points, $\{ ( a ^ { * } , w ^ { * } ) | a ^ { * } = 0 , w ^ { * T } g = \operatorname { \bar { 0 } } \}$ , which are saddle points of ${ J } ( a , w )$ .
+
+We are interested in the nontrivial critical points which satisfy the relations,
+
+It is easy to check that the nontrivial critical points are global minimizers, and the Hessian matrix at each critical point is degenerate. Nevertheless, the saddle points are strict (Details can be found in Appendix), which typically simplifies the analysis of gradient descent on non-convex objectives (Lee et al., 2016; Panageas & Piliouras, 2017).
+
+Consider the gradient descent method to solve the problem (7), which we hereafter call batch normalization gradient descent (BNGD). We set the learning rates for $a$ and $w$ to be $\varepsilon _ { a }$ and $\varepsilon$ respectively. These may be different, for reasons which will become clear in the subsequent analysis. We thus have the following discrete-time dynamical system
+
+$$
+\begin{array} { r l } & { a _ { k + 1 } = a _ { k } + \varepsilon _ { a } \Big ( \frac { w _ { k } ^ { T } g } { \sigma _ { k } } - a _ { k } \Big ) , } \\ & { w _ { k + 1 } = w _ { k } + \varepsilon \frac { a _ { k } } { \sigma _ { k } } \Big ( g - \frac { w _ { k } ^ { T } g } { \sigma _ { k } ^ { 2 } } H w _ { k } \Big ) . } \end{array}
+$$
+
+We now begin a concrete mathematical analysis of the above iteration sequence.
+
+# 3 MATHEMATICAL ANALYSIS OF BNGD ON OLS
+
+In this section, we discuss several mathematical results one can derive concretely for BNGD on the OLS problem (7). First, we establish a simple but useful scaling property, which an important ingredient in allowing us to prove a linear convergence result for arbitrary constant learning rates. We also derive the asymptotic properties of the “effective” learning rate of BNGD (to be precisely defined subsequently), which shows some interesting sensitivity behavior of BNGD on the chosen learning rates. Detailed proofs of all results presented here can be found in the Appendix.
+
+# 3.1 SCALING PROPERTY
+
+In this section, we discuss a straightforward, but useful scaling property that the BNGD iterations possess. Note that the dynamical properties of the BNGD iteration are governed by a set of numbers, or a configuration $\{ H , u , a _ { 0 } , w _ { 0 } , \varepsilon _ { a } , \varepsilon \}$ .
+
+Definition 3.1 (Equivalent configuration). Two configurations, $\{ H , u , a _ { 0 } , w _ { 0 } , \varepsilon _ { a } , \varepsilon \}$ and $\{ H ^ { \prime } , u ^ { \prime } , a _ { 0 } ^ { \prime } , w _ { 0 } ^ { \prime } , \varepsilon _ { a } ^ { \prime } , \varepsilon ^ { \prime } \}$ , are said to be equivalent if for iterates $\{ w _ { k } \}$ , $\{ w _ { k } ^ { \prime } \}$ following these configurations respectively, there is an invertible linear transformation $T$ and a nonzero constant $t$ such that $w _ { k } ^ { \prime } = T w _ { k } , a _ { k } ^ { \prime } = t a _ { k }$ for all $k$ .
+
+The scaling property ensures that equivalent configurations must converge or diverge together, with the same rate up to a constant multiple. Now, it is easy to check the system has the following scaling law.
+
+Proposition 3.2 (Scaling property). Suppose $\mu \neq 0 , \gamma \neq 0 , r \neq 0 , Q ^ { T } Q = I _ { \mathit { i } }$ , then (1) The configurations $\{ \mu Q ^ { T } H Q , \frac { \gamma } { \sqrt { \mu } } Q u , \gamma a _ { 0 } , \gamma Q w _ { 0 } , \varepsilon _ { a } , \varepsilon \} a n d \ \{ H , u , a _ { 0 } , w _ { 0 } , \varepsilon _ { a } , \varepsilon \}$ are equivalent. (2) The configurations $\{ H , u , a _ { 0 } , w _ { 0 } , \varepsilon _ { a } , \varepsilon \}$ and $\{ H , u , a _ { 0 } , r w _ { 0 } , \varepsilon _ { a } , r ^ { 2 } \varepsilon \}$ are equivalent.
+
+It is worth noting that the scaling property (2) in Proposition 3.2 originates from the batchnormalization procedure and is independent of the specific structure of the loss function. Hence, it is valid for general problems where BN is used (Lemma A.9). Despite being a simple result, the scaling property is important in determining the dynamics of BNGD, and is useful in our subsequent analysis of its convergence and stability properties (see the sketch of the proof of Theorem 3.3).
+
+# 3.2 BATCH NORMALIZATION CONVERGES FOR ARBITRARY STEP SIZE
+
+We have the following convergence result.
+
+Theorem 3.3 (Convergence for BNGD). The iteration sequence $( a _ { k } , w _ { k } )$ in equation (9)-(10) converges to a stationary point for any initial value $( a _ { 0 } , w _ { 0 } )$ and any $\varepsilon > 0 _ { i }$ , as long as $\varepsilon _ { a } \in ( 0 , 1 ]$ . Particularly, we have the following sufficient conditions of converging to global minimizers.
+
+(1) If $a _ { 0 } w _ { 0 } ^ { T } g > 0$ (or $a _ { 0 } = 0 , w _ { 0 } ^ { T } g \neq 0 )$ , $\varepsilon _ { a } \in ( 0 , 1 ]$ and $\varepsilon$ is sufficiently small (the smallness is quantified by Lemma A.13), then $( a _ { k } , w _ { k } )$ converges to a global minimizer.
+
+(2) If ${ { \varepsilon } _ { a } } \mathrm { ~ = ~ } 1$ and $\varepsilon > 0$ , then $( a _ { k } , w _ { k } )$ converges to global minimizers for almost all initial values $( a _ { 0 } , w _ { 0 } )$ .
+
+# Sketch of Proof.
+
+We first prove that the algorithm converges for any $\varepsilon _ { a } \in ( 0 , 1 ]$ and small enough $\varepsilon$ , with any initial value $( a _ { 0 } , w _ { 0 } )$ such that $\left\| w _ { 0 } \right\| \ge 1$ (Lemma A.13). Next, we observe that the sequence $\{ \| w _ { k } \| \}$ is monotone increasing, and thus either converges to a finite limit or diverges. The scaling property is then used to exclude the divergent case if $\{ \| w _ { k } \| \}$ diverges, then at some $k$ the norm $\| w _ { k } \|$ should be large enough, and by the scaling property, it is equivalent to a case where $\lVert \boldsymbol { w } _ { k } \rVert = 1$ and $\varepsilon$ is small, which we have proved converges. This shows that $\| w _ { k } \|$ converges to a finite limit, from which the convergence of $w _ { k }$ and the loss function value can be established, after some work. The proof is fully presented in Theorem A.17 and preceding Lemmas.
+
+In addition, using the ’strict saddle point’ arguments in (Lee et al., 2016; Panageas & Piliouras, 2017), we can prove the set of initial value for which $( a _ { k } , w _ { k } )$ converges to saddle points has Lebesgue measure 0, provided some conditions, such as when $\varepsilon _ { a } = 1 , \varepsilon > 0$ (Lemma A.20). It is important to note that BNGD converges for all step size $\varepsilon > 0$ of $w _ { k }$ , independent of the spectral properties of $H$ . This is a significant advantage and is in stark contrast with GD, where the step size is limited by $\lambda _ { \operatorname* { m a x } } ( H )$ , and the condition number of $H$ intimately controls the stability and convergence rate. Although we only prove the almost sure convergence to global minimizer for the case of ${ \varepsilon } _ { a } = 1$ , we have not encountered convergence to saddles in the OLS experiments even for $ { \varepsilon } _ { a } \in ( 0 , 2 )$ with initial values $( a _ { 0 } , w _ { 0 } )$ drawn from typical distributions.
+
+# 3.3 CONVERGENCE RATE, ACCELERATION AND ASYMPTOTIC SENSITIVITY
+
+Now, let us consider the convergence rate of BNGD when it converges to a minimizer. Compared with GD, the update coefficient before $H w _ { k }$ in equation (10) changed from $\varepsilon$ to a complicated term which we named as the effective step size or learning rate $\hat { \varepsilon } _ { k }$
+
+$$
+\begin{array} { r } { \hat { \varepsilon } _ { k } : = \varepsilon \frac { a _ { k } } { \sigma _ { k } } \frac { w _ { k } ^ { T } g } { \sigma _ { k } ^ { 2 } } , } \end{array}
+$$
+
+and the recurrence relation in place of $u - w _ { k }$ is
+
+$$
+\begin{array} { r } { u - \frac { w _ { k } ^ { T } g } { \sigma _ { k } ^ { 2 } } w _ { k + 1 } = ( I - \hat { \varepsilon } _ { k } H ) \Big ( u - \frac { w _ { k } ^ { T } g } { \sigma _ { k } ^ { 2 } } w _ { k } \Big ) . } \end{array}
+$$
+
+Consider the dynamics of the residual $e _ { k } : = u - ( w _ { k } ^ { T } g / \sigma _ { k } ^ { 2 } ) w _ { k }$ , which equals 0 if and only if $w _ { k }$ is a global minimizer. Using the property of $H$ -norm (see section A.1), we observe that the effective learning rate $\hat { \varepsilon } _ { k }$ determines the convergence rate of $e _ { k }$ via
+
+$$
+\begin{array} { r } { \| e _ { k + 1 } \| _ { H } \le \left\| u - \frac { w _ { k } ^ { T } g } { \sigma _ { k } ^ { 2 } } w _ { k + 1 } \right\| _ { H } \le \rho ( I - \hat { \varepsilon } _ { k } H ) \| e _ { k } \| _ { H } , } \end{array}
+$$
+
+where $\rho ( I - \hat { \varepsilon } _ { k } H )$ is spectral radius of the matrix $I { - } \hat { \varepsilon } _ { k } H$ . The inequality (13) shows that the convergence of $e _ { k }$ (and hence the loss function, see Lemma A.23) is linear provided $\hat { \varepsilon } _ { k } \in ( \delta , 2 / \lambda _ { m a x } - \delta )$ for some positive number $\delta$ . In fact, if we enforce $\hat { \varepsilon } _ { k } = 1 / \lambda _ { m a x }$ for each $k$ , which is done in the analysis in Kohler et al. (2018), then one immediately obtains the same linear convergence rate. But this requires knowledge of $\lambda _ { m a x }$ (problem-dependent) and a modification the BNGD algorithm. We instead focus our analysis on the original BNGD algorithm.
+
+Next, let us discuss below an acceleration effect of BNGD over GD. When $( a _ { k } , w _ { k } )$ is close to a minimizer, we can approximate the iteration (9)-(10) by a linearized system. The Hessian matrix for BNGD at a minimizer $( \boldsymbol { a } ^ { * } , \boldsymbol { w } ^ { * } )$ is $\mathrm { d i a g } ( 1 , H ^ { * } / \| w ^ { * } \| ^ { 2 } )$ , where the matrix $H ^ { * }$ is
+
+$$
+\begin{array} { r } { H ^ { * } = H - \frac { H u u ^ { T } H } { u ^ { T } H u } . } \end{array}
+$$
+
+The matrix $H ^ { * }$ is positive semi-definite $( H ^ { * } u = 0 )$ and has better spectral properties than $H$ , such as a lower pseudo-condition number $\begin{array} { r } { \kappa ^ { * } ~ = ~ \frac { \lambda _ { m a x } ^ { * } } { \lambda _ { m i n } ^ { * } } ~ \le ~ \kappa } \end{array}$ , where $\lambda _ { m a x } ^ { * }$ and $\lambda _ { m i n } ^ { * }$ are the maximal and minimal nonzero eigenvalues of $H ^ { * }$ respectively. Particularly, $\kappa ^ { * } < \kappa$ for almost all $u$ (see section A.1 ). This property leads to acceleration effects of BNGD: When $\| e _ { k } \| _ { H }$ is small, the contraction coefficient $\rho$ in (13) can be improved to a lower coefficient. More precisely, we have the following result:
+
+Proposition 3.4. For any positive number $\delta \in ( 0 , 1 )$ , $i f \left( a _ { k } , w _ { k } \right)$ is close to a minimizer, such that λmaxε|ak|σ2 kekkH ≤ δ, then we have
+
+$$
+\begin{array} { r l } & { \| e _ { k + 1 } \| _ { H } \leq \operatorname* { m i n } \{ \frac { \rho ^ { * } ( I - \hat { \varepsilon } _ { k } H ^ { * } ) + \delta } { 1 - \delta } , \rho ( I - \hat { \varepsilon } _ { k } H ) \} \| e _ { k } \| _ { H } , } \\ & { \rho ^ { * } ( I - \hat { \varepsilon } _ { k } H ) = \operatorname* { m a x } \{ | 1 - \hat { \varepsilon } _ { k } \lambda _ { m i n } ^ { * } | , | 1 - \hat { \varepsilon } _ { k } \lambda _ { m a x } ^ { * } | \} . } \end{array}
+$$
+
+Generally, we have $\rho ^ { * } ( I - \hat { \varepsilon } _ { k } H ^ { * } ) \leq \rho ( I - \hat { \varepsilon } _ { k } H )$ provided $\hat { \varepsilon } _ { k } ~ > ~ 0$ , and the optimal rate is $\begin{array} { r } { \rho _ { o p t } ^ { * } : = \frac { \kappa ^ { * } - 1 } { \kappa ^ { * } + 1 } \leq \frac { \kappa - 1 } { \kappa + 1 } = : \rho _ { o p t } } \end{array}$ , where the inequality is strict for almost all $u$ . Hence, the estimate (15) indicates that the optimal BNGD could have a faster convergence rate than the optimal GD, especially when $\kappa ^ { * }$ is much smaller than $\kappa$ .
+
+Finally, we discuss the dependence of the effective learning rate $\hat { \varepsilon } _ { k }$ (and by extension, the effective convergence rate (13) or (15)) on $\varepsilon$ . This is in essence a sensitivity analysis on the performance of BNGD with respect to the choice of learning rate. The explicit dependence of $\hat { \varepsilon } _ { k }$ on $\varepsilon$ is quite complex, but we can nevertheless give the following asymptotic estimates.
+
+Proposition 3.5. Suppose $\varepsilon _ { a } \in ( 0 , 1 ] , a _ { 0 } w _ { 0 } ^ { T } g > 0$ , and $\begin{array} { r } { | | g | | ^ { 2 } \geq \frac { w _ { 0 } ^ { T } g } { \sigma _ { 0 } ^ { 2 } } g ^ { T } H w _ { 0 } } \end{array}$ , then
+
+(1) When $\varepsilon$ is small enough, $\varepsilon \ll 1$ , the effective step size has a same order with $\varepsilon$ , i.e. there are two positive constants, $C _ { 1 } , C _ { 2 }$ , independent on $\varepsilon$ and $k$ , such that $C _ { 1 } \leq \hat { \varepsilon } _ { k } / \varepsilon \leq C _ { 2 }$ .   
+(2) When $\varepsilon$ is large enough, $\varepsilon \gg 1$ , the effective step size has order $O ( \varepsilon ^ { - 1 } )$ , i.e. there are two positive constants, $C _ { 1 } , C _ { 2 }$ , independent on $\varepsilon$ and $k$ , such that $C _ { 1 } \leq \hat { \varepsilon } _ { k } \varepsilon \leq C _ { 2 }$ .
+
+Observe that for finite $k$ , $\hat { \varepsilon } _ { k }$ is a differentiable function of $\varepsilon$ . Therefore, the above result implies, via the mean value theorem, the existence of some $\varepsilon _ { 0 } > 0$ such that $d \hat { \varepsilon } _ { k } / d \varepsilon | _ { \varepsilon = \varepsilon _ { 0 } } = 0$ . Consequently, there is at least some small interval of the choice of learning rates $\varepsilon$ where the performance of BNGD is insensitive to this choice. In fact, empirically this is one commonly observed advantage of BNGD over GD, where the former typically allows for a variety of (large) learning rates to be used without adversely affecting performance. The same is not true for GD, where the convergence rate depends sensitively on the choice of learning rate. We will see later in Section 4 that although we only have a local insensitivity result above, the interval of this insensitivity is actually quite large in practice.
+
+# 4 EXPERIMENTS
+
+Let us first summarize our key findings and insights from the analysis of BNGD on the OLS problem.
+
+1. A scaling law governs BNGD, where certain configurations can be deemed equivalent   
+2. BNGD converges for any learning rate $\varepsilon > 0$ , provided that ${ { \varepsilon } _ { a } } \in \left( 0 , 1 \right]$ . In particular, different learning rates can be used for the BN variables $( a )$ compared with the remaining trainable variables $( w )$ )   
+3. There exists intervals of $\varepsilon$ for which the performance of BNGD is not sensitive to the choice of $\varepsilon$
+
+In the subsequent sections, we first validate numerically these claims on the OLS model, and then show that these insights go beyond the simple OLS model we considered in the theoretical framework. In fact, much of the uncovered properties are observed in general applications of BNGD in deep learning.
+
+# 4.1 EXPERIMENTS ON OLS
+
+Here we test the convergence and stability of BNGD for the OLS model. Consider a diagonal matrix $H = d i a g ( h )$ where $h = ( 1 , . . . , \kappa )$ is a increasing sequence. The scaling property (Proposition 3.2) allows us to set the initial value $w _ { 0 }$ having same 2-norm with $u$ , $\| w _ { 0 } \| = \| u \| = 1$ . Of course, one can verify that the scaling property holds strictly in this case.
+
+Figure 1 gives examples of $H$ with different condition numbers $\kappa$ . We tested the loss function of BNGD, compared with the optimal GD (i.e. GD with the optimal step size $\varepsilon _ { o p t }$ ), in a large range of step sizes $\varepsilon _ { a }$ and $\varepsilon$ , and with different initial values of $a _ { 0 }$ . Another quantity we observe is the effective step size $\hat { \varepsilon } _ { k }$ of BN. The results are encoded by four different colors: whether $\hat { \varepsilon } _ { k }$ is close to the optimal step size $\varepsilon _ { o p t }$ , and whether loss of BNGD is less than the optimal GD. The results indicate that the optimal convergence rate of BNGD can be better than GD in some configurations. This acceleration phenomenon is ascribed to the pseudo-condition number of $H ^ { * }$ (discard the only zero eigenvalue) being less than $\kappa ( H )$ . This advantage of BNGD is significant when the (pseudo)- condition number discrepancy between $H$ and $H ^ { * }$ is large. However, if this difference is small, the acceleration is imperceptible. This is consistent with our analysis in section 3.3.
+
+Another important observation is a region such that $\hat { \varepsilon }$ is close to $\varepsilon _ { o p t }$ , in other words, BNGD significantly extends the range of ‘optimal’ step sizes. Consequently, we can choose step sizes in BNGD at greater liberty to obtain almost the same or better convergence rate than the optimal GD. However, the size of this region is inversely dependent on the initial condition $a _ { 0 }$ . Hence, this suggests that small $a _ { 0 }$ at first steps may improve robustness. On the other hand, small $\varepsilon _ { a }$ will weaken the performance of BN. The phenomenon suggests that improper initialization of the BN parameters weakens the power of BN. This experience is encountered in practice, such as (Cooijmans et al., 2016), where higher initial values of BN parameter are detrimental to the optimization of RNN models.
+
+![](images/f7b9a0d2300589b6a43ee02360f42905e40644a42a566e7e6f0891ea4eca7eb4.jpg)  
+Figure 1: Compare of BNGD and GD on OLS model. The results are encoded by four different colors: whether $\hat { \varepsilon } _ { k }$ is close to the optimal step size $\varepsilon _ { o p t }$ of GD, characterized by the inequality $0 . 8 \varepsilon _ { o p t } < \hat { \varepsilon } _ { k } < \varepsilon _ { o p t } / 0 . 8$ , and whether loss of BNGD is less than the optimal GD. Parameters: $H =$ diag(logspace $\mathrm { 0 , l o g 1 0 } ( \kappa ) , 1 0 0 )$ ), $u$ is randomly chosen uniformly from the unit sphere in $\mathbb { R } ^ { 1 0 0 }$ , $w _ { 0 }$ is set to $H u / \lVert H u \rVert$ . The GD and BNGD iterations are executed for $k = 2 0 0 0$ steps with the same $w _ { 0 }$ . In each image, the range of $\varepsilon _ { a }$ $\mathbf { \bar { X } }$ -axis) is $1 . 9 9 ~ ^ { \ast }$ logspace(-10,0,41), and the range of $\varepsilon$ (y-axis) is logspace(-5,16,43).
+
+# 4.2 EXPERIMENTS ON PRACTICAL DEEP LEARNING PROBLEMS
+
+We conduct experiments on deep learning applied to standard classification datasets: MNIST (LeCun et al., 1998), Fashion MNIST (Xiao et al., 2017) and CIFAR-10 (Krizhevsky & Hinton, 2009). The goal is to explore if the key findings outlined at the beginning of this section continue to hold for more general settings. For the MNIST and Fashion MNIST dataset, we use two different networks: (1) a one-layer fully connected network $7 8 4 \times 1 0 )$ with softmax mean-square loss; (2) a fourlayer convolution network (Conv-MaxPool-Conv-MaxPool-FC-FC) with ReLU activation function and cross-entropy loss. For the CIFAR-10 dataset, we use a five-layer convolution network (ConvMaxPool-Conv-MaxPool-FC-FC-FC). All the trainable parameters are randomly initialized by the Glorot scheme (Glorot & Bengio, 2010) before training. For all three datasets, we use a minibatch size of 100 for computing stochastic gradients. In the BNGD experiments, batch normalization is performed on all layers, the BN parameters are initialized to transform the input to zero mean/unit√ variance distributions, and a small regularization parameter $\epsilon = 1 \mathrm { e } { - 3 }$ is added to variance $\sqrt { \sigma ^ { 2 } + \epsilon }$ to avoid division by zero.
+
+Scaling property Theoretically, the scaling property 3.2 holds for any layer using BN. However, it may be slightly biased by the regularization parameter $\epsilon$ . Here, we test the scaling property in practical settings. Figure 2 gives the loss of network-(2) $( 2 \mathrm { C N N } { + } 2 \mathrm { F C } )$ at epoch $^ { = 1 }$ with different learning rate. The norm of all weights and biases are rescaled by a common factor $\eta$ . We observe that the scaling property remains true for relatively large $\eta$ . However, when √ $\eta$ is small, the norm of weights are small. Therefore, the effect of the $\epsilon$ -regularization in $\sqrt { \sigma ^ { 2 } + \epsilon }$ becomes significant, causing the curves to be shifted.
+
+Stability for large learning rates We use the loss value at the end of the first epoch to characterize the performance of BNGD and GD methods. Although the training of models have generally not converged at this point, it is enough to extract some relative rate information. Figure 3 shows the loss value of the networks on the three datasets. It is observed that GD and BNGD with identical learning rates for weights and BN parameters exhibit a maximum allowed learning rate, beyond which the iterations becomes unstable. On the other hand, BNGD with separate learning rates exhibits a much larger range of stability over learning rate for non-BN parameters, consistent with our theoretical results in Theorem 3.3.
+
+Insensitivity of performance to learning rates Observe that BN accelerates convergence more significantly for deep networks, whereas for one-layer networks, the best performance of BNGD and
+
+![](images/cb43fe810cb54789c944f786043b9a51a1dee42bf02561ca24b1eccc15dbe571.jpg)  
+Figure 2: Tests of scaling property of the $2 \mathrm { C N N } { + } 2 \mathrm { F C }$ network on MNIST dataset. BN is performed on all layers, and $\scriptstyle \epsilon = 1 \mathrm { e } - 3$ is added to variance $\sqrt { \sigma ^ { 2 } + \epsilon }$ . All the trainable parameters (except the BN parameters) are randomly initialized by the Glorot scheme, and then multiplied by a same parameter $\eta$ .
+
+GD are similar. Furthermore, in most cases, the range of optimal learning rates in BNGD is quite large, which is in agreement with the OLS analysis (Proposition 3.5). This phenomenon is potentially crucial for understanding the acceleration of BNGD in deep neural networks. Heuristically, the “optimal” learning rates of GD in distinct layers (depending on some effective notion of “condition number”) may be vastly different. Hence, GD with a shared learning rate across all layers may not achieve the best convergence rates for all layers at the same time. In this case, it is plausible that the acceleration of BNGD is a result of the decreased sensitivity of its convergence rate on the learning rate parameter over a large range of its choice.
+
+![](images/d9a349ba4827a529205dcdf68c18f96aa7f05a755b83ddca159a56da26e398fa.jpg)  
+Figure 3: Performance of BNGD and GD method on MNIST (network-(1), 1FC), Fashion MNIST (network-(2), $2 \mathrm { C N N } { + } 2 \mathrm { F C } )$ and CIFAR-10 $2 \mathrm { C N N } { + } 3 \mathrm { F C }$ ) datasets. The performance is characterized by the loss value at ephoch $^ { = 1 }$ . In the BNGD method, both the shared learning rate schemes and separated learning rate scheme (learning rate lr a for BN parameters) are given. The values are averaged over 5 independent runs.
+
+# 5 CONCLUSION AND OUTLOOK
+
+In this paper, we adopted a modeling approach to investigate the dynamical properties of batch normalization. The OLS problem is chosen as a point of reference, because of its simplicity and the availability of convergence results for gradient descent. Even in such a simple setting, we saw that BNGD exhibits interesting non-trivial behavior, including scaling laws, robust convergence properties, acceleration, as well as the insensitivity of performance to the choice of learning rates. Although these results are derived only for the OLS model, we show via experiments that these are qualitatively valid for general scenarios encountered in deep learning, and points to a concrete way in uncovering the reasons behind the effectiveness of batch normalization.
+
+Interesting future directions include the extension of the results for the OLS model to more general settings of BNGD, where we believe the scaling law (Proposition 3.2) should play a significant role. In addition, we have not touched upon another empirically observed advantage of batch normalization, which is better generalization errors. It will be interesting to see how far the current approach takes us in investigating such probabilistic aspects of BNGD.
+
+# REFERENCES
+
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+Tim Cooijmans, Nicolas Ballas, Cesar Laurent, and Aaron C. Courville. Recurrent batch normal- ´ ization. CoRR, abs/1603.09025, 2016. URL http://arxiv.org/abs/1603.09025.
+
+Xavier Glorot and Yoshua Bengio. Understanding the difficulty of training deep feedforward neural networks. In Proceedings of the thirteenth international conference on artificial intelligence and statistics, pp. 249–256, 2010.
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+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2016.
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+Sergey Ioffe. Batch renormalization: Towards reducing minibatch dependence in batch-normalized models. CoRR, abs/1702.03275, 2017. URL http://arxiv.org/abs/1702.03275.
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+Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. In International Conference on Machine Learning, pp. 448–456, 2015.
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+J. Kohler, H. Daneshmand, A. Lucchi, M. Zhou, K. Neymeyr, and T. Hofmann. Towards a Theoretical Understanding of Batch Normalization. ArXiv e-prints, May 2018.
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+Steven G Krantz and Harold R Parks. A primer of real analytic functions. Springer Science & Business Media, 2002.
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+Alex Krizhevsky and Geoffrey Hinton. Learning multiple layers of features from tiny images. Technical report, Citeseer, 2009.
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+Yann LeCun, Leon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to ´ document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998.
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+J. D. Lee, M. Simchowitz, M. I. Jordan, and B. Recht. Gradient Descent Converges to Minimizers. ArXiv e-prints, February 2016.
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+Yintai Ma and Diego Klabjan. Convergence analysis of batch normalization for deep neural nets. CoRR, 1705.08011, 2017. URL http://arxiv.org/abs/1705.08011.
+
+Ioannis Panageas and Georgios Piliouras. Gradient Descent Only Converges to Minimizers: NonIsolated Critical Points and Invariant Regions. In Christos H. Papadimitriou (ed.), 8th Innovations in Theoretical Computer Science Conference (ITCS 2017), volume 67 of Leibniz International Proceedings in Informatics (LIPIcs), pp. 2:1–2:12, Dagstuhl, Germany, 2017. Schloss Dagstuhl– Leibniz-Zentrum fuer Informatik. ISBN 978-3-95977-029-3. doi: 10.4230/LIPIcs.ITCS.2017.2.
+
+S. P. Ponomarev. Submersions and preimages of sets of measure zero. Siberian Mathematical Journal, 28(1):153–163, Jan 1987. ISSN 1573-9260. doi: 10.1007/BF00970225. URL https: //doi.org/10.1007/BF00970225.
+
+Yousef Saad. Iterative methods for sparse linear systems, volume 82. siam, 2003.
+
+S. Santurkar, D. Tsipras, A. Ilyas, and A. Madry. How Does Batch Normalization Help Optimiza tion? (No, It Is Not About Internal Covariate Shift). ArXiv e-prints, May 2018.
+
+Michael Shub. Global stability of dynamical systems. Springer Science & Business Media, 2013.
+
+Han Xiao, Kashif Rasul, and Roland Vollgraf. Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms, 2017.
+
+# A PROOF OF THEOREMS
+
+# A.1 GRADIENTS AND HESSIAN MATRIX
+
+The objective function in problem (7) has an equivalent form:
+
+$$
+\begin{array} { r } { J ( { a } , { w } ) = \frac { 1 } { 2 } ( { u } - \frac { { a } } { \sigma } { w } ) ^ { T } H ( { u } - \frac { { a } } { \sigma } { w } ) = \frac { 1 } { 2 } \| { u } \| _ { H } ^ { 2 } - \frac { { w } ^ { T } g } { \sigma } { a } + \frac { 1 } { 2 } { a } ^ { 2 } , } \end{array}
+$$
+
+where $u = H ^ { - 1 } g$
+
+The gradients are:
+
+$$
+\begin{array} { r l } & { \frac { \partial J } { \partial a } = - \frac { 1 } { \sigma } ( \boldsymbol { w } ^ { T } H \boldsymbol { u } - \frac { a } { \sigma } \boldsymbol { w } ^ { T } H \boldsymbol { w } ) = - \frac { 1 } { \sigma } \boldsymbol { w } ^ { T } \boldsymbol { g } + \boldsymbol { a } , } \\ & { \frac { \partial J } { \partial \boldsymbol { w } } = - \frac { a } { \sigma } ( H \boldsymbol { u } - \frac { a } { \sigma } H \boldsymbol { w } ) + \frac { a } { \sigma ^ { 3 } } ( \boldsymbol { w } ^ { T } H \boldsymbol { u } - \frac { a } { \sigma } \boldsymbol { w } ^ { T } H \boldsymbol { w } ) H \boldsymbol { w } = - \frac { a } { \sigma } \boldsymbol { g } + \frac { \boldsymbol { a } } { \sigma ^ { 3 } } ( \boldsymbol { w } ^ { T } \boldsymbol { g } ) H \boldsymbol { w } . } \end{array}
+$$
+
+The Hessian matrix is
+
+$$
+\left( \begin{array} { c c } { { \frac { \partial ^ { 2 } J } { \partial a ^ { 2 } } } } & { { \frac { \partial ^ { 2 } J } { \partial a \partial w } } } \\ { { \frac { \partial ^ { 2 } J } { \partial w \partial a } } } & { { \frac { \partial ^ { 2 } J } { \partial w ^ { 2 } } } } \end{array} \right) = \left( \begin{array} { c c } { { 1 } } & { { A _ { 2 1 } ^ { T } } } \\ { { A _ { 2 1 } } } & { { A _ { 2 2 } } } \end{array} \right)
+$$
+
+where
+
+$$
+\begin{array} { r l } & { A _ { 2 2 } = \frac { a } { \sigma ^ { 3 } } ( w ^ { T } g ) \Big [ H + \frac { 1 } { w ^ { T } g } \big ( ( H w ) g ^ { T } + g ( H w ) ^ { T } \big ) - \frac { 3 } { \sigma ^ { 2 } } ( H w ) ( H w ) ^ { T } \Big ] , } \\ & { A _ { 2 1 } = - \frac { 1 } { \sigma } \big ( g - \frac { 1 } { \sigma ^ { 2 } } ( w ^ { T } g ) H w \big ) . } \end{array}
+$$
+
+The objective function ${ J } ( a , w )$ has trivial critical points, $\{ ( a ^ { * } , w ^ { * } ) | a ^ { * } = 0 , w ^ { * T } g = 0 \}$ . It is obvious that $a ^ { * }$ is the minimizer of ${ \cal J } ( a , w ^ { * } )$ , but $( \boldsymbol { a } ^ { * } , \boldsymbol { w } ^ { * } )$ is not a local minimizer of ${ J } ( a , w )$ unless $g = 0$ , hence $( \boldsymbol { a } ^ { * } , \boldsymbol { w } ^ { * } )$ are saddle points of ${ J } ( a , w )$ . The Hessian matrix at those saddle points has at least a negative eigenvalue, i.e. the saddle points are strict. In fact, the eigenvalues at the saddle point $( \boldsymbol { a } ^ { * } , \boldsymbol { w } ^ { * } )$ are $\begin{array} { r } { \Big \{ \frac { 1 } { 2 } ( 1 \pm \sqrt { 1 + 4 \frac { \left\| g \right\| ^ { 2 } } { w ^ { * T } H w ^ { * } } } ) , 0 , . . . , 0 \Big \} } \end{array}$ which contains $d - 2$ repeated zero, a positive and a negative eigenvalue.
+
+On the other hand, the nontrivial critical points satisfies the relations,
+
+$$
+a ^ { * } = \pm \sqrt { u ^ { T } H u } , w ^ { * } / / u ,
+$$
+
+where the sign of $a ^ { * }$ depends on the direction of $u , w ^ { * }$ , i.e. $s i g n ( a ^ { * } ) = s i g n ( u ^ { T } w ^ { * } )$ . It is easy to check that the nontrivial critical points are global minimizers. The Hessian matrix at those minimizers is $d i a g ( 1 , H ^ { * } / \lVert w ^ { * } \rVert ^ { 2 } )$ where the matrix $H ^ { * }$ is
+
+$$
+\begin{array} { r } { H ^ { * } = H - \frac { H u u ^ { T } H } { u ^ { T } H u } } \end{array}
+$$
+
+which is positive semi-definite and has a zero eigenvalue corresponding to the eigenvector $u$ , i.e. $H ^ { * } u = 0$ . The following lemma, similar to the well known Cauchy interlacing theorem, gives an estimate of eigenvalues of $H ^ { * }$ .
+
+ma and If sa $H$ is positive definite and fy the following inequa $H ^ { * }$ is defined as s: $\begin{array} { r } { H ^ { * } = H - \frac { H u u ^ { T } H } { u ^ { T } H u } } \end{array}$ , then the eigenvalues $H$ $H ^ { * }$
+
+$$
+0 = \lambda _ { 1 } ( H ^ { * } ) < \lambda _ { 1 } ( H ) \leq \lambda _ { 2 } ( H ^ { * } ) \leq \lambda _ { 2 } ( H ) \leq \ldots \leq \lambda _ { d } ( H ^ { * } ) \leq \lambda _ { d } ( H ) .
+$$
+
+Here $\lambda _ { i } ( H )$ means the $i$ -th smallest eigenvalue of $H$ .
+
+Proof. (1) According to the definition, we have $H ^ { * } u = 0$ , and for any $\boldsymbol { x } \in \mathbb { R } ^ { d }$ ,
+
+$$
+\begin{array} { r } { x ^ { T } H ^ { \ast } x = x ^ { T } H x - \frac { ( x ^ { T } H u ) ^ { 2 } } { u ^ { T } H u } \in [ 0 , x ^ { T } H x ] , } \end{array}
+$$
+
+which implies $H ^ { * }$ is semi-positive definite, and $\lambda _ { i } ( H ^ { * } ) \geq \lambda _ { 1 } ( H ^ { * } ) = 0$ . Furthermore, we have the following equality:
+
+$$
+x ^ { T } H ^ { * } x = \operatorname* { m i n } _ { t \in \mathbb { R } } \| x - t u \| _ { H } ^ { 2 } .
+$$
+
+(2) We will prove $\lambda _ { i } ( H ^ { * } ) \leq \lambda _ { i } ( H )$ for all i $, 1 \leq i \leq d$ . In fact, using the Min-Max Theorem, we have
+
+$$
+\lambda _ { i } ( H ^ { * } ) = \operatorname* { m i n } _ { d i m V = i } \operatorname* { m a x } _ { x \in V } \frac { x ^ { T } H ^ { * } x } { \| x \| ^ { 2 } } \leq \operatorname* { m i n } _ { d i m V = i } \operatorname* { m a x } _ { x \in V } \frac { x ^ { T } H x } { \| x \| ^ { 2 } } = \lambda _ { i } ( H ) .
+$$
+
+(3) We will prove $\lambda _ { i } ( H ^ { * } ) \geq \lambda _ { i - 1 } ( H )$ for all $i$ , $2 \leq i \leq d$ . In fact, using the Max-Min Theorem, we have
+
+$$
+\begin{array} { r l } & { \lambda _ { i } ( H ^ { * } ) = \underset { d i m V = n - i + 1 } { \operatorname* { m a x } } \underset { x \in V } { \operatorname* { m i n } } \frac { x ^ { T } H ^ { * } x } { \| x \| ^ { 2 } } = \underset { d i m V = n - i + 1 , u \perp V } { \operatorname* { m a x } } \underset { x \in V } { \operatorname* { m i n } } \frac { \underset { \| x - t u \| _ { H } ^ { 2 } } { \operatorname* { m i n } } } { \operatorname* { m i n } } \frac { \| x - t u \| _ { H } ^ { 2 } } { \| x \| ^ { 2 } } } \\ & { \quad \quad \geq \underset { d i m V = n - i + 1 , u \perp V } { \operatorname* { m a x } } \underset { x \in V } { \operatorname* { m i n } } \frac { \underset { \| x - t u \| _ { H } ^ { 2 } } { \operatorname* { m i n } } } { \underset { t \in R } { \operatorname* { m i n } } } \frac { \| x - t u \| _ { H } ^ { 2 } } { \| x - t u \| ^ { 2 } } } \\ & { \quad \quad = \underset { d i m V = n - i + 1 } { \operatorname* { m a x } } \underset { y \in s p a n \{ V , u \} } { \operatorname* { m i n } } \frac { \| y \| _ { H } ^ { 2 } } { \| y \| ^ { 2 } } , y = x - t u } \\ & { \quad \quad \geq \underset { d i m V = n - ( i - 1 ) + 1 } { \operatorname* { m a x } } \underset { y \in V } { \operatorname* { m i n } } \frac { y ^ { T } H y } { \| y \| ^ { 2 } } = \lambda _ { i - 1 } ( H ) , } \end{array}
+$$
+
+where we have used the fact that $\operatorname { \{ \downarrow \downarrow }  u , \| x - t u \| ^ { 2 } = \| x \| ^ { 2 } + t ^ { 2 } \| u \| ^ { 2 } \geq \| x \| ^ { 2 } .$ .
+
+There are several corollaries related to the spectral property of $H ^ { * }$ . We first give some definitions.   
+Since $H ^ { * }$ is positive semi-definite, we can define the $H ^ { * }$ -seminorm.
+
+Definition A.2. The $H ^ { * }$ -seminorm of a vector $x$ is defined as $\| x \| _ { H ^ { * } } : = x ^ { T } H ^ { * } x$ . $\| x \| _ { H ^ { * } } = 0$ if and only if x is parallel to $u$ .
+
+Definition A.3. The pseudo-condition number of $H ^ { * }$ is defined as $\begin{array} { r } { \kappa ^ { * } ( H ^ { * } ) : = \frac { \lambda _ { d } ( H ^ { * } ) } { \lambda _ { 2 } ( H ^ { * } ) } } \end{array}$ .
+
+Definition A.4. For any real number $\varepsilon$ , the pseudo-spectral radius of the matrix $I - \varepsilon H ^ { * }$ is defined as $\rho ^ { * } ( I - \varepsilon H ^ { * } ) : = \operatorname* { m a x } _ { 2 \leq i \leq d } | 1 - \varepsilon \lambda _ { i } ( H ^ { * } ) | .$ .
+
+The following corollaries are direct consequences of Lemma A.1, hence we omit the proofs.
+
+Corollary A.5. The pseudo-condition number of $H ^ { * }$ is less than or equal to the condition number of $H$ :
+
+$$
+\begin{array} { r } { \kappa ^ { * } ( H ^ { * } ) : = \frac { \lambda _ { d } ( H ^ { * } ) } { \lambda _ { 2 } ( H ^ { * } ) } \leq \frac { \lambda _ { d } ( H ) } { \lambda _ { 1 } ( H ) } = : \kappa ( H ) , } \end{array}
+$$
+
+where the equality holds up if and only if $u \perp s p a n \{ v _ { 1 } , v _ { d } \}$ , $v _ { i }$ is the eigenvector of $H$ corresponding to eigenvalue $\lambda _ { i } ( H )$ .
+
+Corollary A.6. For any vector $\boldsymbol { x } ~ \in ~ \mathbb { R } ^ { d }$ and any real number $\varepsilon$ , we have $\| ( I - \varepsilon H ^ { * } ) x \| _ { H ^ { * } } \ \leq$ $\rho ^ { * } ( I - \varepsilon H ^ { * } ) \| x \| _ { H ^ { * } }$ .
+
+Corollary A.7. For any positive number $\varepsilon > 0$ , we have
+
+$$
+\begin{array} { r } { \rho ^ { * } ( I - \varepsilon H ^ { * } ) \leq \rho ( I - \varepsilon H ) , } \end{array}
+$$
+
+where the inequality is strict if $\cdot u ^ { T } v _ { i } \neq 0 f o r i = 1 , d$ .
+
+It is obvious that the inequality in (27) and (28) is strict for almost all $u$ .
+
+# A.2 SCALING PROPERTY
+
+The dynamical system defined in equation (9)-(10) is completely determined by a set of configurations $\{ H , u , a _ { 0 } , w _ { 0 } , \varepsilon _ { a } , \varepsilon \}$ . It is easy to check the system has the following scaling property:
+
+Lemma A.8 (Scaling property). Suppose $\mu \neq 0 , \gamma \neq 0 , r \neq 0 , Q ^ { T } Q = I$ , then
+
+(1) The configurations $\begin{array} { r } { \{ \mu Q ^ { T } H Q , \frac { \gamma } { \sqrt { \mu } } Q u , \gamma a _ { 0 } , \gamma Q w _ { 0 } , \varepsilon _ { a } , \varepsilon \} } \end{array}$ and $\{ H , u , a _ { 0 } , w _ { 0 } , \varepsilon _ { a } , \varepsilon \}$ are equivalent.
+
+(2) The configurations $\{ H , u , a _ { 0 } , w _ { 0 } , \varepsilon _ { a } , \varepsilon \}$ and $\{ H , u , a _ { 0 } , r w _ { 0 } , \varepsilon _ { a } , r ^ { 2 } \varepsilon \}$ are equivalent.
+
+The scaling property is valid for general loss functions provided batch normalization is used. Consider a general problem
+
+$$
+\operatorname* { m i n } _ { w \in \mathbb { R } ^ { d } } J _ { 0 } ( w ) : = E _ { x , y } [ f ( y , x ^ { T } w ) ] ,
+$$
+
+and its BN version
+
+$$
+\operatorname* { m i n } _ { w \in \mathbb { R } ^ { d } , a \in \mathbb { R } } J ( a , w ) : = E _ { x , y } \big [ f \big ( y , a N _ { B N } ( x ^ { T } w ) \big ) \big ] .
+$$
+
+Then the gradient descent method gives the following iteration,
+
+$$
+\begin{array} { r l } & { a _ { k + 1 } = a _ { k } + \varepsilon _ { a } \frac { w _ { k } ^ { T } \tilde { h } } { \sigma _ { k } } , } \\ & { w _ { k + 1 } = w _ { k } + \varepsilon \frac { a _ { k } } { \sigma _ { k } } \Big ( \tilde { h } - \frac { w _ { k } ^ { T } \tilde { h } } { \sigma _ { k } ^ { 2 } } H w _ { k } \Big ) , } \end{array}
+$$
+
+where $\tilde { h } = h ( { a _ { k } w _ { k } } / { \sigma _ { k } } )$ , and $h$ is the gradient of original problem:
+
+$$
+h ( w ) : = E _ { x , y } [ x f _ { 2 } ^ { \prime } ( y , x ^ { T } w ) ] .
+$$
+
+It is easy to check the general BNGD has the following property:
+
+Lemma A.9 (General scaling property). Suppose $r \neq 0 .$ , then the configurations $\{ w _ { 0 } , \varepsilon , * \}$ and $\{ r w _ { 0 } , r ^ { 2 } \varepsilon , * \}$ are equivalent. Here the sign $^ *$ means other parameters.
+
+# A.3 PROOF OF THEOREM 3.3
+
+Recall the BNGD iterations
+
+$$
+\begin{array} { r l } & { a _ { k + 1 } = a _ { k } + \varepsilon _ { a } \Big ( \frac { w _ { k } ^ { T } g } { \sigma _ { k } } - a _ { k } \Big ) , } \\ & { w _ { k + 1 } = w _ { k } + \varepsilon \frac { a _ { k } } { \sigma _ { k } } \Big ( g - \frac { w _ { k } ^ { T } g } { \sigma _ { k } ^ { 2 } } H w _ { k } \Big ) . } \end{array}
+$$
+
+The scaling property simplify our analysis by allowing us to set, for example, $\lVert u \rVert = 1$ and $\| w _ { 0 } \| =$ 1. In the rest of this section, we only set $\lVert u \rVert = 1$ .
+
+For the step size of $a$ , it is easy to check that $a _ { k }$ tends to infinity with $\varepsilon _ { a } \ > \ 2$ and initial value $a _ { 0 } = 1 , w _ { 0 } = u$ . Hence we only consider $0 < \varepsilon _ { a } < 2$ , which make the iteration of $a _ { k }$ bounded by some constant $C _ { a }$ .
+
+Lemma A.10 (Boundedness of $a _ { k }$ ). If the step size $0 < \varepsilon _ { a } < 2$ , then the sequence $a _ { k }$ is bounded for any $\varepsilon > 0$ and any initial value $( a _ { 0 } , w _ { 0 } )$ .
+
+Proof. Define $\begin{array} { r } { \alpha _ { k } : = \frac { w _ { k } ^ { T } g } { \sigma _ { k } } } \end{array}$ , which is bounded by $\vert \alpha _ { k } \vert \le \sqrt { u ^ { T } H u } = : C$ , then
+
+$$
+\begin{array} { l } { a _ { k + 1 } = ( 1 - \varepsilon _ { a } ) a _ { k } + \varepsilon _ { a } \alpha _ { k } } \\ { \qquad = ( 1 - \varepsilon _ { a } ) ^ { k + 1 } a _ { 0 } + ( 1 - \varepsilon _ { a } ) ^ { k } \varepsilon _ { a } \alpha _ { 0 } + \ldots + ( 1 - \varepsilon _ { a } ) \varepsilon _ { a } \alpha _ { k - 1 } + \varepsilon _ { a } \alpha _ { k } . } \end{array}
+$$
+
+$\left| 1 - \varepsilon _ { a } \right| < 1$
+
+According to the iterations (34), we have
+
+$$
+\begin{array} { r } { u - \frac { w _ { k } ^ { T } g } { \sigma _ { k } ^ { 2 } } w _ { k + 1 } = \Big ( I - \varepsilon \frac { a _ { k } } { \sigma _ { k } } \frac { w _ { k } ^ { T } g } { \sigma _ { k } ^ { 2 } } H \Big ) \Big ( u - \frac { w _ { k } ^ { T } g } { \sigma _ { k } ^ { 2 } } w _ { k } \Big ) . } \end{array}
+$$
+
+Define
+
+$$
+\begin{array} { r l } & { { \boldsymbol { e } } _ { k } : = { \boldsymbol { u } } - \frac { w _ { k } ^ { T } g } { \sigma _ { k } ^ { 2 } } { \boldsymbol { w } } _ { k } , } \\ & { { \boldsymbol { q } } _ { k } : = { \boldsymbol { u } } ^ { T } { \boldsymbol { H } } { \boldsymbol { u } } - \frac { ( w _ { k } ^ { T } g ) ^ { 2 } } { \sigma _ { k } ^ { 2 } } = \| { \boldsymbol { e } } _ { k } \| _ { \boldsymbol { H } } ^ { 2 } \geq 0 , } \\ & { \hat { \boldsymbol { \varepsilon } } _ { k } : = \varepsilon \frac { a _ { k } } { \sigma _ { k } } \frac { w _ { k } ^ { T } g } { \sigma _ { k } ^ { 2 } } , } \end{array}
+$$
+
+and using the property $\frac { w ^ { T } g } { \sigma _ { k } ^ { 2 } } = \underset { t } { \operatorname { a r g m i n } } \| u - t w \| _ { H }$ , and the property of $H$ -norm, we have
+
+$$
+\begin{array} { r } { q _ { k + 1 } \leq \left| \left| u - \frac { w _ { k } ^ { T } g } { \sigma _ { k } ^ { 2 } } w _ { k + 1 } \right| \right| _ { H } ^ { 2 } = \| ( I - \hat { \varepsilon } _ { k } H ) e _ { k } \| _ { H } ^ { 2 } \leq \rho ( I - \hat { \varepsilon } _ { k } H ) ^ { 2 } q _ { k } . } \end{array}
+$$
+
+Therefore we have the following lemma to make sure the iteration converge:
+
+Lemma A.11. Let $0 < \varepsilon _ { a } < 2$ . If there are two positive numbers $\varepsilon ^ { - }$ and $\hat { \varepsilon } ^ { + }$ , and the effective step size $\hat { \varepsilon } _ { k }$ satisfies
+
+$$
+\begin{array} { r } { 0 < \frac { \varepsilon ^ { - } } { \| w _ { k } \| ^ { 2 } } \le \hat { \varepsilon } _ { k } \le \hat { \varepsilon } ^ { + } < \frac { 2 } { \lambda _ { m a x } } } \end{array}
+$$
+
+for all $k$ large enough, then the iterations $( 3 4 )$ converge to a minimizer.
+
+Proof. Without loss of generality, we assume $\begin{array} { r } { \frac { \varepsilon ^ { - } } { \| w _ { k } \| ^ { 2 } } < \frac { 1 } { \lambda _ { m a x } } } \end{array}$ and the inequality (39) is satisfied for all $k \geq 0$ . We will prove $\| w _ { k } \|$ converges and the direction of $w _ { k }$ converges to the direction of $u$ .
+
+(1) Since $\| w _ { k } \|$ is always increasing, we only need to prove it is bounded. We have,
+
+$$
+\begin{array} { l } { \displaystyle \| w _ { k + 1 } \| ^ { 2 } = \| w _ { k } \| ^ { 2 } + \varepsilon ^ { 2 } \frac { a _ { k } ^ { 2 } } { \sigma _ { k } ^ { 2 } } \| H e _ { k } \| ^ { 2 } } \\ { \displaystyle = \| w _ { 0 } \| ^ { 2 } + \varepsilon ^ { 2 } \sum _ { i = 0 } ^ { k } \frac { a _ { i } ^ { 2 } } { \sigma _ { i } ^ { 2 } } \| H e _ { i } \| ^ { 2 } } \\ { \displaystyle \le \| w _ { 0 } \| ^ { 2 } + \varepsilon ^ { 2 } \lambda _ { m a x } \sum _ { i = 0 } ^ { k } \frac { a _ { i } ^ { 2 } } { \sigma _ { i } ^ { 2 } } q _ { i } } \\ { \displaystyle \le \| w _ { 0 } \| ^ { 2 } + \varepsilon ^ { 2 } \frac { \lambda _ { m a x } C _ { a } ^ { 2 } } { \lambda _ { m i n } } \sum _ { i = 0 } ^ { k } \frac { q _ { i } } { \| w _ { i } \| ^ { 2 } } . } \end{array}
+$$
+
+The inequality in last lines are based on the fact that $\| H e _ { i } \| ^ { 2 } \leq \lambda _ { m a x } \| e _ { i } \| _ { H } ^ { 2 }$ , and $\left| a _ { k } \right|$ are bounded by a constant $C _ { a }$ . Next, we will prove $\textstyle \sum _ { i = 0 } ^ { \infty } { \frac { q _ { i } } { \| w _ { i } \| ^ { 2 } } } < \infty$ , which implies $\| w _ { k } \|$ are bounded.
+
+According to the estimate (38), we have
+
+$$
+\begin{array} { r l } & { q _ { k + 1 } \leq \operatorname* { m a x } _ { i } \{ | 1 - \hat { \varepsilon } ^ { + } \lambda _ { i } | ^ { 2 } , | 1 - \frac { \varepsilon ^ { - } \lambda _ { i } } { \| w _ { k } \| ^ { 2 } } | ^ { 2 } \} q _ { k } } \\ & { \qquad \leq \operatorname* { m a x } \{ 1 - \gamma ^ { + } , 1 - \frac { \varepsilon ^ { - } \lambda _ { m i n } } { \| w _ { k } \| ^ { 2 } } \} q _ { k } , } \end{array}
+$$
+
+where $\begin{array} { r } { 1 - \gamma ^ { + } = \operatorname* { m a x } _ { i } \{ | 1 - \hat { \varepsilon } ^ { + } \lambda _ { i } | ^ { 2 } \} \in ( 0 , 1 ) } \end{array}$ . Using the definition of $q _ { k }$ , we have
+
+$$
+\begin{array} { r } { q _ { k } - q _ { k + 1 } \geq \frac { \operatorname* { m i n } \{ \gamma ^ { + } \| w _ { 0 } \| ^ { 2 } , \varepsilon ^ { - } \lambda _ { m i n } \} } { \| w _ { k } \| ^ { 2 } } q _ { k } = : \frac { C q _ { k } } { \| w _ { k } \| ^ { 2 } } \geq 0 . } \end{array}
+$$
+
+Since $q _ { k }$ is bounded in $[ 0 , u ^ { T } H u ]$ , summing both side of the inequality, we get the bound of the infinite series $\sum _ { k } \frac { q _ { k } } { \| w _ { k } \| ^ { 2 } } \leq \frac { u ^ { T } H u } { C } < \infty$ .
+
+(2) Since $\lVert w _ { k } \rVert$ is bounded, we denote $\hat { \varepsilon } ^ { - } : = \frac { \varepsilon ^ { - } } { \| w _ { \infty } \| ^ { 2 } }$ , and define $\rho : = \operatorname* { m a x } _ { i } \{ | 1 - \hat { \varepsilon } ^ { \pm } \lambda _ { i } | \} \in ( 0 , 1 )$ then the inequality (38) implies $q _ { k + 1 } \le \rho ^ { 2 } q _ { k }$ . As a consequence, $q _ { k }$ tends to zero, which implies the direction of $w _ { k }$ converges to the direction of $u$ .
+
+(3) The convergence of $a _ { k }$ is a consequence of $w _ { k }$ converging.
+
+Since ak is bounded, we assume |ak| < C˜a uT Hu, C˜a ≥ 1, and define ε0 := 12C˜aκλmax . The following lemma gives the convergence for small step size.
+
+Lemma A.12. If the initial values $( a _ { 0 } , w _ { 0 } )$ satisfies $a _ { 0 } w _ { 0 } ^ { T } g \ > \ 0$ , and step size satisfies $\varepsilon _ { a } ~ \in$ $( 0 , 1 ] , \varepsilon / \| w _ { 0 } \| ^ { 2 } < \varepsilon _ { 0 }$ , then the sequence $( a _ { k } , w _ { k } )$ converges to a global minimizer.
+
+Remark 1: If we set a0 = 0, then we have w1 = w0, a1 = εa σ0 , hence $a _ { 1 } w _ { 1 } ^ { T } g > 0$ provided $w _ { 0 } ^ { T } g \neq 0$ .
+
+Remark 2: For the case of $\varepsilon _ { a } \in ( 1 , 2 )$ , if the initial value satisfies an additional condition $0 ~ <$ $\begin{array} { r } { | a _ { 0 } | \leq \varepsilon _ { a } \frac { | w _ { 0 } ^ { T } g | } { \sigma _ { 0 } } } \end{array}$ , then we have $( a _ { k } , w _ { k } )$ converges to a global minimizer as well.
+
+Proof. Without loss of generality, we only consider the case of $a _ { 0 } > 0 , w _ { 0 } ^ { T } g > 0 , \| w _ { 0 } \| \geq 1$
+
+(1) We will prove $a _ { k } > 0 , w _ { k } ^ { T } g > 0$ for all $k$ . Denote $\begin{array} { r } { y _ { k } : = w _ { k } ^ { T } g , \delta = \frac { \| g \| } { 4 \kappa } } \end{array}$
+
+On the one hand, if $a _ { k } > 0 , 0 < y _ { k } < 2 \delta$ , then
+
+$$
+\begin{array} { r } { y _ { k + 1 } \geq y _ { k } + \varepsilon \frac { a _ { k } } { \sigma _ { k } } \frac { \| g \| ^ { 2 } } { 2 } \geq y _ { k } . } \end{array}
+$$
+
+On the other hand, when $a _ { k } > 0 , y _ { k } > 0 , \varepsilon < \varepsilon _ { 0 }$ , we have
+
+$$
+\begin{array} { r l } & { y _ { k + 1 } \geq \varepsilon \frac { a _ { k } \| g \| ^ { 2 } } { \sigma _ { k } } + y _ { k } \Big ( 1 - \varepsilon \frac { a _ { k } } { \sigma _ { k } ^ { 2 } } \sqrt { g ^ { T } H g } \Big ) \geq \frac { 1 } { 2 } y _ { k } , } \\ & { a _ { k + 1 } \geq \operatorname* { m i n } \{ a _ { k } , y _ { k } / \sigma _ { k } \} . } \end{array}
+$$
+
+As a consequence, we have $a _ { k } > 0 , y _ { k } \geq \delta _ { y } : = \operatorname* { m i n } \{ y _ { 0 } , \delta \}$ for all $k$ by induction.
+
+(2) We will prove the effective step size $\hat { \varepsilon } _ { k }$ satisfying the condition in Lemma A.11.
+
+Since $a _ { k }$ is bounded, $\varepsilon < \varepsilon _ { 0 }$ , we have
+
+$$
+\begin{array} { r } { \hat { \varepsilon } _ { k } : = \varepsilon \frac { a _ { k } } { \sigma _ { k } } \frac { w _ { k } ^ { T } g } { \sigma _ { k } ^ { 2 } } \leq \frac { \varepsilon \tilde { C } _ { a } \lambda _ { m a x } } { \lambda _ { m i n } \| w _ { k } \| ^ { 2 } } \leq \varepsilon \tilde { C } _ { a } \kappa = : \hat { \varepsilon } ^ { + } < \frac { 1 } { 2 \lambda _ { m a x } } , } \end{array}
+$$
+
+and
+
+$$
+q _ { k + 1 } \leq ( 1 - \hat { \varepsilon } _ { k } \lambda _ { m i n } ) ^ { 2 } q _ { k } \leq ( 1 - \hat { \varepsilon } _ { k } \lambda _ { m i n } ) q _ { k } < q _ { k } .
+$$
+
+which implies $\begin{array} { r } { \frac { w _ { k + 1 } ^ { T } g } { \sigma _ { k + 1 } } \geq \frac { w _ { k } ^ { T } g } { \sigma _ { k } } \geq \frac { w _ { 0 } ^ { T } g } { \sigma _ { 0 } } } \end{array}$ . Furthermore, we have $\begin{array} { r } { { a } _ { k } \geq \operatorname* { m i n } \{ { a } _ { 0 } , \frac { w _ { 0 } ^ { T } g } { { \sigma } _ { 0 } } \} } \end{array}$ ， , and there is a positive constant $\varepsilon ^ { - } > 0$ such that
+
+$$
+\begin{array} { r } { \hat { \varepsilon } _ { k } \geq \varepsilon \frac { a _ { k } } { \lambda _ { m a x } \| w _ { k } \| ^ { 2 } } \frac { w _ { k } ^ { T } g } { \sigma _ { k } } \geq \frac { \varepsilon ^ { - } } { \| w _ { k } \| ^ { 2 } } . } \end{array}
+$$
+
+(3) Employing the Lemma A.11, we conclude that $( a _ { k } , w _ { k } )$ converges to a global minimizer.
+
+Lemma A.13. If step size satisfies $\varepsilon _ { a } \in ( 0 , 1 ] , \varepsilon / \| w _ { 0 } \| ^ { 2 } < \varepsilon _ { 0 } ,$ , then the sequence $( a _ { k } , w _ { k } )$ converges.
+
+Proof. Thanks to Lemma A.12, we only need to consider the case of $a _ { k } w _ { k } ^ { T } g \leq 0$ for all $k$ , and we will prove the iteration converges to a saddle point in this case. Since the case of $a _ { k } = 0$ or $w _ { k } ^ { T } g = 0$ is trivial, we assume $a _ { k } w _ { k } ^ { T } g < 0$ below. More specifically , we will prove $| a _ { k + 1 } | < r | a _ { k } |$ for some constant $r \in ( 0 , 1 )$ , which implies convergence to a saddle point.
+
+(1) If $a _ { k }$ and $a _ { k + 1 }$ have same sign, hence different sign with $w _ { k } ^ { T } g$ , then we have $| a _ { k + 1 } | = | 1 -$ $\varepsilon _ { a } \| { a } _ { k } | - \varepsilon _ { a } | w _ { k } ^ { T } g | / \sigma _ { k } \leq | 1 - \varepsilon _ { a } | \| { a } _ { k } |$ .
+
+(2) If $a _ { k }$ and $a _ { k + 1 }$ have different signs, then we have
+
+$$
+\begin{array} { r } { \frac { | w _ { k } ^ { T } g | } { | a _ { k } \sigma _ { k } | } \leq \varepsilon \frac { 1 } { \sigma _ { k } ^ { 2 } } \Big ( \| g \| ^ { 2 } - \frac { w _ { k } ^ { T } g } { \sigma _ { k } ^ { 2 } } g ^ { T } H w _ { k } \Big ) \leq 2 \varepsilon \kappa \lambda _ { m a x } < 1 . } \end{array}
+$$
+
+Consequently, we get
+
+$$
+\begin{array} { r } { \frac { \left| a _ { k + 1 } \right| } { \left| a _ { k } \right| } = \varepsilon _ { a } \frac { \left| w _ { k } ^ { T } g \right| } { \left| a _ { k } \sigma _ { k } \right| } - \left( 1 - \varepsilon _ { a } \right) \leq 2 \varepsilon \varepsilon _ { a } \kappa \lambda _ { m a x } - \left( 1 - \varepsilon _ { a } \right) < \varepsilon _ { a } \leq 1 . } \end{array}
+$$
+
+(3) Setting $r : = \mathrm { m a x } ( | 1 - \varepsilon _ { a } | , 2 \varepsilon \varepsilon _ { a } \kappa \lambda _ { m a x } - ( 1 - \varepsilon _ { a } ) )$ , we finish the proof.
+
+To simplify our proofs for Theorem 3.3, we give two lemmas which are obvious but useful.
+
+Lemma A.14. If positive series $f _ { k } , h _ { k }$ satisfy $f _ { k + 1 } \leq r f _ { k } + h _ { k } , r \in ( 0 , 1 )$ and $\operatorname* { l i m } _ { k \to \infty } h _ { k } = 0 _ { \ast }$ , then $\operatorname* { l i m } _ { k \to \infty } f _ { k } = 0$ .
+
+Proof. It is obvious, because the series $b _ { k }$ defined by $b _ { k + 1 } = r b _ { k } + h _ { k } , b _ { 0 } > 0$ , tends to zeros.
+
+Lemma A.15 (Separation property). For $\delta _ { 0 }$ small enough, the set $S : = \{ w | y ^ { 2 } q < \delta _ { 0 } , \| w \| \ge 1 \}$ is composed by two separated parts: $S _ { 1 }$ and $S _ { 2 }$ , $d i s t ( S _ { 1 } , S _ { 2 } ) > 0$ , where in the set $S _ { 1 }$ one has $y ^ { 2 } < \bar { \delta } _ { 1 } , q > \bar { \delta } _ { 2 }$ , and in $S _ { 2 }$ one has $q < \delta _ { 2 } , y ^ { 2 } > \delta _ { 1 }$ for some $\delta _ { 1 } > 0 , \delta _ { 2 } > 0$ . Here $y : = w ^ { T } g , q : =$ $\begin{array} { r } { u ^ { T } H u - \frac { ( w ^ { T } H u ) ^ { 2 } } { w ^ { T } H w } = u ^ { T } H u - \frac { y ^ { 2 } } { w ^ { T } H w } } \end{array}$ .
+
+Proof. The proof is based on $H$ being positive. The geometric meaning is illustrated in Figure 4. □
+
+![](images/57d99de35f156a48a240c44ddcf9aafd69590674e33f61f21a7d473bf8e3b147.jpg)  
+Figure 4: The geometric meaning of the separation property
+
+Corollary A.16. $I f \operatorname* { l i m } _ { k \to \infty } \left\| w _ { k + 1 } - w _ { k } \right\| = 0$ , and $\operatorname* { l i m } _ { k  \infty } ( w _ { k } ^ { T } g ) ^ { 2 } q _ { k } = 0 ,$ , then either $\operatorname* { l i m } _ { k \to \infty } ( w _ { k } ^ { T } g ) ^ { 2 } = 0$ or $\operatorname* { l i m } _ { k \to \infty } q _ { k } = 0$ .
+
+Proof. Denote $y _ { k } : = w _ { k } ^ { T } g$ . According to the separation property (Lemma A.15), we can chose a $\delta _ { 0 } > 0$ small enough such that the separated parts of the set $\dot { S } : = \dot { \{ w | y ^ { 2 } q < \delta _ { 0 } , \| w \| \ge 1 \} }$ , $S _ { 1 }$ and $S _ { 2 }$ , have $d i s t ( S _ { 1 } , S _ { 2 } ) > 0$ .
+
+Because $y _ { k } ^ { 2 } q _ { k }$ tends to zero, we have $w _ { k }$ belongs to $S$ for $k$ large enough, for instance $k > k _ { 1 }$ . On the other hand, because $\| w _ { k + 1 } - w _ { k } \|$ tends to zero, we have $\lVert w _ { k + 1 } - w _ { k } \rVert < d i s t ( S _ { 1 } , S _ { 2 } )$ for $k$ large enough, for instance $k > k _ { 2 }$ . Then consider $k > k _ { 3 } : = \operatorname* { m a x } ( k _ { 1 } , k _ { 2 } )$ , we have all $w _ { k }$ belongs to the same part $S _ { 1 }$ or $S _ { 2 }$ .
+
+If $w _ { k } \in S _ { 1 }$ , $( q _ { k } > \delta _ { 2 }$ ), for all $k > k _ { 3 }$ , then we have $\operatorname* { l i m } _ { k \to \infty } ( w _ { k } ^ { T } g ) ^ { 2 } = 0$
+
+On the other hand, if $w _ { k } \in S _ { 2 }$ , $( y _ { k } ^ { 2 } > \delta _ { 1 } )$ , for all $k > k _ { 3 }$ , then we have $\operatorname* { l i m } _ { k \to \infty } q _ { k } = 0$
+
+Theorem A.17. Let $\varepsilon _ { a } \in ( 0 , 1 ]$ and $\varepsilon > 0$ . The sequence $( a _ { k } , w _ { k } )$ converges for any initial value $( a _ { 0 } , w _ { 0 } )$ .
+
+Proof. We will prove $\| w _ { k } \|$ converges, then prove $( a _ { k } , w _ { k } )$ converges as well.
+
+(1) We will prove that $\| w _ { k } \|$ is bounded and hence converges.
+
+In fact, according to the Lemma A.13, once $\| w _ { k } \| ^ { 2 } \geq \varepsilon / \varepsilon _ { 0 }$ for some $k$ , the rest of the iteration will converge, hence $\bar { \parallel } w _ { k } \parallel$ is bounded.
+
+(2) We will prove $\operatorname* { l i m } _ { k \to \infty } \| w _ { k + 1 } - w _ { k } \| = 0 , \mathrm { a n d } \operatorname* { l i m } _ { k \to \infty } ( w _ { k } ^ { T } g ) ^ { 2 } q _ { k } = 0 .$
+
+The convergence of $\| w _ { k } \|$ implies $\textstyle \sum _ { k } a _ { k } ^ { 2 } q _ { k }$ is summable. As a consequence,
+
+$$
+\operatorname * { l i m } _ { k \to \infty } a _ { k } ^ { 2 } p _ { k } = 0 , \operatorname * { l i m } _ { k \to \infty } a _ { k } e _ { k } = 0 ,
+$$
+
+and $\operatorname* { l i m } _ { k \to \infty } \left\| w _ { k + 1 } - w _ { k } \right\| = 0$ . In fact, we have
+
+$$
+\begin{array} { r } { \| w _ { k + 1 } - w _ { k } \| ^ { 2 } = \varepsilon ^ { 2 } \frac { a _ { k } ^ { 2 } } { \sigma ^ { 2 } } \| H e _ { k } \| ^ { 2 } \leq \frac { \lambda _ { m a x } \varepsilon ^ { 2 } } { \lambda _ { m i n } ^ { 2 } } a _ { k } ^ { 2 } q _ { k } \to 0 . } \end{array}
+$$
+
+Consider the iteration of series $| a _ { k } - w _ { k } ^ { T } g / \sigma _ { k } |$ ,
+
+$$
+\begin{array} { r l } & { \left| a _ { k + 1 } - \frac { w _ { k + 1 } ^ { T } g } { \sigma _ { k + 1 } } \right| \leq \left| a _ { k + 1 } - \frac { w _ { k + 1 } ^ { T } g } { \sigma _ { k } } \right| + \left| \frac { w _ { k + 1 } ^ { T } g } { \sigma _ { k } } - \frac { w _ { k + 1 } ^ { T } g } { \sigma _ { k + 1 } } \right| } \\ & { \qquad \leq ( 1 - \varepsilon _ { a } ) \Big | a _ { k } - \frac { w _ { k } ^ { T } g } { \sigma _ { k } } \Big | + \varepsilon \frac { | a _ { k } g ^ { T } H e _ { k } | } { \sigma _ { k } ^ { 2 } } + \frac { | w _ { k + 1 } ^ { T } g | } { ( \sigma _ { k } \sigma _ { k + 1 } ) } \big | \sigma _ { k + 1 } - \sigma _ { k } \big | } \\ & { \qquad \leq ( 1 - \varepsilon _ { a } ) \Big | a _ { k } - \frac { w _ { k } ^ { T } g } { \sigma _ { k } } \Big | + \varepsilon \frac { \| g \| _ { H } \| a _ { k } e _ { k } \| _ { H } } { \sigma _ { k } ^ { 2 } } + \frac { | w _ { k + 1 } ^ { T } g | } { ( \sigma _ { k } \sigma _ { k + 1 } ) } \varepsilon \frac { \lambda _ { m a x } } { \sigma _ { k } } \| a _ { k } e _ { k } \| _ { H } } \\ & { \qquad \leq ( 1 - \varepsilon _ { a } ) \Big | a _ { k } - \frac { w _ { k } ^ { T } g } { \sigma _ { k } } \Big | + 2 C \| a _ { k } e _ { k } \| _ { H } . } \end{array}
+$$
+
+The constant $C$ in (57) can be chosen as $\begin{array} { r } { C = \frac { \varepsilon \lambda _ { m a x } \| u \| _ { H } } { \lambda _ { m i n } \| w _ { 0 } \| ^ { 2 } } } \end{array}$ . Since $\| a _ { k } e _ { k } \| _ { H }$ tends to zero, we can use Lemma A.14 to get $\operatorname* { l i m } _ { k \to \infty } | a _ { k } - w _ { k } ^ { T } g / \sigma _ { k } | = 0$ . Combine the equation (55), then we have $\operatorname* { l i m } _ { k  \infty } ( w _ { k } ^ { T } g ) ^ { 2 } p _ { k } = 0 .$
+
+(3) According to the Corollary A.16, we have either $\operatorname* { l i m } _ { k \to \infty } y _ { k } ^ { 2 } = 0$ , or $\operatorname* { l i m } _ { k \to \infty } q _ { k } = 0$ . In the former case, the iteration of $( a _ { k } , w _ { k } )$ converges to a saddle point. However, in the latter case, $( a _ { k } , w _ { k } )$ converges to a global minimizer. In both cases we have $( a _ { k } , w _ { k } )$ converges.
+
+To finish the proof of Theorem 3.3, we have to demonstrate the special case of ${ { \varepsilon } _ { a } } \mathrm { ~ = ~ } 1$ where the set of initial values such that BN iteration converges to saddle points is Lebeguse measure zero. We leave this demonstration in next section where we consider the case of $\varepsilon _ { a } \geq 1$ .
+
+# A.4 IMPOSSIBILITY OF CONVERGING TO STRICT SADDLE POINTS
+
+In this section, we will prove the set of initial values such that BN iteration converges to saddle points is (Lebeguse) measure zero, as long as $\varepsilon _ { a } \geq 1$ . The tools in our proof is similar to the analysis of gradient descent on non-convex objectives (Lee et al., 2016; Panageas & Piliouras, 2017). In addition, we used the real analytic property of the BN loss function (16).
+
+For brevity, here we denote $x : = ( a , w )$ and let $\varepsilon _ { a } = \varepsilon$ , then the BN iteration can be rewrote as
+
+$$
+x _ { n + 1 } = { \cal T } ( x _ { n } ) : = x _ { n } - \varepsilon \nabla J ( x _ { n } ) .
+$$
+
+Lemma A.18. If $A \subset T ( \mathbb { R } ^ { d } / \{ 0 \} )$ is a measure zero set, then the preimage $T ^ { - 1 } ( A )$ is of measure zero as well.
+
+Proof. Since $T$ is smooth enough, according to Theorem 3 of (Ponomarev, 1987), we only need to prove the Jacobian of $T ( x )$ is nonzero for almost all $x \in \mathbb { R } ^ { d }$ . In other words, the set $\{ x :$ $\mathrm { d e f } ( I - \varepsilon \nabla ^ { 2 } J ( x ) ) = 0 \}$ is of measure zero. This is true because the function $\operatorname* { d e t } ( I - \varepsilon \nabla ^ { 2 } J ( x ) )$ is a real analytic function of $x \in \mathbb { R } ^ { d } / \{ 0 \}$ . (Details of properties of real analytic functions can be found in (Krantz & Parks, 2002) for instance).
+
+Lemma A.19. Let $f : X \to \mathbb { R }$ be twice continuously differentiable in an open set $X \subset \mathbb { R } ^ { d }$ and $x ^ { * } \in X$ be a stationary point of $f$ . If $\varepsilon > 0$ , $\operatorname* { d e t } ( I - \varepsilon \nabla ^ { 2 } f ( x ^ { * } ) ) \neq 0$ and the matrix $\nabla ^ { 2 } f ( x ^ { * } )$ has at least a negative eigenvalue, then there exist a neighborhood $U$ of $x ^ { * }$ such that the following set $B$ has measure zero,
+
+$$
+B : = \{ x _ { 0 } \in U : x _ { n + 1 } = x _ { n } - \varepsilon \nabla f ( x _ { n } ) \in U , \forall n \geq 0 \} .
+$$
+
+Proof. The detailed proof is similar to (Lee et al., 2016; Panageas & Piliouras, 2017).
+
+Define the transform function as $F ( x ) : = x - \varepsilon \nabla f ( x )$ . Since $\operatorname* { d e t } ( I - \varepsilon \nabla ^ { 2 } f ( x ^ { * } ) ) \neq 0$ , accorded to the inverse function theorem, there exist a neighborhood $U$ of $x ^ { * }$ such that $T$ has differentiable inverse. Hence $T$ is a local $C ^ { 1 }$ diffeomorphism, which allow us to use the central-stable manifold theorem (Shub, 2013). The negative eigenvalues of $\nabla ^ { 2 } f ( x ^ { * } )$ indicates $\lambda _ { m a x } ( I - \varepsilon \nabla ^ { 2 } f ( x ^ { * } ) ) > 1$ and the dimension of the unstable manifold is at least one, which implies the set $B$ is on a lower dimension manifold hence $B$ is of measure zero.
+
+Lemma A.20. If $\varepsilon _ { a } = \varepsilon \ge 1$ , then the set of initial values such that BN iteration converges to saddle points is of Lebeguse measure zero.
+
+Proof. We will prove this argument using Lemma A.18 and Lemma A.19. Denote the saddle points set as $W : = \{ ( \stackrel { . } { a ^ { * } } , w ^ { * } ) : a ^ { * } \stackrel { - } { = } 0 , w ^ { * T } g = 0 \}$ . The basic point is that the saddle point $x ^ { \ast } : = ( a ^ { \ast } , w ^ { \ast } )$ of the BN loss function (16) has eigenvalues $\begin{array} { r } { \Big \{ \frac { 1 } { 2 } ( 1 \pm \sqrt { 1 + 4 \frac { \| g \| ^ { 2 } } { w ^ { * T } H w ^ { * } } } ) , 0 , . . . , 0 \Big \} } \end{array}$ of the Hessian matrix.
+
+(1) For each saddle point $x ^ { \ast } : = ( a ^ { \ast } , w ^ { \ast } )$ of BN loss function, $\varepsilon \geq 1$ is enough to allow us to use Lemma A.19. Hence there exist a neighborhood $U _ { x ^ { * } }$ of $x ^ { * }$ such that the following set $B _ { x ^ { * } }$ is of measure zero,
+
+$$
+B _ { x ^ { * } } : = \{ x _ { 0 } \in U _ { x ^ { * } } : x _ { n } \in U _ { x ^ { * } } , \forall n \geq 1 \} .
+$$
+
+(2) The neighborhoods $U _ { x ^ { * } }$ of all $x ^ { * } \in W$ forms a cover of $W$ , hence, accorded to Lindelof’s open ¨ cover lemma, there are countable neighborhoods $\{ U _ { i } : i = 1 , 2 , \ldots \}$ cover $W$ , i.e. $U : = \cup _ { i } U _ { i } \supseteq W$ . As a consequence, the following set $A _ { 0 }$ is of measure zero,
+
+$$
+A _ { 0 } : = \cup _ { i } B _ { i } = \cup _ { i } \{ x _ { 0 } \in U _ { i } : x _ { n } \in U _ { i } , \forall n \geq 1 \} .
+$$
+
+(3) Define $A _ { m + 1 } : = T ^ { - 1 } ( A _ { m } ) = \{ x \in \mathbb { R } ^ { d } : T ( x ) \in A _ { m } \} , m \geq 0 .$ . According to Lemma A.18, we have all $A _ { m }$ and $\cup _ { m } A _ { m }$ are of measure zero.
+
+(4) Since each initial value $x _ { 0 }$ such that the iteration converges to a saddle point must be contained in some set $A _ { m }$ , we finish the proof.
+
+Combine the results of Lemma A.20, scaling property 3.2 and the convergence theorem A.17, we have the following theorem directly.
+
+Theorem A.21. If $\varepsilon _ { a } = 1 , \varepsilon \geq 0$ , then the BN iteration (9)-(10) converges to global minimizers for almost all initial values.
+
+# A.5 CONVERGENCE RATE
+
+In the last section, we encountered the following estimate for $\begin{array} { r } { \boldsymbol { e } _ { k } = \boldsymbol { u } - \frac { w _ { k } ^ { T } g } { \sigma _ { k } ^ { 2 } } w _ { k } } \end{array}$
+
+$$
+\| e _ { k + 1 } \| _ { H } \leq \rho ( I - { \hat { \varepsilon } } _ { k } H ) \| e _ { k } \| _ { H } .
+$$
+
+We can improve the convergence rate of the above if $H ^ { * }$ has better spectral property. This is the content of Proposition 3.4 and the following lemma is enough to prove it.
+
+Lemma A.22. The following inequality holds,
+
+$$
+\begin{array} { r } { ( 1 - \delta _ { k } ) \| e _ { k + 1 } \| _ { H } \leq \Big ( \rho ^ { * } ( I - \hat { \varepsilon } _ { k } H ^ { * } ) + \delta _ { k } \Big ) \| e _ { k } \| _ { H } , } \end{array}
+$$
+
+where $\begin{array} { r } { \delta _ { k } : = \frac { \lambda _ { m a x } \varepsilon | { a } _ { k } | } { \sigma _ { k } ^ { 2 } } \| e _ { k } \| _ { H } } \end{array}$
+
+Proof. The case of $w _ { k } ^ { T } g = 0$ is trivial, hence we assume $w _ { k } ^ { T } g \neq 0$ in the following proof. Rewrite the iteration on $w _ { k }$ as the following equality,
+
+$$
+\begin{array} { r } { u - \frac { w _ { k } ^ { T } g } { \sigma _ { k } ^ { 2 } } w _ { k + 1 } = ( I - \hat { \varepsilon } _ { k } H ) e _ { k } = ( I - \hat { \varepsilon } _ { k } H ^ { * } ) e _ { k } - \hat { \varepsilon } _ { k } \Big ( 1 - \frac { ( w _ { k } ^ { T } g ) ^ { 2 } } { u ^ { T } H u \sigma _ { k } ^ { 2 } } \Big ) H u . } \end{array}
+$$
+
+Then we will use the properties of $H ^ { * }$ -seminorm to prove our argument.
+
+(1) Estimate the $H ^ { * }$ -seminorm on the right hand of equation (63).
+
+$$
+\begin{array} { r l } & { \| \mathrm { r i g h t } \| _ { H ^ { * } } \leq \| ( I - \hat { \varepsilon } _ { k } H ^ { * } ) e _ { k } \| _ { H ^ { * } } + | \hat { \varepsilon } _ { k } | \Big ( 1 - \frac { ( w _ { k } ^ { T } g ) ^ { 2 } } { u ^ { T } H u \sigma _ { k } ^ { 2 } } \Big ) \| H u \| _ { H ^ { * } } } \\ & { \qquad \leq \rho ^ { * } ( I - \hat { \varepsilon } _ { k } H ^ { * } ) \| e _ { k } \| _ { H ^ { * } } + \frac { \lambda _ { m a x } | \hat { \varepsilon } _ { k } | } { \sqrt { u ^ { T } H u } } \| e _ { k } \| _ { H } ^ { 2 } } \\ & { \qquad = \rho ^ { * } ( I - \hat { \varepsilon } _ { k } H ^ { * } ) \frac { | w _ { k } ^ { T } g | } { \sqrt { u ^ { T } H u \sigma _ { k } } } \| e _ { k } \| _ { H } + \frac { \lambda _ { m a x } \varepsilon | a _ { k } w _ { k } ^ { T } g | } { \sqrt { u ^ { T } H u \sigma _ { k } ^ { 3 } } } \| e _ { k } \| _ { H } ^ { 2 } } \\ & { \qquad = \frac { | w _ { k } ^ { T } g | } { \sqrt { u ^ { T } H u \sigma _ { k } } } \Big ( \rho ^ { * } ( I - \hat { \varepsilon } _ { k } H ^ { * } ) + \delta _ { k } \Big ) \| e _ { k } \| _ { H } . } \end{array}
+$$
+
+(2) Estimate the $H ^ { * }$ -seminorm on the left hand of equation (63). Using the $H$ -norm on the iteration of $w _ { k }$ , we have
+
+$$
+\begin{array} { r } { \sigma _ { k + 1 } = \| w _ { k } + \varepsilon \frac { a _ { k } } { \sigma _ { k } } H e _ { k } \| _ { H } \ge \sigma _ { k } - \varepsilon \frac { \lambda _ { m a x } | a _ { k } | } { \sigma _ { k } } \| e _ { k } \| _ { H } . } \end{array}
+$$
+
+Consequently, we have
+
+$$
+\begin{array} { r } { \| \mathrm { l e f t } \| _ { H ^ { * } } = \frac { | w _ { k } ^ { T } g | } { \sqrt { u ^ { T } H u \sigma _ { k } } } \frac { \sigma _ { k + 1 } } { \sigma _ { k } } \| e _ { k + 1 } \| _ { H } \geq \frac { | w _ { k } ^ { T } g | } { \sqrt { u ^ { T } H u \sigma _ { k } } } ( 1 - \delta _ { k } ) \| e _ { k + 1 } \| _ { H } . } \end{array}
+$$
+
+(3) Combining (1) and (2), we finish the proof.
+
+Now, we turn to the convergence of the loss function which can be rewritten as $\begin{array} { r } { J _ { k } = \frac { 1 } { 2 } \| \tilde { e } _ { k } \| _ { H } ^ { 2 } } \end{array}$ with $\begin{array} { r } { \tilde { e } _ { k } = u - \frac { a _ { k } } { \sigma _ { k } } w _ { k } } \end{array}$ . There is an useful equality between $\| \tilde { e } _ { k } \| _ { H } ^ { 2 }$ and $\| e _ { k } \| _ { H } ^ { 2 }$ :
+
+$$
+\begin{array} { r } { \| \tilde { e } _ { k } \| _ { H } ^ { 2 } = \| e _ { k } \| _ { H } ^ { 2 } + \Big ( a _ { k } - \frac { w _ { k } ^ { T } g } { \sigma _ { k } } \Big ) ^ { 2 } . } \end{array}
+$$
+
+Recalling the inequality (57) and the boundedness of $a _ { k }$ , we have a constant $C _ { 0 }$ such that
+
+$$
+\begin{array} { r } { \left| { a } _ { k + 1 } - \frac { w _ { k + 1 } ^ { T } g } { \sigma _ { k + 1 } } \right| \leq | 1 - \varepsilon _ { a } | \left| { a } _ { k } - \frac { w _ { k } ^ { T } g } { \sigma _ { k } } \right| + C _ { 0 } \| e _ { k } \| _ { H } , } \end{array}
+$$
+
+which indicates that we can use the convergence of $e _ { k }$ to estimate the convergence of the loss value $J _ { k }$ . In fact we have the following lemma.
+
+Lemma A.23. If $\| e _ { k } \| _ { H } \leq C \rho ^ { k }$ for some constant $C$ and $\rho \in ( 0 , 1 )$ , $\varepsilon _ { a } \in ( 0 , 1 ]$ , then we have
+
+$$
+\| \widetilde { e } _ { k } \| _ { H } ^ { 2 } \le C ^ { 2 } \rho ^ { 2 k } + \Bigl ( C _ { 1 } ( 1 - \varepsilon _ { a } ) ^ { k } + C _ { 2 } k \gamma ^ { k } \Bigr ) ^ { 2 } ,
+$$
+
+where $\gamma = \operatorname* { m a x } ( \rho , 1 - \varepsilon _ { a } )$ , $C _ { 1 } = | a _ { 0 } - w _ { 0 } ^ { T } g / \sigma _ { 0 } |$ and $C _ { 2 } = C C _ { 0 }$ .
+
+Proof. According to the inequality (71), we have
+
+$$
+\left| a _ { k } - \frac { w _ { k } ^ { T } g } { \sigma _ { k } } \right| \leq C _ { 1 } ( 1 - \varepsilon _ { a } ) ^ { k } + C _ { 2 } \sum _ { i = 0 } ^ { k - 1 } ( 1 - \varepsilon _ { a } ) ^ { i } \rho ^ { k - i } \leq C _ { 1 } ( 1 - \varepsilon _ { a } ) ^ { k } + C _ { 2 } k \gamma ^ { k } .
+$$
+
+Put it in the equality (70), then we finish the proof.
+
+# A.6 ESTIMATING THE EFFECTIVE STEP SIZE
+
+Accorded to Lemma A.12, the effective step size $\hat { \varepsilon } _ { k }$ has same order with εwk 2 provided a0wT0 g > $0 , \varepsilon / | | w _ { 0 } | | < \varepsilon _ { 0 }$ . In fact, we have
+
+$$
+\begin{array} { r } { \frac { C _ { 1 } \varepsilon } { \| w _ { k } \| ^ { 2 } } : = \frac { a _ { 0 } w _ { 0 } ^ { T } g } { \sigma _ { 0 } } \frac { \varepsilon } { \lambda _ { m a x } \| w _ { k } \| ^ { 2 } } \le \hat { \varepsilon } _ { k } \le \sqrt { u ^ { T } H u } \frac { C _ { a } \varepsilon } { \lambda _ { m i n } \| w _ { k } \| ^ { 2 } } = : \frac { C _ { 2 } \varepsilon } { \| w _ { k } \| ^ { 2 } } . } \end{array}
+$$
+
+Hence, to prove the Proposition 3.5, we only need to estimate the norm of $w _ { k }$
+
+proof of Proposition 3.5. According to the BNGD iteration, we have (see the proof of Lemma A.11)
+
+$$
+\| w _ { k + 1 } \| ^ { 2 } \leq \| w _ { 0 } \| ^ { 2 } + \varepsilon ^ { 2 } \lambda _ { m a x } \sum _ { i = 0 } ^ { k } \frac { a _ { i } ^ { 2 } } { \sigma _ { i } ^ { 2 } } q _ { i } .
+$$
+
+(1) When $\frac { \varepsilon } { \| w _ { 0 } \| ^ { 2 } } < \varepsilon _ { 0 }$ ( $\scriptstyle \varepsilon _ { 0 }$ is defined in Lemma A.12), the sequence $q _ { k }$ satisfies $q _ { k + 1 } \leq ( 1 -$ $\hat { \varepsilon } _ { k } \lambda _ { m i n } ) q _ { k }$ . Hence the norm of $w _ { k }$ is bounded by
+
+$$
+\begin{array} { r } { \| w _ { k } \| ^ { 2 } \leq \| w _ { 0 } \| ^ { 2 } + \varepsilon \kappa C _ { a } \frac { \sigma _ { 0 } } { w _ { 0 } ^ { T } g } \displaystyle \sum _ { i = 0 } ^ { \infty } ( q _ { i } - q _ { i + 1 } ) \leq \| w _ { 0 } \| ^ { 2 } + C \varepsilon , } \end{array}
+$$
+
+for some constant $C$ . As a consequence,
+
+$$
+\begin{array} { r } { \tilde { C } _ { 1 } \varepsilon : = \frac { C _ { 1 } \varepsilon } { \| w _ { 0 } \| ^ { 2 } ( 1 + C \varepsilon _ { 0 } ) } \leq \hat { \varepsilon } _ { k } \leq \frac { C _ { 2 } \varepsilon } { \| w _ { 0 } \| ^ { 2 } } = : \tilde { C } _ { 2 } \varepsilon . } \end{array}
+$$
+
+(2) When $\varepsilon$ is large enough, the increment of the norm $\| w _ { k } \|$ at the first step is large as well. In fact, we have
+
+$$
+\begin{array} { r } { \| w _ { 1 } \| ^ { 2 } - \| w _ { 0 } \| ^ { 2 } = \varepsilon ^ { 2 } \frac { a _ { 0 } ^ { 2 } } { \sigma _ { 0 } ^ { 2 } } \| H e _ { 0 } \| ^ { 2 } = C _ { 3 } \varepsilon ^ { 2 } . } \end{array}
+$$
+
+Since $\begin{array} { r } { | | g | | ^ { 2 } \geq \frac { w _ { 0 } ^ { T } g } { \sigma _ { 0 } ^ { 2 } } g ^ { T } H w _ { 0 } } \end{array}$ , we have $a _ { 1 } w _ { 1 } ^ { T } g > a _ { 1 } w _ { 0 } ^ { T } g > 0 .$ . Choose $\varepsilon$ to be larger than some value $\varepsilon _ { 1 }$ such that $\frac { \varepsilon } { \| w _ { 1 } \| ^ { 2 } } < \varepsilon _ { 0 }$ , then we can use the argument in (1) on $( a _ { 1 } , w _ { 1 } )$ . More precisely, there are two constants, $C _ { 1 } , C _ { 2 }$ , such that
+
+$$
+\begin{array} { r } { \frac { C _ { 1 } \varepsilon } { \| w _ { 1 } \| ^ { 2 } } \leq \hat { \varepsilon } _ { k } \leq \frac { C _ { 2 } \varepsilon } { \| w _ { 1 } \| ^ { 2 } } . } \end{array}
+$$
+
+Plugging the equation (78) into it, we have
+
+$$
+\begin{array} { r } { \frac { C _ { 1 } \varepsilon _ { 1 } ^ { 2 } } { \| w _ { 0 } \| ^ { 2 } + C _ { 3 } \varepsilon _ { 1 } ^ { 2 } } \leq \frac { C _ { 1 } \varepsilon ^ { 2 } } { \| w _ { 0 } \| ^ { 2 } + C _ { 3 } \varepsilon ^ { 2 } } \leq \hat { \varepsilon } _ { k } \varepsilon \leq \frac { C _ { 2 } \varepsilon ^ { 2 } } { \| w _ { 0 } \| ^ { 2 } + C _ { 3 } \varepsilon ^ { 2 } } \leq \frac { C _ { 2 } } { C _ { 3 } } . } \end{array}
+$$
+
+# B MODIFIED BNGD
+
+Through our analysis, we discovered that a modification of the BNGD, which we call MBNGD, becomes much easier to analyze and possesses better convergence properties. Note that the results in the main paper do not depend on the results in this section.
+
+The modification is simply to enforce $\begin{array} { r } { a _ { k } \ = \ \frac { w _ { k } ^ { T } g } { \sigma _ { k } } } \end{array}$ at every iteration, which yields the modified iterations:
+
+$$
+\begin{array} { r } { \boldsymbol { w } _ { k + 1 } = \boldsymbol { w } _ { k } + \varepsilon \frac { \boldsymbol { w } _ { k } ^ { T } \boldsymbol { g } } { \sigma _ { k } ^ { 2 } } \Big ( \boldsymbol { g } - \frac { \boldsymbol { w } _ { k } ^ { T } \boldsymbol { g } } { \sigma _ { k } ^ { 2 } } H \boldsymbol { w } _ { k } \Big ) . } \end{array}
+$$
+
+In a sense, one can view the above as a limiting version of BNGD where the BN rescaling variable $a$ is adjusted every step to the optimal value based on the current value of the weights $w$ . The MBNGD iterations is governed by the variables: $H , u , a _ { 0 } , w _ { 0 } , \varepsilon$ , where the scaling properties (Lemma 3.2, omit the parameter $\varepsilon _ { a }$ now) remains. More importantly, we find the iteration will converge to a saddle point if and only if it exactly meets the saddle point at a finite step. More precisely, we have the convergence theorem:
+
+Theorem B.1 (Convergence for modified BNGD). The iteration sequence $w _ { k }$ in equation (81) converges for any initial value $w _ { 0 }$ and any step size $\varepsilon > 0$ . It converges to a global minimizer almost sure, in the sense that the set of initial values such that $w _ { k }$ converges to a saddle point is of Lebesgue measure zero. Furthermore, It converges to a saddle point if and only if $w _ { k } ^ { T } g = 0$ for some $k$ .
+
+Particularly, if < 2kw0k2λmaxκ u 2 , wT0 g 6= 0, then wk converges to a global minimizer.
+
+In the following, we assume $\lVert u \rVert = 1$
+
+# B.1 PROOF
+
+Lemma B.2. If $w _ { 0 } ^ { T } g \neq 0$ and $\begin{array} { r } { \frac { \varepsilon } { \| w _ { 0 } \| ^ { 2 } } < \varepsilon _ { 0 } : = \frac { 2 } { \kappa \lambda _ { m a x } } } \end{array}$ , then the sequence $w _ { k }$ converges to a global minimizer.
+
+Proof. Similar to the proof of Lemma A.12, but here the effective step size is always nonnegative which is defined as
+
+$$
+\begin{array} { r } { \hat { \varepsilon } _ { k } : = \varepsilon \bigl ( \frac { w _ { k } ^ { T } g } { \sigma _ { k } ^ { 2 } } \bigr ) ^ { 2 } \leq \frac { \varepsilon \kappa } { \| w _ { k } \| ^ { 2 } } \leq \frac { \varepsilon \kappa } { \| w _ { 0 } \| ^ { 2 } } = : \hat { \varepsilon } ^ { + } < \frac { 2 } { \lambda _ { m a x } } . } \end{array}
+$$
+
+The inequality (38) immediately gives $q _ { k + 1 } \leq q _ { k }$ , which implies $\begin{array} { r } { \frac { ( w _ { k + 1 } ^ { T } g ) ^ { 2 } } { \sigma _ { k + 1 } ^ { 2 } } \geq \frac { ( w _ { k } ^ { T } g ) ^ { 2 } } { \sigma _ { k } ^ { 2 } } \geq \frac { ( w _ { 0 } ^ { T } g ) ^ { 2 } } { \sigma _ { 0 } ^ { 2 } } } \end{array}$ . As a consequence, the effective step size has a lower bound
+
+$$
+\begin{array} { r } { \hat { \varepsilon } _ { k } \geq \varepsilon \frac { ( w _ { 0 } ^ { T } g ) ^ { 2 } } { \sigma _ { 0 } ^ { 2 } } \frac { 1 } { \lambda _ { m a x } \| w _ { k } \| ^ { 2 } } = : \frac { \varepsilon ^ { - } } { \| w _ { k } \| ^ { 2 } } . } \end{array}
+$$
+
+Employing the Lemma A.11, we conclude that $w _ { k }$ converges to a global minimizer.
+
+Proof. Without loss of generality, we assume $\| w _ { 0 } \| \ge 1$ , denote $y _ { k } : = w _ { k } ^ { T } g$ and set $\begin{array} { r } { \delta = \frac { \| g \| } { 4 \kappa } } \end{array}$ . From the iteration of $w _ { k }$ , we have
+
+$$
+\begin{array} { r } { y _ { k + 1 } = y _ { k } + \epsilon \frac { y _ { k } } { \sigma _ { k } ^ { 2 } } \Big ( \| g \| ^ { 2 } - \frac { y _ { k } } { \sigma _ { k } ^ { 2 } } g ^ { T } H w _ { k } \Big ) . } \end{array}
+$$
+
+If $0 < | w _ { k } ^ { T } g | < 2 \delta$ , then we have the inequality:
+
+$$
+\begin{array} { r } { \| g \| ^ { 2 } - \frac { y _ { k } } { \sigma _ { k } ^ { 2 } } g ^ { T } H w _ { k } \geq \| g \| ^ { 2 } - \frac { \kappa \| g \| } { \| w _ { k } \| } | y _ { k } | \geq \frac { 1 } { 2 } \| g \| ^ { 2 } , } \end{array}
+$$
+
+then
+
+$$
+\begin{array} { r } { | y _ { k + 1 } | \geq \left( 1 + \frac { \epsilon } { \lambda _ { m a x } \| w _ { k } \| ^ { 2 } } \frac { \| g \| ^ { 2 } } { 2 } \right) | y _ { k } | > | y _ { k } | > 0 . } \end{array}
+$$
+
+As a consequence, $\scriptstyle \operatorname* { l i m } _ { k \to \infty } w _ { k } ^ { T } g = 0$ is not possible unless $w _ { k } ^ { T } g = 0$ for some $k$ , which implies the results we want.
+
+Theorem B.4. The iteration sequence $w _ { k }$ in equation (81) converges for any initial value $w _ { 0 }$ and any step size $\varepsilon > 0$ . Furthermore, $w _ { k }$ will converge to a global minimizer unless $w _ { k } ^ { T } g = 0$ for some $k$ .
+
+Proof. Obviously, if $w _ { k } ^ { T } g = 0$ for some $k = k _ { 0 }$ , then $w _ { k } = w _ { k _ { 0 } }$ for all $k \geq k _ { 0 }$ , hence $w _ { k }$ converges to $w _ { k _ { 0 } }$ . Without losing generality, we consider $w _ { k } ^ { T } g \neq 0$ for all $k$ and $\| w _ { 0 } \| \ge 1$ below.
+
+(1) Firstly, we will prove that $\| w _ { k } \|$ is bounded and hence converges.
+
+In fact, according to the Lemma B.2, once $\| w _ { k } \| ^ { 2 } \geq \varepsilon / \varepsilon _ { 0 }$ for some $k$ , the rest of the iteration will converge, hence $\| w _ { k } \|$ is bounded.
+
+(2) Secondly, we will prove $w _ { k }$ converges to a vector parallel to $u$
+
+Denote $\begin{array} { r } { y _ { k } : = w _ { k } ^ { T } g , z _ { k } : = \frac { w _ { k } ^ { T } g } { \sigma ^ { 2 } } } \end{array}$ . The convergence of $\| w _ { k } \|$ indicates that $\textstyle \sum _ { k = 0 } ^ { \infty } z _ { k } ^ { 2 } q _ { k }$ is summable, and then $\textstyle \sum _ { k = 0 } ^ { \infty } y _ { k } ^ { 2 } q _ { k }$ is summable as well. Therefore we have
+
+$$
+\begin{array} { r } { \left\| w _ { k + 1 } - w _ { k } \right\| ^ { 2 } = \varepsilon ^ { 2 } \frac { ( w _ { k } ^ { T } g ) ^ { 2 } } { \sigma ^ { 4 } } \bigg \| g - \frac { w _ { k } ^ { T } g } { \sigma _ { k } ^ { 2 } } H w _ { k } \bigg \| ^ { 2 } \leq \lambda _ { m a x } \varepsilon ^ { 2 } z _ { k } ^ { 2 } q _ { k } , } \end{array}
+$$
+
+and the above tends to zero, i.e. $\begin{array} { r } { \operatorname* { l i m } _ { k  \infty } \| w _ { k + 1 } - w _ { k } \| = 0 } \end{array}$
+
+According to the separation property (Lemma A.15), we can chose a $\delta _ { 0 } > 0$ small enough such that the separated parts of the set $S : = \left\{ w | y ^ { 2 } q < \delta _ { 0 } , \| w \| \ge 1 \right\}$ , $S _ { 1 }$ and $S _ { 2 }$ , have $d i s t ( S _ { 1 } , S _ { 2 } ) > 0$ .
+
+Because $y _ { k } ^ { 2 } q _ { k }$ tends to zero, we have $w _ { k }$ belongs to $S$ for $k$ large enough, for instance $k > k _ { 1 }$ . On the other hand, because $\| w _ { k + 1 } - w _ { k } \|$ tends to zero, we have $\lVert w _ { k + 1 } - w _ { k } \rVert < d i s t ( S _ { 1 } , S _ { 2 } )$ for $k$ large enough, for instance $k > k _ { 2 }$ . Then consider $k > k _ { 3 } : = \operatorname* { m a x } ( k _ { 1 } , k _ { 2 } )$ , we have all $w _ { k }$ belongs to the same part $S _ { 1 }$ or $S _ { 2 }$ .
+
+However, Lemma B.3 says $\scriptstyle \sum _ { k = 0 } ^ { \infty } y _ { k } ^ { 2 } \ = \infty$ , hence $w _ { k } \in S _ { 1 }$ $( q _ { k } > \delta _ { 2 } )$ for all $k > k _ { 3 }$ is not true. Therefore $w _ { k } \in S _ { 2 } ( y _ { k } ^ { 2 } > \delta _ { 1 } )$ for all $k > k _ { 3 }$ . Consequently, we can claim that $\textstyle \sum _ { k = 0 } ^ { \infty } q _ { k }$ is summable and $w _ { k }$ converges to a vector parallel to $u$ .
+
+# B.2 EXPERIMENT
+
+Here we test the convergence and stability of MBNGD for OLS model. Consider the diagonal matrix $H = d i a g ( h )$ , where $h = ( 1 , . . . , \kappa )$ is an increasing sequence. The scaling property allows us to set the initial value $w _ { 0 }$ having same 2-norm with $u$ , $\| \bar { w } _ { 0 } \| \overset { - } { = } \| u \| = 1$ .
+
+Figure 5 gives an example of a 5-dimensional $H$ with condition number $\kappa = 2 0 0 0$ . The GD and MBNGD iteration are executed $k = 5 0 0 0$ times where $u$ and $w _ { 0 }$ are randomly chosen from the unit sphere. The values of effective step size, loss $\| e _ { k } \| _ { H } ^ { 2 }$ and error $\| e _ { k } \|$ are plotted. Furthermore, to explore the performance of GD and MBNGD, the mean values over 300 random tests are given. It is worth to note that, the geometric mean (G-mean) is more reliable than the arithmetic mean (Amean), where the geometric mean of $x$ can be defined as $\exp ( \mathbb { E } ( \ln x ) )$ . Here the reliability means that the G-mean converges quickly when the number of tests increase, however the A-mean does not converge as quickly. In this example, the optimal convergence rate of MBNGD is observably better than GD. This acceleration phenomenon is ascribed to the pseudo-condition number of $\kappa ^ { * } ( H ^ { * } )$ being less than $\kappa ( H )$ . However, if the difference between (pseudo-)condition number of $H$ and $H ^ { * }$ is small, the acceleration is imperceptible.
+
+Another important observation is that the BN significantly extends the range of ‘optimal’ step size, which is embodied by the effective step size $\hat { \varepsilon } _ { k }$ having a large constant $C$ in $\hat { \varepsilon } = \stackrel { - } { O } ( C \varepsilon ^ { - 1 } )$ . This means we can chose step size in BN at a large interval to get almost same or better convergence rate than that of the best choice for GD.
+
+Figure 6 gives an example of 100-dimension $H$ with condition number $\kappa = 2 0 0 0$ . Similar results as those in the 5-dimensional case are obtained. However, the best optimal convergence rate of MBNGD here has not noticeably improved compared with GD with the optimal learning rate, which is due to the fact that large $d$ decrease the difference between eigenvalues of $H$ and $H ^ { * }$ .
+
+Additional tests indicate that:
+
+![](images/e2118b1de21777d9d090dfdd9f6f11ed4d624440b4428d09e4f70c518c7f98fc.jpg)  
+Figure 5: Plot of 300 random initial tests. $\mathrm { H } =$ diag(logspace(0,log10(2000),5)).
+
+![](images/0f2250d9076a3b166fba2f8f22fe04145795f432e1c8459b5fb1282d2f0c5cdc.jpg)  
+Figure 6: Plot of 500 random initial tests. $\mathrm { H } =$ diag(linspace(1,2000,100)).
+
+(1) larger dimensions leads to larger intervals of ‘optimal’ step size, (Figure 7)
+
+(2) the effect of condition number on the ‘optimal’ interval is small (Figure 8).
+
+![](images/0a6e31e9cdca96a921d3ab705eef68df7d884908fbbf0960238be533f26474e1.jpg)  
+Figure 7: $\mathrm { H } =$ diag(linspace(1,2000,d)).
+
+![](images/f711f7570a97f8aed4035ade1ceab57535f2f1ed5befeec7ad8bbaf74dea258e.jpg)  
+Figure 8: $\mathrm { H } =$ diag(linspace(1,cond,100)).

parse/train/SkqV-XZRZ/SkqV-XZRZ.md

@@ -0,0 +1,290 @@
+# VARIATIONAL BI-LSTMS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Recurrent neural networks like long short-term memory (LSTM) are important architectures for sequential prediction tasks. LSTMs (and RNNs in general) model sequences along the forward time direction. Bidirectional LSTMs (Bi-LSTMs), which model sequences along both forward and backward directions, generally perform better at such tasks because they capture a richer representation of the data. In the training of Bi-LSTMs, the forward and backward paths are learned independently. We propose a variant of the Bi-LSTM architecture, which we call Variational Bi-LSTM, that creates a dependence between the two paths (during training, but which may be omitted during inference). Our model acts as a regularizer and encourages the two networks to inform each other in making their respective predictions using distinct information. We perform ablation studies to better understand the different components of our model and evaluate the method on various benchmarks, showing state-of-the-art performance.
+
+# 1 INTRODUCTION
+
+Recurrent neural networks (RNNs) have become the standard models for sequential prediction tasks, having achieved state of the art performance in a number of applications that includes sequence prediction, language translation, machine comprehension, and speech synthesis (Arik et al., 2017; Wang et al., 2017; Mehri et al., 2016; Sotelo et al., 2017). RNNs model temporal data by encoding a given arbitrary-length input sequentially, at each time step combining some transformation of the current input with the encoding from the previous time step. This encoding, referred to as the RNN hidden state, summarizes all previous input tokens.
+
+Viewed as “unrolled” feedforward networks, RNNs can become arbitrarily deep depending on the input sequence length, and use a repeating module to combine the input with the previous state at each time step. Consequently, they suffer from the vanishing/exploding gradient problem (Pascanu et al., 2012). This problem has been addressed through architectural variants like the long short-term memory (LSTM) (Hochreiter & Schmidhuber, 1997) and the gated recurrent unit (GRU) (Chung et al., 2014). These architectures add a linear path along the temporal sequence which allows gradients to flow more smoothly back through time.
+
+Various regularization techniques have also been explored to improve RNN performance and generalization. Dropout (Srivastava et al., 2014) regularizes a network by randomly dropping hidden units during training. However, it has been observed that using dropout directly on RNNs is not as effective as in the case of feed-forward networks. To combat this, Zaremba et al. (2014) propose to instead apply dropout on the activations that are not involved in the recurrent connections (Eg. in a multi-layer RNN); Gal & Ghahramani (2016) propose to apply the same dropout mask through an input sequence during training. In a similar spirit to dropout, Zoneout (Krueger et al., 2016) proposes to choose randomly whether to use the previous RNN hidden state.
+
+The aforementioned architectures model sequences along the forward direction of the input sequence. Bidirectional-LSTM, on the other hand, is a variant of LSTM that simultaneously models each sequence in both the forward and backward directions. This enables a richer representation of data, since each token’s encoding contains context information from the past and the future. It has been shown empirically that bidirectional architectures generally outperform unidirectional ones on many sequence-prediction tasks. However, the forward and backward paths in Bi-LSTMs are trained separately and the benefit usually comes from the combined hidden representation from both paths. In this paper, our main idea is to create a dependence between the two paths that acts as a regularization during training, but doesn’t hinder inference even in the absence of the other path (the backward path in practical scenarios). We note that recently proposed methods like TwinNet Serdyuk et al. (2017) and Z-forcing Sordoni et al. (2017) are similar in spirit to this idea. In our approach, we use a variational auto-encoder (VAE; Kingma & Welling (2014)) that takes as input the hidden states from the two paths of the Bi-LSTM and maps them to a shared hidden representation of the VAE at each time step. The samples from the VAE hidden state are then used both for reconstructing the LSTM hidden states and feeding forward to the next hidden state. In this way, we create a channel between the two paths that acts as a regularization for learning better representations. We refer to the resulting model as a Variational Bi-LSTM.
+
+Below, we describe Variational Bi-LSTMs in detail and then demonstrate empirically their ability to model complex sequential distributions. In experiments, we obtain state-of-the-art or competitive performance on the tasks of Penn Treebank, IMDB, TIMIT, Blizzard, and Sequential MNIST.
+
+# 2 VARIATIONAL BI-LSTM
+
+Bi-LSTM is a powerful architecture for sequential tasks because it models temporal data both in the forward and backward directions. For this it uses two LSTMs that are generally learned independently of each other; the richer representation results from combining the hidden states of these LSTMs, where combination is often by concatenation. The idea behind variational Bi-LSTMs is to create a channel of information exchange between the two LSTMs that helps the model to learn better representations. We create this dependence using a variational auto-encoder (VAE). This enables us to take advantage of the fact that VAE allows for sampling from a prior during inference. For sequence prediction tasks like language generation, while one can use Bi-LSTMs during training, there is no straightforward way to employ the full bidirectional model during inference – this would involve, eg, generating a sentence starting at both its beginning and end. In such cases, the VAE allows us to sample from the prior at inference time to make up for the absence of the backward LSTM.
+
+Now we describe our variational Bi-LSTM model formally. Let $\mathbf { X } = \{ \mathbf { x } ^ { ( i ) } \} _ { i = 1 } ^ { N }$ be a dataset consisting of $N$ i.i.d. sequential data samples of continuous or discrete variables. For notational convenience, we will henceforth drop the superscript $i$ indexing samples. For each sample sequence $\mathbf { x } = ( \mathbf { x } _ { 1 } , \ldots , \mathbf { x } _ { T } )$ , the hidden state of the forward LSTM is given by:
+
+$$
+\mathbf { h } _ { t } = \vec { f } ( \mathbf { x } _ { t } , \mathbf { h } _ { t - 1 } , \mathbf { z } _ { t } , \tilde { \mathbf { b } } _ { t } ) .
+$$
+
+The hidden state of the backward LSTM is given by,
+
+$$
+\mathbf { b } _ { t } = \overleftarrow { f } ( \mathbf { x } _ { t } , \mathbf { b } _ { t + 1 } ) .
+$$
+
+In both cases, the function $f$ represents the standard LSTM updates, modified to account for the additional arguments.
+
+In the forward LSTM model, we introduce three latent random variables, $\mathbf { z } _ { t } , \tilde { \mathbf { b } } _ { t }$ , and $\tilde { \mathbf { h } } _ { t - 1 }$ , where $\mathbf { z } _ { t }$ depends on $\mathbf { h } _ { t - 1 }$ and $\mathbf { b } _ { t }$ during training, and $\tilde { \mathbf { b } } _ { t }$ and $\tilde { \mathbf { h } } _ { t - 1 }$ depend only on $\mathbf { z } _ { t }$ (see figure 1-left, for a graphical representation). Note that so far, $\tilde { \mathbf { b } } _ { t }$ and $\tilde { \mathbf { h } } _ { t - 1 }$ are simply latent vectors drawn from conditional distributions $p _ { \psi }$ and $p _ { \xi }$ , respectively, to be defined below. However, as explained in Section 2.1 (see also dashed lines in figure 1-left), we will encourage these to lie near the manifolds of backward and forward LSTM states, respectively.
+
+By design, the joint conditional distribution over latent variables $\mathbf { z } _ { t }$ and $\tilde { \mathbf { b } } _ { t }$ with parameters $\theta$ and $\psi$ factorizes as $p _ { \theta } ( \mathbf { z } _ { t } | \mathbf { x } _ { 1 : t - 1 } , \mathbf { z } _ { 1 : t - 1 } ) p _ { \psi } ( \tilde { \mathbf { b } } _ { t } | \mathbf { z } _ { t } )$ . This factorization enables us to formulate several helpful auxiliary cots, as defined in the next subsection. Further, $p _ { \eta } ( \mathbf { x } _ { t + 1 } | \mathbf { x } _ { 1 : t } , \mathbf { z } _ { 1 : t } , \tilde { \mathbf { b } } _ { t } )$ defines the generating model, which induces the distribution over the next observation given the previous states and the current input.
+
+![](images/161f5732f91f90cba658cac7b9f25b2c774e2166fc933313bfe7d4393dcf6a8f.jpg)  
+(a) Training phase of variational Bi-LSTM
+
+![](images/99ebd8d75a0549959f48953dcef7cd6d9042f86704b858b60aa20dee95753e8b.jpg)  
+(b) Inference phase of variational Bi-LSTM   
+Figure 1: Graphical description of our proposed variational Bi-LSTM model during train phase (left) and inference phase (right). During training, each step $t$ is composed of an encoder which receives both the past and future summary via $\mathbf { h } _ { t - 1 }$ and $\mathbf { b } _ { t }$ respectively, and a decoder that generates $\tilde { \mathbf { h } } _ { t - 1 }$ and $\tilde { \mathbf { b } } _ { t }$ which are forced to be close enough to $\mathbf { h } _ { t - 1 }$ and $\mathbf { b } _ { t }$ using two auxiliary reconstruction costs (dashed lines). This dependence between backward and forward LSTM through the latent random variable encourages the forward LSTM to learn a richer representation. During inference, the backward LSTM is removed. In this case, $\mathbf { z } _ { t }$ is sampled from the prior as in a typical VAE, which in our case, is defined as a function of $\mathbf { h } _ { t - 1 }$ .
+
+Then the marginal likelihood of each individual sequential data sample $\mathbf { x }$ can be written as
+
+$$
+\begin{array} { l } { \displaystyle p ( { \bf x } ; { \bf Y } ) = \prod _ { t = 0 } ^ { T } p ( { \bf x } _ { t + 1 } \vert { \bf x } _ { 1 : t } ) = \prod _ { t = 0 } ^ { T } \int _ { { \bf z } _ { 1 : T } } p ( { \bf x } _ { t + 1 } \vert { \bf x } _ { 1 : t } , { \bf z } _ { 1 : t } ) p _ { \theta } ( { \bf z } _ { t } \vert { \bf x } _ { 1 : t - 1 } , { \bf z } _ { 1 : t - 1 } ) d { \bf z } _ { 1 : T } } \\ { \displaystyle \quad \quad = \prod _ { t = 0 } ^ { T } \int _ { { \bf z } _ { 1 : T } } \int _ { { \bar { \bf b } } _ { t } } \Big [ p _ { \eta } ( { \bf x } _ { t + 1 } \vert { \bf x } _ { 1 : t } , { \bf z } _ { 1 : t } , { \bf \tilde { b } } _ { t } ) p _ { \psi } ( \tilde { { \bf b } } _ { t } \vert { \bf z } _ { 1 : t } ) p _ { \theta } ( { \bf z } _ { t } \vert { \bf x } _ { 1 : t - 1 } , { \bf z } _ { 1 : t - 1 } ) \Big ] d { \tilde { \bf b } } _ { t } d { \bf z } _ { 1 : T } , } \end{array}
+$$
+
+where $q _ { \phi } ( \mathbf { z } _ { t } | \mathbf { x } )$ is the conditional inference model and $\Gamma = \{ \phi , \theta , \psi , \eta \}$ is the set of all parameters of the model. Here, we assume that all conditional distributions belong to parametrized families of distributions which can be evaluated and sampled from efficiently.
+
+Note that the joint distribution in equation (1) is intractable. Kingma & Welling (2014) demonstrated how to maximize a variational lower bound, ${ \mathcal { L } } _ { \Gamma }$ , of the data log likelihood instead, which is given by
+
+$$
+\log p ( \mathbf { x } ; \mathbf { r } ) \ge \mathcal { L } _ { \mathbf { T } } = \sum _ { t = 0 } ^ { T } \underbrace { E } _ { \substack { \mathbf { z } _ { 1 : T } \sim q _ { \phi } ( \mathbf { z } | \mathbf { x } ) \tilde { \mathbf { b } } _ { t } \sim p _ { \psi } ( \tilde { \mathbf { b } } _ { t } | \mathbf { z } _ { t } ) } } \Big [ \log p _ { \eta } ( \mathbf { x } _ { t + 1 } | \mathbf { x } _ { 1 : t } , \mathbf { z } _ { 1 : t } , \tilde { \mathbf { b } } _ { t } ) \Big ]
+$$
+
+where $D _ { K L }$ is the Kullback-Leibler (KL) divergence between the approximate posterior and the conditional prior (see the appendix). This is the approach we take.
+
+# 2.1 TRAINING AND INFERENCE
+
+In the proposed variational Bi-LSTM, the latent variable $\mathbf { z } _ { t }$ is inferred as
+
+$$
+\mathbf { z } _ { t } \sim q _ { \phi } ( \mathbf { z } _ { t } | ( \mathbf { h } _ { t - 1 } , \mathbf { b } _ { t } ) ) = \mathcal { N } ( \pmb { \mu } _ { q , t } , \mathrm { d i a g } ( \pmb { \sigma } _ { q , t } ^ { 2 } ) ) ,
+$$
+
+in which $[ { \pmb \mu } _ { q , t } , { \pmb \sigma } _ { q , t } ^ { 2 } ] = f _ { \phi } ( { \bf h } _ { t - 1 } , { \bf b } _ { t } )$ where $f _ { \phi }$ is a multi-layered feed-forward network with Gaussian outputs. We assume that the prior over $\mathbf { z } _ { t }$ is a diagonal multivariate Gaussian distribution given by
+
+$$
+p _ { \theta } ( \mathbf { z } _ { t } | \mathbf { x } _ { 1 : t - 1 } , \mathbf { z } _ { 1 : t - 1 } ) = \mathcal { N } ( \mu _ { p , t } , \mathrm { d i a g } ( \sigma _ { p , t } ^ { 2 } ) ) , \quad \mathrm { w h e r e } \quad [ \mu _ { p , t } , \sigma _ { p , t } ^ { 2 } ] = f _ { \theta } ( \mathbf { h } _ { t - 1 } ) ,
+$$
+
+for a fully connected network $f _ { \theta }$ . This is important because, during generation (see Figure 1-right, for a graphical representation), we will not have access to the backward LSTM. In this case, as in a
+
+VAE, we will sample from the prior for $\mathbf { z } _ { t }$ . Since we define the prior to be a function of $\mathbf { h } _ { t - 1 }$ , the forward LSTM is encouraged during training to learn the dependency due to the backward hidden state $\mathbf { b } _ { t }$ .
+
+The latent variable $\tilde { \mathbf { b } } _ { t }$ is meant to model information coming from the future of the sequence. Its conditional distribution is given by
+
+$$
+p _ { \psi } ( \tilde { \mathbf { b } } _ { t } | \mathbf { z } _ { t } ) = \mathcal { N } ( \mu _ { \tilde { \mathbf { b } } , t } , \mathrm { d i a g } ( \pmb { \sigma } _ { \tilde { b } , t } ^ { 2 } ) ) ,
+$$
+
+where $[ { \pmb { \mu } } _ { { \tilde { \bf b } } , t } , { \pmb { \sigma } } _ { { \tilde { \bf b } } , t } ^ { 2 } ] = f _ { \psi } ( \mathbf { z } _ { t } )$ for a fully connected neural network $f _ { \psi }$ (See Figure 1(a)). To encourage the encoding of future information in $\tilde { \mathbf { b } } _ { t }$ , we maximize the probability of the true backward hidden state, $\mathbf { b } _ { t }$ , under the distribution $p _ { \psi }$ , as an auxiliary cost during training. In this way we treat $\tilde { \mathbf { b } } _ { t }$ as a predictor of $\mathbf { b } _ { t }$ , similarly to what was done by Sordoni et al. (2017).
+
+To capture information from the past in the latents, we similarly use $\tilde { \mathbf { h } } _ { t - 1 }$ as a predictor of $\mathbf { h } _ { t - 1 }$ . This is accomplished by maximizing the probability of the latter under the conditional distribution of the former, $\log p _ { \xi } ( \tilde { \mathbf { h } } _ { t - 1 } | \mathbf { z } _ { t } )$ , as another auxiliary cost, where
+
+$$
+p _ { \xi } ( \tilde { \mathbf { h } } _ { t - 1 } | \mathbf { z } _ { t } ) = \mathcal { N } ( \mu _ { \tilde { \mathbf { h } } , t } , \mathrm { d i a g } ( \pmb { \sigma } _ { \tilde { \mathbf { h } } , t } ^ { 2 } ) ) .
+$$
+
+Here, $[ \mu _ { \tilde { \mathbf { h } } , t } , \sigma _ { \tilde { \mathbf { h } } , t } ^ { 2 } ]$ is the output of a fully-connected neural network $f _ { \xi }$ taking $\mathbf { z } _ { t }$ as input. The auxiliary costs arising from distributions $p _ { \xi }$ and $p _ { \psi }$ teach the variational Bi-LSTM to encode past and future information into the latent space of $\mathbf { z }$ .
+
+We assume that parameters of the generating distribution $p _ { \eta } ( \mathbf { x } _ { t + 1 } | \mathbf { x } _ { 1 : t } , \mathbf { z } _ { 1 : t } , \tilde { \mathbf { b } } _ { t } )$ are computed via MLP, taking the form of either a Gaussian distribution output in the continuous case or categorical proportions output in the discrete (ie, one-hot) prediction case (See Figure 1(b)).
+
+All the parameters in $\mathbf { \delta T }$ and $\boldsymbol { \xi }$ are updated based on backpropagation through time (Rumelhart et al., 1988) using the reparameterization trick (Kingma & Welling, 2014), where the gradients are computed by differentiating of the following function:
+
+$$
+\begin{array} { r } { \mathcal { L } ( { \bf x } ; { \bf Y } , \pmb { \xi } ) = \displaystyle \sum _ { t = 0 } ^ { T } \sum _ { z _ { 1 } : \tau \sim q _ { \phi } ( { \bf z } | { \bf x } ) } \Big [ \underset { { \tilde { \bf b } } _ { t } \sim p _ { \psi } ( b | { \bf z } _ { 1 } : t ) } { \cal E } \Big [ \log p ( { \bf x } _ { t + 1 } | { \bf x } _ { 1 : t } , { \bf z } _ { 1 : t } , { \tilde { \bf b } } _ { t } ) + } \\ { \alpha \log p _ { \psi } ( { \bf b } _ { t } | { \bf z } _ { t } ) + \beta \log p _ { \xi } ( { \bf h } _ { t - 1 } | { \bf z } _ { t } ) \Big ] \Big ] - } \\ { D _ { K L } \big ( q _ { \phi } ( { \bf z } _ { t } | { \bf x } ) \big ) \big | p _ { \theta } \big ( { \bf z } _ { t } | { \bf x } _ { 1 : t - 1 } , { \bf z } _ { 1 : t - 1 } \big ) \big ) . } \end{array}
+$$
+
+Here, $\alpha$ and $\beta$ are non-negative real numbers. We improve training convergence with a trick for the variational Bi-LSTM, which we refer to as skip gradient, meant to ease learning of the latent variables. It is well known that autoregressive decoder models tend to ignore their stochastic variables (Bowman et al., 2015). Skip gradient is a technique to encourage that relevant summaries of the past and the future are encoded in the latent space. The idea is to skip the gradient of the stochastic operations with respect to the recurrent units through time. To achieve this, at each time step, a mask drawn from a Bernoulli distribution governs whether to skip the gradient or to backpropagate it for a given data point.
+
+# 3 EXPERIMENTAL RESULTS
+
+In this section demonstrate the effectiveness of our proposed model on several tasks. We present experimental results obtained when training the Variational Bi-LSTM on various sequential datasets: Penn Treebank (PTB), IMDB, TIMIT, Blizzard, and Sequential MNIST. Our main goal is to ensure that the model proposed in Section 2 can benefit from a generated relevant summary of the future that yields competitive results. In all experiments, we train all the models using ADAM optimizer (Kingma & Ba, 2014) and we set all MLPs in Section 2 to have one hidden layer with leaky-ReLU hidden activation. All the models are implemented using Theano (Theano Development Team, 2016) and the code is available at https://anonymous.url.
+
+Blizzard: Blizzard is a speech model dataset with 300 hours of English, spoken by a single female speaker. We report the average log-likelihood for half-second sequences (Fraccaro et al., 2016). In our experimental setting, we use 1024 hidden units for MLPs, 1024 LSTM units and 512 latents. Our model is trained using learning rate of 0.001 and minibatches of size 32 and we set $\alpha = \beta = 1$ . A fully factorized multivariate Gaussian distribution is used as the output distribution. The final lower bound estimation on TIMIT can be found in Table 1.
+
+Table 1: The average of log-likelihood per sequence on Blizzard and TIMIT testset   
+
+<table><tr><td>Model</td><td>Blizzard</td><td>TIMIT</td></tr><tr><td></td><td></td><td></td></tr><tr><td>RNN-Gauss</td><td>3539</td><td>-1900</td></tr><tr><td>RNN-GMM</td><td>7413</td><td>26643</td></tr><tr><td>VRNN-I-Gauss</td><td></td><td>28340</td></tr><tr><td>VRNN-Gauss</td><td>9223</td><td>28805</td></tr><tr><td>VRNN-GMM</td><td>9392</td><td>AIAIAI AIAI 28982</td></tr><tr><td>SRNN (smooth+resq)</td><td>11991</td><td>60550</td></tr><tr><td>Z-Forcing (Sordoni et al., 2017)</td><td>&gt;I&gt;I 14315</td><td>68852</td></tr><tr><td>VariationalBi-LSTM</td><td>≥ 17319</td><td>73315</td></tr></table>
+
+TIMIT: Another speech modeling dataset is TIMIT with 6300 English sentences read by 630 speakers. Like the work done in (Fraccaro et al., 2016), our model is trained on raw sequences of 200 dimensional frames. In our experiments, we use 1024 hidden units, 2048 LSTM units and 128 latent variables, and batch size of 128. We train the model using learning rate of 0.0001, $\alpha = 1$ and $\beta = 0$ . The average log-likelihood for the sequences on test can be found in Table 1.
+
+Sequential MNIST: We use the MNIST dataset which is binarized according to (Murray & Salakhutdinov, 2009) and we downloaded in binrized from (Larochelle, 2011). Our best model consists of 1024 hidden units, 1024 LSTM units and 256 latent variables. We train the model using a learning rate of 0.0001 and a batch size of 32. To reach the negative log-likelihood reported in Table 2, we set $\alpha = 0 . 0 0 1$ and $\beta = 0$ .
+
+Table 2: The average of negative log-likelihood on sequential MNIST   
+
+<table><tr><td>Models</td><td>Seq-MNIST</td></tr><tr><td>DBN 2hl (Germain et al., 2015) NADE (Uria et al., 2016) EoNADE-5 2hl (Raiko et al., 2014)</td><td>~ 84.55</td></tr><tr><td>DLGM 8 (Salimans et al., 2014)</td><td>88.33 84.68</td></tr><tr><td></td><td>≈ 85.51</td></tr><tr><td>DARN 1hl (Gregor et al.,2015) BiHM (Bornschein et al., 2015)</td><td>≈84.13</td></tr><tr><td></td><td>≈84.23</td></tr><tr><td>DRAW (Gregor et al., 2015)</td><td>≤ 80.97</td></tr><tr><td>PixelVAE (Gulrajani et al., 2016)</td><td>~ 79.02</td></tr><tr><td>Prof. Forcing (Goyal et al., 2016)</td><td></td></tr><tr><td>PixelRNN(1-layer) (Oord et al., 2016)</td><td>79.58</td></tr><tr><td></td><td>80.75</td></tr><tr><td>PixelRNN(7-layer) (Oord et al., 2016)</td><td></td></tr><tr><td>Z-Forcing (Sordoni et al.,2017)</td><td>79.20</td></tr><tr><td>VariationalBi-LSTM</td><td>≤ 80.09 ≤79.78</td></tr></table>
+
+IMDB: It is a dataset consists of 350000 movie reviews (Diao et al., 2014) in which each sentence has less than 16 words and the vocabulary size is fixed to 16000 words. In this experiment, we use 500 hidden units, 500 LSTM units and latent variables of size 64. The model is trained with a batch size of 32 and a learning rate of 0.001 and we set $\alpha = \beta = 1$ . The word perplexity on valid and test dataset is shown in Table 3.
+
+PTB: Penn Treebank (Marcus et al. (1993)) is a language model dataset consists of 1 million words. We train our model with 1024 LSTM units, 1024 hidden units, and the latent variables of size 128. We train the model using a standard Gaussian prior, a learning rate of 0.001 and batch size of 50 and we set $\alpha = \beta = 1$ . The model is trained to predict the next character in a sequence and the final bits per character on test and valid sets are shown in Table 4
+
+Table 3: Word perplexity on IMDB on valid and test sets   
+
+<table><tr><td>Model</td><td>Valid</td><td>Test</td></tr><tr><td></td><td></td><td></td></tr><tr><td>Gated Word-Char</td><td>70.60</td><td>70.87</td></tr><tr><td>Z-Forcing (Sordoni et al., 2017)</td><td>56.48</td><td>65.68</td></tr><tr><td>Variational Bi-LSTM</td><td>51.43</td><td>51.60</td></tr></table>
+
+![](images/206b55463e83fd04b6ab2796e4d7919dcbedf0b6c96b094d08f5e320a64735e3.jpg)  
+Figure 2: Evolution of the average of log-likelihood during training of Variational Bi-LSTMs with and without using skip gradient and auxiliary costs on PTB and Blizzard.
+
+Table 4: Bits Per Character (BPC) on PTB valid and test sets   
+
+<table><tr><td>Model</td><td>Valid</td><td>Test</td></tr><tr><td>Unregularized LSTM</td><td></td><td>1.36</td></tr><tr><td>Weight noise</td><td>1.47 1.51</td><td>1.34</td></tr><tr><td>Norm stabilizer</td><td>1.46</td><td>1.35</td></tr><tr><td>Stochastic depth</td><td>1.43</td><td>1.34</td></tr><tr><td>Recurrent dropout</td><td></td><td>1.29</td></tr><tr><td>Zoneout (Krueger et al. (2016))</td><td>1.40</td><td></td></tr><tr><td>RBN (Cooijmans et al. (2016))</td><td>1.36</td><td>1.25</td></tr><tr><td>H-LSTM + LN (Ha et al. (2016))</td><td>=</td><td>1.32</td></tr><tr><td>3-HM-LSTM + LN (Chung et al., 2016)</td><td>1.28</td><td>1.25</td></tr><tr><td>2-H-LSTM + LN (Ha et al. (2016))</td><td>=</td><td>1.24</td></tr><tr><td>Z-Forcing</td><td>1.25</td><td>1.22</td></tr><tr><td></td><td>1.29</td><td>1.26</td></tr><tr><td>Variational Bi-LSTM</td><td>1.26</td><td>1.23</td></tr></table>
+
+# 4 ABLATION STUDIES
+
+The goal of this section is to study the importance of the various components in our model to avoid any triviality. The experiments are as follows:
+
+# 1. Reconstruction loss on $\mathbf { h } _ { t }$ vs activity regularization on $\mathbf { h } _ { t }$
+
+![](images/60a747602b77a4e55002528d70589c5448a28766c9d073b54e5170691b29240f.jpg)  
+Figure 3: Evolution of the bits per character on PTB validation with sampling latent variables $\mathbf { z }$ from $\mathcal { N } ( \mathbf { 0 } , \mathbf { I } )$ during training or using a fixed vector which we set to be the mean of latent variables. Interestingly, not sampling from prior during inference does not hurt the final performance on PTB.
+
+Table 5: Perplexity on IMDB using different coefficient $\gamma$ for activity regularization   
+
+<table><tr><td>2</td><td>0.001</td><td>1.</td><td>4.</td><td>8.</td><td>16.</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Test perplexity</td><td>56.07</td><td>60.74</td><td>69.97</td><td>77.24</td><td>86.72</td></tr></table>
+
+The authors of Merity et al. (2017) study the importance of activity regularization (AR) on the hidden states on LSTMs given as,
+
+$$
+\mathcal { R } _ { A R } = \gamma \| \mathbf { h } _ { t } \| _ { 2 } ^ { 2 }
+$$
+
+However, since our model’s reconstruction term on $\mathbf { h } _ { t }$ can be decomposed as,
+
+$$
+\| \mathbf { h } _ { t } - \tilde { \mathbf { h } } _ { t } \| _ { 2 } ^ { 2 } = \| \mathbf { h } _ { t } \| _ { 2 } ^ { 2 } + \| \tilde { \mathbf { h } } _ { t } \| _ { 2 } ^ { 2 } - 2 \mathbf { h } _ { t } ^ { T } \tilde { \mathbf { h } } _ { t }
+$$
+
+we perform experiments to confirm that the gains in our approach is not due to the $\ell ^ { 2 }$ regularization alone since our regularization encapsulates an $\ell ^ { 2 }$ term along with the dot product term.
+
+We use activity regularization using hyperparameter $\alpha \in \{ 0 . 0 0 1 , 1 , 4 , 8 , 1 6 \}$ in place of reconstruction term in our model and study the test perplexity. The results are shown in table 5. We find that in all the cases performance using activity regularization is worse compared with our best model shown in table 3.
+
+Table 6: KL divergence of the Variational Bi-LSTM   
+
+<table><tr><td>Dataset</td><td>PTB</td><td>Seq-MNIST</td><td>IMDB</td><td>TIMIT</td><td>Blizzard</td></tr><tr><td>KL</td><td>0.001</td><td>0.02</td><td>0.18</td><td>3204.71</td><td>3799.79</td></tr></table>
+
+# 2. Use of parametric encoder prior vs. fixed Gaussian prior
+
+In our variational Bi-LSTM model, we propose to have the encoder prior over $\mathbf { z } _ { t }$ as a function of the previous forward LSTM hidden state $\mathbf { h } _ { t - 1 }$ . This is done to omit the need of the backward LSTM during inference because it is unavailable in practical scenarios since predictions are made in the forward direction. However, to study whether the model learns to use this encoder or not, we record the KL divergence value of the best validation model for the various datasets. The results are reported in table 6. We can see that the KL divergence values are large in the case of IMDB, TIMIT and Blizzard datasets, but small in the case of Seq-MNIST and PTB. To further explore, we ran experiments on these datasets with fixed standard Gaussian prior like in the case of traditional VAE.
+
+Interestingly we found that the model with fixed prior performed similarly in the case of PTB, but hurt performance in the other cases, which can be explained given their large KL divergence values in the original experiments.
+
+# 5 RELATED WORK
+
+Variational auto-encoders (Kingma & Welling, 2014) can be easily combined with many deep learning models. They have been applied in the feed-forward setting but they have also found usage in RNNs to better capture variation in sequential data (Sordoni et al., 2017; Fraccaro et al., 2016; Chung et al., 2015; Bayer & Osendorfer, 2014). VAEs consists of several muti-layer neural networks as probabilistic encoders and decoders and training is based on the gradient on log-likelihood lower bound (as the likelihood is in general intractable) of the model parameters $\Gamma$ along with a reparametrization trick. The derived variational lower-bound $\mathcal { L } _ { \Gamma }$ for an observed random variable $\mathbf { x }$ is:
+
+$$
+\log p ( \mathbf { x } ) \ge \mathcal { L } _ { \Gamma } = \underset { q _ { \phi } ( \mathbf { z } | \mathbf { x } ) } { E } \left[ \log \frac { p ( \mathbf { x } , \mathbf { z } ) } { q _ { \phi } ( \mathbf { z } | \mathbf { x } ) } \right] = \underset { q _ { \phi } ( \mathbf { z } | \mathbf { x } ) } { E } \left[ \ln p _ { \theta } ( \mathbf { x } | \mathbf { z } ) \right] - D _ { K L } ( q _ { \phi } ( \mathbf { z } | \mathbf { x } ) | | p _ { \theta } ( \mathbf { z } ) ) ,
+$$
+
+where $D _ { K L }$ denotes the Kullback-Leibler divergence and $p _ { \theta }$ is the prior over a latent variable $\mathbf { z }$ . The KL divergence tem can be expressed as the difference between the entropy of $q _ { \phi }$ and the prior and fortunately, it can be computed and differentiated without estimation for some distribution families like Gaussians. Although maximizing the log-likelihood corresponds to minimizing the KL divergence, we have to ensure that the resulting $q _ { \phi }$ remains far enough from an undesired equilibrium state where $q _ { \phi }$ is almost everywhere equal to the prior over latent variables. Combining recurrent neural networks with variational auto encoders can lead to powerful generative models that are capable of capturing the variations in data, however, they suffer badly from this optimization issue as discussed by Bowman et al. (2015).
+
+Recently, VAEs have also been applied to Bi-LSTMs by Sordoni et al. (2017) through a technique called Z-forcing. It is a powerful generative autoregressive model which is trained using the following variational evidence lower-bound
+
+$$
+\mathcal { L } ( \mathbf { x } ; \theta , \phi , \xi ) = \sum _ { \ell } \underset { q _ { \phi } ( \mathbf { z } _ { \ell } | \mathbf { x } ) } { E } \Big [ \log p _ { \theta } ( \mathbf { x } _ { t + 1 } | \mathbf { x } _ { 1 : t } , \mathbf { z } _ { 1 : t } ) \Big ] - D _ { K L } \big ( q _ { \phi } ( \mathbf { z } _ { t } | \mathbf { x } ) \big | \big | ( p _ { \theta } ( \mathbf { z } _ { t } | \mathbf { x } _ { 1 : t - 1 } , \mathbf { z } _ { 1 : t - 1 } ) \big )
+$$
+
+plus an auxiliary cost as a regularizer which is defined as $\log p _ { \xi } ( \mathbf { b } _ { t } | \mathbf { z } _ { t } )$ . They show that the auxiliary cost helps in improving the final performance; however during inference the backward reconstructions have not been used in their approach. In our ablation study section below, we show experimentally that this connection is important towards improving the performance of Bi-LSTMs as is the case in our model.
+
+# 6 CONCLUSION
+
+Variational Bi-LSTMs are powerful autoregressive generative models that are capable of learning better representations by creating a channel to exchange the information of the past and the future. Moreover, the conditional distribution over the backward LSTM variables is learned that can lead to better learning results in practice. Furthermore, Variational Bi-LSTM model acts as a regularizer and makes both networks be informative enough to perform well on different benchmark problems taken from the literature.
+
+# ACKNOWLEDGMENTS
+
+The authors would like to thank Theano developers (Theano Development Team, 2016) for their great work.
+
+# REFERENCES
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+David Krueger, Tegan Maharaj, Janos Kram ´ ar, Mohammad Pezeshki, Nicolas Ballas, Nan Rose- ´ mary Ke, Anirudh Goyal, Yoshua Bengio, Hugo Larochelle, Aaron C. Courville, and Chris Pal. Zoneout: Regularizing rnns by randomly preserving hidden activations. CoRR, abs/1606.01305, 2016. URL http://arxiv.org/abs/1606.01305.
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+Tim Salimans, Diederik P Kingma, and Max Welling. Markov chain monte carlo and variational inference: Bridging the gap. arXiv preprint arXiv:1410.6460, 2014.
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+Dmitriy Serdyuk, Rosemary Nan Ke, Alessandro Sordoni, Chris Pal, and Yoshua Bengio. Twin networks: Using the future as a regularizer. arXiv preprint arXiv:1708.06742, 2017.
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+Alessandro Sordoni, Anirudh Goyal ALIAS PARTH GOYAL, Marc-Alexandre Cote, Nan Ke, and Yoshua Bengio. Z-forcing: Training stochastic recurrent networks. In Advances in Neural Information Processing Systems. 2017. URL https://nips.cc/Conferences/2017/ Schedule?showEvent $= 9 4 3 9$ .
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+
+# APPENDIX
+
+A: Derivation of variation lower bound ${ \mathcal { L } } _ { \mathbf { T } }$ in equation (2) in more details:
+
+$$
+\begin{array} { r l } & { \quad - \frac { \sqrt { 3 } } { 2 } \left[ \int _ { 0 } ^ { 1 } \sqrt { \alpha } \left( \beta \right) \sin \alpha \sin \beta \right] \left( \frac { \sqrt { 3 } } { 2 } \right) \sin \alpha \sin \beta \sin \left( \beta \right) \sin \left( \frac { \sqrt { 3 } } { 2 } \right) \dot { \alpha } \sin \alpha \sin \beta \sin \left( \frac { \sqrt { 3 } } { 2 } \right) \dot { \alpha } \sin \alpha \sin \beta \sin \left( \frac { \sqrt { 3 } } { 2 } \right) } \\ & { \quad - \frac { \sqrt { 3 } } { 2 } \left[ \int _ { 0 } ^ { 1 } \sqrt { \alpha } \left( \beta \right) \sin \alpha \sin \beta \right] \sin \alpha \sin \beta \sin \left( \frac { \sqrt { 3 } } { 2 } \right) \dot { \alpha } \left[ \int _ { 0 } ^ { 1 } \sqrt { \alpha } \left( \beta \right) \sin \alpha \sin \beta \sin \left( \frac { \sqrt { 3 } } { 2 } \right) \dot { \alpha } \sin \alpha \right] \sin \beta \sin \alpha \sin \beta \sin \alpha \sin \beta \sin \beta \sin \alpha \sin \beta } \\ & { \quad + \frac { \sqrt { 3 } } { 2 } \left[ \int _ { 0 } ^ { 1 } \sqrt { \alpha } \left( \beta \right) \sin \alpha \left( \frac { \sqrt { 3 } } { 2 } \cos \alpha \cos \beta \right) \right] \sin \alpha \sin \beta \sin \alpha \sin \beta \sin \alpha \sin \beta \sin \beta \sin \alpha \sin \beta \sin \alpha \sin \beta \sin \beta \sin \alpha \sin \beta } \\ & { \quad - \frac { \sqrt { 3 } } { 2 } \left[ \int _ { 0 } ^ { 1 } \sqrt { \alpha } \left( \beta \right) \sin ^ { 2 } \alpha \left( \frac { \sqrt { 3 } } { 2 } \cos \alpha \sin \beta \cos \frac { \sqrt { 3 } } { 2 } \right) \right] \cos \alpha \sin \beta \sin \alpha \sin \beta \sin \alpha \sin \beta \sin \alpha \sin \beta \sin \beta \sin \alpha \sin \beta \sin \beta \sin \alpha \beta } \\ & { \quad + \frac { \sqrt { 3 } } { 2 } \left[ \int _ { 0 } ^ { 1 } \sqrt { \alpha } \left( \beta \right) \sin ^ { 2 } \alpha \left( \frac { \sqrt { 3 } } { 2 } \cos \alpha \sin \beta \cos \frac { \sqrt { 3 } } { 2 } \right) \right] \sin \alpha \sin \beta \sin \alpha \sin \beta \sin \alpha \beta \sin \alpha \sin \beta \sin \alpha \gamma \sin \beta \sin \alpha \beta \sin \beta \sin \alpha \gamma } \\ &  \quad - \frac { 3 }  2  \end{array}
+$$

parse/train/SkqV-XZRZ/SkqV-XZRZ_content_list.json

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+    {
+        "type": "text",
+        "text": "VARIATIONAL BI-LSTMS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            485,
+            121
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            210,
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+            224
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recurrent neural networks like long short-term memory (LSTM) are important architectures for sequential prediction tasks. LSTMs (and RNNs in general) model sequences along the forward time direction. Bidirectional LSTMs (Bi-LSTMs), which model sequences along both forward and backward directions, generally perform better at such tasks because they capture a richer representation of the data. In the training of Bi-LSTMs, the forward and backward paths are learned independently. We propose a variant of the Bi-LSTM architecture, which we call Variational Bi-LSTM, that creates a dependence between the two paths (during training, but which may be omitted during inference). Our model acts as a regularizer and encourages the two networks to inform each other in making their respective predictions using distinct information. We perform ablation studies to better understand the different components of our model and evaluate the method on various benchmarks, showing state-of-the-art performance. ",
+        "bbox": [
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+            242,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            453,
+            336,
+            468
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recurrent neural networks (RNNs) have become the standard models for sequential prediction tasks, having achieved state of the art performance in a number of applications that includes sequence prediction, language translation, machine comprehension, and speech synthesis (Arik et al., 2017; Wang et al., 2017; Mehri et al., 2016; Sotelo et al., 2017). RNNs model temporal data by encoding a given arbitrary-length input sequentially, at each time step combining some transformation of the current input with the encoding from the previous time step. This encoding, referred to as the RNN hidden state, summarizes all previous input tokens. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Viewed as “unrolled” feedforward networks, RNNs can become arbitrarily deep depending on the input sequence length, and use a repeating module to combine the input with the previous state at each time step. Consequently, they suffer from the vanishing/exploding gradient problem (Pascanu et al., 2012). This problem has been addressed through architectural variants like the long short-term memory (LSTM) (Hochreiter & Schmidhuber, 1997) and the gated recurrent unit (GRU) (Chung et al., 2014). These architectures add a linear path along the temporal sequence which allows gradients to flow more smoothly back through time. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Various regularization techniques have also been explored to improve RNN performance and generalization. Dropout (Srivastava et al., 2014) regularizes a network by randomly dropping hidden units during training. However, it has been observed that using dropout directly on RNNs is not as effective as in the case of feed-forward networks. To combat this, Zaremba et al. (2014) propose to instead apply dropout on the activations that are not involved in the recurrent connections (Eg. in a multi-layer RNN); Gal & Ghahramani (2016) propose to apply the same dropout mask through an input sequence during training. In a similar spirit to dropout, Zoneout (Krueger et al., 2016) proposes to choose randomly whether to use the previous RNN hidden state. ",
+        "bbox": [
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+            694,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The aforementioned architectures model sequences along the forward direction of the input sequence. Bidirectional-LSTM, on the other hand, is a variant of LSTM that simultaneously models each sequence in both the forward and backward directions. This enables a richer representation of data, since each token’s encoding contains context information from the past and the future. It has been shown empirically that bidirectional architectures generally outperform unidirectional ones on many sequence-prediction tasks. However, the forward and backward paths in Bi-LSTMs are trained separately and the benefit usually comes from the combined hidden representation from both paths. In this paper, our main idea is to create a dependence between the two paths that acts as a regularization during training, but doesn’t hinder inference even in the absence of the other path (the backward path in practical scenarios). We note that recently proposed methods like TwinNet Serdyuk et al. (2017) and Z-forcing Sordoni et al. (2017) are similar in spirit to this idea. In our approach, we use a variational auto-encoder (VAE; Kingma & Welling (2014)) that takes as input the hidden states from the two paths of the Bi-LSTM and maps them to a shared hidden representation of the VAE at each time step. The samples from the VAE hidden state are then used both for reconstructing the LSTM hidden states and feeding forward to the next hidden state. In this way, we create a channel between the two paths that acts as a regularization for learning better representations. We refer to the resulting model as a Variational Bi-LSTM. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+            103,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Below, we describe Variational Bi-LSTMs in detail and then demonstrate empirically their ability to model complex sequential distributions. In experiments, we obtain state-of-the-art or competitive performance on the tasks of Penn Treebank, IMDB, TIMIT, Blizzard, and Sequential MNIST. ",
+        "bbox": [
+            174,
+            236,
+            825,
+            279
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 VARIATIONAL BI-LSTM ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            311,
+            411,
+            329
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Bi-LSTM is a powerful architecture for sequential tasks because it models temporal data both in the forward and backward directions. For this it uses two LSTMs that are generally learned independently of each other; the richer representation results from combining the hidden states of these LSTMs, where combination is often by concatenation. The idea behind variational Bi-LSTMs is to create a channel of information exchange between the two LSTMs that helps the model to learn better representations. We create this dependence using a variational auto-encoder (VAE). This enables us to take advantage of the fact that VAE allows for sampling from a prior during inference. For sequence prediction tasks like language generation, while one can use Bi-LSTMs during training, there is no straightforward way to employ the full bidirectional model during inference – this would involve, eg, generating a sentence starting at both its beginning and end. In such cases, the VAE allows us to sample from the prior at inference time to make up for the absence of the backward LSTM. ",
+        "bbox": [
+            173,
+            352,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Now we describe our variational Bi-LSTM model formally. Let $\\mathbf { X } = \\{ \\mathbf { x } ^ { ( i ) } \\} _ { i = 1 } ^ { N }$ be a dataset consisting of $N$ i.i.d. sequential data samples of continuous or discrete variables. For notational convenience, we will henceforth drop the superscript $i$ indexing samples. For each sample sequence $\\mathbf { x } = ( \\mathbf { x } _ { 1 } , \\ldots , \\mathbf { x } _ { T } )$ , the hidden state of the forward LSTM is given by: ",
+        "bbox": [
+            173,
+            525,
+            825,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/82bf59207de400efbca8528dd3ba1f877af08464996bd529287ee41fa19888f3.jpg",
+        "text": "$$\n\\mathbf { h } _ { t } = \\vec { f } ( \\mathbf { x } _ { t } , \\mathbf { h } _ { t - 1 } , \\mathbf { z } _ { t } , \\tilde { \\mathbf { b } } _ { t } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            410,
+            599,
+            586,
+            622
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The hidden state of the backward LSTM is given by, ",
+        "bbox": [
+            176,
+            641,
+            516,
+            656
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/420571445471dc8fc9f2ca96dfcd8396953e7b2d15a3f26efeb83953c76e7ff0.jpg",
+        "text": "$$\n\\mathbf { b } _ { t } = \\overleftarrow { f } ( \\mathbf { x } _ { t } , \\mathbf { b } _ { t + 1 } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            433,
+            674,
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+            696
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In both cases, the function $f$ represents the standard LSTM updates, modified to account for the additional arguments. ",
+        "bbox": [
+            173,
+            714,
+            823,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In the forward LSTM model, we introduce three latent random variables, $\\mathbf { z } _ { t } , \\tilde { \\mathbf { b } } _ { t }$ , and $\\tilde { \\mathbf { h } } _ { t - 1 }$ , where $\\mathbf { z } _ { t }$ depends on $\\mathbf { h } _ { t - 1 }$ and $\\mathbf { b } _ { t }$ during training, and $\\tilde { \\mathbf { b } } _ { t }$ and $\\tilde { \\mathbf { h } } _ { t - 1 }$ depend only on $\\mathbf { z } _ { t }$ (see figure 1-left, for a graphical representation). Note that so far, $\\tilde { \\mathbf { b } } _ { t }$ and $\\tilde { \\mathbf { h } } _ { t - 1 }$ are simply latent vectors drawn from conditional distributions $p _ { \\psi }$ and $p _ { \\xi }$ , respectively, to be defined below. However, as explained in Section 2.1 (see also dashed lines in figure 1-left), we will encourage these to lie near the manifolds of backward and forward LSTM states, respectively. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "By design, the joint conditional distribution over latent variables $\\mathbf { z } _ { t }$ and $\\tilde { \\mathbf { b } } _ { t }$ with parameters $\\theta$ and $\\psi$ factorizes as $p _ { \\theta } ( \\mathbf { z } _ { t } | \\mathbf { x } _ { 1 : t - 1 } , \\mathbf { z } _ { 1 : t - 1 } ) p _ { \\psi } ( \\tilde { \\mathbf { b } } _ { t } | \\mathbf { z } _ { t } )$ . This factorization enables us to formulate several helpful auxiliary cots, as defined in the next subsection. Further, $p _ { \\eta } ( \\mathbf { x } _ { t + 1 } | \\mathbf { x } _ { 1 : t } , \\mathbf { z } _ { 1 : t } , \\tilde { \\mathbf { b } } _ { t } )$ defines the generating model, which induces the distribution over the next observation given the previous states and the current input. ",
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+    {
+        "type": "image",
+        "img_path": "images/161f5732f91f90cba658cac7b9f25b2c774e2166fc933313bfe7d4393dcf6a8f.jpg",
+        "image_caption": [
+            "(a) Training phase of variational Bi-LSTM "
+        ],
+        "image_footnote": [],
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+    },
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+        "img_path": "images/99ebd8d75a0549959f48953dcef7cd6d9042f86704b858b60aa20dee95753e8b.jpg",
+        "image_caption": [
+            "(b) Inference phase of variational Bi-LSTM ",
+            "Figure 1: Graphical description of our proposed variational Bi-LSTM model during train phase (left) and inference phase (right). During training, each step $t$ is composed of an encoder which receives both the past and future summary via $\\mathbf { h } _ { t - 1 }$ and $\\mathbf { b } _ { t }$ respectively, and a decoder that generates $\\tilde { \\mathbf { h } } _ { t - 1 }$ and $\\tilde { \\mathbf { b } } _ { t }$ which are forced to be close enough to $\\mathbf { h } _ { t - 1 }$ and $\\mathbf { b } _ { t }$ using two auxiliary reconstruction costs (dashed lines). This dependence between backward and forward LSTM through the latent random variable encourages the forward LSTM to learn a richer representation. During inference, the backward LSTM is removed. In this case, $\\mathbf { z } _ { t }$ is sampled from the prior as in a typical VAE, which in our case, is defined as a function of $\\mathbf { h } _ { t - 1 }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Then the marginal likelihood of each individual sequential data sample $\\mathbf { x }$ can be written as ",
+        "bbox": [
+            171,
+            433,
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8328b130f5449326914270d9b91006f783646c860180fd7c58f55706b4211d8f.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle p ( { \\bf x } ; { \\bf Y } ) = \\prod _ { t = 0 } ^ { T } p ( { \\bf x } _ { t + 1 } \\vert { \\bf x } _ { 1 : t } ) = \\prod _ { t = 0 } ^ { T } \\int _ { { \\bf z } _ { 1 : T } } p ( { \\bf x } _ { t + 1 } \\vert { \\bf x } _ { 1 : t } , { \\bf z } _ { 1 : t } ) p _ { \\theta } ( { \\bf z } _ { t } \\vert { \\bf x } _ { 1 : t - 1 } , { \\bf z } _ { 1 : t - 1 } ) d { \\bf z } _ { 1 : T } } \\\\ { \\displaystyle \\quad \\quad = \\prod _ { t = 0 } ^ { T } \\int _ { { \\bf z } _ { 1 : T } } \\int _ { { \\bar { \\bf b } } _ { t } } \\Big [ p _ { \\eta } ( { \\bf x } _ { t + 1 } \\vert { \\bf x } _ { 1 : t } , { \\bf z } _ { 1 : t } , { \\bf \\tilde { b } } _ { t } ) p _ { \\psi } ( \\tilde { { \\bf b } } _ { t } \\vert { \\bf z } _ { 1 : t } ) p _ { \\theta } ( { \\bf z } _ { t } \\vert { \\bf x } _ { 1 : t - 1 } , { \\bf z } _ { 1 : t - 1 } ) \\Big ] d { \\tilde { \\bf b } } _ { t } d { \\bf z } _ { 1 : T } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            186,
+            454,
+            790,
+            544
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $q _ { \\phi } ( \\mathbf { z } _ { t } | \\mathbf { x } )$ is the conditional inference model and $\\Gamma = \\{ \\phi , \\theta , \\psi , \\eta \\}$ is the set of all parameters of the model. Here, we assume that all conditional distributions belong to parametrized families of distributions which can be evaluated and sampled from efficiently. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Note that the joint distribution in equation (1) is intractable. Kingma & Welling (2014) demonstrated how to maximize a variational lower bound, ${ \\mathcal { L } } _ { \\Gamma }$ , of the data log likelihood instead, which is given by ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/266f6c288971c0ab50c83b7813ee474415611a85dd61c336a3b5fb69973bb81a.jpg",
+        "text": "$$\n\\log p ( \\mathbf { x } ; \\mathbf { r } ) \\ge \\mathcal { L } _ { \\mathbf { T } } = \\sum _ { t = 0 } ^ { T } \\underbrace { E } _ { \\substack { \\mathbf { z } _ { 1 : T } \\sim q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) \\tilde { \\mathbf { b } } _ { t } \\sim p _ { \\psi } ( \\tilde { \\mathbf { b } } _ { t } | \\mathbf { z } _ { t } ) } } \\Big [ \\log p _ { \\eta } ( \\mathbf { x } _ { t + 1 } | \\mathbf { x } _ { 1 : t } , \\mathbf { z } _ { 1 : t } , \\tilde { \\mathbf { b } } _ { t } ) \\Big ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $D _ { K L }$ is the Kullback-Leibler (KL) divergence between the approximate posterior and the conditional prior (see the appendix). This is the approach we take. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.1 TRAINING AND INFERENCE ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In the proposed variational Bi-LSTM, the latent variable $\\mathbf { z } _ { t }$ is inferred as ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/20bab4376fd4c920f1a0a23b13e1b5fbedb21a0221b0814fe4da6a5111504843.jpg",
+        "text": "$$\n\\mathbf { z } _ { t } \\sim q _ { \\phi } ( \\mathbf { z } _ { t } | ( \\mathbf { h } _ { t - 1 } , \\mathbf { b } _ { t } ) ) = \\mathcal { N } ( \\pmb { \\mu } _ { q , t } , \\mathrm { d i a g } ( \\pmb { \\sigma } _ { q , t } ^ { 2 } ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            341,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "in which $[ { \\pmb \\mu } _ { q , t } , { \\pmb \\sigma } _ { q , t } ^ { 2 } ] = f _ { \\phi } ( { \\bf h } _ { t - 1 } , { \\bf b } _ { t } )$ where $f _ { \\phi }$ is a multi-layered feed-forward network with Gaussian outputs. We assume that the prior over $\\mathbf { z } _ { t }$ is a diagonal multivariate Gaussian distribution given by ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d40bd57997ef84128f5ad135f93a99bae3915cadaf53a5d3f3fdf473afb0f1df.jpg",
+        "text": "$$\np _ { \\theta } ( \\mathbf { z } _ { t } | \\mathbf { x } _ { 1 : t - 1 } , \\mathbf { z } _ { 1 : t - 1 } ) = \\mathcal { N } ( \\mu _ { p , t } , \\mathrm { d i a g } ( \\sigma _ { p , t } ^ { 2 } ) ) , \\quad \\mathrm { w h e r e } \\quad [ \\mu _ { p , t } , \\sigma _ { p , t } ^ { 2 } ] = f _ { \\theta } ( \\mathbf { h } _ { t - 1 } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "for a fully connected network $f _ { \\theta }$ . This is important because, during generation (see Figure 1-right, for a graphical representation), we will not have access to the backward LSTM. In this case, as in a ",
+        "bbox": [
+            171,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "VAE, we will sample from the prior for $\\mathbf { z } _ { t }$ . Since we define the prior to be a function of $\\mathbf { h } _ { t - 1 }$ , the forward LSTM is encouraged during training to learn the dependency due to the backward hidden state $\\mathbf { b } _ { t }$ . ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The latent variable $\\tilde { \\mathbf { b } } _ { t }$ is meant to model information coming from the future of the sequence. Its conditional distribution is given by ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ed96c186dbbe0737995cd87a5debbfd83a57c2ba833b50dd99715eb369e5332c.jpg",
+        "text": "$$\np _ { \\psi } ( \\tilde { \\mathbf { b } } _ { t } | \\mathbf { z } _ { t } ) = \\mathcal { N } ( \\mu _ { \\tilde { \\mathbf { b } } , t } , \\mathrm { d i a g } ( \\pmb { \\sigma } _ { \\tilde { b } , t } ^ { 2 } ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            383,
+            185,
+            612,
+            209
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $[ { \\pmb { \\mu } } _ { { \\tilde { \\bf b } } , t } , { \\pmb { \\sigma } } _ { { \\tilde { \\bf b } } , t } ^ { 2 } ] = f _ { \\psi } ( \\mathbf { z } _ { t } )$ for a fully connected neural network $f _ { \\psi }$ (See Figure 1(a)). To encourage the encoding of future information in $\\tilde { \\mathbf { b } } _ { t }$ , we maximize the probability of the true backward hidden state, $\\mathbf { b } _ { t }$ , under the distribution $p _ { \\psi }$ , as an auxiliary cost during training. In this way we treat $\\tilde { \\mathbf { b } } _ { t }$ as a predictor of $\\mathbf { b } _ { t }$ , similarly to what was done by Sordoni et al. (2017). ",
+        "bbox": [
+            174,
+            214,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To capture information from the past in the latents, we similarly use $\\tilde { \\mathbf { h } } _ { t - 1 }$ as a predictor of $\\mathbf { h } _ { t - 1 }$ . This is accomplished by maximizing the probability of the latter under the conditional distribution of the former, $\\log p _ { \\xi } ( \\tilde { \\mathbf { h } } _ { t - 1 } | \\mathbf { z } _ { t } )$ , as another auxiliary cost, where ",
+        "bbox": [
+            174,
+            289,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c842c96c40bc84b1db5e146fb3c33371c64b28e01f50f4fa17a03e6478d01824.jpg",
+        "text": "$$\np _ { \\xi } ( \\tilde { \\mathbf { h } } _ { t - 1 } | \\mathbf { z } _ { t } ) = \\mathcal { N } ( \\mu _ { \\tilde { \\mathbf { h } } , t } , \\mathrm { d i a g } ( \\pmb { \\sigma } _ { \\tilde { \\mathbf { h } } , t } ^ { 2 } ) ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            375,
+            339,
+            622,
+            363
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Here, $[ \\mu _ { \\tilde { \\mathbf { h } } , t } , \\sigma _ { \\tilde { \\mathbf { h } } , t } ^ { 2 } ]$ is the output of a fully-connected neural network $f _ { \\xi }$ taking $\\mathbf { z } _ { t }$ as input. The auxiliary costs arising from distributions $p _ { \\xi }$ and $p _ { \\psi }$ teach the variational Bi-LSTM to encode past and future information into the latent space of $\\mathbf { z }$ . ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We assume that parameters of the generating distribution $p _ { \\eta } ( \\mathbf { x } _ { t + 1 } | \\mathbf { x } _ { 1 : t } , \\mathbf { z } _ { 1 : t } , \\tilde { \\mathbf { b } } _ { t } )$ are computed via MLP, taking the form of either a Gaussian distribution output in the continuous case or categorical proportions output in the discrete (ie, one-hot) prediction case (See Figure 1(b)). ",
+        "bbox": [
+            174,
+            422,
+            825,
+            465
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "All the parameters in $\\mathbf { \\delta T }$ and $\\boldsymbol { \\xi }$ are updated based on backpropagation through time (Rumelhart et al., 1988) using the reparameterization trick (Kingma & Welling, 2014), where the gradients are computed by differentiating of the following function: ",
+        "bbox": [
+            176,
+            472,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/69bbd54ac916deadcdfef4f7b8a7c05e3d9af3bcf4c8c0e59acb0e5b735c6075.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { L } ( { \\bf x } ; { \\bf Y } , \\pmb { \\xi } ) = \\displaystyle \\sum _ { t = 0 } ^ { T } \\sum _ { z _ { 1 } : \\tau \\sim q _ { \\phi } ( { \\bf z } | { \\bf x } ) } \\Big [ \\underset { { \\tilde { \\bf b } } _ { t } \\sim p _ { \\psi } ( b | { \\bf z } _ { 1 } : t ) } { \\cal E } \\Big [ \\log p ( { \\bf x } _ { t + 1 } | { \\bf x } _ { 1 : t } , { \\bf z } _ { 1 : t } , { \\tilde { \\bf b } } _ { t } ) + } \\\\ { \\alpha \\log p _ { \\psi } ( { \\bf b } _ { t } | { \\bf z } _ { t } ) + \\beta \\log p _ { \\xi } ( { \\bf h } _ { t - 1 } | { \\bf z } _ { t } ) \\Big ] \\Big ] - } \\\\ { D _ { K L } \\big ( q _ { \\phi } ( { \\bf z } _ { t } | { \\bf x } ) \\big ) \\big | p _ { \\theta } \\big ( { \\bf z } _ { t } | { \\bf x } _ { 1 : t - 1 } , { \\bf z } _ { 1 : t - 1 } \\big ) \\big ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            261,
+            518,
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+            611
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Here, $\\alpha$ and $\\beta$ are non-negative real numbers. We improve training convergence with a trick for the variational Bi-LSTM, which we refer to as skip gradient, meant to ease learning of the latent variables. It is well known that autoregressive decoder models tend to ignore their stochastic variables (Bowman et al., 2015). Skip gradient is a technique to encourage that relevant summaries of the past and the future are encoded in the latent space. The idea is to skip the gradient of the stochastic operations with respect to the recurrent units through time. To achieve this, at each time step, a mask drawn from a Bernoulli distribution governs whether to skip the gradient or to backpropagate it for a given data point. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 EXPERIMENTAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this section demonstrate the effectiveness of our proposed model on several tasks. We present experimental results obtained when training the Variational Bi-LSTM on various sequential datasets: Penn Treebank (PTB), IMDB, TIMIT, Blizzard, and Sequential MNIST. Our main goal is to ensure that the model proposed in Section 2 can benefit from a generated relevant summary of the future that yields competitive results. In all experiments, we train all the models using ADAM optimizer (Kingma & Ba, 2014) and we set all MLPs in Section 2 to have one hidden layer with leaky-ReLU hidden activation. All the models are implemented using Theano (Theano Development Team, 2016) and the code is available at https://anonymous.url. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Blizzard: Blizzard is a speech model dataset with 300 hours of English, spoken by a single female speaker. We report the average log-likelihood for half-second sequences (Fraccaro et al., 2016). In our experimental setting, we use 1024 hidden units for MLPs, 1024 LSTM units and 512 latents. Our model is trained using learning rate of 0.001 and minibatches of size 32 and we set $\\alpha = \\beta = 1$ . A fully factorized multivariate Gaussian distribution is used as the output distribution. The final lower bound estimation on TIMIT can be found in Table 1. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/0d648bdb9d88334583123a7d153647a8b2359a06886f51179259d98c198b9f29.jpg",
+        "table_caption": [
+            "Table 1: The average of log-likelihood per sequence on Blizzard and TIMIT testset "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Blizzard</td><td>TIMIT</td></tr><tr><td></td><td></td><td></td></tr><tr><td>RNN-Gauss</td><td>3539</td><td>-1900</td></tr><tr><td>RNN-GMM</td><td>7413</td><td>26643</td></tr><tr><td>VRNN-I-Gauss</td><td></td><td>28340</td></tr><tr><td>VRNN-Gauss</td><td>9223</td><td>28805</td></tr><tr><td>VRNN-GMM</td><td>9392</td><td>AIAIAI AIAI 28982</td></tr><tr><td>SRNN (smooth+resq)</td><td>11991</td><td>60550</td></tr><tr><td>Z-Forcing (Sordoni et al., 2017)</td><td>&gt;I&gt;I 14315</td><td>68852</td></tr><tr><td>VariationalBi-LSTM</td><td>≥ 17319</td><td>73315</td></tr></table>",
+        "bbox": [
+            310,
+            202,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "TIMIT: Another speech modeling dataset is TIMIT with 6300 English sentences read by 630 speakers. Like the work done in (Fraccaro et al., 2016), our model is trained on raw sequences of 200 dimensional frames. In our experiments, we use 1024 hidden units, 2048 LSTM units and 128 latent variables, and batch size of 128. We train the model using learning rate of 0.0001, $\\alpha = 1$ and $\\beta = 0$ . The average log-likelihood for the sequences on test can be found in Table 1. ",
+        "bbox": [
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+            377,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Sequential MNIST: We use the MNIST dataset which is binarized according to (Murray & Salakhutdinov, 2009) and we downloaded in binrized from (Larochelle, 2011). Our best model consists of 1024 hidden units, 1024 LSTM units and 256 latent variables. We train the model using a learning rate of 0.0001 and a batch size of 32. To reach the negative log-likelihood reported in Table 2, we set $\\alpha = 0 . 0 0 1$ and $\\beta = 0$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/19e5538ba00df72808960543ee01718aa68421e53a4c523e49013e115c6701bf.jpg",
+        "table_caption": [
+            "Table 2: The average of negative log-likelihood on sequential MNIST "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Models</td><td>Seq-MNIST</td></tr><tr><td>DBN 2hl (Germain et al., 2015) NADE (Uria et al., 2016) EoNADE-5 2hl (Raiko et al., 2014)</td><td>~ 84.55</td></tr><tr><td>DLGM 8 (Salimans et al., 2014)</td><td>88.33 84.68</td></tr><tr><td></td><td>≈ 85.51</td></tr><tr><td>DARN 1hl (Gregor et al.,2015) BiHM (Bornschein et al., 2015)</td><td>≈84.13</td></tr><tr><td></td><td>≈84.23</td></tr><tr><td>DRAW (Gregor et al., 2015)</td><td>≤ 80.97</td></tr><tr><td>PixelVAE (Gulrajani et al., 2016)</td><td>~ 79.02</td></tr><tr><td>Prof. Forcing (Goyal et al., 2016)</td><td></td></tr><tr><td>PixelRNN(1-layer) (Oord et al., 2016)</td><td>79.58</td></tr><tr><td></td><td>80.75</td></tr><tr><td>PixelRNN(7-layer) (Oord et al., 2016)</td><td></td></tr><tr><td>Z-Forcing (Sordoni et al.,2017)</td><td>79.20</td></tr><tr><td>VariationalBi-LSTM</td><td>≤ 80.09 ≤79.78</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "IMDB: It is a dataset consists of 350000 movie reviews (Diao et al., 2014) in which each sentence has less than 16 words and the vocabulary size is fixed to 16000 words. In this experiment, we use 500 hidden units, 500 LSTM units and latent variables of size 64. The model is trained with a batch size of 32 and a learning rate of 0.001 and we set $\\alpha = \\beta = 1$ . The word perplexity on valid and test dataset is shown in Table 3. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "PTB: Penn Treebank (Marcus et al. (1993)) is a language model dataset consists of 1 million words. We train our model with 1024 LSTM units, 1024 hidden units, and the latent variables of size 128. We train the model using a standard Gaussian prior, a learning rate of 0.001 and batch size of 50 and we set $\\alpha = \\beta = 1$ . The model is trained to predict the next character in a sequence and the final bits per character on test and valid sets are shown in Table 4 ",
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+    {
+        "type": "table",
+        "img_path": "images/5dc19ada58e73f92f93f5f3812546615e00cac1d2eaa7fe35c5974de5f9b6b1f.jpg",
+        "table_caption": [
+            "Table 3: Word perplexity on IMDB on valid and test sets "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Valid</td><td>Test</td></tr><tr><td></td><td></td><td></td></tr><tr><td>Gated Word-Char</td><td>70.60</td><td>70.87</td></tr><tr><td>Z-Forcing (Sordoni et al., 2017)</td><td>56.48</td><td>65.68</td></tr><tr><td>Variational Bi-LSTM</td><td>51.43</td><td>51.60</td></tr></table>",
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+        "type": "image",
+        "img_path": "images/206b55463e83fd04b6ab2796e4d7919dcbedf0b6c96b094d08f5e320a64735e3.jpg",
+        "image_caption": [
+            "Figure 2: Evolution of the average of log-likelihood during training of Variational Bi-LSTMs with and without using skip gradient and auxiliary costs on PTB and Blizzard. "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "",
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+        "type": "table",
+        "img_path": "images/efd2730830fc6da18aea192673f6e28df5ed9812fa7d6f604a64c483ae50d80b.jpg",
+        "table_caption": [
+            "Table 4: Bits Per Character (BPC) on PTB valid and test sets "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Valid</td><td>Test</td></tr><tr><td>Unregularized LSTM</td><td></td><td>1.36</td></tr><tr><td>Weight noise</td><td>1.47 1.51</td><td>1.34</td></tr><tr><td>Norm stabilizer</td><td>1.46</td><td>1.35</td></tr><tr><td>Stochastic depth</td><td>1.43</td><td>1.34</td></tr><tr><td>Recurrent dropout</td><td></td><td>1.29</td></tr><tr><td>Zoneout (Krueger et al. (2016))</td><td>1.40</td><td></td></tr><tr><td>RBN (Cooijmans et al. (2016))</td><td>1.36</td><td>1.25</td></tr><tr><td>H-LSTM + LN (Ha et al. (2016))</td><td>=</td><td>1.32</td></tr><tr><td>3-HM-LSTM + LN (Chung et al., 2016)</td><td>1.28</td><td>1.25</td></tr><tr><td>2-H-LSTM + LN (Ha et al. (2016))</td><td>=</td><td>1.24</td></tr><tr><td>Z-Forcing</td><td>1.25</td><td>1.22</td></tr><tr><td></td><td>1.29</td><td>1.26</td></tr><tr><td>Variational Bi-LSTM</td><td>1.26</td><td>1.23</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4 ABLATION STUDIES ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The goal of this section is to study the importance of the various components in our model to avoid any triviality. The experiments are as follows: ",
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+    {
+        "type": "text",
+        "text": "1. Reconstruction loss on $\\mathbf { h } _ { t }$ vs activity regularization on $\\mathbf { h } _ { t }$ ",
+        "text_level": 1,
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+        "img_path": "images/60a747602b77a4e55002528d70589c5448a28766c9d073b54e5170691b29240f.jpg",
+        "image_caption": [
+            "Figure 3: Evolution of the bits per character on PTB validation with sampling latent variables $\\mathbf { z }$ from $\\mathcal { N } ( \\mathbf { 0 } , \\mathbf { I } )$ during training or using a fixed vector which we set to be the mean of latent variables. Interestingly, not sampling from prior during inference does not hurt the final performance on PTB. "
+        ],
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+        "type": "table",
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+        "table_caption": [
+            "Table 5: Perplexity on IMDB using different coefficient $\\gamma$ for activity regularization "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>2</td><td>0.001</td><td>1.</td><td>4.</td><td>8.</td><td>16.</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Test perplexity</td><td>56.07</td><td>60.74</td><td>69.97</td><td>77.24</td><td>86.72</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "The authors of Merity et al. (2017) study the importance of activity regularization (AR) on the hidden states on LSTMs given as, ",
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+        "type": "equation",
+        "img_path": "images/9086846d536dd1870d62faec5d4f519706a04a113089a5495facbcae4b2d15cd.jpg",
+        "text": "$$\n\\mathcal { R } _ { A R } = \\gamma \\| \\mathbf { h } _ { t } \\| _ { 2 } ^ { 2 }\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "However, since our model’s reconstruction term on $\\mathbf { h } _ { t }$ can be decomposed as, ",
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+    {
+        "type": "equation",
+        "img_path": "images/209212a396d1ff52a0ed1dd37e11a2dba0072dfc84c29e52271cc340b3ea4bec.jpg",
+        "text": "$$\n\\| \\mathbf { h } _ { t } - \\tilde { \\mathbf { h } } _ { t } \\| _ { 2 } ^ { 2 } = \\| \\mathbf { h } _ { t } \\| _ { 2 } ^ { 2 } + \\| \\tilde { \\mathbf { h } } _ { t } \\| _ { 2 } ^ { 2 } - 2 \\mathbf { h } _ { t } ^ { T } \\tilde { \\mathbf { h } } _ { t }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "we perform experiments to confirm that the gains in our approach is not due to the $\\ell ^ { 2 }$ regularization alone since our regularization encapsulates an $\\ell ^ { 2 }$ term along with the dot product term. ",
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+    {
+        "type": "text",
+        "text": "We use activity regularization using hyperparameter $\\alpha \\in \\{ 0 . 0 0 1 , 1 , 4 , 8 , 1 6 \\}$ in place of reconstruction term in our model and study the test perplexity. The results are shown in table 5. We find that in all the cases performance using activity regularization is worse compared with our best model shown in table 3. ",
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+    {
+        "type": "table",
+        "img_path": "images/53a8b2f380ef5383a341c1f55963d382294818b8a7387d4e10c26c675f5bb6b1.jpg",
+        "table_caption": [
+            "Table 6: KL divergence of the Variational Bi-LSTM "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>PTB</td><td>Seq-MNIST</td><td>IMDB</td><td>TIMIT</td><td>Blizzard</td></tr><tr><td>KL</td><td>0.001</td><td>0.02</td><td>0.18</td><td>3204.71</td><td>3799.79</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "2. Use of parametric encoder prior vs. fixed Gaussian prior ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In our variational Bi-LSTM model, we propose to have the encoder prior over $\\mathbf { z } _ { t }$ as a function of the previous forward LSTM hidden state $\\mathbf { h } _ { t - 1 }$ . This is done to omit the need of the backward LSTM during inference because it is unavailable in practical scenarios since predictions are made in the forward direction. However, to study whether the model learns to use this encoder or not, we record the KL divergence value of the best validation model for the various datasets. The results are reported in table 6. We can see that the KL divergence values are large in the case of IMDB, TIMIT and Blizzard datasets, but small in the case of Seq-MNIST and PTB. To further explore, we ran experiments on these datasets with fixed standard Gaussian prior like in the case of traditional VAE. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Interestingly we found that the model with fixed prior performed similarly in the case of PTB, but hurt performance in the other cases, which can be explained given their large KL divergence values in the original experiments. ",
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+    {
+        "type": "text",
+        "text": "5 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Variational auto-encoders (Kingma & Welling, 2014) can be easily combined with many deep learning models. They have been applied in the feed-forward setting but they have also found usage in RNNs to better capture variation in sequential data (Sordoni et al., 2017; Fraccaro et al., 2016; Chung et al., 2015; Bayer & Osendorfer, 2014). VAEs consists of several muti-layer neural networks as probabilistic encoders and decoders and training is based on the gradient on log-likelihood lower bound (as the likelihood is in general intractable) of the model parameters $\\Gamma$ along with a reparametrization trick. The derived variational lower-bound $\\mathcal { L } _ { \\Gamma }$ for an observed random variable $\\mathbf { x }$ is: ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/36609c39e491ff9c517bbed1c8267f21a85fff75156a2b4792c20519ab71567c.jpg",
+        "text": "$$\n\\log p ( \\mathbf { x } ) \\ge \\mathcal { L } _ { \\Gamma } = \\underset { q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) } { E } \\left[ \\log \\frac { p ( \\mathbf { x } , \\mathbf { z } ) } { q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) } \\right] = \\underset { q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) } { E } \\left[ \\ln p _ { \\theta } ( \\mathbf { x } | \\mathbf { z } ) \\right] - D _ { K L } ( q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) | | p _ { \\theta } ( \\mathbf { z } ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "where $D _ { K L }$ denotes the Kullback-Leibler divergence and $p _ { \\theta }$ is the prior over a latent variable $\\mathbf { z }$ . The KL divergence tem can be expressed as the difference between the entropy of $q _ { \\phi }$ and the prior and fortunately, it can be computed and differentiated without estimation for some distribution families like Gaussians. Although maximizing the log-likelihood corresponds to minimizing the KL divergence, we have to ensure that the resulting $q _ { \\phi }$ remains far enough from an undesired equilibrium state where $q _ { \\phi }$ is almost everywhere equal to the prior over latent variables. Combining recurrent neural networks with variational auto encoders can lead to powerful generative models that are capable of capturing the variations in data, however, they suffer badly from this optimization issue as discussed by Bowman et al. (2015). ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Recently, VAEs have also been applied to Bi-LSTMs by Sordoni et al. (2017) through a technique called Z-forcing. It is a powerful generative autoregressive model which is trained using the following variational evidence lower-bound ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/33a5827738c999d1ae2fdf7be5b4a06a77cd4c7aaa727f0a0b1a50f539194435.jpg",
+        "text": "$$\n\\mathcal { L } ( \\mathbf { x } ; \\theta , \\phi , \\xi ) = \\sum _ { \\ell } \\underset { q _ { \\phi } ( \\mathbf { z } _ { \\ell } | \\mathbf { x } ) } { E } \\Big [ \\log p _ { \\theta } ( \\mathbf { x } _ { t + 1 } | \\mathbf { x } _ { 1 : t } , \\mathbf { z } _ { 1 : t } ) \\Big ] - D _ { K L } \\big ( q _ { \\phi } ( \\mathbf { z } _ { t } | \\mathbf { x } ) \\big | \\big | ( p _ { \\theta } ( \\mathbf { z } _ { t } | \\mathbf { x } _ { 1 : t - 1 } , \\mathbf { z } _ { 1 : t - 1 } ) \\big )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "plus an auxiliary cost as a regularizer which is defined as $\\log p _ { \\xi } ( \\mathbf { b } _ { t } | \\mathbf { z } _ { t } )$ . They show that the auxiliary cost helps in improving the final performance; however during inference the backward reconstructions have not been used in their approach. In our ablation study section below, we show experimentally that this connection is important towards improving the performance of Bi-LSTMs as is the case in our model. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Variational Bi-LSTMs are powerful autoregressive generative models that are capable of learning better representations by creating a channel to exchange the information of the past and the future. Moreover, the conditional distribution over the backward LSTM variables is learned that can lead to better learning results in practice. Furthermore, Variational Bi-LSTM model acts as a regularizer and makes both networks be informative enough to perform well on different benchmark problems taken from the literature. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+            796
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The authors would like to thank Theano developers (Theano Development Team, 2016) for their great work. ",
+        "bbox": [
+            176,
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+        "text": "APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            264,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A: Derivation of variation lower bound ${ \\mathcal { L } } _ { \\mathbf { T } }$ in equation (2) in more details: ",
+        "bbox": [
+            173,
+            132,
+            661,
+            148
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f2352550590b95cf180f6ea83f998b74d821d67c83f5f9aacab271570224c7f8.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\quad - \\frac { \\sqrt { 3 } } { 2 } \\left[ \\int _ { 0 } ^ { 1 } \\sqrt { \\alpha } \\left( \\beta \\right) \\sin \\alpha \\sin \\beta \\right] \\left( \\frac { \\sqrt { 3 } } { 2 } \\right) \\sin \\alpha \\sin \\beta \\sin \\left( \\beta \\right) \\sin \\left( \\frac { \\sqrt { 3 } } { 2 } \\right) \\dot { \\alpha } \\sin \\alpha \\sin \\beta \\sin \\left( \\frac { \\sqrt { 3 } } { 2 } \\right) \\dot { \\alpha } \\sin \\alpha \\sin \\beta \\sin \\left( \\frac { \\sqrt { 3 } } { 2 } \\right) } \\\\ & { \\quad - \\frac { \\sqrt { 3 } } { 2 } \\left[ \\int _ { 0 } ^ { 1 } \\sqrt { \\alpha } \\left( \\beta \\right) \\sin \\alpha \\sin \\beta \\right] \\sin \\alpha \\sin \\beta \\sin \\left( \\frac { \\sqrt { 3 } } { 2 } \\right) \\dot { \\alpha } \\left[ \\int _ { 0 } ^ { 1 } \\sqrt { \\alpha } \\left( \\beta \\right) \\sin \\alpha \\sin \\beta \\sin \\left( \\frac { \\sqrt { 3 } } { 2 } \\right) \\dot { \\alpha } \\sin \\alpha \\right] \\sin \\beta \\sin \\alpha \\sin \\beta \\sin \\alpha \\sin \\beta \\sin \\beta \\sin \\alpha \\sin \\beta } \\\\ & { \\quad + \\frac { \\sqrt { 3 } } { 2 } \\left[ \\int _ { 0 } ^ { 1 } \\sqrt { \\alpha } \\left( \\beta \\right) \\sin \\alpha \\left( \\frac { \\sqrt { 3 } } { 2 } \\cos \\alpha \\cos \\beta \\right) \\right] \\sin \\alpha \\sin \\beta \\sin \\alpha \\sin \\beta \\sin \\alpha \\sin \\beta \\sin \\beta \\sin \\alpha \\sin \\beta \\sin \\alpha \\sin \\beta \\sin \\beta \\sin \\alpha \\sin \\beta } \\\\ & { \\quad - \\frac { \\sqrt { 3 } } { 2 } \\left[ \\int _ { 0 } ^ { 1 } \\sqrt { \\alpha } \\left( \\beta \\right) \\sin ^ { 2 } \\alpha \\left( \\frac { \\sqrt { 3 } } { 2 } \\cos \\alpha \\sin \\beta \\cos \\frac { \\sqrt { 3 } } { 2 } \\right) \\right] \\cos \\alpha \\sin \\beta \\sin \\alpha \\sin \\beta \\sin \\alpha \\sin \\beta \\sin \\alpha \\sin \\beta \\sin \\beta \\sin \\alpha \\sin \\beta \\sin \\beta \\sin \\alpha \\beta } \\\\ & { \\quad + \\frac { \\sqrt { 3 } } { 2 } \\left[ \\int _ { 0 } ^ { 1 } \\sqrt { \\alpha } \\left( \\beta \\right) \\sin ^ { 2 } \\alpha \\left( \\frac { \\sqrt { 3 } } { 2 } \\cos \\alpha \\sin \\beta \\cos \\frac { \\sqrt { 3 } } { 2 } \\right) \\right] \\sin \\alpha \\sin \\beta \\sin \\alpha \\sin \\beta \\sin \\alpha \\beta \\sin \\alpha \\sin \\beta \\sin \\alpha \\gamma \\sin \\beta \\sin \\alpha \\beta \\sin \\beta \\sin \\alpha \\gamma } \\\\ &  \\quad - \\frac { 3 }  2  \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            238,
+            194,
+            883,
+            602
+        ],
+        "page_idx": 11
+    }
+]

parse/train/Sye_OgHFwH/Sye_OgHFwH.md

@@ -0,0 +1,338 @@
+# UNRESTRICTED ADVERSARIAL EXAMPLES VIA SEMANTIC MANIPULATION
+
+Anand Bhattad∗ Min Jin Chong∗ Kaizhao Liang Bo Li D. A. Forsyth
+
+University of Illinois at Urbana-Champaign {bhattad2, mchong6, kl2, lbo, daf}@illinois.edu
+
+# ABSTRACT
+
+Machine learning models, especially deep neural networks (DNNs), have been shown to be vulnerable against adversarial examples which are carefully crafted samples with a small magnitude of the perturbation. Such adversarial perturbations are usually restricted by bounding their ${ \mathcal { L } } _ { p }$ norm such that they are imperceptible, and thus many current defenses can exploit this property to reduce their adversarial impact. In this paper, we instead introduce “unrestricted” perturbations that manipulate semantically meaningful image-based visual descriptors – color and texture – in order to generate effective and photorealistic adversarial examples. We show that these semantically aware perturbations are effective against JPEG compression, feature squeezing and adversarially trained model. We also show that the proposed methods can effectively be applied to both image classification and image captioning tasks on complex datasets such as ImageNet and MSCOCO. In addition, we conduct comprehensive user studies to show that our generated semantic adversarial examples are photorealistic to humans despite large magnitude perturbations when compared to other attacks.
+
+# 1 INTRODUCTION
+
+Machine learning (ML), especially deep neural networks (DNNs) have achieved great success in various tasks, including image recognition (Krizhevsky et al., 2012; He et al., 2016), speech processing (Hinton et al., 2012) and robotics training (Levine et al., 2016). However, recent literature has shown that these widely deployed ML models are vulnerable to adversarial examples – carefully crafted perturbations aiming to mislead learning models (Carlini & Wagner, 2017; Kurakin et al., 2016; Xiao et al., 2018b). The fast growth of DNNs based solutions demands in-depth studies on adversarial examples to help better understand potential vulnerabilities of ML models and therefore improve their robustness.
+
+To date, a variety of different approaches has been proposed to generate adversarial examples (Goodfellow et al., 2014b; Carlini & Wagner, 2017; Kurakin et al., 2016; Xiao et al., 2018a); and many of these attacks search for perturbation within a bounded ${ \mathcal { L } } _ { p }$ norm in order to preserve their photorealism. However, it is known that the ${ \mathcal { L } } _ { p }$ norm distance as a perceptual similarity metric is not ideal (Johnson et al., 2016; Isola et al., 2017). In addition, recent work show that defenses trained on ${ \mathcal { L } } _ { p }$ bounded perturbation are not robust at all against new types of unseen attacks (Kang et al., 2019). Therefore, exploring diverse adversarial examples, especially those with “unrestricted" magnitude of perturbation has acquired a lot of attention in both academia and industries (Brown et al., 2018).
+
+Recent work based on generative adversarial networks (GANs) (Goodfellow et al., 2014a) have introduced unrestricted attacks (Song et al, 2018). However, these attacks are limited to datasets like MNIST, CIFAR and CelebA, and are usually unable to scale up to bigger and more complex datasets such as ImageNet. Xiao et al. (2018b) directly manipulated spatial pixel flow of an image to produce adversarial examples without ${ \mathcal { L } } _ { p }$ bounded constraints on the perturbation. However, the attack does not explicitly control visual semantic representation. More recently, Hosseini & Poovendran (2018) manipulated hue and saturation of an image to create adversarial perturbations. However, these examples are easily distinguishable by human and are also not scalable to complex datasets.
+
+![](images/514af06a88a5eea15c6c9e8bc4ad6c3e460cdc30ee34f96f553dd4860d75f2bd.jpg)  
+Figure 1: An overview of proposed attacks. Top: Colorization attack (cAdv); Bottom: Texture transfer attack (tAdv). Our attacks achieve high attack success rate via semantic manipulation without any constraints on the ${ \mathcal { L } } _ { p }$ norm of the perturbation. Our methods are general and can be used for attacking both classifiers and captioners.
+
+In this work, we propose unrestricted attack strategies that explicitly manipulate semantic visual representations to generate natural-looking adversarial examples that are “far” from the original image in tems of the ${ \mathcal { L } } _ { p }$ norm distance. In particular, we manipulate color (cAdv) and texture (tAdv) to create realistic adversarial examples (see Fig 1). cAdv adaptively chooses locations in an image to change their colors, producing adversarial perturbation that is usually fairly substantial, while tAdv utilizes the texture from other images and adjusts the instance’s texture field using style transfer.
+
+These semantic transformation-based adversarial perturbations shed light upon the understanding of what information is important for DNNs to make predictions. For instance, in one of our case studies, when the road is recolored from gray to blue, the image gets misclassified to tench (a fish) although a car remains evidently visible (Fig. 2b). This indicates that deep learning models can easily be fooled by certain large scale patterns. In addition to image classifiers, the proposed attack methods can be generalized to different machine learning tasks such as image captioning ( Karpathy & Fei-Fei (2015)). Our attacks can either change the entire caption to the target (Chen et al., 2017; Xu et al., 2019) or take on more challenging tasks like changing one or two specific target words from the caption to a target. For example, in Fig. 1, “stop sign” of the original image caption is changed to “cat sitting” and “umbrella is” for cAdv and tAdv respectively.
+
+To ensure our “unrestricted" semantically manipulated images are natural, we conducted extensive user studies with Amazon Mechanical Turk. We also tested our proposed attacks on several state of the art defenses. Rather than just showing the attacks break these defenses (better defenses will come up), we aim to show that cAdv and tAdv are able to produce new types of adversarial examples. Experiments also show that our proposed attacks are more transferable given their large and structured perturbations (Papernot et al., 2016). Our semantic adversarial attacks provide further insights about the vulnerabilities of ML models and therefore encourage new solutions to improve their robustness.
+
+In summary, our contributions are: 1) We propose two novel approaches to generate “unrestricted" adversarial examples via semantic transformation; 2) We conduct extensive experiments to attack both image classification and image captioning models on large scale datasets (ImageNet and MSCOCO); 3) We show that our attacks are equipped with unique properties such as smooth cAdv perturbations and structured tAdv perturbations. 4) We perform comprehensive user studies to show that when compared to other attacks, our generated adversarial examples appear more natural to humans despite their large perturbations; 5) We test different adversarial examples against several state of the art defenses and show that the proposed attacks are more transferable and harder to defend.
+
+# 2 COLORIZATION ATTACK (cADV)
+
+Background. Image Colorization is the task of giving natural colorization to a grayscale image. This is an ill-posed problem as there are multiple viable natural colorizations given a single grayscale image. Deshpande et al. (2017) showed that diverse image colorization can be achieved by using an architecture that combines VAE (Kingma & Welling (2013)) and Mixture Density Network; while Zhang et al. (2017) demonstrated an improved and diverse image colorization by using input hints from users guided colorization process.
+
+![](images/b0804e3a762a7a2f503cf09b86cad9a05b95b04f732e4ddb2fde6c9dd3cdb46a.jpg)  
+Figure 2: Class color affinity. Samples from unconstrained cAdv attacking network weights with zero hints provided. For (a) the ground truth (GT) class is pretzel; and for (b) the GT is car. These new colors added are commonly found in images from target class. For instance, green in Golfcart and blue sea in Tench images.
+
+Our goal is to adversarially color an image by leveraging a pretrained colorization model. We hypothesize that it is possible to find a natural colorization that is adversarial for a target model (e.g., classifier or captioner) by searching in the color space. Since a colorization network learns to color natural colors that conform to boundaries and respect short-range color consistency, we can use it to introduce smooth and consistent adversarial noise with a large magnitude that looks natural to humans. This attack differs from common adversarial attacks which tend to introduce short-scale high-frequency artifacts that are minimized to be invisible for human observers.
+
+We leverage Zhang et al. (2016; 2017) colorization model for our attack. In their work, they produce natural colorizations on ImageNet with input hints from the user. The inputs to their network consist of the L channel of the image in CIELAB color space $X _ { L } \in \mathbb { R } ^ { H \times W \times 1 }$ , the sparse colored input hints $X _ { a b } \in \mathbb { R } ^ { H \times W \times 2 }$ , and the binary mask $M \in \mathbb { B } ^ { \hat { H } \times W \times 1 }$ , indicating the location of the hints.
+
+cAdv Objectives. There are a few ways to leverage the colorization model to achieve adversarial objectives. We experimented with two main methods and achieved varied results.
+
+Network weights. The straightforward approach of producing adversarial colors is to modify Zhang et al. (2017) colorization network, $\mathcal { C }$ directly. To do so, we simply update $\mathcal { C }$ by minimizing the adversarial loss objective ${ J } _ { a d v }$ , which in our case, is the cross entropy loss. $t$ represents target class and $\mathcal { F }$ represents the victim network.
+
+$$
+\boldsymbol { \theta } ^ { * } = \underset { \boldsymbol { \theta } } { \arg \operatorname* { m i n } } \ J _ { a d v } ( \mathcal { F } ( \mathcal { C } ( \boldsymbol { X _ { L } } , \boldsymbol { X _ { a b } } , M ; \boldsymbol { \theta } ) ) , t )
+$$
+
+Hints and mask. We can also vary input hints $X _ { a b }$ and mask $M$ to produce adversarial colorizations. Hints provides the network with ground truth color patches that guides the colorization, while the mask provides its spatial location. By jointly varying both hints and mask, we are able to manipulate the output colorization. We can update the hints and mask as follows:
+
+$$
+\boldsymbol { M } ^ { * } , \boldsymbol { X } _ { a b } ^ { * } = \underset { \boldsymbol { M } , \boldsymbol { X } _ { a b } } { \arg \operatorname* { m i n } } \boldsymbol { J } _ { a d v } \big ( \mathcal { F } ( \mathcal { C } ( \boldsymbol { X } _ { L } , \boldsymbol { X } _ { a b } , \boldsymbol { M } ; \boldsymbol { \theta } ) ) , t \big )
+$$
+
+cAdvAttack Methods. Attacking network weights allows the network to search the color space with no constraints for adversarial colors. This attack is the easiest to optimize, but the output colors are not realistic as shown in Fig. 2. Our various strategies outlined below are ineffective as the model learns to generate the adversarial colors without taking into account color realism. However, colorizations produced often correlate with colors often observed in the target class. This suggests that classifiers associate certain colors with certain classes which we will discuss more in our case study.
+
+Attacking input hints and mask jointly gives us natural results as the pretrained network will not be affected by our optimization. Attacking hints and mask separately also works but takes a long optimization time and give slightly worse results. For our experiments, we use Adam Optimizer (Kingma & Ba (2014)) with a learning rate of $1 0 ^ { - 4 }$ in cAdv. We iteratively update hints and mask until our adversarial image reaches the target class and the confidence change of consecutive iterations does not exceed a threshold of 0.05.
+
+![](images/c7c5f8250772341da4fe8591f5d37ea334cf3298a3d785c6a99babd51fbecadf.jpg)  
+Figure 3: Controlling cAdv. We show a comparison of sampling 50 color hints from k clusters with low-entropy. All images are attacked to misclassify as golf-cart. Second and fourth row visualize our cluster segments, with darker colors representing higher mean entropy and red dots representing the sampled hints location. Sampling hints across more clusters gives less color variety.
+
+Control over colorization. Current attack methods lack control over where the attack occurs, opting to attack all pixels indiscriminately. This lack of control is not important for most attacks where the $\epsilon$ is small but is concerning in cAdv where making unstructured large changes can be jarring. To produce realistic colorization, we need to avoid making large color changes at locations where colors are unambiguous (e.g. roads in general are gray) and focus on those where colors are ambiguous (e.g. an umbrella can have different colors). To do so, we need to segment an image and determine which segments should be attacked or preserved.
+
+To segment the image into meaningful areas, we cluster the image’s ground truth AB space using K-Means. We first use a Gaussian filter of $\sigma = 3$ to smooth the AB channels and then cluster them into 8 clusters. Then, we have to determine which cluster’s colors should be preserved. Fortunately, Zhang et al. (2017) network output a per-pixel color distribution for a given image which we used to calculate the entropy of each pixel. The entropy represents how confident the network is at assigning a color at that location. The average entropy of each cluster represents how ambiguous their color is. We want to avoid making large changes to clusters with low-entropy while allowing our attack to change clusters with high entropy. One way to enforce this behavior is through hints, which are sampled from the ground truth at locations belonging to clusters of low-entropy. We sample hints from the $k$ clusters with the lowest entropy which we refer as $\mathbf { c A d v } _ { k }$ (e.g. $\mathbf { c A d v } _ { 2 }$ samples hints from the 2 lowest entropy clusters).
+
+Number of input hints. Network hints constrain our output to have similar colors as the ground truth, avoiding the possibility of unnatural colorization at the cost of color diversity. This trade-off is controlled by the number of hints given to the network as initialization (Fig. 4). Generally, providing more hints gives us similar colors that are observed in original image. However, having too many hints is also problematic. Too many hints makes the optimization between drawing adversarial colors and matching local color hints difficult. Since the search space for adversarial colors is constrained because of more hints, we may instead generate unrealistic examples.
+
+Number of Clusters. The trade-off between the color diversity and the color realism is also controlled by the number of clusters we sample hints from as shown in Fig. 3. Sampling from multiple clusters gives us realistic colors closer to the ground truth image at the expense of color diversity.
+
+![](images/28c9a7081555c80396d860e35a38f481e641978107e769d4052d98991b1f4987.jpg)  
+Figure 4: Number of color hints required for cAdv. All images are attacked to Merganser with $k = 4$ . When the number of hints increases (from left to right), the output colors are more similar to groundtruth. However, when the number of hints is too high (500), cAdv often generates unrealistic perturbations. This is due to a harder optimization for cAdv to both add adversarial colors and match GT color hints. cAdv is effective and realistic with a balanced number of hints.
+
+Empirically, from our experiments we find that in terms of color diversity, realism, and robustness of attacks, using $k = 4$ and 50 hints gives us better adversarial examples. For the rest of this paper, we fix 50 hints for all $\mathbf { c A d v } _ { k }$ methods.
+
+# 3 TEXTURE ATTACK (tADV)
+
+Background. Texture transfer extracts texture from one image and adds it to another. Transferring texture from one source image to another target image has been widely studied in computer vision ( Efros & Freeman (2001); Gatys et al. (2015)). The Convolutional Neural Network (CNN) based texture transfer from Gatys et al. (2015) led to a series of new ideas in the domain of artistic style transfer ( Gatys et al. (2016); Huang & Belongie (2017); Li et al. (2017); Yeh et al. (2019)). More recently, Geirhos et al. (2018) showed that DNNs trained on ImageNet are biased towards texture for making predictions.
+
+Our goal is to generate adversarial examples by infusing texture from another image without explicit constraints on ${ \mathcal { L } } _ { p }$ norm of the perturbation. For generating our tAdv examples, we used a pretrained VGG19 network (Simonyan & Zisserman, 2014) to extract textural features. We directly optimize our victim image $( I _ { v } )$ by adding texture from a target image $( I _ { t } )$ . A natural strategy to transfer texture is by minimizing within-layer feature correlation statistics (gram matrices) between two images Gatys et al. (2015; 2016). Based on Yeh et al. (2019), we find that optimizing cross-layer gram matrices instead of within-layer gram matrices helps produce more natural looking adversarial examples. The difference between the within-layer and the cross-layer gram matrices is that for a within-layer, the feature’s statistics are computed between the same layer. For a cross-layer, the statistics are computed between two adjacent layers.
+
+tAdv Objectives. tAdv directly attacks the image to create adversarial examples without modifying network parameters. Moreover, there is no additional content loss that is used in style transfer methods (Gatys et al. (2016); Yeh et al. (2019)). Our overall objective function for the texture attack contains a texture transfer loss $( L _ { t } ^ { \mathcal { A } } )$ and an cross-entropy loss $( J _ { a d v } )$ .
+
+$$
+L _ { \mathbf { t A d v } } ^ { \mathcal { A } } = \alpha L _ { t } ^ { \mathcal { A } } ( I _ { v } , I _ { t } ) + \beta J _ { a d v } ( \mathcal { F } ( I _ { v } ) , t )
+$$
+
+Unlike style transfer methods, we do not want the adversarial examples to be artistically pleasing. Our goal is to infuse a reasonable texture from a target class image to the victim image and fool a classifier or captioning network. To ensure a reasonable texture is added without overly perturbing the victim image too much, we introduce an additional constraint on the variation in the gram matrices of the victim image. This constraint helps us to control the image transformation procedure and prevents it from producing artistic images. Let $m$ and $n$ denote two layers of a pretrained VGG-19 with a decreasing spatial resolution and $C$ for number of filter maps in layer $n$ , our texture transfer loss is then given by
+
+$$
+L _ { t } ^ { \mathcal { A } } ( I _ { v } , I _ { t } ) = \sum _ { ( m , n ) \in \mathcal { L } } \frac { 1 } { C ^ { 2 } } \sum _ { i j } \frac { \big \| G _ { i j } ^ { m , n } ( I _ { v } ) - G _ { i j } ^ { m , n } ( I _ { t } ) \big \| ^ { 2 } } { s t d \left( G _ { i j } ^ { m , n } ( I _ { v } ) \right) }
+$$
+
+![](images/ebf1d78bd9d5759fd6fbcdbee62f0234f57921123c53e7a611aa070c3032e6eb.jpg)  
+Figure 5: tAdv strategies. Texture transferred from random “texture source” $( T _ { s } )$ in row 1, random target class $T _ { s }$ (row 2) and from the nearest target class $T _ { s }$ (row 3). All examples are misclassified from Beacon to Nautilus. Images in the last row look photo realistic, while those in the first two rows contain more artifacts as the texture weight $\alpha$ increases (left to right).
+
+Let $f$ be feature maps, $\mathcal { U } f ^ { n }$ be an upsampled $f ^ { n }$ that matches the spatial resolution of layer $m$ . The cross layer gram matrices $G$ between the victim image $( I _ { v } )$ and a target image $( I _ { t } )$ is given as
+
+$$
+G _ { i j } ^ { m , n } ( I ) = \sum _ { p } \left[ f _ { i , p } ^ { m } ( I ) \right] \left[ \mathcal { U } f _ { j , p } ^ { n } ( I ) \right] ^ { T }
+$$
+
+Texture Transfer. To create tAdv adversarial examples, we need to find images to extract the texture from, which we call “texture source” $( T _ { s } )$ . A naive strategy is to randomly select an image from the data bank as $T _ { s }$ . Though this strategy is successful, their perturbations are clearly perceptible. Alternatively, we can randomly select $T _ { s }$ from the adversarial target class. This strategy produces less perceptible perturbations compared to the random $T _ { s }$ method as we are extracting a texture from the known target class. A better strategy to select $T _ { s }$ is to find a target class image that lies closest to the victim image in the feature space using nearest neighbors. This strategy is sensible as we assure our victim image has similar feature statistics as our target image. Consequently, minimizing gram matrices is easier and our attack generates more natural looking images (see Fig. 5).
+
+For texture transfer, we extract cross-layer statistics in Eq. 4 from the R11, R21, R31, R41, and R51 of a pretrained VGG19. We optimize our objective (Eq. 3) using an L-BFGS (Liu & Nocedal (1989)) optimizer. tAdv attacks are sensitive and if not controlled well, images get transformed into artistic images. Since we do not have any constraints over the perturbation norm, it is necessary to decide when to stop the texture transfer procedure. For a successful attack (images look realistic), we limit our L-BFGS to fixed number of small steps and perform two set of experiments: one with only one iteration or round of L-BFGS for 14 steps and another with three iterations of 14 steps. For the three iterations setup, after every iteration, we look at the confidence of our target class and stop if the confidence is greater than 0.9.
+
+Texture and Cross-Entropy Weights. Empirically, we found setting $\alpha$ to be in the range [150, 1000] and $\beta$ in the range $\left[ 1 0 ^ { - 4 } , 1 0 ^ { - 3 } \right]$ to be successful and also produce less perceptible tAdv examples. The additional cross-entropy based adversarial objective $J _ { a d v }$ helps our optimization. We ensure large flow of gradients is from the texture loss and they are sufficiently larger than the adversarial crossentropy objective. The adversarial objective also helps in transforming victim image to adversarial without stylizing the image. All our tabulated results are shown for one iteration, $\alpha = 2 5 0$ and $\beta = 1 0 ^ { - 3 }$ , unless otherwise stated. We use the annotation $\mathbf { t A d v } _ { \alpha } ^ { i t e r }$ for the rest of the paper to denote the texture method that we are using.
+
+<table><tr><td colspan="2">Model</td><td>R50</td><td>D121</td><td>VGG19</td></tr><tr><td colspan="2">Accuracy</td><td>76.15</td><td>74.65</td><td>74.24</td></tr><tr><td rowspan="5">seecss Aarea</td><td>cAdv1</td><td>99.72</td><td>99.89</td><td>99.89</td></tr><tr><td>cAdv4</td><td>99.78</td><td>99.83</td><td>100.00</td></tr><tr><td>tAdv250</td><td>97.99</td><td>99.72</td><td>99.50</td></tr><tr><td>tAdv500</td><td>99.27</td><td>99.83</td><td>99.83</td></tr></table>
+
+# (a) Whitebox Target Attack Success Rate
+
+<table><tr><td>Method</td><td>Model</td><td>R50</td><td>D121</td><td>VGG19</td></tr><tr><td rowspan="3">Kurakin et al. (2016) Carlini&amp;Wagner (2017) Xiao et al. (2018b)</td><td>R50</td><td>100.00</td><td>17.33</td><td>12.95</td></tr><tr><td>R50</td><td>98.85</td><td>16.50</td><td>11.00</td></tr><tr><td>R50</td><td>100</td><td>5.23</td><td>8.90</td></tr><tr><td rowspan="3">cAdv4</td><td>R50</td><td>99.83</td><td>28.56</td><td>31.00</td></tr><tr><td>D121</td><td>18.13</td><td>99.83</td><td>29.43</td></tr><tr><td>VGG19</td><td>22.94</td><td>26.39</td><td>100.00</td></tr><tr><td rowspan="3">tAdv250</td><td>R50</td><td>99.00</td><td>24.51</td><td>34.84</td></tr><tr><td>D121</td><td>21.16</td><td>99.83</td><td>32.56</td></tr><tr><td>VGG19</td><td>20.21</td><td>24.40</td><td>99.89</td></tr></table>
+
+# (b) Transferability
+
+Table 1: Our attacks are highly successful on ResNet50 (R50), DenseNet121 (D121) and VGG19. In (a), for cAdv, we show results for $k = \{ 1 , 4 \}$ when attacked with 50 hints. For tAdv we show results for $\alpha = \{ 2 5 0 , 5 0 0 \}$ and $\beta = 0 . 0 0 1$ . In (b) We show the transferability of our attacks. We attack models from the column and test them on models from the rows.
+
+Control over Texture. The amount of texture that gets added to our victim image is controlled by the texture weight coefficient $( \alpha )$ . Increasing texture weights improves attack success rate at the cost of noticeable perturbation. When compared to within-layer statistics, the cross-layer statistics that we use are not only better at extracting texture, it is also easier to control the texture weight.
+
+# 4 EXPERIMENTAL RESULTS
+
+In this section, we evaluate the two proposed attack methods both quantitatively, via attack success rate under different settings, and qualitatively, based on interesting case studies. We conduct our experiments on ImageNet Deng et al. (2009) by randomly selecting images from 10 sufficiently different classes predicted correctly for the classification attack.
+
+We use a pretrained ResNet 50 classifier (He et al. (2016)) for all our methods. DenseNet 121 and VGG 19 (Huang et al.; Simonyan & Zisserman (2014)) are used for our transferability analysis.
+
+# 4.1 cADV ATTACK
+
+cAdv achieves high targeted attack success rate by adding realistic color perturbation. Our numbers in Table 1 and Table 2 also reveal that cAdv examples with larger color changes (consequently more color diversity) are more robust against transferability and adversarial defenses. However, these big changes are found to be slightly less realistic from our user study (Table 2, Table 4).
+
+Smooth cAdv perturbations. Fig. 8 in our Appendix shows interesting properties of the adversarial colors. We observe that cAdv perturbations are locally smooth and are relatively low-frequency. This is different from most adversarial attacks that generate high-frequency noise-like perturbations. This phenomenon can be explained by the observation that colors are usually smooth within object boundaries. The pretrained colorization model will thus produce smooth, low-frequency adversarial colors that conform to object boundaries.
+
+Importance of color in classification. From Fig. 2, we can compare how different target class affects our colorization results if we relax our constraints on colors (cAdv on Network Weights, 0 hints). In many cases, the images contain strong colors that are related to the target class. In the case of golf-cart, we get a green tint over the entire image. This can push the target classifier to misclassify the image as green grass is usually overabundant in benign golf-cart images. Fig. 2b shows our attack on an image of a car to tench (a type of fish). We observe that the gray road turned blue and that the colors are tinted. We can hypothesize that the blue colors and the tint fooled the classifier into thinking the image is a tench in the sea.
+
+The colorization model is originally trained to produce natural colorization that conforms to object boundaries. By adjusting its parameters, we are able to produce such large and abnormal color change that is impossible with our attack on hints and mask. These colors, however, show us some evidence that colors play a stronger role in classification than we thought. We reserve the exploration of this observation for future works.
+
+While this effect (strong color correlation to target class) is less pronounced for our attack on hints and mask, for all cAdv methods, we observe isoluminant color blobs. Isoluminant colors are characterized
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Res50</td><td rowspan="2">JPEG75</td><td colspan="5">Feature Squeezing</td><td rowspan="2">Res152</td><td rowspan="2">Adv Res152</td><td rowspan="2">User Pref.</td></tr><tr><td>4-bit</td><td>5-bit</td><td>2x2</td><td>3x3</td><td>11-3-4</td></tr><tr><td>Kurakin et al. (2016)</td><td>100</td><td>12.73</td><td>28.62</td><td>86.66</td><td>34.28</td><td>21.56</td><td>29.28</td><td>1.08</td><td>1.38</td><td>0.506</td></tr><tr><td>Carlini &amp;Wagner (2017)</td><td>99.85</td><td>11.50</td><td>12.00</td><td>30.50</td><td>22.00</td><td>14.50</td><td>18.50</td><td>1.08</td><td>1.38</td><td>0.497</td></tr><tr><td>Hosseini &amp; Poovendran (2018)</td><td>1.20</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td></td></tr><tr><td>Xiao et al. (2018b)</td><td>100</td><td>17.61</td><td>22.51</td><td>29.26</td><td>28.71</td><td>23.51</td><td>26.67</td><td>4.13</td><td>1.39</td><td>0.470</td></tr><tr><td>cAdv1</td><td>100</td><td>52.33</td><td>47.78</td><td>76.17</td><td>36.28</td><td>50.50</td><td>61.95</td><td>12.06</td><td>11.62</td><td>0.427</td></tr><tr><td>cAdv2</td><td>99.89</td><td>46.61</td><td>42.78</td><td>72.56</td><td>34.28</td><td>46.45</td><td>59.00</td><td>17.39</td><td>19.4</td><td>0.437</td></tr><tr><td>cAdv4</td><td>99.83</td><td>42.61</td><td>38.39</td><td>69.67</td><td>34.34</td><td>40.78</td><td>54.62</td><td>14.13</td><td>12.5</td><td>0.473</td></tr><tr><td>cAdv8</td><td>99.81</td><td>38.22</td><td>36.62</td><td>67.06</td><td>31.67</td><td>37.67</td><td>49.17</td><td>6.52</td><td>10.04</td><td>0.476</td></tr><tr><td>tAdv250</td><td>99.00</td><td>32.89</td><td>62.79</td><td>89.74</td><td>54.94</td><td>38.92</td><td>40.57</td><td>10.9</td><td>2.10</td><td>0.433</td></tr><tr><td>tAdv250</td><td>100</td><td>36.33</td><td>67.68</td><td>94.11</td><td>58.92</td><td>42.82</td><td>44.56</td><td>15.21</td><td>4.6</td><td>0.425</td></tr><tr><td>tAdv1000</td><td>99.88</td><td>31.49</td><td>52.69</td><td>90.52</td><td>51.24</td><td>34.85</td><td>39.68</td><td>19.12</td><td>5.59</td><td>0.412</td></tr><tr><td>tAdv1000</td><td>100</td><td>35.23</td><td>61.40</td><td>93.18</td><td>56.31</td><td>39.66</td><td>45.59</td><td>22.28</td><td>6.94</td><td>0.406</td></tr></table>
+
+Table 2: Comparison against defense models. Misclassification rate after passing different adversarial examples through the defense models (higher means the attack is stronger). All attacks are performed on ResNet50 with whitebox attack. The highest attack success rate is in bold. We also report the user preference scores from AMT for each attack (last column).
+
+by a change in color without a corresponding change in luminance. As most color changes occur along edges in natural images, it is likely that classifiers trained on ImageNet have never seen isoluminant colors. This suggests that cAdv might be exploiting isoluminant colors to fool classifiers.
+
+# 4.2 tADV ATTACK
+
+tAdv successfully fools the classifiers with a very small weighted adversarial cross-entropy objective $( \beta )$ when combined with texture loss, while remaining realistic to humans. As shown in Table 1, our attacks are highly successful on white-box attacks tested on three different models with the nearest neighbor texture transfer approach. We also show our attacks are more transferable to other models. In our Appendix, we show ablation results for tAdv attacks along with other strategies that we used for generating tAdv adversarial examples.
+
+Structured tAdv Perturbations. Since we extract features across different layers of VGG, the tAdv perturbations follow a textural pattern. They are more structured and organized when compared to others. Our tAdv perturbations are big when compared with existing attack methods in ${ \mathcal { L } } _ { p }$ norm. They are of high-frequency and yet imperceptible (see Fig. 1 and Fig. 8).
+
+Importance of Texture in Classification. Textures are crucial descriptors for image classification and Imagenet trained models can be exploited by altering the texture. Their importance is also shown in the recent work from Geirhos et al. (2018). Our results also shows that even with a small or invisible change in the texture field can break the current state of the art classifiers.
+
+# 4.3 DEFENSE AND TRANSFERABILITY ANALYSIS
+
+We test all our attacks and other existing methods with images attacked from Resnet50. We evaluate them on three defenses – JPEG defense (Das et al., 2017), feature squeezing (Xu et al., 2017) and adversarial training. By leveraging JPEG compression and decompression, adversarial noise may be removed. We tested our methods against JPEG compression of 75. Feature squeezing is a family of simple but surprisingly effective strategies, including reducing color bit depth and spatial smoothing. Adversarial training has been shown as an effective but costly method to defend against adversarial attacks. Mixing adversarial samples into training data of a classifier improves its robustness without affecting the overall accuracy. We were able to obtain an adversarially pretrained Resnet152 model on ImageNet dataset and hence we tested our Resnet50 attacked images with this model.
+
+Robustness. In general, our attacks are more robust to the considered defenses and transferable for targeted attacks. For cAdv, there is a trade-off between more realistic colors (using more hints and sampling from more clusters) and attack robustness. From Table 1 and 2, we show that as we progressively use more clusters, our transferability and defense numbers drop. A similar trend is observed with the change in the number of hints. cAdv is robust to JPEG defense and adversarial training because of their large and spatially smooth perturbations. For tAdv, increasing texture weight $( \alpha )$ does not necessarily perform well with the defense even though it increases attack success rate, but increasing texture flow with more iterations improves attack’s robustness against defenses.
+
+# 5 HUMAN PERCEPTUAL STUDIES
+
+To quantify how realistic tAdv and cAdv examples are, we conducted a user study on Amazon Mechanical Turk (AMT). We follow the same procedure as described in (Zhang et al., 2016; Xiao et al., 2018b). For each attack, we choose the same 200 adversarial images and their corresponding benign ones. During each trial, one random adversarial-benign pair appears for three seconds and workers are given five minutes to identify the realistic one. Each attack has 600 unique pairs of images and each pair is evaluated by at least 10 unique workers. We restrict biases in this process by allowing each unique user up to 5 rounds of trials and also ignore users who complete the study in less than 30 seconds. In total, 598 unique workers completed at least one round of our user study. For each image, we can then calculate the user preference score as the number of times it is chosen divided by the number of times it is displayed. 0.5 represents that users are unable to distinguish if the image is fake. For cAdv and tAdv, user preferences averages at 0.476 and 0.433 respectively, indicating that workers have a hard time distinguishing them. The user preferences for all attacks are summarized in Table 2 and their comparison with ${ \mathcal { L } } _ { p }$ norm is in Table 4 and Table 5.
+
+# 6 ATTACKING CAPTIONING MODEL
+
+Our methods are general and can be easily adapted for other learning tasks. As proof of concept, we test our attacks against image captioning task. Image captioning is the task of generating a sequence of word description for an image. The popular architecture for captioning is a Long-ShortTerm-Memory (LSTM) (Hochreiter & Schmidhuber, 1997) based models (Karpathy & Fei-Fei, 2015; Wang et al., 2017). Recently, (Aneja et al., 2018) proposed a convolutional based captioning model for a fast and accurate caption generation. This convolutional based approach does not suffer from the commonly known problems of vanishing gradients and overly confident predictions of LSTM network. Therefore, we choose to attack the current state of the art convolutional captioning model. We randomly selected images from MSCOCO (Lin et al., 2014) for image captioning attack.
+
+Attacking captioning models is harder than attacking classifiers when the goal is to change exactly one word in the benign image’s caption unlike pixel based attacks (Chen et al., 2017; Xu et al., 2019). We show that our attacks are successful and have no visible artifacts even for this challenging task. In Fig. 6, we change the second word of the caption to dog while keeping the rest of the caption the same. This is a challenging targeted attack because, in many untargeted attacks, the resulted captions do not make sense. More examples are in our Appendix.
+
+![](images/6a956e0df34f6db0ac9fd6dbff2b885fc1fd389ac877e7c3c15a104e82cdbac7.jpg)  
+Figure 6: Captioning attack. Top: cAdv; Bottom: tAdv. We attack the second word to dog and show the corresponding change in attention mask of that word. More examples in Appendix.
+
+Adversarial Cross-Entropy Objective for Captioning. Let $t$ be the target caption, $w$ denote the word position of the caption, $\mathcal { F }$ for the captioning model, $I _ { v }$ for the victim image and $J _ { a d v }$ for the cross-entropy loss
+
+$$
+L _ { c a p t } ^ { \mathcal { A } } = \sum _ { w } J _ { a d v } ( ( \mathcal { F } ( I _ { v } ) ) _ { w } , t _ { w } )
+$$
+
+For cAdv, we give all color hints and optimize to get an adversarial colored image to produce target caption. For tAdv, we add Eqn 6 to Eqn 4 to optimize the image. We select $T _ { S }$ as the nearest neighbor of the victim image from the ones in the adversarial target class using ImageNet dataset. We stop our attack once we reach the target caption and the caption does not change in consecutive iterations. Note we do not change the network weights, we only optimize hints and mask (for cAdv) or the victim image (for tAdv) to achieve our target caption.
+
+# 7 RELATED WORK
+
+Here we briefly summarize existing unrestricted and semantic adversarial attacks. Xiao et al. (2018b) proposed geometric or spatial distortion of pixels in image to create adversarial examples. They distort the input image by optimizing pixel flow instead of pixel values to generate adversarial examples. While this attack leads to “natural” looking adversarial examples with large $\mathcal { L } _ { \infty }$ norm, it does not take image semantics into account. Song et al (2018) and Dunn et al. (2019) considered GANs for adversarial attacks. This attack is unrestricted in ${ \mathcal { L } } _ { p }$ norm but they are restricted to simple datasets as it involves training GANs, which have been known to be unstable and computationally intensive for complex datasets like ImageNet (Karras et al., 2017; Brock et al., 2018).
+
+Hosseini & Poovendran (2018), changes the hue & saturation of an image randomly to create adversarial examples. It is similar to cAdv as they both involve changing colors, however, their search space is limited to two dimensions and their images are unrealistic, Appendix (Fig. 10). Also, while this method has a non-trivial untargeted attack success rate, it performs extremely poorly for targeted attacks $( 1 . 2 0 \%$ success rate in our own experiments on ImageNet). Our work is also related to Joshi et al. (2019) and Qiu et al. (2019), who manipulate images conditioned on face dataset attributes like glasses, beard for their attacks. These work focuses on changing single image visual attribute and are conditionally dependent. Our work focuses on changing visual semantic descriptors to misclassify images and are not conditioned to any semantic attributes.
+
+<table><tr><td>Method</td><td>Semantic Based</td><td></td><td>Unrestricted Photorealistic</td><td>Explainable (eg color affinity)</td><td>Complex Dataset</td><td>Caption Attack</td></tr><tr><td>Kurakin et al. (2016)</td><td>X</td><td>xx&lt;&lt;√</td><td>&lt;&lt;xx&gt;</td><td>X</td><td>√</td><td>X</td></tr><tr><td>Carlini &amp; Wagner (2017)</td><td>X</td><td></td><td></td><td></td><td>√</td><td></td></tr><tr><td>Hosseini &amp; Poovendran (2018)</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Song et al (2018)</td><td>X</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Xiao et al. (2018b)</td><td>X</td><td></td><td></td><td>xxx&gt;</td><td>xx&gt;</td><td></td></tr><tr><td>cAdv (ours)</td><td>√</td><td>√</td><td>√</td><td>√</td><td>√</td><td>xxxx/</td></tr><tr><td>tAdv (ours)</td><td>√</td><td>√</td><td></td><td>√</td><td>√</td><td>√</td></tr></table>
+
+Table 3: Summary of the difference in our work compared to previous work. Unlike previous attack methods, our attacks are unbounded, semantically motivated, realistic, highly successful, and scales to more complex datasets and other ML tasks. They are also robust against tested defenses.
+
+# 8 CONCLUSION
+
+Our proposed two novel unrestricted semantic attacks shed light on the role of texture and color fields in influencing DNN’s predictions. They not only consistently fool human subjects but in general are harder to defend against. We hope by presenting our methods, we encourage future studies on unbounded adversarial attacks, better metrics for measuring perturbations, and more sophisticated defenses.
+
+# ACKNOWLEDGEMENTS
+
+We thank Chaowei Xiao for sharing their code to compare our methods with Xiao et al. (2018b) and helping us setup the user study. We also thank Tianyuan Zhang for providing the AdvRes152 pretrained model. This work was supported by NSF Grant No. 1718221 and ONR MURI Award N00014-16-1-2007.
+
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+
+# A APPENDIX
+
+# A.1 OTHER DETAILS ON HUMAN STUDY
+
+We also chose BIM (Kurakin et al., 2016) and CW (Carlini & Wagner, 2017) for comparing our perturbations. Since these attacks are known to have low ${ \mathcal { L } } _ { p }$ norm, we designed an aggressive version of BIM by relaxing its $\mathcal { L } _ { \infty }$ bound to match the norm of our attacks. We settled with two aggressive versions of BIM with average $\mathcal { L } _ { \infty } = \{ 0 . 2 1 , 0 . 3 4 7 \}$ , which we refer to as $\mathrm { B I M _ { 0 . 2 1 } }$ , $\mathrm { B I M _ { 0 . 3 4 } }$ . The average user preferences for BIM drops drastically from 0.497 to 0.332 when we relax the norm to $\mathrm { B I M _ { 0 . 3 4 } }$ ; the decrease in user preferences for tAdv (0.433 to 0.406) and cAdv (0.476 to 0.437) is not significant. In Fig. 7, we plot a density plot of $\mathcal { L } _ { \infty }$ vs user preference scores.
+
+![](images/d7f388370788c4cbfe65811264c0744fc1d3e1e06e6b15bc94cfad5c0b602a11.jpg)  
+Figure 7: Density Plot. Our methods achieve large $\mathcal { L } _ { \infty }$ norm perturbations without notable reduction in user preference. Each plot is a density plot between perturbation ( $\scriptstyle { \mathcal { L } } _ { \infty }$ norm) on $\mathbf { X }$ axis and $P r$ (user prefers adversarial image) on Y axis. For ideal systems, the density would be a concentrated horizontal line at 0.5. All plots are on the same set of axes. On the left, plots for three baseline methods (Fig a - Fig c). Note the very strong concentration on small norm perturbations, which users like. Right 4 plots shows our methods (Fig d - Fig g). Note strong push into large norm regions, without loss of user preference.
+
+<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>Lo     L2    L8</td><td rowspan=1 colspan=1>Preference</td></tr><tr><td rowspan=1 colspan=1>cw</td><td rowspan=1 colspan=1>0.587 0.026 0.054</td><td rowspan=1 colspan=1>0.506</td></tr><tr><td rowspan=1 colspan=1>BIM0.051</td><td rowspan=1 colspan=1>0.826 0.030 0.051</td><td rowspan=2 colspan=1>0.4970.3550.332</td></tr><tr><td rowspan=1 colspan=1>BIM0.21BIM0.347</td><td rowspan=1 colspan=1>0.893  0.118  0.210.892 0.119 0.347</td></tr><tr><td rowspan=2 colspan=1>cAdv2cAdv4cAdv8</td><td rowspan=1 colspan=1>0.910 0.098  0.489</td><td rowspan=2 colspan=1>0.4370.4730.476</td></tr><tr><td rowspan=1 colspan=1>0.898  0.071  0.4480.896 0.059 0.425</td></tr><tr><td rowspan=4 colspan=1>tAdv250tAdvi000</td><td rowspan=1 colspan=1>0.891  0.041  0.198</td><td rowspan=1 colspan=1>0.433</td></tr><tr><td rowspan=1 colspan=1>0.931 0.056 0.237</td><td rowspan=1 colspan=1>0.412</td></tr><tr><td rowspan=1 colspan=1>0.916 0.061  0.258</td><td rowspan=1 colspan=1>0.425</td></tr><tr><td rowspan=1 colspan=1>0.946  0.08  0.315</td><td rowspan=1 colspan=1>0.406</td></tr></table>
+
+Table 4: User Study. User preference score and ${ \mathcal { L } } _ { p }$ norm of perturbation for different attacks. cAdv and tAdv are imperceptible for humans (score close to 0.5) even with very big ${ \mathcal { L } } _ { p }$ norm perturbation.
+
+<table><tr><td>Attack</td><td></td><td>tarantula</td><td>merganser</td><td>nautilus</td><td>hyena</td><td>beacon</td><td>golfcart</td><td>photocopier</td><td>umbrella</td><td>pretzel</td><td>sandbar</td><td>Mean</td></tr><tr><td rowspan="5">BIM</td><td>Lo</td><td>0.82</td><td>0.81</td><td>0.80</td><td>0.88</td><td>0.80</td><td>0.79</td><td>0.86</td><td>0.80</td><td>0.85</td><td>0.84</td><td>0.83</td></tr><tr><td>L2</td><td>0.05</td><td>0.02</td><td>0.03</td><td>0.04</td><td>0.02</td><td>0.01</td><td>0.04</td><td>0.02</td><td>0.02</td><td>0.04</td><td>0.03</td></tr><tr><td>L8</td><td>0.07</td><td>0.04</td><td>0.06</td><td>0.06</td><td>0.04</td><td>0.03</td><td>0.07</td><td>0.04</td><td>0.04</td><td>0.06</td><td>0.05</td></tr><tr><td>Preference</td><td>0.49</td><td>0.75</td><td>0.44</td><td>0.67</td><td>0.48</td><td>0.48</td><td>0.27</td><td>0.39</td><td>0.38</td><td>0.63</td><td>0.50</td></tr><tr><td>L0</td><td>0.54</td><td>0.55</td><td>0.52</td><td>0.77</td><td>0.48</td><td>0.42</td><td>0.70</td><td>0.48</td><td>0.69</td><td>0.71</td><td>0.59</td></tr><tr><td rowspan="5">CW</td><td>L</td><td>0.05</td><td>0.01</td><td>0.03</td><td>0.03</td><td>0.02</td><td>0.01</td><td>0.04</td><td>0.01</td><td>0.02</td><td>0.04</td><td>0.03</td></tr><tr><td>L</td><td>0.08</td><td>0.04</td><td>0.06</td><td>0.07</td><td>0.04</td><td>0.03</td><td>0.07</td><td>0.04</td><td>0.05</td><td>0.06</td><td>0.05</td></tr><tr><td>Preference</td><td>0.51</td><td>0.68</td><td>0.43</td><td>0.59</td><td>0.52</td><td>0.51</td><td>0.39</td><td>0.40</td><td>0.42</td><td>0.62</td><td>0.51</td></tr><tr><td></td><td>0.94</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>L L</td><td>0.07</td><td>0.95</td><td>0.91</td><td>0.94 0.06</td><td>0.89</td><td>0.94</td><td>0.89</td><td>0.91</td><td>0.94</td><td>0.85</td><td>0.92</td></tr><tr><td rowspan="6">tAdv250</td><td>L8</td><td></td><td>0.08</td><td>0.08</td><td></td><td>0.05</td><td>0.06</td><td>0.03</td><td>0.05</td><td>0.07</td><td>0.05</td><td>0.06</td></tr><tr><td></td><td>0.28</td><td>0.31</td><td>0.30</td><td>0.31</td><td>0.22</td><td>0.27</td><td>0.18</td><td>0.25</td><td>0.27</td><td>0.19</td><td>0.26</td></tr><tr><td>Preference</td><td>0.39</td><td>0.58</td><td>0.43</td><td>0.53</td><td>0.46</td><td>0.35</td><td>0.25</td><td>0.40</td><td>0.33</td><td>0.54</td><td>0.43</td></tr><tr><td>L0</td><td>0.90</td><td>0.90</td><td>0.88</td><td>0.90</td><td>0.90</td><td>0.90</td><td>0.90</td><td>0.89</td><td>0.93</td><td>0.88</td><td>0.90</td></tr><tr><td>L</td><td>0.06</td><td>0.06</td><td>0.07</td><td>0.05</td><td>0.08</td><td>0.07</td><td>0.08</td><td>0.08</td><td>0.09</td><td>0.06</td><td>0.07</td></tr><tr><td>L8 Preference</td><td>0.36 0.46</td><td>0.44 0.65</td><td>0.41 0.46</td><td>0.31 0.58</td><td>0.48 0.45</td><td>0.51 0.48</td><td>0.55 0.30</td><td>0.56 0.43</td><td>0.46 0.35</td><td>0.39 0.59</td><td>0.45 0.47</td></tr></table>
+
+Table 5: Class wise ${ \mathcal { L } } _ { p }$ norm and user preference breakdown. Users are biased and pick a few classes quite often (merganser, sandbar), and do not like a few classes (photocopier) over others.
+
+![](images/0a960c25a339bfbdfd4e118211a67d2ceac05684f40c127d8920823ba900ff03.jpg)  
+Figure 8: Perturbation comparisons. Images are attacked from tarantula to beacon, golf cart, nautilus, photocopier, pretzel from left to right. Our perturbations (cAdv and tAdv) are large, structured and have spatial patterns when compared with other attacks. Perturbations from cAdv are low-frequency and locally smooth while perturbations from tAdv are primarily high-frequency and structured. Note gray color indicates no perturbations.
+
+A.2 ADDITIONAL RESULTS   
+
+<table><tr><td colspan="2">Model</td><td>Resnet50</td><td>Dense121</td><td>VGG19</td></tr><tr><td colspan="2">Accuracy</td><td>76.15</td><td>74.65</td><td>74.24</td></tr><tr><td rowspan="7">Sseeess Arkea</td><td>Random Ts</td><td>99.67</td><td>99.72</td><td>96.16</td></tr><tr><td>Random Target Ts</td><td>99.72</td><td>99.89</td><td>99.94</td></tr><tr><td>Nearest Target Ts</td><td>97.99</td><td>99.72</td><td>99.50</td></tr><tr><td>cAdv4 25 hints</td><td>99.78</td><td>99.83</td><td>99.93</td></tr><tr><td>cAdv4 50 hints</td><td>99.78</td><td>99.83</td><td>100.00</td></tr><tr><td>cAdv4 100 hints</td><td>99.44</td><td>99.50</td><td>99.93</td></tr></table>
+
+Whitebox target attack success rate. Our attacks are highly successful on different models across all strategies. tAdv results are for $\alpha = 2 5 0$ , $\beta = 1 0 ^ { - 3 }$ and iter $= 1$ .
+
+Table 6: Ablation Studies.   
+
+<table><tr><td>α β</td><td>250</td><td>500</td><td>750</td><td>1000</td></tr><tr><td>0</td><td>25.00</td><td>99.61</td><td>98.55</td><td>95.92</td></tr><tr><td>10-4</td><td>99.88</td><td>99.61</td><td>98.55</td><td>95.92</td></tr><tr><td>10-3</td><td>97.99</td><td>99.27</td><td>99.66</td><td>99.50</td></tr><tr><td>10-2</td><td>96.26</td><td>95.42</td><td>96.32</td><td>96.59</td></tr></table>
+
+tAdv ablation study. Whitebox target success rate with nearest target $T _ { s }$ (texture source). In columns, we have increasing texture weight $( \alpha )$ and in rows, we have increasing adversarial cross-entropy weight $( \beta )$ . All attacks are done on Resnet50.
+
+![](images/338ce3b799792303bb653b8fdcfc46b5d5c524075d602fed70070bccd9adb34d.jpg)  
+Figure 9: Additional qualitative examples for controlling cAdv. We show a comparison of sampling 50 color hints from k clusters with low-entropy. All images are attacked to golf-cart. Even numbered rows visualize our cluster segments, with darker colors representing higher mean entropy and red dots representing the location we sample hints from. Sampling hints across more clusters gives less color variety.
+
+![](images/c5de302b9eb8610441b2904e8ef667e2c43474007fbdae082a0c017ff61d4dbb.jpg)  
+Figure 10: Images attacked with the method of (Hosseini & Poovendran, 2018) are not realistic, as these examples show. Compare Fig 12, showing results of our color attack. Similar qualitative results for CIFAR-10 are visible in Hosseini & Poovendran (2018).
+
+![](images/92288996c16f4cc7fda4076186b63e1a00ce1168d673775295b2b020c53eed77.jpg)  
+Figure 11: Additional qualitative examples for tAdv. Texture transferred from random images (row 1), random images from adversarial target class (row 2) and from the nearest neighbor of the victim image from the adversarial target class (row 3). All examples are misclassified from Merganser to Umbrella. Images in the last row look photo realistic, while those in the first two rows contain more artifacts as the texture weight $\alpha$ increases (left to right).
+
+![](images/c4db4a0b5a7776b50c75b3f1b22a265b4b7fe14078b1bff2faa5f499cec851d8.jpg)  
+Figure 12: Captioning attack. We attack the second word of each caption to {dog, bird} and show the corresponding change in attention mask of that word. For tAdv we use the nearest neighbor selection method and for cAdv we initialize with all groundtruth color hints.
+
+![](images/8c8fbb3896dd6b66b4180dcd97895bec534939a21f6e28428e484a34ea71d246.jpg)  
+Figure 13: Randomly sampled, semantically manipulated, unrestricted adversarial examples and their perturbations. (a) Adversarial examples generated by cAdv attacking Hints and mask with 50 hints. (c) Adversarial examples generated by tAdv with $\alpha = 2 5 0$ , $\beta = \bar { 1 0 } ^ { - 3 }$ and iter $= 1$ using nearest target $T _ { s }$ . (b) and (d) are their respective perturbations. Note that diagonal images are groundtruth images and gray pixels indicate no perturbation.

parse/train/Sye_OgHFwH/Sye_OgHFwH_content_list.json

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+    {
+        "type": "text",
+        "text": "UNRESTRICTED ADVERSARIAL EXAMPLES VIA SEMANTIC MANIPULATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            99,
+            689,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anand Bhattad∗ Min Jin Chong∗ Kaizhao Liang Bo Li D. A. Forsyth ",
+        "bbox": [
+            187,
+            169,
+            714,
+            184
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "University of Illinois at Urbana-Champaign {bhattad2, mchong6, kl2, lbo, daf}@illinois.edu ",
+        "bbox": [
+            184,
+            185,
+            598,
+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            250,
+            544,
+            263
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Machine learning models, especially deep neural networks (DNNs), have been shown to be vulnerable against adversarial examples which are carefully crafted samples with a small magnitude of the perturbation. Such adversarial perturbations are usually restricted by bounding their ${ \\mathcal { L } } _ { p }$ norm such that they are imperceptible, and thus many current defenses can exploit this property to reduce their adversarial impact. In this paper, we instead introduce “unrestricted” perturbations that manipulate semantically meaningful image-based visual descriptors – color and texture – in order to generate effective and photorealistic adversarial examples. We show that these semantically aware perturbations are effective against JPEG compression, feature squeezing and adversarially trained model. We also show that the proposed methods can effectively be applied to both image classification and image captioning tasks on complex datasets such as ImageNet and MSCOCO. In addition, we conduct comprehensive user studies to show that our generated semantic adversarial examples are photorealistic to humans despite large magnitude perturbations when compared to other attacks. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            512,
+            334,
+            527
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Machine learning (ML), especially deep neural networks (DNNs) have achieved great success in various tasks, including image recognition (Krizhevsky et al., 2012; He et al., 2016), speech processing (Hinton et al., 2012) and robotics training (Levine et al., 2016). However, recent literature has shown that these widely deployed ML models are vulnerable to adversarial examples – carefully crafted perturbations aiming to mislead learning models (Carlini & Wagner, 2017; Kurakin et al., 2016; Xiao et al., 2018b). The fast growth of DNNs based solutions demands in-depth studies on adversarial examples to help better understand potential vulnerabilities of ML models and therefore improve their robustness. ",
+        "bbox": [
+            174,
+            545,
+            825,
+            656
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "To date, a variety of different approaches has been proposed to generate adversarial examples (Goodfellow et al., 2014b; Carlini & Wagner, 2017; Kurakin et al., 2016; Xiao et al., 2018a); and many of these attacks search for perturbation within a bounded ${ \\mathcal { L } } _ { p }$ norm in order to preserve their photorealism. However, it is known that the ${ \\mathcal { L } } _ { p }$ norm distance as a perceptual similarity metric is not ideal (Johnson et al., 2016; Isola et al., 2017). In addition, recent work show that defenses trained on ${ \\mathcal { L } } _ { p }$ bounded perturbation are not robust at all against new types of unseen attacks (Kang et al., 2019). Therefore, exploring diverse adversarial examples, especially those with “unrestricted\" magnitude of perturbation has acquired a lot of attention in both academia and industries (Brown et al., 2018). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recent work based on generative adversarial networks (GANs) (Goodfellow et al., 2014a) have introduced unrestricted attacks (Song et al, 2018). However, these attacks are limited to datasets like MNIST, CIFAR and CelebA, and are usually unable to scale up to bigger and more complex datasets such as ImageNet. Xiao et al. (2018b) directly manipulated spatial pixel flow of an image to produce adversarial examples without ${ \\mathcal { L } } _ { p }$ bounded constraints on the perturbation. However, the attack does not explicitly control visual semantic representation. More recently, Hosseini & Poovendran (2018) manipulated hue and saturation of an image to create adversarial perturbations. However, these examples are easily distinguishable by human and are also not scalable to complex datasets. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/514af06a88a5eea15c6c9e8bc4ad6c3e460cdc30ee34f96f553dd4860d75f2bd.jpg",
+        "image_caption": [
+            "Figure 1: An overview of proposed attacks. Top: Colorization attack (cAdv); Bottom: Texture transfer attack (tAdv). Our attacks achieve high attack success rate via semantic manipulation without any constraints on the ${ \\mathcal { L } } _ { p }$ norm of the perturbation. Our methods are general and can be used for attacking both classifiers and captioners. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "In this work, we propose unrestricted attack strategies that explicitly manipulate semantic visual representations to generate natural-looking adversarial examples that are “far” from the original image in tems of the ${ \\mathcal { L } } _ { p }$ norm distance. In particular, we manipulate color (cAdv) and texture (tAdv) to create realistic adversarial examples (see Fig 1). cAdv adaptively chooses locations in an image to change their colors, producing adversarial perturbation that is usually fairly substantial, while tAdv utilizes the texture from other images and adjusts the instance’s texture field using style transfer. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "These semantic transformation-based adversarial perturbations shed light upon the understanding of what information is important for DNNs to make predictions. For instance, in one of our case studies, when the road is recolored from gray to blue, the image gets misclassified to tench (a fish) although a car remains evidently visible (Fig. 2b). This indicates that deep learning models can easily be fooled by certain large scale patterns. In addition to image classifiers, the proposed attack methods can be generalized to different machine learning tasks such as image captioning ( Karpathy & Fei-Fei (2015)). Our attacks can either change the entire caption to the target (Chen et al., 2017; Xu et al., 2019) or take on more challenging tasks like changing one or two specific target words from the caption to a target. For example, in Fig. 1, “stop sign” of the original image caption is changed to “cat sitting” and “umbrella is” for cAdv and tAdv respectively. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To ensure our “unrestricted\" semantically manipulated images are natural, we conducted extensive user studies with Amazon Mechanical Turk. We also tested our proposed attacks on several state of the art defenses. Rather than just showing the attacks break these defenses (better defenses will come up), we aim to show that cAdv and tAdv are able to produce new types of adversarial examples. Experiments also show that our proposed attacks are more transferable given their large and structured perturbations (Papernot et al., 2016). Our semantic adversarial attacks provide further insights about the vulnerabilities of ML models and therefore encourage new solutions to improve their robustness. ",
+        "bbox": [
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+        "type": "text",
+        "text": "In summary, our contributions are: 1) We propose two novel approaches to generate “unrestricted\" adversarial examples via semantic transformation; 2) We conduct extensive experiments to attack both image classification and image captioning models on large scale datasets (ImageNet and MSCOCO); 3) We show that our attacks are equipped with unique properties such as smooth cAdv perturbations and structured tAdv perturbations. 4) We perform comprehensive user studies to show that when compared to other attacks, our generated adversarial examples appear more natural to humans despite their large perturbations; 5) We test different adversarial examples against several state of the art defenses and show that the proposed attacks are more transferable and harder to defend. ",
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+    {
+        "type": "text",
+        "text": "2 COLORIZATION ATTACK (cADV) ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Background. Image Colorization is the task of giving natural colorization to a grayscale image. This is an ill-posed problem as there are multiple viable natural colorizations given a single grayscale image. Deshpande et al. (2017) showed that diverse image colorization can be achieved by using an architecture that combines VAE (Kingma & Welling (2013)) and Mixture Density Network; while Zhang et al. (2017) demonstrated an improved and diverse image colorization by using input hints from users guided colorization process. ",
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+    {
+        "type": "image",
+        "img_path": "images/b0804e3a762a7a2f503cf09b86cad9a05b95b04f732e4ddb2fde6c9dd3cdb46a.jpg",
+        "image_caption": [
+            "Figure 2: Class color affinity. Samples from unconstrained cAdv attacking network weights with zero hints provided. For (a) the ground truth (GT) class is pretzel; and for (b) the GT is car. These new colors added are commonly found in images from target class. For instance, green in Golfcart and blue sea in Tench images. "
+        ],
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Our goal is to adversarially color an image by leveraging a pretrained colorization model. We hypothesize that it is possible to find a natural colorization that is adversarial for a target model (e.g., classifier or captioner) by searching in the color space. Since a colorization network learns to color natural colors that conform to boundaries and respect short-range color consistency, we can use it to introduce smooth and consistent adversarial noise with a large magnitude that looks natural to humans. This attack differs from common adversarial attacks which tend to introduce short-scale high-frequency artifacts that are minimized to be invisible for human observers. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We leverage Zhang et al. (2016; 2017) colorization model for our attack. In their work, they produce natural colorizations on ImageNet with input hints from the user. The inputs to their network consist of the L channel of the image in CIELAB color space $X _ { L } \\in \\mathbb { R } ^ { H \\times W \\times 1 }$ , the sparse colored input hints $X _ { a b } \\in \\mathbb { R } ^ { H \\times W \\times 2 }$ , and the binary mask $M \\in \\mathbb { B } ^ { \\hat { H } \\times W \\times 1 }$ , indicating the location of the hints. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "cAdv Objectives. There are a few ways to leverage the colorization model to achieve adversarial objectives. We experimented with two main methods and achieved varied results. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Network weights. The straightforward approach of producing adversarial colors is to modify Zhang et al. (2017) colorization network, $\\mathcal { C }$ directly. To do so, we simply update $\\mathcal { C }$ by minimizing the adversarial loss objective ${ J } _ { a d v }$ , which in our case, is the cross entropy loss. $t$ represents target class and $\\mathcal { F }$ represents the victim network. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/13b03bec04a0f0f782942344d9319b7b1f597bd062d0202a754f42ba3ae81e53.jpg",
+        "text": "$$\n\\boldsymbol { \\theta } ^ { * } = \\underset { \\boldsymbol { \\theta } } { \\arg \\operatorname* { m i n } } \\ J _ { a d v } ( \\mathcal { F } ( \\mathcal { C } ( \\boldsymbol { X _ { L } } , \\boldsymbol { X _ { a b } } , M ; \\boldsymbol { \\theta } ) ) , t )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Hints and mask. We can also vary input hints $X _ { a b }$ and mask $M$ to produce adversarial colorizations. Hints provides the network with ground truth color patches that guides the colorization, while the mask provides its spatial location. By jointly varying both hints and mask, we are able to manipulate the output colorization. We can update the hints and mask as follows: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fda6b92ba29f99d98b00479b20d952826d173a102069df0648a3547f7e5d06e0.jpg",
+        "text": "$$\n\\boldsymbol { M } ^ { * } , \\boldsymbol { X } _ { a b } ^ { * } = \\underset { \\boldsymbol { M } , \\boldsymbol { X } _ { a b } } { \\arg \\operatorname* { m i n } } \\boldsymbol { J } _ { a d v } \\big ( \\mathcal { F } ( \\mathcal { C } ( \\boldsymbol { X } _ { L } , \\boldsymbol { X } _ { a b } , \\boldsymbol { M } ; \\boldsymbol { \\theta } ) ) , t \\big )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "cAdvAttack Methods. Attacking network weights allows the network to search the color space with no constraints for adversarial colors. This attack is the easiest to optimize, but the output colors are not realistic as shown in Fig. 2. Our various strategies outlined below are ineffective as the model learns to generate the adversarial colors without taking into account color realism. However, colorizations produced often correlate with colors often observed in the target class. This suggests that classifiers associate certain colors with certain classes which we will discuss more in our case study. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Attacking input hints and mask jointly gives us natural results as the pretrained network will not be affected by our optimization. Attacking hints and mask separately also works but takes a long optimization time and give slightly worse results. For our experiments, we use Adam Optimizer (Kingma & Ba (2014)) with a learning rate of $1 0 ^ { - 4 }$ in cAdv. We iteratively update hints and mask until our adversarial image reaches the target class and the confidence change of consecutive iterations does not exceed a threshold of 0.05. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/c7c5f8250772341da4fe8591f5d37ea334cf3298a3d785c6a99babd51fbecadf.jpg",
+        "image_caption": [
+            "Figure 3: Controlling cAdv. We show a comparison of sampling 50 color hints from k clusters with low-entropy. All images are attacked to misclassify as golf-cart. Second and fourth row visualize our cluster segments, with darker colors representing higher mean entropy and red dots representing the sampled hints location. Sampling hints across more clusters gives less color variety. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Control over colorization. Current attack methods lack control over where the attack occurs, opting to attack all pixels indiscriminately. This lack of control is not important for most attacks where the $\\epsilon$ is small but is concerning in cAdv where making unstructured large changes can be jarring. To produce realistic colorization, we need to avoid making large color changes at locations where colors are unambiguous (e.g. roads in general are gray) and focus on those where colors are ambiguous (e.g. an umbrella can have different colors). To do so, we need to segment an image and determine which segments should be attacked or preserved. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To segment the image into meaningful areas, we cluster the image’s ground truth AB space using K-Means. We first use a Gaussian filter of $\\sigma = 3$ to smooth the AB channels and then cluster them into 8 clusters. Then, we have to determine which cluster’s colors should be preserved. Fortunately, Zhang et al. (2017) network output a per-pixel color distribution for a given image which we used to calculate the entropy of each pixel. The entropy represents how confident the network is at assigning a color at that location. The average entropy of each cluster represents how ambiguous their color is. We want to avoid making large changes to clusters with low-entropy while allowing our attack to change clusters with high entropy. One way to enforce this behavior is through hints, which are sampled from the ground truth at locations belonging to clusters of low-entropy. We sample hints from the $k$ clusters with the lowest entropy which we refer as $\\mathbf { c A d v } _ { k }$ (e.g. $\\mathbf { c A d v } _ { 2 }$ samples hints from the 2 lowest entropy clusters). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Number of input hints. Network hints constrain our output to have similar colors as the ground truth, avoiding the possibility of unnatural colorization at the cost of color diversity. This trade-off is controlled by the number of hints given to the network as initialization (Fig. 4). Generally, providing more hints gives us similar colors that are observed in original image. However, having too many hints is also problematic. Too many hints makes the optimization between drawing adversarial colors and matching local color hints difficult. Since the search space for adversarial colors is constrained because of more hints, we may instead generate unrealistic examples. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Number of Clusters. The trade-off between the color diversity and the color realism is also controlled by the number of clusters we sample hints from as shown in Fig. 3. Sampling from multiple clusters gives us realistic colors closer to the ground truth image at the expense of color diversity. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/28c9a7081555c80396d860e35a38f481e641978107e769d4052d98991b1f4987.jpg",
+        "image_caption": [
+            "Figure 4: Number of color hints required for cAdv. All images are attacked to Merganser with $k = 4$ . When the number of hints increases (from left to right), the output colors are more similar to groundtruth. However, when the number of hints is too high (500), cAdv often generates unrealistic perturbations. This is due to a harder optimization for cAdv to both add adversarial colors and match GT color hints. cAdv is effective and realistic with a balanced number of hints. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Empirically, from our experiments we find that in terms of color diversity, realism, and robustness of attacks, using $k = 4$ and 50 hints gives us better adversarial examples. For the rest of this paper, we fix 50 hints for all $\\mathbf { c A d v } _ { k }$ methods. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 TEXTURE ATTACK (tADV) ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Background. Texture transfer extracts texture from one image and adds it to another. Transferring texture from one source image to another target image has been widely studied in computer vision ( Efros & Freeman (2001); Gatys et al. (2015)). The Convolutional Neural Network (CNN) based texture transfer from Gatys et al. (2015) led to a series of new ideas in the domain of artistic style transfer ( Gatys et al. (2016); Huang & Belongie (2017); Li et al. (2017); Yeh et al. (2019)). More recently, Geirhos et al. (2018) showed that DNNs trained on ImageNet are biased towards texture for making predictions. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our goal is to generate adversarial examples by infusing texture from another image without explicit constraints on ${ \\mathcal { L } } _ { p }$ norm of the perturbation. For generating our tAdv examples, we used a pretrained VGG19 network (Simonyan & Zisserman, 2014) to extract textural features. We directly optimize our victim image $( I _ { v } )$ by adding texture from a target image $( I _ { t } )$ . A natural strategy to transfer texture is by minimizing within-layer feature correlation statistics (gram matrices) between two images Gatys et al. (2015; 2016). Based on Yeh et al. (2019), we find that optimizing cross-layer gram matrices instead of within-layer gram matrices helps produce more natural looking adversarial examples. The difference between the within-layer and the cross-layer gram matrices is that for a within-layer, the feature’s statistics are computed between the same layer. For a cross-layer, the statistics are computed between two adjacent layers. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "tAdv Objectives. tAdv directly attacks the image to create adversarial examples without modifying network parameters. Moreover, there is no additional content loss that is used in style transfer methods (Gatys et al. (2016); Yeh et al. (2019)). Our overall objective function for the texture attack contains a texture transfer loss $( L _ { t } ^ { \\mathcal { A } } )$ and an cross-entropy loss $( J _ { a d v } )$ . ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/180d29bf642c245b808af8eeb571d2bb2aac660cfb0bd7e27d26d9b6c1d72626.jpg",
+        "text": "$$\nL _ { \\mathbf { t A d v } } ^ { \\mathcal { A } } = \\alpha L _ { t } ^ { \\mathcal { A } } ( I _ { v } , I _ { t } ) + \\beta J _ { a d v } ( \\mathcal { F } ( I _ { v } ) , t )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Unlike style transfer methods, we do not want the adversarial examples to be artistically pleasing. Our goal is to infuse a reasonable texture from a target class image to the victim image and fool a classifier or captioning network. To ensure a reasonable texture is added without overly perturbing the victim image too much, we introduce an additional constraint on the variation in the gram matrices of the victim image. This constraint helps us to control the image transformation procedure and prevents it from producing artistic images. Let $m$ and $n$ denote two layers of a pretrained VGG-19 with a decreasing spatial resolution and $C$ for number of filter maps in layer $n$ , our texture transfer loss is then given by ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/b63cfb5d2edbcb82179969507cb13407c700072371e748efba768c24788c4280.jpg",
+        "text": "$$\nL _ { t } ^ { \\mathcal { A } } ( I _ { v } , I _ { t } ) = \\sum _ { ( m , n ) \\in \\mathcal { L } } \\frac { 1 } { C ^ { 2 } } \\sum _ { i j } \\frac { \\big \\| G _ { i j } ^ { m , n } ( I _ { v } ) - G _ { i j } ^ { m , n } ( I _ { t } ) \\big \\| ^ { 2 } } { s t d \\left( G _ { i j } ^ { m , n } ( I _ { v } ) \\right) }\n$$",
+        "text_format": "latex",
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/ebf1d78bd9d5759fd6fbcdbee62f0234f57921123c53e7a611aa070c3032e6eb.jpg",
+        "image_caption": [
+            "Figure 5: tAdv strategies. Texture transferred from random “texture source” $( T _ { s } )$ in row 1, random target class $T _ { s }$ (row 2) and from the nearest target class $T _ { s }$ (row 3). All examples are misclassified from Beacon to Nautilus. Images in the last row look photo realistic, while those in the first two rows contain more artifacts as the texture weight $\\alpha$ increases (left to right). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Let $f$ be feature maps, $\\mathcal { U } f ^ { n }$ be an upsampled $f ^ { n }$ that matches the spatial resolution of layer $m$ . The cross layer gram matrices $G$ between the victim image $( I _ { v } )$ and a target image $( I _ { t } )$ is given as ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2a4fd9f001a4e56214b0e98f84a8a09ba4fc2de23326b5770f634b8c39ad5072.jpg",
+        "text": "$$\nG _ { i j } ^ { m , n } ( I ) = \\sum _ { p } \\left[ f _ { i , p } ^ { m } ( I ) \\right] \\left[ \\mathcal { U } f _ { j , p } ^ { n } ( I ) \\right] ^ { T }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Texture Transfer. To create tAdv adversarial examples, we need to find images to extract the texture from, which we call “texture source” $( T _ { s } )$ . A naive strategy is to randomly select an image from the data bank as $T _ { s }$ . Though this strategy is successful, their perturbations are clearly perceptible. Alternatively, we can randomly select $T _ { s }$ from the adversarial target class. This strategy produces less perceptible perturbations compared to the random $T _ { s }$ method as we are extracting a texture from the known target class. A better strategy to select $T _ { s }$ is to find a target class image that lies closest to the victim image in the feature space using nearest neighbors. This strategy is sensible as we assure our victim image has similar feature statistics as our target image. Consequently, minimizing gram matrices is easier and our attack generates more natural looking images (see Fig. 5). ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For texture transfer, we extract cross-layer statistics in Eq. 4 from the R11, R21, R31, R41, and R51 of a pretrained VGG19. We optimize our objective (Eq. 3) using an L-BFGS (Liu & Nocedal (1989)) optimizer. tAdv attacks are sensitive and if not controlled well, images get transformed into artistic images. Since we do not have any constraints over the perturbation norm, it is necessary to decide when to stop the texture transfer procedure. For a successful attack (images look realistic), we limit our L-BFGS to fixed number of small steps and perform two set of experiments: one with only one iteration or round of L-BFGS for 14 steps and another with three iterations of 14 steps. For the three iterations setup, after every iteration, we look at the confidence of our target class and stop if the confidence is greater than 0.9. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Texture and Cross-Entropy Weights. Empirically, we found setting $\\alpha$ to be in the range [150, 1000] and $\\beta$ in the range $\\left[ 1 0 ^ { - 4 } , 1 0 ^ { - 3 } \\right]$ to be successful and also produce less perceptible tAdv examples. The additional cross-entropy based adversarial objective $J _ { a d v }$ helps our optimization. We ensure large flow of gradients is from the texture loss and they are sufficiently larger than the adversarial crossentropy objective. The adversarial objective also helps in transforming victim image to adversarial without stylizing the image. All our tabulated results are shown for one iteration, $\\alpha = 2 5 0$ and $\\beta = 1 0 ^ { - 3 }$ , unless otherwise stated. We use the annotation $\\mathbf { t A d v } _ { \\alpha } ^ { i t e r }$ for the rest of the paper to denote the texture method that we are using. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/874f1ae539e452fab2cc898d9f606ee3f118d14f6f5a1ca46fc6cb81261a7ff1.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Model</td><td>R50</td><td>D121</td><td>VGG19</td></tr><tr><td colspan=\"2\">Accuracy</td><td>76.15</td><td>74.65</td><td>74.24</td></tr><tr><td rowspan=\"5\">seecss Aarea</td><td>cAdv1</td><td>99.72</td><td>99.89</td><td>99.89</td></tr><tr><td>cAdv4</td><td>99.78</td><td>99.83</td><td>100.00</td></tr><tr><td>tAdv250</td><td>97.99</td><td>99.72</td><td>99.50</td></tr><tr><td>tAdv500</td><td>99.27</td><td>99.83</td><td>99.83</td></tr></table>",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "(a) Whitebox Target Attack Success Rate ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/6a505079fc5cbbedaa93baf7b239f71f376ac1abad3344f099cb4f926881883c.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Model</td><td>R50</td><td>D121</td><td>VGG19</td></tr><tr><td rowspan=\"3\">Kurakin et al. (2016) Carlini&amp;Wagner (2017) Xiao et al. (2018b)</td><td>R50</td><td>100.00</td><td>17.33</td><td>12.95</td></tr><tr><td>R50</td><td>98.85</td><td>16.50</td><td>11.00</td></tr><tr><td>R50</td><td>100</td><td>5.23</td><td>8.90</td></tr><tr><td rowspan=\"3\">cAdv4</td><td>R50</td><td>99.83</td><td>28.56</td><td>31.00</td></tr><tr><td>D121</td><td>18.13</td><td>99.83</td><td>29.43</td></tr><tr><td>VGG19</td><td>22.94</td><td>26.39</td><td>100.00</td></tr><tr><td rowspan=\"3\">tAdv250</td><td>R50</td><td>99.00</td><td>24.51</td><td>34.84</td></tr><tr><td>D121</td><td>21.16</td><td>99.83</td><td>32.56</td></tr><tr><td>VGG19</td><td>20.21</td><td>24.40</td><td>99.89</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "(b) Transferability ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Table 1: Our attacks are highly successful on ResNet50 (R50), DenseNet121 (D121) and VGG19. In (a), for cAdv, we show results for $k = \\{ 1 , 4 \\}$ when attacked with 50 hints. For tAdv we show results for $\\alpha = \\{ 2 5 0 , 5 0 0 \\}$ and $\\beta = 0 . 0 0 1$ . In (b) We show the transferability of our attacks. We attack models from the column and test them on models from the rows. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Control over Texture. The amount of texture that gets added to our victim image is controlled by the texture weight coefficient $( \\alpha )$ . Increasing texture weights improves attack success rate at the cost of noticeable perturbation. When compared to within-layer statistics, the cross-layer statistics that we use are not only better at extracting texture, it is also easier to control the texture weight. ",
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+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTAL RESULTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we evaluate the two proposed attack methods both quantitatively, via attack success rate under different settings, and qualitatively, based on interesting case studies. We conduct our experiments on ImageNet Deng et al. (2009) by randomly selecting images from 10 sufficiently different classes predicted correctly for the classification attack. ",
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+        "type": "text",
+        "text": "We use a pretrained ResNet 50 classifier (He et al. (2016)) for all our methods. DenseNet 121 and VGG 19 (Huang et al.; Simonyan & Zisserman (2014)) are used for our transferability analysis. ",
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+    {
+        "type": "text",
+        "text": "4.1 cADV ATTACK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "cAdv achieves high targeted attack success rate by adding realistic color perturbation. Our numbers in Table 1 and Table 2 also reveal that cAdv examples with larger color changes (consequently more color diversity) are more robust against transferability and adversarial defenses. However, these big changes are found to be slightly less realistic from our user study (Table 2, Table 4). ",
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+    {
+        "type": "text",
+        "text": "Smooth cAdv perturbations. Fig. 8 in our Appendix shows interesting properties of the adversarial colors. We observe that cAdv perturbations are locally smooth and are relatively low-frequency. This is different from most adversarial attacks that generate high-frequency noise-like perturbations. This phenomenon can be explained by the observation that colors are usually smooth within object boundaries. The pretrained colorization model will thus produce smooth, low-frequency adversarial colors that conform to object boundaries. ",
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+    {
+        "type": "text",
+        "text": "Importance of color in classification. From Fig. 2, we can compare how different target class affects our colorization results if we relax our constraints on colors (cAdv on Network Weights, 0 hints). In many cases, the images contain strong colors that are related to the target class. In the case of golf-cart, we get a green tint over the entire image. This can push the target classifier to misclassify the image as green grass is usually overabundant in benign golf-cart images. Fig. 2b shows our attack on an image of a car to tench (a type of fish). We observe that the gray road turned blue and that the colors are tinted. We can hypothesize that the blue colors and the tint fooled the classifier into thinking the image is a tench in the sea. ",
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+    {
+        "type": "text",
+        "text": "The colorization model is originally trained to produce natural colorization that conforms to object boundaries. By adjusting its parameters, we are able to produce such large and abnormal color change that is impossible with our attack on hints and mask. These colors, however, show us some evidence that colors play a stronger role in classification than we thought. We reserve the exploration of this observation for future works. ",
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+    {
+        "type": "text",
+        "text": "While this effect (strong color correlation to target class) is less pronounced for our attack on hints and mask, for all cAdv methods, we observe isoluminant color blobs. Isoluminant colors are characterized ",
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+    {
+        "type": "table",
+        "img_path": "images/01671e255d42d7bd9db41006d3f56951fe579c06ae25f352edb97c176ca6311b.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">Res50</td><td rowspan=\"2\">JPEG75</td><td colspan=\"5\">Feature Squeezing</td><td rowspan=\"2\">Res152</td><td rowspan=\"2\">Adv Res152</td><td rowspan=\"2\">User Pref.</td></tr><tr><td>4-bit</td><td>5-bit</td><td>2x2</td><td>3x3</td><td>11-3-4</td></tr><tr><td>Kurakin et al. (2016)</td><td>100</td><td>12.73</td><td>28.62</td><td>86.66</td><td>34.28</td><td>21.56</td><td>29.28</td><td>1.08</td><td>1.38</td><td>0.506</td></tr><tr><td>Carlini &amp;Wagner (2017)</td><td>99.85</td><td>11.50</td><td>12.00</td><td>30.50</td><td>22.00</td><td>14.50</td><td>18.50</td><td>1.08</td><td>1.38</td><td>0.497</td></tr><tr><td>Hosseini &amp; Poovendran (2018)</td><td>1.20</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td></td></tr><tr><td>Xiao et al. (2018b)</td><td>100</td><td>17.61</td><td>22.51</td><td>29.26</td><td>28.71</td><td>23.51</td><td>26.67</td><td>4.13</td><td>1.39</td><td>0.470</td></tr><tr><td>cAdv1</td><td>100</td><td>52.33</td><td>47.78</td><td>76.17</td><td>36.28</td><td>50.50</td><td>61.95</td><td>12.06</td><td>11.62</td><td>0.427</td></tr><tr><td>cAdv2</td><td>99.89</td><td>46.61</td><td>42.78</td><td>72.56</td><td>34.28</td><td>46.45</td><td>59.00</td><td>17.39</td><td>19.4</td><td>0.437</td></tr><tr><td>cAdv4</td><td>99.83</td><td>42.61</td><td>38.39</td><td>69.67</td><td>34.34</td><td>40.78</td><td>54.62</td><td>14.13</td><td>12.5</td><td>0.473</td></tr><tr><td>cAdv8</td><td>99.81</td><td>38.22</td><td>36.62</td><td>67.06</td><td>31.67</td><td>37.67</td><td>49.17</td><td>6.52</td><td>10.04</td><td>0.476</td></tr><tr><td>tAdv250</td><td>99.00</td><td>32.89</td><td>62.79</td><td>89.74</td><td>54.94</td><td>38.92</td><td>40.57</td><td>10.9</td><td>2.10</td><td>0.433</td></tr><tr><td>tAdv250</td><td>100</td><td>36.33</td><td>67.68</td><td>94.11</td><td>58.92</td><td>42.82</td><td>44.56</td><td>15.21</td><td>4.6</td><td>0.425</td></tr><tr><td>tAdv1000</td><td>99.88</td><td>31.49</td><td>52.69</td><td>90.52</td><td>51.24</td><td>34.85</td><td>39.68</td><td>19.12</td><td>5.59</td><td>0.412</td></tr><tr><td>tAdv1000</td><td>100</td><td>35.23</td><td>61.40</td><td>93.18</td><td>56.31</td><td>39.66</td><td>45.59</td><td>22.28</td><td>6.94</td><td>0.406</td></tr></table>",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 2: Comparison against defense models. Misclassification rate after passing different adversarial examples through the defense models (higher means the attack is stronger). All attacks are performed on ResNet50 with whitebox attack. The highest attack success rate is in bold. We also report the user preference scores from AMT for each attack (last column). ",
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+    {
+        "type": "text",
+        "text": "by a change in color without a corresponding change in luminance. As most color changes occur along edges in natural images, it is likely that classifiers trained on ImageNet have never seen isoluminant colors. This suggests that cAdv might be exploiting isoluminant colors to fool classifiers. ",
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+    {
+        "type": "text",
+        "text": "4.2 tADV ATTACK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "tAdv successfully fools the classifiers with a very small weighted adversarial cross-entropy objective $( \\beta )$ when combined with texture loss, while remaining realistic to humans. As shown in Table 1, our attacks are highly successful on white-box attacks tested on three different models with the nearest neighbor texture transfer approach. We also show our attacks are more transferable to other models. In our Appendix, we show ablation results for tAdv attacks along with other strategies that we used for generating tAdv adversarial examples. ",
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+    {
+        "type": "text",
+        "text": "Structured tAdv Perturbations. Since we extract features across different layers of VGG, the tAdv perturbations follow a textural pattern. They are more structured and organized when compared to others. Our tAdv perturbations are big when compared with existing attack methods in ${ \\mathcal { L } } _ { p }$ norm. They are of high-frequency and yet imperceptible (see Fig. 1 and Fig. 8). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Importance of Texture in Classification. Textures are crucial descriptors for image classification and Imagenet trained models can be exploited by altering the texture. Their importance is also shown in the recent work from Geirhos et al. (2018). Our results also shows that even with a small or invisible change in the texture field can break the current state of the art classifiers. ",
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+    {
+        "type": "text",
+        "text": "4.3 DEFENSE AND TRANSFERABILITY ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We test all our attacks and other existing methods with images attacked from Resnet50. We evaluate them on three defenses – JPEG defense (Das et al., 2017), feature squeezing (Xu et al., 2017) and adversarial training. By leveraging JPEG compression and decompression, adversarial noise may be removed. We tested our methods against JPEG compression of 75. Feature squeezing is a family of simple but surprisingly effective strategies, including reducing color bit depth and spatial smoothing. Adversarial training has been shown as an effective but costly method to defend against adversarial attacks. Mixing adversarial samples into training data of a classifier improves its robustness without affecting the overall accuracy. We were able to obtain an adversarially pretrained Resnet152 model on ImageNet dataset and hence we tested our Resnet50 attacked images with this model. ",
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+    {
+        "type": "text",
+        "text": "Robustness. In general, our attacks are more robust to the considered defenses and transferable for targeted attacks. For cAdv, there is a trade-off between more realistic colors (using more hints and sampling from more clusters) and attack robustness. From Table 1 and 2, we show that as we progressively use more clusters, our transferability and defense numbers drop. A similar trend is observed with the change in the number of hints. cAdv is robust to JPEG defense and adversarial training because of their large and spatially smooth perturbations. For tAdv, increasing texture weight $( \\alpha )$ does not necessarily perform well with the defense even though it increases attack success rate, but increasing texture flow with more iterations improves attack’s robustness against defenses. ",
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+    {
+        "type": "text",
+        "text": "5 HUMAN PERCEPTUAL STUDIES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "To quantify how realistic tAdv and cAdv examples are, we conducted a user study on Amazon Mechanical Turk (AMT). We follow the same procedure as described in (Zhang et al., 2016; Xiao et al., 2018b). For each attack, we choose the same 200 adversarial images and their corresponding benign ones. During each trial, one random adversarial-benign pair appears for three seconds and workers are given five minutes to identify the realistic one. Each attack has 600 unique pairs of images and each pair is evaluated by at least 10 unique workers. We restrict biases in this process by allowing each unique user up to 5 rounds of trials and also ignore users who complete the study in less than 30 seconds. In total, 598 unique workers completed at least one round of our user study. For each image, we can then calculate the user preference score as the number of times it is chosen divided by the number of times it is displayed. 0.5 represents that users are unable to distinguish if the image is fake. For cAdv and tAdv, user preferences averages at 0.476 and 0.433 respectively, indicating that workers have a hard time distinguishing them. The user preferences for all attacks are summarized in Table 2 and their comparison with ${ \\mathcal { L } } _ { p }$ norm is in Table 4 and Table 5. ",
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "6 ATTACKING CAPTIONING MODEL ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our methods are general and can be easily adapted for other learning tasks. As proof of concept, we test our attacks against image captioning task. Image captioning is the task of generating a sequence of word description for an image. The popular architecture for captioning is a Long-ShortTerm-Memory (LSTM) (Hochreiter & Schmidhuber, 1997) based models (Karpathy & Fei-Fei, 2015; Wang et al., 2017). Recently, (Aneja et al., 2018) proposed a convolutional based captioning model for a fast and accurate caption generation. This convolutional based approach does not suffer from the commonly known problems of vanishing gradients and overly confident predictions of LSTM network. Therefore, we choose to attack the current state of the art convolutional captioning model. We randomly selected images from MSCOCO (Lin et al., 2014) for image captioning attack. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Attacking captioning models is harder than attacking classifiers when the goal is to change exactly one word in the benign image’s caption unlike pixel based attacks (Chen et al., 2017; Xu et al., 2019). We show that our attacks are successful and have no visible artifacts even for this challenging task. In Fig. 6, we change the second word of the caption to dog while keeping the rest of the caption the same. This is a challenging targeted attack because, in many untargeted attacks, the resulted captions do not make sense. More examples are in our Appendix. ",
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+        "page_idx": 8
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+    {
+        "type": "image",
+        "img_path": "images/6a956e0df34f6db0ac9fd6dbff2b885fc1fd389ac877e7c3c15a104e82cdbac7.jpg",
+        "image_caption": [
+            "Figure 6: Captioning attack. Top: cAdv; Bottom: tAdv. We attack the second word to dog and show the corresponding change in attention mask of that word. More examples in Appendix. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Adversarial Cross-Entropy Objective for Captioning. Let $t$ be the target caption, $w$ denote the word position of the caption, $\\mathcal { F }$ for the captioning model, $I _ { v }$ for the victim image and $J _ { a d v }$ for the cross-entropy loss ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/102810fa5337660b10267a4e844284b9a19c65863d3a2262348535571d3fbbdb.jpg",
+        "text": "$$\nL _ { c a p t } ^ { \\mathcal { A } } = \\sum _ { w } J _ { a d v } ( ( \\mathcal { F } ( I _ { v } ) ) _ { w } , t _ { w } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For cAdv, we give all color hints and optimize to get an adversarial colored image to produce target caption. For tAdv, we add Eqn 6 to Eqn 4 to optimize the image. We select $T _ { S }$ as the nearest neighbor of the victim image from the ones in the adversarial target class using ImageNet dataset. We stop our attack once we reach the target caption and the caption does not change in consecutive iterations. Note we do not change the network weights, we only optimize hints and mask (for cAdv) or the victim image (for tAdv) to achieve our target caption. ",
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+    {
+        "type": "text",
+        "text": "7 RELATED WORK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Here we briefly summarize existing unrestricted and semantic adversarial attacks. Xiao et al. (2018b) proposed geometric or spatial distortion of pixels in image to create adversarial examples. They distort the input image by optimizing pixel flow instead of pixel values to generate adversarial examples. While this attack leads to “natural” looking adversarial examples with large $\\mathcal { L } _ { \\infty }$ norm, it does not take image semantics into account. Song et al (2018) and Dunn et al. (2019) considered GANs for adversarial attacks. This attack is unrestricted in ${ \\mathcal { L } } _ { p }$ norm but they are restricted to simple datasets as it involves training GANs, which have been known to be unstable and computationally intensive for complex datasets like ImageNet (Karras et al., 2017; Brock et al., 2018). ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Hosseini & Poovendran (2018), changes the hue & saturation of an image randomly to create adversarial examples. It is similar to cAdv as they both involve changing colors, however, their search space is limited to two dimensions and their images are unrealistic, Appendix (Fig. 10). Also, while this method has a non-trivial untargeted attack success rate, it performs extremely poorly for targeted attacks $( 1 . 2 0 \\%$ success rate in our own experiments on ImageNet). Our work is also related to Joshi et al. (2019) and Qiu et al. (2019), who manipulate images conditioned on face dataset attributes like glasses, beard for their attacks. These work focuses on changing single image visual attribute and are conditionally dependent. Our work focuses on changing visual semantic descriptors to misclassify images and are not conditioned to any semantic attributes. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/e2c43dc6804b3a03ec4cb6a333e957ee2973eb5a70e083f308e2049dae343ffa.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Semantic Based</td><td></td><td>Unrestricted Photorealistic</td><td>Explainable (eg color affinity)</td><td>Complex Dataset</td><td>Caption Attack</td></tr><tr><td>Kurakin et al. (2016)</td><td>X</td><td>xx&lt;&lt;√</td><td>&lt;&lt;xx&gt;</td><td>X</td><td>√</td><td>X</td></tr><tr><td>Carlini &amp; Wagner (2017)</td><td>X</td><td></td><td></td><td></td><td>√</td><td></td></tr><tr><td>Hosseini &amp; Poovendran (2018)</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Song et al (2018)</td><td>X</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Xiao et al. (2018b)</td><td>X</td><td></td><td></td><td>xxx&gt;</td><td>xx&gt;</td><td></td></tr><tr><td>cAdv (ours)</td><td>√</td><td>√</td><td>√</td><td>√</td><td>√</td><td>xxxx/</td></tr><tr><td>tAdv (ours)</td><td>√</td><td>√</td><td></td><td>√</td><td>√</td><td>√</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 3: Summary of the difference in our work compared to previous work. Unlike previous attack methods, our attacks are unbounded, semantically motivated, realistic, highly successful, and scales to more complex datasets and other ML tasks. They are also robust against tested defenses. ",
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+    {
+        "type": "text",
+        "text": "8 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Our proposed two novel unrestricted semantic attacks shed light on the role of texture and color fields in influencing DNN’s predictions. They not only consistently fool human subjects but in general are harder to defend against. We hope by presenting our methods, we encourage future studies on unbounded adversarial attacks, better metrics for measuring perturbations, and more sophisticated defenses. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENTS ",
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+    {
+        "type": "text",
+        "text": "We thank Chaowei Xiao for sharing their code to compare our methods with Xiao et al. (2018b) and helping us setup the user study. We also thank Tianyuan Zhang for providing the AdvRes152 pretrained model. This work was supported by NSF Grant No. 1718221 and ONR MURI Award N00014-16-1-2007. ",
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+        "text": "A APPENDIX ",
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+        "text": "A.1 OTHER DETAILS ON HUMAN STUDY ",
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+    {
+        "type": "text",
+        "text": "We also chose BIM (Kurakin et al., 2016) and CW (Carlini & Wagner, 2017) for comparing our perturbations. Since these attacks are known to have low ${ \\mathcal { L } } _ { p }$ norm, we designed an aggressive version of BIM by relaxing its $\\mathcal { L } _ { \\infty }$ bound to match the norm of our attacks. We settled with two aggressive versions of BIM with average $\\mathcal { L } _ { \\infty } = \\{ 0 . 2 1 , 0 . 3 4 7 \\}$ , which we refer to as $\\mathrm { B I M _ { 0 . 2 1 } }$ , $\\mathrm { B I M _ { 0 . 3 4 } }$ . The average user preferences for BIM drops drastically from 0.497 to 0.332 when we relax the norm to $\\mathrm { B I M _ { 0 . 3 4 } }$ ; the decrease in user preferences for tAdv (0.433 to 0.406) and cAdv (0.476 to 0.437) is not significant. In Fig. 7, we plot a density plot of $\\mathcal { L } _ { \\infty }$ vs user preference scores. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/d7f388370788c4cbfe65811264c0744fc1d3e1e06e6b15bc94cfad5c0b602a11.jpg",
+        "image_caption": [
+            "Figure 7: Density Plot. Our methods achieve large $\\mathcal { L } _ { \\infty }$ norm perturbations without notable reduction in user preference. Each plot is a density plot between perturbation ( $\\scriptstyle { \\mathcal { L } } _ { \\infty }$ norm) on $\\mathbf { X }$ axis and $P r$ (user prefers adversarial image) on Y axis. For ideal systems, the density would be a concentrated horizontal line at 0.5. All plots are on the same set of axes. On the left, plots for three baseline methods (Fig a - Fig c). Note the very strong concentration on small norm perturbations, which users like. Right 4 plots shows our methods (Fig d - Fig g). Note strong push into large norm regions, without loss of user preference. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/c59de58b09ddc1adb8e06cf2da10986e190056b448a07bce2b6d6064476488ab.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 4: User Study. User preference score and ${ \\mathcal { L } } _ { p }$ norm of perturbation for different attacks. cAdv and tAdv are imperceptible for humans (score close to 0.5) even with very big ${ \\mathcal { L } } _ { p }$ norm perturbation. "
+        ],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>Lo     L2    L8</td><td rowspan=1 colspan=1>Preference</td></tr><tr><td rowspan=1 colspan=1>cw</td><td rowspan=1 colspan=1>0.587 0.026 0.054</td><td rowspan=1 colspan=1>0.506</td></tr><tr><td rowspan=1 colspan=1>BIM0.051</td><td rowspan=1 colspan=1>0.826 0.030 0.051</td><td rowspan=2 colspan=1>0.4970.3550.332</td></tr><tr><td rowspan=1 colspan=1>BIM0.21BIM0.347</td><td rowspan=1 colspan=1>0.893  0.118  0.210.892 0.119 0.347</td></tr><tr><td rowspan=2 colspan=1>cAdv2cAdv4cAdv8</td><td rowspan=1 colspan=1>0.910 0.098  0.489</td><td rowspan=2 colspan=1>0.4370.4730.476</td></tr><tr><td rowspan=1 colspan=1>0.898  0.071  0.4480.896 0.059 0.425</td></tr><tr><td rowspan=4 colspan=1>tAdv250tAdvi000</td><td rowspan=1 colspan=1>0.891  0.041  0.198</td><td rowspan=1 colspan=1>0.433</td></tr><tr><td rowspan=1 colspan=1>0.931 0.056 0.237</td><td rowspan=1 colspan=1>0.412</td></tr><tr><td rowspan=1 colspan=1>0.916 0.061  0.258</td><td rowspan=1 colspan=1>0.425</td></tr><tr><td rowspan=1 colspan=1>0.946  0.08  0.315</td><td rowspan=1 colspan=1>0.406</td></tr></table>",
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+        "type": "table",
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+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Attack</td><td></td><td>tarantula</td><td>merganser</td><td>nautilus</td><td>hyena</td><td>beacon</td><td>golfcart</td><td>photocopier</td><td>umbrella</td><td>pretzel</td><td>sandbar</td><td>Mean</td></tr><tr><td rowspan=\"5\">BIM</td><td>Lo</td><td>0.82</td><td>0.81</td><td>0.80</td><td>0.88</td><td>0.80</td><td>0.79</td><td>0.86</td><td>0.80</td><td>0.85</td><td>0.84</td><td>0.83</td></tr><tr><td>L2</td><td>0.05</td><td>0.02</td><td>0.03</td><td>0.04</td><td>0.02</td><td>0.01</td><td>0.04</td><td>0.02</td><td>0.02</td><td>0.04</td><td>0.03</td></tr><tr><td>L8</td><td>0.07</td><td>0.04</td><td>0.06</td><td>0.06</td><td>0.04</td><td>0.03</td><td>0.07</td><td>0.04</td><td>0.04</td><td>0.06</td><td>0.05</td></tr><tr><td>Preference</td><td>0.49</td><td>0.75</td><td>0.44</td><td>0.67</td><td>0.48</td><td>0.48</td><td>0.27</td><td>0.39</td><td>0.38</td><td>0.63</td><td>0.50</td></tr><tr><td>L0</td><td>0.54</td><td>0.55</td><td>0.52</td><td>0.77</td><td>0.48</td><td>0.42</td><td>0.70</td><td>0.48</td><td>0.69</td><td>0.71</td><td>0.59</td></tr><tr><td rowspan=\"5\">CW</td><td>L</td><td>0.05</td><td>0.01</td><td>0.03</td><td>0.03</td><td>0.02</td><td>0.01</td><td>0.04</td><td>0.01</td><td>0.02</td><td>0.04</td><td>0.03</td></tr><tr><td>L</td><td>0.08</td><td>0.04</td><td>0.06</td><td>0.07</td><td>0.04</td><td>0.03</td><td>0.07</td><td>0.04</td><td>0.05</td><td>0.06</td><td>0.05</td></tr><tr><td>Preference</td><td>0.51</td><td>0.68</td><td>0.43</td><td>0.59</td><td>0.52</td><td>0.51</td><td>0.39</td><td>0.40</td><td>0.42</td><td>0.62</td><td>0.51</td></tr><tr><td></td><td>0.94</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>L L</td><td>0.07</td><td>0.95</td><td>0.91</td><td>0.94 0.06</td><td>0.89</td><td>0.94</td><td>0.89</td><td>0.91</td><td>0.94</td><td>0.85</td><td>0.92</td></tr><tr><td rowspan=\"6\">tAdv250</td><td>L8</td><td></td><td>0.08</td><td>0.08</td><td></td><td>0.05</td><td>0.06</td><td>0.03</td><td>0.05</td><td>0.07</td><td>0.05</td><td>0.06</td></tr><tr><td></td><td>0.28</td><td>0.31</td><td>0.30</td><td>0.31</td><td>0.22</td><td>0.27</td><td>0.18</td><td>0.25</td><td>0.27</td><td>0.19</td><td>0.26</td></tr><tr><td>Preference</td><td>0.39</td><td>0.58</td><td>0.43</td><td>0.53</td><td>0.46</td><td>0.35</td><td>0.25</td><td>0.40</td><td>0.33</td><td>0.54</td><td>0.43</td></tr><tr><td>L0</td><td>0.90</td><td>0.90</td><td>0.88</td><td>0.90</td><td>0.90</td><td>0.90</td><td>0.90</td><td>0.89</td><td>0.93</td><td>0.88</td><td>0.90</td></tr><tr><td>L</td><td>0.06</td><td>0.06</td><td>0.07</td><td>0.05</td><td>0.08</td><td>0.07</td><td>0.08</td><td>0.08</td><td>0.09</td><td>0.06</td><td>0.07</td></tr><tr><td>L8 Preference</td><td>0.36 0.46</td><td>0.44 0.65</td><td>0.41 0.46</td><td>0.31 0.58</td><td>0.48 0.45</td><td>0.51 0.48</td><td>0.55 0.30</td><td>0.56 0.43</td><td>0.46 0.35</td><td>0.39 0.59</td><td>0.45 0.47</td></tr></table>",
+        "bbox": [
+            173,
+            710,
+            825,
+            872
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Table 5: Class wise ${ \\mathcal { L } } _ { p }$ norm and user preference breakdown. Users are biased and pick a few classes quite often (merganser, sandbar), and do not like a few classes (photocopier) over others. ",
+        "bbox": [
+            173,
+            882,
+            821,
+            911
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/0a960c25a339bfbdfd4e118211a67d2ceac05684f40c127d8920823ba900ff03.jpg",
+        "image_caption": [
+            "Figure 8: Perturbation comparisons. Images are attacked from tarantula to beacon, golf cart, nautilus, photocopier, pretzel from left to right. Our perturbations (cAdv and tAdv) are large, structured and have spatial patterns when compared with other attacks. Perturbations from cAdv are low-frequency and locally smooth while perturbations from tAdv are primarily high-frequency and structured. Note gray color indicates no perturbations. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            223,
+            94,
+            772,
+            833
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/d0d387c0dcec061636b182102c8c415a7fd77b3743afef98fb38dc20448785e0.jpg",
+        "table_caption": [
+            "A.2 ADDITIONAL RESULTS "
+        ],
+        "table_footnote": [
+            "Whitebox target attack success rate. Our attacks are highly successful on different models across all strategies. tAdv results are for $\\alpha = 2 5 0$ , $\\beta = 1 0 ^ { - 3 }$ and iter $= 1$ . "
+        ],
+        "table_body": "<table><tr><td colspan=\"2\">Model</td><td>Resnet50</td><td>Dense121</td><td>VGG19</td></tr><tr><td colspan=\"2\">Accuracy</td><td>76.15</td><td>74.65</td><td>74.24</td></tr><tr><td rowspan=\"7\">Sseeess Arkea</td><td>Random Ts</td><td>99.67</td><td>99.72</td><td>96.16</td></tr><tr><td>Random Target Ts</td><td>99.72</td><td>99.89</td><td>99.94</td></tr><tr><td>Nearest Target Ts</td><td>97.99</td><td>99.72</td><td>99.50</td></tr><tr><td>cAdv4 25 hints</td><td>99.78</td><td>99.83</td><td>99.93</td></tr><tr><td>cAdv4 50 hints</td><td>99.78</td><td>99.83</td><td>100.00</td></tr><tr><td>cAdv4 100 hints</td><td>99.44</td><td>99.50</td><td>99.93</td></tr></table>",
+        "bbox": [
+            173,
+            143,
+            517,
+            243
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/b32659907e3942f3f1b389f1349e6b14b4af30afcb464df9e3a8a58600fc4388.jpg",
+        "table_caption": [
+            "Table 6: Ablation Studies. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>α β</td><td>250</td><td>500</td><td>750</td><td>1000</td></tr><tr><td>0</td><td>25.00</td><td>99.61</td><td>98.55</td><td>95.92</td></tr><tr><td>10-4</td><td>99.88</td><td>99.61</td><td>98.55</td><td>95.92</td></tr><tr><td>10-3</td><td>97.99</td><td>99.27</td><td>99.66</td><td>99.50</td></tr><tr><td>10-2</td><td>96.26</td><td>95.42</td><td>96.32</td><td>96.59</td></tr></table>",
+        "bbox": [
+            521,
+            132,
+            782,
+            214
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "tAdv ablation study. Whitebox target success rate with nearest target $T _ { s }$ (texture source). In columns, we have increasing texture weight $( \\alpha )$ and in rows, we have increasing adversarial cross-entropy weight $( \\beta )$ . All attacks are done on Resnet50. ",
+        "bbox": [
+            526,
+            214,
+            779,
+            310
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/338ce3b799792303bb653b8fdcfc46b5d5c524075d602fed70070bccd9adb34d.jpg",
+        "image_caption": [
+            "Figure 9: Additional qualitative examples for controlling cAdv. We show a comparison of sampling 50 color hints from k clusters with low-entropy. All images are attacked to golf-cart. Even numbered rows visualize our cluster segments, with darker colors representing higher mean entropy and red dots representing the location we sample hints from. Sampling hints across more clusters gives less color variety. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            212,
+            353,
+            787,
+            830
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/c5de302b9eb8610441b2904e8ef667e2c43474007fbdae082a0c017ff61d4dbb.jpg",
+        "image_caption": [
+            "Figure 10: Images attacked with the method of (Hosseini & Poovendran, 2018) are not realistic, as these examples show. Compare Fig 12, showing results of our color attack. Similar qualitative results for CIFAR-10 are visible in Hosseini & Poovendran (2018). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            148,
+            825,
+            337
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/92288996c16f4cc7fda4076186b63e1a00ce1168d673775295b2b020c53eed77.jpg",
+        "image_caption": [
+            "Figure 11: Additional qualitative examples for tAdv. Texture transferred from random images (row 1), random images from adversarial target class (row 2) and from the nearest neighbor of the victim image from the adversarial target class (row 3). All examples are misclassified from Merganser to Umbrella. Images in the last row look photo realistic, while those in the first two rows contain more artifacts as the texture weight $\\alpha$ increases (left to right). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            483,
+            823,
+            803
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/c4db4a0b5a7776b50c75b3f1b22a265b4b7fe14078b1bff2faa5f499cec851d8.jpg",
+        "image_caption": [
+            "Figure 12: Captioning attack. We attack the second word of each caption to {dog, bird} and show the corresponding change in attention mask of that word. For tAdv we use the nearest neighbor selection method and for cAdv we initialize with all groundtruth color hints. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            189,
+            199,
+            808,
+            775
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/8c8fbb3896dd6b66b4180dcd97895bec534939a21f6e28428e484a34ea71d246.jpg",
+        "image_caption": [
+            "Figure 13: Randomly sampled, semantically manipulated, unrestricted adversarial examples and their perturbations. (a) Adversarial examples generated by cAdv attacking Hints and mask with 50 hints. (c) Adversarial examples generated by tAdv with $\\alpha = 2 5 0$ , $\\beta = \\bar { 1 0 } ^ { - 3 }$ and iter $= 1$ using nearest target $T _ { s }$ . (b) and (d) are their respective perturbations. Note that diagonal images are groundtruth images and gray pixels indicate no perturbation. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            207,
+            222,
+            789,
+            726
+        ],
+        "page_idx": 18
+    }
+]

parse/train/SyuWNMZ0W/SyuWNMZ0W.md

@@ -0,0 +1,202 @@
+# DIRECTING GENERATIVE NETWORKS WITH WEIGHTED MAXIMUM MEAN DISCREPANCY
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+The maximum mean discrepancy (MMD) between two probability measures $\mathbb { P }$ and $\mathbb { Q }$ is a metric that is zero if and only if all moments of the two measures are equal, making it an appealing statistic for two-sample tests. Given i.i.d. samples from $\mathbb { P }$ and $\mathbb { Q }$ , Gretton et al. (2012) show that we can construct an unbiased estimator for the square of the MMD between the two distributions. If $\mathbb { P }$ is a distribution of interest and $\mathbb { Q }$ is the distribution implied by a generative neural network with stochastic inputs, we can use this estimator to train our neural network. However, in practice we do not always have i.i.d. samples from our target of interest. Data sets often exhibit biases—for example, under-representation of certain demographics—and if we ignore this fact our machine learning algorithms will propagate these biases. Alternatively, it may be useful to assume our data has been gathered via a biased sample selection mechanism in order to manipulate properties of the estimating distribution $\mathbb { Q }$ .
+
+In this paper, we construct an estimator for the MMD between $\mathbb { P }$ and $\mathbb { Q }$ when we only have access to $\mathbb { P }$ via some biased sample selection mechanism, and suggest methods for estimating this sample selection mechanism when it is not already known. We show that this estimator can be used to train generative neural networks on a biased data sample, to give a simulator that reverses the effect of that bias.
+
+# 1 INTRODUCTION
+
+Neural networks with stochastic input layers can be trained to approximately sample from an arbitrary probability distribution $\mathbb { P }$ based on samples from $\mathbb { P }$ (Goodfellow et al., 2014). Generating simulations from complex distributions has applications in a large number of fields: We can automatically generate illustrations for text (Zhang et al., 2017) or streams of video (Vondrick et al., 2016); we can simulate novel molecular fingerprints to aid scientific exploration (Kadurin et al., 2017); and, we can synthesize medical time-series data that can be shared without violating patient privacy (Esteban et al., 2017).
+
+In this paper, we consider the setting of a feedforward neural network (referred to as the generator) that maps random noise inputs $z \in \mathbb { R } ^ { d }$ to some observation space $\mathcal { X }$ . The weights of the neural network are trained to minimize some loss function between the resulting simulations and exemplars of real data. The general form of the resulting distribution $\mathbb { Q }$ over simulations $G ( z )$ is determined by the architecture of the generator—which governs the form of the mapping $G$ —and by the loss function used to train the generator. Generative adversarial networks (Goodfellow et al., 2014) use dynamically varying, adversarially learned loss functions specified in terms of the output of a classifier. Other generative networks use a loss function defined using a distributional distance or divergence between the simulation distribution $\mathbb { Q }$ and a target distribution $\mathbb { P }$ (Arjovsky et al., 2017; Nowozin et al., 2016; Zhao et al., 2017a), requiring the generator to mimic the variance in a collection of data points rather than simply converge to a single mode. In particular, the maximum mean discrepancy (Gretton et al., 2012) has demonstrated good performance as a loss function in this setting (Sutherland et al., 2016; Li et al., 2017; Sutherland et al., 2017), since it reduces to zero if and only if all moments of two distributions are equal, requiring the generator to reproduce the full range of variation found in the data.
+
+These approaches, like most machine learning methods, assume our data is a representative sample from the distribution of interest. If this assumption is correct, minimizing the distributional distance between the simulations and the data is equivalent to learning a distribution that is indistinguishable under an appropriate two-sample test from our target distribution. However, we run into problems if our data is not in fact a representative sample from our target distribution—for instance, if our data gathering mechanism is susceptible to sample selection bias. The problem of machine learning algorithms replicating and even magnifying human biases is gathering increasing awareness (Bolukbasi et al., 2016; Zhao et al., 2017b), and if we believe our dataset suffers from such biases—for example, if our audio dataset contains primarily male speakers or our image dataset contains primarily white faces – we will typically want to take steps to correct this bias.
+
+Even if our data is representative of the underlying distribution, we might want to generate samples from a modified version of this distribution. For example, we might want to alter the demographics of characters in a scene to fit a story-line. In this setting, we can treat our desired modified distribution as our target distribution, and treat our data as if they were sampled from this distribution subject to an appropriately biased sample selection mechanism.
+
+If we know the form of our sample selection bias, we can reformulate our loss function to penalize the generator based on the difference between simulated data and the unbiased distribution of interest, which we will refer to as our target distribution. After a review of relevant background information in Section 2, we show in Section 3 that, given a function that describes how our observed data deviates from this target distribution, we can construct an estimator of the MMD between the generator and the target distribution.
+
+In practice, we will not know the function linking the target distribution and the empirical data distribution. However, we can approximate this function based on user-provided examples of data points that are over- and under-represented. In Section 4, we discuss ways to estimate this function, and in Section 5 we discuss related work in survey sampling statistics and bias reduction. We demonstrate the efficacy and applicability of our approach in Section 6.
+
+# 2 BACKGROUND
+
+# 2.1 GENERATIVE NETWORKS
+
+Generative networks are a class of models which take a collection of data lying in some observation space $\mathcal { X }$ as input, and aim to generate simulations that are similar to that data – or more generally, whose empirical distribution is close to the distribution underlying the data. These models do not use or require an explicit probability distribution for the data, but instead rely on the fact that sufficiently complex neural networks have the capacity to approximate any arbitrary function (Hornik et al., 1989). We can therefore construct a method of simulating from an arbitrarily complex probability distribution $\mathbb { Q }$ on $\mathcal { X }$ by using a neural network generator $G : \mathbb { R } ^ { d }  \mathcal { X }$ to transform random $d$ - dimensional inputs $z$ into simulations $G ( z ) \in \mathcal { X }$ . In order to minimize the difference between the probability distribution $\mathbb { Q }$ over simulations $G ( z )$ and the target distribution $\mathbb { P }$ , we train the neural network to minimize a loss function between simulations $G ( \bar { z } ) \sim \mathbb { Q }$ and data $x \sim \mathbb { P }$
+
+The most common forms of generative network are generative adversarial networks (GANs, Goodfellow et al., 2014), so called because the loss function is dynamically defined in terms of the output of an adversarially learned classifier. This classifier—itself a neural network—is trained to differentiate between two classes, data and simulations, and for a given observation returns a score under the two classes (loosely corresponding to a probability of belonging to that class). The generator’s loss function is a function of the score assigned to simulations under the true data class, so that reducing the loss function leads to an increased chance of fooling the classifier.
+
+While this adversarial framework has the advantage of a dynamically evolving and increasingly discriminative loss function, a disadvantage is that the generator can successfully minimize this loss function by mimicking only a subset of the data, leading to a phenomenon known as mode collapse (Salimans et al., 2016; Che et al., 2017). To avoid this, recent works have incorporated estimators of distributional distances between $\mathbb { P }$ and $\mathbb { Q }$ , that consider the overall distributions rather than just element-wise distances between samples. For example, the maximum mean discrepancy (Gretton et al., 2012; Sutherland et al., 2016; Li et al., 2017; Sutherland et al., 2017) between two distributions, which we explore further in Section 2.2, reduces to zero if and only if all moments of the two distributions are the same. Other distributional distances that have been used include the Wasserstein distance and the Cramer distance (Arjovsky et al., 2017; Bellemare et al., 2017). In some cases, these distance-based generative networks include adversarial components in their loss functions; for example the MMD-GAN of Li et al. (2017) adversarially learns parameters of the maximum mean discrepancy metric.
+
+# .2 MAXIMUM MEAN DISCREPANCY BETWEEN TWO DISTRIBUTIONS
+
+The maximum mean discrepancy (MMD, Gretton et al., 2012) projects two distributions $\mathbb { P }$ and $\mathbb { Q }$ into a reproducing kernel Hilbert space (RKHS) $\mathcal { H }$ , and looks at the maximum mean distance between the two projections, i.e.
+
+$$
+\mathbf { M M D } [ \mathbb { P } , \mathbb { Q } ] : = \operatorname* { s u p } _ { f \in \mathcal { H } } \left( { \mathbf { E } } _ { x \sim \mathbb { P } } [ f ( x ) ] - { \mathbf { E } } _ { y \sim \mathbb { Q } } [ f ( y ) ] \right) .
+$$
+
+If we specify the kernel mean embedding $\mu _ { \mathbb { P } }$ of $\mathbb { P }$ as $\begin{array} { r } { \mu _ { \mathbb { P } } = \int k ( x , \cdot ) d \mathbb { P } ( x ) } \end{array}$ , where $k ( \cdot , \cdot )$ is the characteristic kernel defining the RKHS, then we can write the square of this distance as
+
+$$
+\mathbf { M M D } ^ { 2 } [ \mathbb { P } , \mathbb { Q } ] = | | \mu _ { \mathbb { P } } - \mu _ { \mathbb { Q } } | | _ { \mathcal { H } } ^ { 2 } = \mathbb { E } _ { \mathbb { P } } [ k ( x , x ^ { \prime } ) ] - 2 \mathbb { E } _ { \mathbb { P } , \mathbb { Q } } [ k ( x , y ) ] + \mathbb { E } _ { \mathbb { Q } } [ k ( y , y ^ { \prime } ) ] .
+$$
+
+Since we have projected the distributions into an infinite-dimensional space, the distance between the two distributions is zero if and only if all their moments are the same.
+
+In order to be a useful loss function for training a neural network, we must be able to estimate the MMD from data, and also take derivatives of this estimate with respect to the network parameters. We can construct an unbiased estimator of square of the MMD (Gretton et al., 2012) using $m$ samples $x _ { i } \sim \mathbb { P }$ and $n$ samples $y _ { i } \sim \mathbb { Q }$ as
+
+$$
+\begin{array} { r l r } { \widehat { \boldsymbol { \boldsymbol { \mathrm { M M D } } } } ^ { 2 } [ \mathbb { P } , \mathbb { Q } ] = } & { = } & { \displaystyle \frac { 1 } { m ( m - 1 ) } \sum _ { i = 1 } ^ { m } \sum _ { j \neq i } ^ { m } k ( x _ { i } , x _ { j } ) + \frac { 1 } { n ( n - 1 ) } \sum _ { i = 1 } ^ { n } \sum _ { j \neq i } ^ { n } k ( y _ { i } , y _ { j } ) } \\ & { } & { \displaystyle - \frac { 2 } { m n } \sum _ { i = 1 } ^ { m } \sum _ { j = 1 } ^ { n } k ( x _ { i } , y _ { j } ) . } \end{array}
+$$
+
+If $\mathbb { P }$ is the distribution underlying our data and $\mathbb { Q }$ is an approximating distribution represented using a neural network, we can differentiate the individual kernel terms in Equation 1 with respect to the simulated data $y _ { i }$ , and hence (via the chain rule) with respect to the parameters of the neural networks.
+
+The MMD has been successfully used as a loss function in several generative adversarial networks. Dziugaite et al. (2015) and Li et al. (2015) propose training a feedforward neural network to minimize the MMD between simulations and data; Li et al. (2015) also propose minimizing the MMD between simulations and data that has been encoded using a pre-trained autoencoder. The generator in the MMD-GAN of Li et al. (2017) also aims to reduce the MMD between simulations and data, but learns the characteristic kernel in an adversarial manner by combining the kernel with a dynamically learned autoencoder.
+
+# 3 WEIGHTED MMD
+
+If we have unbiased samples from two distributions $\mathbb { P }$ and $\mathbb { Q }$ , the estimator described in Equation 1 gives an unbiased estimate of the MMD between those two distributions. In a generative network context, we can therefore use this estimator as a loss function in order to modify the generator associated with $\mathbb { Q }$ so that the MMD between the two distributions is minimized.
+
+However, this relies on having access to unbiased samples from our target distribution $\mathbb { P }$ . In practice, our data may have been gathered using biased sampling practices: A dataset of images of faces may over-represent white faces; datasets generated from medical experiments may over-represent male patients; datasets generated from on-campus studies may over-represent college-aged students. If our data is a biased sample of our target distribution, this estimator will estimate the difference between our simulations and the biased empirical distribution, so our simulations will recreate the biases therein.
+
+In this section, we propose an estimator for the MMD between two distributions $\mathbb { P }$ and $\mathbb { Q }$ when we have access to $\mathbb { P }$ only via some biased sample selection mechanism. Concretely, we assume that $\mathbb { P }$ is our target of interest, but our observed data are actually sampled from a related distribution $T ( x ) \mathbb { P } ( x )$ . We can think of $T ( x )$ as an appropriately scaled “thinning function” $\widetilde { T } ( x ) = Z T ( x )$ that characterizes the sample selection mechanism. In other words, we assume that candidate samples $x ^ { * }$ are sampled from $\mathbb { P }$ , and these candidates are selected into our pool with probability $\tilde { T } ( \hat { x } ^ { * } )$ ; the normalizing constant $Z$ ensures that $\begin{array} { r } { T ( x ) \mathbb { P } ( x ) = \frac { \widetilde { T } ( x ) \mathbb { P } ( x ) } { Z } } \end{array}$ is a valid probability distribution. While in the remainder of this paper we will continue to use the language of biased sample selection mechanisms, we note that this framework can also be used if our data are unbiased samples but we want to explicitly modulate our target distribution via some function $F ( x )$ so that we generate simulations from $F ( x ) \mathbb { P } ( x )$ ; in this setting we can treat the transformed target as $\mathbb { P }$ and let $T ( x ) =$ $1 / F ( x )$ .
+
+For now, we assume that our thinning function $\widetilde { T } ( x )$ is known; we discuss ways to estimate or specify it in Section 4. Our estimation problem becomes an importance sampling problem: We have samples from $T ( \cdot ) \mathbb { P } ( \cdot )$ , and we want to use these to estimate $\scriptstyle \mathbf { \dot { M } M D } ^ { 2 } [ \mathbb { P } , \mathbb { Q } ]$ , which is a function of the target distribution $\mathbb { P }$ . Importance sampling provides a method for constructing an estimator for the expectation of a function $\phi ( x )$ with respect to a distribution $\mathbb { P }$ , by taking an appropriately weighted sum of evaluations of $\phi$ at values sampled from a different distribution $\mathbb { P } ^ { \prime }$ . If we knew the normalizing constant $Z$ , we could construct an unbiased estimator of the MMD between $\mathbb { P }$ and $\mathbb { Q }$ by weighting each function evaluation associated with sample $x$ from $T ( x ) \mathbb { P } ( x )$ with the likelihood ratio $\mathbb { P } ( x ) / T ( x ) \mathbb { P } ( x ) = 1 / T ( x )$ , i.e.
+
+$$
+\begin{array} { c } { { \displaystyle \widehat { M } _ { u } = \frac { 1 } { m ( m - 1 ) } \sum _ { i = 1 } ^ { m } \sum _ { j \neq i } ^ { m } \frac { 1 } { T ( x _ { i } ) T ( x _ { j } ) } k ( x _ { i } , x _ { j } ) + \frac { 1 } { n ( n - 1 ) } \sum _ { i = 1 } ^ { n } \sum _ { j \neq i } ^ { n } k ( y _ { i } , y _ { j } ) } } \\ { { - \frac { 2 } { m n } \sum _ { i = 1 } ^ { m } \sum _ { j = 1 } ^ { n } \frac { 1 } { T ( x _ { i } ) } k ( x _ { i } , y _ { j } ) . } } \end{array}
+$$
+
+However, the normalizing constant $Z$ depends on both $\widetilde { T }$ and $\mathbb { P }$ . We will not, in general, know an analytic form for $\mathbb { P }$ , so we will not be able to calculate $Z$ . Since we will only be able to evaluate $T ( x )$ up to a normalizing constant, we cannot work with Equation 2 directly. Instead, we can construct a biased estimator $\widehat { M _ { d } }$ by using self-normalized importance weights,
+
+$$
+\widehat { M } _ { b } = \sum _ { i = 1 } ^ { m } \sum _ { j \neq i } ^ { m } w ( x _ { i } ) w ( x _ { j } ) k ( x _ { i } , x _ { j } ) + \frac { 1 } { n ( n - 1 ) } \sum _ { i = 1 } ^ { n } \sum _ { j \neq i } ^ { n } k ( y _ { i } , y _ { j } ) - \frac { 2 } { n } \sum _ { i = 1 } ^ { m } \sum _ { j = 1 } ^ { n } w ( x _ { i } ) k ( x _ { i } , y _ { j } )
+$$
+
+where $\begin{array} { r } { w ( x _ { i } ) = \frac { 1 / \widetilde { T } ( x _ { i } ) } { \sum _ { j = 1 } ^ { m } 1 / \widetilde { T } ( x _ { j } ) } } \end{array}$ .
+
+We refer to the estimator $\widehat { M _ { b } }$ in Equation 3 as the weighted MMD estimator. While this estimator is biased due to the self-normalized weights, this bias will decrease as $1 / m$ where $m$ is the number of samples from $T ( \cdot ) \mathbb { P } ( \cdot )$ . Further, this biased estimator will often have lower variance than the unbiased estimator in Equation 2 (Robert & Casella, 2004).
+
+# 4 LEARNING THE THINNING FUNCTION $\widetilde { T } ( x )$
+
+In Section 3, we assumed that our target distribution $\mathbb { P }$ is only accessible via a biased sample selection mechanism characterized by a thinning function $\widetilde { T } ( x )$ , so that our samples are actually distributed (up to a normalizing constant) according to $\widetilde { T } ( \boldsymbol { x } ) \mathbb { P } ( \boldsymbol { x } )$ . If we know the thinning function $\widetilde { T } ( x )$ that describes our sample selection mechanism, we can use Equation 3 directly in the loss function of our generator.
+
+However, in practice we will not have access to this thinning function. Rather, we are likely to be in a situation where a practitioner has noticed that one or more classes is over- or under-represented, either in the dataset or in simulations from a generative network trained on the data. If our dataset is fully labeled, we could either manually re-balance our dataset to match the target distribution, or use these labels to estimate $\tilde { T }$ by comparing the number of data points in various subsets of $\mathcal { X }$ with the expected number under our target distribution.
+
+Even if we do not have a fully labeled dataset, we might be able to label a subset of examples and use these to estimate $T ( x )$ . For example, assume we have an image dataset with more pictures of men than women, and that we select and label a random subset of this dataset. A reasonable estimate for our thinning function would be to set $T ( \mathrm { m e n } ) = 1$ and approximate $T ( \mathrm { w o m e n } )$ by the sample ratio of men to women in our labeled subset. In this simple two-class setting, to extrapolate values of $T$ across our observation space $X$ we can run a logistic regression using the labeled subset.
+
+In a more complicated problem, we may need to deploy more sophisticated regression tools, but the problem remains one of function estimation from labeled data. Given a set of labeled exemplars and user-specified estimates of the thinning function evaluated at those exemplars (which could be based on demographic statistics or domain knowledge, or desired statistics of the simulation distribution), we could learn a thinning function $T ( x )$ using techniques such as neural network function estimation or Gaussian process regression.
+
+# 5 RELATED WORK
+
+While it has received little attention in the deep learning literature, the problem of correcting for biased sample selection mechanisms is familiar in the field of survey statistics. Our estimator is related to inverse probability weighting (IPW), originally proposed by Horvitz & Thompson (1952) and still in wide use today (e.g. Mansournia & Altman, 2016). IPW is used in stratified random sampling study designs to correct parameter estimates for intentional bias in sampling. Under IPW, each data point is assigned a weight proportionate to the inverse of its selection probability, so that samples from classes which are disproportionately unlikely to be seen in the sample are assigned increased weights to counteract the undersampling of their classes. This is the same form as the weights in our sampling scheme, and serves a similar purpose, although it is not placed in the context of the MMD between two distributions and assumes that the probability of selection is known for each observation.
+
+This work also follows increased awareness of the effects of biased data on machine learning outcomes, and interest in mitigating these effects. For example, Caliskan et al. (2017) and Bolukbasi et al. (2016) explore how biases and prejudices expressed in our language manifest in the word embeddings and representations found. Zhao et al. (2017b) find that popular image labeling datasets exhibit gender stereotypes and that algorithms trained on these datasets tend to amplify these stereotypes.
+
+# 6 EVALUATION
+
+We consider two experimental settings: One where the form of the sampling bias is known, and one where it is estimated from data.
+
+# 6.1 CORRECTING FOR BIASED SAMPLE SELECTION WITH A KNOWN THINNING FUNCTION
+
+In Section 3, we discussed how, for a given thinning function $\widetilde { T } ( x )$ , we can construct an estimator for the MMD between the distribution $\mathbb { Q }$ implied by our generator and the underlying unbiased data distribution $\mathbb { P }$ . To demonstrate the efficacy of this in practice, we assume $\mathbb { P }$ to be a mixture of two Gaussians (Figure 1a), and let $\begin{array} { r } { \widetilde { T } = \frac { \mathsf { \bar { \rho } } _ { 0 . 5 } } { 1 + \exp ( 1 0 ( x - 1 ) ) } + \mathsf { \bar { 0 } } . 5 } \end{array}$ be a scaled logistic function as shown in Figure 1b. The resulting data is distributed according to Figure 1c.
+
+We construct a simple generator network taking univariate random noise as input, and consisting of six fully-connected layers, each with three nodes and exponential linear unit activations, and with a univariate output with no activation. We train this network using both the standard unbiased MMD
+
+![](images/7aacc16a56bf12eeb6be717b6613a40a417beb992777bd5af63e5666555011da.jpg)
+
+![](images/3f5c0eac359a26d961b834e96bec7b0a6c4fed4d50c6297945c38ed15c4d5107.jpg)
+
+![](images/06ae327a5fad1bf87dc87eb272572d8d59131835adeb160506ba22f947b9b10b.jpg)  
+(c) Observed distribution
+
+(a) Target distribution $x \sim \bar { \mathbb { P } } ( x ) = 0 . 5 \mathcal { N } ( 0 , 0 . 5 ) +$ $0 . 5 \mathcal { N } ( 2 , 0 . 5 )$
+
+(d) Simulations using standard MMD estimator (with PDF of $T ( x ) \mathbb { P } ( x )$ for comparison)
+
+![](images/f5def123fb06980fd1e88eb246a74795dc78e139e8f49ff7122923770c7a2b62.jpg)  
+Figure 1: Output of GANs trained on data sampled according to $T ( x ) \mathbb { P } ( x )$ . Simulation plots PDFs are computed using NumPy’s random normal sampler, mixed to create the appropriate mode ratios.
+
+![](images/a64c49077e37ff61c4666331367883dd199fc765c973539e475a5db6635f88c0.jpg)  
+(e) Simulations using weighted MMD estimator (with PDF of $\mathbb { P } ( x )$ for comparison)
+
+estimator of Equation 1, and the weighted MMD estimator proposed in Equation 3, using ADAM optimization with a learning rate of 0.001.
+
+Figures 1e and 1d show histograms of samples generated using the two estimators. We see that, as expected, the standard MMD estimator does a good job of replicating the empirical distribution of the data presented to it. The weighted MMD estimator, however, is able to replicate the target distribution $\mathbb { P }$ , even though it only has access to samples from $T ( x ) \mathbb { P } ( x )$ .
+
+# 6.2 LEARNING AND CORRECTING FOR A BIASED SAMPLE SELECTION MECHANISM
+
+In practice we are unlikely to know a functional form for $\widetilde { T } ( x )$ , the function that describes the form of the sampling bias. Here we consider the case where we must estimate this function from data. We consider as an example a dataset containing zeros and ones from the MNIST dataset. We assume our target distribution contains $50 \%$ zeros and $50 \%$ ones, but that due to a biased selection procedure our data contains $80 \%$ zeros and $20 \%$ ones.
+
+We hypothesise that the practitioner using this dataset is aware that there is a discrepancy, but does not know how to translate this into a functional form for $\widetilde { T }$ . Instead, the practitioner labels examples of each class, and supplies an estimate of how much each class is under-represented. In our example, we assume the practitioner has labeled 800 zeros and 200 ones, and estimates that ones are undersampled according to a thinning probability of 0.75. We note that this number can easily be estimated from a data sample if the practitioner knows the global ratio of zeros and ones.
+
+Table 1: Comparison of different estimators on the proportion of generated images classified as ones.   
+
+<table><tr><td>Standard MMD</td><td>Weighted MMD (initialized to standard MMD)</td><td>Weighted MMD (random initialization)</td></tr><tr><td>0.2201 ±0.0027</td><td>0.3316 ± 0.0028</td><td>0.3888 ± 0.0027</td></tr></table>
+
+![](images/c49db018f7cfa2478f872ec32baa2858bfd3b4aeba60a03d7965a5d270163485.jpg)  
+(a) Subset of real images used to train GANs (50 zeros, 14 ones)
+
+![](images/e0998658d6585d8b00805e69f5602390d4bed21b0d816186957e74df04e982a6.jpg)  
+(b) Simulations from GAN trained with weighted MMD, randomly initialized (40 zeros, 24 ones)
+
+![](images/d4667bf229d1d82687a71753d56e4725067be1a52846039cfe8874faa1cf5bf5.jpg)
+
+(c) Simulations from GAN trained with MMD (53 zeros, 11 ones)
+
+![](images/8695a4a363b021bb1b0eb97740068f60209475352eb4162f7fe864e89c64bc32.jpg)  
+(d) Simulations from GAN trained with weighted MMD, initialized to MMD (47 zeros, 17 ones)   
+Figure 2: Data used to train the GANs, and simulations from GANs trained using unweighted and weighted MMD.
+
+![](images/00331afd357bd0ed43a608eb27ddfec7d4dcc7fd8fa6aca331817beadcc138e4.jpg)  
+Figure 3: Proportion of images classified as ones, over time, for a network initially learned using the standard MMD estimator. The loss function switches to the weighted MMD estimator at iteration 10000, leading to an increase in the proportion of images classified as ones.
+
+We use an architecture based on the MMD-GAN of Li et al. (2017), which incorporates an autoencoder, and trains a generator based on the MMD between the encoded representations of the data and of the simulations. The autoencoder is simultaneously learned in an adversarial manner, and serves to optimize the kernel used in MMD to maximally discriminate between the two distributions. Our experiments are run on an adapted version of the original MMD-GAN Torch code, which will be made available after publication.
+
+Since the autoencoder provides a low-dimensional embedding of our images, we specify our thinning function $\widetilde { T } ( x )$ on the space spanned by the encoder. After each update of the autoencoder, we estimate $\widetilde { T }$ with the labeled set, using appropriately scaled logistic regression. We then replace the standard MMD estimator in the loss function with our weighted estimator, calculating using the estimated $\widetilde { T }$ .
+
+One could imagine using the weighted estimator in two scenarios. We might be already aware of the bias in our data, in which case we would simply train our GAN using the weighted MMD estimator. Alternatively, we might only become aware of the bias after already training a GAN using the standard estimator, in which case we might initialize our network weights to the resulting pretrained values. To simulate these scenarios, we trained two networks using the weighted estimator: one initialized to a pre-trained MMD-GAN, and one randomly initialized.
+
+Figure 2a shows a random subset of the data used to train the GANs. Due to the biased sample selection method, there are significantly more zeros than ones. Figure 2c shows simulations generated by minimizing the standard MMD estimator using this biased data; as expected, it reflects the greater proportion of zeros compared to ones. By contrast, the simulations trained by minimizing the weighted MMD estimator, which estimates the MMD from the underlying target distribution as estimated by our thinning function, shows a marked increase in the number of ones, without any obvious difference in simulation quality. All networks were trained for 10,000 iterations.
+
+To further quantify these results, we used the (unscaled) logistic regression used to specify the thinning function to classify the simulated images as zeros or ones. Table 1 shows the proportion of ones obtained using the three scenarios (standard MMD estimator, weighted MMD estimator initialized with a pre-trained network, and weighted MMD estimator with random initialization). We see that, as expected, the network trained using the standard MMD estimator produces around $20 \%$ ones, reflecting the proportion in the dataset. The networks trained using the weighted MMD estimator achieve a significanly higher percent of ones, with the randomly initialized network performing better than the pre-trained network.
+
+While both the networks trained using the weighted MMD estimator simulate a higher proportion of ones than is present in our dataset, neither reaches the 50:50 ratio in our assumed target distribution. We believe this is due to inability of the network to fully converge to a solution that gives zero loss. This interpretation is supported by the fact that the randomly initialized network converges to a better solution than the pre-initialized network, despite the architecture and objective function being the same in both cases. To visualize the changes introduced by using the weighted MMD estimator, in Figure 3 we plot the proportion of ones on a network that is trained for 10,000 iterations using the standard MMD estimator, then for 10,000 iterations using the weighted MMD estimator. We see that the proportion quickly rises after the loss function is altered.
+
+# 7 DISCUSSION AND FUTURE WORK
+
+We have presented an asymptotically unbiased estimator for the MMD between two distributions $\mathbb { P }$ and $\mathbb { Q }$ , for use when we only have access to $\mathbb { P }$ via a biased sampling mechanism. This mechanism can be specified by a known or estimated thinning function $\tilde { T } ( \bar { x } )$ , where samples are then assumed to come from a distribution ${ \widetilde { T } } ( x ) \mathbb { P } ( x ) / Z$ . We show that this estimator can be used to manipulate the distribution of simulations learned by a generative network, in order to correct for sampling bias or to explicitly change the distribution according to a user-specified function.
+
+When the thinning function is unknown, it can be estimated from labeled data. We demonstrate this in an interpretable experiment using partially labeled images, where we jointly estimate the thinning function alongside the generator weights. An obvious next step is to explore the use of more sophisticated thinning functions appropriate for complex, multimodal settings.
+
+# REFERENCES
+
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+T. Bolukbasi, K.-W. Chang, J.Y. Zou, V. Saligrama, and A.T. Kalai. Man is to computer programmer as woman is to homemaker? debiasing word embeddings. In NIPS, 2016.   
+A. Caliskan, J.J. Bryson, and A. Narayanan. Semantics derived automatically from language corpora contain human-like biases. Science, 356(6334):183–186, 2017. doi: 10.1126/science.aal4230.   
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+I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio. Generative adversarial nets. In NIPS, 2014.   
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+C. Robert and G. Casella. Monte Carlo Statistical Methods. Springer Texts in Statistics. Springer, 2 edition, 2004.   
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+        "text": "DIRECTING GENERATIVE NETWORKS WITH WEIGHTED MAXIMUM MEAN DISCREPANCY ",
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+        "text": "Anonymous authors Paper under double-blind review ",
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+    {
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+        "text": "ABSTRACT ",
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+        "text": "The maximum mean discrepancy (MMD) between two probability measures $\\mathbb { P }$ and $\\mathbb { Q }$ is a metric that is zero if and only if all moments of the two measures are equal, making it an appealing statistic for two-sample tests. Given i.i.d. samples from $\\mathbb { P }$ and $\\mathbb { Q }$ , Gretton et al. (2012) show that we can construct an unbiased estimator for the square of the MMD between the two distributions. If $\\mathbb { P }$ is a distribution of interest and $\\mathbb { Q }$ is the distribution implied by a generative neural network with stochastic inputs, we can use this estimator to train our neural network. However, in practice we do not always have i.i.d. samples from our target of interest. Data sets often exhibit biases—for example, under-representation of certain demographics—and if we ignore this fact our machine learning algorithms will propagate these biases. Alternatively, it may be useful to assume our data has been gathered via a biased sample selection mechanism in order to manipulate properties of the estimating distribution $\\mathbb { Q }$ . ",
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+        "text": "In this paper, we construct an estimator for the MMD between $\\mathbb { P }$ and $\\mathbb { Q }$ when we only have access to $\\mathbb { P }$ via some biased sample selection mechanism, and suggest methods for estimating this sample selection mechanism when it is not already known. We show that this estimator can be used to train generative neural networks on a biased data sample, to give a simulator that reverses the effect of that bias. ",
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+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "text": "Neural networks with stochastic input layers can be trained to approximately sample from an arbitrary probability distribution $\\mathbb { P }$ based on samples from $\\mathbb { P }$ (Goodfellow et al., 2014). Generating simulations from complex distributions has applications in a large number of fields: We can automatically generate illustrations for text (Zhang et al., 2017) or streams of video (Vondrick et al., 2016); we can simulate novel molecular fingerprints to aid scientific exploration (Kadurin et al., 2017); and, we can synthesize medical time-series data that can be shared without violating patient privacy (Esteban et al., 2017). ",
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+        "text": "In this paper, we consider the setting of a feedforward neural network (referred to as the generator) that maps random noise inputs $z \\in \\mathbb { R } ^ { d }$ to some observation space $\\mathcal { X }$ . The weights of the neural network are trained to minimize some loss function between the resulting simulations and exemplars of real data. The general form of the resulting distribution $\\mathbb { Q }$ over simulations $G ( z )$ is determined by the architecture of the generator—which governs the form of the mapping $G$ —and by the loss function used to train the generator. Generative adversarial networks (Goodfellow et al., 2014) use dynamically varying, adversarially learned loss functions specified in terms of the output of a classifier. Other generative networks use a loss function defined using a distributional distance or divergence between the simulation distribution $\\mathbb { Q }$ and a target distribution $\\mathbb { P }$ (Arjovsky et al., 2017; Nowozin et al., 2016; Zhao et al., 2017a), requiring the generator to mimic the variance in a collection of data points rather than simply converge to a single mode. In particular, the maximum mean discrepancy (Gretton et al., 2012) has demonstrated good performance as a loss function in this setting (Sutherland et al., 2016; Li et al., 2017; Sutherland et al., 2017), since it reduces to zero if and only if all moments of two distributions are equal, requiring the generator to reproduce the full range of variation found in the data. ",
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+        "text": "These approaches, like most machine learning methods, assume our data is a representative sample from the distribution of interest. If this assumption is correct, minimizing the distributional distance between the simulations and the data is equivalent to learning a distribution that is indistinguishable under an appropriate two-sample test from our target distribution. However, we run into problems if our data is not in fact a representative sample from our target distribution—for instance, if our data gathering mechanism is susceptible to sample selection bias. The problem of machine learning algorithms replicating and even magnifying human biases is gathering increasing awareness (Bolukbasi et al., 2016; Zhao et al., 2017b), and if we believe our dataset suffers from such biases—for example, if our audio dataset contains primarily male speakers or our image dataset contains primarily white faces – we will typically want to take steps to correct this bias. ",
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+    {
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+        "text": "Even if our data is representative of the underlying distribution, we might want to generate samples from a modified version of this distribution. For example, we might want to alter the demographics of characters in a scene to fit a story-line. In this setting, we can treat our desired modified distribution as our target distribution, and treat our data as if they were sampled from this distribution subject to an appropriately biased sample selection mechanism. ",
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+    {
+        "type": "text",
+        "text": "If we know the form of our sample selection bias, we can reformulate our loss function to penalize the generator based on the difference between simulated data and the unbiased distribution of interest, which we will refer to as our target distribution. After a review of relevant background information in Section 2, we show in Section 3 that, given a function that describes how our observed data deviates from this target distribution, we can construct an estimator of the MMD between the generator and the target distribution. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In practice, we will not know the function linking the target distribution and the empirical data distribution. However, we can approximate this function based on user-provided examples of data points that are over- and under-represented. In Section 4, we discuss ways to estimate this function, and in Section 5 we discuss related work in survey sampling statistics and bias reduction. We demonstrate the efficacy and applicability of our approach in Section 6. ",
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+    },
+    {
+        "type": "text",
+        "text": "2 BACKGROUND ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "2.1 GENERATIVE NETWORKS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Generative networks are a class of models which take a collection of data lying in some observation space $\\mathcal { X }$ as input, and aim to generate simulations that are similar to that data – or more generally, whose empirical distribution is close to the distribution underlying the data. These models do not use or require an explicit probability distribution for the data, but instead rely on the fact that sufficiently complex neural networks have the capacity to approximate any arbitrary function (Hornik et al., 1989). We can therefore construct a method of simulating from an arbitrarily complex probability distribution $\\mathbb { Q }$ on $\\mathcal { X }$ by using a neural network generator $G : \\mathbb { R } ^ { d }  \\mathcal { X }$ to transform random $d$ - dimensional inputs $z$ into simulations $G ( z ) \\in \\mathcal { X }$ . In order to minimize the difference between the probability distribution $\\mathbb { Q }$ over simulations $G ( z )$ and the target distribution $\\mathbb { P }$ , we train the neural network to minimize a loss function between simulations $G ( \\bar { z } ) \\sim \\mathbb { Q }$ and data $x \\sim \\mathbb { P }$ ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The most common forms of generative network are generative adversarial networks (GANs, Goodfellow et al., 2014), so called because the loss function is dynamically defined in terms of the output of an adversarially learned classifier. This classifier—itself a neural network—is trained to differentiate between two classes, data and simulations, and for a given observation returns a score under the two classes (loosely corresponding to a probability of belonging to that class). The generator’s loss function is a function of the score assigned to simulations under the true data class, so that reducing the loss function leads to an increased chance of fooling the classifier. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "While this adversarial framework has the advantage of a dynamically evolving and increasingly discriminative loss function, a disadvantage is that the generator can successfully minimize this loss function by mimicking only a subset of the data, leading to a phenomenon known as mode collapse (Salimans et al., 2016; Che et al., 2017). To avoid this, recent works have incorporated estimators of distributional distances between $\\mathbb { P }$ and $\\mathbb { Q }$ , that consider the overall distributions rather than just element-wise distances between samples. For example, the maximum mean discrepancy (Gretton et al., 2012; Sutherland et al., 2016; Li et al., 2017; Sutherland et al., 2017) between two distributions, which we explore further in Section 2.2, reduces to zero if and only if all moments of the two distributions are the same. Other distributional distances that have been used include the Wasserstein distance and the Cramer distance (Arjovsky et al., 2017; Bellemare et al., 2017). In some cases, these distance-based generative networks include adversarial components in their loss functions; for example the MMD-GAN of Li et al. (2017) adversarially learns parameters of the maximum mean discrepancy metric. ",
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": ".2 MAXIMUM MEAN DISCREPANCY BETWEEN TWO DISTRIBUTIONS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The maximum mean discrepancy (MMD, Gretton et al., 2012) projects two distributions $\\mathbb { P }$ and $\\mathbb { Q }$ into a reproducing kernel Hilbert space (RKHS) $\\mathcal { H }$ , and looks at the maximum mean distance between the two projections, i.e. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5bfe543663ff7d4664d4381f649cae8355f1a0ea204f9e96f6bde691aaf620c8.jpg",
+        "text": "$$\n\\mathbf { M M D } [ \\mathbb { P } , \\mathbb { Q } ] : = \\operatorname* { s u p } _ { f \\in \\mathcal { H } } \\left( { \\mathbf { E } } _ { x \\sim \\mathbb { P } } [ f ( x ) ] - { \\mathbf { E } } _ { y \\sim \\mathbb { Q } } [ f ( y ) ] \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "If we specify the kernel mean embedding $\\mu _ { \\mathbb { P } }$ of $\\mathbb { P }$ as $\\begin{array} { r } { \\mu _ { \\mathbb { P } } = \\int k ( x , \\cdot ) d \\mathbb { P } ( x ) } \\end{array}$ , where $k ( \\cdot , \\cdot )$ is the characteristic kernel defining the RKHS, then we can write the square of this distance as ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/54b08162859845b45204bec9fe976190865d2f0e4aefb2e4374c161633aeb166.jpg",
+        "text": "$$\n\\mathbf { M M D } ^ { 2 } [ \\mathbb { P } , \\mathbb { Q } ] = | | \\mu _ { \\mathbb { P } } - \\mu _ { \\mathbb { Q } } | | _ { \\mathcal { H } } ^ { 2 } = \\mathbb { E } _ { \\mathbb { P } } [ k ( x , x ^ { \\prime } ) ] - 2 \\mathbb { E } _ { \\mathbb { P } , \\mathbb { Q } } [ k ( x , y ) ] + \\mathbb { E } _ { \\mathbb { Q } } [ k ( y , y ^ { \\prime } ) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Since we have projected the distributions into an infinite-dimensional space, the distance between the two distributions is zero if and only if all their moments are the same. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In order to be a useful loss function for training a neural network, we must be able to estimate the MMD from data, and also take derivatives of this estimate with respect to the network parameters. We can construct an unbiased estimator of square of the MMD (Gretton et al., 2012) using $m$ samples $x _ { i } \\sim \\mathbb { P }$ and $n$ samples $y _ { i } \\sim \\mathbb { Q }$ as ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f2f7525a1b195325eeac4d352150b9eff9746a714d4e14ef43157c932254a2a6.jpg",
+        "text": "$$\n\\begin{array} { r l r } { \\widehat { \\boldsymbol { \\boldsymbol { \\mathrm { M M D } } } } ^ { 2 } [ \\mathbb { P } , \\mathbb { Q } ] = } & { = } & { \\displaystyle \\frac { 1 } { m ( m - 1 ) } \\sum _ { i = 1 } ^ { m } \\sum _ { j \\neq i } ^ { m } k ( x _ { i } , x _ { j } ) + \\frac { 1 } { n ( n - 1 ) } \\sum _ { i = 1 } ^ { n } \\sum _ { j \\neq i } ^ { n } k ( y _ { i } , y _ { j } ) } \\\\ & { } & { \\displaystyle - \\frac { 2 } { m n } \\sum _ { i = 1 } ^ { m } \\sum _ { j = 1 } ^ { n } k ( x _ { i } , y _ { j } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "If $\\mathbb { P }$ is the distribution underlying our data and $\\mathbb { Q }$ is an approximating distribution represented using a neural network, we can differentiate the individual kernel terms in Equation 1 with respect to the simulated data $y _ { i }$ , and hence (via the chain rule) with respect to the parameters of the neural networks. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The MMD has been successfully used as a loss function in several generative adversarial networks. Dziugaite et al. (2015) and Li et al. (2015) propose training a feedforward neural network to minimize the MMD between simulations and data; Li et al. (2015) also propose minimizing the MMD between simulations and data that has been encoded using a pre-trained autoencoder. The generator in the MMD-GAN of Li et al. (2017) also aims to reduce the MMD between simulations and data, but learns the characteristic kernel in an adversarial manner by combining the kernel with a dynamically learned autoencoder. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 WEIGHTED MMD",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "If we have unbiased samples from two distributions $\\mathbb { P }$ and $\\mathbb { Q }$ , the estimator described in Equation 1 gives an unbiased estimate of the MMD between those two distributions. In a generative network context, we can therefore use this estimator as a loss function in order to modify the generator associated with $\\mathbb { Q }$ so that the MMD between the two distributions is minimized. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "However, this relies on having access to unbiased samples from our target distribution $\\mathbb { P }$ . In practice, our data may have been gathered using biased sampling practices: A dataset of images of faces may over-represent white faces; datasets generated from medical experiments may over-represent male patients; datasets generated from on-campus studies may over-represent college-aged students. If our data is a biased sample of our target distribution, this estimator will estimate the difference between our simulations and the biased empirical distribution, so our simulations will recreate the biases therein. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section, we propose an estimator for the MMD between two distributions $\\mathbb { P }$ and $\\mathbb { Q }$ when we have access to $\\mathbb { P }$ only via some biased sample selection mechanism. Concretely, we assume that $\\mathbb { P }$ is our target of interest, but our observed data are actually sampled from a related distribution $T ( x ) \\mathbb { P } ( x )$ . We can think of $T ( x )$ as an appropriately scaled “thinning function” $\\widetilde { T } ( x ) = Z T ( x )$ that characterizes the sample selection mechanism. In other words, we assume that candidate samples $x ^ { * }$ are sampled from $\\mathbb { P }$ , and these candidates are selected into our pool with probability $\\tilde { T } ( \\hat { x } ^ { * } )$ ; the normalizing constant $Z$ ensures that $\\begin{array} { r } { T ( x ) \\mathbb { P } ( x ) = \\frac { \\widetilde { T } ( x ) \\mathbb { P } ( x ) } { Z } } \\end{array}$ is a valid probability distribution. While in the remainder of this paper we will continue to use the language of biased sample selection mechanisms, we note that this framework can also be used if our data are unbiased samples but we want to explicitly modulate our target distribution via some function $F ( x )$ so that we generate simulations from $F ( x ) \\mathbb { P } ( x )$ ; in this setting we can treat the transformed target as $\\mathbb { P }$ and let $T ( x ) =$ $1 / F ( x )$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For now, we assume that our thinning function $\\widetilde { T } ( x )$ is known; we discuss ways to estimate or specify it in Section 4. Our estimation problem becomes an importance sampling problem: We have samples from $T ( \\cdot ) \\mathbb { P } ( \\cdot )$ , and we want to use these to estimate $\\scriptstyle \\mathbf { \\dot { M } M D } ^ { 2 } [ \\mathbb { P } , \\mathbb { Q } ]$ , which is a function of the target distribution $\\mathbb { P }$ . Importance sampling provides a method for constructing an estimator for the expectation of a function $\\phi ( x )$ with respect to a distribution $\\mathbb { P }$ , by taking an appropriately weighted sum of evaluations of $\\phi$ at values sampled from a different distribution $\\mathbb { P } ^ { \\prime }$ . If we knew the normalizing constant $Z$ , we could construct an unbiased estimator of the MMD between $\\mathbb { P }$ and $\\mathbb { Q }$ by weighting each function evaluation associated with sample $x$ from $T ( x ) \\mathbb { P } ( x )$ with the likelihood ratio $\\mathbb { P } ( x ) / T ( x ) \\mathbb { P } ( x ) = 1 / T ( x )$ , i.e. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/690472041e35bcb42b6f24f16e8c326256ec44231c3e5be974e8000d83110b87.jpg",
+        "text": "$$\n\\begin{array} { c } { { \\displaystyle \\widehat { M } _ { u } = \\frac { 1 } { m ( m - 1 ) } \\sum _ { i = 1 } ^ { m } \\sum _ { j \\neq i } ^ { m } \\frac { 1 } { T ( x _ { i } ) T ( x _ { j } ) } k ( x _ { i } , x _ { j } ) + \\frac { 1 } { n ( n - 1 ) } \\sum _ { i = 1 } ^ { n } \\sum _ { j \\neq i } ^ { n } k ( y _ { i } , y _ { j } ) } } \\\\ { { - \\frac { 2 } { m n } \\sum _ { i = 1 } ^ { m } \\sum _ { j = 1 } ^ { n } \\frac { 1 } { T ( x _ { i } ) } k ( x _ { i } , y _ { j } ) . } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "However, the normalizing constant $Z$ depends on both $\\widetilde { T }$ and $\\mathbb { P }$ . We will not, in general, know an analytic form for $\\mathbb { P }$ , so we will not be able to calculate $Z$ . Since we will only be able to evaluate $T ( x )$ up to a normalizing constant, we cannot work with Equation 2 directly. Instead, we can construct a biased estimator $\\widehat { M _ { d } }$ by using self-normalized importance weights, ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b93d0e3449dd8883d6b33cc875f981cc86942050e2a2d5528bd38e273a5451fd.jpg",
+        "text": "$$\n\\widehat { M } _ { b } = \\sum _ { i = 1 } ^ { m } \\sum _ { j \\neq i } ^ { m } w ( x _ { i } ) w ( x _ { j } ) k ( x _ { i } , x _ { j } ) + \\frac { 1 } { n ( n - 1 ) } \\sum _ { i = 1 } ^ { n } \\sum _ { j \\neq i } ^ { n } k ( y _ { i } , y _ { j } ) - \\frac { 2 } { n } \\sum _ { i = 1 } ^ { m } \\sum _ { j = 1 } ^ { n } w ( x _ { i } ) k ( x _ { i } , y _ { j } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { w ( x _ { i } ) = \\frac { 1 / \\widetilde { T } ( x _ { i } ) } { \\sum _ { j = 1 } ^ { m } 1 / \\widetilde { T } ( x _ { j } ) } } \\end{array}$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We refer to the estimator $\\widehat { M _ { b } }$ in Equation 3 as the weighted MMD estimator. While this estimator is biased due to the self-normalized weights, this bias will decrease as $1 / m$ where $m$ is the number of samples from $T ( \\cdot ) \\mathbb { P } ( \\cdot )$ . Further, this biased estimator will often have lower variance than the unbiased estimator in Equation 2 (Robert & Casella, 2004). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 LEARNING THE THINNING FUNCTION $\\widetilde { T } ( x )$ ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In Section 3, we assumed that our target distribution $\\mathbb { P }$ is only accessible via a biased sample selection mechanism characterized by a thinning function $\\widetilde { T } ( x )$ , so that our samples are actually distributed (up to a normalizing constant) according to $\\widetilde { T } ( \\boldsymbol { x } ) \\mathbb { P } ( \\boldsymbol { x } )$ . If we know the thinning function $\\widetilde { T } ( x )$ that describes our sample selection mechanism, we can use Equation 3 directly in the loss function of our generator. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "However, in practice we will not have access to this thinning function. Rather, we are likely to be in a situation where a practitioner has noticed that one or more classes is over- or under-represented, either in the dataset or in simulations from a generative network trained on the data. If our dataset is fully labeled, we could either manually re-balance our dataset to match the target distribution, or use these labels to estimate $\\tilde { T }$ by comparing the number of data points in various subsets of $\\mathcal { X }$ with the expected number under our target distribution. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Even if we do not have a fully labeled dataset, we might be able to label a subset of examples and use these to estimate $T ( x )$ . For example, assume we have an image dataset with more pictures of men than women, and that we select and label a random subset of this dataset. A reasonable estimate for our thinning function would be to set $T ( \\mathrm { m e n } ) = 1$ and approximate $T ( \\mathrm { w o m e n } )$ by the sample ratio of men to women in our labeled subset. In this simple two-class setting, to extrapolate values of $T$ across our observation space $X$ we can run a logistic regression using the labeled subset. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In a more complicated problem, we may need to deploy more sophisticated regression tools, but the problem remains one of function estimation from labeled data. Given a set of labeled exemplars and user-specified estimates of the thinning function evaluated at those exemplars (which could be based on demographic statistics or domain knowledge, or desired statistics of the simulation distribution), we could learn a thinning function $T ( x )$ using techniques such as neural network function estimation or Gaussian process regression. ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "5 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "While it has received little attention in the deep learning literature, the problem of correcting for biased sample selection mechanisms is familiar in the field of survey statistics. Our estimator is related to inverse probability weighting (IPW), originally proposed by Horvitz & Thompson (1952) and still in wide use today (e.g. Mansournia & Altman, 2016). IPW is used in stratified random sampling study designs to correct parameter estimates for intentional bias in sampling. Under IPW, each data point is assigned a weight proportionate to the inverse of its selection probability, so that samples from classes which are disproportionately unlikely to be seen in the sample are assigned increased weights to counteract the undersampling of their classes. This is the same form as the weights in our sampling scheme, and serves a similar purpose, although it is not placed in the context of the MMD between two distributions and assumes that the probability of selection is known for each observation. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "This work also follows increased awareness of the effects of biased data on machine learning outcomes, and interest in mitigating these effects. For example, Caliskan et al. (2017) and Bolukbasi et al. (2016) explore how biases and prejudices expressed in our language manifest in the word embeddings and representations found. Zhao et al. (2017b) find that popular image labeling datasets exhibit gender stereotypes and that algorithms trained on these datasets tend to amplify these stereotypes. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "6 EVALUATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We consider two experimental settings: One where the form of the sampling bias is known, and one where it is estimated from data. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "6.1 CORRECTING FOR BIASED SAMPLE SELECTION WITH A KNOWN THINNING FUNCTION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In Section 3, we discussed how, for a given thinning function $\\widetilde { T } ( x )$ , we can construct an estimator for the MMD between the distribution $\\mathbb { Q }$ implied by our generator and the underlying unbiased data distribution $\\mathbb { P }$ . To demonstrate the efficacy of this in practice, we assume $\\mathbb { P }$ to be a mixture of two Gaussians (Figure 1a), and let $\\begin{array} { r } { \\widetilde { T } = \\frac { \\mathsf { \\bar { \\rho } } _ { 0 . 5 } } { 1 + \\exp ( 1 0 ( x - 1 ) ) } + \\mathsf { \\bar { 0 } } . 5 } \\end{array}$ be a scaled logistic function as shown in Figure 1b. The resulting data is distributed according to Figure 1c. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We construct a simple generator network taking univariate random noise as input, and consisting of six fully-connected layers, each with three nodes and exponential linear unit activations, and with a univariate output with no activation. We train this network using both the standard unbiased MMD ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/7aacc16a56bf12eeb6be717b6613a40a417beb992777bd5af63e5666555011da.jpg",
+        "image_caption": [],
+        "image_footnote": [],
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/3f5c0eac359a26d961b834e96bec7b0a6c4fed4d50c6297945c38ed15c4d5107.jpg",
+        "image_caption": [],
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+        "img_path": "images/06ae327a5fad1bf87dc87eb272572d8d59131835adeb160506ba22f947b9b10b.jpg",
+        "image_caption": [
+            "(c) Observed distribution "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "(a) Target distribution $x \\sim \\bar { \\mathbb { P } } ( x ) = 0 . 5 \\mathcal { N } ( 0 , 0 . 5 ) +$ $0 . 5 \\mathcal { N } ( 2 , 0 . 5 )$ ",
+        "bbox": [
+            186,
+            227,
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+            265
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "(d) Simulations using standard MMD estimator (with PDF of $T ( x ) \\mathbb { P } ( x )$ for comparison) ",
+        "bbox": [
+            295,
+            390,
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/f5def123fb06980fd1e88eb246a74795dc78e139e8f49ff7122923770c7a2b62.jpg",
+        "image_caption": [
+            "Figure 1: Output of GANs trained on data sampled according to $T ( x ) \\mathbb { P } ( x )$ . Simulation plots PDFs are computed using NumPy’s random normal sampler, mixed to create the appropriate mode ratios. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/a64c49077e37ff61c4666331367883dd199fc765c973539e475a5db6635f88c0.jpg",
+        "image_caption": [
+            "(e) Simulations using weighted MMD estimator (with PDF of $\\mathbb { P } ( x )$ for comparison) "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "estimator of Equation 1, and the weighted MMD estimator proposed in Equation 3, using ADAM optimization with a learning rate of 0.001. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Figures 1e and 1d show histograms of samples generated using the two estimators. We see that, as expected, the standard MMD estimator does a good job of replicating the empirical distribution of the data presented to it. The weighted MMD estimator, however, is able to replicate the target distribution $\\mathbb { P }$ , even though it only has access to samples from $T ( x ) \\mathbb { P } ( x )$ . ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "6.2 LEARNING AND CORRECTING FOR A BIASED SAMPLE SELECTION MECHANISM",
+        "text_level": 1,
+        "bbox": [
+            171,
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+    },
+    {
+        "type": "text",
+        "text": "In practice we are unlikely to know a functional form for $\\widetilde { T } ( x )$ , the function that describes the form of the sampling bias. Here we consider the case where we must estimate this function from data. We consider as an example a dataset containing zeros and ones from the MNIST dataset. We assume our target distribution contains $50 \\%$ zeros and $50 \\%$ ones, but that due to a biased selection procedure our data contains $80 \\%$ zeros and $20 \\%$ ones. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We hypothesise that the practitioner using this dataset is aware that there is a discrepancy, but does not know how to translate this into a functional form for $\\widetilde { T }$ . Instead, the practitioner labels examples of each class, and supplies an estimate of how much each class is under-represented. In our example, we assume the practitioner has labeled 800 zeros and 200 ones, and estimates that ones are undersampled according to a thinning probability of 0.75. We note that this number can easily be estimated from a data sample if the practitioner knows the global ratio of zeros and ones. ",
+        "bbox": [
+            173,
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/4d1e806f18db7ec294af1efd9f6aadc7d2a77f05052b86648baf061fdbe26181.jpg",
+        "table_caption": [
+            "Table 1: Comparison of different estimators on the proportion of generated images classified as ones. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Standard MMD</td><td>Weighted MMD (initialized to standard MMD)</td><td>Weighted MMD (random initialization)</td></tr><tr><td>0.2201 ±0.0027</td><td>0.3316 ± 0.0028</td><td>0.3888 ± 0.0027</td></tr></table>",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/c49db018f7cfa2478f872ec32baa2858bfd3b4aeba60a03d7965a5d270163485.jpg",
+        "image_caption": [
+            "(a) Subset of real images used to train GANs (50 zeros, 14 ones) "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/e0998658d6585d8b00805e69f5602390d4bed21b0d816186957e74df04e982a6.jpg",
+        "image_caption": [
+            "(b) Simulations from GAN trained with weighted MMD, randomly initialized (40 zeros, 24 ones) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            513,
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/d4667bf229d1d82687a71753d56e4725067be1a52846039cfe8874faa1cf5bf5.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            245,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "(c) Simulations from GAN trained with MMD (53 zeros, 11 ones) ",
+        "bbox": [
+            245,
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+            483,
+            584
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/8695a4a363b021bb1b0eb97740068f60209475352eb4162f7fe864e89c64bc32.jpg",
+        "image_caption": [
+            "(d) Simulations from GAN trained with weighted MMD, initialized to MMD (47 zeros, 17 ones) ",
+            "Figure 2: Data used to train the GANs, and simulations from GANs trained using unweighted and weighted MMD. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            514,
+            375,
+            748,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/00331afd357bd0ed43a608eb27ddfec7d4dcc7fd8fa6aca331817beadcc138e4.jpg",
+        "image_caption": [
+            "Figure 3: Proportion of images classified as ones, over time, for a network initially learned using the standard MMD estimator. The loss function switches to the weighted MMD estimator at iteration 10000, leading to an increase in the proportion of images classified as ones. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            708,
+            812,
+            832
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We use an architecture based on the MMD-GAN of Li et al. (2017), which incorporates an autoencoder, and trains a generator based on the MMD between the encoded representations of the data and of the simulations. The autoencoder is simultaneously learned in an adversarial manner, and serves to optimize the kernel used in MMD to maximally discriminate between the two distributions. Our experiments are run on an adapted version of the original MMD-GAN Torch code, which will be made available after publication. ",
+        "bbox": [
+            174,
+            103,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Since the autoencoder provides a low-dimensional embedding of our images, we specify our thinning function $\\widetilde { T } ( x )$ on the space spanned by the encoder. After each update of the autoencoder, we estimate $\\widetilde { T }$ with the labeled set, using appropriately scaled logistic regression. We then replace the standard MMD estimator in the loss function with our weighted estimator, calculating using the estimated $\\widetilde { T }$ . ",
+        "bbox": [
+            174,
+            194,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "One could imagine using the weighted estimator in two scenarios. We might be already aware of the bias in our data, in which case we would simply train our GAN using the weighted MMD estimator. Alternatively, we might only become aware of the bias after already training a GAN using the standard estimator, in which case we might initialize our network weights to the resulting pretrained values. To simulate these scenarios, we trained two networks using the weighted estimator: one initialized to a pre-trained MMD-GAN, and one randomly initialized. ",
+        "bbox": [
+            174,
+            279,
+            823,
+            362
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Figure 2a shows a random subset of the data used to train the GANs. Due to the biased sample selection method, there are significantly more zeros than ones. Figure 2c shows simulations generated by minimizing the standard MMD estimator using this biased data; as expected, it reflects the greater proportion of zeros compared to ones. By contrast, the simulations trained by minimizing the weighted MMD estimator, which estimates the MMD from the underlying target distribution as estimated by our thinning function, shows a marked increase in the number of ones, without any obvious difference in simulation quality. All networks were trained for 10,000 iterations. ",
+        "bbox": [
+            174,
+            369,
+            825,
+            467
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To further quantify these results, we used the (unscaled) logistic regression used to specify the thinning function to classify the simulated images as zeros or ones. Table 1 shows the proportion of ones obtained using the three scenarios (standard MMD estimator, weighted MMD estimator initialized with a pre-trained network, and weighted MMD estimator with random initialization). We see that, as expected, the network trained using the standard MMD estimator produces around $20 \\%$ ones, reflecting the proportion in the dataset. The networks trained using the weighted MMD estimator achieve a significanly higher percent of ones, with the randomly initialized network performing better than the pre-trained network. ",
+        "bbox": [
+            174,
+            473,
+            825,
+            585
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "While both the networks trained using the weighted MMD estimator simulate a higher proportion of ones than is present in our dataset, neither reaches the 50:50 ratio in our assumed target distribution. We believe this is due to inability of the network to fully converge to a solution that gives zero loss. This interpretation is supported by the fact that the randomly initialized network converges to a better solution than the pre-initialized network, despite the architecture and objective function being the same in both cases. To visualize the changes introduced by using the weighted MMD estimator, in Figure 3 we plot the proportion of ones on a network that is trained for 10,000 iterations using the standard MMD estimator, then for 10,000 iterations using the weighted MMD estimator. We see that the proportion quickly rises after the loss function is altered. ",
+        "bbox": [
+            173,
+            592,
+            825,
+            717
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "7 DISCUSSION AND FUTURE WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            739,
+            478,
+            755
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We have presented an asymptotically unbiased estimator for the MMD between two distributions $\\mathbb { P }$ and $\\mathbb { Q }$ , for use when we only have access to $\\mathbb { P }$ via a biased sampling mechanism. This mechanism can be specified by a known or estimated thinning function $\\tilde { T } ( \\bar { x } )$ , where samples are then assumed to come from a distribution ${ \\widetilde { T } } ( x ) \\mathbb { P } ( x ) / Z$ . We show that this estimator can be used to manipulate the distribution of simulations learned by a generative network, in order to correct for sampling bias or to explicitly change the distribution according to a user-specified function. ",
+        "bbox": [
+            174,
+            771,
+            825,
+            861
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "When the thinning function is unknown, it can be estimated from labeled data. We demonstrate this in an interpretable experiment using partially labeled images, where we jointly estimate the thinning function alongside the generator weights. An obvious next step is to explore the use of more sophisticated thinning functions appropriate for complex, multimodal settings. ",
+        "bbox": [
+            174,
+            867,
+            823,
+            924
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            287,
+            118
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "M. Arjovsky, S. Chintala, and L. Bottou. Wasserstein GAN. arXiv:1701.07875, 2017.   \nM.G. Bellemare, I. Danihelka, W. Dabney, S. Mohamed, B. Lakshminarayanan, S. Hoyer, and R. Munos. The Cramer distance as a solution to biased Wasserstein gradients. arXiv:1705.10743, 2017.   \nT. Bolukbasi, K.-W. Chang, J.Y. Zou, V. Saligrama, and A.T. Kalai. Man is to computer programmer as woman is to homemaker? debiasing word embeddings. In NIPS, 2016.   \nA. Caliskan, J.J. Bryson, and A. Narayanan. Semantics derived automatically from language corpora contain human-like biases. Science, 356(6334):183–186, 2017. doi: 10.1126/science.aal4230.   \nT. Che, Y. Li, A.P. Jacob, Y. Bengio, and W. Li. Mode regularized generative adversarial networks. In ICLR, 2017.   \nG.K. Dziugaite, D.M. Roy, and Z. Ghahramani. Training generative neural networks via maximum mean discrepancy optimization. In UAI, 2015.   \nC. Esteban, S.L. Hyland, and G. Ratsch. Real-valued (medical) time series generation with recurrent ¨ conditional GANs. arXiv:1706.02633, 2017.   \nI. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio. Generative adversarial nets. In NIPS, 2014.   \nA. Gretton, K.M. Borgwardt, M.J. Rasch, B. Scholkopf, and A. Smola. A kernel two-sample test. ¨ JMLR, 13(Mar):723–773, 2012.   \nK. Hornik, M. Stinchcombe, and H. White. Multilayer feedforward networks are universal approximators. Neural Networks, 2(5):359–366, 1989.   \nD.G. Horvitz and D.J. Thompson. A generalization of sampling without replacement from a finite universe. JASA, 47(260):663–685, 1952.   \nA. Kadurin, A. Aliper, A. Kazennov, P. Mamoshina, Q. Vanhaelen, K. Khrabrov, and A. Zhavoronkov. The cornucopia of meaningful leads: Applying deep adversarial autoencoders for new molecule development in oncology. Oncotarget, 8(7):10883, 2017.   \nC.-L. Li, W.-C. Chang, Y. Cheng, Y. Yang, and B. Poczos. MMD GAN: Towards deeper under- ´ standing of moment matching network. In NIPS, 2017.   \nY. Li, K. Swersky, and R. Zemel. Generative moment matching networks. In ICML, 2015.   \nM.A. Mansournia and D.G. Altman. Inverse probability weighting. BMJ, 352:i189, 2016.   \nS. Nowozin, B. Cseke, and R. Tomioka. f-GAN: Training generative neural samplers using variational divergence minimization. In NIPS, 2016.   \nC. Robert and G. Casella. Monte Carlo Statistical Methods. Springer Texts in Statistics. Springer, 2 edition, 2004.   \nT. Salimans, I. Goodfellow, W. Zaremba, V. Cheung, A. Radford, and X. Chen. Improved techniques for training GANs. In NIPS, 2016.   \nD.J. Sutherland, H.-Y. Tung, H. Strathmann, S. De, A. Ramdas, A. Smola, and A. Gretton. Generative models and model criticism via optimized maximum mean discrepancy. In ICLR, 2016.   \nD.J. Sutherland, H.-Y. Tung, H. Strathmann, S. De, A. Ramdas, A. Smola, and A. Gretton. Generative models and model criticism via optimized maximum mean discrepancy. In ICLR, 2017.   \nC. Vondrick, H. Pirsiavash, and A. Torralba. Generating videos with scene dynamics. In NIPS, 2016.   \nH. Zhang, T. Xu, H. Li, S. Zhang, X. Huang, X. Wang, and D. Metaxas. Stackgan: Text to photorealistic image synthesis with stacked generative adversarial networks. In ICCV, 2017.   \nJ. Zhao, M. Mathieu, and Y. LeCun. Energy-based generative adversarial network. In ICLR, 2017a.   \nJ. Zhao, T. Wang, M. Yatskar, V. Ordonez, and K.-W. Chang. Men also like shopping: Reducing gender bias amplification using corpus-level constraints. In EMNLP, 2017b. ",
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+]

parse/train/bMzj6hXL2VJ/bMzj6hXL2VJ.md

@@ -0,0 +1,403 @@
+# ORDERING-BASED CAUSAL DISCOVERY WITH REINFORCEMENT LEARNING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+It is a long-standing question to discover causal relations among a set of variables in many empirical sciences. Recently, Reinforcement Learning (RL) has achieved promising results in causal discovery. However, searching the space of directed graphs directly and enforcing acyclicity by implicit penalties tend to be inefficient and restrict the method to small problems. In this work, we alternatively consider searching an ordering by RL from the variable ordering space that is much smaller than that of directed graphs, which also helps avoid dealing with acyclicity. Specifically, we formulate the ordering search problem as a Markov decision process, and then use different reward designs to optimize the ordering generating model. A generated ordering is then processed using variable selection methods to obtain the final directed acyclic graph. In contrast to other causal discovery methods, our method can also utilize a pretrained model to accelerate training. We conduct experiments on both synthetic and real-world datasets, and show that the proposed method outperforms other baselines on important metrics even on large graph tasks.
+
+# 1 INTRODUCTION
+
+Identifying causal structure from observational data is an important but also challenging task in many applications. This problem can be formulated as that of finding a Directed Acyclic Graph (DAG) that minimizes some score function defined w.r.t. observed data. Though there exist well-studied score functions like Bayesian Information Criterion (BIC) or Minimum Description Length (MDL) (Schwarz et al., 1978; Chickering, 2002), searching over the space of DAGs is known to be NP-hard, even if each node has at most two parents (Chickering, 1996). Consequently, traditional methods mostly rely on local heuristics to perform the search, including greedy hill climbing and Greedy Equivalence Search (GES) that explore the Markov equivalence classes (Chickering, 1996).
+
+Along with various search strategies, existing methods have also considered to reduce the search space while meeting the DAG constraint. A useful practice is to cast the causal structure learning problem as that of learning an optimal ordering of variables (Koller & Friedman, 2009). Because the ordering space is significantly smaller than the space of directed graphs and searching over ordering space can avoid the problem of dealing with acyclic constraints (Teyssier & Koller, 2005). Many algorithms are used to search for ordering such as genetic algorithm (Larranaga et al., 1996), Markov chain Monte Carlo (Friedman & Koller, 2003) and greedy local hill-climbing (Teyssier & Koller, 2005). However, these algorithms often cannot find the best ordering effectively.
+
+Recently, with differentiable score functions, several gradient-based methods have been proposed based on a smooth characterization of acyclicity, including NOTEARS (Zheng et al., 2018) for linear causal models and several subsequent works, e.g., Yu et al. (2019); Lachapelle et al. (2020); $\mathrm { N g }$ et al. (2019b;a), which use neural networks to model nonlinear causal relationships. As another attempt, Zhu et al. (2020) utilize Reinforcement Learning (RL) as a search strategy to find the best DAG from the graph space and it can be incorporated with a wide range of score functions. Unfortunately, its good performance is achieved only with around 30 variables, for at least two reasons: 1) the action space, consisting of directed graphs, is tremendous for large scale problems and hard to be explored efficiently; and 2) it has to compute scores for many non-DAGs generated during training but computing scores w.r.t. data is generally be time-consuming. It appears that the RL-based approach may not be able to achieve a close performance to other gradient-based methods that directly optimize the same (differentiable) score function for large causal discovery problems, due to its search nature.
+
+In this work, we propose a RL-based approach for causal discovery, named Causal discovery with Ordering-based Reinforcement Learning (CORL), which combines RL with the ordering based paradigm so that we can exploit the powerful search ability of RL to search the best ordering efficiently. To achieve this, we formulate the ordering search problem as a Markov Decision Process (MDP), and then use different reward designs for RL to optimize the ordering generating model. In addition, we notice that pretrained model can be incorporated into our model to accelerate training. For a generated ordering, we prune it to the final DAG by variable selection. The proposed approach is evaluated on both synthetic and real datasets to validate its effectiveness. In particular, the proposed method can achieve a much improved performance than the previous RL-based method on both linear and non-linear data models, even outperforms NOTEARS, a gradient-based method, on 150-node linear data models, and is competitive with Causal Additive Model (CAM) on non-linear data models.
+
+# 2 RELATED WORKS
+
+Existing causal discovery methods roughly fall into three classes. The first class, as described in the introduction, are the so-called score-based methods. Besides the mentioned BIC/MDL scores, other score functions include the Bayesian Gaussian equivalent score (Geiger & Heckerman, 1994), the generalized score based on (conditional) independence relationship (Huang et al., 2018), and a recently proposed meta-transfer score (Bengio et al., 2020). Another class of methods such as fast causal inference and PC (Spirtes et al., 2000; Zhang, 2008) which first find causal skeleton and then decide the orientations of the edges up to the Markov equivalence class, are viewed as constraint-based. Such methods usually involve multiple independent testing problems; the testing results may have conflicts and handling them is not easy. The last class of methods relies on properly defined functional causal models, includeing Linear Non-Gaussian Acyclic Model (LiNGAM), nonlinear Additive Noise Model (ANM) (Hoyer et al., 2009; Peters et al., 2014), and the post-nonlinear causal model (Zhang & Hyvärinen, 2009). By placing certain assumptions on the class of causal functions and/or noise distributions, these methods can distinguish different DAGs in the same Markov equivalence class.
+
+Since we particularly consider ordering-based approaches, here we present a more detailed review of such methods. Most of the ordering-based methods belong to the class of score-based methods. Besides the mentioned heuristic search algorithms, Schmidt et al. (2007) proposed L1OBS to conduct variable selection using $\ell _ { 1 }$ -regularization paths based on Teyssier & Koller (2005). Scanagatta et al. (2015) further proposed an ordering exploration method on the basis of an approximated score function so as to scale to thousands of variables. The Causal Additive Model (CAM) was developed by Bühlmann et al. (2014) for nonlinear data models. Some recent ordering-based methods such as sparsest permutation (Raskutti & Uhler, 2018) and greedy sparest permutation (Solus et al., 2017) can guarantee consistency of Markov equivalence class, based on some conditional independence relations and certain assumptions like faithfulness. A variant of greedy sparest permutation was also proposed in Bernstein et al. (2020) for the setting with latent variables. In the present work, we mainly work on identifiable cases which may have different assumptions from theirs.
+
+In addition, several exact algorithms such as dynamic programming (Silander & Myllymäki, 2006; Perrier et al., 2008) and integer or linear programming (Jaakkola et al., 2010; Cussens, 2011; Bartlett & Cussens, 2017) are used to solve causal discovery problem. However, these exact algorithms usually work on small graphs efficiently (De Campos & Ji, 2011), and in order to handle larger problems with hundreds of variables, they need to incorporate heuristics search (Xiang & Kim, 2013) or limit the maximum number of parents.
+
+Recently, RL has been used to tackle several combinatorial problems such as maximum cut and the traveling salesman problem, due to their relatively simple reward mechanisms (Khalil et al., 2017). In combination with the encoder-decoder based pointer networks (Vinyals et al., 2015), Bello et al. (2016) showed that RL can have a better generalization even when the optimal solutions are used as labeled data in a supervised way. Kool et al. (2019) further used an attention based encoder-decoder model for an improved performance. These works aim to learn a policy as a combinatorial solver based on the same structure of a particular type of combinatorial problems. However, various causal discovery tasks generally have different causal relationships, data types, graph structures, etc, and moreover, are typically off-line with focus on a causal graph. As such, we use RL as a search strategy, similar to Zhu et al. (2020); Zoph & Le (2017); nevertheless, a pretrained model or a policy can offer a good starting point to speed up training, as shown in our evaluation results (Figure 3).
+
+# 3 BACKGROUND
+
+# 3.1 CAUSAL STRUCTURE LEARNING
+
+Let $\mathcal { G } = ( d , V , E )$ denotes a DAG, with $d$ the number of nodes, $V = \{ v _ { 1 } , \ldots , v _ { d } \}$ the set of nodes, and $E = \{ ( v _ { i } , v _ { j } ) | i , j = 1 , \dots , d \}$ the set of directed edges from $v _ { i }$ to $v _ { j }$ . Each node $v _ { j }$ is associated with a random variable $X _ { j }$ . The probability model associated with $\mathcal { G }$ factorizes as $p ( \check { X _ { 1 } } , \dots , X _ { d } ) =$ $\begin{array} { r } { \prod _ { j = 1 } ^ { d } p ( X _ { j } | \mathrm { P a } ( X _ { j } ) ) } \end{array}$ , where $p ( X _ { j } | \mathrm { P a } ( X _ { j } ) )$ is the conditional probability distribution for $X _ { j }$ given its parents $\operatorname { P a } ( X _ { j } ) : = \{ X _ { k } | ( v _ { k } , v _ { j } ) \in E \}$ . We assume that the observed data $\mathbf { x } _ { j }$ is obtained by the Structural Equation Model (SEM) with addtive noise: $X _ { j } : = f _ { \theta _ { j } } ( \mathrm { P a } ( X _ { j } ) ) + \epsilon _ { j } , \overset { \cdot } { j } = 1 , \dots , d$ , where $f _ { \theta _ { j } }$ parameterized by $\theta _ { j }$ is used to represent the functional relationship between $X _ { j }$ and its parents, and $\epsilon _ { j }$ ’s denote jointly independent additive noise variables. We assume causal minimality, which is equivalent to that each $f _ { j }$ is not a constant for any $X _ { k } \in \operatorname { P a } ( X _ { j } )$ in this SEM (Peters et al., 2014).
+
+Given a sample $\mathbf { X } = [ \mathbf { x } _ { 1 } , \ldots \mathbf { \mu } , \mathbf { x } _ { d } ]$ , where $\mathbf { X } \in \mathbb { R } ^ { m \times d }$ and $\mathbf { x } _ { j }$ is a vector of $m$ observations for random variables $X _ { j }$ . Our goal in this paper is to find the DAG $\mathcal { G }$ that maximizes the BIC score (or other well studied scores), defined as
+
+$$
+\mathrm { S c o r e } _ { \mathrm { B I C } } ( \mathcal { G } ) = \sum _ { j = 1 } ^ { d } \left[ \sum _ { k = 1 } ^ { m } \log p ( X _ { j } ^ { k } | \mathbf { P a } ( X _ { j } ^ { k } ) , \theta _ { j } ) - \frac { | \theta _ { j } | } { 2 } \log m \right] ,
+$$
+
+where $| \theta _ { j } |$ is the number of free parameters in $p ( X _ { j } ^ { k } | \mathrm { P a } ( X _ { j } ^ { k } ) , \theta _ { j } )$ . For linear causal relationships, $| \theta _ { j } | = | \mathrm { P a } ( X _ { j } ^ { k } ) |$ the number of parents, up to some constant factor.
+
+The problem of finding a directed graph that satisfies the ayclicity constraint can be cast as that of finding an variable ordering (Teyssier & Koller, 2005; Schmidt et al., 2007). The score of an ordering is usually defined as the score of the best DAG that is consistent with the given ordering. Specifically, let $\Pi$ denote an ordering of the nodes in $V$ , where the length of the ordering $| \Pi | = | V |$ and $\Pi$ is indexed from 1. If node $v _ { j } \in V$ lies in the $p$ -th position, then $\Pi ( p ) = v _ { j }$ . Notation $\Pi _ { \prec v _ { j } }$ denotes the set of nodes in $V$ that precede node $v _ { j }$ in $\Pi$ . One can establish a canonical correspondence between an ordering $\Pi$ and a fully-connected DAG $\mathcal { G } ^ { \Pi }$ ; an example with four nodes is presented in Figure 1. For a given DAG $\mathcal { G }$ , it can be consistent with more than one orderings and the set of these orderings is given by
+
+$$
+\Phi ( \Pi ) : = \{ \Pi : { \mathrm { t h e ~ f u l l y - c o n n e c t e d ~ D A G ~ \mathcal { G } ^ { I I } ~ i s ~ a ~ s u p e r { \mathrm { - } } D A G ~ o f ~ \mathcal { G } } } \} ,
+$$
+
+where a super-DAG of $\mathcal { G }$ is a DAG whose edge set is a superset of that of $\mathcal { G }$ . The score of an ordering is usually defined as the score of the best DAG that is consistent with the given ordering (Teyssier & Koller, 2005; Peters et al., 2014). We provide a formal description in Proposition 1 to show that it is possible to find the correct ordering with high probability in the large sample
+
+$$
+\boldsymbol { \Pi } : = \{ v _ { 2 } , v _ { 4 } , v _ { 1 } , v _ { 3 } \}
+$$
+
+![](images/95fc68b0ba24d471bd4374d41713fc742059254749010acf6c0f082930e53e93.jpg)  
+Figure 1: An example of the correspondence between an ordering (down) and a fully-connected DAG (top).
+
+limit. Therefore, the search for the true DAG $\mathcal { G } ^ { * }$ can be decomposed to two phases: finding the correct ordering and performing variable selection (feature selection); the latter is to find the optimal DAG that is consistent with the ordering found in the first step.
+
+Proposition 1. Suppose that an identifiable SEM with true causal DAG $\mathcal { G } ^ { * }$ on $X = \{ X _ { j } \} _ { j = 1 } ^ { d }$ induces distribution $P ( X )$ . Let $\mathcal { G } ^ { \Pi }$ be the fully-connected DAG that corresponds to an ordering Π. If there is an SEM with $\mathcal { G } _ { \Pi }$ inducing the same distribution $P ( X )$ , then $\mathcal { G } ^ { \Pi }$ must be a super-graph of $\mathcal { G } ^ { * }$ , i.e., every edge in $\mathcal { G } ^ { * }$ is covered in $\mathcal { G } ^ { \Pi }$ .
+
+Proof. The SEM with $\mathcal { G } ^ { \Pi }$ may not be causally minimal but can be reduced to an SEM satisfying the causal minimality condition (Peters et al., 2014). Let $\tilde { \mathcal { G } } ^ { \Pi }$ denotes the causal graph in the reduced SEM with the same distribution $P ( X )$ . Since we have assumed that original SEM is identifiable, i.e., the distribution $P ( X )$ corresponds to a unique true graph, $\tilde { \mathcal { G } } ^ { \Pi }$ is then identical to $\mathcal { G } ^ { * }$ . The proof is complete by noticing that $\mathcal { G } ^ { \Pi }$ is a super-graph of $\tilde { \mathcal { G } } ^ { \Pi }$ . □
+
+# 3.2 REINFORCEMENT LEARNING
+
+Standard RL is usually formulated as an MDP over the environment state $s \in S$ and agent action $a \in { \mathcal { A } }$ , under an (unknown) environmental dynamics defined by a transition probability $\mathcal { T } ( s ^ { \prime } | s , a )$ . We use $\pi _ { \phi } ( a | s )$ to denote the policy, parameterized by $\phi$ , which outputs a discrete (or continuous) distribution used to select an action from action space $\mathcal { A }$ based on state $s$ . For episodic tasks, a trajectory ${ \boldsymbol { \tau } } = \{ s _ { t } , a _ { t } \} _ { t = 0 } ^ { T }$ , where $T$ is the finite time horizon, can be collected by executing the policy repeatedly. In many cases, an immediate reward $r ( s , a )$ can be received when agent executes an action. The objective of RL is to learn a policy which can maximize the expected cumulative reward along a trajectory, i.e., a discount factor. For some s $J ( \phi ) = \mathbb { E } _ { \pi _ { \phi } } [ R _ { 0 } ]$ with ard is $\begin{array} { r } { R _ { 0 } = \sum _ { t = 0 } ^ { T } \gamma ^ { t } r _ { t } ( s _ { t } , a _ { t } ) } \end{array}$ and nal t $\gamma \in ( 0 , 1 ]$ being called episodic reward), and $J ( \phi ) = \mathbb { E } _ { \pi _ { \phi } } [ R ( \tau ) ]$ with $R ( \tau ) = r _ { T } ( s _ { T } , a _ { T } )$ .
+
+# 4 METHOD
+
+In this section, we first introduce how to model the ordering search problem as an MDP, then we show how to use RL to find the optimal ordering, and we introduce how to process the searched ordering to obtain the final DAG, finally we provide a discussion on computational complexity.
+
+# 4.1 ORDERING SEARCH AS A MARKOV DECISION PROCESS
+
+We can regard the variable ordering search problem as a muti-stage decision problem with a variable selected at each decision step. We sort the selected variables according to decision steps to obtain a variable ordering, which is defined as the searched ordering. The decision-making process is Markovian, and its elements in the problem can be defined as follows:
+
+State One can directly take the sample data $\mathbf { x } _ { j }$ of each variable $X _ { j }$ as a state $s _ { j }$ . However, preliminary experiments show that it is difficult for feed-forward neural network models to capture the underlying causal relationships directly using observed data as states, and the data pre-processed by a module conventionally named encoder is helpful to finding the better ordering, see Appendix A.1. The encoder module embeds each $\mathbf { x } _ { j }$ to $s _ { j }$ and all the embedded states constitute the space $s : =$ $\{ s _ { 1 } , \ldots , s _ { d } \}$ . Adding initial state $s _ { 0 }$ to the space constitutes the complete state space $\hat { S } : = \mathcal { S } \cup { s } _ { 0 }$ . We use $\hat { s } _ { t }$ to denote the state encountered at the $t$ -th decision step.
+
+Action We select an action (variable) from the action space constituted by all the variables at each decision step, and the space size is equal to the number of variables, $| { \mathcal { A } } | = d$ . Compared to the previous RL-based method searching from the graph space with size ${ \mathcal { O } } ( 2 ^ { d \times d } )$ (Zhu et al., 2020), the search space is smaller. Note that according to the definition of ordering, the first selected variable is the source node and the last selected node is the sink node.
+
+State transition The specified state transition is related to the action selected at the current decision step. Specifically, if the selected variable is $v _ { j }$ at the $t$ -th decision step, then the state is transferred to the state $s _ { j } \in { \mathcal { S } }$ which is the $j$ -th output from Transformer encoder, i.e., $\hat { s } _ { t + 1 } = s _ { j }$ .
+
+Reward As we described in Section 3.1, only the variables that have been selected in previous decision steps can be the potential parents of the currently selected variable. Hence, we can design the rewards in the following cases: dense reward and episodic reward. For dense reward case, we can exploit the decomposability of the score function (BIC score) to calculate an immediate reward for the current decision step based on the potential parent variables (that have been selected), i.e., if $v _ { j }$ is selected at time step $t$ , the immediate reward is calculated by
+
+$$
+r _ { t } = \sum _ { k = 1 } ^ { m } \log p ( X _ { j } ^ { k } | U ( X _ { j } ) , \theta _ { j } ) - \frac { | \theta _ { j } | } { 2 } \log m ,
+$$
+
+where $U ( X _ { j } )$ denotes the potential parent variable set of $X _ { j }$ and the set consists of the variables associated with the nodes in $\Pi _ { \prec v _ { j } }$ . For episodic reward case, we directly calculate a score as an episodic reward for a complete variable ordering regardless of whether the scoring function is decomposable or not, i.e.,
+
+$$
+R ( \tau ) = r _ { T } ( \hat { s } , a ) = \mathrm { S c o r e } _ { \mathrm { B I C } } ( \mathcal { G } ^ { \Pi } )
+$$
+
+where ScoreBIC has been defined in Equation (1), with the set $\mathrm { P a } ( X _ { j } )$ of each variable $X _ { j }$ replaced by the potential parent variable set $U ( X _ { j } )$ here.
+
+# 4.2 IMPLEMENTATION AND OPTIMIZATION WITH REINFORCEMENT LEARNING
+
+We describe the neural network architectures implemented in our method, which consists of an encoder and a decoder as shown in Figure 2. Here we briefly describe the model architectures and leave details regarding model parameters to Appendix A.
+
+Encoder f enc : X˜ 7→ S is used to map the observed data to the embedding space ${ \boldsymbol { s } } \ =$ $\{ s _ { 1 } , \ldots , s _ { d } \}$ . For sample efficiency, we follow Zhu et al. (2020) to randomly draw $n$ samples from the dataset $\mathbf { X }$ to construct $\dot { \tilde { \mathbf { X } } } \in \mathbb { R } ^ { n \times d }$ at each episode and use $\tilde { \mathbf { X } }$ instead of $\mathbf { X }$ . We also set the embedding $s _ { j }$ to be in the same dimension, i.e., $s _ { j } \in \mathbb { R } ^ { n }$ . For encoder choice, we conduct an empirical comparison among several possible structures such as a selfattention based encoder (Vaswani et al., 2017) and an LSTM structure. Empirically, we confirm that the self-attention based encoder in the Transformer structure performs the best, which is also used in Zhu et al. (2020). Please find more details in Appendix A.1.
+
+![](images/bed038f6c2e3baa97c1dee3df9116cfab13aff53c55a96f7a77a859376bed0db.jpg)  
+Figure 2: Illustration of the policy model. The encoder embeds the observed data $\mathbf { x } _ { j }$ into the state $s _ { j }$ . An action $a _ { t }$ can be selected by the decoder according to the given state $\hat { s } _ { t }$ at each time step $t$ .
+
+Decoder $f _ { \phi _ { d } } ^ { \mathrm { d e c } } : \hat { S } \mapsto A$ maps the state space $\hat { S }$ to the action space $\mathcal { A }$ . Among several decoder choices (please see also Appendix A.1 for empirical comparison), we specifically pick an LSTM based structure that proves effective in our experiments. Although the initial state is usually generated randomly, we pick it as $\begin{array} { r } { s _ { 0 } = \frac { 1 } { d } \sum _ { i = 1 } ^ { d } s _ { i } } \end{array}$ , considering that the source node is fixed in correct ordering. We then restrict each node only be selected once by masking the selected nodes so as to generate a valid variable ordering (Vinyals et al., 2015).
+
+Optimization The optimization objective is to learn a policy which maximizes $J ( \phi ) = \mathbb { E } _ { \pi _ { \phi } } [ R ]$ , where $\pi _ { \phi }$ denotes the policy model parameterized by the paprameters $\{ \phi _ { e } , \phi _ { d } \}$ of encoder $f ^ { \mathrm { e n c } }$ and decoder $f ^ { \mathrm { d e c } }$ . Based on the above definition, policy gradient (Sutton $\&$ Barto, 2018) is used to optimize the ordering generation model parameters. For the dense reward case, policy gradient can be written as $\begin{array} { r } { \nabla J ( \phi ) = \mathbb { E } _ { \pi _ { \phi } } \left[ \sum _ { t = 0 } ^ { T } R _ { t } \hat { \nabla } _ { \phi } \log \pi _ { \phi } \left( a _ { t } \lvert \hat { s } _ { t } \right) \right] } \end{array}$ , where $\begin{array} { r } { R _ { t } = \sum _ { l = 0 } ^ { T - t } \gamma ^ { l } r _ { t + l } } \end{array}$ denotes the return at time step $t$ . We denote the algorithm using this reward design as CORL-1. For the episodic reward case, we have the following policy gradient $\begin{array} { r } { \nabla J ( \phi ) = \mathbb { E } _ { \pi _ { \phi } } \left[ R ( \tau ) \sum _ { t = 0 } ^ { T } \nabla _ { \phi } \log \pi _ { \phi } \left( a _ { t } | \hat { s } _ { t } \right) \right] } \end{array}$ , the algorithm using this reward design is denoted as CORL-2. Using a parametric baseline to estimate the expected score typically improves learning (Konda $\&$ Tsitsiklis, 2000). Therefore, we introduce a critic network $V _ { \phi _ { v } } ( \hat { s } _ { t } )$ parameterized by $\phi _ { v }$ , which learns the expected return given a state $\hat { s } _ { t }$ . It is trained with stochastic gradient descent using Adam optimizer on a mean squared error objective between its predicted value and the actual return. The details about the parameters of critic network are described in Appendix A.2.
+
+Inspired by the benefits of pretrained models in other tasks (Hinton & Salakhutdinov, 2012), we also consider to incorporate it to our method to accelerate training. Usually, one can obtain some observed data with known causal structure or correct ordering, e.g., by simulation or real data with labeled graph. Hence, we pretrain a policy model with such data in a supervised way.
+
+So far we presented CORL in a general manner without specifying explicitly which distribution family is during the evaluation of rewards. In principle, any distribution family could be employed as long as its log-likelihood can be computed and no differentiability is required. However, it is not always clear whether the maximization of the accumulated reward recovers the correct ordering. It will depend on both the modelling choice of reward and the underlying SEM; in fact, if the causal relationships fall into the chosen model functions and a right distribution family is assumed, then given infinite samples the optimal accumulated reward, corresponding to the negative log-likelihood, must be achieved by a super-DAG of the underlying graph according to Proposition 1. In practice, we
+
+# Algorithm 1 CORL.
+
+Require: observed data $\mathbf { X }$ , initial parameters $\theta _ { e } , \theta _ { d }$ and $\theta _ { v }$ , two empty buffers $\mathcal { D }$ and $\mathcal { D } _ { s c o r e }$ , initial   
+value (negative infinite) BestScore and a random ordering BestOrdeing.   
+1: while not terminated do   
+2: draw a batch of samples from $\mathbf { X }$ , encode them to $s$ and calculate the initial state $\hat { s } _ { 0 }$   
+3: for $t = 0 , 1 , \ldots , T$ do   
+4: collect a batch of data $\left. \hat { s } _ { t } , a _ { t } , r _ { t } \right.$ with $\pi _ { \theta }$ $\therefore D = \mathcal { D } \cup \{ \langle \hat { s } _ { t } , a _ { t } , r _ { t } \rangle \}$   
+5: if $\left. v _ { t } , \Pi _ { \prec v _ { t } } , r _ { t } \right.$ is not in $\mathcal { D } _ { s c o r e }$ then   
+6: store $\left. v _ { t } , \Pi _ { \prec v _ { t } } , r _ { t } \right.$ in $\mathcal { D } _ { s c o r e }$ to avoid repeated computations   
+7: end if   
+8: end for   
+9: update $\theta _ { e } , \theta _ { d } , \theta _ { v }$ according to Section 4.2   
+10: if $\textstyle \sum _ { t = 0 } ^ { T } r _ { t } >$ BestScore then   
+11: update the BestScore and BestOrdering   
+12: end if   
+13: end while   
+14: get the final DAG by pruning the BestOrdering
+
+can only apply approximate model functions and also need to assume certain distribution family for caculating the reward.
+
+Our method is summarized in Algorithm 1. In addition, we record the decomposed scores for each variable $v _ { j }$ with different parental sets $\Pi _ { \prec v _ { j } }$ to avoid repeated computations which are generally time-consuming (see Section 4.4). Although we cannot guarantee to find the optimal ordering because policy gradient can at most guarantee local convergence (Sutton et al., 2000) and also we only have access to the empirical log-likelihood, we remark that the ordering obtained from CORL still enjoy a good performance in the experiments, compared with consistent methods like GES and PC.
+
+# 4.3 VARIABLE SELECTION
+
+If an estimated ordering $\hat { \Pi }$ is consistent, then we obtain a fully-connected DAG (super-DAG) $\mathcal { G } ^ { \hat { \Pi } }$ of the underlying DAG $\mathcal { G }$ . One can then pursue consistent estimation of intervention distributions based on $\hat { \Pi }$ without any additional need to find the true underlying DAG $\mathcal { G } ^ { * }$ (Bühlmann et al., 2014). For other purposes, however, we need to recover the true graph from the fully-connected DAG. There exist several efficient methods such as sparse candidate (Teyssier & Koller, 2005), significance testing of covariates (Bühlmann et al., 2014), the group Lasso (Ravikumar et al., 2009), or its improved version with a sparsity-smoothness penalty proposed in Meier et al. (2009).
+
+For linear data models, we apply linear regression to the obtained fully-connected DAG, followed by thresholding to prune edges with small weights, as similarly used by Zheng et al. (2018); Yu et al. (2019); Zhu et al. (2020). For the nonlinear model, we follow the pruning process used by Bühlmann et al. (2014); Lachapelle et al. (2020). Specifically, for each variable $X _ { j }$ , one can fit a generalized additive model against the current parents of $X _ { j }$ and then apply significance testing of covariates, declaring significance if the reported p-values are lower or equal to 0.001.
+
+# 4.4 COMPUTATIONAL COMPLEXITY
+
+To learn an ordering, CORL relies on the proper training of RL model. Policy gradient and stochastic gradient are adopted to train the actor and critic respectively, which are the standard choice in RL (Konda & Tsitsiklis, 2000). Similar to RL-BIC2 (Zhu et al., 2020), CORL requires the evaluation of the rewards at each episode with $O ( d m ^ { 2 } + d ^ { 3 } )$ computational cost if linear functions are adopted to model the causal relations, but does not need to compute the matrix exponential term with $O ( d ^ { \mathbf { \hat { 3 } } } )$ cost due to the use of ordering search. In addition, CORL formulates causal discovery as a multi-stage decision process and we observe that CORL performs fewer episodes than RL-BIC2 before the episode reward converges (see Appendix C). We suspect that due to a significant reduction in the size of action space, the model complexity of the RL policy is reduced, thus leading to higher sample efficiency. The evaluation of Transformer encoder and LSTM decoder in CORL take $\bar { \mathcal { O } } ( n d ^ { 2 } )$ and $\mathcal { O } ( d n ^ { 2 } )$ , respectively. However, we find that computing rewards is more dominating in the total runing time (e.g., around $9 5 \%$ and $8 7 \%$ for 30- and 100-node linear data models, respectively). Speeding up the calculation of rewards would be helpful in extend our approach to a larger problem, which is left as a future work.
+
+In contrast with typical RL applications, we treat RL here as a search strategy for causal discovery, aiming to find an ordering that achieves the best score and then applying a variable selection method to remove redundant edges. Nevertheless, for the pretraining part with the goal of a good initialization, we may want sufficient generalization ability and hence consider diverse datasets with different number of nodes, noise types, causal relationships, etc.
+
+# 5 EXPERIMENTS
+
+In this section, we evaluate our methods against a number of methods on synthetic datasets with linear and non-linear causal relationships and also a real dataset. Specifically, these baselines include ICA-LiNGAM (Shimizu et al., 2006), three ordering-based approaches L1OBS (Schmidt et al., 2007), CAM (Bühlmann et al., 2014) and $\mathbf { A } ^ { * }$ Lasso (Xiang & Kim, 2013), some recent gradient-based approaches NOTEARS (Zheng et al., 2018), DAG-GNN (Yu et al., 2019) and GraN-DAG (Lachapelle et al., 2020), and the RL-based approach RL-BIC2 (Zhu et al., 2020). For all the compared algorithms, we use their original implementations (see Appendix B.1 for details) and pick the recommended hyper-parameters unless otherwise stated.
+
+Different types of data are generated in synthetic datasets which vary along five dimensions: level of edge sparsity, graph type, number of nodes, causal functions and sample size. Two types of graph sampling schemes: Erdös–Rényi (ER) and Scale-free (SF) are considered in our experiments. We denote $d$ -node ER or SF graphs with on average hd edges as $\mathrm { E R } h$ or $\mathrm { S F } h$ . Two common metrics are considered: True Positive Rate (TPR) and Structural Hamming Distance (SHD). The former indicates the probability of correctly finding the positive edges among the discoveries (Jain et al., 2017). Hence, it can be used to measure the quality of an ordering, the higher the better. The latter counts the total number of missing, falsely detected or reversed edges, the smaller the better.
+
+# 5.1 LINEAR DATA MODELS WITH GAUSSIAN AND NON-GAUSSIAN NOISE
+
+We evaluate the proposed methods on Linear Gaussian (LG) with equal variance Gaussian noise and LiNGAM data models, and the true DAGs in both cases are known to be identifiable (Peters & Bühlmann, 2014; Shimizu et al., 2006). We set $h \in \{ 2 , 5 \}$ and $d \in \{ 3 0 , 5 0 , 1 0 0 \}$ to generate observed data following the procedure done in Zheng et al. (2018) (see Appendix B.2 for details). For variable selection, we set the thresholding as 0.3 and apply it to the estimated coefficients.
+
+Table 1: Empirical results for ER and SF graphs of 30 nodes with LG data.   
+
+<table><tr><td></td><td></td><td>RANDOM</td><td>NOTEARS</td><td>DAG-GNN</td><td>RL-BIC2</td><td>L1OBS</td><td>A* Lasso</td><td>CORL-1</td><td>CORL-2</td></tr><tr><td>ER2</td><td>TPR</td><td>0.41 (0.04)</td><td>0.95 (0.03)</td><td>0.91 (0.05)</td><td>0.94 (0.05)</td><td>0.78 (0.06)</td><td>0.88 (0.04)</td><td>0.99 (0.02)</td><td>0.99 (0.01)</td></tr><tr><td></td><td>SHD</td><td>140.4 (36.7)</td><td>14.2 (9.4)</td><td>26.5 (12.4)</td><td>17.8 (22.5)</td><td>85.2 (23.8)</td><td>35.3 (14.3)</td><td>5.2 (7.4)</td><td>4.4 (3.5)</td></tr><tr><td>ER5</td><td>TPR</td><td>0.43 (0.03)</td><td>0.93 (0.01)</td><td>0.85 (0.11)</td><td>0.91 (0.03)</td><td>0.74 (0.04)</td><td>0.84 (0.05)</td><td>0.94 (0.03)</td><td>0.95 (0.03)</td></tr><tr><td></td><td>SHD</td><td>210.2 (43.5)</td><td>35.4 (7.3)</td><td>68.0 (39.8)</td><td>45.6 (13.3)</td><td>98.6 (32.7)</td><td>71.2 (21.5)</td><td>37.4 (16.9)</td><td>37.6 (14.5)</td></tr><tr><td>SF2</td><td>TPR</td><td>0.58 (0.02)</td><td>0.98 (0.02)</td><td>0.92 (0.09)</td><td>0.99 (0.02)</td><td>0.83 (0.04)</td><td>0.93 (0.02)</td><td>1.0 (0.01)</td><td>1.0 (0.01)</td></tr><tr><td></td><td>SHD</td><td>118.4 (12.3)</td><td>6.1 (2.3)</td><td>36.8 (33.1)</td><td>3.2 (1.7)</td><td>49.7 (28.1)</td><td>27.3 (18.4)</td><td>0.0 (0.0)</td><td>0.0 (0.0)</td></tr><tr><td>SF5</td><td>TPR</td><td>0.44 (0.03)</td><td>0.94 (0.03)</td><td>0.89 (0.09)</td><td>0.96 (0.03)</td><td>0.79 (0.04)</td><td>0.88 (0.03)</td><td>1.00 (0.00)</td><td>1.00 (0.00)</td></tr><tr><td></td><td>SHD</td><td>165.4 (10.6)</td><td>23.3 (6.9)</td><td>47.8 (35.2)</td><td>11.3 (5.2)</td><td>89.3 (25.7)</td><td>40.5 (19.8)</td><td>0.0 (0.0)</td><td>0.0 (0.0)</td></tr></table>
+
+Tables 1 &2 present results only for 30- and 100-node LG data models since the conclusions do not change with graphs of 50 nodes (see Appendix D for 50-node graphs). We report the performance of the popular ICA-LiNGAM, GraN-DAG and CAM in Appendix D since they are almost never on par with the best methods presented in this section. CORL-1 and CORL-2 achieve consistently good results on the LiNGAM datasets which are reported in Appendix E due to the space limit.
+
+We now examine Tables 1 &2 (the values in parentheses represent the standard deviation across datasets per task). We can see that, across all settings, CORL-1 and CORL-2 are the best performing methods, both in terms of TPR and SHD, while NOTEARS and DAG-GNN are not too far behind.
+
+Table 2: Empirical results for ER and SF graphs of 100 nodes with LG data.   
+
+<table><tr><td></td><td></td><td>RANDOM</td><td>NOTEARS</td><td>DAG-GNN</td><td>RL-BIC2</td><td>L1OBS</td><td>A*Lasso</td><td>CORL-1</td><td>CORL-2</td></tr><tr><td>ER2</td><td>TPR SHD</td><td>0.33 (0.05) 491.4 (17.6)</td><td>0.93 (0.02) 72.6 (23.5)</td><td>0.93 (0.03) 66.2 (19.2)</td><td>0.02 (0.01) 270.8 (13.5)</td><td>0.54 (0.02) 481.2 (49.9)</td><td>0.86 (0.04) 128.5 (38.4)</td><td>0.98 (0.02) 24.8 (10.1)</td><td>0.98 (0.01) 18.6 (5.7)</td></tr><tr><td>ER5</td><td>TPR</td><td>0.34 (0.04)</td><td>0.91 (0.01)</td><td>0.86 (0.16)</td><td>0.08 (0.03)</td><td>0.53 (0.02)</td><td>0.82 (0.05)</td><td>0.93 (0.02)</td><td>0.94 (0.03)</td></tr><tr><td></td><td>SHD</td><td>984.4 (35.7)</td><td>170.3 (34.2)</td><td>236.4 (36.8)</td><td>421.2 (46.2)</td><td>547.9 (63.4)</td><td>244.0 (42.3)</td><td>175.3 (18.9)</td><td>164.8 (17.1)</td></tr><tr><td>SF2</td><td>TPR SHD</td><td>0.48 (0.03)</td><td>0.98 (0.01)</td><td>0.89 (0.14)</td><td>0.04 (0.02)</td><td>0.57 (0.03)</td><td>0.92 (0.03)</td><td>1.00 (0.00)</td><td>1.00 (0.00)</td></tr><tr><td></td><td></td><td>503.4 (23.8)</td><td>2.3 (1.3)</td><td>156.8 (21.2)</td><td>281.2 (17.4)</td><td>377.3 (53.4)</td><td>54.0 (22.3)</td><td>0.0 (0.0)</td><td>0.0 (0.0)</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>SF5</td><td>TPR</td><td>0.47 (0.04)</td><td>0.95 (0.01)</td><td>0.87 (0.15)</td><td>0.05 (0.03)</td><td>0.55 (0.04)</td><td>0.89 (0.03)</td><td>0.97 (0.02)</td><td>0.98 (0.01)</td></tr><tr><td></td><td>SHD</td><td>891.3 (19.4)</td><td>90.2 (34.5)</td><td>165.2 (22.0)</td><td>405.2 (77.4)</td><td>503.7 (56.4)</td><td>114.0 (36.4)</td><td>19.4 (5.2)</td><td>10.8 (6.1)</td></tr></table>
+
+As we discussed previously, RL-BIC2 only achieves satisfactory results on graphs with 30 nodes. The TPR of L1OBS is lower than that of $\mathbf { A } ^ { * }$ Lasso, which indicates that L1OBS using greedy hill-climbing with tabu lists may not find a good ordering. Note that SHD of L1OBS and $\mathbf { A } ^ { * }$ Lasso reported here are the results of applying our pruning method. We observe that SHD of them is greatly improved by pruning. For example, specifically, SHD of L1OBS decreases from 171.6 (29.5), 588.0 (66.2) and 1964.5 (136.6) to 85.2 (23.8), 215.4 (26.3) and 481.2 (49.9) in LG ER2 with 30, 50 and 100 nodes respectively, while TPR almost did not degraded. It shows the effectiveness of pruning by model fitting with a thresholding.
+
+We further evaluate our method on 150-node LG ER2 data models. On the datasets, CORL-1 has TPR and SHD being 0.95 (0.01) and 63.7 (9.1), while CORL-2 has 0.97 (0.01) and 38.3 (14.3), respectively. CORL-2 outperforms NOTEARS with 0.94 (0.02) and 50.8 (21.8).
+
+Although both CORL-1 and CORL-2 achieve the desired performance, one can notice that CORL-2 achieves slightly better SHD than CORL-1. This is a little different from the usual understanding that RL is usually easier to learn from dense rewards than from episodic reward case. We take the 100-node LG ER2 data models as an example to show the training reward curves of the CORL-1 and CORL-2 in Figure 3. One can notice that CORL-2 converges faster to a better result than CORL-1, which corresponds to the fact that CORL-2 achieves the slightly better TPR and SHD than CORL-1. We conjecture that this is because it is difficult for the critic to learn to predict the score accurately for each state; in the episodic reward case, however, it is only required to learn to predict the score accurately for the initial state.
+
+![](images/a87bc88da2f55a24fad2cb7cdf2033e7b87ddf2445e53542a8c1ef896ada9006.jpg)  
+Figure 3: Learning curves of CORL-1, CORL-2 and CORL-2-pretrain on 100- node linear Gaussian dataset.
+
+Pretaining We show the training reward curve of   
+CORL-2-pretrain in Figure 3, which is CORL-2 based on a pretrained model trained in the supervised way. The testing data task is unseen in the datasets for pretraining which contain 30-node LiNGAM ER2, 50-node LiNGAM ER2, 30-node LiNGAM SF2 and 30-node GP ER1. Obviously compared to that of CORL-2 using random initialization (CORL-2), the use of a pretrained model can accelerate the model learning process. Consistent conclusion can be drawn from the experiment of CORL-1, see Appendix G. In addition, we consider using the policy model learned only on 30-node LiNGAM data model as the pretrained model on 100-node LG task. We observe that the performance is similar to that of using the pretrained model obtained in the supervised way, we do not report the results repeatedly here.
+
+# 5.2 NON-LINEAR MODEL WITH GAUSSIAN PROCESS
+
+In this experiment, we consider to use Gaussian Process (GP) to model causal relationships in which each causal relation $f _ { j }$ is a function sampled from a GP with radial basis function kernel of bandwidth one and normal Gaussian noises, which is known to be identifiable according to Peters et al. (2014). We set $h = 1$ and 4 to get ER1 and ER4 graphs, respectively, and generate data by exploiting them (see Appendix B.2 for details). Our method is evaluated on different sample numbers, the results when $m = 5 0 0$ are reported here (see Appendix F for additional results). Note that due to the efficiency of the reward calculation, we only experimented on nonlinear data models of up to 30 nodes scale. The pruning method for variable selection used here is the CAM pruning from Bühlmann et al. (2014).
+
+![](images/dec2b597023fe891b9174b627d83ad061518ff0f568f3cc35a562476a61eaa3e.jpg)  
+Figure 4: The empirical results on GP data models with 10 and 30 nodes.
+
+The results on 10-node and 30-node datasets with ER1 and ER4 graphs are shown in Figure 3. Here we only consider some baselines that are competitive in the nonlinear data models, where CAM is a very strong ordering based baseline. Although GraN-DAG achieves better results than DAG-GNN, they are worse than CAM overall. We believe this is because 500 samples are so small that GraN-DAG and DAG-GNN have not learned the good model. RL-BIC2 performs well on 10-node datasets, but achieves poor results on 30-node datasets, probably due to its lack of scalability. CAM, CORL-1 and CORL-2 have good results, with CORL-2 achieving the best results on 10-node graphs and slightly worse than CAM on 30-node graphs. All of these methods have better results on ER1 than on ER4, especially on 30-node graphs.
+
+# 5.3 REAL DATA
+
+The Sachs dataset (Sachs et al., 2005), with 11-node and 17-edge true graph, is widely used for research on graphical models. The expression levels of protein and phospholipid in the dataset can be used to discover the implicit protein signal network. The observational dataset has $m = 8 5 3$ samples and is used to discover the causal structure. GP is used to model the causal relationship in our method. In this experiment, CORL-1, CORL-2 and RL-BIC2 achieve the best SHD 11. CAM, GraN-DAG, and ICA-LiNGAM achieve the SHD 12, 13 and 14, respectively. Particularly, DAG-GNN and NOTEARS result in SHD 16 and 19, respectively, whereas an empty graph has an SHD 17.
+
+# 6 CONCLUSION
+
+In this work, we have proposed a RL-based approach for causal discovery named CORL. It searches the space of variable orderings, instead of the space of directed graphs. We formulate ordering search as an MDP and have further proposed CORL-1 and CORL-2 for training the ordering generating model. For a generated ordering, it can be pruned by variable selection to the final DAG. The empirical results on the synthetic and the real datasets show that our approach is promising.
+
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+
+# APPENDIX
+
+# A NETWORK ARCHITECTURES AND HYPER-PARAMETERS
+
+A.1 MULTIPLE NETWORK ARCHITECTURE DESIGNS FOR ENCODER AND DECODER
+
+There are a variety of neural network modules for encoder and decoder structures. Here we consider some representative modules: including , Multi-layer Perceptrons (MLP) module, an LSTM based recurrent neural network module, and the self-attention based encoder in the Transformer. In addition, we use the original observation data as the state directly, i.e., no encoder module is used, to show the necessity of encoder, which is denoted as Null. MLP consists of 3-layer feed-forward neural networks with 256, 512 and 256 units. The architecture of LSTM and Transformer are that introduced in Appendix A.2.
+
+The empirical results of CORL-2 on 30-node LG ER2 datasets are reported in Table 3. We observe that LSTM decoder achieves a better performance than that of MLP decoder. This shows that LSTM is more effective than MLP in sequential decision tasks. Besides, the overall performance of neural network encoder is better than that of Null, which shows that the data pre-processed by encoder module is necessary. Among all these encoders, Transformer encoder achieves the best results. Similar conclusion was drawned in Zhu et al. (2020). We hypothesize that the performance of Transformer encoder is benefit from the self-attention scheme.
+
+Table 3: Empirical results of CORL-2 with different choices of encoder and decoder on 30-node LG ER2 datasets. The smaller SHD the better, the higher TPR the better.   
+
+<table><tr><td rowspan="2" colspan="2"></td><td colspan="4">Encoder</td></tr><tr><td>Null</td><td>LSTM</td><td>MLP</td><td>Transformer</td></tr><tr><td rowspan="2">MLP Decoder</td><td>TPR</td><td>0.81 (0.07)</td><td>0.86 (0.10)</td><td>0.96 (0.02)</td><td>0.98 (0.02)</td></tr><tr><td>SHD</td><td>54.2 (25.1)</td><td>36.0 (26.7)</td><td>11.0 (5.3)</td><td>5.0 (3.3)</td></tr><tr><td rowspan="2">LSTMDecoder</td><td>TPR</td><td>0.94 (0.04)</td><td>0.88 (0.09)</td><td>0.97 (0.01)</td><td>0.99 (0.01)</td></tr><tr><td>SHD</td><td>20.6 (20.0)</td><td>29.0 (17.8)</td><td>8.6 (4.3)</td><td>4.4 (3.5)</td></tr></table>
+
+A.2 NETWORK ARCHITECTURE AND HYPER-PARAMETERS OF CORL
+
+![](images/9f88b5954cb118a9e912f65a4df0f134a845ef9eb629d40c85ad8ee23c5b0fd6.jpg)  
+Figure 5: Illustration of the Transformer encoder. The encoder embeds the observed data $\mathrm { x } _ { j }$ of each variable $j$ into the state $s _ { j }$ . Notation block $\textcircled { \alpha } 3$ denotes three blocks here.
+
+Both CORL-1 and CORL-2 use the actor-critic algorithm. We use the Adam optimizer with learning rate $1 e { - 4 }$ and $1 e { - 3 }$ for actor and critic respectively. The discount factor $\gamma$ is set to 0.98. The actor consists of an encoder and a decoder. We illustrate the neural network structure of the Transformer encoder used in our experiments in Figure 5. It consists of a feed-forward layer with 256 units and three blocks. Each block is composed of a multi-head attention network with 8 heads and 2-layer feed-forward neural networks with 1024 and 256 units. Each feed-forward layer is followed by a normalization layer. Given a batch of observed samples with shape $b \times d \times n$ , where $b$ denotes the batch size, $d$ the node number and $n$ the number of observed data for each variable in a batch, the final output of the encoder is a batch of embedded state with shape $b \times d \times 2 5 6$ .
+
+![](images/a193ed85b4b7d459c37b36cf01884ec7dfb4cfcc588f9a89f2d43e101e0c1bce.jpg)  
+Figure 6: Illustration of the LSTM decoder. At each time step, it maps the state $\hat { s } _ { t }$ to a distribution over action space $\mathcal { A } : = \{ a _ { 1 } , \ldots , a _ { d } \}$ , then an action (variable) can be selected randomly according to the distribution.
+
+We illustrate the neural network structure of the decoder in Figure 6, which is mainly a LSTM similar to the decoder proposed by Vinyals et al. (2015). The LSTM takes a state as input and outputs a embedding. The embedding is mapped to the action space $\mathcal { A }$ by using some feed-forward neural networks, a soft-max module and the pointer mechanism (Vinyals et al., 2015). The outputs of the encoder are processed as the initial hidden state $h _ { 0 }$ of the decoder. The LSTM with 256 hidden units is used here. All of the feed-forward neural networks used in decoder have 256 units.
+
+The critic uses 3-layer feed-forward neural networks with 512, 256 and 1 units. It takes a state $\hat { s }$ as input and outputs a predicted value for the current policy given state $\hat { s }$ . For CORL-1, the critic needs to predict the score for each state. For CORL-2, the critic takes the initial state $\hat { s } _ { 0 }$ as input and outputs a predicted value directly for a complete ordering.
+
+# B BASELINES AND DATE SETS
+
+# B.1 DETAILS OF BASELINES
+
+The details of all the baselines considered in our experiments are listed as follows:
+
+• ICA-LiNGAM assumes linear non-Gaussian additive model for data generating procedure and applies independent component analysis to recover the weighted adjacency matrix. This method can usually achieve good performance on LiNGAM datasets. However, it does not provide guarantee for linear Gaussian datasets. 1
+
+• NOTEARS recovers the causal graph by estimating the weighted adjacency matrix with the least squares loss and the smooth characterization for acyclicity constraint.
+
+• DAG-GNN formulates causal discovery in the framework of variational autoencoder and optimizes a weighted adjacency matrix with the evidence lower bound and a modified smooth characterization on acyclicity as loss function.
+
+• GraN-DAG models the conditional distribution of each variable given its parents with feedforward neural networks. It uses the smooth directed acyclic constraint from NOTEARS to find a DAG that maximizes the log-likelihood of the observed samples. 4 RL-BIC2 formulates the causal discovery as a one-step decision making process and uses score function and acyclic constraint from NOTEARS to calculate the reward for the recovered directed graph. 5 CAM conducts a greedy estimation procedure that starts with an empty DAG and adds at each iteration the edge $( v _ { k } , v _ { j } )$ between nodes $v _ { k }$ and $v _ { j }$ that corresponds to the largest gain in log-likelihood. For a searched ordering, they prune it to the final DAG by applying significance testing of covariates. They perform the preliminary neighborhood selection to reduce the ordering space size searched. 6   
+L1OBS performs heuristic search (greedy hill-climbing with tabu lists) through the space of topological orderings to search a ordering with the best score, then it uses L1 variable selection to pruning the searched ordering (fully-connected DAG) to the final DAG. 7 $\mathbf { A } ^ { * }$ Lasso with a limited queue size incorporates a heuristic scheme into a dynamic programming based method. It first prunes the search space by using $\mathbf { A } ^ { * }$ Lasso, then it further pruning the search space by limiting the size of the priority queue in the OPEN list of $\mathbf { A } ^ { * }$ Lasso. The queue size usually needs to be adjusted to balance the time cost and the quality of the solution. 8
+
+# B.2 DETAILS ON DATASETS GENERATION
+
+We generate synthetic datasets which vary along five dimensions: level of edge sparsity, graph type, number of nodes, causal functions and sample size. We sampled 5 datasets of the required number examples for each task as follows: a ground truth DAG $\mathcal { G }$ is drawn randomly from either the Erdös–Rényi (ER) or Scale-free (SF) graph model; then, the data is generated according to a specific sampling scheme.
+
+Specifically, for Linear Gaussian (LG) case, we set $h \in \{ 2 , 5 \}$ and $d \in \{ 3 0 , 5 0 , 1 0 0 \}$ to obtain the ER graph and SF graph (different types) with different levels of edge sparsity and different number of nodes, respectively. Then 5 datasets of 3000 examples are generated for each task following $\mathbf { X } = W ^ { T } \mathbf { X } + \mathbf { \bar { \Gamma } } \epsilon$ , where $W \in \mathbb { R } ^ { d \times d }$ denotes the weight matrix which is obtained by assigning edge weights independently from $\operatorname { U n i f } ( [ - 2 , - 0 . 5 ] \cup [ 0 . 5 , 2 ] )$ . Note that $\epsilon$ ’s are standard Gaussian noises with equal variances for each variable so as to LG data model is identifiable (Peters & Bühlmann, 2014).
+
+For LiNGAM data model, the datasets are generated in the similar way with LG but the sampling for $\epsilon$ are different. The non-Gaussian noises are obtained by following Shimizu et al. (2006) which passes the noise samples from Gaussian distribution through a power nonlinearity to make them non-Gaussian. LiNGAM is identifiable shown in Shimizu et al. (2006).
+
+Another data generating process is GP. We first obtain graph with different density and different number of nodes. Then the datasets with different sample sizes are generated following $X _ { j } ~ =$ $f _ { j } ( \mathbf { P a } ( X _ { j } ) ) + \epsilon _ { j }$ with jointly independent Gaussian noises for all $j$ where the function $f _ { j }$ is a function sampled from a GP with radial basis function kernel of bandwidth one and normal Gaussian noises. This setting is known to be identifiable according to Peters et al. (2014). Note that due to the efficiency of the reward calculation, we only experimented on nonlinear data models of up to 30 nodes.
+
+# C TOTAL NUMBER OF EPISODES BEFORE CONVERGENCE
+
+Table 4 reports the total number of episodes required for CORL-2 and RL-BIC2 to be converged, averaged over five datasets. Note that the episodic reward is evaluated once per episode. CORL formulates causal discovery as a multi-stage decision process and we observe that CORL performs fewer episodes than RL-BIC2 before the episode reward converges. We suspect that due to a significant reduction in the size of action space, the model complexity of the RL policy is reduced thus leading to higher sample efficiency. Some runtimes about them are also provided here (CORL-2 total runtime $\approx 1 5$ minutes against RL- $\mathrm { B I C } 2 \approx 3$ hours for 30-node ER2 graphs, $\approx 4$ hours against $\approx 1 4$ hour for 50-node ER2 graphs, and CORL- ${ \mathrm { } } 2 \approx 7$ hours for 100-node ER2 graphs). We set the maximal runtime up to 24 hours, but RL-BIC2 did not converge within that time on 100-node graphs, hence we did not report it here. Note that these runtime may be significantly reduced by parallelizing the evaluation of reward.
+
+Table 4: Total number of iterations $( \times 1 0 ^ { 3 } )$ before RL converge on LG data.   
+
+<table><tr><td rowspan="2"></td><td colspan="2">30 nodes</td><td colspan="2">50 nodes</td><td colspan="2">100 nodes</td></tr><tr><td>ER2</td><td>ER5</td><td>ER2</td><td>ER5</td><td>ER2</td><td>ER5</td></tr><tr><td>CORL-2</td><td>1.0 (0.3)</td><td>1.1 (0.4)</td><td>1.9 (0.3)</td><td>2.4 (0.3)</td><td>2.3 (0.5)</td><td>2.9 (0.4)</td></tr><tr><td>RL-BIC2</td><td>3.9 (0.5)</td><td>4.1 (0.6)</td><td>3.4 (0.4)</td><td>3.5 (0.5)</td><td>×</td><td>×</td></tr></table>
+
+# D ADDITIONAL RESULTS ON LINEAR GAUSSIAN DATASETS
+
+The results for 50-node LG data models are presented in Table 5. The conclusion is similar to the 30- and 100-node experiments. The results of ICA-LiNGAM, GraN-DAG and CAM on LG data models are presented in Table 6. Their performances do not compare favorably to CORL-1 nor CORL-2 in LG datasets.
+
+It is not surprising that ICA-LiNGAM does not perform well because the algorithm is specifically designed for non-Gaussian noise and does not provide guarantee for LG data models. We hypothesize that CAM’s poor performance on LG data models is because it uses nonlinear regression instead of linear regression. As for GraN-DAG, it uses 2-layer feed-forward neural networks to model the causal relationships, which may not be able to learn a good linear relationship in this experiment.
+
+Table 5: Empirical results for ER and SF graphs of 50 nodes with LG data. The higher TPR the better, the smaller SHD the better.   
+
+<table><tr><td></td><td></td><td>RANDOM</td><td>NOTEARS</td><td>DAG-GNN</td><td>RL-BIC2</td><td>L1OBS</td><td>A*Lasso</td><td>CORL-1</td><td>CORL-2</td></tr><tr><td>ER2</td><td>TPR</td><td>0.31(0.03) 295.4 (28.5)</td><td>0.94 (0.02) 38.6 (10.8)</td><td>0.94 (0.04)</td><td>0.79 (0.10)</td><td>0.56 (0.02)</td><td>0.88 (0.03)</td><td>0.97 (0.04) 24.0 (32.3</td><td>0.97 (0.02)</td></tr><tr><td></td><td>SHD TPR</td><td>0.32 (0.02)</td><td></td><td>30.6 (8.3)</td><td>88.5 (49.3)</td><td>288.0 (66.2)</td><td>154.3 (27.6)</td><td></td><td>17.9 (10.6)</td></tr><tr><td>ER5</td><td>SHD</td><td>378.4 (24.2)</td><td>0.90 (0.01) 67.8 (7.5)</td><td>0.87(0.14) 93.2 (109.4)</td><td>0.74 (0.03) 128.9 (40.4)</td><td>0.57 (0.03) 299.4 (53.6)</td><td>0.82 (0.03) 104.0 (28.3)</td><td>0.90 (0.02) 68.3 (10.2)</td><td>0.92 (0.02) 64.8 (13.1)</td></tr><tr><td>SF2</td><td>TPR</td><td>0.49 (0.04)</td><td>0.99 (0.01)</td><td>0.90 (0.13)</td><td>0.84 (0.05)</td><td>0.67 (0.02)</td><td></td><td>1.00 (0.00)</td><td>1.00 (0.00)</td></tr><tr><td></td><td>SHD</td><td>215.6 (14.7)</td><td>3.5 (1.6)</td><td>79.3 (93.2)</td><td>115.2 (57.4)</td><td>182.3 (33.4)</td><td>0.89 (0.03) 124.0 (35.2)</td><td>0.0 (0.0)</td><td>0.0 (0.0)</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>SF5</td><td>TPR</td><td>0.51 (0.03)</td><td>0.94 (0.12)</td><td>0.88 (0.12)</td><td>0.75 (0.05)</td><td>0.61 (0.03)</td><td>0.81 (0.02)</td><td>0.94 (0.03)</td><td>0.95 (0.02)</td></tr><tr><td></td><td>SHD</td><td>345.6 (24.3)</td><td>20.1 (14.3)</td><td>89.2 (99.2)</td><td>115.2 (57.4)</td><td>217.3 (36.4)</td><td>131.0 (25.3)</td><td>24.3 (11.2)</td><td>20.8 (10.1)</td></tr></table>
+
+# E RESULTS ON 30-, 50- AND 100-NODE LINGAM DATASETS
+
+Here, we report the empirical results on 30-, 50- and 100-node LiNGAM datasets in Table 7. The observed samples are generated according to the same procedure with linear Gaussian datasets (see Appendix B.2 for details). The non-Gaussian noise is obtained by passing the noise samples from Gaussian distribution through a power nonlinearity to make them non-Gaussian. For L1OBS, we increased the authors’ recommended number of evaluations 2500 to 10000. For $\mathbf { A } ^ { * }$ Lasso, we set the queue size to 10, 500 and 1000, and report the best result for all of these different parameter settings. The results of L1OBS and $\mathbf { A } ^ { * }$ Lasso reported here are that of pruning using our pruning method. For other baselines, we pick the recommended hyper-parameters.
+
+Among all these algorithms, ICA-LiNGAM can recover the true graph on most of the LiNGAM data models. This is because ICA-LiNGAM is specifically designed for non-Gaussian noise. CORL-1 and CORL-2 achieve consistently good results than other baselines.
+
+Table 6: Empirical results of ICA-LiNGAM, GraN-DAG and CAM (against CORL-2 for reference) for ER and SF graphs with LG data. The higher TPR the better, the smaller SHD the better.   
+
+<table><tr><td></td><td></td><td></td><td>ICA-LiNGAM</td><td>GraN-DAG</td><td>CAM</td><td>CORL-2</td></tr><tr><td>30 nodes</td><td>ER2</td><td>TPR</td><td>0.75 (0.03)</td><td>0.51 (0.17)</td><td>0.47 (0.05)</td><td>0.99 (0.01)</td></tr><tr><td></td><td></td><td>SHD</td><td>112.3 (12.8)</td><td>96.0 (11.3)</td><td>110.8 (10.3)</td><td>4.4 (3.5))</td></tr><tr><td></td><td>ER5</td><td>TPR SHD</td><td>0.57 (0.03) 161.8 (9.2)</td><td>0.52 (0.03) 175.2 (27.4)</td><td>0.46 (0.02) 191.3 (32.5)</td><td>0.95 (0.03) 37.6 (14.5)</td></tr><tr><td></td><td>SF2</td><td>TPR</td><td>0.58 (0.05)</td><td>0.61 (0.04)</td><td>0.63 (0.02)</td><td>1.0 (0.0)</td></tr><tr><td></td><td></td><td>SHD</td><td>149.0 (19,8)</td><td>136.4 (21.2)</td><td>115.2 (26.7)</td><td>0.0 (0.0)</td></tr><tr><td></td><td>SF5</td><td>TPR</td><td>0.56 (0.04)</td><td>0.58 (0.02)</td><td>0.60 (0.03)</td><td>1.0 (0.0)</td></tr><tr><td></td><td></td><td>SHD</td><td>160.5 (8.9)</td><td>142.4 (24.3)</td><td>122.2 (17.4)</td><td>0.0 (0.0)</td></tr><tr><td>50 nodes</td><td>ER2</td><td>TPR</td><td>0.73 (0.03)</td><td>0.11 (0.04)</td><td>0.55 (0.06)</td><td>0.97 (0.02)</td></tr><tr><td></td><td></td><td>SHD</td><td>108.8 (11.3)</td><td>173.0 (22.9)</td><td>140.8 (35.4)</td><td>17.9 (10.6)</td></tr><tr><td></td><td>ER5</td><td>TPR</td><td>0.57 (0.01)</td><td>0.64 (0.03)</td><td>0.61 (0.02)</td><td>0.92 (0.02)</td></tr><tr><td></td><td></td><td>SHD</td><td>199.8 (90.7)</td><td>154.2 (36.4)</td><td>178.3 (34.8)</td><td>64.8 (13.1)</td></tr><tr><td></td><td>SF2</td><td></td><td>0.59 (0.04)</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td>TPR SHD</td><td>208.5 (83.2)</td><td>0.44 (0.05) 158.6 (34.5)</td><td>0.57 (0.02) 131.2 (24.4)</td><td>1.00 (0.00) 0.0 (0.0)</td></tr><tr><td></td><td>SF5</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td>TPR SHD</td><td>0.57 (0.01) 216.6 (88.4)</td><td>0.49 (0.04) 243.9 (27.2)</td><td>0.53 (0.03) 235.2 (34.2)</td><td>0.95 (0.02) 20.8 (10.1)</td></tr><tr><td>100 nodes</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>ER2</td><td>TPR</td><td>0.73 (0.02) 268.4 (28.5)</td><td>0.38 (0.02)</td><td>0.43 (0.02)</td><td>0.98 (0.01)</td></tr><tr><td></td><td></td><td>SHD</td><td></td><td>191.3 (31.9)</td><td>126.4 (27.8)</td><td>18.6 (5.7)</td></tr><tr><td></td><td>ER5</td><td>TPR</td><td>0.57 (0.05)</td><td>0.42 (0.03)</td><td>0.47 (0.02)</td><td>0.94 (0.03)</td></tr><tr><td></td><td></td><td>SHD</td><td>311.1 (63.7)</td><td>208.2 (54.4)</td><td>182.3 (34.9)</td><td>164.8 (17.1)</td></tr><tr><td></td><td>SF2</td><td>TPR</td><td>0.69 (0.03)</td><td>0.40 (0.03)</td><td>0.44 (0.02)</td><td>1.00 (0.00)</td></tr><tr><td></td><td></td><td>SHD</td><td>367.6 (67.5)</td><td>239.9 (43.2)</td><td>35.2 (37.4)</td><td>0.0 (0.0)</td></tr><tr><td></td><td>SF5</td><td>TPR</td><td>0.57 (0.05)</td><td>0.39 (0.03)</td><td>0.48 (0.04)</td><td>0.98 (0.01)</td></tr><tr><td></td><td></td><td>SHD</td><td>362.3 (82.8)</td><td>219.3 (32.2)</td><td>125.2 (24.7)</td><td>10.8 (6.1)</td></tr></table>
+
+Table 7: Empirical results on 30-, 50- and 100-node LiNGAM ER2 datasets. The smaller SHD the better, the higher TPR the better.   
+
+<table><tr><td></td><td colspan="2">30 nodes ER2</td><td colspan="2">50 nodes ER2</td><td colspan="2">100 nodes ER2</td></tr><tr><td>Method</td><td>TPR</td><td>SHD</td><td>TPR</td><td>SHD</td><td>TPR</td><td>SHD</td></tr><tr><td>ICA-LiNGAM</td><td>1.00 (0.00)</td><td>0.0 (0.0)</td><td>1.00 (0.00)</td><td>0.0 (0.0)</td><td>1.00 (0.00)</td><td>1.0 (0.9)</td></tr><tr><td>NOTEARS</td><td>0.94 (0.04)</td><td>17.2 (13.2)</td><td>0.95 (0.02)</td><td>33.2 (16.5)</td><td>0.94 (0.03)</td><td>69.2 (23.2)</td></tr><tr><td>DAG-GNN</td><td>0.94 (0.03)</td><td>19.6 (10.5)</td><td>0.96 (0.01)</td><td>24.6 (2.9)</td><td>0.93 (0.03)</td><td>66.2 (19.2)</td></tr><tr><td>GraN-DAG</td><td>0.28 (0.09)</td><td>100.8 (14.6)</td><td>0.20 (0.01)</td><td>177.0 (25.9)</td><td>0.16 (0.04)</td><td>312.8 (25.2)</td></tr><tr><td>RL-BIC2</td><td>0.94 (0.07)</td><td>19.8 (23.0)</td><td>0.80 (0.08)</td><td>86.0 (51.9)</td><td>0.13 (0.12)</td><td>291.3 (24.1)</td></tr><tr><td>CAM</td><td>0.60 (0.11)</td><td>310.0 (34.0)</td><td>0.33 (0.07)</td><td>178.0 (31.9)</td><td>0.53 (0.05)</td><td>247.2 (32.1)</td></tr><tr><td>L1OBS</td><td>0.72 (0.04)</td><td>85.3 (23.3)</td><td>0.47 (0.02)</td><td>212.6 (24.6)</td><td>0.41 (0.03)</td><td>470.5 (48.1)</td></tr><tr><td>A* Lasso</td><td>0.87 (0.03)</td><td>42.3 (16.3)</td><td>0.88 (0.03)</td><td>82.6 (17.6)</td><td>0.85 (0.04)</td><td>102.5 (22.6)</td></tr><tr><td>CORL-1</td><td>0.99 (0.01)</td><td>3.8 (6.4)</td><td>0.96 (0.06)</td><td>24.6 (37.7)</td><td>0.98 (0.01)</td><td>20.0 (7.9)</td></tr><tr><td>CORL-2</td><td>0.99 (0.01)</td><td>3.9 (5.6)</td><td>0.96 (0.08)</td><td>20.2 (11.3)</td><td>0.99 (0.01)</td><td>13.8 (7.2)</td></tr></table>
+
+# F RESULTS ON 20-NODE GP DATASETS WITH DIFFERENT SAMPLE SIZES
+
+We take the 20-node GP data models as an example to show the performance of our method on different sample numbers. We set $h = 4$ to get ER4 graphs and generate data by using them. We illustrate the empirical results in Table 8. Since previous experiments have shown that CORL-2 is slightly better than CORL-1, we only report the results of CORL-2 here. CAM as the most competitive baseline, we also report its results on these datasets. TPR reported here is calculated based on the variable ordering. We can see that, as the sample size decreases, CORL-2 ends up outperforming CAM. We believe this is because CORL-2 benefits from the exploratory ability of RL.
+
+# G CORL-1 WITH A PRETRAINED MODEL
+
+We show the training reward curves of CORL-1 and CORL-1-pretrain which is CORL-1 based on a pretrained model in Figure 7. To obtain a pretraining model with good generalization ability, we combine various types data described in Appendix B.2 with different levels of edge sparsity, graph
+
+Table 8: Empirical results on 20-node GP ER1 datasets with different sample sizes. The smaller SHD the better, the higher TPR the better.   
+
+<table><tr><td>Sample size</td><td>Name</td><td>TPR</td><td>SHD</td></tr><tr><td>1000</td><td>CAM CORL-2</td><td>0.91 (0.03) 0.87 (0.03)</td><td>30.0 (3.7) 36.5 (3.1)</td></tr><tr><td>500</td><td>CAM CORL-2</td><td>0.86 (0.03) 0.85 (0.03)</td><td>45.0 (2.5) 46.3 (2.3)</td></tr><tr><td>400</td><td>CAM CORL-2</td><td>0.83 (0.02) 0.84 (0.03)</td><td>51.0 (2.7) 50.5 (3.0)</td></tr><tr><td>200</td><td>CAM CORL-2</td><td>0.60 (0.03) 0.75 (0.02)</td><td>66.3 (1.9) 63.1 (1.5)</td></tr></table>
+
+![](images/818183bee85f0d445744cdd88e28f3a9495545ef84ab66bb468d11891c255001.jpg)  
+Figure 7: Learning curves of CORL-1 and CORL-1-pretrain on 100-node LG datasets.
+
+type and number of nodes and data generating process to construct observation samples. Next, we train a policy model by supervised learning on the mixed datasets. Finally, we use the pretrained model as a start point on the task that has never been seen before during pretraining.
+
+From Figure 7, we can observe that although the pretrained model is trained on other types of datasets, it can still accelerate the training as the initial model on the LG dataset task. This shows that our policy model may learn some implicit knowledge across different tasks.

parse/train/eo6U4CAwVmg/eo6U4CAwVmg.md

@@ -0,0 +1,438 @@
+# TRAINING GANS WITH STRONGER AUGMENTATIONS VIA CONTRASTIVE DISCRIMINATOR
+
+Jongheon Jeong1 & Jinwoo $\mathbf { S h i n } ^ { 2 , 1 }$   
+1School of Electrical Engineering 2Graduate School of AI   
+Korea Advanced Institute of Science and Technology (KAIST)   
+Daejeon 34141, South Korea   
+{jongheonj,jinwoos}@kaist.ac.kr
+
+# ABSTRACT
+
+Recent works in Generative Adversarial Networks (GANs) are actively revisiting various data augmentation techniques as an effective way to prevent discriminator overfitting. It is still unclear, however, that which augmentations could actually improve GANs, and in particular, how to apply a wider range of augmentations in training. In this paper, we propose a novel way to address these questions by incorporating a recent contrastive representation learning scheme into the GAN discriminator, coined ContraD. This “fusion” enables the discriminators to work with much stronger augmentations without increasing their training instability, thereby preventing the discriminator overfitting issue in GANs more effectively. Even better, we observe that the contrastive learning itself also benefits from our GAN training, i.e., by maintaining discriminative features between real and fake samples, suggesting a strong coherence between the two worlds: good contrastive representations are also good for GAN discriminators, and vice versa. Our experimental results show that GANs with ContraD consistently improve FID and IS compared to other recent techniques incorporating data augmentations, still maintaining highly discriminative features in the discriminator in terms of the linear evaluation. Finally, as a byproduct, we also show that our GANs trained in an unsupervised manner (without labels) can induce many conditional generative models via a simple latent sampling, leveraging the learned features of ContraD. Code is available at https://github.com/jh-jeong/ContraD.
+
+# 1 INTRODUCTION
+
+Generative adversarial networks (GANs) (Goodfellow et al., 2014) have become one of the most prominent approaches for generative modeling with a wide range of applications (Ho & Ermon, 2016; Zhu et al., 2017; Karras et al., 2019; Rott Shaham et al., 2019). In general, a GAN is defined by a minimax game between two neural networks: a generator network that maps a random vector into the data domain, and a discriminator network that classifies whether a given sample is real (from the training dataset) or fake (from the generator). Provided that both generator and discriminator attain their optima at each minimax objective alternatively, it is theoretically guaranteed that the generator implicitly converges to model the data generating distribution (Goodfellow et al., 2014).
+
+Due to the non-convex/stationary nature of the minimax game, however, training GANs in practice is often very unstable with an extreme sensitivity to many hyperparameters (Salimans et al., 2016; Lucic et al., 2018; Kurach et al., 2019). Stabilizing the GAN dynamics has been extensively studied in the literature (Arjovsky et al., 2017; Gulrajani et al., 2017; Miyato et al., 2018; Wei et al., 2018; Jolicoeur-Martineau, 2019; Chen et al., 2019; Schonfeld et al., 2020), and the idea of incorporating data augmentation techniques has recently gained a particular attention on this line of research: more specifically, Zhang et al. (2020) have shown that consistency regularization between discriminator outputs of clean and augmented samples could greatly stabilize GAN training, and Zhao et al. (2020c) further improved this idea. The question of which augmentations are good for GANs has been investigated very recently in several works (Zhao et al., 2020d; Tran et al., 2021; Karras et al., 2020a; Zhao et al., 2020a), while they unanimously conclude only a limited range of augmentations (e.g., flipping and spatial translation) were actually helpful for the current form of training GANs.
+
+Meanwhile, not only for GANs, data augmentation has also been played a key role in the literature of self-supervised representation learning (Doersch et al., 2015; Gidaris et al., 2018; Wu et al., 2018), especially with the recent advances in contrastive learning (Bachman et al., 2019; Oord et al., 2018; Chen et al., 2020a;b; Grill et al., 2020): e.g., Chen et al. (2020a) have shown that the performance gap between supervised- and unsupervised learning can be significantly closed with large-scale contrastive learning over strong data augmentations. In this case, contrastive learning aims to extract the mutual information shared across augmentations, so good augmentations for contrastive learning should keep information relevant to downstream tasks (e.g., classification), while discarding nuisances for generalization. Finding such augmentations is still challenging, yet in some sense, it is more tangible than the case of GANs, as there are some known ways to formulate the goal rigourously, e.g., InfoMax (Linsker, 1988) or InfoMin principles (Tian et al., 2020).
+
+Contribution. In this paper, we propose Contrastive Discriminator (ContraD), a new way of training discriminators of GAN that incorporates the principle of contrastive learning. Specifically, instead of directly optimizing the discriminator network for the GAN loss, ContraD uses the network mainly to extract a contrastive representation from a given set of data augmentations and (real or generated) samples. The actual discriminator that minimizes the GAN loss is defined independently upon the contrastive representation, which turns out that a simple 2-layer network is sufficient to work as a complete GAN. By design, ContraD can be naturally trained with augmentations used in the literature of contrastive learning, e.g., those proposed by SimCLR (Chen et al., 2020a), which are in fact much stronger than typical practices in the context of GAN training (Zhang et al., 2020; Zhao et al., 2020c;a; Karras et al., 2020a). Our key observation here is that, the task of contrastive learning (to discriminate each of independent real samples) and that of GAN discriminator (to discriminate fake samples from the reals) benefit each other when jointly trained with a shared representation.
+
+Self-supervised learning, including contrastive learning, have been recently applied in GAN as an auxiliary task upon the GAN loss (Chen et al., 2019; Tran et al., 2019; Lee et al., 2021; Zhao et al., 2020d), mainly in attempt to alleviate catastopic forgetting in discriminators (Chen et al., 2019). For conditional GANs, Kang & Park (2020) have proposed a contrastive form of loss to efficiently incorporate a given conditional information into discriminators. Our work can be differentiated to these prior works in a sense that, to the best of our knowledge, it is the first method that successfully leverage contrastive learning alone to incorporate a wide range of data augmentations in GAN training. Indeed, for example, Zhao et al. (2020d) recently reported that simply regularizing auxiliary SimCLR loss (Chen et al., 2020a) improves GAN training, but could not outperform existing methods based on simple data augmentations, e.g., bCR (Zhao et al., 2020c).
+
+# 2 BACKGROUND
+
+Generative adversarial networks. We consider a problem of learning a generative model $p _ { g }$ from a given dataset $\{ { \pmb x } _ { i } \} _ { i = 1 } ^ { N }$ , where $\mathbf { \boldsymbol { x } } _ { i } \sim p _ { \mathrm { d a t a } }$ and $\mathbf { { \boldsymbol { x } } } _ { i } \in { \mathcal { X } }$ . To this end, generative adversarial network (GAN) (Goodfellow et al., 2014) considers two neural networks: (a) a generator network $G : { \mathcal { Z } } $ $\mathcal { X }$ that maps a latent variable $\mathbf { z } \sim p ( \mathbf { z } )$ into $\mathcal { X }$ , where $p ( \mathbf { z } )$ is a specific prior distribution, and (b) a discriminator network $D : \mathcal { X }  [ 0 , 1 ]$ that discriminates samples from $p _ { \mathrm { d a t a } }$ and those from the implicit distribution $p _ { g }$ derived from $G ( \mathbf { z } )$ . The primitive form of training $G$ and $D$ is the following:
+
+$$
+\operatorname* { m i n } _ { G } \operatorname* { m a x } _ { D } V ( G , D ) : = \mathbb { E } _ { \mathbf { x } \sim p _ { \mathrm { d a t a } } } [ \log ( D ( \mathbf { x } ) ) ] + \mathbb { E } _ { \mathbf { z } \sim p ( \mathbf { z } ) } [ \log ( 1 - D ( G ( \mathbf { z } ) ) ) ] .
+$$
+
+For a fixed $G$ , the inner maximization objective (1) with respect to $D$ leads to the following optimal discriminator $D _ { G } ^ { * }$ , and consequently the outer minimization objective with respect to $G$ becomes to minimize the Jensen-Shannon divergence between $p _ { \mathrm { d a t a } }$ and $\begin{array} { r } { p _ { g } \colon D _ { G } ^ { * } : = \operatorname* { m a x } _ { D } V ( G , D ) = \frac { p _ { \mathrm { d a t a } } } { p _ { \mathrm { d a t a } } + p _ { g } } } \end{array}$ Although this formulation (1) theoretically guarantees $p _ { g } ^ { * } = p _ { \mathrm { d a t a } }$ as the global optimum, the nonsaturating loss (Goodfellow et al., 2014) is more favored in practice for better optimization stability:
+
+$$
+\operatorname* { m a x } _ { D } L ( D ) : = V ( G , D ) , { \mathrm { ~ a n d ~ } } \operatorname* { m i n } _ { G } L ( G ) : = - \mathbb { E } _ { \mathbf { z } } [ \log ( D ( G ( \mathbf { z } ) ) ) ] .
+$$
+
+Here, compared to (1), $G$ is now optimized to let $D$ to classify $G ( \mathbf { z } )$ as 1, i.e., the “real”.
+
+Contrastive representation learning. Consider two random variables $\mathbf { v } ^ { ( 1 ) }$ and $\mathbf { v } ^ { ( 2 ) }$ , which are often referred as views. Generally speaking, contrastive learning aims to extract a useful representation of $\mathbf { v } ^ { ( 1 ) }$ and $\mathbf { v } ^ { ( 2 ) }$ from learning a function that identifies whether a given sample is from $p ( \mathbf { v } ^ { ( 1 ) } ) p ( \mathbf { v } ^ { ( 2 ) } | \mathbf { v } ^ { ( 1 ) } )$ or $p ( \mathbf { v } ^ { ( 1 ) } ) p ( \mathbf { v } ^ { ( 2 ) } )$ , i.e., whether two views are dependent or not. More specifically, the function estimates the mutual information $I ( \mathbf { v } ^ { ( 1 ) } ; \mathbf { v } ^ { ( 2 ) } )$ between the two views. To this end, Oord et al. (2018) proposed to minimize InfoNCE loss, which turns out to maximize a lower bound of $I ( \mathbf { v } ^ { ( 1 ) } ; \mathbf { v } ^ { ( 2 ) } )$ . Formally, for a given $\pmb { v } _ { i } ^ { ( 1 ) } \sim p ( \mathbf { v } ^ { ( 1 ) } )$ and $v _ { i } ^ { ( 2 ) } \sim p ( \mathbf { v } ^ { ( 2 ) } | \pmb { v } _ { i } ^ { ( 1 ) } )$ while assuming $v _ { j } ^ { ( 2 ) } \sim p ( \mathbf { v } ^ { ( 2 ) } )$ for $j = 1 , \cdots , K$ , the InfoNCE loss is defined by:
+
+$$
+L _ { \mathtt { N C E } } ( { \pmb v } _ { i } ^ { ( 1 ) } ; { \pmb v } ^ { ( 2 ) } , s ) : = - \log \frac { \exp ( s ( { \pmb v } _ { i } ^ { ( 1 ) } , { \pmb v } _ { i } ^ { ( 2 ) } ) ) } { \sum _ { j = 1 } ^ { K } \exp ( s ( { \pmb v } _ { i } ^ { ( 1 ) } , { \pmb v } _ { j } ^ { ( 2 ) } ) ) } ,
+$$
+
+where $s ( \cdot , \cdot )$ is the score function that models the log-density ratio of $p ( \mathbf { v } ^ { ( 2 ) } | \mathbf { v } ^ { ( 1 ) } )$ to $p ( \mathbf { v } ^ { ( 2 ) } )$ , possibly including some parametrized encoders for $\mathbf { v } ^ { ( 1 ) }$ and $\mathbf { v } ^ { ( 2 ) }$ .
+
+Many of recent unsupervised representation learning methods are based on this general framework of contrastive learning (Wu et al., 2018; Bachman et al., 2019; Henaff et al. ´ , 2020; He et al., 2020; Chen et al., 2020a). In this paper, we focus on the one called SimCLR (Chen et al., 2020a), that adopts a wide range of independent data augmentations to define views: specifically, for a given set of data samples $\mathbf { \bar { x } } = [ \mathbf { x } _ { i } ] _ { i = 1 } ^ { N }$ , SimCLR applys two independent augmentations, namely $\mathbf { t } _ { 1 }$ and $\mathbf { t } _ { 2 }$ , to the given data to obtain $\mathbf { v } ^ { ( 1 ) }$ and $\mathbf { v } ^ { ( 2 ) }$ , i.e., $( \mathbf { v } ^ { ( 1 ) } , \mathbf { v } ^ { ( 2 ) } ) : = ( \mathbf { t } _ { 1 } ( \mathbf { x } ) , \mathbf { t } _ { 2 } ( \mathbf { x } ) )$ . The actual loss of SimCLR is slightly different to InfoNCE, mainly due to practical considerations for sample efficiency:
+
+$$
+\operatorname { \widetilde { \operatorname { s i n c I } } } \ ( \boldsymbol { v } ^ { ( 1 ) } , \boldsymbol { v } ^ { ( 2 ) } ) : = \frac { 1 } { 2 N } \sum _ { i = 1 } ^ { N } \left( L _ { \mathtt { N C E } } ( \boldsymbol { v } _ { i } ^ { ( 1 ) } ; [ \boldsymbol { v } ^ { ( 2 ) } ; \boldsymbol { v } _ { - i } ^ { ( 1 ) } ] , \boldsymbol { s } _ { \mathtt { S i n c I R } } ) + L _ { \mathtt { N C E } } ( \boldsymbol { v } _ { i } ^ { ( 2 ) } ; [ \boldsymbol { v } ^ { ( 1 ) } ; \boldsymbol { v } _ { - i } ^ { ( 2 ) } ] , \boldsymbol { s } _ { \mathtt { S i n c I R } } ) \right) ,
+$$
+
+where ${ \pmb v } _ { - i } : = { \pmb v } \setminus \{ { \pmb v } _ { i } \}$ . For $s _ { \tt S i m C L R }$ , SimCLR specifies to use (a) an encoder network $f : \mathcal { X } $ $\mathbb { R } ^ { d _ { e } }$ , (b) a small neural network called projection head $h : \mathbb { R } ^ { d _ { e } }  \mathbb { R } ^ { d _ { p } }$ , and (c) the normalized temperature-scaled cross entropy (NT-Xent). Putting altogether, $s _ { \mathsf { S i m C L R } }$ is defined by:
+
+$$
+s _ { \mathtt { S i m C L R } } ( \mathbf { v } ^ { ( 1 ) } , \mathbf { v } ^ { ( 2 ) } ; f , h ) : = \frac { h ( f ( \mathbf { v } ^ { ( 1 ) } ) ) \cdot h ( f ( \mathbf { v } ^ { ( 2 ) } ) ) } { \tau \cdot | | h ( f ( \mathbf { v } ^ { ( 1 ) } ) ) | | _ { 2 } | | h ( f ( \mathbf { v } ^ { ( 2 ) } ) ) | | _ { 2 } } ,
+$$
+
+where $\tau$ is a temperature hyperparameter. Once the training is done with respect to $L _ { \tt S i m C L R }$ , the projection head $h$ is discarded and $f$ is served as the learned representation for downstream tasks.
+
+# 3 CONTRAD: CONTRASTIVE DISCRIMINATOR FOR GANS
+
+We aim to develop a training scheme for GANs that is capable to extract more useful information under stronger data augmentation beyond the existing yet limited practices, e.g., random translations up to few pixels (Zhang et al., 2020; Zhao et al., 2020c;a). For examples, one may consider to apply augmentations introduced in SimCLR (Chen et al., 2020a) for training GANs, i.e., a stochastic composition of random resizing, cropping, horizontal flipping, color distortion, and Gaussian blurring. Indeed, existing approaches that handle data augmentation in GANs are not guranteed to work best in this harsh case (as also observed empirically in Table 5), and some of them are even expected to harm the performance: e.g., applying consistency regularization (Zhang et al., 2020; Zhao et al., 2020c), which regularizes a discriminator to be invariant to these augmentations, could overly restrict the expressivity of the discriminator to learn meaningful features.
+
+Our proposed architecture of Contrastive Discriminator (ContraD) is designed to alleviate such potential risks from using stronger data augmentations by incorporating the contrastive learning scheme of SimCLR (Chen et al., 2020a), which arguably works with those augmentations, and its design practices into the discriminator. More concretely, the main objective of ContraD is not to minimize the discriminator loss of GAN, but to learn a contrastive representation that is compatible to GAN. This means that the objective does not break the contrastive learning, while the representation still contains sufficient information to discriminate real and fake samples, so that a small neural network discriminator is enough to perform its task upon the representation. We describe how to learn such a representation of ContraD in Section 3.1, and how to incorporate this in the actual GAN training in Section 3.2. Finally, Section 3.3 introduces a natural application of ContraD for deriving conditional generative models from an unconditionally trained GAN. Figure 1 illustrates ContraD.
+
+![](images/5fdb334767df6c9358787fa0fab5130822f2ab0676c4a3153c20a3b458ea6d6a.jpg)  
+Figure 1: An overview of Contrastive Discriminator (ContraD). Overall, the representation of ContraD is not learned from the discriminator loss $\left( L _ { \mathrm { d i s } } \right)$ , but from two contrastive losses $L _ { \mathrm { c o n } } ^ { + }$ and $L _ { \mathsf { c o n } } ^ { - }$ , each is for the real and fake samples, respectively. Here, $\mathsf { s g } ( \cdot )$ denotes the stop-gradient operation.
+
+# 3.1 CONTRASTIVE REPRESENTATION FOR GANS
+
+Recall the standard training of GANs, i.e., we train $G : { \mathcal { Z } } \to { \mathcal { X } }$ and $D : \mathcal { X }  [ 0 , 1 ]$ via optimizing (2) from a given dataset $\{ { \bf x } _ { i } \} _ { i = 1 } ^ { N }$ , and let $\tau$ be a family of possible transforms for data augmentation, which we assume in this paper to be those used in SimCLR (Chen et al., 2020a). In order to define the training objective of ContraD, we start by re-defining the scalar-valued $D$ to have vector-valued outputs, i.e., $\dot { D } : \mathcal { X }  \mathbb { R } ^ { d _ { e } }$ , to represent an encoder network of contrastive learning. Overall, the encoder network $D$ of ContraD is trained by minimizing two different contrastive losses: (a) the SimCLR loss (4) on the real samples, and (b) the supervised contrastive loss (Khosla et al., 2020) on the fake samples, which will be explained one-by-one in what follows.
+
+Loss for real samples. By default, ContraD is built upon the SimCLR training: in fact, the training becomes equivalent to SimCLR if the loss (b) is missing, i.e., without considering the fake samples. Here, we attempt to simply follow the contrastive training scheme for real samples $_ { \textbf { \em x } }$ , to keep open the possibility to improve the method by adopting other self-supervised learning methods with data augmentations (Chen et al., 2020b; Grill et al., 2020). More concretely, we first compute two independent views ${ \pmb v } _ { \mathrm { r } } ^ { ( 1 ) } , { \pmb v } _ { \mathrm { r } } ^ { ( 2 ) } : = { \pmb t } _ { 1 } ( { \bf x } ) , { \pmb t } _ { 2 } ( { \bf x } )$ for real samples using $t _ { 1 } , t _ { 2 } \sim \tau$ , and minimize:
+
+$$
+\begin{array} { r } { L _ { \mathrm { c o n } } ^ { + } ( D , h _ { \mathrm { r } } ) : = L _ { \mathrm { S i m C L R } } ( \pmb { v } _ { \mathrm { r } } ^ { ( 1 ) } , \pmb { v } _ { \mathrm { r } } ^ { ( 2 ) } ; D , h _ { \mathrm { r } } ) , } \end{array}
+$$
+
+where $h _ { \tt r } : \mathbb { R } ^ { d _ { e } }  \mathbb { R } ^ { d _ { p } }$ is a projection head for this loss.
+
+Loss for fake samples. Although the representation learned via $L _ { \mathrm { c o n } } ^ { + }$ (6) may be enough, e.g., for discriminating two independent real samples, it does not give any guarantee that this representation would still work for discriminating real and fake samples, which is needed as a GAN discriminator. We compensate this by considering an auxiliary loss $L _ { \mathsf { c o n } } ^ { - }$ that regularizes the encoder to keep necessary information to discriminate real and fake samples. Here, several design choices can be possible: nevertheless, we observe that one should choose the loss deliberately, otherwise the contrastive representation from $L _ { \mathrm { c o n } } ^ { + }$ could be significantly affected. For example, we found using the original GAN loss (2) for $L _ { \mathsf { c o n } } ^ { - }$ could completely negate the effectiveness of ContraD.
+
+In this respect, we propose to use the supervised contrastive loss (Khosla et al., 2020) over fake samples to this end, an extended version of contrastive loss to support supervised learning by allowing more than one view to be positive, so that views of the same label can be attracted to each other in the embedding space. In our case, we assume all the views from fake samples have the same label against those from real samples. Formally, for each $\pmb { v } _ { i } ^ { ( 1 ) }$ , let $V _ { i + } ^ { ( 2 ) }$ be a subset of $\mathbf { \boldsymbol { v } } ^ { ( 2 ) }$ that represent the positive views for v(1)i . Then, the supervised contrastive loss is defined by:
+
+$$
+L _ { \mathrm { S u p C o n } } ( v _ { i } ^ { ( 1 ) } , v ^ { ( 2 ) } , V _ { i + } ^ { ( 2 ) } ) : = - \frac { 1 } { | V _ { i + } ^ { ( 2 ) } | } \sum _ { v _ { i + } ^ { ( 2 ) } \in V _ { i + } ^ { ( 2 ) } } \log \frac { \exp ( s _ { \mathrm { S i n C L R } } ( v _ { i } ^ { ( 1 ) } , v _ { i + } ^ { ( 2 ) } ) ) } { \sum _ { j } \exp ( s _ { \mathrm { S i n C L R } } ( v _ { i } ^ { ( 1 ) } , v _ { j } ^ { ( 2 ) } ) ) } .
+$$
+
+Using the notation, we define the ContraD loss for fake samples as follows:
+
+$$
+L _ { \mathrm { c o n } } ^ { - } ( D , h _ { \textup { \bf f } } ) : = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } L _ { \mathrm { S u p c o n } } ( v _ { \textup { \bf f } , i } , [ v _ { \textup { \bf f } , - i } ; v _ { \textup { \bf r } } ^ { ( 1 ) } ; { v _ { \textup { \bf r } } ^ { ( 2 ) } } ] , [ v _ { \textup { \bf f } , - i } ] ; D , h _ { \textup { \bf f } } ) ,
+$$
+
+where again $v _ { \mathrm { f } } : = t _ { 3 } ( G ( \mathbf { z } ) )$ is a random view of fake samples, and ${ \pmb v } _ { - i } : = { \pmb v } \setminus \{ { \pmb v } _ { i } \}$ . Remark that we use an independent projection header $h _ { \mathbf { f } } : \mathbb { R } ^ { d _ { e } }  \mathbb { R } ^ { d _ { p } }$ instead of $h _ { \mathbf { r } }$ in (6) for this loss.
+
+There can be other variants for $L _ { \mathrm { c o n } } ^ { - }$ , as long as (a) it leads $D$ to discriminate real and fake samples, while (b) not compromising the representation from $L _ { \mathrm { c o n } } ^ { + }$ . For example, one can incorporate another view of fake samples to additionally perform SimCLR similarly to $L _ { \mathrm { c o n } } ^ { + }$ (6). Nevertheless, we found the current design (8) is favorable among such variants considered, both in terms of performance under practice of GANs, and computational efficiency from using only a single view.
+
+o sum up, ContraD learns its contrastive representation by minimizing the following loss:
+
+$$
+\begin{array} { r } { L _ { \mathsf { c o n } } ( D , h _ { \mathbf { r } } , h _ { \mathbf { f } } ) : = L _ { \mathsf { c o n } } ^ { + } ( D , h _ { \mathbf { r } } ) + \lambda _ { \mathsf { c o n } } L _ { \mathsf { c o n } } ^ { - } ( D , h _ { \mathbf { f } } ) , } \end{array}
+$$
+
+where $\lambda _ { \tt c o n } > 0$ is a hyperparameter. Nevertheless, we simply use $\lambda _ { \tt c o n } = 1$ in our experiments.
+
+# 3.2 TRAINING GANS WITH CONTRAD
+
+The contrastive loss defined in (9) is jointly trained under the standard framework of GAN, e.g., we present in this paper the case of non-saturating loss (2), for training the generator network $G$ . To obtain a (scalar) discriminator score needed to define the GAN loss, we simply use an additional discriminator head $h _ { \mathsf { d } } : \mathbb { R } ^ { d _ { e } }  \mathbb { R }$ upon the contrastive representation of $D$ . Here, a key difference of ContraD training from other GANs is that ContraD only optimizes the parameters of $h _ { \mathrm { d } }$ for minimizing the GAN loss: in practical situations of using stochastic gradient descent for the training, this can be implemented by stopping gradient before feeding inputs to $h _ { \mathrm { d } }$ . Therefore, the discriminator loss of ContraD is defined as follows:
+
+$$
+L _ { \mathrm { d i s } } ( h _ { \mathrm { d } } ) : = - \mathbb { E } _ { \mathbf { r } _ { \mathrm { r } } } [ \log ( \sigma ( h _ { \mathrm { d } } ( \mathbf { r } _ { \mathbf { r } } ) ) ) ] - \mathbb { E } _ { \mathbf { r } _ { \mathrm { f } } } [ \log ( 1 - \sigma ( h _ { \mathrm { d } } ( \mathbf { r } _ { \mathbf { f } } ) ) ) ] ,
+$$
+
+where $\mathbf { r } _ { \mathbf { r } } = \mathbf { s g } ( D ( \mathbf { v } _ { \mathbf { r } } ) )$ and $\mathbf { r _ { f } } = \mathbf { s g } ( D ( \mathbf { v _ { f } } ) )$ . Here, $\sigma ( \cdot )$ denotes the sigmoid function, $\mathbf { v _ { r } }$ and $\mathbf { v _ { f } }$ are random views of real and fake samples augmented via $\tau$ , respectively, and $\mathsf { s g } ( \cdot )$ is the stopgradient operation. Combined with the contrastive training loss (9), the joint training loss $L _ { D }$ for a single discriminator update is the following:
+
+$$
+L _ { D } : = L _ { \mathsf { c o n } } + \lambda _ { \mathsf { d i s } } L _ { \mathsf { d i s } } = L _ { \mathsf { c o n } } ^ { + } + \lambda _ { \mathsf { c o n } } L _ { \mathsf { c o n } } ^ { - } + \lambda _ { \mathsf { d i s } } L _ { \mathsf { d i s } } ,
+$$
+
+where $\lambda _ { \mathrm { d i s } } > 0$ is a hyperparameter. Again, as like $\lambda _ { \mathsf { c o n } }$ , we simply use $\lambda _ { \dot { \mathrm { d i s } } } = 1$ in our experiments.
+
+Finally, the loss for the generator $G$ can be defined simply like other standard GANs using the discriminator score, except that we also augment the generated samples $G ( \mathbf { z } )$ to $\mathbf { v _ { f } }$ , in a similar way to DiffAug (Zhao et al., 2020a) or ADA (Karras et al., 2020a): namely, in case of the non-saturating loss (Goodfellow et al., 2014), we have:
+
+$$
+L _ { G } : = - \mathbb { E } _ { \mathbf { v _ { f } } } [ \log ( \sigma ( h _ { \mathsf { d } } ( D ( \mathbf { v _ { f } } ) ) ) ) ] .
+$$
+
+Note that the form (12) is equivalent to the original non-saturating loss (2) if we regard $\sigma ( h _ { \mathsf { d } } ( D ( \cdot ) ) ) :$ $\mathcal { X }  [ 0 , 1 ]$ as a single discriminator. Algorithm 1 in Appendix A describes a concrete training procedure of GANs with ContraD using Adam optimizer (Kingma & Ba, 2014).
+
+# 3.3 SELF-CONDITIONAL SAMPLING WITH CONTRAD
+
+Apart from the improved training of GANs, the learned contrastive representation of ContraD could offer some additional benefits in practical scenarios compared to the standard GANs. In this section, we present an example of how one could further utilize this additional information to derive a conditional generative model from an unconditionally-trained GAN with ContraD. More specifically, here we consider a variant of discriminator-driven latent sampling (DDLS) (Che et al., 2020) to incorporate a given representation vector from the ContraD encoder as a conditional information. Originally, DDLS attempts to improve the sample quality of a pre-trained GAN via a Langevin sampling (Welling & Teh, 2011) on the latent space of the following form:
+
+$$
+\mathbf { z } _ { t + 1 } : = \mathbf { z } _ { t } - \frac { \varepsilon } { 2 } \nabla _ { \mathbf { z } _ { t } } E ( \mathbf { z } _ { t } ) + \sqrt { \varepsilon } \mathbf { n } _ { t } , \mathbf { n } _ { t } \sim \mathcal { N } ( 0 , 1 ) ,
+$$
+
+Table 1: Comparison of the best FID score and IS on unconditional image generation of CIFAR-10 and CIFAR-100. Values in the rows marked by \* are from those reported in its reference.   
+
+<table><tr><td rowspan="2">Architecture</td><td rowspan="2">Method</td><td rowspan="2">Augment.</td><td colspan="2">CIFAR-10</td><td colspan="2">CIFAR-100</td></tr><tr><td>FID↓</td><td>IS个</td><td>FID↓</td><td>IS个</td></tr><tr><td rowspan="5">G: SNDCGAN D:SNDCGAN</td><td></td><td>=</td><td>26.6</td><td>7.38</td><td>28.5</td><td>7.25</td></tr><tr><td>CR (Zhang et al., 2020)</td><td>HFlip, Trans</td><td>19.5</td><td>7.87</td><td>22.2</td><td>7.91</td></tr><tr><td>bCR (Zhao et al., 2020c)</td><td>HFlip,Trans</td><td>14.0</td><td>8.35</td><td>19.2</td><td>8.46</td></tr><tr><td>DiffAug (Zhao et al., 2020a)</td><td>Trans, CutOut</td><td>22.9</td><td>7.64</td><td>27.0</td><td>7.47</td></tr><tr><td>ContraD (ours)</td><td>SimCLR</td><td>10.9</td><td>8.78</td><td>15.2</td><td>9.09</td></tr><tr><td rowspan="5">G: SNDCGAN D:SNResNet-18</td><td></td><td></td><td>41.3</td><td>6.33</td><td>52.3</td><td>5.24</td></tr><tr><td>CR (Zhang et al., 2020)</td><td>HFlip, Trans</td><td>32.1</td><td>7.08</td><td>36.5</td><td>6.55</td></tr><tr><td>bCR (Zhao et al.,2020c)</td><td>HFlip, Trans</td><td>22.8</td><td>7.29</td><td>28.2</td><td>7.30</td></tr><tr><td>DiffAug (Zhao et al., 2020a)</td><td>Trans, CutOut</td><td>59.5</td><td>5.62</td><td>58.7</td><td>5.39</td></tr><tr><td>ContraD (ours)</td><td>SimCLR</td><td>9.86</td><td>9.09</td><td>15.0</td><td>9.56</td></tr><tr><td rowspan="2">G: StyleGAN2</td><td></td><td></td><td>11.1</td><td>9.18</td><td>16.5</td><td>9.51</td></tr><tr><td>DiffAug* (Zhao et al.,2020a)</td><td>Trans, CutOut</td><td>9.89</td><td>9.40</td><td>15.2</td><td>10.0</td></tr><tr><td>D: StyleGAN2</td><td>ContraD (ours)</td><td>SimCLR</td><td>9.80</td><td>9.47</td><td>14.1</td><td>10.0</td></tr></table>
+
+Table 2: Comparison of the best FID score and IS on unconditional image generation of CelebAHQ-128 with SNDCGAN. We report the mean and standard deviation of best scores across 3 trials.   
+
+<table><tr><td>CelebA-HQ</td><td>W/0</td><td>Hinge</td><td>CR</td><td>bCR</td><td>ContraD (ours)</td></tr><tr><td>FID←</td><td>24.8±1.69</td><td>26.4±1.32</td><td>20.9±0.37</td><td>19.5±0.08</td><td>17.1±1.38</td></tr><tr><td>IS↑</td><td>2.06±0.06</td><td>2.05±0.06</td><td>2.11±0.09</td><td>2.21±0.10</td><td>2.44±0.06</td></tr></table>
+
+where $E ( \mathbf { z } ) : = - \log p ( \mathbf { z } ) - d ( G ( \mathbf { z } ) )$ and $d ( \cdot )$ denotes the logit of $D$ . In this manner, for our ContraD, we propose to use the following “conditional” version of energy function $E ( \mathbf { z } , v )$ for an arbitrary vector $\dot { v } \in \mathbb { R } ^ { d _ { e } }$ in the ContraD representation space, called conditional DDLS (cDDLS):
+
+$$
+\begin{array} { r } { E ( \mathbf { z } , v ) : = - \log p ( \mathbf { z } ) - h _ { \mathrm { d } } ( D ( G ( \mathbf { z } ) ) ) - \lambda ( v \cdot D ( G ( \mathbf { z } ) ) ) . } \end{array}
+$$
+
+In our experiments, for instance but not limited to, we show that using the learned weights obtained from linear evaluation as conditional vectors successfully recovers class-conditional generations from an unconditional ContraD model.
+
+# 4 EXPERIMENTS
+
+We verify the effectiveness of our ContraD in three different aspects: (a) its performance on image generation compared to other techniques for data augmentations in GANs, (b) the quality of representations learned from ContraD in terms of linear evaluation, and (c) its ability of self-conditional sampling via cDDLS. Overall, we constantly observe integrating contrastive learning into GANs and vice versa have positive effects to each other: (a) for image generation, ContraD enables a simple SNDCGAN to outperform a state-of-the-art architecture of StyleGAN2 on CIFAR-10/100; (b) in case of linear evaluation, on the other hand, we observe the representation from ContraD could outperform those learned from SimCLR. Finally, we perform an ablation study to further understand each of the components we propose. We provide the detailed specification on the experimental setups, e.g., architectures, training configurations and hyperparameters in Appendix F.
+
+We consider a variety of datasets including CIFAR-10/100 (Krizhevsky, 2009), CelebA-HQ-128 (Lee et al., 2020), AFHQ (Choi et al., 2020) and ImageNet (Russakovsky et al., 2015) in our experiments, mainly with three well-known GAN architectures: SNDCGAN (Miyato et al., 2018), StyleGAN2 (Karras et al., 2020b) and BigGAN (Brock et al., 2019).1 We measure Frechet Incep- ´ tion distance (FID) (Heusel et al., 2017) and Inception score (IS) (Salimans et al., 2016) as quantitative metrics to evaluate generation quality.2 We compute FIDs between the test set and generated samples of the same size. We follow the best hyperparameter practices explored by Kurach et al. (2019) for SNDCGAN models on CIFAR and CelebA-HQ-128 datasets. For StyleGAN2, on the other hand, we follow the training details of Zhao et al. (2020a) in their CIFAR experiments. All the training details are shared across methods per experiment, but the choice of batch size for training ContraD: we generally observe that ContraD greatly benefits from using larger batch (Appendix E), similarly to SimCLR (Chen et al., 2020a), but in contrast to the common practice of GAN training. Therefore, in our experiments, we consider to use $2 \times$ to $8 \times$ larger batch sizes for training ContraD, where the details are specified in Appendix F.
+
+Table 3: Comparison of classification accuracy under linear evaluation protocol on CIFAR-10 and CIFAR-100. We report the mean and standard deviation across 3 runs of the evaluation.   
+
+<table><tr><td>Dataset</td><td>Training</td><td>SNDCGAN</td><td>SNResNet-18</td><td>StyleGAN2</td></tr><tr><td>CIFAR-10</td><td>SimCLR(入con = 入dis = 0) ContraD (ours)</td><td>72.9±0.02</td><td>80.3±0.05</td><td>86.2±0.06</td></tr><tr><td rowspan="2">CIFAR-100</td><td></td><td>77.5±0.20</td><td>85.7±0.10</td><td>88.6±0.06</td></tr><tr><td>SimCLR(Xcon = Xdis = 0)</td><td>30.8±0.11</td><td>41.2±0.06</td><td>61.1±0.06</td></tr><tr><td></td><td>ContraD (ours)</td><td>37.4±0.06</td><td>51.1±0.18</td><td>68.1±0.07</td></tr></table>
+
+# 4.1 RESULTS
+
+Image generation on CIFAR-10/100. CIFAR-10 and CIFAR-100 (Krizhevsky, 2009) consist of 60K images of size $3 2 \times 3 2$ in 10 and 100 classes, respectively, 50K for training and 10K for testing. We start by comparing ContraD on CIFAR-10/100 with three recent methods that applies data augmentation in GANs: (a) Consistency Regularization (CR) (Zhang et al., 2020), (b) the “balanced” Consistency Regularization (bCR) (Zhao et al., 2020c), and (c) Differentiable Augmentation (DiffAug) (Zhao et al., 2020a) as baselines.3 We follow their best pratices on which data augmentations to use: specifically, we use a combination of horizontal flipping (HFlip) and random translation up to 4 pixels (Trans) for CR and bCR, and Trans $^ +$ CutOut (DeVries & Taylor, 2017) for DiffAug for CIFAR datasets. We mainly consider SNDCGAN and StyleGAN2 for training, but in addition, we further consider a scenario that the discriminator architecture of SNDCGAN is scaled up into SNResNet-18, which is identical to ResNet-18 (He et al., 2016) without batch normalization (Ioffe & Szegedy, 2015) but with spectral normalization (Miyato et al., 2018), in order to see that how each training method behaves when the capacity of generator and discriminator are significantly different.
+
+Table 1 shows the results: overall, we observe our ContraD uniformly improves the baseline GAN training by a large margin, significantly outperforming other baselines as well. Indeed, our results on SNDCGAN already achieve better FID than those from the baseline training of StyleGAN2. Moreover, when the discriminator of SNDCGAN is increased to SNResNet-18, ContraD could further improve its results, without even modifying any of the hyperparameters. It is particularly remarkable that all the other baselines could not improve their results in this setup: A further ablation study presented in Section 4.2 reveals that stopping gradients before the discriminator head in our design is an important factor to benefit from SNResNet-18. Finally, our ContraD can also be applied to a larger, sophisticated StyleGAN2, further improving both FID and IS compared to DiffAug.
+
+Image generation on CelebA-HQ-128. We also perform experiments on CelebA-HQ-128 (Lee et al., 2020), which consist of 30K human faces of size $1 2 8 \times 1 2 8$ , to see that our design of ContraD still applies well for high-resolution datasets. We split 3K images out of CelebA-HQ-128 for testing, and use the remaining for training. We use SNDCGAN for this experiment, and compare FID and IS to (a) the default non-saturating loss (W/O) (Goodfellow et al., 2014), (b) the hinge loss (Hinge) (Lim & Ye, 2017; Tran et al., 2017), (c) CR and (d) bCR. Similarly to the CIFAR experiments, we use the “HFlip $^ +$ Trans” augmentation for both CR and bCR, but for CelebA-HQ-128 we allow them to translate up to 16 pixels, following (Zhang et al., 2020). Again, as shown in Table 2, we confirm that ContraD could still improve training of GANs compared to other methods, even with this high-resolution, yet with limited samples and a straightforward choice of architecture.
+
+Linear evaluation. Recall that the training objective of ContraD we propose in (11) can be simply reduced into the SimCLR loss (Chen et al., 2020a) without considering the fake samples, i.e., by letting $\lambda _ { \tt c o n } = \lambda _ { \tt d i s } = 0$ . A natural question from here is whether those fake samples could help to improve the representation of SimCLR: we verify this by comparing the linear evaluation performance (Kolesnikov et al., 2019; Chen et al., 2020a; Grill et al., 2020) of the learned representation of ContraD to those from SimCLR: specifically, linear evaluation measures the test accuracy that a linear classifier on the top of given (frozen) representation could maximally achieve for a given (downstream) labeled data. Following this protocol, we compare the linear evaluation of a ContraD model with its ablation that $\lambda _ { \mathsf { c o n } }$ and $\lambda _ { \mathrm { d i s } }$ are set to 0 during its training. We consider three discriminator architectures and two datasets for this experiments: namely, we train SNDCGAN, SNResNet-18, and StyleGAN2 models on CIFAR-10 and CIFAR-100. In this experiment, we use batch size 256 for training all the SNDCGAN and SNResNet-18 models, while in case of StyleGAN2 we use batch size 64 instead. Table 3 summarizes the results, and somewhat interestingly, it shows that ContraD significantly improves the SimCLR counterpart in terms of linear evaluation for all the models tested: in other words, we observe that keeping discriminative features between real and fake samples in their representation, which is what ContraD does in addition to SimCLR, can be beneficial to improve the contrastive learning itself. Again, our observation supports a great potential of ContraD not only in the context of GANs, but also in the contrastive learning.
+
+Table 4: Comparison of FID (lower is better) from the class-wise subsets of CIFAR-10 test set (1K images per class). “Random” indicates unconditional generation, and “Train-1K” indicates a 1K random subsamples from the CIFAR-10 train set of the same class.   
+
+<table><tr><td>D Arch.</td><td>Sampling</td><td>Plane</td><td>Car</td><td>Bird</td><td>Cat</td><td>Deer</td><td>Dog</td><td>Frog</td><td>Horse</td><td>Ship</td><td>Truck</td><td>Mean</td></tr><tr><td>SNDCGAN</td><td>Random + cDDLS</td><td>110.6 65.3</td><td>125.9 72.1</td><td>103.8 63.9</td><td>104.4 69.4</td><td>107.0 57.5</td><td>121.0 69.0</td><td>134.1 107.0</td><td>120.1 61.5</td><td>128.8 58.2</td><td>130.0 53.6</td><td>118.6 67.8</td></tr><tr><td>SNResNet-18</td><td>Random + cDDLS</td><td>117.6 67.1</td><td>136.3 65.8</td><td>100.5 59.8</td><td>99.6 63.8</td><td>98.6 59.7</td><td>115.0 61.9</td><td>128.2 68.8</td><td>111.7 59.9</td><td>140.9 58.9</td><td>140.3 51.6</td><td>118.9 61.7</td></tr><tr><td>-</td><td>Train-1K</td><td>38.9</td><td>28.4</td><td>41.4</td><td>49.7</td><td>36.4</td><td>40.2</td><td>41.7</td><td>33.6</td><td>30.7</td><td>24.6</td><td>36.6</td></tr></table>
+
+Table 5: Comparison of the best FID and IS on CIFAR-10 with SNDCGAN when stronger (bCR and DiffAug) or weaker augmentations (ContraD) are used for each method.   
+
+<table><tr><td>Method</td><td>Augment.</td><td>FID↓</td><td>IS个</td></tr><tr><td>bCR</td><td>HFlip,Trans</td><td>14.0</td><td>8.35</td></tr><tr><td>+ Aug.</td><td>SimCLR</td><td>20.6</td><td>7.44</td></tr><tr><td>DiffAug</td><td>Trans, CutOut</td><td>22.9</td><td>7.64</td></tr><tr><td>+Aug.</td><td>SimCLR</td><td>21.9</td><td>7.42</td></tr><tr><td>ContraD</td><td>SimCLR</td><td>10.9</td><td>8.78</td></tr><tr><td>-Aug.</td><td>HFlip, Trans</td><td>13.7</td><td>8.54</td></tr></table>
+
+Table 6: Comparison of the best FID score and IS on CIFAR-10 for ablations of our proposed components. All the models are trained with batch size 256. For the ablation of “MLP $h _ { \mathrm { d } } ^ { \ \mathrm { ~ \prime ~ } }$ , we replace $h _ { \mathrm { d } }$ with a linear model. “ $\mathfrak { s g } ( \cdot )$ ” indicates the use of stop-gradient operation.   
+
+<table><tr><td></td><td></td><td></td><td></td><td colspan="2">D: SNDCGAN</td><td colspan="2">D:SNResNet-18</td></tr><tr><td>MLP ha</td><td>sg(.)</td><td>L+n</td><td>Lcon</td><td>FID↓</td><td>IS↑</td><td>FID↓</td><td>IS↑</td></tr><tr><td></td><td>【</td><td>√</td><td>√</td><td>11.1</td><td>8.62</td><td>10.6</td><td>8.99</td></tr><tr><td>x//</td><td>√</td><td>√</td><td>√</td><td>185</td><td>3.43</td><td>274</td><td>2.09</td></tr><tr><td></td><td>X</td><td>√</td><td></td><td>11.6</td><td>8.61</td><td>28.0</td><td>7.52</td></tr><tr><td></td><td></td><td>X</td><td></td><td>11.9</td><td>8.45</td><td>182</td><td>2.02</td></tr><tr><td>√</td><td></td><td></td><td>X</td><td>210</td><td>1.93</td><td>232</td><td>2.56</td></tr></table>
+
+Conditional DDLS with ContraD. We further evaluate the performance of conditional DDLS (cDDLS) (14) proposed for ContraD, with a task of deriving class-conditional generative models for a given (unconditionally-trained) GAN model. Here, we measure the performance of this conditional generation by computing the class-wise FIDs to the test set, i.e., FIDs computed for each subsets of the same class in the test set. We test cDDLS on SNDCGAN and SNResNet-18 ContraD models trained on CIFAR-10. To perform a conditional generation for a specific class, we simply take a learned linear weight obtained from the linear evaluation protocol for that class. The results summarized in Table 4 show that applying cDDLS upon unconditional generation significantly improves class-wise FIDs for all the classes in CIFAR-10. Some of the actual samples conditionally generated from an SNResNet-18 ContraD model can be found in Figure 5 of Appendix D.
+
+# 4.2 ABLATION STUDY
+
+We conduct an ablation study for a more detailed analysis on the proposed method. For this section, unless otherwise specified, we perform experiments on CIFAR-10 with SNDCGAN architecture.
+
+Stronger augmentations for other methods. One of the key characteristics of ContraD compared to the prior works is the use of stronger data augmentations for training GANs, e.g., we use the augmentation pipeline of SimCLR (Chen et al., 2020a) in our experiments. In Table 5, we examine the performance bCR (Zhao et al., 2020c) and DiffAug (Zhao et al., 2020a) when this stronger augmentation is applied, and shows that the SimCLR augmentation could not meaningfully improve FID or IS for both methods. Unlike these methods, our ContraD effectively handles the SimCLR augmentations without overly regularizing the discriminator, possibly leveraging many components of contrastive learning: e.g., the normalized loss, or the use of separate projection heads.
+
+Weaker augmentations for ContraD. On the other side, we also consider a case when ContraD is rather trained with a weaker data augmentation, here we consider “HFlip, Trans” instead of “SimCLR”, in Table 5: even in this case, we observe ContraD is still as good as (or better in terms of IS) a strong baseline of bCR. The degradation in FID (and IS) compared to $\mathrm { ^ { * } C o n t r a D + S i m C L R ^ { , } }$ is possibly due to that SimCLR requires strong augmentations to learn a good representation (Chen et al., 2020a), e.g., we also observe there is a degradation in the linear evaluation performances, $7 7 . 5  7 2 . 9$ , when “HFlip, Trans” is used.
+
+MLP discriminator head. In our experiments, we use a 2-layer network for the discriminator head $h _ { \mathrm { d } }$ . As shown in Table 6, this choice can be minimal: replacing $h _ { \mathrm { d } }$ with a linear model severely breaks the training. Conversely, although not presented here, we have also observed that MLPs of more than two layers neither give significant gains on the performance in our experimental setup.
+
+Stopping gradients. We use the stop-gradient operation $\mathsf { s g } ( \cdot )$ before the discriminator head $h _ { \mathrm { d } }$ in attempt to decouple the effect of GAN loss to the ContraD representation. Table 6 also reports the ablation of this component: overall, not using $\mathsf { s g } ( \cdot )$ does not completely break the training, but it does degrade the performance especially when a deeper discriminator is used, e.g., SNResNet-18.
+
+Contrastive losses. Recall that the representation of ContraD is trained by two losses, namely $L _ { \mathrm { c o n } } ^ { + }$ and $L _ { \mathrm { c o n } } ^ { - }$ (Section 3.1). From an ablation study on each loss, as presented in Table 6, we observe that both of the losses are indispensable for our training to work in a larger discriminator architecture.
+
+Separate projection headers. As mentioned in Section 3.1, we use two independent projection headers, namely $h _ { \mathbf { r } }$ and $h _ { \mathrm { f } }$ , to define $L _ { \mathsf { c o n } } ^ { + }$ and $L _ { \mathrm { c o n } } ^ { - }$ , respectively. We have observed this choice is empirically more stable than sharing them: e.g., under $h _ { \tt r } = h _ { \tt f }$ , we could not obtain a reasonable performance when only $D$ is increased to SNResNet-18 on CIFAR-10 with SNDCGAN. Intuitively, an optimal embedding (after projection) from $L _ { \mathrm { c o n } } ^ { - }$ can be harmful to SimCLR, as it encourages the embedding of real samples to degenerate into a single point, i.e., the opposite direction of SimCLR.
+
+# 5 CONCLUSION AND FUTURE WORK
+
+In this paper, we explore a novel combination of GAN and contrastive learning, two important, yet seemingly different paradigms in unsupervised representation learning. They both have been independently observed the crucial importance of maintaining consistency to data augmentations in their representations (Zhang et al., 2020; Tian et al., 2020), and here we further observe that these two representations complement each other when combined upon the shared principle of viewinvariant representations: although we have put more efforts in this work to verify the effectiveness of contrastive learning on GANs, we do observe the opposite direction through our experiments, and scaling up our method to compete with state-of-the-art self-supervised learning benchmarks (Chen et al., 2020b; Grill et al., 2020; Caron et al., 2020) would be an important future work.
+
+We believe our work also suggests several interesting ideas to explore in future research in both sides of GAN and contrastive learning: for example, our new design of introducing a small header to minimize the GAN loss upon other (e.g., contrastive) representation is a promising yet unexplored way of designing a new GAN architecture. For the SimCLR side, on the other hand, we suggest a new idea of incorporating “fake” samples for contrastive learning, which is also an interesting direction along with other recent attempts to improve the efficiency of negative sampling in contrastive learning, e.g., via hard negative mining (Kalantidis et al., 2020; Robinson et al., 2021).
+
+# ACKNOWLEDGMENTS
+
+This work was supported by Samsung Advanced Institute of Technology (SAIT). This work was also partly supported by Institute of Information & Communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No.2019-0-00075, Artificial Intelligence Graduate School Program (KAIST)). The authors would like to thank Minkyu Kim for helping additional experiments in preparation of the camera-ready revision, and thank the anonymous reviewers for their valuable comments to improve our paper.
+
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+Shengyu Zhao, Zhijian Liu, Ji Lin, Jun-Yan Zhu, and Song Han. Differentiable augmentation for data-efficient GAN training. In H. Larochelle, M. Ranzato, R. Hadsell, M. F. Balcan, and H. Lin (eds.), Advances in Neural Information Processing Systems, volume 33, pp. 7559–7570. Curran Associates, Inc., 2020a. URL https://proceedings.neurips.cc/paper/2020/ file/55479c55ebd1efd3ff125f1337100388-Paper.pdf.
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+
+# A TRAINING PROCEDURE OF CONTRAD
+
+<table><tr><td colspan="2">Algorithm 1 GANs with Contrastive Discriminator (ContraD)</td></tr><tr><td colspan="2">Require: generator G, discriminator D,Adam hyperparameters α,β1, β2, number of D updates</td></tr><tr><td>per G update ND,family of data augmentations T,入 &gt; 0. 1:for # training iterations do</td><td></td></tr><tr><td>2:</td><td>for t= 1 to ND do</td></tr><tr><td>3:</td><td>Sample x ~ Pdata(x) and z ~ p(z)</td></tr><tr><td>4:</td><td>Samplet1,t2,t3~T</td></tr><tr><td>5:</td><td>v①）,u（2）,Uf←t(x）,t(x）,t3(G(z))</td></tr><tr><td>6:</td><td>L+on ← LsimCLR(U1），v2）;D,hr）</td></tr><tr><td>7:</td><td>L←∑pon(,[t,i2][;Dh</td></tr><tr><td>8:</td><td>rr,rf ← sg(D(v(2）)),sg(D(uf))</td></tr><tr><td>9:</td><td>Ldis←-Er,[log(σ(ha(rr)))]-Er[log(1-σ(ha(rf)))]</td></tr><tr><td>10:</td><td>LD←L+on+XconLcon+XdisLdis</td></tr><tr><td>11:</td><td>D,hr,hf,ha ←Adam([D,hr,hf,hd],LD;α,β1,β2)</td></tr><tr><td>12:</td><td>end for</td></tr><tr><td>13:</td><td>LG←-Ev[log(σ(ha(D(vf))))]</td></tr><tr><td>14:</td><td></td></tr><tr><td>15: end for</td><td>G ←Adam(G,LG;α,β1,β2)</td></tr><tr><td></td><td></td></tr></table>
+
+# B RESULTS ON AFHQ DATASETS
+
+We evaluate our training method on the Animal Faces-HQ (AFHQ) dataset (Choi et al., 2020), which consists of ${ \sim } 1 5 { , } 0 0 0$ samples of animal-face images at $5 1 2 \times 5 1 2$ resolution, to further verify the effectiveness of ContraD on higher-resolution, yet limited-sized datasets. We partition the dataset by class labels into three sub-datasets, namely AFHQ-Dog (4,739 samples), AFHQ-Cat (5,153 samples) and AFHQ-Wild (4,738 samples), and evaluate FIDs on these datasets. In this experiment, we compare our method with ADA (Karras et al., 2020a), another recent data augmentation scheme for GANs (concurrently to DiffAug (Zhao et al., 2020a) considered in Section 4), which includes a dynamic adaptation of augmentation pipeline during training. We follow the training details of ADA particularly for this experiment: specifically, we train StyleGAN2 with batch size 64 for 25M training samples, using Adam with $( \bar { \alpha } , \beta _ { 1 } , \beta _ { 2 } ) \overset { \cdot } { = } ( 0 . 0 0 2 5 , 0 . 0 , 0 . 9 9 )$ , $R _ { 1 }$ regularization with $\gamma = 0 . 5$ , and exponential moving average on the generator weights with half-life of 20K samples. For ContraD training on AFHQ datasets, we consider to additionally use CutOut (DeVries & Taylor, 2017) of $p = 0 . 5$ upon the SimCLR augmentation, which we observe an additional gain in FID. We also follow the way of computing FIDs as per Karras et al. (2020a) here for a fair comparison, i.e., by comparing 50K of generated samples and all the training samples ${ \sim } 5 \mathrm { K }$ per dataset).
+
+Table 7 compares our ContraD models with the results reported by Karras et al. (2020b) where the models are available at https://github.com/NVlabs/stylegan2-ada. Overall, we consistently observe that ContraD improves the baseline StyleGAN2, even achieving significantly better FIDs than ADA on AFHQ-Dog and AFHQ-Wild. These results confirms the effectiveness of ContraD on higher-resolution datasets, especially under the regime of limited data. Qualitative comparisons can be found in Figure 6, 7, and 8 of Appendix D.
+
+Table 7: Comparison of the best FID score (lower is better) on unconditional image generation of AFHQ datasets (Choi et al., 2020) with StyleGAN2. Values in the rows marked by \* are from those reported in its reference. We set our results bold-faced whenever the value improves the StyleGAN2 baseline (“Baseline”). Underlined indicates the best score among tested for each dataset.   
+
+<table><tr><td>Architecture</td><td>Method</td><td>DOG</td><td>CAT</td><td>WILD</td></tr><tr><td rowspan="2">StyleGAN2</td><td>Baseline* (Karras et al., 2020b) ADA* (Karras et al., 2020a)</td><td>19.4</td><td>5.13</td><td>3.48</td></tr><tr><td></td><td>7.40</td><td>3.55</td><td>3.05</td></tr><tr><td></td><td>ContraD (ours)</td><td>7.16</td><td>3.82</td><td>2.54</td></tr></table>
+
+# C RESULTS ON IMAGENET DATASET
+
+Table 8: Comparison of the best FID score and IS on conditional generation of ImageNet $( 6 4 \times 6 4 )$ with BigGAN architecture. Values in the rows marked by $^ *$ are from those reported in its reference.   
+
+<table><tr><td>ImageNet</td><td>FID↓</td><td>IS↑</td></tr><tr><td>BigGAN* (Brock et al., 2019)</td><td>10.6</td><td>25.43±0.15</td></tr><tr><td>FQ-BigGAN* (Zhao et al.,2020b)</td><td>9.67</td><td>25.96±0.24</td></tr><tr><td>ContraD-BigGAN (ours)</td><td>8.32</td><td>26.94±0.40</td></tr></table>
+
+Table 9: Comparison linear evaluation and transfer learning performance across 6 natural image classification datasets for BigGAN discriminators pretrained on ImageNet $( 6 4 \times 6 4 )$ . We report the top-1 accuracy except for ImageNet and SUN397, which we instead report the top-5 accuracy.   
+
+<table><tr><td>Training (BigGAN)</td><td>ImageNet</td><td>CIFAR10</td><td>CIFAR100</td><td>DTD</td><td>SUN397</td><td>Flowers</td><td>Food</td></tr><tr><td>Supervised (ImageNet)</td><td>63.5</td><td>76.1</td><td>55.2</td><td>45.4</td><td>31.7</td><td>78.1</td><td>44.5</td></tr><tr><td>SimCLR(λcon = Xdis = 0)</td><td>43.4</td><td>81.2</td><td>55.3</td><td>43.9</td><td>37.6</td><td>69.8</td><td>38.8</td></tr><tr><td>ContraD (ours)</td><td>51.5</td><td>84.5</td><td>61.1</td><td>50.6</td><td>44.4</td><td>78.6</td><td>44.5</td></tr></table>
+
+We also apply ContraD to the state-of-the-art BigGAN (Brock et al., 2019) architecture on ImageNet (Russakovsky et al., 2015), to examine the effectiveness of our method on this large-scale, classconditional GAN model. For this experiment, we exactly follow the training setups done in FQBigGAN (Zhao et al., 2020b), a recent work showing that feature quantization in discriminator could improve state-of-the-art GANs, which is based on the official PyTorch (Paszke et al., 2019) implementation of BigGAN.4 Following Zhao et al. (2020b), we down-scale the ImageNet dataset into $6 4 \times 6 4$ , and reduce the default channel width of 96 into 64. We train the model with batch size 512 for 100 epochs $\sim 2 5 0 \mathrm { K }$ generator steps). Unlike other experiments, we compute FIDs between the training set and 50K of generated samples for a fair comparison to the baseline scores reported by Zhao et al. (2020b). In Table 8, we show that ContraD further improves the baseline BigGAN training and FQ-BigGAN (Zhao et al., 2020b) both in FID and IS under the same training setups.
+
+Next, we evaluate the representation of the learned BigGAN encoder $D$ from ContraD5 under linear evaluation (on ImageNet) and several transfer learning benchmarks of natural image classification: namely, we consider CIFAR-10 and CIFAR-100 (Krizhevsky, 2009), the Describable Textures Dataset (DTD) (Cimpoi et al., 2014), SUN397 (Xiao et al., 2010), Oxford 102 Flowers (Flowers) (Nilsback & Zisserman, 2008), and Food-101 (Bossard et al., 2014). As done in Table 3, we compare ContraD with an ablation when $\lambda _ { \tt c o n } = \lambda _ { \tt d i s } = 0$ , which is equivalent to the SimCLR (Chen et al., 2020a) training. In addition, we also compare against the supervised baseline, where the representation is pre-trained on ImageNet with standard cross-entropy loss using the same architecture of BigGAN discriminator. Similarly to the ContraD training, this baseline is also trained with the SimCLR augmentation, using Adam optimizer (Kingma & Ba, 2014) for 120 epochs.
+
+Table 9 summarizes the results. We first note that the supervised training achieves $6 3 . 5 \%$ top-5 accuracy on ImageNet, while the SimCLR baseline achieves $4 3 . 4 \%$ (in linear evaluation). Again, as also observed in Table 3, ContraD significantly improves the linear evaluation performance from the baseline SimCLR, namely to $5 1 . 5 \%$ . Although the results are somewhat far behind compared to those reported in the original SimCLR (Chen et al., 2020a), this is possibly due to the use of model with much limited capacity, i.e., the BigGAN discriminator: its essential depth is only ${ \sim } 1 0$ , and it also uses spectral normalization (Miyato et al., 2018) for every layer unlike the ResNet-50 architecture considered by Chen et al. (2020a). Nevertheless, we still observe a similar trend in the accuracy gap between supervised and (unsupervised) contrastive learning. On the other hand, the remaining results in Table 9, those from the transfer learning benchmarks, confirm a clear advantage of ContraD, compared to both the SimCLR and supervised baselines: we found the two baselines perform similarly on transfer learning, while our ContraD training further improves SimCLR consistently.
+
+![](images/2803a19d7148673684f3a61f5c48939276fcff7966c6d0202f225462a15fb966.jpg)  
+Figure 2: Qualitative comparison of unconditionally generated samples from GANs with different training methods. All the models are trained on CIFAR-10 with an SNDCGAN architecture.
+
+![](images/8f10f0a4cefb14366c386e6b394004a11293bdcebbed43433e8a85bd84c5168f.jpg)  
+Figure 3: Qualitative comparison of unconditionally generated samples from GANs with different training methods. All the models are trained on CIFAR-100 with an SNDCGAN architecture.
+
+![](images/565bf6b5cf21dc3e3234e8279bd49157ad927d1d16e70bdac73f4da4d1f6ac18.jpg)  
+Figure 4: Qualitative comparison of unconditionally generated samples from GANs with different training methods. All the models are trained on CelebA-HQ-128 with an SNDCGAN architecture.
+
+![](images/49648ae7bc3e6ec618e8000ad8a3ed53e716c93129f6ae98838e349ceccb8a69.jpg)  
+Figure 5: Visualization of conditionally generated samples via conditional DDLS (Section 3.3) with ContraD. We use an SNDCGAN generator trained with an SNResNet-18 ContraD for the generation.
+
+![](images/68083929ad3443b94e7ae29b393dd0097254c53feade8b38553414bf78967ad6.jpg)  
+Figure 6: Qualitative comparison of unconditionally generated samples from GANs with different training methods, along with real images from the training set. All the models are trained on AFHQDog (4,739 images) with StyleGAN2. We apply the truncation trick (Karras et al., 2019) with $\psi = 0 . 7$ to produce the images at the bottom row. We use the pre-trained models officially released by the authors to obtain the results of the baseline StyleGAN2 and ADA.
+
+![](images/41afd1b49d3fa3f188e153074f2345bf1da01dcd36dda4431fece8688ed18740.jpg)  
+Figure 7: Qualitative comparison of unconditionally generated samples from GANs with different training methods, along with real images from the training set. All the models are trained on AFHQCat (5,153 images) with StyleGAN2. We apply the truncation trick (Karras et al., 2019) with $\psi =$ 0.7 to produce the images at the bottom row. We use the pre-trained models officially released by the authors to obtain the results of the baseline StyleGAN2 and ADA.
+
+![](images/5334a13c1b6ad161356252ae7fe4b3b4aa7d64d024d490908f18933d91092f0f.jpg)  
+Figure 8: Qualitative comparison of unconditionally generated samples from GANs with different training methods, along with real images from the training set. All the models are trained on AFHQWild (4,738 images) with StyleGAN2. We apply the truncation trick (Karras et al., 2019) with $\psi = 0 . 7$ to produce the images at the bottom row. We use the pre-trained models officially released by the authors to obtain the results of the baseline StyleGAN2 and ADA.
+
+![](images/831a68a5a7ad640f8312c5449910b7a6b99801db205ba64be4311b3e38853e51.jpg)  
+Figure 9: Comparion of the FID distribution of the top $2 5 \%$ of trained models (Kurach et al., 2019) on CIFAR-10 (SNDCGAN) for different batch sizes.
+
+Contrary to existing practices for training GANs, we observe in our experiments that ContraD typically needs a larger batch size in training to perform best. We further confirm this effect of batch sizes in Figure 9: overall, we observe the stanard training (“W/O”) is not significantly affected by a particular batch size, while ContraD offers much better FIDs on larger batch sizes. We also observe bCR (Zhao et al., 2020c) slightly benefits from larger batches, but not as much as ContraD does. This observation is consistent with Chen et al. (2020a) that contrastive learning benefits from using larger batch sizes, and it also supports our key hypothesis on the close relationship between contrastive and discriminator representations.
+
+# F EXPERIMENTAL DETAILS
+
+# F.1 ARCHITECTURE
+
+Overall, we consider four architectures in our experiments: SNDCGAN (Miyato et al., 2018), StyleGAN-2 (Karras et al., 2020b), and BigGAN (Brock et al., 2019) for GAN architectures, and additionally SNResNet-18 for a discriminator architecture only. For all the architectures considered, we have modified the last linear layer into a multi-layer perceptron (MLP) having the same input and output size, for a fair comparison to ContraD that requires such an MLP for the discriminator header $h _ { \mathrm { d } }$ . When a GAN model is trained via ContraD, we additionally introduce two 2-layer MLP projection heads of the output size 128 upon the penutimate representation in the discriminator to stand $h _ { \mathbf { r } }$ and $h _ { \mathbf { f } }$ . All the models are implemented in PyTorch (Paszke et al., 2019) framework.
+
+SNDCGAN. We adopt a modified version of SNDCGAN architecture by Kurach et al. (2019) following the experimental setups of other baselines (Zhang et al., 2020; Zhao et al., 2020c). The official implementation can be found in compare gan codebase,6 and we have re-implemented this TensorFlow implementation into PyTorch framework. The detailed structures of the generator and discriminator of SNDCGAN are summarized in Table 10 and 11, respectively.
+
+SNResNet-18. We modify the ResNet-18 (He et al., 2016) architecuture, where a PyTorch implementation is available from torchvision7 library, to not include batch normalization (Ioffe & Szegedy, 2015) layers inside the network. Instead, we apply spectral normalization (Miyato et al., 2018) for all the convolutional and linear layers. We made such a modification to adapt this model as a stable GAN discriminator, as batch normalization layers often harm the GAN dynamics in discriminator in practice.
+
+StyleGAN2. We follow the StyleGAN2 architecture used in the DiffAug baseline (Zhao et al., 2020a). Specifically, we generally follow the official StyleGAN2, but the number of channels at $3 2 \times 3 2$ resolution is reduced to 128 and doubled at each coarser level with a maximum of 512 channels, in order to optimize the model for CIFAR datasets, as suggested by Zhao et al. (2020a).
+
+Table 10: SNDCGAN generator   
+
+<table><tr><td>Layer</td><td>Kernel</td><td>Output</td></tr><tr><td>Latent z</td><td></td><td>128</td></tr><tr><td>Linear, BN, ReLU</td><td></td><td>h/8 × w/8 × 512</td></tr><tr><td>DeConv, BN,ReLU</td><td>[4,4,2]</td><td>h/4×w/4× 256</td></tr><tr><td>DeConv,BN,ReLU</td><td>[4,4,2]</td><td>h/2 ×w/2×128</td></tr><tr><td>DeConv,BN,ReLU</td><td>[4,4,2]</td><td>×w × 64</td></tr><tr><td>DeConv, Tanh</td><td>[3,3,1]</td><td>×w ×</td></tr></table>
+
+Table 11: SNDCGAN discriminator   
+
+<table><tr><td>Layer</td><td>Kernel</td><td>Output</td></tr><tr><td>Conv,LeakyReLU</td><td>[3,3,1]</td><td>h ×w ×64</td></tr><tr><td>Conv,LeakyReLU</td><td>[4,4,2]</td><td>h/2 × w/2 × 128</td></tr><tr><td>Conv,LeakyReLU</td><td>[3,3,1]</td><td>h/2 × w/2 × 128</td></tr><tr><td>Conv,LeakyReLU</td><td>[4,4,2]</td><td>h/4 × w/4× 256</td></tr><tr><td>Conv,LeakyReLU</td><td>[3,3,1]</td><td>h/4 × w/4× 256</td></tr><tr><td>Conv,LeakyReLU</td><td>[4,4,2]</td><td>h/8×w/8× 512</td></tr><tr><td>Conv,LeakyReLU</td><td>[3,3,1]</td><td></td></tr><tr><td></td><td></td><td>h/8 × w/8× 512</td></tr></table>
+
+BigGAN. We use the official PyToch implementation of BigGAN (https://github.com/ ajbrock/BigGAN-PyTorch). In order to adapt ContraD into the class-conditional BigGAN architecture, we apply the stop-gradient operation not only on the penultimate representation before computing $h _ { \mathrm { d } }$ , but also before applying the class-conditional projection (Miyato & Koyama, 2018).
+
+# F.2 TRAINING DETAILS
+
+SNDCGAN. Overall, we follow the best hyperparameter practices explored by Kurach et al. (2019) for SNDCGAN models on CIFAR and CelebA-HQ-128 datasets: We use Adam (Kingma & Ba, 2014) with $( \alpha , \beta _ { 1 } , \beta _ { 2 } ) = ( 0 . 0 0 0 2 , 0 . 5 , 0 . 9 9 9 )$ for optimization with batch size of 64, while in case of ContraD on CIFAR datasets the batch size of 512 is used instead. For CelebA-HQ-128, we keep the default batch size of 64 even for ContraD. All models but the “Hinge” baselines in Table 2 are trained with the non-saturating loss (Goodfellow et al., 2014). We stop training after 200K generator updates for CIFAR-10/100 and 100K for CelebA-HQ-128. When CR or bCR is used, we set their regularization strengths $\lambda = 1 0$ for both real and fake images.
+
+StyleGAN2. We follow the training details of DiffAug (Zhao et al., 2020a) in their CIFAR experiments: we use Adam with $( \alpha , \beta _ { 1 } , \beta _ { 2 } ) = ( 0 . 0 0 2 , 0 . 0 , 0 . 9 9 )$ for optimization with batch size of 32, but 64 for ContraD models. We use non-saturating loss for training, and use $R _ { 1 }$ regularization (Mescheder et al., 2018) with $\gamma = 0 . 1$ . We do not use, however, the path length regularization and the lazy regularization (Karras et al., 2020b) in training. We take exponential moving average on the generator weights with half-life of $1 0 ^ { 6 }$ . We stop training after 800K generator updates. When CR or bCR is used, we set their regularization strengths $\lambda = 1 0$ for both real and fake images.
+
+Linear evaluation and transfer learning. For a single linear evaluation (and transfer learning) run, we train a linear classifier upon a given (frozen) representation. The results in Table 3 are trained via stochastic gradient descent for 100 epochs: the initial learning rate of 0.1, and it is decayed by 0.1 at 60, 75, and 90-th epoch. In case of Table 9, on the other hand, we use Adam optimizer (Kingma & Ba, 2014) for 90 epochs: the initial learning rate if 0.0002, and it is decayed by 0.3 at 60 and 75-th epoch. We augment every training dataset via random crop, resizing and horizontal flipping.
+
+Conditional DDLS. We run Langevin sampling of step size $\varepsilon ~ = ~ 0 . 1$ for 1,000 iterations. We follow all the practical considerations proposed by Che et al. (2020) to improve the sample quality of DDLS: specifically, (a) we separately set the standard deviation of the Gaussian noise (13) to 0.1, and (b) to alleviate mode dropping issues, instead of running DDLS on $G$ itself, we run the sampling with $G ^ { * } ( \mathbf { z } , \mathbf { z } ^ { \prime } ) : = G ( \mathbf { z } ) + \varepsilon \bar { \mathbf { z } } ^ { \prime }$ , where $\mathbf { z } ^ { \prime }$ is also jointly updated via Langevin dynamics.
+
+Hyperparameters in ContraD. Unless otherwise specified, we use $\lambda _ { \tt c o n } = \lambda _ { \tt d i s } = 1$ , and $\tau = 0 . 1$ in our experiments, where $\tau$ is the temperature hyperparameter defined in (5). We use hidden layer of size 512 for $h _ { \mathbf { r } }$ , $h _ { \mathbf { f } }$ , $h _ { \mathrm { d } }$ in SNDCGAN and StyleGAN2, and 1024 in SNResNet-18 and BigGAN. For ContraD models, we use linear warm-up strategy on learning rate up to 3K steps of training for both generator and discriminator, following practices in contrastive learning (Chen et al., 2020a).
+
+SimCLR augmentations. We use the augmentation pipeline proposed in SimCLR (Chen et al., 2020a) for training ContraD. Specifically, for CIFAR datasets, we sequentially transform a given image by: (a) random crop and resizing, (b) horizontal flipping, (c) color jittering with probability of $p = 0 . 8$ , and (d) graying the image with probability of $p = 0 . 2$ . We additionally apply (f) Gaussian blurring with probability of $p = 0 . 5$ for higher resolution images such as CelebA-HQ-128 and ImageNet. We ensure that every transformation in this pipelines are differentiably implemented, so that the augmentation still can be used for training the generator of GAN.

parse/train/lwwEh0OM61b/lwwEh0OM61b.md

@@ -0,0 +1,278 @@
+# A-NeRF: Articulated Neural Radiance Fields for Learning Human Shape, Appearance, and Pose
+
+Shih-Yang Su1 Frank Yu1 Michael Zollhöfer2 Helge Rhodin1
+
+1University of British Columbia
+
+2Facebook Reality Labs
+
+# Abstract
+
+While deep learning reshaped the classical motion capture pipeline with feedforward networks, generative models are required to recover fine alignment via iterative refinement. Unfortunately, the existing models are usually hand-crafted or learned in controlled conditions, only applicable to limited domains. We propose a method to learn a generative neural body model from unlabelled monocular videos by extending Neural Radiance Fields (NeRFs). We equip them with a skeleton to apply to time-varying and articulated motion. A key insight is that implicit models require the inverse of the forward kinematics used in explicit surface models. Our reparameterization defines spatial latent variables relative to the pose of body parts and thereby overcomes ill-posed inverse operations with an overparameterization. This enables learning volumetric body shape and appearance from scratch while jointly refining the articulated pose; all without ground truth labels for appearance, pose, or 3D shape on the input videos. When used for novel-view-synthesis and motion capture, our neural model improves accuracy on diverse datasets. Project website: https://lemonatsu.github.io/anerf/.
+
+# 1 Introduction
+
+Generative models have evolved from Generative Adversarial Networks (GANs) recreating images [14, 21] to neural scene representations [39, 56, 57] providing control and image understanding for downstream tasks via structured latent variables. However, most 3D models require 3D labels that cannot be crowd-sourced on natural images and require dedicated depth sensors. It is hence an important research problem to learn 3D representations from 2D observations, which is particularly challenging for humans with diverse body shapes and appearances and their non-rigid motion.
+
+Modern human motion capture techniques typically combine the advantages of discriminative and generative approaches. A feed-forward 3D human pose estimation approach provides a rough initial estimate of human pose. Afterward, a generative approach based on either a high-quality 3D scan of the person [17], or a parametric human body model learned from laser scans [4] refines the estimate iteratively based on the image evidence. Although achieving unprecedented accuracy, existing models require a low-dimensional, restrictive, shape body model or a personalized 3D scan of the user.
+
+We introduce Articulated Neural Radiance Fields (A-NeRF) for learning a user-specific neural 3D body model and underlying skeleton pose from unlabelled videos (see Figure 1). When applied to motion capture, it alleviates the need for template models while maintaining the advantages and accuracy of current generative approaches. A-NeRF extends Neural Radiance Fields (NeRF) [39] to work with single videos and articulated motion. NeRF parameterizes the scene implicitly as
+
+$$
+F _ { \phi } ( \Gamma ( \mathbf { q } ) , \Gamma ( \mathbf { d } ) ) \mapsto ( \sigma , \mathbf { c } ) , \quad \mathrm { ~ w i t h ~ } \sigma \in \mathbb { R } , \mathbf { c } \in \mathbb { R } ^ { 3 } , \mathbf { q } \in \mathbb { R } ^ { 3 } , \mathrm { ~ a n d ~ } \mathbf { d } \in \mathbb { R } ^ { 3 } ,
+$$
+
+by chaining $F _ { \phi }$ , a Multi-layer Perceptron (MLP), with $\Gamma$ , the Positional Encoding (PE) [66]. First, the PE maps the input scene point q and view direction $\mathbf { d }$ to a higher dimensional space that enables the
+
+![](images/1a46586f02de34fa53c421f18f47e2d0004b50edec411cd88126f9b5f48e35dd.jpg)  
+Figure 1: Our A-NeRF jointly learns a neural body model of the user and works with diverse body poses (left) while also refining the initial 3D articulated skeleton pose estimate from a single or, if available, multiple views without tedious camera calibration (center). Underlying is a templatefree neural representation (right) and skeleton-based embedding coupled with volume volumetric rendering. Real faces and their reconstructions are blurred in all figures for anonymity.
+
+MLP to learn a meaningful scene representation function $F _ { \phi }$ that subsequently outputs the radiance c and opacity $\sigma$ at every point in space. Second, the implicitly described scene (via conditioning on query locations) is rendered via classical ray-marching from computer graphics. The advantage of the MLP representation is that it avoids the complexity of volumetric grids [29], circumvents the artifacts induced by the implicit bias of screen-space convolution [40, 54], and, unlike surface meshes, can have flexible topology. However, the original NeRF only works for static scenes captured from dozens of calibrated cameras such that each 3D point is seen from multiple views.
+
+Our conceptual contribution lies in learning a neural latent representation relative to an articulated skeleton. While explicit models such as the popular SMPL body model [30] deform a surface via forwards kinematics, the implicit form of A-NeRF makes us re-think how skeletons can be integrated—implicit networks require the inverse transformation from 3D world coordinates to the reference skeleton, a significantly harder task that has not been fully explored. Our core technical novelty is to come up with and evaluate different parameterizations of $\Gamma ( \mathbf { q } ) , \Gamma ( \mathbf { d } )$ in Eq. 1 to create local coordinates relative to the articulated skeleton. Since a point in 3D world coordinates cannot uniquely be associated with a body part, we resolve the mentioned ill-posed inverse problem by overparameterizing with one embedding per bone. This embeds domain knowledge of how humans move and provides a common frame for the neural network to combine body shape and appearance constraints across the entire captured sequence (see Figure 2).
+
+We demonstrate that all our contributions together enable learning of a neural body model from monocular video, requiring only rough 3D pose estimates for initialization, that reaches a level of detail previously only attained with parametric surface models or multi-view approaches [51].
+
+Scope. We apply the model to motion capture, character animation, and appearance and motion transfer and demonstrate that the pose refinement improves on existing monocular skeleton reconstruction. We learn in the transductive setting, for a specific target video that is known at training time but has no ground truth. A-NeRF enables novel view synthesis of dynamic motions, with plausible however non-physical illumination. Additional steps are needed to enable relighting applications.
+
+General impact. Building a self-supervised approach for personalized human body modelling promises to be more inclusive to people and activities that are not well represented in supervised datasets. However, it bears the risk that 3D models of people are created without consent. We urge users to only use datasets collected for developing and validating motion capture algorithms.
+
+# 2 Related Work
+
+Our approach builds upon and is related to the following work on human pose and shape estimation, human modeling, and neural scene representations [62].
+
+Discriminative Human Pose Estimation. While feed-forward estimation of the 3D joint positions [26, 32, 34, 35, 43, 44, 52, 59, 64, 69, 72] or joint angles and bone lengths [53, 73] of the skeleton is highly accurate, such discriminative estimates are prone to misalignment when overlayed onto the input image due to the generalization gap. The skeleton pose can be refined to better match the 2D pose estimates, but this usually leads to larger errors in 3D [36, 37]. We use [23] for initializing skeleton pose and combine it with a neural body model.
+
+Surface-based Generative Body Models. These are obtained by either constraining template meshes via deformation energies [17, 70] or learning parametric human body models from a large collection of laser scans [6, 9, 30]. Their low-dimensional parameters constrain the space of plausible human shapes and motions. This enables real-time reconstructions from single images [7, 15], detailed texturing and displacement mapping [3, 5], and alleviates manual rigging [2]. It also enables optimization within the bounds of the learned prior [11, 16, 25] and weak-supervision when integrated in a differentiable form [27] into neural training processes [4, 20, 23, 42, 45, 65]. Closest to our approach in this category are the model fitting methods by [74] that textures and geometrically refines an untextured parametric quadruped model to zebra images and to [68] that uses optical flow to refine human pose. Although in a similar setting, our surface-free neural body model and volumetric rendering is fundamentally different to their textured triangle mesh that is rigged with forward kinematics and needs to be obtained a-priori.
+
+Implicit Body Models. A-NeRF bears close similarities with body models defined implicitly in terms of level-sets [58] and density of a sum of Gaussians [18, 48, 49] that are used for refining human pose, shape, and appearance via differentiable ray-tracing. However, sum of Gaussians and other primitives only provide rough approximations.
+
+Neural Scene Representations. Recent neural scene representations learn low-dimensional nonlinear representations of meshes [28, 63], point clouds [1, 38, 67], sphere sets [24], and dense volumetric grids [29, 55]. Their respective geometric output representations enable rendering with classical rendering techniques but have limited expressiveness, e.g., due to the fixed connectivity of a surface mesh and large memory footprint of discretized volumes. This limitation is overcome by using unconstrained MLPs [39] paired with positional encoding [60] to characterize an arbitrary point in 3D space. Common are surface definitions via level-sets of the MLP that are rendered with sphere tracing [56] and density representations rendered with ray-marching [39]. The rendering step is required for maximum likelihood estimation, to define a likelihood over observable variables—real images—while learning a 3D model. The rendering can be learned too [40, 50, 51] but usually leads to inconsistencies, particularly when training data is scarce. Some concurrent works also use neural scene representations for refining camera motion [71], video reenactment [47], and facial models [12, 13]. Orthogonal to these works, our A-NeRF learns an articulated body model from estimated poses and uncalibrated cameras.
+
+Closely related to ours is the NASA surface body model [10] that also defines an implicit function as the minimum of individual implicit functions that are rigidly attached to the bones of a skeleton, each conditioned on the entire human pose to model dependencies and learned from 3D scans. By contrast, we learn a volumetric model instead of a surface model and include appearance and rendering. Even more similar is the recent NeuralBody [46] representation, which combines a NeRF with a surface body model and underlying skeleton. By contrast to both approaches, we do not require surface supervision or initialization, condition pose differently, and refine pose, which enables us to learn from single videos in unconstrained environments.
+
+# 3 Formulation
+
+Objective. Given a sequence $[ \mathbf { I } _ { k } ] _ { k = 1 } ^ { N }$ of N images $\mathbf { I } _ { k } \in \mathbb { R } ^ { H \times W \times 3 }$ stemming from one or several videos of the same person, our goal is to simultaneously estimate the time-varying skeleton poses $[ \theta _ { k } ] _ { k = 1 } ^ { N }$ and learn a detailed body model. Our A-NeRF body model $C _ { \phi }$ is parametrized by neural network parameters $\phi$ that define volumetric shape and color while the skeleton captures motion over time. Figure 2 gives an overview of this generative model. It enables a rendering of the virtual body model in unseen poses and optimizes its parameters $\theta$ and $\phi$ on the image reconstruction objective
+
+![](images/0dd8ad9ad3adccd8b797a7e65d2fc737423d8d9d961ba23092b4e4926956105c.jpg)  
+Figure 2: Overview. A-NeRF is a generative model that can be rendered and optimized on a photometric loss $\mathcal { L } ^ { \mathrm { S V } }$ (white). First, the skeleton pose is initialized with an off-the-shelf estimator (orange). Second, this pose is refined via a skeleton-relative embedding (blue) that, when fed to NeRF (green), drives the implicit body model that is rendered by ray-marching (red). A key property of the skeleton-relative embedding is that a single 3D query location maps to an overcomplete reparametrization, with the same point represented relative to each skeleton bone (right).
+
+![](images/c2eb2fdaaaeb577984bf32b25a8c52d647df88f1d3195d6358d2678e7dd03b97.jpg)  
+Figure 3: Importance of our skeleton-relative encodings. The original NeRF breaks (a) when training on a diverse set of poses and (b) further degrades when the poses are rotated. Even if (e) conditioned directly on $\theta _ { k }$ , the NeRF still suffers from artifacts due to the complexity and ambiguity of human articulation. With our skeleton-relative encoding (f, g), the geometry for the subject is consistent under rotation, and the quality is greatly improved, with the full model working best.
+
+$$
+\mathcal { L } ^ { \mathrm { S V } } ( \theta , \phi ) = \sum _ { k } \underbrace { \| C _ { \phi } ( \theta _ { k } ) - \mathbf { I } _ { k } \| _ { 1 } } _ { \mathrm { d a t a : t e r m } } + \underbrace { \lambda _ { \theta } d ( \theta _ { k } - \hat { \theta } _ { k } ) } _ { \mathrm { p o s e r e g u l a r i z e r } } + \underbrace { \lambda _ { t } \left\| \frac { \partial ^ { 2 } \theta _ { k } } { \partial t ^ { 2 } } \right\| _ { 2 } ^ { 2 } } _ { \mathrm { s m o o t h n e s s ~ p r i o r } } .
+$$
+
+with the influence of all three terms balanced by hyperparameters $\lambda _ { t }$ and $\lambda _ { \theta }$ . The data term measures the distance between the images generated by $C _ { \phi }$ and the input image with the L1 distance. The pose regularizer encourages the solution to stay close to an initial pose estimate $\hat { \theta }$ obtained from an offthe-shelf predictor [23], tolerating small shifts up to $\epsilon = 0 . 0 1$ with $d ( \boldsymbol { x } ) = \operatorname* { m i n } ( \left\| \boldsymbol { x } \right\| _ { 2 } ^ { 2 } - \epsilon , 0 )$ . Lastly, the smoothness prior penalizes acceleration ∂2θk∂t2 between poses of consecutive frames. Minimizing Eq. 2 can be seen as maximizing a corresponding probabilistic model, with the quadratic energy terms being the log-likelihoods of Gaussian distributions. Our focus is on formalizing the neural body model. For simplicity, we continue to write equations in terms of the objective functions used during inference with stochastic gradient descent.
+
+# 3.1 NeRF and A-NeRF Image Formation Model
+
+Instead of modeling the scene as a collection of triangles or other primitives, we define the human implicitly by a neural network as a function (Eq. 1) defined over all possible 3D points and view
+
+directions in space [39]. Similar to NeRF, we render the image of the human subject via ray marching
+
+$$
+C _ { \phi } ( u , v ; \theta _ { k } ) = \sum _ { i = 1 } ^ { Q } T _ { i } ( 1 - \exp ( - \sigma _ { i } \delta _ { i } ) ) \mathbf { c } _ { i } , \quad T _ { i } = \exp \left( - \sum _ { j = 1 } ^ { i - 1 } \sigma _ { j } \delta _ { j } \right) ,
+$$
+
+with $( u , v )$ the 2D pixel location on the image, $i$ the index to 3D query positions $\mathbf { q } _ { i }$ sampled along d and $\delta _ { i }$ the distance to neighboring samples—a constant if samples would be taken at regular intervals. The $T _ { i }$ is the accumulated transmittance for the ray traveling from the near plane to $\mathbf { q } _ { i }$ —the fraction of light reaching the sensor from sample point $i$ . The $\mathbf { c } _ { i }$ is the light color emitted or reflected at $i$ . The final pixel color is the sum over all $Q$ samples, with the last sample taking the special role of the background. The background color is easily inferred as the median pixel color over the entire video for static camera setups. The ray direction $\mathbf { d } = \mathbf { K } _ { k } ^ { - 1 } ( u , v )$ is computed using the estimated camera intrinsics [23]. In the following, we introduce our skeleton parametrization $\theta _ { k }$ and how to use it to effectively model dynamic articulated human motion.
+
+# 3.2 Articulated Skeleton Pose Model
+
+Our skeleton representation encodes the connectivity and static bone lengths via a rest pose of 3D joint locations. Dynamics are modeled with per-frame skeleton poses $\theta _ { k }$ , which define an affine transformation $T ( \theta _ { k } , m )$ for each bone $m$ . Specifically, $T ( \theta _ { k } , m )$ maps a 3D position $\mathbf { p } _ { k , m } \in \mathbb { R } ^ { 3 }$ in the $m$ -th local bone coordinates to world coordinates $\mathbf { q } \in \mathbb { R } ^ { 3 }$ using homogeneous coordinates,
+
+$$
+{ \Bigg [ } { \begin{array} { c } { \mathbf { q } } \\ { \mathbf { 1 } } \end{array} } { \Bigg ] } = T ( \theta _ { k } , m ) \left[ { \begin{array} { c } { \mathbf { p } _ { k , m } } \\ { 1 } \end{array} } \right] ,
+$$
+
+where subscript $^ { k , m }$ indicates that a variable is related to the $m$ -th joint of image $\mathbf { I } _ { k }$ . Conversely, $T ( \theta _ { k } , m ) ^ { - 1 }$ maps world to local bone coordinates. Note that our skeleton is equivalent to SMPL [30] and others, but without their parametric surface model, and can therefore be initialized with any skeleton pose estimator. We include more details of our skeleton representation in the supplementary.
+
+# 3.3 A-NeRF Skeleton-Relative Encoding
+
+Our core contribution is to transform the query locations $\mathbf { q }$ and view direction d relative to the skeleton before determining the color and opacity at that transformed point via NeRF. It is a form of reparameterization that explicitly incorporates domain knowledge of how the human body parts are linked and transformed relative to each other. Intuitively, our implicit formulation turns explicit models, such as SMPL [30], on its head. Instead of deforming the output surface via skinning, the query location is mapped in the inverse direction to local bone-relative coordinate systems before processing through the NeRF network. Our final model uses a combined encoding $\dot { \mathbf { e } _ { k } } = [ h ( \tilde { \mathbf { v } } _ { k } ) \Gamma ( \bar { \tilde { \mathbf { v } } } _ { k } ) , \tilde { \mathbf { r } } _ { k } , h ( \tilde { \mathbf { v } } _ { k } ) \Gamma ( \bar { \mathbf { d } } _ { k } ) ]$ as input to the NeRF $F _ { \phi }$ . Note that the skeleton embedding introduces the desired time dependency, denoted by subscript $k$ . The inlet shows our most crucial contribution, the relative distance encoding $\tilde { \mathbf { v } } _ { k }$ followed by PE with Cutoff to reduce the influence of irrelevant bones. We derive the components of our encoding $\mathbf { e } _ { k }$ and the other alternatives below.
+
+![](images/faacd7f0218a007608aa3244fd5f0457a5d1bcc9b52b31fb173fda2fa1d7f54f.jpg)
+
+• Reference Pose Encoding One could compensate motion by attaching the query $\mathbf { q }$ in world coordinates at frame $k$ to the closest bone $m$ and transforming it with
+
+$$
+\mathbf { a } _ { k } = T ( \theta _ { 0 } , m ) T ^ { - 1 } ( \theta _ { k } , m ) \mathbf { q } .
+$$
+
+This puts the query relative to the bone $m$ as in frame $k$ but with the skeleton in rest pose $\theta _ { 0 }$ . NeRF could then learn without change in the 3D space of the rest pose as done before for surfaces [61]. However, this cannot capture non-rigid pose-dependent effects, such as muscle bulging, and has ambiguities when $\mathbf { q }$ is at equal distance to two bones.
+
+• Bone-relative Position (Rel. Pos.) To remove these ambiguities and the ill-posed association to a single part, we map q relative to each bone $m$ with,
+
+$$
+\tilde { \mathbf { q } } _ { k } = [ \tilde { \mathbf { q } } _ { k , 1 } , \cdots , \tilde { \mathbf { q } } _ { k , 2 4 } ] \mathrm { a n d } \tilde { \mathbf { q } } _ { k , m } = T ^ { - 1 } ( \theta _ { k } , m ) \mathbf { q } .
+$$
+
+The resulting individual bone coordinates are well suited to model the overwhelmingly rigid motion of the corresponding body part. Moreover, the overparameterization of position by concatenating all local encodings enables learning when complex interactions are necessary. However, such an embedding for all bones increases the dimensionality by an order of magnitude.
+
+• Relative Distance (Rel. Dist.) Much simpler to compute are distances from $\mathbf { q }$ to all bones $m$
+
+$$
+\tilde { \mathbf { v } } _ { k } = [ \tilde { \mathbf { v } } _ { k , 1 } , \cdots , \tilde { \mathbf { v } } _ { k , 2 4 } ] , \mathrm { ~ w i t h ~ } \tilde { \mathbf { v } } _ { k , m } = \| \tilde { \mathbf { q } } _ { k , m } \| _ { 2 } \in \mathbb { R } .
+$$
+
+This radial encoding is used in our final model in favor of $\tilde { \mathbf { q } }$ because it naturally captures spherically shaped limbs, is lower-dimensional, and thereby improves reconstruction accuracy.
+
+• Relative Direction (Rel. Dir.) Since the distance encoding is invariant to direction, we additionally obtain the direction vector to capture the orientation information of $\mathbf { q }$ ,
+
+$$
+\tilde { \mathbf { r } } _ { k } = [ \tilde { \mathbf { r } } _ { k , 1 } , \cdots , \tilde { \mathbf { r } } _ { k , 2 4 } ] , \quad \tilde { \mathbf { r } } _ { k , m } = \frac { \tilde { \mathbf { q } } _ { k , m } } { \left\| \tilde { \mathbf { q } } _ { k , m } \right\| _ { 2 } } \in \mathbb { R } ^ { 3 } .
+$$
+
+Note that by contrast to all other embeddings, this direction encoding did not profit from subsequent PE. We therefore pass it directly into $\mathbf { e } _ { k }$ .
+
+• Relative Ray Direction (Rel. Ray.) NeRF models the illumination effects in a static 3D scene using the position and view direction. By contrast, our goal is to learn a body model that produces plausible colors with dynamic skeleton poses. Therefore, we transform $\mathbf { d }$ to obtain $\tilde { \mathbf { d } }$ , the outgoing ray direction relative to each bone, similar to query position,
+
+$$
+\tilde { \mathbf { d } } _ { k } = [ \tilde { \mathbf { d } } _ { k , 1 } , \boldsymbol { \cdot \cdot \cdot } , \tilde { \mathbf { d } } _ { k , 2 4 } ] , \quad \tilde { \mathbf { d } } _ { k , m } = [ T ^ { - 1 } ( \theta _ { k } , m ) ] _ { 3 \times 3 } \mathbf { d } \in \mathbb { R } ^ { 3 } ,
+$$
+
+with $[ T ^ { - 1 } ( \theta _ { k } , m ) ] _ { 3 \times 3 }$ the rotational part of the bone-to-world transformation $T ^ { - 1 } ( \theta _ { k } , m )$ . Following concurrent works [33, 46], we also optimize an appearance code for each image to handle dynamic light effects. The combination of $\tilde { \mathbf { d } }$ and the per-image code enables A-NeRF to approximate the light effects in $\mathbf { I } _ { k }$ plausibly. See the supplemental material for detailed discussions on modeling view-dependent effects in our setting.
+
+• Cutoff. We desire a local embedding where points should not be influenced by all but only nearby bones. To this end, we introduce a windowed version of positional encoding by multiplying the encoding with respect to bone $m$ by $h ( \tilde { \mathbf { v } } _ { k , m } ) = 1 - S ( \tau ( \tilde { \mathbf { v } } _ { k , m } - t ) )$ , with $S$ the sigmoid step function, $t$ the cutoff point, $\tau$ the sharpness. This leaves queries unaffected by distant bones.
+
+Our embedding choice of $\mathbf { e } _ { k } = [ h ( \tilde { \mathbf { v } } _ { k } ) \Gamma ( \tilde { \mathbf { v } } _ { k } ) , \tilde { \mathbf { r } } _ { k } , h ( \tilde { \mathbf { v } } _ { k } ) \Gamma ( \mathbf { d } _ { k } ) ]$ has the advantage of being invariant to the global shift and rotation of the person and preserves the piece-wise rigidity of articulated motion while still allowing for pose-dependent deformation (see Figure 3). In addition to $\mathbf { e } _ { k }$ , we also consider other embedding variants. See the supplementary and Section 4 for a detailed discussion.
+
+# 4 Evaluation
+
+We performed experiments to validate that A-NeRF learns accurate body models and poses, with fewer assumptions (single view, uncalibrated, and w/o a parametric surface model) than the related works. This makes it applicable to fine-grained pose refinement that improves the estimates of stateof-the-art methods. The supplements provide the implementation details, additional comparisons and ablation studies.
+
+Inference and Implementation Details Our A-NeRF model is learned without supervision on a single or multiple videos of the same person. Camera intrinsics, bone lengths for setting $\mathbf { a } _ { m }$ , and pose $\theta _ { k }$ are initialized with [23] for every frame $k$ . These poses are then optimized on objective Eq.2, alongside the generative A-NeRF model. See supplementary for more details.
+
+Datasets. We evaluate on the following benchmarks, and additionally on synthetic data created from SURREAL and Mixamo characters, which are listed in the supplement.
+
+• Human 3.6M [19] The dataset consists of 5 training and 2 testing subjects (S9/S11) with ground truth 3D joint locations. We follow two widely adopted test protocols denoted as Protocol I [20, 22, 23] and Protocol II [41, 59], in which we evaluate 14/17-joint estimation error on every $5 ^ { t h } / 6 4 ^ { { \bar { t } } h }$ frame of the test videos, respectively. See the supplement for details.
+
+• MPI-INF-3DHP [35] This dataset is a standard benchmark for human pose estimation. It consists of 4 indoor and 2 outdoor subjects with challenging human poses. The number of frames per subject range from 276 to 603.
+
+• MonoPerfCap [70] The dataset consists of human performance video captured with a monocular camera in both indoor and outdoor settings. We use two subjects, Weipeng_outdoor and Nadia_outdoor, for our qualitative experiments. The two subjects have 1151 and 1635 frames, respectively, for training.
+
+Pose Metrics. We report the PA-MPJPE metric, the Euclidean distance between Procrustes-aligned (PA) predictions and ground truth 3D joint position averaged over all frames and joints of the test set. The PA alignment in scale and orientation is essential for comparing approaches that do not assume knowledge of the ground truth calibration and are, hence, ill-posed to the factors that the alignment removes. Following prior work [23, 22, 35], we also report percentange of correct keypoints (PCK) for MPI-INF-3DHP; the percentage of joints that lie within a distance of $1 5 0 \mathrm { m m }$ to the ground truth.
+
+Visual Metrics. We quantify the visual quality on MonoPerfCap and Human $3 . 6 \mathbf { M }$ datasets by training on a subset and testing on a held-out test set of the same character. Image quality is quantified via the PSNR and SSIM of the rendering compared with the reference image within the character bounding boxes. Because no ground truth pose is available in the required skeleton format on these datasets, we train our model once on the entire dataset to get reliable skeleton pose estimates as pseudo ground truth, and a second time with a part withheld to learn the body model for visual quality evaluation. Since MonoPerfCap has one sequence per actor, we exclude the last $20 \%$ of each video. For Human 3.6M we exclude entire actions, namely Geeting-1,2, Posing-1,2 and Walking-1,2. The body model is then transferred to the held-out portion by using the pseudo ground truth poses as the driving motion. Thereby, we can still test the generalization of different models to new poses and viewpoints, irrespective of the underlying skeleton model provided in the dataset.
+
+![](images/77235a6a06d5ac66c46a74745fcf55edc869ad3bc1d83569510a8397a6885b1c.jpg)  
+Figure 4: Novel view synthesis from all models. All models are trained with SPIN [23] estimated human pose and camera parameters. A-NeRF renderings (ours) align better with the reference images (top row), and the rendered novel views (2nd and 3rd rows) show better details (limbs, facial features).
+
+Novel-View-Synthesis and Character Animation. Our body model is generative, which allows us to train on a single or multiple uncalibrated videos and alter viewpoint and human pose. Figure 4 shows renderings of the same persons from a new camera angle; novel-view-synthesis. Likewise, Figure 5 demonstrates character animation, where the view is fixed, and the underlying skeleton is re-posed by manually changing joint angles or by transferring the motion between characters. A-NeRF is the first model that learns a detailed human body model with such capabilities without needing a 3D surface or multi-view supervision. While NeuralBody [46] can also learn photo-realistic human models from monocular images, their model anchors its representation on the SMPL 3D surface. When trained on our single-view setup with the same noisy estimated poses $\hat { \theta }$ as input, NeuralBody suffers from artifacts and less detail when training, as it assumes 3D ground truth, which is only available in controlled conditions. The importance of our joint body and pose optimization is further validated by the ablation with refinement disabled (Ours w/o rf.), which similarly produces ghosting artifacts around extremities.
+
+![](images/a8e5445a54eeaa430908d4d28914b76732bdf3195d9e59a0ce6f40af5d2f1486.jpg)  
+Figure 5: Motion retargeting and animation. Top rows: Pose transfer, with the source pose reconstructed by A-NeRF and rendered with different target body models. Last row: Animating the A-NeRF model while keeping the lower or upper part of the body fixed.
+
+Table 1: Quantitative evaluation on Human3.6M [19] and MPI-INF-3DHP [35]. Our test-time pose refinement improves consistently upon the SPIN baseline, with largest improvements for extremities (PA-Wrist).   
+
+<table><tr><td rowspan="2"></td><td colspan="4">Human 3.6M</td><td rowspan="2">MPI-INF-3DHP</td></tr><tr><td>Protocol I</td><td>Protocol II</td><td></td><td>Prot.I Wrist Prot.II Multi-view</td></tr><tr><td>Method</td><td>PA-MPJPE↓</td><td>PA-MPJPE↓</td><td>PA-Wrist↓</td><td>PA-MPJPE↓</td><td>PA-MPJPE↓</td><td>PCK↑</td></tr><tr><td>MotioNet [53]</td><td>54.6</td><td>-</td><td>-</td><td>1</td><td>-</td><td>-</td></tr><tr><td>HoloPose [16]</td><td>46.5</td><td>-</td><td>=</td><td>-</td><td></td><td></td></tr><tr><td>VIBE [22]</td><td>41.4</td><td>n/a</td><td>n/a</td><td>n/a</td><td>64.6</td><td>89.3</td></tr><tr><td>SPIN [23]</td><td> 41.1</td><td>-</td><td>-</td><td>-</td><td>67.5</td><td>76.4</td></tr><tr><td>Baseline (SPIN github [23])</td><td>42.7*</td><td>41.9*</td><td>66.5*</td><td>34.0*</td><td>68.2*</td><td>79.3*</td></tr><tr><td>SPIN-SMPLify+[7, 23]</td><td>57.7</td><td>59.2</td><td>100.9</td><td>=</td><td></td><td>=</td></tr><tr><td> Ours (w/o smoothness prior)</td><td>39.4</td><td>39.6</td><td>57.3</td><td>28.0</td><td>66.9</td><td>80.4</td></tr><tr><td>Ours</td><td> 39.3</td><td>n/a</td><td>n/a</td><td>n/a</td><td>66.8</td><td>80.4</td></tr></table>
+
+? Reevaluation of publicly-available model because missing evaluation protocols or not reproducible. † We refine the SPIN estimated pose using SMPLify. We adopt the implementation from the SPIN repository.
+
+Human Pose Estimation. Training A-NeRF includes a form of test-time optimization (see Figure 1, right), only the initialization from [23] is trained supervised on the Human3.6M training set. Table 1 shows that on Human 3.6M, A-NeRF reaches comparable results with other single-view approaches, and achieves a $8 . 0 \%$ improvement in PA-MPJPE upon the baseline used for pose initialization on Protocol I $4 2 . 7  3 9 . 3 $ ) and $5 . 5 \%$ on Protocol II $( 4 1 . 9  3 9 . 6 ) $ ). Note that these are average numbers across all joints, including easy-to-predict hip, shoulder, and head joints. Our largest gains are on the extremities, e.g., with an improvement of $1 4 \%$ $( 9 . 2 \mathrm { m m } )$ for the wrist joint on Protocol II (PA-Wrist). We also compare to applying SMPLify [7], a method that refines 3D poses using 2D joint locations (estimated with [8]) as constraints, at test time. It tends to explain the 2D joint estimates perfectly but degrades the 3D pose. The final pose estimations become less accurate than the initial ones. In contrast, A-NeRF optimizes the poses by implicitly minimizing the disagreement among the 3D body representation in different images, and thus achieves better performance. On MPI-INF-3DHP, Table 1 shows the results averaged over the 6 test subjects from MPI-INF-3DHP. Despite having a low number of frames and human poses available for learning the skeleton-relative encoding, A-NeRF still provides moderate improvements over the baseline estimations.
+
+Table 2: Visual quality evaluation on the Human3.6M [19] and MonoPerfCap [70] held-out sets. Our full A-NeRF model significantly improves the visual quality. The body model itself attains a higher quality than NeuralBody (2nd vs. 3rd row), and additional detail is gained with the proposed pose refinement (last row).   
+
+<table><tr><td rowspan=1 colspan=1>MonoPerfCap   Human 3.6M</td></tr><tr><td rowspan=1 colspan=1>PSNR ↑SSIM↑PSNR↑SSIM↑</td></tr><tr><td rowspan=1 colspan=1>NeuralBody, driving motion from [23]                21.80 0.8476 22.08 0.8766</td></tr><tr><td rowspan=1 colspan=1>NeuralBody, driving motion from A-NeRF refinement21.75 0.8468 22.55 0.8782</td></tr><tr><td rowspan=1 colspan=1>A-NeRF w/o pose refinement                         21.99 0.8405 23.33 0.8776</td></tr><tr><td rowspan=1 colspan=1>A-NeRF (Our full model)                            24.39 0.8851 27.45 0.9277</td></tr></table>
+
+![](images/b90719df2ea1c9f918c0ab07c25b4a45b990fd9512e57715fe9cfa8540593914.jpg)  
+Figure 6: Geometry. The isosurface of our density model is rendered from multiple unseen views. A-NeRF learns plausible geometry without using explicit surface templates and accurate initial poses.
+
+Video-based volumetric reconstruction. Figure 6 visualizes the learned density using Marching Cubes [31], with voxel grid resolution of 256 and density threshold 10. Despite only learned from monocular videos (no stereo, depth camera, or multi-view constraints) and without using a predefined template model, A-NeRF reconstructs a detailed volumetric body with details that could not be captured by offsets to a parametric surface model, such as the head phones and basebal cap in the last row of Figure 6. Note that no geometric smoothness term is enforced.
+
+Multi-view extension. A-NeRF can also leverage multi-view refinement (Table 1, Protocol II Multi-view), even without access to ground truth camera calibration (see supplementary).
+
+Visual Quality Comparison. We report the results in Table 2. We compare to NeuralBody with both the initial estimates from [23] and our refined pose since NeuralBody has no refinement step. Compared to both variants, our A-NeRF shows significantly better reconstruction performance on held-out poses. We observe that NeuralBody models can retain the facial features and hands as they anchor their representation on a 3D surface model. However, the rendered limbs and faces are blurry and distorted. Results are similar to training A-NeRF without refinement (see Figure 7). As the estimation for these joints is often inaccurate and noisy, the models without pose refinement simply learn to predict mean pixel values. To conclude, it is important to train with pose refinement, with which A-NeRF suffers less from artifacts with overall better visual quality.
+
+![](images/d42017362181a1cd5719808aa12373f2bfb5f1d1c9e046d24161ffc6a68f3df1.jpg)  
+Figure 7: A-NeRF with pose refinement generalizes better to test sequences. We visualize both the rendered images, as well as the photometric error (squared distance, normalized to $[ 0 , 1 ] )$ between the rendered and the ground truth images (warmer color indicates higher error) from our MonoPerfCap (1-2th rows) and Human 3.6M (3-4th rows) held-out sets. NeuralBody models produce artifacts around body contours. A-NeRF w/o refinement cannot reproduce facial features and limbs. With pose refinement, A-NeRF can produce both appearances and shapes more plausibly.
+
+Ablation study. Our detailed ablation studies are reported in the supplemental document. In summary, they reveal: i) Embedding relative 3D position, $\tilde { \mathbf { q } } _ { k }$ , instead of our proposed radial embedding $\tilde { \mathbf { v } }$ yields only half as good pose refinements. ii) Our embedding choices keep the dimensionality moderate while improving on or matching the PSNR and SSIM of higher-dimensional variants. iii) For a fixed number of images with accurate poses, learning from a long video with diverse poses has visual quality comparable to learning from multiple shorter clips.
+
+Limitations and Failure Cases. Our computation time is the biggest bottleneck in extending A-NeRF to long sequences and multiple actors. Although a single static camera suffices, A-NeRF requires to see the person from all sides in varying poses to learn pose dependencies from data. The inlet shows a rendering of an extreme breakdance pose retargeted to a model trained on normal walking motions. Hence, the source pose is unseen during training and far from the data distribution, which leads to artifacts.
+
+![](images/f72296b8b5fb945fc5a96f6035cb96a97b7e69d3f741535d709a518fede6072e.jpg)
+
+# 5 Conclusion
+
+We propose a new way for integrating articulated skeleton models and implicit functions via an overcomplete re-parametrization. It includes learning an interpretable 3D representation from 2D images; a personalized volumetric density field with texture detail and time-varying poses of the actor depicted in the input. The underlying ill-posed problem of mapping a single query location to multiple parts is addressed with an overparametrization over nearby parts. To the best of our knowledge, ANeRF is the first approach to define NeRF models for extreme and articulated motion on unconstrained video and this new approach scores high on the Human 3.6M benchmark. Importantly, it works from a single video and naturally extends to multi-view and does not require camera calibration in either scenario. This is an important step towards making motion capture more accurate and practical. In future work, we will learn a general human model from a database of subjects instead of individuals.
+
+Funding in direct support of this work: NSERC Discovery grant, UBC Advanced Research Computing (ARC) GPU cluster, Compute Canada GPU servers, and a gift by Facebook Reality Labs.
+
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+        "text": "While deep learning reshaped the classical motion capture pipeline with feedforward networks, generative models are required to recover fine alignment via iterative refinement. Unfortunately, the existing models are usually hand-crafted or learned in controlled conditions, only applicable to limited domains. We propose a method to learn a generative neural body model from unlabelled monocular videos by extending Neural Radiance Fields (NeRFs). We equip them with a skeleton to apply to time-varying and articulated motion. A key insight is that implicit models require the inverse of the forward kinematics used in explicit surface models. Our reparameterization defines spatial latent variables relative to the pose of body parts and thereby overcomes ill-posed inverse operations with an overparameterization. This enables learning volumetric body shape and appearance from scratch while jointly refining the articulated pose; all without ground truth labels for appearance, pose, or 3D shape on the input videos. When used for novel-view-synthesis and motion capture, our neural model improves accuracy on diverse datasets. Project website: https://lemonatsu.github.io/anerf/. ",
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+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+        "text": "Generative models have evolved from Generative Adversarial Networks (GANs) recreating images [14, 21] to neural scene representations [39, 56, 57] providing control and image understanding for downstream tasks via structured latent variables. However, most 3D models require 3D labels that cannot be crowd-sourced on natural images and require dedicated depth sensors. It is hence an important research problem to learn 3D representations from 2D observations, which is particularly challenging for humans with diverse body shapes and appearances and their non-rigid motion. ",
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+        "text": "Modern human motion capture techniques typically combine the advantages of discriminative and generative approaches. A feed-forward 3D human pose estimation approach provides a rough initial estimate of human pose. Afterward, a generative approach based on either a high-quality 3D scan of the person [17], or a parametric human body model learned from laser scans [4] refines the estimate iteratively based on the image evidence. Although achieving unprecedented accuracy, existing models require a low-dimensional, restrictive, shape body model or a personalized 3D scan of the user. ",
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+        "text": "We introduce Articulated Neural Radiance Fields (A-NeRF) for learning a user-specific neural 3D body model and underlying skeleton pose from unlabelled videos (see Figure 1). When applied to motion capture, it alleviates the need for template models while maintaining the advantages and accuracy of current generative approaches. A-NeRF extends Neural Radiance Fields (NeRF) [39] to work with single videos and articulated motion. NeRF parameterizes the scene implicitly as ",
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+        "img_path": "images/7de33391d06fded249034edb0b96c308d1b1c44ad2fc22fba8e02b387f71cea2.jpg",
+        "text": "$$\nF _ { \\phi } ( \\Gamma ( \\mathbf { q } ) , \\Gamma ( \\mathbf { d } ) ) \\mapsto ( \\sigma , \\mathbf { c } ) , \\quad \\mathrm { ~ w i t h ~ } \\sigma \\in \\mathbb { R } , \\mathbf { c } \\in \\mathbb { R } ^ { 3 } , \\mathbf { q } \\in \\mathbb { R } ^ { 3 } , \\mathrm { ~ a n d ~ } \\mathbf { d } \\in \\mathbb { R } ^ { 3 } ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "by chaining $F _ { \\phi }$ , a Multi-layer Perceptron (MLP), with $\\Gamma$ , the Positional Encoding (PE) [66]. First, the PE maps the input scene point q and view direction $\\mathbf { d }$ to a higher dimensional space that enables the ",
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+        "img_path": "images/1a46586f02de34fa53c421f18f47e2d0004b50edec411cd88126f9b5f48e35dd.jpg",
+        "image_caption": [
+            "Figure 1: Our A-NeRF jointly learns a neural body model of the user and works with diverse body poses (left) while also refining the initial 3D articulated skeleton pose estimate from a single or, if available, multiple views without tedious camera calibration (center). Underlying is a templatefree neural representation (right) and skeleton-based embedding coupled with volume volumetric rendering. Real faces and their reconstructions are blurred in all figures for anonymity. "
+        ],
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+        "type": "text",
+        "text": "MLP to learn a meaningful scene representation function $F _ { \\phi }$ that subsequently outputs the radiance c and opacity $\\sigma$ at every point in space. Second, the implicitly described scene (via conditioning on query locations) is rendered via classical ray-marching from computer graphics. The advantage of the MLP representation is that it avoids the complexity of volumetric grids [29], circumvents the artifacts induced by the implicit bias of screen-space convolution [40, 54], and, unlike surface meshes, can have flexible topology. However, the original NeRF only works for static scenes captured from dozens of calibrated cameras such that each 3D point is seen from multiple views. ",
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+        "type": "text",
+        "text": "Our conceptual contribution lies in learning a neural latent representation relative to an articulated skeleton. While explicit models such as the popular SMPL body model [30] deform a surface via forwards kinematics, the implicit form of A-NeRF makes us re-think how skeletons can be integrated—implicit networks require the inverse transformation from 3D world coordinates to the reference skeleton, a significantly harder task that has not been fully explored. Our core technical novelty is to come up with and evaluate different parameterizations of $\\Gamma ( \\mathbf { q } ) , \\Gamma ( \\mathbf { d } )$ in Eq. 1 to create local coordinates relative to the articulated skeleton. Since a point in 3D world coordinates cannot uniquely be associated with a body part, we resolve the mentioned ill-posed inverse problem by overparameterizing with one embedding per bone. This embeds domain knowledge of how humans move and provides a common frame for the neural network to combine body shape and appearance constraints across the entire captured sequence (see Figure 2). ",
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+        "type": "text",
+        "text": "We demonstrate that all our contributions together enable learning of a neural body model from monocular video, requiring only rough 3D pose estimates for initialization, that reaches a level of detail previously only attained with parametric surface models or multi-view approaches [51]. ",
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+        "type": "text",
+        "text": "Scope. We apply the model to motion capture, character animation, and appearance and motion transfer and demonstrate that the pose refinement improves on existing monocular skeleton reconstruction. We learn in the transductive setting, for a specific target video that is known at training time but has no ground truth. A-NeRF enables novel view synthesis of dynamic motions, with plausible however non-physical illumination. Additional steps are needed to enable relighting applications. ",
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+        "text": "General impact. Building a self-supervised approach for personalized human body modelling promises to be more inclusive to people and activities that are not well represented in supervised datasets. However, it bears the risk that 3D models of people are created without consent. We urge users to only use datasets collected for developing and validating motion capture algorithms. ",
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+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our approach builds upon and is related to the following work on human pose and shape estimation, human modeling, and neural scene representations [62]. ",
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+    {
+        "type": "text",
+        "text": "Discriminative Human Pose Estimation. While feed-forward estimation of the 3D joint positions [26, 32, 34, 35, 43, 44, 52, 59, 64, 69, 72] or joint angles and bone lengths [53, 73] of the skeleton is highly accurate, such discriminative estimates are prone to misalignment when overlayed onto the input image due to the generalization gap. The skeleton pose can be refined to better match the 2D pose estimates, but this usually leads to larger errors in 3D [36, 37]. We use [23] for initializing skeleton pose and combine it with a neural body model. ",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Surface-based Generative Body Models. These are obtained by either constraining template meshes via deformation energies [17, 70] or learning parametric human body models from a large collection of laser scans [6, 9, 30]. Their low-dimensional parameters constrain the space of plausible human shapes and motions. This enables real-time reconstructions from single images [7, 15], detailed texturing and displacement mapping [3, 5], and alleviates manual rigging [2]. It also enables optimization within the bounds of the learned prior [11, 16, 25] and weak-supervision when integrated in a differentiable form [27] into neural training processes [4, 20, 23, 42, 45, 65]. Closest to our approach in this category are the model fitting methods by [74] that textures and geometrically refines an untextured parametric quadruped model to zebra images and to [68] that uses optical flow to refine human pose. Although in a similar setting, our surface-free neural body model and volumetric rendering is fundamentally different to their textured triangle mesh that is rigged with forward kinematics and needs to be obtained a-priori. ",
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+    {
+        "type": "text",
+        "text": "Implicit Body Models. A-NeRF bears close similarities with body models defined implicitly in terms of level-sets [58] and density of a sum of Gaussians [18, 48, 49] that are used for refining human pose, shape, and appearance via differentiable ray-tracing. However, sum of Gaussians and other primitives only provide rough approximations. ",
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+    {
+        "type": "text",
+        "text": "Neural Scene Representations. Recent neural scene representations learn low-dimensional nonlinear representations of meshes [28, 63], point clouds [1, 38, 67], sphere sets [24], and dense volumetric grids [29, 55]. Their respective geometric output representations enable rendering with classical rendering techniques but have limited expressiveness, e.g., due to the fixed connectivity of a surface mesh and large memory footprint of discretized volumes. This limitation is overcome by using unconstrained MLPs [39] paired with positional encoding [60] to characterize an arbitrary point in 3D space. Common are surface definitions via level-sets of the MLP that are rendered with sphere tracing [56] and density representations rendered with ray-marching [39]. The rendering step is required for maximum likelihood estimation, to define a likelihood over observable variables—real images—while learning a 3D model. The rendering can be learned too [40, 50, 51] but usually leads to inconsistencies, particularly when training data is scarce. Some concurrent works also use neural scene representations for refining camera motion [71], video reenactment [47], and facial models [12, 13]. Orthogonal to these works, our A-NeRF learns an articulated body model from estimated poses and uncalibrated cameras. ",
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+    {
+        "type": "text",
+        "text": "Closely related to ours is the NASA surface body model [10] that also defines an implicit function as the minimum of individual implicit functions that are rigidly attached to the bones of a skeleton, each conditioned on the entire human pose to model dependencies and learned from 3D scans. By contrast, we learn a volumetric model instead of a surface model and include appearance and rendering. Even more similar is the recent NeuralBody [46] representation, which combines a NeRF with a surface body model and underlying skeleton. By contrast to both approaches, we do not require surface supervision or initialization, condition pose differently, and refine pose, which enables us to learn from single videos in unconstrained environments. ",
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+    {
+        "type": "text",
+        "text": "3 Formulation ",
+        "text_level": 1,
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+        "text": "Objective. Given a sequence $[ \\mathbf { I } _ { k } ] _ { k = 1 } ^ { N }$ of N images $\\mathbf { I } _ { k } \\in \\mathbb { R } ^ { H \\times W \\times 3 }$ stemming from one or several videos of the same person, our goal is to simultaneously estimate the time-varying skeleton poses $[ \\theta _ { k } ] _ { k = 1 } ^ { N }$ and learn a detailed body model. Our A-NeRF body model $C _ { \\phi }$ is parametrized by neural network parameters $\\phi$ that define volumetric shape and color while the skeleton captures motion over time. Figure 2 gives an overview of this generative model. It enables a rendering of the virtual body model in unseen poses and optimizes its parameters $\\theta$ and $\\phi$ on the image reconstruction objective ",
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+    {
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+        "img_path": "images/0dd8ad9ad3adccd8b797a7e65d2fc737423d8d9d961ba23092b4e4926956105c.jpg",
+        "image_caption": [
+            "Figure 2: Overview. A-NeRF is a generative model that can be rendered and optimized on a photometric loss $\\mathcal { L } ^ { \\mathrm { S V } }$ (white). First, the skeleton pose is initialized with an off-the-shelf estimator (orange). Second, this pose is refined via a skeleton-relative embedding (blue) that, when fed to NeRF (green), drives the implicit body model that is rendered by ray-marching (red). A key property of the skeleton-relative embedding is that a single 3D query location maps to an overcomplete reparametrization, with the same point represented relative to each skeleton bone (right). "
+        ],
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+    {
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+        "img_path": "images/c2eb2fdaaaeb577984bf32b25a8c52d647df88f1d3195d6358d2678e7dd03b97.jpg",
+        "image_caption": [
+            "Figure 3: Importance of our skeleton-relative encodings. The original NeRF breaks (a) when training on a diverse set of poses and (b) further degrades when the poses are rotated. Even if (e) conditioned directly on $\\theta _ { k }$ , the NeRF still suffers from artifacts due to the complexity and ambiguity of human articulation. With our skeleton-relative encoding (f, g), the geometry for the subject is consistent under rotation, and the quality is greatly improved, with the full model working best. "
+        ],
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "equation",
+        "img_path": "images/484f5aec0392746bd77b7a537089355f9faef4770be1096c51639822981812d3.jpg",
+        "text": "$$\n\\mathcal { L } ^ { \\mathrm { S V } } ( \\theta , \\phi ) = \\sum _ { k } \\underbrace { \\| C _ { \\phi } ( \\theta _ { k } ) - \\mathbf { I } _ { k } \\| _ { 1 } } _ { \\mathrm { d a t a : t e r m } } + \\underbrace { \\lambda _ { \\theta } d ( \\theta _ { k } - \\hat { \\theta } _ { k } ) } _ { \\mathrm { p o s e r e g u l a r i z e r } } + \\underbrace { \\lambda _ { t } \\left\\| \\frac { \\partial ^ { 2 } \\theta _ { k } } { \\partial t ^ { 2 } } \\right\\| _ { 2 } ^ { 2 } } _ { \\mathrm { s m o o t h n e s s ~ p r i o r } } .\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "with the influence of all three terms balanced by hyperparameters $\\lambda _ { t }$ and $\\lambda _ { \\theta }$ . The data term measures the distance between the images generated by $C _ { \\phi }$ and the input image with the L1 distance. The pose regularizer encourages the solution to stay close to an initial pose estimate $\\hat { \\theta }$ obtained from an offthe-shelf predictor [23], tolerating small shifts up to $\\epsilon = 0 . 0 1$ with $d ( \\boldsymbol { x } ) = \\operatorname* { m i n } ( \\left\\| \\boldsymbol { x } \\right\\| _ { 2 } ^ { 2 } - \\epsilon , 0 )$ . Lastly, the smoothness prior penalizes acceleration ∂2θk∂t2 between poses of consecutive frames. Minimizing Eq. 2 can be seen as maximizing a corresponding probabilistic model, with the quadratic energy terms being the log-likelihoods of Gaussian distributions. Our focus is on formalizing the neural body model. For simplicity, we continue to write equations in terms of the objective functions used during inference with stochastic gradient descent. ",
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+    {
+        "type": "text",
+        "text": "3.1 NeRF and A-NeRF Image Formation Model ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Instead of modeling the scene as a collection of triangles or other primitives, we define the human implicitly by a neural network as a function (Eq. 1) defined over all possible 3D points and view ",
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+    {
+        "type": "text",
+        "text": "directions in space [39]. Similar to NeRF, we render the image of the human subject via ray marching ",
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+        "page_idx": 4
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+    {
+        "type": "equation",
+        "img_path": "images/7509bd3aa6a3da56375d1ae923d8024efe590f5905f40137127df65020c884d2.jpg",
+        "text": "$$\nC _ { \\phi } ( u , v ; \\theta _ { k } ) = \\sum _ { i = 1 } ^ { Q } T _ { i } ( 1 - \\exp ( - \\sigma _ { i } \\delta _ { i } ) ) \\mathbf { c } _ { i } , \\quad T _ { i } = \\exp \\left( - \\sum _ { j = 1 } ^ { i - 1 } \\sigma _ { j } \\delta _ { j } \\right) ,\n$$",
+        "text_format": "latex",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "with $( u , v )$ the 2D pixel location on the image, $i$ the index to 3D query positions $\\mathbf { q } _ { i }$ sampled along d and $\\delta _ { i }$ the distance to neighboring samples—a constant if samples would be taken at regular intervals. The $T _ { i }$ is the accumulated transmittance for the ray traveling from the near plane to $\\mathbf { q } _ { i }$ —the fraction of light reaching the sensor from sample point $i$ . The $\\mathbf { c } _ { i }$ is the light color emitted or reflected at $i$ . The final pixel color is the sum over all $Q$ samples, with the last sample taking the special role of the background. The background color is easily inferred as the median pixel color over the entire video for static camera setups. The ray direction $\\mathbf { d } = \\mathbf { K } _ { k } ^ { - 1 } ( u , v )$ is computed using the estimated camera intrinsics [23]. In the following, we introduce our skeleton parametrization $\\theta _ { k }$ and how to use it to effectively model dynamic articulated human motion. ",
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+    {
+        "type": "text",
+        "text": "3.2 Articulated Skeleton Pose Model ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our skeleton representation encodes the connectivity and static bone lengths via a rest pose of 3D joint locations. Dynamics are modeled with per-frame skeleton poses $\\theta _ { k }$ , which define an affine transformation $T ( \\theta _ { k } , m )$ for each bone $m$ . Specifically, $T ( \\theta _ { k } , m )$ maps a 3D position $\\mathbf { p } _ { k , m } \\in \\mathbb { R } ^ { 3 }$ in the $m$ -th local bone coordinates to world coordinates $\\mathbf { q } \\in \\mathbb { R } ^ { 3 }$ using homogeneous coordinates, ",
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+    {
+        "type": "equation",
+        "img_path": "images/83322dd3419c63f84144e0827545b3732903f00c03cba9ebc9092a61ff759891.jpg",
+        "text": "$$\n{ \\Bigg [ } { \\begin{array} { c } { \\mathbf { q } } \\\\ { \\mathbf { 1 } } \\end{array} } { \\Bigg ] } = T ( \\theta _ { k } , m ) \\left[ { \\begin{array} { c } { \\mathbf { p } _ { k , m } } \\\\ { 1 } \\end{array} } \\right] ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where subscript $^ { k , m }$ indicates that a variable is related to the $m$ -th joint of image $\\mathbf { I } _ { k }$ . Conversely, $T ( \\theta _ { k } , m ) ^ { - 1 }$ maps world to local bone coordinates. Note that our skeleton is equivalent to SMPL [30] and others, but without their parametric surface model, and can therefore be initialized with any skeleton pose estimator. We include more details of our skeleton representation in the supplementary. ",
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+    {
+        "type": "text",
+        "text": "3.3 A-NeRF Skeleton-Relative Encoding ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our core contribution is to transform the query locations $\\mathbf { q }$ and view direction d relative to the skeleton before determining the color and opacity at that transformed point via NeRF. It is a form of reparameterization that explicitly incorporates domain knowledge of how the human body parts are linked and transformed relative to each other. Intuitively, our implicit formulation turns explicit models, such as SMPL [30], on its head. Instead of deforming the output surface via skinning, the query location is mapped in the inverse direction to local bone-relative coordinate systems before processing through the NeRF network. Our final model uses a combined encoding $\\dot { \\mathbf { e } _ { k } } = [ h ( \\tilde { \\mathbf { v } } _ { k } ) \\Gamma ( \\bar { \\tilde { \\mathbf { v } } } _ { k } ) , \\tilde { \\mathbf { r } } _ { k } , h ( \\tilde { \\mathbf { v } } _ { k } ) \\Gamma ( \\bar { \\mathbf { d } } _ { k } ) ]$ as input to the NeRF $F _ { \\phi }$ . Note that the skeleton embedding introduces the desired time dependency, denoted by subscript $k$ . The inlet shows our most crucial contribution, the relative distance encoding $\\tilde { \\mathbf { v } } _ { k }$ followed by PE with Cutoff to reduce the influence of irrelevant bones. We derive the components of our encoding $\\mathbf { e } _ { k }$ and the other alternatives below. ",
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+        "type": "text",
+        "text": "• Reference Pose Encoding One could compensate motion by attaching the query $\\mathbf { q }$ in world coordinates at frame $k$ to the closest bone $m$ and transforming it with ",
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+        "text": "$$\n\\mathbf { a } _ { k } = T ( \\theta _ { 0 } , m ) T ^ { - 1 } ( \\theta _ { k } , m ) \\mathbf { q } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "type": "text",
+        "text": "This puts the query relative to the bone $m$ as in frame $k$ but with the skeleton in rest pose $\\theta _ { 0 }$ . NeRF could then learn without change in the 3D space of the rest pose as done before for surfaces [61]. However, this cannot capture non-rigid pose-dependent effects, such as muscle bulging, and has ambiguities when $\\mathbf { q }$ is at equal distance to two bones. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "• Bone-relative Position (Rel. Pos.) To remove these ambiguities and the ill-posed association to a single part, we map q relative to each bone $m$ with, ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/96b445eae26554ebac7795a303b6d685f28f694453775681aed2d0762754f4f5.jpg",
+        "text": "$$\n\\tilde { \\mathbf { q } } _ { k } = [ \\tilde { \\mathbf { q } } _ { k , 1 } , \\cdots , \\tilde { \\mathbf { q } } _ { k , 2 4 } ] \\mathrm { a n d } \\tilde { \\mathbf { q } } _ { k , m } = T ^ { - 1 } ( \\theta _ { k } , m ) \\mathbf { q } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "The resulting individual bone coordinates are well suited to model the overwhelmingly rigid motion of the corresponding body part. Moreover, the overparameterization of position by concatenating all local encodings enables learning when complex interactions are necessary. However, such an embedding for all bones increases the dimensionality by an order of magnitude. ",
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+    {
+        "type": "text",
+        "text": "• Relative Distance (Rel. Dist.) Much simpler to compute are distances from $\\mathbf { q }$ to all bones $m$ ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/1ecb04ca702ebf743671353eeef5f45a472e75f0031b096515c5bd70fff48ad0.jpg",
+        "text": "$$\n\\tilde { \\mathbf { v } } _ { k } = [ \\tilde { \\mathbf { v } } _ { k , 1 } , \\cdots , \\tilde { \\mathbf { v } } _ { k , 2 4 } ] , \\mathrm { ~ w i t h ~ } \\tilde { \\mathbf { v } } _ { k , m } = \\| \\tilde { \\mathbf { q } } _ { k , m } \\| _ { 2 } \\in \\mathbb { R } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This radial encoding is used in our final model in favor of $\\tilde { \\mathbf { q } }$ because it naturally captures spherically shaped limbs, is lower-dimensional, and thereby improves reconstruction accuracy. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "• Relative Direction (Rel. Dir.) Since the distance encoding is invariant to direction, we additionally obtain the direction vector to capture the orientation information of $\\mathbf { q }$ , ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/317902f589d867344ecf1aca193dbf13b6686b7217911622c5a30a45ed43ae54.jpg",
+        "text": "$$\n\\tilde { \\mathbf { r } } _ { k } = [ \\tilde { \\mathbf { r } } _ { k , 1 } , \\cdots , \\tilde { \\mathbf { r } } _ { k , 2 4 } ] , \\quad \\tilde { \\mathbf { r } } _ { k , m } = \\frac { \\tilde { \\mathbf { q } } _ { k , m } } { \\left\\| \\tilde { \\mathbf { q } } _ { k , m } \\right\\| _ { 2 } } \\in \\mathbb { R } ^ { 3 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Note that by contrast to all other embeddings, this direction encoding did not profit from subsequent PE. We therefore pass it directly into $\\mathbf { e } _ { k }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "• Relative Ray Direction (Rel. Ray.) NeRF models the illumination effects in a static 3D scene using the position and view direction. By contrast, our goal is to learn a body model that produces plausible colors with dynamic skeleton poses. Therefore, we transform $\\mathbf { d }$ to obtain $\\tilde { \\mathbf { d } }$ , the outgoing ray direction relative to each bone, similar to query position, ",
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+    {
+        "type": "equation",
+        "img_path": "images/563e484cc82fb5fb68e21d168cda52277502cbb74416ae62dbe1d65abd860d72.jpg",
+        "text": "$$\n\\tilde { \\mathbf { d } } _ { k } = [ \\tilde { \\mathbf { d } } _ { k , 1 } , \\boldsymbol { \\cdot \\cdot \\cdot } , \\tilde { \\mathbf { d } } _ { k , 2 4 } ] , \\quad \\tilde { \\mathbf { d } } _ { k , m } = [ T ^ { - 1 } ( \\theta _ { k } , m ) ] _ { 3 \\times 3 } \\mathbf { d } \\in \\mathbb { R } ^ { 3 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "with $[ T ^ { - 1 } ( \\theta _ { k } , m ) ] _ { 3 \\times 3 }$ the rotational part of the bone-to-world transformation $T ^ { - 1 } ( \\theta _ { k } , m )$ . Following concurrent works [33, 46], we also optimize an appearance code for each image to handle dynamic light effects. The combination of $\\tilde { \\mathbf { d } }$ and the per-image code enables A-NeRF to approximate the light effects in $\\mathbf { I } _ { k }$ plausibly. See the supplemental material for detailed discussions on modeling view-dependent effects in our setting. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "• Cutoff. We desire a local embedding where points should not be influenced by all but only nearby bones. To this end, we introduce a windowed version of positional encoding by multiplying the encoding with respect to bone $m$ by $h ( \\tilde { \\mathbf { v } } _ { k , m } ) = 1 - S ( \\tau ( \\tilde { \\mathbf { v } } _ { k , m } - t ) )$ , with $S$ the sigmoid step function, $t$ the cutoff point, $\\tau$ the sharpness. This leaves queries unaffected by distant bones. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Our embedding choice of $\\mathbf { e } _ { k } = [ h ( \\tilde { \\mathbf { v } } _ { k } ) \\Gamma ( \\tilde { \\mathbf { v } } _ { k } ) , \\tilde { \\mathbf { r } } _ { k } , h ( \\tilde { \\mathbf { v } } _ { k } ) \\Gamma ( \\mathbf { d } _ { k } ) ]$ has the advantage of being invariant to the global shift and rotation of the person and preserves the piece-wise rigidity of articulated motion while still allowing for pose-dependent deformation (see Figure 3). In addition to $\\mathbf { e } _ { k }$ , we also consider other embedding variants. See the supplementary and Section 4 for a detailed discussion. ",
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+    {
+        "type": "text",
+        "text": "4 Evaluation ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We performed experiments to validate that A-NeRF learns accurate body models and poses, with fewer assumptions (single view, uncalibrated, and w/o a parametric surface model) than the related works. This makes it applicable to fine-grained pose refinement that improves the estimates of stateof-the-art methods. The supplements provide the implementation details, additional comparisons and ablation studies. ",
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+    {
+        "type": "text",
+        "text": "Inference and Implementation Details Our A-NeRF model is learned without supervision on a single or multiple videos of the same person. Camera intrinsics, bone lengths for setting $\\mathbf { a } _ { m }$ , and pose $\\theta _ { k }$ are initialized with [23] for every frame $k$ . These poses are then optimized on objective Eq.2, alongside the generative A-NeRF model. See supplementary for more details. ",
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+    {
+        "type": "text",
+        "text": "Datasets. We evaluate on the following benchmarks, and additionally on synthetic data created from SURREAL and Mixamo characters, which are listed in the supplement. ",
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+    {
+        "type": "text",
+        "text": "• Human 3.6M [19] The dataset consists of 5 training and 2 testing subjects (S9/S11) with ground truth 3D joint locations. We follow two widely adopted test protocols denoted as Protocol I [20, 22, 23] and Protocol II [41, 59], in which we evaluate 14/17-joint estimation error on every $5 ^ { t h } / 6 4 ^ { { \\bar { t } } h }$ frame of the test videos, respectively. See the supplement for details. ",
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+    {
+        "type": "text",
+        "text": "• MPI-INF-3DHP [35] This dataset is a standard benchmark for human pose estimation. It consists of 4 indoor and 2 outdoor subjects with challenging human poses. The number of frames per subject range from 276 to 603. ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "• MonoPerfCap [70] The dataset consists of human performance video captured with a monocular camera in both indoor and outdoor settings. We use two subjects, Weipeng_outdoor and Nadia_outdoor, for our qualitative experiments. The two subjects have 1151 and 1635 frames, respectively, for training. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Pose Metrics. We report the PA-MPJPE metric, the Euclidean distance between Procrustes-aligned (PA) predictions and ground truth 3D joint position averaged over all frames and joints of the test set. The PA alignment in scale and orientation is essential for comparing approaches that do not assume knowledge of the ground truth calibration and are, hence, ill-posed to the factors that the alignment removes. Following prior work [23, 22, 35], we also report percentange of correct keypoints (PCK) for MPI-INF-3DHP; the percentage of joints that lie within a distance of $1 5 0 \\mathrm { m m }$ to the ground truth. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Visual Metrics. We quantify the visual quality on MonoPerfCap and Human $3 . 6 \\mathbf { M }$ datasets by training on a subset and testing on a held-out test set of the same character. Image quality is quantified via the PSNR and SSIM of the rendering compared with the reference image within the character bounding boxes. Because no ground truth pose is available in the required skeleton format on these datasets, we train our model once on the entire dataset to get reliable skeleton pose estimates as pseudo ground truth, and a second time with a part withheld to learn the body model for visual quality evaluation. Since MonoPerfCap has one sequence per actor, we exclude the last $20 \\%$ of each video. For Human 3.6M we exclude entire actions, namely Geeting-1,2, Posing-1,2 and Walking-1,2. The body model is then transferred to the held-out portion by using the pseudo ground truth poses as the driving motion. Thereby, we can still test the generalization of different models to new poses and viewpoints, irrespective of the underlying skeleton model provided in the dataset. ",
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+    {
+        "type": "image",
+        "img_path": "images/77235a6a06d5ac66c46a74745fcf55edc869ad3bc1d83569510a8397a6885b1c.jpg",
+        "image_caption": [
+            "Figure 4: Novel view synthesis from all models. All models are trained with SPIN [23] estimated human pose and camera parameters. A-NeRF renderings (ours) align better with the reference images (top row), and the rendered novel views (2nd and 3rd rows) show better details (limbs, facial features). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Novel-View-Synthesis and Character Animation. Our body model is generative, which allows us to train on a single or multiple uncalibrated videos and alter viewpoint and human pose. Figure 4 shows renderings of the same persons from a new camera angle; novel-view-synthesis. Likewise, Figure 5 demonstrates character animation, where the view is fixed, and the underlying skeleton is re-posed by manually changing joint angles or by transferring the motion between characters. A-NeRF is the first model that learns a detailed human body model with such capabilities without needing a 3D surface or multi-view supervision. While NeuralBody [46] can also learn photo-realistic human models from monocular images, their model anchors its representation on the SMPL 3D surface. When trained on our single-view setup with the same noisy estimated poses $\\hat { \\theta }$ as input, NeuralBody suffers from artifacts and less detail when training, as it assumes 3D ground truth, which is only available in controlled conditions. The importance of our joint body and pose optimization is further validated by the ablation with refinement disabled (Ours w/o rf.), which similarly produces ghosting artifacts around extremities. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/a8e5445a54eeaa430908d4d28914b76732bdf3195d9e59a0ce6f40af5d2f1486.jpg",
+        "image_caption": [
+            "Figure 5: Motion retargeting and animation. Top rows: Pose transfer, with the source pose reconstructed by A-NeRF and rendered with different target body models. Last row: Animating the A-NeRF model while keeping the lower or upper part of the body fixed. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/0f25cc8231101a3c02e97a7449386b3b091c50925f26c55313d8a0c9a115047e.jpg",
+        "table_caption": [
+            "Table 1: Quantitative evaluation on Human3.6M [19] and MPI-INF-3DHP [35]. Our test-time pose refinement improves consistently upon the SPIN baseline, with largest improvements for extremities (PA-Wrist). "
+        ],
+        "table_footnote": [
+            "? Reevaluation of publicly-available model because missing evaluation protocols or not reproducible. † We refine the SPIN estimated pose using SMPLify. We adopt the implementation from the SPIN repository. "
+        ],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"4\">Human 3.6M</td><td rowspan=\"2\">MPI-INF-3DHP</td></tr><tr><td>Protocol I</td><td>Protocol II</td><td></td><td>Prot.I Wrist Prot.II Multi-view</td></tr><tr><td>Method</td><td>PA-MPJPE↓</td><td>PA-MPJPE↓</td><td>PA-Wrist↓</td><td>PA-MPJPE↓</td><td>PA-MPJPE↓</td><td>PCK↑</td></tr><tr><td>MotioNet [53]</td><td>54.6</td><td>-</td><td>-</td><td>1</td><td>-</td><td>-</td></tr><tr><td>HoloPose [16]</td><td>46.5</td><td>-</td><td>=</td><td>-</td><td></td><td></td></tr><tr><td>VIBE [22]</td><td>41.4</td><td>n/a</td><td>n/a</td><td>n/a</td><td>64.6</td><td>89.3</td></tr><tr><td>SPIN [23]</td><td> 41.1</td><td>-</td><td>-</td><td>-</td><td>67.5</td><td>76.4</td></tr><tr><td>Baseline (SPIN github [23])</td><td>42.7*</td><td>41.9*</td><td>66.5*</td><td>34.0*</td><td>68.2*</td><td>79.3*</td></tr><tr><td>SPIN-SMPLify+[7, 23]</td><td>57.7</td><td>59.2</td><td>100.9</td><td>=</td><td></td><td>=</td></tr><tr><td> Ours (w/o smoothness prior)</td><td>39.4</td><td>39.6</td><td>57.3</td><td>28.0</td><td>66.9</td><td>80.4</td></tr><tr><td>Ours</td><td> 39.3</td><td>n/a</td><td>n/a</td><td>n/a</td><td>66.8</td><td>80.4</td></tr></table>",
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+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "Human Pose Estimation. Training A-NeRF includes a form of test-time optimization (see Figure 1, right), only the initialization from [23] is trained supervised on the Human3.6M training set. Table 1 shows that on Human 3.6M, A-NeRF reaches comparable results with other single-view approaches, and achieves a $8 . 0 \\%$ improvement in PA-MPJPE upon the baseline used for pose initialization on Protocol I $4 2 . 7  3 9 . 3 $ ) and $5 . 5 \\%$ on Protocol II $( 4 1 . 9  3 9 . 6 ) $ ). Note that these are average numbers across all joints, including easy-to-predict hip, shoulder, and head joints. Our largest gains are on the extremities, e.g., with an improvement of $1 4 \\%$ $( 9 . 2 \\mathrm { m m } )$ for the wrist joint on Protocol II (PA-Wrist). We also compare to applying SMPLify [7], a method that refines 3D poses using 2D joint locations (estimated with [8]) as constraints, at test time. It tends to explain the 2D joint estimates perfectly but degrades the 3D pose. The final pose estimations become less accurate than the initial ones. In contrast, A-NeRF optimizes the poses by implicitly minimizing the disagreement among the 3D body representation in different images, and thus achieves better performance. On MPI-INF-3DHP, Table 1 shows the results averaged over the 6 test subjects from MPI-INF-3DHP. Despite having a low number of frames and human poses available for learning the skeleton-relative encoding, A-NeRF still provides moderate improvements over the baseline estimations. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/0d02bdfe56d33b0f028250f8449450cf2225aefc483132798493c61b6a4f8d61.jpg",
+        "table_caption": [
+            "Table 2: Visual quality evaluation on the Human3.6M [19] and MonoPerfCap [70] held-out sets. Our full A-NeRF model significantly improves the visual quality. The body model itself attains a higher quality than NeuralBody (2nd vs. 3rd row), and additional detail is gained with the proposed pose refinement (last row). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>MonoPerfCap   Human 3.6M</td></tr><tr><td rowspan=1 colspan=1>PSNR ↑SSIM↑PSNR↑SSIM↑</td></tr><tr><td rowspan=1 colspan=1>NeuralBody, driving motion from [23]                21.80 0.8476 22.08 0.8766</td></tr><tr><td rowspan=1 colspan=1>NeuralBody, driving motion from A-NeRF refinement21.75 0.8468 22.55 0.8782</td></tr><tr><td rowspan=1 colspan=1>A-NeRF w/o pose refinement                         21.99 0.8405 23.33 0.8776</td></tr><tr><td rowspan=1 colspan=1>A-NeRF (Our full model)                            24.39 0.8851 27.45 0.9277</td></tr></table>",
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/b90719df2ea1c9f918c0ab07c25b4a45b990fd9512e57715fe9cfa8540593914.jpg",
+        "image_caption": [
+            "Figure 6: Geometry. The isosurface of our density model is rendered from multiple unseen views. A-NeRF learns plausible geometry without using explicit surface templates and accurate initial poses. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
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+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Video-based volumetric reconstruction. Figure 6 visualizes the learned density using Marching Cubes [31], with voxel grid resolution of 256 and density threshold 10. Despite only learned from monocular videos (no stereo, depth camera, or multi-view constraints) and without using a predefined template model, A-NeRF reconstructs a detailed volumetric body with details that could not be captured by offsets to a parametric surface model, such as the head phones and basebal cap in the last row of Figure 6. Note that no geometric smoothness term is enforced. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Multi-view extension. A-NeRF can also leverage multi-view refinement (Table 1, Protocol II Multi-view), even without access to ground truth camera calibration (see supplementary). ",
+        "bbox": [
+            176,
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+            768
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Visual Quality Comparison. We report the results in Table 2. We compare to NeuralBody with both the initial estimates from [23] and our refined pose since NeuralBody has no refinement step. Compared to both variants, our A-NeRF shows significantly better reconstruction performance on held-out poses. We observe that NeuralBody models can retain the facial features and hands as they anchor their representation on a 3D surface model. However, the rendered limbs and faces are blurry and distorted. Results are similar to training A-NeRF without refinement (see Figure 7). As the estimation for these joints is often inaccurate and noisy, the models without pose refinement simply learn to predict mean pixel values. To conclude, it is important to train with pose refinement, with which A-NeRF suffers less from artifacts with overall better visual quality. ",
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+        "page_idx": 8
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+    {
+        "type": "image",
+        "img_path": "images/d42017362181a1cd5719808aa12373f2bfb5f1d1c9e046d24161ffc6a68f3df1.jpg",
+        "image_caption": [
+            "Figure 7: A-NeRF with pose refinement generalizes better to test sequences. We visualize both the rendered images, as well as the photometric error (squared distance, normalized to $[ 0 , 1 ] )$ between the rendered and the ground truth images (warmer color indicates higher error) from our MonoPerfCap (1-2th rows) and Human 3.6M (3-4th rows) held-out sets. NeuralBody models produce artifacts around body contours. A-NeRF w/o refinement cannot reproduce facial features and limbs. With pose refinement, A-NeRF can produce both appearances and shapes more plausibly. "
+        ],
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+    {
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+        "text": "Ablation study. Our detailed ablation studies are reported in the supplemental document. In summary, they reveal: i) Embedding relative 3D position, $\\tilde { \\mathbf { q } } _ { k }$ , instead of our proposed radial embedding $\\tilde { \\mathbf { v } }$ yields only half as good pose refinements. ii) Our embedding choices keep the dimensionality moderate while improving on or matching the PSNR and SSIM of higher-dimensional variants. iii) For a fixed number of images with accurate poses, learning from a long video with diverse poses has visual quality comparable to learning from multiple shorter clips. ",
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+        "text": "Limitations and Failure Cases. Our computation time is the biggest bottleneck in extending A-NeRF to long sequences and multiple actors. Although a single static camera suffices, A-NeRF requires to see the person from all sides in varying poses to learn pose dependencies from data. The inlet shows a rendering of an extreme breakdance pose retargeted to a model trained on normal walking motions. Hence, the source pose is unseen during training and far from the data distribution, which leads to artifacts. ",
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+        "text": "5 Conclusion ",
+        "text_level": 1,
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+        "text": "We propose a new way for integrating articulated skeleton models and implicit functions via an overcomplete re-parametrization. It includes learning an interpretable 3D representation from 2D images; a personalized volumetric density field with texture detail and time-varying poses of the actor depicted in the input. The underlying ill-posed problem of mapping a single query location to multiple parts is addressed with an overparametrization over nearby parts. To the best of our knowledge, ANeRF is the first approach to define NeRF models for extreme and articulated motion on unconstrained video and this new approach scores high on the Human 3.6M benchmark. Importantly, it works from a single video and naturally extends to multi-view and does not require camera calibration in either scenario. This is an important step towards making motion capture more accurate and practical. In future work, we will learn a general human model from a database of subjects instead of individuals. ",
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+        "type": "text",
+        "text": "Funding in direct support of this work: NSERC Discovery grant, UBC Advanced Research Computing (ARC) GPU cluster, Compute Canada GPU servers, and a gift by Facebook Reality Labs. ",
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+        "text": "References   \n[1] K.-A. Aliev, A. Sevastopolsky, M. Kolos, D. Ulyanov, and V. Lempitsky. Neural point-based graphics. arXiv preprint arXiv:1906.08240, 2019.   \n[2] T. Alldieck, M. Magnor, W. Xu, C. Theobalt, and G. Pons-moll. Video Based Reconstruction of 3D People Models. In CVPR, 2018.   \n[3] T. Alldieck, M. Magnor, W. Xu, C. Theobalt, and G. Pons-Moll. Detailed human avatars from monocular video. In 3DV, pages 98–109. IEEE, 2018.   \n[4] T. Alldieck, M. Magnor, B. L. Bhatnagar, C. Theobalt, and G. Pons-Moll. Learning to reconstruct people in clothing from a single rgb camera. In CVPR, pages 1175–1186, 2019.   \n[5] T. Alldieck, G. Pons-Moll, C. Theobalt, and M. Magnor. Tex2shape: Detailed full human body geometry from a single image. In ICCV, 2019.   \n[6] A. O. Balan, L. Sigal, M. J. Black, J. E. Davis, and H. W. Haussecker. Detailed human shape and pose from images. In CVPR, pages 1–8, 2007.   \n[7] F. Bogo, A. Kanazawa, C. Lassner, P. Gehler, J. 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parse/train/rJgz8sA5F7/rJgz8sA5F7.md

@@ -0,0 +1,278 @@
+# HC-NET: MEMORY-BASED INCREMENTAL DUALNETWORK SYSTEM FOR CONTINUAL LEARNING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Training a neural network for a classification task typically assumes that the data to train are given from the beginning. However, in the real world, additional data accumulate gradually and the model requires additional training without accessing the old training data. This usually leads to the catastrophic forgetting problem which is inevitable for the traditional training methodology of neural networks. In this paper, we propose a memory-based continual learning method that is able to learn additional tasks while retaining the performance of previously learned tasks. Composed of two complementary networks, the Hippocampus-Net (H-Net) and the Cortex-Net (C-Net), our model estimates the index of the corresponding task for an input sample and utilizes a particular portion of itself with the estimated index. The C-Net guarantees no degradation in the performance of the previously learned tasks and the H-Net shows high confidence in finding the origin of an input sample.
+
+# 1 INTRODUCTION
+
+The main difference between the human brain and the machine learning methodology is the ability to evolve. Using neurophysiological processing principles, human brains can achieve and organize knowledges throughout their lifespan. Having the neuroplasticity, human brains can transfer an activating region of a given function to a different location or control the creation and destruction of synapses according to its experiences. Usually, artificial neural network (ANN) models consist of finite numbers of filters. Also, parameters and operations in the ANN do not possess the ability corresponding to the memory system of human brains. This structural limit leads to the problem called catastrophic forgetting, i.e., the newly coming information diverts the model from previously learned knowledge. The field of researches trying to solve this problem is referred to as continual or lifelong learning.
+
+Kirkpatrick et al. (2017) suggested a method applying a penalty on the quadratic distance between the old parameters and the new ones. Called elastic weight consolidation (EWC), this method can be applied to either supervised or reinforcement learning scenarios. Jung et al. (2018) proposed a method using feature regularization referred as less-forgetful learning (LF). When training a new dataset, the feature just before the fully connected layer is regularized to become similar to that of the previous model. A considerably small learning rate is used in order to prevent a drastic change of the model. Despite of such efforts, as more and more tasks are added, the model eventually collapses with poor performances in old datasets. Li & Hoiem (2017) proposed a method using a similar regularization method, referred as learning without forgetting (LwF). Continual learning is conducted to a single convolutional neural network (CNN), adding an additional classifier whenever a dataset is newly trained. Classifiers in the newly trained network are regularized to reproduce the outputs of the old network while the new classifier is trained in a classical way. LwF shows better performance than LF, but still suffers from the catastrophic forgetting with multiple tasks. These approaches using regularization basically focus on alleviating the deformation of the original network. Therefore, the more the new knowledge flows in the model, the more the performance on the original task degrades. To make a single network deal with a large number of datasets, instead of using all parameters of a single network for each task, PackNet suggested a model which allocates the datasets to specific weights of filters (Mallya & Lazebnik, 2018). As only the specialized weights for a particular task are involved in its classification process, PackNet shows a remarkable result in multiple tasks. However, the information about from which task the input originates and which group of weights should be used must be given in advance. Typically, images do not contain such prior knowledges and this makes the PackNet hard to apply in real world situation.
+
+![](images/eec4e95f50f3656571df72b4b26b98e1b4b036bdaf04a02263b2b5ce95f1d31a.jpg)  
+Figure 1: The overall architecture of our method. Blue arrows depicts the training process for the new task (Task C) of our method, where Task A and Task B have been already trained. H-Net is trained each time a dataset for a new task is provided and then the C-Net is trained in a standard way such as using the cross-entropy loss. We allow only the trainable filters (Blue parameters) to be updated. Red arrows shows the inference procedure of our method. H-Net finds out the task index $\mathcal { I }$ (Task C) from which the input came and let the C-Net to use the filters specialized for task $\mathcal { I }$ (Blue $+ \mathrm { O r a n g e + G r e y } )$ . Class labels are also switched to those of task $\mathcal { I }$ .
+
+In this paper, inspired by the human brain system, we propose a network which efficiently increases its complexity without degrading the performance of previous tasks. Instead of saving all the memory in the cortex and trying to access it directly, the hippocampus finds the index of the memory scattered over the cortex where knowledges are well organized inside. Under the inspiration from the quote in (Buzsaki, 2006), “Think of the cortex as a huge library and the hippocampus as its librarian”, we have built our network using two complementary sub-networks as shown in Figure 1. The Cortex-Net (C-Net) keeps the knowledge from several independent tasks. After training a network from the previous task, additional filters in the convolutional modules are attached and trained with the next task. To infer the newly learned data, the model utilizes the newly trained filters along with the previously learned filters, whereas the data from the old task is inferred only using the previously learned filters. In order to determine which combination of filters to use, we adopted a memory network which can distinguish the origin of a given input sample. Referred as the Hippocampus-Net (H-Net), it can recall the information not only which group of filters to use but also which group of class labels to use. This endows our method label-expandability, which enables our method to be practically applied in real world situation. We suggest three methods, each of which respectively using generative adversarial networks (GAN) (Goodfellow et al., 2014), variational auto-encoder (VAE) (Kingma & Welling, 2013), and principal component analysis (PCA) (Jolliffe, 2011) for H-Net and report the properties and performance of each method. Our H-Net can be combined with any other works such as LwF or PackNet. The combined H-Net and C-Net referred as HC-Net can prevent the catastrophic forgetting using a densely organized architecture and a highly reliable memory system. The contributions of this paper can be summarized as follows:
+
+• Filters in the convolutional layers are allocated to each task and parameters from the previously learned tasks are shared among post tasks and used for the initialization of post tasks parameters, which makes the overall model compact and efficient.   
+• Attaching filters whenever a new dataset is trained allows the model to be expandable as far as the physical constraints, such as memory and processing time constraints, allow.   
+• To identify which portion of convolutional filters should be used to classify a data, a memory network (H-Net) is proposed with several methodologies.   
+• The overall method is applicable to multi-task continual learning whose number of class labels increases over tasks.
+
+![](images/da0df10bb0079357dd40b2e4e7473a7490a92408d4a5087fefd1e8725576404d.jpg)  
+Figure 2: Trainable parameters $W _ { k } ^ { T }$ (Orange) and frozen parameters $W _ { k } ^ { \mathcal { P } }$ (Grey) in the C-Net training process. In the training process, the parameters (Orange), which are trainable parameters except frozen parameters (Grey), are updated. $x$ denotes the spatial dimension of filters.
+
+# 2 METHOD
+
+We propose a memory-based incremental dual network system for continual learning that satisfies three properties as stated below.
+
+Property 1. The Hippocampus-Net (H-Net) offers the index about which part of the Cortex-Net (C-Net) should be used.
+
+Property 2. The C-Net learns the knowledge on the new task under the presence of the previously learned knowledge.
+
+Property 3. H-Net and C-Net are flexible to the expansion of the network and as many new tasks can be learned in one network as physical constraints allow.
+
+In this respect, our proposed H-Net works as a hippocampus of a human brain in that it gives a summarized information to the C-Net. Also, the C-Net corresponds to the human cortex in terms of its ability to learn semantic knowledge.
+
+# 2.1 CORTEX NETWORK (C-NET)
+
+We propose an efficient way to allocate the capacity to a network according to the given tasks. C-Net is an expandable network where additional filters can be attached whenever a new task is given. It can be applied to many typical networks such as VGG and ResNet (Simonyan & Zisserman, 2014; He et al., 2016). The only difference with the original versions of those networks is that it uses different parts of filters for different tasks. As shown in (Li et al., 2016), typical CNN are capable of maintaining their performance even when the majority of their parameters are pruned. This implies that filters in a CNN contains a lot of unnecessary or redundant information. Also, sharing the filters among different tasks inevitably leads to the catastrophic forgetting of the network since the parameters trained by an old task should be replaced with those trained by a new task. For these reasons, we utilize a network divided into several parts and refer it as C-Net where each part takes charge of a particular task.
+
+While training the C-Net with several tasks, each task uses a different part of the C-Net. For the convenience of referring which segments of the C-Net are to be activated, we introduce a filter index $I ^ { \mathcal { I } }$ for the $\mathcal { I }$ -th task. The filter index $I ^ { \mathcal { I } }$ defines the range of filters to use for the specific task $\mathcal { I }$ in every layers. For example, if the filter index $I ^ { 1 }$ equals to ten, the first task uses filters from the first to the tenth filter.
+
+Depicted in Figure 2, $I _ { k } ^ { \mathcal { I } }$ denotes the filter index in the $k$ -th layer corresponding to the task $\mathcal { I }$ . This equals to the number of output channels in the sequent feature map. $I _ { k } ^ { \mathcal { P } }$ denotes the filter index in the $k$ -th layer for the previous task. As the same principle is applied to the precedent $( k - 1 )$ -th layer, the number of input channels for the filters in the $k$ -th layer will change likewise. That is, total of $I _ { k } ^ { \mathcal { P } }$ filters have the channel length of $I _ { k - 1 } ^ { \mathcal { P } }$ when the previous task is under training. As the new task is given, the length of the filters become $I _ { k - 1 } ^ { \mathcal { I } }$ since the filters in the prior layer will deliver a feature
+
+Input: $W ^ { \mathcal { I } } , I ^ { \mathcal { P } } , I ^ { \mathcal { I } }$
+
+Output: W J ∗
+
+1: Freeze the $W ^ { \mathcal { P } } \subset W ^ { \mathcal { I } }$ using the information $I ^ { \mathcal { P } }$ 2: Initialize the $W ^ { \tau }$ with $W ^ { \mathcal { P } }$ and a random noise 3: $W ^ { \mathcal { I } * }  \arg \operatorname* { m i n } ( \mathcal { L } _ { c } )$ {Update $W ^ { \breve { T } }$ using backpropagation}
+
+map whose length is $I _ { k - 1 } ^ { \mathcal { I } }$ . Therefore, both the number of input and output channels increases as the number of tasks increases until the physical constraints allow.
+
+Weight parameters of $\mathcal { I }$ -th task in filters of the $k$ -th layer are referred as $W _ { k } ^ { \mathcal { I } } \in R ^ { x \times x \times I _ { k - 1 } ^ { \mathcal { I } } \times I _ { k } ^ { \mathcal { I } } }$ , where $x$ means the size of the kernel while $\begin{array} { r } { W _ { k } ^ { \mathcal { P } } \in R ^ { x \times x \times I _ { k - 1 } ^ { \mathcal { P } } \times I _ { k } ^ { \mathcal { P } } } } \end{array}$ are the weight parameters of the previous task. When training the new task, $W _ { k } ^ { \mathcal { I } }$ is used for inference but only the parameters $w \in W _ { k } ^ { \mathcal { T } }$ $( W _ { k } ^ { \mathcal { T } } = W _ { k } ^ { \mathcal { I } } \setminus W _ { k } ^ { \mathcal { P } } )$ are updated while leaving $W _ { k } ^ { \mathcal { P } }$ fixed. Note that biases in the filters can also be trained likewise as explained above. The fully connected layer located just before the linear classifier can be divided as well into the sections of $\dot { I } ^ { \mathcal { I } }$ and $I ^ { \mathcal { P } }$ .
+
+To enhance the performance, slices of trained parameters $W ^ { \mathcal { P } }$ from the old task are duplicated to the newly available task parameters $W ^ { T }$ and Gaussian noise is added to them for the sake of good initialization. Then, the network is trained in a standard way, e.g., using a cross-entropy loss $\mathcal { L } _ { c }$ . Independent linear classifiers are used for every tasks. This scheme of C-Net can also be applied to structures using shortcut connections such as ResNet. The overall training process is shown in Algorithm 1.
+
+# 2.2 HIPPOCAMPUS NETWORK (H-NET)
+
+Even if the C-Net is well trained separately with several tasks, the task index $\mathcal { I }$ must be given at the time of inference. Especially when the class labels expand, this prior information $\mathcal { I }$ is necessary to estimate which group of class labels to use. In the testing step, existing methods needs the exact origination of the input data. However in the real world, a given data for classification normally does not offer any information about its origination. Therefore, this prior knowledge should be estimated by an independent network.
+
+To solve this problem, we introduce the Hippocampus network (H-Net) using several different methods (GAN, VAE and PCA), which is able to estimate the task $\mathcal { I }$ of the given input data and inform this to the C-Net to specify which filters to use. In this paper, we explain the GAN method as a representative and other methods are explained in Appendix A.
+
+# 2.2.1 GENERATIVE ADVERSARIAL NETWORKS (GAN)
+
+This method is inspired by the on-line replay method used for continual learning (Shin et al., 2017). We train a task-specific generative model $G _ { \mathcal { I } }$ using a GAN to generate pseudo-samples of task $\mathcal { I }$ . The generator is trained using the adversarial loss as follows:
+
+$$
+\operatorname* { m i n } _ { G , \mathcal { T } } \operatorname* { m a x } _ { D , \mathcal { T } } V ( D _ { \mathcal { T } } , G _ { \mathcal { T } } ) = \mathbb { E } _ { \boldsymbol { x } \sim P _ { d a t a } ^ { \mathcal { T } } } [ \mathrm { l o g } ( D _ { \mathcal { T } } ( \boldsymbol { x } ) ] + \mathbb { E } _ { \boldsymbol { z } \sim p _ { \mathcal { z } } ( \boldsymbol { z } ) } [ \mathrm { l o g } ( 1 - D _ { \mathcal { T } } ( G _ { \mathcal { T } } ( \boldsymbol { z } ) ) ] .
+$$
+
+Here, $D _ { \mathcal { I } }$ is a discriminator for the task $\mathcal { I }$ and $x$ is a sample from the task $\mathcal { I }$ .
+
+After generating samples of all tasks, a task-wise binary classifier $B _ { \mathcal { I } }$ is trained to classify whether the given input is from the task $\mathcal { I }$ or not. Note that only the generated samples are involved for the training of each binary classifier $B _ { \mathcal { I } }$ . This means that positive samples are from the generator $G _ { \mathcal { I } }$ for the task $\mathcal { I }$ and negative samples are from all the other task generators $G _ { \kappa }$ $( \forall \boldsymbol { K } \neq \mathcal { T }$ ).
+
+In the testing phase, each classifier produces the probability of how likely the given unknown input data is from the task $\mathcal { I }$ . The classifier $B _ { i }$ with the highest probability assumes that the input data is from the task $i$ . H-Net can figure out the task $\mathcal { I }$ of input data with maximum probability across the set of classifiers. H-Net gives this estimated task index $\mathcal { I }$ to the C-Net. The H-Net using GAN can be summarized as below:
+
+$$
+\begin{array} { r l } & { \mathrm { I n i t i a l i z a t i o n : } \qquad B = \{ B _ { 1 } , B _ { 2 } , . . . , B _ { N } \} } \\ & { \mathrm { T r a i n i n g : } \quad B _ { i } ( x ) = \left\{ 1 \quad \mathrm { i f } x \mathrm { i s f r o m } G _ { i } \right. } \\ & { \mathrm { I n f e r e n c e : } \qquad \mathcal { I } = \arg \underset { i } { \operatorname { i f } } x \mathrm { i s f r o m } G _ { j } \forall j \neq i } \end{array}
+$$
+
+# 3 EXPERIMENT
+
+We evaluate our method on several image classification datasets. First, we verify the effectiveness of our method with MNIST (LeCun et al., 1998), SVHN (Netzer et al., 2011) and CIFAR-10 (Krizhevsky & Hinton, 2009) which are widely used to evaluate image classification performance. Then, we evaluate our method on two subsets of ImageNet (Russakovsky et al., 2015) which is a realistic image dataset. More details are summarized in Table 1.
+
+We compare our method to various methods such as Elastic Weight Consolidation (EWC), Learning without forgetting (LwF), less-forgetful learning (LF) and PackNet as well as networks trained for a single target task which shows the performance without conducting continual learning. We use the GAN method for H-Net in all the experiments as a representative. To solely examine the influence of H-Net on our method (HC-Net), we have conducted experiments with the same experimental scenarios on the HC-Net experiments. In Table 2, each row shows the result of one among GAN, VAE and PCA. In Table 3 and Table 4, the result of ‘HC-Net (without H-Net)’ shows the performance of the C-Net only. That is, it assumes that the H-Net never fails and the task index $\mathcal { I }$ is always given correctly. Likewise, LwF and PackNet need prior knowledge of a given input to know which classifier to use. As there is no structure like H-Net in the original paper, The actual comparison must be done with ‘C-Net’.
+
+# 3.1 CONTINUAL LEARNING USING BASIC IMAGE CLASSIFICATION DATASETS
+
+We have built three experimental scenarios to evaluate our method using basic image classification datasets. We compare the performance of learning a consecutive two task pair among MNIST, SVHN and CIFAR-10 dataset. After that, we conduct multiple-task continual learning with these datasets. In these experiments, we use the same learning hyperparameters and network architectures suggested in (Jung et al., 2018) used for ‘Tiny image classification’. Training details are reported in Appendix B.
+
+# 3.1.1 MNIST → SVHN
+
+After selecting a subset of SVHN to equalize the number of training data between MNIST and SVHN, images are resized to $2 8 \times 2 8$ as in (Jung et al., 2018). The results of Modified LwF, EWC and LF are referred from (Jung et al., 2018). Modified LwF is sligthly different from the original LwF in that the old output $\hat { Y } _ { o }$ and new outputs $\hat { Y _ { n } }$ is united into a single output. The ‘single network’ in Table 3 is a network of full capacity trained using a single dataset. The column ‘Old’ represents the test accuracy on the previously learnt task and ‘New’ is that of the newly trained task . The value $\gamma$ for EWC is determined as in (Kirkpatrick et al., 2017). A knowledge distillation loss with hyper-parameters of $T = 2$ and $T = 1$ is used for LwF, referred as temperature in (Hinton et al., 2015).
+
+As MNIST and SVHN highly resemble each other, the accuracy of all methods are relatively high. LF demonstrate better average classification rates than Modified LwF and EWC. LwF can control the performance trade-off between the old and new task by changing the temperature parameter $T$ . The bigger $T$ is, the lower the accuracy drops but it still outperforms those of the methods mentioned above. PackNet almost maintains the performance of the single network for the first task with less than $2 \%$ drop in accuracy. HC-Net outperforms all other methods in the table. Note that the prior knowledge to select which task is given in experiments of other methods. Therefore, our HC-Net without H-Net (i.e., C-Net), which shows the highest in performance, should be compared to other baseline methods. Nonetheless, our full model, HC-Net also outperforms PackNet, especially on the second task by around $0 . 7 \%$ .
+
+Table 1: Details of datasets.   
+
+<table><tr><td>Datasets</td><td>#ofTrain data</td><td>#of Test data</td><td>#of Class</td></tr><tr><td>MNIST</td><td>60,000</td><td>10,000</td><td>10</td></tr><tr><td>SVHN</td><td>73,275</td><td>26.032</td><td>10</td></tr><tr><td>CIFAR-10</td><td>50,000</td><td>10.000</td><td>10</td></tr><tr><td>ImageNet-A</td><td>64,750</td><td>2.500</td><td>50</td></tr><tr><td>ImageNet-B</td><td>64,497</td><td>2,500</td><td>50</td></tr></table>
+
+Table 2: Experimental results of H-Net on the basic datasets   
+
+<table><tr><td>Datasets</td><td>Methods</td><td>Old(%)</td><td>New(%)</td></tr><tr><td>MNIST √ SVHN</td><td>GAN VAE PCA</td><td>100 100 100</td><td>99.94 99.85 100</td></tr><tr><td>SVHN √ CIFAR-10</td><td>GAN VAE PCA</td><td>97.00 71.07 93.28</td><td>99.90 97.06 92.5</td></tr><tr><td>Datasets</td><td></td><td>Methods MNIST(%)</td><td>SVHN(%) CIFAR(%)</td></tr><tr><td>MNIST</td><td>GAN</td><td>100 96.65</td><td>99.83</td></tr><tr><td>→ SVHN →CIFAR-10</td><td>VAE PCA 100</td><td>100 71.22 93.28</td><td>97.07 92.5</td></tr></table>
+
+Table 3: Mean classification results on the basic datasets (5 runs).   
+
+<table><tr><td>Datasets</td><td>Methods</td><td>Old(%)</td><td></td><td>New(%) Avg.(%)</td></tr><tr><td rowspan="2">MNIST √ SVHN SVHN</td><td>single network (MNIST) single network (SVHN) ModifiedLwF EWC(y = 2.32 × 104) LF(λe = 1.6 × 10-3)</td><td>99.49 1 94.78 94.15 97.37</td><td>92.82 83.77 79.31 83.79</td><td>1 1 89.28 86.73 90.58</td></tr><tr><td>LwF(T=1) LwF(T=2) PackNet HC-Net HC-Net (without H-Net) single network (SVHN) single network (CIFAR)</td><td>98.27 97.33 99.45 99.43 99.43 92.94</td><td>86.40 86.97 91.49 92.20 92.37 一 79.69</td><td>92.34 92.15 95.47 95.82 95.90 1</td></tr><tr><td colspan="2">√ CIFAR-10</td><td>1 LF(λe=2.0 ×10-3) 83.18 91.76 90.19 92.84 90.05</td><td>48.99 70.57 71.94 76.78 76.62</td><td>1 66.09 81.17 81.07 84.81 83.34</td></tr><tr><td>Datasets</td><td colspan="2">HC-Net (without H-Net) Methods</td><td>92.21 76.70 MNIST(%)SVHN(%)CIFAR(%)Avg(%)</td><td>84.46</td></tr><tr><td>MNIST ↓ SVHN √ CIFAR-10</td><td colspan="2">single network (MNIST) 99.44 single network (SVHN) 二 single network (CIFAR) = LF(λe=1.0×10-) 89.57 LwF(T=1) 95.06 LwF(T= 2) 86.09 PackNet 99.38</td><td colspan="2">一 92.94 79.69 72.1 45.43</td></tr></table>
+
+# 3.1.2
+
+In this experiment, we want to see the performance of the proposed method when the statistics of the two tasks are quite different. The datasets SVHN and CIFAR-10 have a clear distinction in image styles and colors. For LF whose classifier must be shared between the old and new tasks, this inevitably leads to a significant decrease in performance. LwF and PackNet show similar trends as that of ‘MNIST to SVHN’. Higher temperature $( T )$ induces better performance in the new task but a bit of degradation in average performance still occurs. PackNet experiences almost no performance degradation. The pruning method of PackNet acts like a generalization method and helps the model to maintain its capacity. The average performance of C-Net is slightly lower than that of PackNet by $0 . 3 5 \%$ . However, note that PackNet requires additional finetuning after the pruning process and takes additional time for training. On the other hand, C-Net requires no additional pruning or finetuning precedure and yet retains the performance.
+
+# 3.1.3
+
+Training three or more tasks is unmanageable for the models using regularization methods. LF scored an average accuracy of no more than $6 9 . 0 3 \%$ , which is much more lower than any other methods. No matter how we change the temperature $( T )$ , LwF suffers from a drop in average performance, especially in the third task. Also, even though the proportion allocated to each task is equivalent between the PackNet and our methods (HC-Net and C-Net), our method highly outperforms the PackNet in the third task. This implies that HC-Net is more suitable for multi-task sequential learning situation.
+
+# 3.2 CONTINUAL LEARNING USING REALISTIC DATASETS
+
+ImageNet contains images more realistic than other datasets used in this paper. Higher resolution and complex backgrounds make the classification even harder. To save the training time, two subsets of the ImageNet dataset each having 50 randomly chosen classes have been used for evaluation. They are referred to as ImageNet-A and ImageNet-B respectively in this paper. To show the adaptability of our method on structures having shortcut connections, ResNet-50 is used for the experiment. We use the same experimental setting as in (He et al., 2016) with a batch size of 128. We compare our C-Net with LwF and PackNet.
+
+Table 4: Mean classification results for the realistic dataset (2 runs)   
+
+<table><tr><td>datasets</td><td>Methods</td><td>Old(%)</td><td>New(%)</td><td>Avg.(%)</td></tr><tr><td rowspan="4">ImageNet-A √ ImageNet-B</td><td>single network(ImageNet-A)</td><td>83.28</td><td>一</td><td>1</td></tr><tr><td>single network (ImageNet-B)</td><td>1 82.2</td><td>85.28</td><td>1</td></tr><tr><td>LwF(T=1)</td><td></td><td>86.72</td><td>84.46</td></tr><tr><td>LwF(T= 2)</td><td>80.92</td><td>86.96</td><td>83.94</td></tr><tr><td rowspan="2"></td><td>PackNet</td><td>82.16</td><td>88.72</td><td>85.44</td></tr><tr><td>HC-Net (without H-Net)</td><td>83.30</td><td>88.66</td><td>85.98</td></tr></table>
+
+Table 5: Performances of the old (MNIST) and new (SVHN) tasks with different initialization methods: 1) initialization with only random Gaussian noise and 2) the pretrained parameter added by random Gaussian noise   
+
+<table><tr><td>Initialization</td><td>0ld(%)</td><td>New(%)</td><td>Avg.(%)</td></tr><tr><td>Random</td><td>99.42</td><td>91.67</td><td>95.55</td></tr><tr><td>Pretrained parameters</td><td>99.43</td><td>92.37</td><td>95.90</td></tr></table>
+
+Table 4 shows the results on the ImageNet dataset. The ‘Old’ accuracy of all methods slightly dropped from those of single networks. Like in the ‘SVHN to CIFAR- $1 0 ^ { \circ }$ experiment, PackNet shows a better result than the LwF regardless of the value of $T$ . The result of C-Net is almost identical to PackNet with a slight increase in the average accuracy.
+
+PackNet actually utilizes more parameters for each task than C-Net does. Since the masks force weights with no influence to be zero and utilize the well trained remaining weights, PackNet can make good use of the entire network. However in C-Net, just adding filters gradually without any other post-processing performs well enough compared to PackNet. When the model of an initially designed size is fully occupied by several tasks, C-Net can just add more filters and train them along with the trained filters while this is not the case of PackNet. Furthermore, PackNet requires additional memories to store the masks for all filters in it. C-Net needs only one integer per layer for each task. For these reasons, C-Net is far more efficient than the PackNet with no loss in performance.
+
+# 3.3 ABLATION STUDY ON THE EFFICIENCY OF C-NET
+
+As mentioned in Section 2.1, our C-Net appends filters whenever a new task is to be trained. In this process, parameters learned from previous tasks are utilized altogether and the parameters which are to be trained newly are initialized using the existing ones. We conduct experiments to show the effect of these methods. Experiments are carried out in ‘MNIST to SVHN’ case as a representative.
+
+# 3.3.1 INITIALIZATION WITH PRETRAINED PARAMETERS
+
+To analyse the effectiveness of initialization using pretrained old task parameters with random Gaussian noise, we compare the accuracy between a model initializing the new parameters with just a random Gaussian distribution and a model initializing the new task parameters using the pretrained old task parameters with additional random Gaussian noise. In Table 5, the model initialized by our method obviously shows higher accuracy on the SVHN task. This result implies that a good initialization prevents the model from going through a local minima.
+
+# 3.3.2 PARAMETER SHARING
+
+We verify the effect of the parameter sharing between the old task and the new task. The baseline method is a model where the old parameters are filled with random Gaussian initialization and no further training is done. The new parameters in the model have to learn knowledges from the new task without the aid of old parameters since they are fixed from the beginning. On the other hand, the old parameters in our method is trained by the old task and the new parameters can make use of these learnt knowledges to learn the new task. To solely observe the effect of the parameter sharing, both methods do not use our parameter initialization method mentioned in the previous section. The task indices $I ^ { 1 }$ and $I ^ { 2 }$ are $\{ 1 6 , 1 6 , 3 2 \}$ and $\{ 1 8 , 1 8 , 3 4 \}$ respectively. The model using parameter sharing converges far more faster than the model with no parameter sharing which is shown in Figure 3. Also the final error rate of our method is lower than the other. Increasing the number of indices elevation enhances our method by $6 . 3 8 \%$ and the other model by $7 . 9 1 \%$ shown in Table 6. This implies that the parameter sharing improves the model and allow it to be compact with fewer numbers of filters.
+
+![](images/d23170eb7767a0fb74e8d8a3b06991296184d1f4092acd3844393ea1fb9e66e7.jpg)  
+Figure 3: Error rate comparison between sharing the parameters which learned previous task and random parameters.
+
+Table 6: The performances on the new task while increasing the filter index of the model using parameter sharing and the model with fixed random parameters.   
+
+<table><tr><td>#ofincreasingeach index</td><td>Ours(%)</td><td>Random(%)</td></tr><tr><td>2</td><td>80.60</td><td>78.10</td></tr><tr><td>4</td><td>85.31</td><td>83.85</td></tr><tr><td>6</td><td>86.98</td><td>86.01</td></tr></table>
+
+# 3.4 H-NET
+
+The results of H-Net are reported in Table 2. In ‘MNIST to SVHN’, all methods approximates the results to perfection. Despite of the similar appearance of these two datasets, VAE has been able to generate distinct outputs. If a sample in the MNIST dataset is given, the result from the VAE trained for SVHN resembles those of the SVHN dataset. In other words, auto-encoders of each task tends to memorize only what they used for training. PCA is completely able to discriminate the origin of the images from MNIST and SVHN. Particularly in this experiment, even when using a single eigenvector with the least eigen-value showed a result of $100 \%$ and $9 9 . 7 \%$ . According to the dataset, PCA can highly reduce the computational cost as well as the memory resource.
+
+In ‘SVHN to CIFAR- $\mathbf { \Omega } _ { 1 0 } ^ { } ,$ , GAN showed the best discriminating performance. VAE shows a poor result on SVHN while the performance on CIFAR-10 remains more than $9 7 \%$ . This means that the auto-encoder trained for CIFAR-10 reconstructs SVHN images well, leading to low reconstruction losses in both of the auto-encoders. With PCA, both datasets are able to separate from each other with chances more than $92 \%$ . The experiments using all three datasets showed almost the same results as the experiments above. GAN definitely shows the best results among all methods introduced.
+
+# 3.4.1 LIMITATION
+
+Generative models usually have difficulties in generating realistic images with high resolution and there are few researches experimented on ImageNet data. This incompetence makes the H-Net hard to be applied to the ImageNet data. Also, as the subgroups share the same statistics, PCA hardly catch the characteristic of each set. Experiments have shown poor results on ImageNet and we leave this for a future work.
+
+# 4 CONCLUSION
+
+In this paper, we have proposed a novel framework which is able to divide its capacity into several parts and utilize them according to the given input. Composed of two networks, H-Net is responsible for memorizing “where the data came from” and complicated knowledges such as “what the data is” are engraved to the C-Net. To the best of our knowledge, our work, H-Net, is the first attempt to estimate the origin of data which used to be assumed as given in the previous works. As well as overcoming the catastrophic forgetting, HC-Net allows extra class labels when training a new dataset and also has expandability to the network architecture.
+
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+Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, Alexander C. Berg, and Li Fei-Fei. ImageNet Large Scale Visual Recognition Challenge. International Journal of Computer Vision (IJCV), 115(3):211–252, 2015. doi: 10.1007/s11263-015-0816-y.
+
+Hanul Shin, Jung Kwon Lee, Jaehong Kim, and Jiwon Kim. Continual learning with deep generative replay. In Advances in Neural Information Processing Systems, pp. 2990–2999, 2017.
+
+Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. arXiv preprint arXiv:1409.1556, 2014.
+
+# APPENDIX A H-NET
+
+# A.1 VARIATIONAL AUTO-ENCODER (VAE)
+
+This approach is based on the variational auto-encoder. VAE estimates the data distribution by approximating it to a Gaussian distribution. We trained auto-encoder for each task separately with the following reconstruction loss:
+
+$$
+\mathcal { L } _ { \boldsymbol { r } } ( A E _ { \mathcal { I } } ) = \| \boldsymbol { x } - A E _ { \mathcal { I } } ( \boldsymbol { x } ) \| ^ { 2 }
+$$
+
+where $A E _ { \mathcal { I } } ( x )$ is the reconstructed data from the auto-encoder for the task $\mathcal { I }$ . As $A E _ { \mathcal { I } }$ is trained only with samples of the task $\mathcal { I }$ , samples from other tasks will produce large reconstruction loss by $A E _ { \mathcal { I } }$ since they will not follow the trained distribution. In the testing phase, an unknown input data is given to all auto-encoders. Among the results from the trained auto-encoders, the task $\mathcal { I }$ is determined to the one that has the smallest reconstruction error:
+
+$$
+\mathcal { T } = \mathop { \arg \operatorname* { m i n } } _ { i } \| x - A E _ { i } ( x ) \| ^ { 2 } .
+$$
+
+# A.2 PRINCIPAL COMPONENT ANALYSIS (PCA)
+
+Another approach to separate a dataset from another is using PCA. Typically, PCA conducts dimensionality reduction over a given dataset and the projected features that have the biggest eigen-values are used to represent each sample. The bigger the eigen-value is, the more the corresponding eigenvector explains the variance of the dataset and the PCA-based reconstruction error similar to (4) can be used for the determination of the appropriate task.
+
+In this paper, instead of using principal components of top- $k$ eigen-values, we utilized those of the bottom- $k$ eigen-values. A small eigen-value indicates low variance over the projected coordinate. Therefore, a sample from other tasks is likely to be far from the origin in this projection while a sample from the original dataset is near the origin. If the matrix of the eigen-vectors corresponding to the top $L$ smallest eigen-values is denoted as $\mathbf { W } _ { L } ^ { i }$ which is extracted from the $i$ -th dataset, a sample $x$ from an unknown origin can be projected to produce the projection vector $\mathbf { T } _ { L } ^ { i } = x ^ { T } \mathbf { W } _ { L } ^ { i }$ and the task can be found as the one that minimize the $L _ { 1 }$ or $L _ { 2 }$ norm of the vector as
+
+$$
+\mathcal { T } = \arg \operatorname* { m i n } _ { i } \| \mathbf { T } _ { L } ^ { i } \| _ { p }
+$$
+
+where $p = 1$ or 2. In all our experiments, the number of eigen-vectors $\mathbf { L }$ is set empirically to 1,000.
+
+# APPENDIX B TRAINING DETAILS
+
+# B.1 TRAINING DETAILS OF C-NET
+
+We use the same learning parameters and network architectures suggested in (Jung et al., 2018) used for ‘Tiny image classification’. It is composed of three convolutional layers using $5 \times 5$ kernels with the size of 32, 32 and 64 channels respectively, three max pooling layers, one fully connected layer with 200 nodes and the last softmax classifier layer producing 10 outputs. ReLU is used as the activation function in all the experiments. Also an SGD optimizer with a mini-batch size of 100 has been used for model optimization. When training the old task, the weight decay and the momentum were set to 0.004 and 0.9 respectively. The learning rate starts from 0.01 and the decay of the learning rate with a factor 0.1 is done at the time of 20,000 iterations. After 40,000 iterations, the training is terminated in all the experiments. When training a new task, the training of several approaches including Modified LwF, EWC and LF starts with a learning rate of 0.0001 and is terminated at 10,000 iterations without learning rate decay as in (Jung et al., 2018). Unlike above methods, LwF, PackNet and HC-Net have the same settings for the old and new task because they need more iterations as the new parameters are to be trained from the scratch. Note that PackNet needs additional pruning and finetuning step. In the new task training of LwF, we start with a learning rate of 0.0002 and 0.001 which are $0 . 1 \sim 0 . 0 2$ times less than that of old task for the ‘MNIST to SVHN’ and ‘SVHN to CIFAR-10’ respectively, as recommended in (Li & Hoiem, 2017). The channel split ratio which determines $I ^ { \mathcal { I } }$ follows the setting of PackNet.
+
+# B.2 TRAINING DETAILS OF H-NET
+
+# B.2.1 GAN
+
+We use the deep convolutional generative adversarial networks (DCGAN) (Radford et al., 2015). In all experiments of the basic image classification dataset, we use the size of latent vector to 200. The generator use the four transposed convolution layers which are using 512,256,128 and 3 channels with $4 \times 4$ kernels, respectively and we use ReLU activation for all layers except for the output, which uses Tanh. Transposed convolution layers use 2 stride except the first layer. we use LeakyReLU activation in the discriminator for all layers. For the discriminator, the last convolution layer is flattened and then fed into a sigmoid output to discriminate. The architecture of discriminator is symmetrical to it of the generator. We use batchnorm in both generator and the discriminator.
+
+We use the Adam optimizer with learning rate of 0.0002 using the batch size of 64. The momentum term $\beta 1$ set to 0.5 and we train the generator until 200 epochs. The generated samples of basic image classification dataset are shown in Figure 4(a), 5(a) and 6(a).
+
+For the binary classifier, we use the same network and experimental settings in (Jung et al., 2018) used for ‘Tiny image classification’.
+
+# B.2.2 VAE
+
+We use the simplified VAE. we use two fully connected layer with a hidden layer with 400 units and 20 units for mean and variance of the latent state distributions. The decoder have an equal number of hidden units.
+
+We train the VAE with ReLU activation, sigmoid activation and Adam optimizer with a learning rate of 0.0001 using batch size of 128. We train the VAE until 100 epochs.
+
+![](images/df24de6f3f2e16dc39da1f7505c3e24fd85d1617dcdf92b6edef025073d06245.jpg)  
+Figure 4: MNIST dataset
+
+# APPENDIX C RELATED WORK
+
+Dual memory network system has been suggested to overcome the low performance of the regularization methods and the inefficiency of the dynamic methods. Most of these researches try to mimic the brain system of mammals whose recognition process is believed to be done by the complementary contribution of the hippocampus and the cortex. In several researches, the hippocampus has been realized by networks replaying the previous dataset or determining whether the input is from the old dataset (Gepperth & Karaoguz, 2016; Kemker & Kanan, 2017). Shin et al. (2017) proposed a method of replaying previously encoded experiences using GAN and training the new task without the presence of the old dataset.
+
+![](images/f9d7243ae53471a2e450d0a88b40fec1ea48dc187666e09955e120a3532a86a6.jpg)  
+Figure 5: SVHN dataset
+
+![](images/327670c06f2530c08e7677d7718b3b3d91979ae74143a9c635efd6c3e832e619.jpg)  
+Figure 6: CIFAR-10 dataset
+
+In the data mining field, multi-task learning can be seen as learning from data streams. Therefore, our method can be considered to be one of researches on learning methods from data streams (Gama & Gaber, 2007; Gama, 2010).

parse/train/rJgz8sA5F7/rJgz8sA5F7_content_list.json

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+    {
+        "type": "text",
+        "text": "HC-NET: MEMORY-BASED INCREMENTAL DUALNETWORK SYSTEM FOR CONTINUAL LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            234,
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+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Training a neural network for a classification task typically assumes that the data to train are given from the beginning. However, in the real world, additional data accumulate gradually and the model requires additional training without accessing the old training data. This usually leads to the catastrophic forgetting problem which is inevitable for the traditional training methodology of neural networks. In this paper, we propose a memory-based continual learning method that is able to learn additional tasks while retaining the performance of previously learned tasks. Composed of two complementary networks, the Hippocampus-Net (H-Net) and the Cortex-Net (C-Net), our model estimates the index of the corresponding task for an input sample and utilizes a particular portion of itself with the estimated index. The C-Net guarantees no degradation in the performance of the previously learned tasks and the H-Net shows high confidence in finding the origin of an input sample. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            469,
+            336,
+            486
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The main difference between the human brain and the machine learning methodology is the ability to evolve. Using neurophysiological processing principles, human brains can achieve and organize knowledges throughout their lifespan. Having the neuroplasticity, human brains can transfer an activating region of a given function to a different location or control the creation and destruction of synapses according to its experiences. Usually, artificial neural network (ANN) models consist of finite numbers of filters. Also, parameters and operations in the ANN do not possess the ability corresponding to the memory system of human brains. This structural limit leads to the problem called catastrophic forgetting, i.e., the newly coming information diverts the model from previously learned knowledge. The field of researches trying to solve this problem is referred to as continual or lifelong learning. ",
+        "bbox": [
+            174,
+            500,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Kirkpatrick et al. (2017) suggested a method applying a penalty on the quadratic distance between the old parameters and the new ones. Called elastic weight consolidation (EWC), this method can be applied to either supervised or reinforcement learning scenarios. Jung et al. (2018) proposed a method using feature regularization referred as less-forgetful learning (LF). When training a new dataset, the feature just before the fully connected layer is regularized to become similar to that of the previous model. A considerably small learning rate is used in order to prevent a drastic change of the model. Despite of such efforts, as more and more tasks are added, the model eventually collapses with poor performances in old datasets. Li & Hoiem (2017) proposed a method using a similar regularization method, referred as learning without forgetting (LwF). Continual learning is conducted to a single convolutional neural network (CNN), adding an additional classifier whenever a dataset is newly trained. Classifiers in the newly trained network are regularized to reproduce the outputs of the old network while the new classifier is trained in a classical way. LwF shows better performance than LF, but still suffers from the catastrophic forgetting with multiple tasks. These approaches using regularization basically focus on alleviating the deformation of the original network. Therefore, the more the new knowledge flows in the model, the more the performance on the original task degrades. To make a single network deal with a large number of datasets, instead of using all parameters of a single network for each task, PackNet suggested a model which allocates the datasets to specific weights of filters (Mallya & Lazebnik, 2018). As only the specialized weights for a particular task are involved in its classification process, PackNet shows a remarkable result in multiple tasks. However, the information about from which task the input originates and which group of weights should be used must be given in advance. Typically, images do not contain such prior knowledges and this makes the PackNet hard to apply in real world situation. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/eec4e95f50f3656571df72b4b26b98e1b4b036bdaf04a02263b2b5ce95f1d31a.jpg",
+        "image_caption": [
+            "Figure 1: The overall architecture of our method. Blue arrows depicts the training process for the new task (Task C) of our method, where Task A and Task B have been already trained. H-Net is trained each time a dataset for a new task is provided and then the C-Net is trained in a standard way such as using the cross-entropy loss. We allow only the trainable filters (Blue parameters) to be updated. Red arrows shows the inference procedure of our method. H-Net finds out the task index $\\mathcal { I }$ (Task C) from which the input came and let the C-Net to use the filters specialized for task $\\mathcal { I }$ (Blue $+ \\mathrm { O r a n g e + G r e y } )$ . Class labels are also switched to those of task $\\mathcal { I }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            101,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this paper, inspired by the human brain system, we propose a network which efficiently increases its complexity without degrading the performance of previous tasks. Instead of saving all the memory in the cortex and trying to access it directly, the hippocampus finds the index of the memory scattered over the cortex where knowledges are well organized inside. Under the inspiration from the quote in (Buzsaki, 2006), “Think of the cortex as a huge library and the hippocampus as its librarian”, we have built our network using two complementary sub-networks as shown in Figure 1. The Cortex-Net (C-Net) keeps the knowledge from several independent tasks. After training a network from the previous task, additional filters in the convolutional modules are attached and trained with the next task. To infer the newly learned data, the model utilizes the newly trained filters along with the previously learned filters, whereas the data from the old task is inferred only using the previously learned filters. In order to determine which combination of filters to use, we adopted a memory network which can distinguish the origin of a given input sample. Referred as the Hippocampus-Net (H-Net), it can recall the information not only which group of filters to use but also which group of class labels to use. This endows our method label-expandability, which enables our method to be practically applied in real world situation. We suggest three methods, each of which respectively using generative adversarial networks (GAN) (Goodfellow et al., 2014), variational auto-encoder (VAE) (Kingma & Welling, 2013), and principal component analysis (PCA) (Jolliffe, 2011) for H-Net and report the properties and performance of each method. Our H-Net can be combined with any other works such as LwF or PackNet. The combined H-Net and C-Net referred as HC-Net can prevent the catastrophic forgetting using a densely organized architecture and a highly reliable memory system. The contributions of this paper can be summarized as follows: ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "• Filters in the convolutional layers are allocated to each task and parameters from the previously learned tasks are shared among post tasks and used for the initialization of post tasks parameters, which makes the overall model compact and efficient.   \n• Attaching filters whenever a new dataset is trained allows the model to be expandable as far as the physical constraints, such as memory and processing time constraints, allow.   \n• To identify which portion of convolutional filters should be used to classify a data, a memory network (H-Net) is proposed with several methodologies.   \n• The overall method is applicable to multi-task continual learning whose number of class labels increases over tasks. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/da0df10bb0079357dd40b2e4e7473a7490a92408d4a5087fefd1e8725576404d.jpg",
+        "image_caption": [
+            "Figure 2: Trainable parameters $W _ { k } ^ { T }$ (Orange) and frozen parameters $W _ { k } ^ { \\mathcal { P } }$ (Grey) in the C-Net training process. In the training process, the parameters (Orange), which are trainable parameters except frozen parameters (Grey), are updated. $x$ denotes the spatial dimension of filters. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 METHOD ",
+        "text_level": 1,
+        "bbox": [
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+            348
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We propose a memory-based incremental dual network system for continual learning that satisfies three properties as stated below. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Property 1. The Hippocampus-Net (H-Net) offers the index about which part of the Cortex-Net (C-Net) should be used. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Property 2. The C-Net learns the knowledge on the new task under the presence of the previously learned knowledge. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Property 3. H-Net and C-Net are flexible to the expansion of the network and as many new tasks can be learned in one network as physical constraints allow. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this respect, our proposed H-Net works as a hippocampus of a human brain in that it gives a summarized information to the C-Net. Also, the C-Net corresponds to the human cortex in terms of its ability to learn semantic knowledge. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.1 CORTEX NETWORK (C-NET) ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            570,
+            411,
+            585
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We propose an efficient way to allocate the capacity to a network according to the given tasks. C-Net is an expandable network where additional filters can be attached whenever a new task is given. It can be applied to many typical networks such as VGG and ResNet (Simonyan & Zisserman, 2014; He et al., 2016). The only difference with the original versions of those networks is that it uses different parts of filters for different tasks. As shown in (Li et al., 2016), typical CNN are capable of maintaining their performance even when the majority of their parameters are pruned. This implies that filters in a CNN contains a lot of unnecessary or redundant information. Also, sharing the filters among different tasks inevitably leads to the catastrophic forgetting of the network since the parameters trained by an old task should be replaced with those trained by a new task. For these reasons, we utilize a network divided into several parts and refer it as C-Net where each part takes charge of a particular task. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "While training the C-Net with several tasks, each task uses a different part of the C-Net. For the convenience of referring which segments of the C-Net are to be activated, we introduce a filter index $I ^ { \\mathcal { I } }$ for the $\\mathcal { I }$ -th task. The filter index $I ^ { \\mathcal { I } }$ defines the range of filters to use for the specific task $\\mathcal { I }$ in every layers. For example, if the filter index $I ^ { 1 }$ equals to ten, the first task uses filters from the first to the tenth filter. ",
+        "bbox": [
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+            758,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Depicted in Figure 2, $I _ { k } ^ { \\mathcal { I } }$ denotes the filter index in the $k$ -th layer corresponding to the task $\\mathcal { I }$ . This equals to the number of output channels in the sequent feature map. $I _ { k } ^ { \\mathcal { P } }$ denotes the filter index in the $k$ -th layer for the previous task. As the same principle is applied to the precedent $( k - 1 )$ -th layer, the number of input channels for the filters in the $k$ -th layer will change likewise. That is, total of $I _ { k } ^ { \\mathcal { P } }$ filters have the channel length of $I _ { k - 1 } ^ { \\mathcal { P } }$ when the previous task is under training. As the new task is given, the length of the filters become $I _ { k - 1 } ^ { \\mathcal { I } }$ since the filters in the prior layer will deliver a feature ",
+        "bbox": [
+            173,
+            834,
+            825,
+            925
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Input: $W ^ { \\mathcal { I } } , I ^ { \\mathcal { P } } , I ^ { \\mathcal { I } }$ ",
+        "bbox": [
+            174,
+            123,
+            312,
+            138
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Output: W J ∗ ",
+        "bbox": [
+            174,
+            140,
+            272,
+            152
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "1: Freeze the $W ^ { \\mathcal { P } } \\subset W ^ { \\mathcal { I } }$ using the information $I ^ { \\mathcal { P } }$ 2: Initialize the $W ^ { \\tau }$ with $W ^ { \\mathcal { P } }$ and a random noise 3: $W ^ { \\mathcal { I } * }  \\arg \\operatorname* { m i n } ( \\mathcal { L } _ { c } )$ {Update $W ^ { \\breve { T } }$ using backpropagation} ",
+        "bbox": [
+            181,
+            151,
+            521,
+            210
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "map whose length is $I _ { k - 1 } ^ { \\mathcal { I } }$ . Therefore, both the number of input and output channels increases as the number of tasks increases until the physical constraints allow. ",
+        "bbox": [
+            171,
+            242,
+            823,
+            271
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Weight parameters of $\\mathcal { I }$ -th task in filters of the $k$ -th layer are referred as $W _ { k } ^ { \\mathcal { I } } \\in R ^ { x \\times x \\times I _ { k - 1 } ^ { \\mathcal { I } } \\times I _ { k } ^ { \\mathcal { I } } }$ , where $x$ means the size of the kernel while $\\begin{array} { r } { W _ { k } ^ { \\mathcal { P } } \\in R ^ { x \\times x \\times I _ { k - 1 } ^ { \\mathcal { P } } \\times I _ { k } ^ { \\mathcal { P } } } } \\end{array}$ are the weight parameters of the previous task. When training the new task, $W _ { k } ^ { \\mathcal { I } }$ is used for inference but only the parameters $w \\in W _ { k } ^ { \\mathcal { T } }$ $( W _ { k } ^ { \\mathcal { T } } = W _ { k } ^ { \\mathcal { I } } \\setminus W _ { k } ^ { \\mathcal { P } } )$ are updated while leaving $W _ { k } ^ { \\mathcal { P } }$ fixed. Note that biases in the filters can also be trained likewise as explained above. The fully connected layer located just before the linear classifier can be divided as well into the sections of $\\dot { I } ^ { \\mathcal { I } }$ and $I ^ { \\mathcal { P } }$ . ",
+        "bbox": [
+            173,
+            280,
+            825,
+            375
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To enhance the performance, slices of trained parameters $W ^ { \\mathcal { P } }$ from the old task are duplicated to the newly available task parameters $W ^ { T }$ and Gaussian noise is added to them for the sake of good initialization. Then, the network is trained in a standard way, e.g., using a cross-entropy loss $\\mathcal { L } _ { c }$ . Independent linear classifiers are used for every tasks. This scheme of C-Net can also be applied to structures using shortcut connections such as ResNet. The overall training process is shown in Algorithm 1. ",
+        "bbox": [
+            174,
+            380,
+            825,
+            464
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.2 HIPPOCAMPUS NETWORK (H-NET) ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            483,
+            457,
+            498
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Even if the C-Net is well trained separately with several tasks, the task index $\\mathcal { I }$ must be given at the time of inference. Especially when the class labels expand, this prior information $\\mathcal { I }$ is necessary to estimate which group of class labels to use. In the testing step, existing methods needs the exact origination of the input data. However in the real world, a given data for classification normally does not offer any information about its origination. Therefore, this prior knowledge should be estimated by an independent network. ",
+        "bbox": [
+            174,
+            511,
+            825,
+            594
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To solve this problem, we introduce the Hippocampus network (H-Net) using several different methods (GAN, VAE and PCA), which is able to estimate the task $\\mathcal { I }$ of the given input data and inform this to the C-Net to specify which filters to use. In this paper, we explain the GAN method as a representative and other methods are explained in Appendix A. ",
+        "bbox": [
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+            659
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.2.1 GENERATIVE ADVERSARIAL NETWORKS (GAN) ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This method is inspired by the on-line replay method used for continual learning (Shin et al., 2017). We train a task-specific generative model $G _ { \\mathcal { I } }$ using a GAN to generate pseudo-samples of task $\\mathcal { I }$ . The generator is trained using the adversarial loss as follows: ",
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ac1556454e211a62fbafd3b50e62e15ccd436f9fa15792c42534487bd513252b.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { G , \\mathcal { T } } \\operatorname* { m a x } _ { D , \\mathcal { T } } V ( D _ { \\mathcal { T } } , G _ { \\mathcal { T } } ) = \\mathbb { E } _ { \\boldsymbol { x } \\sim P _ { d a t a } ^ { \\mathcal { T } } } [ \\mathrm { l o g } ( D _ { \\mathcal { T } } ( \\boldsymbol { x } ) ] + \\mathbb { E } _ { \\boldsymbol { z } \\sim p _ { \\mathcal { z } } ( \\boldsymbol { z } ) } [ \\mathrm { l o g } ( 1 - D _ { \\mathcal { T } } ( G _ { \\mathcal { T } } ( \\boldsymbol { z } ) ) ] .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Here, $D _ { \\mathcal { I } }$ is a discriminator for the task $\\mathcal { I }$ and $x$ is a sample from the task $\\mathcal { I }$ . ",
+        "bbox": [
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+        "type": "text",
+        "text": "After generating samples of all tasks, a task-wise binary classifier $B _ { \\mathcal { I } }$ is trained to classify whether the given input is from the task $\\mathcal { I }$ or not. Note that only the generated samples are involved for the training of each binary classifier $B _ { \\mathcal { I } }$ . This means that positive samples are from the generator $G _ { \\mathcal { I } }$ for the task $\\mathcal { I }$ and negative samples are from all the other task generators $G _ { \\kappa }$ $( \\forall \\boldsymbol { K } \\neq \\mathcal { T }$ ). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In the testing phase, each classifier produces the probability of how likely the given unknown input data is from the task $\\mathcal { I }$ . The classifier $B _ { i }$ with the highest probability assumes that the input data is from the task $i$ . H-Net can figure out the task $\\mathcal { I }$ of input data with maximum probability across the set of classifiers. H-Net gives this estimated task index $\\mathcal { I }$ to the C-Net. The H-Net using GAN can be summarized as below: ",
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+        "type": "text",
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+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/54462def6348af8ef6b3cfa2568cc95dd246e2aa8026e93195df1caf695bb239.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathrm { I n i t i a l i z a t i o n : } \\qquad B = \\{ B _ { 1 } , B _ { 2 } , . . . , B _ { N } \\} } \\\\ & { \\mathrm { T r a i n i n g : } \\quad B _ { i } ( x ) = \\left\\{ 1 \\quad \\mathrm { i f } x \\mathrm { i s f r o m } G _ { i } \\right. } \\\\ & { \\mathrm { I n f e r e n c e : } \\qquad \\mathcal { I } = \\arg \\underset { i } { \\operatorname { i f } } x \\mathrm { i s f r o m } G _ { j } \\forall j \\neq i } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "3 EXPERIMENT ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We evaluate our method on several image classification datasets. First, we verify the effectiveness of our method with MNIST (LeCun et al., 1998), SVHN (Netzer et al., 2011) and CIFAR-10 (Krizhevsky & Hinton, 2009) which are widely used to evaluate image classification performance. Then, we evaluate our method on two subsets of ImageNet (Russakovsky et al., 2015) which is a realistic image dataset. More details are summarized in Table 1. ",
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+    {
+        "type": "text",
+        "text": "We compare our method to various methods such as Elastic Weight Consolidation (EWC), Learning without forgetting (LwF), less-forgetful learning (LF) and PackNet as well as networks trained for a single target task which shows the performance without conducting continual learning. We use the GAN method for H-Net in all the experiments as a representative. To solely examine the influence of H-Net on our method (HC-Net), we have conducted experiments with the same experimental scenarios on the HC-Net experiments. In Table 2, each row shows the result of one among GAN, VAE and PCA. In Table 3 and Table 4, the result of ‘HC-Net (without H-Net)’ shows the performance of the C-Net only. That is, it assumes that the H-Net never fails and the task index $\\mathcal { I }$ is always given correctly. Likewise, LwF and PackNet need prior knowledge of a given input to know which classifier to use. As there is no structure like H-Net in the original paper, The actual comparison must be done with ‘C-Net’. ",
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+    {
+        "type": "text",
+        "text": "3.1 CONTINUAL LEARNING USING BASIC IMAGE CLASSIFICATION DATASETS ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "We have built three experimental scenarios to evaluate our method using basic image classification datasets. We compare the performance of learning a consecutive two task pair among MNIST, SVHN and CIFAR-10 dataset. After that, we conduct multiple-task continual learning with these datasets. In these experiments, we use the same learning hyperparameters and network architectures suggested in (Jung et al., 2018) used for ‘Tiny image classification’. Training details are reported in Appendix B. ",
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+    {
+        "type": "text",
+        "text": "3.1.1 MNIST → SVHN ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "After selecting a subset of SVHN to equalize the number of training data between MNIST and SVHN, images are resized to $2 8 \\times 2 8$ as in (Jung et al., 2018). The results of Modified LwF, EWC and LF are referred from (Jung et al., 2018). Modified LwF is sligthly different from the original LwF in that the old output $\\hat { Y } _ { o }$ and new outputs $\\hat { Y _ { n } }$ is united into a single output. The ‘single network’ in Table 3 is a network of full capacity trained using a single dataset. The column ‘Old’ represents the test accuracy on the previously learnt task and ‘New’ is that of the newly trained task . The value $\\gamma$ for EWC is determined as in (Kirkpatrick et al., 2017). A knowledge distillation loss with hyper-parameters of $T = 2$ and $T = 1$ is used for LwF, referred as temperature in (Hinton et al., 2015). ",
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+    {
+        "type": "text",
+        "text": "As MNIST and SVHN highly resemble each other, the accuracy of all methods are relatively high. LF demonstrate better average classification rates than Modified LwF and EWC. LwF can control the performance trade-off between the old and new task by changing the temperature parameter $T$ . The bigger $T$ is, the lower the accuracy drops but it still outperforms those of the methods mentioned above. PackNet almost maintains the performance of the single network for the first task with less than $2 \\%$ drop in accuracy. HC-Net outperforms all other methods in the table. Note that the prior knowledge to select which task is given in experiments of other methods. Therefore, our HC-Net without H-Net (i.e., C-Net), which shows the highest in performance, should be compared to other baseline methods. Nonetheless, our full model, HC-Net also outperforms PackNet, especially on the second task by around $0 . 7 \\%$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/5e49759128a4d4657fbbbcc0041f7120038451332ca2aca6a4e75cef8bb92a40.jpg",
+        "table_caption": [
+            "Table 1: Details of datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Datasets</td><td>#ofTrain data</td><td>#of Test data</td><td>#of Class</td></tr><tr><td>MNIST</td><td>60,000</td><td>10,000</td><td>10</td></tr><tr><td>SVHN</td><td>73,275</td><td>26.032</td><td>10</td></tr><tr><td>CIFAR-10</td><td>50,000</td><td>10.000</td><td>10</td></tr><tr><td>ImageNet-A</td><td>64,750</td><td>2.500</td><td>50</td></tr><tr><td>ImageNet-B</td><td>64,497</td><td>2,500</td><td>50</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/6c36330d92565e178500de8eba2a1658296de5a4815c477f8101eece353a9ecc.jpg",
+        "table_caption": [
+            "Table 2: Experimental results of H-Net on the basic datasets "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Datasets</td><td>Methods</td><td>Old(%)</td><td>New(%)</td></tr><tr><td>MNIST √ SVHN</td><td>GAN VAE PCA</td><td>100 100 100</td><td>99.94 99.85 100</td></tr><tr><td>SVHN √ CIFAR-10</td><td>GAN VAE PCA</td><td>97.00 71.07 93.28</td><td>99.90 97.06 92.5</td></tr><tr><td>Datasets</td><td></td><td>Methods MNIST(%)</td><td>SVHN(%) CIFAR(%)</td></tr><tr><td>MNIST</td><td>GAN</td><td>100 96.65</td><td>99.83</td></tr><tr><td>→ SVHN →CIFAR-10</td><td>VAE PCA 100</td><td>100 71.22 93.28</td><td>97.07 92.5</td></tr></table>",
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+    },
+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 3: Mean classification results on the basic datasets (5 runs). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Datasets</td><td>Methods</td><td>Old(%)</td><td></td><td>New(%) Avg.(%)</td></tr><tr><td rowspan=\"2\">MNIST √ SVHN SVHN</td><td>single network (MNIST) single network (SVHN) ModifiedLwF EWC(y = 2.32 × 104) LF(λe = 1.6 × 10-3)</td><td>99.49 1 94.78 94.15 97.37</td><td>92.82 83.77 79.31 83.79</td><td>1 1 89.28 86.73 90.58</td></tr><tr><td>LwF(T=1) LwF(T=2) PackNet HC-Net HC-Net (without H-Net) single network (SVHN) single network (CIFAR)</td><td>98.27 97.33 99.45 99.43 99.43 92.94</td><td>86.40 86.97 91.49 92.20 92.37 一 79.69</td><td>92.34 92.15 95.47 95.82 95.90 1</td></tr><tr><td colspan=\"2\">√ CIFAR-10</td><td>1 LF(λe=2.0 ×10-3) 83.18 91.76 90.19 92.84 90.05</td><td>48.99 70.57 71.94 76.78 76.62</td><td>1 66.09 81.17 81.07 84.81 83.34</td></tr><tr><td>Datasets</td><td colspan=\"2\">HC-Net (without H-Net) Methods</td><td>92.21 76.70 MNIST(%)SVHN(%)CIFAR(%)Avg(%)</td><td>84.46</td></tr><tr><td>MNIST ↓ SVHN √ CIFAR-10</td><td colspan=\"2\">single network (MNIST) 99.44 single network (SVHN) 二 single network (CIFAR) = LF(λe=1.0×10-) 89.57 LwF(T=1) 95.06 LwF(T= 2) 86.09 PackNet 99.38</td><td colspan=\"2\">一 92.94 79.69 72.1 45.43</td></tr></table>",
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "3.1.2 ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this experiment, we want to see the performance of the proposed method when the statistics of the two tasks are quite different. The datasets SVHN and CIFAR-10 have a clear distinction in image styles and colors. For LF whose classifier must be shared between the old and new tasks, this inevitably leads to a significant decrease in performance. LwF and PackNet show similar trends as that of ‘MNIST to SVHN’. Higher temperature $( T )$ induces better performance in the new task but a bit of degradation in average performance still occurs. PackNet experiences almost no performance degradation. The pruning method of PackNet acts like a generalization method and helps the model to maintain its capacity. The average performance of C-Net is slightly lower than that of PackNet by $0 . 3 5 \\%$ . However, note that PackNet requires additional finetuning after the pruning process and takes additional time for training. On the other hand, C-Net requires no additional pruning or finetuning precedure and yet retains the performance. ",
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+    {
+        "type": "text",
+        "text": "3.1.3 ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Training three or more tasks is unmanageable for the models using regularization methods. LF scored an average accuracy of no more than $6 9 . 0 3 \\%$ , which is much more lower than any other methods. No matter how we change the temperature $( T )$ , LwF suffers from a drop in average performance, especially in the third task. Also, even though the proportion allocated to each task is equivalent between the PackNet and our methods (HC-Net and C-Net), our method highly outperforms the PackNet in the third task. This implies that HC-Net is more suitable for multi-task sequential learning situation. ",
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+    {
+        "type": "text",
+        "text": "3.2 CONTINUAL LEARNING USING REALISTIC DATASETS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "ImageNet contains images more realistic than other datasets used in this paper. Higher resolution and complex backgrounds make the classification even harder. To save the training time, two subsets of the ImageNet dataset each having 50 randomly chosen classes have been used for evaluation. They are referred to as ImageNet-A and ImageNet-B respectively in this paper. To show the adaptability of our method on structures having shortcut connections, ResNet-50 is used for the experiment. We use the same experimental setting as in (He et al., 2016) with a batch size of 128. We compare our C-Net with LwF and PackNet. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/7c006c6def734db8b0de491adc26138e2c28247ead1b018621f668de1265eb3a.jpg",
+        "table_caption": [
+            "Table 4: Mean classification results for the realistic dataset (2 runs) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>datasets</td><td>Methods</td><td>Old(%)</td><td>New(%)</td><td>Avg.(%)</td></tr><tr><td rowspan=\"4\">ImageNet-A √ ImageNet-B</td><td>single network(ImageNet-A)</td><td>83.28</td><td>一</td><td>1</td></tr><tr><td>single network (ImageNet-B)</td><td>1 82.2</td><td>85.28</td><td>1</td></tr><tr><td>LwF(T=1)</td><td></td><td>86.72</td><td>84.46</td></tr><tr><td>LwF(T= 2)</td><td>80.92</td><td>86.96</td><td>83.94</td></tr><tr><td rowspan=\"2\"></td><td>PackNet</td><td>82.16</td><td>88.72</td><td>85.44</td></tr><tr><td>HC-Net (without H-Net)</td><td>83.30</td><td>88.66</td><td>85.98</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/8729df3689b888c044728f7949f1257e9f76135613d71476bab1de8b46b7083b.jpg",
+        "table_caption": [
+            "Table 5: Performances of the old (MNIST) and new (SVHN) tasks with different initialization methods: 1) initialization with only random Gaussian noise and 2) the pretrained parameter added by random Gaussian noise "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Initialization</td><td>0ld(%)</td><td>New(%)</td><td>Avg.(%)</td></tr><tr><td>Random</td><td>99.42</td><td>91.67</td><td>95.55</td></tr><tr><td>Pretrained parameters</td><td>99.43</td><td>92.37</td><td>95.90</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Table 4 shows the results on the ImageNet dataset. The ‘Old’ accuracy of all methods slightly dropped from those of single networks. Like in the ‘SVHN to CIFAR- $1 0 ^ { \\circ }$ experiment, PackNet shows a better result than the LwF regardless of the value of $T$ . The result of C-Net is almost identical to PackNet with a slight increase in the average accuracy. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "PackNet actually utilizes more parameters for each task than C-Net does. Since the masks force weights with no influence to be zero and utilize the well trained remaining weights, PackNet can make good use of the entire network. However in C-Net, just adding filters gradually without any other post-processing performs well enough compared to PackNet. When the model of an initially designed size is fully occupied by several tasks, C-Net can just add more filters and train them along with the trained filters while this is not the case of PackNet. Furthermore, PackNet requires additional memories to store the masks for all filters in it. C-Net needs only one integer per layer for each task. For these reasons, C-Net is far more efficient than the PackNet with no loss in performance. ",
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+    {
+        "type": "text",
+        "text": "3.3 ABLATION STUDY ON THE EFFICIENCY OF C-NET ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As mentioned in Section 2.1, our C-Net appends filters whenever a new task is to be trained. In this process, parameters learned from previous tasks are utilized altogether and the parameters which are to be trained newly are initialized using the existing ones. We conduct experiments to show the effect of these methods. Experiments are carried out in ‘MNIST to SVHN’ case as a representative. ",
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+    {
+        "type": "text",
+        "text": "3.3.1 INITIALIZATION WITH PRETRAINED PARAMETERS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To analyse the effectiveness of initialization using pretrained old task parameters with random Gaussian noise, we compare the accuracy between a model initializing the new parameters with just a random Gaussian distribution and a model initializing the new task parameters using the pretrained old task parameters with additional random Gaussian noise. In Table 5, the model initialized by our method obviously shows higher accuracy on the SVHN task. This result implies that a good initialization prevents the model from going through a local minima. ",
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+    {
+        "type": "text",
+        "text": "3.3.2 PARAMETER SHARING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We verify the effect of the parameter sharing between the old task and the new task. The baseline method is a model where the old parameters are filled with random Gaussian initialization and no further training is done. The new parameters in the model have to learn knowledges from the new task without the aid of old parameters since they are fixed from the beginning. On the other hand, the old parameters in our method is trained by the old task and the new parameters can make use of these learnt knowledges to learn the new task. To solely observe the effect of the parameter sharing, both methods do not use our parameter initialization method mentioned in the previous section. The task indices $I ^ { 1 }$ and $I ^ { 2 }$ are $\\{ 1 6 , 1 6 , 3 2 \\}$ and $\\{ 1 8 , 1 8 , 3 4 \\}$ respectively. The model using parameter sharing converges far more faster than the model with no parameter sharing which is shown in Figure 3. Also the final error rate of our method is lower than the other. Increasing the number of indices elevation enhances our method by $6 . 3 8 \\%$ and the other model by $7 . 9 1 \\%$ shown in Table 6. This implies that the parameter sharing improves the model and allow it to be compact with fewer numbers of filters. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/d23170eb7767a0fb74e8d8a3b06991296184d1f4092acd3844393ea1fb9e66e7.jpg",
+        "image_caption": [
+            "Figure 3: Error rate comparison between sharing the parameters which learned previous task and random parameters. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/5758681bd04fa6f2f9e2cb16451cb34c07ca0bc823114ddfd246f7e80dc823c9.jpg",
+        "table_caption": [
+            "Table 6: The performances on the new task while increasing the filter index of the model using parameter sharing and the model with fixed random parameters. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>#ofincreasingeach index</td><td>Ours(%)</td><td>Random(%)</td></tr><tr><td>2</td><td>80.60</td><td>78.10</td></tr><tr><td>4</td><td>85.31</td><td>83.85</td></tr><tr><td>6</td><td>86.98</td><td>86.01</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.4 H-NET ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The results of H-Net are reported in Table 2. In ‘MNIST to SVHN’, all methods approximates the results to perfection. Despite of the similar appearance of these two datasets, VAE has been able to generate distinct outputs. If a sample in the MNIST dataset is given, the result from the VAE trained for SVHN resembles those of the SVHN dataset. In other words, auto-encoders of each task tends to memorize only what they used for training. PCA is completely able to discriminate the origin of the images from MNIST and SVHN. Particularly in this experiment, even when using a single eigenvector with the least eigen-value showed a result of $100 \\%$ and $9 9 . 7 \\%$ . According to the dataset, PCA can highly reduce the computational cost as well as the memory resource. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In ‘SVHN to CIFAR- $\\mathbf { \\Omega } _ { 1 0 } ^ { } ,$ , GAN showed the best discriminating performance. VAE shows a poor result on SVHN while the performance on CIFAR-10 remains more than $9 7 \\%$ . This means that the auto-encoder trained for CIFAR-10 reconstructs SVHN images well, leading to low reconstruction losses in both of the auto-encoders. With PCA, both datasets are able to separate from each other with chances more than $92 \\%$ . The experiments using all three datasets showed almost the same results as the experiments above. GAN definitely shows the best results among all methods introduced. ",
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+    },
+    {
+        "type": "text",
+        "text": "3.4.1 LIMITATION",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Generative models usually have difficulties in generating realistic images with high resolution and there are few researches experimented on ImageNet data. This incompetence makes the H-Net hard to be applied to the ImageNet data. Also, as the subgroups share the same statistics, PCA hardly catch the characteristic of each set. Experiments have shown poor results on ImageNet and we leave this for a future work. ",
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+    },
+    {
+        "type": "text",
+        "text": "4 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            792,
+            318,
+            809
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we have proposed a novel framework which is able to divide its capacity into several parts and utilize them according to the given input. Composed of two networks, H-Net is responsible for memorizing “where the data came from” and complicated knowledges such as “what the data is” are engraved to the C-Net. To the best of our knowledge, our work, H-Net, is the first attempt to estimate the origin of data which used to be assumed as given in the previous works. As well as overcoming the catastrophic forgetting, HC-Net allows extra class labels when training a new dataset and also has expandability to the network architecture. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+    },
+    {
+        "type": "text",
+        "text": "APPENDIX A H-NET ",
+        "text_level": 1,
+        "bbox": [
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+            102,
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+            118
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.1 VARIATIONAL AUTO-ENCODER (VAE) ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            133,
+            480,
+            148
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "This approach is based on the variational auto-encoder. VAE estimates the data distribution by approximating it to a Gaussian distribution. We trained auto-encoder for each task separately with the following reconstruction loss: ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7e949789c3cc151068fbbdff33ec55ab87c1b2eaeb0ec152f56c16b47d1fcc25.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\boldsymbol { r } } ( A E _ { \\mathcal { I } } ) = \\| \\boldsymbol { x } - A E _ { \\mathcal { I } } ( \\boldsymbol { x } ) \\| ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            397,
+            208,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "where $A E _ { \\mathcal { I } } ( x )$ is the reconstructed data from the auto-encoder for the task $\\mathcal { I }$ . As $A E _ { \\mathcal { I } }$ is trained only with samples of the task $\\mathcal { I }$ , samples from other tasks will produce large reconstruction loss by $A E _ { \\mathcal { I } }$ since they will not follow the trained distribution. In the testing phase, an unknown input data is given to all auto-encoders. Among the results from the trained auto-encoders, the task $\\mathcal { I }$ is determined to the one that has the smallest reconstruction error: ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/dda9f033e218657c002d56434b72fae661c11df6dc948b919559518ad518910f.jpg",
+        "text": "$$\n\\mathcal { T } = \\mathop { \\arg \\operatorname* { m i n } } _ { i } \\| x - A E _ { i } ( x ) \\| ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.2 PRINCIPAL COMPONENT ANALYSIS (PCA) ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Another approach to separate a dataset from another is using PCA. Typically, PCA conducts dimensionality reduction over a given dataset and the projected features that have the biggest eigen-values are used to represent each sample. The bigger the eigen-value is, the more the corresponding eigenvector explains the variance of the dataset and the PCA-based reconstruction error similar to (4) can be used for the determination of the appropriate task. ",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In this paper, instead of using principal components of top- $k$ eigen-values, we utilized those of the bottom- $k$ eigen-values. A small eigen-value indicates low variance over the projected coordinate. Therefore, a sample from other tasks is likely to be far from the origin in this projection while a sample from the original dataset is near the origin. If the matrix of the eigen-vectors corresponding to the top $L$ smallest eigen-values is denoted as $\\mathbf { W } _ { L } ^ { i }$ which is extracted from the $i$ -th dataset, a sample $x$ from an unknown origin can be projected to produce the projection vector $\\mathbf { T } _ { L } ^ { i } = x ^ { T } \\mathbf { W } _ { L } ^ { i }$ and the task can be found as the one that minimize the $L _ { 1 }$ or $L _ { 2 }$ norm of the vector as ",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2167207f11172b273ed34356b07c5c52ab2f9602bb000c4fbdf9d1b7f6b1e3ea.jpg",
+        "text": "$$\n\\mathcal { T } = \\arg \\operatorname* { m i n } _ { i } \\| \\mathbf { T } _ { L } ^ { i } \\| _ { p }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            428,
+            560,
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+            587
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "where $p = 1$ or 2. In all our experiments, the number of eigen-vectors $\\mathbf { L }$ is set empirically to 1,000. ",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX B TRAINING DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            628,
+            465,
+            646
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "B.1 TRAINING DETAILS OF C-NET ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            661,
+            426,
+            676
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We use the same learning parameters and network architectures suggested in (Jung et al., 2018) used for ‘Tiny image classification’. It is composed of three convolutional layers using $5 \\times 5$ kernels with the size of 32, 32 and 64 channels respectively, three max pooling layers, one fully connected layer with 200 nodes and the last softmax classifier layer producing 10 outputs. ReLU is used as the activation function in all the experiments. Also an SGD optimizer with a mini-batch size of 100 has been used for model optimization. When training the old task, the weight decay and the momentum were set to 0.004 and 0.9 respectively. The learning rate starts from 0.01 and the decay of the learning rate with a factor 0.1 is done at the time of 20,000 iterations. After 40,000 iterations, the training is terminated in all the experiments. When training a new task, the training of several approaches including Modified LwF, EWC and LF starts with a learning rate of 0.0001 and is terminated at 10,000 iterations without learning rate decay as in (Jung et al., 2018). Unlike above methods, LwF, PackNet and HC-Net have the same settings for the old and new task because they need more iterations as the new parameters are to be trained from the scratch. Note that PackNet needs additional pruning and finetuning step. In the new task training of LwF, we start with a learning rate of 0.0002 and 0.001 which are $0 . 1 \\sim 0 . 0 2$ times less than that of old task for the ‘MNIST to SVHN’ and ‘SVHN to CIFAR-10’ respectively, as recommended in (Li & Hoiem, 2017). The channel split ratio which determines $I ^ { \\mathcal { I } }$ follows the setting of PackNet. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "B.2 TRAINING DETAILS OF H-NET ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            428,
+            117
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B.2.1 GAN ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            128,
+            269,
+            143
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We use the deep convolutional generative adversarial networks (DCGAN) (Radford et al., 2015). In all experiments of the basic image classification dataset, we use the size of latent vector to 200. The generator use the four transposed convolution layers which are using 512,256,128 and 3 channels with $4 \\times 4$ kernels, respectively and we use ReLU activation for all layers except for the output, which uses Tanh. Transposed convolution layers use 2 stride except the first layer. we use LeakyReLU activation in the discriminator for all layers. For the discriminator, the last convolution layer is flattened and then fed into a sigmoid output to discriminate. The architecture of discriminator is symmetrical to it of the generator. We use batchnorm in both generator and the discriminator. ",
+        "bbox": [
+            173,
+            155,
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+            266
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We use the Adam optimizer with learning rate of 0.0002 using the batch size of 64. The momentum term $\\beta 1$ set to 0.5 and we train the generator until 200 epochs. The generated samples of basic image classification dataset are shown in Figure 4(a), 5(a) and 6(a). ",
+        "bbox": [
+            174,
+            273,
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+            315
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "For the binary classifier, we use the same network and experimental settings in (Jung et al., 2018) used for ‘Tiny image classification’. ",
+        "bbox": [
+            174,
+            321,
+            818,
+            351
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B.2.2 VAE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            367,
+            266,
+            382
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We use the simplified VAE. we use two fully connected layer with a hidden layer with 400 units and 20 units for mean and variance of the latent state distributions. The decoder have an equal number of hidden units. ",
+        "bbox": [
+            176,
+            392,
+            823,
+            434
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We train the VAE with ReLU activation, sigmoid activation and Adam optimizer with a learning rate of 0.0001 using batch size of 128. We train the VAE until 100 epochs. ",
+        "bbox": [
+            173,
+            441,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/df24de6f3f2e16dc39da1f7505c3e24fd85d1617dcdf92b6edef025073d06245.jpg",
+        "image_caption": [
+            "Figure 4: MNIST dataset "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            204,
+            489,
+            794,
+            747
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX C RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            808,
+            436,
+            824
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Dual memory network system has been suggested to overcome the low performance of the regularization methods and the inefficiency of the dynamic methods. Most of these researches try to mimic the brain system of mammals whose recognition process is believed to be done by the complementary contribution of the hippocampus and the cortex. In several researches, the hippocampus has been realized by networks replaying the previous dataset or determining whether the input is from the old dataset (Gepperth & Karaoguz, 2016; Kemker & Kanan, 2017). Shin et al. (2017) proposed a method of replaying previously encoded experiences using GAN and training the new task without the presence of the old dataset. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/f9d7243ae53471a2e450d0a88b40fec1ea48dc187666e09955e120a3532a86a6.jpg",
+        "image_caption": [
+            "Figure 5: SVHN dataset "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/327670c06f2530c08e7677d7718b3b3d91979ae74143a9c635efd6c3e832e619.jpg",
+        "image_caption": [
+            "Figure 6: CIFAR-10 dataset "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the data mining field, multi-task learning can be seen as learning from data streams. Therefore, our method can be considered to be one of researches on learning methods from data streams (Gama & Gaber, 2007; Gama, 2010). ",
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