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+# IQ-Learn: Inverse soft-Q Learning for Imitation
+
+Divyansh Garg Shuvam Chakraborty Chris Cundy Jiaming Song Stefano Ermon Stanford University   
+{divgarg, shuvamc, cundy, tsong, ermon}@stanford.edu
+
+# Abstract
+
+In many sequential decision-making problems (e.g., robotics control, game playing, sequential prediction), human or expert data is available containing useful information about the task. However, imitation learning (IL) from a small amount of expert data can be challenging in high-dimensional environments with complex dynamics. Behavioral cloning is a simple method that is widely used due to its simplicity of implementation and stable convergence but doesn’t utilize any information involving the environment’s dynamics. Many existing methods that exploit dynamics information are difficult to train in practice due to an adversarial optimization process over reward and policy approximators or biased, high variance gradient estimators. We introduce a method for dynamics-aware IL which avoids adversarial training by learning a single Q-function, implicitly representing both reward and policy. On standard benchmarks, the implicitly learned rewards show a high positive correlation with the ground-truth rewards, illustrating our method can also be used for inverse reinforcement learning (IRL). Our method, Inverse soft-Q learning (IQ-Learn) obtains state-of-the-art results in offline and online imitation learning settings, significantly outperforming existing methods both in the number of required environment interactions and scalability in high-dimensional spaces, often by more than $3 \mathbf { x }$ .
+
+# 1 Introduction
+
+Imitation of an expert has long been recognized as a powerful approach for sequential decisionmaking [29, 1], with applications as diverse as healthcare [39], autonomous driving [41], and playing complex strategic games [8]. In the imitation learning (IL) setting, we are given a set of expert trajectories, with the goal of learning a policy which induces behavior similar to the expert’s. The learner has no access to the reward, and no explicit knowledge of the dynamics.
+
+The simple behavioural cloning [34] approach simply maximizes the probability of the expert’s actions under the learned policy, approaching the IL problem as a supervised learning problem. While this can work well in simple environments and with large quantities of data, it ignores the sequential nature of the decision-making problem, and small errors can quickly compound when the learned policy departs from the states observed under the expert. A natural way of introducing the environment dynamics is by framing the IL problem as an Inverse RL (IRL) problem, aiming to learn a reward function under which the expert’s trajectory is optimal, and from which the learned imitation policy can be trained [1]. This framing has inspired several approaches which use rewards either explicitly or implicitly to incorporate dynamics while learning an imitation policy [17, 10, 33, 22]. However, these dynamics-aware methods are typically hard to put into practice due to unstable learning which can be sensitive to hyperparameter choice or minor implementation details [21].
+
+In this work, we introduce a dynamics-aware imitation learning method which has stable, nonadversarial training, allowing us to achieve state-of-the-art performance on imitation learning benchmarks. Our key insight is that much of the difficulty with previous $\mathrm { I L }$ methods arises from the IRL-motivated representation of the $\mathrm { I L }$ problem as a min-max problem over reward and policy [17, 1].
+
+Table 1: A comparison of various algorithms for imitation learning. “Convergence Guarantees” refers to if a proof is given that the algorithm converges to the correct policy with sufficient data. We consider an algorithm “directly optimized” if it consists of an optimization algorithm (such as gradient descent) applied to the parameters of a single function   
+
+<table><tr><td colspan="2">Method</td><td>Reference</td><td>Dynamics Aware</td><td>Non- Adversarial Training</td><td>Guarantees</td><td>Convergence Non-restrictive Reward</td><td>Direct Optimizatior</td></tr><tr><td rowspan="7">nniai ASAF SQIL</td><td rowspan="7">Max Margin IRL Max Entropy IRL</td><td></td><td></td><td></td><td></td><td>×</td><td></td></tr><tr><td>[29, 1] [43]</td><td></td><td>√</td><td></td><td></td><td>×</td></tr><tr><td>[17, 10]</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>GAIL/AIRL [4]</td><td></td><td></td><td></td><td>×</td><td></td></tr><tr><td>[33]</td><td></td><td>√</td><td>×</td><td>×</td><td>√</td></tr><tr><td>Ours (Online)</td><td>√</td><td>√</td><td>√</td><td>√</td><td>√</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="7">Duiiii</td><td>MaxMargin IRL Max Likelihood IRL</td><td>[24, 20] g</td><td></td><td></td><td></td><td>× ×</td><td>× ×</td></tr><tr><td>Max Entropy IRL</td><td></td><td></td><td></td><td></td><td>×</td><td>×</td></tr><tr><td>ValueDICE</td><td></td><td></td><td></td><td></td><td>×</td><td>×</td></tr><tr><td>Behavioral Cloning</td><td></td><td></td><td></td><td></td><td>×</td><td>√</td></tr><tr><td>Regularized BC</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>EDM</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Ours (Offline)</td><td></td><td>√</td><td>√</td><td>√</td><td>√</td><td>√</td></tr></table>
+
+This introduces a requirement to separately model the reward and policy, and train these two functions jointly, often in an adversarial fashion. Drawing on connections between RL and energy-based models [13, 14], we propose learning a single model for the $Q$ -value. The $Q$ -value then implicitly defines both a reward and policy function. This turns a difficult min-max problem over policy and reward functions into a simpler minimization problem over a single function, the $Q$ -value. Since our problem has a one-to-one correspondence with the min-max problem studied in adversarial IL [17], we maintain the generality and guarantees of these previous approaches, resulting in a meaningful reward that may be used for inverse reinforcement learning. Furthermore, our method may be used to minimize a variety of statistical divergences between the expert and learned policy. We show that we recover several previously-described approaches as special cases of particular divergences, such as the regularized behavioural cloning of [30], and the conservative Q-learning of [23].
+
+In our experiments, we find that our method is performant even with very sparse data - surpassing prior methods using one expert demonstration in the completely offline setting - and can scale to complex image-based tasks like Atari reaching expert performance. Moreover, our learnt rewards are highly predictive of the original environment rewards.
+
+Concretely, our contributions are as follows:
+
+• We present a modified $Q$ -learning update rule for imitation learning that can be implemented on top of soft-Q learning or soft actor-critic (SAC) algorithms in fewer than 15 lines of code. • We introduce a simple framework to minimize a wide range of statistical distances: Integral Probability Metrics (IPMs) and f-divergences, between the expert and learned distributions. • We empirically show state-of-art results in a variety of imitation learning settings: online and offline IL. On the complex Atari suite, we outperform prior methods by $\mathbf { 3 - 7 x }$ while requiring $\mathbf { 3 x }$ less environment steps. • We characterize our learnt rewards and show a high positive correlation with the ground-truth rewards, justifying the use of our method for Inverse Reinforcement Learning.
+
+# 2 Background
+
+Preliminaries We consider environments represented as a Markov decision process (MDP), which is defined by a tuple $( S , \mathcal { A } , p _ { 0 } , \mathcal { P } , r , \gamma )$ . $s , A$ represent state and action spaces, $p _ { 0 }$ and $\mathcal { P } ( s ^ { \prime } | s , a )$ represent the initial state distribution and the dynamics, $r ( s , a )$ represents the reward function, and $\gamma \in ( 0 , 1 )$ represents the discount factor. $\mathbb { R } ^ { S \times A } = \{ x : \mathcal { S } \times \mathcal { A }  \mathbb { R } \}$ will denote the set of all functions in the state-action space and $\overline { { \mathbb { R } } }$ will denote the extended real numbers $\mathbb { R } \cup \{ \infty \}$ . Section 3 and 4 will work with finite state and action spaces $S$ and $A$ , but our algorithms and experiments later in the paper use continuous environments. $\Pi$ is the set of all stationary stochastic policies that take actions in $A$ given states in $S$ . We work in the $\gamma$ -discounted infinite horizon setting, and we will use an expectation with respect to a policy $\pi \in \Pi$ to denote an expectation with respect to the trajectory it generates: $\begin{array} { r } { \mathbb { E } _ { \pi } [ r ( s , a ) ] \triangleq \mathbb { E } [ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } r ( s _ { t } , a _ { t } ) ] } \end{array}$ , where $s _ { 0 } \sim p _ { 0 }$ , $a _ { t } \sim \pi ( \cdot | s _ { t } )$ , and $s _ { t + 1 } \sim \mathcal { P } ( \cdot | s _ { t } , a _ { t } )$ $t \geq 0$ $\pi \in \Pi$ , we define its occupancy me refer to the expert policy as e a $\rho _ { \pi } : \mathcal { S } \times \mathcal { A } \to \mathbb { R }$ $\begin{array} { r } { \rho _ { \pi } ( s , a ) \stackrel { \cdot } { = } \pi ( a | s ) \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } P \stackrel { \cdot } { s _ { t } } = \dot { s | \pi } ) } \end{array}$ $\pi _ { E }$ measure as $\rho _ { E }$ . In practice, $\pi _ { E }$ is unknown and we have access to a sampled dataset of demonstrations. For brevity, we refer to $\rho _ { \pi }$ as $\rho$ for a learnt policy in the paper.
+
+Soft $Q$ -functions For a reward $\boldsymbol { r } ~ \in ~ \mathbb { R } ^ { S \times A }$ and $\pi \in \Pi$ , the soft Bellman operator $B ^ { \pi }$ : $\mathbb { R } ^ { S \times A ^ { * } } \to \mathbb { R } ^ { S \times A }$ defined as $( \mathcal { B } ^ { \pi } Q ) ( s , a ) \ = \ r ( s , a ) + \gamma \mathbb { E } _ { s ^ { \prime } \sim P ( s , a ) } V ^ { \pi } ( s ^ { \prime } )$ with $\begin{array} { r l } { V ^ { \pi } ( s ) } & { { } = } \end{array}$ $\mathbb { E } _ { a \sim \pi ( \cdot | s ) } \left[ Q ( s , a ) - \log \pi ( a | s ) \right]$ . The soft Bellman operator is contractive [13] and defines a unique soft $Q$ -function for $r$ , given as $Q = B ^ { \pi } Q$ .
+
+Max Entropy Reinforcement Learning For a given reward function $\boldsymbol { r } ~ \in ~ \mathbb { R } ^ { S \times A }$ , maximum entropy RL [14, 5] aims to learn a policy that maximizes the expected cumulative discounted reward along with the entropy in each state: $\begin{array} { r } { \operatorname* { m a x } _ { \pi \in \Pi } \mathbb { E } _ { \pi } [ r ( s , a ) ] + H ( \pi ) } \end{array}$ . Where $H ( \pi ) \triangleq \mathbb { E } _ { \pi } [ - \log \pi ( a | s ) ]$ is the discounted causal entropy of the policy $\pi$ . The optimal policy satisfies [42, 5]:
+
+$$
+\pi ^ { * } ( a | s ) = \frac { 1 } { Z _ { s } } \exp { ( Q ( s , a ) ) } ,
+$$
+
+where $Q$ is the soft $Q$ -function and $Z _ { s }$ is the normalization factor given as $\begin{array} { r } { \sum _ { a ^ { \prime } } \exp \left( Q \left( s , a ^ { \prime } \right) \right) } \end{array}$ .
+
+$Q$ satisfies the soft-Bellman equation:
+
+$$
+Q ( s , a ) = r ( s , a ) + \gamma \mathbb { E } _ { s ^ { \prime } \sim \mathcal { P } ( \cdot \mid s , a ) } \Big [ \log \sum _ { a ^ { \prime } } \exp ( Q ( s ^ { \prime } , a ^ { \prime } ) ) \Big ]
+$$
+
+In continuous action spaces, $Z _ { s }$ becomes intractable and soft actor-critic methods like SAC [13] can be used to learn an explicit policy.
+
+Max Entropy Inverse Reinforcement Learning Given demonstrations sampled using the policy $\pi _ { E }$ , maximum entropy Inverse RL aims to recover the reward function in a family of functions $\mathcal { R }$ that rationalizes the expert behavior by solving the optimization problem: $\begin{array} { r } { \operatorname* { m a x } _ { r \in { \mathcal { R } } } \operatorname* { m i n } _ { \pi \in \Pi } \mathbb { E } _ { \pi _ { E } } [ r ( s , a ) ] - ( \mathbb { E } _ { \pi } [ r ( s , a ) ] + H ( \pi ) ) } \end{array}$ , where the expected reward of $\pi _ { E }$ is empirically approximated. It looks for a reward function that assigns high reward to the expert policy and a low reward to other policies, while searching for the best policy for the reward function in an inner loop.
+
+The Inverse RL objective can be reformulated in terms of its occupancy measure, and with a convex reward regularizer $\psi : \mathbb { R } ^ { S \times A }  \overline { { \mathbb { R } } }$ [17]
+
+$$
+\operatorname* { m a x } _ { r \in { \mathcal R } } \operatorname* { m i n } _ { \pi \in \Pi } L ( \pi , r ) = \mathbb { E } _ { \rho _ { E } } [ r ( s , a ) ] - \mathbb { E } _ { \rho } [ r ( s , a ) ] - H ( \pi ) - \psi ( r )
+$$
+
+In general, we can exchange the max-min resulting in an objective that minimizes the statistical distance parameterized by $\psi$ , between the expert and the policy [17]
+
+$$
+\operatorname* { m i n } _ { \pi \in \Pi } \operatorname* { m a x } _ { r \in \mathcal { R } } L ( \pi , r ) = \operatorname* { m i n } _ { \pi \in \Pi } d _ { \psi } ( \rho , \rho _ { E } ) - H ( \pi ) ,
+$$
+
+with $d _ { \psi } \triangleq \psi ^ { * } ( \rho _ { E } - \rho )$ , where $\psi ^ { * }$ is the convex conjugate of $\psi$ .
+
+# 3 Inverse soft Q-learning (IQ-Learn) Framework
+
+A naive solution to the IRL problem in (Eq. 3) involves (1) an outer loop learning rewards and (2) executing RL in an inner loop to find an optimal policy for them. However, we know that this optimal policy can be obtained analytically in terms of soft $Q$ -functions (Eq. 1). Interestingly, as we will show later, the rewards can also be represented in terms of $Q$ (Eq. 2). Together, these observations suggest it might be possible to directly solve the IRL problem by optimizing only over the $Q$ -function.
+
+To motivate the search of an imitation learning algorithm that depends only on the $Q$ -function, we characterize the space of $Q$ -functions and policies obtained using Inverse RL. We will study $\pi \in \Pi$ , $r \in \mathcal { R }$ and $Q$ -functions $Q \in \Omega$ where $\mathcal { R } \overset { \cdot } { = } \Omega = \mathbb { R } ^ { S \times A }$ . We assume $\Pi$ is convex, compact and that $\pi _ { E } \in \Pi ^ { 1 }$ . We define $\begin{array} { r } { V ^ { \pi } ( s ) = \mathbb { E } _ { a \sim \pi ( \cdot | s ) } \left[ Q ( s , a ) - \log \pi ( a | s ) \right] . } \end{array}$ .
+
+We start with analysis developed in [17]: The regularized IRL objective $L ( \pi , r )$ given by Eq. 3, is concave in the policy and convex in rewards. And has a unique saddle point where it is optimized.
+
+To characterize the $Q$ -functions it is useful to transform the optimization problem over rewards to a problem over $Q$ -functions. We can get a one-to-one correspondence between $r$ and $Q$ :
+
+Define the inverse soft bellman operator $\mathcal { T } ^ { \pi } : \mathbb { R } ^ { S \times A }  \mathbb { R } ^ { S \times A }$ such that
+
+$$
+( \mathcal T ^ { \pi } Q ) ( s , a ) = Q ( s , a ) - \gamma \mathbb { E } _ { s ^ { \prime } \sim P ( s , a ) } V ^ { \pi } ( s ^ { \prime } ) ,
+$$
+
+Lemma 3.1. The inverse soft bellman operator $\mathcal { T } ^ { \pi }$ is bijective, and for any $r _ { \mathrm { { \ell } } }$ $( T ^ { \pi } ) ^ { - 1 } r$ is the unique contraction of $B ^ { \pi }$ .
+
+The proof of this lemma is in Appendix A.1. For a policy $\pi$ , we are thus justified in changing between rewards and their corresponding soft-Q functions. We can freely transform functions from the reward-policy space: $\Pi \times \mathcal { R }$ to the $Q$ -policy space: $\Pi \times \Omega$ , giving us the lemma:
+
+Lemma 3.2. If $\begin{array} { r } { { ^ { \mathnormal { \cdot } } L } ( \pi , r ) = \mathbb { E } _ { \rho _ { E } } [ r ( s , a ) ] - \mathbb { E } _ { \rho } [ r ( s , a ) ] - H ( \pi ) - \psi ( r ) } \end{array}$ and $\mathcal { I } ( \pi , Q ) = \mathbb { E } _ { \rho _ { E } } [ ( \mathcal { T } ^ { \pi } Q ) ( s , a ) ] - \mathbb { E } _ { \rho } [ ( \mathcal { T } ^ { \pi } Q ) ( s , a ) ] - H ( \pi ) - \psi ( \mathcal { T } ^ { \pi } Q )$ ), then for all policies $\pi \in \Pi$ , $L ( \pi , r ) = \mathcal { I } ( \pi , ( \mathcal { T } ^ { \pi } ) ^ { - 1 } r )$ for all $r \in \mathcal { R }$ , and ${ \mathcal { I } } ( \pi , Q ) = L ( \pi , T ^ { \pi } Q ) ,$ , for all $Q \in \Omega$ .
+
+Lemma 3.1 and 3.2 allow us to adapt the Inverse RL objective $L ( \pi , r )$ to learning $Q$ through ${ \mathcal { I } } ( \pi , Q )$
+
+Simplifying our new objective (using Lemma A.3 in Appendix):
+
+$$
+\mathcal { I } ( \pi , Q ) = \mathbb { E } _ { s , a \sim \rho _ { E } } [ Q - \gamma \mathbb { E } _ { s ^ { \prime } \sim \mathcal { P } ( \cdot \vert s , a ) } V ^ { \pi } ( s ^ { \prime } ) ] - ( 1 - \gamma ) \mathbb { E } _ { s _ { 0 } \sim p _ { 0 } } [ V ^ { \pi } ( s _ { 0 } ) ] - \psi ( T ^ { \pi } Q ) ,
+$$
+
+We are now ready to study ${ \mathcal { I } } ( \pi , Q )$ , the Inverse RL optimization problem in the $Q$ -policy space. As the regularizer $\psi$ depends on both $Q$ and $\pi$ , a general analysis over all functions in $\mathbb { R } ^ { S \times \lambda }$ becomes too difficult. We restrict ourselves to regularizers induced by a convex function $g : \mathbb { R }  \overline { { \mathbb { R } } }$ such that
+
+$$
+\psi _ { g } ( r ) = \mathbb { E } _ { \rho _ { E } } [ g ( r ( s , a ) ) ]
+$$
+
+This allows us to simplify our analysis to the set of all real functions while retaining generality2. We further motivate this choice in Section 4.
+
+Proposition 3.3. In the $Q$ -policy space, there exists a unique saddle point $( \pi ^ { * } , Q ^ { * } )$ that optimizes $\mathcal { I }$ . i.e. $Q ^ { * } = \mathrm { a r g m a x } _ { Q \in \Omega } \operatorname* { m i n } _ { \pi \in \Pi } \mathcal { I } ( \pi , Q )$ and $\pi ^ { * } = \operatorname { a r g m i n } _ { \pi \in \Pi } \operatorname* { m a x } _ { Q \in \Omega } \mathcal { I } ( \pi , Q ) .$ . Furthermore, $\pi ^ { * }$ and $r ^ { * } = \tau ^ { \pi ^ { * } } Q ^ { * }$ are the solution to the Inverse RL objective $L ( \pi , r )$ .
+
+Thus we have, $\begin{array} { r } { \operatorname* { m a x } _ { Q \in \Omega } \operatorname* { m i n } _ { \pi \in \Pi } \mathcal { I } ( \pi , Q ) = \operatorname* { m a x } _ { r \in \mathcal { R } } \operatorname* { m i n } _ { \pi \in \Pi } L ( \pi , r ) . } \end{array}$
+
+This tells us, even after transforming to $Q$ -functions we have retained the saddle point property of the original IRL objective and optimizing ${ \mathcal { I } } ( \pi , Q )$ recovers this saddle point. In the $Q$ -policy space, we can get an additional property:
+
+Proposition 3.4. For a fixed $Q$ , $\mathrm { a r g m i n } _ { \pi \in \Pi } ~ { \mathcal { I } } ( \pi , Q )$ is the solution to max entropy $R L$ with rewards $r = \mathcal { T } ^ { \pi } Q$ . Thus, this forms a manifold in the $Q$ -policy space, that satisfies
+
+$$
+\pi _ { Q } ( a | s ) = \frac { 1 } { Z _ { s } } \exp ( Q ( s , a ) ) ,
+$$
+
+with normalization factor $\begin{array} { r } { Z _ { s } = \sum _ { a } \exp Q ( s , a ) } \end{array}$ and $\pi _ { Q }$ defined as the ⇡ corresponding to $Q$ .
+
+Proposition 3.3 and 3.4 are telling us that if we know $Q$ , then the inner optimization problem in terms of policy is trivial, and obtained in a closed form! Thus, we can recover an objective that only requires learning $Q$ :
+
+$$
+\operatorname* { m a x } _ { Q \in \Omega } \operatorname* { m i n } _ { \pi \in \Pi } \mathcal { I } ( \pi , Q ) = \operatorname* { m a x } _ { Q \in \Omega } \mathcal { I } \left( \pi _ { Q } , Q \right)
+$$
+
+Furthermore, we have:
+
+Proposition 3.5. Let ${ \mathcal { T } } ^ { * } ( Q ) = { \mathcal { T } } \left( \pi _ { Q } , Q \right)$ . Then $\mathcal { T } ^ { * }$ is concave in $Q$ .
+
+Thus, this new optimization objective is well-behaved and is maximized only at the saddle point.
+
+In Appendix C, we expand on our analysis and characterize the behavior for different choices of regularizer $\psi$ , while giving proofs of all our propositions. Figure 1 summarizes the properties for the IRL objective: there exists a optimal policy manifold depending on $Q$ , allowing optimization along it (using ${ \mathcal { T } } ^ { * }$ ) to converge to
+
+![](images/e1630f6d8490830dd99da2e72bf9290d94e43da2f7a16794cd79eaa494de062d.jpg)  
+Figure 1: Properties of IRL objective in reward-policy space and Q-policy space.
+
+the saddle point. We further present analysis of IL methods that learn $Q$ -functions like SQIL [33] and ValueDICE [22] and find subtle fallacies affecting their learning.
+
+Note that although the same analysis holds in the reward-policy space, the optimal policy manifold depends on $Q$ , which isn’t trivially known unlike when in the Q-policy space.
+
+# 4 Approach
+
+In this section, we develop our inverse soft-Q learning (IQ-Learn) algorithm, such that it recovers the optimal soft $Q$ -function for a MDP from a given expert distribution. We start by learning energy-based models for the policy similar to soft $Q$ -learning and later learn an explicit policy similar to actor-critic methods.
+
+# 4.1 General Inverse RL Objective
+
+For designing a practical algorithm using regularizers of the form $\psi _ { g }$ (from Eq. 6), we define $g$ using a concave function $\phi : \mathcal { R } _ { \psi } \to \mathbb { R }$ , such that $g ( x ) = { \left\{ \begin{array} { l l } { x - \phi ( x ) } & { { \mathrm { i f ~ } } x \in { \mathcal { R } } _ { \psi } } \\ { + \infty } & { { \mathrm { o t h e r w i s e } } } \end{array} \right. }$ with the rewards constrained in $R _ { \psi }$ .
+
+For this choice of $\psi$ , the Inverse RL objective $L ( \pi , r )$ takes the form of Eq. 4 with a distance measure:
+
+$$
+d _ { \psi } ( \rho , \rho _ { E } ) = \operatorname* { m a x } _ { r \in \mathcal { R } _ { \psi } } \mathbb { E } _ { \rho _ { E } } [ \phi ( r ( s , a ) ) ] - \mathbb { E } _ { \rho } [ r ( s , a ) ] ,
+$$
+
+This forms a general learning objective that allows the use of a wide-range of statistical distances including Integral Probability Metrics (IPMs) and f-divergences (see Appendix B). 3
+
+# 4.2 Choice of Statistical Distances
+
+While choosing a practical regularizer, it can be useful to obtain certain properties on the reward functions we recover. Some (natural) nice properties are: having rewards bounded in a range, learning smooth functions or enforcing a norm-penalty.
+
+In fact, we find these properties correspond to the Total Variation distance, the Wasserstein-1 distance and the $\chi ^ { 2 }$ -divergence respectively. The regularizers and the induced statistical distances are summarized in Table 2:
+
+Table 2: Enforced reward property, corresponding regularizer $\psi$ and statistical distance $( R _ { \operatorname* { m a x } } , K , \alpha \in \mathbb { R } ^ { + }$ )   
+
+<table><tr><td>Reward Property</td><td>4</td><td>d</td></tr><tr><td>Bound range Smoothness</td><td>=O if |rl ≤Rmax and+oo otherwise =Oif |lrllLip ≤Kand+oo otherwise</td><td>2Rmax ·TV(p,pE) K·Wi(p,pε)</td></tr></table>
+
+3 We recover IPMs when using identity $\phi$ and restricted reward family $\mathcal { R }$
+
+We find that these choice of regularizers4 work very well in our experiments. In Appendix B, we further give a table for the well known $f$ -divergences, the corresponding $\phi$ and the learnt reward estimators, along with a result ablation on using different divergences. Compared to $\chi ^ { 2 }$ , we find other $f$ -divergences like Jensen-Shannon result in similar performances but are not as readily interpretable.
+
+# 4.3 Inverse soft-Q update (Discrete control)
+
+Optimization along the optimal policy manifold gives the concave objective (Prop 3.5):
+
+with $\begin{array} { r } { V ^ { * } ( s ) = \log \sum _ { a } \exp Q ( s , a ) } \end{array}$
+
+For each $Q$ , we get a corresponding reward $\begin{array} { r } { r ( s , a ) = Q ( s , a ) - \gamma \mathbb { E } _ { s ^ { \prime } \sim \mathcal { P } ( \cdot \vert s , a ) } [ \log \sum _ { a ^ { \prime } } \exp Q \left( s ^ { \prime } , a ^ { \prime } \right) ] } \end{array}$ This correspondence is unique (Lemma A.1 in Appendix), and every update step can be seen as finding a better reward for IRL.
+
+Note that estimating $V ^ { \ast } ( s )$ exactly is only possible in discrete action spaces. Our objective forms a variant of soft-Q learning: to learn the optimal $Q$ -function given an expert distribution.
+
+# 4.4 Inverse soft actor-critic update (Continuous control)
+
+In continuous action spaces, it might not be possible to exactly obtain the optimal policy $\pi _ { Q }$ , which forms an energy-based model of the $Q$ -function, and we use an explicit policy $\pi$ to approximate $\pi _ { Q }$
+
+For any policy $\pi$ , we have a objective (from Eq. 5):
+
+$$
+\mathcal { I } ( \pi , Q ) = \mathbb { E } _ { \rho _ { E } } [ \phi ( Q - \gamma \mathbb { E } _ { s ^ { \prime } \sim \mathcal { P } ( \cdot \vert s , a ) } V ^ { \pi } ( s ^ { \prime } ) ) ] - ( 1 - \gamma ) \mathbb { E } _ { \rho _ { 0 } } [ V ^ { \pi } ( s _ { 0 } ) ]
+$$
+
+For a fixed $Q$ , soft actor-critic (SAC) update: $\displaystyle \operatorname* { m i n } _ { \pi } \mathbb { E } _ { s \sim \mathcal { D } , a \sim \pi ( \cdot | s ) } [ Q ( s , a ) - \log \pi ( a | s ) ]$ , brings $\pi$ closer to $\pi _ { Q }$ while always minimizing Eq. 10 (Lemma A.4 in Appendix). Here $\mathcal { D }$ is the distribution of previously sampled states, or a replay buffer.
+
+Thus, we obtain the modified actor-critic update rule to learn $Q$ -functions from the expert distribution:
+
+1. For a fixed $\pi$ , optimize $Q$ by maximizing ${ \mathcal { I } } ( \pi , Q )$ .   
+2. For a fixed $Q$ , apply SAC update to optimize $\pi$ towards $\pi _ { Q }$
+
+This differs from ValueDICE [22], where the actor is updated adverserially and the objective may not always converge (Appendix C).
+
+# 5 Practical Algorithm
+
+Pseudocode in Algorithm 1, shows our $Q$ -learning and actor-critic variants, with differences with conventional RL algorithms in red (we optimize $^ { - \mathcal { J } }$ to use gradient descent). We can implement our algorithm IQ-Learn in 15 lines of code on top of standard implementations of (soft) DQN [14] for discrete control or soft actor-critic (SAC) [13] for continuous control, with a change on the objective for the $Q$ -function. Default hyperparameters from [14, 13] work well, except for tuning the entropy regularization. Target networks were helpful for continuous control. We elaborate details in Appendix D.
+
+# 5.1 Training methodology
+
+Corollary 2.1 in Appendix A states $\mathbb { E } _ { ( s , a ) \sim \mu } [ V ^ { \pi } ( s ) - \gamma ] \mathbb { E } _ { s ^ { \prime } \sim \mathcal { P } ( \cdot \vert s , a ) } V ^ { \pi } ( s ^ { \prime } ) ] = ( 1 - \gamma ) \mathbb { E } _ { s \sim p _ { 0 } } [ V ^ { \pi } ( s ) ]$ , where $\mu$ is any policy’s occupancy. We use this to stabilize training instead of using Eq. 9 directly.
+
+Online: Instead of directly estimating $\mathbb { E } _ { p _ { 0 } } [ V ^ { \pi } ( s _ { 0 } ) ]$ in our algorithm, we can sample $( s , a , s ^ { \prime } )$ from a replay buffer and get a single-sample estimate $\bar { \mathbb { E } } _ { ( s , a , s ^ { \prime } ) \sim \mathrm { r e p l a y } } [ V ^ { \pi } ( s ) - \gamma V ^ { \pi } ( s ^ { \bar { \prime } } ) ]$ . This removes the issue where we are only optimizing $Q$ in the inital states resulting in overfitting of $V ^ { \pi } ( s _ { 0 } )$ , and improves the stability for convergence in our experiments. We find sampling half from the policy buffer and half from the expert distribution gives the best performances. Note that this is makes our learning online, requiring environment interactions.
+
+Offline: Although $\mathbb { E } _ { p _ { 0 } } [ V ^ { \pi } ( s _ { 0 } ) ]$ can be estimated offline we still observe an overfitting issue. Instead of requiring policy samples we use only expert samples to estimate $\overline { { { \mathbb { E } } } } _ { ( s , a , s ^ { \prime } ) \sim \mathrm { e x p e r t } } [ V ^ { \bar { \pi } } ( s ) ~ - ~$ $\gamma V ^ { \pi } ( s ^ { \prime } ) ]$ to sufficiently approximate the term. This methodology gives us state-of-art results for offline IL.
+
+# 5.2 Recovering rewards
+
+Instead of the conventional reward function $r ( s , a )$ on state and action pairs, our algorithm allows recovering rewards for each transition $( s , a , s ^ { \prime } )$ using the learnt $Q$ -values as follows:
+
+# Algorithm 1 Inverse soft Q-Learning (both variants)
+
+1: Initialize Q-function $Q _ { \theta }$ , and optionally a policy $\pi _ { \phi }$   
+2: for step $t$ in $\{ 1 . . . \mathrm { N } \}$ do   
+3: Train Q-function using objective from Equation 9: $\begin{array} { r } { \theta _ { t + 1 }  \theta _ { t } - \alpha _ { Q } \nabla _ { \theta } \big [ \qquad \big ] } \end{array}$ (Use $V ^ { * }$ for Q-learning and $V ^ { \pi _ { \phi } }$ for actor-critic)   
+4: (only with actor-critic) Improve policy $\pi _ { \phi }$ with SAC style actor update: $\begin{array} { r } { \underset { \bullet } { \phi } _ { t + 1 }  \phi _ { t } - \alpha _ { \pi } \nabla _ { \phi } \mathbb { E } _ { s \sim \mathcal { D } , a \sim \pi _ { \phi } ( \cdot | s ) } [ Q ( s , a ) - \log \pi _ { \phi } ( a | s ) ] } \end{array}$
+
+5: end for
+
+# Algorithm 2 Recover policy and reward
+
+1: Given trained Q-function $Q _ { \theta }$ , and optionally a trained policy $\pi _ { \phi }$   
+2: Recover policy $\pi$ : (Q-learning) $\textstyle \pi : = { \frac { 1 } { Z } } \exp Q _ { \theta }$ (actor-critic) $\pi : = \pi _ { \phi }$   
+3: For state s, action a and $\mathbf { s } ^ { \prime } \sim \mathcal { P } ( \cdot | \mathbf { s } , \mathbf { a } )$   
+4: Recover reward $r ( \mathbf { s } , \mathbf { a } , \mathbf { s } ^ { \prime } ) = Q _ { \theta } ( \mathbf { \bar { s } } , \mathbf { a } ) - \gamma V ^ { \pi } \left( \mathbf { s } ^ { \prime } \right)$
+
+$$
+r ( s , a , s ^ { \prime } ) = Q ( s , a ) - \gamma V ^ { \pi } \left( s ^ { \prime } \right)
+$$
+
+Now, $\begin{array} { r } { { \mathbb E } _ { s ^ { \prime } \sim \mathcal { P } ( \cdot \vert s , a ) } [ Q ( s , a ) - \gamma V ^ { \pi } \left( s ^ { \prime } \right) ] = Q ( s , a ) - \gamma { \mathbb E } _ { s ^ { \prime } \sim \mathcal { P } ( \cdot \vert s , a ) } [ V ^ { \pi } \left( s ^ { \prime } \right) ] = \mathcal { T } ^ { \pi } Q ( s , a ) } \end{array}$ . This is just the reward function $r ( s , a )$ we want. So by marginalizing over next-states, our expression correctly recovers the reward over state-actions. Thus, Eq. 11 gives the reward over transitions.
+
+Our rewards require $s ^ { \prime }$ which can be sampled from the environment, or by using a dynamics model.
+
+# 5.3 Implementation of Statistical Distances
+
+Implementing TV and $W _ { 1 }$ distances is fairly trivial and we give details in Appendix B. For the $\chi ^ { 2 }$ -divergence, we note that it corresponds to $\begin{array} { r } { \phi ( x ) = x - \frac { 1 } { 4 \alpha } \hat { x ^ { 2 } } } \end{array}$ . On substituting in Eq. 9, we get
+
+$$
+\begin{array} { r l } & { \underset { 2 \leq \infty } { \mathrm { a a x } } \mathbb { E } _ { \rho _ { \mathrm { E } } } [ ( Q ( s , a ) - \gamma \mathbb { E } _ { s ^ { \prime } \sim \mathcal { P } ( \cdot \vert s , a ) } V ^ { * } ( s ^ { \prime } ) ) ] - ( 1 - \gamma ) \mathbb { E } _ { p _ { 0 } } [ V ^ { * } ( s _ { 0 } ) ] - \frac { 1 } { 4 \alpha } \mathbb { E } _ { \rho _ { \mathrm { E } } } [ ( Q ( s , a ) - \gamma \mathbb { E } _ { s ^ { \prime } \sim \mathcal { P } ( \cdot \vert s , a ) } V ^ { * } ( s ^ { \prime } ) ) ^ { 2 } ] } \end{array}
+$$
+
+In a fully offline setting, this can be further simplified as (using the offline methodology in Sec 5.1):
+
+$$
+\operatorname* { m i n } _ { Q \in \Omega } - \mathbb { E } _ { \rho _ { E } } [ ( Q ( s , a ) - V ^ { * } ( s ) ) ] + \frac { 1 } { 4 \alpha } \mathbb { E } _ { \rho _ { E } } [ ( Q ( s , a ) - \gamma \mathbb { E } _ { s ^ { \prime } \sim \mathcal { P } ( \cdot \vert s , a ) } V ^ { * } ( s ^ { \prime } ) ) ^ { 2 } ]
+$$
+
+This is interestingly the same as the $Q$ -learning objective in CQL [23], an state-of-art method for offline RL (using 0 rewards), and shares similarities with regularized behavior cloning [33] 5.
+
+# 5.4 Learning state-only reward functions
+
+Previous works like AIRL [10] propose learning rewards that are only function of the state, and claim that these form of reward functions generalize between different MDPs. We find our method can predict state-only rewards by using the policy and expert state-marginals with a modification to Eq. 9:
+
+$$
+\operatorname* { m a x } _ { \mathcal { O } \in \Omega } \mathcal { I } ^ { * } ( Q ) = \mathbb { E } _ { s \sim \rho _ { E } ( s ) } [ \mathbb { E } _ { a \sim \pi ( \cdot | s ) } [ \phi ( Q ( s , a ) - \gamma \mathbb { E } _ { s ^ { \prime } \sim \mathcal { P } ( \cdot | s , a ) } V ^ { * } ( s ^ { \prime } ) ) ] ] - ( 1 - \gamma ) \mathbb { E } _ { p _ { 0 } } [ V ^ { * } ( s _ { 0 } ) ]
+$$
+
+Interestingly, our objective no longer depends on the the expert actions $\pi _ { E }$ and can be used for $\mathrm { I L }$ using only observations. For the sake of brevity, we expand on this in Section 1 in Appendix A.
+
+# 6 Related Work
+
+Classical IL: Imitation learning has a long history, with early works using supervised learning to match a policy’s actions to those of the expert [15, 35]. A significant advance was made with the formulation of $\mathrm { I L }$ as the composition of RL and IRL [29, 1, 43], recovering the expert’s policy by inferring the expert’s reward function, then finding the policy which maximizes reward under this reward function. These early approaches required a hand-designed featurization of the MDP, limiting their applicability to complex MDPs. In this setting, early approaches [9, 31] noted a formal equivalence between IRL and IL using an inverse Bellman operator similar to our own.
+
+Online IL: More recent work aims to leverage the power of modern machine learning approaches to learn good featurizations and extend IL to complex settings. Recent work generally falls into one of two settings: online or offline. In the online setting, the IL algorithm is able to interact with the environment to obtain dynamics information. GAIL [17] takes the nested RL/IRL formulation of earlier work , optimizing over all reward functions with a convex regularizer. This results in the objective in Eq. (3), with a max-min adversarial problem similar to a GAN [11]. A variety of further work has built on this adversarial approach [21, 10, 3]. A separate line of work aims to simplify the problem in Eq. (3) by using a fixed $r$ or $\pi$ . In SQIL [33], $r$ is chosen to be the 1-0 indicator on the expert demonstrations, while ASAF [4] takes the GAN approach and uses a discriminator (with role similar to $r$ ) of fixed form, consisting of a ratio of expert and learner densities. AdRIL [38] is a recent extension of SQIL, additionally assigning decaying negative reward to previous policy rollouts.
+
+Offline IL: In the offline setting, the learner has no access to the environment. The simple behavioural cloning (BC) [34] approach is offline, but doesn’t use any dynamics information. ValueDICE [22] is a dynamics-aware offline approach with an objective somewhat similar to ours, motivated from minimization of a variational representation of the KL-divergence between expert and learner policies. ValueDICE requires adversarial optimization to learn the policy and Q-functions, with a biased gradient estimator for training. We show a way to recover a unbiased gradient estimate for the KL-divergence in Appendix C. The O-NAIL algorithm [2] builds on ValueDICE and combines with an SAC update to obtain a method that is similar to our algorithm described in section 4.4, with the specific choice of reverse KL-divergence as the relevant statistical distance. The EDM method [19] incorporates dynamics via learning an explicit energy based model for the expert state occupancy, although some theoretical details have been called into question (see [37] for details). The recent AVRIL approach [6] uses a variational method to solve a probabilistic formulation of IL, finding a posterior distribution over $r$ and $\pi$ . Illustrating the potential benefits of alternative distances for $\mathrm { I L }$ , the PWIL [7] algorithm gives a non-adversarial procedure to minimize the Wasserstein distance between expert and learned occupancies. The approach is specific to the primal form of the $\mathcal { W } _ { 1 }$ -distance, while our method (when used with the Wasserstein distance) targets the dual form.
+
+# 7 Experiments
+
+# 7.1 Experimental Setup
+
+We compare IQ-Learn ("IQ") to prior work on a diverse collection of RL tasks and environments - ranging from low-dimensional control tasks: CartPole, Acrobot, LunarLander - to more challenging continuous control MuJoCo tasks: HalfCheetah, Hopper, Walker and Ant. Furthermore, we test on the visually challenging Atari Suite with high-dimensional image inputs. We compare on offline IL - with no access to the the environment while training, and online IL - with environment access. We show results on $W _ { 1 }$ and $\chi ^ { 2 }$ as our statistical distances, as we found them more effective than TV distance. In all cases, we train until convergence and average over multiple seeds. Hyperparameter settings and training details are detailed in Appendix D.
+
+# 7.2 Benchmarks
+
+Offline $\mathbf { I L }$ We compare to the state-of-art IL methods EDM and AVRIL, following the same experimental setting as [6]. Furthermore, we compare with ValueDICE which also learns Q-functions, albeit with drawbacks such as adversarial optimization. We also experimented with SQIL, but found that it was not competitive in the offline setting. Finally, we utilize BC as an additional IL baseline.
+
+Online IL We use MuJoCo and Atari environments and compare against state-of-art online IL methods: ValueDICE, SQIL and GAIL. We only show results on $\chi ^ { 2 }$ as $W _ { 1 }$ was harder to stabilize on complex environments6. Using target updates stabilizes the $Q$ -learning on MuJoCo. For brevity, further online IL results are shown in the Appendix D.
+
+![](images/2991437dc0b464a8ab137fb0cb4318c35069cc219e9829f40272ad9e62707a2f.jpg)  
+Figure 2: Offline IL results. We plot the average environment returns vs the number of expert trajectories.
+
+Offline IL We present results on the three offline control tasks in Figure 2. On all tasks, IQ strongly outperforms prior works we compare to in performance and sample efficiency. Using just one expert trajectory, we achieve expert performance on Acrobot and reach near expert on Cartpole.
+
+Mujoco Control We present our results on the MuJoCo tasks using a single expert demo in Table 3. IQ achieves expert-level performance in all the tasks while outperforming prior methods like ValueDICE and GAIL. We did not find SQIL competitive in this setting, and skip it for brevity.
+
+Table 3: Mujoco Results. We show our performance on MuJoCo control tasks using a single expert trajectory.   
+
+<table><tr><td>Task</td><td>GAIL</td><td>DAC</td><td>ValueDICE</td><td>IQ (Ours)</td><td>Expert</td></tr><tr><td>Hopper</td><td>3252.5</td><td>3305.1</td><td>3312.1</td><td>3546.4</td><td>3532.7</td></tr><tr><td>Half-Cheetah</td><td>3080.0</td><td>4080.6</td><td>3835.6</td><td>5076.6</td><td>5098.3</td></tr><tr><td>Walker</td><td>4013.7</td><td>4107.9</td><td>3842.6</td><td>5134.0</td><td>5274.5</td></tr><tr><td>Ant</td><td>2299.1</td><td>1437.5</td><td>1806.3</td><td>4362.9</td><td>4700.0</td></tr><tr><td>Humanoid</td><td>232.6</td><td>380.5</td><td>644.5</td><td>5227.1</td><td>5312.8</td></tr></table>
+
+Atari We present our results on
+
+Atari using 20 expert demos in Figure 3. We reach expert performance on Space Invaders while being near expert on Pong and Breakout. Compared to prior methods like SQIL, IQ obtains $\mathbf { 3 - 7 x }$ normalized score7 and converges in ${ \sim } 3 0 0 \mathrm { k }$ steps, being 3x faster compared to Q-learning based RL methods that take more than 1M steps to converge. Other popular methods like GAIL and ValueDICE perform near random even with 1M env steps.
+
+![](images/ab324e15c2691ab2c4c5c8daf067b32caf4f046d5a1756f3a015e4acf22e3a89.jpg)  
+Figure 3: Atari Results. We show the returns vs the number of env steps. (Averaged over 5 seeds)
+
+# 7.4 Recovered Rewards
+
+IQ has the added benefit of recovering rewards and can be used for IRL. On Hopper task, our learned rewards have a Pearson correlation of 0.99 with the true rewards. In Figure 4, we visualize our recovered rewards in a simple grid environment. We elaborate details in Appendix D.
+
+# 8 Discussion and Outlook
+
+We present a new principled framework for learning soft- $Q$ functions for IL and recovering the optimal policy and the reward, building on past works in IRL [43]. Our algorithm IQ-Learn outperforms prior methods with very sparse expert data and scales to complex image-based environments. We also recover rewards highly correlated with actual rewards. It has applications in autonomous driving and complex decision-making, but proper considerations need to be taken into account to ensure safety and reduce uncertainty, before any deployment. Finally, human or expert data can have errors that can propagate. A limitation of our method is that our recovered rewards depend on the environment dynamics, preventing trivial use on reward transfer settings. One direction of future work could be to learn a reward model from the trained soft- $Q$ model to make the rewards explicit.
+
+![](images/2e92c4cecb167956b2f3f30af2237262fa266f72969b8f686abf4df516850f9f.jpg)  
+Figure 4: Reward Visualization. We use a discrete GridWorld environment with 5 possible actions: up, down, left, right, stay. Agent starts in a random state. (With 30 expert demos)
+
+# 9 Acknowledgements
+
+We thank Kuno Kim and John Schulman for helpful discussions. We also thank Ian Goodfellow as some initial motivations for this work were developed under an internship with him.
+
+# 10 Funding Transparency
+
+This research was supported in part by NSF (#1651565, #1522054, #1733686), ONR (N00014-19-1- 2145), AFOSR (FA9550-19-1-0024) and FLI.
+
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+
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+        "text": "IQ-Learn: Inverse soft-Q Learning for Imitation ",
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+        "text": "Divyansh Garg Shuvam Chakraborty Chris Cundy Jiaming Song Stefano Ermon Stanford University   \n{divgarg, shuvamc, cundy, tsong, ermon}@stanford.edu ",
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+        "text": "Abstract ",
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+        "text": "In many sequential decision-making problems (e.g., robotics control, game playing, sequential prediction), human or expert data is available containing useful information about the task. However, imitation learning (IL) from a small amount of expert data can be challenging in high-dimensional environments with complex dynamics. Behavioral cloning is a simple method that is widely used due to its simplicity of implementation and stable convergence but doesn’t utilize any information involving the environment’s dynamics. Many existing methods that exploit dynamics information are difficult to train in practice due to an adversarial optimization process over reward and policy approximators or biased, high variance gradient estimators. We introduce a method for dynamics-aware IL which avoids adversarial training by learning a single Q-function, implicitly representing both reward and policy. On standard benchmarks, the implicitly learned rewards show a high positive correlation with the ground-truth rewards, illustrating our method can also be used for inverse reinforcement learning (IRL). Our method, Inverse soft-Q learning (IQ-Learn) obtains state-of-the-art results in offline and online imitation learning settings, significantly outperforming existing methods both in the number of required environment interactions and scalability in high-dimensional spaces, often by more than $3 \\mathbf { x }$ . ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+        "text": "Imitation of an expert has long been recognized as a powerful approach for sequential decisionmaking [29, 1], with applications as diverse as healthcare [39], autonomous driving [41], and playing complex strategic games [8]. In the imitation learning (IL) setting, we are given a set of expert trajectories, with the goal of learning a policy which induces behavior similar to the expert’s. The learner has no access to the reward, and no explicit knowledge of the dynamics. ",
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+        "text": "The simple behavioural cloning [34] approach simply maximizes the probability of the expert’s actions under the learned policy, approaching the IL problem as a supervised learning problem. While this can work well in simple environments and with large quantities of data, it ignores the sequential nature of the decision-making problem, and small errors can quickly compound when the learned policy departs from the states observed under the expert. A natural way of introducing the environment dynamics is by framing the IL problem as an Inverse RL (IRL) problem, aiming to learn a reward function under which the expert’s trajectory is optimal, and from which the learned imitation policy can be trained [1]. This framing has inspired several approaches which use rewards either explicitly or implicitly to incorporate dynamics while learning an imitation policy [17, 10, 33, 22]. However, these dynamics-aware methods are typically hard to put into practice due to unstable learning which can be sensitive to hyperparameter choice or minor implementation details [21]. ",
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+        "text": "In this work, we introduce a dynamics-aware imitation learning method which has stable, nonadversarial training, allowing us to achieve state-of-the-art performance on imitation learning benchmarks. Our key insight is that much of the difficulty with previous $\\mathrm { I L }$ methods arises from the IRL-motivated representation of the $\\mathrm { I L }$ problem as a min-max problem over reward and policy [17, 1]. ",
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+        "table_caption": [
+            "Table 1: A comparison of various algorithms for imitation learning. “Convergence Guarantees” refers to if a proof is given that the algorithm converges to the correct policy with sufficient data. We consider an algorithm “directly optimized” if it consists of an optimization algorithm (such as gradient descent) applied to the parameters of a single function "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Method</td><td>Reference</td><td>Dynamics Aware</td><td>Non- Adversarial Training</td><td>Guarantees</td><td>Convergence Non-restrictive Reward</td><td>Direct Optimizatior</td></tr><tr><td rowspan=\"7\">nniai ASAF SQIL</td><td rowspan=\"7\">Max Margin IRL Max Entropy IRL</td><td></td><td></td><td></td><td></td><td>×</td><td></td></tr><tr><td>[29, 1] [43]</td><td></td><td>√</td><td></td><td></td><td>×</td></tr><tr><td>[17, 10]</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>GAIL/AIRL [4]</td><td></td><td></td><td></td><td>×</td><td></td></tr><tr><td>[33]</td><td></td><td>√</td><td>×</td><td>×</td><td>√</td></tr><tr><td>Ours (Online)</td><td>√</td><td>√</td><td>√</td><td>√</td><td>√</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=\"7\">Duiiii</td><td>MaxMargin IRL Max Likelihood IRL</td><td>[24, 20] g</td><td></td><td></td><td></td><td>× ×</td><td>× ×</td></tr><tr><td>Max Entropy IRL</td><td></td><td></td><td></td><td></td><td>×</td><td>×</td></tr><tr><td>ValueDICE</td><td></td><td></td><td></td><td></td><td>×</td><td>×</td></tr><tr><td>Behavioral Cloning</td><td></td><td></td><td></td><td></td><td>×</td><td>√</td></tr><tr><td>Regularized BC</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>EDM</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Ours (Offline)</td><td></td><td>√</td><td>√</td><td>√</td><td>√</td><td>√</td></tr></table>",
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+        "text": "This introduces a requirement to separately model the reward and policy, and train these two functions jointly, often in an adversarial fashion. Drawing on connections between RL and energy-based models [13, 14], we propose learning a single model for the $Q$ -value. The $Q$ -value then implicitly defines both a reward and policy function. This turns a difficult min-max problem over policy and reward functions into a simpler minimization problem over a single function, the $Q$ -value. Since our problem has a one-to-one correspondence with the min-max problem studied in adversarial IL [17], we maintain the generality and guarantees of these previous approaches, resulting in a meaningful reward that may be used for inverse reinforcement learning. Furthermore, our method may be used to minimize a variety of statistical divergences between the expert and learned policy. We show that we recover several previously-described approaches as special cases of particular divergences, such as the regularized behavioural cloning of [30], and the conservative Q-learning of [23]. ",
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+        "type": "text",
+        "text": "In our experiments, we find that our method is performant even with very sparse data - surpassing prior methods using one expert demonstration in the completely offline setting - and can scale to complex image-based tasks like Atari reaching expert performance. Moreover, our learnt rewards are highly predictive of the original environment rewards. ",
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+    {
+        "type": "text",
+        "text": "Concretely, our contributions are as follows: ",
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+        "text": "• We present a modified $Q$ -learning update rule for imitation learning that can be implemented on top of soft-Q learning or soft actor-critic (SAC) algorithms in fewer than 15 lines of code. • We introduce a simple framework to minimize a wide range of statistical distances: Integral Probability Metrics (IPMs) and f-divergences, between the expert and learned distributions. • We empirically show state-of-art results in a variety of imitation learning settings: online and offline IL. On the complex Atari suite, we outperform prior methods by $\\mathbf { 3 - 7 x }$ while requiring $\\mathbf { 3 x }$ less environment steps. • We characterize our learnt rewards and show a high positive correlation with the ground-truth rewards, justifying the use of our method for Inverse Reinforcement Learning. ",
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+    {
+        "type": "text",
+        "text": "2 Background ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Preliminaries We consider environments represented as a Markov decision process (MDP), which is defined by a tuple $( S , \\mathcal { A } , p _ { 0 } , \\mathcal { P } , r , \\gamma )$ . $s , A$ represent state and action spaces, $p _ { 0 }$ and $\\mathcal { P } ( s ^ { \\prime } | s , a )$ represent the initial state distribution and the dynamics, $r ( s , a )$ represents the reward function, and $\\gamma \\in ( 0 , 1 )$ represents the discount factor. $\\mathbb { R } ^ { S \\times A } = \\{ x : \\mathcal { S } \\times \\mathcal { A }  \\mathbb { R } \\}$ will denote the set of all functions in the state-action space and $\\overline { { \\mathbb { R } } }$ will denote the extended real numbers $\\mathbb { R } \\cup \\{ \\infty \\}$ . Section 3 and 4 will work with finite state and action spaces $S$ and $A$ , but our algorithms and experiments later in the paper use continuous environments. $\\Pi$ is the set of all stationary stochastic policies that take actions in $A$ given states in $S$ . We work in the $\\gamma$ -discounted infinite horizon setting, and we will use an expectation with respect to a policy $\\pi \\in \\Pi$ to denote an expectation with respect to the trajectory it generates: $\\begin{array} { r } { \\mathbb { E } _ { \\pi } [ r ( s , a ) ] \\triangleq \\mathbb { E } [ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } r ( s _ { t } , a _ { t } ) ] } \\end{array}$ , where $s _ { 0 } \\sim p _ { 0 }$ , $a _ { t } \\sim \\pi ( \\cdot | s _ { t } )$ , and $s _ { t + 1 } \\sim \\mathcal { P } ( \\cdot | s _ { t } , a _ { t } )$ $t \\geq 0$ $\\pi \\in \\Pi$ , we define its occupancy me refer to the expert policy as e a $\\rho _ { \\pi } : \\mathcal { S } \\times \\mathcal { A } \\to \\mathbb { R }$ $\\begin{array} { r } { \\rho _ { \\pi } ( s , a ) \\stackrel { \\cdot } { = } \\pi ( a | s ) \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } P \\stackrel { \\cdot } { s _ { t } } = \\dot { s | \\pi } ) } \\end{array}$ $\\pi _ { E }$ measure as $\\rho _ { E }$ . In practice, $\\pi _ { E }$ is unknown and we have access to a sampled dataset of demonstrations. For brevity, we refer to $\\rho _ { \\pi }$ as $\\rho$ for a learnt policy in the paper. ",
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+        "type": "text",
+        "text": "Soft $Q$ -functions For a reward $\\boldsymbol { r } ~ \\in ~ \\mathbb { R } ^ { S \\times A }$ and $\\pi \\in \\Pi$ , the soft Bellman operator $B ^ { \\pi }$ : $\\mathbb { R } ^ { S \\times A ^ { * } } \\to \\mathbb { R } ^ { S \\times A }$ defined as $( \\mathcal { B } ^ { \\pi } Q ) ( s , a ) \\ = \\ r ( s , a ) + \\gamma \\mathbb { E } _ { s ^ { \\prime } \\sim P ( s , a ) } V ^ { \\pi } ( s ^ { \\prime } )$ with $\\begin{array} { r l } { V ^ { \\pi } ( s ) } & { { } = } \\end{array}$ $\\mathbb { E } _ { a \\sim \\pi ( \\cdot | s ) } \\left[ Q ( s , a ) - \\log \\pi ( a | s ) \\right]$ . The soft Bellman operator is contractive [13] and defines a unique soft $Q$ -function for $r$ , given as $Q = B ^ { \\pi } Q$ . ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Max Entropy Reinforcement Learning For a given reward function $\\boldsymbol { r } ~ \\in ~ \\mathbb { R } ^ { S \\times A }$ , maximum entropy RL [14, 5] aims to learn a policy that maximizes the expected cumulative discounted reward along with the entropy in each state: $\\begin{array} { r } { \\operatorname* { m a x } _ { \\pi \\in \\Pi } \\mathbb { E } _ { \\pi } [ r ( s , a ) ] + H ( \\pi ) } \\end{array}$ . Where $H ( \\pi ) \\triangleq \\mathbb { E } _ { \\pi } [ - \\log \\pi ( a | s ) ]$ is the discounted causal entropy of the policy $\\pi$ . The optimal policy satisfies [42, 5]: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/436fa8b9f2bf7c4f92c9d15f36bf301c083094a1157bc2b86870b9fadec7e1a7.jpg",
+        "text": "$$\n\\pi ^ { * } ( a | s ) = \\frac { 1 } { Z _ { s } } \\exp { ( Q ( s , a ) ) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $Q$ is the soft $Q$ -function and $Z _ { s }$ is the normalization factor given as $\\begin{array} { r } { \\sum _ { a ^ { \\prime } } \\exp \\left( Q \\left( s , a ^ { \\prime } \\right) \\right) } \\end{array}$ . ",
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+    {
+        "type": "text",
+        "text": "$Q$ satisfies the soft-Bellman equation: ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/4d698fe88dcc3263f08b652e88cddc7c1e6403505fe99425c0964e5d010690b9.jpg",
+        "text": "$$\nQ ( s , a ) = r ( s , a ) + \\gamma \\mathbb { E } _ { s ^ { \\prime } \\sim \\mathcal { P } ( \\cdot \\mid s , a ) } \\Big [ \\log \\sum _ { a ^ { \\prime } } \\exp ( Q ( s ^ { \\prime } , a ^ { \\prime } ) ) \\Big ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "In continuous action spaces, $Z _ { s }$ becomes intractable and soft actor-critic methods like SAC [13] can be used to learn an explicit policy. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Max Entropy Inverse Reinforcement Learning Given demonstrations sampled using the policy $\\pi _ { E }$ , maximum entropy Inverse RL aims to recover the reward function in a family of functions $\\mathcal { R }$ that rationalizes the expert behavior by solving the optimization problem: $\\begin{array} { r } { \\operatorname* { m a x } _ { r \\in { \\mathcal { R } } } \\operatorname* { m i n } _ { \\pi \\in \\Pi } \\mathbb { E } _ { \\pi _ { E } } [ r ( s , a ) ] - ( \\mathbb { E } _ { \\pi } [ r ( s , a ) ] + H ( \\pi ) ) } \\end{array}$ , where the expected reward of $\\pi _ { E }$ is empirically approximated. It looks for a reward function that assigns high reward to the expert policy and a low reward to other policies, while searching for the best policy for the reward function in an inner loop. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "The Inverse RL objective can be reformulated in terms of its occupancy measure, and with a convex reward regularizer $\\psi : \\mathbb { R } ^ { S \\times A }  \\overline { { \\mathbb { R } } }$ [17] ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/41a9a807d65e46285b946fe9f7d174b7fc16929dbee1d297295a298b65013d94.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { r \\in { \\mathcal R } } \\operatorname* { m i n } _ { \\pi \\in \\Pi } L ( \\pi , r ) = \\mathbb { E } _ { \\rho _ { E } } [ r ( s , a ) ] - \\mathbb { E } _ { \\rho } [ r ( s , a ) ] - H ( \\pi ) - \\psi ( r )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In general, we can exchange the max-min resulting in an objective that minimizes the statistical distance parameterized by $\\psi$ , between the expert and the policy [17] ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/61e65f046fc0a31ddde640fe7b28a5f746b7d2e5fa25b312d05b1c9cd002b640.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\pi \\in \\Pi } \\operatorname* { m a x } _ { r \\in \\mathcal { R } } L ( \\pi , r ) = \\operatorname* { m i n } _ { \\pi \\in \\Pi } d _ { \\psi } ( \\rho , \\rho _ { E } ) - H ( \\pi ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "with $d _ { \\psi } \\triangleq \\psi ^ { * } ( \\rho _ { E } - \\rho )$ , where $\\psi ^ { * }$ is the convex conjugate of $\\psi$ . ",
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+    {
+        "type": "text",
+        "text": "3 Inverse soft Q-learning (IQ-Learn) Framework ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A naive solution to the IRL problem in (Eq. 3) involves (1) an outer loop learning rewards and (2) executing RL in an inner loop to find an optimal policy for them. However, we know that this optimal policy can be obtained analytically in terms of soft $Q$ -functions (Eq. 1). Interestingly, as we will show later, the rewards can also be represented in terms of $Q$ (Eq. 2). Together, these observations suggest it might be possible to directly solve the IRL problem by optimizing only over the $Q$ -function. ",
+        "bbox": [
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "To motivate the search of an imitation learning algorithm that depends only on the $Q$ -function, we characterize the space of $Q$ -functions and policies obtained using Inverse RL. We will study $\\pi \\in \\Pi$ , $r \\in \\mathcal { R }$ and $Q$ -functions $Q \\in \\Omega$ where $\\mathcal { R } \\overset { \\cdot } { = } \\Omega = \\mathbb { R } ^ { S \\times A }$ . We assume $\\Pi$ is convex, compact and that $\\pi _ { E } \\in \\Pi ^ { 1 }$ . We define $\\begin{array} { r } { V ^ { \\pi } ( s ) = \\mathbb { E } _ { a \\sim \\pi ( \\cdot | s ) } \\left[ Q ( s , a ) - \\log \\pi ( a | s ) \\right] . } \\end{array}$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We start with analysis developed in [17]: The regularized IRL objective $L ( \\pi , r )$ given by Eq. 3, is concave in the policy and convex in rewards. And has a unique saddle point where it is optimized. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To characterize the $Q$ -functions it is useful to transform the optimization problem over rewards to a problem over $Q$ -functions. We can get a one-to-one correspondence between $r$ and $Q$ : ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Define the inverse soft bellman operator $\\mathcal { T } ^ { \\pi } : \\mathbb { R } ^ { S \\times A }  \\mathbb { R } ^ { S \\times A }$ such that ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/344a04514349fe5ac32c0ac551326a7e8957e3a491427b2a95c47094c3cf820c.jpg",
+        "text": "$$\n( \\mathcal T ^ { \\pi } Q ) ( s , a ) = Q ( s , a ) - \\gamma \\mathbb { E } _ { s ^ { \\prime } \\sim P ( s , a ) } V ^ { \\pi } ( s ^ { \\prime } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Lemma 3.1. The inverse soft bellman operator $\\mathcal { T } ^ { \\pi }$ is bijective, and for any $r _ { \\mathrm { { \\ell } } }$ $( T ^ { \\pi } ) ^ { - 1 } r$ is the unique contraction of $B ^ { \\pi }$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The proof of this lemma is in Appendix A.1. For a policy $\\pi$ , we are thus justified in changing between rewards and their corresponding soft-Q functions. We can freely transform functions from the reward-policy space: $\\Pi \\times \\mathcal { R }$ to the $Q$ -policy space: $\\Pi \\times \\Omega$ , giving us the lemma: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Lemma 3.2. If $\\begin{array} { r } { { ^ { \\mathnormal { \\cdot } } L } ( \\pi , r ) = \\mathbb { E } _ { \\rho _ { E } } [ r ( s , a ) ] - \\mathbb { E } _ { \\rho } [ r ( s , a ) ] - H ( \\pi ) - \\psi ( r ) } \\end{array}$ and $\\mathcal { I } ( \\pi , Q ) = \\mathbb { E } _ { \\rho _ { E } } [ ( \\mathcal { T } ^ { \\pi } Q ) ( s , a ) ] - \\mathbb { E } _ { \\rho } [ ( \\mathcal { T } ^ { \\pi } Q ) ( s , a ) ] - H ( \\pi ) - \\psi ( \\mathcal { T } ^ { \\pi } Q )$ ), then for all policies $\\pi \\in \\Pi$ , $L ( \\pi , r ) = \\mathcal { I } ( \\pi , ( \\mathcal { T } ^ { \\pi } ) ^ { - 1 } r )$ for all $r \\in \\mathcal { R }$ , and ${ \\mathcal { I } } ( \\pi , Q ) = L ( \\pi , T ^ { \\pi } Q ) ,$ , for all $Q \\in \\Omega$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Lemma 3.1 and 3.2 allow us to adapt the Inverse RL objective $L ( \\pi , r )$ to learning $Q$ through ${ \\mathcal { I } } ( \\pi , Q )$ ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Simplifying our new objective (using Lemma A.3 in Appendix): ",
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/174fd559c02e34744fbf48165c74dbc19fcb86afa889e8f0f647bbd0628927c8.jpg",
+        "text": "$$\n\\mathcal { I } ( \\pi , Q ) = \\mathbb { E } _ { s , a \\sim \\rho _ { E } } [ Q - \\gamma \\mathbb { E } _ { s ^ { \\prime } \\sim \\mathcal { P } ( \\cdot \\vert s , a ) } V ^ { \\pi } ( s ^ { \\prime } ) ] - ( 1 - \\gamma ) \\mathbb { E } _ { s _ { 0 } \\sim p _ { 0 } } [ V ^ { \\pi } ( s _ { 0 } ) ] - \\psi ( T ^ { \\pi } Q ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We are now ready to study ${ \\mathcal { I } } ( \\pi , Q )$ , the Inverse RL optimization problem in the $Q$ -policy space. As the regularizer $\\psi$ depends on both $Q$ and $\\pi$ , a general analysis over all functions in $\\mathbb { R } ^ { S \\times \\lambda }$ becomes too difficult. We restrict ourselves to regularizers induced by a convex function $g : \\mathbb { R }  \\overline { { \\mathbb { R } } }$ such that ",
+        "bbox": [
+            173,
+            493,
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+            537
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/edb54dbc2d5233a1d446f9799113f9e116aa5260437d918a4c7a412e76768ee3.jpg",
+        "text": "$$\n\\psi _ { g } ( r ) = \\mathbb { E } _ { \\rho _ { E } } [ g ( r ( s , a ) ) ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            415,
+            540,
+            581,
+            558
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This allows us to simplify our analysis to the set of all real functions while retaining generality2. We further motivate this choice in Section 4. ",
+        "bbox": [
+            174,
+            560,
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+            588
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 3.3. In the $Q$ -policy space, there exists a unique saddle point $( \\pi ^ { * } , Q ^ { * } )$ that optimizes $\\mathcal { I }$ . i.e. $Q ^ { * } = \\mathrm { a r g m a x } _ { Q \\in \\Omega } \\operatorname* { m i n } _ { \\pi \\in \\Pi } \\mathcal { I } ( \\pi , Q )$ and $\\pi ^ { * } = \\operatorname { a r g m i n } _ { \\pi \\in \\Pi } \\operatorname* { m a x } _ { Q \\in \\Omega } \\mathcal { I } ( \\pi , Q ) .$ . Furthermore, $\\pi ^ { * }$ and $r ^ { * } = \\tau ^ { \\pi ^ { * } } Q ^ { * }$ are the solution to the Inverse RL objective $L ( \\pi , r )$ . ",
+        "bbox": [
+            173,
+            590,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Thus we have, $\\begin{array} { r } { \\operatorname* { m a x } _ { Q \\in \\Omega } \\operatorname* { m i n } _ { \\pi \\in \\Pi } \\mathcal { I } ( \\pi , Q ) = \\operatorname* { m a x } _ { r \\in \\mathcal { R } } \\operatorname* { m i n } _ { \\pi \\in \\Pi } L ( \\pi , r ) . } \\end{array}$ ",
+        "bbox": [
+            176,
+            645,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This tells us, even after transforming to $Q$ -functions we have retained the saddle point property of the original IRL objective and optimizing ${ \\mathcal { I } } ( \\pi , Q )$ recovers this saddle point. In the $Q$ -policy space, we can get an additional property: ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 3.4. For a fixed $Q$ , $\\mathrm { a r g m i n } _ { \\pi \\in \\Pi } ~ { \\mathcal { I } } ( \\pi , Q )$ is the solution to max entropy $R L$ with rewards $r = \\mathcal { T } ^ { \\pi } Q$ . Thus, this forms a manifold in the $Q$ -policy space, that satisfies ",
+        "bbox": [
+            171,
+            710,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/08273cb9296fa3aeff1f595d48ba0ce5f0cc73396bf44673e3ee05d405c7d4f6.jpg",
+        "text": "$$\n\\pi _ { Q } ( a | s ) = \\frac { 1 } { Z _ { s } } \\exp ( Q ( s , a ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            400,
+            741,
+            594,
+            772
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "with normalization factor $\\begin{array} { r } { Z _ { s } = \\sum _ { a } \\exp Q ( s , a ) } \\end{array}$ and $\\pi _ { Q }$ defined as the ⇡ corresponding to $Q$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 3.3 and 3.4 are telling us that if we know $Q$ , then the inner optimization problem in terms of policy is trivial, and obtained in a closed form! Thus, we can recover an objective that only requires learning $Q$ : ",
+        "bbox": [
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+            797,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4be89384b480aaf4b506df1fe1d4c468859ae6470fcee037ea2cfe65759c4824.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { Q \\in \\Omega } \\operatorname* { m i n } _ { \\pi \\in \\Pi } \\mathcal { I } ( \\pi , Q ) = \\operatorname* { m a x } _ { Q \\in \\Omega } \\mathcal { I } \\left( \\pi _ { Q } , Q \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            370,
+            838,
+            627,
+            862
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, we have: ",
+        "bbox": [
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+            862,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 3.5. Let ${ \\mathcal { T } } ^ { * } ( Q ) = { \\mathcal { T } } \\left( \\pi _ { Q } , Q \\right)$ . Then $\\mathcal { T } ^ { * }$ is concave in $Q$ . ",
+        "bbox": [
+            173,
+            90,
+            633,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Thus, this new optimization objective is well-behaved and is maximized only at the saddle point. ",
+        "bbox": [
+            171,
+            117,
+            805,
+            133
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In Appendix C, we expand on our analysis and characterize the behavior for different choices of regularizer $\\psi$ , while giving proofs of all our propositions. Figure 1 summarizes the properties for the IRL objective: there exists a optimal policy manifold depending on $Q$ , allowing optimization along it (using ${ \\mathcal { T } } ^ { * }$ ) to converge to ",
+        "bbox": [
+            173,
+            138,
+            356,
+            304
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/e1630f6d8490830dd99da2e72bf9290d94e43da2f7a16794cd79eaa494de062d.jpg",
+        "image_caption": [
+            "Figure 1: Properties of IRL objective in reward-policy space and Q-policy space. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            366,
+            143,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "the saddle point. We further present analysis of IL methods that learn $Q$ -functions like SQIL [33] and ValueDICE [22] and find subtle fallacies affecting their learning. ",
+        "bbox": [
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+            304,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Note that although the same analysis holds in the reward-policy space, the optimal policy manifold depends on $Q$ , which isn’t trivially known unlike when in the Q-policy space. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 Approach ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            388,
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+            406
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section, we develop our inverse soft-Q learning (IQ-Learn) algorithm, such that it recovers the optimal soft $Q$ -function for a MDP from a given expert distribution. We start by learning energy-based models for the policy similar to soft $Q$ -learning and later learn an explicit policy similar to actor-critic methods. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1 General Inverse RL Objective ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For designing a practical algorithm using regularizers of the form $\\psi _ { g }$ (from Eq. 6), we define $g$ using a concave function $\\phi : \\mathcal { R } _ { \\psi } \\to \\mathbb { R }$ , such that $g ( x ) = { \\left\\{ \\begin{array} { l l } { x - \\phi ( x ) } & { { \\mathrm { i f ~ } } x \\in { \\mathcal { R } } _ { \\psi } } \\\\ { + \\infty } & { { \\mathrm { o t h e r w i s e } } } \\end{array} \\right. }$ with the rewards constrained in $R _ { \\psi }$ . ",
+        "bbox": [
+            173,
+            523,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For this choice of $\\psi$ , the Inverse RL objective $L ( \\pi , r )$ takes the form of Eq. 4 with a distance measure: ",
+        "bbox": [
+            176,
+            589,
+            823,
+            606
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a20470aebc67d4d3cc8c1b3383812e4063862a19f7c5d7719cf9da5193f73042.jpg",
+        "text": "$$\nd _ { \\psi } ( \\rho , \\rho _ { E } ) = \\operatorname* { m a x } _ { r \\in \\mathcal { R } _ { \\psi } } \\mathbb { E } _ { \\rho _ { E } } [ \\phi ( r ( s , a ) ) ] - \\mathbb { E } _ { \\rho } [ r ( s , a ) ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            334,
+            613,
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+            638
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "This forms a general learning objective that allows the use of a wide-range of statistical distances including Integral Probability Metrics (IPMs) and f-divergences (see Appendix B). 3 ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.2 Choice of Statistical Distances ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            695,
+            421,
+            712
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "While choosing a practical regularizer, it can be useful to obtain certain properties on the reward functions we recover. Some (natural) nice properties are: having rewards bounded in a range, learning smooth functions or enforcing a norm-penalty. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In fact, we find these properties correspond to the Total Variation distance, the Wasserstein-1 distance and the $\\chi ^ { 2 }$ -divergence respectively. The regularizers and the induced statistical distances are summarized in Table 2: ",
+        "bbox": [
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+            770,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/a814cc3a0ac2f7b13cacd11a9016feecb7f16f5f304f20584f9449b554950a37.jpg",
+        "table_caption": [
+            "Table 2: Enforced reward property, corresponding regularizer $\\psi$ and statistical distance $( R _ { \\operatorname* { m a x } } , K , \\alpha \\in \\mathbb { R } ^ { + }$ ) "
+        ],
+        "table_footnote": [
+            "3 We recover IPMs when using identity $\\phi$ and restricted reward family $\\mathcal { R }$ "
+        ],
+        "table_body": "<table><tr><td>Reward Property</td><td>4</td><td>d</td></tr><tr><td>Bound range Smoothness</td><td>=O if |rl ≤Rmax and+oo otherwise =Oif |lrllLip ≤Kand+oo otherwise</td><td>2Rmax ·TV(p,pE) K·Wi(p,pε)</td></tr></table>",
+        "bbox": [
+            225,
+            828,
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+            895
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We find that these choice of regularizers4 work very well in our experiments. In Appendix B, we further give a table for the well known $f$ -divergences, the corresponding $\\phi$ and the learnt reward estimators, along with a result ablation on using different divergences. Compared to $\\chi ^ { 2 }$ , we find other $f$ -divergences like Jensen-Shannon result in similar performances but are not as readily interpretable. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.3 Inverse soft-Q update (Discrete control) ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            161,
+            488,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Optimization along the optimal policy manifold gives the concave objective (Prop 3.5): ",
+        "bbox": [
+            173,
+            188,
+            743,
+            202
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "with $\\begin{array} { r } { V ^ { * } ( s ) = \\log \\sum _ { a } \\exp Q ( s , a ) } \\end{array}$ ",
+        "bbox": [
+            174,
+            229,
+            403,
+            246
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For each $Q$ , we get a corresponding reward $\\begin{array} { r } { r ( s , a ) = Q ( s , a ) - \\gamma \\mathbb { E } _ { s ^ { \\prime } \\sim \\mathcal { P } ( \\cdot \\vert s , a ) } [ \\log \\sum _ { a ^ { \\prime } } \\exp Q \\left( s ^ { \\prime } , a ^ { \\prime } \\right) ] } \\end{array}$ This correspondence is unique (Lemma A.1 in Appendix), and every update step can be seen as finding a better reward for IRL. ",
+        "bbox": [
+            173,
+            251,
+            823,
+            292
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Note that estimating $V ^ { \\ast } ( s )$ exactly is only possible in discrete action spaces. Our objective forms a variant of soft-Q learning: to learn the optimal $Q$ -function given an expert distribution. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.4 Inverse soft actor-critic update (Continuous control) ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            342,
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In continuous action spaces, it might not be possible to exactly obtain the optimal policy $\\pi _ { Q }$ , which forms an energy-based model of the $Q$ -function, and we use an explicit policy $\\pi$ to approximate $\\pi _ { Q }$ ",
+        "bbox": [
+            173,
+            367,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For any policy $\\pi$ , we have a objective (from Eq. 5): ",
+        "bbox": [
+            176,
+            402,
+            508,
+            416
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/587841f3cc254e7759a847e6f562e96822ae2e9913b9992410c71eff587ba352.jpg",
+        "text": "$$\n\\mathcal { I } ( \\pi , Q ) = \\mathbb { E } _ { \\rho _ { E } } [ \\phi ( Q - \\gamma \\mathbb { E } _ { s ^ { \\prime } \\sim \\mathcal { P } ( \\cdot \\vert s , a ) } V ^ { \\pi } ( s ^ { \\prime } ) ) ] - ( 1 - \\gamma ) \\mathbb { E } _ { \\rho _ { 0 } } [ V ^ { \\pi } ( s _ { 0 } ) ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            267,
+            417,
+            730,
+            435
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For a fixed $Q$ , soft actor-critic (SAC) update: $\\displaystyle \\operatorname* { m i n } _ { \\pi } \\mathbb { E } _ { s \\sim \\mathcal { D } , a \\sim \\pi ( \\cdot | s ) } [ Q ( s , a ) - \\log \\pi ( a | s ) ]$ , brings $\\pi$ closer to $\\pi _ { Q }$ while always minimizing Eq. 10 (Lemma A.4 in Appendix). Here $\\mathcal { D }$ is the distribution of previously sampled states, or a replay buffer. ",
+        "bbox": [
+            174,
+            438,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Thus, we obtain the modified actor-critic update rule to learn $Q$ -functions from the expert distribution: ",
+        "bbox": [
+            173,
+            491,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "1. For a fixed $\\pi$ , optimize $Q$ by maximizing ${ \\mathcal { I } } ( \\pi , Q )$ .   \n2. For a fixed $Q$ , apply SAC update to optimize $\\pi$ towards $\\pi _ { Q }$ ",
+        "bbox": [
+            210,
+            513,
+            624,
+            547
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "This differs from ValueDICE [22], where the actor is updated adverserially and the objective may not always converge (Appendix C). ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5 Practical Algorithm ",
+        "text_level": 1,
+        "bbox": [
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+            602,
+            372,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Pseudocode in Algorithm 1, shows our $Q$ -learning and actor-critic variants, with differences with conventional RL algorithms in red (we optimize $^ { - \\mathcal { J } }$ to use gradient descent). We can implement our algorithm IQ-Learn in 15 lines of code on top of standard implementations of (soft) DQN [14] for discrete control or soft actor-critic (SAC) [13] for continuous control, with a change on the objective for the $Q$ -function. Default hyperparameters from [14, 13] work well, except for tuning the entropy regularization. Target networks were helpful for continuous control. We elaborate details in Appendix D. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.1 Training methodology ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            744,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Corollary 2.1 in Appendix A states $\\mathbb { E } _ { ( s , a ) \\sim \\mu } [ V ^ { \\pi } ( s ) - \\gamma ] \\mathbb { E } _ { s ^ { \\prime } \\sim \\mathcal { P } ( \\cdot \\vert s , a ) } V ^ { \\pi } ( s ^ { \\prime } ) ] = ( 1 - \\gamma ) \\mathbb { E } _ { s \\sim p _ { 0 } } [ V ^ { \\pi } ( s ) ]$ , where $\\mu$ is any policy’s occupancy. We use this to stabilize training instead of using Eq. 9 directly. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Online: Instead of directly estimating $\\mathbb { E } _ { p _ { 0 } } [ V ^ { \\pi } ( s _ { 0 } ) ]$ in our algorithm, we can sample $( s , a , s ^ { \\prime } )$ from a replay buffer and get a single-sample estimate $\\bar { \\mathbb { E } } _ { ( s , a , s ^ { \\prime } ) \\sim \\mathrm { r e p l a y } } [ V ^ { \\pi } ( s ) - \\gamma V ^ { \\pi } ( s ^ { \\bar { \\prime } } ) ]$ . This removes the issue where we are only optimizing $Q$ in the inital states resulting in overfitting of $V ^ { \\pi } ( s _ { 0 } )$ , and improves the stability for convergence in our experiments. We find sampling half from the policy buffer and half from the expert distribution gives the best performances. Note that this is makes our learning online, requiring environment interactions. ",
+        "bbox": [
+            174,
+            804,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Offline: Although $\\mathbb { E } _ { p _ { 0 } } [ V ^ { \\pi } ( s _ { 0 } ) ]$ can be estimated offline we still observe an overfitting issue. Instead of requiring policy samples we use only expert samples to estimate $\\overline { { { \\mathbb { E } } } } _ { ( s , a , s ^ { \\prime } ) \\sim \\mathrm { e x p e r t } } [ V ^ { \\bar { \\pi } } ( s ) ~ - ~$ $\\gamma V ^ { \\pi } ( s ^ { \\prime } ) ]$ to sufficiently approximate the term. This methodology gives us state-of-art results for offline IL. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.2 Recovering rewards ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Instead of the conventional reward function $r ( s , a )$ on state and action pairs, our algorithm allows recovering rewards for each transition $( s , a , s ^ { \\prime } )$ using the learnt $Q$ -values as follows: ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 Inverse soft Q-Learning (both variants) ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "1: Initialize Q-function $Q _ { \\theta }$ , and optionally a policy $\\pi _ { \\phi }$   \n2: for step $t$ in $\\{ 1 . . . \\mathrm { N } \\}$ do   \n3: Train Q-function using objective from Equation 9: $\\begin{array} { r } { \\theta _ { t + 1 }  \\theta _ { t } - \\alpha _ { Q } \\nabla _ { \\theta } \\big [ \\qquad \\big ] } \\end{array}$ (Use $V ^ { * }$ for Q-learning and $V ^ { \\pi _ { \\phi } }$ for actor-critic)   \n4: (only with actor-critic) Improve policy $\\pi _ { \\phi }$ with SAC style actor update: $\\begin{array} { r } { \\underset { \\bullet } { \\phi } _ { t + 1 }  \\phi _ { t } - \\alpha _ { \\pi } \\nabla _ { \\phi } \\mathbb { E } _ { s \\sim \\mathcal { D } , a \\sim \\pi _ { \\phi } ( \\cdot | s ) } [ Q ( s , a ) - \\log \\pi _ { \\phi } ( a | s ) ] } \\end{array}$ ",
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+    },
+    {
+        "type": "text",
+        "text": "5: end for ",
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+    },
+    {
+        "type": "text",
+        "text": "Algorithm 2 Recover policy and reward ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "1: Given trained Q-function $Q _ { \\theta }$ , and optionally a trained policy $\\pi _ { \\phi }$   \n2: Recover policy $\\pi$ : (Q-learning) $\\textstyle \\pi : = { \\frac { 1 } { Z } } \\exp Q _ { \\theta }$ (actor-critic) $\\pi : = \\pi _ { \\phi }$   \n3: For state s, action a and $\\mathbf { s } ^ { \\prime } \\sim \\mathcal { P } ( \\cdot | \\mathbf { s } , \\mathbf { a } )$   \n4: Recover reward $r ( \\mathbf { s } , \\mathbf { a } , \\mathbf { s } ^ { \\prime } ) = Q _ { \\theta } ( \\mathbf { \\bar { s } } , \\mathbf { a } ) - \\gamma V ^ { \\pi } \\left( \\mathbf { s } ^ { \\prime } \\right)$ ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b2ca25d40583a9431621bc58498e5774648dcee562afdf7670546efbf7db31d9.jpg",
+        "text": "$$\nr ( s , a , s ^ { \\prime } ) = Q ( s , a ) - \\gamma V ^ { \\pi } \\left( s ^ { \\prime } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            354,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Now, $\\begin{array} { r } { { \\mathbb E } _ { s ^ { \\prime } \\sim \\mathcal { P } ( \\cdot \\vert s , a ) } [ Q ( s , a ) - \\gamma V ^ { \\pi } \\left( s ^ { \\prime } \\right) ] = Q ( s , a ) - \\gamma { \\mathbb E } _ { s ^ { \\prime } \\sim \\mathcal { P } ( \\cdot \\vert s , a ) } [ V ^ { \\pi } \\left( s ^ { \\prime } \\right) ] = \\mathcal { T } ^ { \\pi } Q ( s , a ) } \\end{array}$ . This is just the reward function $r ( s , a )$ we want. So by marginalizing over next-states, our expression correctly recovers the reward over state-actions. Thus, Eq. 11 gives the reward over transitions. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Our rewards require $s ^ { \\prime }$ which can be sampled from the environment, or by using a dynamics model. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5.3 Implementation of Statistical Distances ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Implementing TV and $W _ { 1 }$ distances is fairly trivial and we give details in Appendix B. For the $\\chi ^ { 2 }$ -divergence, we note that it corresponds to $\\begin{array} { r } { \\phi ( x ) = x - \\frac { 1 } { 4 \\alpha } \\hat { x ^ { 2 } } } \\end{array}$ . On substituting in Eq. 9, we get ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4ae8851a1423069c0b8420bc16491ef4b0808d120576fe0f72bd06287eacd4c1.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\underset { 2 \\leq \\infty } { \\mathrm { a a x } } \\mathbb { E } _ { \\rho _ { \\mathrm { E } } } [ ( Q ( s , a ) - \\gamma \\mathbb { E } _ { s ^ { \\prime } \\sim \\mathcal { P } ( \\cdot \\vert s , a ) } V ^ { * } ( s ^ { \\prime } ) ) ] - ( 1 - \\gamma ) \\mathbb { E } _ { p _ { 0 } } [ V ^ { * } ( s _ { 0 } ) ] - \\frac { 1 } { 4 \\alpha } \\mathbb { E } _ { \\rho _ { \\mathrm { E } } } [ ( Q ( s , a ) - \\gamma \\mathbb { E } _ { s ^ { \\prime } \\sim \\mathcal { P } ( \\cdot \\vert s , a ) } V ^ { * } ( s ^ { \\prime } ) ) ^ { 2 } ] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In a fully offline setting, this can be further simplified as (using the offline methodology in Sec 5.1): ",
+        "bbox": [
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+            546,
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fe0d0b5e44a302fe6f03ad11881ce2e856bf3b82b6717ae0c45df625d46bb3fc.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { Q \\in \\Omega } - \\mathbb { E } _ { \\rho _ { E } } [ ( Q ( s , a ) - V ^ { * } ( s ) ) ] + \\frac { 1 } { 4 \\alpha } \\mathbb { E } _ { \\rho _ { E } } [ ( Q ( s , a ) - \\gamma \\mathbb { E } _ { s ^ { \\prime } \\sim \\mathcal { P } ( \\cdot \\vert s , a ) } V ^ { * } ( s ^ { \\prime } ) ) ^ { 2 } ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "This is interestingly the same as the $Q$ -learning objective in CQL [23], an state-of-art method for offline RL (using 0 rewards), and shares similarities with regularized behavior cloning [33] 5. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.4 Learning state-only reward functions ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Previous works like AIRL [10] propose learning rewards that are only function of the state, and claim that these form of reward functions generalize between different MDPs. We find our method can predict state-only rewards by using the policy and expert state-marginals with a modification to Eq. 9: ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7f61992cdcda56b963b5770898e781028a12697477493d9f18955eb161560f07.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { \\mathcal { O } \\in \\Omega } \\mathcal { I } ^ { * } ( Q ) = \\mathbb { E } _ { s \\sim \\rho _ { E } ( s ) } [ \\mathbb { E } _ { a \\sim \\pi ( \\cdot | s ) } [ \\phi ( Q ( s , a ) - \\gamma \\mathbb { E } _ { s ^ { \\prime } \\sim \\mathcal { P } ( \\cdot | s , a ) } V ^ { * } ( s ^ { \\prime } ) ) ] ] - ( 1 - \\gamma ) \\mathbb { E } _ { p _ { 0 } } [ V ^ { * } ( s _ { 0 } ) ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Interestingly, our objective no longer depends on the the expert actions $\\pi _ { E }$ and can be used for $\\mathrm { I L }$ using only observations. For the sake of brevity, we expand on this in Section 1 in Appendix A. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "6 Related Work ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Classical IL: Imitation learning has a long history, with early works using supervised learning to match a policy’s actions to those of the expert [15, 35]. A significant advance was made with the formulation of $\\mathrm { I L }$ as the composition of RL and IRL [29, 1, 43], recovering the expert’s policy by inferring the expert’s reward function, then finding the policy which maximizes reward under this reward function. These early approaches required a hand-designed featurization of the MDP, limiting their applicability to complex MDPs. In this setting, early approaches [9, 31] noted a formal equivalence between IRL and IL using an inverse Bellman operator similar to our own. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Online IL: More recent work aims to leverage the power of modern machine learning approaches to learn good featurizations and extend IL to complex settings. Recent work generally falls into one of two settings: online or offline. In the online setting, the IL algorithm is able to interact with the environment to obtain dynamics information. GAIL [17] takes the nested RL/IRL formulation of earlier work , optimizing over all reward functions with a convex regularizer. This results in the objective in Eq. (3), with a max-min adversarial problem similar to a GAN [11]. A variety of further work has built on this adversarial approach [21, 10, 3]. A separate line of work aims to simplify the problem in Eq. (3) by using a fixed $r$ or $\\pi$ . In SQIL [33], $r$ is chosen to be the 1-0 indicator on the expert demonstrations, while ASAF [4] takes the GAN approach and uses a discriminator (with role similar to $r$ ) of fixed form, consisting of a ratio of expert and learner densities. AdRIL [38] is a recent extension of SQIL, additionally assigning decaying negative reward to previous policy rollouts. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Offline IL: In the offline setting, the learner has no access to the environment. The simple behavioural cloning (BC) [34] approach is offline, but doesn’t use any dynamics information. ValueDICE [22] is a dynamics-aware offline approach with an objective somewhat similar to ours, motivated from minimization of a variational representation of the KL-divergence between expert and learner policies. ValueDICE requires adversarial optimization to learn the policy and Q-functions, with a biased gradient estimator for training. We show a way to recover a unbiased gradient estimate for the KL-divergence in Appendix C. The O-NAIL algorithm [2] builds on ValueDICE and combines with an SAC update to obtain a method that is similar to our algorithm described in section 4.4, with the specific choice of reverse KL-divergence as the relevant statistical distance. The EDM method [19] incorporates dynamics via learning an explicit energy based model for the expert state occupancy, although some theoretical details have been called into question (see [37] for details). The recent AVRIL approach [6] uses a variational method to solve a probabilistic formulation of IL, finding a posterior distribution over $r$ and $\\pi$ . Illustrating the potential benefits of alternative distances for $\\mathrm { I L }$ , the PWIL [7] algorithm gives a non-adversarial procedure to minimize the Wasserstein distance between expert and learned occupancies. The approach is specific to the primal form of the $\\mathcal { W } _ { 1 }$ -distance, while our method (when used with the Wasserstein distance) targets the dual form. ",
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+    },
+    {
+        "type": "text",
+        "text": "7 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "7.1 Experimental Setup ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We compare IQ-Learn (\"IQ\") to prior work on a diverse collection of RL tasks and environments - ranging from low-dimensional control tasks: CartPole, Acrobot, LunarLander - to more challenging continuous control MuJoCo tasks: HalfCheetah, Hopper, Walker and Ant. Furthermore, we test on the visually challenging Atari Suite with high-dimensional image inputs. We compare on offline IL - with no access to the the environment while training, and online IL - with environment access. We show results on $W _ { 1 }$ and $\\chi ^ { 2 }$ as our statistical distances, as we found them more effective than TV distance. In all cases, we train until convergence and average over multiple seeds. Hyperparameter settings and training details are detailed in Appendix D. ",
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+    {
+        "type": "text",
+        "text": "7.2 Benchmarks ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Offline $\\mathbf { I L }$ We compare to the state-of-art IL methods EDM and AVRIL, following the same experimental setting as [6]. Furthermore, we compare with ValueDICE which also learns Q-functions, albeit with drawbacks such as adversarial optimization. We also experimented with SQIL, but found that it was not competitive in the offline setting. Finally, we utilize BC as an additional IL baseline. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Online IL We use MuJoCo and Atari environments and compare against state-of-art online IL methods: ValueDICE, SQIL and GAIL. We only show results on $\\chi ^ { 2 }$ as $W _ { 1 }$ was harder to stabilize on complex environments6. Using target updates stabilizes the $Q$ -learning on MuJoCo. For brevity, further online IL results are shown in the Appendix D. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/2991437dc0b464a8ab137fb0cb4318c35069cc219e9829f40272ad9e62707a2f.jpg",
+        "image_caption": [
+            "Figure 2: Offline IL results. We plot the average environment returns vs the number of expert trajectories. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            169,
+            117,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Offline IL We present results on the three offline control tasks in Figure 2. On all tasks, IQ strongly outperforms prior works we compare to in performance and sample efficiency. Using just one expert trajectory, we achieve expert performance on Acrobot and reach near expert on Cartpole. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Mujoco Control We present our results on the MuJoCo tasks using a single expert demo in Table 3. IQ achieves expert-level performance in all the tasks while outperforming prior methods like ValueDICE and GAIL. We did not find SQIL competitive in this setting, and skip it for brevity. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/30591f68fc93c39d36de160b81df4e55c5a9e3d65f303688d83af8dbc3106322.jpg",
+        "table_caption": [
+            "Table 3: Mujoco Results. We show our performance on MuJoCo control tasks using a single expert trajectory. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Task</td><td>GAIL</td><td>DAC</td><td>ValueDICE</td><td>IQ (Ours)</td><td>Expert</td></tr><tr><td>Hopper</td><td>3252.5</td><td>3305.1</td><td>3312.1</td><td>3546.4</td><td>3532.7</td></tr><tr><td>Half-Cheetah</td><td>3080.0</td><td>4080.6</td><td>3835.6</td><td>5076.6</td><td>5098.3</td></tr><tr><td>Walker</td><td>4013.7</td><td>4107.9</td><td>3842.6</td><td>5134.0</td><td>5274.5</td></tr><tr><td>Ant</td><td>2299.1</td><td>1437.5</td><td>1806.3</td><td>4362.9</td><td>4700.0</td></tr><tr><td>Humanoid</td><td>232.6</td><td>380.5</td><td>644.5</td><td>5227.1</td><td>5312.8</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Atari We present our results on ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Atari using 20 expert demos in Figure 3. We reach expert performance on Space Invaders while being near expert on Pong and Breakout. Compared to prior methods like SQIL, IQ obtains $\\mathbf { 3 - 7 x }$ normalized score7 and converges in ${ \\sim } 3 0 0 \\mathrm { k }$ steps, being 3x faster compared to Q-learning based RL methods that take more than 1M steps to converge. Other popular methods like GAIL and ValueDICE perform near random even with 1M env steps. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/ab324e15c2691ab2c4c5c8daf067b32caf4f046d5a1756f3a015e4acf22e3a89.jpg",
+        "image_caption": [
+            "Figure 3: Atari Results. We show the returns vs the number of env steps. (Averaged over 5 seeds) "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "7.4 Recovered Rewards ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "IQ has the added benefit of recovering rewards and can be used for IRL. On Hopper task, our learned rewards have a Pearson correlation of 0.99 with the true rewards. In Figure 4, we visualize our recovered rewards in a simple grid environment. We elaborate details in Appendix D. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "8 Discussion and Outlook ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We present a new principled framework for learning soft- $Q$ functions for IL and recovering the optimal policy and the reward, building on past works in IRL [43]. Our algorithm IQ-Learn outperforms prior methods with very sparse expert data and scales to complex image-based environments. We also recover rewards highly correlated with actual rewards. It has applications in autonomous driving and complex decision-making, but proper considerations need to be taken into account to ensure safety and reduce uncertainty, before any deployment. Finally, human or expert data can have errors that can propagate. A limitation of our method is that our recovered rewards depend on the environment dynamics, preventing trivial use on reward transfer settings. One direction of future work could be to learn a reward model from the trained soft- $Q$ model to make the rewards explicit. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/2e92c4cecb167956b2f3f30af2237262fa266f72969b8f686abf4df516850f9f.jpg",
+        "image_caption": [
+            "Figure 4: Reward Visualization. We use a discrete GridWorld environment with 5 possible actions: up, down, left, right, stay. Agent starts in a random state. (With 30 expert demos) "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "9 Acknowledgements ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We thank Kuno Kim and John Schulman for helpful discussions. We also thank Ian Goodfellow as some initial motivations for this work were developed under an internship with him. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "10 Funding Transparency ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This research was supported in part by NSF (#1651565, #1522054, #1733686), ONR (N00014-19-1- 2145), AFOSR (FA9550-19-1-0024) and FLI. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            457,
+            267,
+            473
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
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Of moments and matching: Trade-offs and treatments in imitation learning. arXiv preprint arXiv:2103.03236, 2021. 8   \n[39] Lu Wang, Wenchao Yu, Xiaofeng He, Wei Cheng, Martin Renqiang Ren, Wei Wang, Bo Zong, Haifeng Chen, and Hongyuan Zha. Adversarial cooperative imitation learning for dynamic treatment regimes. In Proceedings of The Web Conference 2020, pages 1785–1795, 2020. 1   \n[40] Lantao Yu, Tianhe Yu, Chelsea Finn, and Stefano Ermon. Meta-inverse reinforcement learning with probabilistic context variables. In NeurIPS, 2019. 26   \n[41] Jinyun Zhou, Rui Wang, Xu Liu, Yifei Jiang, Shu Jiang, Jiaming Tao, Jinghao Miao, and Shiyu Song. Exploring imitation learning for autonomous driving with feedback synthesizer and differentiable rasterization. arXiv preprint arXiv:2103.01882, 2021. 1   \n[42] Brian D Ziebart. Modeling purposeful adaptive behavior with the principle of maximum causal entropy. 2010. 3   \n[43] Brian D. Ziebart, Andrew L. Maas, J. Bagnell, and A. Dey. 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parse/train/BJgGhiR5KX/BJgGhiR5KX.md

@@ -0,0 +1,279 @@
+# LEARNING CROSS-LINGUAL SENTENCE REPRESENTATIONS VIA A MULTI-TASK DUAL-ENCODER MODEL
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+A significant roadblock in multilingual neural language modeling is the lack of labeled non-English data. One potential method for overcoming this issue is learning cross-lingual text representations that can be used to transfer the performance from training on English tasks to non-English tasks, despite little to no task-specific non-English data. In this paper, we explore a natural setup for learning crosslingual sentence representations: the dual-encoder. We provide a comprehensive evaluation of our cross-lingual representations on a number of monolingual, crosslingual, and zero-shot/few-shot learning tasks, and also give an analysis of different learned cross-lingual embedding spaces.
+
+# 1 INTRODUCTION
+
+There has been a significant amount of recent work on developing models that can produce sentence representations that are useful for a number of language processing tasks (Kiros et al., 2015; Conneau et al., 2017; Subramanian et al., 2018; Logeswaran & Lee, 2018; Cer et al., 2018). However, these models are trained on largely monolingual data, and can thus only be used for tasks in a single language. A promising direction for extending the previous models to multiple languages is learning cross-lingual embedding spaces (Schwenk et al., 2017; Eriguchi et al., 2018; Singla et al., 2018), which could be used to transfer performance in one language to others.
+
+We develop a novel approach for cross-lingual representation learning by combining the dualencoder architectures used for learning sentence representations (Logeswaran & Lee, 2018; Cer et al., 2018) and for bi-text retrieval (Guo et al., 2018). By doing so, we learn representations that maintain state-of-the-art performance in tasks for a source language while simultaneously obtaining state-of-the-art performance in zero-shot learning tasks for a target language. For a given sourcetarget language pair, we construct a multi-task training scheme using native source language tasks, native target language tasks, and a bridging source-target translation retrieval task to learn sentence representations that are aligned between the source and target languages. We then evaluate the learned representations on several monolingual and cross-lingual tasks, and also provide a graphbased analysis of the learned representations.
+
+We find that multi-task training using additional monolingual tasks improves performance over models that only make use of parallel data on both cross-lingual semantic textual similarity (STS) (Cer et al., 2017) and Søgaard et al. (2018)’s cross-lingual eigen-similarity metric. The results show that the addition of monolingual data improves the embedding alignment of sentences and their translations. Furthermore, we find that cross-lingual training with additional monolingual data leads to far better transfer learning performance, and we show that our cross-lingual representations outperform state-of-the-art zero-shot learning models in sentiment classification and natural language inference.
+
+# 2 MULTI-TASK DUAL-ENCODER MODEL
+
+The core of our approach is the idea of modeling various tasks as ranking input-response pairs by encoding them via two encoders, with the crucial task for learning cross-lingual representations being translation ranking. For translation ranking, as well as for our other tasks, we take an input sentence $s _ { i } ^ { I }$ and an associated response sentence $s _ { i } ^ { R }$ , and we seek to rank $s _ { i } ^ { R }$ over all other possible response sentences $s _ { j } ^ { R } \in \mathcal S ^ { R }$ . To do so, we model the conditional probability $P ( s _ { i } ^ { R } \mid s _ { i } ^ { I } )$ as:
+
+![](images/b1f75398f482a5e36c4d08125507fc51b9dfc4aca41e1aa374a6e53486c83078.jpg)  
+Figure 1: Multi-task dual-encoder model. It consists of a group of native tasks in each language and a bridging task using translation pair data. The encoders in the gray box all share their parameters, and thus constitute $g$ .
+
+$$
+P ( s _ { i } ^ { R } \mid s _ { i } ^ { I } ) = \frac { e ^ { \phi ( s _ { i } ^ { I } , s _ { i } ^ { R } ) } } { \sum _ { s _ { j } ^ { R } \in \mathcal { S } ^ { R } } e ^ { \phi ( s _ { i } ^ { R } , s _ { j } ^ { R } ) } } , \quad \phi ( s _ { i } ^ { I } , s _ { j } ^ { R } ) = g ^ { I } ( s _ { i } ^ { I } ) ^ { \top } g ^ { R } ( s _ { j } ^ { R } )
+$$
+
+Where $g ^ { I }$ and $g ^ { R }$ are the input and response sentence encoding functions that compose the dualencoder. Since the normalization term in equation 1 is computationally intractable, we follow the approaches of Henderson et al. (2017) and instead choose to model an approximate conditional probability $\mathcal { \widetilde { P } } ( s _ { i } ^ { R } \mid s _ { i } ^ { I } )$ :
+
+$$
+\displaystyle \widetilde { P } ( s _ { i } ^ { R } \mid s _ { i } ^ { I } ) = \frac { e ^ { \phi ( s _ { i } ^ { I } , s _ { i } ^ { R } ) } } { \sum _ { j = 1 , j \neq i } ^ { K } e ^ { \phi ( s _ { i } ^ { R } , s _ { j } ^ { R } ) } }
+$$
+
+Where $K$ denotes the size of a single batch of training examples, and the $s _ { j } ^ { R }$ correspond to the response sentences associated with the other input sentences in the same batch as $s _ { i } ^ { I }$ . We parametrize $g ^ { I }$ and $g ^ { R }$ as deep neural networks that are trained to minimize the negative log-likelihood of $\widetilde { P } ( s _ { i } ^ { R } \mid$ $s _ { i } ^ { I }$ ) for each task.
+
+In order to produce a single sentence encoding function $g$ that can be evaluated on downstream tasks, we share several layers between the input and response encoders and treat the final output of these shared layers as $g$ . Additionally, these layers are modeled after the Universal Sentence Encoder (USE) model of Cer et al. (2018), since it is the state-of-the-art model that is most amenable to our setup. To learn cross-lingual representations, we train $g$ on several tasks from mirrored corpora across languages1 for the source-target language pairs English-French (en-fr), English-Spanish (enes), and English-German (en-de). The resulting model structure is illustrated in Figure 1.
+
+# 2.1 ENCODER ARCHITECTURE
+
+Word and Character Embeddings. As part of the training process for learning the cross-lingual sentence encoding function $g$ , we learn embeddings for the words and characters present in the training data for a given source-target language pair. Word embeddings are learned end-to-end, as we noticed that pre-trained embeddings did not make a difference for final performance. Character embeddings are learned in a similar manner, but with the added stipulation that we consider character n-gram embeddings instead of single character embeddings by using a single feedforward layer with tanh activation on top of character n-grams. Each word in an input sentence then obtains a character representation by having its character n-gram representations summed together. To have the sentence encoder $g$ leverage the word and character representations together without drastically increasing its number of parameters, we sum the word and character embeddings before using them as input to $g$ .
+
+Transformer Encoder. The actual architecture of the shared encoder $g$ consists of three2 layers of transformer stacks, which contain the feed-forward and multi-head attention sub-layers described in Vaswani et al. (2017). The transformer encoder output is a variable-length sequence at each stack. We average encodings of all sequence positions in the final layer as the final sentence encoding. This embedding is then fed into different sets of feedforward layers that are used for each task. For our transformer layers, we use 8 attentions heads, a hidden size of 512, and a filter size of 2048.
+
+# 2.2 MULTI-TASK TRAINING SETUP
+
+To learn a function $g$ that is capable of strong cross-lingual matching and transfer learning performance for a source-target language pair while also maintaining monolingual downstream task performance, we employ four unique task types for each language pair. Specifically, we employ $a$ conversation response prediction task, a quick thought task, a natural language inference task, and a bridging task – translation ranking. Six total tasks are used in training, as the first two tasks are mirrored across languages.
+
+Conversation Response Prediction. We model the conversation response prediction task in the same manner as Yang et al. (2018). We minimize the negative log-likelihood of $\mathcal { P } ( s _ { i } ^ { R } \mid s _ { i } ^ { I } )$ , where $s _ { i } ^ { I }$ is a single comment and $s _ { i } ^ { R }$ is its associated response comment. For the response side, we model $g ^ { \dot { R } } ( s _ { i } ^ { R } )$ as two fully-connected feedforward layers of size 320 and 512 with tanh activation on top of $g ( s _ { i } ^ { R } )$ . For the input side, however, we simply let $g ^ { I } ( s _ { i } ^ { I } ) = g ( s _ { i } ^ { I } )$ , as we noticed in early experiments that letting the optimization of the conversational response task more directly influence the parameters of the underlying sentence encoder $g$ led to better downstream task performance.
+
+Quick Thought. We use a modified version of the Quick Thought task detailed by Logeswaran & Lee (2018). We minimize the sum of the negative log-likelihoods of $\widetilde { P } ( s _ { i } ^ { R } \mid s _ { i } ^ { I } )$ and $\bar { \widetilde { P } } ( s _ { i } ^ { P } \mid s _ { i } ^ { I } )$ , where $s _ { i } ^ { I }$ is a sentence taken from an article and $s _ { i } ^ { P }$ and $s _ { i } ^ { R }$ are its predecessor and successor sentences respectively. For this task, we model all three of $g ^ { P } ( s _ { i } ^ { P } )$ , $g ^ { I } ( s _ { i } ^ { I } )$ , and $g ^ { R } ( s _ { i } ^ { R } )$ using separate, fully-connected feedforward layers of size 320 and 512 with tanh activation on top of $g$ , as we did for $\dot { \boldsymbol g } ^ { R } ( s _ { i } ^ { R } )$ in our conversational modeling task.
+
+Natural Language Inference (NLI). We also include an English-only natural language inference task based on Bowman et al. (2015). For this task, we first encode an input sentence $s _ { i } ^ { I }$ and its corresponding response hypothesis $s _ { i } ^ { R }$ into vectors $u _ { 1 }$ and $u _ { 2 }$ using $g$ . The vectors $u _ { 1 } , u _ { 2 }$ are then used to construct a feature vector $( u _ { 1 } , u _ { 2 } , | u _ { 1 } - u _ { 2 } | , u _ { 1 } * u _ { 2 } )$ , where $( \cdot )$ represents concatenation and $^ *$ represents element-wise multiplication. The form of this feature vector is derived from the original experiments of Bowman et al. (2015). This feature vector is then fed into a single feedforward layer of size 512 that is used to perform the 3-way NLI classification.
+
+Translation Ranking. Our translation task setup is identical to the one used by Guo et al. (2018) for bi-text retrieval. We minimize the negative log-likelihood of $\widetilde { P } ( s _ { i } \mid t _ { i } )$ , where $( s _ { i } , t _ { i } )$ is a sourcetarget translation pair. Since the translation task is intended to align the sentence representations for the source and target languages, we do not use any kind of task-specific feedforward layers and instead use $g$ as both $g ^ { I }$ and $g ^ { R }$ . Following Guo et al. (2018), we append 5 translations that are similar to the correct translation to each training example as “hard-negatives”. Similarity is determined via a version of our model trained only on the translation ranking task. We did not see additional gains from using more than 5 hard-negatives.
+
+# 3 EXPERIMENTS
+
+# 3.1 CORPORA
+
+We draw upon multiple, openly available data sources and training corpora for the training of the tasks mentioned above. For each of our datasets, we use $90 \%$ of the data for training, and the remaining $10 \%$ for development/validation. Our data preprocessing procedures are described in detail in the supplementary material.
+
+Reddit. We preprocess the Reddit data extracted by Al-Rfou et al. (2016) into 600 million inputresponse comment pairs for training our conversation response prediction task. We also translate this data using the Google neural machine translation (NMT) system of Wu et al. (2016).
+
+Wikipedia. To get native, non-English data, we extract triplets of contiguous sentences from English, French, Spanish, and German articles take from Wikipedia. Our final extracted corpus of Wikipedia sentence triplets consists of 127.9, 49.5, 29.8, and 49.3 million triplets for English, French, Spanish, and German respectively, which we use to train our Quick Thought task.
+
+Stanford Natural Language Inference (SNLI). The NLI data we use is taken from the Stanford Natural Language Inference (SNLI) dataset of Bowman et al. (2015), which consists of 570K sentence pairs associated with one of three labels: entailment, contradiction, or neutral. The corpus is split into training (550K), validation (10K), and testing sets (10K).
+
+Translation. The data for training the translation task is constructed using a system similar to the approach described by Guo et al. (2018). The final constructed corpus contains around 600M en-fr pairs, 470M en-es pairs and 500M en-de pairs.
+
+# 3.2 MODEL CONFIGURATION
+
+In all of our experiments, multi-task training is done by cycling through the different tasks (translation pairs, Reddit, Wikipedia, NLI) and performing an optimization step for a single task at a time. We train all of our models with a batch size of 100 using stochastic gradient descent with a learning rate of 0.008. All of our models are trained for 30 million steps. All input text is tokenized prior to being used for training. We build a vocab containing 200 thousand unigram tokens with 10 thousand hash buckets for out-of-vocabulary tokens. The character n-gram vocab contains 200 thousand hash buckets used for 3 and 4 grams. Both the word and character n-gram embedding sizes are 320. All hyperparameters are tuned based on performance on the development portion (random $10 \%$ slice) of our datasets. Finally, as an additional training heuristic, we multiply the gradients to the word and character embeddings by a factor of $1 0 0 ^ { 3 }$ . We found that using this embedding gradient multiplier alleviated vanishing gradient issues and greatly improved training.
+
+We compare the proposed cross-lingual multi-task (referred to simply as multi-task) models with baseline models that are trained using only the translation ranking task, which we dub as the “translation-ranking” models.
+
+# 3.3 MODEL PERFORMANCE ON ENGLISH DOWNSTREAM TASKS
+
+We first evaluated all of our cross-lingual models on several downstream English tasks taken from SentEval (Conneau & Kiela, 2018) to verify the impact of cross-lingual training. Each task is described in the supplementary material. Results on the tasks are summarized in Table 1. We note that cross-lingual training does not hinder the effectiveness of our encoder on English tasks, as the multi-task models are close to state-of-the-art in each of the downstream tasks. For the Text REtrieval Conference (TREC) eval, we actually find that our multi-task models outperform the previous state-of-the-art models by a sizable amount.
+
+# 3.4 CROSS-LINGUAL RETRIEVAL
+
+We also evaluate both the multi-task and translation-ranking models’ efficacy in performing crosslingual retrieval by using held-out translation pair data. Following Henderson et al. (2017), we use precision at $\mathrm { ~ N ~ } ( \mathrm { P } @ \mathrm { N } )$ as the evaluation metric by checking if a source sentence’s target translation ranks (where ranking is done using dot product) in the top $N$ scored candidates when considering $K$ other randomly selected target sentences. Unlike Henderson et al. (2017), we set $K$ to be 999 instead of 99 because using $K = 9 9$ results in all metrics quickly shooting up to $9 9 \%$ .
+
+The translation-ranking model remains as a strong baseline for finding the true translation, with $9 5 . 4 \%$ , $8 7 . 5 \%$ , $9 7 . 5 \%$ $\mathrm { P @ 1 }$ for en-fr, en-es and en-de retrieval tasks respectively. The multi-task model performs almost identical with $9 5 . 1 \%$ , $8 8 . 8 \%$ and $9 7 . 8 \%$ , which provides empirical justification that it is possible to maintain embedding space alignment despite optimizing for native tasks in each individual language. We also experimented with $\mathrm { P @ 3 }$ and $\mathrm { P @ 1 0 }$ , the results are identical.
+
+Table 1: Performance on classification transfer tasks.   
+
+<table><tr><td>Model</td><td>MR</td><td>CR</td><td>SUBJ</td><td>MPQA</td><td>TREC</td><td>SST</td><td>STS Bench (dev /test)</td></tr><tr><td colspan="7">Cross-lingual Multi-taskModels</td></tr><tr><td>en-fr</td><td>77.9</td><td>82.9</td><td>95.5</td><td>89.3</td><td>95.3</td><td>84.0</td><td>0.803/0.763</td></tr><tr><td>en-es</td><td>80.1</td><td>85.9</td><td>94.6</td><td>86.5</td><td>96.2</td><td>85.2</td><td>0.809 / 0.770</td></tr><tr><td>en-de</td><td>78.8</td><td>84.0</td><td>95.9</td><td>87.6</td><td>96.1</td><td>85.0</td><td>0.802 /0.764</td></tr><tr><td colspan="8">Translation-ranking Models</td></tr><tr><td>en-fr</td><td>68.7</td><td>79.3</td><td>87.0</td><td>81.8</td><td>89.4</td><td>74.2</td><td>0.668/ 0.558</td></tr><tr><td>en-es</td><td>67.7</td><td>75.7</td><td>83.5</td><td>86.0</td><td>94.4</td><td>72.6</td><td>0.669 / 0.631</td></tr><tr><td>en-de</td><td>67.8</td><td>75.2</td><td>84.4</td><td>83.6</td><td>86.8</td><td>74.6</td><td>0.673 / 0.632</td></tr><tr><td colspan="8">State-of-the-art Models</td></tr><tr><td>InferSent Skip-Thought LN</td><td>81.1 79.4</td><td>86.3 83.1</td><td>92.4 93.7</td><td>90.2 89.3</td><td>88.2 1</td><td>84.6 1</td><td>0.801/0.758</td></tr><tr><td>Quick-Thought</td><td>82.4</td><td>86.0</td><td>94.8</td><td>90.2</td><td>92.4</td><td>87.6</td><td>1 1</td></tr><tr><td></td><td></td><td></td><td>93.9</td><td>87.0</td><td>92.5</td><td>85.4</td><td></td></tr><tr><td>USE Transformer</td><td>81.4</td><td>87.4</td><td></td><td></td><td></td><td></td><td>0.814 /0.782</td></tr></table>
+
+# 3.5 MULTILINGUAL STS
+
+We further test whether our learned cross-lingual representations can also perform well in their associated non-English language tasks by evaluating semantic textual similarity (STS) performance on French, Spanish, and German.
+
+To evaluate Spanish-Spanish (es-es) STS, we use SemEval-2017 task 1 (STS17) track 3 of Cer et al. (2017), which contains 250 Spanish sentence pairs with human labeled similarity scores. We also evaluate es-en STS by using the track 4(a) task4, which contains 250 en-es sentence pairs.
+
+Beyond English and Spanish, however, there are no standard STS datasets available for other languages. As such, we evaluate on a translated version of the STS Benchmark dataset from Cer et al. (2017) for French, Spanish, and German. We use Google’s translation system to translate the STS Benchmark sentences to French, Spanish and German. We believe that the results on our pseudomultilingual STS Benchmark dataset are expected to still be a reasonable indicator of multilingual semantic similarly performance, since the NMT encoder-decoder architecture for translation differs significantly from our dual-encoder approach.
+
+Following Cer et al. (2018), we first compute the sentence encodings $u , v$ of an STS sentence pair, and then score the sentence pair similarity based on the angular distance between the two vectors, $- \operatorname { a r c c o s } { \left( { \frac { \boldsymbol { u v } } { | | \boldsymbol { u } | | \ | | | \boldsymbol { v } | | } } \right) }$ . Table 2 shows the Pearson’s correlation coefficient of the STS tasks for all models. The first column shows the trained model performance on original English STS Benchmark data. Columns 2 to 4 shows the the performance on the other languages. All multi-task models remain strong on the translated STS tasks, with around 0.77 for dev and 0.74 for test in all languages. Lastly, columns 5 and 6 shows the results of en-es models on STS17 tasks. The un-tuned multi-task models achieve 0.827 for the es-es task and 0.769 for the es-en task. As a point of reference, we also list the two best performing STS systems, Tian et al. (2017) (ECNU) and Wu et al. (2017) (BIT), reported from Cer et al. (2017). Our results are very close to these state-of-the-art feature engineered and mixed systems, which we describe in greater detail in the supplementary material.
+
+# 4 ZERO-SHOT CLASSIFICATION
+
+To evaluate the transfer learning capabilities of our models, we examine how well the multi-task and translation-ranking encoders perform on zero-shot and few-shot classification tasks.
+
+Table 2: Pearson’s correlation coefficients on translated STS Benchmark and STS17 tasks. The first column shows the results on the original STS Benchmark data in English.   
+
+<table><tr><td rowspan="2">Model</td><td colspan="4">Translated STSBenchmark (dev/test)</td><td colspan="2">STS17</td></tr><tr><td>en-en</td><td>fr-fr</td><td>es-es</td><td>de-de</td><td>es-es</td><td>es-en</td></tr><tr><td>Multi-task en-fr</td><td>0.803/0.763</td><td>0.777/0.738</td><td>1</td><td></td><td></td><td>1</td></tr><tr><td>Trans.-ranking en-fr</td><td>0.668 / 0.558</td><td>0.641 /0.579</td><td>1</td><td></td><td>1</td><td>1</td></tr><tr><td>Multi-task en-es</td><td>0.809/0.770</td><td></td><td>0.779/0.744</td><td></td><td>0.827</td><td>0.769</td></tr><tr><td>Trans.-ranking en-es</td><td>0.669 /0.631</td><td></td><td>0.622 / 0.611</td><td>1</td><td>0.642</td><td>0.587</td></tr><tr><td>Multi-task en-de</td><td>0.802/0.764</td><td></td><td></td><td>0.768/0.722</td><td></td><td>1</td></tr><tr><td>Trans.-ranking en-de</td><td>0.673 /0.632</td><td></td><td></td><td>0.630 / 0.526</td><td>1</td><td>1</td></tr><tr><td>ECNU</td><td></td><td></td><td></td><td>一</td><td>0.856</td><td>0.813</td></tr><tr><td>BIT</td><td></td><td></td><td></td><td></td><td>0.846</td><td>0.749</td></tr></table>
+
+# 4.1 MULTILINGUAL NLI
+
+We evaluate the zero-shot classification performance of our multi-task models on two multilingual natural language inference (NLI) tasks. However, prior to doing so, we first train a modified version5 of our multi-task models that also includes training on the English Multi-genre NLI (MultiNLI) dataset of Williams et al. (2018) in addition to SNLI. We train with MultiNLI to be consistent with the baselines we compare to, which also use MultiNLI.
+
+First, we make use of the professionally translated French and Spanish subsets of SNLI created by Agic & Schluter (2017) for an initial cross-lingual zero-shot evaluation of French and Spanish. We ´ refer to these translated subsets as SNLI-X. There are 1000 examples in the translated subsets for each language. To evaluate, we simply feed the French and Spanish examples into the pre-trained English NLI sub-network of our multi-task models.
+
+We also use the more recent dataset (XNLI) of Conneau et al. (2018), which provides a means for multilingual NLI evaluation in Spanish, French, German, Chinese and more. Since XNLI provides non-European-language evaluations, we also train an English-Chinese (en-zh) version of our multitask model. There are 5000 examples in each XNLI test set, and zero-shot evaluation is once again done by feeding non-English examples into the pre-trained English NLI sub-network.
+
+Table 3 lists the accuracy on the English SNLI test set as well as on SNLI-X and XNLI for all of our multi-task models. The original English SNLI accuracies are around $84 \%$ for all of our multitask models, indicating that English SNLI performance remains stable in the multi-task training setting. The zero-shot accuracy on SNLI-X is around $74 \%$ for both the en-fr and en-es models. The zero-shot accuracy on XNLI is around $65 \%$ for en-es, en-fr, and en-de, and around $63 \%$ for en-zh, thereby significantly outperforming the pretrained sentence encoding baselines (X-CBOW) described in Conneau et al. (2018). The X-CBOW baselines use fixed sentence encoders that are the result of averaging tuned multilingual word embeddings.
+
+Row 4 of Table 3 shows the zero-shot French NLI performance of Eriguchi et al. (2018), which is a state-of-the-art zero-shot NLI classifiers based on multilingual NMT embeddings. Our multi-task model shows comparable performance to the NMT-based model in both English and French.
+
+# 4.2 AMAZON REVIEW
+
+Zero-shot Learning. We also conduct a zero-shot evaluation based on the Amazon review data extracted by Prettenhofer & Stein (2010). We preprocess the Amazon reviews and convert the data into a sentiment classification task by considering reviews with strictly more than three stars as positive and strictly less than three stars as negative, in the same manner as Prettenhofer & Stein (2010). Each review contains a summary field and a text field, which we concatenate to produce a single input. As the multi-task models are trained with sentence lengths clipped to 64, we only take the first 64 tokens from the the concatenated text as the input. There are 6000 training reviews in English, which we split into $90 \%$ for training and $10 \%$ for development.
+
+Table 3: Zero-shot classification accuracy $( \% )$ on SNLI-X and XNLI datasets.   
+
+<table><tr><td rowspan=2 colspan=1>Model</td><td rowspan=1 colspan=3>SNLI-X</td><td rowspan=1 colspan=5>XNLI</td></tr><tr><td rowspan=1 colspan=1>en</td><td rowspan=1 colspan=1>fr</td><td rowspan=1 colspan=1>es</td><td rowspan=1 colspan=1>en</td><td rowspan=1 colspan=1>fr</td><td rowspan=1 colspan=1>es</td><td rowspan=1 colspan=1>de</td><td rowspan=1 colspan=1>zh</td></tr><tr><td rowspan=1 colspan=1>Multi-task en-frMulti-task en-esMulti-task en-deMulti-task en-zh</td><td rowspan=1 colspan=1>84.283.984.183.7</td><td rowspan=1 colspan=1>74.0111</td><td rowspan=1 colspan=1>175.911</td><td rowspan=1 colspan=1>71.670.271.569.2</td><td rowspan=1 colspan=1>64.4111</td><td rowspan=1 colspan=1>165.211</td><td rowspan=1 colspan=1>1165.01</td><td rowspan=1 colspan=1>11162.8</td></tr><tr><td rowspan=1 colspan=1>Eriguchi et al. (2018) (NMT en-fr)XNLI-CBOW zero-shot</td><td rowspan=1 colspan=1>84.41</td><td rowspan=1 colspan=1>73.91</td><td rowspan=1 colspan=1>11</td><td rowspan=1 colspan=1>64.5</td><td rowspan=1 colspan=1>160.3</td><td rowspan=1 colspan=1>160.7</td><td rowspan=1 colspan=1>161.0</td><td rowspan=1 colspan=1>158.8</td></tr><tr><td rowspan=1 colspan=3>Nonzero-shot baselines</td><td rowspan=1 colspan=6>aselines</td></tr><tr><td rowspan=1 colspan=1>XNLI-BiLSTM-lastXNLI-BiLSTM-max</td><td rowspan=1 colspan=1>11</td><td rowspan=1 colspan=1>11</td><td rowspan=1 colspan=1>11</td><td rowspan=1 colspan=1>71.073.7</td><td rowspan=1 colspan=1>65.267.7</td><td rowspan=1 colspan=1>67.868.7</td><td rowspan=1 colspan=1>66.667.7</td><td rowspan=1 colspan=1>63.765.8</td></tr></table>
+
+We first encode inputs using the pre-trained multi-task and translation-ranking encoders and feed the encoded vectors into a 2-layer feed-forward network culminating in a softmax layer. We use layers of size 512 and tanh activation functions in each layer. We use Adam for optimization with an initial learning rate of 0.0005 and a learning rate decay of 0.9 at every epoch during training. We use a batch size of 16 and train for 20 total epochs in all experiments. We freeze the cross-lingual encoder during training. The model architecture and parameters are tuned on the development set.
+
+We first train the classifier on English data, and then evaluate it on the 6000 French and German Amazon review test examples. The results are summarized in Table 4. The accuracy on the English test set is $8 7 . 4 \%$ for the en-fr model and $8 7 . 1 \%$ for the en-de model, with the zero-shot accuracy being above $80 \%$ for both models. The translation-ranking models again perform worse on all metrics. Once again we compare the proposed model with Eriguchi et al. (2018), and find that our zero-shot performance has a reasonable gain on the fr test set6.
+
+Table 4: Zero-shot sentiment classification accuracy $\% )$ on target language Amazon review test data after training on only English Amazon review data.
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>en</td><td rowspan=1 colspan=1>fr</td><td rowspan=1 colspan=1>de</td></tr><tr><td rowspan=1 colspan=1>Multi-task en-frTranslation-ranking en-fr</td><td rowspan=1 colspan=1>87.474.4</td><td rowspan=1 colspan=1>82.366.3</td><td rowspan=1 colspan=1>11</td></tr><tr><td rowspan=1 colspan=1>Multi-task en-deTranslation-ranking en-de</td><td rowspan=1 colspan=1>87.173.8</td><td rowspan=1 colspan=1>11</td><td rowspan=1 colspan=1>81.067.0</td></tr><tr><td rowspan=1 colspan=1>Eriguchi et al. (2018) (NMT en-fr)</td><td rowspan=1 colspan=1>83.2</td><td rowspan=1 colspan=1>81.3</td><td rowspan=1 colspan=1>1</td></tr></table>
+
+Few-shot Learning. We further evaluate the proposed multi-task models via few-shot learning, by training on English reviews and only a portion of French and German reviews. Our few-shot models are compared with baselines of training on French and German reviews only. Table 5 shows the classification accuracy of the few-shot models, where the second row shows the percent of French and German data that is used when training each model. With as little as $20 \%$ of the French or German training data, the few-shot models perform nearly as good as the baseline models trained on $100 \%$ of the French and German data. Adding more French and German training data leads to further improvements in few-shot model performance, with the few-shot models reaching $8 5 . 8 \%$ accuracy in French and $8 4 . 5 \%$ accuracy in German when using all of the French and German data.
+
+# 5 ANALYSIS OF CROSS-LINGUAL EMBEDDING SPACES
+
+Motivated by the recent work of Søgaard et al. (2018) studying the graph structure of multilingual word representations, we perform a similar analysis for our learned cross-lingual sentence representations. To do so, we take $N$ samples of size $K$ from en-fr, en-es, and en-de translation data and then encode these samples using the corresponding multi-task and translation-ranking models. We then compute pairwise distance matrices within each sampled set of encodings, and use these distance matrices to construct graph Laplacians7. Finally, we obtain the similarity $\Psi ( S , T )$ between each model’s source and target language embedding subsets by comparing the eigenvalues of the source language graph Laplacians to the eigenvalues of the target language graph Laplacians as follows:
+
+Table 5: Sentiment classification accuracy $( \% )$ on target language Amazon review test data after training on English Amazon review data and a portion of French of German data. The second row shows the percent of French (fr) or German (de) data is used for training in each model.   
+
+<table><tr><td rowspan="2">Model</td><td colspan="6">fr</td><td colspan="6">de</td></tr><tr><td>0%</td><td>10%</td><td>20%</td><td>40%</td><td>80%</td><td>100%</td><td>0%</td><td>10%</td><td>20%</td><td>40%</td><td>80%</td><td>100%</td></tr><tr><td>Few-shot</td><td>82.3</td><td>84.4</td><td>84.4</td><td>84.8</td><td>85.2</td><td>85.8</td><td>81.0</td><td>81.6</td><td>83.3</td><td>84.0</td><td>84.7</td><td>84.5</td></tr><tr><td>Baseline</td><td>1</td><td>79.2</td><td>80.0</td><td>82.7</td><td>84.3</td><td>84.9</td><td>1</td><td>75.5</td><td>77.7</td><td>81.6</td><td>83.5</td><td>84.4</td></tr></table>
+
+$$
+\Psi ( S , T ) = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \sum _ { j = 1 } ^ { K } ( \lambda _ { j } ( L _ { i } ^ { ( s ) } ) - \lambda _ { j } ( L _ { i } ^ { ( t ) } ) ) ^ { 2 }
+$$
+
+Where L(s)i a nd $\boldsymbol { L } _ { i } ^ { ( t ) }$ refer to the graph Laplacians of the source language and target language sentences obtained from the $i ^ { t h }$ sample of source-target translation pairs. A smaller value of $\Psi ( S , T )$ indicates higher eigen-similarity of the source language and target language embedding subsets. Following Søgaard et al. (2018) we use a sample size of $K = 1 0$ translation pairs, but we choose to use $N = 1 0 0 0$ samples instead of $N = 1 0$ (as was done in their work) since we found $\Psi ( S , T )$ to have very high variance at $N = 1 0$ . The computed values of $\Psi ( S , T )$ for our multi-task and translation-ranking models are summarized in Table 6.
+
+Table 6: Average eigen-similarity values of source and target embedding subsets.   
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>en-fr</td><td rowspan=1 colspan=1>en-es</td><td rowspan=1 colspan=1>en-de</td></tr><tr><td rowspan=1 colspan=1>multi-tasktranslation-ranking</td><td rowspan=1 colspan=1>0.5921.036</td><td rowspan=1 colspan=1>0.5260.572</td><td rowspan=1 colspan=1>0.7612.187</td></tr></table>
+
+We find that the source and target embedding subsets constructed from the multi-task models exhibit greater average eigen-similarity than those resulting from the translation-ranking models for all source-target language pairs. This result is not necessarily intuitive, since one might expect the translation-ranking model to optimize more for alignment. Given that eigen-similarity correlates with the better performance of the multi-task models in almost all tasks, a potential direction for future work could be to introduce regularization penalties based on graph similarity in multitask training. Interestingly, we also observe that the eigen-similarity gaps between the multi-task and translation-ranking models are not uniform across language pairs (although it may be that translation-ranking requires even more training). Thus, another direction could be to further study differences in the difficulty of aligning different source-target language embeddings.
+
+# 6 CONCLUSION
+
+In this work, we explored a straightforward framework for training cross-lingual, multi-task dualencoder models. We showed that by training English-French, English-Spanish, and English-German multi-task models using our setup, we can achieve near-state-of-the-art or state-of-the-art performance in a variety of English tasks while also being able to produce similar caliber results in zeroshot transfer learning tasks for other languages. Finally, we note that the fact that multi-task training can actually improve performance on some downstream English tasks (TREC) is particularly interesting, and believe that there are many possibilities for future explorations of cross-lingual model training.
+
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+
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+
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+
+# A SUPPLEMENTARY MATERIAL
+
+# A.1 DATA PREPROCESSING
+
+In order to effectively use Reddit, Wikipedia, and translation data, we do a significant amount of preprocessing on the raw data. We describe our preprocessing procedures for each dataset in the following paragraphs.
+
+Reddit. The raw Reddit corpus extracted by Al-Rfou et al. (2016) consists of 2.4 billion posts and comments from Reddit between 2007 and 2016, making it our largest data source by an order of magnitude. To preprocess this data for training our conversation response prediction task, we follow the same procedure as Yang et al. (2018). Essentially, we consider Reddit comments and their children (response comments) as input-response pairs. We filter out comments that have $\geq 3 5 0$ characters, due to the limitations on the number of input tokens our models can accept. We also remove comments that start with “https”, “ $@$ ”, or $\mathrm { ^ { 6 6 } } / \mathrm { r } / \mathrm { ^ { 9 } }$ , and also remove comments whose authors have “bot” in their usernames. Lastly, we find that Reddit comments also contain a small mix of nonEnglish text, and we filter out comments where the percentage of alphabetic characters is $\leq 7 0 \%$ . As mentioned in the paper, the final, processed Reddit data consists of 600 million input-response pairs.
+
+Since Reddit comments are predominantly English, we found the original data to be unsuitable for training encoders for non-English languages. To create conversational corpora for other languages, we translate the entire Reddit dataset using Google’s neural machine translation (NMT) system of Wu et al. (2016).
+
+Wikipedia. One concern with using the translated Reddit corpus as a monolingual task for nonEnglish languages is that the translated data will propagate any errors made by the NMT system used for translation. To ameliorate this problem, we crawl Wikipedia and extract triplets of contiguous sentences from Wikipedia articles. We use Wikipedia due to it having well-formed articles with a broad coverage of several languages. We extract all Wikipedia articles for English, French, Spanish and German from the Wikipedia dump of May 5, 2018, and use the sentence splitter model of Gillick (2009) to split the articles into sentences. All article titles and article section names are treated as sentences inline. As mentioned in the paper, our final extracted Wikipedia corpus consists of 127.9 million English sentence triplets, 49.5 million French triplets, 29.8 million Spanish triplets, and 49.3 million German triplets for French, Spanish and German respectively.
+
+Stanford Natural Language Inference (SNLI). We use the English Stanford Natural Language Inference (SNLI) dataset consisting of 570K sentence pairs of Bowman et al. (2015) as is, without any further preprocessing. We also use the provided splits for training (550K), validation (10K), and testing (10K).
+
+Translation. The data for training the translation task is crawled from the public web using a system similar to the approach described by Uszkoreit et al. (2010). The extracted data is further cleaned by a pre-trained translation pair scoring system. We then generate “hard-negatives” following Guo et al. (2018) by using a pre-trained coarse translation-ranking model to determine translations that are close to a correct translation. As mentioned in the paper, the final constructed corpus contains around 600M en-fr pairs, 470M en-es pairs and 500M en-de pairs.
+
+# A.2 DOWNSTREAM TASK DESCRIPTIONS
+
+Each of the English downstream tasks shown in Table 1 in the main body of the paper are described briefly below:
+
+• Movie Reviews (MR): Multi-class prediction (scale of one to five stars) on movie review snippet data from Pang & Lee (2005).   
+Customer Reviews (CR): Binary sentiment classification of sentences from customer reviews mined by Hu & Liu (2004).   
+• Subjectivity (SUBJ): Binary classification of sentences from movie reviews (Pang & Lee, 2004) as either objective or subjective.   
+• Multi-Perspective Question Answering (MPQA): Multi-class opinion polarity prediction on news data from Wiebe et al. (2005). Text REtrieval Conference (TREC): Multi-class classification of questions obtained from TREC by Li & Roth (2002).   
+Stanford Sentiment Treebank (SST): Phrase-level binary sentiment classification of text from Socher et al. (2013).   
+• Semantic Text Similarity (STS): Semantic textual similarity of sentence pairs in the form of Pearson correlation with human judgments (Cer et al., 2017).
+
+# A.3 STATE-OF-THE-ART STS SYSTEM DESCRIPTIONS
+
+For the STS evaluations provided in the body of our paper, we compared to the state-of-the-art systems of Tian et al. (2017) and Wu et al. (2017). We refer the reader to Cer et al. (2017) for a detailed summary of the characteristics of these systems, but we provide a brief characterization below for convenience:
+
+• ECNU: The ECNU model is an ensemble of random forest, gradient boosted decision trees, and deep learning models that leverages several manually constructed features such as ngram overlap, edit distance, longest common prefix, common substrings, word alignments, and more. BIT: The BIT model relies mostly on a feature called information content (IC), which is based on the likelihood of occurrence of a concept (in this case, a word). Information content is then computed and combined for sentences through a hierarchical approach, which Wu et al. (2017) also augment with word embedding alignment to produce a final ensemble model.
+
+# A.4 ABLATION TESTS AND FURTHER EXPERIMENTS
+
+No NLI. Given that the multi-task models detailed in the body of the paper are trained with an English NLI task but no non-English NLI tasks, we evaluate how much training without this English NLI task affects model performance on downstream English tasks. The performance of these no-NLI models is summarized under the “Cross-lingual Multitask Transformer No SNLI” section of Table 7. We find that training without SNLI leads to comparable or better performance on all English downstream tasks except for STS, where we find training with SNLI provides a significant bump in performance.
+
+No Wikipedia. We similarly investigate whether there is a significant advantage gained from using the only native target language data source, Wikipedia. To do so, we train en-fr, en-es, and en-de multi-task models without the Wikipedia-based Quick Thought task. The performance of these nowiki models is shown under the “Cross-lingual Multitask Transformer No Wiki” section of Table 7. We find that training without the Wikipedia Quick Thought task does not have a strong impact on model performance, with some tasks being marginally better and others being marginally worse.
+
+No target language native tasks. Additionally, the cross-lingual, multi-task models discussed in the body of our paper use a largely parallel task setup, where for each source-target language pair the monolingual target tasks are mirrored from the monolingual source tasks (the only exception being NLI, which is English-only). To test how useful this mirroring of monolingual tasks is, we train several models without target language monolingual tasks. We label these models as the “no non-English native task” models, and also summarize their downstream English task performance in Table 7. We note that removing the non-English native tasks actually leads to significant decreases in performance on TREC, SST, and STS. We find this quite interesting, as it provides some more empirical justification for the notion that cross-lingual training for a source-target language pair can actually improve monolingual source task performance.
+
+Finally, we also evaluate the zero-shot performance of the no non-English native task models on Amazon review sentiment classification. As can be seen in Table 8, training without the non-English native tasks leads to lower performance on English and roughly the same zero-shot classification performance as training with only translation pair data.
+
+Table 7: English task performance of different model configurations. Notably, removing non-English tasks in encoder training actually hurts performance on downstream English tasks.   
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>MR</td><td rowspan=1 colspan=1>CR</td><td rowspan=1 colspan=1>SUBJ</td><td rowspan=1 colspan=1>MPQA</td><td rowspan=1 colspan=1>TREC</td><td rowspan=1 colspan=1>SST</td><td rowspan=1 colspan=1>STSBenchmark(dev /test)</td></tr><tr><td rowspan=1 colspan=8>Cross-lingualMultitaskTransformerSNLI + Native Tasks (report above)</td></tr><tr><td rowspan=1 colspan=1>en-fren-esen-de</td><td rowspan=1 colspan=1>77.980.178.8</td><td rowspan=1 colspan=1>82.985.984.0</td><td rowspan=1 colspan=1>95.594.695.9</td><td rowspan=1 colspan=1>89.386.587.6</td><td rowspan=1 colspan=1>95.396.296.1</td><td rowspan=1 colspan=1>84.085.285.0</td><td rowspan=1 colspan=1>0.803/0.7630.809 /0.7700.802/0.764</td></tr><tr><td rowspan=1 colspan=8>Cross-lingual Multitask TransformerNo SNLI</td></tr><tr><td rowspan=1 colspan=1>en-fren-esen-de</td><td rowspan=1 colspan=1>80.580.080.5</td><td rowspan=1 colspan=1>86.589.485.5</td><td rowspan=1 colspan=1>94.095.694.0</td><td rowspan=1 colspan=1>89.190.590.0</td><td rowspan=1 colspan=1>96.694.592.8</td><td rowspan=1 colspan=1>85.085.782.6</td><td rowspan=1 colspan=1>0.747/0.7220.754 /0.7300.737 /0.723</td></tr><tr><td rowspan=1 colspan=8>Cross-lingual Multitask TransformerNo Wiki</td></tr><tr><td rowspan=1 colspan=1>en-fren-esen-de</td><td rowspan=1 colspan=1>78.979.179.6</td><td rowspan=1 colspan=1>85.887.485.9</td><td rowspan=1 colspan=1>92.194.594.1</td><td rowspan=1 colspan=1>90.089.689.7</td><td rowspan=1 colspan=1>94.794.392.2</td><td rowspan=1 colspan=1>82.783.082.5</td><td rowspan=1 colspan=1>0.804/0.7740.809 / 0.7690.801/0.741</td></tr><tr><td rowspan=1 colspan=8>Cross-lingual Multitask TransformerNo non-English native tasks</td></tr><tr><td rowspan=1 colspan=1>en-fren-esen-de</td><td rowspan=1 colspan=1>79.477.479.3</td><td rowspan=1 colspan=1>84.081.783.7</td><td rowspan=1 colspan=1>93.594.994.0</td><td rowspan=1 colspan=1>89.789.488.1</td><td rowspan=1 colspan=1>93.894.091.2</td><td rowspan=1 colspan=1>83.482.382.5</td><td rowspan=1 colspan=1>0.797/0.7580.796 /0.7610.760 / 0.732</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Translo</td><td rowspan=1 colspan=1>tion-rankin</td><td rowspan=1 colspan=1>gModels</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>en-fren-esen-de</td><td rowspan=1 colspan=1>68.767.767.8</td><td rowspan=1 colspan=1>79.375.775.2</td><td rowspan=1 colspan=1>87.083.584.4</td><td rowspan=1 colspan=1>81.886.083.6</td><td rowspan=1 colspan=1>89.494.486.8</td><td rowspan=1 colspan=1>74.272.674.6</td><td rowspan=1 colspan=1>0.668/0.5580.669 / 0.6310.673 /0.632</td></tr></table>
+
+Table 8: Zero-shot sentiment classification accuracy $\% )$ on target language Amazon review test data after training on only English Amazon review data.
+
+<table><tr><td>Model</td><td>en</td><td>fr</td><td>de</td></tr><tr><td>Multi-task en-fr Multi-task No non-English native tasks en-fr</td><td>87.4 85.6</td><td>82.3 65.4</td><td>1</td></tr><tr><td>Translation-ranking en-fr</td><td>74.4</td><td>66.3</td><td>1 1</td></tr><tr><td>Multi-task en-de</td><td>87.1</td><td>1</td><td>81.0</td></tr><tr><td>Multi-task No non-English native tasks en-fr</td><td>85.1</td><td>1</td><td>67.2</td></tr><tr><td>Translation-ranking en-de</td><td>73.8</td><td>1</td><td>67.0</td></tr></table>

parse/train/BJgGhiR5KX/BJgGhiR5KX_content_list.json

@@ -0,0 +1,1512 @@
+[
+    {
+        "type": "text",
+        "text": "LEARNING CROSS-LINGUAL SENTENCE REPRESENTATIONS VIA A MULTI-TASK DUAL-ENCODER MODEL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            171,
+            398,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            234,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A significant roadblock in multilingual neural language modeling is the lack of labeled non-English data. One potential method for overcoming this issue is learning cross-lingual text representations that can be used to transfer the performance from training on English tasks to non-English tasks, despite little to no task-specific non-English data. In this paper, we explore a natural setup for learning crosslingual sentence representations: the dual-encoder. We provide a comprehensive evaluation of our cross-lingual representations on a number of monolingual, crosslingual, and zero-shot/few-shot learning tasks, and also give an analysis of different learned cross-lingual embedding spaces. ",
+        "bbox": [
+            233,
+            267,
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+            392
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            420,
+            336,
+            435
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "There has been a significant amount of recent work on developing models that can produce sentence representations that are useful for a number of language processing tasks (Kiros et al., 2015; Conneau et al., 2017; Subramanian et al., 2018; Logeswaran & Lee, 2018; Cer et al., 2018). However, these models are trained on largely monolingual data, and can thus only be used for tasks in a single language. A promising direction for extending the previous models to multiple languages is learning cross-lingual embedding spaces (Schwenk et al., 2017; Eriguchi et al., 2018; Singla et al., 2018), which could be used to transfer performance in one language to others. ",
+        "bbox": [
+            174,
+            452,
+            825,
+            549
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We develop a novel approach for cross-lingual representation learning by combining the dualencoder architectures used for learning sentence representations (Logeswaran & Lee, 2018; Cer et al., 2018) and for bi-text retrieval (Guo et al., 2018). By doing so, we learn representations that maintain state-of-the-art performance in tasks for a source language while simultaneously obtaining state-of-the-art performance in zero-shot learning tasks for a target language. For a given sourcetarget language pair, we construct a multi-task training scheme using native source language tasks, native target language tasks, and a bridging source-target translation retrieval task to learn sentence representations that are aligned between the source and target languages. We then evaluate the learned representations on several monolingual and cross-lingual tasks, and also provide a graphbased analysis of the learned representations. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We find that multi-task training using additional monolingual tasks improves performance over models that only make use of parallel data on both cross-lingual semantic textual similarity (STS) (Cer et al., 2017) and Søgaard et al. (2018)’s cross-lingual eigen-similarity metric. The results show that the addition of monolingual data improves the embedding alignment of sentences and their translations. Furthermore, we find that cross-lingual training with additional monolingual data leads to far better transfer learning performance, and we show that our cross-lingual representations outperform state-of-the-art zero-shot learning models in sentiment classification and natural language inference. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "2 MULTI-TASK DUAL-ENCODER MODEL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            820,
+            524,
+            837
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The core of our approach is the idea of modeling various tasks as ranking input-response pairs by encoding them via two encoders, with the crucial task for learning cross-lingual representations being translation ranking. For translation ranking, as well as for our other tasks, we take an input sentence $s _ { i } ^ { I }$ and an associated response sentence $s _ { i } ^ { R }$ , and we seek to rank $s _ { i } ^ { R }$ over all other possible response sentences $s _ { j } ^ { R } \\in \\mathcal S ^ { R }$ . To do so, we model the conditional probability $P ( s _ { i } ^ { R } \\mid s _ { i } ^ { I } )$ as: ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/b1f75398f482a5e36c4d08125507fc51b9dfc4aca41e1aa374a6e53486c83078.jpg",
+        "image_caption": [
+            "Figure 1: Multi-task dual-encoder model. It consists of a group of native tasks in each language and a bridging task using translation pair data. The encoders in the gray box all share their parameters, and thus constitute $g$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            99,
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+        ],
+        "page_idx": 1
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+    {
+        "type": "equation",
+        "img_path": "images/6f6115400158993c3f347c2ca0927896779986f6ef45dd9b0673b3b7cd8ccbd0.jpg",
+        "text": "$$\nP ( s _ { i } ^ { R } \\mid s _ { i } ^ { I } ) = \\frac { e ^ { \\phi ( s _ { i } ^ { I } , s _ { i } ^ { R } ) } } { \\sum _ { s _ { j } ^ { R } \\in \\mathcal { S } ^ { R } } e ^ { \\phi ( s _ { i } ^ { R } , s _ { j } ^ { R } ) } } , \\quad \\phi ( s _ { i } ^ { I } , s _ { j } ^ { R } ) = g ^ { I } ( s _ { i } ^ { I } ) ^ { \\top } g ^ { R } ( s _ { j } ^ { R } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            279,
+            361,
+            718,
+            406
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Where $g ^ { I }$ and $g ^ { R }$ are the input and response sentence encoding functions that compose the dualencoder. Since the normalization term in equation 1 is computationally intractable, we follow the approaches of Henderson et al. (2017) and instead choose to model an approximate conditional probability $\\mathcal { \\widetilde { P } } ( s _ { i } ^ { R } \\mid s _ { i } ^ { I } )$ : ",
+        "bbox": [
+            173,
+            417,
+            825,
+            478
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/08f8bdffb0fd8ff8a0e65f60c962b0346c26f6a90195905a1c08fcebb2dac861.jpg",
+        "text": "$$\n\\displaystyle \\widetilde { P } ( s _ { i } ^ { R } \\mid s _ { i } ^ { I } ) = \\frac { e ^ { \\phi ( s _ { i } ^ { I } , s _ { i } ^ { R } ) } } { \\sum _ { j = 1 , j \\neq i } ^ { K } e ^ { \\phi ( s _ { i } ^ { R } , s _ { j } ^ { R } ) } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            382,
+            494,
+            614,
+            537
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Where $K$ denotes the size of a single batch of training examples, and the $s _ { j } ^ { R }$ correspond to the response sentences associated with the other input sentences in the same batch as $s _ { i } ^ { I }$ . We parametrize $g ^ { I }$ and $g ^ { R }$ as deep neural networks that are trained to minimize the negative log-likelihood of $\\widetilde { P } ( s _ { i } ^ { R } \\mid$ $s _ { i } ^ { I }$ ) for each task. ",
+        "bbox": [
+            173,
+            547,
+            825,
+            612
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In order to produce a single sentence encoding function $g$ that can be evaluated on downstream tasks, we share several layers between the input and response encoders and treat the final output of these shared layers as $g$ . Additionally, these layers are modeled after the Universal Sentence Encoder (USE) model of Cer et al. (2018), since it is the state-of-the-art model that is most amenable to our setup. To learn cross-lingual representations, we train $g$ on several tasks from mirrored corpora across languages1 for the source-target language pairs English-French (en-fr), English-Spanish (enes), and English-German (en-de). The resulting model structure is illustrated in Figure 1. ",
+        "bbox": [
+            174,
+            618,
+            825,
+            717
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 ENCODER ARCHITECTURE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            734,
+            398,
+            748
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Word and Character Embeddings. As part of the training process for learning the cross-lingual sentence encoding function $g$ , we learn embeddings for the words and characters present in the training data for a given source-target language pair. Word embeddings are learned end-to-end, as we noticed that pre-trained embeddings did not make a difference for final performance. Character embeddings are learned in a similar manner, but with the added stipulation that we consider character n-gram embeddings instead of single character embeddings by using a single feedforward layer with tanh activation on top of character n-grams. Each word in an input sentence then obtains a character representation by having its character n-gram representations summed together. To have the sentence encoder $g$ leverage the word and character representations together without drastically increasing its number of parameters, we sum the word and character embeddings before using them as input to $g$ . ",
+        "bbox": [
+            174,
+            760,
+            825,
+            898
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Transformer Encoder. The actual architecture of the shared encoder $g$ consists of three2 layers of transformer stacks, which contain the feed-forward and multi-head attention sub-layers described in Vaswani et al. (2017). The transformer encoder output is a variable-length sequence at each stack. We average encodings of all sequence positions in the final layer as the final sentence encoding. This embedding is then fed into different sets of feedforward layers that are used for each task. For our transformer layers, we use 8 attentions heads, a hidden size of 512, and a filter size of 2048. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            186
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 MULTI-TASK TRAINING SETUP ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            208,
+            429,
+            222
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To learn a function $g$ that is capable of strong cross-lingual matching and transfer learning performance for a source-target language pair while also maintaining monolingual downstream task performance, we employ four unique task types for each language pair. Specifically, we employ $a$ conversation response prediction task, a quick thought task, a natural language inference task, and a bridging task – translation ranking. Six total tasks are used in training, as the first two tasks are mirrored across languages. ",
+        "bbox": [
+            174,
+            234,
+            825,
+            319
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Conversation Response Prediction. We model the conversation response prediction task in the same manner as Yang et al. (2018). We minimize the negative log-likelihood of $\\mathcal { P } ( s _ { i } ^ { R } \\mid s _ { i } ^ { I } )$ , where $s _ { i } ^ { I }$ is a single comment and $s _ { i } ^ { R }$ is its associated response comment. For the response side, we model $g ^ { \\dot { R } } ( s _ { i } ^ { R } )$ as two fully-connected feedforward layers of size 320 and 512 with tanh activation on top of $g ( s _ { i } ^ { R } )$ . For the input side, however, we simply let $g ^ { I } ( s _ { i } ^ { I } ) = g ( s _ { i } ^ { I } )$ , as we noticed in early experiments that letting the optimization of the conversational response task more directly influence the parameters of the underlying sentence encoder $g$ led to better downstream task performance. ",
+        "bbox": [
+            173,
+            325,
+            825,
+            430
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Quick Thought. We use a modified version of the Quick Thought task detailed by Logeswaran & Lee (2018). We minimize the sum of the negative log-likelihoods of $\\widetilde { P } ( s _ { i } ^ { R } \\mid s _ { i } ^ { I } )$ and $\\bar { \\widetilde { P } } ( s _ { i } ^ { P } \\mid s _ { i } ^ { I } )$ , where $s _ { i } ^ { I }$ is a sentence taken from an article and $s _ { i } ^ { P }$ and $s _ { i } ^ { R }$ are its predecessor and successor sentences respectively. For this task, we model all three of $g ^ { P } ( s _ { i } ^ { P } )$ , $g ^ { I } ( s _ { i } ^ { I } )$ , and $g ^ { R } ( s _ { i } ^ { R } )$ using separate, fully-connected feedforward layers of size 320 and 512 with tanh activation on top of $g$ , as we did for $\\dot { \\boldsymbol g } ^ { R } ( s _ { i } ^ { R } )$ in our conversational modeling task. ",
+        "bbox": [
+            174,
+            436,
+            825,
+            525
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Natural Language Inference (NLI). We also include an English-only natural language inference task based on Bowman et al. (2015). For this task, we first encode an input sentence $s _ { i } ^ { I }$ and its corresponding response hypothesis $s _ { i } ^ { R }$ into vectors $u _ { 1 }$ and $u _ { 2 }$ using $g$ . The vectors $u _ { 1 } , u _ { 2 }$ are then used to construct a feature vector $( u _ { 1 } , u _ { 2 } , | u _ { 1 } - u _ { 2 } | , u _ { 1 } * u _ { 2 } )$ , where $( \\cdot )$ represents concatenation and $^ *$ represents element-wise multiplication. The form of this feature vector is derived from the original experiments of Bowman et al. (2015). This feature vector is then fed into a single feedforward layer of size 512 that is used to perform the 3-way NLI classification. ",
+        "bbox": [
+            174,
+            531,
+            825,
+            631
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Translation Ranking. Our translation task setup is identical to the one used by Guo et al. (2018) for bi-text retrieval. We minimize the negative log-likelihood of $\\widetilde { P } ( s _ { i } \\mid t _ { i } )$ , where $( s _ { i } , t _ { i } )$ is a sourcetarget translation pair. Since the translation task is intended to align the sentence representations for the source and target languages, we do not use any kind of task-specific feedforward layers and instead use $g$ as both $g ^ { I }$ and $g ^ { R }$ . Following Guo et al. (2018), we append 5 translations that are similar to the correct translation to each training example as “hard-negatives”. Similarity is determined via a version of our model trained only on the translation ranking task. We did not see additional gains from using more than 5 hard-negatives. ",
+        "bbox": [
+            174,
+            637,
+            825,
+            752
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            776,
+            326,
+            791
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 CORPORA ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We draw upon multiple, openly available data sources and training corpora for the training of the tasks mentioned above. For each of our datasets, we use $90 \\%$ of the data for training, and the remaining $10 \\%$ for development/validation. Our data preprocessing procedures are described in detail in the supplementary material. ",
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+    },
+    {
+        "type": "text",
+        "text": "Reddit. We preprocess the Reddit data extracted by Al-Rfou et al. (2016) into 600 million inputresponse comment pairs for training our conversation response prediction task. We also translate this data using the Google neural machine translation (NMT) system of Wu et al. (2016). ",
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+    },
+    {
+        "type": "text",
+        "text": "Wikipedia. To get native, non-English data, we extract triplets of contiguous sentences from English, French, Spanish, and German articles take from Wikipedia. Our final extracted corpus of Wikipedia sentence triplets consists of 127.9, 49.5, 29.8, and 49.3 million triplets for English, French, Spanish, and German respectively, which we use to train our Quick Thought task. ",
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+    },
+    {
+        "type": "text",
+        "text": "Stanford Natural Language Inference (SNLI). The NLI data we use is taken from the Stanford Natural Language Inference (SNLI) dataset of Bowman et al. (2015), which consists of 570K sentence pairs associated with one of three labels: entailment, contradiction, or neutral. The corpus is split into training (550K), validation (10K), and testing sets (10K). ",
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+    },
+    {
+        "type": "text",
+        "text": "Translation. The data for training the translation task is constructed using a system similar to the approach described by Guo et al. (2018). The final constructed corpus contains around 600M en-fr pairs, 470M en-es pairs and 500M en-de pairs. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.2 MODEL CONFIGURATION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In all of our experiments, multi-task training is done by cycling through the different tasks (translation pairs, Reddit, Wikipedia, NLI) and performing an optimization step for a single task at a time. We train all of our models with a batch size of 100 using stochastic gradient descent with a learning rate of 0.008. All of our models are trained for 30 million steps. All input text is tokenized prior to being used for training. We build a vocab containing 200 thousand unigram tokens with 10 thousand hash buckets for out-of-vocabulary tokens. The character n-gram vocab contains 200 thousand hash buckets used for 3 and 4 grams. Both the word and character n-gram embedding sizes are 320. All hyperparameters are tuned based on performance on the development portion (random $10 \\%$ slice) of our datasets. Finally, as an additional training heuristic, we multiply the gradients to the word and character embeddings by a factor of $1 0 0 ^ { 3 }$ . We found that using this embedding gradient multiplier alleviated vanishing gradient issues and greatly improved training. ",
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+    },
+    {
+        "type": "text",
+        "text": "We compare the proposed cross-lingual multi-task (referred to simply as multi-task) models with baseline models that are trained using only the translation ranking task, which we dub as the “translation-ranking” models. ",
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+    },
+    {
+        "type": "text",
+        "text": "3.3 MODEL PERFORMANCE ON ENGLISH DOWNSTREAM TASKS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We first evaluated all of our cross-lingual models on several downstream English tasks taken from SentEval (Conneau & Kiela, 2018) to verify the impact of cross-lingual training. Each task is described in the supplementary material. Results on the tasks are summarized in Table 1. We note that cross-lingual training does not hinder the effectiveness of our encoder on English tasks, as the multi-task models are close to state-of-the-art in each of the downstream tasks. For the Text REtrieval Conference (TREC) eval, we actually find that our multi-task models outperform the previous state-of-the-art models by a sizable amount. ",
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+    },
+    {
+        "type": "text",
+        "text": "3.4 CROSS-LINGUAL RETRIEVAL ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We also evaluate both the multi-task and translation-ranking models’ efficacy in performing crosslingual retrieval by using held-out translation pair data. Following Henderson et al. (2017), we use precision at $\\mathrm { ~ N ~ } ( \\mathrm { P } @ \\mathrm { N } )$ as the evaluation metric by checking if a source sentence’s target translation ranks (where ranking is done using dot product) in the top $N$ scored candidates when considering $K$ other randomly selected target sentences. Unlike Henderson et al. (2017), we set $K$ to be 999 instead of 99 because using $K = 9 9$ results in all metrics quickly shooting up to $9 9 \\%$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The translation-ranking model remains as a strong baseline for finding the true translation, with $9 5 . 4 \\%$ , $8 7 . 5 \\%$ , $9 7 . 5 \\%$ $\\mathrm { P @ 1 }$ for en-fr, en-es and en-de retrieval tasks respectively. The multi-task model performs almost identical with $9 5 . 1 \\%$ , $8 8 . 8 \\%$ and $9 7 . 8 \\%$ , which provides empirical justification that it is possible to maintain embedding space alignment despite optimizing for native tasks in each individual language. We also experimented with $\\mathrm { P @ 3 }$ and $\\mathrm { P @ 1 0 }$ , the results are identical. ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "table",
+        "img_path": "images/be94470b27be940d4e38fe9748038c652d19c017257baaaae1a345b3fc7e1f54.jpg",
+        "table_caption": [
+            "Table 1: Performance on classification transfer tasks. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>MR</td><td>CR</td><td>SUBJ</td><td>MPQA</td><td>TREC</td><td>SST</td><td>STS Bench (dev /test)</td></tr><tr><td colspan=\"7\">Cross-lingual Multi-taskModels</td></tr><tr><td>en-fr</td><td>77.9</td><td>82.9</td><td>95.5</td><td>89.3</td><td>95.3</td><td>84.0</td><td>0.803/0.763</td></tr><tr><td>en-es</td><td>80.1</td><td>85.9</td><td>94.6</td><td>86.5</td><td>96.2</td><td>85.2</td><td>0.809 / 0.770</td></tr><tr><td>en-de</td><td>78.8</td><td>84.0</td><td>95.9</td><td>87.6</td><td>96.1</td><td>85.0</td><td>0.802 /0.764</td></tr><tr><td colspan=\"8\">Translation-ranking Models</td></tr><tr><td>en-fr</td><td>68.7</td><td>79.3</td><td>87.0</td><td>81.8</td><td>89.4</td><td>74.2</td><td>0.668/ 0.558</td></tr><tr><td>en-es</td><td>67.7</td><td>75.7</td><td>83.5</td><td>86.0</td><td>94.4</td><td>72.6</td><td>0.669 / 0.631</td></tr><tr><td>en-de</td><td>67.8</td><td>75.2</td><td>84.4</td><td>83.6</td><td>86.8</td><td>74.6</td><td>0.673 / 0.632</td></tr><tr><td colspan=\"8\">State-of-the-art Models</td></tr><tr><td>InferSent Skip-Thought LN</td><td>81.1 79.4</td><td>86.3 83.1</td><td>92.4 93.7</td><td>90.2 89.3</td><td>88.2 1</td><td>84.6 1</td><td>0.801/0.758</td></tr><tr><td>Quick-Thought</td><td>82.4</td><td>86.0</td><td>94.8</td><td>90.2</td><td>92.4</td><td>87.6</td><td>1 1</td></tr><tr><td></td><td></td><td></td><td>93.9</td><td>87.0</td><td>92.5</td><td>85.4</td><td></td></tr><tr><td>USE Transformer</td><td>81.4</td><td>87.4</td><td></td><td></td><td></td><td></td><td>0.814 /0.782</td></tr></table>",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.5 MULTILINGUAL STS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We further test whether our learned cross-lingual representations can also perform well in their associated non-English language tasks by evaluating semantic textual similarity (STS) performance on French, Spanish, and German. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To evaluate Spanish-Spanish (es-es) STS, we use SemEval-2017 task 1 (STS17) track 3 of Cer et al. (2017), which contains 250 Spanish sentence pairs with human labeled similarity scores. We also evaluate es-en STS by using the track 4(a) task4, which contains 250 en-es sentence pairs. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Beyond English and Spanish, however, there are no standard STS datasets available for other languages. As such, we evaluate on a translated version of the STS Benchmark dataset from Cer et al. (2017) for French, Spanish, and German. We use Google’s translation system to translate the STS Benchmark sentences to French, Spanish and German. We believe that the results on our pseudomultilingual STS Benchmark dataset are expected to still be a reasonable indicator of multilingual semantic similarly performance, since the NMT encoder-decoder architecture for translation differs significantly from our dual-encoder approach. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Following Cer et al. (2018), we first compute the sentence encodings $u , v$ of an STS sentence pair, and then score the sentence pair similarity based on the angular distance between the two vectors, $- \\operatorname { a r c c o s } { \\left( { \\frac { \\boldsymbol { u v } } { | | \\boldsymbol { u } | | \\ | | | \\boldsymbol { v } | | } } \\right) }$ . Table 2 shows the Pearson’s correlation coefficient of the STS tasks for all models. The first column shows the trained model performance on original English STS Benchmark data. Columns 2 to 4 shows the the performance on the other languages. All multi-task models remain strong on the translated STS tasks, with around 0.77 for dev and 0.74 for test in all languages. Lastly, columns 5 and 6 shows the results of en-es models on STS17 tasks. The un-tuned multi-task models achieve 0.827 for the es-es task and 0.769 for the es-en task. As a point of reference, we also list the two best performing STS systems, Tian et al. (2017) (ECNU) and Wu et al. (2017) (BIT), reported from Cer et al. (2017). Our results are very close to these state-of-the-art feature engineered and mixed systems, which we describe in greater detail in the supplementary material. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 ZERO-SHOT CLASSIFICATION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To evaluate the transfer learning capabilities of our models, we examine how well the multi-task and translation-ranking encoders perform on zero-shot and few-shot classification tasks. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/798664b1b9f4c234de4dea39849ead6c8b8a4a3a0332db572d89c247774c50ec.jpg",
+        "table_caption": [
+            "Table 2: Pearson’s correlation coefficients on translated STS Benchmark and STS17 tasks. The first column shows the results on the original STS Benchmark data in English. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td colspan=\"4\">Translated STSBenchmark (dev/test)</td><td colspan=\"2\">STS17</td></tr><tr><td>en-en</td><td>fr-fr</td><td>es-es</td><td>de-de</td><td>es-es</td><td>es-en</td></tr><tr><td>Multi-task en-fr</td><td>0.803/0.763</td><td>0.777/0.738</td><td>1</td><td></td><td></td><td>1</td></tr><tr><td>Trans.-ranking en-fr</td><td>0.668 / 0.558</td><td>0.641 /0.579</td><td>1</td><td></td><td>1</td><td>1</td></tr><tr><td>Multi-task en-es</td><td>0.809/0.770</td><td></td><td>0.779/0.744</td><td></td><td>0.827</td><td>0.769</td></tr><tr><td>Trans.-ranking en-es</td><td>0.669 /0.631</td><td></td><td>0.622 / 0.611</td><td>1</td><td>0.642</td><td>0.587</td></tr><tr><td>Multi-task en-de</td><td>0.802/0.764</td><td></td><td></td><td>0.768/0.722</td><td></td><td>1</td></tr><tr><td>Trans.-ranking en-de</td><td>0.673 /0.632</td><td></td><td></td><td>0.630 / 0.526</td><td>1</td><td>1</td></tr><tr><td>ECNU</td><td></td><td></td><td></td><td>一</td><td>0.856</td><td>0.813</td></tr><tr><td>BIT</td><td></td><td></td><td></td><td></td><td>0.846</td><td>0.749</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.1 MULTILINGUAL NLI ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We evaluate the zero-shot classification performance of our multi-task models on two multilingual natural language inference (NLI) tasks. However, prior to doing so, we first train a modified version5 of our multi-task models that also includes training on the English Multi-genre NLI (MultiNLI) dataset of Williams et al. (2018) in addition to SNLI. We train with MultiNLI to be consistent with the baselines we compare to, which also use MultiNLI. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "First, we make use of the professionally translated French and Spanish subsets of SNLI created by Agic & Schluter (2017) for an initial cross-lingual zero-shot evaluation of French and Spanish. We ´ refer to these translated subsets as SNLI-X. There are 1000 examples in the translated subsets for each language. To evaluate, we simply feed the French and Spanish examples into the pre-trained English NLI sub-network of our multi-task models. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We also use the more recent dataset (XNLI) of Conneau et al. (2018), which provides a means for multilingual NLI evaluation in Spanish, French, German, Chinese and more. Since XNLI provides non-European-language evaluations, we also train an English-Chinese (en-zh) version of our multitask model. There are 5000 examples in each XNLI test set, and zero-shot evaluation is once again done by feeding non-English examples into the pre-trained English NLI sub-network. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 3 lists the accuracy on the English SNLI test set as well as on SNLI-X and XNLI for all of our multi-task models. The original English SNLI accuracies are around $84 \\%$ for all of our multitask models, indicating that English SNLI performance remains stable in the multi-task training setting. The zero-shot accuracy on SNLI-X is around $74 \\%$ for both the en-fr and en-es models. The zero-shot accuracy on XNLI is around $65 \\%$ for en-es, en-fr, and en-de, and around $63 \\%$ for en-zh, thereby significantly outperforming the pretrained sentence encoding baselines (X-CBOW) described in Conneau et al. (2018). The X-CBOW baselines use fixed sentence encoders that are the result of averaging tuned multilingual word embeddings. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Row 4 of Table 3 shows the zero-shot French NLI performance of Eriguchi et al. (2018), which is a state-of-the-art zero-shot NLI classifiers based on multilingual NMT embeddings. Our multi-task model shows comparable performance to the NMT-based model in both English and French. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2 AMAZON REVIEW ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Zero-shot Learning. We also conduct a zero-shot evaluation based on the Amazon review data extracted by Prettenhofer & Stein (2010). We preprocess the Amazon reviews and convert the data into a sentiment classification task by considering reviews with strictly more than three stars as positive and strictly less than three stars as negative, in the same manner as Prettenhofer & Stein (2010). Each review contains a summary field and a text field, which we concatenate to produce a single input. As the multi-task models are trained with sentence lengths clipped to 64, we only take the first 64 tokens from the the concatenated text as the input. There are 6000 training reviews in English, which we split into $90 \\%$ for training and $10 \\%$ for development. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/0314c6cc153bd417fe24ec18960afcba9c1ee6fdc92c4da10bc4ad7e186b4745.jpg",
+        "table_caption": [
+            "Table 3: Zero-shot classification accuracy $( \\% )$ on SNLI-X and XNLI datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=2 colspan=1>Model</td><td rowspan=1 colspan=3>SNLI-X</td><td rowspan=1 colspan=5>XNLI</td></tr><tr><td rowspan=1 colspan=1>en</td><td rowspan=1 colspan=1>fr</td><td rowspan=1 colspan=1>es</td><td rowspan=1 colspan=1>en</td><td rowspan=1 colspan=1>fr</td><td rowspan=1 colspan=1>es</td><td rowspan=1 colspan=1>de</td><td rowspan=1 colspan=1>zh</td></tr><tr><td rowspan=1 colspan=1>Multi-task en-frMulti-task en-esMulti-task en-deMulti-task en-zh</td><td rowspan=1 colspan=1>84.283.984.183.7</td><td rowspan=1 colspan=1>74.0111</td><td rowspan=1 colspan=1>175.911</td><td rowspan=1 colspan=1>71.670.271.569.2</td><td rowspan=1 colspan=1>64.4111</td><td rowspan=1 colspan=1>165.211</td><td rowspan=1 colspan=1>1165.01</td><td rowspan=1 colspan=1>11162.8</td></tr><tr><td rowspan=1 colspan=1>Eriguchi et al. (2018) (NMT en-fr)XNLI-CBOW zero-shot</td><td rowspan=1 colspan=1>84.41</td><td rowspan=1 colspan=1>73.91</td><td rowspan=1 colspan=1>11</td><td rowspan=1 colspan=1>64.5</td><td rowspan=1 colspan=1>160.3</td><td rowspan=1 colspan=1>160.7</td><td rowspan=1 colspan=1>161.0</td><td rowspan=1 colspan=1>158.8</td></tr><tr><td rowspan=1 colspan=3>Nonzero-shot baselines</td><td rowspan=1 colspan=6>aselines</td></tr><tr><td rowspan=1 colspan=1>XNLI-BiLSTM-lastXNLI-BiLSTM-max</td><td rowspan=1 colspan=1>11</td><td rowspan=1 colspan=1>11</td><td rowspan=1 colspan=1>11</td><td rowspan=1 colspan=1>71.073.7</td><td rowspan=1 colspan=1>65.267.7</td><td rowspan=1 colspan=1>67.868.7</td><td rowspan=1 colspan=1>66.667.7</td><td rowspan=1 colspan=1>63.765.8</td></tr></table>",
+        "bbox": [
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+            281
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We first encode inputs using the pre-trained multi-task and translation-ranking encoders and feed the encoded vectors into a 2-layer feed-forward network culminating in a softmax layer. We use layers of size 512 and tanh activation functions in each layer. We use Adam for optimization with an initial learning rate of 0.0005 and a learning rate decay of 0.9 at every epoch during training. We use a batch size of 16 and train for 20 total epochs in all experiments. We freeze the cross-lingual encoder during training. The model architecture and parameters are tuned on the development set. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We first train the classifier on English data, and then evaluate it on the 6000 French and German Amazon review test examples. The results are summarized in Table 4. The accuracy on the English test set is $8 7 . 4 \\%$ for the en-fr model and $8 7 . 1 \\%$ for the en-de model, with the zero-shot accuracy being above $80 \\%$ for both models. The translation-ranking models again perform worse on all metrics. Once again we compare the proposed model with Eriguchi et al. (2018), and find that our zero-shot performance has a reasonable gain on the fr test set6. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 4: Zero-shot sentiment classification accuracy $\\% )$ on target language Amazon review test data after training on only English Amazon review data. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/f077ebf42e1ce12f7f263bf78850658a5602a0400fdc77deee2965b962adb6ab.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>en</td><td rowspan=1 colspan=1>fr</td><td rowspan=1 colspan=1>de</td></tr><tr><td rowspan=1 colspan=1>Multi-task en-frTranslation-ranking en-fr</td><td rowspan=1 colspan=1>87.474.4</td><td rowspan=1 colspan=1>82.366.3</td><td rowspan=1 colspan=1>11</td></tr><tr><td rowspan=1 colspan=1>Multi-task en-deTranslation-ranking en-de</td><td rowspan=1 colspan=1>87.173.8</td><td rowspan=1 colspan=1>11</td><td rowspan=1 colspan=1>81.067.0</td></tr><tr><td rowspan=1 colspan=1>Eriguchi et al. (2018) (NMT en-fr)</td><td rowspan=1 colspan=1>83.2</td><td rowspan=1 colspan=1>81.3</td><td rowspan=1 colspan=1>1</td></tr></table>",
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+        "text": "Few-shot Learning. We further evaluate the proposed multi-task models via few-shot learning, by training on English reviews and only a portion of French and German reviews. Our few-shot models are compared with baselines of training on French and German reviews only. Table 5 shows the classification accuracy of the few-shot models, where the second row shows the percent of French and German data that is used when training each model. With as little as $20 \\%$ of the French or German training data, the few-shot models perform nearly as good as the baseline models trained on $100 \\%$ of the French and German data. Adding more French and German training data leads to further improvements in few-shot model performance, with the few-shot models reaching $8 5 . 8 \\%$ accuracy in French and $8 4 . 5 \\%$ accuracy in German when using all of the French and German data. ",
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+    {
+        "type": "text",
+        "text": "5 ANALYSIS OF CROSS-LINGUAL EMBEDDING SPACES ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Motivated by the recent work of Søgaard et al. (2018) studying the graph structure of multilingual word representations, we perform a similar analysis for our learned cross-lingual sentence representations. To do so, we take $N$ samples of size $K$ from en-fr, en-es, and en-de translation data and then encode these samples using the corresponding multi-task and translation-ranking models. We then compute pairwise distance matrices within each sampled set of encodings, and use these distance matrices to construct graph Laplacians7. Finally, we obtain the similarity $\\Psi ( S , T )$ between each model’s source and target language embedding subsets by comparing the eigenvalues of the source language graph Laplacians to the eigenvalues of the target language graph Laplacians as follows: ",
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+        "type": "table",
+        "img_path": "images/497d2857f70d14380f60f0610f620ed5459e4e7cc4512bf7f0fc2f0e16dc0c94.jpg",
+        "table_caption": [
+            "Table 5: Sentiment classification accuracy $( \\% )$ on target language Amazon review test data after training on English Amazon review data and a portion of French of German data. The second row shows the percent of French (fr) or German (de) data is used for training in each model. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td colspan=\"6\">fr</td><td colspan=\"6\">de</td></tr><tr><td>0%</td><td>10%</td><td>20%</td><td>40%</td><td>80%</td><td>100%</td><td>0%</td><td>10%</td><td>20%</td><td>40%</td><td>80%</td><td>100%</td></tr><tr><td>Few-shot</td><td>82.3</td><td>84.4</td><td>84.4</td><td>84.8</td><td>85.2</td><td>85.8</td><td>81.0</td><td>81.6</td><td>83.3</td><td>84.0</td><td>84.7</td><td>84.5</td></tr><tr><td>Baseline</td><td>1</td><td>79.2</td><td>80.0</td><td>82.7</td><td>84.3</td><td>84.9</td><td>1</td><td>75.5</td><td>77.7</td><td>81.6</td><td>83.5</td><td>84.4</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/dafa36ebcdf31b6c20cb955ac5ae0fe133c7e647ed9cd0b6741fa6c84a2d911e.jpg",
+        "text": "$$\n\\Psi ( S , T ) = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\sum _ { j = 1 } ^ { K } ( \\lambda _ { j } ( L _ { i } ^ { ( s ) } ) - \\lambda _ { j } ( L _ { i } ^ { ( t ) } ) ) ^ { 2 }\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "Where L(s)i a nd $\\boldsymbol { L } _ { i } ^ { ( t ) }$ refer to the graph Laplacians of the source language and target language sentences obtained from the $i ^ { t h }$ sample of source-target translation pairs. A smaller value of $\\Psi ( S , T )$ indicates higher eigen-similarity of the source language and target language embedding subsets. Following Søgaard et al. (2018) we use a sample size of $K = 1 0$ translation pairs, but we choose to use $N = 1 0 0 0$ samples instead of $N = 1 0$ (as was done in their work) since we found $\\Psi ( S , T )$ to have very high variance at $N = 1 0$ . The computed values of $\\Psi ( S , T )$ for our multi-task and translation-ranking models are summarized in Table 6. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/e1df6c19aaaf92d5cfefe4bc57a1af678bf665526470fd863810931e6194a67f.jpg",
+        "table_caption": [
+            "Table 6: Average eigen-similarity values of source and target embedding subsets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>en-fr</td><td rowspan=1 colspan=1>en-es</td><td rowspan=1 colspan=1>en-de</td></tr><tr><td rowspan=1 colspan=1>multi-tasktranslation-ranking</td><td rowspan=1 colspan=1>0.5921.036</td><td rowspan=1 colspan=1>0.5260.572</td><td rowspan=1 colspan=1>0.7612.187</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "We find that the source and target embedding subsets constructed from the multi-task models exhibit greater average eigen-similarity than those resulting from the translation-ranking models for all source-target language pairs. This result is not necessarily intuitive, since one might expect the translation-ranking model to optimize more for alignment. Given that eigen-similarity correlates with the better performance of the multi-task models in almost all tasks, a potential direction for future work could be to introduce regularization penalties based on graph similarity in multitask training. Interestingly, we also observe that the eigen-similarity gaps between the multi-task and translation-ranking models are not uniform across language pairs (although it may be that translation-ranking requires even more training). Thus, another direction could be to further study differences in the difficulty of aligning different source-target language embeddings. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this work, we explored a straightforward framework for training cross-lingual, multi-task dualencoder models. We showed that by training English-French, English-Spanish, and English-German multi-task models using our setup, we can achieve near-state-of-the-art or state-of-the-art performance in a variety of English tasks while also being able to produce similar caliber results in zeroshot transfer learning tasks for other languages. Finally, we note that the fact that multi-task training can actually improve performance on some downstream English tasks (TREC) is particularly interesting, and believe that there are many possibilities for future explorations of cross-lingual model training. ",
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+    {
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+    {
+        "type": "text",
+        "text": "A SUPPLEMENTARY MATERIAL ",
+        "text_level": 1,
+        "bbox": [
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+            102,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "A.1 DATA PREPROCESSING ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In order to effectively use Reddit, Wikipedia, and translation data, we do a significant amount of preprocessing on the raw data. We describe our preprocessing procedures for each dataset in the following paragraphs. ",
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+    {
+        "type": "text",
+        "text": "Reddit. The raw Reddit corpus extracted by Al-Rfou et al. (2016) consists of 2.4 billion posts and comments from Reddit between 2007 and 2016, making it our largest data source by an order of magnitude. To preprocess this data for training our conversation response prediction task, we follow the same procedure as Yang et al. (2018). Essentially, we consider Reddit comments and their children (response comments) as input-response pairs. We filter out comments that have $\\geq 3 5 0$ characters, due to the limitations on the number of input tokens our models can accept. We also remove comments that start with “https”, “ $@$ ”, or $\\mathrm { ^ { 6 6 } } / \\mathrm { r } / \\mathrm { ^ { 9 } }$ , and also remove comments whose authors have “bot” in their usernames. Lastly, we find that Reddit comments also contain a small mix of nonEnglish text, and we filter out comments where the percentage of alphabetic characters is $\\leq 7 0 \\%$ . As mentioned in the paper, the final, processed Reddit data consists of 600 million input-response pairs. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Since Reddit comments are predominantly English, we found the original data to be unsuitable for training encoders for non-English languages. To create conversational corpora for other languages, we translate the entire Reddit dataset using Google’s neural machine translation (NMT) system of Wu et al. (2016). ",
+        "bbox": [
+            174,
+            372,
+            825,
+            429
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Wikipedia. One concern with using the translated Reddit corpus as a monolingual task for nonEnglish languages is that the translated data will propagate any errors made by the NMT system used for translation. To ameliorate this problem, we crawl Wikipedia and extract triplets of contiguous sentences from Wikipedia articles. We use Wikipedia due to it having well-formed articles with a broad coverage of several languages. We extract all Wikipedia articles for English, French, Spanish and German from the Wikipedia dump of May 5, 2018, and use the sentence splitter model of Gillick (2009) to split the articles into sentences. All article titles and article section names are treated as sentences inline. As mentioned in the paper, our final extracted Wikipedia corpus consists of 127.9 million English sentence triplets, 49.5 million French triplets, 29.8 million Spanish triplets, and 49.3 million German triplets for French, Spanish and German respectively. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Stanford Natural Language Inference (SNLI). We use the English Stanford Natural Language Inference (SNLI) dataset consisting of 570K sentence pairs of Bowman et al. (2015) as is, without any further preprocessing. We also use the provided splits for training (550K), validation (10K), and testing (10K). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Translation. The data for training the translation task is crawled from the public web using a system similar to the approach described by Uszkoreit et al. (2010). The extracted data is further cleaned by a pre-trained translation pair scoring system. We then generate “hard-negatives” following Guo et al. (2018) by using a pre-trained coarse translation-ranking model to determine translations that are close to a correct translation. As mentioned in the paper, the final constructed corpus contains around 600M en-fr pairs, 470M en-es pairs and 500M en-de pairs. ",
+        "bbox": [
+            174,
+            645,
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+            728
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.2 DOWNSTREAM TASK DESCRIPTIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            751,
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+            765
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Each of the English downstream tasks shown in Table 1 in the main body of the paper are described briefly below: ",
+        "bbox": [
+            173,
+            779,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• Movie Reviews (MR): Multi-class prediction (scale of one to five stars) on movie review snippet data from Pang & Lee (2005).   \nCustomer Reviews (CR): Binary sentiment classification of sentences from customer reviews mined by Hu & Liu (2004).   \n• Subjectivity (SUBJ): Binary classification of sentences from movie reviews (Pang & Lee, 2004) as either objective or subjective.   \n• Multi-Perspective Question Answering (MPQA): Multi-class opinion polarity prediction on news data from Wiebe et al. (2005). Text REtrieval Conference (TREC): Multi-class classification of questions obtained from TREC by Li & Roth (2002).   \nStanford Sentiment Treebank (SST): Phrase-level binary sentiment classification of text from Socher et al. (2013).   \n• Semantic Text Similarity (STS): Semantic textual similarity of sentence pairs in the form of Pearson correlation with human judgments (Cer et al., 2017). ",
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+            217,
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+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            215,
+            103,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.3 STATE-OF-THE-ART STS SYSTEM DESCRIPTIONS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            246,
+            566,
+            261
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For the STS evaluations provided in the body of our paper, we compared to the state-of-the-art systems of Tian et al. (2017) and Wu et al. (2017). We refer the reader to Cer et al. (2017) for a detailed summary of the characteristics of these systems, but we provide a brief characterization below for convenience: ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• ECNU: The ECNU model is an ensemble of random forest, gradient boosted decision trees, and deep learning models that leverages several manually constructed features such as ngram overlap, edit distance, longest common prefix, common substrings, word alignments, and more. BIT: The BIT model relies mostly on a feature called information content (IC), which is based on the likelihood of occurrence of a concept (in this case, a word). Information content is then computed and combined for sentences through a hierarchical approach, which Wu et al. (2017) also augment with word embedding alignment to produce a final ensemble model. ",
+        "bbox": [
+            215,
+            340,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.4 ABLATION TESTS AND FURTHER EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            488,
+            550,
+            502
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "No NLI. Given that the multi-task models detailed in the body of the paper are trained with an English NLI task but no non-English NLI tasks, we evaluate how much training without this English NLI task affects model performance on downstream English tasks. The performance of these no-NLI models is summarized under the “Cross-lingual Multitask Transformer No SNLI” section of Table 7. We find that training without SNLI leads to comparable or better performance on all English downstream tasks except for STS, where we find training with SNLI provides a significant bump in performance. ",
+        "bbox": [
+            173,
+            513,
+            825,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "No Wikipedia. We similarly investigate whether there is a significant advantage gained from using the only native target language data source, Wikipedia. To do so, we train en-fr, en-es, and en-de multi-task models without the Wikipedia-based Quick Thought task. The performance of these nowiki models is shown under the “Cross-lingual Multitask Transformer No Wiki” section of Table 7. We find that training without the Wikipedia Quick Thought task does not have a strong impact on model performance, with some tasks being marginally better and others being marginally worse. ",
+        "bbox": [
+            174,
+            618,
+            825,
+            702
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "No target language native tasks. Additionally, the cross-lingual, multi-task models discussed in the body of our paper use a largely parallel task setup, where for each source-target language pair the monolingual target tasks are mirrored from the monolingual source tasks (the only exception being NLI, which is English-only). To test how useful this mirroring of monolingual tasks is, we train several models without target language monolingual tasks. We label these models as the “no non-English native task” models, and also summarize their downstream English task performance in Table 7. We note that removing the non-English native tasks actually leads to significant decreases in performance on TREC, SST, and STS. We find this quite interesting, as it provides some more empirical justification for the notion that cross-lingual training for a source-target language pair can actually improve monolingual source task performance. ",
+        "bbox": [
+            173,
+            709,
+            825,
+            848
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Finally, we also evaluate the zero-shot performance of the no non-English native task models on Amazon review sentiment classification. As can be seen in Table 8, training without the non-English native tasks leads to lower performance on English and roughly the same zero-shot classification performance as training with only translation pair data. ",
+        "bbox": [
+            176,
+            856,
+            823,
+            911
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/51777982fdfee926e9a588c90906c635d7b023b00b8bf8fe4baddb3323992829.jpg",
+        "table_caption": [
+            "Table 7: English task performance of different model configurations. Notably, removing non-English tasks in encoder training actually hurts performance on downstream English tasks. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>MR</td><td rowspan=1 colspan=1>CR</td><td rowspan=1 colspan=1>SUBJ</td><td rowspan=1 colspan=1>MPQA</td><td rowspan=1 colspan=1>TREC</td><td rowspan=1 colspan=1>SST</td><td rowspan=1 colspan=1>STSBenchmark(dev /test)</td></tr><tr><td rowspan=1 colspan=8>Cross-lingualMultitaskTransformerSNLI + Native Tasks (report above)</td></tr><tr><td rowspan=1 colspan=1>en-fren-esen-de</td><td rowspan=1 colspan=1>77.980.178.8</td><td rowspan=1 colspan=1>82.985.984.0</td><td rowspan=1 colspan=1>95.594.695.9</td><td rowspan=1 colspan=1>89.386.587.6</td><td rowspan=1 colspan=1>95.396.296.1</td><td rowspan=1 colspan=1>84.085.285.0</td><td rowspan=1 colspan=1>0.803/0.7630.809 /0.7700.802/0.764</td></tr><tr><td rowspan=1 colspan=8>Cross-lingual Multitask TransformerNo SNLI</td></tr><tr><td rowspan=1 colspan=1>en-fren-esen-de</td><td rowspan=1 colspan=1>80.580.080.5</td><td rowspan=1 colspan=1>86.589.485.5</td><td rowspan=1 colspan=1>94.095.694.0</td><td rowspan=1 colspan=1>89.190.590.0</td><td rowspan=1 colspan=1>96.694.592.8</td><td rowspan=1 colspan=1>85.085.782.6</td><td rowspan=1 colspan=1>0.747/0.7220.754 /0.7300.737 /0.723</td></tr><tr><td rowspan=1 colspan=8>Cross-lingual Multitask TransformerNo Wiki</td></tr><tr><td rowspan=1 colspan=1>en-fren-esen-de</td><td rowspan=1 colspan=1>78.979.179.6</td><td rowspan=1 colspan=1>85.887.485.9</td><td rowspan=1 colspan=1>92.194.594.1</td><td rowspan=1 colspan=1>90.089.689.7</td><td rowspan=1 colspan=1>94.794.392.2</td><td rowspan=1 colspan=1>82.783.082.5</td><td rowspan=1 colspan=1>0.804/0.7740.809 / 0.7690.801/0.741</td></tr><tr><td rowspan=1 colspan=8>Cross-lingual Multitask TransformerNo non-English native tasks</td></tr><tr><td rowspan=1 colspan=1>en-fren-esen-de</td><td rowspan=1 colspan=1>79.477.479.3</td><td rowspan=1 colspan=1>84.081.783.7</td><td rowspan=1 colspan=1>93.594.994.0</td><td rowspan=1 colspan=1>89.789.488.1</td><td rowspan=1 colspan=1>93.894.091.2</td><td rowspan=1 colspan=1>83.482.382.5</td><td rowspan=1 colspan=1>0.797/0.7580.796 /0.7610.760 / 0.732</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Translo</td><td rowspan=1 colspan=1>tion-rankin</td><td rowspan=1 colspan=1>gModels</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>en-fren-esen-de</td><td rowspan=1 colspan=1>68.767.767.8</td><td rowspan=1 colspan=1>79.375.775.2</td><td rowspan=1 colspan=1>87.083.584.4</td><td rowspan=1 colspan=1>81.886.083.6</td><td rowspan=1 colspan=1>89.494.486.8</td><td rowspan=1 colspan=1>74.272.674.6</td><td rowspan=1 colspan=1>0.668/0.5580.669 / 0.6310.673 /0.632</td></tr></table>",
+        "bbox": [
+            246,
+            218,
+            750,
+            551
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Table 8: Zero-shot sentiment classification accuracy $\\% )$ on target language Amazon review test data after training on only English Amazon review data. ",
+        "bbox": [
+            171,
+            704,
+            825,
+            733
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/a09b4b07dc6caabbc2b798f36ee3b7e33750694305509cf4e692d8ca50560684.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>en</td><td>fr</td><td>de</td></tr><tr><td>Multi-task en-fr Multi-task No non-English native tasks en-fr</td><td>87.4 85.6</td><td>82.3 65.4</td><td>1</td></tr><tr><td>Translation-ranking en-fr</td><td>74.4</td><td>66.3</td><td>1 1</td></tr><tr><td>Multi-task en-de</td><td>87.1</td><td>1</td><td>81.0</td></tr><tr><td>Multi-task No non-English native tasks en-fr</td><td>85.1</td><td>1</td><td>67.2</td></tr><tr><td>Translation-ranking en-de</td><td>73.8</td><td>1</td><td>67.0</td></tr></table>",
+        "bbox": [
+            287,
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+            844
+        ],
+        "page_idx": 13
+    }
+]

parse/train/HyGIdiRqtm/HyGIdiRqtm.md

@@ -0,0 +1,502 @@
+# EVALUATING ROBUSTNESS OF NEURAL NETWORKS WITH MIXED INTEGER PROGRAMMING
+
+Vincent Tjeng, Kai Xiao, Russ Tedrake Massachusetts Institute of Technology {vtjeng, kaix, russt}@mit.edu
+
+# ABSTRACT
+
+Neural networks trained only to optimize for training accuracy can often be fooled by adversarial examples — slightly perturbed inputs misclassified with high confidence. Verification of networks enables us to gauge their vulnerability to such adversarial examples. We formulate verification of piecewise-linear neural networks as a mixed integer program. On a representative task of finding minimum adversarial distortions, our verifier is two to three orders of magnitude quicker than the state-of-the-art. We achieve this computational speedup via tight formulations for non-linearities, as well as a novel presolve algorithm that makes full use of all information available. The computational speedup allows us to verify properties on convolutional and residual networks with over 100,000 ReLUs — several orders of magnitude more than networks previously verified by any complete verifier. In particular, we determine for the first time the exact adversarial accuracy of an MNIST classifier to perturbations with bounded $l _ { \infty }$ norm $\epsilon = 0 . 1$ : for this classifier, we find an adversarial example for $4 . 3 8 \%$ of samples, and a certificate of robustness to norm-bounded perturbations for the remainder. Across all robust training procedures and network architectures considered, and for both the MNIST and CIFAR-10 datasets, we are able to certify more samples than the state-of-the-art and find more adversarial examples than a strong first-order attack.
+
+# 1 INTRODUCTION
+
+Neural networks trained only to optimize for training accuracy have been shown to be vulnerable to adversarial examples: perturbed inputs that are very similar to some regular input but for which the output is radically different (Szegedy et al., 2014). There is now a large body of work proposing defense methods to produce classifiers that are more robust to adversarial examples. However, as long as a defense is evaluated only via heuristic attacks (such as the Fast Gradient Sign Method (FGSM) (Goodfellow et al., 2015) or Carlini & Wagner (2017b)’s attack (CW)), we have no guarantee that the defense actually increases the robustness of the classifier produced. Defense methods thought to be successful when published have often later been found to be vulnerable to a new class of attacks. For instance, multiple defense methods are defeated in Carlini & Wagner (2017a) by constructing defense-specific loss functions and in Athalye et al. (2018) by overcoming obfuscated gradients.
+
+Fortunately, we can evaluate robustness to adversarial examples in a principled fashion. One option is to determine (for each test input) the minimum distance to the closest adversarial example, which we call the minimum adversarial distortion (Carlini et al., 2017). Alternatively, we can determine the adversarial test accuracy (Bastani et al., 2016), which is the proportion of the test set for which no perturbation in some bounded class causes a misclassification. An increase in the mean minimum adversarial distortion or in the adversarial test accuracy indicates an improvement in robustness.1
+
+We present an efficient implementation of a mixed-integer linear programming (MILP) verifier for properties of piecewise-linear feed-forward neural networks. Our tight formulation for nonlinearities and our novel presolve algorithm combine to minimize the number of binary variables in the MILP problem and dramatically improve its numerical conditioning. Optimizations in our MILP implementation improve performance by several orders of magnitude when compared to a na¨ıve MILP implementation, and we are two to three orders of magnitude faster than the state-of-the-art Satisfiability Modulo Theories (SMT) based verifier, Reluplex (Katz et al., 2017)
+
+We make the following key contributions:
+
+• We demonstrate that, despite considering the full combinatorial nature of the network, our verifier can succeed at evaluating the robustness of larger neural networks, including those with convolutional and residual layers. We identify why we can succeed on larger neural networks with hundreds of thousands of units. First, a large fraction of the ReLUs can be shown to be either always active or always inactive over the bounded input domain. Second, since the predicted label is determined by the unit in the final layer with the maximum activation, proving that a unit never has the maximum activation over all bounded perturbations eliminates it from consideration. We exploit both phenomena, reducing the overall number of non-linearities considered. We determine for the first time the exact adversarial accuracy for MNIST classifiers to perturbations with bounded $l _ { \infty }$ norm . We are also able to certify more samples than the state-of-the-art and find more adversarial examples across MNIST and CIFAR-10 classifiers with different architectures trained with a variety of robust training procedures.
+
+Our code is available at https://github.com/vtjeng/MIPVerify.jl.
+
+# 2 RELATED WORK
+
+Our work relates most closely to other work on verification of piecewise-linear neural networks; Bunel et al. (2018) provides a good overview of the field. We categorize verification procedures as complete or incomplete. To understand the difference between these two types of procedures, we consider the example of evaluating adversarial accuracy.
+
+As in Kolter & Wong (2017), we call the exact set of all final-layer activations that can be achieved by applying a bounded perturbation to the input the adversarial polytope. Incomplete verifiers reason over an outer approximation of the adversarial polytope. As a result, when using incomplete verifiers, the answer to some queries about the adversarial polytope may not be decidable. In particular, incomplete verifiers can only certify robustness for a fraction of robust input; the status for the remaining input is undetermined. In contrast, complete verifiers reason over the exact adversarial polytope. Given sufficient time, a complete verifier can provide a definite answer to any query about the adversarial polytope. In the context of adversarial accuracy, complete verifiers will obtain a valid adversarial example or a certificate of robustness for every input. When a time limit is set, complete verifiers behave like incomplete verifiers, and resolve only a fraction of queries. However, complete verifiers do allow users to answer a larger fraction of queries by extending the set time limit.
+
+Incomplete verifiers for evaluating network robustness employ a range of techniques, including duality (Dvijotham et al., 2018; Kolter & Wong, 2017; Raghunathan et al., 2018), layer-by-layer approximations of the adversarial polytope (Xiang et al., 2018), discretizing the search space (Huang et al., 2017), abstract interpretation (Gehr et al., 2018), bounding the local Lipschitz constant (Weng et al., 2018), or bounding the activation of the ReLU with linear functions (Weng et al., 2018).
+
+Complete verifiers typically employ either MILP solvers as we do (Cheng et al., 2017; Dutta et al., 2018; Fischetti & Jo, 2018; Lomuscio & Maganti, 2017) or SMT solvers (Carlini et al., 2017; Ehlers, 2017; Katz et al., 2017; Scheibler et al., 2015). Our approach improves upon existing MILP-based approaches with a tighter formulation for non-linearities and a novel presolve algorithm that makes full use of all information available, leading to solve times several orders of magnitude faster than a na¨ıvely implemented MILP-based approach. When comparing our approach to the state-of-the-art SMT-based approach (Reluplex) on the task of finding minimum adversarial distortions, we find that our verifier is two to three orders of magnitude faster. Crucially, these improvements in performance allow our verifier to verify a network with over 100,000 units — several orders of magnitude larger than the largest MNIST classifier previously verified with a complete verifier.
+
+A complementary line of research to verification is in robust training procedures that train networks designed to be robust to bounded perturbations. Robust training aims to minimize the “worst-case loss” for each example — that is, the maximum loss over all bounded perturbations of that example (Kolter & Wong, 2017). Since calculating the exact worst-case loss can be computationally costly, robust training procedures typically minimize an estimate of the worst-case loss: either a lower bound as is the case for adversarial training (Goodfellow et al., 2015), or an upper bound as is the case for certified training approaches (Hein & Andriushchenko, 2017; Kolter & Wong, 2017; Raghunathan et al., 2018). Complete verifiers such as ours can augment robust training procedures by resolving the status of input for which heuristic attacks cannot find an adversarial example and incomplete verifiers cannot guarantee robustness, enabling more accurate comparisons between different training procedures.
+
+# 3 BACKGROUND AND NOTATION
+
+We denote a neural network by a function $f ( \cdot ; \theta ) : \mathbb { R } ^ { m } \to \mathbb { R } ^ { n }$ parameterized by a (fixed) vector of weights $\theta$ . For a classifier, the output layer has a neuron for each target class.
+
+Verification as solving an MILP. The general problem of verification is to determine whether some property $P$ on the output of a neural network holds for all input in a bounded input domain ${ \mathcal { C } } \subseteq \mathbb { R } ^ { m }$ . For the verification problem to be expressible as solving an MILP, $P$ must be expressible as the conjunction or disjunction of linear properties $P _ { i , j }$ over some set of polyhedra $\mathcal { C } _ { i }$ , where ${ \mathcal { C } } = \cup { \mathcal { C } } _ { i }$ .
+
+In addition, $f ( \cdot )$ must be composed of piecewise-linear layers. This is not a particularly restrictive requirement: piecewise-linear layers include linear transformations (such as fully-connected, convolution, and average-pooling layers) and layers that use piecewise-linear functions (such as ReLU or maximum-pooling layers). We provide details on how to express these piecewise-linear functions in the MILP framework in Section 4.1. The “shortcut connections” used in architectures such as ResNet (He et al., 2016) are also linear, and batch normalization (Ioffe & Szegedy, 2015) or dropout (Srivastava et al., 2014) are linear transformations at evaluation time (Bunel et al., 2018).
+
+# 4 FORMULATING ROBUSTNESS EVALUATION OF CLASSIFIERS AS AN MILP
+
+Evaluating Adversarial Accuracy. Let $\mathcal G ( x )$ denote the region in the input domain corresponding to all allowable perturbations of a particular input $x$ . In general, perturbed inputs must also remain in the domain of valid inputs $\mathcal { X } _ { v a l i d }$ . For example, for normalized images with pixel values ranging from 0 to 1, $\chi _ { v a l i d } = [ 0 , 1 ] ^ { m }$ . As in Madry et al. (2018), we say that a neural network is robust to perturbations on $x$ if the predicted probability of the true label $\lambda ( x )$ exceeds that of every other label for all perturbations:
+
+$$
+\begin{array} { r } { \forall x ^ { \prime } \in ( \mathcal G ( x ) \cap \mathcal X _ { v a l i d } ) : \ \mathrm { a r g m a x } _ { i } ( f _ { i } ( x ^ { \prime } ) ) = \lambda ( x ) } \end{array}
+$$
+
+quivalently, the network is robust to perturbations on $x$ if and only if Equation 2 is infeasible for $x ^ { \prime }$ .
+
+$$
+( x ^ { \prime } \in ( \mathcal { G } ( x ) \cap \mathcal { X } _ { v a l i d } ) ) \land \left( f _ { \lambda ( x ) } ( x ^ { \prime } ) < \operatorname* { m a x } _ { \mu \in [ 1 , n ] \setminus \{ \lambda ( x ) \} } f _ { \mu } ( x ^ { \prime } ) \right)
+$$
+
+where $f _ { i } ( \cdot )$ is the $i ^ { \mathrm { { t h } } }$ output of the network. For conciseness, we call $x$ robust with respect to the network if $f ( \cdot )$ is robust to perturbations on $x$ . If $x$ is not robust, we call any $x ^ { \prime }$ satisfying the constraints a valid adversarial example to $x$ . The adversarial accuracy of a network is the fraction of the test set that is robust; the adversarial error is the complement of the adversarial accuracy.
+
+As long as $\mathcal { G } ( x ) \cap \mathcal { X } _ { v a l i d }$ can be expressed as the union of a set of polyhedra, the feasibility problem can be expressed as an MILP. The four robust training procedures we consider (Kolter $\&$ Wong, 2017; Wong et al., 2018; Madry et al., 2018; Raghunathan et al., 2018) are designed to be robust to perturbations with bounded $l _ { \infty }$ norm, and the $l _ { \infty }$ -ball of radius $\epsilon$ around each input $x$ can be succinctly represented by the set of linear constraints $\mathcal { G } ( x ) = \{ x ^ { \prime } | \forall i : - \epsilon \leq ( x - x ^ { \prime } ) \bar { \iota } \leq \epsilon \}$ .
+
+Evaluating Mean Minimum Adversarial Distortion. Let $d ( \cdot , \cdot )$ denote a distance metric that measures the perceptual similarity between two input images. The minimum adversarial distortion under $d$ for input $x$ with true label $\lambda ( x )$ corresponds to the solution to the optimization:
+
+$$
+{ \begin{array} { r l } & { \operatorname* { m i n } _ { x ^ { \prime } } d ( x ^ { \prime } , x ) } \\ { \operatorname { s u b j e c t \ t o } } & { { \mathrm { a r g m a x } } _ { i } ( f _ { i } ( x ^ { \prime } ) ) \neq \lambda ( x ) } \\ & { x ^ { \prime } \in { \mathcal { X } } _ { v a l i d } } \end{array} }
+$$
+
+We can target the attack to generate an adversarial example that is classified in one of a set of target labels $T$ by replacing Equation 4 with $\mathrm { a r g m a x } _ { i } ( f _ { i } ( x ^ { \prime } ) ) \in T$ .
+
+The most prevalent distance metrics in the literature for generating adversarial examples are the $l _ { 1 }$ (Carlini & Wagner, 2017b; Chen et al., 2018), $l _ { 2 }$ (Szegedy et al., 2014), and $l _ { \infty }$ (Goodfellow et al., 2015; Papernot et al., 2016) norms. All three can be expressed in the objective without adding any additional integer variables to the model (Boyd & Vandenberghe, 2004); details are in Appendix A.3.
+
+4.1 FORMULATING PIECEWISE-LINEAR FUNCTIONS IN THE MILP FRAMEWORK
+
+Tight formulations of the ReLU and maximum functions are critical to good performance of the MILP solver; we thus present these formulations in detail with accompanying proofs.2
+
+Formulating ReLU Let $y = \operatorname* { m a x } ( x , 0 )$ , and $l \leq x \leq u$ . There are three possibilities for the phase of the ReLU. If $u \leq 0$ , we have $y \equiv 0$ . We say that such a unit is stably inactive. Similarly, if $l \geq 0$ , we have $y \equiv x$ . We say that such a unit is stably active. Otherwise, the unit is unstable. For unstable units, we introduce an indicator decision variable $a = \mathbb { 1 } _ { x \geq 0 }$ . As we prove in Appendix A.1, $y = \operatorname* { m a x } ( x , 0 )$ is equivalent to the set of linear and integer constraints in Equation 6.3
+
+$$
+( y \leq x - l ( 1 - a ) ) \land ( y \geq x ) \land ( y \leq u \cdot a ) \land ( y \geq 0 ) \land ( a \in \{ 0 , 1 \} )
+$$
+
+Formulating the Maximum Function Let $y = \operatorname* { m a x } ( x _ { 1 } , x _ { 2 } , \dots , x _ { m } )$ , and $l _ { i } \le x _ { i } \le u _ { i }$ .
+
+Proposition 1. Let $l _ { m a x } \triangleq \operatorname* { m a x } ( l _ { 1 } , l _ { 2 } , \ldots , l _ { m } )$ . We can eliminate from consideration all $x _ { i }$ where $u _ { i } \leq l _ { m a x }$ , since we know that $y \ge l _ { m a x } \ge u _ { i } \ge x _ { i }$ .
+
+We introduce an indicator decision variable $a _ { i }$ for each of our input variables, where $a _ { i } = 1 \implies y =$ $x _ { i }$ . Furthermore, we define $u _ { m a x , - i } \triangleq \operatorname* { m a x } _ { j \neq i } ( u _ { j } )$ . As we prove in Appendix A.2, the constraint $y = \operatorname* { m a x } ( x _ { 1 } , x _ { 2 } , \dots , x _ { m } )$ is equivalent to the set of linear and integer constraints in Equation 7.
+
+$$
+\bigwedge _ { i = 1 } ^ { m } ( ( y \leq x _ { i } + ( 1 - a _ { i } ) ( u _ { m a x , - i } - l _ { i } ) ) \wedge ( y \geq x _ { i } ) ) \wedge \left( \sum _ { i = 1 } ^ { m } a _ { i } = 1 \right) \wedge ( a _ { i } \in \{ 0 , 1 \} )
+$$
+
+# 4.2 PROGRESSIVE BOUNDS TIGHTENING
+
+We previously assumed that we had some element-wise bounds on the inputs to non-linearities. In practice, we have to carry out a presolve step to determine these bounds. Determining tight bounds is critical for problem tractability: tight bounds strengthen the problem formulation and thus improve solve times (Vielma, 2015). For instance, if we can prove that the phase of a ReLU is stable, we can avoid introducing a binary variable. More generally, loose bounds on input to some unit will propagate downstream, leading to units in later layers having looser bounds.
+
+We used two procedures to determine bounds: INTERVAL ARITHMETIC (IA), also used in Cheng et al. (2017); Dutta et al. (2018), and the slower but tighter LINEAR PROGRAMMING (LP) approach. Implementation details are in Appendix B.
+
+Since faster procedures achieve efficiency by compromising on tightness of bounds, we face a tradeoff between higher build times (to determine tighter bounds to inputs to non-linearities), and higher solve times (to solve the main MILP problem in Equation 2 or Equation 3-5). While a degree of compromise is inevitable, our knowledge of the non-linearities used in our network allows us to reduce average build times without affecting the strength of the problem formulation.
+
+The key observation is that, for piecewise-linear non-linearities, there are thresholds beyond which further refining a bound will not improve the problem formulation. With this in mind, we adopt a progressive bounds tightening approach: we begin by determining coarse bounds using fast procedures and only spend time refining bounds using procedures with higher computational complexity if doing so could provide additional information to improve the problem formulation.4 Pseudocode demonstrating how to efficiently determine bounds for the tightest possible formulations for the ReLU and maximum function is provided below and in Appendix C respectively.
+
+GETBOUNDSFORRELU $( x , f s )$
+
+$1 \triangleright f s$ are the procedures to determine bounds, sorted in increasing computational complexity.   
+2 $l _ { b e s t } = - \infty$ ; $u _ { b e s t } = \infty$ $\vartriangleright$ initialize best known upper and lower bounds on $x$   
+3 for $f$ in $f s$ : $\vartriangleright$ carrying out progressive bounds tightening   
+4 do $u = f ( x , b o u n d T y p e = u p p e r )$ ; $u _ { b e s t } = \operatorname* { m i n } ( u _ { b e s t } , u )$   
+5 if $u _ { b e s t } \leq 0$ return $( l _ { b e s t } , u _ { b e s t } )$ $\vartriangleright$ Early return: $x \leq u _ { b e s t } \leq 0$ ; thus $\operatorname* { m a x } ( x , 0 ) \equiv 0 .$ .   
+6 $l = f ( x , b o u n d T y p e = l o w e r )$ ; $l _ { b e s t } = \operatorname* { m a x } ( l _ { b e s t } , l )$   
+7 if $l _ { b e s t } \geq 0$ return $( l _ { b e s t } , u _ { b e s t } ) \quad \triangleright$ Early return: $x \geq l _ { b e s t } \geq 0$ ; thus $\operatorname* { m a x } ( x , 0 ) \equiv x$   
+8 return $( l _ { b e s t } , u _ { b e s t } ) \quad \triangleright x$ could be either positive or negative.
+
+The process of progressive bounds tightening is naturally extensible to more procedures. Kolter & Wong (2017); Wong et al. (2018); Dvijotham et al. (2018); Weng et al. (2018) each discuss procedures to determine bounds with computational complexity and tightness intermediate between IA and LP. Using one of these procedures in addition to IA and LP has the potential to further reduce build times.
+
+# 5 EXPERIMENTS
+
+Dataset. All experiments are carried out on classifiers for the MNIST dataset of handwritten digits or the CIFAR-10 dataset of color images.
+
+Architectures. We conduct experiments on a range of feed-forward networks. In all cases, ReLUs follow each layer except the output layer. MLP- $m \times [ n ]$ refers to a multilayer perceptron with $m$ hidden layers and $n$ units per hidden layer. We further abbreviate MLP- $\cdot 1 \times [ 5 0 0 ]$ and MLP- $\cdot 2 \times [ 2 0 0 ]$ as $\mathbf { M L P _ { A } }$ and $\mathbf { M L P _ { B } }$ respectively. $\mathbf { C N N _ { A } }$ and $\mathbf { C N N _ { B } }$ refer to the small and large ConvNet architectures in Wong et al. (2018). $\mathbf { C N N _ { A } }$ has two convolutional layers (stride length 2) with 16 and 32 filters (size $4 \times 4 )$ ) respectively, followed by a fully-connected layer with 100 units. $\mathbf { C N N _ { B } }$ has four convolutional layers with 32, 32, 64, and 64 filters, followed by two fully-connected layers with 512 units. RES refers to the ResNet architecture used in Wong et al. (2018), with 9 convolutional layers in four blocks, followed by two fully-connected layers with 4096 and 1000 units respectively.
+
+Training Methods. We conduct experiments on networks trained with a regular loss function and networks trained to be robust. Networks trained to be robust are identified by a prefix corresponding to the method used to approximate the worst-case loss: $\mathbf { L P _ { d } } ^ { 5 }$ when the dual of a linear program is used, as in Kolter & Wong (2017); $\mathbf { S D P _ { d } }$ when the dual of a semidefinite relaxation is used, as in Raghunathan et al. (2018); and Adv when adversarial examples generated via Projected Gradient Descent (PGD) are used, as in Madry et al. (2018). Full details on each network are in Appendix D.1.
+
+Experimental Setup. We run experiments on a modest $8 ~ { \mathrm { C P U s } } \ @ 2 . 2 0 ~ { \mathrm { G H z } }$ with 8GB of RAM. Appendix D.2 provides additional details about the computational environment. Maximum build effort is LP. Unless otherwise noted, we report a timeout if solve time for some input exceeds 1200s.
+
+# 5.1 PERFORMANCE COMPARISONS
+
+# 5.1.1 COMPARISONS TO OTHER MILP-BASED COMPLETE VERIFIERS
+
+Our MILP approach implements three key optimizations: we use progressive tightening, make use of the information provided by the restricted input domain $\mathcal G ( x )$ , and use asymmetric bounds in the ReLU formulation in Equation 6. None of the four other MILP-based complete verifiers implement progressive tightening or use the restricted input domain, and only Fischetti & Jo (2018) uses asymmetric bounds. Since none of the four verifiers have publicly available code, we use ablation tests to provide an idea of the difference in performance between our verifier and these existing ones.
+
+When removing progressive tightening, we directly use LP rather than doing IA first. When removing using restricted input domain, we determine bounds under the assumption that our perturbed input could be anywhere in the full input domain $\mathcal { X } _ { v a l i d }$ , imposing the constraint $x ^ { \prime } \in \mathcal { G } ( \bar { x } )$ only after all bounds are determined. Finally, when removing using asymmetric bounds, we replace $l$ and $u$ in Equation 6 with $- M$ and $M$ respectively, where $M \triangleq \operatorname* { m a x } ( - l , u )$ , as is done in Cheng et al. (2017); Dutta et al. (2018); Lomuscio & Maganti (2017). We carry out experiments on an MNIST classifier; results are reported in Table 1.
+
+Table 1: Results of ablation testing on our verifier, where each test removes a single optimization. The task was to determine the adversarial accuracy of the MNIST classifier $\mathrm { L P _ { d } \mathrm { - } C N N _ { A } }$ to perturbations with $l _ { \infty }$ norm-bound $\epsilon = 0 . 1$ . Build time refers to time used to determine bounds, while solve time refers to time used to solve the main MILP problem in Equation 2 once all bounds have been determined. During solve time, we solve a linear program for each of the nodes explored in the MILP search tree.
+
+†We exclude the initial build time required (3593s) to determine reusable bounds.
+
+<table><tr><td rowspan="2">Optimization Removed</td><td colspan="3">Mean Time /s</td><td colspan="2">Nodes Explored</td><td rowspan="2">Fraction Timed Out</td></tr><tr><td>Build</td><td>Solve</td><td>Total</td><td>Mean</td><td>Median</td></tr><tr><td>(Control)</td><td>3.44</td><td>0.08</td><td>3.52</td><td>1.91</td><td>00</td><td></td></tr><tr><td>Progressive tightening</td><td>7.66</td><td>0.11</td><td>7.77</td><td>1.91</td><td>00</td><td></td></tr><tr><td>Using restricted input domaint</td><td>1.49</td><td>56.47</td><td>57.96</td><td>649.63</td><td>65</td><td>0.0047</td></tr><tr><td>Using asymmetric bounds</td><td>4465.11</td><td>133.03</td><td>4598.15</td><td>1279.06</td><td>105</td><td>0.0300</td></tr></table>
+
+The ablation tests demonstrate that each optimization is critical to the performance of our verifier. In terms of performance comparisons, we expect our verifier to have a runtime several orders of magnitude faster than any of the three verifiers not using asymmetric bounds. While Fischetti & Jo (2018) do use asymmetric bounds, they do not use information from the restricted input domain; we thus expect our verifier to have a runtime at least an order of magnitude faster than theirs.
+
+# 5.1.2 COMPARISONS TO OTHER COMPLETE AND INCOMPLETE VERIFIERS
+
+We also compared our verifier to other verifiers on the task of finding minimum targeted adversarial distortions for MNIST test samples. Verifiers included for comparison are 1) Reluplex (Katz et al., 2017), a complete verifier also able to find the true minimum distortion; and 2) $\mathrm { L P } ^ { 6 }$ , Fast-Lip, Fast-Lin (Weng et al., 2018), and LP-full (Kolter & Wong, 2017), incomplete verifiers that provide a certified lower bound on the minimum distortion.
+
+Verification Times, vis-a-vis the state-of-the-art SMT-based complete verifier \` Reluplex. Figure 1 presents average verification times per sample. All solves for our method were run to completion. On the $l _ { \infty }$ norm, we improve on the speed of Reluplex by two to three orders of magnitude.
+
+![](images/340f365b8b0aa336f2f1c6de1caa670c4538c3f9c23e543ec847b3deca13fc0b.jpg)  
+Figure 1: Average times for determining bounds on or exact values of minimum targeted adversarial distortion for MNIST test samples. We improve on the speed of the state-of-the-art complete verifier Reluplex by two to three orders of magnitude. Results for methods other than ours are from Weng et al. (2018); results for Reluplex were only available in Weng et al. (2018) for the $l _ { \infty }$ norm.
+
+Minimum Targeted Adversarial Distortions, vis-a-vis incomplete verifiers. \` Figure 2 compares lower bounds from the incomplete verifiers to the exact value we obtain. The gap between the best lower bound and the true minimum adversarial distortion is significant even on these small networks. This corroborates the observation in Raghunathan et al. (2018) that incomplete verifiers provide weak bounds if the network they are applied to is not optimized for that verifier. For example, under the $l _ { \infty }$ norm, the best certified lower bound is less than half of the true minimum distortion. In context: a network robust to perturbations with $l _ { \infty }$ norm-bound $\epsilon = 0 . 1$ would only be verifiable to $\epsilon = 0 . 0 5$ .
+
+![](images/335e2887c4a21a40d9a55d2d84e643e746f0a2531e0c1971692cd27a6babad7b.jpg)  
+Figure 2: Bounds on or exact values of minimum targeted adversarial distortion for MNIST test samples. The gap between the true minimum adversarial distortion and the best lower bound is significant in all cases, increasing for deeper networks. We report mean values over 100 samples.
+
+# 5.2 DETERMINING ADVERSARIAL ACCURACY OF MNIST AND CIFAR-10 CLASSIFIERS
+
+We use our verifier to determine the adversarial accuracy of classifiers trained by a range of robust training procedures on the MNIST and CIFAR-10 datasets. Table 2 presents the test error and estimates of the adversarial error for these classifiers.7 For MNIST, we verified a range of networks trained to be robust to attacks with bounded $l _ { \infty }$ norm $\epsilon = 0 . 1$ , as well as networks trained to be robust to larger attacks of $\epsilon = 0 . 2 , 0 . 3$ and 0.4. Lower bounds on the adversarial error are proven by providing adversarial examples for input that is not robust. We compare the number of samples for which we successfully find adversarial examples to the number for PGD, a strong first-order attack. Upper bounds on the adversarial error are proven by providing certificates of robustness for input that is robust. We compare our upper bounds to the previous state-of-the-art for each network.
+
+While performance depends on the training method and architecture, we improve on both the lower and upper bounds for every network tested.8 For lower bounds, we successfully find an adversarial example for every test sample that PGD finds an adversarial example for. In addition, we observe that PGD ‘misses’ some valid adversarial examples: it fails to find these adversarial examples even though they are within the norm bounds. As the last three rows of Table 2 show, PGD misses for a larger fraction of test samples when $\epsilon$ is larger. We also found that PGD is far more likely to miss for some test sample if the minimum adversarial distortion for that sample is close to $\epsilon$ ; this observation is discussed in more depth in Appendix G. For upper bounds, we improve on the bound on adversarial error even when the upper bound on the worst-case loss — which is used to generate the certificate of robustness — is explicitly optimized for during training (as is the case for $\mathrm { L P _ { d } }$ and $\mathrm { S D P _ { d } }$ training). Our method also scales well to the more complex CIFAR-10 dataset and the larger $\mathrm { L P _ { d } }$ -RES network (which has 107,496 units), with the solver reaching the time limit for only $0 . 3 1 \%$ of samples.
+
+Most importantly, we are able to determine the exact adversarial accuracy for Adv- ${ \bf \cdot M L P _ { B } }$ and $\mathrm { L P _ { d } \mathrm { - } C N N _ { A } }$ for all $\epsilon$ tested, finding either a certificate of robustness or an adversarial example for every test sample. For $\mathbf { A d v - M L P _ { B } }$ and $\mathrm { L P _ { d } \mathrm { - } C N N _ { A } }$ , running our verifier over the full test set takes approximately 10 hours on 8 CPUs — the same order of magnitude as the time to train each network on a single GPU. Better still, verification of individual samples is fully parallelizable — so verification time can be reduced with more computational power.
+
+Table 2: Adversarial accuracy of MNIST and CIFAR-10 classifiers to perturbations with $l _ { \infty }$ normbound $\epsilon$ . In every case, we improve on both 1) the lower bound on the adversarial error, found by PGD, and 2) the previous state-of-the-art (SOA) for the upper bound, generated by the following methods: [1] Kolter & Wong (2017), [2] Dvijotham et al. (2018), [3] Raghunathan et al. (2018). For classifiers marked with a $\checkmark$ , we have a guarantee of robustness or a valid adversarial example for every test sample. Gaps between our bounds correspond to cases where the solver reached the time limit for some samples. Solve statistics on nodes explored are in Appendix F.1.   
+
+<table><tr><td rowspan="2">Dataset</td><td rowspan="2">Network</td><td rowspan="2">E</td><td rowspan="2">Test Error</td><td colspan="5">Certified Bounds on Adversarial Error</td><td rowspan="2">Mean Time /s</td></tr><tr><td>Lower Bound PGD</td><td>Ours</td><td>Upper Bound SOA</td><td>Ours</td><td>No Gap?</td></tr><tr><td rowspan="9">MNIST</td><td>LPd-CNNB</td><td>0.1</td><td>1.19%</td><td>2.62%</td><td>2.73%</td><td>4.45%[1]</td><td>2.74%</td><td></td><td>46.33</td></tr><tr><td>LPd-CNNA</td><td>0.1</td><td>1.89%</td><td>4.11%</td><td>4.38%</td><td>5.82%[1]</td><td>4.38%</td><td>√</td><td>3.52</td></tr><tr><td>Adv-CNNA</td><td>0.1</td><td>0.96%</td><td>4.10%</td><td>4.21%</td><td></td><td>7.21%</td><td></td><td>135.74</td></tr><tr><td>Adv-MLPB</td><td>0.1</td><td>4.02%</td><td>9.03%</td><td>9.74%</td><td>15.41%[2]</td><td>9.74%</td><td>厂</td><td>3.69</td></tr><tr><td>SDPd-MLPA</td><td>0.1</td><td>4.18%</td><td>11.51%</td><td>14.36%</td><td>34.77%[3]</td><td>30.81%</td><td></td><td>312.43</td></tr><tr><td>LPd-CNNA</td><td>0.2</td><td>4.23%</td><td>9.54%</td><td>10.68%</td><td>17.50%[1]</td><td>10.68%</td><td>√</td><td>7.32</td></tr><tr><td>LPd-CNNB</td><td>0.3</td><td>11.16%</td><td>19.70%</td><td>24.12%</td><td>41.98%[1]</td><td>24.19%</td><td></td><td>98.79</td></tr><tr><td>LPd-CNNA</td><td>0.3</td><td>11.40%</td><td>22.70%</td><td>25.79%</td><td>35.03%[1]</td><td>25.79%</td><td>√</td><td>5.13</td></tr><tr><td>LPd-CNNA</td><td>0.4</td><td>26.13%</td><td>39.22%</td><td>48.98%</td><td>62.49%[1]</td><td>48.98%</td><td>√</td><td>5.07</td></tr><tr><td rowspan="2">CIFAR-10</td><td>LPd-CNNA</td><td>2 营</td><td>39.14%</td><td>48.23%</td><td> 49.84%</td><td>53.59%[1] 78.52%[1]</td><td>50.20%</td><td></td><td>22.41</td></tr><tr><td>LPd-RES</td><td>255</td><td>72.93%</td><td>76.52%</td><td>77.29%</td><td></td><td>77.60%</td><td></td><td>15.23</td></tr></table>
+
+# 5.2.1 OBSERVATIONS ON DETERMINANTS OF VERIFICATION TIME
+
+All else being equal, we might expect verification time to be correlated to the total number of ReLUs, since the solver may need to explore both possibilities for the phase of each ReLU. However, there is clearly more at play: even though $\mathrm { L P _ { d } \mathrm { - } C N N _ { A } }$ and Adv- $\mathrm { C N N _ { A } }$ have identical architectures, verification time for $\mathbf { A d v - C N N _ { A } }$ is two orders of magnitude higher.
+
+Table 3: Determinants of verification time: mean verification time is 1) inversely correlated to the number of labels that can be eliminated from consideration and 2) correlated to the number of ReLUs that are not provably stable. Results are for $\epsilon = 0 . 1$ on MNIST; results for other networks are in Appendix F.2.   
+
+<table><tr><td rowspan="2">Network</td><td rowspan="2">Mean Time /s</td><td rowspan="2">Number of Labels Eliminated</td><td colspan="4">Numberof ReLUs</td></tr><tr><td>Possibly Unstable</td><td>Provably Stable</td><td></td><td>Total</td></tr><tr><td></td><td>46.33</td><td>6.87</td><td></td><td>Active</td><td>Inactive</td><td></td></tr><tr><td>LPd-CNNB</td><td>3.52</td><td>6.57</td><td>311.96 121.18</td><td>30175.65 1552.52</td><td>17576.39 3130.30</td><td>48064 4804</td></tr><tr><td>LPd-CNNA</td><td>135.74</td><td>3.14</td><td>545.90</td><td>3383.30</td><td>874.80</td><td>4804</td></tr><tr><td>Adv-CNNA Adv-MLPB</td><td>3.69</td><td>4.77</td><td>55.21</td><td>87.31</td><td>257.48</td><td>400</td></tr><tr><td>SDPd-MLPA</td><td>312.43</td><td>0.00</td><td>297.66</td><td>73.85</td><td>128.50</td><td>500</td></tr></table>
+
+The key lies in the restricted input domain $\mathcal { G } ( x )$ for each test sample $x$ . When input is restricted to $\mathcal G ( x )$ , we can prove that many ReLUs are stable (with respect to $\mathcal { G }$ ). Furthermore, we can eliminate some labels from consideration by proving that the upper bound on the output neuron corresponding to that label is lower than the lower bound for some other output neuron. As the results in Table 3 show, a significant number of ReLUs can be proven to be stable, and a significant number of labels can be eliminated from consideration. Rather than being correlated to the total number of ReLUs, solve times are instead more strongly correlated to the number of ReLUs that are not provably stable, as well as the number of labels that cannot be eliminated from consideration.
+
+# 6 DISCUSSION
+
+This paper presents an efficient complete verifier for piecewise-linear neural networks.
+
+While we have focused on evaluating networks on the class of perturbations they are designed to be robust to, defining a class of perturbations that generates images perceptually similar to the original remains an important direction of research. Our verifier is able to handle new classes of perturbations (such as convolutions applied to the original image) as long as the set of perturbed images is a union of polytopes in the input space.
+
+We close with ideas on improving verification of neural networks. First, our improvements can be combined with other optimizations in solving MILPs. For example, Bunel et al. (2018) discusses splitting on the input domain, producing two sub-MILPs where the input in each sub-MILP is restricted to be from a half of the input domain. Splitting on the input domain could be particularly useful where the split selected tightens bounds sufficiently to significantly reduce the number of unstable ReLUs that need to be considered in each sub-MILP. Second, as previously discussed, taking advantage of locally stable ReLUs speeds up verification; network verifiability could be improved during training via a regularizer that increases the number of locally stable ReLUs. Finally, we observed (see Appendix H) that sparsifying weights promotes verifiability. Adopting a principled sparsification approach (for example, $l _ { 1 }$ regularization during training, or pruning and retraining (Han et al., 2016)) could potentially further increase verifiability without compromising on the true adversarial accuracy.
+
+# ACKNOWLEDGMENTS
+
+This work was supported by Lockheed Martin Corporation under award number RPP2016-002 and the NSF Graduate Research Fellowship under grant number 1122374. We would like to thank Eric Wong, Aditi Raghunathan, Jonathan Uesato, Huan Zhang and Tsui-Wei Weng for sharing the networks verified in this paper, and Guy Katz, Nicholas Carlini and Matthias Hein for discussing their results.
+
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+
+# A FORMULATING THE MILP MODEL
+
+A.1 FORMULATING RELU IN AN MILP MODEL
+
+We reproduce our formulation for the ReLU below.
+
+$$
+\begin{array} { l } { y \leq x - l ( 1 - a ) } \\ { y \geq x } \\ { y \leq u \cdot a } \\ { y \geq 0 } \\ { a \in \{ 0 , 1 \} } \end{array}
+$$
+
+We consider two cases.
+
+Recall that $a$ is the indicator variable $a = \mathbb { 1 } _ { x \geq 0 }$
+
+When $a = 0$ , the constraints in Equation 10 and 11 are binding, and together imply that $y = 0$ . The other two constraints are not binding, since Equation 9 is no stricter than Equation 11 when $x < 0$ , while Equation 8 is no stricter than Equation 10 since $x - l \ge 0$ . We thus have $a = 0 \implies y = 0$ .
+
+When $a = 1$ , the constraints in Equation 8 and 9 are binding, and together imply that $y = x$ . The other two constraints are not binding, since Equation 11 is no stricter than Equation 9 when $x > 0$ , while Equation 10 is no stricter than Equation 8 since $x \leq u$ . We thus have $a = 1 \implies y = x$ .
+
+This formulation for rectified linearities is sharp (Vielma, 2015) if we have no further information about $x$ . This is the case since relaxing the integrality constraint on $a$ leads to $( x , y )$ being restricted to an area that is the convex hull of $y = \operatorname* { m a x } ( x , 0 )$ . However, if $x$ is an affine expression $\overline { { x } } = w ^ { T } z + b$ the formulation is no longer sharp, and we can add more constraints using bounds we have on $z$ to improve the problem formulation.
+
+# A.2 FORMULATING THE MAXIMUM FUNCTION IN AN MILP MODEL
+
+We reproduce our formulation for the maximum function below.
+
+$$
+\begin{array} { c } { y \leq x _ { i } + ( 1 - a _ { i } ) ( u _ { m a x , - i } - l _ { i } ) \forall i } \\ { y \geq x _ { i } \forall i } \\ { \displaystyle \sum _ { i = 1 } ^ { m } a _ { i } = 1 } \\ { a _ { i } \in \{ 0 , 1 \} } \end{array}
+$$
+
+Equation 15 ensures that exactly one of the $a _ { i }$ is 1. It thus suffices to consider the value of $a _ { i }$ for a single variable.
+
+When $a _ { i } = 1$ , Equations 13 and 14 are binding, and together imply that $y = x _ { i }$ . We thus have $a _ { i } = 1 \implies y = x _ { i }$ .
+
+When $a _ { i } = 0$ , we simply need to show that the constraints involving $x _ { i }$ are never binding regardless of the values of $x _ { 1 } , x _ { 2 } , \ldots , x _ { m }$ . Equation 14 is not binding since $a _ { i } = 0$ implies $x _ { i }$ is not the (unique) maximum value. Furthermore, we have chosen the coefficient of $a _ { i }$ such that Equation 13 is not binding, since $x _ { i } + u _ { m a x , - i } - l _ { i } \geq u _ { m a x , - i } \geq y .$ . This completes our proof.
+
+A.3 EXPRESSING $l _ { p }$ NORMS AS THE OBJECTIVE OF AN MIP MODEL
+
+# A.3.1 $l _ { 1 }$
+
+When $d ( x ^ { \prime } , x ) = \| x ^ { \prime } - x \| _ { 1 }$ , we introduce the auxiliary variable $\delta$ , which bounds the elementwise absolute value from above: $\delta _ { j } \ : \geq \ : x _ { j } ^ { \prime } - x _ { j } , \delta _ { j } \ : \geq \ : x _ { j } ^ { \prime } - x _ { j } ^ { \prime }$ . The optimization in Equation 3-5 is
+
+equivalent to
+
+$$
+\begin{array} { l } { \displaystyle \operatorname* { m i n } _ { x ^ { \prime } } \sum _ { j } \delta _ { j } } \\ { \displaystyle \operatornamewithlimits { a r g m a x } _ { i } ( f _ { i } ( x ^ { \prime } ) ) \neq \lambda ( x ) } \\ { \displaystyle x ^ { \prime } \in \mathcal { X } _ { v a l i d } } \\ { \displaystyle \delta _ { j } \geq x _ { j } ^ { \prime } - x _ { j } } \\ { \displaystyle \delta _ { j } \geq x _ { j } - x _ { j } ^ { \prime } } \end{array}
+$$
+
+A.3.2 $l _ { \infty }$
+
+When $d ( x ^ { \prime } , x ) = \| x ^ { \prime } - x \| _ { \infty }$ , we introduce the auxiliary variable $\epsilon$ , which bounds the $l _ { \infty }$ norm from above: $\epsilon \geq x _ { j } ^ { \prime } - x _ { j } , \epsilon \geq x _ { j } - x _ { j } ^ { \prime }$ . The optimization in Equation 3-5 is equivalent to
+
+$$
+\begin{array} { l l } & { \underset { x ^ { \prime } } { \mathrm { m i n } } \epsilon } \\ { \mathrm { s u b j e c t ~ t o } } & { \mathrm { a r g m a x } _ { i } ( f _ { i } ( x ^ { \prime } ) ) \not = \lambda ( x ) } \\ & { x ^ { \prime } \in \mathcal { X } _ { v a l i d } } \\ & { \epsilon \geq x _ { j } ^ { \prime } - x _ { j } } \\ & { \epsilon \geq x _ { j } - x _ { j } ^ { \prime } } \end{array}
+$$
+
+# A.3.3 $l _ { 2 }$
+
+When $d ( x ^ { \prime } , x ) = \| x ^ { \prime } - x \| _ { 2 }$ , the objective becomes quadratic, and we have to use a Mixed Integer Quadratic Program (MIQP) solver. However, no auxiliary variables are required: the optimization in Equation 3-5 is simply equivalent to
+
+$$
+\begin{array} { r l } { \displaystyle } & { \underset { x ^ { \prime } } { \operatorname* { m i n } } \sum _ { j } ( x _ { j } ^ { \prime } - x _ { j } ) ^ { 2 } } \\ { \mathrm { s u b j e c t ~ t o } } & { \mathrm { a r g m a x } _ { i } ( f _ { i } ( x ^ { \prime } ) ) \neq \lambda ( x ) } \\ & { x ^ { \prime } \in \mathscr { X } _ { v a l i d } } \end{array}
+$$
+
+# B DETERMINING TIGHT BOUNDS ON DECISION VARIABLES
+
+Our framework for determining bounds on decision variables is to view the neural network as a computation graph $G$ . Directed edges point from function input to output, and vertices represent variables. Source vertices in $G$ correspond to the input of the network, and sink vertices in $G$ correspond to the output of the network. The computation graph begins with defined bounds on the input variables (based on the input domain $( \mathcal G ( x ) \cap \chi _ { v a l i d } ) \rangle$ ), and is augmented with bounds on intermediate variables as we determine them. The computation graph is acyclic for the feed-forward networks we consider.
+
+Since the networks we consider are piecewise-linear, any subgraph of $G$ can be expressed as an MILP, with constraints derived from 1) input-output relationships along edges and 2) bounds on the values of the source nodes in the subgraph. Integer constraints are added whenever edges describe a non-linear relationship.
+
+We focus on computing an upper bound on some variable $v$ ; computing lower bounds follows a similar process. All the information required to determine the best possible bounds on $v$ is contained in the subtree of $G$ rooted at $v$ , $G _ { v }$ . (Other variables that are not ancestors of $v$ in the computation graph cannot affect its value.) Maximizing the value of $v$ in the MILP $M _ { v }$ corresponding to $G _ { v }$ gives the optimal upper bound on $v$ .
+
+We can reduce computation time in two ways. Firstly, we can prune some edges and vertices of $G _ { v }$ Specifically, we select a set of variables with existing bounds $V _ { I }$ that we assume to be independent (that is, we assume that they each can take on any value independent of the value of the other variables in $V _ { I }$ ). We remove all in-edges to vertices in $V _ { I }$ , and eliminate variables without children, resulting in the smaller computation graph $G _ { v , V _ { I } }$ . Maximizing the value of $v$ in the MILP $M _ { v , V _ { I } }$ corresponding to $G _ { v , V _ { I } }$ gives a valid upper bound on $v$ that is optimal if the independence assumption holds.
+
+We can also reduce computation time by relaxing some of the integer constraints in $M _ { v }$ to obtain a MILP with fewer integer variables $M _ { v } ^ { \prime }$ . Relaxing an integer constraint corresponds to replacing the relevant non-linear relationship with its convex relaxation. Again, the objective value returned by maximizing the value of $v$ over $M _ { v } ^ { \prime }$ may not be the optimal upper bound, but is guaranteed to be a valid bound.
+
+# B.1 FULL
+
+FULL considers the full subtree $G _ { v }$ and does not relax any integer constraints. The upper and lower bound on $v$ is determined by maximizing and minimizing the value of $v$ in $M _ { v }$ respectively. FULL is also used in Cheng et al. (2017) and Fischetti & Jo (2018).
+
+If solves proceed to optimality, FULL is guaranteed to find the optimal bounds on the value of a single variable $v$ . The trade-off is that, for deeper layers, using FULL can be relatively inefficient, since solve times in the worst case are exponential in the number of binary variables in $M _ { v }$ .
+
+Nevertheless, contrary to what is asserted in Cheng et al. (2017), we can terminate solves early and still obtain useful bounds. For example, to determine an upper bound on $v$ , we set the objective of $M _ { v }$ to be to maximize the value of $v$ . As the solve process proceeds, we obtain progressively better certified upper bounds on the maximum value of $v$ . We can thus terminate the solve process and extract the best upper bound found at any time, using this upper bound as a valid (but possibly loose) bound on the value of $v$ .
+
+# B.2 LINEAR PROGRAMMING (LP)
+
+LP considers the full subtree $G _ { v }$ but relaxes all integer constraints. This results in the optimization problem becoming a linear program that can be solved more efficiently. LP represents a good middle ground between the optimality of FULL and the performance of IA.
+
+# B.3 INTERVAL ARITHMETIC (IA)
+
+IA selects $V _ { I }$ to be the parents of $v$ . In other words, bounds on $v$ are determined solely by considering the bounds on the variables in the previous layer. We note that this is simply interval arithmetic Moore et al. (2009).
+
+Consider the example of computing bounds on the variable $\hat { z } _ { i } = W _ { i } z _ { i - 1 } + b _ { i }$ , where $l _ { z _ { i - 1 } } \leq z _ { i - 1 } \leq$ $u _ { z _ { i - 1 } }$ . We have
+
+$$
+\begin{array} { r l } & { \hat { z } _ { i } \geq W _ { i } ^ { - } u _ { z _ { i - 1 } } + W _ { i } ^ { + } l _ { z _ { i - 1 } } } \\ & { \hat { z } _ { i } \leq W _ { i } ^ { + } u _ { z _ { i - 1 } } + W _ { i } ^ { - } l _ { z _ { i - 1 } } } \\ & { W _ { i } ^ { + } \triangleq \operatorname* { m a x } ( W _ { i } , 0 ) } \\ & { W _ { i } ^ { - } \triangleq \operatorname* { m i n } ( W _ { i } , 0 ) } \end{array}
+$$
+
+IA is efficient (since it only involves matrix operations for our applications). However, for deeper layers, using interval arithmetic can lead to overly conservative bounds.
+
+# C PROGRESSIVE BOUNDS TIGHTENING
+
+GETBOUNDSFORMAX finds the tightest bounds required for specifying the constraint $y = \operatorname* { m a x } ( x s )$ . Using the observation in Proposition 1, we stop tightening the bounds on a variable if its maximum possible value is lower than the minimum value of some other variable. GETBOUNDSFORMAX returns a tuple containing the set of elements in xs that can still take on the maximum value, as well as a dictionary of upper and lower bounds.
+
+# GETBOUNDSFORMAX $( x s , f s )$
+
+$1 \triangleright f s$ are the procedures to determine bounds, sorted in increasing computational complexi   
+2 $d _ { l } = \{ x : - \infty$ for $x$ in $x s \}$   
+3 $d _ { u } = \{ x : \infty$ for $x$ in $x s \}$   
+$4 \triangleright$ initialize dictionaries containing best known upper and lower bounds on xs   
+5 $l _ { m a x } = - \infty \quad \triangleright l _ { m a x }$ is the maximum known lower bound on any of the xs   
+6 $a = \{ x s \}$   
+$7 \ \triangleright a$ is a set of active elements in $x s$ that can still potentially take on the maximum value.   
+8 for $f$ in $f s$ : $\vartriangleright$ carrying out progressive bounds tightening   
+9 do for $x$ in $x s$ :   
+10 if $d _ { u } [ x ] < l _ { m a x }$   
+11 then a.remov $\begin{array} { r l r l } { { } \mathcal { \ L } _ { \mathcal { S } } ( x ) } & { { } \vartriangle } & { \vartriangle } & { \vartriangle { \Sigma } _ { \mathcal { X } } } \end{array}$ cannot take on the maximum value   
+12 else $u = f ( x , b o u n d T y p e = u p p e r )$   
+13 $d _ { u } [ x ] = \operatorname* { m i n } ( d _ { u } [ x ] , u )$   
+14 $l = f ( x , b o u n d T y p e = l o w e r )$   
+15 $d _ { l } [ x ] = \operatorname* { m a x } ( d _ { l } [ x ] , l )$   
+16 $l _ { m a x } = \operatorname* { m a x } ( l _ { m a x } , l )$   
+17 return $( a , d _ { l } , d _ { u } )$
+
+# D ADDITIONAL EXPERIMENTAL DETAILS
+
+D.1 NETWORKS USED
+
+The source of the weights for each of the networks we present results for in the paper are provided below.
+
+• MNIST classifiers not designed to be robust: – MLP- $\cdot 2 \times [ 2 0 ]$ and MLP- $3 \times [ 2 0 ]$ are the MNIST classifiers in Weng et al. (2018), and can be found at https://github.com/huanzhang12/ CertifiedReLURobustness.
+
+• MNIST classifiers designed for robustness to perturbations with $l _ { \infty }$ norm-bound $\epsilon = 0 . 1$ :
+
+– $\mathrm { L P _ { d } \mathrm { - } C N N _ { B } }$ is the large MNIST classifier for $\epsilon = 0 . 1$ in Wong et al. (2018), and can be found at https://github.com/locuslab/convex_adversarial/ blob/master/models_scaled/mnist_large_0_1.pth.
+
+– $\mathrm { L P _ { d } \mathrm { - } C N N _ { A } }$ is the MNIST classifier in Kolter & Wong (2017), and can be found at https://github.com/locuslab/convex_adversarial/ blob/master/models/mnist.pth.
+
+– Adv- $\mathrm { . C N N _ { A } }$ was trained with adversarial examples generated by PGD. PGD attacks were carried out with $l _ { \infty }$ norm-bound $\epsilon = 0 . 1$ , 8 steps per sample, and a step size of 0.334. An $l _ { 1 }$ regularization term was added to the objective with a weight of 0.0015625 on the first convolution layer and 0.003125 for the remaining layers.
+
+– Adv-MLP- $\phantom { - } 2 \times [ 2 0 0 ]$ was trained with adversarial examples generated by PGD. PGD attacks were carried out with with $l _ { \infty }$ norm-bound $\epsilon = 0 . 1 5$ , 200 steps per sample, and a step size of 0.1. An $l _ { 1 }$ regularization term was added to the objective with a weight of 0.003 on the first layer and 0 for the remaining layers.
+
+– $\mathrm { S D P _ { d } }$ -MLP- $\mathrm { 1 \times [ 5 0 0 ] }$ is the classifier in Raghunathan et al. (2018).
+
+• MNIST classifiers designed for robustness to perturbations with $l _ { \infty }$ norm-bound $\epsilon =$ 0.2, 0.3, 0.4:
+
+– $\mathrm { L P _ { d } \mathrm { - } C N N _ { A } }$ was trained with the code available at https://github.com/ locuslab/convex_adversarial at commit 4e9377f. Parameters selected were batch siz $e { = } 2 0$ , starting epsilon $\mathord {  \downarrow } 0 . 0 1$ , epochs $= 2 0 0$ , seed ${ } = 0$ .
+
+– $\mathrm { L P _ { d } \mathrm { - } C N N _ { B } }$ is the large MNIST classifier for $\epsilon = 0 . 3$ in Wong et al. (2018), and can be found at https://github.com/locuslab/convex_adversarial/ blob/master/models_scaled/mnist_large_0_3.pth.
+
+• CIFAR-10 classifiers designed for robustness to perturbations with $l _ { \infty }$ norm-bound $\begin{array} { r } { \epsilon = \frac { 2 } { 2 5 5 } } \end{array}$ – $\mathrm { L P _ { d } \mathrm { - } C N N _ { A } }$ is the small CIFAR classifier in Wong et al. (2018), courtesy of the authors.
+
+• CIFAR-10 classifiers designed for robustness to perturbations with $l _ { \infty }$ norm-bound $\begin{array} { r } { \epsilon = \frac { 8 } { 2 5 5 } } \end{array}$ – $\mathrm { L P _ { d } }$ -RES is the resnet CIFAR classifier in Wong et al. (2018), and can be found at https://github.com/locuslab/convex_adversarial/ blob/master/models_scaled/cifar_resnet_8px.pth.
+
+# D.2 COMPUTATIONAL ENVIRONMENT
+
+We construct the MILP models in Julia (Bezanson et al., 2017) using JuMP (Dunning et al., 2017), with the model solved by the commercial solver Gurobi 7.5.2 (Gurobi Optimization, 2017). All experiments were run on a KVM virtual machine with 8 virtual CPUs running on shared hardware, with Intel(R) Xeon(R) CPU E5-2630 v4 $@$ 2.20GHz processors, and 8GB of RAM.
+
+# E PERFORMANCE OF VERIFIER WITH OTHER MILP SOLVERS
+
+To give a sense for how our verifier performs with other solvers, we ran a comparison with the Cbc (Forrest et al., 2018) and GLPK (Makhorin, 2012) solvers, two open-source MILP solvers.
+
+Table 4: Performance of verifier with different MILP solvers on MNIST $\mathrm { L P _ { d } \mathrm { - } C N N _ { A } }$ network with $\epsilon = 0 . 1$ . Verifier performance is best with Gurobi, but our verifier outperforms both the lower bound from PGD and the upper bound generated by the SOA method in Kolter & Wong (2017) when using the Cbc solver too.
+
+<table><tr><td rowspan="2">Approach</td><td>Adversarial Error</td><td>Mean</td></tr><tr><td>Lower Bound Upper Bound</td><td>Time / s</td></tr><tr><td>Ours w/ Gurobi</td><td>4.38% 4.38%</td><td>3.52</td></tr><tr><td>Ours w/ Cbc</td><td>4.30% 4.82%</td><td>18.92</td></tr><tr><td>Ours w/ GLPK</td><td>3.50% 7.30%</td><td>35.78</td></tr><tr><td>PGD /SOA</td><td>4.11% 5.82%</td><td>1</td></tr></table>
+
+When we use GLPK as the solver, our performance is significantly worse than when using Gurobi, with the solver timing out on almost $4 \%$ of samples. While we time out on some samples with Cbc, our verifier still provides a lower bound better than PGD and an upper bound significantly better than the state-of-the-art for this network. Overall, verifier performance is affected by the underlying MILP solver used, but we are still able to improve on existing bounds using an open-source solver.
+
+# F ADDITIONAL SOLVE STATISTICS
+
+# F.1 NODES EXPLORED
+
+Table 5 presents solve statistics on nodes explored to supplement the results reported in Table 2. If the solver explores zero nodes for a particular sample, it proved that the sample was robust (or found an adversarial example) without branching on any binary variables. This occurs when the bounds we find during the presolve step are sufficiently tight. We note that this occurs for over $9 5 \%$ of samples for $\mathrm { L P _ { d } \mathrm { - } C N N _ { A } }$ for $\epsilon = 0 . 1$ .
+
+# F.2 DETERMINANTS OF VERIFICATION TIME
+
+Table 6 presents additional information on the determinants of verification time for networks we omit in Table 3.
+
+Table 5: Solve statistics on nodes explored when determining adversarial accuracy of MNIST and CIFAR-10 classifiers to perturbations with $l _ { \infty }$ norm-bound $\epsilon$ . We solve a linear program for each of the nodes explored in the MILP search tree.   
+
+<table><tr><td rowspan="3">Dataset</td><td rowspan="3">Network</td><td rowspan="3">E</td><td rowspan="3">Mean Time /s</td><td colspan="7">Nodes Explored</td></tr><tr><td rowspan="2">Mean</td><td rowspan="2">Median</td><td colspan="3">Percentile</td><td rowspan="2">99.9</td><td rowspan="2">Max</td></tr><tr><td>90</td><td>95</td><td>99</td></tr><tr><td rowspan="9">MNIST</td><td>LPd-CNNB</td><td>0.1</td><td>46.33</td><td>8.18</td><td>0</td><td>0</td><td>0</td><td>1</td><td>2385</td><td>20784</td></tr><tr><td>LPd-CNNA</td><td>0.1</td><td>3.52</td><td>1.91</td><td>0</td><td>0</td><td>0</td><td>1</td><td>698</td><td>1387</td></tr><tr><td>Adv-CNNA</td><td>0.1</td><td>135.74</td><td>749.55</td><td>0</td><td>201</td><td>3529</td><td>20195</td><td>31559</td><td>50360</td></tr><tr><td>Adv-MLPB</td><td>0.1</td><td>3.69</td><td>87.17</td><td>0</td><td>1</td><td>3</td><td>2129</td><td>11625</td><td>103481</td></tr><tr><td>SDPd-MLPA</td><td>0.1</td><td>312.43</td><td>4641.33</td><td>39</td><td>17608</td><td>21689</td><td>27120</td><td>29770</td><td>29887</td></tr><tr><td>LPd-CNNA</td><td>0.2</td><td>7.32</td><td>15.71</td><td>0</td><td>1</td><td>1</td><td>540</td><td>2151</td><td>7105</td></tr><tr><td>LPd-CNNB</td><td>0.3</td><td>98.79</td><td>305.82</td><td>0</td><td>607</td><td>1557</td><td>4319</td><td>28064</td><td>185500</td></tr><tr><td>LPd-CNNA</td><td>0.3</td><td>5.13</td><td>31.58</td><td>0</td><td>5</td><td>119</td><td>788</td><td>2123</td><td>19650</td></tr><tr><td>LPd-CNNA</td><td>0.4</td><td>5.07</td><td>57.32</td><td>1</td><td>79</td><td>320</td><td>932</td><td>3455</td><td>43274</td></tr><tr><td rowspan="2">CIFAR-10</td><td>LPd-CNNA</td><td>2</td><td>22.41</td><td>195.67</td><td>0</td><td>1</td><td>1</td><td>4166</td><td>29774</td><td>51010</td></tr><tr><td>LPd-RES</td><td>动</td><td>15.23</td><td>41.38</td><td>0</td><td>1</td><td>3</td><td>1339</td><td>4239</td><td>5022</td></tr></table>
+
+Table 6: Solve statistics on number of labels that can be eliminated from consideration, and number of ReLUs that are provably stable, when determining adversarial accuracy of MNIST and CIFAR-10 classifiers to perturbations with $l _ { \infty }$ norm-bound .   
+
+<table><tr><td rowspan="3">Dataset</td><td rowspan="3">Network</td><td rowspan="3">E</td><td rowspan="3">Mean Time /s</td><td rowspan="3">Number ofLabels Eliminated</td><td colspan="4">Number of ReLUs</td></tr><tr><td rowspan="2">Possibly Unstable</td><td colspan="2">Provably Stable</td><td rowspan="2">Total</td></tr><tr><td>Active</td><td>Inactive</td></tr><tr><td>MNIST</td><td>LPd-CNNA</td><td>0.2</td><td>7.32</td><td>5.84</td><td>115.51</td><td>3202.67</td><td>1485.82</td><td>4804</td></tr><tr><td></td><td>LPd-CNNB</td><td>0.3</td><td>98.79</td><td>5.43</td><td>575.61</td><td>34147.99</td><td>13340.40</td><td>48064</td></tr><tr><td></td><td>LPd-CNNA</td><td>0.3</td><td>5.13</td><td>4.57</td><td>150.90</td><td>3991.38</td><td>661.72</td><td>4804</td></tr><tr><td></td><td>LPd-CNNA</td><td>0.4</td><td>5.07</td><td>2.67</td><td>172.63</td><td>4352.60</td><td>278.78</td><td>4804</td></tr><tr><td>CIFAR-10 LPd-CNNA</td><td></td><td>2</td><td>22.41</td><td>7.06</td><td>371.36</td><td>4185.35</td><td>1687.29</td><td>6244</td></tr><tr><td></td><td>LPd-RES</td><td>喜 255</td><td>15.23</td><td>6.94</td><td>1906.15</td><td>96121.50</td><td>9468.35</td><td>107496</td></tr></table>
+
+![](images/e216b0c28d5a91da3b750cb4f68cd59e9fb4be69a1e1f9d23baf661b9342381b.jpg)  
+Figure 3: Fraction of samples in the MNIST test set vulnerable to attack for which PGD succeeds at finding an adversarial example. Samples are binned by their minimum adversarial distortion (as measured under the $l _ { \infty }$ norm), with bins of size 0.01. Each of these are $\mathrm { L P _ { d } \mathrm { - } C N N _ { A } }$ networks, and were trained to optimize for robustness to attacks with $l _ { \infty }$ norm-bound $\epsilon$ . For any given network, the success rate of PGD declines as the minimum adversarial distortion increases. Comparing networks, success rates decline for networks with larger $\epsilon$ even at the same minimum adversarial distortion.
+
+PGD succeeds in finding an adversarial example if and only if the starting point for the gradient descent is in the basin of attraction of some adversarial example. Since PGD initializes the gradient descent with a randomly chosen starting point within $\mathcal G ( x ) \cap \bar { \mathcal X } _ { v a l i d }$ , the success rate (with a single random start) corresponds to the fraction of $\mathcal { G } ( x ) \cap \chi _ { v a l i d }$ that is in the basin of attraction of some adversarial example.
+
+Intuitively, the success rate of PGD should be inversely related to the magnitude of the minimum adversarial distortion $\hat { \delta }$ : if $\hat { \delta }$ is small, we expect more of $\mathcal { G } ( x ) \cap \mathcal { X } _ { v a l i d }$ to correspond to adversarial examples, and thus the union of the basins of attraction of the adversarial examples is likely to be larger. We investigate here whether our intuition is substantiated.
+
+To obtain the best possible empirical estimate of the success rate of PGD for each sample, we would need to re-run PGD initialized with multiple different randomly chosen starting points within $\mathcal { G } ( x ) \cap \mathcal { X } _ { v a l i d }$ .
+
+However, since we are simply interested in the relationship between success rate and minimum adversarial distortion, we obtained a coarser estimate by binning the samples based on their minimum adversarial distortion, and then calculating the fraction of samples in each bin for which PGD with a single randomly chosen starting point succeeds at finding an adversarial example.
+
+Figure 3 plots this relationship for four networks using the $\mathrm { C N N _ { A } }$ architecture and trained with the same training method $\mathrm { L P _ { d } }$ but optimized for attacks of different size. Three features are clearly discernible:
+
+• PGD is very successful at finding adversarial examples when the magnitude of the minimum adversarial distortion, $\hat { \delta }$ , is small. • The success rate of PGD declines significantly for all networks as $\hat { \delta }$ approaches . • For a given value of $\hat { \delta }$ , and two networks $a$ and $b$ trained to be robust to attacks with $l _ { \infty }$ norm-bound $\epsilon _ { a }$ and $\epsilon _ { b }$ respectively (where $\epsilon _ { a } < \epsilon _ { b }$ ), PGD is consistently more successful at attacking the network trained to be robust to smaller attacks, $a$ , as long as $\hat { \delta } \ll \epsilon _ { a }$ .
+
+The sharp decline in the success rate of PGD as $\hat { \delta }$ approaches $\epsilon$ is particularly interesting, especially since it is suggests a pathway to generating networks that appear robust when subject to PGD attacks of bounded $l _ { \infty }$ norm but are in fact vulnerable to such bounded attacks: we simply train the network to maximize the total number of adversarial examples with minimum adversarial distortion close to $\epsilon$
+
+# H SPARSIFICATION AND VERIFIABILITY
+
+When verifying the robustness of $\mathrm { S D P _ { d } \mathrm { - M L P _ { A } } }$ , we observed that a significant proportion of kernel weights were close to zero. Many of these tiny weights are unlikely to be contributing significantly to the final classification of any input image. Having said that, setting these tiny weights to zero could potentially reduce verification time, by 1) reducing the size of the MILP formulation, and by 2) ameliorating numerical issues caused by the large range of numerical coefficients in the network (Gurobi, 2017).
+
+We generated sparse versions of the original network to study the impact of sparseness on solve times. Our heuristic sparsification algorithm is as follows: for each fully-connected layer $i$ , we set a fraction $f _ { i }$ of the weights with smallest absolute value in the kernel to 0, and rescale the rest of the weights such that the $l _ { 1 }$ norm of the kernel remains the same.9 Note that ${ \bf M L P _ { A } }$ consists of only two layers: one hidden layer (layer 1) and one output layer (layer 2).
+
+Table 7: Effect of sparsification of $\mathrm { S D P _ { d } \mathrm { - M L P _ { A } } }$ on verifiability. Test error increases slightly as larger fractions of kernel weights are set to zero, but the certified upper bound on adversarial error decreases significantly as the solver reaches the time limit for fewer samples.
+
+<table><tr><td colspan="2">Fraction zeroed</td><td rowspan="2">Test Error</td><td colspan="2">Certified Bounds on Adversarial Error</td><td rowspan="2">Mean Time / s</td><td rowspan="2">Fraction Timed Out</td></tr><tr><td>f1</td><td>f2</td><td>Lower Bound</td><td>Upper Bound</td></tr><tr><td>0.0</td><td>0.00</td><td>4.18%</td><td>14.36%</td><td>30.81%</td><td>312.4</td><td>0.1645</td></tr><tr><td>0.5</td><td>0.25</td><td>4.22%</td><td>14.60%</td><td>25.25%</td><td>196.0</td><td>0.1065</td></tr><tr><td>0.8</td><td>0.25</td><td>4.22%</td><td>15.03%</td><td>18.26%</td><td>69.7</td><td>0.0323</td></tr><tr><td>0.9</td><td>0.25</td><td>4.93%</td><td>17.97%</td><td>18.76%</td><td>22.2</td><td>0.0079</td></tr></table>
+
+Table 7 summarizes the results of verifying sparse versions of $\mathrm { S D P _ { d } \mathrm { - M L P _ { A } } }$ ; the first row presents results for the original network, and the subsequent rows present results when more and more of the kernel weights are set to zero.
+
+When comparing the first and last rows, we observe an improvement in both mean time and fraction timed out by an order of magnitude. As expected, sparsifying weights increases the test error, but the impact is not significant until $f _ { 1 }$ exceeds 0.8. We also find that sparsification significantly improves our upper bound on adversarial error — to a point: the upper bound on adversarial error for $f _ { 1 } = 0 . 9$ is higher than that for $f _ { 1 } = 0 . 8$ , likely because the true adversarial error has increased significantly.
+
+Starting with a network that is robust, we have demonstrated that a simple sparsification approach can already generate a sparsified network with an upper bound on adversarial error significantly lower than the best upper bound that can be determined for the original network. Adopting a more principled sparsification approach could achieve the same improvement in verifiability but without compromising on the true adversarial error as much.
+
+# I ROBUST TRAINING AND RELU STABILITY
+
+Networks that are designed to be robust need to balance two competing objectives. Locally, they need to be robust to small perturbations to the input. However, they also need to retain sufficient global expressiveness to maintain a low test error.
+
+For the networks in Table 3, even though each robust training approach estimates the worst-case error very differently, all approaches lead to a significant fraction of the ReLUs in the network being provably stable with respect to perturbations with bounded $l _ { \infty }$ norm. In other words, for the input domain $\mathcal { G } ( x )$ consisting of all bounded perturbations of the sample $x$ , we can show that, for many ReLUs, the input to the unit is always positive (and thus the output is linear in the input) or always negative (and thus the output is always zero). As discussed in the main text, we believe that the need for the network to be robust to perturbations in $\mathcal { G }$ drives more ReLUs to be provably stable with respect to $\mathcal { G }$ .
+
+To better understand how networks can retain global expressiveness even as many ReLUs are provably stable with respect to perturbations with bounded $l _ { \infty }$ norm , we study how the number of ReLUs that are provably stable changes as we vary the size of $\mathcal G ( x )$ by changing the maximum allowable $l _ { \infty }$ norm of perturbations. The results are presented in Figure 4.
+
+As expected, the number of ReLUs that cannot be proven to be stable increases as the maximum allowable $l _ { \infty }$ norm of perturbations increases. More interestingly, $\mathrm { L P _ { d } \mathrm { - } C N N _ { A } }$ is very sensitive to the $\epsilon = 0 . 1$ threshold, with a sharp increase in the number of ReLUs that cannot be proven to be stable when the maximum allowable $l _ { \infty }$ norm of perturbations increases beyond 0.102. An increase of the same abruptness is not seen for the other two networks.
+
+![](images/2058624d5e3210e7ef776c1c30b630d343c1104da3fd6ef0b5b2ed1a707ef45b.jpg)  
+ReLU stability for LPd-CNN
+
+![](images/7f8ec00aadca26e2ba8f875ea6b7ca069147bf835e92bf31a70344d3b8c5fc81.jpg)  
+ReLU stability for SDPd-MLP-A
+
+(a) $\mathrm { L P _ { d } \mathrm { - } C N N _ { A } }$ . Note the sharp increase in the number of ReLUs that cannot be proven to be stable when the maximum $l _ { \infty }$ norm increases beyond 0.102.
+
+![](images/d775581f50ef9ce67c809e30e413c727190a9c0b8af5396c4249324ed1b32a4c.jpg)  
+ReLU stability for Adv-MLP-B   
+(c) Adv- ${ \bf \cdot M L P _ { B } }$ . Adversarial training alone is sufficient to significantly increase the number of ReLUs that are provably stable.   
+Figure 4: Comparison of provably ReLU stability for networks trained via different robust training procedures to be robust at $\epsilon = 0 . 1$ , when varying the maximum allowable $l _ { \infty }$ norm of the perturbation. The results reported in Table 3 are marked by a dotted line. As we increase the maximum allowable $l _ { \infty }$ norm of perturbations, the number of ReLUs that cannot be proven to be stable increases across all networks (as expected), but $\mathrm { L P _ { d } \mathrm { - } C N N _ { A } }$ is far more sensitive to the $\epsilon = 0 . 1$ threshold.

parse/train/rkxtNaNKwr/rkxtNaNKwr.md

@@ -0,0 +1,328 @@
+# EVOLUTIONARY REINFORCEMENT LEARNING FORSAMPLE-EFFICIENT MULTIAGENT COORDINATION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Many cooperative multiagent reinforcement learning environments provide agents with a sparse team-based reward, as well as a dense agent-specific reward that incentivizes learning basic skills. Training policies solely on the team-based reward is often difficult due to its sparsity. Also, relying solely on the agent-specific reward is sub-optimal because it usually does not capture the team coordination objective. A common approach is to use reward shaping to construct a proxy reward by combining the individual rewards. However, this requires manual tuning for each environment. We introduce Multiagent Evolutionary Reinforcement Learning (MERL), a split-level training platform that handles the two objectives separately through two optimization processes. An evolutionary algorithm maximizes the sparse team-based objective through neuroevolution on a population of teams. Concurrently, a gradient-based optimizer trains policies to only maximize the dense agent-specific rewards. The gradient-based policies are periodically added to the evolutionary population as a way of information transfer between the two optimization processes. This enables the evolutionary algorithm to use skills learned via the agent-specific rewards toward optimizing the global objective. Results demonstrate that MERL significantly outperforms state-of-the-art methods, such as MADDPG, on a number of difficult coordination benchmarks.
+
+# 1 INTRODUCTION
+
+Cooperative multiagent reinforcement learning (MARL) studies how multiple agents can learn to coordinate as a team toward maximizing a global objective. Cooperative MARL has been applied to many real world applications such as air traffic control (Tumer and Agogino, 2007), multi-robot coordination (Sheng et al., 2006; Yliniemi et al., 2014), communication and language (Lazaridou et al., 2016; Mordatch and Abbeel, 2018), and autonomous driving (Shalev-Shwartz et al., 2016).
+
+Many such environments endow agents with a team reward that reflects the team’s coordination objective, as well as an agent-specific local reward that rewards basic skills. For instance, in soccer, dense local rewards could capture agent-specific skills such as passing, dribbling and running. The agents must then coordinate when and where to use these skills in order to optimize the team objective, which is winning the game. Usually, the agent-specific reward is dense and easy to learn from, while the team reward is sparse and requires the cooperation of all or most agents.
+
+Having each agent directly optimize the team reward and ignore the agent-specific reward usually fails or is sample-inefficient for complex tasks due to the sparsity of the team reward. Conversely, having each agent directly optimize the agent-specific reward also fails because it does not capture the team’s objective, even with state of the art multiagent RL algorithms such as MADDPG (Lowe et al., 2017).
+
+One solution to this problem is to use reward shaping, where extensive domain knowledge about the task is used to create a proxy reward function (Rahmattalabi et al., 2016). Constructing this proxy reward function is difficult in complex environments, and is domain-dependent. Apart from requiring domain knowledge and manual tuning, this approach also poses risks of changing the underlying problem itself (Ng et al., 1999). Simple approaches to creating a proxy reward via linear combinations of the two objectives also fail to solve or generalize to complex coordination tasks (Devlin et al., 2011; Williamson et al., 2009).
+
+In this paper, we introduce Multiagent Evolutionary Reinforcement Learning (MERL), a state-ofthe-art algorithm for cooperative MARL that does not require reward shaping. MERL is a split-level training platform that combines gradient-based and gradient-free optimization. The gradient-free optimizer is an evolutionary algorithm that maximizes the team objective through neuroevolution. The gradient-based optimizer is a policy gradient algorithm that maximizes each agent’s dense, local rewards. These gradient-based policies are periodically copied into the evolutionary population. The two processes operate concurrently and share information through a shared replay buffer.
+
+A key strength of MERL is that it is a general method which does not require domain-specific reward shaping. This is because MERL optimizes the team objective directly while simultaneously leveraging agent-specific rewards to learn basic skills. We test MERL in a number of multiagent coordination benchmarks. Results demonstrate that MERL significantly outperforms state-of-the-art methods such as MADDPG, while using the same observations and reward functions. We also demonstrate that MERL scales gracefully to increasing complexity of coordination objectives where MADDPG and its variants fail to learn entirely.
+
+# 2 BACKGROUND AND RELATED WORK
+
+Markov Games: A standard reinforcement learning (RL) setting is often formalized as a Markov Decision Process (MDP) and consists of an agent interacting with an environment over a finite number of discrete time steps. This formulation can be extended to multiagent systems in the form of partially observable Markov games (Littman, 1994; Lowe et al., 2017). An $N$ -agent Markov game is defined by a global state of the world, $s$ , and a set of $N$ observations $\{ \mathcal { O } _ { i } \}$ and $N$ actions $\bar { \{ \mathcal { A } _ { i } \} }$ corresponding to the $N$ agents. At each time step $t$ , each agent observes its corresponding observation $O _ { i } ^ { t }$ and maps it to an action $A _ { i } ^ { t }$ using its policy $\pi _ { i }$ .
+
+Each agent receives a scalar reward $r _ { i } ^ { t }$ based on the global state $S _ { t }$ and joint action of the team. The   
+world then transitioprocess continues u ext state inal stat $\boldsymbol { S } _ { t + 1 }$ which ached. et of observations is the total return $\{ \mathcal { O } _ { i } \}$ . Thgent $\begin{array} { r } { R _ { i } = \sum _ { t = 0 } ^ { T } \gamma ^ { t } r _ { i } ^ { t } } \end{array}$ $i$ $\gamma \in ( 0 , 1 ]$
+
+TD3: Policy gradient (PG) methods frame the goal of maximizing the expected return as the minimization of a loss function. A widely used PG method for continuous, high-dimensional action spaces is DDPG (Lillicrap et al., 2015). Recently, (Fujimoto et al., 2018) extended DDPG to Twin Delayed DDPG (TD3), addressing its well-known overestimation problem. TD3 is the state-of-the-art, off-policy algorithm for model-free DRL in continuous action spaces.
+
+TD3 uses an actor-critic architecture (Sutton and Barto, 1998) maintaining a deterministic policy (actor) $\pi : { \mathcal { S } }  A$ , and two distinct critics $\mathcal { Q } : \mathcal { S } \times \mathcal { A }  \mathbb { R } _ { i }$ . Each critic independently approximates the actor’s action-value function ${ \mathcal { Q } } ^ { \pi }$ . A separate copy of the actor and critics are kept as target networks for stability and are updated periodically. A noisy version of the actor is used to explore the environment during training. The actor is trained using a noisy version of the sampled policy gradient computed by backpropagation through the combined actor-critic networks. This mitigates overfitting of the deterministic policy by smoothing the policy gradient updates.
+
+Evolutionary Reinforcement Learning (ERL) is a hybrid algorithm that combines Evolutionary Algorithms (EAs) (Floreano et al., 2008; Lüders et al., 2017; Fogel, 2006; Spears et al., 1993), with policy gradient methods (Khadka and Tumer, 2018). Instead of discarding the data generated during a standard EA rollout, ERL stores this data in a central replay buffer shared with the policy gradient’s own rollouts - thereby increasing the diversity of the data available for the policy gradient learners. Since the EA directly optimizes for episode-wide return, it biases exploration towards states with higher long-term returns. The policy gradient algorithm which learns using this state distribution inherits this implicit bias towards long-term optimization. Concurrently, the actor trained by the policy gradient algorithm is inserted into the evolutionary population allowing the EA to benefit from the fast gradient-based learning.
+
+Related Work: Lowe et al. (2017) introduced MADDPG which tackled the inherent non-stationarity of a multiagent learning environment by leveraging a critic which had full access to the joint state and action during training. Foerster et al. (2018b) utilized a similar setup with a centralized critic across agents to tackle StarCraft micromanagement tasks. An algorithm that could explicitly model other agents’ learning was investigated in Foerster et al. (2018a). However, all these approaches rely on a dense agent reward that properly captures the team objective. Methods to solve for these types of agent-specific reward functions were investigated in Li et al. (2012) but were limited to tasks with strong simulators where tree-based planning could be used.
+
+A closely related work to MERL is (Liu et al., 2019) where Population-Based Training (PBT) (Jaderberg et al., 2017) is used to optimize the relative importance between a collection of dense, shaped rewards automatically during training. This can be interpreted as a singular central reward function constructed by scalarizing a collection of reward signals where the scalarization coefficients are adaptively learned during training. In contrast, MERL optimizes its reward functions independently with information transfer across them facilitated through shared replay buffers and policy migration directly. This form of information transfer through a shared replay buffer has been explored extensively in recent literature (Colas et al., 2018; Khadka et al., 2019).
+
+# 3 MULTIAGENT EVOLUTIONARY REINFORCEMENT LEARNING
+
+MERL leverages both agent-specific and team objectives through a hybrid algorithm that combines gradient-free and gradient-based optimization. The gradient-free optimizer is an evolutionary algorithm that maximizes the team objective through neuroevolution. The gradient-based optimizer trains policies to maximize agent-specific rewards. These gradient-based policies are periodically added to the evolutionary population and participate in evolution. This enables the evolutionary algorithm to use agent-specific skills learned by training on the agent-specific rewards toward optimizing the team objective without needing to resort to reward shaping.
+
+# Algorithm 1 Multiagent Evolutionary Reinforcement Learning
+
+1: Initialize a population of $k$ multi-head teams $p o p _ { \pi }$ , each with weights $\theta ^ { \pi }$ initialized randomly   
+2: Initialize a shared critic $\mathcal { Q }$ with weights $\theta ^ { \mathcal { Q } }$   
+3: Initialize an ensemble of $N$ empty cyclic replay buffers $\mathcal { R } ^ { k }$ , one for each agent   
+4: Define a white Gaussian noise generator ${ \mathcal { W } } _ { g }$ random number generator $r ( ) \in [ 0 , 1 )$   
+5: for generation $= 1$ , $\infty$ do   
+6: for team $\pi \in p o p _ { \pi }$ do   
+7: $g$ , R = Rollout $( \pi , \mathcal { R }$ , noise None, $\xi$ )   
+8: _, R = Rollout (π, R, noise=Wg, $\xi = 1 \AA$ )   
+9: Assign $g$ as $\pi$ ’s fitness   
+10: end for   
+11: Rank the population $p o p _ { \pi }$ based on fitness scores   
+12: Select the first $e$ teams $\pi \in p o p _ { \pi }$ as elites   
+13: Select the remaining $( k - e )$ teams $\pi$ from $p o p _ { \pi }$ , to form Set $S$ using tournament selection   
+14: while $\left. S \right. < \left( k - e \right)$ do   
+15: Single-point crossover between a randomly sampled $\pi \in e$ and $\pi \in S$ and append to $S$   
+16: end while   
+17: for Agent $k { = } 1 , N$ do   
+18: Randomly sample a minibatch of $T$ transitions $\left( o _ { i } , a _ { i } , l _ { i } , o _ { i + 1 } \right)$ from $R ^ { k }$   
+19: Compute $y _ { i } = l _ { i } + \gamma \operatorname* { m i n } _ { j = 1 , 2 } \mathcal { Q } _ { j } ^ { \prime } \left( o _ { i + 1 } , a ^ { \sim } | \theta ^ { \mathcal { Q } _ { j } ^ { \prime } } \right)$   
+20: where $a ^ { \sim } = \pi _ { p g } ^ { \prime } ( k , o _ { i + 1 } | \theta ^ { \pi _ { p g } ^ { \prime } } )$ [action sampled from the $k ^ { t h }$ head of $\pi _ { p g } ^ { \prime } ] + \epsilon$   
+21: Update $\mathcal { Q }$ by minimizing the loss: $\begin{array} { r } { L = { \frac { 1 } { T } } \sum _ { i } ( y _ { i } - { \mathcal { Q } } ( o _ { i } , a _ { i } | \theta ^ { \mathcal { Q } } ) ^ { 2 } } \end{array}$   
+22: Update $\pi _ { p g } ^ { k }$ using the sampled policy gradient   
+23: Soft update $\begin{array} { r l } & { \nabla _ { \theta _ { p g } ^ { \pi } } J \sim \frac { 1 } { T } \sum \nabla _ { a } Q ( o , a | \theta ^ { \mathcal { Q } } ) | _ { o = o _ { i } , a = a _ { i } } \nabla _ { \theta _ { p g } ^ { \pi } } \pi _ { p g } ^ { k } ( s | \theta _ { p g } ^ { \pi } ) | _ { o = o _ { i } } } \\ & { \mathrm { t a r g e t n e t w o r k s : ~ } \theta ^ { \pi ^ { \prime } } \Leftarrow \tau \theta ^ { \pi } + ( 1 - \tau ) \theta ^ { \pi ^ { \prime } } \mathrm { ~ a n d ~ } \theta ^ { \mathcal { Q } ^ { \prime } } \Leftarrow \tau \theta ^ { \mathcal { Q } } + ( 1 - \tau ) \theta ^ { \mathcal { Q } ^ { \prime } } } \end{array}$   
+24: end for   
+25: Migrate the policy gradient team $p o p _ { j } :$ for weakest $\pi \in p o p _ { \pi } ^ { j } : \theta ^ { \pi } \Leftarrow \theta ^ { \pi _ { p g } }$   
+26: end for
+
+agents act independently based on their own observations while sharing weights (and by extension, the features) in the lower layers (trunk). This is commonly used to improve learning speed (Silver et al., 2017). Further, each agent $k$ also has its own replay buffer $( R ^ { k } )$ which stores its experience defined by the tuple (state, action, next state, local reward) for each interaction with the environment (rollout) involving that agent.
+
+Team Reward Optimization: Figure 2 illustrates the MERL algorithm. A population of multi-headed teams, each with the same topology, is initialized with random weights. The replay buffer $\mathcal { R } ^ { k }$ is shared by the $k$ -th agent across all teams. The population is then evaluated for each rollout. The team reward for each team is disbursed at the end of the episode and is considered as its fitness score. A selection operator selects a portion of the population for survival with probability proportionate to their fitness scores. The weights of the teams in the population are probabilistically perturbed through mutation and crossover operators to create the next generation of teams. A portion of the teams with the highest relative fitness are preserved as elites. At any given time, the team with the highest fitness, or the champion, represents the best solution for the task.
+
+![](images/eb3f28e8a0fec07cd2f6e77c01e5ccf1add948a76802c35d1e49a8f4adf218f2.jpg)  
+Figure 1: Team represented as multi-headed policy net $\pi$
+
+Policy Gradient: The procedure described so far resembles a standard EA except that each agent $k$ stores each of its experiences in its associated replay buffer $( R ^ { \dot { k } } )$ instead of just discarding it. However, unlike EA, which only learns based on the low-fidelity global reward, MERL also learns from the experiences within episodes of a rollout using policy gradients. To enable this kind of "local learning", MERL initializes one multi-headed policy network $\pi _ { p g }$ and one critic $\mathcal { Q }$ . A noisy version of $\pi _ { p g }$ is then used to conduct its own set of rollouts in the environment, storing each agent $k$ ’s experiences in its corresponding buffer $( R ^ { k } )$ similar to the evolutionary rollouts.
+
+![](images/5b69ac49b6df2a62148c62848c51ca1c209b99d74212f1183ffe91b5a0f8b374.jpg)  
+Figure 2: High level schematic of MERL highlighting the integration of local and global reward functions
+
+# Agent-Specific Reward Optimization:
+
+Crucially, each agent’s replay buffer is kept separate from that of every other agent to ensure diversity amongst the agents. The shared critic samples a random mini-batch unifand uses it to update its parameters using gradient descent. Each agent $\pi _ { p g } ^ { k }$ y from each replay bufferthen draws a mini-batch of experiences from its corresponding buffer $( R ^ { k } )$ and uses it to sample a policy gradient from the shared critic. Unlike the teams in the evolutionary population which directly seek to optimize the team reward, $\pi _ { p g }$ seeks to maximize the agent-specific local reward while exploiting the experiences collected via evolution.
+
+Skill Migration: Periodically, the $\pi _ { p g }$ network is copied into the evolving population of teams and can propagate its features by participating in evolution. This is the core mechanism that combines policies learned via agent-specific and team rewards. Regardless of whether the two rewards are aligned, evolution ensures that only the performant derivatives of the migrated network are retained. This mechanism guarantees protection against destructive interference commonly seen when a direct scalarization between two reward functions is attempted. Further, the level of information exchange is automatically adjusted during the process of learning, in contrast to being manually tuned by an expert designer.
+
+Algorithm 1 provides a detailed pseudo-code of the MERL algorithm. The choice of hyperparameters is explained in the Appendix. Additionally, our source code 1 is available online.
+
+![](images/f3b9dc7d018d352f954e20cef7776fd9dff6f3ca08a097a0cbcc007f35687ad7.jpg)  
+Figure 3: Illustration of environments tested (Lowe et al., 2017; Rahmattalabi et al., 2016)
+
+We adopt environments from (Lowe et al., 2017) and (Rahmattalabi et al., 2016) to perform our experiments. Each environment consists of multiple agents and landmarks in a two-dimensional world. Agents take continuous control actions to move about the world. Figure 3 illustrates the four environments which are described in more detail below.
+
+Predator-Prey: In this environment, $N$ slower cooperating agents (predators) must chase the faster adversary (prey) around an environment with $L$ large landmarks in randomly-generated locations. The predators get a reward when they catch (touch) the prey while the prey is penalized. The team reward for the predators is the cumulative number of prey-touches in an episode. Each predator can also compute the average distance to the prey and use it as its agent-specific reward. All agents observe the relative positions and velocities of the other agents as well as the positions of the landmarks. The prey can accelerate $3 3 \%$ faster than the predator and has a higher top speed. We tests two versions termed simple and hard predator-prey where the prey is $3 0 \%$ and $1 0 0 \%$ faster, respectively. Additionally, the prey itself learns dynamically during training. We use DDPG (Lillicrap et al., 2015) as a learning algorithm for training the prey policy. All of our candidate algorithms are tested on their ability to train the team of predators in catching this prey.
+
+Physical Deception: $N$ agents cooperate to reach a single target Point of Interest (POI) among $N$ POIs. They are rewarded based on the closest distance of any agent to the target. A lone adversary also desires to reach the target POI. However, the adversary does not know which of the POIs is the correct one. Thus the cooperating agents must learn to spread out and cover all POIs so as to deceive the adversary as they are penalized based on the adversary’s distance to the target. The team reward for the agents is then the cumulative reward in an episode. We use DDPG (Lillicrap et al., 2015) to train the adversary policy.
+
+Keep-Away: In this scenario, a team of $N$ cooperating agents must reach a target POI out of $L$ total POIs. Each agent is rewarded based on its distance to the target. We construct the team reward as simply the sum of the agent-specific rewards in an episode. An adversary also has to occupy the target while keeping the cooperating agents from reaching the target by pushing them away. To incentivize this behavior, the adversary is rewarded based on its distance to the target POI and penalized based on the distance of the target from the nearest cooperating agent. Additionally, it does not know which of the POIs is the target and must infer this from the movement of the agents. DDPG (Lillicrap et al., 2015) is used to train the adversary policy.
+
+Rover Domain: This environment is adapted from (Rahmattalabi et al., 2016). Here, $N$ agents must cooperate to reach a set of $K$ POIs. Multiple agents need to simultaneously go to the same POI in order to observe it. The number of agents required to observe a POI is termed the coupling requirement. Agents do not know and must infer the coupling factor from the rewards obtained. If a team with fewer agents than this number go to a POI, no reward is observed. The team’s reward is the percentage of POIs observed at the end of an episode.
+
+Each agent can also locally compute its distance to its closest POI and use it as its agent-specific reward. Its observation comprises two channels to detect POIs and rovers, respectively. Each channel receives intensity information over $1 0 ^ { \circ }$ resolution spanning the $3 6 0 ^ { \circ }$ around the agent’s position loosely based on the characteristic of a Pioneer robot (Thrun et al., 2000). This is similar to a LIDAR. Since it returns the closest reflector, occlusions make the problem partially-observable. A coupling factor of 1 is similar to the cooperative navigation task in Lowe et al. (2017). We test coupling factors from 1 to 7 to capture extremely complex coordination objectives.
+
+Compared Baselines: We compare the performance of MERL with a standard neuroevolutionary algorithm (EA) (Fogel, 2006), MADDPG (Lowe et al., 2017) and MATD3, a variant of MADDPG that integrates the improvements described within TD3 (Fujimoto et al., 2018) over DDPG. Internally, MERL uses EA and TD3 as its team-reward and agent-specific reward optimizer, respectively. MADDPG was chosen as it is the state-of-the-art multiagent RL algorithm. We implemented MATD3 to ensure that the differences between MADDPG and MERL do not originate from having the more stable TD3 over DDPG.
+
+Methodology for Reported Metrics: For MATD3 and MADDPG, the team network was periodically tested on 10 task instances without any exploratory noise. The average score was logged as its performance. For MERL and EA, the team with the highest fitness was chosen as the champion for each generation. The champion was then tested on 10 task instances, and the average score was logged. This protocol shielded the reported metrics from any bias of the population size. We conduct 5 statistically independent runs with random seeds from $\lbrace 2 0 1 9 , 2 0 2 3 \rbrace$ and report the average with error bars showing a $9 5 \%$ confidence interval. All scores reported are compared against the number of environment steps (frames). A step is defined as the multiagent team taking a joint action and receiving a feedback from the environment. To make the comparisons fair across single-team and population-based algorithms, all steps taken by all teams in the population are counted cumulatively.
+
+# 5 RESULTS
+
+Predator-Prey: Figure 4 shows the comparative performance in controlling the team of predators in the Predator-Prey environment. Note that this is an adversarial environment where the prey dynamically adapts against the predators. The prey (considered as part of the environment in this analysis) uses DDPG to learn constantly against our team of predators. This is why predator performance (measured as number of prey
+
+![](images/0cb4a4c8aeb4b34c2205565ce137e4925afd1c7f9e7a013cbf367d6a4e7f9e1d.jpg)  
+Figure 4: Performance on Predator-Prey where the prey is $3 0 \%$ faster (left) and $1 0 0 \%$ faster (right), respectively.
+
+touches) exhibits ebb and flow during learning. MERL outperforms MATD3, EA, and MADDPG across both simple and hard variations of the task. EA seems to be approaching MERL’s performance but is significantly slower to learn. This is an expected behavior for neuroevolutionary methods which are known to be sample-inefficient. In contrast, MERL, by virtue of its fast policy-gradient components, learns significantly faster.
+
+Physical Deception: Figure 5 (left) shows the comparative performance in controlling the team of agents in the Physical Deception environment. The performance here is largely based on how close the adversary comes to the target POI. Since the adversary starts out untrained, all compared algorithms start out with a fairly high score. As the adversary gradually learns to infer and move towards the target POI, MATD3 and MADDPG
+
+![](images/4864eb6072b1511223e15b3197f11c097b65f9808b70b7c4a3311a5e6f03c190.jpg)  
+Figure 5: Performance on Physical Deception (left) and Keep-Away (right)
+
+demonstrate a gradual decline in performance. However, MERL and EA are able to hold their performance by concocting effective counter-strategies in deceiving the adversary. EA reaches the same performance as MERL but is slower to learn.
+
+Keep-Away: Figure 5 (right) show the comparative performance in Keep-Away. Similar to Physical Deception, MERL and EA are able to hold performance by attaining good counter-measures against the adversary while MATD3 and MADDPG fail to do so. However, EA slightly outperforms MERL on this task.
+
+Rover Domain: Figure 6 shows the comparative performance of MERL, MADDPG, MATD3, and EA tested in the rover domain with coupling factors $1 , 3$ and 7. In order to benchmark against the proxy reward functions that use scalarized linear combinations, we test MADDPG and MATD3 with two variations of reward functions. Global represents the scenario where only the sparse team reward is used. Mixed represents the scenario where a linear combination of the team-reward and agent-specific reward is used. Each reward is normalized before being combined. A weighing coefficient of 10 is used to amplify the teamreward’s influence in order to counter its sparsity. The weighing coefficient was tuned using a grid search (more details in Figure 7).
+
+MERL significantly outperforms all baselines across all coupling requirements. The tested baselines clearly degrade quickly beyond a coupling of 3. The increasing coupling requirement is equiv
+
+![](images/b593e65cf96e99547954c9251a5b9203d02d03ea91735f2ea45476a8207d0515.jpg)  
+Figure 6: Performance on the Rover Domain.
+
+alent to increasing difficulty in joint-space exploration and entanglement in the team objective. However, it does not increase the size of the state-space, complexity of perception, or navigation. This indicates that the degradation in performance is strictly due to the increase in complexity of the team objective.
+
+Notably, MERL is able to learn on coupling greater than $n = 6$ where methods without explicit reward shaping have been shown to fail entirely (Rahmattalabi et al., 2016). MERL successfully completes the task using the same set of information and coarse, unshaped reward functions as the other algorithms. The primary mechanism that enables this is MERL’s split-level approach that allows it to leverage the agent-specific reward function to solve navigation and perception while concurrently using the team-reward function to learn team formation and effective coordination.
+
+Scalarization Coefficients for Mixed Rewards: Figure 7 shows the performance of MATD3 in optimizing mixed rewards computed with different coefficients used to amplify the team-reward relative to the agent-reward. The results demonstrate that finding a good balance between these two rewards through linear scalarization is difficult, as all values tested fail to make any progress in the task. This is because a static scalarization cannot capture the dynamic properties of which reward is important when and instead leads to an ineffective proxy. In contrast, MERL is able to leverage both reward functions without the need to explicitly combine them either linearly or via more complex mixing functions.
+
+![](images/ba37acf16c8bc52690b2989a0832adf3ea0f962cf5473aef09e9dc91832868da.jpg)  
+Figure 7: MATD3’s performance for different scalarization coefficients
+
+Team Behaviors: Figure 8 illustrates the trajectories generated for the Rover Domain with a coupling of $n = 3$ . The trajectories for partially and fully trained MERL are shown in Figure 8 (a) and (b), respectively. During training, when MERL has not discovered team success (no POIs are successfully observed), MERL simply optimizes the agent-specific reward for each agent. This allows it to reach trajectories such as the ones shown in 8(a) where each agent learns to go towards a POI.
+
+Since each agent explicitly aims to reach a POI, the probability 3 agents congregating to the same POI is higher compared to random undirected exploration by each agent without the dense agent-specific reward. Once this scenario is discovered, the team reward optimizer (EA) within MERL explicitly selects for agent policies that jointly lead to such team-forming behaviors. Eventually it succeeds as shown in Figure 8(b). Here, team formation and collaborative pursuit of the POIs is immediately apparent. Two teams of 3 agents each form at the start of the episode. Further, the two teams also coordinate to pursue different POIs in order to maximize the team reward. While not perfect (the bottom POI is left unobserved), they do succeed in observing 3 out of the 4 POIs.
+
+In contrast, MATD3-mixed fails to observe any POI. From the trajectories, it is apparent that the agents have successfully learned to perceive and navigate to reach POIs. However, they are unable to use this skill towards fulfilling the team objective. Instead each agent is rather split on the objective that it is optimizing. Some agents seem
+
+![](images/6dc013826d99b17b97f74e86c24af82539916517c50656be71043609e41829bd.jpg)  
+Figure 8: Agent trajectories for couplingto be in sole pursuit of POIs $= 3$ . Red/black squares are observed/unobserved POIs respectivelywithout any regard for team formation or collaboration while others seem to exhibit random movements.
+
+The primary reason for this is the mixed reward function that directly combines the agent-specific and team reward functions. Since the two reward functions have no guarantees of alignment across the state-space of the task, they invariably lead to learning these sub-optimal joint-behaviors that solve a certain form of scalarized mixed objective. In contrast, MERL by virtue of its bi-level optimization framework is able to leverage both reward functions without the need to explicitly combine them. This enables MERL to avoid these sub-optimal policies and solve the task without any reward shaping or manual tuning.
+
+Selection Rate: We ran experiments tracking whether the policies migrated from the policy gradient learners to the evolutionary population were selected or discarded during the subsequent selection process (Figure 9). Note that the expected selection rate if chosen at random is 0.1 as 1 policy is migrated into a population of 10. In contrast, the selection rate for migrated policies is significantly higher across all benchmarks with the exception of Keep-Away. This is consistent with the performance results seen in Keep-Away where EA initially outperforms MERL. However, in general, these results indicate that MERL’s integrative approach in combining the two optimization processes towards optimizing the team objective is crucial.
+
+![](images/e7821c81bad78633dfa494753f5f4c47ed78a7244362ff762a44fe016c551a0e.jpg)  
+Figure 9: Selection rate for migrating policies
+
+# 6 CONCLUSION
+
+In this paper, we introduced MERL, a split-level algorithm that leverages both agent-specific and team objectives by combining gradient-based and gradient-free optimization. MERL achieves this by using a fast policy-gradient optimizer to exploit dense agent-specific rewards while concurrently leveraging neuroevolution to tackle the team-objective.
+
+Results demonstrate that MERL significantly outperforms MADDPG, the state-of-the-art multiagent RL method, in a wide array of benchmarks. We also tested a modification of MADDPG to integrate TD3 - the state-of-the-art single-agent RL algorithm. These experiments demonstrated that the core improvements of MERL originate from its ability to leverage both team and agent-specific reward functions without the need to explicitly combine them. This differentiates MERL from other approaches like reward scalarization and reward shaping that either require extensive manual tuning or can detrimentally change the MDP ( $\mathrm { N g }$ et al., 1999) itself.
+
+Future work will explore MERL for adversarial settings such as Pommerman (Resnick et al., 2018), StarCraft (Justesen and Risi, 2017; Vinyals et al., 2017) and RoboCup (Kitano et al., 1995; Liu et al., 2019). Further, extending MERL to general multi-reward settings such as is the case for multitask learning, is another promising area for future work.
+
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+
+# A HYPERPARAMETERS DESCRIPTION
+
+Table 1: Hyperparameters used for Predator-Prey, Keep-away and Physical Deception   
+
+<table><tr><td rowspan=1 colspan=2>Hyperparameter</td><td rowspan=1 colspan=2>MERL</td><td rowspan=1 colspan=1>MATD3/MADDPG</td></tr><tr><td></td><td></td><td rowspan=1 colspan=2>10</td><td rowspan=1 colspan=1>N/A</td></tr><tr><td></td><td></td><td rowspan=1 colspan=2>10</td><td rowspan=1 colspan=1>10</td></tr><tr><td></td><td></td><td rowspan=1 colspan=2>0.01</td><td rowspan=1 colspan=1>0.01</td></tr><tr><td></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>0.01</td><td rowspan=1 colspan=1>0.01</td></tr><tr><td></td><td></td><td rowspan=1 colspan=2>0.01</td><td rowspan=1 colspan=1>0.01</td></tr><tr><td></td><td></td><td rowspan=1 colspan=2>0.95</td><td rowspan=1 colspan=1>0.95</td></tr><tr><td></td><td></td><td rowspan=1 colspan=2>1e6</td><td rowspan=1 colspan=1>1e6</td></tr><tr><td></td><td></td><td rowspan=1 colspan=2>1024</td><td rowspan=1 colspan=1>1024</td></tr><tr><td></td><td></td><td rowspan=1 colspan=2>0.9</td><td rowspan=1 colspan=1>N/A</td></tr><tr><td></td><td></td><td rowspan=1 colspan=2>0.1</td><td rowspan=9 colspan=1>N/AN/AN/AN/AN/AN(0,σ)0.4N/A[100,100][300,300]0.20.52</td></tr><tr><td></td><td></td><td rowspan=1 colspan=2>0.1</td></tr><tr><td></td><td></td><td rowspan=1 colspan=2>0.05</td></tr><tr><td></td><td></td><td rowspan=1 colspan=2>0.05</td></tr><tr><td></td><td></td><td rowspan=1 colspan=2>4</td></tr><tr><td></td><td></td><td rowspan=2 colspan=2>N(0,σ)0.410[100,100][100,100]0.2</td></tr><tr><td></td><td></td><td rowspan=1 colspan=1>0.2</td></tr><tr><td></td><td></td><td rowspan=1 colspan=2>0.5</td></tr><tr><td></td><td></td><td rowspan=1 colspan=2>2</td></tr></table>
+
+Table 1 details the hyperparameters used for MERL, MATD3, and MADDPG in tackling predatorprey and cooperative navigation. The hyperparmaeters were inherited from Lowe et al. (2017) to match the original experiments for MADDPG and MATD3. The only exception to this was the use of hyperbolic tangent instead of Relu activation functions. Table 2: Hyperparameters used for Rover Domain
+
+<table><tr><td rowspan=1 colspan=6>Hyperparameter</td><td rowspan=1 colspan=1>MERL</td><td rowspan=1 colspan=1>MATD3/MADDPG</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>N/A</td></tr><tr><td></td><td rowspan=1 colspan=5></td><td rowspan=1 colspan=1>50</td><td rowspan=1 colspan=1>50</td></tr><tr><td></td><td></td><td rowspan=2 colspan=3></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td rowspan=1 colspan=2></td><td rowspan=1 colspan=1>le-5</td><td rowspan=1 colspan=1>le-5</td></tr><tr><td></td><td></td><td rowspan=1 colspan=4></td><td rowspan=1 colspan=2></td><td rowspan=1 colspan=1>5e-5</td><td rowspan=1 colspan=1>5e-5</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan=1 colspan=1>le-5</td><td rowspan=1 colspan=1>le-5</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan=1 colspan=1>0.5</td><td rowspan=1 colspan=1>0.97</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan=1 colspan=1>1e5</td><td rowspan=1 colspan=1>1e5</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan=1 colspan=1>512</td><td rowspan=1 colspan=1>512</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan=1 colspan=1>0.9</td><td rowspan=1 colspan=1>N/A</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan=1 colspan=1>0.1</td><td rowspan=1 colspan=1>N/A</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan=1 colspan=1>0.1</td><td rowspan=9 colspan=1>N/AN/AN/AN/AN(0,σ)0.4N/A[100,100][300,300]0.20.52</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan=1 colspan=1>0.05</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan=1 colspan=1>0.05</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan=1 colspan=1>4</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan=1 colspan=1>N(0,σ)</td><td rowspan=1 colspan=1></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan=1 colspan=1>0.4</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan=1 colspan=1>10[100,100][100,100]</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan=1 colspan=1>0.2</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan=1 colspan=1>0.52</td></tr></table>
+
+Table 2 details the hyperparameters used for MERL, MATD3, and MADDPG in the rover domain. The hyperparameters themselves are defined below:
+
+# • Optimizer $=$ Adam
+
+Adam optimizer was used to update both the actor and critic networks for all learners.
+
+# • Population size $k$
+
+This parameter controls the number of different actors (policies) that are present in the evolutionary population.
+
+# • Rollout size
+
+This parameter controls the number of rollout workers (each running an episode of the task) per generation.
+
+Note: The two parameters above (population size $k$ and rollout size) collectively modulates the proportion of exploration carried out through noise in the actor’s parameter space and its action space.
+
+# • Target weight $\tau$
+
+This parameter controls the magnitude of the soft update between the actors and critic networks, and their target counterparts.
+
+• Actor Learning Rate This parameter controls the learning rate of the actor network.
+
+• Critic Learning Rate This parameter controls the learning rate of the critic network.
+
+# • Discount Rate
+
+This parameter controls the discount rate used to compute the return optimized by policy gradient.
+
+# • Replay Buffer Size
+
+This parameter controls the size of the replay buffer. After the buffer is filled, the oldest experiences are deleted in order to make room for new ones.
+
+• Batch Size This parameters controls the batch size used to compute the gradients.
+
+• Actor Activation Function Hyperbolic tangent was used as the activation function.
+
+• Critic Activation Function Hyperbolic tangent was used as the activation function.
+
+# • Number of Elites
+
+This parameter controls the fraction of the population that are categorized as elites. Since an elite individual (actor) is shielded from the mutation step and preserved as it is, the elite fraction modulates the degree of exploration/exploitation within the evolutionary population.
+
+# • Mutation Probability
+
+This parameter represents the probability that an actor goes through a mutation operation between generation.
+
+# • Mutation Fraction
+
+This parameter controls the fraction of the weights in a chosen actor (neural network) that are mutated, once the actor is chosen for mutation.
+
+# • Mutation Strength
+
+This parameter controls the standard deviation of the Gaussian operation that comprises mutation.
+
+# • Super Mutation Probability
+
+This parameter controls the probability that a super mutation (larger mutation) happens in place of a standard mutation.
+
+# • Reset Mutation Probability
+
+This parameter controls the probability a neural weight is instead reset between $\mathcal { N } ( 0 , 1 )$ rather than being mutated.
+
+# • Exploration Noise
+
+This parameter controls the standard deviation of the Gaussian operation that comprise the noise added to the actor’s actions during exploration by the learners (learner roll-outs).
+
+# • TD3 Policy Noise Variance
+
+This parameter controls the standard deviation of the Gaussian operation that comprise the noise added to the policy output before applying the Bellman backup. This is often referred to as the magnitude of policy smoothing in TD3.
+
+# • TD3 Policy Noise Clip
+
+This parameter controls the maximum norm of the policy noise used to smooth the policy.
+
+# • TD3 Policy Update Frequency
+
+This parameter controls the number of critic updates per policy update in TD3.
+
+# B ROLLOUT METHODOLOGY
+
+Algorithm 2 describes an episode of rollout under MERL detailing the connections between the local reward, global reward, and the associated replay buffer.
+
+Algorithm 2 Function Rollout   
+
+<table><tr><td colspan="2">1: procedure RoLLOUT(π,R, noise, $)</td></tr><tr><td colspan="2"></td></tr><tr><td>2: 3:</td><td>fitness =0</td></tr><tr><td></td><td>for j= 1:g do</td></tr><tr><td>4:</td><td>Reset environment and get initial joint state js</td></tr><tr><td>5:</td><td>while env is not done do</td></tr><tr><td>6:</td><td>Initialize an empty list of joint action ja = []</td></tr><tr><td>7:</td><td>for Each agent (actor head) πk ∈ Tand sk in js do</td></tr><tr><td>8:</td><td>ja← jaUπk(sklθπ²)+noiset</td></tr><tr><td>9:</td><td>end for</td></tr><tr><td>10:</td><td>Execute ja and observe joint local reward jl, global reward g and joint next state js&#x27;</td></tr><tr><td>11:</td><td>for Each Replay BufferRk ∈R and sk,ak,lk,s&#x27; in js,ja,jl,js&#x27; do</td></tr><tr><td>12:</td><td>Append transition (sk,ak,lk,sk） to Rk</td></tr><tr><td>13:</td><td>end for</td></tr><tr><td>14:</td><td>js=js&#x27;</td></tr><tr><td>15:</td><td>if env is done: then</td></tr><tr><td>16:</td><td>fitness ←g</td></tr><tr><td>17:</td><td>end if</td></tr><tr><td>18:</td><td>end while</td></tr><tr><td>19:</td><td>end for</td></tr><tr><td>20:</td><td>Return fitness ,R m</td></tr><tr><td colspan="2">21: end procedure</td></tr><tr><td colspan="2"></td></tr></table>
+
+# C EVOLUTIONARY ALGORITHM POPULATION RUNS
+
+![](images/31897184e1e3225ecb0a3f141721e52e606ebebf4df50503614d680deaa0ec62.jpg)  
+Figure 10: Evolutionary Algorithm Population size sweep on the rover domain with a coupling of 3. MERL was run for 2-million steps while the other EA runs were ran for 100-million steps.
+
+Figure 10 compares EA with varying population sizes in the rover domain with a coupling of 3. Among the EA runs, a population size of 100 yields the best results converging to 0.3 in 100-millions frames. MERL (red) on the other hand is ran for 2-million frames and converges to 0.48. This is due to MERL’s ability to leverage gradient descent from its policy gradient components that lead to significantly faster learning performance.
+
+# D EVOLUTIONARY STRATEGIES (ES)
+
+# D.1 ES POPULATION SWEEP
+
+![](images/4ee99d83da66f487dc427af2cf5d71394e3d805f0a87ea6dbe09fa4ccd6b0763.jpg)  
+Figure 11: Evolutionary Strategies population size sweep on the rover domain with a coupling of 3
+
+Figure 11 compares ES with varying population sizes in the rover domain with a coupling of 3. Sigma for all ES runs are set at 0.1. Among the ES runs, a population size of 100 yields the best results converging to 0.1 in 100-millions frames. MERL (red) on the other hand is ran for 2-million frames and converges to 0.48.
+
+![](images/49f1480299040c90148f478fe4c358fa4511e7bd4d91fbcee05c70b0018be32a.jpg)  
+Figure 12: Evolutionary Strategies Noise magnitude (sigma) sweep on the rover domain with a coupling of 3
+
+Figure 12 compares ES with varying variance of noises (sigma) that control the magnitude of each perturbation. The experiments are conducted in the rover domain with a coupling of 3 with a population size of 100. Among the ES runs, a sigma of 0.1 yields the best results converging to 0.1 in 100-millions frames. MERL (red) on the other hand is ran for 2-million frames and converges to 0.48.
+
+![](images/1a1de3f5dc93c1b14b2219f95cf247cb476716aaecda946c312f88a54da57da8.jpg)  
+E PREDATOR-PREY WITH 3 PREY   
+Figure 13: Predator-prey with varying numbers of prey. Prey are $3 0 \%$ faster than the predators
+
+Figure 13 shows the results of running MATD3 with varying number of prey in the predator-prey domain. The experiments are ongoing.

parse/train/rkxtNaNKwr/rkxtNaNKwr_content_list.json

@@ -0,0 +1,1550 @@
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+    {
+        "type": "text",
+        "text": "EVOLUTIONARY REINFORCEMENT LEARNING FORSAMPLE-EFFICIENT MULTIAGENT COORDINATION",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            170,
+            398,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            234,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Many cooperative multiagent reinforcement learning environments provide agents with a sparse team-based reward, as well as a dense agent-specific reward that incentivizes learning basic skills. Training policies solely on the team-based reward is often difficult due to its sparsity. Also, relying solely on the agent-specific reward is sub-optimal because it usually does not capture the team coordination objective. A common approach is to use reward shaping to construct a proxy reward by combining the individual rewards. However, this requires manual tuning for each environment. We introduce Multiagent Evolutionary Reinforcement Learning (MERL), a split-level training platform that handles the two objectives separately through two optimization processes. An evolutionary algorithm maximizes the sparse team-based objective through neuroevolution on a population of teams. Concurrently, a gradient-based optimizer trains policies to only maximize the dense agent-specific rewards. The gradient-based policies are periodically added to the evolutionary population as a way of information transfer between the two optimization processes. This enables the evolutionary algorithm to use skills learned via the agent-specific rewards toward optimizing the global objective. Results demonstrate that MERL significantly outperforms state-of-the-art methods, such as MADDPG, on a number of difficult coordination benchmarks. ",
+        "bbox": [
+            233,
+            267,
+            766,
+            517
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            549,
+            336,
+            565
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Cooperative multiagent reinforcement learning (MARL) studies how multiple agents can learn to coordinate as a team toward maximizing a global objective. Cooperative MARL has been applied to many real world applications such as air traffic control (Tumer and Agogino, 2007), multi-robot coordination (Sheng et al., 2006; Yliniemi et al., 2014), communication and language (Lazaridou et al., 2016; Mordatch and Abbeel, 2018), and autonomous driving (Shalev-Shwartz et al., 2016). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Many such environments endow agents with a team reward that reflects the team’s coordination objective, as well as an agent-specific local reward that rewards basic skills. For instance, in soccer, dense local rewards could capture agent-specific skills such as passing, dribbling and running. The agents must then coordinate when and where to use these skills in order to optimize the team objective, which is winning the game. Usually, the agent-specific reward is dense and easy to learn from, while the team reward is sparse and requires the cooperation of all or most agents. ",
+        "bbox": [
+            174,
+            659,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Having each agent directly optimize the team reward and ignore the agent-specific reward usually fails or is sample-inefficient for complex tasks due to the sparsity of the team reward. Conversely, having each agent directly optimize the agent-specific reward also fails because it does not capture the team’s objective, even with state of the art multiagent RL algorithms such as MADDPG (Lowe et al., 2017). ",
+        "bbox": [
+            174,
+            750,
+            825,
+            819
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "One solution to this problem is to use reward shaping, where extensive domain knowledge about the task is used to create a proxy reward function (Rahmattalabi et al., 2016). Constructing this proxy reward function is difficult in complex environments, and is domain-dependent. Apart from requiring domain knowledge and manual tuning, this approach also poses risks of changing the underlying problem itself (Ng et al., 1999). Simple approaches to creating a proxy reward via linear combinations of the two objectives also fail to solve or generalize to complex coordination tasks (Devlin et al., 2011; Williamson et al., 2009). ",
+        "bbox": [
+            174,
+            825,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we introduce Multiagent Evolutionary Reinforcement Learning (MERL), a state-ofthe-art algorithm for cooperative MARL that does not require reward shaping. MERL is a split-level training platform that combines gradient-based and gradient-free optimization. The gradient-free optimizer is an evolutionary algorithm that maximizes the team objective through neuroevolution. The gradient-based optimizer is a policy gradient algorithm that maximizes each agent’s dense, local rewards. These gradient-based policies are periodically copied into the evolutionary population. The two processes operate concurrently and share information through a shared replay buffer. ",
+        "bbox": [
+            174,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "A key strength of MERL is that it is a general method which does not require domain-specific reward shaping. This is because MERL optimizes the team objective directly while simultaneously leveraging agent-specific rewards to learn basic skills. We test MERL in a number of multiagent coordination benchmarks. Results demonstrate that MERL significantly outperforms state-of-the-art methods such as MADDPG, while using the same observations and reward functions. We also demonstrate that MERL scales gracefully to increasing complexity of coordination objectives where MADDPG and its variants fail to learn entirely. ",
+        "bbox": [
+            174,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 BACKGROUND AND RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            325,
+            511,
+            342
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Markov Games: A standard reinforcement learning (RL) setting is often formalized as a Markov Decision Process (MDP) and consists of an agent interacting with an environment over a finite number of discrete time steps. This formulation can be extended to multiagent systems in the form of partially observable Markov games (Littman, 1994; Lowe et al., 2017). An $N$ -agent Markov game is defined by a global state of the world, $s$ , and a set of $N$ observations $\\{ \\mathcal { O } _ { i } \\}$ and $N$ actions $\\bar { \\{ \\mathcal { A } _ { i } \\} }$ corresponding to the $N$ agents. At each time step $t$ , each agent observes its corresponding observation $O _ { i } ^ { t }$ and maps it to an action $A _ { i } ^ { t }$ using its policy $\\pi _ { i }$ . ",
+        "bbox": [
+            174,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Each agent receives a scalar reward $r _ { i } ^ { t }$ based on the global state $S _ { t }$ and joint action of the team. The   \nworld then transitioprocess continues u ext state inal stat $\\boldsymbol { S } _ { t + 1 }$ which ached. et of observations is the total return $\\{ \\mathcal { O } _ { i } \\}$ . Thgent $\\begin{array} { r } { R _ { i } = \\sum _ { t = 0 } ^ { T } \\gamma ^ { t } r _ { i } ^ { t } } \\end{array}$ $i$ $\\gamma \\in ( 0 , 1 ]$ ",
+        "bbox": [
+            174,
+            460,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "TD3: Policy gradient (PG) methods frame the goal of maximizing the expected return as the minimization of a loss function. A widely used PG method for continuous, high-dimensional action spaces is DDPG (Lillicrap et al., 2015). Recently, (Fujimoto et al., 2018) extended DDPG to Twin Delayed DDPG (TD3), addressing its well-known overestimation problem. TD3 is the state-of-the-art, off-policy algorithm for model-free DRL in continuous action spaces. ",
+        "bbox": [
+            174,
+            526,
+            825,
+            597
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "TD3 uses an actor-critic architecture (Sutton and Barto, 1998) maintaining a deterministic policy (actor) $\\pi : { \\mathcal { S } }  A$ , and two distinct critics $\\mathcal { Q } : \\mathcal { S } \\times \\mathcal { A }  \\mathbb { R } _ { i }$ . Each critic independently approximates the actor’s action-value function ${ \\mathcal { Q } } ^ { \\pi }$ . A separate copy of the actor and critics are kept as target networks for stability and are updated periodically. A noisy version of the actor is used to explore the environment during training. The actor is trained using a noisy version of the sampled policy gradient computed by backpropagation through the combined actor-critic networks. This mitigates overfitting of the deterministic policy by smoothing the policy gradient updates. ",
+        "bbox": [
+            173,
+            603,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Evolutionary Reinforcement Learning (ERL) is a hybrid algorithm that combines Evolutionary Algorithms (EAs) (Floreano et al., 2008; Lüders et al., 2017; Fogel, 2006; Spears et al., 1993), with policy gradient methods (Khadka and Tumer, 2018). Instead of discarding the data generated during a standard EA rollout, ERL stores this data in a central replay buffer shared with the policy gradient’s own rollouts - thereby increasing the diversity of the data available for the policy gradient learners. Since the EA directly optimizes for episode-wide return, it biases exploration towards states with higher long-term returns. The policy gradient algorithm which learns using this state distribution inherits this implicit bias towards long-term optimization. Concurrently, the actor trained by the policy gradient algorithm is inserted into the evolutionary population allowing the EA to benefit from the fast gradient-based learning. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Related Work: Lowe et al. (2017) introduced MADDPG which tackled the inherent non-stationarity of a multiagent learning environment by leveraging a critic which had full access to the joint state and action during training. Foerster et al. (2018b) utilized a similar setup with a centralized critic across agents to tackle StarCraft micromanagement tasks. An algorithm that could explicitly model other agents’ learning was investigated in Foerster et al. (2018a). However, all these approaches rely on a dense agent reward that properly captures the team objective. Methods to solve for these types of agent-specific reward functions were investigated in Li et al. (2012) but were limited to tasks with strong simulators where tree-based planning could be used. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A closely related work to MERL is (Liu et al., 2019) where Population-Based Training (PBT) (Jaderberg et al., 2017) is used to optimize the relative importance between a collection of dense, shaped rewards automatically during training. This can be interpreted as a singular central reward function constructed by scalarizing a collection of reward signals where the scalarization coefficients are adaptively learned during training. In contrast, MERL optimizes its reward functions independently with information transfer across them facilitated through shared replay buffers and policy migration directly. This form of information transfer through a shared replay buffer has been explored extensively in recent literature (Colas et al., 2018; Khadka et al., 2019). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 MULTIAGENT EVOLUTIONARY REINFORCEMENT LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            290,
+            705,
+            308
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "MERL leverages both agent-specific and team objectives through a hybrid algorithm that combines gradient-free and gradient-based optimization. The gradient-free optimizer is an evolutionary algorithm that maximizes the team objective through neuroevolution. The gradient-based optimizer trains policies to maximize agent-specific rewards. These gradient-based policies are periodically added to the evolutionary population and participate in evolution. This enables the evolutionary algorithm to use agent-specific skills learned by training on the agent-specific rewards toward optimizing the team objective without needing to resort to reward shaping. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 Multiagent Evolutionary Reinforcement Learning ",
+        "text_level": 1,
+        "bbox": [
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+            439,
+            589,
+            455
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "1: Initialize a population of $k$ multi-head teams $p o p _ { \\pi }$ , each with weights $\\theta ^ { \\pi }$ initialized randomly   \n2: Initialize a shared critic $\\mathcal { Q }$ with weights $\\theta ^ { \\mathcal { Q } }$   \n3: Initialize an ensemble of $N$ empty cyclic replay buffers $\\mathcal { R } ^ { k }$ , one for each agent   \n4: Define a white Gaussian noise generator ${ \\mathcal { W } } _ { g }$ random number generator $r ( ) \\in [ 0 , 1 )$   \n5: for generation $= 1$ , $\\infty$ do   \n6: for team $\\pi \\in p o p _ { \\pi }$ do   \n7: $g$ , R = Rollout $( \\pi , \\mathcal { R }$ , noise None, $\\xi$ )   \n8: _, R = Rollout (π, R, noise=Wg, $\\xi = 1 \\AA$ )   \n9: Assign $g$ as $\\pi$ ’s fitness   \n10: end for   \n11: Rank the population $p o p _ { \\pi }$ based on fitness scores   \n12: Select the first $e$ teams $\\pi \\in p o p _ { \\pi }$ as elites   \n13: Select the remaining $( k - e )$ teams $\\pi$ from $p o p _ { \\pi }$ , to form Set $S$ using tournament selection   \n14: while $\\left. S \\right. < \\left( k - e \\right)$ do   \n15: Single-point crossover between a randomly sampled $\\pi \\in e$ and $\\pi \\in S$ and append to $S$   \n16: end while   \n17: for Agent $k { = } 1 , N$ do   \n18: Randomly sample a minibatch of $T$ transitions $\\left( o _ { i } , a _ { i } , l _ { i } , o _ { i + 1 } \\right)$ from $R ^ { k }$   \n19: Compute $y _ { i } = l _ { i } + \\gamma \\operatorname* { m i n } _ { j = 1 , 2 } \\mathcal { Q } _ { j } ^ { \\prime } \\left( o _ { i + 1 } , a ^ { \\sim } | \\theta ^ { \\mathcal { Q } _ { j } ^ { \\prime } } \\right)$   \n20: where $a ^ { \\sim } = \\pi _ { p g } ^ { \\prime } ( k , o _ { i + 1 } | \\theta ^ { \\pi _ { p g } ^ { \\prime } } )$ [action sampled from the $k ^ { t h }$ head of $\\pi _ { p g } ^ { \\prime } ] + \\epsilon$   \n21: Update $\\mathcal { Q }$ by minimizing the loss: $\\begin{array} { r } { L = { \\frac { 1 } { T } } \\sum _ { i } ( y _ { i } - { \\mathcal { Q } } ( o _ { i } , a _ { i } | \\theta ^ { \\mathcal { Q } } ) ^ { 2 } } \\end{array}$   \n22: Update $\\pi _ { p g } ^ { k }$ using the sampled policy gradient   \n23: Soft update $\\begin{array} { r l } & { \\nabla _ { \\theta _ { p g } ^ { \\pi } } J \\sim \\frac { 1 } { T } \\sum \\nabla _ { a } Q ( o , a | \\theta ^ { \\mathcal { Q } } ) | _ { o = o _ { i } , a = a _ { i } } \\nabla _ { \\theta _ { p g } ^ { \\pi } } \\pi _ { p g } ^ { k } ( s | \\theta _ { p g } ^ { \\pi } ) | _ { o = o _ { i } } } \\\\ & { \\mathrm { t a r g e t n e t w o r k s : ~ } \\theta ^ { \\pi ^ { \\prime } } \\Leftarrow \\tau \\theta ^ { \\pi } + ( 1 - \\tau ) \\theta ^ { \\pi ^ { \\prime } } \\mathrm { ~ a n d ~ } \\theta ^ { \\mathcal { Q } ^ { \\prime } } \\Leftarrow \\tau \\theta ^ { \\mathcal { Q } } + ( 1 - \\tau ) \\theta ^ { \\mathcal { Q } ^ { \\prime } } } \\end{array}$   \n24: end for   \n25: Migrate the policy gradient team $p o p _ { j } :$ for weakest $\\pi \\in p o p _ { \\pi } ^ { j } : \\theta ^ { \\pi } \\Leftarrow \\theta ^ { \\pi _ { p g } }$   \n26: end for ",
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+    {
+        "type": "text",
+        "text": "agents act independently based on their own observations while sharing weights (and by extension, the features) in the lower layers (trunk). This is commonly used to improve learning speed (Silver et al., 2017). Further, each agent $k$ also has its own replay buffer $( R ^ { k } )$ which stores its experience defined by the tuple (state, action, next state, local reward) for each interaction with the environment (rollout) involving that agent. ",
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+        "type": "text",
+        "text": "Team Reward Optimization: Figure 2 illustrates the MERL algorithm. A population of multi-headed teams, each with the same topology, is initialized with random weights. The replay buffer $\\mathcal { R } ^ { k }$ is shared by the $k$ -th agent across all teams. The population is then evaluated for each rollout. The team reward for each team is disbursed at the end of the episode and is considered as its fitness score. A selection operator selects a portion of the population for survival with probability proportionate to their fitness scores. The weights of the teams in the population are probabilistically perturbed through mutation and crossover operators to create the next generation of teams. A portion of the teams with the highest relative fitness are preserved as elites. At any given time, the team with the highest fitness, or the champion, represents the best solution for the task. ",
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+    {
+        "type": "image",
+        "img_path": "images/eb3f28e8a0fec07cd2f6e77c01e5ccf1add948a76802c35d1e49a8f4adf218f2.jpg",
+        "image_caption": [
+            "Figure 1: Team represented as multi-headed policy net $\\pi$ "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "type": "text",
+        "text": "Policy Gradient: The procedure described so far resembles a standard EA except that each agent $k$ stores each of its experiences in its associated replay buffer $( R ^ { \\dot { k } } )$ instead of just discarding it. However, unlike EA, which only learns based on the low-fidelity global reward, MERL also learns from the experiences within episodes of a rollout using policy gradients. To enable this kind of \"local learning\", MERL initializes one multi-headed policy network $\\pi _ { p g }$ and one critic $\\mathcal { Q }$ . A noisy version of $\\pi _ { p g }$ is then used to conduct its own set of rollouts in the environment, storing each agent $k$ ’s experiences in its corresponding buffer $( R ^ { k } )$ similar to the evolutionary rollouts. ",
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+    {
+        "type": "image",
+        "img_path": "images/5b69ac49b6df2a62148c62848c51ca1c209b99d74212f1183ffe91b5a0f8b374.jpg",
+        "image_caption": [
+            "Figure 2: High level schematic of MERL highlighting the integration of local and global reward functions "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Agent-Specific Reward Optimization: ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Crucially, each agent’s replay buffer is kept separate from that of every other agent to ensure diversity amongst the agents. The shared critic samples a random mini-batch unifand uses it to update its parameters using gradient descent. Each agent $\\pi _ { p g } ^ { k }$ y from each replay bufferthen draws a mini-batch of experiences from its corresponding buffer $( R ^ { k } )$ and uses it to sample a policy gradient from the shared critic. Unlike the teams in the evolutionary population which directly seek to optimize the team reward, $\\pi _ { p g }$ seeks to maximize the agent-specific local reward while exploiting the experiences collected via evolution. ",
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+        "type": "text",
+        "text": "Skill Migration: Periodically, the $\\pi _ { p g }$ network is copied into the evolving population of teams and can propagate its features by participating in evolution. This is the core mechanism that combines policies learned via agent-specific and team rewards. Regardless of whether the two rewards are aligned, evolution ensures that only the performant derivatives of the migrated network are retained. This mechanism guarantees protection against destructive interference commonly seen when a direct scalarization between two reward functions is attempted. Further, the level of information exchange is automatically adjusted during the process of learning, in contrast to being manually tuned by an expert designer. ",
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+    {
+        "type": "text",
+        "text": "Algorithm 1 provides a detailed pseudo-code of the MERL algorithm. The choice of hyperparameters is explained in the Appendix. Additionally, our source code 1 is available online. ",
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+        "page_idx": 3
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+    {
+        "type": "image",
+        "img_path": "images/f3b9dc7d018d352f954e20cef7776fd9dff6f3ca08a097a0cbcc007f35687ad7.jpg",
+        "image_caption": [
+            "Figure 3: Illustration of environments tested (Lowe et al., 2017; Rahmattalabi et al., 2016) "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We adopt environments from (Lowe et al., 2017) and (Rahmattalabi et al., 2016) to perform our experiments. Each environment consists of multiple agents and landmarks in a two-dimensional world. Agents take continuous control actions to move about the world. Figure 3 illustrates the four environments which are described in more detail below. ",
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+        "type": "text",
+        "text": "Predator-Prey: In this environment, $N$ slower cooperating agents (predators) must chase the faster adversary (prey) around an environment with $L$ large landmarks in randomly-generated locations. The predators get a reward when they catch (touch) the prey while the prey is penalized. The team reward for the predators is the cumulative number of prey-touches in an episode. Each predator can also compute the average distance to the prey and use it as its agent-specific reward. All agents observe the relative positions and velocities of the other agents as well as the positions of the landmarks. The prey can accelerate $3 3 \\%$ faster than the predator and has a higher top speed. We tests two versions termed simple and hard predator-prey where the prey is $3 0 \\%$ and $1 0 0 \\%$ faster, respectively. Additionally, the prey itself learns dynamically during training. We use DDPG (Lillicrap et al., 2015) as a learning algorithm for training the prey policy. All of our candidate algorithms are tested on their ability to train the team of predators in catching this prey. ",
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+    {
+        "type": "text",
+        "text": "Physical Deception: $N$ agents cooperate to reach a single target Point of Interest (POI) among $N$ POIs. They are rewarded based on the closest distance of any agent to the target. A lone adversary also desires to reach the target POI. However, the adversary does not know which of the POIs is the correct one. Thus the cooperating agents must learn to spread out and cover all POIs so as to deceive the adversary as they are penalized based on the adversary’s distance to the target. The team reward for the agents is then the cumulative reward in an episode. We use DDPG (Lillicrap et al., 2015) to train the adversary policy. ",
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+        "type": "text",
+        "text": "Keep-Away: In this scenario, a team of $N$ cooperating agents must reach a target POI out of $L$ total POIs. Each agent is rewarded based on its distance to the target. We construct the team reward as simply the sum of the agent-specific rewards in an episode. An adversary also has to occupy the target while keeping the cooperating agents from reaching the target by pushing them away. To incentivize this behavior, the adversary is rewarded based on its distance to the target POI and penalized based on the distance of the target from the nearest cooperating agent. Additionally, it does not know which of the POIs is the target and must infer this from the movement of the agents. DDPG (Lillicrap et al., 2015) is used to train the adversary policy. ",
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+        "type": "text",
+        "text": "Rover Domain: This environment is adapted from (Rahmattalabi et al., 2016). Here, $N$ agents must cooperate to reach a set of $K$ POIs. Multiple agents need to simultaneously go to the same POI in order to observe it. The number of agents required to observe a POI is termed the coupling requirement. Agents do not know and must infer the coupling factor from the rewards obtained. If a team with fewer agents than this number go to a POI, no reward is observed. The team’s reward is the percentage of POIs observed at the end of an episode. ",
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+    {
+        "type": "text",
+        "text": "Each agent can also locally compute its distance to its closest POI and use it as its agent-specific reward. Its observation comprises two channels to detect POIs and rovers, respectively. Each channel receives intensity information over $1 0 ^ { \\circ }$ resolution spanning the $3 6 0 ^ { \\circ }$ around the agent’s position loosely based on the characteristic of a Pioneer robot (Thrun et al., 2000). This is similar to a LIDAR. Since it returns the closest reflector, occlusions make the problem partially-observable. A coupling factor of 1 is similar to the cooperative navigation task in Lowe et al. (2017). We test coupling factors from 1 to 7 to capture extremely complex coordination objectives. ",
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+        "type": "text",
+        "text": "Compared Baselines: We compare the performance of MERL with a standard neuroevolutionary algorithm (EA) (Fogel, 2006), MADDPG (Lowe et al., 2017) and MATD3, a variant of MADDPG that integrates the improvements described within TD3 (Fujimoto et al., 2018) over DDPG. Internally, MERL uses EA and TD3 as its team-reward and agent-specific reward optimizer, respectively. MADDPG was chosen as it is the state-of-the-art multiagent RL algorithm. We implemented MATD3 to ensure that the differences between MADDPG and MERL do not originate from having the more stable TD3 over DDPG. ",
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+    {
+        "type": "text",
+        "text": "Methodology for Reported Metrics: For MATD3 and MADDPG, the team network was periodically tested on 10 task instances without any exploratory noise. The average score was logged as its performance. For MERL and EA, the team with the highest fitness was chosen as the champion for each generation. The champion was then tested on 10 task instances, and the average score was logged. This protocol shielded the reported metrics from any bias of the population size. We conduct 5 statistically independent runs with random seeds from $\\lbrace 2 0 1 9 , 2 0 2 3 \\rbrace$ and report the average with error bars showing a $9 5 \\%$ confidence interval. All scores reported are compared against the number of environment steps (frames). A step is defined as the multiagent team taking a joint action and receiving a feedback from the environment. To make the comparisons fair across single-team and population-based algorithms, all steps taken by all teams in the population are counted cumulatively. ",
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+    {
+        "type": "text",
+        "text": "5 RESULTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Predator-Prey: Figure 4 shows the comparative performance in controlling the team of predators in the Predator-Prey environment. Note that this is an adversarial environment where the prey dynamically adapts against the predators. The prey (considered as part of the environment in this analysis) uses DDPG to learn constantly against our team of predators. This is why predator performance (measured as number of prey ",
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+    {
+        "type": "image",
+        "img_path": "images/0cb4a4c8aeb4b34c2205565ce137e4925afd1c7f9e7a013cbf367d6a4e7f9e1d.jpg",
+        "image_caption": [
+            "Figure 4: Performance on Predator-Prey where the prey is $3 0 \\%$ faster (left) and $1 0 0 \\%$ faster (right), respectively. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "touches) exhibits ebb and flow during learning. MERL outperforms MATD3, EA, and MADDPG across both simple and hard variations of the task. EA seems to be approaching MERL’s performance but is significantly slower to learn. This is an expected behavior for neuroevolutionary methods which are known to be sample-inefficient. In contrast, MERL, by virtue of its fast policy-gradient components, learns significantly faster. ",
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+    {
+        "type": "text",
+        "text": "Physical Deception: Figure 5 (left) shows the comparative performance in controlling the team of agents in the Physical Deception environment. The performance here is largely based on how close the adversary comes to the target POI. Since the adversary starts out untrained, all compared algorithms start out with a fairly high score. As the adversary gradually learns to infer and move towards the target POI, MATD3 and MADDPG ",
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/4864eb6072b1511223e15b3197f11c097b65f9808b70b7c4a3311a5e6f03c190.jpg",
+        "image_caption": [
+            "Figure 5: Performance on Physical Deception (left) and Keep-Away (right) "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "demonstrate a gradual decline in performance. However, MERL and EA are able to hold their performance by concocting effective counter-strategies in deceiving the adversary. EA reaches the same performance as MERL but is slower to learn. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Keep-Away: Figure 5 (right) show the comparative performance in Keep-Away. Similar to Physical Deception, MERL and EA are able to hold performance by attaining good counter-measures against the adversary while MATD3 and MADDPG fail to do so. However, EA slightly outperforms MERL on this task. ",
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+    {
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+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Rover Domain: Figure 6 shows the comparative performance of MERL, MADDPG, MATD3, and EA tested in the rover domain with coupling factors $1 , 3$ and 7. In order to benchmark against the proxy reward functions that use scalarized linear combinations, we test MADDPG and MATD3 with two variations of reward functions. Global represents the scenario where only the sparse team reward is used. Mixed represents the scenario where a linear combination of the team-reward and agent-specific reward is used. Each reward is normalized before being combined. A weighing coefficient of 10 is used to amplify the teamreward’s influence in order to counter its sparsity. The weighing coefficient was tuned using a grid search (more details in Figure 7). ",
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+    {
+        "type": "text",
+        "text": "MERL significantly outperforms all baselines across all coupling requirements. The tested baselines clearly degrade quickly beyond a coupling of 3. The increasing coupling requirement is equiv",
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+    {
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+        "img_path": "images/b593e65cf96e99547954c9251a5b9203d02d03ea91735f2ea45476a8207d0515.jpg",
+        "image_caption": [
+            "Figure 6: Performance on the Rover Domain. "
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+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "alent to increasing difficulty in joint-space exploration and entanglement in the team objective. However, it does not increase the size of the state-space, complexity of perception, or navigation. This indicates that the degradation in performance is strictly due to the increase in complexity of the team objective. ",
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+        "type": "text",
+        "text": "Notably, MERL is able to learn on coupling greater than $n = 6$ where methods without explicit reward shaping have been shown to fail entirely (Rahmattalabi et al., 2016). MERL successfully completes the task using the same set of information and coarse, unshaped reward functions as the other algorithms. The primary mechanism that enables this is MERL’s split-level approach that allows it to leverage the agent-specific reward function to solve navigation and perception while concurrently using the team-reward function to learn team formation and effective coordination. ",
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+    {
+        "type": "text",
+        "text": "Scalarization Coefficients for Mixed Rewards: Figure 7 shows the performance of MATD3 in optimizing mixed rewards computed with different coefficients used to amplify the team-reward relative to the agent-reward. The results demonstrate that finding a good balance between these two rewards through linear scalarization is difficult, as all values tested fail to make any progress in the task. This is because a static scalarization cannot capture the dynamic properties of which reward is important when and instead leads to an ineffective proxy. In contrast, MERL is able to leverage both reward functions without the need to explicitly combine them either linearly or via more complex mixing functions. ",
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+    {
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+        "img_path": "images/ba37acf16c8bc52690b2989a0832adf3ea0f962cf5473aef09e9dc91832868da.jpg",
+        "image_caption": [
+            "Figure 7: MATD3’s performance for different scalarization coefficients "
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+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Team Behaviors: Figure 8 illustrates the trajectories generated for the Rover Domain with a coupling of $n = 3$ . The trajectories for partially and fully trained MERL are shown in Figure 8 (a) and (b), respectively. During training, when MERL has not discovered team success (no POIs are successfully observed), MERL simply optimizes the agent-specific reward for each agent. This allows it to reach trajectories such as the ones shown in 8(a) where each agent learns to go towards a POI. ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Since each agent explicitly aims to reach a POI, the probability 3 agents congregating to the same POI is higher compared to random undirected exploration by each agent without the dense agent-specific reward. Once this scenario is discovered, the team reward optimizer (EA) within MERL explicitly selects for agent policies that jointly lead to such team-forming behaviors. Eventually it succeeds as shown in Figure 8(b). Here, team formation and collaborative pursuit of the POIs is immediately apparent. Two teams of 3 agents each form at the start of the episode. Further, the two teams also coordinate to pursue different POIs in order to maximize the team reward. While not perfect (the bottom POI is left unobserved), they do succeed in observing 3 out of the 4 POIs. ",
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+        "type": "text",
+        "text": "In contrast, MATD3-mixed fails to observe any POI. From the trajectories, it is apparent that the agents have successfully learned to perceive and navigate to reach POIs. However, they are unable to use this skill towards fulfilling the team objective. Instead each agent is rather split on the objective that it is optimizing. Some agents seem ",
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/6dc013826d99b17b97f74e86c24af82539916517c50656be71043609e41829bd.jpg",
+        "image_caption": [
+            "Figure 8: Agent trajectories for couplingto be in sole pursuit of POIs $= 3$ . Red/black squares are observed/unobserved POIs respectivelywithout any regard for team formation or collaboration while others seem to exhibit random movements. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            397,
+            174,
+            841,
+            304
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The primary reason for this is the mixed reward function that directly combines the agent-specific and team reward functions. Since the two reward functions have no guarantees of alignment across the state-space of the task, they invariably lead to learning these sub-optimal joint-behaviors that solve a certain form of scalarized mixed objective. In contrast, MERL by virtue of its bi-level optimization framework is able to leverage both reward functions without the need to explicitly combine them. This enables MERL to avoid these sub-optimal policies and solve the task without any reward shaping or manual tuning. ",
+        "bbox": [
+            173,
+            367,
+            825,
+            465
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Selection Rate: We ran experiments tracking whether the policies migrated from the policy gradient learners to the evolutionary population were selected or discarded during the subsequent selection process (Figure 9). Note that the expected selection rate if chosen at random is 0.1 as 1 policy is migrated into a population of 10. In contrast, the selection rate for migrated policies is significantly higher across all benchmarks with the exception of Keep-Away. This is consistent with the performance results seen in Keep-Away where EA initially outperforms MERL. However, in general, these results indicate that MERL’s integrative approach in combining the two optimization processes towards optimizing the team objective is crucial. ",
+        "bbox": [
+            174,
+            472,
+            589,
+            638
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/e7821c81bad78633dfa494753f5f4c47ed78a7244362ff762a44fe016c551a0e.jpg",
+        "image_caption": [
+            "Figure 9: Selection rate for migrating policies "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            606,
+            479,
+            816,
+            599
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            665,
+            318,
+            681
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we introduced MERL, a split-level algorithm that leverages both agent-specific and team objectives by combining gradient-based and gradient-free optimization. MERL achieves this by using a fast policy-gradient optimizer to exploit dense agent-specific rewards while concurrently leveraging neuroevolution to tackle the team-objective. ",
+        "bbox": [
+            174,
+            700,
+            825,
+            756
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Results demonstrate that MERL significantly outperforms MADDPG, the state-of-the-art multiagent RL method, in a wide array of benchmarks. We also tested a modification of MADDPG to integrate TD3 - the state-of-the-art single-agent RL algorithm. These experiments demonstrated that the core improvements of MERL originate from its ability to leverage both team and agent-specific reward functions without the need to explicitly combine them. This differentiates MERL from other approaches like reward scalarization and reward shaping that either require extensive manual tuning or can detrimentally change the MDP ( $\\mathrm { N g }$ et al., 1999) itself. ",
+        "bbox": [
+            174,
+            763,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Future work will explore MERL for adversarial settings such as Pommerman (Resnick et al., 2018), StarCraft (Justesen and Risi, 2017; Vinyals et al., 2017) and RoboCup (Kitano et al., 1995; Liu et al., 2019). Further, extending MERL to general multi-reward settings such as is the case for multitask learning, is another promising area for future work. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES   \nC. Colas, O. Sigaud, and P.-Y. Oudeyer. Gep-pg: Decoupling exploration and exploitation in deep reinforcement learning algorithms. arXiv preprint arXiv:1802.05054, 2018.   \nS. Devlin, M. Grzes, and D. Kudenko. Multi-agent, reward shaping for robocup keepaway. In ´ The 10th International Conference on Autonomous Agents and Multiagent Systems-Volume 3, pages 1227–1228. International Foundation for Autonomous Agents and Multiagent Systems, 2011.   \nD. Floreano, P. Dürr, and C. Mattiussi. Neuroevolution: from architectures to learning. Evolutionary Intelligence, 1(1):47–62, 2008.   \nJ. Foerster, R. Y. Chen, M. Al-Shedivat, S. Whiteson, P. Abbeel, and I. Mordatch. Learning with opponent-learning awareness. In Proceedings of the 17th International Conference on Autonomous Agents and MultiAgent Systems, pages 122–130. International Foundation for Autonomous Agents and Multiagent Systems, 2018a.   \nJ. N. Foerster, G. Farquhar, T. Afouras, N. Nardelli, and S. Whiteson. Counterfactual multi-agent policy gradients. In Thirty-Second AAAI Conference on Artificial Intelligence, 2018b.   \nD. B. Fogel. Evolutionary computation: toward a new philosophy of machine intelligence, volume 1. John Wiley & Sons, 2006.   \nS. Fujimoto, H. van Hoof, and D. Meger. Addressing function approximation error in actor-critic methods. arXiv preprint arXiv:1802.09477, 2018.   \nM. Jaderberg, V. Dalibard, S. Osindero, W. M. Czarnecki, J. Donahue, A. Razavi, O. Vinyals, T. Green, I. Dunning, K. Simonyan, et al. Population based training of neural networks. arXiv preprint arXiv:1711.09846, 2017.   \nN. Justesen and S. Risi. Learning macromanagement in starcraft from replays using deep learning. In 2017 IEEE Conference on Computational Intelligence and Games (CIG), pages 162–169. IEEE, 2017.   \nS. Khadka and K. Tumer. Evolution-guided policy gradient in reinforcement learning. In Advances in Neural Information Processing Systems, pages 1196–1208, 2018.   \nS. Khadka, S. Majumdar, T. Nassar, Z. Dwiel, E. Tumer, S. Miret, Y. Liu, and K. Tumer. Collaborative evolutionary reinforcement learning. arXiv preprint arXiv:1905.00976v2, 2019.   \nH. Kitano, M. Asada, Y. Kuniyoshi, I. Noda, and E. Osawa. Robocup: The robot world cup initiative, 1995.   \nA. Lazaridou, A. Peysakhovich, and M. Baroni. Multi-agent cooperation and the emergence of (natural) language. arXiv preprint arXiv:1612.07182, 2016.   \nF.-D. Li, M. Wu, Y. He, and X. Chen. Optimal control in microgrid using multi-agent reinforcement learning. ISA transactions, 51(6):743–751, 2012.   \nT. P. Lillicrap, J. J. Hunt, A. Pritzel, N. Heess, T. Erez, Y. Tassa, D. Silver, and D. Wierstra. Continuous control with deep reinforcement learning. arXiv preprint arXiv:1509.02971, 2015.   \nM. L. Littman. Markov games as a framework for multi-agent reinforcement learning. In Machine learning proceedings 1994, pages 157–163. Elsevier, 1994.   \nS. Liu, G. Lever, J. Merel, S. Tunyasuvunakool, N. Heess, and T. Graepel. Emergent coordination through competition. arXiv preprint arXiv:1902.07151, 2019.   \nR. Lowe, Y. Wu, A. Tamar, J. Harb, O. P. Abbeel, and I. Mordatch. Multi-agent actor-critic for mixed cooperative-competitive environments. In Advances in Neural Information Processing Systems, pages 6379–6390, 2017.   \nB. Lüders, M. Schläger, A. Korach, and S. Risi. Continual and one-shot learning through neural networks with dynamic external memory. In European Conference on the Applications of Evolutionary Computation, pages 886–901. Springer, 2017. ",
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+        "bbox": [
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+            114,
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+            710
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "A HYPERPARAMETERS DESCRIPTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            501,
+            118
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/8a694ec12d39f618b766321d01e2737f0c1bb2f30a0d0d7d1d5d2d5415a23e8a.jpg",
+        "table_caption": [
+            "Table 1: Hyperparameters used for Predator-Prey, Keep-away and Physical Deception "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=2>Hyperparameter</td><td rowspan=1 colspan=2>MERL</td><td rowspan=1 colspan=1>MATD3/MADDPG</td></tr><tr><td></td><td></td><td rowspan=1 colspan=2>10</td><td rowspan=1 colspan=1>N/A</td></tr><tr><td></td><td></td><td rowspan=1 colspan=2>10</td><td rowspan=1 colspan=1>10</td></tr><tr><td></td><td></td><td rowspan=1 colspan=2>0.01</td><td rowspan=1 colspan=1>0.01</td></tr><tr><td></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>0.01</td><td rowspan=1 colspan=1>0.01</td></tr><tr><td></td><td></td><td rowspan=1 colspan=2>0.01</td><td rowspan=1 colspan=1>0.01</td></tr><tr><td></td><td></td><td rowspan=1 colspan=2>0.95</td><td rowspan=1 colspan=1>0.95</td></tr><tr><td></td><td></td><td rowspan=1 colspan=2>1e6</td><td rowspan=1 colspan=1>1e6</td></tr><tr><td></td><td></td><td rowspan=1 colspan=2>1024</td><td rowspan=1 colspan=1>1024</td></tr><tr><td></td><td></td><td rowspan=1 colspan=2>0.9</td><td rowspan=1 colspan=1>N/A</td></tr><tr><td></td><td></td><td rowspan=1 colspan=2>0.1</td><td rowspan=9 colspan=1>N/AN/AN/AN/AN/AN(0,σ)0.4N/A[100,100][300,300]0.20.52</td></tr><tr><td></td><td></td><td rowspan=1 colspan=2>0.1</td></tr><tr><td></td><td></td><td rowspan=1 colspan=2>0.05</td></tr><tr><td></td><td></td><td rowspan=1 colspan=2>0.05</td></tr><tr><td></td><td></td><td rowspan=1 colspan=2>4</td></tr><tr><td></td><td></td><td rowspan=2 colspan=2>N(0,σ)0.410[100,100][100,100]0.2</td></tr><tr><td></td><td></td><td rowspan=1 colspan=1>0.2</td></tr><tr><td></td><td></td><td rowspan=1 colspan=2>0.5</td></tr><tr><td></td><td></td><td rowspan=1 colspan=2>2</td></tr></table>",
+        "bbox": [
+            174,
+            156,
+            712,
+            483
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Table 1 details the hyperparameters used for MERL, MATD3, and MADDPG in tackling predatorprey and cooperative navigation. The hyperparmaeters were inherited from Lowe et al. (2017) to match the original experiments for MADDPG and MATD3. The only exception to this was the use of hyperbolic tangent instead of Relu activation functions. Table 2: Hyperparameters used for Rover Domain ",
+        "bbox": [
+            173,
+            486,
+            825,
+            555
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/7c437a0908eb8552726dd302084fcca1d5434e98fa9886190f7918a9012e5202.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=6>Hyperparameter</td><td rowspan=1 colspan=1>MERL</td><td rowspan=1 colspan=1>MATD3/MADDPG</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>N/A</td></tr><tr><td></td><td rowspan=1 colspan=5></td><td rowspan=1 colspan=1>50</td><td rowspan=1 colspan=1>50</td></tr><tr><td></td><td></td><td rowspan=2 colspan=3></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td rowspan=1 colspan=2></td><td rowspan=1 colspan=1>le-5</td><td rowspan=1 colspan=1>le-5</td></tr><tr><td></td><td></td><td rowspan=1 colspan=4></td><td rowspan=1 colspan=2></td><td rowspan=1 colspan=1>5e-5</td><td rowspan=1 colspan=1>5e-5</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan=1 colspan=1>le-5</td><td rowspan=1 colspan=1>le-5</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan=1 colspan=1>0.5</td><td rowspan=1 colspan=1>0.97</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan=1 colspan=1>1e5</td><td rowspan=1 colspan=1>1e5</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan=1 colspan=1>512</td><td rowspan=1 colspan=1>512</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan=1 colspan=1>0.9</td><td rowspan=1 colspan=1>N/A</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan=1 colspan=1>0.1</td><td rowspan=1 colspan=1>N/A</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan=1 colspan=1>0.1</td><td rowspan=9 colspan=1>N/AN/AN/AN/AN(0,σ)0.4N/A[100,100][300,300]0.20.52</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan=1 colspan=1>0.05</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan=1 colspan=1>0.05</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan=1 colspan=1>4</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan=1 colspan=1>N(0,σ)</td><td rowspan=1 colspan=1></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan=1 colspan=1>0.4</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan=1 colspan=1>10[100,100][100,100]</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan=1 colspan=1>0.2</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td rowspan=1 colspan=1>0.52</td></tr></table>",
+        "bbox": [
+            173,
+            565,
+            712,
+            893
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Table 2 details the hyperparameters used for MERL, MATD3, and MADDPG in the rover domain. The hyperparameters themselves are defined below: ",
+        "bbox": [
+            171,
+            103,
+            825,
+            132
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• Optimizer $=$ Adam ",
+        "text_level": 1,
+        "bbox": [
+            217,
+            146,
+            366,
+            160
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Adam optimizer was used to update both the actor and critic networks for all learners. ",
+        "bbox": [
+            220,
+            160,
+            792,
+            174
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• Population size $k$ ",
+        "text_level": 1,
+        "bbox": [
+            217,
+            180,
+            354,
+            194
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "This parameter controls the number of different actors (policies) that are present in the evolutionary population. ",
+        "bbox": [
+            215,
+            194,
+            823,
+            222
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• Rollout size ",
+        "text_level": 1,
+        "bbox": [
+            215,
+            228,
+            316,
+            241
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "This parameter controls the number of rollout workers (each running an episode of the task) per generation. ",
+        "bbox": [
+            220,
+            241,
+            823,
+            270
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Note: The two parameters above (population size $k$ and rollout size) collectively modulates the proportion of exploration carried out through noise in the actor’s parameter space and its action space. ",
+        "bbox": [
+            222,
+            272,
+            825,
+            313
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• Target weight $\\tau$ ",
+        "text_level": 1,
+        "bbox": [
+            217,
+            319,
+            343,
+            333
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "This parameter controls the magnitude of the soft update between the actors and critic networks, and their target counterparts. ",
+        "bbox": [
+            220,
+            334,
+            823,
+            361
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• Actor Learning Rate This parameter controls the learning rate of the actor network. ",
+        "bbox": [
+            217,
+            367,
+            638,
+            395
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• Critic Learning Rate This parameter controls the learning rate of the critic network. ",
+        "bbox": [
+            217,
+            401,
+            638,
+            429
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• Discount Rate ",
+        "text_level": 1,
+        "bbox": [
+            217,
+            435,
+            333,
+            448
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "This parameter controls the discount rate used to compute the return optimized by policy gradient. ",
+        "bbox": [
+            222,
+            449,
+            825,
+            476
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• Replay Buffer Size ",
+        "text_level": 1,
+        "bbox": [
+            217,
+            482,
+            366,
+            496
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "This parameter controls the size of the replay buffer. After the buffer is filled, the oldest experiences are deleted in order to make room for new ones. ",
+        "bbox": [
+            223,
+            496,
+            825,
+            523
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• Batch Size This parameters controls the batch size used to compute the gradients. ",
+        "bbox": [
+            222,
+            531,
+            691,
+            558
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• Actor Activation Function Hyperbolic tangent was used as the activation function. ",
+        "bbox": [
+            217,
+            563,
+            594,
+            592
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• Critic Activation Function Hyperbolic tangent was used as the activation function. ",
+        "bbox": [
+            217,
+            597,
+            594,
+            625
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• Number of Elites ",
+        "text_level": 1,
+        "bbox": [
+            217,
+            631,
+            354,
+            643
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "This parameter controls the fraction of the population that are categorized as elites. Since an elite individual (actor) is shielded from the mutation step and preserved as it is, the elite fraction modulates the degree of exploration/exploitation within the evolutionary population. ",
+        "bbox": [
+            217,
+            645,
+            825,
+            686
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• Mutation Probability ",
+        "text_level": 1,
+        "bbox": [
+            217,
+            691,
+            382,
+            705
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "This parameter represents the probability that an actor goes through a mutation operation between generation. ",
+        "bbox": [
+            217,
+            707,
+            823,
+            733
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• Mutation Fraction ",
+        "text_level": 1,
+        "bbox": [
+            217,
+            739,
+            364,
+            752
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "This parameter controls the fraction of the weights in a chosen actor (neural network) that are mutated, once the actor is chosen for mutation. ",
+        "bbox": [
+            222,
+            753,
+            823,
+            781
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• Mutation Strength ",
+        "text_level": 1,
+        "bbox": [
+            217,
+            787,
+            364,
+            800
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "This parameter controls the standard deviation of the Gaussian operation that comprises mutation. ",
+        "bbox": [
+            222,
+            801,
+            825,
+            828
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• Super Mutation Probability ",
+        "text_level": 1,
+        "bbox": [
+            217,
+            834,
+            429,
+            848
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "This parameter controls the probability that a super mutation (larger mutation) happens in place of a standard mutation. ",
+        "bbox": [
+            220,
+            848,
+            825,
+            876
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• Reset Mutation Probability ",
+        "text_level": 1,
+        "bbox": [
+            217,
+            882,
+            424,
+            896
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "This parameter controls the probability a neural weight is instead reset between $\\mathcal { N } ( 0 , 1 )$ rather than being mutated. ",
+        "bbox": [
+            217,
+            896,
+            823,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• Exploration Noise ",
+        "text_level": 1,
+        "bbox": [
+            217,
+            104,
+            361,
+            117
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "This parameter controls the standard deviation of the Gaussian operation that comprise the noise added to the actor’s actions during exploration by the learners (learner roll-outs). ",
+        "bbox": [
+            223,
+            118,
+            823,
+            146
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• TD3 Policy Noise Variance ",
+        "text_level": 1,
+        "bbox": [
+            217,
+            204,
+            421,
+            218
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "This parameter controls the standard deviation of the Gaussian operation that comprise the noise added to the policy output before applying the Bellman backup. This is often referred to as the magnitude of policy smoothing in TD3. ",
+        "bbox": [
+            227,
+            218,
+            825,
+            260
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• TD3 Policy Noise Clip ",
+        "text_level": 1,
+        "bbox": [
+            217,
+            316,
+            387,
+            330
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "This parameter controls the maximum norm of the policy noise used to smooth the policy. ",
+        "bbox": [
+            215,
+            332,
+            820,
+            345
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• TD3 Policy Update Frequency ",
+        "text_level": 1,
+        "bbox": [
+            217,
+            404,
+            444,
+            417
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "This parameter controls the number of critic updates per policy update in TD3. ",
+        "bbox": [
+            228,
+            417,
+            746,
+            431
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B ROLLOUT METHODOLOGY ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            512,
+            431,
+            527
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 2 describes an episode of rollout under MERL detailing the connections between the local reward, global reward, and the associated replay buffer. ",
+        "bbox": [
+            174,
+            579,
+            823,
+            607
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/e1f6353e69e18020dcb8b95f45dadc2c88866a2602cbab3bb2ce8a9bcb9e7942.jpg",
+        "table_caption": [
+            "Algorithm 2 Function Rollout "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">1: procedure RoLLOUT(π,R, noise, $)</td></tr><tr><td colspan=\"2\"></td></tr><tr><td>2: 3:</td><td>fitness =0</td></tr><tr><td></td><td>for j= 1:g do</td></tr><tr><td>4:</td><td>Reset environment and get initial joint state js</td></tr><tr><td>5:</td><td>while env is not done do</td></tr><tr><td>6:</td><td>Initialize an empty list of joint action ja = []</td></tr><tr><td>7:</td><td>for Each agent (actor head) πk ∈ Tand sk in js do</td></tr><tr><td>8:</td><td>ja← jaUπk(sklθπ²)+noiset</td></tr><tr><td>9:</td><td>end for</td></tr><tr><td>10:</td><td>Execute ja and observe joint local reward jl, global reward g and joint next state js&#x27;</td></tr><tr><td>11:</td><td>for Each Replay BufferRk ∈R and sk,ak,lk,s&#x27; in js,ja,jl,js&#x27; do</td></tr><tr><td>12:</td><td>Append transition (sk,ak,lk,sk） to Rk</td></tr><tr><td>13:</td><td>end for</td></tr><tr><td>14:</td><td>js=js&#x27;</td></tr><tr><td>15:</td><td>if env is done: then</td></tr><tr><td>16:</td><td>fitness ←g</td></tr><tr><td>17:</td><td>end if</td></tr><tr><td>18:</td><td>end while</td></tr><tr><td>19:</td><td>end for</td></tr><tr><td>20:</td><td>Return fitness ,R m</td></tr><tr><td colspan=\"2\">21: end procedure</td></tr><tr><td colspan=\"2\"></td></tr></table>",
+        "bbox": [
+            174,
+            626,
+            825,
+            928
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C EVOLUTIONARY ALGORITHM POPULATION RUNS ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            102,
+            620,
+            118
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/31897184e1e3225ecb0a3f141721e52e606ebebf4df50503614d680deaa0ec62.jpg",
+        "image_caption": [
+            "Figure 10: Evolutionary Algorithm Population size sweep on the rover domain with a coupling of 3. MERL was run for 2-million steps while the other EA runs were ran for 100-million steps. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            276,
+            131,
+            718,
+            382
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Figure 10 compares EA with varying population sizes in the rover domain with a coupling of 3. Among the EA runs, a population size of 100 yields the best results converging to 0.3 in 100-millions frames. MERL (red) on the other hand is ran for 2-million frames and converges to 0.48. This is due to MERL’s ability to leverage gradient descent from its policy gradient components that lead to significantly faster learning performance. ",
+        "bbox": [
+            173,
+            436,
+            826,
+            507
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D EVOLUTIONARY STRATEGIES (ES) ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            529,
+            501,
+            546
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D.1 ES POPULATION SWEEP ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            561,
+            387,
+            577
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/4ee99d83da66f487dc427af2cf5d71394e3d805f0a87ea6dbe09fa4ccd6b0763.jpg",
+        "image_caption": [
+            "Figure 11: Evolutionary Strategies population size sweep on the rover domain with a coupling of 3 "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            276,
+            584,
+            720,
+            827
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Figure 11 compares ES with varying population sizes in the rover domain with a coupling of 3. Sigma for all ES runs are set at 0.1. Among the ES runs, a population size of 100 yields the best results converging to 0.1 in 100-millions frames. MERL (red) on the other hand is ran for 2-million frames and converges to 0.48. ",
+        "bbox": [
+            174,
+            867,
+            825,
+            924
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/49f1480299040c90148f478fe4c358fa4511e7bd4d91fbcee05c70b0018be32a.jpg",
+        "image_caption": [
+            "Figure 12: Evolutionary Strategies Noise magnitude (sigma) sweep on the rover domain with a coupling of 3 "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            277,
+            125,
+            720,
+            367
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Figure 12 compares ES with varying variance of noises (sigma) that control the magnitude of each perturbation. The experiments are conducted in the rover domain with a coupling of 3 with a population size of 100. Among the ES runs, a sigma of 0.1 yields the best results converging to 0.1 in 100-millions frames. MERL (red) on the other hand is ran for 2-million frames and converges to 0.48. ",
+        "bbox": [
+            173,
+            421,
+            826,
+            491
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/1a1de3f5dc93c1b14b2219f95cf247cb476716aaecda946c312f88a54da57da8.jpg",
+        "image_caption": [
+            "E PREDATOR-PREY WITH 3 PREY ",
+            "Figure 13: Predator-prey with varying numbers of prey. Prey are $3 0 \\%$ faster than the predators "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            282,
+            540,
+            715,
+            790
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Figure 13 shows the results of running MATD3 with varying number of prey in the predator-prey domain. The experiments are ongoing. ",
+        "bbox": [
+            174,
+            833,
+            823,
+            863
+        ],
+        "page_idx": 14
+    }
+]

parse/train/ry4S90l0b/ry4S90l0b.md

@@ -0,0 +1,213 @@
+# A SELF-TRAINING METHOD FOR SEMI-SUPERVISED GANS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Since the creation of Generative Adversarial Networks (GANs), much work has been done to improve their training stability, their generated image quality, their range of application but nearly none of them explored their self-training potential. Self-training has been used before the advent of deep learning in order to allow training on limited labelled training data and has shown impressive results in semi-supervised learning. In this work, we combine these two ideas and make GANs self-trainable for semi-supervised learning tasks by exploiting their infinite data generation potential. Results show that using even the simplest form of self-training yields an improvement. We also show results for a more complex self-training scheme that performs at least as well as the basic self-training scheme but with significantly less data augmentation.
+
+# 1 INTRODUCTION
+
+Generative Adversarial Networks (GANs) have shown impressive results in a variety of image synthesis tasks (Denton et al., 2015; Radford et al., 2015; Im et al., 2016; Yoo et al., 2016). Their adversarial training has also been used to make supervised learning more robust (Salimans et al., 2016). However, not much work has been done to use the generated data as a source of knowledge. The use of generated data offers a new dimension to data augmentation and model training (Patki et al., 2016).
+
+This is especially useful when data is limited and costly as in the semi-supervised learning paradigm. This paradigm has known much success allowing systems with much less labelled data and much more unlabelled data to perform nearly as well as if the data were all labelled.
+
+In this work, we propose a self-training meta-learning method that can be applied to GANs that make use of their generated data in order to increase their classification performance. We base ourselves on the Improved GAN (Salimans et al., 2016) since it already has mechanisms to cope with semi-supervised learning. We compare a simple baseline self-training method with a more advanced scheme inspired from Self-training with selection-by-rejection by Zhou et al. (2012).
+
+In Section 2, we first present related work done in the areas of semi-supervised learning and of self-training. In Section 3, we then present the basic theory behind GANs, the Improved GAN and the self-training method upon which we base our work. In Section 4, we present the two selftraining algorithms we compare. In Section 5, we present the results of our self-training experiments compared with the vanilla Improved GAN and we finally conclude in Section 6.
+
+# 2 RELATED WORK
+
+The idea of semi-supervised learning is not new: as labelled training data can be expensive to obtain, semi-supervised learning (Zhu, 2005; Chapelle et al., 2009) becomes an important topic when one has only access to a small amount of labelled training data and a large amount of unlabelled training data. Works regarding semi-supervised learning have been previously done for word sense disambiguation (Yarowsky, 1995; Abney, 2004), for Gaussian random fields (Zhu et al., 2003) and for learning word representation (Turian et al., 2010).
+
+When paired with deep learning, semi-supervised learning has had a few success stories from CatGAN (Springenberg, 2015), from Sutskever et al. (2015) introducing a new cost function, from the Improved GAN by Salimans et al. (2016) and from Triple GAN by Li et al. (2017). However, most of the work in GANs has been focused on improving the visual quality of the generated images (Denton et al., 2015; Radford et al., 2015; Im et al., 2016; Warde-Farley & Bengio, 2016; Yoo et al., 2016; Arjovsky & Bottou, 2017) and their training stability (Salimans et al., 2016; Arjovsky & Bottou, 2017) while not much work has been done on improving the GAN’s performance using its own generated data. The work we base ourselves on, Improved GAN (Salimans et al., 2016), will be explained later with the Technical Background Section 3.
+
+On the self-training side, before the advent of deep learning, Hearst (1991) and Yarowsky (1995) used self-training for word sense disambiguation, Riloff et al. (1999) used self-training in the form of bootstrapping for information extraction and later for learning subjective nouns (Riloff et al., 2003) with Nigam et al. (2000) using EM for text classification. More recently, self-training has been used for object recognition (Rosenberg et al., 2005; Zhou et al., 2012). These techniques allowed semi-supervised learning models to train themselves through multiple rounds while seeing their performance increase as the rounds go.
+
+# 3 TECHNICAL BACKGROUND
+
+GANs are composed of two agents: a discriminator $\mathcal { D }$ and a generator $\mathcal { G }$ . They are typically both deep neural networks with each their set of parameters. The goal of $\mathcal { D }$ is to distinguish between real images and generated images coming from $\mathcal { G }$ . Each is trained successively through many rounds to reach an equilibrium where $\mathcal { G }$ produces images indistinguishable from real images and where $\mathcal { D }$ can only randomly guess if the image is real or generated. The minimax objective they are trying to optimize is
+
+$$
+\operatorname* { m i n } _ { \mathcal { G } } \operatorname* { m a x } _ { \mathcal { D } } \mathbb { E } _ { x \sim p _ { \mathrm { d a t a } } ( x ) } \left[ \log \mathcal { D } ( x ) \right] + \mathbb { E } _ { z \sim p _ { z } ( z ) } \left[ \log ( 1 - \mathcal { D } ( \mathcal { G } ( z ) ) ) \right]
+$$
+
+where $p _ { \mathrm { d a t a } }$ is the real data distributionm, where $p _ { z }$ is a noise distribution to be used by the generator and where ${ \mathcal { D } } ( x )$ must be as close to 1 as possible when $x$ is a real image.
+
+The Improved GAN brings a modification to the traditional objective to allow semi-supervised training with the use of the generated data for classification tasks. First, the discriminator $\mathcal { D }$ the authors use is a classifier. Thus, instead of outputting a single number representing the probability of a true example, $\mathcal { D }$ outputs a softmaxed vector for each classes. Additionally, they introduce a fake class for the generated images, the $K + 1$ th class. As a result, $\mathcal { D }$ must be able to predict the class $x$ belongs to, including the “generated” class. The new loss function for $\mathcal { D }$ is thus
+
+$$
+\mathcal { L } = \mathcal { L } _ { \mathrm { s u p e r v i s e d } } + \mathcal { L } _ { \mathrm { u n s u p e r v i s e d } }
+$$
+
+where
+
+$\begin{array} { r l } & { \mathcal { L } _ { \mathrm { s u p e r i s e d } } = - \mathbb { E } _ { x , y \sim p _ { \mathrm { d a t } } ( x , y ) } \log p _ { \mathrm { m o d e l } } ( y \mid x , y < K + 1 ) } \\ & { \mathcal { L } _ { \mathrm { u n s u p e r i s e d } } = - \{ \mathbb { E } _ { x \sim p _ { \mathrm { d a t } } ( x ) } \log \left[ 1 - p _ { \mathrm { m o d e l } } ( y = K + 1 \mid x ) \right] + \mathbb { E } _ { x \sim \mathcal { G } } \log \left[ p _ { \mathrm { m o d e l } } ( y = K + 1 \mid x ) \right] \} } \end{array}$ and where $p _ { \mathrm { m o d e l } }$ is the predicted probability of the discriminator $\mathcal { D }$ .
+
+For the generator $\mathcal { G }$ , the authors introduce the feature matching loss which goes as follows:
+
+$$
+\| \mathbb { E } _ { x \sim p _ { \mathrm { d a t a } } } f ( x ) - \mathbb { E } _ { z \sim p _ { z } } f ( G ( z ) ) \| _ { 2 } ^ { 2 }
+$$
+
+where $f ( x )$ is the activation of an intermediate layer of $\mathcal { D }$ when given the input $x$ . This should prevent mode collapse and allow for a more stable training for the GAN.
+
+We also take inspiration from the self-training method Self-training with selection-by-rejection proposed by Zhou et al. (2012). They aim to add unlabelled data in which the classifier is confident and which are influential to its decision boundary.
+
+They start with a labelled dataset $L$ and an unlabelled dataset $U$ and train classifier $h$ on $L$ . Samples far from $h$ ’s decision boundary are considered ones where $h$ is confident about their labels. The authors use the negative entropy as a proxy to the distance of an example to the decision boundary. Now associate each sample in $U$ with a weight that will be used for sampling and label the samples in $U$ with $h$ . Take the half of $U$ with samples furthest to the decision boundary to be candidates for addition to the labelled dataset, call this set $U _ { \delta }$ . From $U _ { \delta }$ , subsets $U _ { i } ~ \subseteq ~ U _ { \delta }$ are randomly sampled according to the weights attributed to each sample and each of those subsets $U _ { i }$ have a corrupted counterpart $U _ { i } ^ { \prime }$ where the samples are identical to those in $U _ { i }$ except for their labels which are randomly changed to another. The idea is that the set $U _ { i }$ which is the most influential to the decision boundary will yield a hypothesis $h _ { i }$ trained on $L \cup ( U \setminus U _ { i } ) \cup U _ { i } ^ { \prime }$ that disagrees the most in predictions with a hypothesis $\hat { h }$ which is simply trained on $L \cup U$ . This set is added to $L$ and the weights associated with those added samples are decayed. We note that because of the weighted sampling, examples that were previously added can be re-added with another label in further rounds of self-training.
+
+# 4 METHODOLOGY
+
+In this work, we compare two self-training schemes with the vanilla Improved GAN which does not use self-training. This section details the different algorithms we use.
+
+The first self-training scheme is a basic one which simply adds unlabelled data according to some confidence threshold. This Basic Self-Training algorithm is detailed in Algorithm 1.
+
+The second is a scheme nearly identical to the Self-Training through Selection-by-Rejection explained in Section 3 with a few distinctions. The first distinction is that we substitute the weighted sampling of $U _ { i }$ with a fixed sized uniform random sampling of $U _ { \delta }$ . We also keep track of the unlabelled examples that were added to the labelled dataset to make sure that they are never added again in further self-training rounds. Instead, we allow the main hypothesis to relabel all added examples after every self-training round. We have found that these modifications work better in the GAN case with the addition of generated data. This Improved Self-Training algorithm is detailed in Algorithm 2.
+
+The network architectures used in this work are the same as those used in the original Improved GAN paper and the feature matching loss is used for the generator.
+
+Input : • Labelled dataset L • Unlabelled dataset U • Confidence threshold threshold Number of self-training rounds num rounds
+
+Output: Self-Trained GAN via Basic Self-Training Scheme
+
+1 function BasicSelfTrain(L, U):
+
+2 $\hat { \mathbf { U } } \gets \mathsf { U } \backslash \mathsf { L }$   
+3 for iter $= 1$ to num rounds do   
+4 $\mathsf { h } \gets$ new GAN trained on L and U   
+5 for x in Uˆ do   
+6 $\mathsf { P } ( \mathsf { y } | \mathsf { x } ) \gets \mathsf { h }$ .predict (x)   
+7 if max $\mathsf { P } ( \mathsf { y } | \mathsf { x } ) >$ threshold then   
+8 $\begin{array} { r l } & { \mathsf { L } \gets \dot { \mathsf { L } } \dot { \cup } \left\{ ( \mathsf { x } , \mathsf { a r g m a x P } ( \mathsf { y } | \mathsf { x } ) ) \right\} } \\ & { \hat { \mathbf { U } } \gets \hat { \mathbf { U } } \setminus \{ \mathsf { x } \} } \end{array}$   
+9   
+10 end   
+11 relabel all added data from the beginning in L   
+12 gen data $ \mathsf { h }$ .generate()   
+13 $\mathsf { \bar { U } } \gets \mathsf { U } \cup$ gen data   
+14 $\hat { \mathbf { U } }  \hat { \mathbf { U } } \cup$ gen data   
+15 end   
+16 $\mathsf { h } \gets$ new GAN trained on L and U   
+17 return best trained GAN
+
+Input : • Labelled dataset L • Unlabelled dataset U • Number of subsets n • $\mathsf { U } _ { \mathbf { r } }$ sample fraction sample frac • Number of self-training rounds num rounds
+
+Output: Self-trained GAN
+
+1 function CalculateDisagreement $( \mathsf { h } _ { 1 } , \mathsf { h } _ { 2 } , \mathsf { S } )$ :
+
+disagreement $ \Vert \mathsf { S } \Vert - \Vert \{ x \in \mathsf { S } \mid \mathsf { h } _ { 1 } . \mathsf { p r e d i c t } ( \mathsf { x } )$ is equal to h2.predict(x)}k return disagreement/ $\lvert \lvert \mathsf { S } \rvert \rvert$
+
+4 function SelfTrain(L, U):
+
+5 $\hat { \mathbf { U } } \gets \mathsf { U } \backslash \mathsf { L }$   
+6 for iter $= 1$ to num rounds do   
+7 $\mathsf { h } \gets$ new GAN trained on L and U   
+8 $\hat { \mathbf { h } } \gets$ new GAN trained on L and U labelled by h   
+9 foreach $\mathbf { \boldsymbol { x } } \in \hat { \mathbf { U } } \mathbf { \delta d o } { \mathsf { d } } [ { \mathsf { x } } ] \gets \mathrm { ~ \sum ~ } \mathsf { P } ( { \mathsf { I } } | \mathbf { \boldsymbol { x } } ) \log \mathsf { P } ( { \mathsf { I } } | \mathbf { \boldsymbol { x } } )$   
+labels l   
+10   
+11 $\delta \gets$ median(d)   
+12 $\mathsf { U } _ { \delta } \gets \{ \mathsf { x } \in \hat { \mathbf { U } } \ \mathrm { s u c h } \operatorname { t h a t } \mathsf { d } [ \mathsf { x } ] > \delta \}$   
+13 for $i = 1$ to n do   
+14 $\mathsf { U } _ { \mathbf { i } } \gets$ randomly sample sample frac of $\mathsf { U } _ { \delta }$   
+15 $\mathrm { U _ { i } ^ { \prime } } $ randomly change the labels associated with examples in $\mathrm { \Delta U _ { i } }$ as labelled by h   
+16 $\mathbf { h _ { i } } $ new GAN trained on $\mathsf { L } \cup \mathsf { U } _ { \mathrm { i } } ^ { \prime } \cup ( \mathsf { U } \setminus \mathsf { U } _ { \mathrm { i } } ^ { \prime } )$ labelled by $\mathsf { h }$   
+17 end   
+18 Ur ← arg max CalculateDisagreement $( \hat { \mathbf { h } } , \mathbf { h _ { i } } , \mathsf { U } )$   
+Ui   
+19 relabel all added data from the beginning in L   
+20 $\mathsf { L }  \mathsf { L } \cup \mathsf { U } _ { \mathbf { r } }$   
+21 gen data $ \mathsf { h }$ .generate()   
+22 $\mathsf { \bar { U } } \gets \mathsf { U } \cup$ gen data   
+23 $\hat { \mathbf { U } } \gets ( \hat { \mathbf { U } } \setminus \mathsf { U } _ { \mathbf { r } } ) \cup$ gen data   
+24 end   
+25 $\mathsf { h } \gets$ new GAN trained on L and U   
+26 return best trained GAN
+
+The running time for one round of self-training is much higher in the improved self-training scheme. The most significant part is the training of the hypotheses $\hat { h } , h _ { 1 } , \cdots , h _ { n }$ which are trained for the same number of epochs as the main GAN.
+
+Dataset Following the evaluation scheme from Improved GAN (Salimans et al., 2016), our methods are tested against the MNIST dataset of handwritten digits (Lecun et al., 1998) and CIFAR-10 (Krizhevsky & Hinton, 2009). MNIST contains a total of 80 000 labelled images of size 32 by 32 pixels, black and white where 60 000 of them are reserved as the training set and 20 000 are for the testing set. CIFAR-10 is a dataset of natural coloured images also of size 32 by 32 pixels with 50 000 train images and 10 000 test images. The experiments presented in the later Section 5 on Experimental Results had GANs trained on the whole training datasets and the test results were on the test sets. During the experiments, a subset of the training data has been kept labelled to be taken as the labelled dataset while the rest had their labels removed for the unsupervised part of the training. The hyperparameter search was done on a validation set that was about a third of MNIST’s training set when there were 10 labelled examples for each of the 10 classes making 100 labelled examples in total.
+
+Hyperparameters In both of these self-training algorithms, we have found experimentally that retraining the classifier from scratch at the beginning of every self-training iteration yielded better results than continuing the training with the added data. Following this, the hypotheses $\hat { h }$ and $h _ { i }$ in our version of the self-training through rejection are also retrained from scratch at every round. The number of rounds to run the self-training can be tuned but good improvements are already observed after 2 or 3 rounds. The confidence threshold for addition of data to the labelled training set in the basic self-training scheme was set to 0.95, i.e. unlabelled examples $x$ were added if max $P ( y \mid x ) > 0 . 9 5$ , with $P ( y \mid x )$ being the softmaxed output of the classifier. This results in a large proportion of data being added even on the first self-training round but we found that setting this threshold higher, e.g. to the median or to the mean of max $\bar { P } ( y \mid x )$ over all $x$ , lead to a worse performance. For the other self-training scheme, the number of subsets $U _ { i }$ to draw from $U _ { \delta }$ has been set to 4. While the authors claim that having a higher number of sets is better to select more permutations of data points it increases the computation time. To make each $U _ { i }$ , we randomly sampled one fifth of $U _ { \delta }$ . In other words, one fifth of $U _ { \delta }$ corresponding to $U _ { i }$ is corrupted and the rest is uncorrupted when training $h _ { i }$ . The disagreement calculation between hypotheses was done on the whole unlabelled dataset rather than only on a select subset of data as in the original paper. This is a choice we made with hopes that having more data to compute disagreement on would lead to a better estimate of what affects the decision boundary. An exhaustive search of hyperparameters was not performed but we have tested a few combinations through a held-out validation set and found that these parameters worked well.
+
+# 5 EXPERIMENTAL RESULTS
+
+The self-training algorithms were tested against two datasets: MNIST and CIFAR-10. For MNIST, the GANs were trained for 550 epochs everytime they required training and the results for the selftraining schemes were after 2 rounds of self-training iterations, i.e. data augmentation was performed twice so the GANs were retrained three times in total. For CIFAR-10, each GAN was trained for 300 epochs and also for 2 rounds of self-training. The count parameter indicates how many samples per class are labelled in MNIST while the count remained at 400 for CIFAR-10. Each MNIST experiment was done over three seeds and the results were averaged; the results for CIFAR10 are averaged over two seeds. The error bounds correspond to one standard deviation. MNIST results can be seen in Table 1 and CIFAR-10 results can be seen in Table 2.
+
+We notice that having some kind of self-training yields better results than no self-training at all. Most importantly, this improvement was observed after only 1 or 2 rounds of data augmentation. The results shown in the tables are the best ones after 2 rounds, but in some cases, the best results were obtained immediately after 1 round.
+
+In the case of MNIST, We observe that the basic self-training scheme performs well but we must keep in mind that in the basic self-training scheme, a significant amount of data is added at every round. Indeed, for the counts of 10 and 20, over $9 8 \%$ of the unlabelled examples was added (about 59 000) while for the more complex self-training scheme, only one fifth of the unlabelled data is
+
+<table><tr><td>Counts per class</td><td>5</td><td>10</td><td>20</td></tr><tr><td colspan="4">BestErrorRates</td></tr><tr><td>Vanilla</td><td>0.1359 ± 0.1295</td><td>0.0085 ± 0.0003</td><td>0.0102 ± 0.0011</td></tr><tr><td>Basic Self-Training</td><td>0.1019 ± 0.1255</td><td>0.0080 ±0.0003</td><td>0.0098 ± 0.0013</td></tr><tr><td>Improved Self-Training</td><td>0.1201 ± 0.1202</td><td>0.0081 ± 0.0001</td><td>0.0097± 0.0009</td></tr><tr><td colspan="4">Improvements over Vanilla</td></tr><tr><td>Vanilla</td><td>0</td><td>0</td><td>0</td></tr><tr><td>Basic Self-Training</td><td>0.0340 ± 0.0245</td><td>0.0005 ± 0.0002</td><td>0.0004± 0.0003</td></tr><tr><td>Improved Self-Training</td><td>0.0158 ± 0.0103</td><td>0.0004±0.0003</td><td>0.0004 ± 0.0005</td></tr></table>
+
+Table 1: Experimental results for the error rates (lower is better) and relative improvements over the vanilla GAN training without self-training (higher is better) for different counts of labelled data: 5, 10 and 20 labelled examples per class for each of the first, second and third columns respectively. All experiments are averaged over 3 different seeds and the mean is shown with the error bounds being one standard deviation.
+
+<table><tr><td></td><td>BestError Rates</td><td>ImprovementsoverVanilla</td></tr><tr><td>Vanilla</td><td>0.2513± 0.0037</td><td>0</td></tr><tr><td>Basic Self-Training</td><td>0.2471 ± 0.0002</td><td>0.0042 ± 0.0039</td></tr><tr><td>Improved Self-Training</td><td>0.2231 ± 0.0029</td><td>0.0282 ± 0.0008</td></tr></table>
+
+Table 2: Experimental results for the error rates (lower is better) and relative improvements over the vanilla GAN for CIFAR-10 (higher is better) for a count of 400. All experiments averaged over 2 different seeds and the mean is shown with the error bounds being one standard deviation.
+
+added (around 5 000) but this still yields a similar result to the basic self-training scheme. When the basic self-training scheme adds a little less data, e.g. about 40 000, no improvement is seen. We think that this is due to the addition of data poor in information or even contradictory data compared to the improved self-training case, where fewer data yielded in an improvement. This shows that adding based on a confidence score is not enough and that we also need more information. Thus the examples that were added in this more complex schemes were informative, as informative as adding nearly all of the unlabelled training set when using the basic self-training scheme. In all cases, the added unlabelled data were assigned the correct label more than $9 7 \%$ of the time.
+
+# 6 CONCLUSION
+
+This paper proposed a self-training meta-learning scheme that makes use of the generated data from GANs in order to improve the classification accuracy in the MNIST and the CIFAR-10 datasets. We noticed important improvements even after two rounds of self-training and even more so in the CIFAR-10 dataset. Our improved method of data augmentation performs similarly to the basic scheme although it uses much less labelled data. On the other hand, it is much more computationally costly. When one is looking for a quick gain in accuracy, the basic self-training scheme can be enough but when looking for more, the improved method should be used as it adds less data at each round but does a more thorough analysis of what data is best to add.
+
+Future steps for our self-training algorithm include a more thorough theoretical analysis of the selftraining and trying different knobs in the algorithm. For example, the label inversion scheme can be changed to corrupt the label to the least likely label instead of randomly corrupting it. Preliminary analyses seem to indicate that this performs worst than the random scheme but it might be worth exploring more to see the reason. The calculation of the disagreement can also be changed to the mean squared error of the prediction matrices instead of simply counting the number of differing predictions. This mean squared error might help in quantifying the disagreement. The amount of corrupted data fed to the $h _ { i }$ can also be changed and its effect studied. The selection of $U _ { \delta }$ was done through calculating the negative entropy which should be a proxy for the distance to the boundary but maybe other meaures can be used for the initial selection of $U _ { \delta }$ . One should also look for ways to decrease the computationl cost of the improved method; the key as we have seen is to try different subsets and to check their effect on the decision boundary. Maybe another similar testing diffrent subsets can be used without having to retrain the whole GANs.
+
+All in all, different choices can affect the performance of self-training but our initial analysis shows that we have been successful in implementing a self-training method to GANs making use of their generated data improving the performance on semi-supervised learning tasks.
+
+# ACKNOWLEDGMENTS
+
+We would like to thank the National Natural Science Foundation of China for previously supporting the authors to prepare for the knowledge and skills demanded by this work.
+
+# REFERENCES
+
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+Jost Tobias Springenberg. Unsupervised and semi-supervised learning with categorical generative adversarial networks. arXiv preprint arXiv:1511.06390, 2015.
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+Yan Zhou, Murat Kantarcioglu, and Bhavani Thuraisingham. Self-training with selection-byrejection. In Data Mining (ICDM), 2012 IEEE 12th International Conference on, pp. 795–803. IEEE, 2012.
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+    {
+        "type": "text",
+        "text": "A SELF-TRAINING METHOD FOR SEMI-SUPERVISED GANS ",
+        "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,
+            236,
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+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Since the creation of Generative Adversarial Networks (GANs), much work has been done to improve their training stability, their generated image quality, their range of application but nearly none of them explored their self-training potential. Self-training has been used before the advent of deep learning in order to allow training on limited labelled training data and has shown impressive results in semi-supervised learning. In this work, we combine these two ideas and make GANs self-trainable for semi-supervised learning tasks by exploiting their infinite data generation potential. Results show that using even the simplest form of self-training yields an improvement. We also show results for a more complex self-training scheme that performs at least as well as the basic self-training scheme but with significantly less data augmentation. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            458,
+            336,
+            474
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Generative Adversarial Networks (GANs) have shown impressive results in a variety of image synthesis tasks (Denton et al., 2015; Radford et al., 2015; Im et al., 2016; Yoo et al., 2016). Their adversarial training has also been used to make supervised learning more robust (Salimans et al., 2016). However, not much work has been done to use the generated data as a source of knowledge. The use of generated data offers a new dimension to data augmentation and model training (Patki et al., 2016). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This is especially useful when data is limited and costly as in the semi-supervised learning paradigm. This paradigm has known much success allowing systems with much less labelled data and much more unlabelled data to perform nearly as well as if the data were all labelled. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this work, we propose a self-training meta-learning method that can be applied to GANs that make use of their generated data in order to increase their classification performance. We base ourselves on the Improved GAN (Salimans et al., 2016) since it already has mechanisms to cope with semi-supervised learning. We compare a simple baseline self-training method with a more advanced scheme inspired from Self-training with selection-by-rejection by Zhou et al. (2012). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In Section 2, we first present related work done in the areas of semi-supervised learning and of self-training. In Section 3, we then present the basic theory behind GANs, the Improved GAN and the self-training method upon which we base our work. In Section 4, we present the two selftraining algorithms we compare. In Section 5, we present the results of our self-training experiments compared with the vanilla Improved GAN and we finally conclude in Section 6. ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+            804,
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+            821
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The idea of semi-supervised learning is not new: as labelled training data can be expensive to obtain, semi-supervised learning (Zhu, 2005; Chapelle et al., 2009) becomes an important topic when one has only access to a small amount of labelled training data and a large amount of unlabelled training data. Works regarding semi-supervised learning have been previously done for word sense disambiguation (Yarowsky, 1995; Abney, 2004), for Gaussian random fields (Zhu et al., 2003) and for learning word representation (Turian et al., 2010). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "When paired with deep learning, semi-supervised learning has had a few success stories from CatGAN (Springenberg, 2015), from Sutskever et al. (2015) introducing a new cost function, from the Improved GAN by Salimans et al. (2016) and from Triple GAN by Li et al. (2017). However, most of the work in GANs has been focused on improving the visual quality of the generated images (Denton et al., 2015; Radford et al., 2015; Im et al., 2016; Warde-Farley & Bengio, 2016; Yoo et al., 2016; Arjovsky & Bottou, 2017) and their training stability (Salimans et al., 2016; Arjovsky & Bottou, 2017) while not much work has been done on improving the GAN’s performance using its own generated data. The work we base ourselves on, Improved GAN (Salimans et al., 2016), will be explained later with the Technical Background Section 3. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "On the self-training side, before the advent of deep learning, Hearst (1991) and Yarowsky (1995) used self-training for word sense disambiguation, Riloff et al. (1999) used self-training in the form of bootstrapping for information extraction and later for learning subjective nouns (Riloff et al., 2003) with Nigam et al. (2000) using EM for text classification. More recently, self-training has been used for object recognition (Rosenberg et al., 2005; Zhou et al., 2012). These techniques allowed semi-supervised learning models to train themselves through multiple rounds while seeing their performance increase as the rounds go. ",
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+    {
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+        "text": "3 TECHNICAL BACKGROUND ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "GANs are composed of two agents: a discriminator $\\mathcal { D }$ and a generator $\\mathcal { G }$ . They are typically both deep neural networks with each their set of parameters. The goal of $\\mathcal { D }$ is to distinguish between real images and generated images coming from $\\mathcal { G }$ . Each is trained successively through many rounds to reach an equilibrium where $\\mathcal { G }$ produces images indistinguishable from real images and where $\\mathcal { D }$ can only randomly guess if the image is real or generated. The minimax objective they are trying to optimize is ",
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+    {
+        "type": "equation",
+        "img_path": "images/38f88ce122df5a6984de98274311f869816694fad5da19a56ac8d6fa2a3adf1a.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\mathcal { G } } \\operatorname* { m a x } _ { \\mathcal { D } } \\mathbb { E } _ { x \\sim p _ { \\mathrm { d a t a } } ( x ) } \\left[ \\log \\mathcal { D } ( x ) \\right] + \\mathbb { E } _ { z \\sim p _ { z } ( z ) } \\left[ \\log ( 1 - \\mathcal { D } ( \\mathcal { G } ( z ) ) ) \\right]\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where $p _ { \\mathrm { d a t a } }$ is the real data distributionm, where $p _ { z }$ is a noise distribution to be used by the generator and where ${ \\mathcal { D } } ( x )$ must be as close to 1 as possible when $x$ is a real image. ",
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+        "type": "text",
+        "text": "The Improved GAN brings a modification to the traditional objective to allow semi-supervised training with the use of the generated data for classification tasks. First, the discriminator $\\mathcal { D }$ the authors use is a classifier. Thus, instead of outputting a single number representing the probability of a true example, $\\mathcal { D }$ outputs a softmaxed vector for each classes. Additionally, they introduce a fake class for the generated images, the $K + 1$ th class. As a result, $\\mathcal { D }$ must be able to predict the class $x$ belongs to, including the “generated” class. The new loss function for $\\mathcal { D }$ is thus ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/94f148b196b77a4d9aa78463dc0c804a723ff2a087278ac3831302b8cd1a8885.jpg",
+        "text": "$$\n\\mathcal { L } = \\mathcal { L } _ { \\mathrm { s u p e r v i s e d } } + \\mathcal { L } _ { \\mathrm { u n s u p e r v i s e d } }\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where ",
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+    {
+        "type": "text",
+        "text": "$\\begin{array} { r l } & { \\mathcal { L } _ { \\mathrm { s u p e r i s e d } } = - \\mathbb { E } _ { x , y \\sim p _ { \\mathrm { d a t } } ( x , y ) } \\log p _ { \\mathrm { m o d e l } } ( y \\mid x , y < K + 1 ) } \\\\ & { \\mathcal { L } _ { \\mathrm { u n s u p e r i s e d } } = - \\{ \\mathbb { E } _ { x \\sim p _ { \\mathrm { d a t } } ( x ) } \\log \\left[ 1 - p _ { \\mathrm { m o d e l } } ( y = K + 1 \\mid x ) \\right] + \\mathbb { E } _ { x \\sim \\mathcal { G } } \\log \\left[ p _ { \\mathrm { m o d e l } } ( y = K + 1 \\mid x ) \\right] \\} } \\end{array}$ and where $p _ { \\mathrm { m o d e l } }$ is the predicted probability of the discriminator $\\mathcal { D }$ . ",
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+    {
+        "type": "text",
+        "text": "For the generator $\\mathcal { G }$ , the authors introduce the feature matching loss which goes as follows: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/5426258b8e8c01f56ed9f7fb680b31d04470af7461cbed9d64f83acf0513d450.jpg",
+        "text": "$$\n\\| \\mathbb { E } _ { x \\sim p _ { \\mathrm { d a t a } } } f ( x ) - \\mathbb { E } _ { z \\sim p _ { z } } f ( G ( z ) ) \\| _ { 2 } ^ { 2 }\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "where $f ( x )$ is the activation of an intermediate layer of $\\mathcal { D }$ when given the input $x$ . This should prevent mode collapse and allow for a more stable training for the GAN. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We also take inspiration from the self-training method Self-training with selection-by-rejection proposed by Zhou et al. (2012). They aim to add unlabelled data in which the classifier is confident and which are influential to its decision boundary. ",
+        "bbox": [
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+        "type": "text",
+        "text": "They start with a labelled dataset $L$ and an unlabelled dataset $U$ and train classifier $h$ on $L$ . Samples far from $h$ ’s decision boundary are considered ones where $h$ is confident about their labels. The authors use the negative entropy as a proxy to the distance of an example to the decision boundary. Now associate each sample in $U$ with a weight that will be used for sampling and label the samples in $U$ with $h$ . Take the half of $U$ with samples furthest to the decision boundary to be candidates for addition to the labelled dataset, call this set $U _ { \\delta }$ . From $U _ { \\delta }$ , subsets $U _ { i } ~ \\subseteq ~ U _ { \\delta }$ are randomly sampled according to the weights attributed to each sample and each of those subsets $U _ { i }$ have a corrupted counterpart $U _ { i } ^ { \\prime }$ where the samples are identical to those in $U _ { i }$ except for their labels which are randomly changed to another. The idea is that the set $U _ { i }$ which is the most influential to the decision boundary will yield a hypothesis $h _ { i }$ trained on $L \\cup ( U \\setminus U _ { i } ) \\cup U _ { i } ^ { \\prime }$ that disagrees the most in predictions with a hypothesis $\\hat { h }$ which is simply trained on $L \\cup U$ . This set is added to $L$ and the weights associated with those added samples are decayed. We note that because of the weighted sampling, examples that were previously added can be re-added with another label in further rounds of self-training. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "4 METHODOLOGY ",
+        "text_level": 1,
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+        "text": "In this work, we compare two self-training schemes with the vanilla Improved GAN which does not use self-training. This section details the different algorithms we use. ",
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+        "type": "text",
+        "text": "The first self-training scheme is a basic one which simply adds unlabelled data according to some confidence threshold. This Basic Self-Training algorithm is detailed in Algorithm 1. ",
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+        "text": "The second is a scheme nearly identical to the Self-Training through Selection-by-Rejection explained in Section 3 with a few distinctions. The first distinction is that we substitute the weighted sampling of $U _ { i }$ with a fixed sized uniform random sampling of $U _ { \\delta }$ . We also keep track of the unlabelled examples that were added to the labelled dataset to make sure that they are never added again in further self-training rounds. Instead, we allow the main hypothesis to relabel all added examples after every self-training round. We have found that these modifications work better in the GAN case with the addition of generated data. This Improved Self-Training algorithm is detailed in Algorithm 2. ",
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+    {
+        "type": "text",
+        "text": "The network architectures used in this work are the same as those used in the original Improved GAN paper and the feature matching loss is used for the generator. ",
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+        "type": "text",
+        "text": "Input : • Labelled dataset L • Unlabelled dataset U • Confidence threshold threshold Number of self-training rounds num rounds ",
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+        "type": "text",
+        "text": "Output: Self-Trained GAN via Basic Self-Training Scheme ",
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+        "type": "text",
+        "text": "1 function BasicSelfTrain(L, U): ",
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+        "type": "text",
+        "text": "2 $\\hat { \\mathbf { U } } \\gets \\mathsf { U } \\backslash \\mathsf { L }$   \n3 for iter $= 1$ to num rounds do   \n4 $\\mathsf { h } \\gets$ new GAN trained on L and U   \n5 for x in Uˆ do   \n6 $\\mathsf { P } ( \\mathsf { y } | \\mathsf { x } ) \\gets \\mathsf { h }$ .predict (x)   \n7 if max $\\mathsf { P } ( \\mathsf { y } | \\mathsf { x } ) >$ threshold then   \n8 $\\begin{array} { r l } & { \\mathsf { L } \\gets \\dot { \\mathsf { L } } \\dot { \\cup } \\left\\{ ( \\mathsf { x } , \\mathsf { a r g m a x P } ( \\mathsf { y } | \\mathsf { x } ) ) \\right\\} } \\\\ & { \\hat { \\mathbf { U } } \\gets \\hat { \\mathbf { U } } \\setminus \\{ \\mathsf { x } \\} } \\end{array}$   \n9   \n10 end   \n11 relabel all added data from the beginning in L   \n12 gen data $ \\mathsf { h }$ .generate()   \n13 $\\mathsf { \\bar { U } } \\gets \\mathsf { U } \\cup$ gen data   \n14 $\\hat { \\mathbf { U } }  \\hat { \\mathbf { U } } \\cup$ gen data   \n15 end   \n16 $\\mathsf { h } \\gets$ new GAN trained on L and U   \n17 return best trained GAN ",
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+    {
+        "type": "text",
+        "text": "Input : • Labelled dataset L • Unlabelled dataset U • Number of subsets n • $\\mathsf { U } _ { \\mathbf { r } }$ sample fraction sample frac • Number of self-training rounds num rounds ",
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+    },
+    {
+        "type": "text",
+        "text": "Output: Self-trained GAN ",
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+    },
+    {
+        "type": "text",
+        "text": "1 function CalculateDisagreement $( \\mathsf { h } _ { 1 } , \\mathsf { h } _ { 2 } , \\mathsf { S } )$ : ",
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+        "type": "text",
+        "text": "disagreement $ \\Vert \\mathsf { S } \\Vert - \\Vert \\{ x \\in \\mathsf { S } \\mid \\mathsf { h } _ { 1 } . \\mathsf { p r e d i c t } ( \\mathsf { x } )$ is equal to h2.predict(x)}k return disagreement/ $\\lvert \\lvert \\mathsf { S } \\rvert \\rvert$ ",
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+        "type": "text",
+        "text": "4 function SelfTrain(L, U): ",
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+        "type": "text",
+        "text": "5 $\\hat { \\mathbf { U } } \\gets \\mathsf { U } \\backslash \\mathsf { L }$   \n6 for iter $= 1$ to num rounds do   \n7 $\\mathsf { h } \\gets$ new GAN trained on L and U   \n8 $\\hat { \\mathbf { h } } \\gets$ new GAN trained on L and U labelled by h   \n9 foreach $\\mathbf { \\boldsymbol { x } } \\in \\hat { \\mathbf { U } } \\mathbf { \\delta d o } { \\mathsf { d } } [ { \\mathsf { x } } ] \\gets \\mathrm { ~ \\sum ~ } \\mathsf { P } ( { \\mathsf { I } } | \\mathbf { \\boldsymbol { x } } ) \\log \\mathsf { P } ( { \\mathsf { I } } | \\mathbf { \\boldsymbol { x } } )$   \nlabels l   \n10   \n11 $\\delta \\gets$ median(d)   \n12 $\\mathsf { U } _ { \\delta } \\gets \\{ \\mathsf { x } \\in \\hat { \\mathbf { U } } \\ \\mathrm { s u c h } \\operatorname { t h a t } \\mathsf { d } [ \\mathsf { x } ] > \\delta \\}$   \n13 for $i = 1$ to n do   \n14 $\\mathsf { U } _ { \\mathbf { i } } \\gets$ randomly sample sample frac of $\\mathsf { U } _ { \\delta }$   \n15 $\\mathrm { U _ { i } ^ { \\prime } } $ randomly change the labels associated with examples in $\\mathrm { \\Delta U _ { i } }$ as labelled by h   \n16 $\\mathbf { h _ { i } } $ new GAN trained on $\\mathsf { L } \\cup \\mathsf { U } _ { \\mathrm { i } } ^ { \\prime } \\cup ( \\mathsf { U } \\setminus \\mathsf { U } _ { \\mathrm { i } } ^ { \\prime } )$ labelled by $\\mathsf { h }$   \n17 end   \n18 Ur ← arg max CalculateDisagreement $( \\hat { \\mathbf { h } } , \\mathbf { h _ { i } } , \\mathsf { U } )$   \nUi   \n19 relabel all added data from the beginning in L   \n20 $\\mathsf { L }  \\mathsf { L } \\cup \\mathsf { U } _ { \\mathbf { r } }$   \n21 gen data $ \\mathsf { h }$ .generate()   \n22 $\\mathsf { \\bar { U } } \\gets \\mathsf { U } \\cup$ gen data   \n23 $\\hat { \\mathbf { U } } \\gets ( \\hat { \\mathbf { U } } \\setminus \\mathsf { U } _ { \\mathbf { r } } ) \\cup$ gen data   \n24 end   \n25 $\\mathsf { h } \\gets$ new GAN trained on L and U   \n26 return best trained GAN ",
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+    {
+        "type": "text",
+        "text": "The running time for one round of self-training is much higher in the improved self-training scheme. The most significant part is the training of the hypotheses $\\hat { h } , h _ { 1 } , \\cdots , h _ { n }$ which are trained for the same number of epochs as the main GAN. ",
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+        "type": "text",
+        "text": "Dataset Following the evaluation scheme from Improved GAN (Salimans et al., 2016), our methods are tested against the MNIST dataset of handwritten digits (Lecun et al., 1998) and CIFAR-10 (Krizhevsky & Hinton, 2009). MNIST contains a total of 80 000 labelled images of size 32 by 32 pixels, black and white where 60 000 of them are reserved as the training set and 20 000 are for the testing set. CIFAR-10 is a dataset of natural coloured images also of size 32 by 32 pixels with 50 000 train images and 10 000 test images. The experiments presented in the later Section 5 on Experimental Results had GANs trained on the whole training datasets and the test results were on the test sets. During the experiments, a subset of the training data has been kept labelled to be taken as the labelled dataset while the rest had their labels removed for the unsupervised part of the training. The hyperparameter search was done on a validation set that was about a third of MNIST’s training set when there were 10 labelled examples for each of the 10 classes making 100 labelled examples in total. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Hyperparameters In both of these self-training algorithms, we have found experimentally that retraining the classifier from scratch at the beginning of every self-training iteration yielded better results than continuing the training with the added data. Following this, the hypotheses $\\hat { h }$ and $h _ { i }$ in our version of the self-training through rejection are also retrained from scratch at every round. The number of rounds to run the self-training can be tuned but good improvements are already observed after 2 or 3 rounds. The confidence threshold for addition of data to the labelled training set in the basic self-training scheme was set to 0.95, i.e. unlabelled examples $x$ were added if max $P ( y \\mid x ) > 0 . 9 5$ , with $P ( y \\mid x )$ being the softmaxed output of the classifier. This results in a large proportion of data being added even on the first self-training round but we found that setting this threshold higher, e.g. to the median or to the mean of max $\\bar { P } ( y \\mid x )$ over all $x$ , lead to a worse performance. For the other self-training scheme, the number of subsets $U _ { i }$ to draw from $U _ { \\delta }$ has been set to 4. While the authors claim that having a higher number of sets is better to select more permutations of data points it increases the computation time. To make each $U _ { i }$ , we randomly sampled one fifth of $U _ { \\delta }$ . In other words, one fifth of $U _ { \\delta }$ corresponding to $U _ { i }$ is corrupted and the rest is uncorrupted when training $h _ { i }$ . The disagreement calculation between hypotheses was done on the whole unlabelled dataset rather than only on a select subset of data as in the original paper. This is a choice we made with hopes that having more data to compute disagreement on would lead to a better estimate of what affects the decision boundary. An exhaustive search of hyperparameters was not performed but we have tested a few combinations through a held-out validation set and found that these parameters worked well. ",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The self-training algorithms were tested against two datasets: MNIST and CIFAR-10. For MNIST, the GANs were trained for 550 epochs everytime they required training and the results for the selftraining schemes were after 2 rounds of self-training iterations, i.e. data augmentation was performed twice so the GANs were retrained three times in total. For CIFAR-10, each GAN was trained for 300 epochs and also for 2 rounds of self-training. The count parameter indicates how many samples per class are labelled in MNIST while the count remained at 400 for CIFAR-10. Each MNIST experiment was done over three seeds and the results were averaged; the results for CIFAR10 are averaged over two seeds. The error bounds correspond to one standard deviation. MNIST results can be seen in Table 1 and CIFAR-10 results can be seen in Table 2. ",
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+        "type": "text",
+        "text": "We notice that having some kind of self-training yields better results than no self-training at all. Most importantly, this improvement was observed after only 1 or 2 rounds of data augmentation. The results shown in the tables are the best ones after 2 rounds, but in some cases, the best results were obtained immediately after 1 round. ",
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+        "type": "text",
+        "text": "In the case of MNIST, We observe that the basic self-training scheme performs well but we must keep in mind that in the basic self-training scheme, a significant amount of data is added at every round. Indeed, for the counts of 10 and 20, over $9 8 \\%$ of the unlabelled examples was added (about 59 000) while for the more complex self-training scheme, only one fifth of the unlabelled data is ",
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+    {
+        "type": "table",
+        "img_path": "images/fc2395134bac93161c951f8bd134e571c2daf1f07b13015a003426e61e9234e0.jpg",
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+        "table_body": "<table><tr><td>Counts per class</td><td>5</td><td>10</td><td>20</td></tr><tr><td colspan=\"4\">BestErrorRates</td></tr><tr><td>Vanilla</td><td>0.1359 ± 0.1295</td><td>0.0085 ± 0.0003</td><td>0.0102 ± 0.0011</td></tr><tr><td>Basic Self-Training</td><td>0.1019 ± 0.1255</td><td>0.0080 ±0.0003</td><td>0.0098 ± 0.0013</td></tr><tr><td>Improved Self-Training</td><td>0.1201 ± 0.1202</td><td>0.0081 ± 0.0001</td><td>0.0097± 0.0009</td></tr><tr><td colspan=\"4\">Improvements over Vanilla</td></tr><tr><td>Vanilla</td><td>0</td><td>0</td><td>0</td></tr><tr><td>Basic Self-Training</td><td>0.0340 ± 0.0245</td><td>0.0005 ± 0.0002</td><td>0.0004± 0.0003</td></tr><tr><td>Improved Self-Training</td><td>0.0158 ± 0.0103</td><td>0.0004±0.0003</td><td>0.0004 ± 0.0005</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 1: Experimental results for the error rates (lower is better) and relative improvements over the vanilla GAN training without self-training (higher is better) for different counts of labelled data: 5, 10 and 20 labelled examples per class for each of the first, second and third columns respectively. All experiments are averaged over 3 different seeds and the mean is shown with the error bounds being one standard deviation. ",
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+    {
+        "type": "table",
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+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>BestError Rates</td><td>ImprovementsoverVanilla</td></tr><tr><td>Vanilla</td><td>0.2513± 0.0037</td><td>0</td></tr><tr><td>Basic Self-Training</td><td>0.2471 ± 0.0002</td><td>0.0042 ± 0.0039</td></tr><tr><td>Improved Self-Training</td><td>0.2231 ± 0.0029</td><td>0.0282 ± 0.0008</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Table 2: Experimental results for the error rates (lower is better) and relative improvements over the vanilla GAN for CIFAR-10 (higher is better) for a count of 400. All experiments averaged over 2 different seeds and the mean is shown with the error bounds being one standard deviation. ",
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+        "type": "text",
+        "text": "added (around 5 000) but this still yields a similar result to the basic self-training scheme. When the basic self-training scheme adds a little less data, e.g. about 40 000, no improvement is seen. We think that this is due to the addition of data poor in information or even contradictory data compared to the improved self-training case, where fewer data yielded in an improvement. This shows that adding based on a confidence score is not enough and that we also need more information. Thus the examples that were added in this more complex schemes were informative, as informative as adding nearly all of the unlabelled training set when using the basic self-training scheme. In all cases, the added unlabelled data were assigned the correct label more than $9 7 \\%$ of the time. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "This paper proposed a self-training meta-learning scheme that makes use of the generated data from GANs in order to improve the classification accuracy in the MNIST and the CIFAR-10 datasets. We noticed important improvements even after two rounds of self-training and even more so in the CIFAR-10 dataset. Our improved method of data augmentation performs similarly to the basic scheme although it uses much less labelled data. On the other hand, it is much more computationally costly. When one is looking for a quick gain in accuracy, the basic self-training scheme can be enough but when looking for more, the improved method should be used as it adds less data at each round but does a more thorough analysis of what data is best to add. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Future steps for our self-training algorithm include a more thorough theoretical analysis of the selftraining and trying different knobs in the algorithm. For example, the label inversion scheme can be changed to corrupt the label to the least likely label instead of randomly corrupting it. Preliminary analyses seem to indicate that this performs worst than the random scheme but it might be worth exploring more to see the reason. The calculation of the disagreement can also be changed to the mean squared error of the prediction matrices instead of simply counting the number of differing predictions. This mean squared error might help in quantifying the disagreement. The amount of corrupted data fed to the $h _ { i }$ can also be changed and its effect studied. The selection of $U _ { \\delta }$ was done through calculating the negative entropy which should be a proxy for the distance to the boundary but maybe other meaures can be used for the initial selection of $U _ { \\delta }$ . One should also look for ways to decrease the computationl cost of the improved method; the key as we have seen is to try different subsets and to check their effect on the decision boundary. Maybe another similar testing diffrent subsets can be used without having to retrain the whole GANs. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "All in all, different choices can affect the performance of self-training but our initial analysis shows that we have been successful in implementing a self-training method to GANs making use of their generated data improving the performance on semi-supervised learning tasks. ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We would like to thank the National Natural Science Foundation of China for previously supporting the authors to prepare for the knowledge and skills demanded by this work. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
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