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+# SINKHORN AUTOENCODERS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Optimal Transport offers an alternative to maximum likelihood for learning generative autoencoding models. We show how this principle dictates the minimization of the Wasserstein distance between the encoder aggregated posterior and the prior, plus a reconstruction error. We prove that in the non-parametric limit the autoencoder generates the data distribution if and only if the two distributions match exactly, and that the optimum can be obtained by deterministic autoencoders. We then introduce the Sinkhorn AutoEncoder (SAE), which casts the problem into Optimal Transport on the latent space. The resulting Wasserstein distance is minimized by backpropagating through the Sinkhorn algorithm. SAE models the aggregated posterior as an implicit distribution and therefore does not need a reparameterization trick for gradients estimation. Moreover, it requires virtually no adaptation to different prior distributions. We demonstrate its flexibility by considering models with hyperspherical and Dirichlet priors, as well as a simple case of probabilistic programming. SAE matches or outperforms other autoencoding models in visual quality and FID scores.
+
+# 1 INTRODUCTION
+
+Unsupervised learning aims to find the underlying rules that govern a given data distribution. It can be approached by learning to mimic the data generation process, or by finding an adequate representation of the data. Generative Adversarial Networks (GAN) (Goodfellow et al., 2014) belong to the former class, by learning to transform noise into a distribution that matches the given one. AutoEncoders (AE) (Hinton & Salakhutdinov, 2006) are of the latter type, by learning a representation that maximizes the mutual information between the data and its reconstruction, subject to an information bottleneck. Variational AutoEncoders (VAE) (Kingma & Welling, 2013; Rezende et al., 2014), provide both a generative model — i.e. a prior distribution on the latent space with a decoder that models the conditional likelihood — and an encoder — approximating the posterior distribution of the generative model. Optimizing the exact marginal likelihood is intractable in latent variable models such as VAE’s. Instead one maximizes the Evidence Lower BOund (ELBO) as a surrogate. This objective trades off a reconstruction error of the input and a regularization term that aims at minimizing the Kullback-Leibler (KL) divergence from the approximate posterior to the prior.
+
+An alternative principle for learning generative autoencoders is proposed by Tolstikhin et al. (2018). The theory of Optimal Transport (OT) (Villani, 2008) prescribes a different regularizer: one that matches the prior with the aggregated posterior — the average (approximate) posterior over the training data. In Wasserstein AutoEncoders (WAE) (Tolstikhin et al., 2018), this is enforced by the choice of either the Maximum Mean Discrepancy (MMD) (Gretton et al., 2012)), or by adversarial training on the latent space. WAE empirically improves upon VAE. More recently, a family of Wasserstein divergences has been used by Ambrogioni et al. (2018) in the context of variational inference. The particular choice of Wasserstein distances may be crucial for convergence, due to the induced weaker topology as compared to other divergences, such as the KL (Arjovsky et al., 2017).
+
+We contribute to the formal analysis of autoencoders with OT. First, we prove that in order to minimize the Wasserstein distance between the generative model and the data distribution, we can miniminize the usual reconstruction-plus-regularizer cost, where the regularizer is the Wasserstein distance between the encoder aggregated posterior and the prior. Second, in the non-parametric limit, the model learns the data distribution if and only if the aggregated posterior matches the prior exactly. Third, as a consequence of the Monge-Kontorovich equivalence (Villani, 2008), the functional space of this learning problem can be limited to that of deterministic autoencoders.
+
+The theory supports practical innovations. We learn deterministic autoencoders by minimizing a reconstruction error and the Wasserstein distance on the latent space between samples of the aggregated posterior and the prior. The latter is known to be costly, but a fast approximate solution is provided by the Sinkhorn algorithm (Cuturi, 2013). We follow Frogner et al. (2015) and Genevay et al. (2018), by exploiting the differentiability of the Sinkhorn iterations, and unroll it for backpropagation. Altogether, we call our method the Sinkhorn AutoEncoder (SAE).
+
+The Sinkhorn AutoEncoder is agnostic to the analytical form of the prior, as it optimizes a samplebased cost function which is aware of the geometry of the latent space. Furthermore, as a byproduct of using deterministic networks, it models the aggregated posterior as an implicit distribution (Mohamed & Lakshminarayanan, 2016) with no need of the reparametrization trick for learning the encoder (Kingma & Welling, 2013). Therefore, with essentially no change in the algorithm, we can learn models with Normally distributed priors and aggregated posteriors, as well as distributions living on manifolds such as hyperspheres (Davidson et al., 2018) and probability simplices.
+
+We start our experiments by studying unsupervised representation learning by training an encoder in isolation. Our results demonstrate the capability of the Sinkhorn algorithm to produce embeddings that conserve the local geometry of the data, echoing results from Bojanowski & Joulin (2017). Next we move to the autoencoder. In an ablation study, we compare with the exact Hungarian algorithm in place of the Sinkhorn and show that our method performs equally well, while converging faster. We then compare against prior work on autoencoders with Normal and spherical priors on MNIST, CIFAR10 and CelebA. SAE with a spherical prior produces visually more appealing interpolations, crisper samples and comparable or lower FID (Heusel et al., 2017). Finally, we further show the flexibility of SAE with qualitative results by using a Dirichlet prior, which defines the latent space on a probability simplex, as well as with a simple probabilistic programming task.
+
+# 2 BACKGROUND
+
+# 2.1 WASSERSTEIN DISTANCE AND WASSERSTEIN AUTOENCODERS
+
+We follow Tolstikhin et al. (2018) and denote with $x , y , z$ the sample spaces and with $X , Y , Z$ and $P _ { X } , P _ { Y } , P _ { Z }$ the corresponding random variables and distributions. Given a map $F : \mathcal { X }  \mathcal { V }$ we denote by $F _ { \# }$ the push-forward map acting on a distribution $P$ as $P \circ F ^ { - 1 }$ . I f $F ( Y | X )$ is non-deterministic we define the push-forward of a distribution $P$ as the induced marginal of the joint distribution $F ( Y | X ) P _ { X }$ (denoted by $F ( Y | X ) _ { \# } P _ { X } )$ . For any measurable non-negative cost $c : \mathcal { X } \times \mathcal { Y }  \mathbb { R } ^ { + } \cup \{ \infty \}$ , one can define the following $O T$ -cost between marginal distributions $P _ { X }$ and $P _ { Y }$ via:
+
+$$
+W _ { c } ( P _ { X } , P _ { Y } ) = \operatorname* { i n f } _ { \Gamma \in \Pi ( P _ { X } , P _ { Y } ) } \mathbb { E } _ { ( X , Y ) \sim \Gamma } [ c ( X , Y ) ] ,
+$$
+
+where $\Pi ( P _ { X } , P _ { Y } )$ is the set of all joint distributions that have as marginals the given $P _ { X }$ and $P _ { Y }$ . The elements from $\Pi ( P _ { X } , P _ { Y } )$ are called couplings from $P _ { X }$ to $P _ { Y }$ . From now on we will assume that $\mathcal { X } = \mathcal { y }$ and $c ( x , y )$ is a distance. In this case $W _ { c } ( P _ { X } , P _ { Y } )$ is the Wasserstein distance w.r.t the cost $c$ . If $c ( x , y ) = \| x - y \| _ { p } ^ { p }$ for $p \geq 1$ then $W _ { p } = \sqrt [ p ] { W _ { c } }$ is called the $p$ -th Wasserstein distance.
+
+Let $P _ { X }$ denote the true data distribution on $\mathcal { X }$ . We define a latent variable model given as follows: we fix a latent space $\mathcal { Z }$ and a prior distribution $P _ { Z }$ on $\mathcal { Z }$ and consider the conditional distribution $G ( X | Z )$ (the decoder) parameterized by a neural network $G$ . Together they specify a generative model as $G ( X | Z ) P _ { Z }$ . The induced marginal will be denoted by $P _ { G }$ . Learning $P _ { G }$ to approximate the true $P _ { X }$ is then defined as:
+
+$$
+\operatorname* { m i n } _ { G } W _ { c } ( P _ { X } , P _ { G } ) .
+$$
+
+Because of the infimum over $\Pi ( P _ { X } , P _ { G } )$ inside $W _ { c }$ , this is intractable. To rewrite this objective we consider the posterior distribution $Q ( Z | X )$ (the encoder) and its aggregated posterior $Q _ { Z }$ :
+
+$$
+Q _ { Z } = Q ( Z | X ) _ { \# } P _ { X } = \mathbb { E } _ { X \sim P _ { X } } Q ( Z | X ) ,
+$$
+
+the induced marginal of the joint distribution $Q ( Z | X ) P _ { X }$ . Tolstikhin et al. (2018) show that, if the decoder $G ( X | Z )$ is deterministic, i.e. $P _ { G } = G _ { \# } P _ { Z }$ , or in other words, if all stochasticity of the generative model is captured by $Z$ , then:
+
+$$
+W _ { c } ( P _ { X } , P _ { G } ) = \operatorname* { i n f } _ { Q ( Z | X ) : Q _ { Z } = P _ { Z } } \mathbb { E } _ { X \sim P _ { X } } \mathbb { E } _ { Z \sim Q ( Z | X ) } [ c ( X , G ( Z ) ) ] .
+$$
+
+Learning the generative model $G$ with the Wasserstein AutoEncoder amounts to:
+
+$$
+\operatorname* { m i n } _ { G } \operatorname* { m i n } _ { Q ( Z | X ) } \mathbb { E } _ { X \sim P _ { X } } \mathbb { E } _ { Z \sim Q ( Z | X ) } [ c ( X , G ( Z ) ) ] \ + \ \beta \cdot D _ { Z } ( Q _ { Z } , P _ { Z } ) ,
+$$
+
+where $\beta > 0$ is a Lagrange multiplier and $D _ { Z }$ is any divergence measure on probability distributions on $\mathcal { Z }$ , which choice is left open. WAE uses either MMD or a discriminator trained adversarially for $D _ { Z }$ . As discussed in Bousquet et al. (2017), Equation 4 is a lower bound of Equation 3 for any value of $\beta > 0$ . Minimizing this lower bound does not ensure a minimization of the original objective of Equation 3.
+
+# 2.2 THE SINKHORN ALGORITHM
+
+In place of any choice of $D _ { Z }$ , in Section 3 we formally support the minimization of a Wasserstein distance on latent space. The distance is notoriously hard to compute, which is the reason why the rewriting of Equation 3 is of practical interest. When restricting to discrete distributions, the problem becomes more amenable and efficient approximations exist. To motivate this direction, recall that we can always see samples of a continuous distribution as Dirac deltas, whose expectation defines a discrete distribution. Let two discrete distributions with support on $M$ points be $\hat { P } =$ $\begin{array} { r } { \frac { 1 } { M } \sum _ { i = 1 } ^ { M } \delta _ { z _ { i } } , \hat { Q } = \frac { 1 } { M } \sum _ { i = 1 } ^ { M } \delta _ { z _ { i } ^ { \prime } } } \end{array}$ . Given a cost $c ^ { \prime }$ , their (empirical) Wasserstein distance is:
+
+$$
+W _ { c ^ { \prime } } ( \hat { Q } , \hat { P } ) = \operatorname* { m i n } _ { R \in \mathrm { S } _ { M } } { \textstyle \frac { 1 } { M } } \langle R , C ^ { \prime } \rangle _ { \cal F } ,
+$$
+
+where $C _ { i j } ^ { \prime } = c ^ { \prime } ( z _ { i } ^ { \prime } , z _ { j } )$ is the matrix associated to the cost $c , R$ is a doubly stochastic matrix as defined in $\mathsf { S } _ { M } = \{ R \in \mathbb { R } _ { \geq 0 } ^ { M \times M } \mid R \mathbf { 1 } = \mathbf { 1 } , R ^ { T } \mathbf { 1 } = \mathbf { 1 } \}$ , and $\langle \cdot , \cdot \rangle _ { F }$ denotes the Frobenius inner product; 1 is the vector of ones. Eq. (5) is known to converge to the Wasserstein distance between the continuous distributions as $M$ tends to infinity (Weed & Bach, 2017). This linear program has solutions on the vertices of $\mathrm { S } _ { M }$ , which is the set of permutation matrices (Peyre & Cuturi, 2018). ´ The Hungarian algorithm finds an optimal solution in $O ( M ^ { 3 } )$ time (Kuhn, 1955).
+
+An entropy-regularized version of problem (5) can be solved more efficiently. Let the entropy of $R$ be $\begin{array} { r } { H ( R ) = - \sum _ { i , j = 1 } ^ { M } R _ { i , j } \log R _ { i , j } } \end{array}$ . For $\varepsilon > 0$ , Cuturi (2013) defines the Sinkhorn distance $S _ { c ^ { \prime } }$ :
+
+$$
+\begin{array} { r } { R ^ { * } = \underset { R \in \mathrm { S } _ { M } } { \arg \operatorname* { m i n } } \frac { 1 } { M } \langle R , C ^ { \prime } \rangle _ { F } - \varepsilon H ( R ) , \qquad S _ { c ^ { \prime } } ( \hat { Q } , \hat { P } ) = \langle R ^ { * } , C ^ { \prime } \rangle _ { F } , } \end{array}
+$$
+
+and shows that the (Sinkhorn, 1964)’s algorithm returns its regularized optimum — that is also unique due to strong convexity of the entropy. The Sinkhorn is a fixed point algorithm that runs nearly in $M ^ { 2 }$ time (Altschuler et al., 2017) and can be efficiently implemented with matrix multiplications; see Algorithm 1. Its convergence to the Wasserstein distance is studied by Weed (2018).
+
+The smaller the $\varepsilon$ , the smaller the entropy and the better the approximation of the Wasserstein distance. At the same time, a larger number of steps $O ( L )$ is needed to converge. Conversely, high entropy encourages the solution to lie far from a permutation matrix. Note that all Sinkhorn operations are differentiable. So when the distance is used as a cost function, we can unroll $O ( L )$ iterations and backpropagate (Genevay et al., 2018). In conclusion, we obtain a differentiable surrogate for Wasserstein distances between empirical distributions; the approximation arises from sampling, entropy regularization and the finite amount of steps in place of convergence.
+
+# 2.3 NOISE AS TARGETS
+
+Bojanowski & Joulin (2017) introduce Noise As Targets (NAT), an algorithm for unsupervised representation learning. The method learns a neural network $f _ { \theta }$ by embedding images into a uniform hypersphere. A sample $z$ is drawn from the sphere for each training image and fixed. The goal is to learn $\theta$ such that 1-to-1 matching between images and samples is improved: matching is coded with a permutation matrix $R$ , and updated with the Hungarian algorithm. The objective is:
+
+$$
+{ \underset { \theta } { \operatorname* { m a x } } } \ { \underset { R \in P _ { M } } { \operatorname* { m a x } } } \ \operatorname { T r } ( R Z f _ { \theta } ( X ) ^ { \top } ) ,
+$$
+
+where $\operatorname { T r } ( \cdot )$ is the trace operator, $Z$ and $X$ are respectively prior samples and images stacked in a matrix and $P _ { M } \subset S _ { M }$ is the set of $M$ -dimensional permutations. NAT learns by alternating SGD and the Hungarian. One can interpret this problem as supervised learning, where the samples are targets (sampled only once) but their assignment is learned; notice that freely learnable $Z$ would make the problem ill-defined. The authors relate NAT to OT, a link that we make formal below.
+
+# 3 PRINCIPLES OF WASSERSTEIN AUTOENCODING
+
+With Equation 3, Tolstikhin et al. (2018) reformulate the Wasserstein distance in image space in terms of autoencoders. The hard constraint $Q _ { Z } = P _ { Z }$ is in practice replaced with a soft constraint by adding a penalty in the form of a divergence $D _ { Z } ( Q _ { Z } , P _ { Z } )$ . The resulting objective (4) is a lower bound to the Wasserstein difference and the choice of a divergence $D _ { Z }$ is left open. In contrast, we show that one should opt for minimizing a Wasserstein distance in latent space and that this leads to an equality with — not a bound for — the original Wasserstein distance in image space.
+
+More precisely, Theorem 3.1 first proves that the Wasserstein distance between the generative model and data distribution is bounded from above by a quantity consisting of the reconstruction error and the Wasserstein distance between $P _ { Z }$ and $Q _ { Z }$ . Theorem 3.2 shows that we can restrict learning to the class of deterministic (auto)encoders. Put together, Corollary 3.3 provides a principled learning objective in the framework of Optimal Transport by rewriting the Wasserstein distance in image space into an equivalent tractable form. We start with the following bound:
+
+Theorem 3.1. If $G ( X | Z )$ is deterministic and $\gamma$ -Lipschitz then:
+
+$$
+W _ { p } ( P _ { X } , P _ { G } ) \leq W _ { p } ( P _ { X } , G _ { \# } Q _ { Z } ) + \gamma \cdot W _ { p } ( Q _ { Z } , P _ { Z } ) .
+$$
+
+The proof (A.2) exploits the triangle inequality of the Wasserstein distance and its behaviour under composition with Lipschitz maps – a property not shared with divergences such as the KL. To effectively minimize the right-hand side in Theorem 3.1 over a class of encoders we need to further upper bound the reconstruction term with the following1:
+
+$$
+W _ { p } ( P _ { X } , G _ { \# } Q _ { Z } ) \leq \sqrt { \mathbb { E } _ { X \sim P _ { X } } \mathbb { E } _ { Z \sim Q ( Z | X ) } \mathbb { E } _ { X ^ { \prime } \sim G ( X | Z ) } [ \| X - X ^ { \prime } \| _ { p } ^ { p } ] } ,
+$$
+
+which reduces to the $p$ -th root of $\mathbb { E } _ { X \sim P _ { X } } [ \| X - G ( Q ( X ) ) \| _ { p } ^ { p } ]$ if both $G$ and $Q$ are deterministic. The tightness of this bound and its use as an objective function for learning are discussed below.
+
+We now improve the characterization of Equation 3, which is formulated in terms of stochastic encoders $Q ( Z | X )$ and deterministic decoders $G ( X | Z )$ . In fact, it is possible to restrict the learning class to that of deterministic autoencoders:
+
+Theorem 3.2. Let $P _ { X }$ be not atomic2 and $G ( X | Z )$ deterministic. Then for every continuous cost $c$ :
+
+$$
+W _ { c } ( P _ { X } , P _ { G } ) = \operatorname* { i n f } _ { \substack { Q ( Z | X ) d e t e r m i n i s t i c : Q _ { Z } = P _ { Z } } } \mathbb { E } _ { X \sim P _ { X } } [ c ( X , G ( Q ( X ) ) ) ] .
+$$
+
+Using the cost $c ( x , y ) = \| x - y \| _ { p } ^ { p } ,$ , the equation holds with $W _ { p } ^ { p } ( P _ { X } , P _ { G } )$ in place of $W _ { c } ( P _ { X } , P _ { G } )$ .
+
+The statement is a direct consequence of the equivalence between the Kantorovich and Monge formulations of OT (Villani, 2008); see the proof in A.3. We remark that this result is stronger than, and can be used to deduce Equation 3; see A.4 for a proof. Combining the two previous results, we are now in position to prove that the bound in Theorem 3.1 is tight for deterministic (auto)encoders:
+
+$$
+\begin{array} { r l } { W _ { p } ( P _ { X } , P _ { G } ) \overset { T h \ \mathrm { . 3 . 1 } } { \leq } } & { \underset { Q \ \mathrm { d e t . } } { \operatorname* { i n f } } \ \overset { \ell } { \sqrt { \mathbb { E } _ { X \sim P _ { X } } [ \| X - G ( Q ( X ) ) \| _ { p } ^ { p } ] } } + \gamma \cdot W _ { p } ( Q _ { Z } , P _ { Z } ) } \\ { \ } & { \qquad \overset { \mathrm { i n f } } { \leq } } \\ & { \ Q \underset { Q \ \mathrm { d e t . } , Q _ { Z } = P _ { Z } } { \operatorname * { i n f } } \ \overset { \ell } { \sqrt { \mathbb { E } _ { X \sim P _ { X } } [ \| X - G ( Q ( X ) ) \| _ { p } ^ { p } ] } } + \gamma \cdot \underset { = 0 } { \underbrace { W _ { p } ( Q _ { Z } , P _ { Z } ) } } } \\ { \ } & { \qquad T \underset { = } { \overset { h \ \cdot 3 \cdot 2 } { = } } \ W _ { p } ( P _ { X } , P _ { G } ) . } \end{array}
+$$
+
+Inequality in Step 10 holds because we restrict the domain of the infimum, which in turns implies $W _ { p } ( Q _ { Z } , P _ { Z } ) = 0$ . As a consequence we obtain the following Corollary, which provides us with an objective for learning generative autoencoders:
+
+Corollary 3.3. Let $P _ { X }$ be non-atomic and $G ( X | Z )$ be deterministic and $\gamma$ -Lipschitz. Then we have the equality:
+
+$$
+W _ { p } ( P _ { X } , P _ { G } ) = \operatorname* { i n f } _ { \substack { Q ( Z | X ) } } \operatorname* { i n f } _ { \substack { d e t e r m i n i s t i c } } \big \{ \big \langle \mathbb { E } _ { X \sim P _ { X } } [ \| X - G ( Q ( X ) ) \| _ { p } ^ { p } ] + \gamma \cdot W _ { p } ( Q _ { Z } , P _ { Z } ) .
+$$
+
+More precisely, we can now formulate our learning problem as the minimization of the right-hand side of Equation 12 over deterministic decoders:
+
+$$
+\operatorname* { m i n } _ { G } \operatorname* { m i n } _ { Q } \sqrt [ \gamma ] { \mathbb { E } _ { X \sim P _ { X } } [ \| X - G ( Q ( X ) ) \| _ { p } ^ { p } ] } + \gamma \cdot W _ { p } ( Q _ { Z } , P _ { Z } )
+$$
+
+When the encoder is a neural network of limited capacity, enforcing $Q _ { Z } \approx P _ { Z }$ might not be feasible in the general case of dimension mismatch (Rubenstein et al., 2018). In fact, since the class of deterministic neural networks is much smaller than the class of deterministic measurable maps, one might consider adding noise to the output, i.e. use stochastic networks instead. Nonetheless, neural networks can approximate any measurable map up to arbitrarily small error (Hornik, 1991), and we prove a related bound for the Wasserstein distance in A.5. It follows that learning deterministic autoencoders is sufficient to approach the theoretical upper bound and thus it will be our empirical choice.
+
+Finally, Theorems 3.1 and 3.2 strengthen the relevance of matching aggregated posterior and prior, which we show to be a sufficient and necessary condition for generative autoencoding. Justified by the previous results, we state it for deterministic autoencoders (proof in A.6).
+
+Theorem 3.4 (Sufficiency and necessity for generative autoencoding). Suppose perfect reconstruction, that is, $P _ { X } = ( G \circ { \dot { Q } } ) _ { \# } P _ { X }$ . Then:
+
+$$
+i ) ~ P _ { Z } = Q _ { Z } \implies P _ { X } = P _ { G } , \qquad i i ) ~ P _ { Z } \neq Q _ { Z } \implies P _ { X } \neq P _ { G } .
+$$
+
+In particular, Theorem 3.4 ii) certifies that, under perfect reconstruction, failing to match aggregated posterior and prior makes learning the data distribution impossible. Matching in latent space should be seen as fundamental as minimizing the reconstruction error, a fact known about the performance of VAE (Hoffman & Johnson, 2016; Higgins et al., 2017; Alemi et al., 2018; Rosca et al., 2018).
+
+# 4 SINKHORN AUTOENCODERS
+
+In light of our theory, we minimize the Wasserstein distance between the aggregated posterior and the prior, and we do so by running the Sinkhorn on their empirical samples. Let $\bar { \{ } x _ { i } \} _ { i = 1 } ^ { M }$ be the data input to the deterministic encoder $Q ( z _ { i } ^ { \prime } | x _ { i } ) = \delta _ { z _ { i } ^ { \prime } }$ and $\{ z _ { i } \} _ { i = 1 } ^ { M }$ the samples from the prior $P _ { Z }$ . The empirical distributions are $\begin{array} { r } { \hat { Q } _ { Z } = \frac { 1 } { M } \sum _ { i = 1 } ^ { M } \delta _ { z _ { i } ^ { \prime } } } \end{array}$ and $\begin{array} { r } { \hat { P } _ { Z } = \frac { 1 } { M } \sum _ { i = 1 } ^ { M } \delta _ { z _ { i } } } \end{array}$ . With $C _ { i j } ^ { \prime } = c ( z _ { i } ^ { \prime } , z _ { j } )$ , the Sinkhorn distance is $S _ { c ^ { \prime } } ( \hat { Q } _ { Z } , \hat { P } _ { Z } )$ as defined in Equation 6.
+
+We compute the Sinkhorn distance in two steps: first obtain the optimal regularized coupling $R ^ { * }$ and then multiply it with the cost, i.e. set $\varepsilon = 0$ :
+
+# Algorithm 1 SINKHORN
+
+$$
+\begin{array} { r } { R ^ { * } = \underset { R \in \mathrm { S } _ { M } } { \arg \operatorname* { m i n } } \frac { 1 } { M } \langle R , C ^ { \prime } \rangle _ { F } - \varepsilon H ( R ) } \\ { S _ { c ^ { \prime } } ( \hat { Q } _ { Z } , \hat { P } _ { Z } ) = \frac { 1 } { M } \langle R ^ { * } , C ^ { \prime } \rangle _ { F } \ . \ } \end{array}
+$$
+
+<table><tr><td>Input: {zi}=1~ Pz,{2}m=1 ~ Qz,ε,L ∀i,j,Cij=c(zi,zj) K=e-C/ε,u←1</td></tr><tr><td># elem-wise exp repeat L times:</td></tr><tr><td>v ←1/(KTu) # elem-wise division</td></tr><tr><td>u ←1/(Kv) R* ←Diag(u)KDiag(u)</td></tr><tr><td>Output: M(R*,C) F</td></tr><tr><td></td></tr></table>
+
+See Algorithm 1. Note that we do not sacrifice differentiability: we stack $O ( L )$ Sinkhorn operations on top of the encoder, without additional learnable parameters, and run auto-differentiation.
+
+With a deterministic decoder $G$ and encoder $Q$ , we arrive at the objective for the Sinkhorn AutoEncoder (SAE):
+
+$$
+\operatorname* { m i n } _ { G } \operatorname* { m i n } _ { Q } \mathbb { E } _ { X \sim \hat { P } _ { X } } [ \| X - G ( Q ( X ) ) \| _ { p } ^ { p } ] + \beta \cdot S _ { c ^ { \prime } } ( \hat { Q } _ { Z } , \hat { P } _ { Z } ) .
+$$
+
+In practice, we drop the $p$ -th root and tune a $\beta > 0$ hyper-parameter to mix the two terms. Small $\varepsilon$ and hence large $L$ worsen the numerical stability of the Sinkhorn; thus it is more convenient to scale $\boldsymbol { S _ { c ^ { \prime } } }$ by $\beta$ to explore the trade off, as in the WAE. In most experiments, both $c$ and $c ^ { \prime }$ will be $\| \cdot \| _ { 2 } ^ { 2 }$ . This objective is minimized by mini-batch SGD, which requires the re-calculation of an optimal regularized coupling $R ^ { * }$ at each iteration. Experimentally we found that this is not a significant overhead, unless a large $L$ is needed for convergence due to a small $\varepsilon$ . In practice, Algorithm 1 loops for $L$ iterations but can exit earlier if the updates of $u$ reach a fixed point.
+
+We have not specified our distribution $P _ { Z }$ yet. In fact, SAE can work in principle with arbitrary priors. The only requirement coming from the Sinkhorn is the ability to generate samples. The choice should be motivated by the desired geometric properties of the latent space; Theorem 3.4 stresses the importance of such choice for the generative model. For quantitative comparison with prior work, we focus primarily on hyperspheres, as in the Hyperspherical VAE (HVAE) (Davidson et al., 2018). Moreover, considering the Wasserstein distance $\varepsilon = 0$ ) from a uniform hyperspherical prior with squared Euclidean cost, we recover the NAT objective as a special case of ours (see Appendix A.7); yet, our method enjoys lower complexity and differentiability. The remarkable performance of NAT on representation learning on ImageNet confirms the value of the spherical prior. Other distributions are also considered in the paper, in particular the Dirichlet prior — with a tunable bias towards the simplex vertices — as a choice for controlling latent space clustering.
+
+Deterministic encoders model implicit distributions. Distributions are said to be implicit when their probability density may be intractable or even unknown, but it is possible to obtain samples and gradients for their parameters; GANs are examples of models with implicit distributions. Implicit distributions can give more flexibility as they are not limited by families of distributions with tractable density (Mohamed & Lakshminarayanan, 2016; Huszar, 2017). Moreover, by encoding with deter- ´ ministic neural networks, we bypass the use of reparametrization tricks for gradient estimation.
+
+# 5 RELATED WORK
+
+The normal prior is common in VAE for the reason of tractability. In fact, changing the prior and/or the approximate posterior distributions requires the use of tractable densities and the appropriate reparametrization trick. A hyperspherical prior is used by Davidson et al. (2018) with improved experimental performance; the algorithm models a Von Mises-Fisher posterior, with a non-trivial posterior sampling procedure and a reparametrization trick based on rejection sampling. Our implicit encoder distribution sidesteps these difficulties; recent advances on variable reparametrization can also simplify these requirements (Figurnov et al., 2018). We are not aware of methods embedding on probability simplices, except the use of Dirichlet priors by the same Figurnov et al. (2018).
+
+Hoffman & Johnson (2016) showed that the objective of a VAE does not force the aggregated posterior and prior to match, and that the mutual information of input and codes may be minimized instead. Just like the WAE, SAE avoids this effect by construction. Makhzani et al. (2015) and WAE improve latent matching by GAN/MMD. With the same goal, Alemi et al. (2017), Tomczak & Welling (2017) introduce learnable priors in the form of a mixture of approximate posteriors, which can be used in SAE as well.
+
+The Sinkhorn (1964) algorithm gained interest after Cuturi (2013) showed its application for fast computation of Wasserstein distances. The algorithm has been applied to ranking (Adams & Zemel, 2011), domain adaptation (Courty et al., 2014), multi-label classification (Frogner et al., 2015), metric learning (Huang et al., 2016) and ecological inference (Muzellec et al., 2017). Santa Cruz et al. (2017); Linderman et al. (2018) used it for supervised combinatorial losses. Our use of the Sinkhorn for generative modeling is akin to that of Genevay et al. (2018), which matches data and model samples with adversarial training, and to Ambrogioni et al. (2018), which matches samples from the model joint distribution and a variational joint approximation. WAE and WGAN objectives are linked respectively to primal and dual formulations of OT (Tolstikhin et al., 2018).
+
+Our approach for training the encoder alone qualifies as self-supervised representation learning (Donahue et al., 2017; Noroozi & Favaro, 2016; Noroozi et al., 2017). As in NAT (Bojanowski & Joulin, 2017) and in constrast to most other methods, we can sample pseudo labels (from the prior) independently from the input. In Appendix A.7 we show a formal connection with NAT.
+
+![](images/2e5ebccd10b19ece54456afe2a68b4d6d991a487ad6d505e519353d0bf2f25fb.jpg)  
+Figure 1: a) Swiss Roll and its b) squared and c) spherical embeddings learned by Sinkhorn encoders. MNIST embedded onto a 10D sphere viewed through $t$ -SNE, with classes by colours: d) encoder only or e) encoder $^ +$ decoder.
+
+# 6 EXPERIMENTS
+
+We start our empirical analysis with a qualitative assessment of the representation learned with the Sinkhorn algorithm. In the rest we focus on the autoencoder. We compare with NAT and confirm the Sinkhorn to be a better choice than the Hungarian. We display interpolations and samples of SAE and compare numerically with AE, $( \beta )$ -VAE, HVAE and WAE-MMD. We further show the flexibility of SAE by using a Dirichlet prior and on a toy probabilistic programming task.
+
+We experiment on MNIST, CIFAR10 (Krizhevsky & Hinton, 2009) and CelebA (Liu et al., 2015). MNIST is dynamically binarized and the reconstruction error is the binary cross-entropy (although not a distance, it is a commonly used divergence for binary data). For CIFAR10 and CelebA the reconstruction is the squared Euclidean distance; in every experiment, the latent cost is also squared Euclidean. We train fully connected neural networks for MNIST and the convolutional architectures from Tolstikhin et al. (2018) for the rest; the latent space dimensions are respectively 10, 64, 64. We run Adam (Kingma & Ba, 2014) with mini-batches of 128. Hyperspherical embedding is hardcoded in the architectures by $L 2$ normalization of the encoder output as in Bojanowski & Joulin (2017). The Sinkhorn runs with $\epsilon = 0 . 1$ , $L = 5 0$ , except when otherwise stated. FID scores for CIFAR10 and CelebA are calculated as in Heusel et al. (2017), while for MNIST we train a 2-layer convolutional network to extract features for the Frechet distance, similarly to Odena et al. (2018). Notice ´ that the FID score is a Wasserstein-2 distance and hence our theory applies directly.
+
+# 6.1 REPRESENTATION LEARNING WITH SINKHORN ENCODERS
+
+We demonstrate qualitatively that the Sinkhorn distance is a valid objective for unsupervised feature learning by showing that we can learn the encoder in isolation. The task is to embed the input distribution in a lower dimensional space, preserving the local data geometry, by solving Problem 14 with no reconstruction cost. We display the representation of a 3D Swiss Roll and MNIST. For the Swiss Roll we set $\varepsilon = 1 0 ^ { - 3 }$ , while for MNIST it is set to 0.5, and $L$ is picked to ensure convergence. For the Swiss roll (Figure 1a), we use a 50-50 fully connected network with ReLUs.
+
+Figures 1b, 1c show that the local geometry of the Swiss Roll is conserved in the new representational spaces — a square and a sphere. While the global shape is not necessarily more unfolded than the original, it looks qualitatively more amenable for further computation. Figure 1d shows the $t$ -SNE visualization (Maaten & Hinton, 2008) of the learned representation of the MNIST test set. With neither labels nor reconstruction error, we learn an embedding that is aware of class-wise clusters. Minimization of the Sinkhorn distance achieves this by encoding onto a $d$ -dimensional uniform sphere, such that points are encouraged to map far apart; in particular, in high dimension we can prove (see A.8) that the collapse probability decreases with $d$ :
+
+Proposition 6.1. Let $z , z ^ { \prime }$ be two uniform samples from a $d$ -dimensional sphere. In the high dimensional regime, for any δ < 2 we have P (kz − z0k2 > δ) ≥ 1 − 14d(√2−δ)2 .
+
+Other than this repulsive effect — the uniform distribution has max-entropy on any compact space —, a contractive force is present due to the inductive prior of neural networks, which are known to be Lipschitz functions (Balan et al., 2017). On the one hand, points in the latent space disperse in order to fill up the sphere; on the other hand, points close on image space cannot be mapped too far from each other. As a result, local distances are conserved while the overall distribution is spread. When the encoder is combined with a decoder $G$ — the topic of the experiments below —, the contractive force strenghtens: they collaborate in learning a latent space which makes reconstruction possible despite finite capacity and hence favours the conservation of local similarities; see Figure 1e.
+
+Table 1: Ablation for spherical SAE: Sinkhorn vs. Hungarian, fixed targets vs. sampling. MMD are scaled up by 1000. We compute a baseline for the MMD between two independent set of 10K samples (same as the test set size) from the prior. The baseline is 0.2 for both datasets.   
+
+<table><tr><td colspan="8">MNIST</td><td colspan="3">CIFAR10</td></tr><tr><td>method</td><td>prior</td><td>β</td><td>MMD</td><td>RE</td><td>FID</td><td>β</td><td>MMD</td><td></td><td>RE</td><td>FID</td></tr><tr><td>Hungarian</td><td>sample</td><td>10</td><td>0.37</td><td>65.9</td><td>10.3</td><td>10</td><td>0.25</td><td></td><td>22.4</td><td>98.5</td></tr><tr><td>Hungarian</td><td>targets</td><td>10</td><td>0.32</td><td>68.5</td><td>10.0</td><td>10</td><td></td><td>0.26</td><td>22.8</td><td>98.4</td></tr><tr><td>Hungarian</td><td>sample</td><td>100</td><td>0.60</td><td>85.0</td><td>9.7</td><td>100</td><td></td><td>0.23</td><td>23.8</td><td>98.6</td></tr><tr><td>Hungarian</td><td>targets</td><td>100</td><td>0.21</td><td>67.2</td><td>7.1</td><td>100</td><td></td><td>0.24</td><td>23.5</td><td>102.0</td></tr><tr><td>Sinkhorn</td><td>sample</td><td>10</td><td>0.35</td><td>66.2</td><td>9.4</td><td>10</td><td></td><td>0.25</td><td>22.5</td><td>97.5</td></tr><tr><td>Sinkhorn</td><td>targets</td><td>10</td><td>0.29</td><td>65.3</td><td>9.4</td><td>10</td><td></td><td>0.25</td><td>22.4</td><td>97.0</td></tr><tr><td>Sinkhorn</td><td>sample</td><td>100</td><td>0.30</td><td>66.8</td><td>6.8</td><td>100</td><td></td><td>0.21</td><td>23.7</td><td>100.4</td></tr><tr><td>Sinkhorn</td><td>targets</td><td>100</td><td>0.30</td><td>66.8</td><td>6.8</td><td>100</td><td></td><td>0.24</td><td>23.1</td><td>107.5</td></tr></table>
+
+Table 2: SAE vs. prior work. In boldface the best two FID per dataset. Note that MMD are not comparable if the prior is different. †The ‘spherical’ AE amounts to normalizing the encoder output.   
+
+<table><tr><td></td><td></td><td></td><td colspan="4">MNIST</td><td colspan="4">CIFAR10</td><td colspan="4">CelebA</td></tr><tr><td>method</td><td>prior</td><td>cost</td><td>β</td><td>MMD</td><td>RE</td><td>FID</td><td>β</td><td>MMD</td><td>RE</td><td>FID</td><td>β</td><td>MMD</td><td>RE</td><td>FID</td></tr><tr><td>AE</td><td>-</td><td>-</td><td>：</td><td>-</td><td>62.6</td><td>45.2</td><td>·</td><td>-</td><td>22.6</td><td>375.6</td><td>·</td><td>-</td><td>61.8</td><td>357.0</td></tr><tr><td>VAE</td><td>normal</td><td>KL</td><td>1</td><td>0.63</td><td>66.4</td><td>7.2</td><td>1</td><td>4.6</td><td>40.6</td><td>161.0</td><td>1</td><td>0.35</td><td>75.1</td><td>51.4</td></tr><tr><td>β-VAE</td><td>normal</td><td>KL</td><td>0.1</td><td>2.3</td><td>62.8</td><td>15.2</td><td>0.1</td><td>0.23</td><td>22.8</td><td>106.6</td><td>0.1</td><td>0.21</td><td>63.7</td><td>56.5</td></tr><tr><td>WAE</td><td>normal</td><td>MMD</td><td>100</td><td>0.69</td><td>63.1</td><td>9.0</td><td>100</td><td>0.29</td><td>22.9</td><td>105.3</td><td>100</td><td>0.21</td><td>62.6</td><td>61.6</td></tr><tr><td>AE</td><td>sphere↑</td><td>-</td><td>-</td><td>4.7</td><td>66.2</td><td>22.0</td><td>-</td><td>1.8</td><td>22.4</td><td>107.8</td><td>-</td><td>1.1</td><td>62.4</td><td>83.9</td></tr><tr><td>HVAE</td><td>sphere</td><td>KL</td><td>1</td><td>0.33</td><td>72.2</td><td>9.5</td><td>-</td><td>-</td><td>=</td><td>-</td><td>-</td><td>=</td><td>-</td><td>-</td></tr><tr><td>WAE</td><td>sphere</td><td>MMD</td><td>100</td><td>0.25</td><td>65.7</td><td>8.9</td><td>100</td><td>0.24</td><td>22.4</td><td>99.7</td><td>100</td><td>0.23</td><td>61.9</td><td>61.3</td></tr><tr><td>SAE</td><td>sphere</td><td>Sinkhorn</td><td>100</td><td>0.30</td><td>66.8</td><td>6.8</td><td>10</td><td>0.23</td><td>22.5</td><td>97.2</td><td>10</td><td>0.26</td><td>63.4</td><td>56.5</td></tr></table>
+
+# 6.2 AUTOENCODING WITH THE SINKHORN DISTANCE AND NAT
+
+We investigate the advantages of the Sinkhorn with respect to NAT in training autoencoders; this is an ablation study for our method. First, Sinkhorn has a lower complexity than the Hungarian. In both cases, the complexity can be reduced by mini-batch optimization. Yet, training with large minibatches $( > 2 0 0 )$ becomes quickly impractical with the Hungarian. Second, the differentiability of the Sinkhorn allows us to avoid the alternating minimization and instead backpropagate on the joint parameter space of encoder and doubly stochastic matrices. Third, the Sinkhorn approximates the empirical Wasserstein distance, while the Hungarian is optimal. Last, NAT draws samples once and uses them as targets throughout learning; their assignment to training points is updated by optimizing a permutation matrix over mini-batches and storing the local optimal result. We term NAT in this context Hungarian-targets and our method Sinkhorn-sample. We can design two hybrid methods. Hungarian-sample: a permutation $R$ can be used to compute the cost $\langle R , C ^ { \prime } \rangle _ { F }$ and backpropagate. Sinkhorn-targets: a doubly stochastic matrix $R$ solution of the Sinkhorn can be used for sampling a permutation3 and targets can be re-assigned. We test the impact of these choices experimentally by test set reconstruction error and FID score on MNIST and CIFAR10; we measure latent space mismatch by the MMD with Gaussian kernel over the test set.
+
+Table 1 shows the results. From the FID scores, we conclude that there is no significant difference in generative performance between either Sinkhorn vs. Hungarian, or samples vs. targets. The parameter $\beta$ trading off reconstruction and latent space cost is more influential than any of these choices. On MNIST, MMD is often lower with fixed targets; this is a sign that the FID does not fully account for all model qualities. Due to the additional overhead of the Hungarian and the targets updating, our algorithm implements the Sinkhorn with mini-batch sampling. In the rest, we also fix $\beta$ for MNIST and CIFAR as the best found here.
+
+![](images/3f9baee55f57d78ac3dd9e528dd531c7a6f3eb1f20542a706cd100fb41baf6be.jpg)  
+Figure 2: From left to right: CIFAR10 interpolations, CelebA interpolations and samples. Models from Table 2: $( \beta$ -)VAE (top) and SAE (bottom).
+
+![](images/7f9d92eb546d597a7cde15ae76f87a21f7cf2c0d949585ebb9b3ae81b36734b1.jpg)  
+Figure 3: $t$ -SNEs of SAE latent spaces on MNIST: a) 10-dimensional $\operatorname { D i r } ( 1 / 2 )$ and b) 16- dimensional $\operatorname { D i r } ( 1 / 5 )$ priors. For the latter: c) aggr. posterior (red) vs. prior (blue), d) interpolation between vertices and e) samples from the prior.
+
+# 6.3 COMPARISON WITH OTHER AUTOENCODERS
+
+We compare with AE, $( \beta \mathrm { - } ) \mathrm { V A E }$ , $\mathrm { H V A E ^ { 4 } }$ and WAE. Figures 2 shows interpolations and samples of SAE and VAE from CIFAR10 and CelebA. SAE interpolations are defined on geodesics connecting points on the hypersphere. SAE tends to produce crisper images, with higher contrast, and avoids averaging effects as particularly evident in the CelebA interpolations. The CelebA samples are also interesting: while SAE generally maintains a crisper look than VAE’s, faces appear more often malformed. Table 2 reports a quantitative comparison. Each baseline model has a version with normal and spherical prior. FID scores of SAE are on par or superior to that of VAE and consistently better than WAE. The spherical prior appears to reduce FID scores in several cases.
+
+# 6.4 DIRICHLET PRIORS
+
+We further demonstrate the flexibility of SAE by using Dirichlet priors on MNIST. The prior draws samples on the probability simplex; hence, here we constrain the encoder by a final softmax layer. We use priors that concentrate on the vertices, by the intuition that digits would naturally cluster around them. A 10-dimensional $\operatorname { D i r } ( 1 / 2 )$ prior (Figure 3a) results in an embedding qualitatively similar to the uniform sphere (1e). With a more skewed prior $\operatorname { D i r } ( 1 / 5 )$ , we could expect an organization in latent space where each digit is mapped to a vertex, as little mass lies in the center. We found that in dimension 10 this is seldom the case, as multiple vertices can be taken by the same digit to model different styles, while other digits share the same vertex.
+
+![](images/a13489c808243c33f9abfa49450c2f1c6bedd6c7a64407266288ac780653be70.jpg)  
+Figure 4: Toy probabilistic programming: data and localization (left), reconstructions (center) and samples (right). AIR (top) and SAE (bottom).
+
+We thus experiment with a 16-dimensional $\operatorname { D i r } ( 1 / 5 )$ , which yields more disconnected clusters (3b); the effect is evident when showing the prior and the aggregated posterior that tries to cover it (3c). Figure 3d (leftmost and rightmost columns) shows that every digit $0 - 9$ is indeed represented on one of the 16 vertices, while some digits are present with multiple styles, e.g. the 7. The central samples in the Figure are the interpolations obtained by sampling on edges connecting vertices – no real data is autoencoded. Samples from the vertices appear much crisper than other prior samples (3e), a sign of mismatch between prior and aggregated posterior on areas with lower probability mass. Finally, we point out that we could even learn the Dirichlet hyperparameter(s) with a reparametrization trick (Figurnov et al., 2018) and let the data inform the model on the best prior.
+
+# 6.5 TOY PROBABILISTIC PROGRAMMING
+
+We run a final experiment to showcase that SAE can handle more complex implicit distributions, on a toy example of probabilistic programming. The goal is to learn a generative model for MNIST digits positioned on a larger canvas; the data is corrupted with salt noise that we do not model explicitly and which are model is thus required to ignore. The generative model samples from a factored prior distribution for $z _ { w h a t }$ — the digit appearance — from a 10-dimensional sphere and for zwhere — the location and scale — from a 3-dimensional Normal. A decoder network is fed with $z _ { w h a t }$ and generates the digit; the digit is then positioned on the black canvas on the coordinates given by a spatial transformer (Jaderberg et al., 2015) which is fed with $z _ { w h e r e }$ . The inference model produces $z _ { w h a t } , z _ { w h e r e }$ from the canvas, by using a spatial transformer and a encoder mirroring the generator.
+
+Our autoencoder is fully deterministic. The cost in latent space amounts to the sum of the Sinkhorn distances in the two prior components, Normal and hyperspherical. Figure 4 compares qualitatively with a simplified version of AIR (Eslami et al., 2016), that is built on variational inference with an explicit modelling of the approximate posterior distribution for this program. SAE is able to replicate the behaviour of AIR by locating the digit on the canvas, ignoring the noise in reconstruction and generating realistic samples.
+
+# 7 CONCLUSIONS
+
+We introduced a new generative model built on the principles of Optimal Transport. Working with empirical Wasserstein distances and deterministic networks provides us with a flexible likelihoodfree framework for latent variable modeling. Besides, the theory suggests improving matching in latent space which could be achieved by the use of parametric implicit prior distributions.
+
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+
+# A APPENDIX
+
+# A.1 LEMMA
+
+As a useful helper Lemma, we prove a Lipschitz property for the Wasserstein distance $W _ { p }$ .
+
+Lemma A.1. For every $P _ { X } , P _ { Y }$ distributions on a sample space $s$ and a Lipschitz map $F$ we have that
+
+$$
+W _ { p } ( F _ { \# } P _ { X } , F _ { \# } P _ { Y } ) \leq \gamma \cdot W _ { p } ( P _ { X } , P _ { Y } ) ,
+$$
+
+where $\gamma$ is the Lipschitz constant of $F$ .
+
+Proof. Recall that
+
+$$
+W _ { p } ( F _ { \# } P _ { X } , F _ { \# } P _ { Y } ) ^ { p } = \operatorname* { i n f } _ { \substack { \Gamma \in \Pi ( F _ { \# } P _ { X } , F _ { \# } P _ { Y } ) } } \int _ { S \times S } \| x - y \| _ { p } ^ { p } d \Gamma ( x , y ) .
+$$
+
+Notice then that for every $\Gamma \in \Pi ( P _ { X } , P _ { Y } )$ we have that $( F \times F ) _ { \# } \Gamma \in \Pi ( F _ { \# } P _ { X } , F _ { \# } P _ { Y } )$ . Hence
+
+$$
+\{ ( F \times F ) _ { \# } \Gamma : \Gamma \in \Pi ( P _ { X } , P _ { Y } ) \} \subset \Pi ( F _ { \# } P _ { X } , F _ { \# } P _ { Y } ) .
+$$
+
+From (16) we deduce that
+
+$$
+\begin{array} { l } { { \displaystyle W _ { p } ( F _ { \# } P _ { X } , F _ { \# } P _ { Y } ) ^ { p } \leq \operatorname* { i n f } _ { \Gamma \in \Pi ( P _ { X } , P _ { Y } ) } \int _ { S \times S } \| x - y \| _ { p } ^ { p } d ( F \times F ) _ { \# } \Gamma } } \\ { ~ = \operatorname* { i n f } _ { \Gamma \in \Pi ( P _ { X } , P _ { Y } ) } \int _ { S \times S } \| F ( x ) - F ( y ) \| _ { p } ^ { p } d \Gamma } \\ { ~ \leq \gamma ^ { p } \cdot ( W _ { p } ( P _ { X } , P _ { Y } ) ) ^ { p } . } \end{array}
+$$
+
+Taking the $p$ -root on both sides we conclude.
+
+# A.2 PROOF OF THEOREM 3.1
+
+Proof. Using the triangle inequality of the Wasserstein distance we obtain
+
+$$
+\begin{array} { r l } & { W _ { p } ( P _ { X } , G _ { \# } P _ { Z } ) \leq W _ { p } ( P _ { X } , G _ { \# } Q _ { Z } ) + W _ { p } ( G _ { \# } Q _ { Z } , G _ { \# } P _ { Z } ) } \\ & { \qquad \leq W _ { p } ( P _ { X } , G _ { \# } Q _ { Z } ) + \gamma \cdot W _ { p } ( Q _ { Z } , P _ { Z } ) , } \end{array}
+$$
+
+where in line (17) we have used Lemma A.1.
+
+In case $G$ is not deterministic, defining
+
+$$
+\gamma = \operatorname* { s u p } _ { \mathcal { P } , \mathcal { Q } } \frac { W _ { p } ( G ( X \vert Z ) _ { \# } \mathcal { P } , G ( X \vert Z ) _ { \# } \mathcal { Q } ) } { W _ { p } ( \mathcal { P } , \mathcal { Q } ) }
+$$
+
+exp we can still formulate a bound. The result follows directly from the first line of (17).
+
+# A.3 PROOF OF THEOREM 3.2
+
+The basic tool to prove Theorem 3.2 is the equivalence between Monge and Kantorovich formulation of optimal transport. For convenience we formulate its statement and we refer to Villani (2008) for a more detailed explanation.
+
+Theorem A.2 (Monge-Kontorovich equivalence). Given $P _ { X }$ and $P _ { Y }$ probability distributions on $\mathcal { X }$ such that $P _ { X }$ is not atomic, $c : \mathcal { X } \times \mathcal { X }  \mathbb { R }$ continuous, we have
+
+$$
+W _ { c } ( P _ { X } , P _ { Y } ) = \operatorname* { i n f } _ { \stackrel { T : \mathcal { X } \to \mathcal { X } _ { Y } } { T _ { \# } P _ { X } = P _ { Y } } } \int _ { \mathcal { X } } c ( x , T ( x ) ) d P _ { X } ( x ) .
+$$
+
+We are now in position to prove Theorem 3.2. We will prove it for a general continuous cost $c$
+
+Proof. Notice that as the encoder $Q ( Z | X )$ is deterministic there exists $Q : \mathcal { X }  \mathcal { Z }$ such that $Q _ { Z } = Q _ { \# } P _ { X }$ and $Q ( Z | X ) = \delta _ { \{ Q ( x ) = z \} }$ . Hence
+
+$$
+\begin{array} { l l l } { { \mathbb { E } _ { X \sim P _ { X } } \mathbb { E } _ { Z \sim Q ( Z | X ) } [ c ( X , G ( Z ) ) ] } } & { { = } } & { { \displaystyle \int _ { \mathcal { X } \times \mathcal { Z } } c ( x , G ( z ) ) d P _ { X } ( x ) d \delta _ { \{ Q ( x ) = z \} } ( z ) } } \\ { { } } & { { = } } & { { \displaystyle \int _ { \mathcal { X } } d P _ { X } ( x ) \int _ { \mathcal { Z } } c ( x , G ( z ) ) d \delta _ { \{ Q ( x ) = z \} } ( z ) } } \\ { { } } & { { = } } & { { \displaystyle \int _ { \mathcal { X } } c ( x , G ( Q ( x ) ) ) d P _ { X } ( x ) . } } \end{array}
+$$
+
+Therefore
+
+$$
+\operatorname* { i n f } _ { \substack { 2 ( Z | X ) \mathrm { \scriptsize ~ d e t e r m i n i s t i c : } Q _ { Z } = P _ { Z } } } \mathbb { E } _ { X \sim P _ { X } } \mathbb { E } _ { Z \sim Q ( Z | X ) } [ e ( X , G ( Z ) ) ] = \operatorname* { i n f } _ { \substack { Q _ { Z } \colon \mathcal { X } \to Z } \atop \mathcal { Q } _ { Z } = P _ { Z } } \int _ { \mathcal X } c ( x , G ( Q ( x ) ) ) d P _ { X } .
+$$
+
+We now want to prove that
+
+$$
+\left\{ G \circ Q : Q _ { \# } P _ { X } = P _ { Z } \right\} = \left\{ T : \mathcal { X } \to \mathcal { X } : T _ { \# } P _ { X } = P _ { G } \right\} .
+$$
+
+For the first inclusion $\subset$ notice that for every $Q : \mathcal { X }  \mathcal { Z }$ such that $Q _ { Z } = P _ { Z }$ we have that $G \circ Q : \mathcal { X } \to \mathcal { X }$ and
+
+$$
+( G \circ Q ) _ { \# } P _ { X } = G _ { \# } Q _ { \# } P _ { X } = G _ { \# } P _ { Z } .
+$$
+
+For the other inclusion $\supset$ consider $T : \mathcal { X }  \mathcal { X }$ such that $T _ { \# } P _ { X } = P _ { G } = G _ { \# } P _ { Z }$ . We want first to prove that there exists a set $A \subset { \mathcal { X } }$ with $P _ { X } ( A ) = 1$ such that $G : { \mathcal { Z } } \to T ( A )$ is surjective. Indeed if it does not hold there exists $B \subset { \mathcal { X } }$ with $P _ { X } ( B ) > 0$ and $G ^ { - 1 } ( T ( B ) ) = \mathrm { \hat { \varnothing } }$ . Hence
+
+$$
+0 = G _ { \# } P _ { Z } ( T ( B ) ) = T _ { \# } P _ { X } ( T ( B ) ) = P _ { X } ( B ) > 0
+$$
+
+that is a contraddiction. Therefore by standard set theory the map $G : { \mathcal { Z } } \to T ( A )$ has a right inverse that we denote by $\widetilde { G }$ . Then define $Q = \widetilde { G } \circ T$ . Notice that $G \circ Q = G \circ \widetilde { G } \circ T = T$ almost surely in $P _ { X }$ and also
+
+$$
+( \widetilde G \circ T ) _ { \# } P _ { X } = P _ { Z } .
+$$
+
+Indeed for any $A \subset { \mathcal { Z } }$ Borel we have
+
+$$
+( \widetilde G \circ T ) _ { \# } P _ { X } ( A ) = ( \widetilde G \circ G ) _ { \# } P _ { Z } ( A ) = P _ { Z } ( \widetilde G ^ { - 1 } ( G ^ { - 1 } ( A ) ) = P _ { Z } ( A ) .
+$$
+
+This concludes the proof of the claim in (20). Now we have
+
+$$
+\operatorname* { i n f } _ { Q : \mathcal { X }  \mathcal { Z } \atop Q _ { \# } ( P _ { X } ) = P _ { Z } } \int _ { \mathcal X } c ( x , G ( Q ( x ) ) ) d P _ { X } ( x ) = \operatorname* { i n f } _ { T : \mathcal { X }  \mathcal { X } _ { G } } \int _ { \mathcal X } c ( x , T ( x ) ) d P _ { X } ( x ) .
+$$
+
+Notice that this is exactly the Monge formulation of optimal transport. Therefore by Theorem A.2 we conclude that
+
+${ \underset { { \substack { \mathrm { 2 ( } } Z \mid X ) \mathrm { ~ d e t e r m i n i s t i c : } } \ Q _ { Z } = P _ { Z } } { \operatorname* { l i m } } } \mathbb { E } _ { X \sim P _ { X } } \mathbb { E } _ { Z \sim Q ( Z \mid X ) } [ e ( X , G ( Z ) ) ] = { \underset { { \substack { \mathrm { ~ \tiny ~ \mathrm { 1 ( } } P _ { X } , P _ { G } ) } } } { \operatorname* { i n f } } } \mathbb { E } _ { ( X , Y ) \sim \Gamma } [ e ( X , Y ) ]$ as we aimed.
+
+A.4 TOLSTIKHIN ET AL. (2018)’S THEOREM AS A CONSEQUENCE
+
+Proof. Thanks to Theorem 3.2 we have that
+
+$$
+\begin{array} { r l r } { W _ { c } ( P _ { X } , P _ { G } ) } & { = } & { \underset { Q ( Z | X ) \mathrm { ~ d e t e r m i n i s t i c : ~ } Q _ { Z } = P _ { Z } } { \operatorname* { i n f } } \mathbb { E } _ { X \sim P _ { X } } \mathbb { E } _ { Z \sim Q ( Z | X ) } [ c ( X , G ( Z ) ) ] } \\ & { \geq } & { \underset { Q ( Z | X ) \colon Q _ { Z } = P _ { Z } } { \operatorname* { i n f } } \mathbb { E } _ { X \sim P _ { X } } \mathbb { E } _ { Z \sim Q ( Z | X ) } [ c ( X , G ( Z ) ) ] . } \end{array}
+$$
+
+For the opposite inequality given $Q ( Z | X )$ such that $\begin{array} { r } { P _ { Z } \ = \ \int Q ( Z | X ) d P _ { X } } \end{array}$ define $Q ( X , Y ) =$ $P _ { X } \times \left[ G _ { \# } Q ( Z | X ) \right]$ . It is a distribution on $\mathcal { X } \times \mathcal { X }$ and it is easy to check that $\pi _ { \# } ^ { 1 } Q ( X , Y ) = P _ { X }$ and $\pi _ { \# } ^ { 2 } Q ( X , Y ) = G _ { \# } P _ { Z }$ , where $\pi ^ { 1 }$ and $\pi ^ { 2 }$ are the projection on the first and the second component. Therefore
+
+$$
+\{ Q ( X , Z ) : Q ( Z | X ) \mathrm { s u c h t h a t } Q _ { Z } = P _ { Z } \} \subset \Pi ( P _ { X } , P _ { G } )
+$$
+
+and so
+
+$$
+\begin{array} { l } { { W _ { c } ( P _ { X } , P _ { G } ) \leq \underset { Q ( Z | X ) : Q _ { Z } = P _ { Z } } { \operatorname* { i n f } } \int _ { \mathcal { X } \times \mathcal { X } } c ( x , y ) d Q ( x , y ) } } \\ { { \ } } \\ { { \displaystyle \quad = \int _ { \mathcal { X } } \left[ \int _ { \mathcal { X } } c ( x , y ) d G _ { \# } Q ( Z | X ) ( y ) \right] d P _ { X } } } \\ { { \ \displaystyle \quad = \int _ { \mathcal { X } } \left[ \int _ { \mathcal { X } } c ( x , G ( z ) ) d Q ( Z | X ) ( z ) \right] d P _ { X } . } } \end{array}
+$$
+
+# A.5 BOUNDS FOR NEURAL NETWORKS
+
+Theorem A.3. Let $P _ { X }$ be non-atomic and $G ( X | Z )$ be deterministic and $\gamma$ -Lipschitz. If $Q ^ { N N }$ is a neural network approximating a near optimal deterministic encoder up to an error of $\varepsilon \geq 0$ in $L _ { p }$ -norm then we have the inequality:
+
+$$
+0 \leq \left\{ \sqrt { \mathbb { E } _ { X \sim P _ { X } } [ \| X - G ( Q ^ { N N } ( X ) ) \| _ { p } ^ { p } ] } + \gamma \cdot W _ { p } ( Q _ { Z } ^ { N N } , P _ { Z } ) \right\} - W _ { p } ( P _ { X } , P _ { G } ) \leq 3 \gamma \varepsilon .
+$$
+
+Proof. Let $Q ^ { * }$ be an optimal measurable deterministic encoder that optimizes the right-hand side of Theorem 3.1 among measurable deterministic encoder (or at least $\delta \leq \gamma \varepsilon$ close to it) and $Q ^ { \mathrm { N N } }$ a
+
+neural network approximation of $Q ^ { * }$ such that $\sqrt { \mathbb { E } _ { X \sim P _ { X } } [ \| Q ^ { * } ( X ) - Q ^ { \mathrm { N N } } ( X ) ) \| _ { p } ^ { p } ] } \leq \varepsilon$ (existence by Hornik (1991)). Then we get:
+
+$$
+\begin{array} { r l } & { \sqrt { | \mathbf { E } \| _ { X \sim \mathcal { P } _ { x } } \| | X - G ( Q ^ { \mathrm { W } } ( X ) ) \| _ { \mathcal { F } } ^ { 2 } \Big ) + \gamma \cdot W _ { \mathcal { F } } ( Q _ { Z } ^ { \mathrm { W Z } } , P _ { Z } ) } } \\ & { \operatorname* { m a x } _ { \phi \leq \infty } \sqrt { | \mathbf { E } \| - G ( Q ^ { \mathrm { W } } ( X ) ) \| _ { \mathcal { F } } ^ { 2 } \Big ) + \gamma \cdot W _ { \mathcal { F } } ( Q _ { Z } ^ { \mathrm { W Z } } , P _ { Z } ) } } \\ & { \qquad + \underbrace { \sqrt { | \mathbf { E } _ { X \sim \mathcal { P } _ { x } } | | G ( Q ^ { \mathrm { W } } ( X ) ) - G ( Q ^ { \mathrm { W } } ( X ) ) | | _ { \mathcal { F } } ^ { 2 } | } } _ { \mathrm { C r e ~ } } } \\ & { \qquad + \frac { \gamma } { \gamma } \cdot \underbrace { \| \mathbf { U } _ { \mathcal { F } } ( Q _ { X } ^ { \mathrm { W } } , Q _ { Z } ^ { \mathrm { W Z } } ) + \gamma \cdot W _ { \mathcal { F } } ( Q _ { Z } ^ { \mathrm { W Z } } , P _ { Z } ) } _ { \leq 2 } } \\ & { \qquad \lesssim \sqrt { | \mathbf { E } _ { X \sim \mathcal { P } _ { x } } | | X - G ( Q ^ { \mathrm { W } } ( X ) ) | _ { \mathcal { F } } ^ { 2 } | } + \gamma \cdot W _ { \mathcal { S } } ( Q _ { Z } ^ { \mathrm { W } } , P _ { Z } ) + 2 \gamma \varepsilon } \\ & { \qquad \times \mathrm { d } _ { \mathcal { F } } ^ { \mathrm { a d } } \mathrm { ~ s i n c ~ } \Big \{ \sqrt { | \mathbf { E } _ { X \sim \mathcal { P } _ { x } } | | X - G ( Q ^ { \mathrm { W } } ( X ) ) | | _ { \mathcal { F } } ^ { 2 } | } + \gamma \cdot W _ { \mathcal { F } } ( Q _ { Z } , P _ { Z } ) \Big \} + \delta + 2 \gamma \varepsilon } \\ & { \qquad \leq \frac { \gamma } { \alpha } W _ { \mathcal { F } } ( P _ { X } , P _ { Z } ) + 3 \gamma \varepsilon , } \end{array}
+$$
+
+where in the last inequality we use additionally that the upper bound in Theorem 3.1 is sharp.
+
+Finally, we can also formulate a version of Corollary 3.3 restricted to deterministic neural networks as follows:
+
+Theorem A.4. Let $P _ { X }$ be non-atomic and $G ( X | Z )$ be deterministic and $\gamma$ -Lipschitz. Then we have the equality:
+
+$$
+W _ { p } ( P _ { X } , P _ { G } ) = \operatorname* { i n f } _ { Q N N } \sqrt { \mathbb { E } _ { X \sim P _ { X } } [ \| X - G ( Q ( X ) ) \| _ { p } ^ { p } ] } + \gamma \cdot W _ { p } ( Q _ { Z } , P _ { Z } ) ,
+$$
+
+where $Q$ runs through all deterministic neural network encoders (or any other class of universal approximators).
+
+Proof. This directly follows from A.3 in combination with Hornik (1991).
+
+# A.6 PROOF OF THEOREM 3.4
+
+Proof. Statement $i$ ) follows directly from the definition of push-forward of a measure.
+
+For $_ { i i }$ ) notice that if $P _ { Z } \neq Q _ { Z }$ then there exists $A \subset { \mathcal { Z } }$ a Borel set such that $P _ { Z } ( A ) \neq Q _ { \# } P _ { X } ( A )$ . Then
+
+$$
+\begin{array} { r l } & { G _ { \# } P _ { Z } ( G ( A ) ) = P _ { Z } ( ( G ^ { - 1 } \circ G ) ( A ) ) = P _ { Z } ( A ) \neq Q _ { \# } P _ { X } ( A ) } \\ & { \qquad = Q _ { \# } P _ { X } ( A ) ( ( G ^ { - 1 } \circ G ) ( A ) ) = ( G \circ Q ) _ { \# } P _ { X } ( G ( A ) ) . } \end{array}
+$$
+
+Hence as $P _ { X } = ( G \circ Q ) _ { \# } P _ { X }$ by hypothesis, we immediately deduce that $P _ { X } \neq P _ { G }$ .
+
+# A.7 COMPARISON WITH BOJANOWSKI & JOULIN (2017)
+
+We prove that the cost function of NAT is equivalent to ours when the encoder output is $L _ { 2 }$ normalized, $c ^ { \prime }$ is squared Euclidean and the Sinkhorn distance is considered with $\varepsilon = 0$ :
+
+$$
+\begin{array} { r l } & { \begin{array} { r l } & { \mathrm { s u r g ~ } _ { \theta } ^ { \mathrm { m a x } } \underset { \theta \in \mathcal { P } _ { n } } { \operatorname { I m } } } \\ & { = \mathrm { a r g ~ } _ { \theta } ^ { \mathrm { m a x } } \underset { \theta \in \mathcal { P } _ { n } } { \operatorname { I m } } } \end{array} \{ R \mathcal { L } \ell _ { 2 } f _ { \theta } ( X ) \} ^ { T } ) } \\ & { = \begin{array} { r l } & { ( R + \underset { \theta \in \mathcal { P } _ { n } } { \operatorname { I m } } ) _ { \theta \in \mathcal { P } _ { n } } } \\ & { = \mathrm { a r g ~ } _ { \theta } ^ { \mathrm { i n } \operatorname* { i n } } \underset { \theta \in \mathcal { P } _ { n } } { \operatorname { I m } } , } \end{array} } \\ & { = \begin{array} { r l } & { ( R + \underset { \theta \in \mathcal { P } _ { n } } { \operatorname { I m } } ) _ { \theta \in \mathcal { P } _ { n } } } \\ & { = \mathrm { a r g ~ } _ { \theta } ^ { \mathrm { i n } \operatorname* { i n } } \underset { \theta \in \mathcal { P } _ { n } } { \operatorname { I m } } } \end{array} \| R \mathbb { Z } \| _ { \theta } ^ { 2 } + \| \mathcal { L } \mu ( X ) \| _ { \theta } ^ { 2 } - 2 ( R \mathbb { Z } , \hat { \rho } _ { \theta } ( X ) ) F } \\ & { = \underset { \theta \in \mathcal { P } _ { n } } { \operatorname { I m } } \underset { \theta \in \mathcal { P } _ { n } } { \operatorname { I m } } } \end{array}  \\ &  = \begin{array} { r l } & { ( R + \underset { \theta \in \mathcal { P } _ { n } } { \operatorname { I m } } ) _ { \theta \in \mathcal { P } _ { n } } } \\ & { = \mathrm { a r g ~ } _ { \theta } ^ { \mathrm { i n } \operatorname* { i n } } \underset { \theta \in \mathcal { P } _ { n } } { \operatorname { I m } } } \end{array}
+$$
+
+Step 24 holds because both $R$ and $f _ { \theta } ( X )$ are row normalized. Step 25 exploits $R$ being a permutation matrix. The inclusion in Step 28 extend to degenerate solutions of the linear program that may not lie on vertices. We have discussed several differences between our Sinkhorn encoder and NAT. There are other minor ones with Bojanowski $\&$ Joulin (2017): ImageNet inputs are first converted to grey and passed through Sobel filters and the permutations are updated with the Hungarian only every 3 epochs. Preliminary experiments ruled our any clear gain of those choices in our setting.
+
+# A.8 PROOF OF PROPOSITION 6.1
+
+Proof. Let $z , z ^ { \prime }$ two points sampled uniformrly from a $d$ -dimensional sphere. Let $\alpha$ be the Euclidean distance between the two points. $\alpha$ has an analytical form (Wu et al., 2017) :
+
+$$
+p ( \| z - z ^ { \prime } \| _ { 2 } ) = p ( \alpha ) = \frac { \alpha ^ { d - 2 } } { c ( d ) } \left[ 1 - \frac { 1 } { 4 } \alpha ^ { 2 } \right] ^ { \frac { d - 3 } { 2 } } , \quad \mathrm { w h e r e } \quad c ( d ) = \sqrt { \pi } \frac { \Gamma \left( \frac { d - 1 } { 2 } \right) } { \Gamma \left( \frac { d } { 2 } \right) } .
+$$
+
+For high dimension, it approaches a Gaussian: $\begin{array} { r } { p ( \alpha ) \approx \mathcal { N } ( \sqrt { 2 } , \frac { 1 } { 2 d } ) } \end{array}$ as $d \to + \infty$ . By the Chebischev inequality, for every $t > 0$
+
+$$
+P ( | \alpha - \sqrt { 2 } | \geq t ) \leq \frac { 1 } { 2 d t ^ { 2 } } .
+$$
+
+Choosing $t = - \delta + \sqrt { 2 }$ for $\delta < \sqrt { 2 }$ and using the symmetry of the Gaussian around the expectation we obtain
+
+$$
+\begin{array} { c } { { \displaystyle \frac 1 { 2 d ( \sqrt 2 - \delta ) ^ { 2 } } \geq P ( | \alpha - \sqrt 2 | \geq - \delta + \sqrt 2 ) } } \\ { { { } } } \\ { { { } = 2 P ( \alpha \leq \sqrt 2 + \delta - \sqrt 2 ) } } \\ { { { } } } \\ { { { } = 2 \left( 1 - P ( \alpha \geq \delta ) \right) . } } \end{array}
+$$
+
+Hence
+
+$$
+P ( \alpha \ge \delta ) \ge 1 - { \frac { 1 } { 4 d ( \sqrt { 2 } - \delta ) ^ { 2 } } } .
+$$

parse/train/SJcKhk-Ab/SJcKhk-Ab_content_list.json

@@ -0,0 +1,1703 @@
+[
+    {
+        "type": "text",
+        "text": "CAN RECURRENT NEURAL NETWORKS WARP TIME?",
+        "text_level": 1,
+        "bbox": [
+            173,
+            102,
+            790,
+            121
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Corentin Tallec ",
+        "text_level": 1,
+        "bbox": [
+            184,
+            145,
+            294,
+            159
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yann Ollivier ",
+        "text_level": 1,
+        "bbox": [
+            516,
+            145,
+            611,
+            159
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Laboratoire de Recherche en Informatique Université Paris Sud Gif-sur-Yvette, 91190, France corentin.tallec@u-psud.fr ",
+        "bbox": [
+            184,
+            160,
+            462,
+            214
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Facebook Articial Intelligence Research   \nParis, France   \nyol@fb.com ",
+        "bbox": [
+            516,
+            160,
+            779,
+            200
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            251,
+            544,
+            267
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Successful recurrent models such as long short-term memories (LSTMs) and gated recurrent units (GRUs) use ad hoc gating mechanisms. Empirically these models have been found to improve the learning of medium to long term temporal dependencies and to help with vanishing gradient issues. ",
+        "bbox": [
+            233,
+            285,
+            764,
+            342
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We prove that learnable gates in a recurrent model formally provide quasiinvariance to general time transformations in the input data. We recover part of the LSTM architecture from a simple axiomatic approach. ",
+        "bbox": [
+            232,
+            345,
+            764,
+            387
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "This result leads to a new way of initializing gate biases in LSTMs and GRUs. Experimentally, this new chrono initialization is shown to greatly improve learning of long term dependencies, with minimal implementation effort. ",
+        "bbox": [
+            232,
+            391,
+            766,
+            433
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recurrent neural networks (e.g. (Jaeger, 2002)) are a standard machine learning tool to model and represent temporal data; mathematically they amount to learning the parameters of a parameterized dynamical system so that its behavior optimizes some criterion, such as the prediction of the next data in a sequence. ",
+        "bbox": [
+            176,
+            455,
+            823,
+            510
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Handling long term dependencies in temporal data has been a classical issue in the learning of recurrent networks. Indeed, stability of a dynamical system comes at the price of exponential decay of the gradient signals used for learning, a dilemma known as the vanishing gradient problem (Pascanu et al., 2012; Hochreiter, 1991; Bengio et al., 1994). This has led to the introduction of recurrent models specifically engineered to help with such phenomena. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Use of feedback connections (Hochreiter & Schmidhuber, 1997) and control of feedback weights through gating mechanisms (Gers et al., 1999) partly alleviate the vanishing gradient problem. The resulting architectures, namely long short-term memories (LSTMs (Hochreiter & Schmidhuber, 1997; Gers et al., 1999)) and gated recurrent units (GRUs (Chung et al., 2014)) have become a standard for treating sequential data. ",
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+        "type": "text",
+        "text": "Using orthogonal weight matrices is another proposed solution to the vanishing gradient problem, thoroughly studied in (Saxe et al., 2013; Le et al., 2015; Arjovsky et al., 2016; Wisdom et al., 2016; Henaff et al., 2016). This comes with either computational overhead, or limitation in representational power. Furthermore, restricting the weight matrices to the set of orthogonal matrices makes forgetting of useless information difficult. ",
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+        "type": "text",
+        "text": "The contribution of this paper is threefold: ",
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+        "type": "text",
+        "text": "∙ We show that postulating invariance to time transformations in the data (taking invariance to time warping as an axiom) necessarily leads to a gate-like mechanism in recurrent models (Section 1). This provides a clean derivation of part of the popular LSTM and GRU architectures from first principles. In this framework, gate values appear as time contraction or time dilation coefficients, similar in spirit to the notion of time constant introduced in (Mozer, 1992). ∙ From these insights, we provide precise prescriptions on how to initialize gate biases (Section 2) depending on the range of time dependencies to be captured. It has previously been advocated that setting the bias of the forget gate of LSTMs to 1 or 2 provides overall good performance (Gers & Schmidhuber, 2000; Jozefowicz et al., 2015). The viewpoint here explains why this is reasonable in most cases, when facing medium term dependencies, but fails when facing long to very long term dependencies. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "∙ We test the empirical benefits of the new initialization on both synthetic and real world data (Section 3). We observe substantial improvement with long-term dependencies, and slight gains or no change when short-term dependencies dominate. ",
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+    {
+        "type": "text",
+        "text": "1 FROM TIME WARPING INVARIANCE TO GATING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "When tackling sequential learning problems, being resilient to a change in time scale is crucial. Lack of resilience to time rescaling implies that we can make a problem arbitrarily difficult simply by changing the unit of measurement of time. Ordinary recurrent neural networks are highly nonresilient to time rescaling: a task can be rendered impossible for an ordinary recurrent neural network to learn, simply by inserting a fixed, small number of zeros or whitespaces between all elements of the input sequence. An explanation is that, with a given number of recurrent units, the class of functions representable by an ordinary recurrent network is not invariant to time rescaling. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Ideally, one would like a recurrent model to be able to learn from time-warped input data $x ( c ( t ) )$ as easily as it learns from data $x ( t )$ , at least if the time warping $c ( t )$ is not overly complex. The change of time $c$ may represent not only time rescalings, but, for instance, accelerations or decelerations of the phenomena in the input data. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We call a class of models invariant to time warping, if for any model in the class with input data $x ( t )$ , and for any time warping $c ( t )$ , there is another (or the same) model in the class that behaves on data $x ( c ( t ) )$ in the same way the original model behaves on $x ( t )$ . (In practice, this will only be possible if the warping $c$ is not too complex.) We will show that this is deeply linked to having gating mechanisms in the model. ",
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+    {
+        "type": "text",
+        "text": "Invariance to time rescaling ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Let us first discuss the simpler case of a linear time rescaling. Formally, this is a linear transformation of time, that is ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "equation",
+        "img_path": "images/8220d05219a09914df46d3b4371bed17b41164db0828154e0a546ddd10704e85.jpg",
+        "text": "$$\n\\begin{array} { r c c c c } { c } & { : } & { \\mathbb { R } _ { + } } & { \\longrightarrow } & { \\mathbb { R } _ { + } } \\\\ & & { \\ t } & { \\longmapsto } & { \\alpha t } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "with $\\alpha > 0$ . For instance, receiving a new input character every 10 time steps only, would correspond to $\\alpha = 0 . 1$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Studying time transformations is easier in the continuous-time setting. The discrete time equation of a basic recurrent network with hidden state $h _ { t }$ , ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "equation",
+        "img_path": "images/49ddb52e7711425d138b4ff1720cdb20a98714ce320e7997ada45090dd14d6de.jpg",
+        "text": "$$\nh _ { t + 1 } = \\operatorname { t a n h } \\left( W _ { x } x _ { t } + W _ { h } h _ { t } + b \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "can be seen as a time-discretized version of the continuous-time equation1 ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f2e8636e96edb0836e90cf6bd241b9486c343d36ced6aea5afef2f3b92b87cd4.jpg",
+        "text": "$$\n\\frac { \\mathrm { d } h ( t ) } { \\mathrm { d } t } = \\operatorname { t a n h } { \\left( W _ { x } x ( t ) + W _ { h } h ( t ) + b \\right) } - h ( t )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "namely, (2) is the Taylor expansion $\\begin{array} { r } { h ( t + \\delta t ) \\approx h ( t ) + \\delta t \\frac { \\mathrm { d } h ( t ) } { \\mathrm { d } t } } \\end{array}$ ?? dℎ(??)d?? with discretization step ???? = 1. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Now imagine that we want to describe time-rescaled data $x ( \\alpha t )$ with a model from the same class. Substituting $t \\gets c ( t ) = \\alpha t$ , $x ( t ) \\gets x ( \\alpha t )$ and $h ( t ) \\gets h ( \\alpha t )$ and rewriting (3) in terms of the new variables, the time-rescaled model satisfies2 ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/379eedb4f5a8429ab3a8e804974dfe035ef8abcb335b701e4179e9d4aa44614d.jpg",
+        "text": "$$\n\\frac { \\mathrm { d } h ( t ) } { \\mathrm { d } t } = \\alpha \\operatorname { t a n h } { \\left( W _ { x } x ( t ) + W _ { h } h ( t ) + b \\right) } - \\alpha h ( t ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "However, when translated back to a discrete-time model, this no longer describes an ordinary RNN but a leaky RNN (Jaeger, 2002, $\\ S 8 . 1 \\AA$ ). Indeed, taking the Taylor expansion of $h ( t + \\delta t )$ with $\\delta t = 1$ in (4) yields the recurrent model ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/52d572b082dc47e1c8a2ff5fbccace2c8d5c4551cfdb1efd559597b85597dff6.jpg",
+        "text": "$$\nh _ { t + 1 } = \\alpha \\operatorname { t a n h } { \\left( W _ { x } x _ { t } + W _ { h } h _ { t } + b \\right) } + ( 1 - \\alpha ) h _ { t }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Thus, a straightforward way to ensure that a class of (continuous-time) models is able to represent input data $x ( \\alpha t )$ in the same way that it can represent input data $x ( t )$ , is to take a leaky model in which $\\alpha > 0$ is a learnable parameter, corresponding to the coefficient of the time rescaling. Namely, the class of ordinary recurrent networks is not invariant to time rescaling, while the class of leaky RNNs (5) is. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Learning $\\alpha$ amounts to learning the global characteristic timescale of the problem at hand. More precisely, $1 / \\alpha$ ought to be interpreted as the characteristic forgetting time of the neural network.3 ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Invariance to time warpings ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In all generality, we would like recurrent networks to be resilient not only to time rescaling, but to all sorts of time transformations of the inputs, such as variable accelerations or decelerations. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "An eligible time transformation, or time warping, is any increasing differentiable function $c$ from $\\mathbb { R } _ { + }$ to $\\mathbb { R } _ { + }$ . This amounts to facing input data $x ( c ( t ) )$ instead of $x ( t )$ . Applying a time warping $t \\gets c ( t )$ to the model and data in equation (3) and reasoning as above yields ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4109234ccec89906fb0cf557a55ab8e19e0a3af3b61f0df59aa7f1db3b911aee.jpg",
+        "text": "$$\n\\frac { \\mathrm { d } h ( t ) } { \\mathrm { d } t } = \\frac { \\mathrm { d } c ( t ) } { \\mathrm { d } t } \\operatorname { t a n h } { \\left( W _ { x } x ( t ) + W _ { h } h ( t ) + b \\right) } - \\frac { \\mathrm { d } c ( t ) } { \\mathrm { d } t } h ( t ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Ideally, one would like a model to be able to learn from input data $x ( c ( t ) )$ as easily as it learns from data $x ( t )$ , at least if the time warping $c ( t )$ is not overly complex. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To be invariant to time warpings, a class of (continuous-time) models has to be able to represent Equation (6) for any time warping $c ( t )$ . Moreover, the time warping is unknown a priori, so would have to be learned. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Ordinary recurrent networks do not constitute a model class that is invariant to time rescalings, as seen above. A fortiori, this model class is not invariant to time warpings either. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "For time warping invariance, one has to introduce a learnable function $g$ that will represent the derivative4 of the time warping, d??(??)d?? in (6). For instance ?? may be a recurrent neural network taking the $x$ ’s as input.5 Thus we get a class of recurrent networks defined by the equation ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3b2238bebe207baea7f7062ed3a57e50f025117fd639181619d564b5db6911d5.jpg",
+        "text": "$$\n\\frac { \\mathrm { d } h ( t ) } { \\mathrm { d } t } = g ( t ) \\operatorname { t a n h } { \\left( W _ { x } x ( t ) + W _ { h } h ( t ) + b \\right) } - g ( t ) h ( t )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $g$ belongs to a large class (universal approximator) of functions of the inputs. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The class of recurrent models (7) is quasi-invariant to time warpings. The quality of the invariance will depend on the learning power of the learnable function $g$ : a function $g$ that can represent any function of the data would define a class of recurrent models that is perfectly invariant to time warpings; however, a specific model for $g$ (e.g., neural networks of a given size) can only represent a specific, albeit large, class of time warpings, and so will only provide quasi-invariance. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Heuristically, $g ( t )$ acts as a time-dependent version of the fixed $\\alpha$ in (4). Just like $1 / \\alpha$ above, $1 / g ( t _ { 0 } )$ represents the local forgetting time of the network at time $t _ { 0 }$ : the network will effectively retain information about the inputs at $t _ { 0 }$ for a duration of the order of magnitude of $1 / g ( t _ { 0 } )$ (assuming $g ( t )$ does not change too much around $t _ { 0 }$ ). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Let us translate back this equation to the more computationally realistic case of discrete time, using a Taylor expansion with step size $\\delta t = 1$ , so that $\\begin{array} { r } { \\frac { \\mathrm { d } \\bar { h } ( t ) } { \\mathrm { d } t } = \\cdots } \\end{array}$ becomes $h _ { t + 1 } = h _ { t } + \\cdots .$ Then the model (7) becomes ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a07a6bfd364068157f4c05b4c6052755d6df77a7b27f9fde44fba4c840babdf7.jpg",
+        "text": "$$\nh _ { t + 1 } = g _ { t } \\operatorname { t a n h } \\left( W _ { x } x _ { t } + W _ { h } h _ { t } + b \\right) + \\left( 1 - g _ { t } \\right) h _ { t } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $g _ { t }$ itself is a function of the inputs. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This model is the simplest extension of the RNN model that provides invariance to time warpings.6 It is a basic gated recurrent network, with input gating $g _ { t }$ and forget gating $( 1 - g _ { t } )$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Here $g _ { t }$ has to be able to learn an arbitrary function of the past inputs $x$ ; for instance, take for $g _ { t }$ the output of a recurrent network with hidden state $h ^ { g }$ : ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b734e7ed9a1c840c8af3e68c59d8bde4ca46ee30f22847ff9e8ad0fd09c90f58.jpg",
+        "text": "$$\ng _ { t } = \\sigma ( W _ { g x } x _ { t } + W _ { g h } h _ { t } ^ { g } + b _ { g } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            196,
+            606,
+            214
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "with sigmoid activation function $\\sigma$ (more on the choice of sigmoid below). Current architectures just reuse for $h ^ { g }$ the states $h$ of the main network (or, equivalently, relabel $h \\gets ( h , h ^ { g } )$ to be the union of both recurrent networks and do not make the distinction). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The model (8) provides invariance to global time warpings, making all units face the same dilation/contraction of time. One might, instead, endow every unit $i$ with its own local contraction/dilation function $g ^ { i }$ . This offers more flexibility (gates have been introduced for several reasons beyond time warpings (Hochreiter, 1991)), especially if several unknown timescales coexist in the signal: for instance, in a multilayer model, each layer may have its own characteristic timescales corresponding to different levels of abstraction from the signal. This yields a model ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6dba5e0215b5563daa60d8ec6c8e77521ebfafbefb864557f62d1d541e24549e.jpg",
+        "text": "$$\nh _ { t + 1 } ^ { i } = g _ { t } ^ { i } \\operatorname { t a n h } { \\left( W _ { x } ^ { i } x _ { t } + W _ { h } ^ { i } h _ { t } + b ^ { i } \\right) } + \\left( 1 - g _ { t } ^ { i } \\right) h _ { t } ^ { i }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            320,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "with $h ^ { i }$ and $( W _ { x } ^ { i } , W _ { h } ^ { i } , b ^ { i } )$ being respectively the activation and the incoming parameters of unit $i$ , and with each $g ^ { i }$ a function of both inputs and units. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Equation 10 defines a simple form of gated recurrent network, that closely resembles the evolution equation of cell units in LSTMs, and of hidden units in GRUs. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In (10), the forget gate is tied to the input gate $( g _ { t } ^ { i }$ and $1 - g _ { t } ^ { i } )$ . Such a setting has been successfully used before (e.g. (Lample et al., 2016)) and saves some parameters, but we are not aware of systematic comparisons. Below, we initialize LSTMs this way but do not enforce the constraint throughout training. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Continuous time versus discrete time ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Of course, the analogy between continuous and discrete time breaks down if the Taylor expansion is not valid. The Taylor expansion is valid when the derivative of the time warping is not too large, say, when $\\alpha \\lesssim 1$ or $g _ { t } \\lesssim 1$ (then (8) and (7) are close). Intuitively, for continuous-time data, the physical time increment corresponding to each time step $t  t + 1$ of the discrete-time recurrent model should be smaller than the speed at which the data changes, otherwise the situation is hopeless. So discrete-time gated models are invariant to time warpings that stretch time (such as interspersing the data with blanks or having long-term dependencies), but obviously not to those that make things happen too fast for the model. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Besides, since time warpings are monotonous, we hav e d??(??)d?? > 0, i.e., ???? > 0. The two constraints $g _ { t } > 0$ and $g _ { t } < 1$ square nicely with the use of a sigmoid for the gate function $g$ . ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2 TIME WARPINGS AND GATE INITIALIZATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "If we happen to know that the sequential data we are facing have temporal dependencies in an approximate range $[ T _ { \\mathrm { m i n } } , T _ { \\mathrm { m a x } } ]$ , it seems reasonable to use a model with memory (forgetting time) lying approximately in the same temporal range. As mentioned in Section 1, this amounts to having values of $g$ in the range $\\left[ \\frac { 1 } { T _ { \\mathrm { m a x } } } , \\frac { 1 } { T _ { \\mathrm { m i n } } } \\right] ^ { }$ ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The biases $b _ { g }$ of the gates $g$ greatly impact the order of magnitude of the values of $g ( t )$ over time. If the values of both inputs and hidden layers are centered over time, $g ( t )$ will typically take values centered around $\\sigma ( b _ { g } )$ . Values of $\\sigma ( b _ { g } )$ in the desired range $\\left[ \\frac { 1 } { T _ { \\mathrm { m a x } } } , \\frac { 1 } { T _ { \\mathrm { m i n } } } \\right]$ are obtained by choosing the biases $b _ { g }$ between $- \\log ( T _ { \\mathrm { m a x } } - 1 )$ and $- \\log ( T _ { \\mathrm { m i n } } - 1 )$ . This is a loose prescription: we only want to control the order of magnitude of the memory range of the neural networks. Furthermore, we don’t want to bound $g ( t )$ too tightly to some value forever: if rare events occur, abruplty changing the time scale can be useful. Therefore we suggest to use these values as initial values only. ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This suggests a practical initialization for the bias of the gates of recurrent networks such as (10): when characteristic timescales of the sequential data at hand are expected to lie between $T _ { \\mathrm { m i n } }$ and $T _ { \\mathrm { m a x } }$ , initialize the biases of $g$ a $\\mathrm { ~ : ~ } \\log ( \\bar { \\mathcal { U } } ( [ T _ { \\operatorname* { m i n } } , T _ { \\operatorname* { m a x } } ] ) - 1 )$ where $\\mathcal { U }$ is the uniform distribution7. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For LSTMs, using a variant of (Graves et al., 2013): ",
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+    {
+        "type": "equation",
+        "img_path": "images/07048735bfcd11b70b98f0f80befdeaabeb5b49441b0cbb7adcdd9e3ab6a6731.jpg",
+        "text": "$$\n\\begin{array} { r l } & { i _ { t } = \\sigma ( W _ { x i } x _ { t } + W _ { h i } h _ { t - 1 } + b _ { i } ) } \\\\ & { f _ { t } = \\sigma ( W _ { x f } x _ { t } + W _ { h f } h _ { t - 1 } + b _ { f } ) } \\\\ & { c _ { t } = f _ { t } c _ { t - 1 } + i _ { t } \\operatorname { t a n h } ( W _ { x c } x _ { t } + W _ { h c } h _ { t - 1 } + b _ { c } ) } \\\\ & { o _ { t } = \\sigma ( W _ { x o } x _ { t } + W _ { h o } h _ { t - 1 } + b _ { o } ) } \\\\ & { h _ { t } = o _ { t } \\operatorname { t a n h } ( c _ { t } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "the correspondence between between the gates in (10) and those in (13) is as follows: $1 - g _ { t }$ corresponds to $f _ { t }$ , and $g _ { t }$ to $i _ { t }$ . To obtain a time range around $T$ for unit $i$ , we must both ensure that $f _ { t } ^ { i }$ lies around $1 - 1 / T$ , and that $i _ { t }$ lies around $1 / \\bar { T }$ . When facing time dependencies with largest time range $T _ { \\mathrm { m a x } }$ , this suggests to initialize LSTM gate biases to ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/a138f71246f19696cb1a6d9d7822d56ee7fe9c0d8d2e20ae51859c90a55d277f.jpg",
+        "text": "$$\n\\begin{array} { r } { b _ { f } \\sim \\log ( \\mathcal { U } ( [ 1 , T _ { \\operatorname* { m a x } } - 1 ] ) ) } \\\\ { b _ { i } = - b _ { f } } \\end{array}\n$$",
+        "text_format": "latex",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "with $\\mathcal { U }$ the uniform distribution and $T _ { \\mathrm { m a x } }$ the expected range of long-term dependencies to be captured. ",
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+    {
+        "type": "text",
+        "text": "Hereafter, we refer to this as the chrono initialization. ",
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+    {
+        "type": "text",
+        "text": "3 EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "First, we test the theoretical arguments by explicitly introducing random time warpings in some data, and comparing the robustness of gated and ungated architectures. ",
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+    {
+        "type": "text",
+        "text": "Next, the chrono LSTM initialization is tested against the standard initialization on a variety of both synthetic and real world problems. It heavily outperforms standard LSTM initialization on all synthetic tasks, and outperforms or competes with it on real world problems. ",
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+    {
+        "type": "text",
+        "text": "The synthetic tasks are taken from previous test suites for RNNs, specifically designed to test the efficiency of learning when faced with long term dependencies (Hochreiter & Schmidhuber, 1997; Le et al., 2015; Graves et al., 2014; Martens & Sutskever, 2011; Arjovsky et al., 2016). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In addition (Appendix A), we test the chrono initialization on next character prediction on the Text8 (Mahoney, 2011) dataset, and on next word prediction on the Penn Treebank dataset (Mikolov et al., 2012). Single layer LSTMs with various layer sizes are used for all experiments, except for the word level prediction, where we use the best model from (Zilly et al., 2016), a 10 layer deep recurrent highway network (RHN). ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Pure warpings and paddings. To test the theoretical relationship between gating and robustness to time warpings, various recurrent architectures are compared on a task where the only challenge comes from warping. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The unwarped task is simple: remember the previous character of a random sequence of characters. Without time warping or padding, this is an extremely easy task and all recurrent architectures are successful. The only difficulty will come from warping; this way, we explicitly test the robustness of various architectures to time warping and nothing else. ",
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+    {
+        "type": "image",
+        "img_path": "images/6c46e8ef8c79df872b5e3f55469fc5df0264e2d77e62f944d2edfe0ecee3e4aa.jpg",
+        "image_caption": [
+            "Figure 1: Performance of different recurrent architectures on warped and padded sequences sequences. From top left to bottom right: uniform time warping of length maximum_warping, uniform padding of length maximum_warping, variable time warping and variable time padding, from 1 to maximum_warping. (For uniform padding/warpings, the leaky RNN and gated RNN curves overlap, with loss 0.) Lower is better. ",
+            "Figure 2: A task involving pure warping. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "Unwarped task example: ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "Input: All human beings are born free and equal Output: All human beings are born free and equa Uniform warping example (warping $\\times 4 _ { , }$ ): Input: AAAAllllllll hhhhuuuummmmaaaannnn Output: AAAAllllllll hhhhuuuummmmaaaa Variable warping example (random warping $\\times 1 \\mathrm { - } \\times 4 )$ ): Input: Allllll hhhummmmaannn bbbbeeiiingssss Output: AAAlllll huuuummaaan bbeeeingggg ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Uniformly time-warped tasks are produced by repeating each character maximum_warping times both in the input and output sequence, for some fixed number maximum_warping. ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "Variably time-warped tasks are produced similarly, but each character is repeated a random number of times uniformly drawn between 1 and maximum_warping. The same warping is used for the input and output sequence (so that the desired output is indeed a function of the input). This exactly corresponds to transforming input $x ( t )$ into $x ( c ( t ) )$ with $c$ a random, piecewise affine time warping. Fig. 2 gives an illustration. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For each value of maximum_warping, the train dataset consists of 50, 000 length-500 randomly warped random sequences, with either uniform or variable time warpings. The alphabet is of size 10 (including a dummy symbol). Contiguous characters are enforced to be different. After warping, each sequence is truncated to length 500. Test datasets of 10, 000 sequences are generated similarily. The criterion to be minimized is the cross entropy in predicting the next character of the output sequence. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/bd4adb4baf30faee6b6bf7d3dbb5f7a7b738cab56b48d4587698d03c65a35b90.jpg",
+        "image_caption": [
+            "Figure 3: Standard initialization (blue) vs. chrono initialization (red) on the copy and variable copy task. From left to right, top to bottom, standard copy $T \\ = \\ 5 0 0$ and $T \\ : = \\ : 2 0 0 0$ , variable copy $T = 5 0 0$ and $T = 1 0 0 0$ . Chrono initialization heavily outperforms standard initialization, except for variable length copy with the smaller $T$ where both perform well. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/3e9e06a6aa7d4d6793068804cfa3dbf2e4cb33f7e5fab941e86d271e77ce89a6.jpg",
+        "image_caption": [
+            "Figure 4: Standard initialization (blue) vs. chrono initialization (red) on the adding task. From left to right, $T = 2 0 0$ , and $T = 7 5 0$ . Chrono initialization heavily outperforms standard initialization. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Note that each sample in the dataset uses a new random sequence from a fixed alphabet, and (for variable warpings) a new random warping. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "A similar, slightly more difficult task uses padded sequences instead of warped sequences, obtained by padding each element in the input sequence with a fixed or variable number of 0’s (in continuoustime, this amounts to a time warping of a continuous-time input sequence that is nonzero at certain points in time only). Each time the input is nonzero, the network has to output the previous nonzero character seen. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We compare three recurrent architectures: RNNs (Eq. (2), a simple, ungated recurrent network), leaky RNNs (Eq. (5), where each unit has a constant learnable “gate” between 0 and 1) and gated RNNs, with one gate per unit, described by (10). All networks contain 64 recurrent units. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The point of using gated RNNs (10) (“LSTM-lite” with tied input and forget gates), rather than full LSTMs, is to explicitly test the relevance of the arguments in Section 1 for time warpings. Indeed these LSTM-lite already exhibit perfect robustness to warpings in these tasks. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "RMSprop with an $\\alpha$ parameter of 0.9 and a batch size of 32 is used. For faster convergence, learning rates are divided by 2 each time the evaluation loss has not decreased after 100 batches. All architectures are trained for 3 full passes through the dataset, and their evaluation losses are compared. Each setup is run 5 times, and mean, maximum and minimum results among the five trials are reported. Results on the test set are summarized in Fig. 1. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Gated architectures significantly outperform RNNs as soon as moderate warping coefficients are involved. As expected from theory, leaky RNNs perfectly solve uniform time warpings, but fail to achieve optimal behavior with variable warpings, to which they are not invariant. Gated RNNs, which are quasi invariant to general time warpings, achieve perfect performance in both setups for all values of maximum_warping. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Synthetic tasks. For synthetic tasks, optimization is performed using RMSprop (Tieleman & Hinton, 2012) with a learning rate of $1 0 ^ { - 3 }$ and a moving average parameter of 0.9. No gradient clipping is performed; this results in a few short-lived spikes in the plots below, which do not affect final performance. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "COPY TASKS. The copy task checks whether a model is able to remember information for arbitrarily long durations. We use the setup from (Hochreiter & Schmidhuber, 1997; Arjovsky et al., 2016), which we summarize here. Consider an alphabet of 10 characters. The ninth character is a dummy character and the tenth character is a signal character. For a given $T$ , input sequences consist of $T + 2 0$ characters. The first 10 characters are drawn uniformly randomly from the first 8 letters of the alphabet. These first characters are followed by $T - 1$ dummy characters, a signal character, whose aim is to signal the network that it has to provide its outputs, and the last 10 characters are dummy characters. The target sequence consists of $T + 1 0$ dummy characters, followed by the first 10 characters of the input. This dataset is thus about remembering an input sequence for exactly $T$ timesteps. We also provide results for the variable copy task setup presented in (Henaff et al., 2016), where the number of characters between the end of the sequence to copy and the signal character is drawn at random between 1 and $T$ . ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The best that a memoryless model can do on the copy task is to predict at random from among possible characters, yielding a loss of $\\frac { 1 0 \\log ( 8 ) } { T + 2 0 }$ (Arjovsky et al., 2016). ",
+        "bbox": [
+            176,
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "On those tasks we use LSTMs with 128 units. For the standard initialization (baseline), the forget gate biases are set to 1. For the new initialization, the forget gate and input gate biases are chosen according to the chrono initialization (16), with $\\begin{array} { r } { T _ { \\mathrm { m a x } } = \\frac { 3 \\breve { T } } { 2 } } \\end{array}$ for the copy task, thus a bit larger than input length, and $T _ { \\mathrm { m a x } } = T$ for the variable copy task. The results are provided in Figure 3. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Importantly, our LSTM baseline (with standard initialization) already performs better than the LSTM baseline of (Arjovsky et al., 2016), which did not outperform random prediction. This is presumably due to slightly larger network size, increased training time, and our using the bias initialization from (Gers & Schmidhuber, 2000). ",
+        "bbox": [
+            174,
+            667,
+            823,
+            723
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "On the copy task, for all the selected $T$ ’s, chrono initialization largely outperforms the standard initialization. Notably, it does not plateau at the memoryless optimum. On the variable copy task, chrono initialization is even with standard initialization for $T = 5 0 0$ , but largely outperforms it for $T = 1 0 0 0$ . ",
+        "bbox": [
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+            729,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "ADDING TASK. The adding task also follows a setup from (Hochreiter & Schmidhuber, 1997; Arjovsky et al., 2016). Each training example consists of two input sequences of length $T$ . The first one is a sequence of numbers drawn from $\\mathcal { U } ( [ 0 , 1 ] )$ , the second is a sequence containing zeros everywhere, except for two locations, one in the first half and another in the second half of the sequence. The target is a single number, which is the sum of the numbers contained in the first sequence at the positions marked in the second sequence. ",
+        "bbox": [
+            174,
+            804,
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+            888
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The best a memoryless model can do on this task is to predict the mean of $2 \\times \\mathcal { U } ( [ 0 , 1 ] )$ , namely 1 (Arjovsky et al., 2016). Such a model reaches a mean squared error of 0.167. ",
+        "bbox": [
+            174,
+            895,
+            820,
+            924
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "LSTMs with 128 hidden units are used. The baseline (standard initialization) initializes the forget biases to 1. The chrono initialization uses $T _ { \\mathrm { m a x } } = T$ . Results are provided in Figure 4. For all $T$ ’s, chrono initialization significantly speeds up learning. Notably it converges 7 times faster for $T = 7 5 0$ . ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            180,
+            289,
+            196
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The self loop feedback gating mechanism of recurrent networks has been derived from first principles via a postulate of invariance to time warpings. Gated connections appear to regulate the local time constants in recurrent models. With this in mind, the chrono initialization, a principled way of initializing gate biases in LSTMs, has been introduced. Experimentally, chrono initialization is shown to bring notable benefits when facing long term dependencies. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+    {
+        "type": "text",
+        "text": "A ADDITIONAL EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            442,
+            117
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/3ac950df571f739848b4bae6e2688b7ee5dbe948de1a1a51a09c8071e469f0ea.jpg",
+        "image_caption": [
+            "Figure 5: Generalization performances of different recurrent architectures on the warping problem. Networks are trained with uniform warps between 1 and 50 and evaluated on uniform warps between 100 and a variable maximum warp. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            359,
+            140,
+            633,
+            313
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/673116037ba3f32ce6d33c0e571cdf60028c2d523081bfe42a5e67ad7544ebb5.jpg",
+        "image_caption": [
+            "Figure 6: Standard initialization (blue) vs. chrono initialization (red) on pixel level classification tasks. From left to right, MNIST and pMNIST. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            391,
+            772,
+            563
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/174f6dead2e3ba16ae4b102d9c82860c795865024e865927602475d9670919ea.jpg",
+        "image_caption": [
+            "Figure 7: Standard initialization (blue) vs. chrono initialization (red) on the word level PTB (left) and on the character level text8 (right) validation sets. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            627,
+            771,
+            799
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "On the generalization capacity of recurrent architectures. We proceeded to test the generalization properties of RNNs, leaky RNNs and chrono RNNs on the pure warping experiments presented in Section 3. For each of the architectures, a recurrent network with 64 recurrent units is trained for 3 epochs on a variable warping task with warps between 1 and 50. Each network is then tested on warped sequences, with warps between 100 and an increasingly big maximum warping. Results are summarized in Figure 5. ",
+        "bbox": [
+            174,
+            868,
+            825,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "All networks display reasonably good, but not perfect, generalization. Even with warps 10 times longer than the training set warps, the networks still have decent accuracy, decreasing from $1 0 0 \\%$ to around $7 5 \\%$ . ",
+        "bbox": [
+            176,
+            138,
+            823,
+            180
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Interestingly, plain RNNs and gated RNNs display a different pattern: overall, gated RNNs perform better but their generalization performance decreases faster with warps eight to ten times longer than those seen during training, while plain RNN never have perfect accuracy, below $8 0 \\%$ even within the training set range, but have a flatter performance when going beyond the training set warp range. ",
+        "bbox": [
+            174,
+            188,
+            825,
+            244
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Pixel level classification: MNIST and pMNIST. This task, introduced in (Le et al., 2015), consists in classifying images using a recurrent model. The model is fed pixels one by one, from top to bottom, left to right, and has to output a probability distribution for the class of the object in the image. ",
+        "bbox": [
+            174,
+            258,
+            823,
+            315
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We evaluate standard and chrono initialization on two image datasets: MNIST (LeCun et al., 1999) and permuted MNIST, that is, MNIST where all images have undergone the same pixel permutation. ",
+        "bbox": [
+            174,
+            321,
+            823,
+            351
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "LSTMs with 512 hidden units are used. Once again, standard initialization sets forget biases to 1, and the chrono initialization parameter is set to the length of the input sequences, $T _ { \\mathrm { m a x } } = 7 8 4$ . Results on the validation set are provided in Figure 6. On non-permuted MNIST, there is no clear difference, even though the best validation error is obtained with chrono initialization. On permuted MNIST, chrono initialization performs better, with a best validation result of $9 6 . 3 \\%$ , while standard initialization obtains a best validation result of $9 5 . 4 \\%$ . ",
+        "bbox": [
+            174,
+            357,
+            825,
+            440
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Next character prediction on text8. Chrono initialization is benchmarked against standard initialization on the character level text8 dataset (Mahoney, 2011). Text8 is a 100M character formatted text sample from Wikipedia. (Mikolov et al., 2012)’s train-valid-test split is used: the first 90M characters are used as training set, the next 5M as validation set and the last 5M as test set. ",
+        "bbox": [
+            174,
+            457,
+            823,
+            512
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The exact same setup as in (Cooijmans et al., 2016) is used, with the code directly taken from there. Namely: LSTMs with 2000 units, trained with Adam (Kingma & Ba, 2014) with learning rate $1 0 ^ { - 3 }$ , batches of size 128 made of non-overlapping sequences of length 180, and gradient clipping at 1.0. Weights are orthogonally initialized, and recurrent batch normalization (Cooijmans et al., 2016) is used. ",
+        "bbox": [
+            174,
+            518,
+            823,
+            588
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Chrono initialization with $T _ { \\mathrm { m a x } } = 8$ is compared to standard $b _ { f } = 1$ initialization. Results are presented in Figure 7. On the validation set, chrono initialization uniformly outperforms standard initialization by a small margin. On the test set, the compression rate is 1.37 with chrono initialization, versus 1.38 for standard initialization.8 This same slight difference is observed on two independent runs. ",
+        "bbox": [
+            174,
+            595,
+            825,
+            665
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Our guess is that, on next character prediction, with moderately sized networks, short term dependencies dominate, making the difference between standard and chrono initialization relatively small. ",
+        "bbox": [
+            176,
+            672,
+            823,
+            700
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Next word prediction on Penn Treebank. To attest for the resilience of chrono initialization to more complex models than simple LSTMs, we train on word level Penn Treebank (Mikolov et al., 2012) using the best deep RHN network from (Zilly et al., 2016). All hyperparameters are taken from of (Zilly et al., 2016). For the chrono bias initialization, a single bias vector $b$ is sampled according to $b \\sim \\log ( \\mathcal { U } ( 1 , T _ { \\operatorname* { m a x } } ) )$ , the carry gate bias vectors of all layers are initialized to $- b$ , and the transform gate biases to $b$ . $T _ { \\mathrm { m a x } }$ is chosen to be 11 (because this gives an average bias initialization close to the value 2 from (Zilly et al., 2016)).9. Without further hyperparameter search and with a single run, we obtain test results similar to (Zilly et al., 2016), with a test perplexity of 6.54. ",
+        "bbox": [
+            174,
+            715,
+            825,
+            840
+        ],
+        "page_idx": 12
+    }
+]

parse/train/lmTWnm3coJJ/lmTWnm3coJJ.md

@@ -0,0 +1,394 @@
+# ROBUST CURRICULUM LEARNING: FROM CLEAN LABEL DETECTION TO NOISY LABEL SELF-CORRECTION
+
+Tianyi $\mathbf { Z } \mathbf { h } \mathbf { o } \mathbf { u } ^ { * }$ , Shengjie Wang∗, Jeff A. Bilmes University of Washington, Seattle {tianyizh,wangsj,bilmes}@uw.edu
+
+# ABSTRACT
+
+Neural network training can easily overfit noisy labels resulting in poor generalization performance. Existing methods address this problem by (1) filtering out the noisy data and only using the clean data for training or (2) relabeling the noisy data by the model during training or by another model trained only on a clean dataset. However, the former does not leverage the features’ information of wrongly-labeled data, while the latter may produce wrong pseudo-labels for some data and introduce extra noises. In this paper, we propose a smooth transition and interplay between these two strategies as a curriculum that selects training samples dynamically. In particular, we start with learning from clean data and then gradually move to learn noisy-labeled data with pseudo labels produced by a time-ensemble of the model and data augmentations. Instead of using the instantaneous loss computed at the current step, our data selection is based on the dynamics of both the loss and output consistency for each sample across historical steps and different data augmentations, resulting in more precise detection of both clean labels and correct pseudo labels. On multiple benchmarks of noisy labels, we show that our curriculum learning strategy can significantly improve the test accuracy without any auxiliary model or extra clean data.
+
+# 1 INTRODUCTION
+
+The expressive power and high capacity of deep neural networks (DNNs) result in accurate modeling and promising generalization if provided with sufficient data and clean(correct) labels. However, recent studies show that the training process is fragile and can easily overfit on noisy labels (Zhang et al., 2017), which commonly appear in real-world data since precise annotation is not always available or affordable. Hence, it is important to study the training dynamics affected by imperfect labels and develop robust learning strategies that ideally eliminate the negative impact of noisy labels while fully exploiting the information from all the available data.
+
+Numerous approaches have been developed to address this challenge from various perspectives, e.g., loss correction (Xiao et al., 2015; Vahdat, 2017; Lee et al., 2018; Veit et al., 2017; Li et al., 2017b), robust loss functions (Ghosh et al., 2017; Zhang & Sabuncu, 2018; Wang et al., 2019; Ma et al., 2020) with provable noise tolerance, sample re-weighting (Patrini et al., 2017), curriculum learning (Kumar et al., 2010; Jiang et al., 2018; Guo et al., 2018), model co-teaching (Han et al., 2018), etc. A principal methodology behind a variety of methods is to detect clean labels while discard/downweigh the data with wrong labels, so the model mainly learns from correct labels. A broadly applied criterion is to select the samples with small losses and treat them as clean data. It is inspired by empirical observations that DNNs learn simple patterns first before overfitting on the noisy labels (Zhang et al., 2017; Arpit et al., 2017). Several curriculum learning methods utilize this criterion (Kumar et al., 2010; Jiang et al., 2014), and in each step, select/upweigh samples with small losses. Robust loss functions also suppress the large losses associated with the possibly wrong labels. More recent approaches use mixture models (Arazo et al., 2019) to estimate the distribution of losses for clean and noisy data.
+
+However, the instantaneous loss (i.e., the loss evaluated at the current step) of an individual sample is an unstable signal that can rapidly fluctuate due to DNN training’s randomness. The error generated by such an unstable metric accumulates when the selected samples are used to train the model producing the losses. Co-teaching methods alleviate this problem by training two DNNs and using the loss computed on one model to guild the other. Also, as the model changes during training, each sample’s loss needs to be re-evaluated even when it is not selected, which requires extra inference cost. MentorNet (Jiang et al., 2018) and Data Parameters (Saxena et al., 2019) train an extra model to produce the sample weights or selection results without computing the loss. Furthermore, it may not be efficient to repeatedly train the model only on clean data that consistently have small losses, since the model have already learned, well memorized or overfitted to them.
+
+A primary drawback of training only on clean labels detected is that discarding the whole data pairs $( x , y )$ with wrong labels $y$ removes potentially useful information about the data distribution $p ( x )$ (Arazo et al., 2019). Hence, there has been growing interest in leveraging noisy data. Loss correction methods aim to correct the predicted class probabilities based on an estimated mislabeling probability between classes. Some other methods seek to relabel them by using the model itself (e.g., bootstrapping loss (Reed et al., 2014)) or another model/mechanism (e.g., directed graphical models, conditional random fields, or CNNs) trained on an additional set of clean data, which, however, is not always available. Self-training and unsupervised learning techniques (Rasmus et al., 2015; Berthelot et al., 2019) have also been employed to generate pseudo labels to replace noisy labels (Arazo et al., 2019). The pseudo labels are optimized together with the model or generated by the model with data augmentations to encourage the output consistency on the same sample’s augmentations. Unfortunately, the pseudo labels’ quality may vary across different samples and significantly degenerate when the noise ratio is high, or the model fails to produce stable and correct predictions. In such a case, the relabeling error on some samples can be accumulated during training.
+
+In this paper, we address the aforementioned problems of noise-label learning by developing a curriculum learning strategy called Robust Curriculum Learning (RoCL) that smoothly transitions between two phases: (1) detection and supervised training on clean data; and (2) relabeling and self-supervision on noisy data. Specifically, we train the model for multiple episodes, each starting from phase(1) and gradually moving to phase(2). Unlike existing approaches, we only select samples with accurate given/pseudo labels that are most informative to the current model training. Our data selection criterion takes both the dynamics of per-sample loss and output consistency (across multiple data augmentations) into account. Using an exponential moving average of the loss and consistency over training history, it overcomes the instability of instantaneous losses and does not incur any additional inference cost. In addition, by adjusting a temperature parameter, the criterion can interpolate between the two phases and keep the training focusing on the data that the model mostly needs to improve on, e.g., clean data with unsatisfying output consistency or wrongly-labeled data with accurate pseudo labels. Thus, we can fully exploit both clean and noisy data more efficiently with less risk of introducing extra noise or error accumulation. We further show that our data selection can be derived from a novel optimization formulation for robust curriculum learning. We evaluate our method on multiple noisy learning benchmarks and show that our method outperforms a diverse set of recent noisy-label learning approaches.
+
+# 1.1 RELATED WORK
+
+Early curriculum learning (CL) (Khan et al., 2011; Basu & Christensen, 2013; Spitkovsky et al., 2009; Zhou et al., 2021) seeks an optimized sequence of training samples (i.e., a curriculum, which can be designed by human experts) to improve model performance. Self-paced learning (SPL)(Kumar et al., 2010; Tang et al., 2012a; Supancic III & Ramanan, 2013; Tang et al., 2012b) selects easy samples with smaller losses. It starts with selecting a few samples of small loss and gradually increases the selection size to cover all the training data. Self-paced curriculum learning (Jiang et al., 2015) combines the human expert in CL and loss-adaptation in SPL. SPL with diversity (SPLD) (Jiang et al., 2014) applies a negative group sparse regularization to SPL to promote the diversity of selected samples. Minimax curriculum learning Zhou & Bilmes (2018) promotes the diversity of samples during early learning to encourage exploration and focus on hard samples in later stages.
+
+In the context of robust learning with noisy labels, label correction methods aim to identify the wrong labels and possibly correct them to get more consistent labels for training. Previous work often apply an extra noise model (directed graphical model (Xiao et al., 2015), conditional random fields (Vahdat, 2017), neural network (Lee et al., 2018; Veit et al., 2017), knowledge graph (Li et al., 2017b)) to correct the noisy labels, which often require extra clean data and as well as training/inference of the noise model. Another line of research focuses on loss correction, which modifies the loss or prediction probabilities during training to correct the misinformation from the noisy labels. Patrini et al. (2017) uses two noise transition (backward and forward) matrices to correct the prediction probabilities. Label Smoothing Regularization (Szegedy et al., 2016; Pereyra et al., 2017) alleviates the overfitting to noisy labels by using soft labels instead of one-hot labels. Reed et al. (2014) augments the loss with a notion of perceptual consistency. Jiang et al. (2018) trains a mentor network to reweigh samples duri‘ng the training of a student network. Guo et al. (2018) designs a curriculum by ranking the complexity of data using its distribution density in a feature space. Ren et al. (2018) proposes a meta-learning algorithm that learns to assign weights to samples based on their gradients in training compared to those of validation data, which requires extra clean data. Co-teaching (Han et al., 2018) feeds in the network with the most confident samples of another network to reduce confirmation bias. Amid et al. (2019) generalizes the logistic loss and the exponents in the softmax by applying a temperature to each of them and makes the training more robust to noise. Hu et al. (2019) trains a network on noisy labels in the weakly supervised setting and uses it as a regularization term to improve the training on clean data.
+
+Some approaches focus on designing loss functions that have robust behaviors and provable tolerance to label noise. Ghosh et al. (2017) theoretically proves that the Mean Absolute Error(MAE) is a robust loss. The Generalized Cross Entropy (Zhang & Sabuncu, 2018) uses a negative Box-Cox transformation to obtain a loss function that generalizes MAE and Cross Entropy loss. Wang et al. (2019) proposes a Symmetric Cross Entropy that combines Cross Entropy loss and Reverse Cross Entropy loss. Ma et al. (2020) proposes a loss normalization method and shows that any loss can be made robust to noisy labels.
+
+RoCL shares similar ideas with some CL methods in that RoCL starts with learning easy and clean samples and gradually moves to hard and noisy ones. RoCL is more related to the loss correction approach in noisy-label learning literature as RoCL generates a curriculum dynamically assigning weight (probability) to each sample. RoCL differs from existing methods in: (1) it only selects a subset of informative and reliable labels for training in each epoch; (2) it is a smooth transition not only from clean data to noisy data but also from supervised learning to self-supervision; (3) it runs multiple episodes of the curriculum to avoid getting in a local minimum dominated by a small set of clean/noisy data or a specific type of loss; (4) it does not assume the availability of an extra set of clean data; (5) it does not require extra computation or any modification to the model.
+
+# 2 DYNAMIC PATTERNS OF CLEAN/NOISY LABELS IN TRAINING 2.1 LOSS DYNAMICS AND CLEAN LABEL DETECTION
+
+A key challenge for most noise-label learning methods is to design a reliable criterion to select/reweigh clean data and distinguish them from the noisy data, so all the clean data can be fully exploited while most noisy labels are filtered out of the training process. Loss computed at an instantaneous step have been widely used for this purpose according to the observation that the loss on clean data is usually smaller than noisy data. One important reason is that the clean labels are mutually consistent with each other in producing gradient updates, and therefore, the model can fit them better and faster. On the other hand, the noisy labels may contain mutually inconsistent information, creating a form of long-lasting “tug of war” amongst themselves. For example, it can be hard for the model to find consistent visual patterns from images with noisy labels to make the desired predictions. However, instantaneous loss suffers from high variance across training epochs (as shown in the first plot of Figure 1) and is inaccurate for clean data detection under high noise ratio (i.e., the proportion of wrong labels is high) and the randomness of DNN training,
+
+![](images/972989ee42c7ed8245747a4ebabe033ccdda9bbea1f681487b6eb9b558bda3ba.jpg)  
+Figure 1: Dynamic patterns (mean±std) of instantaneous metrics (top) and exponential moving average (EMA) metrics (bottom) when applying Alg. 2 that alternates between supervised learning on given labels and self-supervision on pseudo labels. Larger gap between curves in each plot is better. Symmetric noise is defined in the beginning of Section 4. We use cross entropy for supervised loss $\ell ( \cdot , \cdot )$ in Eq. (1) and 0-1 loss for $\ell ( \cdot , \cdot )$ in consistency loss Eq. (2) with $m = 7$ .
+
+e.g., random initialization, random data augmentation, etc. Moreover, it needs to evaluate the instantaneous loss for all samples in each step, resulting in extra inference cost on unselected samples.
+
+The dynamic patterns of losses (Zhou et al., 2020b) over the course of training give us a new insight for better clean data detection even when the noise ratio (proportion of wrong labels) is high. In particular, we hypothesize that a sample’s label is more likely to be correct if its losses persistently retain low values over training steps. Given a sample $( x _ { i } , y _ { i } )$ with $x _ { i }$ being the features and $y _ { i }$ being the label, we describe its loss dynamics using a simple exponential moving average (EMA) of the instantaneous loss $\ell ( f ( x _ { i } ; \theta _ { t } ) , \overset { \cdot } { y } _ { i } )$ (where $f ( x _ { i } ; \theta _ { t } )$ denotes the model output and $\theta _ { t }$ is the model parameters at step $t ^ { * }$ along the training history, which is defined and computed recursively as
+
+$$
+l _ { t + 1 } ( i ) = \left\{ \begin{array} { l l } { \gamma \times \ell ( f ( x _ { i } ; \theta _ { t } ) , y _ { i } ) + ( 1 - \gamma ) \times l _ { t } ( i ) } & { \mathrm { ~ i f ~ } i \in S _ { t } } \\ { l _ { t } ( i ) } & { \mathrm { ~ e l s e ~ , ~ } } \end{array} \right.
+$$
+
+Where $\gamma \in [ 0 , 1 ]$ is a discounting factor, $V$ is the set of all $n$ training samples, and $S _ { t } \subseteq V$ is the set of samples selected (by a certain curriculum) for training at epoch $t$ . We only update the EMA loss for selected samples using the byproduct $l _ { t } ( i )$ of training without requiring extra inference.
+
+In the first and the third plots of Figure 1, we show how the losses and EMA losses associated with clean/noisy data change throughout the training process. Specifically, we train a ResNet34 (He et al., 2016) model on CIFAR10 with $6 0 \%$ of the original labels randomly changed to a wrong class. To avoid quick overfitting to the noise, we train the model for multiple episodes (each composed of several epochs over all data with a cosine annealing learning rate) and alternate between the supervised learning episode that minimizes the cross-entropy loss against the given noisy labels and the self-supervision episode that minimizes the consistency loss Eq.(2) against the pseudo labels. Comparing the shaded areas (std) of instantaneous loss and EMA loss and the gap between curves in the two plots, we see that EMA leads to smaller variance within each group and larger gap between the clean and noisy groups, demonstrating the effectiveness of EMA loss for clean data detection.
+
+# 2.2 CONSISTENCY DYNAMICS AND PSEUDO LABEL SELECTION
+
+Simply removing noisy data $( x , y )$ with wrong label $y$ discards important information about the data distribution $p ( x )$ (Arazo et al., 2019). Including all the noisy data for self-supervision, bootstrapping or relabeling can also harm the training since the pseudo labels’ quality is not equal across samples and much depends on the model generating them, where the model’s predictions may contain errors that can accumulate if adopted for training. Hence, a careful selection of noisy data is necessary. However, without access to a purely clean dataset or a reliable pre-trained model, it is nontrivial to evaluate pseudo labels’ correctness. By analyzing the training dynamics of model outputs in the above experiment, we discover that the model output for a sample tends to be an accurate pseudo label if the output remains consistent over training steps and across different augmentations of the sample.
+
+We first define the instantaneous consistency loss of a sample $x _ { i }$ at step $t$ as the discrepancy of the model output $f ( x _ { i } ; \theta )$ between step $t$ and $t - 1$ on $x _ { i }$ and its $m$ data augmentations $\{ x _ { i } ^ { ( j ) } \} _ { j = 1 } ^ { m }$ where the discrepancy can be measured by any loss function $\ell ( \cdot , \cdot )$ . However, the discrepancy can be small if the models of epoch $t$ and $t - 1$ are too similar and make the same errors. Therefore, we use an exponential moving average of the model parameters (according to mean teacher (Tarvainen & Valpola, 2017)) and compute the prediction at step $t - 1$ by averaging over multiple data augmentations (according to MixMatch (Berthelot et al., 2019)): $\begin{array} { r } { \overline { { f } } _ { t } ( x _ { i } ) \triangleq \mathbb { 1 } / m \sum _ { j = 1 } ^ { m } f ( x _ { i } ^ { ( j ) } ; \overline { { \theta } } _ { t } ) , \overline { { \theta } } _ { t } \triangleq \gamma \theta _ { t - 1 } + } \end{array}$ $( 1 - \gamma ) \overline { { \theta } } _ { t - 1 }$ . Computing pseudo labels on augmented data and a time averaging ensemble of models is commonly-adopted for semi-supervised learning (Sajjadi et al., 2016; Laine $\&$ Aila, 2016; Zhou et al., 2020a). The instantaneous consistency loss† $\zeta _ { t } ( i )$ of sample $x _ { i }$ at step $t$ is then defined as
+
+$$
+\zeta _ { t } ( i ) \triangleq \frac { 1 } { m } \sum _ { j = 1 } ^ { m } \ell ( f ( x _ { i } ; \theta _ { t } ) , \overline { { f } } _ { t } ( x _ { i } ^ { ( j ) } ) ) ,
+$$
+
+Which can also be minimized as a consistency loss for $x _ { i }$ in self-supervised learning when no label is given. Similar to the EMA loss in Eq. (1), we define the EMA consistency loss over training history
+
+$$
+c _ { t + 1 } ( i ) = \left\{ \begin{array} { l l } { \gamma \times \zeta _ { t } ( i ) + ( 1 - \gamma ) \times c _ { t } ( i ) } & { \mathrm { ~ i f ~ } i \in S _ { t } } \\ { c _ { t } ( i ) } & { \mathrm { ~ e l s e . ~ } } \end{array} \right.
+$$
+
+Note we use the same $\gamma$ value as in Eq. (1). The EMA consistency loss $c _ { t } ( i )$ measures both the time-consistency (Zhou et al., 2020a) over multiple training steps and the spatial consistency over different augmentations of sample $x _ { i }$ . If the output prediction is wrong and contradicts other samples’ labels, it will be inconsistent over time and across augmentations since it can easily change or flip after the next training step. In the last plot of Figure 1, we report the mean and standard deviation (the middle line and the shaded area) of the EMA consistency loss for two groups of data at each epoch, i.e., the ones with correct pseudo labels and the ones with incorrect pseudo labels. Comparing to the instantaneous consistency loss in the second plot, EMA consistency loss is a more reliable criterion for allocating correct pseudo labels. Thus, we can safely learn the noisy data by using their pseudo labels as training targets and avoid introducing harmful noises.
+
+# 3 ROBUST CURRICULUM LEARNING
+
+In this section, we first introduce the selection criterion for both the clean label detection and pseudo label selection. By adjusting two temperature parameters $\tau _ { 1 }$ and $\tau _ { 2 }$ , it can smoothly interpolate between the two criteria and control their trade-off. We then show that the criterion is derived from a novel optimization formulation for robust curriculum learning. We finally present the RoCL algorithm.
+
+# 3.1 DATA SELECTION CRITERION
+
+To combine of clean label detection and pseudo label selection, we use the two criteria from the previous sections and apply a temperature parameter to control the preference for small/large loss or consistency loss and their trade-off. We sample $x _ { i }$ at training step $t$ with probability:
+
+$$
+\begin{array} { r } { \mathcal { P } _ { t } ( i ) = \lambda \times p _ { t } ( i ) + ( 1 - \lambda ) \times q _ { t } ( i ) , } \end{array}
+$$
+
+where $p _ { t } ( i )$ and $q _ { t } ( i )$ are defined as softmax probabilities, i.e.,
+
+$$
+p _ { t } ( i ) \triangleq \frac { \exp [ \tau _ { 1 } l _ { t } ( i ) ] } { \sum _ { j = 1 } ^ { n } \exp [ \tau _ { 1 } l _ { t } ( j ) ] } , q _ { t } ( i ) \triangleq \frac { \exp [ \tau _ { 2 } c _ { t } ( i ) ] } { \sum _ { j = 1 } ^ { n } \exp [ \tau _ { 2 } c _ { t } ( j ) ] } ,
+$$
+
+and $\lambda \in [ 0 , 1 ]$ controls trade-off between them. Here, $p _ { t } ( i )$ is a softmax probability computed from EMA losses for clean data detection: samples with smaller (larger) EMA losses and thus clean (noisy) labels tend to have high probability $\bar { p } _ { t } ( i )$ when $\tau _ { 1 }$ is negative (positive). Similarly, $q _ { t } ( i )$ is computed from EMA consistency loss for pseudo label selection: samples with smaller (larger) EMA consistency loss and thus correct (wrong) pseudo labels tend to have high probability $q _ { t } ( i )$ when $\tau _ { 2 }$ is negative (positive). When $\tau _ { 1 } , \tau _ { 2 } = 0$ , the probabilities are uniform, and when $\tau _ { 1 } , \tau _ { 2 }  + \infty /$ $- \infty$ , the probabilities approximate the max (min) operator.
+
+Either $p _ { t } ( i )$ or $q _ { t } ( i )$ can be independently employed to select or reweigh samples for noise-label learning. However, selecting samples with high probabilities $p _ { t } ( i )$ when $\tau _ { 1 } \ll 0$ tends to make the training focus on clean data that the model has already learned, which carries limited new information and prevents exploration. Similarly, selecting samples with high probabilities $q _ { t } ( i )$ when $\tau _ { 2 } \ll 0$ is not informative since the model outputs are consistently correct for those data, and little progress can be made.We can encourage exploration by manipulating the temperature parameters. By setting $\tau _ { 1 }$ and $\tau _ { 2 }$ close to zero, we move towards uniform exploration of all data. A more effective strategy is to couple the values of $\tau _ { 1 }$ and $\tau _ { 2 }$ . For example, a negative $\tau _ { 1 }$ and a positive $\tau _ { 2 }$ strengthen the preference for clean data that have not been fully exploited and learned by the model. Alternatively, a positive $\tau _ { 1 }$ with a negative $\tau _ { 2 }$ emphasizes the noisy data with correct pseudo labels, so relabeling them provides new information in addition to the clean data.
+
+We apply the data selection criterion of Eq. (4) in each step and gradually change its parameters $( \tau _ { 1 } , \tau _ { 2 } , \lambda )$ over the course of training according to the properties discussed above. We start from a negative $\tau _ { 1 }$ associated with a positive $\tau _ { 2 }$ and a large $\lambda$ ${ p } _ { t } ( i )$ dominates), then gradually increase $\tau _ { 1 }$ while decrease $\tau _ { 2 }$ and $\lambda$ , and end with a positive $\tau _ { 1 }$ , a negative $\tau _ { 2 }$ and a small $\lambda$ . In this way, we get a curriculum with a smooth transition between supervised learning of clean data using correct given labels and self-supervision of noisy data using reliable pseudo labels (i.e., minimizing Eq. (2). Moreover, the coupling strategy on $\tau _ { 1 }$ and $\tau _ { 2 }$ encourages selecting informative samples that the model mostly needs to improve on, i.e., clean data with inconsistent model outputs or noisy data with correct pseudo labels. In our experiments, we can further reduce the hyperparameters: (1) given the sequence for $\tau _ { 1 }$ in the curriculum as $\tau _ { 1 : T }$ , we can reverse it as the sequence for $\tau _ { 2 }$ , i.e., $\tau _ { T : 1 }$ ; (2) we can make $\lambda$ monotone increase with $\tau _ { 1 }$ , e.g., setting the initial value $\lambda _ { 1 } = a \tau _ { 1 } + b$ and ending value $\lambda _ { T } = a \tau _ { T } + b$ , solving this linear system for $a$ and $b$ , which generate $\lambda _ { 1 : T } = a \tau _ { 1 : T } + b$ .
+
+# 3.2 ROBUST CURRICULUM LEARNING AS AN OPTIMIZATION
+
+The data selection criterion is derived from an optimization formulation of our robust curriculum learning (RoCL), in which we aim to minimize a combination of supervised loss and
+
+consistency(self-supervised) loss in the following form.
+
+$$
+\operatorname* { m i n } _ { \theta } F ( \theta ) \triangleq \frac { \lambda } { \tau _ { 1 } } \log \left( \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \exp [ \tau _ { 1 } \ell ( f ( x _ { i } ; \theta ) , y _ { i } ) ] \right) + \frac { 1 - \lambda } { \tau _ { 2 } } \log \left( \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \exp [ \tau _ { 2 } \zeta ( i ) ] \right) ,
+$$
+
+where the consistency loss $\zeta ( i )$ is defined in Eq. (2) (with $t$ removed). For the first term of Eq. (6), $\textstyle 1 / n \sum _ { i = 1 } ^ { n } \ell ( f ( x _ { i } ; \theta ) , y _ { i } )$ l empirical risk minimization (ERM) with arithmetic average loss, i.e.,, we use LogSumExp loss with an additional temperature parameter (i.e., $\begin{array} { r l } & { \frac { 1 } { \tau _ { 1 } } \log \left( \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \exp [ \tau _ { 1 } \ell ( f ( x _ { i } ; \theta ) , y _ { i } ) ] \right) } \end{array}$ ), so it can approximate both the min-loss (when $\tau _ { 1 } \to - \infty ,$ max-loss (when $\tau _ { 1 } \to + \infty ,$ ), and any interpolation between them, where the min-loss focuses on the easiest samples with the smallest losses and the max-loss focuses on the hardest ones. Note it reduces to the arithmetic average loss when $\tau  0$ . It is called “tilted loss” in a recent work (Li et al., 2020), which shows several intriguing properties in different learning settings. The second term of Eq. (6) focuses on the consistency loss $\zeta ( i )$ . We use $\lambda$ to control the trade-off between the supervised loss and the consistency loss. By simple algebra, the gradient of $F ( \theta )$ at step $t$ is
+
+$$
+\begin{array} { l } { \displaystyle \nabla _ { \theta } F ( \theta _ { t } ) = \lambda \sum _ { i = 1 } ^ { n } p _ { t } ^ { \prime } ( i ) \nabla _ { \theta } \ell ( f ( x _ { i } ; \theta _ { t } ) , y _ { i } ) + ( 1 - \lambda ) \sum _ { i = 1 } ^ { n } q _ { t } ^ { \prime } ( i ) \nabla _ { \theta } \zeta _ { t } ( i ) } \\ { \displaystyle = \sum _ { i = 1 } ^ { n } \mathcal { P } _ { t } ^ { \prime } ( i ) \left[ \frac { \lambda p _ { t } ^ { \prime } ( i ) } { \mathcal { P } _ { t } ^ { \prime } ( i ) } \nabla _ { \theta } \ell ( f ( x _ { i } ; \theta _ { t } ) , y _ { i } ) + \frac { ( 1 - \lambda ) q _ { t } ^ { \prime } ( i ) } { \mathcal { P } _ { t } ^ { \prime } ( i ) } \nabla _ { \theta } \zeta _ { t } ( i ) \right] = \mathbb { E } _ { i \sim \mathcal { P } _ { t } ^ { \prime } ( i ) } G _ { t } ( i ) , } \end{array}
+$$
+
+where
+
+$$
+G _ { t } ( i ) \triangleq \frac { \lambda p _ { t } ^ { \prime } ( i ) } { \mathcal { P } _ { t } ^ { \prime } ( i ) } \nabla _ { \theta } \ell ( f ( x _ { i } ; \theta _ { t } ) , y _ { i } ) + \frac { ( 1 - \lambda ) q _ { t } ^ { \prime } ( i ) } { \mathcal { P } _ { t } ^ { \prime } ( i ) } \nabla _ { \theta } \zeta _ { t } ( i ) .
+$$
+
+Here, $p _ { t } ^ { \prime } ( i )$ and $q _ { t } ^ { \prime } ( i )$ are similar to $p _ { t } ( i )$ and $q _ { t } ( i )$ defined in Eq. (5) except that the EMA loss and EMA consistency loss are replaced by their instantaneous counterparts respectively, i.e., $p _ { t } ^ { \prime } ( i ) \triangleq$ $\frac { \exp [ \tau _ { 1 } \ell ( f ( x _ { i } ; \theta _ { t } ) , y _ { i } ) ] } { \sum _ { j = 1 } ^ { n } \exp [ \tau _ { 1 } \ell ( f ( x _ { j } ; \theta _ { t } ) , y _ { j } ) ] }$ , P exp[τ2ζt(i)]nj=1 exp[τ2ζt(j)] . Similarly, we also denote P 0t(i) = λ × p0t(i) + $( 1 - \lambda ) \times q _ { t } ^ { \prime } ( i )$ . Note in Section 2, we already discussed that the EMA metrics are better alternatives to the instantaneous metrics when used for data selection in noisy-label learning. In our experiments, we use the EMA metrics $\{ p _ { t } ( i ) , q _ { t } ( i ) , \mathcal { P } _ { t } ( i ) \}$ instead of the instantaneous ones $\{ p _ { t } ^ { \prime } ( i ) , q _ { t } ^ { \prime } ( \dot { i } ) , \mathcal { P } _ { t } ^ { \prime } ( i ) \}$ . For every training step, an unbiased estimator of the gradient in Eq. (7) can be achieved by drawing a subset of samples $S _ { t }$ according to $\mathcal { P } _ { t } ( i )$ and averaging their gradients $G _ { t } ( i )$ in Eq. (8).
+
+# 3.3 ROBUST CURRICULUM LEARNING ALGORITHM
+
+# Algorithm 1 Robust Curriculum Learning (RoCL)
+
+We describe our curriculum learning algorithm RoCL in Alg. 1 based on the data selection criterion in Section 3.1 and the optimization formulation in Section 3.2. We denote the update of $\theta _ { t }$ produced by the adopted optimizer as $\mathsf { \bar { h } } ( \nabla _ { \theta } F ( \theta _ { t } ) , \eta )$ , where $\eta$ contains all hyperparameters of the optimizer at step $t$ , e.g., the learning rate. We denote $\ell _ { t } ( i )$ as a shorthand notation for $\ell ( f ( x _ { i } ; \theta _ { t - 1 } ) , y _ { i } )$ . We apply a warm starting episode of a few epochs (e.g., 5-10) over all the data and given labels with label smoothing to obtain stable EMA metrics. After that, we apply multiple episodes of curriculum learning, each including a sequence of steps following the curriculum at the end of Section 3.1 for data selection per step (Line 6-14). We repeat the transition between clean data learning to noisy data learning for $K$ episodes to avoid getting trapped in a local
+
+1: input: $\{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { n } , h ( \cdot ; \eta ) , \ell ( \cdot , \cdot ) , f ( \cdot ; \theta ) , T _ { 0 : K } ; \tau _ { 1 } <$   
+$0 , \tau _ { T } > 0 ; \lambda _ { 1 } , \lambda _ { T } \in [ 0 , 1 ] ; \gamma , \gamma _ { b } \in [ 0 , 1 ]$   
+2: initialize: $\theta _ { 0 }$ , $b _ { 0 } \in ( 0 , n )$ , $l _ { 0 } ( i ) = \bar { c } _ { 0 } ( i ) \bar { = } 0 \forall i \in [ n ]$   
+3: for $k \in \{ 0 , \cdots , K \}$ do   
+4: Schedule $\tau _ { 1 : T _ { k } }$ and $\lambda _ { 1 : T _ { k } }$ by Eq. (9)-(10);   
+5: for $t ^ { \prime } \in \{ 1 , \cdots , T _ { k } \}$ do   
+6: $t \gets t ^ { \prime } + T _ { k - 1 }$ ;   
+7: if $k = 0$ then   
+8: $\begin{array} { r l } & { { S _ { t } } \gets [ n ] ; } \\ & { \theta _ { t } \gets \theta _ { t - 1 } + h \left( \nabla _ { \theta } \frac { 1 } { n } \sum _ { i = 0 } ^ { n } \ell _ { t } ( i ) ; \eta \right) ; } \end{array}$   
+9:   
+10: else   
+11: Draw a subset $S _ { t } \subseteq [ n ]$ of $b _ { k }$ samples according   
+to probability $\mathcal { P } _ { t }$ in Eq. (4) with $\tau _ { 1 } = \tau _ { t ^ { \prime } } , \tau _ { 2 } =$   
+$\tau _ { T _ { k } - t ^ { \prime } } , \lambda = \lambda _ { t ^ { \prime } }$ ;   
+12: $\begin{array} { r l } & { \theta _ { t }  \theta _ { t - 1 } + h ( \frac { 1 } { b _ { k } } \sum _ { i \in S _ { t } } G _ { t } ( i ) ; \eta ) } \end{array}$ (Eq. (8));   
+13: end if   
+14: Update $l _ { t + 1 } ( i )$ and $c _ { t + 1 } ( i )$ by Eq. (1) and Eq. (3);   
+15: end for   
+16: $b _ { k + 1 }  ( 1 + \gamma _ { b } ) \times b _ { k } ;$   
+17: end for
+
+the memorization of clean labels and correct pseudo labels learned in previous episodes. Moreover, under the coupling strategy of $\tau _ { 1 }$ and $\tau _ { 2 }$ , each episode is encouraged to explore the clean/noisy data that the previous episode fails to learn. Considering the undertrained model (producing inaccurate pseudo labels) and the relatively high variance of the EMA metrics at the earlier episodes, we start from a small budget for the selected subset size and gradually increase in later episodes.
+
+To generate the whole schedule of $\tau _ { 1 : T }$ in each episode, we can apply any monotone interpolation between $\tau _ { 1 }$ and $\tau _ { T }$ whose values are predefined. Let $g : { \mathcal { R } } \mapsto [ - \sigma , \sigma ]$ be an invertible monotone continuous function. We define the interpolation between $\tau _ { 1 }$ and $\tau _ { T }$ as follows, $\forall t \in [ T ]$ ,
+
+$$
+\tau _ { t } = \frac { \tau _ { T } - \tau _ { m } } { \sigma } \times \left[ g ( \sigma _ { t } ) - \frac { g ( - \sigma ) + g ( \sigma ) } { 2 } \right] + \tau _ { m } , \sigma _ { t } = g ^ { - 1 } ( - \sigma ) + \frac { 2 t } { T } g ^ { - 1 } ( \sigma ) , \tau _ { m } = \frac { \tau _ { 1 } + \tau _ { T } } { 2 } .
+$$
+
+Note the $g ( \sigma _ { t } )$ produces interpolation values between $[ - \sigma , \sigma ]$ that correspond to $T$ evenly spaced input $\sigma _ { t } \in \mathring { [ g ^ { - 1 } ( - \sigma ) , g ^ { - 1 } ( \sigma ) ] }$ . In our curriculum for each episode, we need to keep a high quality of the selected clean (pseudo) labels in earlier (later) stages and make the exploration stages in between shorter since their selected labels contain more noise. Therefore, we choose “s”-shaped functions such as tanh or the logistic function for the interpolation. In this paper, we use $g ( \cdot ) \stackrel { - } { = } \operatorname { t a n h } ( \cdot )$ and pick $\sigma = 0 . 9 5$ . We illustrate Eq. (9) and visualize our choice of $g ( \cdot )$ and the resulting $\tau _ { t }$ in Figure 6 (Appendix). The corresponding schedule for $\lambda$ can then be defined as an affine transformation of $\tau _ { 1 : T }$
+
+$$
+\forall t \in [ T ] , \lambda _ { t } = a _ { \lambda } ( \tau _ { t } - \tau _ { 1 } ) + \lambda _ { 1 } , a _ { \lambda } = \frac { \lambda _ { T } - \lambda _ { 1 } } { \tau _ { T } - \tau _ { 1 } } .
+$$
+
+# 4 EXPERIMENTS
+
+We evaluate RoCL with other approaches for noisylabel learning on three widely used benchmarks, i.e., CIFAR10/100 with two types of synthetic noises (i.e., symmetric and asymmetric), and mini-WebVision (Li et al., 2017a) (the first 50 classes) containing unknown noises from web labels. Symmetric noise flips each label randomly to an incorrect class with probability $\rho$ (i.e., noise rate), and our experiments cover $\rho = \{ 0 . 4 , 0 . 6 , 0 . 8 \}$ . Asymmetric noise flips the labels within a specific set of classes. For CIFAR10, flipping TRUCK AUTOMOBILE, BIRD AIRPLANE, DEER HORSE, $\mathrm { C A T } {  } \mathrm { D O G }$ . In CIFAR100, the 100 classes are grouped into 20 super-classes with each has 5 sub-classes, we then flip each class within the same super-class to the next in a circular fashion with probability $\rho$ . Our experiments cover $\rho = \{ 0 . 2 , 0 . 3 , 0 . 4 \}$ .
+
+Table 1: Accuracy $( \% )$ evaluated on WebVision and ILSVRC2012 validation sets for DNNs trained by noisy-label learning methods on mini-WebVision training set (first 50 classes), which contains real-world web-label noises.   
+
+<table><tr><td>Val. set</td><td>WebVision</td><td>ILSVRC2012</td></tr><tr><td> Accuracy</td><td>Top-1 Top-5</td><td>Top-1 Top-5</td></tr><tr><td>F-correct +*</td><td>61.12 82.68</td><td>57.36 82.36</td></tr><tr><td>Decoupling **</td><td>62.54 84.74</td><td>58.26 82.26</td></tr><tr><td>Co-teaching *</td><td>63.58 85.20</td><td>61.48 84.70</td></tr><tr><td>MentorNet **</td><td>63.00 81.40</td><td>57.80 79.92</td></tr><tr><td>MentorMix ***</td><td>76.00 90.20</td><td>72.90 91.10</td></tr><tr><td>D2L *</td><td>62.68 84.00</td><td>57.80 81.36</td></tr><tr><td>INCV *</td><td>65.24 85.34</td><td>61.60 84.98</td></tr><tr><td>RoCL (ours) *†~</td><td>80.04 92.68</td><td>75.81 92.28</td></tr></table>
+
+Practical Modifications In the experiments, we follow previous work and apply the techniques below. We will present an ablation study of their effectiveness in Table 5.
+
+• We apply the class-balance regularization used in (Tanaka et al., 2018), which prevents the model from predicting the same class for all the samples within a mini-batch $B$ . We add $\textstyle \bigl ( 1 / B \bigr ) \sum _ { i \in B } \ell \bigl ( f ( x _ { i } ; \theta ) , \overline { { 1 } } / C \cdot { \mathbf { 1 } } \bigr )$ (where $C$ is the number of classes) to the objective for a minibatch $B$ with regularization weight of 1. • We apply label smoothing whose effectiveness in noisy-label learning has been studied in (Lukasik et al., 2020). We modify each one-hot label $y _ { i }$ to be $\bar { y _ { i } }  ( 1 - \alpha ) y _ { i } \dot { + } \alpha / C$ (e.g., we use $\alpha = 0 . 5$ ). • We apply Mix-Up (Zhang et al., 2018) to all the selected data. However, Mix-Up of two (soft) pseudo labels can significantly increase the entropy of the mixed label if both pseudo labels are under-confident. Hence, we apply a curriculum to the beta distribution’s parameter $\alpha$ of Mix-Up and gradually reduce it (e.g., from 8.0 to 0.2 in our experiments) within each episode.
+
+Hyperparameter Setting We apply $\scriptstyle \mathrm { R o C L }$ to train ResNet34 on CIFAR10/100 and ResNet50 on WebVision, which are the most widely used models in other baseline papers. We apply SGD with momentum of 0.9, weight decay of $1 0 ^ { - 4 }$ and cosine annealing learning rate in each training episode. The initial learning rate is set to 0.1 for CIFAR10/100 and 1.0 for WebVision. In all RoCL experiments, we apply $T _ { 0 } = 1 0$ warm starting epochs followed by $K = 1 0$ episodes of curriculum learning, whose lengths start from $T _ { 1 } = 1 0$ and increase by 10 for every episode afterwards. We initialize the subset size $b _ { 0 } = 0 . 2 n$ and set $\gamma = \gamma _ { b } = 0 . 1$ , which are common choices for discounting/augmenting factors. We use Cubuk et al. (2020) for data augmentations. We did not heavily tune $\lambda _ { 1 } , \lambda _ { T }$ and $\tau _ { 1 } , \tau _ { T }$ and followed a principle that the resulting curriculum should have a transition from supervised learning on clean data to self-supervised learning on noisy data with correct pseudo labels.
+
+• For $\lambda$ , we start from $\lambda _ { 1 }$ close to 1 and end with $\lambda _ { T }$ close to 0 because our curriculum is a transition from supervised learning $( \lambda = 1 )$ ) to self-supervised learning $\lambda = 0$ ).   
+• As explained in Section 3.1, our curriculum requires $\tau _ { 1 }$ for $p _ { t } ( i )$ changing from negative to positive values and an inverse sequence for $\tau _ { 2 }$ in $q _ { t } ( i )$ to gain the above transition and encourage learning on more informative samples. Hence, we set the starting value $\tau _ { 1 }$ to be negative and $\tau _ { T }$ to be positive for the sequence $\tau _ { 1 : T }$ .   
+• We set their exact values based on observations in Figure 1: clean data detection is easier but the detection of correct pseudo-labels is harder. So we can be more confident on the former than the latter and set the starting $\tau _ { 1 }$ larger than $\tau _ { T }$ in magnitude. For the same reason, we set the starting value $\lambda _ { 1 }$ to be closer to 1 than the ending value $\lambda _ { T }$ to 0. In experiments, we tried $\tau _ { 1 } = \{ - 4 , - 3 \}$ and $\tau _ { T } = \{ 1 , 2 \}$ and finally chose $\tau _ { 1 } = - 4$ and $\tau _ { T } = 1$ since this choice performs consistently well on all experiments, though it might not be the best choice for all; we set $\lambda _ { 1 } = 0 . 9$ and did not try other values; we tried $\lambda _ { T } = \{ 0 . 1 , \bar { 0 . 2 } , 0 . 3 \}$ and on some experiments the first two choices lead to slightly worse performance, so we chose $\lambda _ { T } = 0 . 3$ .
+
+We compare RoCL to the following baselines: Fcorrect (Patrini et al., 2017), Decoupling (Malach & Shalev-Shwartz, 2017), Co-teaching (Han et al., 2018), D2L (Ma et al., 2018), INCV (Chen et al., 2019), MDDYR-SH (Arazo et al., 2019), MentorNet (Jiang et al., 2018), MentorMix (Jiang et al., 2020), O2U-net (Huang et al., 2019), $\mathrm { R o G + D 2 L }$ (Lee et al., 2019), PENCIL (Yi & Wu, 2019), GCE (Zhang & Sabuncu, 2018), SCE (Wang et al., 2019), NFL/NCE variants (Ma et al., 2020), and Bootstrap (Reed et al., 2014). To better compare and categorize different baseline methods, we use the following symbols to denote the techniques used: $^ +$ for additional clean training data; $^ *$ for training additional auxiliary models; $^ \ddag$ for using mixup; $\star$ for using data augmentations; $\dagger$ for class-balance regularization; o for label-smoothing. We report the results and comparisons to baselines in three tables: real-world noise in Table. 1, symmetric noise in Table. 3 and asymmetric noise in Table. 4. RoCL achieves the best performance in every setting, and for most of the cases, improves upon the existing methods by large margins. The closest rival to RoCL is MentorMix, which utilizes MentorNet and Mix-Up to assign weights to each sample. We note that MentorMix requires training of an extra mentor network to generate the sample weights, while RoCL is more flexible and only makes changes to the training process without modifying the model. Table. 2 reports RoCL’s performance when applied with different loss functions on
+
+Table 2: Test accuracy $( \% )$ of $\mathbf { R o C L }$ applied with different loss functions on CIFAR10 corrupted by $\{ 6 0 \% , 8 0 \% \}$ symmetric(uniform) noises (CE-cross entropy).   
+
+<table><tr><td>Noise Rate</td><td>60%</td><td>80%</td></tr><tr><td>CE</td><td>90.22 ± 0.24</td><td>77.47± 0.67</td></tr><tr><td>GCE</td><td>89.30 ± 0.68</td><td>79.84± 1.12</td></tr><tr><td>SCE</td><td>92.06 ±0.23</td><td>74.25 ±0.86</td></tr><tr><td>NFL+MAE</td><td>88.73±0.47</td><td>85.76±0.26</td></tr><tr><td>NFL+RCE</td><td>87.68 ± 0.35</td><td>80.09 ± 0.41</td></tr><tr><td>NCE+MAE</td><td>90.37 ± 0.43</td><td>82.16 ± 0.93</td></tr><tr><td></td><td></td><td></td></tr><tr><td>NCE+RCE</td><td>88.03 ±0.39</td><td>80.33 ± 0.80</td></tr></table>
+
+Table 3: Test accuracy $( \% )$ of noisy-label learning methods on CIFAR10/100 corrupted by symmetric(uniform) label noises of different levels. All the baselines’ results are from the original papers or the following-up works. There are two formats of these reported results: “mean±variance” of 5 trials and single-trial accuracy.   
+
+<table><tr><td>Dataset</td><td colspan="3">CIFAR10</td><td colspan="3">CIFAR100</td></tr><tr><td>Noise Rate</td><td>40%</td><td>60%</td><td>80%</td><td>40%</td><td>60%</td><td>80%</td></tr><tr><td>MD-DYR-SH **†+</td><td>92.3</td><td>86.1</td><td>74.1</td><td>70.1</td><td>59.5</td><td>39.5</td></tr><tr><td>MentorNet **</td><td>91.2</td><td>74.2</td><td>60.0</td><td>68.5</td><td>61.2</td><td>35.5</td></tr><tr><td>MentorMix ***</td><td>94.2</td><td>91.3</td><td>81.0</td><td>71.3</td><td>64.6</td><td>41.2</td></tr><tr><td>O2U-net *</td><td>90.3</td><td>=</td><td>43.4</td><td>69.2</td><td>-</td><td>39.4</td></tr><tr><td>RoG+D2L **</td><td>87.0</td><td>78.0</td><td>=</td><td>64.9</td><td>40.6</td><td>=</td></tr><tr><td>PENCIL *</td><td>=</td><td>=</td><td>=</td><td>69.12 ±0.62</td><td>57.79± 3.86</td><td>fail</td></tr><tr><td>GCE*</td><td>87.62 ±0.26</td><td>82.70±0.23</td><td>67.92 ± 0.60</td><td>62.64± 0.33</td><td>54.04± 0.56</td><td>29.60±0.51</td></tr><tr><td>SCE *</td><td>85.34 ± 0.07</td><td>80.07 ±0.02</td><td>53.81 ± 0.27</td><td>53.69 ±0.07</td><td>41.47 ± 0.04</td><td>15.00 ± 0.04</td></tr><tr><td>NFL+MAE *</td><td>83.81 ±0.06</td><td>76.36 ± 0.31</td><td>45.23 ± 0.52</td><td>58.18 ±0.08</td><td>46.10±0.50</td><td>24.78±0.82</td></tr><tr><td>NCE+RCE *</td><td>86.02 ±0.09</td><td>79.78 ±0.50</td><td>52.71 ±1.90</td><td>59.48 ± 0.56</td><td>47.12 ± 0.62</td><td>25.80±1.12</td></tr><tr><td>RoCL (ours) ‡*†1</td><td>94.55 ±0.12 92.06 ±0.23 85.76 ±0.26</td><td></td><td></td><td></td><td>74.64 ± 0.43 66.79 ± 0.58 53.89 ± 0.62</td><td></td></tr></table>
+
+Table 4: Test accuracy $( \% )$ of noisy-label learning methods on CIFAR10/100 corrupted by asymmetric(classdependent) noises of 3 levels. All the baselines’ results are from the original papers or the following-up works.   
+
+<table><tr><td>Dataset</td><td colspan="3">CIFAR10</td><td colspan="3">CIFAR100</td></tr><tr><td>Noise Rate</td><td>20%</td><td>30%</td><td>40%</td><td>20%</td><td>30%</td><td>40%</td></tr><tr><td>PENCIL *</td><td>92.43</td><td>91.84</td><td>91.01</td><td>74.70 ±0.56</td><td>72.52 ± 0.38</td><td>63.61 ±0.23</td></tr><tr><td>Bootstrap *</td><td>86.57 ±0.08</td><td>84.86±0.05</td><td>79.76±0.07</td><td>63.44 ± 0.35</td><td>63.18 ±0.35</td><td>62.08 ±0.22</td></tr><tr><td>F-correct +*</td><td>89.09 ±0.47</td><td>86.79±0.36</td><td>83.55 ± 0.58</td><td>42.46± 2.16</td><td>38.13 ± 2.97</td><td>34.44 ± 1.93</td></tr><tr><td>GCE*</td><td>86.07 ± 0.31</td><td>80.78±0.21</td><td>74.98 ± 0.32</td><td>59.99 ± 0.83</td><td>53.99 ± 0.29</td><td>41.49 ± 0.79</td></tr><tr><td>SCE *</td><td>83.92 ± 0.07</td><td>79.70±0.27</td><td>78.20 ± 0.03</td><td>58.22 ± 0.47</td><td>49.85 ± 0.91</td><td>42.19 ±0.19</td></tr><tr><td>NFL+MAE *</td><td>86.81 ± 0.32</td><td>83.91 ± 0.34</td><td>77.16 ± 0.10</td><td>63.10 ± 0.22</td><td>56.19 ± 0.61</td><td>43.51 ± 0.42</td></tr><tr><td>NCE+RCE *</td><td>88.56 ± 0.17</td><td>85.58±0.44</td><td>79.59 ± 0.40</td><td>62.68 ± 0.79</td><td>57.82 ± 0.41</td><td>46.79 ±0.96</td></tr><tr><td>RoCL (ours) t*†</td><td>95.38±0.219</td><td>94.19 ±0.28 92.31±0.35</td><td></td><td></td><td>80.03 ± 0.34 77.59 ± 0.45 73.28 ± 0.83</td><td></td></tr></table>
+
+CIFAR10 under high noise rates, i.e., $60 \%$ and $80 \%$ . We observe significant improvements over their performance without using RoCL in Table. 3. It indicates that RoCL is compatible with any loss function and can further enhance their performance.
+
+To analyze the effect of each component in RoCL, we conduct a thorough ablation study of 10 variants of RoCL, each removing/changing one component of the original RoCL. In Table 5, we report their test accuracies on CIFAR10/100 with noise rates of $\{ 6 0 \% , 8 0 \% \}$ . In Figure 7-10 in Appendix, we report how their test accuracies change during the training to study their learning efficiency and convergence. Among them, “no ClassBalance” removes the class-balance regularization; “no RandAugment” replaces the strong data augmentation RandAugment Cubuk et al. (2020) with random crop and random horizontal flip; “no RandSampling” replaces the weighted sampling in Line 11 of Algorithm 1 by selecting the top- $\boldsymbol { \cdot } \boldsymbol { b } _ { k }$ samples with the largest $\mathcal { P } _ { t } ( i )$ ; “no EMA metrics” replaces EMA loss and EMA
+
+Table 5: Ablation study: Test accuracy $( \% )$ of $\scriptstyle \mathrm { R o C L }$ variants with one part removed/changed when applied to CIFAR10/100 corrupted by symmetric(uniform) label noise.   
+
+<table><tr><td>Dataset</td><td>CIFAR10</td><td>CIFAR100</td><td></td></tr><tr><td>Noise Rate</td><td>60%</td><td>80%</td><td>60% 80%</td></tr><tr><td>RoCL: no MixUp</td><td>92.98</td><td>88.18</td><td>69.72 58.72</td></tr><tr><td>RoCL:no LabelSmooth</td><td>91.94</td><td>85.05</td><td>62.92 42.95</td></tr><tr><td>RoCL:no ClassBalance</td><td>93.08</td><td>74.91</td><td>62.66 43.94</td></tr><tr><td>RoCL: no RandAugment</td><td>86.59</td><td>72.35</td><td>64.84 44.06</td></tr><tr><td>RoCL: no RandSampling</td><td>92.31</td><td>85.99</td><td>64.09 57.00</td></tr><tr><td>RoCL: no EMA metrics</td><td>92.84</td><td>87.79</td><td>65.99 53.10</td></tr><tr><td>RoCL: pt(i) = 1/n</td><td>92.42</td><td>86.05</td><td>62.69 44.35</td></tr><tr><td>RoCL: qt(i)=1/n</td><td>92.59</td><td>86.93</td><td>64.71 50.79</td></tr><tr><td>RoCL: pt(i)= qt(i) =1/n</td><td>92.07</td><td>85.77</td><td>64.18 47.88</td></tr><tr><td>RoCLBase: no curriculum</td><td>87.83</td><td>66.93</td><td>61.84 41.92</td></tr><tr><td>RoCL: original version</td><td>92.82</td><td>88.00</td><td>66.79 54.22</td></tr><tr><td>MentorMix:+RandAugment</td><td>85.45</td><td>20.68</td><td>52.70 8.02</td></tr><tr><td>MentorMix:+RandAugment-MixUp</td><td>84.31</td><td>38.21</td><td>58.31 8.18</td></tr><tr><td>MentorMix:original version</td><td>91.30</td><td>81.00</td><td>64.60 41.20</td></tr><tr><td></td><td></td><td></td><td></td></tr></table>
+
+consistency loss with their instantaneous counterparts; $\dot { \boldsymbol { p } } _ { t } ( i ) = 1 / n ^ { \mathfrak { N } }$ samples the clean data using uniform probabilities; $\mathsf { \bar { q } } _ { t } ( i ) = 1 / n ^ { \prime \prime }$ samples the correct pseudo-labels using uniform probabilities; $\begin{array} { r } { \dot { \mathbf { \sigma } } p _ { t } ( i ) = \mathbf { \bar { { q } } } _ { t } ( i ) = 1 / n ^ { \prime } } \end{array}$ uses uniform probabilities for both. Note for the final three variants, we still have the curriculum of $\lambda$ . We keep the same hyperparameter settings as the original RoCL. We give brief conclusions here and leave a detailed analysis to Appendix: (1) Except $\mathrm { R o C L } _ { B a s e }$ in Algorithm 2, “no RandAugment” and “no ClassBalance”, most variants perform similarly as the original RoCL and outperform the previous SoTA achieved by MentorMix. The removed components are more important under higher noise rates. (2) $\mathrm { R o C L } _ { B a s e }$ removes our proposed curriculum and preserves all other techniques but shows significant degradation on accuracies, indicating that the curriculum is essential to RoCL’s appealing performance. (3) A strong data augmentation is critical to effective self-supervision and accurate EMA consistency loss estimation in RoCL, while a weak one may lead to error accumulation. However, applying RandAugment in MentorMix degrades its original performance. (4) Class-balance regularization is only important under very high noise rates. (5) Removing Mix-Up can improve RoCL’s performance since it damages information when mixing soft pseudo labels. (6) Compared to other variants, “no RandSampling” or “no EMA metrics” causes less degeneration on the final accuracies but can slow down the convergence and learning speed in the early stages when exploration is insufficient. (7) Changing $p _ { t } ( i ) , q _ { t } ( i )$ or both to uniform probabilities reduces the final accuracies in all cases and significantly slows down the learning process.
+
+# 5 CONCLUSION
+
+We propose a novel curriculum learning method RoCL for robust learning under label noises. RoCL features a smooth transition from learning with clean data to noisy data, and from learning with supervised loss to self-supervised loss. Based on observations of training dynamics, RoCL can select samples with reliable labels/pseudo labels and most informative to training. RoCL does not require availability of extra clean data or training of extra auxiliary models. On multiple benchmarks of noisy label learning, RoCL significantly improves upon existing baselines.
+
+# ACKNOWLEDGMENTS
+
+This research is based upon work supported by the National Science Foundation under Grant No. IIS1162606, the National Institutes of Health under award R01GM103544, and by a Google, a Microsoft, and an Intel research award. It is also supported by the CONIX Research Center, one of six centers in JUMP, a Semiconductor Research Corporation (SRC) program sponsored by DARPA. Some GPUs used to produce the experimental results are donated by NVIDIA. We would like to thank ICLR area chairs and anonymous reviewers for their efforts in reviewing this paper and their constructive comments! We also thank all the MELODI lab members for their helpful discussions and feedback.
+
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+A APPENDIX
+
+A.1 ${ \mathrm { R o C L } } _ { B a s e }$ (NO CURRICULUM) IN SECTION 2 AND FIGURE 2-10
+
+# Algorithm $2 \mathrm { R o C L } _ { B a s e }$ (no curriculum)
+
+1: input: $\{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { n } , h ( \cdot ; \eta ) , \ell ( \cdot , \cdot ) , f ( \cdot ; \theta ) , T _ { 0 : K } ; \gamma \in [ 0 ,$ 1]   
+2: initialize: $\theta _ { 0 }$ , $l _ { 0 } ( i ) = c _ { 0 } ( i ) = 0 \forall i \in [ n ] , T _ { - 1 } = 0$   
+3: for $k \in \{ 0 , \cdots , K \}$ do   
+4: for $t ^ { \prime } \in \{ 1 , \cdots , T _ { k } \}$ do   
+5: $t \gets t ^ { \prime } + T _ { k - 1 }$ ;   
+6: $S _ { t } \gets [ n ]$ ;   
+7: if $k \% 2 = 0$ then   
+8: $\begin{array} { r } { \theta _ { t } \gets \theta _ { t - 1 } + h \left( \nabla _ { \theta } \frac { 1 } { n } \sum _ { i = 0 } ^ { n } \ell _ { t } ( i ) ; \eta \right) } \end{array}$ ; {supervised learning using given labels}   
+9: Update $l _ { t + 1 } ( i )$ by Eq. (1); {update EMA loss}   
+10: else   
+11: $\begin{array} { r } { \theta _ { t } \gets \theta _ { t - 1 } + h \left( \nabla _ { \theta } \frac { 1 } { n } \sum _ { i = 0 } ^ { n } \zeta _ { t } ( i ) ; \eta \right) } \end{array}$ ; {self-supervised learning using pseudo labels}   
+12: Update $c _ { t + 1 } ( i )$ by Eq. (3); {update EMA consistency loss}   
+13: end if   
+14: end for   
+15 : end for
+
+Note the EMA metrics in line 9 and line 12 are not used for training in $\mathrm { R o C L } _ { B a s e }$ . They have been updated and recorded for the purpose of empirical study presented in Section 2.
+
+# A.2 ADDITIONAL EXPERIMENTS
+
+![](images/06c8dd0bcc7e6faaef96f0775f95c2e10c991f3379c829db0ca57ebe9e2cb77c.jpg)  
+Figure 2: RoCL (Algorithm 1) vs. $\mathrm { R o C L } _ { B a s e }$ without any curriculum (Algorithm 2 in Appendix) during the training of ResNet34 on CIFAR10 containing $60 \%$ symmetric noises on labels.
+
+![](images/9c7f03c78842b7bc96db77c13df6ed85ce43bf7258bebeecb9ee0ededbd5d85a.jpg)  
+Figure 3: RoCL (Algorithm 1) vs. $\mathrm { R o C L } _ { B a s e }$ without any curriculum (Algorithm 2 in Appendix) during the training of ResNet34 on CIFAR10 containing $80 \%$ symmetric noises on labels.
+
+![](images/e4afeffb81135d02191a255b9b7bd31cf60cc2592a43f0e608b211f9b6c0ead6.jpg)  
+Figure 4: RoCL (Algorithm 1) vs. $\mathrm { R o C L } _ { B a s e }$ without any curriculum (Algorithm 2 in Appendix) during the training of ResNet34 on CIFAR100 containing $60 \%$ symmetric noises on labels.
+
+![](images/7dc0c5b0ce8c820312eab61aee251af0225840b317146c9e338d0de04698e07d.jpg)  
+Figure 5: RoCL (Algorithm 1) vs. $\mathrm { R o C L } _ { B a s e }$ without any curriculum (Algorithm 2 in Appendix) during the training of ResNet34 on CIFAR100 containing $80 \%$ symmetric noises on labels.
+
+Table 6: Extended version of Table 3 with two more baselines: $\mathrm { N F L + R C E }$ and $\mathbf { N C E { + } M A E }$ .   
+
+<table><tr><td>Dataset</td><td colspan="3">CIFAR10</td><td colspan="3">CIFAR100</td></tr><tr><td>Noise Rate</td><td>40%</td><td>60%</td><td>80%</td><td>40%</td><td>60%</td><td>80%</td></tr><tr><td>MD-DYR-SH</td><td>92.3</td><td>86.1</td><td>74.1</td><td>70.1</td><td>59.5</td><td>39.5</td></tr><tr><td>MentorNet</td><td>91.2</td><td>74.2</td><td>60.0</td><td>68.5</td><td>61.2</td><td>35.5</td></tr><tr><td>MentorMix</td><td>94.2</td><td>91.3</td><td>81.0</td><td>71.3</td><td>64.6</td><td>41.2</td></tr><tr><td>O2U-net</td><td>90.3</td><td>=</td><td>43.4</td><td>69.2</td><td>=</td><td>39.4</td></tr><tr><td>RoG+D2L</td><td>87.0</td><td>78.0</td><td>1</td><td>64.9</td><td>40.6</td><td>-</td></tr><tr><td>PENCIL</td><td></td><td></td><td></td><td>69.12 ± 0.62</td><td>57.79 ± 3.86</td><td>fail</td></tr><tr><td>GCE</td><td>87.62 ± 0.26</td><td>82.70± 0.23</td><td>67.92 ±0.60</td><td>62.64± 0.33</td><td>54.04± 0.56</td><td>29.60 ± 0.51</td></tr><tr><td>SCE</td><td>85.34 ± 0.07</td><td>80.07 ±0.02</td><td>53.81 ± 0.27</td><td>53.69 ± 0.07</td><td>41.47 ± 0.04</td><td>15.00 ± 0.04</td></tr><tr><td>NFL+MAE</td><td>83.81±0.06</td><td>76.36 ± 0.31</td><td>45.23 ±0.52</td><td>58.18 ±0.08</td><td>46.10 ±0.50</td><td>24.78 ±0.82</td></tr><tr><td>NFL+RCE</td><td>86.05 ± 0.12</td><td>79.78 ± 0.13</td><td>55.06 ±1.08</td><td>58.20 ± 0.31</td><td>46.30 ± 0.45</td><td>25.16 ± 0.55</td></tr><tr><td>NCE+MAE</td><td>84.19 ± 0.43</td><td>77.61 ± 0.05</td><td>49.62 ±0.72</td><td>59.22 ±0.36</td><td>48.06 ±0.34</td><td>25.50±0.76</td></tr><tr><td>NCE+RCE</td><td>86.02 ±0.09</td><td>79.78 ± 0.50</td><td>52.71 ±1.90</td><td>59.48±0.56</td><td>47.12 ± 0.62</td><td>25.80 ±1.12</td></tr><tr><td>RoCL (ours) t*+</td><td>94.55 ± 0.12</td><td>92.06 ±0.23</td><td>85.76±0.26</td><td>74.64±0.43</td><td>66.79± 0.58</td><td>53.89±0.62</td></tr></table>
+
+Table 7: Extended version of Table 4 with two more baselines: $\mathrm { N F L + R C E }$ and $\mathbf { N C E + M A E }$   
+
+<table><tr><td>Dataset</td><td></td><td>CIFAR10</td><td></td><td></td><td>CIFAR100</td><td></td></tr><tr><td>Noise Rate</td><td>20%</td><td>30%</td><td>40%</td><td>20%</td><td>30%</td><td>40%</td></tr><tr><td>PENCIL</td><td>92.43</td><td>91.84</td><td>91.01</td><td>74.70±0.56</td><td>72.52 ± 0.38</td><td>63.61 ±0.23</td></tr><tr><td>Bootstrap</td><td>86.57 ±0.08</td><td>84.86 ±0.05</td><td>79.76 ± 0.07</td><td>63.44±0.35</td><td>63.18 ±0.35</td><td>62.08 ±0.22</td></tr><tr><td>F-correct</td><td>89.09 ±0.47</td><td>86.79±0.36</td><td>83.55 ± 0.58</td><td>42.46 ±2.16</td><td>38.13 ± 2.97</td><td>34.44 ± 1.93</td></tr><tr><td>GCE</td><td>86.07 ±0.31</td><td>80.78 ±0.21</td><td>74.98 ±0.32</td><td>59.99 ± 0.83</td><td>53.99 ±0.29</td><td>41.49 ± 0.79</td></tr><tr><td>SCE</td><td>83.92 ± 0.07</td><td>79.70±0.27</td><td>78.20± 0.03</td><td>58.22 ± 0.47</td><td>49.85 ± 0.91</td><td>42.19 ±0.19</td></tr><tr><td>NFL+MAE</td><td>86.81±0.32</td><td>83.91 ± 0.34</td><td>77.16 ± 0.10</td><td>63.10±0.22</td><td>56.19 ± 0.61</td><td>43.51±0.42</td></tr><tr><td>NFL+RCE</td><td>88.73± 0.29</td><td>85.74±0.22</td><td>79.27 ± 0.43</td><td>63.12 ± 0.41</td><td>54.72 ± 0.38</td><td>42.97 ±1.03</td></tr><tr><td>NCE+MAE</td><td>86.44 ± 0.23</td><td>83.98 ± 0.52</td><td>78.23 ± 0.42</td><td>62.38 ± 0.60</td><td>58.02 ±0.48</td><td>47.22 ± 0.30</td></tr><tr><td>NCE+RCE</td><td>88.56 ± 0.17</td><td>85.58 ± 0.44</td><td>79.59 ± 0.40</td><td>62.68 ±0.79</td><td>57.82 ± 0.41</td><td>46.79±0.96</td></tr><tr><td>RoCL (ours)</td><td>95.38 ± 0.21</td><td>94.19±0.28</td><td>92.31 ± 0.35</td><td>80.03±0.34</td><td>77.59 ± 0.45</td><td>73.28 ±0.83</td></tr></table>
+
+![](images/2a9acfa10628cb94cf84939d89b39acde8f22f947cba09d06e35d285b0913bcb.jpg)  
+Figure 6: Illustration of Eq. (9) and visualization of our choice for $g ( \cdot )$ and the resulted $\tau _ { t }$ when $T = 5 0$ . We use $g ( \cdot ) = \operatorname { t a n h } ( \cdot )$ (which can be other “S”-shape functions) and $\sigma = 0 . 9 5$ in our experiments. Here, we map the points on the black curve in the left plot to the points on the red curve in the right plot. Each gray point on the bottom of the left plot is from the $T$ evenly spaced $\mathbf { X }$ -coordinates between the $\mathbf { X }$ -interval $[ g ^ { - 1 } ( { - } \sigma ) , { \dot { g } ^ { - 1 } } ( \sigma ) ]$ . We scale them to the $T$ t-coordinates in the bottom of the right plot (i.e., $t = 1 , 2 , \cdots , 5 0 )$ , which associates with $T \tau _ { t }$ values represented by the red points between $[ \tau _ { 1 } , \bar { \tau _ { T } } ]$ on the red curve.
+
+![](images/bf4eee76003616bf450689dc0ec2a65c7f3f789ad6159e5e49843b635f5fa482.jpg)  
+Figure 7: Ablation study: RoCL vs. its variants during the training of ResNet34 on CIFAR10 containing $60 \%$ symmetric noises on labels.
+
+![](images/41cfc89692fac02b1556e5932b674367a602de5bbecf3badb160dc39291e2214.jpg)  
+Figure 8: Ablation study: RoCL vs. its variants during the training of ResNet34 on CIFAR10 containing $80 \%$ symmetric noises on labels.
+
+![](images/73e0492fb528379505f2b183152ca4e4a50ef79d149f12ed776a04eb84333841.jpg)  
+Figure 9: Ablation study: RoCL vs. its variants during the training of ResNet34 on CIFAR100 containing $60 \%$ symmetric noises on labels.
+
+![](images/f3d366d85ae407d1c54297b05f3eb0d4aa8322390d1e39adc19c500cabbca6cf.jpg)  
+Figure 10: Ablation study: RoCL vs. its variants during the training of ResNet34 on CIFAR100 containing $80 \%$ symmetric noises on labels.
+
+We present a more detailed analysis of the ablation study results with explanations of the observed phenomenons below.
+
+• Most variants (except $\mathrm { R o C L } _ { B a s e }$ , no RandAugment, and no ClassBalance) have similar performance as the original RoCL and perform better than or competitive with the SoTA results achieved by MentorMix. The differences compared to original RoCL become smaller under the lower noise rate setting $( 6 0 \% )$ . $\mathrm { R o C L } _ { B a s e }$ uses all data for training in each step without applying any curriculum, showing that our proposed curriculum is the most critical component of RoCL in achieving the appealing improvements. Note $\mathrm { R o C L } _ { B a s e }$ already outperforms most methods in Table 3, which verifies the effectiveness of multi-episode training that alternates between supervised learning with the given labels and self-supervision with the pseudo labels.
+
+• Removing RandAugment degrades the performance, especially when the noise rate is very high (e.g., $80 \%$ ) because strong data augmentations are required by the self-supervision and the EMA consistency loss in RoCL, while trivial data augmentations can result in error accumulation or over-confidence in pseudo labels and inaccurate EMA consistency loss. The self-supervision aims to encourage the model output consistency over different augmentations of the same sample. Without augmentations with sufficient variations, self-supervision reduces to reinforcing the same outputs on similar samples and thus carries little information and can even magnify/accumulate errors (if any) in the original outputs. Also, the EMA consistency loss cannot generate meaningful consistency measures if computed on the same data or its trivial augmentations. Note a strong data augmentation is not always beneficial in all noisy label learning methods since it can increase the uncertainty in the presence of wrong labels, making the detection of clean data and noise correction more challenging. For example, we tried applying RandAugment to MentorMix (using the official implementations of both) but observed inferior performance compared to the results using its original data augmentations.
+
+• Class balance regularization is useful for the very high noise rate setting $( 8 0 \% )$ , in which a wrong label may dominate the learning on a mini-batch by a large chance. However, when the noise rate is not that high (e.g., $60 \%$ on CIFAR10), removing it results in better performance.
+
+• Although Mix-Up has been proved effective in previous methods, and for this reason, we followed MentorMix by starting with a relatively strong Mix-Up $( a l p h a = 8 . 0 $ ) and then gradually reducing it to $\alpha = 0 . 2$ . In the ablation study, we find that completely removing Mix-Up significantly improves performance. Mix-Up is helpful when applied to mix a clean label with a noisy label since the latter can be mediated with the former and thus softened. However, this is rarely the case for RoCL since RoCL either mainly learns from clean data or wrongly-labeled data with correct pseudo labels, and the transition between the two phases is short. When applied to two correct labels/pseudo labels, Mix-Up weakens each label’s confidence, and we may lose information from the inter-class probabilities in the soft pseudo labels.
+
+• Replacing weighted sampling with top-k selection (“no RandSampling”) or replacing EMA metrics with instantaneous metrics (“no EMA metrics”) causes less degeneration on the final test accuracies. However, they are important to the early-stage exploration and accurate estimation of EMA metrics on less-visited samples. In Figure 7-10, these two variants usually suffer from low accuracy and convergence speed during early stages. The only exception is “no RandSampling” in Figure 10, which performs better than the original RoCL. A possible reason is that the randomness brought by high uniform label noises already bring sufficient randomness for exploration.
+
+• Replacing $p _ { t } ( i ) , q _ { t } ( i )$ or both with uniform probabilities over all samples reduces the final test accuracies in all cases, e.g., the degradation is significant on CIFAR100 with $80 \%$ noise. In Figure 7-10, we can see that by setting $q _ { t } ( i ) = 1 / n$ results in less degradation than the other two. This is due to the more accurate pseudo labels generated for more data (even the ones with larger EMA consistency loss) as training proceeds. Moreover, since we are conservative in setting $\lambda _ { T }$ and $\tau _ { T }$ , the performance is not very sensitive to wrong pseudo labels.

parse/train/r1eVMnA9K7/r1eVMnA9K7.md

@@ -0,0 +1,321 @@
+# UNSUPERVISED CONTROL THROUGH NON-PARAMETRIC DISCRIMINATIVE REWARDS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Learning to control an environment without hand-crafted rewards or expert data remains challenging and is at the frontier of reinforcement learning research. We present an unsupervised learning algorithm to train agents to achieve perceptuallyspecified goals using only a stream of observations and actions. Our agent simultaneously learns a goal-conditioned policy and a goal achievement reward function that measures how similar a state is to the goal state. This dual optimization leads to a co-operative game, giving rise to a learned reward function that reflects similarity in controllable aspects of the environment instead of distance in the space of observations. We demonstrate the efficacy of our agent to learn, in an unsupervised manner, to reach a diverse set of goals on three domains – Atari, the DeepMind Control Suite and DeepMind Lab.
+
+# 1 INTRODUCTION
+
+Currently, the best performing methods on many reinforcement learning benchmark problems combine model-free reinforcement learning methods with policies represented using deep neural networks (Horgan et al., 2018; Espeholt et al., 2018). Despite reaching or surpassing human-level performance on many challenging tasks, deep model-free reinforcement learning methods that learn purely from the reward signal learn in a way that differs greatly from the manner in which humans learn. In the case of learning to play a video game, a human player not only acquires a strategy for achieving a high score, but also gains a degree of mastery of the environment in the process. Notably, a human player quickly learns which aspects of the environment are under their control as well as how to control them, as evidenced by their ability to rapidly adapt to novel reward functions (Lake et al., 2017).
+
+Focusing learning on mastery of the environment instead of optimizing a single scalar reward function has many potential benefits. One benefit is that learning is possible even in the absence of an extrinsic reward signal or with an extrinsic reward signal that is very sparse. Another benefit is that an agent that has fully mastered its environment should be able to reach arbitrary achievable goals, which would allow it to generalize to tasks on which it wasn’t explicitly trained. Building reinforcement learning agents that aim for environment mastery instead of or in addition to learning about a scalar reward signal is currently an open challenge.
+
+One way to represent such knowledge about an environment is using an environment model. Modelbased reinforcement learning methods aim to learn accurate environment models and use them either for planning or for training a policy. While learning accurate environment models of some visually rich environments is now possible (Oh et al., 2015; Chiappa et al., 2018; Ha & Schmidhuber, 2018) using learned models in model-based reinforcement learning has proved to be challenging and model-free approaches still dominate common benchmarks.
+
+We present a new model-free agent architecture of Discriminative Embedding Reward Networks, or DISCERN for short. DISCERN learns to control an environment in an unsupervised way by learning purely from the stream of observations and actions. The aim of our agent is to learn a goal-conditioned policy $\pi _ { \boldsymbol { \theta } } ( a | s ; s _ { g } )$ (Kaelbling, 1993; Schaul et al., 2015) which can reach any goal state $s _ { g }$ that is reachable from the current state $s$ . We show how to learn a goal achievement reward function $\dot { \boldsymbol { r } } ( s ; s _ { g } )$ that measures how similar state $s$ is to state $s _ { g }$ using a mutual information objective at the same time as learning $\pi _ { \boldsymbol { \theta } } ( a | s ; s _ { g } )$ . The resulting learned reward function $r ( s ; s _ { g } )$ measures similarity in the space of controllable aspects of the environment instead of in the space of raw observations.
+
+Crucially, the DISCERN architecture is able to deal with goal states that are not perfectly reachable, for example, due to the presence of distractor objects that are not under the agent’s control. In such cases the goal-conditioned policy learned by DISCERN tends to seek states where the controllable elements match those in the goal state as closely as possible.
+
+We demonstrate the effectiveness of our approach on three domains – Atari games, continuous control tasks from the DeepMind Control Suite, and DeepMind Lab. We show that our agent learns to successfully achieve a wide variety of visually-specified goals, discovering underlying degrees of controllability of an environment in a purely unsupervised manner and without access to an extrinsic reward signal.
+
+# 2 PROBLEM FORMULATION
+
+In the standard reinforcement learning setup an agent interacts with an environment over discrete time steps. At each time step $t$ the agent observes the current state $s _ { t }$ and selects an action $a _ { t }$ according to a policy $\pi ( \boldsymbol { a } _ { t } | \boldsymbol { s } _ { t } )$ . The agent then receives a reward $r _ { t } = r ( s _ { t } , a _ { t } )$ and transitions to the next state $s _ { t + 1 }$ . The aim of learning is to maximize the expected discounted return $\textstyle R = \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } r _ { t }$ of policy $\pi$ where $\gamma \in [ 0 , 1 )$ is a discount factor.
+
+In this work we focus on learning only from the stream of actions and observations in order to forego the need for an extrinsic reward function. Motivated by the idea that an agent capable of reaching any reachable goal state $s _ { g }$ from the current state $s$ has complete mastery of its environment, we pose the problem of learning in the absence of rewards as one of learning a goal-conditioned policy $\pi _ { \boldsymbol { \theta } } ( a | s ; s _ { g } )$ with parameters $\theta$ . More specifically, we assume that the agent interacts with an environment defined by a transition distribution $p ( s _ { t + 1 } | s _ { t } , a _ { t } )$ . We define a goal-reaching problem as follows. At the beginning of each episode, the agent receives a goal $s _ { g }$ sampled from a distribution over possible goals $p _ { g o a l }$ . For example, $p _ { g o a l }$ could be the uniform distribution over all previously visited states. The agent then acts for $T$ steps according to the goal-conditioned policy $\pi _ { \boldsymbol { \theta } } ( a | s ; \bar { s } _ { g } )$ receiving a reward of 0 for each of the first $T - 1$ actions and a reward of $r ( s _ { T } ; s _ { g } )$ after the last action, where $r ( s ; s _ { g } ) \in [ 0 , 1 ]$ for all $s$ and $s _ { g }$ 1. The goal achievement reward function $r ( s ; s _ { g } )$ measures the degree to which being in state $s$ achieves goal $s _ { g }$ . The episode terminates upon the agent receiving the reward $r ( s _ { T } ; s _ { g } )$ and a new episode begins.
+
+It is straightforward to train $\pi _ { \boldsymbol { \theta } } ( a | s ; s _ { g } )$ in a tabular environment using the indicator reward $r ( s ; s _ { g } ) ~ \bar { = } ~ \mathbb { 1 } \{ s ~ = ~ s _ { g } \}$ . We are, however, interested in environments with continuous highdimensional observation spaces. While there is extensive prior work on learning goal-conditioned policies (Kaelbling, 1993; Schaul et al., 2015; Andrychowicz et al., 2017; Held et al., 2017; Pathak et al., 2018), the reward function is often hand-crafted, limiting generality of the approaches. In the few cases where the reward is learned, the learning objective is typically tied to a pre-specified notion of visual similarity. Learning to achieve goals based purely on visual similarity is unlikely to work in complex, real world environments due to the possible variations in appearance of objects, or goal-irrelevant perceptual context. We now turn to the problem of learning a goal achievement reward function $r _ { \phi } ( s ; s _ { g } )$ with parameters $\phi$ for high-dimensional state spaces.
+
+# 3 LEARNING A REWARD FUNCTION BY MAXIMIZING MUTUAL INFORMATION
+
+We aim to simultaneously learn a goal-conditioned policy $\pi _ { \theta }$ and a goal achievement reward function $r _ { \phi }$ by maximizing the mutual information between the goal state $s _ { g }$ and the achieved state $s _ { T }$ as shown in (1).
+
+$$
+I ( s _ { g } , s _ { T } ) = H ( s _ { g } ) + \mathbb { E } _ { s _ { g } , s _ { T } \sim p ( s _ { g } , s _ { T } ) } \log p ( s _ { g } | s _ { T } )
+$$
+
+Note that we are slightly overloading notation by treating $s _ { g }$ as a random variable distributed according to $p _ { g o a l }$ . Similarly, $s _ { T }$ is a random variable distributed according to the state distribution induced by running $\pi _ { \theta }$ for $T$ steps for goal states sampled from $p _ { g o a l }$ .
+
+The prior work of Gregor et al. (2016) showed how to learn a set of abstract options by optimizing a similar objective, namely the mutual information between an abstract option and the achieved state. Following their approach, we simplify (1) in two ways. First, we rewrite the expectation in terms of the goal distribution $p _ { g o a l }$ and the goal conditioned policy $\pi _ { \theta }$ . Second, we lower bound the expectation term by replacing $\check { p } ( s _ { g } | s _ { T } )$ with a variational distribution $q _ { \phi } ( s _ { g } | s _ { T } )$ with parameters $\phi$ following Barber $\&$ Agakov (2004), leading to
+
+$$
+I ( s _ { g } , s _ { T } ) \geq H ( s _ { g } ) + \mathbb { E } _ { s _ { g } \sim p _ { g o a l } , s _ { 1 } , \dots s _ { T } \sim \pi _ { \theta } ( \dots | s _ { g } ) } \log q _ { \phi } ( s _ { g } | s _ { T } ) .
+$$
+
+Finally, we discard the entropy term $H ( s _ { g } )$ from (2) because it does not depend on either the policy parameters $\theta$ or the variational distribution parameters $\phi$ , giving our overall objective
+
+$$
+O _ { \mathrm { D I S C E R N } } = \mathbb { E } _ { s _ { g } \sim p _ { g o a l } , s _ { 1 } , \ldots s _ { T } \sim \pi _ { \theta } ( \cdots | s _ { g } ) } \log q _ { \phi } ( s _ { g } | s _ { T } ) .
+$$
+
+This objective may seem difficult to work with because the variational distribution $q _ { \phi }$ is a distribution over possible goals $s _ { g }$ , which in our case are high-dimensional observations, such as images. We sidestep the difficulty of directly modelling the density of high-dimensional observations by restricting the set of possible goals to be a finite subset of previously encountered states that evolves over time (Lin, 1993). Restricting the support of $q _ { \phi }$ to a finite set of goals turns the problem of learning $q _ { \phi }$ into a problem of modelling the conditional distribution of possible intended goals given an achieved state, which obviates the requirement of modelling arbitrary statistical dependencies in the observations.2
+
+Optimization: The expectation in the DISCERN objective is with respect to the distribution of trajectories generated by the goal-conditioned policy $\pi _ { \theta }$ acting in the environment against goals drawn from the goal distribution $p _ { g o a l }$ . We can therefore optimize this objective with respect to policy parameters $\theta$ by repeatedly generating trajectories and performing reinforcement learning updates on $\pi _ { \theta }$ with a reward of $\log { \bar { q _ { \phi } } ( s _ { g } | s _ { T } ) }$ given at time $T$ and 0 for other time steps. Optimizing the objective with respect to the variational distribution parameters $\phi$ is also straightforward since it is equivalent to a maximum likelihood classification objective. As will be discussed in the next section, we found that using a reward that is a non-linear transformation mapping $\log q _ { \phi } ( s _ { g } | s _ { T } )$ to $[ 0 , 1 ]$ worked better in practice. Nevertheless, since the reward for the goal conditioned-policy is a function of $\log q _ { \phi } ( s _ { g } \bar { | } s _ { T } )$ , training the variational distribution function $q _ { \phi }$ amounts to learning a reward function.
+
+Communication Game Interpretation: Dual optimization of the DISCERN objective has an appealing interpretation as a cooperative communication game between two players – an imitator that corresponds to the goal-conditioned policy and a teacher that corresponds to the variational distribution. At the beginning of each round or episode of the game the imitator is provided with a goal state. The aim of the imitator is to communicate the goal state to the teacher by taking $T$ actions in the environment. After the imitator takes $T$ actions, the teacher has to guess which state from a set of possible goals was given to the imitator purely from observing the final state $s _ { T }$ reached by the imitator. The teacher does this by assigning a probability to each candidate goal state that it was the goal given to the imitator at the start of the episode, i.e. it produces a distribution $p ( s _ { g } | s _ { T } )$ . The objective of both players is for the teacher to guess the goal given to the imitator correctly as measured by the log probability assigned by the teacher to the correct goal.
+
+# 4 DISCRIMINATIVE EMBEDDING REWARD NETWORKS
+
+We now describe the DISCERN algorithm – a practical instantiation of the approach for jointly learning $\pi _ { \boldsymbol { \theta } } ( a | s ; s _ { g } )$ and $r ( s ; s _ { g } )$ outlined in the previous section.
+
+Goal distribution: We adopt a non-parametric approach to the problem of proposing goals, whereby we maintain a fixed size buffer $\mathcal { G }$ of past observations from which we sample goals during training. We update $\mathcal { G }$ by replacing the contents of an existing buffer slot with an observation from the agent’s recent experience according to some substitution strategy; in this work we considered two such strategies, detailed in Appendix A3. This means that the space of goals available for training drifts as a function of the agent’s experience, and states which may not have been reachable under a poorly trained policy become reachable and available for substitution into the goal buffer, leading to a naturally induced curriculum. In this work, we sample training goals for our agent uniformly at random from the goal buffer, leaving the incorporation of more explicitly instantiated curricula to future work.
+
+Goal achievement reward: We train a goal achievement reward function $r ( s ; s _ { g } )$ used to compute rewards for the goal-conditioned policy based on a learned measure of state similarity. We parameterize $r ( s ; s _ { g } )$ as the positive part of the cosine similarity between $s$ and $s _ { g }$ in a learned embedding space, although shaping functions other than rectification could be explored. The state embedding in which we measure cosine similarity is the composition of a feature transformation $h ( \cdot )$ and a learned $L ^ { 2 }$ -normalized mapping $\xi _ { \phi } ( \cdot )$ . In our implementation, where states and goals are represented as 2-D RGB images, we take $h ( \cdot )$ to be the final layer features of the convolutional network learned by the policy in order to avoid learning a second convolutional network. We find this works well provided that while training $r$ , we treat $h ( \cdot )$ as fixed and do not adapt the convolutional network’s parameters with respect to the reward learner’s loss. This has the effect of regularizing the reward learner by limiting its adaptive capacity while avoiding the need to introduce a hyperparameter weighing the two losses against one another.
+
+We train $\xi _ { \phi } ( \cdot )$ according to a goal-discrimination objective suggested by (3). However, rather than using the set of all goals in the buffer $\mathcal { G }$ as the set of possible classes in the goal discriminator, we sample a small subset for each trajectory. Specifically, the set of possible classes includes the goal $g$ for the trajectory and $K$ decoy observations $d _ { 1 } , d _ { 2 } , \dots , d _ { K }$ from the same distribution as $s _ { g }$ . Letting
+
+$$
+\boldsymbol { \ell } _ { g } = \xi _ { \phi } ( h ( \boldsymbol { s } _ { T } ) ) ^ { \mathsf { T } } \xi _ { \phi } ( h ( \boldsymbol { g } ) )
+$$
+
+we maximize the log likelihood given by
+
+$$
+\log \hat { q } ( s _ { g } = g | s _ { T } ; d _ { 1 } , . . . d _ { K } , \pi _ { \theta } ) = \log \frac { \exp { ( \beta \ell _ { g } ) } } { \exp { ( \beta \ell _ { g } ) } + \sum _ { k = 1 } ^ { K } \exp { ( \beta \xi _ { \phi } ( h ( s _ { T } ) ) ^ { \top } \xi _ { \phi } ( h ( d _ { k } ) ) ) } }
+$$
+
+where $\beta$ is an inverse temperature hyperparameter which we fix to $K + 1$ in all experiments. Note that (5) is a maximum log likelihood training objective for a softmax nearest neighbour classifier in a learned embedding space, making it similar to a matching network (Vinyals et al., 2016). Intuitively, updating the embedding $\xi _ { \phi }$ using the objective in (5) aims to increase the cosine similarity between $e { \left( s _ { T } \right) }$ and $e ( g )$ and to decrease the cosine similarity between $e { \left( s _ { T } \right) }$ and the decoy embeddings $e ( d ) , \ldots , e ( d _ { K } )$ . Subsampling the set of possible classes as we do is a known method for approximate maximum likelihood training of a softmax classifier with many classes (Bengio & Sen´ ecal, 2003). ´
+
+We use $\operatorname* { m a x } ( 0 , \ell _ { g } )$ as the reward for reaching state $s _ { T }$ when given goal $g$ . We found that this reward function is better behaved than the reward $\log \hat { q } ( s _ { g } = g \vert s _ { T } ; d _ { 1 } , . . . d _ { K } , \pi _ { \theta } )$ suggested by the DISCERN objective in Section 3 since it is scaled to lie in $[ 0 , 1 ]$ . The reward we use is also less noisy since, unlike $\log { \hat { q } }$ , it does not depend on the decoy states.
+
+Goal-conditioned policy: The goal-conditioned policy $\pi _ { \boldsymbol { \theta } } ( a | s ; s _ { g } )$ is trained to optimize the goal achievement reward $r ( s ; s _ { g } )$ . In this paper, $\pi _ { \theta } ( a | s ; s _ { g } )$ is an $\epsilon$ -greedy policy of a goal-conditioned action-value function $\mathrm { Q }$ with parameters $\theta$ . $\mathrm { Q }$ is trained using Q-learning and minibatch experience replay; specifically, we use the variant of $Q ( \lambda )$ due to Peng (see Chapter 7, Sutton & Barto (1998)).
+
+Goal relabelling: We use a form of goal relabelling (Kaelbling, 1993) or hindsight experience replay (Andrychowicz et al., 2017; Nair $\&$ Hinton, 2006) as a source successfully achieved goals as well as to regularize the embedding $e ( \cdot )$ . Specifically, for the purposes of parameter updates (in both the policy and the reward learner) we substitute, with probability $p _ { \mathrm { H E R } }$ the goal with an observation selected from the final $H$ steps of the trajectory, and consider the agent to have received a reward of 1. The motivation, in the case of the policy, is similar to that of previous work, i.e. that being in state $s _ { t }$ should correspond to having achieved the goal of reaching $s _ { t }$ . When employed in the reward learner, it amounts to encouraging temporally consistent state embeddings (Mobahi et al., 2009; Sermanet et al., 2017), i.e. encouraging observations which are nearby in time to have similar embeddings.
+
+Pseudocode for the DISCERN algorithm, decomposed into an experience-gathering (possibly distributed) actor process and a centralized learner process, is given in Algorithm 1.
+
+# 5 RELATED WORK
+
+The problem of reinforcement learning in the context of multiple goals dates at least to Kaelbling (1993), where the problem was examined in the context of grid worlds where the state space is
+
+# Algorithm 1: DISCERN
+
+<table><tr><td colspan="2">procedure ACTOR Input: Time budget T, policy parameters 0, goal embedding parameters Φ, shared goal</td></tr><tr><td colspan="2">π ←BEHAVIOR-POLICY(0) /* e.g. E-greedy */</td></tr><tr><td colspan="2">g~g r1:T-1←0 fort←1...Tdo Take action at ~ πθ(St; g) obtaining St+1 from p(St+1|St, at)</td></tr><tr><td colspan="2">9 ← PROPOSE-GOAL-SUBSTITUTION(G,St) /* See Appendix A3 */ end</td></tr><tr><td colspan="2">with probability PHER, Sample SHER uniformly from {sT-H,...,sT} and set g ← SHER,rT ←1 otherwise</td></tr><tr><td colspan="2">Compute lg using (4) rT ←max(0,lg) Send (s1:T,a1:T,r1:T,g) to the learner.</td></tr><tr><td colspan="2">Poll the learner periodically for updated values of 0,. Reset the environment if the episode has terminated.</td></tr><tr><td colspan="2">until termination procedure LEARNER</td></tr><tr><td colspan="2">Input:Batch size B, number of decoys K,initial policy parameters 0,initial goal embedding parametersΦ</td></tr><tr><td colspan="2">repeat Assemble batchof experienceB={(s:,:T,b)}1</td></tr><tr><td colspan="2">forb←1...Bdo</td></tr><tr><td colspan="2">Sample K decoy goals d,d,...,dk ~ g end</td></tr><tr><td colspan="2">Use an off-policy reinforcement learning algorithm to update θ based on B</td></tr><tr><td colspan="2">Update Φ to maximize B∑b=1 log@(sg = gʰ|sr;d1,..dk,πθ) computed by (5) B</td></tr></table>
+
+small and enumerable. Sutton et al. (2011) proposed generalized value functions (GVFs) as a way of representing knowledge about sub-goals, or as a basis for sub-policies or options. Universal Value Function Approximators (UVFAs) (Schaul et al., 2015) extend this idea by using a function approximator to parameterize a joint function of states and goal representations, allowing compact representation of an entire class of conditional value functions and generalization across classes of related goals.
+
+While the above works assume a goal achievement reward to be available a priori, our work includes an approach to learning a reward function for goal achievement jointly with the policy. Several recent works have examined reward learning for goal achievement in the context of the Generative Adversarial Networks (GAN) paradigm (Goodfellow et al., 2014). The SPIRAL (Ganin et al., 2018) algorithm trains a goal conditioned policy with a reward function parameterized by a Wasserstein GAN (Arjovsky et al., 2017) discriminator. Similarly, AGILE (Bahdanau et al., 2018) learns an instruction-conditional policy where goals in a grid-world are specified in terms of predicates which should be satisfied, and a reward function is learned using a discriminator trained to distinguish states achieved by the policy from a dataset of instruction, goal state pairs.
+
+Reward learning has also been used in the context of imitation. Ho & Ermon (2016) derives an adversarial network algorithm for imitation, while time-contrastive networks (Sermanet et al., 2017) leverage pre-trained ImageNet classifier representations to learn a reward function for robotics skills from video demonstrations, including robotic imitation of human poses. Universal Planning Networks (UPNs) (Srinivas et al., 2018) learn a state representation by training a differentiable planner to imitate expert trajectories. Experiments showed that once a UPN is trained the state representation it learned can be used to construct a reward function for visually specified goals. Bridging goal-conditioned policy learning and imitation learning, Pathak et al. (2018) learns a goal-conditioned policy and a dynamics model with supervised learning without expert trajectories, and present zero-shot imitation of trajectories from a sequence of images of a desired task.
+
+A closely related body of work to that of goal-conditioned reinforcement learning is that of unsupervised option or skill discovery. Machado & Bowling (2016) proposes a method based on an eigendecomposition of differences in features between successive states, further explored and extended in Machado et al. (2017). Variational Intrinsic Control (VIC) (Gregor et al., 2016) leverages the same lower bound on the mutual information as the present work in an unsupervised control setting, in the space of abstract options rather than explicit perceptual goals. VIC aims to jointly maximize the entropy of the set of options while making the options maximally distinguishable from their final states according to a parametric predictor. Recently, Eysenbach et al. (2018) showed that a special case of the VIC objective can scale to significantly more complex tasks and provide a useful basis for low-level control in a hierarchical reinforcement learning context.
+
+Other work has explored learning policies in tandem with a task policy, where the task or environment rewards are assumed to be sparse. Florensa et al. (2017) propose a framework in which low-level skills are discovered in a pre-training phase of a hierarchial system based on simple-to-design proxy rewards, while Riedmiller et al. (2018) explore a suite of auxiliary tasks through simultaneous off-policy learning.
+
+Several authors have explored a pre-training stage, sometimes paired with fine-tuning, based on unsupervised representation learning. Per´ e et al. (2018) and Laversanne-Finot et al. (2018) employ a ´ two-stage framework wherein unsupervised representation learning is used to learn a model of the observations from which to sample goals for control in simple simulated environments. Nair et al. (2018) propose a similar approach in the context of model-free Q-learning applied to 3-dimensional simulations and robots. Goals for training the policy are sampled from the model’s prior, and a reward function is derived from the latent codes. This contrasts with our non-parametric approach to selecting goals, as well as our method for learning the goal space online and jointly with the policy.
+
+An important component of our method is a form of goal relabelling, introduced to the reinforcement learning literature as hindsight experience replay by Andrychowicz et al. (2017), based on the intuition that any trajectory constitutes a valid trajectory which achieves the goal specified by its own terminal observation. Earlier, Nair & Hinton (2006) employed a related scheme in the context of supervised learning of motor programs, where a program encoder is trained on pairs of trajectory realizations and programs obtained by expanding outwards from a pre-specified prototypical motor program through the addition of noise. Veeriah et al. (2018) expands upon hindsight replay and the all-goal update strategy proposed by Kaelbling (1993), generalizing the latter to non-tabular environments and exploring related strategies for skill discovery, unsupervised pre-training and auxiliary tasks. Levy et al. (2018) propose a hierarchical Q-learning system which employs hindsight replay both conventionally in the lower-level controller and at higher levels in the hierarchy. Nair et al. (2018) also employ a generalized goal relabeling scheme whereby the policy is trained based on a trajectory’s achievement not just of its own terminal observation, but a variety of retrospectively considered possible goals.
+
+# 6 EXPERIMENTS
+
+We evaluate, both qualitatively and quantitatively, the ability of DISCERN to achieve visuallyspecified goals in three diverse domains – the Arcade Learning Environment (Bellemare et al., 2013), continuous control tasks in the DeepMind Control Suite (Tassa et al., 2018), and DeepMind Lab, a 3D first person environment (Beattie et al., 2016). Experimental details including architecture details, details of distributed training, and hyperparameters can be found in the Appendix. We compared DISCERN to several baseline methods for learning goal-conditioned policies:
+
+Conditioned Autoencoder (AE): In order to specifically interrogate the role of the discriminative reward learning criterion, we replace the discriminative criterion for embedding learning with an $L ^ { 2 }$ reconstruction loss on $h _ { t }$ ; that is, in addition to $\xi _ { \phi } ( \cdot )$ , we learn an inverse mapping $\xi _ { \phi } ^ { - 1 } ( \cdot )$ with a separate set of parameters, and train both with the criterion $\| h _ { t } - \xi _ { \phi } ^ { - 1 } ( \xi _ { \phi } ( h _ { t } ) ) \| ^ { 2 }$ .
+
+Conditioned WGAN Discriminator: We compare to an adversarial reward on the domains considered according to the protocol of Ganin et al. (2018), who successfully used a WGAN discriminator as a reward for training agents to perform inverse graphics tasks. The discriminator takes two pairs of images – (1) a real pair of goal images $( s _ { g } , s _ { g } )$ and (2) a fake pair consisting of the terminal state of the agent and the goal frame $( s _ { t } , s _ { g } )$ . The output of the discriminator is used as the reward function for the policy. Unlike our DISCERN implementation and the conditioned autoencoder baseline, we train the WGAN discriminator as a separate convolutional network directly from pixels, as in previous work.
+
+Pixel distance reward (L2): Finally, we directly compare to a reward based on $L ^ { 2 }$ distance in pixel space, equal to e $\mathbf { \boldsymbol { x } } \mathbf { \boldsymbol { p } } \left( - \lVert \mathbf { \boldsymbol { s } } _ { t } - \mathbf { \boldsymbol { s } } _ { g } \rVert ^ { 2 } / \sigma _ { \mathrm { p i x e l } } \right)$ where $\sigma _ { \mathrm { p i x e l } }$ is a hyperparameter which we tuned on a per-environment basis.
+
+All the baselines use the same goal-conditioned policy architecture as DISCERN. The baselines also used hindsight experience replay in the same way as DISCERN. They can therefore be seen as ablations of DISCERN’s goal-achievement reward learning mechanism.
+
+# 6.1 ATARI
+
+The suite of 57 Atari games provided by the Arcade Learning Environment (Bellemare et al., 2013) is a widely used benchmark in the deep reinforcement learning literature. We compare DISCERN to other methods on the task of achieving visually specified goals on the games of Seaquest and Montezuma’s Revenge. The relative simplicity of these domains makes it possible to handcraft a detector in order to localize the controllable aspects of the environment, namely the submarine in Seaquest and Panama Joe, the character controlled by the player in Montezuma’s Revenge.
+
+We evaluated the methods by running the learned goal policies on a fixed set of goals and measured the percentage of goals it was able to reach successfully. We evaluated both DISCERN and the baselines with two different goal buffer substitution strategies, uniform and diverse, which are described in the Appendix. A goal was deemed to be successfully achieved if the position of the avatar in the last frame was within $10 \%$ of the playable area of the position of the avatar in the goal for each controllable dimension. The controllable dimensions in Atari were considered to be the $\mathbf { X } ^ { - }$ and y-coordinates of the avatar. The results are displayed in Figure 1a. DISCERN learned to achieve a large fraction of goals in both Seaquest and Montezuma’s Revenge while none of the baselines learned to reliably achieve goals in either game. We hypothesize that the baselines failed to learn to control the avatars because their objectives are too closely tied to visual similarity. Figure 1b shows examples of goal achievement on Seaquest and Montezuma’s Revenge. In Seaquest, DISCERN learned to match the position of the submarine in the goal image while ignoring the position of the fish, since the fish are not directly controllable. We have provided videos of the goal-conditioned policies learned by DISCERN on Seaquest and Montezuma’s Revenge at the following anonymous URL https://sites.google.com/view/discern-anonymous/home.
+
+# 6.2 DEEPMIND CONTROL SUITE TASKS
+
+The DeepMind Control Suite (Tassa et al., 2018) is a suite of continuous control tasks built on the MuJoCo physics engine (Todorov et al., 2012). While most frequently used to evaluate agents which receive the underlying state variables as observations, we train our agents on pixel renderings of the scene using the default environment-specified camera, and do not directly observe the state variables.
+
+Agents acting greedily with respect to a state-action value function require the ability to easily maximize $Q$ over the candidate actions. For ease of implementation, as well as comparison to other considered environments, we discretize the space of continuous actions to no more than 11 unique actions per environment (see Appendix A4.1).
+
+The availability of an underlying representation of the physical state, while not used by the learner, provides a useful basis for comparison of achieved states to goals. We mask out state variables relating to entities in the scene not under the control of the agent; for example, the position of the target in the reacher or manipulator domains.
+
+DISCERN is compared to the baselines on a fixed set of 100 goals with 20 trials for each goal. The goals are generated by acting randomly for 25 environment steps after initialization. In the case of cartpole, we draw the goals from a random policy acting in the environment set to the balance task, where the pole is initialized upwards, in order to generate a more diverse set of goals against which to measure. Figure 3 compares learning progress of 5 independent seeds for the “uniform” goal replacement strategy (see Appendix A5 for results with “diverse” goal replacement) for 6 domains. We adopt the same definition of achievement as in Section 6.1. Figure 2 summarizes averaged goal achievement frames on these domains except for the cartpole domain for policies learned by DISCERN. Performance on cartpole is discussed in more detail in Figure 7 of the Appendix.
+
+![](images/5b3e12a48cafd1721f4333b127759482b9454525058189dba01de4eb721472a3.jpg)  
+Figure 1: a) Percentage of goals successfully achieved on Seaquest and Montezuma’s Revenge. b) Examples of goals achieved by DISCERN on the games of Seaquest (top) and Montezuma’s Revenge (bottom). For each game, the four goal states are shown in the top row. Below each goal is the averaged (over 5 trials) final state achieved by the goal-conditioned policy learned by DISCERN after $T = 5 0$ steps for the goal above.
+
+![](images/7b3edb4b948bf794500d2ba25ac3f4200f8eb96222c803675d051d10fe952eb2.jpg)  
+Figure 2: Average achieved frames for point mass (task easy), reacher (task hard), manipulator (task bring ball), pendulum (task swingup), finger (task spin) and ball in cup (task catch) environments. The goal is shown in the top row and the achieved frame is shown in the bottom row.
+
+The results show that in aggregate, DISCERN outperforms baselines in terms of goal achievement on several, but not all, of the considered Control Suite domains. In order to obtain a more nuanced understanding of DISCERN’s behaviour when compared with the baselines, we also examined achievement in terms of the individual dimensions of the controllable state. Figure 4 shows goal achievement separately for each dimension of the underlying state on four domains. The perdimension results show that on difficult goal-achievement tasks such as those posed in cartpole (where most proposed goal states are unstable due to the effect of gravity) and finger (where a free-spinning piece is only indirectly controllable) DISCERN learns to reliably match the major dimensions of controllability such as the cart position and finger pose while ignoring the other dimensions, whereas none of the baselines learned to reliably match any of the controllable state dimensions on the difficult tasks cartpole and finger.
+
+![](images/114229ab7fa3bfb129b7f218526a96105037f7769482bc7693d43f7f5a857239.jpg)  
+Figure 3: Quantitative evaluation of goal achievement on continuous control domains using the “uniform” goal substitution scheme (see Appendix A3). For each method, we show the fraction of goals achieved over a fixed goal set (100 images per domain).
+
+![](images/c0e9f48e1d12f29d0e502308dbf0bbc4c8472777cc3ae707af685e34eedf153c.jpg)  
+Figure 4: Per-dimension quantitative evaluation of goal achievement on continuous control domains using the “uniform” goal substitution scheme (Appendix A3). Each subplot corresponds to a domain, with each group of colored rows representing a method. Each individual row represents a dimension of the controllable state (such as a joint angle). The color of each cell indicates the fraction of goal states for which the method was able to match the ground truth value for that dimension to within $1 0 \%$ of the possible range. The position along the $x$ -axis indicates the point in training in millions of frames. For example, on the reacher domain DISCERN learns to match both dimensions of the controllable state, but on the cartpole domain it learns to match the first dimension (cart position) but not the second dimension (pole angle).
+
+We omitted the manipulator domain from these figures as none of the methods under consideration achieved non-negligible goal achievement performance on this domain, however a video showing the policy learned by DISCERN on this domain can be found at https://sites.google.com/view/discern-anonymous/home. The policy learned on the manipulator domain shows that DISCERN was able to discover several major dimensions of controllability even on such a challenging task, as further evidenced by the per-dimension analysis on the manipulator domain in Figure 8 in the Appendix.
+
+![](images/4989701958c232e7bd7a0ac51efe637aad8935114d605e32c3234a2672c05347.jpg)  
+Figure 5: Average achieved frames over 30 trials from a random initialization on the rooms watermaze task. Goals are shown in the top row while the corresponding average achieved frames are in the bottom row.
+
+# 6.3 DEEPMIND LAB
+
+DeepMind Lab (Beattie et al., 2016) is a platform for 3D first person reinforcement learning environments. We trained DISCERN on the watermaze level and found that it learned to approximately achieve the same wall and horizon position as in the goal image. While the agent did not learn to achieve the position and viewpoint shown in a goal image as one may have expected, it is encouraging that our approach learns a reasonable space of goals on a first-person 3D domain in addition to domains with third-person viewpoints like Atari and the DM Control Suite.
+
+# 7 DISCUSSION
+
+We have presented a system that can learn to achieve goals, specified in the form of observations from the environment, in a purely unsupervised fashion, i.e. without any extrinsic rewards or expert demonstrations. Integral to this system is a powerful and principled discriminative reward learning objective, which we have demonstrated can recover the dominant underlying degrees of controllability in a variety of visual domains.
+
+In this work, we have adopted a fixed episode length of $T$ in the interest of simplicity and computational efficiency. This implicitly assumes not only that all sampled goals are approximately achievable in $T$ steps, but that the policy need not be concerned with finishing in less than the allotted number of steps. Both of these limitations could be addressed by considering schemes for early termination based on the embedding, though care must be taken not to deleteriously impact training by terminating episodes too early based on a poorly trained reward embedding. Relatedly, our goal selection strategy is agnostic to both the state of the environment at the commencement of the goal episode and the current skill profile of the policy, utilizing at most the content of the goal itself to drive the evolution of the goal buffer $\mathcal { G }$ . We view it as highly encouraging that learning proceeds using such a naive goal selection strategy, however more sophisticated strategies, such as tracking and sampling from the frontier of currently achievable goals (Held et al., 2017), may yield substantial improvements.
+
+DISCERN’s ability to automatically discover controllable aspects of the observation space is a highly desirable property in the pursuit of robust low-level control. A natural next step is the incorporation of DISCERN into a deep hierarchical reinforcement learning setup (Vezhnevets et al., 2017; Levy et al., 2018; Nachum et al., 2018) where a meta-policy for proposing goals is learned after or in tandem with a low-level controller, i.e. by optimizing an extrinsic reward signal.
+
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+
+# APPENDIX
+
+# A1 DISTRIBUTED TRAINING
+
+We employ a distributed reinforcement learning architecture inspired by the IMPALA reinforcement learning architecture (Espeholt et al., 2018), with a centralized GPU learner batching parameter updates on experience collected by a large number of CPU-based parallel actors. While Espeholt et al. (2018) learns a stochastic policy through the use of an actor-critic architecture, we instead learn a goal-conditioned state-action value function with Q-learning. Each actor acts $\epsilon$ -greedily with respect to a local copy of the $Q$ network, and sends observations $s _ { t }$ , actions $a _ { t }$ , rewards $r _ { t }$ and discounts $\gamma _ { t }$ for a trajectory to the learner. Following Horgan et al. (2018), we use a different value of $\epsilon$ for each actor, as this has been shown to improve exploration. The learner batches re-evaluation of the convolutional network and LSTM according to the action trajectories supplied and performs parameter updates, periodically broadcasting updated model parameters to the actors. As Q-learning is an off-policy algorithm, the experience traces sent to the learner can be used in the usual $n$ -step Q-learning update without the need for an off-policy correction as in Espeholt et al. (2018). We also maintain actor-local replay buffers of previous actor trajectories and use them to perform both standard experience replay (Lin, 1993) and our variant of hindsight experience replay (Andrychowicz et al., 2017).
+
+# A2 ARCHITECTURE DETAILS
+
+Our network architectures closely resemble those in Espeholt et al. (2018), with policy and value heads replaced with a $Q$ -function. We apply the same convolutional network to both $s _ { t }$ and $s _ { g }$ and concatenate the final layer outputs. Note that the convolutional network outputs for $s _ { g }$ need only be computed once per episode. We include a periodic representation $( \sin ( 2 \pi t / T ) , \cos ( 2 \pi t / T ) )$ of the current time step, with period equal to the goal length achievement period $T$ , as an extra input to the network. The periodic representation is processed by a single hidden layer of rectified linear units and is concatenated with the visual representations fed to the LSTM. While not strictly necessary, we find that this allows the agent to become better at achieving goal states which may be unmaintainable due to their instability in the environment dynamics.
+
+The output of the LSTM is the input to a dueling action-value output network (Wang et al., 2016). In all of our experiments, both branches of the dueling network are linear mappings. That is, given LSTM outputs $\psi _ { t }$ , we compute the action values for the current time step $t$ as
+
+$$
+Q ( a _ { t } | \psi _ { t } ) = { \psi _ { t } } ^ { \mathsf { T } } \mathbf { v } + \left( \psi _ { t } ^ { \mathsf { T } } \mathbf { w } _ { a _ { t } } - { \frac { 1 } { n } } \sum _ { a _ { t } ^ { \prime } } { \psi _ { t } } ^ { \mathsf { T } } \mathbf { w } _ { a _ { t } ^ { \prime } } \right) + b
+$$
+
+# A3 GOAL BUFFER
+
+We experimented with two strategies for updating the goal buffer. In the first strategy, which we call uniform, the current observation replaces a uniformly selected entry in the goal buffer with probability $p _ { \mathrm { r e p l a c e } }$ . The second strategy, which we refer to as diverse goal sampling attempts to maintain a goal buffer that more closely approximates the uniform distribution over all observation. In the diverse goal strategy, we consider the current observation for addition to the goal buffer with probability $p _ { \mathrm { r e p l a c e } }$ at each step during acting. If the current observation $s$ is considered for addition to the goal buffer, then we select a random removal candidate $s _ { r }$ by sampling uniformly from the goal buffer and replace it with $s$ if $s _ { r }$ is closer to the rest of the goal buffer than $s$ . If $s$ is closer to the rest of the goal buffer than $s _ { r }$ then we still replace $s _ { r }$ with $s$ with probability padd−non−diverse. We used $L _ { 2 }$ distance in pixel space for the diverse sampling strategy and found it to greatly increase the coverage of states in the goal buffer, especially early during training. This bears some relationship to Determinantal Point Processes (Kulesza et al., 2012), and goal-selection strategies with a more explicit theoretical foundation are a promising future direction.
+
+# A4 EXPERIMENTAL DETAILS
+
+The following hyper-parameters were used in all of the experiments described in Section 6. All weight matrices are initialized using a standard truncated normal initializer, with the standard deviation inversely proportional to the square root of the fan-in. We maintain a goal buffer of size 1024 and use $p _ { \mathrm { r e p l a c e } } \overset { \cdot } { = } 1 0 ^ { - 3 }$ . We also use $p _ { \mathrm { a d d - n o n - d i v e r s e } } = 1 0 ^ { - 3 }$ . For the teacher, we choose $\xi _ { \phi } ( \cdot )$ to be an $\bar { \mathcal { L } } _ { 2 }$ -normalized single layer of 32 tanh units, trained in all experiments with 4 decoys (and thus, according to our heuristic, $\beta$ equal to 5). For hindsight experience replay, a highsight goal is substituted $2 5 \%$ of the time. These goals are chosen uniformly at random from the last 3 frames of the trajectory. Trajectories were set to be 50 steps long for Atari and DeepMind Lab and 100 for the DeepMind control suite. It is important to note that the environment was not reset after each trajectory, but rather the each new trajectory begins where the previous one ended. We train the agent and teacher jointly with RMSProp (Tieleman & Hinton, 2012) with a learning rate of $1 0 ^ { - 4 }$ . We follow the preprocessing protocol of Mnih et al. (2015), resizing to $8 4 \times 8 4$ pixels and scaling 8-bit pixel values to lie in the range [0, 1]. While originally designed for Atari, we apply this preprocessing pipeline across all environments used in this paper.
+
+# A4.1 CONTROL SUITE
+
+In the point mass domain we use a control step equal to 5 times the task-specified default, i.e. the agent acts on every fifth environment step (Mnih et al., 2015). In all other Control Suite domains, we use the default. We use the “easy” version of the task where actuator semantics are fixed across environment episodes.
+
+Discrete action spaces admit function approximators which simultaneously compute the action values for all possible actions, as popularized in Mnih et al. (2015). The action with maximal $Q$ -value can thus be identified in time proportional to the cardinality of the action space. An enumeration of possible actions is no longer possible in the continuous setting. While approaches exist to enable continuous maximization in closed form (Gu et al., 2016), they come at the cost of greatly restricting the functional form of $Q$ .
+
+For ease of implementation, as well as comparison to other considered environments, we instead discretize the space of continuous actions. For all Control Suite environments considered except manipulator, we discretize an $A$ -dimensional continuous action space into $3 ^ { A }$ discrete actions, consisting of the Cartesian product over action dimensions with values in $\{ - 1 , 0 , 1 \}$ . In the case of manipulator, we adopt a “diagonal” discretization where each action consists of setting one actuator to $\pm 1$ , and all other actuators to 0, with an additional action consisting of every actuator being set to 0. This is a reasonable choice for manipulator because any position can be achieved by a concatenation of actuator actions, which may not be true of more complex Control Suite environments such as humanoid, where the agent’s body is subject to gravity and successful trajectories may require multi-joint actuation in a single control time step. The subset of the Control Suite considered in this work was chosen primarily such that the discretized action space would be of a reasonable size. We leave extensions to continuous domains to future work.
+
+# A5 ADDITIONAL EXPERIMENTAL RESULTS
+
+# A5.1 ATARI
+
+We ran two additional baselines on Seaquest and Montezuma’s Revenge, ablating our use of hindsight experience replay in opposite ways. One involved training the goal-conditioned policy only in hindsight, without any learned goal achievement reward, i.e. $p _ { \mathrm { H E R } } = 1$ . This approach achieved $1 2 \%$ of goals on Seaquest and $1 \bar { 1 } . 4 \%$ of goals on Montezuma’s Revenge, making it comparable to a uniform random policy. This result underscores the importance of learning a goal achievement reward. The second baseline consisted of DISCERN learning a goal achievement reward without hindsight experience replay, i.e. $p _ { \mathrm { H E R } } = 0$ . This also performed poorly, achieving $1 1 . 4 \%$ of goals on Seaquest and $8 \%$ of goals on Montezuma’s Revenge. Taken together, these preliminary results suggest that the combination of hindsight experience replay and a learned goal achievement reward is important.
+
+# A5.2 CONTROL SUITE
+
+For the sake of completeness, Figure 6 reports goal achievement curves on Control Suite domains using the “diverse” goal selection scheme.
+
+![](images/a7ca0c9ad1c291d0cbfaddf7270977c0c4deeebe096c75c589562b2587954197.jpg)  
+Figure 6: Results for Control Suite tasks using the “diverse” goal substitution scheme.
+
+![](images/4242284d4a82a40cfc6cafcd2d5920b721064c931f25cfaf54bc1a0196440044.jpg)  
+Figure 7: Average goal achievement on cartpole. Top row shows the goals. Middle row shows achievement by the Autoencoder baseline. Bottom row shows average goal achievement by DISCERN. Shading of columns is for emphasis. DISCERN always matches the cart position. The autoencoder baseline matches both cart and pole position when the pole is pointing down, but fails to match either when the pole is pointing up.
+
+Figure 7 displays goal achievements for DISCERN and the Autoencoder baseline, highlighting DISCERN’s preference for communicating with the cart position, and robustness to the pole positions unseen during training.
+
+![](images/b29cc81a794b8f87d7a0a882d3bf3bc279517f9a447430ce3aaed586759b12c3.jpg)  
+Figure 8: Per-dimension quantitative evaluation on the manipulator domain. See Figure 4 for a description of the visualization. DISCERN learns to reliably control more dimensions of the underlying state than any of the baselines.

parse/train/r1eVMnA9K7/r1eVMnA9K7_content_list.json

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+    {
+        "type": "text",
+        "text": "UNSUPERVISED CONTROL THROUGH NON-PARAMETRIC DISCRIMINATIVE REWARDS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            99,
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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,
+            236,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Learning to control an environment without hand-crafted rewards or expert data remains challenging and is at the frontier of reinforcement learning research. We present an unsupervised learning algorithm to train agents to achieve perceptuallyspecified goals using only a stream of observations and actions. Our agent simultaneously learns a goal-conditioned policy and a goal achievement reward function that measures how similar a state is to the goal state. This dual optimization leads to a co-operative game, giving rise to a learned reward function that reflects similarity in controllable aspects of the environment instead of distance in the space of observations. We demonstrate the efficacy of our agent to learn, in an unsupervised manner, to reach a diverse set of goals on three domains – Atari, the DeepMind Control Suite and DeepMind Lab. ",
+        "bbox": [
+            233,
+            265,
+            766,
+            416
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            440,
+            336,
+            457
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Currently, the best performing methods on many reinforcement learning benchmark problems combine model-free reinforcement learning methods with policies represented using deep neural networks (Horgan et al., 2018; Espeholt et al., 2018). Despite reaching or surpassing human-level performance on many challenging tasks, deep model-free reinforcement learning methods that learn purely from the reward signal learn in a way that differs greatly from the manner in which humans learn. In the case of learning to play a video game, a human player not only acquires a strategy for achieving a high score, but also gains a degree of mastery of the environment in the process. Notably, a human player quickly learns which aspects of the environment are under their control as well as how to control them, as evidenced by their ability to rapidly adapt to novel reward functions (Lake et al., 2017). ",
+        "bbox": [
+            174,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Focusing learning on mastery of the environment instead of optimizing a single scalar reward function has many potential benefits. One benefit is that learning is possible even in the absence of an extrinsic reward signal or with an extrinsic reward signal that is very sparse. Another benefit is that an agent that has fully mastered its environment should be able to reach arbitrary achievable goals, which would allow it to generalize to tasks on which it wasn’t explicitly trained. Building reinforcement learning agents that aim for environment mastery instead of or in addition to learning about a scalar reward signal is currently an open challenge. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "One way to represent such knowledge about an environment is using an environment model. Modelbased reinforcement learning methods aim to learn accurate environment models and use them either for planning or for training a policy. While learning accurate environment models of some visually rich environments is now possible (Oh et al., 2015; Chiappa et al., 2018; Ha & Schmidhuber, 2018) using learned models in model-based reinforcement learning has proved to be challenging and model-free approaches still dominate common benchmarks. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We present a new model-free agent architecture of Discriminative Embedding Reward Networks, or DISCERN for short. DISCERN learns to control an environment in an unsupervised way by learning purely from the stream of observations and actions. The aim of our agent is to learn a goal-conditioned policy $\\pi _ { \\boldsymbol { \\theta } } ( a | s ; s _ { g } )$ (Kaelbling, 1993; Schaul et al., 2015) which can reach any goal state $s _ { g }$ that is reachable from the current state $s$ . We show how to learn a goal achievement reward function $\\dot { \\boldsymbol { r } } ( s ; s _ { g } )$ that measures how similar state $s$ is to state $s _ { g }$ using a mutual information objective at the same time as learning $\\pi _ { \\boldsymbol { \\theta } } ( a | s ; s _ { g } )$ . The resulting learned reward function $r ( s ; s _ { g } )$ measures similarity in the space of controllable aspects of the environment instead of in the space of raw observations. ",
+        "bbox": [
+            174,
+            811,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Crucially, the DISCERN architecture is able to deal with goal states that are not perfectly reachable, for example, due to the presence of distractor objects that are not under the agent’s control. In such cases the goal-conditioned policy learned by DISCERN tends to seek states where the controllable elements match those in the goal state as closely as possible. ",
+        "bbox": [
+            174,
+            103,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We demonstrate the effectiveness of our approach on three domains – Atari games, continuous control tasks from the DeepMind Control Suite, and DeepMind Lab. We show that our agent learns to successfully achieve a wide variety of visually-specified goals, discovering underlying degrees of controllability of an environment in a purely unsupervised manner and without access to an extrinsic reward signal. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "2 PROBLEM FORMULATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            256,
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+            272
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In the standard reinforcement learning setup an agent interacts with an environment over discrete time steps. At each time step $t$ the agent observes the current state $s _ { t }$ and selects an action $a _ { t }$ according to a policy $\\pi ( \\boldsymbol { a } _ { t } | \\boldsymbol { s } _ { t } )$ . The agent then receives a reward $r _ { t } = r ( s _ { t } , a _ { t } )$ and transitions to the next state $s _ { t + 1 }$ . The aim of learning is to maximize the expected discounted return $\\textstyle R = \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } r _ { t }$ of policy $\\pi$ where $\\gamma \\in [ 0 , 1 )$ is a discount factor. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this work we focus on learning only from the stream of actions and observations in order to forego the need for an extrinsic reward function. Motivated by the idea that an agent capable of reaching any reachable goal state $s _ { g }$ from the current state $s$ has complete mastery of its environment, we pose the problem of learning in the absence of rewards as one of learning a goal-conditioned policy $\\pi _ { \\boldsymbol { \\theta } } ( a | s ; s _ { g } )$ with parameters $\\theta$ . More specifically, we assume that the agent interacts with an environment defined by a transition distribution $p ( s _ { t + 1 } | s _ { t } , a _ { t } )$ . We define a goal-reaching problem as follows. At the beginning of each episode, the agent receives a goal $s _ { g }$ sampled from a distribution over possible goals $p _ { g o a l }$ . For example, $p _ { g o a l }$ could be the uniform distribution over all previously visited states. The agent then acts for $T$ steps according to the goal-conditioned policy $\\pi _ { \\boldsymbol { \\theta } } ( a | s ; \\bar { s } _ { g } )$ receiving a reward of 0 for each of the first $T - 1$ actions and a reward of $r ( s _ { T } ; s _ { g } )$ after the last action, where $r ( s ; s _ { g } ) \\in [ 0 , 1 ]$ for all $s$ and $s _ { g }$ 1. The goal achievement reward function $r ( s ; s _ { g } )$ measures the degree to which being in state $s$ achieves goal $s _ { g }$ . The episode terminates upon the agent receiving the reward $r ( s _ { T } ; s _ { g } )$ and a new episode begins. ",
+        "bbox": [
+            173,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "It is straightforward to train $\\pi _ { \\boldsymbol { \\theta } } ( a | s ; s _ { g } )$ in a tabular environment using the indicator reward $r ( s ; s _ { g } ) ~ \\bar { = } ~ \\mathbb { 1 } \\{ s ~ = ~ s _ { g } \\}$ . We are, however, interested in environments with continuous highdimensional observation spaces. While there is extensive prior work on learning goal-conditioned policies (Kaelbling, 1993; Schaul et al., 2015; Andrychowicz et al., 2017; Held et al., 2017; Pathak et al., 2018), the reward function is often hand-crafted, limiting generality of the approaches. In the few cases where the reward is learned, the learning objective is typically tied to a pre-specified notion of visual similarity. Learning to achieve goals based purely on visual similarity is unlikely to work in complex, real world environments due to the possible variations in appearance of objects, or goal-irrelevant perceptual context. We now turn to the problem of learning a goal achievement reward function $r _ { \\phi } ( s ; s _ { g } )$ with parameters $\\phi$ for high-dimensional state spaces. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3 LEARNING A REWARD FUNCTION BY MAXIMIZING MUTUAL INFORMATION ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            712,
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+    },
+    {
+        "type": "text",
+        "text": "We aim to simultaneously learn a goal-conditioned policy $\\pi _ { \\theta }$ and a goal achievement reward function $r _ { \\phi }$ by maximizing the mutual information between the goal state $s _ { g }$ and the achieved state $s _ { T }$ as shown in (1). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f8115495a99fbf160078d0df5ef708cd8e3e95937f1718cc8a5269ed3effd918.jpg",
+        "text": "$$\nI ( s _ { g } , s _ { T } ) = H ( s _ { g } ) + \\mathbb { E } _ { s _ { g } , s _ { T } \\sim p ( s _ { g } , s _ { T } ) } \\log p ( s _ { g } | s _ { T } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Note that we are slightly overloading notation by treating $s _ { g }$ as a random variable distributed according to $p _ { g o a l }$ . Similarly, $s _ { T }$ is a random variable distributed according to the state distribution induced by running $\\pi _ { \\theta }$ for $T$ steps for goal states sampled from $p _ { g o a l }$ . ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The prior work of Gregor et al. (2016) showed how to learn a set of abstract options by optimizing a similar objective, namely the mutual information between an abstract option and the achieved state. Following their approach, we simplify (1) in two ways. First, we rewrite the expectation in terms of the goal distribution $p _ { g o a l }$ and the goal conditioned policy $\\pi _ { \\theta }$ . Second, we lower bound the expectation term by replacing $\\check { p } ( s _ { g } | s _ { T } )$ with a variational distribution $q _ { \\phi } ( s _ { g } | s _ { T } )$ with parameters $\\phi$ following Barber $\\&$ Agakov (2004), leading to ",
+        "bbox": [
+            176,
+            859,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
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+        "img_path": "images/74d6b74ea56623ce74e0ca2f236162884c7ef739d63bc0e616773a175f544601.jpg",
+        "text": "$$\nI ( s _ { g } , s _ { T } ) \\geq H ( s _ { g } ) + \\mathbb { E } _ { s _ { g } \\sim p _ { g o a l } , s _ { 1 } , \\dots s _ { T } \\sim \\pi _ { \\theta } ( \\dots | s _ { g } ) } \\log q _ { \\phi } ( s _ { g } | s _ { T } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Finally, we discard the entropy term $H ( s _ { g } )$ from (2) because it does not depend on either the policy parameters $\\theta$ or the variational distribution parameters $\\phi$ , giving our overall objective ",
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+        "page_idx": 2
+    },
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+        "type": "equation",
+        "img_path": "images/6a5974b27a2b14e72cb9b39451c111792b35ed7d228ef7e32e47f4943fbb4bcc.jpg",
+        "text": "$$\nO _ { \\mathrm { D I S C E R N } } = \\mathbb { E } _ { s _ { g } \\sim p _ { g o a l } , s _ { 1 } , \\ldots s _ { T } \\sim \\pi _ { \\theta } ( \\cdots | s _ { g } ) } \\log q _ { \\phi } ( s _ { g } | s _ { T } ) .\n$$",
+        "text_format": "latex",
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+        "text": "This objective may seem difficult to work with because the variational distribution $q _ { \\phi }$ is a distribution over possible goals $s _ { g }$ , which in our case are high-dimensional observations, such as images. We sidestep the difficulty of directly modelling the density of high-dimensional observations by restricting the set of possible goals to be a finite subset of previously encountered states that evolves over time (Lin, 1993). Restricting the support of $q _ { \\phi }$ to a finite set of goals turns the problem of learning $q _ { \\phi }$ into a problem of modelling the conditional distribution of possible intended goals given an achieved state, which obviates the requirement of modelling arbitrary statistical dependencies in the observations.2 ",
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+        "text": "Optimization: The expectation in the DISCERN objective is with respect to the distribution of trajectories generated by the goal-conditioned policy $\\pi _ { \\theta }$ acting in the environment against goals drawn from the goal distribution $p _ { g o a l }$ . We can therefore optimize this objective with respect to policy parameters $\\theta$ by repeatedly generating trajectories and performing reinforcement learning updates on $\\pi _ { \\theta }$ with a reward of $\\log { \\bar { q _ { \\phi } } ( s _ { g } | s _ { T } ) }$ given at time $T$ and 0 for other time steps. Optimizing the objective with respect to the variational distribution parameters $\\phi$ is also straightforward since it is equivalent to a maximum likelihood classification objective. As will be discussed in the next section, we found that using a reward that is a non-linear transformation mapping $\\log q _ { \\phi } ( s _ { g } | s _ { T } )$ to $[ 0 , 1 ]$ worked better in practice. Nevertheless, since the reward for the goal conditioned-policy is a function of $\\log q _ { \\phi } ( s _ { g } \\bar { | } s _ { T } )$ , training the variational distribution function $q _ { \\phi }$ amounts to learning a reward function. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Communication Game Interpretation: Dual optimization of the DISCERN objective has an appealing interpretation as a cooperative communication game between two players – an imitator that corresponds to the goal-conditioned policy and a teacher that corresponds to the variational distribution. At the beginning of each round or episode of the game the imitator is provided with a goal state. The aim of the imitator is to communicate the goal state to the teacher by taking $T$ actions in the environment. After the imitator takes $T$ actions, the teacher has to guess which state from a set of possible goals was given to the imitator purely from observing the final state $s _ { T }$ reached by the imitator. The teacher does this by assigning a probability to each candidate goal state that it was the goal given to the imitator at the start of the episode, i.e. it produces a distribution $p ( s _ { g } | s _ { T } )$ . The objective of both players is for the teacher to guess the goal given to the imitator correctly as measured by the log probability assigned by the teacher to the correct goal. ",
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+    {
+        "type": "text",
+        "text": "4 DISCRIMINATIVE EMBEDDING REWARD NETWORKS ",
+        "text_level": 1,
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+        "text": "We now describe the DISCERN algorithm – a practical instantiation of the approach for jointly learning $\\pi _ { \\boldsymbol { \\theta } } ( a | s ; s _ { g } )$ and $r ( s ; s _ { g } )$ outlined in the previous section. ",
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+        "text": "Goal distribution: We adopt a non-parametric approach to the problem of proposing goals, whereby we maintain a fixed size buffer $\\mathcal { G }$ of past observations from which we sample goals during training. We update $\\mathcal { G }$ by replacing the contents of an existing buffer slot with an observation from the agent’s recent experience according to some substitution strategy; in this work we considered two such strategies, detailed in Appendix A3. This means that the space of goals available for training drifts as a function of the agent’s experience, and states which may not have been reachable under a poorly trained policy become reachable and available for substitution into the goal buffer, leading to a naturally induced curriculum. In this work, we sample training goals for our agent uniformly at random from the goal buffer, leaving the incorporation of more explicitly instantiated curricula to future work. ",
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+        "text": "Goal achievement reward: We train a goal achievement reward function $r ( s ; s _ { g } )$ used to compute rewards for the goal-conditioned policy based on a learned measure of state similarity. We parameterize $r ( s ; s _ { g } )$ as the positive part of the cosine similarity between $s$ and $s _ { g }$ in a learned embedding space, although shaping functions other than rectification could be explored. The state embedding in which we measure cosine similarity is the composition of a feature transformation $h ( \\cdot )$ and a learned $L ^ { 2 }$ -normalized mapping $\\xi _ { \\phi } ( \\cdot )$ . In our implementation, where states and goals are represented as 2-D RGB images, we take $h ( \\cdot )$ to be the final layer features of the convolutional network learned by the policy in order to avoid learning a second convolutional network. We find this works well provided that while training $r$ , we treat $h ( \\cdot )$ as fixed and do not adapt the convolutional network’s parameters with respect to the reward learner’s loss. This has the effect of regularizing the reward learner by limiting its adaptive capacity while avoiding the need to introduce a hyperparameter weighing the two losses against one another. ",
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+    {
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+        "text": "We train $\\xi _ { \\phi } ( \\cdot )$ according to a goal-discrimination objective suggested by (3). However, rather than using the set of all goals in the buffer $\\mathcal { G }$ as the set of possible classes in the goal discriminator, we sample a small subset for each trajectory. Specifically, the set of possible classes includes the goal $g$ for the trajectory and $K$ decoy observations $d _ { 1 } , d _ { 2 } , \\dots , d _ { K }$ from the same distribution as $s _ { g }$ . Letting ",
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+    {
+        "type": "equation",
+        "img_path": "images/b126a2d569eb79d1d04a551c7dfbb36e693ba7cde1c9a37a3dccd60a7f2ab9d1.jpg",
+        "text": "$$\n\\boldsymbol { \\ell } _ { g } = \\xi _ { \\phi } ( h ( \\boldsymbol { s } _ { T } ) ) ^ { \\mathsf { T } } \\xi _ { \\phi } ( h ( \\boldsymbol { g } ) )\n$$",
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+    {
+        "type": "text",
+        "text": "we maximize the log likelihood given by ",
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+        "text": "$$\n\\log \\hat { q } ( s _ { g } = g | s _ { T } ; d _ { 1 } , . . . d _ { K } , \\pi _ { \\theta } ) = \\log \\frac { \\exp { ( \\beta \\ell _ { g } ) } } { \\exp { ( \\beta \\ell _ { g } ) } + \\sum _ { k = 1 } ^ { K } \\exp { ( \\beta \\xi _ { \\phi } ( h ( s _ { T } ) ) ^ { \\top } \\xi _ { \\phi } ( h ( d _ { k } ) ) ) } }\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where $\\beta$ is an inverse temperature hyperparameter which we fix to $K + 1$ in all experiments. Note that (5) is a maximum log likelihood training objective for a softmax nearest neighbour classifier in a learned embedding space, making it similar to a matching network (Vinyals et al., 2016). Intuitively, updating the embedding $\\xi _ { \\phi }$ using the objective in (5) aims to increase the cosine similarity between $e { \\left( s _ { T } \\right) }$ and $e ( g )$ and to decrease the cosine similarity between $e { \\left( s _ { T } \\right) }$ and the decoy embeddings $e ( d ) , \\ldots , e ( d _ { K } )$ . Subsampling the set of possible classes as we do is a known method for approximate maximum likelihood training of a softmax classifier with many classes (Bengio & Sen´ ecal, 2003). ´ ",
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+    {
+        "type": "text",
+        "text": "We use $\\operatorname* { m a x } ( 0 , \\ell _ { g } )$ as the reward for reaching state $s _ { T }$ when given goal $g$ . We found that this reward function is better behaved than the reward $\\log \\hat { q } ( s _ { g } = g \\vert s _ { T } ; d _ { 1 } , . . . d _ { K } , \\pi _ { \\theta } )$ suggested by the DISCERN objective in Section 3 since it is scaled to lie in $[ 0 , 1 ]$ . The reward we use is also less noisy since, unlike $\\log { \\hat { q } }$ , it does not depend on the decoy states. ",
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+        "text": "Goal-conditioned policy: The goal-conditioned policy $\\pi _ { \\boldsymbol { \\theta } } ( a | s ; s _ { g } )$ is trained to optimize the goal achievement reward $r ( s ; s _ { g } )$ . In this paper, $\\pi _ { \\theta } ( a | s ; s _ { g } )$ is an $\\epsilon$ -greedy policy of a goal-conditioned action-value function $\\mathrm { Q }$ with parameters $\\theta$ . $\\mathrm { Q }$ is trained using Q-learning and minibatch experience replay; specifically, we use the variant of $Q ( \\lambda )$ due to Peng (see Chapter 7, Sutton & Barto (1998)). ",
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+        "text": "Goal relabelling: We use a form of goal relabelling (Kaelbling, 1993) or hindsight experience replay (Andrychowicz et al., 2017; Nair $\\&$ Hinton, 2006) as a source successfully achieved goals as well as to regularize the embedding $e ( \\cdot )$ . Specifically, for the purposes of parameter updates (in both the policy and the reward learner) we substitute, with probability $p _ { \\mathrm { H E R } }$ the goal with an observation selected from the final $H$ steps of the trajectory, and consider the agent to have received a reward of 1. The motivation, in the case of the policy, is similar to that of previous work, i.e. that being in state $s _ { t }$ should correspond to having achieved the goal of reaching $s _ { t }$ . When employed in the reward learner, it amounts to encouraging temporally consistent state embeddings (Mobahi et al., 2009; Sermanet et al., 2017), i.e. encouraging observations which are nearby in time to have similar embeddings. ",
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+        "type": "text",
+        "text": "Pseudocode for the DISCERN algorithm, decomposed into an experience-gathering (possibly distributed) actor process and a centralized learner process, is given in Algorithm 1. ",
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+    {
+        "type": "text",
+        "text": "5 RELATED WORK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The problem of reinforcement learning in the context of multiple goals dates at least to Kaelbling (1993), where the problem was examined in the context of grid worlds where the state space is ",
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+    {
+        "type": "text",
+        "text": "Algorithm 1: DISCERN ",
+        "text_level": 1,
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+    {
+        "type": "table",
+        "img_path": "images/fbe163367cddd7518bdf20ed5a24e58f997557ee0c491e7f82afb3792f9aa4dc.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">procedure ACTOR Input: Time budget T, policy parameters 0, goal embedding parameters Φ, shared goal</td></tr><tr><td colspan=\"2\">π ←BEHAVIOR-POLICY(0) /* e.g. E-greedy */</td></tr><tr><td colspan=\"2\">g~g r1:T-1←0 fort←1...Tdo Take action at ~ πθ(St; g) obtaining St+1 from p(St+1|St, at)</td></tr><tr><td colspan=\"2\">9 ← PROPOSE-GOAL-SUBSTITUTION(G,St) /* See Appendix A3 */ end</td></tr><tr><td colspan=\"2\">with probability PHER, Sample SHER uniformly from {sT-H,...,sT} and set g ← SHER,rT ←1 otherwise</td></tr><tr><td colspan=\"2\">Compute lg using (4) rT ←max(0,lg) Send (s1:T,a1:T,r1:T,g) to the learner.</td></tr><tr><td colspan=\"2\">Poll the learner periodically for updated values of 0,. Reset the environment if the episode has terminated.</td></tr><tr><td colspan=\"2\">until termination procedure LEARNER</td></tr><tr><td colspan=\"2\">Input:Batch size B, number of decoys K,initial policy parameters 0,initial goal embedding parametersΦ</td></tr><tr><td colspan=\"2\">repeat Assemble batchof experienceB={(s:,:T,b)}1</td></tr><tr><td colspan=\"2\">forb←1...Bdo</td></tr><tr><td colspan=\"2\">Sample K decoy goals d,d,...,dk ~ g end</td></tr><tr><td colspan=\"2\">Use an off-policy reinforcement learning algorithm to update θ based on B</td></tr><tr><td colspan=\"2\">Update Φ to maximize B∑b=1 log@(sg = gʰ|sr;d1,..dk,πθ) computed by (5) B</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "small and enumerable. Sutton et al. (2011) proposed generalized value functions (GVFs) as a way of representing knowledge about sub-goals, or as a basis for sub-policies or options. Universal Value Function Approximators (UVFAs) (Schaul et al., 2015) extend this idea by using a function approximator to parameterize a joint function of states and goal representations, allowing compact representation of an entire class of conditional value functions and generalization across classes of related goals. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "While the above works assume a goal achievement reward to be available a priori, our work includes an approach to learning a reward function for goal achievement jointly with the policy. Several recent works have examined reward learning for goal achievement in the context of the Generative Adversarial Networks (GAN) paradigm (Goodfellow et al., 2014). The SPIRAL (Ganin et al., 2018) algorithm trains a goal conditioned policy with a reward function parameterized by a Wasserstein GAN (Arjovsky et al., 2017) discriminator. Similarly, AGILE (Bahdanau et al., 2018) learns an instruction-conditional policy where goals in a grid-world are specified in terms of predicates which should be satisfied, and a reward function is learned using a discriminator trained to distinguish states achieved by the policy from a dataset of instruction, goal state pairs. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Reward learning has also been used in the context of imitation. Ho & Ermon (2016) derives an adversarial network algorithm for imitation, while time-contrastive networks (Sermanet et al., 2017) leverage pre-trained ImageNet classifier representations to learn a reward function for robotics skills from video demonstrations, including robotic imitation of human poses. Universal Planning Networks (UPNs) (Srinivas et al., 2018) learn a state representation by training a differentiable planner to imitate expert trajectories. Experiments showed that once a UPN is trained the state representation it learned can be used to construct a reward function for visually specified goals. Bridging goal-conditioned policy learning and imitation learning, Pathak et al. (2018) learns a goal-conditioned policy and a dynamics model with supervised learning without expert trajectories, and present zero-shot imitation of trajectories from a sequence of images of a desired task. ",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "A closely related body of work to that of goal-conditioned reinforcement learning is that of unsupervised option or skill discovery. Machado & Bowling (2016) proposes a method based on an eigendecomposition of differences in features between successive states, further explored and extended in Machado et al. (2017). Variational Intrinsic Control (VIC) (Gregor et al., 2016) leverages the same lower bound on the mutual information as the present work in an unsupervised control setting, in the space of abstract options rather than explicit perceptual goals. VIC aims to jointly maximize the entropy of the set of options while making the options maximally distinguishable from their final states according to a parametric predictor. Recently, Eysenbach et al. (2018) showed that a special case of the VIC objective can scale to significantly more complex tasks and provide a useful basis for low-level control in a hierarchical reinforcement learning context. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Other work has explored learning policies in tandem with a task policy, where the task or environment rewards are assumed to be sparse. Florensa et al. (2017) propose a framework in which low-level skills are discovered in a pre-training phase of a hierarchial system based on simple-to-design proxy rewards, while Riedmiller et al. (2018) explore a suite of auxiliary tasks through simultaneous off-policy learning. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Several authors have explored a pre-training stage, sometimes paired with fine-tuning, based on unsupervised representation learning. Per´ e et al. (2018) and Laversanne-Finot et al. (2018) employ a ´ two-stage framework wherein unsupervised representation learning is used to learn a model of the observations from which to sample goals for control in simple simulated environments. Nair et al. (2018) propose a similar approach in the context of model-free Q-learning applied to 3-dimensional simulations and robots. Goals for training the policy are sampled from the model’s prior, and a reward function is derived from the latent codes. This contrasts with our non-parametric approach to selecting goals, as well as our method for learning the goal space online and jointly with the policy. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "An important component of our method is a form of goal relabelling, introduced to the reinforcement learning literature as hindsight experience replay by Andrychowicz et al. (2017), based on the intuition that any trajectory constitutes a valid trajectory which achieves the goal specified by its own terminal observation. Earlier, Nair & Hinton (2006) employed a related scheme in the context of supervised learning of motor programs, where a program encoder is trained on pairs of trajectory realizations and programs obtained by expanding outwards from a pre-specified prototypical motor program through the addition of noise. Veeriah et al. (2018) expands upon hindsight replay and the all-goal update strategy proposed by Kaelbling (1993), generalizing the latter to non-tabular environments and exploring related strategies for skill discovery, unsupervised pre-training and auxiliary tasks. Levy et al. (2018) propose a hierarchical Q-learning system which employs hindsight replay both conventionally in the lower-level controller and at higher levels in the hierarchy. Nair et al. (2018) also employ a generalized goal relabeling scheme whereby the policy is trained based on a trajectory’s achievement not just of its own terminal observation, but a variety of retrospectively considered possible goals. ",
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+    {
+        "type": "text",
+        "text": "6 EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We evaluate, both qualitatively and quantitatively, the ability of DISCERN to achieve visuallyspecified goals in three diverse domains – the Arcade Learning Environment (Bellemare et al., 2013), continuous control tasks in the DeepMind Control Suite (Tassa et al., 2018), and DeepMind Lab, a 3D first person environment (Beattie et al., 2016). Experimental details including architecture details, details of distributed training, and hyperparameters can be found in the Appendix. We compared DISCERN to several baseline methods for learning goal-conditioned policies: ",
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+        "type": "text",
+        "text": "Conditioned Autoencoder (AE): In order to specifically interrogate the role of the discriminative reward learning criterion, we replace the discriminative criterion for embedding learning with an $L ^ { 2 }$ reconstruction loss on $h _ { t }$ ; that is, in addition to $\\xi _ { \\phi } ( \\cdot )$ , we learn an inverse mapping $\\xi _ { \\phi } ^ { - 1 } ( \\cdot )$ with a separate set of parameters, and train both with the criterion $\\| h _ { t } - \\xi _ { \\phi } ^ { - 1 } ( \\xi _ { \\phi } ( h _ { t } ) ) \\| ^ { 2 }$ . ",
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+    {
+        "type": "text",
+        "text": "Conditioned WGAN Discriminator: We compare to an adversarial reward on the domains considered according to the protocol of Ganin et al. (2018), who successfully used a WGAN discriminator as a reward for training agents to perform inverse graphics tasks. The discriminator takes two pairs of images – (1) a real pair of goal images $( s _ { g } , s _ { g } )$ and (2) a fake pair consisting of the terminal state of the agent and the goal frame $( s _ { t } , s _ { g } )$ . The output of the discriminator is used as the reward function for the policy. Unlike our DISCERN implementation and the conditioned autoencoder baseline, we train the WGAN discriminator as a separate convolutional network directly from pixels, as in previous work. ",
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "Pixel distance reward (L2): Finally, we directly compare to a reward based on $L ^ { 2 }$ distance in pixel space, equal to e $\\mathbf { \\boldsymbol { x } } \\mathbf { \\boldsymbol { p } } \\left( - \\lVert \\mathbf { \\boldsymbol { s } } _ { t } - \\mathbf { \\boldsymbol { s } } _ { g } \\rVert ^ { 2 } / \\sigma _ { \\mathrm { p i x e l } } \\right)$ where $\\sigma _ { \\mathrm { p i x e l } }$ is a hyperparameter which we tuned on a per-environment basis. ",
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+    {
+        "type": "text",
+        "text": "All the baselines use the same goal-conditioned policy architecture as DISCERN. The baselines also used hindsight experience replay in the same way as DISCERN. They can therefore be seen as ablations of DISCERN’s goal-achievement reward learning mechanism. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "6.1 ATARI ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The suite of 57 Atari games provided by the Arcade Learning Environment (Bellemare et al., 2013) is a widely used benchmark in the deep reinforcement learning literature. We compare DISCERN to other methods on the task of achieving visually specified goals on the games of Seaquest and Montezuma’s Revenge. The relative simplicity of these domains makes it possible to handcraft a detector in order to localize the controllable aspects of the environment, namely the submarine in Seaquest and Panama Joe, the character controlled by the player in Montezuma’s Revenge. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We evaluated the methods by running the learned goal policies on a fixed set of goals and measured the percentage of goals it was able to reach successfully. We evaluated both DISCERN and the baselines with two different goal buffer substitution strategies, uniform and diverse, which are described in the Appendix. A goal was deemed to be successfully achieved if the position of the avatar in the last frame was within $10 \\%$ of the playable area of the position of the avatar in the goal for each controllable dimension. The controllable dimensions in Atari were considered to be the $\\mathbf { X } ^ { - }$ and y-coordinates of the avatar. The results are displayed in Figure 1a. DISCERN learned to achieve a large fraction of goals in both Seaquest and Montezuma’s Revenge while none of the baselines learned to reliably achieve goals in either game. We hypothesize that the baselines failed to learn to control the avatars because their objectives are too closely tied to visual similarity. Figure 1b shows examples of goal achievement on Seaquest and Montezuma’s Revenge. In Seaquest, DISCERN learned to match the position of the submarine in the goal image while ignoring the position of the fish, since the fish are not directly controllable. We have provided videos of the goal-conditioned policies learned by DISCERN on Seaquest and Montezuma’s Revenge at the following anonymous URL https://sites.google.com/view/discern-anonymous/home. ",
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+    {
+        "type": "text",
+        "text": "6.2 DEEPMIND CONTROL SUITE TASKS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The DeepMind Control Suite (Tassa et al., 2018) is a suite of continuous control tasks built on the MuJoCo physics engine (Todorov et al., 2012). While most frequently used to evaluate agents which receive the underlying state variables as observations, we train our agents on pixel renderings of the scene using the default environment-specified camera, and do not directly observe the state variables. ",
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+        "type": "text",
+        "text": "Agents acting greedily with respect to a state-action value function require the ability to easily maximize $Q$ over the candidate actions. For ease of implementation, as well as comparison to other considered environments, we discretize the space of continuous actions to no more than 11 unique actions per environment (see Appendix A4.1). ",
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+    {
+        "type": "text",
+        "text": "The availability of an underlying representation of the physical state, while not used by the learner, provides a useful basis for comparison of achieved states to goals. We mask out state variables relating to entities in the scene not under the control of the agent; for example, the position of the target in the reacher or manipulator domains. ",
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+        "type": "text",
+        "text": "DISCERN is compared to the baselines on a fixed set of 100 goals with 20 trials for each goal. The goals are generated by acting randomly for 25 environment steps after initialization. In the case of cartpole, we draw the goals from a random policy acting in the environment set to the balance task, where the pole is initialized upwards, in order to generate a more diverse set of goals against which to measure. Figure 3 compares learning progress of 5 independent seeds for the “uniform” goal replacement strategy (see Appendix A5 for results with “diverse” goal replacement) for 6 domains. We adopt the same definition of achievement as in Section 6.1. Figure 2 summarizes averaged goal achievement frames on these domains except for the cartpole domain for policies learned by DISCERN. Performance on cartpole is discussed in more detail in Figure 7 of the Appendix. ",
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+        "type": "image",
+        "img_path": "images/5b3e12a48cafd1721f4333b127759482b9454525058189dba01de4eb721472a3.jpg",
+        "image_caption": [
+            "Figure 1: a) Percentage of goals successfully achieved on Seaquest and Montezuma’s Revenge. b) Examples of goals achieved by DISCERN on the games of Seaquest (top) and Montezuma’s Revenge (bottom). For each game, the four goal states are shown in the top row. Below each goal is the averaged (over 5 trials) final state achieved by the goal-conditioned policy learned by DISCERN after $T = 5 0$ steps for the goal above. "
+        ],
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+        "image_caption": [
+            "Figure 2: Average achieved frames for point mass (task easy), reacher (task hard), manipulator (task bring ball), pendulum (task swingup), finger (task spin) and ball in cup (task catch) environments. The goal is shown in the top row and the achieved frame is shown in the bottom row. "
+        ],
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+    {
+        "type": "text",
+        "text": "The results show that in aggregate, DISCERN outperforms baselines in terms of goal achievement on several, but not all, of the considered Control Suite domains. In order to obtain a more nuanced understanding of DISCERN’s behaviour when compared with the baselines, we also examined achievement in terms of the individual dimensions of the controllable state. Figure 4 shows goal achievement separately for each dimension of the underlying state on four domains. The perdimension results show that on difficult goal-achievement tasks such as those posed in cartpole (where most proposed goal states are unstable due to the effect of gravity) and finger (where a free-spinning piece is only indirectly controllable) DISCERN learns to reliably match the major dimensions of controllability such as the cart position and finger pose while ignoring the other dimensions, whereas none of the baselines learned to reliably match any of the controllable state dimensions on the difficult tasks cartpole and finger. ",
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+    {
+        "type": "image",
+        "img_path": "images/114229ab7fa3bfb129b7f218526a96105037f7769482bc7693d43f7f5a857239.jpg",
+        "image_caption": [
+            "Figure 3: Quantitative evaluation of goal achievement on continuous control domains using the “uniform” goal substitution scheme (see Appendix A3). For each method, we show the fraction of goals achieved over a fixed goal set (100 images per domain). "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "image",
+        "img_path": "images/c0e9f48e1d12f29d0e502308dbf0bbc4c8472777cc3ae707af685e34eedf153c.jpg",
+        "image_caption": [
+            "Figure 4: Per-dimension quantitative evaluation of goal achievement on continuous control domains using the “uniform” goal substitution scheme (Appendix A3). Each subplot corresponds to a domain, with each group of colored rows representing a method. Each individual row represents a dimension of the controllable state (such as a joint angle). The color of each cell indicates the fraction of goal states for which the method was able to match the ground truth value for that dimension to within $1 0 \\%$ of the possible range. The position along the $x$ -axis indicates the point in training in millions of frames. For example, on the reacher domain DISCERN learns to match both dimensions of the controllable state, but on the cartpole domain it learns to match the first dimension (cart position) but not the second dimension (pole angle). "
+        ],
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "We omitted the manipulator domain from these figures as none of the methods under consideration achieved non-negligible goal achievement performance on this domain, however a video showing the policy learned by DISCERN on this domain can be found at https://sites.google.com/view/discern-anonymous/home. The policy learned on the manipulator domain shows that DISCERN was able to discover several major dimensions of controllability even on such a challenging task, as further evidenced by the per-dimension analysis on the manipulator domain in Figure 8 in the Appendix. ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/4989701958c232e7bd7a0ac51efe637aad8935114d605e32c3234a2672c05347.jpg",
+        "image_caption": [
+            "Figure 5: Average achieved frames over 30 trials from a random initialization on the rooms watermaze task. Goals are shown in the top row while the corresponding average achieved frames are in the bottom row. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "6.3 DEEPMIND LAB ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "DeepMind Lab (Beattie et al., 2016) is a platform for 3D first person reinforcement learning environments. We trained DISCERN on the watermaze level and found that it learned to approximately achieve the same wall and horizon position as in the goal image. While the agent did not learn to achieve the position and viewpoint shown in a goal image as one may have expected, it is encouraging that our approach learns a reasonable space of goals on a first-person 3D domain in addition to domains with third-person viewpoints like Atari and the DM Control Suite. ",
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+    {
+        "type": "text",
+        "text": "7 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We have presented a system that can learn to achieve goals, specified in the form of observations from the environment, in a purely unsupervised fashion, i.e. without any extrinsic rewards or expert demonstrations. Integral to this system is a powerful and principled discriminative reward learning objective, which we have demonstrated can recover the dominant underlying degrees of controllability in a variety of visual domains. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this work, we have adopted a fixed episode length of $T$ in the interest of simplicity and computational efficiency. This implicitly assumes not only that all sampled goals are approximately achievable in $T$ steps, but that the policy need not be concerned with finishing in less than the allotted number of steps. Both of these limitations could be addressed by considering schemes for early termination based on the embedding, though care must be taken not to deleteriously impact training by terminating episodes too early based on a poorly trained reward embedding. Relatedly, our goal selection strategy is agnostic to both the state of the environment at the commencement of the goal episode and the current skill profile of the policy, utilizing at most the content of the goal itself to drive the evolution of the goal buffer $\\mathcal { G }$ . We view it as highly encouraging that learning proceeds using such a naive goal selection strategy, however more sophisticated strategies, such as tracking and sampling from the frontier of currently achievable goals (Held et al., 2017), may yield substantial improvements. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "DISCERN’s ability to automatically discover controllable aspects of the observation space is a highly desirable property in the pursuit of robust low-level control. A natural next step is the incorporation of DISCERN into a deep hierarchical reinforcement learning setup (Vezhnevets et al., 2017; Levy et al., 2018; Nachum et al., 2018) where a meta-policy for proposing goals is learned after or in tandem with a low-level controller, i.e. by optimizing an extrinsic reward signal. ",
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+    {
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+    {
+        "type": "text",
+        "text": "APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            263,
+            117
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A1 DISTRIBUTED TRAINING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            132,
+            385,
+            146
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We employ a distributed reinforcement learning architecture inspired by the IMPALA reinforcement learning architecture (Espeholt et al., 2018), with a centralized GPU learner batching parameter updates on experience collected by a large number of CPU-based parallel actors. While Espeholt et al. (2018) learns a stochastic policy through the use of an actor-critic architecture, we instead learn a goal-conditioned state-action value function with Q-learning. Each actor acts $\\epsilon$ -greedily with respect to a local copy of the $Q$ network, and sends observations $s _ { t }$ , actions $a _ { t }$ , rewards $r _ { t }$ and discounts $\\gamma _ { t }$ for a trajectory to the learner. Following Horgan et al. (2018), we use a different value of $\\epsilon$ for each actor, as this has been shown to improve exploration. The learner batches re-evaluation of the convolutional network and LSTM according to the action trajectories supplied and performs parameter updates, periodically broadcasting updated model parameters to the actors. As Q-learning is an off-policy algorithm, the experience traces sent to the learner can be used in the usual $n$ -step Q-learning update without the need for an off-policy correction as in Espeholt et al. (2018). We also maintain actor-local replay buffers of previous actor trajectories and use them to perform both standard experience replay (Lin, 1993) and our variant of hindsight experience replay (Andrychowicz et al., 2017). ",
+        "bbox": [
+            173,
+            159,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A2 ARCHITECTURE DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            382,
+            390,
+            397
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Our network architectures closely resemble those in Espeholt et al. (2018), with policy and value heads replaced with a $Q$ -function. We apply the same convolutional network to both $s _ { t }$ and $s _ { g }$ and concatenate the final layer outputs. Note that the convolutional network outputs for $s _ { g }$ need only be computed once per episode. We include a periodic representation $( \\sin ( 2 \\pi t / T ) , \\cos ( 2 \\pi t / T ) )$ of the current time step, with period equal to the goal length achievement period $T$ , as an extra input to the network. The periodic representation is processed by a single hidden layer of rectified linear units and is concatenated with the visual representations fed to the LSTM. While not strictly necessary, we find that this allows the agent to become better at achieving goal states which may be unmaintainable due to their instability in the environment dynamics. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The output of the LSTM is the input to a dueling action-value output network (Wang et al., 2016). In all of our experiments, both branches of the dueling network are linear mappings. That is, given LSTM outputs $\\psi _ { t }$ , we compute the action values for the current time step $t$ as ",
+        "bbox": [
+            176,
+            540,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/72d050cad22f22b360269c810da2599ab1c92207391396b327211e1f883ff6de.jpg",
+        "text": "$$\nQ ( a _ { t } | \\psi _ { t } ) = { \\psi _ { t } } ^ { \\mathsf { T } } \\mathbf { v } + \\left( \\psi _ { t } ^ { \\mathsf { T } } \\mathbf { w } _ { a _ { t } } - { \\frac { 1 } { n } } \\sum _ { a _ { t } ^ { \\prime } } { \\psi _ { t } } ^ { \\mathsf { T } } \\mathbf { w } _ { a _ { t } ^ { \\prime } } \\right) + b\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            583,
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+            633
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A3 GOAL BUFFER",
+        "text_level": 1,
+        "bbox": [
+            174,
+            646,
+            313,
+            661
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We experimented with two strategies for updating the goal buffer. In the first strategy, which we call uniform, the current observation replaces a uniformly selected entry in the goal buffer with probability $p _ { \\mathrm { r e p l a c e } }$ . The second strategy, which we refer to as diverse goal sampling attempts to maintain a goal buffer that more closely approximates the uniform distribution over all observation. In the diverse goal strategy, we consider the current observation for addition to the goal buffer with probability $p _ { \\mathrm { r e p l a c e } }$ at each step during acting. If the current observation $s$ is considered for addition to the goal buffer, then we select a random removal candidate $s _ { r }$ by sampling uniformly from the goal buffer and replace it with $s$ if $s _ { r }$ is closer to the rest of the goal buffer than $s$ . If $s$ is closer to the rest of the goal buffer than $s _ { r }$ then we still replace $s _ { r }$ with $s$ with probability padd−non−diverse. We used $L _ { 2 }$ distance in pixel space for the diverse sampling strategy and found it to greatly increase the coverage of states in the goal buffer, especially early during training. This bears some relationship to Determinantal Point Processes (Kulesza et al., 2012), and goal-selection strategies with a more explicit theoretical foundation are a promising future direction. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A4 EXPERIMENTAL DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            869,
+            388,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The following hyper-parameters were used in all of the experiments described in Section 6. All weight matrices are initialized using a standard truncated normal initializer, with the standard deviation inversely proportional to the square root of the fan-in. We maintain a goal buffer of size 1024 and use $p _ { \\mathrm { r e p l a c e } } \\overset { \\cdot } { = } 1 0 ^ { - 3 }$ . We also use $p _ { \\mathrm { a d d - n o n - d i v e r s e } } = 1 0 ^ { - 3 }$ . For the teacher, we choose $\\xi _ { \\phi } ( \\cdot )$ to be an $\\bar { \\mathcal { L } } _ { 2 }$ -normalized single layer of 32 tanh units, trained in all experiments with 4 decoys (and thus, according to our heuristic, $\\beta$ equal to 5). For hindsight experience replay, a highsight goal is substituted $2 5 \\%$ of the time. These goals are chosen uniformly at random from the last 3 frames of the trajectory. Trajectories were set to be 50 steps long for Atari and DeepMind Lab and 100 for the DeepMind control suite. It is important to note that the environment was not reset after each trajectory, but rather the each new trajectory begins where the previous one ended. We train the agent and teacher jointly with RMSProp (Tieleman & Hinton, 2012) with a learning rate of $1 0 ^ { - 4 }$ . We follow the preprocessing protocol of Mnih et al. (2015), resizing to $8 4 \\times 8 4$ pixels and scaling 8-bit pixel values to lie in the range [0, 1]. While originally designed for Atari, we apply this preprocessing pipeline across all environments used in this paper. ",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A4.1 CONTROL SUITE ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In the point mass domain we use a control step equal to 5 times the task-specified default, i.e. the agent acts on every fifth environment step (Mnih et al., 2015). In all other Control Suite domains, we use the default. We use the “easy” version of the task where actuator semantics are fixed across environment episodes. ",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Discrete action spaces admit function approximators which simultaneously compute the action values for all possible actions, as popularized in Mnih et al. (2015). The action with maximal $Q$ -value can thus be identified in time proportional to the cardinality of the action space. An enumeration of possible actions is no longer possible in the continuous setting. While approaches exist to enable continuous maximization in closed form (Gu et al., 2016), they come at the cost of greatly restricting the functional form of $Q$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "For ease of implementation, as well as comparison to other considered environments, we instead discretize the space of continuous actions. For all Control Suite environments considered except manipulator, we discretize an $A$ -dimensional continuous action space into $3 ^ { A }$ discrete actions, consisting of the Cartesian product over action dimensions with values in $\\{ - 1 , 0 , 1 \\}$ . In the case of manipulator, we adopt a “diagonal” discretization where each action consists of setting one actuator to $\\pm 1$ , and all other actuators to 0, with an additional action consisting of every actuator being set to 0. This is a reasonable choice for manipulator because any position can be achieved by a concatenation of actuator actions, which may not be true of more complex Control Suite environments such as humanoid, where the agent’s body is subject to gravity and successful trajectories may require multi-joint actuation in a single control time step. The subset of the Control Suite considered in this work was chosen primarily such that the discretized action space would be of a reasonable size. We leave extensions to continuous domains to future work. ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A5 ADDITIONAL EXPERIMENTAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            178,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A5.1 ATARI ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We ran two additional baselines on Seaquest and Montezuma’s Revenge, ablating our use of hindsight experience replay in opposite ways. One involved training the goal-conditioned policy only in hindsight, without any learned goal achievement reward, i.e. $p _ { \\mathrm { H E R } } = 1$ . This approach achieved $1 2 \\%$ of goals on Seaquest and $1 \\bar { 1 } . 4 \\%$ of goals on Montezuma’s Revenge, making it comparable to a uniform random policy. This result underscores the importance of learning a goal achievement reward. The second baseline consisted of DISCERN learning a goal achievement reward without hindsight experience replay, i.e. $p _ { \\mathrm { H E R } } = 0$ . This also performed poorly, achieving $1 1 . 4 \\%$ of goals on Seaquest and $8 \\%$ of goals on Montezuma’s Revenge. Taken together, these preliminary results suggest that the combination of hindsight experience replay and a learned goal achievement reward is important. ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A5.2 CONTROL SUITE ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "For the sake of completeness, Figure 6 reports goal achievement curves on Control Suite domains using the “diverse” goal selection scheme. ",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/a7ca0c9ad1c291d0cbfaddf7270977c0c4deeebe096c75c589562b2587954197.jpg",
+        "image_caption": [
+            "Figure 6: Results for Control Suite tasks using the “diverse” goal substitution scheme. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/4242284d4a82a40cfc6cafcd2d5920b721064c931f25cfaf54bc1a0196440044.jpg",
+        "image_caption": [
+            "Figure 7: Average goal achievement on cartpole. Top row shows the goals. Middle row shows achievement by the Autoencoder baseline. Bottom row shows average goal achievement by DISCERN. Shading of columns is for emphasis. DISCERN always matches the cart position. The autoencoder baseline matches both cart and pole position when the pole is pointing down, but fails to match either when the pole is pointing up. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            362,
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Figure 7 displays goal achievements for DISCERN and the Autoencoder baseline, highlighting DISCERN’s preference for communicating with the cart position, and robustness to the pole positions unseen during training. ",
+        "bbox": [
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+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/b29cc81a794b8f87d7a0a882d3bf3bc279517f9a447430ce3aaed586759b12c3.jpg",
+        "image_caption": [
+            "Figure 8: Per-dimension quantitative evaluation on the manipulator domain. See Figure 4 for a description of the visualization. DISCERN learns to reliably control more dimensions of the underlying state than any of the baselines. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    }
+]