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+# PREDICT THEN PROPAGATE: GRAPH NEURAL NETWORKS MEET PERSONALIZED PAGERANK
+
+Johannes Gasteiger, Aleksandar Bojchevski & Stephan Gunnemann ¨
+
+Technical University of Munich, Germany {j.gasteiger,a.bojchevski,guennemann}@in.tum.de
+
+# ABSTRACT
+
+Neural message passing algorithms for semi-supervised classification on graphs have recently achieved great success. However, for classifying a node these methods only consider nodes that are a few propagation steps away and the size of this utilized neighborhood is hard to extend. In this paper, we use the relationship between graph convolutional networks (GCN) and PageRank to derive an improved propagation scheme based on personalized PageRank. We utilize this propagation procedure to construct a simple model, personalized propagation of neural predictions (PPNP), and its fast approximation, APPNP. Our model’s training time is on par or faster and its number of parameters on par or lower than previous models. It leverages a large, adjustable neighborhood for classification and can be easily combined with any neural network. We show that this model outperforms several recently proposed methods for semi-supervised classification in the most thorough study done so far for GCN-like models. Our implementation is available online. 1
+
+# 1 INTRODUCTION
+
+Graphs are ubiquitous in the real world and its description through scientific models. They are used to study the spread of information, to optimize delivery, to recommend new books, to suggest friends, or to find a party’s potential voters. Deep learning approaches have achieved great success on many important graph problems such as link prediction (Grover & Leskovec, 2016; Bojchevski et al., 2018), graph classification (Duvenaud et al., 2015; Niepert et al., 2016; Gilmer et al., 2017) and semi-supervised node classification (Yang et al., 2016; Kipf & Welling, 2017).
+
+There are many approaches for leveraging deep learning algorithms on graphs. Node embedding methods use random walks or matrix factorization to directly train individual node embeddings, often without using node features and usually in an unsupervised manner, i.e. without leveraging node classes (Perozzi et al., 2014; Tang et al., 2015; Nandanwar & Murty, 2016; Grover & Leskovec, 2016; Qiu et al., 2018). Many other approaches use both graph structure and node features in a supervised setting. Examples for these include spectral graph convolutional neural networks (Bruna et al., 2014; Defferrard et al., 2016), message passing (or neighbor aggregation) algorithms (Kearnes et al., 2016; Kipf & Welling, 2017; Hamilton et al., 2017; Pham et al., 2017; Monti et al., 2017; Gilmer et al., 2017), and neighbor aggregation via recurrent neural networks (Scarselli et al., 2009; Li et al., 2016; Dai et al., 2018). Among these categories, the class of message passing algorithms has garnered particular attention recently due to its flexibility and good performance.
+
+Several works have been aimed at improving the basic neighborhood aggregation scheme by using attention mechanisms (Kearnes et al., 2016; Hamilton et al., 2017; Velickovi ˇ c et al., 2018), random ´ walks (Abu-El-Haija et al., 2018a; Ying et al., 2018; Li et al., 2018), edge features (Kearnes et al., 2016; Gilmer et al., 2017; Schlichtkrull et al., 2018) and making it more scalable on large graphs (Chen et al., 2018; Ying et al., 2018). However, all of these methods only use the information of a very limited neighborhood for each node. A larger neighborhood would be desirable to provide the model with more information, especially for nodes in the periphery or in a sparsely labelled setting.
+
+Increasing the size of the neighborhood used by these algorithms, i.e. their range, is not trivial since neighborhood aggregation in this scheme is essentially a type of Laplacian smoothing and too many layers lead to oversmoothing (Li et al., 2018). Xu et al. (2018) highlighted the same problem by establishing a relationship between the message passing algorithm termed Graph Convolutional Network (GCN) by Kipf & Welling (2017) and a random walk. Using this relationship we see that GCN converges to this random walk’s limit distribution as the number of layers increases. The limit distribution is a property of the graph as a whole and does not take the random walk’s starting (root) node into account. As such it is unsuited to describe the root node’s neighborhood. Hence, GCN’s performance necessarily deteriorates for a high number of layers (or aggregation/propagation steps).
+
+To solve this issue, in this paper, we first highlight the inherent connection between the limit distribution and PageRank (Page et al., 1998). We then propose an algorithm that utilizes a propagation scheme derived from personalized PageRank instead. This algorithm adds a chance of teleporting back to the root node, which ensures that the PageRank score encodes the local neighborhood for every root node (Page et al., 1998). The teleport probability allows us to balance the needs of preserving locality (i.e. staying close to the root node to avoid oversmoothing) and leveraging the information from a large neighborhood. We show that this propagation scheme permits the use of far more (in fact, infinitely many) propagation steps without leading to oversmoothing.
+
+Moreover, while propagation and classification are inherently intertwined in message passing, our proposed algorithm separates the neural network from the propagation scheme. This allows us to achieve a much higher range without changing the neural network, whereas in the message passing scheme every additional propagation step would require an additional layer. It also permits the independent development of the propagation algorithm and the neural network generating predictions from node features. That is, we can combine any state-of-the-art prediction method with our propagation scheme. We even found that adding our propagation scheme during inference significantly improves the accuracy of networks that were trained without using any graph information.
+
+Our model achieves state-of-the-art results while requiring fewer parameters and less training time compared to most competing models, with a computational complexity that is linear in the number of edges. We show these results in the most thorough study (including significance testing) of message passing models using graphs with text-based features that has been done so far.
+
+# 2 GRAPH CONVOLUTIONAL NETWORKS AND THEIR LIMITED RANGE
+
+We first introduce our notation and explain the problem our model solves. Let $G = ( V , E )$ be a graph with nodes $V$ and edges $E$ . Let $n$ denote the number of nodes and $m$ the number of edges. The nodes are described by the feature matrix $\ b { X } \in \mathbb { R } ^ { n \times f }$ , with the number of features $f$ per node, and the class (or label) matrix $\ b { Y } \in \mathbb { R } ^ { n \times c }$ , with the number of classes $c$ . The graph $G$ is described by the adjacency matrix $A \in \mathbb { R } ^ { n \times n }$ . $\tilde { \pmb { A } } = \pmb { A } + \pmb { I _ { n } }$ denotes the adjacency matrix with added self-loops.
+
+One simple and widely used message passing algorithm for semi-supervised classification is the Graph Convolutional Network (GCN). In the case of two message passing layers its equation is
+
+$$
+\boldsymbol { Z } _ { \mathrm { G C N } } = \mathrm { s o f t m a x } \left( \hat { \tilde { A } } \mathrm { R e L U } \left( \hat { \tilde { A } } \boldsymbol { X } \boldsymbol { W } _ { 0 } \right) \boldsymbol { W } _ { 1 } \right) ,
+$$
+
+where $\ b { Z } \in \mathbb { R } ^ { n \times c }$ are the predicted node labels, $\hat { \tilde { A } } = \tilde { D } ^ { - 1 / 2 } \tilde { A } \tilde { D } ^ { - 1 / 2 }$ is the symmetrically normalized adjacency matrix with self-loops, with the diagonal degree matrix $\begin{array} { r } { \tilde { D } _ { i j } = \sum _ { k } \tilde { A } _ { i k } \delta _ { i j } } \end{array}$ , and $W _ { 0 }$ and $W _ { 1 }$ are trainable weight matrices (Kipf & Welling, 2017).
+
+With two GCN-layers, only neighbors in the two-hop neighborhood are considered. There are essentially two reasons why a message passing algorithm like GCN cannot be trivially expanded to use a larger neighborhood. First, aggregation by averaging causes oversmoothing if too many layers are used. It, therefore, loses its focus on the local neighborhood (Li et al., 2018). Second, most common aggregation schemes use learnable weight matrices in each layer. Therefore, using a larger neighborhood necessarily increases the depth and number of learnable parameters of the neural network (the second aspect can be circumvented by using weight sharing, which is typically not the case, though). However, the required neighborhood size and neural network depth are two completely orthogonal aspects. This fixed relationship is a strong limitation and leads to bad compromises.
+
+We will start by concentrating on the filayer GCN the influence score of node $x$ isson $y$ ., $\begin{array} { r } { I ( x , y ) ~ = ~ \sum _ { i } \sum _ { j } \frac { \partial Z _ { y i } } { \partial X _ { x j } } } \end{array}$ $\mathrm { X u }$ hown that for a , is proportional $k$ expectation to a slightly modified $k$ -step random walk distribution starting at the root node $x$ , $P _ { \mathrm { r w } } , ( x \to y , k )$ . Hence, the information of node $x$ spreads to node $y$ in a random walk-like manner. If we take the limit $k  \infty$ and the graph is irreducible and aperiodic, this random walk probability distribution $P _ { \mathrm { r w } } , ( x \to y , k )$ converges to the limit (or stationary) distribution $P _ { \mathrm { l i m } } ( \to y )$ . This distribution can be obtained by solving the equation $\pi _ { \mathrm { l i m } } = \hat { \tilde { A } } \pi _ { \mathrm { l i m } }$ . Obviously, the result only depends on the graph as a whole and is independent of the random walk’s starting (root) node $x$ . This global property is therefore unsuitable for describing the root node’s neighborhood.
+
+![](images/b4aa525e1c873f19349a9967b8eab74ff37d89d4cc7d33d209b29d100026a523.jpg)  
+Figure 1: Illustration of (approximate) personalized propagation of neural predictions (PPNP, APPNP). Predictions are first generated from each node’s own features by a neural network and then propagated using an adaptation of personalized PageRank. The model is trained end-to-end.
+
+# 3 PERSONALIZED PROPAGATION OF NEURAL PREDICTIONS
+
+From message passing to personalized PageRank. We can solve the problem of lost focus by recognizing the connection between the limit distribution and PageRank (Page et al., 1998). The only differences between these two are the added self-loops and the adjacency matrix normalization in $\hat { \tilde { A } }$ . Original PageRank is calculated via $\pi _ { \mathrm { p r } } = A _ { \mathrm { r w } } \pi _ { \mathrm { p r } }$ , with $A _ { \mathrm { r w } } = A D ^ { - 1 }$ . Having made this connection we can now consider using a variant of PageRank that takes the root node into account – personalized PageRank (Page et al., 1998). We define the root node $x$ via the teleport vector $i _ { x }$ , which is a one-hot indicator vector. Our adaptation of personalized PageRank can be obtained for node $x$ using the recurrent equation $\pi _ { \mathrm { p p r } } ( i _ { x } ) \stackrel { - } { = } ( 1 - \alpha ) \mathring { \hat { A } } \pi _ { \mathrm { p p r } } ( i _ { x } ) + \alpha i _ { x }$ , with the teleport (or restart) probability $\alpha \in \mathsf { \Gamma } ( 0 , 1 ]$ . By solving this equation, we obtain
+
+$$
+\pi _ { \mathrm { p p r } } ( i _ { x } ) = \alpha \left( I _ { n } - ( 1 - \alpha ) \hat { \tilde { A } } \right) ^ { - 1 } i _ { x } .
+$$
+
+Introducing the teleport vector $i _ { x }$ allows us to preserve the node’s local neighborhood even in the limit distribution. In this model the influence score of root node $x$ on node $y$ , $I ( x , y )$ , is proportional to the $y$ -th element of our personalized PageRank $\pi _ { \mathrm { p p r } } ( i _ { x } )$ . This value is different for every root node. How fast it decreases as we move away from the root node can be adjusted via $\alpha$ . By substituting the indicator vector $i _ { x }$ with the unit matrix ${ { I } _ { n } }$ we obtain our fully personalized PageRank matrix $\Pi _ { \mathrm { p p r } } = \alpha ( I _ { n } - ( 1 - \alpha ) \hat { \tilde { { \bf A } } } ) ^ { - 1 }$ , whose element $( y x )$ specifies the influence score of node $x$ on node $y$ , $I ( x , y ) \propto \mathbf { I } _ { \mathrm { p p r } } ^ { ( y x ) }$ . Note that due to symmetry $\mathbf { \Pi } \mathbf { \Pi } _ { \mathrm { p p r } } ^ { ( y x ) } = \mathbf { \Pi } \mathbf { I } _ { \mathrm { p p r } } ^ { ( x y ) }$ , i.e. the influence of $x$ on $y$ is equal to the influence of $y$ on . This inverse always exists since $\begin{array} { r } { \frac { 1 } { 1 - \alpha } > 1 } \end{array}$ and therefore cannot be an eigenvalue of $\hat { \tilde { A } }$ (see Appendix A).
+
+Personalized propagation of neural predictions (PPNP). To utilize the above influence scores for semi-supervised classification we generate predictions for each node based on its own features and then propagate them via our fully personalized PageRank scheme to generate the final predictions. This is the foundation of personalized propagation of neural predictions. PPNP’s model equation is
+
+$$
+\begin{array} { r l r } { Z _ { \mathrm { P P N P } } = \operatorname { s o f t m a x } \left( \alpha \left( \boldsymbol { I } _ { n } - ( 1 - \alpha ) \hat { \boldsymbol { A } } \right) ^ { - 1 } \boldsymbol { H } \right) , } & { } & { \boldsymbol { H } _ { i , : } = f _ { \boldsymbol { \theta } } ( X _ { i , : } ) , } \end{array}
+$$
+
+where $\boldsymbol { X }$ is the feature matrix and $f _ { \theta }$ a neural network with parameter set $\theta$ generating the predictions $\pmb { H } \in \mathbb { R } ^ { n \times c }$ . Note that $f _ { \theta }$ operates on each node’s features independently, allowing for parallelization. Furthermore, one could substitute $\hat { \tilde { A } }$ with any propagation matrix, such as $\boldsymbol { A } _ { \mathrm { r w } }$ .
+
+As a consequence, PPNP separates the neural network used for generating predictions from the propagation scheme. This separation additionally solves the second issue mentioned above: the depth of the neural network is now fully independent of the propagation algorithm. As we saw when connecting GCN to PageRank, personalized PageRank can effectively use even infinitely many neighborhood aggregation layers, which is clearly not possible in the classical message passing framework. Furthermore, the separation gives us the flexibility to use any method for generating predictions, e.g. deep convolutional neural networks for graphs of images.
+
+While generating predictions and propagating them happen consecutively during inference, it is important to note that the model is trained end-to-end. That is, the gradient flows through the propagation scheme during backpropagation (implicitly considering infinitely many neighborhood aggregation layers). Adding these propagation effects significantly improves the model’s accuracy.
+
+Efficiency analysis. Directly calculating the fully personalized PageRank matrix $\Pi _ { \mathrm { p p r } }$ , is computationally inefficient and results in a dense $\mathbb { R } ^ { n \times n }$ matrix. Using this matrix would lead to a computational complexity and memory requirement of $\mathcal { O } ( n ^ { 2 } )$ for training and inference.
+
+To solve this issue, reconsider the equation ${ \cal Z } = \alpha ( { \cal I } _ { n } - ( 1 - \alpha ) \hat { \tilde { A } } ) ^ { - 1 } H$ . Instead of viewing this equation as a combination of a dense fully personalized PageRank matrix with the prediction matrix, we can also view it as a variant of topic-sensitive PageRank, with each class corresponding to one topic (Haveliwala, 2002). In this view every column of $\pmb { H }$ defines an (unnormalized) distribution over nodes that acts as a teleport set. Hence, we can approximate PPNP via an approximate computation of topic-sensitive PageRank.
+
+Approximate personalized propagation of neural predictions (APPNP). More precisely, APPNP achieves linear computational complexity by approximating topic-sensitive PageRank via power iteration. While PageRank’s power iteration is connected to the regular random walk, the power iteration of topic-sensitive PageRank is related to a random walk with restarts. Each power iteration (random walk/propagation) step of our topic-sensitive PageRank variant is, thus, calculated via
+
+$$
+\begin{array} { r c l } { { } } & { { } } & { { { \pmb Z } ^ { ( 0 ) } = { \pmb H } = f _ { \theta } ( { \pmb X } ) , } } \\ { { } } & { { } } & { { { \pmb Z } ^ { ( k + 1 ) } = ( 1 - \alpha ) \hat { \hat { \bf A } } { \pmb Z } ^ { ( k ) } + \alpha { \pmb H } , } } \\ { { } } & { { } } & { { { \pmb Z } ^ { ( K ) } = \mathrm { s o f t m a x } \left( ( 1 - \alpha ) \hat { \hat { \bf A } } { \pmb Z } ^ { ( K - 1 ) } + \alpha { \pmb H } \right) , } } \end{array}
+$$
+
+where the prediction matrix $\pmb { H }$ acts as both the starting vector and the teleport set, $K$ defines the number of power iteration steps and $k \in [ 0 , K - 2 ]$ . Note that this method retains the graph’s sparsity and never constructs an $\mathbb { R } ^ { n \times n }$ matrix. The convergence of this iterative scheme can be shown by investigating the resulting series (see Appendix B).
+
+Note that the propagation scheme of this model does not require any additional parameters to train – as opposed to models like GCN, which typically require more parameters for each additional propagation layer. We can therefore propagate very far with very few parameters. Our experiments show that this ability is indeed very beneficial (see Section 6). A similar model expressed in the message passing framework would therefore not be able to achieve the same level of performance.
+
+The reformulation of PPNP via fixed-point iterations illustrates a connection to the original graph neural network (GNN) model (Scarselli et al., 2009). While the latter uses a learned fixed-point iteration, our approach uses a predetermined iteration (adapted personalized PageRank) and applies a learned feature transformation before propagation.
+
+In both PPNP and APPNP, the size of the neighborhood influencing each node can be adjusted via the teleport probability $\alpha$ . The freedom to choose $\alpha$ allows us to adjust the model for different types of networks, since varying graph types require the consideration of different neighborhood sizes, as shown in Section 6 and described by Grover & Leskovec (2016) and Abu-El-Haija et al. (2018b).
+
+Table 1: Dataset statistics. Shortest path length is denoted by SP.   
+
+<table><tr><td>Dataset</td><td>Type</td><td>Classes</td><td>Features</td><td>Nodes</td><td>Edges</td><td>Label rate</td><td>Avg. SP</td></tr><tr><td>CITESEER</td><td>Citation</td><td>6</td><td>3703</td><td>2110</td><td>3668</td><td>0.036</td><td>9.31</td></tr><tr><td>CORA-ML</td><td>Citation</td><td>7</td><td>2879</td><td>2810</td><td>7981</td><td>0.047</td><td>5.27</td></tr><tr><td>PUBMED</td><td>Citation</td><td>3</td><td>500</td><td>19 717</td><td>44324</td><td>0.003</td><td>6.34</td></tr><tr><td>MS ACADEMIC</td><td>Co-author</td><td>15</td><td>6805</td><td>18 333</td><td>81894</td><td>0.016</td><td>5.43</td></tr></table>
+
+# 4 RELATED WORK
+
+Several works have tried to improve the training of message passing algorithms and increase the neighborhood available at each node by adding skip connections (Li et al., 2016; Pham et al., 2017; Hamilton et al., 2017; Ying et al., 2018). One recent approach combined skip connection with aggregation schemes (Xu et al., 2018). However, the range of these models is still limited, as apparent in the low number of message passing layers used. While it is possible to add skip connections in the neural network used by our algorithm, this would not influence the propagation scheme. Our approach to solving the range problem is therefore unrelated to these models.
+
+Li et al. (2018) facilitated training by combining message passing with co- and self-training. The improvements achieved by this combination are similar to results reported with other semi-supervised classification models (Buchnik & Cohen, 2018). Note that most algorithms, including ours, can be improved using self- and co-training. However, each additional step used by these methods corresponds to a full training cycle and therefore significantly increases the training time.
+
+Deep GNNs that avoid the oversmoothing issue have been proposed in recent works by combining residual (skip) connections with batch normalization (Kawamoto et al., 2018; Chen et al., 2019). However, our model solves this issue by simplifying the architecture via decoupling prediction and propagation and does not rely on ad-hoc techniques that further complicate the model and introduce additional hyperparameters. Furthermore, since PPNP increases the range without introducing additional layers and parameters it is easier and faster to train compared to a deep GNN.
+
+# 5 EXPERIMENTAL SETUP
+
+Recently, many experimental evaluations have suffered from superficial statistical evaluation and experimental bias from using varying training setups and overfitting. The latter is caused by experiments using a single training/validation/test split, by not distinguishing clearly between the validation and test set, and by finetuning hyperparameters to each dataset or even data split separately. Message-passing algorithms are very sensitive to both data splits and weight initialization (as clearly shown by our evaluation). Thus, a carefully designed evaluation protocol is extremely important. Our work aims to establish such a thorough evaluation protocol. First, we run each experiment 100 times on multiple random splits and initializations. Second, we split the data into a visible and a test set, which do not change. The test set was only used once to report the final performance; and in particular, has never been used to perform hyperparameter and model selection. To further prevent overfitting we use the same number of layers and hidden units, dropout rate $d$ , $L _ { 2 }$ regularization parameter $\lambda$ , and learning rate $l$ across datasets, since all datasets use bag-of-words as features. To prevent experimental bias we optimized the hyperparameters of all models individually using a grid search on CITESEER and CORA-ML and use the same early stopping criterion across models.
+
+Finally, to ensure the statistical robustness of our experimental setup, we calculate confidence intervals via bootstrapping and report the p-values of a paired $t$ -test for our main claims. To our knowledge, this is the most rigorous study on GCN-like models which has been done so far. More details about the experimental setup are provided in Appendix C.
+
+Datasets. We use four text-classification datasets for evaluation. CITESEER (Sen et al., 2008), CORA-ML (McCallum et al., 2000; Bojchevski & Gunnemann, 2018) and P ¨ UBMED (Namata et al., 2012) are citation graphs, where each node represents a paper and the edges represent citations between them. In the MICROSOFT ACADEMIC graph (Shchur et al., 2018) edges represent coauthorship. We use the largest connected component of each graph. All graphs use a bag-of-words representation of the papers’ abstracts as features. While large graphs do not necessarily have a larger diameter (Leskovec et al., 2005), note that these graphs indeed have average shortest path lengths between 5 and 10 and therefore a regular two-layer GCN cannot cover the entire graph. Table 1 reports the dataset statistics.
+
+Table 2: Average accuracy with uncertainties showing the $9 5 \%$ confidence level calculated by bootstrapping. Previously reported improvements vanish on our rigorous experimental setup, while PPNP and APPNP significantly outperform the compared models on all datasets.   
+
+<table><tr><td>Model</td><td>CITESEER</td><td>CORA-ML</td><td>PUBMED</td><td>MS ACADEMIC</td></tr><tr><td>V. GCN</td><td>73.51± 0.48</td><td>82.30±0.34</td><td>77.65± 0.40</td><td>91.65±0.09</td></tr><tr><td>GCN</td><td>75.40 ± 0.30</td><td>83.41 ± 0.39</td><td>78.68 ± 0.38</td><td>92.10±0.08</td></tr><tr><td>N-GCN</td><td>74.25 ± 0.40</td><td>82.25 ±0.30</td><td>77.43± 0.42</td><td>92.86 ±0.11</td></tr><tr><td>GAT</td><td>75.39 ± 0.27</td><td>84.37 ±0.24</td><td>77.76± 0.44</td><td>91.22 ± 0.07</td></tr><tr><td>JK</td><td>73.03 ± 0.47</td><td>82.69 ± 0.35</td><td>77.88 ±0.38</td><td>91.71 ± 0.10</td></tr><tr><td>Bt. FP</td><td>73.55 ± 0.57</td><td>80.84 ± 0.97</td><td>72.94 ± 1.00</td><td>91.61 ± 0.24</td></tr><tr><td>PPNP*</td><td>75.83 ± 0.27</td><td>85.29 ± 0.25</td><td></td><td></td></tr><tr><td>APPNP</td><td>75.73 ±0.30</td><td>85.09 ± 0.25</td><td>79.73 ± 0.31</td><td>93.27 ± 0.08</td></tr></table>
+
+∗out of memory on PUBMED, MS ACADEMIC (see efficiency analysis in Section 3)
+
+![](images/450c3c6b12966fe24757d76c94f47567c73eff0ace75efe4410c70428228806c.jpg)  
+Figure 2: Accuracy distributions of different models. The high standard deviation between data splits and initializations shows the importance of a rigorous evaluation, which is often omitted.
+
+Baseline models. We compare to five state-of-the-art models: GCN (Kipf & Welling, 2017), network of GCNs (N-GCN) (Abu-El-Haija et al., 2018a), graph attention networks (GAT) (Velickovi ˇ c´ et al., 2018), bootstrapped feature propagation (bt. FP) (Buchnik & Cohen, 2018) and jumping knowledge networks with concatenation (JK) (Xu et al., 2018). For GCN we also show the results of the (unoptimized) vanilla version (V. GCN) to demonstrate the strong impact of early stopping and hyperparameter optimization. The hyperparameters of all models are listed in Appendix D.
+
+Model hyperparameters. To ensure a fair model comparison we used a neural network for PPNP that is structurally very similar to GCN and has the same number of parameters. We use two layers with $h = 6 4$ hidden units. We apply $L _ { 2 }$ regularization with $\lambda = 0 . 0 0 5$ on the weights of the first layer and use dropout with dropout rate $d = 0 . 5$ on both layers and the adjacency matrix. For APPNP, adjacency dropout is resampled for each power iteration step. For propagation we use the teleport probability $\alpha = 0 . 1$ and $K = 1 0$ power iteration steps for APPNP. We use $\alpha = 0 . 2$ on the MICROSOFT ACADEMIC graph due to its structural difference (see Figure 5 and its discussion). The combination of this shallow neural network with a comparatively high number of power iteration steps achieved the best results during hyperparameter optimization (see Appendix G).
+
+# 6 RESULTS
+
+Overall accuracy. The results for the accuracy (micro F1-score) are summarized in Table 2. Similar trends are observed for the macro F1-score (see Appendix E). Both models significantly outperform the state-of-the-art baseline models on all datasets. Our rigorous setup might understate the improvements achieved by PPNP and APPNP – this result is statistically significant $p < 0 . 0 5$ , as tested via a paired $t$ -test (see Appendix F). This thorough setup furthermore shows that the advantages reported by recent works practically vanish when training is harmonized, hyperparameters are properly optimized and multiple data splits are considered. A simple GCN with optimized hyperparameters outperforms several recently proposed models on our setup.
+
+Table 3: Average training time per epoch. PPNP and APPNP are only slightly slower than GCN and much faster than more sophisticated methods like GAT.   
+
+<table><tr><td>Graph</td><td>V. GCN</td><td>GCN</td><td>N-GCN</td><td>GAT</td><td>JK</td><td>Bt.FP*</td><td>PPNP**</td><td>APPNP</td></tr><tr><td>CITESEER</td><td>37.6ms</td><td>35.3ms</td><td>115.9 ms</td><td>187.0 ms</td><td>57.5 ms</td><td>=</td><td>49.2 ms</td><td>43.3ms</td></tr><tr><td>CORA-ML</td><td>32.4 ms</td><td>36.5ms</td><td>118.9 ms</td><td>217.4 ms</td><td>43.6 ms</td><td></td><td>55.3ms</td><td>42.7ms</td></tr><tr><td>PUBMED</td><td>48.6ms</td><td>48.3ms</td><td>342.6 ms</td><td>1029.8 ms</td><td>77.8ms</td><td></td><td>-</td><td>64.1ms</td></tr><tr><td>MS ACADEMIC</td><td>45.5ms</td><td>39.2 ms</td><td>328.5ms</td><td>772.2 ms</td><td>61.9 ms</td><td>=</td><td>=</td><td>59.8ms</td></tr></table>
+
+∗not applicable, since core method not trainable ∗∗out of memory on PUBMED, MS ACADEMIC (see efficiency analysis in Section 3)
+
+![](images/88566eae2ce6f160a22712efc36611a1a63fdd2342b12dae2ed67301a9c7b259.jpg)  
+Figure 3: Accuracy for different training set sizes (number of labeled nodes per class) on CORA-ML. PPNP’s dominance increases further for smaller training set sizes.
+
+Figure 2 shows how broad the accuracy distribution of each model is. This is caused by both random initialization and different data splits (train / early stopping / test). This demonstrates how crucial a statistically rigorous evaluation is for a conclusive model comparison. Moreover, it shows the sensitivity (robustness) of each method, e.g. PPNP, APPNP and GAT typically have lower variance.
+
+Training time per epoch. We report the average training time per epoch in Table 3. We decided to only compare the training time per epoch since all hyperparameters were solely optimized for accuracy and the used early stopping criterion is very generous. Obviously, (exact) PPNP can only be applied to moderately sized graphs, while APPNP scales to large data. On average, APPNP is around $2 5 \%$ slower than GCN due to its higher number of matrix multiplications. It scales similarly with graph size as GCN and is therefore significantly faster than other more sophisticated models like GAT. This is observed even though our implementation improved GAT’s training time roughly by a factor of 2 compared to the reference implementation.
+
+Training set size. Since the labeling rate is often very small for real world datasets, investigating how the models perform with a small number of training samples is very important. Figure 3 shows how the number of training nodes per class $n _ { \mathrm { t r a i n } }$ , per class impacts the accuracy on CORA-ML (for other datasets see Appendix H). The dominance of PPNP and APPNP increases further in this sparsely labelled setting. This can be attributed to their higher range, which allows them to better propagate the information further away from the (few) training nodes. We see further evidence for this when comparing the accuracy of APPNP and GCN depending on the distance between a node and the training set (in terms of shortest path). Appendix I shows that the performance gap between APPNP and GCN tends to increase for nodes that are far away from the training nodes. That is, nodes further away from the training set benefit more from the increase in range.
+
+![](images/b5ef1be3d27841710cb335d16bfccf768880bcfa79885bfbb88b30e0d3582494.jpg)  
+Figure 4: Accuracy depending on the number of propagation steps $K$ . The accuracy breaks down for the GCN-like propagation $( \alpha = 0$ ), while it increases and stabilizes when using APPNP $\alpha = 0 . 1 $ ).
+
+![](images/62154be5f91cc09d9d69baa2c807574a402caacda004d7ce1b31b0a43cf0d3ec.jpg)  
+Figure 5: Accuracy depending on teleport probability $\alpha$ . The optimum typically lies within $\alpha \in$ [0.05, 0.2], but changes for different types of datasets.
+
+![](images/97478f5dbeec637506fb81f6d63e7bc17303dab521e5c1b780080f8ab5e41d8d.jpg)  
+Figure 6: Accuracy of APPNP with propagation used only during training/inference. Best results are achieved with full propagation, but propagating only during inference also achieves good results.
+
+Number of power iteration steps. Figure 4 shows how the accuracy depends on the number of power iterations for two different propagation schemes. The first mimics the standard propagation as known from GCNs (i.e. $\alpha = 0$ in APPNP). As clearly shown the performance breaks down as we increase the number of power iterations $K$ (since we approach the global PageRank solution). However, when using personalized propagation (with $\alpha = 0 . 1$ ) the accuracy increases and converges to exact PPNP with infinitely many propagation steps, thus demonstrating the personalized propagation principle is indeed beneficial. As also shown in the figure, it is enough to use a moderate number of power iterations (e.g. $K = 1 0$ ) to effectively approximate exact PPNP. Interestingly, we’ve found that this number coincides with the highest shortest path distance of any node to the training set.
+
+Teleport probability $\alpha$ . Figure 5 shows the effect of the hyperparameter $\alpha$ on the accuracy on the validation set. While the optimum differs slightly for every dataset, we consistently found a teleport probability of around $\alpha \in [ 0 . 0 5 , 0 . 2 ]$ to perform best. This probability should be adjusted for the dataset under investigation, since different graphs exhibit different neighborhood structures (Grover & Leskovec, 2016; Abu-El-Haija et al., 2018b). Note that a higher $\alpha$ improves convergence speed.
+
+Neural network without propagation. PPNP and APPNP are trained end-to-end, with the propagation scheme affecting (i) the neural network $f _ { \theta }$ during training, and (ii) the classification decision during inference. Investigating how the model performs without propagation shows if and how valuable this addition is. Figure 6 shows how propagation affects both training and inference. ”Never” denotes the case where no propagation is used; essentially we train and apply a standard multilayer perceptron (MLP) $f _ { \theta }$ using the features only. ”Training” denotes the case where we use APPNP during training to learn $f _ { \theta }$ ; at inference time, however, only $f _ { \theta }$ is used to predict the class labels. ”Inference”, in contrast, denotes the case where $f _ { \theta }$ is trained without APPNP (i.e. standard MLP on features). This pretrained network with fixed weights is then used with APPNP’s propagation for inference. Finally, ”Inf. & Training” denotes the regular APPNP, which always uses propagation.
+
+The best results are achieved with regular APPNP, which validates our approach. However, on most datasets the accuracy decreases surprisingly little when propagating only during inference. Skipping propagation during training can significantly reduce training time for large graphs as all nodes can be handled independently. This also shows that our model can be combined with pretrained neural networks that do not incorporate any graph information and still significantly improve their accuracy. Moreover, Figure 6 shows that just propagating during training can also lead to large improvements. This indicates that our model can also be applied to online/inductive learning where only the features and not the neighborhood information of an incoming (previously unobserved) node are available.
+
+# 7 CONCLUSION
+
+In this paper we have introduced personalized propagation of neural predictions (PPNP) and its fast approximation, APPNP. We derived this model by considering the relationship between GCN and PageRank and extending it to personalized PageRank. This simple model decouples prediction and propagation and solves the limited range problem inherent in many message passing models without introducing any additional parameters. It uses the information from a large, adjustable (via the teleport probability $\alpha$ ) neighborhood for classifying each node. The model is computationally efficient and outperforms several state-of-the-art methods for semi-supervised classification on multiple graphs in the most thorough study which has been done for GCN-like models so far.
+
+For future work it would be interesting to combine PPNP with more complex neural networks used e.g. in computer vision or natural language processing. Furthermore, faster or incremental approximations of personalized PageRank (Bahmani et al., 2010; 2011; Lofgren et al., 2014) and more sophisticated propagation schemes would also benefit the method.
+
+# ACKNOWLEDGEMENTS
+
+This research was supported by the German Research Foundation, grant GU 1409/2-1.
+
+# REFERENCES
+
+Sami Abu-El-Haija, Amol Kapoor, Bryan Perozzi, and Joonseok Lee. N-GCN: Multi-scale Graph Convolution for Semi-supervised Node Classification. In International Workshop on Mining and Learning with Graphs (MLG), 2018a.   
+Sami Abu-El-Haija, Bryan Perozzi, Rami Al-Rfou, and Alex Alemi. Watch Your Step: Learning Node Embeddings via Graph Attention. In NeurIPS, 2018b.   
+Bahman Bahmani, Abdur Chowdhury, and Ashish Goel. Fast Incremental and Personalized PageRank. VLDB, 2010.   
+Bahman Bahmani, Kaushik Chakrabarti, and Dong Xin. Fast Personalized PageRank on MapReduce. In SIGMOD, 2011.   
+Aleksandar Bojchevski and Stephan Gunnemann. Deep Gaussian Embedding of Graphs: Unsuper- ¨ vised Inductive Learning via Ranking. ICLR, 2018.   
+Aleksandar Bojchevski, Oleksandr Shchur, Daniel Zugner, and Stephan G ¨ unnemann. NetGAN: ¨ Generating Graphs via Random Walks. In ICML, 2018.   
+Joan Bruna, Wojciech Zaremba, Arthur Szlam, and Yann LeCun. Spectral Networks and Deep Locally Connected Networks on Graphs. ICLR, 2014.   
+Eliav Buchnik and Edith Cohen. Bootstrapped Graph Diffusions: Exposing the Power of Nonlinearity. Proceedings of the ACM on Measurement and Analysis of Computing Systems (POMACS), 2 (1):1–19, April 2018.   
+Jie Chen, Tengfei Ma, and Cao Xiao. FastGCN: Fast Learning with Graph Convolutional Networks via Importance Sampling. ICLR, 2018.   
+Zhengdao Chen, Lisha Li, and Joan Bruna. Supervised Community Detection with Line Graph Neural Networks. In ICLR, 2019.   
+Hanjun Dai, Zornitsa Kozareva, Bo Dai, Alexander J. Smola, and Le Song. Learning Steady-States of Iterative Algorithms over Graphs. In ICML, 2018.   
+Michael Defferrard, Xavier Bresson, and Pierre Vandergheynst. Convolutional Neural Networks on ¨ Graphs with Fast Localized Spectral Filtering. In NIPS, 2016.   
+David K. Duvenaud, Dougal Maclaurin, Jorge Aguilera-Iparraguirre, Rafael Gomez-Bombarelli, ´ Timothy Hirzel, Alan Aspuru-Guzik, and Ryan P. Adams. Convolutional Networks on Graphs for ´ Learning Molecular Fingerprints. In NIPS, 2015.
+
+Justin Gilmer, Samuel S. Schoenholz, Patrick F. Riley, Oriol Vinyals, and George E. Dahl. Neural Message Passing for Quantum Chemistry. In ICML, 2017.
+
+Xavier Glorot and Yoshua Bengio. Understanding the difficulty of training deep feedforward neural networks. In AISTATS, 2010.
+
+Aditya Grover and Jure Leskovec. node2vec: Scalable Feature Learning for Networks. In KDD, 2016.
+
+William L. Hamilton, Zhitao Ying, and Jure Leskovec. Inductive Representation Learning on Large Graphs. In NIPS, 2017.
+
+Taher H. Haveliwala. Topic-sensitive PageRank. In WWW, 2002.
+
+Tatsuro Kawamoto, Masashi Tsubaki, and Tomoyuki Obuchi. Mean-field theory of graph neural networks in graph partitioning. In NeurIPS, 2018.
+
+Steven M. Kearnes, Kevin McCloskey, Marc Berndl, Vijay S. Pande, and Patrick Riley. Molecular graph convolutions: moving beyond fingerprints. Journal of Computer-Aided Molecular Design, 30(8):595–608, 2016.
+
+Diederik P. Kingma and Jimmy Ba. Adam: A Method for Stochastic Optimization. ICLR, 2015.
+
+Thomas N. Kipf and Max Welling. Semi-Supervised Classification with Graph Convolutional Networks. ICLR, 2017.
+
+Jure Leskovec, Jon Kleinberg, and Christos Faloutsos. Graphs over Time: Densification Laws, Shrinking Diameters and Possible Explanations. In KDD, 2005.
+
+Qimai Li, Zhichao Han, and Xiao-Ming Wu. Deeper Insights Into Graph Convolutional Networks for Semi-Supervised Learning. In AAAI, 2018.
+
+Yujia Li, Daniel Tarlow, Marc Brockschmidt, and Richard S. Zemel. Gated Graph Sequence Neural Networks. In ICLR, 2016.
+
+Peter Lofgren, Siddhartha Banerjee, Ashish Goel, and Seshadhri Comandur. FAST-PPR: scaling personalized pagerank estimation for large graphs. In KDD, 2014.
+
+Mart´ın Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S. Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Ian Goodfellow, Andrew Harp, Geoffrey Irving, Michael Isard, Yangqing Jia, Rafal Jozefowicz, Lukasz Kaiser, Manjunath Kudlur, Josh Levenberg, Dandelion Mane, Rajat Monga, Sherry Moore, Derek Murray, ´ Chris Olah, Mike Schuster, Jonathon Shlens, Benoit Steiner, Ilya Sutskever, Kunal Talwar, Paul Tucker, Vincent Vanhoucke, Vijay Vasudevan, Fernanda Viegas, Oriol Vinyals, Pete Warden, ´ Martin Wattenberg, Martin Wicke, Yuan Yu, and Xiaoqiang Zheng. TensorFlow: Large-Scale Machine Learning on Heterogeneous Systems, 2015.
+
+Andrew Kachites McCallum, Kamal Nigam, Jason Rennie, and Kristie Seymore. Automating the construction of internet portals with machine learning. Information Retrieval, 3(2):127–163, 2000.
+
+Federico Monti, Davide Boscaini, Jonathan Masci, Emanuele Rodola, Jan Svoboda, and Michael M. Bronstein. Geometric Deep Learning on Graphs and Manifolds Using Mixture Model CNNs. In CVPR, 2017.
+
+Galileo Namata, Ben London, Lise Getoor, and Bert Huang. Query-driven Active Surveying for Collective Classification. In International Workshop on Mining and Learning with Graphs (MLG), 2012.
+
+Sharad Nandanwar and M. N. Murty. Structural Neighborhood Based Classification of Nodes in a Network. In KDD, 2016.
+
+Mathias Niepert, Mohamed Ahmed, and Konstantin Kutzkov. Learning Convolutional Neural Networks for Graphs. In ICML, 2016.
+
+Lawrence Page, Sergey Brin, Rajeev Motwani, and Terry Winograd. The pagerank citation ranking: Bringing order to the web. Technical report, Stanford InfoLab, 1998.
+
+Bryan Perozzi, Rami Al-Rfou, and Steven Skiena. DeepWalk: online learning of social representations. In KDD, 2014.
+
+Trang Pham, Truyen Tran, Dinh Q. Phung, and Svetha Venkatesh. Column Networks for Collective Classification. In AAAI, 2017.
+
+Jiezhong Qiu, Yuxiao Dong, Hao Ma, Jian Li, Kuansan Wang, and Jie Tang. Network Embedding as Matrix Factorization: Unifying DeepWalk, LINE, PTE, and node2vec. In ACM International Conference on Web Search and Data Mining (WSDM), 2018.
+
+F. Scarselli, M. Gori, Ah Chung Tsoi, M. Hagenbuchner, and G. Monfardini. The Graph Neural Network Model. IEEE Transactions on Neural Networks, 20(1):61–80, January 2009.
+
+Michael Sejr Schlichtkrull, Thomas N. Kipf, Peter Bloem, Rianne van den Berg, Ivan Titov, and Max Welling. Modeling Relational Data with Graph Convolutional Networks. In Extended Semantic Web Conference (ESWC), 2018.
+
+Prithviraj Sen, Galileo Namata, Mustafa Bilgic, Lise Getoor, Brian Gallagher, and Tina Eliassi-Rad. Collective Classification in Network Data. AI Magazine, 29(3):93–106, 2008.
+
+Oleksandr Shchur, Maximilian Mumme, Aleksandar Bojchevski, and Stephan Gunnemann. Pitfalls¨ of Graph Neural Network Evaluation. In Relational Representation Learning Workshop (R2L 2018), NeurIPS, 2018.
+
+Jian Tang, Meng Qu, Mingzhe Wang, Ming Zhang, Jun Yan, and Qiaozhu Mei. LINE: Large-scale Information Network Embedding. In WWW, 2015.
+
+Petar Velickovi ˇ c, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Li ´ o, and Yoshua \` Bengio. Graph Attention Networks. ICLR, 2018.
+
+Keyulu Xu, Chengtao Li, Yonglong Tian, Tomohiro Sonobe, Ken-ichi Kawarabayashi, and Stefanie Jegelka. Representation Learning on Graphs with Jumping Knowledge Networks. In ICML, 2018.
+
+Zhilin Yang, William W. Cohen, and Ruslan Salakhutdinov. Revisiting Semi-Supervised Learning with Graph Embeddings. In ICML, 2016.
+
+Rex Ying, Ruining He, Kaifeng Chen, Pong Eksombatchai, William L. Hamilton, and Jure Leskovec. Graph Convolutional Neural Networks for Web-Scale Recommender Systems. KDD, 2018.
+
+# A EXISTENCE OF $\Pi _ { \mathrm { P P R } }$
+
+The matrix
+
+$$
+\Pi _ { \mathrm { p p r } } = \alpha \left( I _ { n } - ( 1 - \alpha ) \hat { \tilde { A } } \right) ^ { - 1 }
+$$
+
+exists iff the determinant $\operatorname* { d e t } ( I _ { n } - ( 1 - \alpha ) { \hat { \tilde { A } } } ) \neq 0$ , which is the case iff $\operatorname* { d e t } ( \hat { \tilde { A } } - \frac { 1 } { 1 - \alpha } I _ { n } ) \neq 0$ , i.e. iff $\frac { 1 } { 1 - \alpha }$ is not an eigenvalue of $\hat { \tilde { A } }$ . This value is always larger than 1 since the teleport probability $\alpha \in \ ( 0 , 1 ]$ . Furthermore, the symmetrically normalized matrix $\hat { \tilde { A } }$ has the same eigenvalues as the row-stochastic matrix $\tilde { A } _ { \mathrm { r w } }$ . This can be shown by multiplying the eigenvalue equation $\hat { \tilde { A } } v = \lambda v$ with $\tilde { D } ^ { - 1 / 2 }$ from left and substituting $\pmb { w } = \tilde { \pmb { D } } ^ { - 1 / 2 } \pmb { v }$ . This also shows that the eigenvectors of $\hat { \tilde { A } }$ are the eigenvectors of $\tilde { A } _ { \mathrm { r w } }$ scaled by $\tilde { D } ^ { 1 / 2 }$ . The largest eigenvalue of a row-stochastic matrix is 1, as can be proven using the Gershgorin circle theorem. Hence, $\frac { 1 } { 1 - \alpha }$ cannot be an eigenvalue and $\Pi _ { \mathrm { p p r } }$ always exists.
+
+# B CONVERGENCE OF APPNP
+
+APPNP uses the iterative equation
+
+$$
+{ \cal Z } ^ { ( k + 1 ) } = ( 1 - \alpha ) \hat { \tilde { \cal A } } { \cal Z } ^ { ( k ) } + \alpha { \cal H } .
+$$
+
+After the $k$ -th propagation step, the resulting predictions are
+
+$$
+\pmb { Z } ^ { ( k ) } = \left( ( 1 - \alpha ) ^ { k } \hat { \tilde { \mathbf { A } } } ^ { k } + \alpha \sum _ { i = 0 } ^ { k - 1 } ( 1 - \alpha ) ^ { i } \hat { \tilde { \mathbf { A } } } ^ { i } \right) \pmb { H } .
+$$
+
+If we take the limit $k  \infty$ the left term tends to 0 and the right term becomes a geometric series. The series converges since $\alpha \in ( 0 , 1 ]$ and $\hat { \tilde { A } }$ is symmetrically normalized and therefore $\operatorname* { d e t } ( \hat { \tilde { A } } ) \leq 1$ , resulting in
+
+$$
+\pmb { Z } ^ { ( \infty ) } = \alpha \left( \pmb { I _ { n } } - ( 1 - \alpha ) \hat { \tilde { A } } \right) ^ { - 1 } \pmb { H } ,
+$$
+
+which is the equation for calculating (exact) PPNP.
+
+# C EXPERIMENTAL DETAILS
+
+![](images/922f154318507fafbeb2f3ed5f38a53906d5fadd66a576d6fed279871fc59744.jpg)  
+Figure 7: Illustration of the node sampling procedure.
+
+The sampling procedure is illustrated in Figure 7. The data is first split into a visible and a test set. For the visible set 1500 nodes were sampled for the citation graphs and 5000 for MICROSOFT ACADEMIC. The test set contains all remaining nodes. We use three different label sets in each experiment: A training set of 20 nodes per class, an early stopping set of 500 nodes and either a validation or test set. The validation set contains the remaining nodes of the visible set. We use 20 random seeds for determining the splits. These seeds are drawn once and fixed across runs to facilitate comparisons. We use one set of seeds for the validation splits and a different set for the test splits. Each experiment is run with 5 random initializations on each data split, leading to a total of 100 runs per experiment.
+
+The early stopping criterion uses a patience of $p = 1 0 0$ and an (unreachably high) maximum of $n = 1 0 0 0 0$ epochs. The patience is reset whenever the accuracy increases or the loss decreases on the early stopping set. We choose the parameter set achieving the highest accuracy and break ties by selecting the lowest loss on this set. This criterion was inspired by GAT (Velickovi ˇ c et al., 2018). ´
+
+We used TensorFlow (Mart´ın Abadi et al., 2015) for all experiments except bootstrapped feature propagation. All uncertainties and confidence intervals correspond to a confidence level of $9 5 \%$ and were calculated by bootstrapping with 1000 samples.
+
+We use the Adam optimizer with a learning rate of $l = 0 . 0 1$ and cross-entropy loss for all models (Kingma & Ba, 2015). Weights are initialized as described in Glorot & Bengio (2010). The feature matrix is $L _ { 1 }$ normalized per row.
+
+# D BASELINE HYPERPARAMETERS
+
+Vanilla GCN uses the original settings of two layers with $h = 1 6$ hidden units, no dropout on the adjacency matrix, $L _ { 2 }$ regularization parameter $\lambda { \stackrel { \cdot } { = } } 5 \times 1 0 ^ { - 4 }$ and the original early stopping with a maximum of 200 steps and a patience of 10 steps based on the loss.
+
+The optimized GCN uses two layers with $h = 6 4$ hidden units, dropout on the adjacency matrix with $d = 0 . 5$ and $L _ { 2 }$ regularization parameter $\lambda = 0 . 0 2$ .
+
+N-GCN uses $h = 1 6$ hidden units, $R = 4$ heads per random walk length and random walks of up to $K - 1 = 4$ steps. It uses $L _ { 2 }$ regularization on all layers with $\lambda = \overline { { 1 } } \times 1 0 ^ { - 5 }$ and the attention variant for merging the predictions (Abu-El-Haija et al., 2018a). Note that this model effectively uses $R K h = 3 2 0$ hidden units, which is 5 times as many units compared to GCN, GAT, and PPNP.
+
+For GAT we use the (well optimized) original hyperparameters, except the $L _ { 2 }$ regularization parameter $\lambda = 0 . 0 0 1$ and learning rate $l = 0 . 0 1$ . As opposed to the original paper, we do not use different hyperparameters on PUBMED, as described in our experimental setup.
+
+Bootstrapped feature propagation uses a return probability of $\alpha = 0 . 2$ , 10 propagation steps, 10 bootstrapping (self-training) steps with $r = 0 . 1 n$ training nodes added per step. We add the training nodes with the lowest entropy on the predictions. The number of nodes added per class is based on the class proportions estimated using the predictions. Note that this model does not include any stochasticity in its initialization. We therefore only run it once per train/early stopping/test split.
+
+For the jumping knowledge networks we use the concatenation variant with three layers and $h = 6 4$ hidden units per layer. We apply $L _ { 2 }$ regularization with $\lambda = 0 . 0 0 1$ on all layers and perform dropout with $d = 0 . 5$ on all layers but not on the adjacency matrix.
+
+# E F1 SCORE
+
+Table 4: Average macro F1 score with uncertainties showing the $9 5 \%$ confidence level calculated by bootstrapping. PPNP achieves the highest F1 score on all datasets investigated.   
+
+<table><tr><td>Model</td><td>CITESEER</td><td>CORA-ML</td><td>PUBMED</td><td>MS ACADEMIC</td></tr><tr><td>V. GCN</td><td>0.7002±0.0043</td><td>0.8205±0.0027</td><td>0.7801 ± 0.0038</td><td>0.9000±0.0008</td></tr><tr><td>GCN</td><td>0.7065 ± 0.0037</td><td>0.8289 ± 0.0030</td><td>0.7883 ± 0.0032</td><td>0.9045 ± 0.0008</td></tr><tr><td>N-GCN</td><td>0.7021 ± 0.0035</td><td>0.8183 ± 0.0024</td><td>0.7773 ± 0.0040</td><td>0.9144 ± 0.0012</td></tr><tr><td>GAT</td><td>0.7062 ± 0.0029</td><td>0.8359 ± 0.0025</td><td>0.7777 ± 0.0040</td><td>0.8917 ± 0.0007</td></tr><tr><td>JK</td><td>0.6914 ± 0.0043</td><td>0.8202 ± 0.0026</td><td>0.7799 ± 0.0039</td><td>0.8985 ±0.0012</td></tr><tr><td>Bt.FP</td><td>0.6789 ± 0.0055</td><td>0.8026 ± 0.0082</td><td>0.7448 ± 0.0079</td><td>0.8997 ± 0.0018</td></tr><tr><td>PPNP*</td><td>0.7102 ± 0.0041</td><td>0.8454 ± 0.0021</td><td></td><td></td></tr><tr><td>APPNP</td><td>0.7105 ± 0.0038</td><td>0.8429 ±0.0022</td><td>0.7966 ± 0.0031</td><td>0.9184 ± 0.0009</td></tr></table>
+
+∗out of memory on PUBMED, MS ACADEMIC (see efficiency analysis in Section 3)
+
+# F PAIRED $t$ -TEST
+
+Table 5: p-value of the paired $t$ -test with respect to accuracy.   
+
+<table><tr><td>Model</td><td>CITESEER</td><td>CORA-ML</td><td>PUBMED</td><td>MS ACADEMIC</td></tr><tr><td>PPNP</td><td>1.02 ×10-3</td><td>5.96×10-14</td><td></td><td></td></tr><tr><td>APPNP</td><td>1.77 × 10-2</td><td>4.27 ×10-9</td><td>2.19 ×10-15</td><td>5.93 × 10-13</td></tr></table>
+
+Table 6: p-value of the paired $t$ -test with respect to F1 score.   
+
+<table><tr><td>Model</td><td>CITESEER</td><td>CORA-ML</td><td>PUBMED</td><td>MS ACADEMIC</td></tr><tr><td>PPNP</td><td>4.49×10-2</td><td>4.50×10-14</td><td></td><td></td></tr><tr><td>APPNP</td><td>2.32 ×10-2</td><td>1.07 × 10-8</td><td>8.70 × 10-14</td><td>1.99 × 10-8</td></tr></table>
+
+![](images/f5436eb85eb29bfa0226b787ea9e7661e7b18c50f0879781bcd8952e22d5f919.jpg)  
+Figure 8: Validation accuracy of APPNP for varying numbers of neural network (NN) layers. Deep NNs do not improve the accuracy, which is probably due to the simple bag-of-words features and the small training set size.
+
+# H TRAINING SET SIZE
+
+![](images/1df3e15b9e80568ac6bbedbc6c5497b5b9982cec9d08137aa5b32a1273986c1f.jpg)  
+Figure 9: Accuracy for different training set sizes on CITESEER.
+
+![](images/c9a16536a4e516b2d58581e5824a4760c7222b772af1594436a217beb5a80ac1.jpg)  
+Figure 10: Accuracy for different training set sizes on PUBMED.
+
+![](images/5389f5b7b8ef0d13b1c87be6e2841f435778ee204063bc1b91ca6aa3f22410a4.jpg)  
+Figure 11: Accuracy for different training set sizes on MICROSOFT ACADEMIC.
+
+![](images/0072cf936b72a26218ba2729a695270a0013a7756cf97608c3cdfb41adef98a8.jpg)  
+Figure 12: $\Delta$ Accuracy $( \% )$ denotes the average improvement in percentage points of APPNP over GCN depending on the distance (number of hops) from the training nodes on CORA-ML. $\bar { n }$ denotes the average number of nodes at each distance. The improvement increases with distance.
+
+![](images/2fd5099893deec0cdde802e9feb63de47fc5c089a3e5393abf3aaeca4685822b.jpg)  
+Figure 13: $\Delta$ Accuracy $( \% )$ denotes the average improvement in percentage points of APPNP over GCN depending on the distance (number of hops) from the training nodes on different graphs. $\bar { n }$ denotes the average number of nodes at each distance over different splits.

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+    {
+        "type": "text",
+        "text": "PREDICT THEN PROPAGATE: GRAPH NEURAL NETWORKS MEET PERSONALIZED PAGERANK ",
+        "text_level": 1,
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+        "text": "Johannes Gasteiger, Aleksandar Bojchevski & Stephan Gunnemann ¨ ",
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+        "text": "Technical University of Munich, Germany {j.gasteiger,a.bojchevski,guennemann}@in.tum.de ",
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+        "text": "ABSTRACT ",
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+        "text": "Neural message passing algorithms for semi-supervised classification on graphs have recently achieved great success. However, for classifying a node these methods only consider nodes that are a few propagation steps away and the size of this utilized neighborhood is hard to extend. In this paper, we use the relationship between graph convolutional networks (GCN) and PageRank to derive an improved propagation scheme based on personalized PageRank. We utilize this propagation procedure to construct a simple model, personalized propagation of neural predictions (PPNP), and its fast approximation, APPNP. Our model’s training time is on par or faster and its number of parameters on par or lower than previous models. It leverages a large, adjustable neighborhood for classification and can be easily combined with any neural network. We show that this model outperforms several recently proposed methods for semi-supervised classification in the most thorough study done so far for GCN-like models. Our implementation is available online. 1 ",
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+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "text": "Graphs are ubiquitous in the real world and its description through scientific models. They are used to study the spread of information, to optimize delivery, to recommend new books, to suggest friends, or to find a party’s potential voters. Deep learning approaches have achieved great success on many important graph problems such as link prediction (Grover & Leskovec, 2016; Bojchevski et al., 2018), graph classification (Duvenaud et al., 2015; Niepert et al., 2016; Gilmer et al., 2017) and semi-supervised node classification (Yang et al., 2016; Kipf & Welling, 2017). ",
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+        "text": "There are many approaches for leveraging deep learning algorithms on graphs. Node embedding methods use random walks or matrix factorization to directly train individual node embeddings, often without using node features and usually in an unsupervised manner, i.e. without leveraging node classes (Perozzi et al., 2014; Tang et al., 2015; Nandanwar & Murty, 2016; Grover & Leskovec, 2016; Qiu et al., 2018). Many other approaches use both graph structure and node features in a supervised setting. Examples for these include spectral graph convolutional neural networks (Bruna et al., 2014; Defferrard et al., 2016), message passing (or neighbor aggregation) algorithms (Kearnes et al., 2016; Kipf & Welling, 2017; Hamilton et al., 2017; Pham et al., 2017; Monti et al., 2017; Gilmer et al., 2017), and neighbor aggregation via recurrent neural networks (Scarselli et al., 2009; Li et al., 2016; Dai et al., 2018). Among these categories, the class of message passing algorithms has garnered particular attention recently due to its flexibility and good performance. ",
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+        "text": "Several works have been aimed at improving the basic neighborhood aggregation scheme by using attention mechanisms (Kearnes et al., 2016; Hamilton et al., 2017; Velickovi ˇ c et al., 2018), random ´ walks (Abu-El-Haija et al., 2018a; Ying et al., 2018; Li et al., 2018), edge features (Kearnes et al., 2016; Gilmer et al., 2017; Schlichtkrull et al., 2018) and making it more scalable on large graphs (Chen et al., 2018; Ying et al., 2018). However, all of these methods only use the information of a very limited neighborhood for each node. A larger neighborhood would be desirable to provide the model with more information, especially for nodes in the periphery or in a sparsely labelled setting. ",
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+        "type": "text",
+        "text": "Increasing the size of the neighborhood used by these algorithms, i.e. their range, is not trivial since neighborhood aggregation in this scheme is essentially a type of Laplacian smoothing and too many layers lead to oversmoothing (Li et al., 2018). Xu et al. (2018) highlighted the same problem by establishing a relationship between the message passing algorithm termed Graph Convolutional Network (GCN) by Kipf & Welling (2017) and a random walk. Using this relationship we see that GCN converges to this random walk’s limit distribution as the number of layers increases. The limit distribution is a property of the graph as a whole and does not take the random walk’s starting (root) node into account. As such it is unsuited to describe the root node’s neighborhood. Hence, GCN’s performance necessarily deteriorates for a high number of layers (or aggregation/propagation steps). ",
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+        "type": "text",
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+        "text": "To solve this issue, in this paper, we first highlight the inherent connection between the limit distribution and PageRank (Page et al., 1998). We then propose an algorithm that utilizes a propagation scheme derived from personalized PageRank instead. This algorithm adds a chance of teleporting back to the root node, which ensures that the PageRank score encodes the local neighborhood for every root node (Page et al., 1998). The teleport probability allows us to balance the needs of preserving locality (i.e. staying close to the root node to avoid oversmoothing) and leveraging the information from a large neighborhood. We show that this propagation scheme permits the use of far more (in fact, infinitely many) propagation steps without leading to oversmoothing. ",
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+        "type": "text",
+        "text": "Moreover, while propagation and classification are inherently intertwined in message passing, our proposed algorithm separates the neural network from the propagation scheme. This allows us to achieve a much higher range without changing the neural network, whereas in the message passing scheme every additional propagation step would require an additional layer. It also permits the independent development of the propagation algorithm and the neural network generating predictions from node features. That is, we can combine any state-of-the-art prediction method with our propagation scheme. We even found that adding our propagation scheme during inference significantly improves the accuracy of networks that were trained without using any graph information. ",
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+        "type": "text",
+        "text": "Our model achieves state-of-the-art results while requiring fewer parameters and less training time compared to most competing models, with a computational complexity that is linear in the number of edges. We show these results in the most thorough study (including significance testing) of message passing models using graphs with text-based features that has been done so far. ",
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+        "type": "text",
+        "text": "2 GRAPH CONVOLUTIONAL NETWORKS AND THEIR LIMITED RANGE ",
+        "text_level": 1,
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+        "text": "We first introduce our notation and explain the problem our model solves. Let $G = ( V , E )$ be a graph with nodes $V$ and edges $E$ . Let $n$ denote the number of nodes and $m$ the number of edges. The nodes are described by the feature matrix $\\ b { X } \\in \\mathbb { R } ^ { n \\times f }$ , with the number of features $f$ per node, and the class (or label) matrix $\\ b { Y } \\in \\mathbb { R } ^ { n \\times c }$ , with the number of classes $c$ . The graph $G$ is described by the adjacency matrix $A \\in \\mathbb { R } ^ { n \\times n }$ . $\\tilde { \\pmb { A } } = \\pmb { A } + \\pmb { I _ { n } }$ denotes the adjacency matrix with added self-loops. ",
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+        "type": "text",
+        "text": "One simple and widely used message passing algorithm for semi-supervised classification is the Graph Convolutional Network (GCN). In the case of two message passing layers its equation is ",
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+        "type": "equation",
+        "img_path": "images/2ca77a0438d85565dfbd33375f652c8cc3b1101eb60bceefdd54ecda12330734.jpg",
+        "text": "$$\n\\boldsymbol { Z } _ { \\mathrm { G C N } } = \\mathrm { s o f t m a x } \\left( \\hat { \\tilde { A } } \\mathrm { R e L U } \\left( \\hat { \\tilde { A } } \\boldsymbol { X } \\boldsymbol { W } _ { 0 } \\right) \\boldsymbol { W } _ { 1 } \\right) ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where $\\ b { Z } \\in \\mathbb { R } ^ { n \\times c }$ are the predicted node labels, $\\hat { \\tilde { A } } = \\tilde { D } ^ { - 1 / 2 } \\tilde { A } \\tilde { D } ^ { - 1 / 2 }$ is the symmetrically normalized adjacency matrix with self-loops, with the diagonal degree matrix $\\begin{array} { r } { \\tilde { D } _ { i j } = \\sum _ { k } \\tilde { A } _ { i k } \\delta _ { i j } } \\end{array}$ , and $W _ { 0 }$ and $W _ { 1 }$ are trainable weight matrices (Kipf & Welling, 2017). ",
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+        "type": "text",
+        "text": "With two GCN-layers, only neighbors in the two-hop neighborhood are considered. There are essentially two reasons why a message passing algorithm like GCN cannot be trivially expanded to use a larger neighborhood. First, aggregation by averaging causes oversmoothing if too many layers are used. It, therefore, loses its focus on the local neighborhood (Li et al., 2018). Second, most common aggregation schemes use learnable weight matrices in each layer. Therefore, using a larger neighborhood necessarily increases the depth and number of learnable parameters of the neural network (the second aspect can be circumvented by using weight sharing, which is typically not the case, though). However, the required neighborhood size and neural network depth are two completely orthogonal aspects. This fixed relationship is a strong limitation and leads to bad compromises. ",
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+        "type": "text",
+        "text": "We will start by concentrating on the filayer GCN the influence score of node $x$ isson $y$ ., $\\begin{array} { r } { I ( x , y ) ~ = ~ \\sum _ { i } \\sum _ { j } \\frac { \\partial Z _ { y i } } { \\partial X _ { x j } } } \\end{array}$ $\\mathrm { X u }$ hown that for a , is proportional $k$ expectation to a slightly modified $k$ -step random walk distribution starting at the root node $x$ , $P _ { \\mathrm { r w } } , ( x \\to y , k )$ . Hence, the information of node $x$ spreads to node $y$ in a random walk-like manner. If we take the limit $k  \\infty$ and the graph is irreducible and aperiodic, this random walk probability distribution $P _ { \\mathrm { r w } } , ( x \\to y , k )$ converges to the limit (or stationary) distribution $P _ { \\mathrm { l i m } } ( \\to y )$ . This distribution can be obtained by solving the equation $\\pi _ { \\mathrm { l i m } } = \\hat { \\tilde { A } } \\pi _ { \\mathrm { l i m } }$ . Obviously, the result only depends on the graph as a whole and is independent of the random walk’s starting (root) node $x$ . This global property is therefore unsuitable for describing the root node’s neighborhood. ",
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+    {
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+        "img_path": "images/b4aa525e1c873f19349a9967b8eab74ff37d89d4cc7d33d209b29d100026a523.jpg",
+        "image_caption": [
+            "Figure 1: Illustration of (approximate) personalized propagation of neural predictions (PPNP, APPNP). Predictions are first generated from each node’s own features by a neural network and then propagated using an adaptation of personalized PageRank. The model is trained end-to-end. "
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+        "type": "text",
+        "text": "3 PERSONALIZED PROPAGATION OF NEURAL PREDICTIONS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "From message passing to personalized PageRank. We can solve the problem of lost focus by recognizing the connection between the limit distribution and PageRank (Page et al., 1998). The only differences between these two are the added self-loops and the adjacency matrix normalization in $\\hat { \\tilde { A } }$ . Original PageRank is calculated via $\\pi _ { \\mathrm { p r } } = A _ { \\mathrm { r w } } \\pi _ { \\mathrm { p r } }$ , with $A _ { \\mathrm { r w } } = A D ^ { - 1 }$ . Having made this connection we can now consider using a variant of PageRank that takes the root node into account – personalized PageRank (Page et al., 1998). We define the root node $x$ via the teleport vector $i _ { x }$ , which is a one-hot indicator vector. Our adaptation of personalized PageRank can be obtained for node $x$ using the recurrent equation $\\pi _ { \\mathrm { p p r } } ( i _ { x } ) \\stackrel { - } { = } ( 1 - \\alpha ) \\mathring { \\hat { A } } \\pi _ { \\mathrm { p p r } } ( i _ { x } ) + \\alpha i _ { x }$ , with the teleport (or restart) probability $\\alpha \\in \\mathsf { \\Gamma } ( 0 , 1 ]$ . By solving this equation, we obtain ",
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+        "type": "equation",
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+        "text": "$$\n\\pi _ { \\mathrm { p p r } } ( i _ { x } ) = \\alpha \\left( I _ { n } - ( 1 - \\alpha ) \\hat { \\tilde { A } } \\right) ^ { - 1 } i _ { x } .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Introducing the teleport vector $i _ { x }$ allows us to preserve the node’s local neighborhood even in the limit distribution. In this model the influence score of root node $x$ on node $y$ , $I ( x , y )$ , is proportional to the $y$ -th element of our personalized PageRank $\\pi _ { \\mathrm { p p r } } ( i _ { x } )$ . This value is different for every root node. How fast it decreases as we move away from the root node can be adjusted via $\\alpha$ . By substituting the indicator vector $i _ { x }$ with the unit matrix ${ { I } _ { n } }$ we obtain our fully personalized PageRank matrix $\\Pi _ { \\mathrm { p p r } } = \\alpha ( I _ { n } - ( 1 - \\alpha ) \\hat { \\tilde { { \\bf A } } } ) ^ { - 1 }$ , whose element $( y x )$ specifies the influence score of node $x$ on node $y$ , $I ( x , y ) \\propto \\mathbf { I } _ { \\mathrm { p p r } } ^ { ( y x ) }$ . Note that due to symmetry $\\mathbf { \\Pi } \\mathbf { \\Pi } _ { \\mathrm { p p r } } ^ { ( y x ) } = \\mathbf { \\Pi } \\mathbf { I } _ { \\mathrm { p p r } } ^ { ( x y ) }$ , i.e. the influence of $x$ on $y$ is equal to the influence of $y$ on . This inverse always exists since $\\begin{array} { r } { \\frac { 1 } { 1 - \\alpha } > 1 } \\end{array}$ and therefore cannot be an eigenvalue of $\\hat { \\tilde { A } }$ (see Appendix A). ",
+        "bbox": [
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+        "type": "text",
+        "text": "Personalized propagation of neural predictions (PPNP). To utilize the above influence scores for semi-supervised classification we generate predictions for each node based on its own features and then propagate them via our fully personalized PageRank scheme to generate the final predictions. This is the foundation of personalized propagation of neural predictions. PPNP’s model equation is ",
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+        "img_path": "images/62deb9b7b97bdabbc698dd47825aa1c31be99565ce8bbf2200f8f4cfbd5d284f.jpg",
+        "text": "$$\n\\begin{array} { r l r } { Z _ { \\mathrm { P P N P } } = \\operatorname { s o f t m a x } \\left( \\alpha \\left( \\boldsymbol { I } _ { n } - ( 1 - \\alpha ) \\hat { \\boldsymbol { A } } \\right) ^ { - 1 } \\boldsymbol { H } \\right) , } & { } & { \\boldsymbol { H } _ { i , : } = f _ { \\boldsymbol { \\theta } } ( X _ { i , : } ) , } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\boldsymbol { X }$ is the feature matrix and $f _ { \\theta }$ a neural network with parameter set $\\theta$ generating the predictions $\\pmb { H } \\in \\mathbb { R } ^ { n \\times c }$ . Note that $f _ { \\theta }$ operates on each node’s features independently, allowing for parallelization. Furthermore, one could substitute $\\hat { \\tilde { A } }$ with any propagation matrix, such as $\\boldsymbol { A } _ { \\mathrm { r w } }$ . ",
+        "bbox": [
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+        "type": "text",
+        "text": "As a consequence, PPNP separates the neural network used for generating predictions from the propagation scheme. This separation additionally solves the second issue mentioned above: the depth of the neural network is now fully independent of the propagation algorithm. As we saw when connecting GCN to PageRank, personalized PageRank can effectively use even infinitely many neighborhood aggregation layers, which is clearly not possible in the classical message passing framework. Furthermore, the separation gives us the flexibility to use any method for generating predictions, e.g. deep convolutional neural networks for graphs of images. ",
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+    {
+        "type": "text",
+        "text": "While generating predictions and propagating them happen consecutively during inference, it is important to note that the model is trained end-to-end. That is, the gradient flows through the propagation scheme during backpropagation (implicitly considering infinitely many neighborhood aggregation layers). Adding these propagation effects significantly improves the model’s accuracy. ",
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+    {
+        "type": "text",
+        "text": "Efficiency analysis. Directly calculating the fully personalized PageRank matrix $\\Pi _ { \\mathrm { p p r } }$ , is computationally inefficient and results in a dense $\\mathbb { R } ^ { n \\times n }$ matrix. Using this matrix would lead to a computational complexity and memory requirement of $\\mathcal { O } ( n ^ { 2 } )$ for training and inference. ",
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+    {
+        "type": "text",
+        "text": "To solve this issue, reconsider the equation ${ \\cal Z } = \\alpha ( { \\cal I } _ { n } - ( 1 - \\alpha ) \\hat { \\tilde { A } } ) ^ { - 1 } H$ . Instead of viewing this equation as a combination of a dense fully personalized PageRank matrix with the prediction matrix, we can also view it as a variant of topic-sensitive PageRank, with each class corresponding to one topic (Haveliwala, 2002). In this view every column of $\\pmb { H }$ defines an (unnormalized) distribution over nodes that acts as a teleport set. Hence, we can approximate PPNP via an approximate computation of topic-sensitive PageRank. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Approximate personalized propagation of neural predictions (APPNP). More precisely, APPNP achieves linear computational complexity by approximating topic-sensitive PageRank via power iteration. While PageRank’s power iteration is connected to the regular random walk, the power iteration of topic-sensitive PageRank is related to a random walk with restarts. Each power iteration (random walk/propagation) step of our topic-sensitive PageRank variant is, thus, calculated via ",
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+        "img_path": "images/345a40dc400f9132fe6fa440007a4ce1f551838991b5b87f4b091afde88da86e.jpg",
+        "text": "$$\n\\begin{array} { r c l } { { } } & { { } } & { { { \\pmb Z } ^ { ( 0 ) } = { \\pmb H } = f _ { \\theta } ( { \\pmb X } ) , } } \\\\ { { } } & { { } } & { { { \\pmb Z } ^ { ( k + 1 ) } = ( 1 - \\alpha ) \\hat { \\hat { \\bf A } } { \\pmb Z } ^ { ( k ) } + \\alpha { \\pmb H } , } } \\\\ { { } } & { { } } & { { { \\pmb Z } ^ { ( K ) } = \\mathrm { s o f t m a x } \\left( ( 1 - \\alpha ) \\hat { \\hat { \\bf A } } { \\pmb Z } ^ { ( K - 1 ) } + \\alpha { \\pmb H } \\right) , } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "where the prediction matrix $\\pmb { H }$ acts as both the starting vector and the teleport set, $K$ defines the number of power iteration steps and $k \\in [ 0 , K - 2 ]$ . Note that this method retains the graph’s sparsity and never constructs an $\\mathbb { R } ^ { n \\times n }$ matrix. The convergence of this iterative scheme can be shown by investigating the resulting series (see Appendix B). ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Note that the propagation scheme of this model does not require any additional parameters to train – as opposed to models like GCN, which typically require more parameters for each additional propagation layer. We can therefore propagate very far with very few parameters. Our experiments show that this ability is indeed very beneficial (see Section 6). A similar model expressed in the message passing framework would therefore not be able to achieve the same level of performance. ",
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "The reformulation of PPNP via fixed-point iterations illustrates a connection to the original graph neural network (GNN) model (Scarselli et al., 2009). While the latter uses a learned fixed-point iteration, our approach uses a predetermined iteration (adapted personalized PageRank) and applies a learned feature transformation before propagation. ",
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+    {
+        "type": "text",
+        "text": "In both PPNP and APPNP, the size of the neighborhood influencing each node can be adjusted via the teleport probability $\\alpha$ . The freedom to choose $\\alpha$ allows us to adjust the model for different types of networks, since varying graph types require the consideration of different neighborhood sizes, as shown in Section 6 and described by Grover & Leskovec (2016) and Abu-El-Haija et al. (2018b). ",
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+    {
+        "type": "table",
+        "img_path": "images/dd772eccfe4e8fb719bfcb306dc2425831eff18677efde174b6f43a5b6b8e331.jpg",
+        "table_caption": [
+            "Table 1: Dataset statistics. Shortest path length is denoted by SP. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>Type</td><td>Classes</td><td>Features</td><td>Nodes</td><td>Edges</td><td>Label rate</td><td>Avg. SP</td></tr><tr><td>CITESEER</td><td>Citation</td><td>6</td><td>3703</td><td>2110</td><td>3668</td><td>0.036</td><td>9.31</td></tr><tr><td>CORA-ML</td><td>Citation</td><td>7</td><td>2879</td><td>2810</td><td>7981</td><td>0.047</td><td>5.27</td></tr><tr><td>PUBMED</td><td>Citation</td><td>3</td><td>500</td><td>19 717</td><td>44324</td><td>0.003</td><td>6.34</td></tr><tr><td>MS ACADEMIC</td><td>Co-author</td><td>15</td><td>6805</td><td>18 333</td><td>81894</td><td>0.016</td><td>5.43</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4 RELATED WORK ",
+        "text_level": 1,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Several works have tried to improve the training of message passing algorithms and increase the neighborhood available at each node by adding skip connections (Li et al., 2016; Pham et al., 2017; Hamilton et al., 2017; Ying et al., 2018). One recent approach combined skip connection with aggregation schemes (Xu et al., 2018). However, the range of these models is still limited, as apparent in the low number of message passing layers used. While it is possible to add skip connections in the neural network used by our algorithm, this would not influence the propagation scheme. Our approach to solving the range problem is therefore unrelated to these models. ",
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+    {
+        "type": "text",
+        "text": "Li et al. (2018) facilitated training by combining message passing with co- and self-training. The improvements achieved by this combination are similar to results reported with other semi-supervised classification models (Buchnik & Cohen, 2018). Note that most algorithms, including ours, can be improved using self- and co-training. However, each additional step used by these methods corresponds to a full training cycle and therefore significantly increases the training time. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Deep GNNs that avoid the oversmoothing issue have been proposed in recent works by combining residual (skip) connections with batch normalization (Kawamoto et al., 2018; Chen et al., 2019). However, our model solves this issue by simplifying the architecture via decoupling prediction and propagation and does not rely on ad-hoc techniques that further complicate the model and introduce additional hyperparameters. Furthermore, since PPNP increases the range without introducing additional layers and parameters it is easier and faster to train compared to a deep GNN. ",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTAL SETUP ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Recently, many experimental evaluations have suffered from superficial statistical evaluation and experimental bias from using varying training setups and overfitting. The latter is caused by experiments using a single training/validation/test split, by not distinguishing clearly between the validation and test set, and by finetuning hyperparameters to each dataset or even data split separately. Message-passing algorithms are very sensitive to both data splits and weight initialization (as clearly shown by our evaluation). Thus, a carefully designed evaluation protocol is extremely important. Our work aims to establish such a thorough evaluation protocol. First, we run each experiment 100 times on multiple random splits and initializations. Second, we split the data into a visible and a test set, which do not change. The test set was only used once to report the final performance; and in particular, has never been used to perform hyperparameter and model selection. To further prevent overfitting we use the same number of layers and hidden units, dropout rate $d$ , $L _ { 2 }$ regularization parameter $\\lambda$ , and learning rate $l$ across datasets, since all datasets use bag-of-words as features. To prevent experimental bias we optimized the hyperparameters of all models individually using a grid search on CITESEER and CORA-ML and use the same early stopping criterion across models. ",
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+        "type": "text",
+        "text": "Finally, to ensure the statistical robustness of our experimental setup, we calculate confidence intervals via bootstrapping and report the p-values of a paired $t$ -test for our main claims. To our knowledge, this is the most rigorous study on GCN-like models which has been done so far. More details about the experimental setup are provided in Appendix C. ",
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+        "type": "text",
+        "text": "Datasets. We use four text-classification datasets for evaluation. CITESEER (Sen et al., 2008), CORA-ML (McCallum et al., 2000; Bojchevski & Gunnemann, 2018) and P ¨ UBMED (Namata et al., 2012) are citation graphs, where each node represents a paper and the edges represent citations between them. In the MICROSOFT ACADEMIC graph (Shchur et al., 2018) edges represent coauthorship. We use the largest connected component of each graph. All graphs use a bag-of-words representation of the papers’ abstracts as features. While large graphs do not necessarily have a larger diameter (Leskovec et al., 2005), note that these graphs indeed have average shortest path lengths between 5 and 10 and therefore a regular two-layer GCN cannot cover the entire graph. Table 1 reports the dataset statistics. ",
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+        "table_caption": [
+            "Table 2: Average accuracy with uncertainties showing the $9 5 \\%$ confidence level calculated by bootstrapping. Previously reported improvements vanish on our rigorous experimental setup, while PPNP and APPNP significantly outperform the compared models on all datasets. "
+        ],
+        "table_footnote": [
+            "∗out of memory on PUBMED, MS ACADEMIC (see efficiency analysis in Section 3) "
+        ],
+        "table_body": "<table><tr><td>Model</td><td>CITESEER</td><td>CORA-ML</td><td>PUBMED</td><td>MS ACADEMIC</td></tr><tr><td>V. GCN</td><td>73.51± 0.48</td><td>82.30±0.34</td><td>77.65± 0.40</td><td>91.65±0.09</td></tr><tr><td>GCN</td><td>75.40 ± 0.30</td><td>83.41 ± 0.39</td><td>78.68 ± 0.38</td><td>92.10±0.08</td></tr><tr><td>N-GCN</td><td>74.25 ± 0.40</td><td>82.25 ±0.30</td><td>77.43± 0.42</td><td>92.86 ±0.11</td></tr><tr><td>GAT</td><td>75.39 ± 0.27</td><td>84.37 ±0.24</td><td>77.76± 0.44</td><td>91.22 ± 0.07</td></tr><tr><td>JK</td><td>73.03 ± 0.47</td><td>82.69 ± 0.35</td><td>77.88 ±0.38</td><td>91.71 ± 0.10</td></tr><tr><td>Bt. FP</td><td>73.55 ± 0.57</td><td>80.84 ± 0.97</td><td>72.94 ± 1.00</td><td>91.61 ± 0.24</td></tr><tr><td>PPNP*</td><td>75.83 ± 0.27</td><td>85.29 ± 0.25</td><td></td><td></td></tr><tr><td>APPNP</td><td>75.73 ±0.30</td><td>85.09 ± 0.25</td><td>79.73 ± 0.31</td><td>93.27 ± 0.08</td></tr></table>",
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+        "img_path": "images/450c3c6b12966fe24757d76c94f47567c73eff0ace75efe4410c70428228806c.jpg",
+        "image_caption": [
+            "Figure 2: Accuracy distributions of different models. The high standard deviation between data splits and initializations shows the importance of a rigorous evaluation, which is often omitted. "
+        ],
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+    {
+        "type": "text",
+        "text": "Baseline models. We compare to five state-of-the-art models: GCN (Kipf & Welling, 2017), network of GCNs (N-GCN) (Abu-El-Haija et al., 2018a), graph attention networks (GAT) (Velickovi ˇ c´ et al., 2018), bootstrapped feature propagation (bt. FP) (Buchnik & Cohen, 2018) and jumping knowledge networks with concatenation (JK) (Xu et al., 2018). For GCN we also show the results of the (unoptimized) vanilla version (V. GCN) to demonstrate the strong impact of early stopping and hyperparameter optimization. The hyperparameters of all models are listed in Appendix D. ",
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+    {
+        "type": "text",
+        "text": "Model hyperparameters. To ensure a fair model comparison we used a neural network for PPNP that is structurally very similar to GCN and has the same number of parameters. We use two layers with $h = 6 4$ hidden units. We apply $L _ { 2 }$ regularization with $\\lambda = 0 . 0 0 5$ on the weights of the first layer and use dropout with dropout rate $d = 0 . 5$ on both layers and the adjacency matrix. For APPNP, adjacency dropout is resampled for each power iteration step. For propagation we use the teleport probability $\\alpha = 0 . 1$ and $K = 1 0$ power iteration steps for APPNP. We use $\\alpha = 0 . 2$ on the MICROSOFT ACADEMIC graph due to its structural difference (see Figure 5 and its discussion). The combination of this shallow neural network with a comparatively high number of power iteration steps achieved the best results during hyperparameter optimization (see Appendix G). ",
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+    {
+        "type": "text",
+        "text": "6 RESULTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Overall accuracy. The results for the accuracy (micro F1-score) are summarized in Table 2. Similar trends are observed for the macro F1-score (see Appendix E). Both models significantly outperform the state-of-the-art baseline models on all datasets. Our rigorous setup might understate the improvements achieved by PPNP and APPNP – this result is statistically significant $p < 0 . 0 5$ , as tested via a paired $t$ -test (see Appendix F). This thorough setup furthermore shows that the advantages reported by recent works practically vanish when training is harmonized, hyperparameters are properly optimized and multiple data splits are considered. A simple GCN with optimized hyperparameters outperforms several recently proposed models on our setup. ",
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+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 3: Average training time per epoch. PPNP and APPNP are only slightly slower than GCN and much faster than more sophisticated methods like GAT. "
+        ],
+        "table_footnote": [
+            "∗not applicable, since core method not trainable ∗∗out of memory on PUBMED, MS ACADEMIC (see efficiency analysis in Section 3) "
+        ],
+        "table_body": "<table><tr><td>Graph</td><td>V. GCN</td><td>GCN</td><td>N-GCN</td><td>GAT</td><td>JK</td><td>Bt.FP*</td><td>PPNP**</td><td>APPNP</td></tr><tr><td>CITESEER</td><td>37.6ms</td><td>35.3ms</td><td>115.9 ms</td><td>187.0 ms</td><td>57.5 ms</td><td>=</td><td>49.2 ms</td><td>43.3ms</td></tr><tr><td>CORA-ML</td><td>32.4 ms</td><td>36.5ms</td><td>118.9 ms</td><td>217.4 ms</td><td>43.6 ms</td><td></td><td>55.3ms</td><td>42.7ms</td></tr><tr><td>PUBMED</td><td>48.6ms</td><td>48.3ms</td><td>342.6 ms</td><td>1029.8 ms</td><td>77.8ms</td><td></td><td>-</td><td>64.1ms</td></tr><tr><td>MS ACADEMIC</td><td>45.5ms</td><td>39.2 ms</td><td>328.5ms</td><td>772.2 ms</td><td>61.9 ms</td><td>=</td><td>=</td><td>59.8ms</td></tr></table>",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/88566eae2ce6f160a22712efc36611a1a63fdd2342b12dae2ed67301a9c7b259.jpg",
+        "image_caption": [
+            "Figure 3: Accuracy for different training set sizes (number of labeled nodes per class) on CORA-ML. PPNP’s dominance increases further for smaller training set sizes. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Figure 2 shows how broad the accuracy distribution of each model is. This is caused by both random initialization and different data splits (train / early stopping / test). This demonstrates how crucial a statistically rigorous evaluation is for a conclusive model comparison. Moreover, it shows the sensitivity (robustness) of each method, e.g. PPNP, APPNP and GAT typically have lower variance. ",
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+    },
+    {
+        "type": "text",
+        "text": "Training time per epoch. We report the average training time per epoch in Table 3. We decided to only compare the training time per epoch since all hyperparameters were solely optimized for accuracy and the used early stopping criterion is very generous. Obviously, (exact) PPNP can only be applied to moderately sized graphs, while APPNP scales to large data. On average, APPNP is around $2 5 \\%$ slower than GCN due to its higher number of matrix multiplications. It scales similarly with graph size as GCN and is therefore significantly faster than other more sophisticated models like GAT. This is observed even though our implementation improved GAT’s training time roughly by a factor of 2 compared to the reference implementation. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Training set size. Since the labeling rate is often very small for real world datasets, investigating how the models perform with a small number of training samples is very important. Figure 3 shows how the number of training nodes per class $n _ { \\mathrm { t r a i n } }$ , per class impacts the accuracy on CORA-ML (for other datasets see Appendix H). The dominance of PPNP and APPNP increases further in this sparsely labelled setting. This can be attributed to their higher range, which allows them to better propagate the information further away from the (few) training nodes. We see further evidence for this when comparing the accuracy of APPNP and GCN depending on the distance between a node and the training set (in terms of shortest path). Appendix I shows that the performance gap between APPNP and GCN tends to increase for nodes that are far away from the training nodes. That is, nodes further away from the training set benefit more from the increase in range. ",
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+        "img_path": "images/b5ef1be3d27841710cb335d16bfccf768880bcfa79885bfbb88b30e0d3582494.jpg",
+        "image_caption": [
+            "Figure 4: Accuracy depending on the number of propagation steps $K$ . The accuracy breaks down for the GCN-like propagation $( \\alpha = 0$ ), while it increases and stabilizes when using APPNP $\\alpha = 0 . 1 $ ). "
+        ],
+        "image_footnote": [],
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+        "image_caption": [
+            "Figure 5: Accuracy depending on teleport probability $\\alpha$ . The optimum typically lies within $\\alpha \\in$ [0.05, 0.2], but changes for different types of datasets. "
+        ],
+        "image_footnote": [],
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+    {
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+        "image_caption": [
+            "Figure 6: Accuracy of APPNP with propagation used only during training/inference. Best results are achieved with full propagation, but propagating only during inference also achieves good results. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Number of power iteration steps. Figure 4 shows how the accuracy depends on the number of power iterations for two different propagation schemes. The first mimics the standard propagation as known from GCNs (i.e. $\\alpha = 0$ in APPNP). As clearly shown the performance breaks down as we increase the number of power iterations $K$ (since we approach the global PageRank solution). However, when using personalized propagation (with $\\alpha = 0 . 1$ ) the accuracy increases and converges to exact PPNP with infinitely many propagation steps, thus demonstrating the personalized propagation principle is indeed beneficial. As also shown in the figure, it is enough to use a moderate number of power iterations (e.g. $K = 1 0$ ) to effectively approximate exact PPNP. Interestingly, we’ve found that this number coincides with the highest shortest path distance of any node to the training set. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Teleport probability $\\alpha$ . Figure 5 shows the effect of the hyperparameter $\\alpha$ on the accuracy on the validation set. While the optimum differs slightly for every dataset, we consistently found a teleport probability of around $\\alpha \\in [ 0 . 0 5 , 0 . 2 ]$ to perform best. This probability should be adjusted for the dataset under investigation, since different graphs exhibit different neighborhood structures (Grover & Leskovec, 2016; Abu-El-Haija et al., 2018b). Note that a higher $\\alpha$ improves convergence speed. ",
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "Neural network without propagation. PPNP and APPNP are trained end-to-end, with the propagation scheme affecting (i) the neural network $f _ { \\theta }$ during training, and (ii) the classification decision during inference. Investigating how the model performs without propagation shows if and how valuable this addition is. Figure 6 shows how propagation affects both training and inference. ”Never” denotes the case where no propagation is used; essentially we train and apply a standard multilayer perceptron (MLP) $f _ { \\theta }$ using the features only. ”Training” denotes the case where we use APPNP during training to learn $f _ { \\theta }$ ; at inference time, however, only $f _ { \\theta }$ is used to predict the class labels. ”Inference”, in contrast, denotes the case where $f _ { \\theta }$ is trained without APPNP (i.e. standard MLP on features). This pretrained network with fixed weights is then used with APPNP’s propagation for inference. Finally, ”Inf. & Training” denotes the regular APPNP, which always uses propagation. ",
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+    {
+        "type": "text",
+        "text": "The best results are achieved with regular APPNP, which validates our approach. However, on most datasets the accuracy decreases surprisingly little when propagating only during inference. Skipping propagation during training can significantly reduce training time for large graphs as all nodes can be handled independently. This also shows that our model can be combined with pretrained neural networks that do not incorporate any graph information and still significantly improve their accuracy. Moreover, Figure 6 shows that just propagating during training can also lead to large improvements. This indicates that our model can also be applied to online/inductive learning where only the features and not the neighborhood information of an incoming (previously unobserved) node are available. ",
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+    {
+        "type": "text",
+        "text": "7 CONCLUSION ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "In this paper we have introduced personalized propagation of neural predictions (PPNP) and its fast approximation, APPNP. We derived this model by considering the relationship between GCN and PageRank and extending it to personalized PageRank. This simple model decouples prediction and propagation and solves the limited range problem inherent in many message passing models without introducing any additional parameters. It uses the information from a large, adjustable (via the teleport probability $\\alpha$ ) neighborhood for classifying each node. The model is computationally efficient and outperforms several state-of-the-art methods for semi-supervised classification on multiple graphs in the most thorough study which has been done for GCN-like models so far. ",
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+    {
+        "type": "text",
+        "text": "For future work it would be interesting to combine PPNP with more complex neural networks used e.g. in computer vision or natural language processing. Furthermore, faster or incremental approximations of personalized PageRank (Bahmani et al., 2010; 2011; Lofgren et al., 2014) and more sophisticated propagation schemes would also benefit the method. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This research was supported by the German Research Foundation, grant GU 1409/2-1. ",
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            392,
+            285,
+            409
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Sami Abu-El-Haija, Amol Kapoor, Bryan Perozzi, and Joonseok Lee. N-GCN: Multi-scale Graph Convolution for Semi-supervised Node Classification. In International Workshop on Mining and Learning with Graphs (MLG), 2018a.   \nSami Abu-El-Haija, Bryan Perozzi, Rami Al-Rfou, and Alex Alemi. Watch Your Step: Learning Node Embeddings via Graph Attention. In NeurIPS, 2018b.   \nBahman Bahmani, Abdur Chowdhury, and Ashish Goel. Fast Incremental and Personalized PageRank. VLDB, 2010.   \nBahman Bahmani, Kaushik Chakrabarti, and Dong Xin. Fast Personalized PageRank on MapReduce. In SIGMOD, 2011.   \nAleksandar Bojchevski and Stephan Gunnemann. Deep Gaussian Embedding of Graphs: Unsuper- ¨ vised Inductive Learning via Ranking. ICLR, 2018.   \nAleksandar Bojchevski, Oleksandr Shchur, Daniel Zugner, and Stephan G ¨ unnemann. NetGAN: ¨ Generating Graphs via Random Walks. In ICML, 2018.   \nJoan Bruna, Wojciech Zaremba, Arthur Szlam, and Yann LeCun. Spectral Networks and Deep Locally Connected Networks on Graphs. ICLR, 2014.   \nEliav Buchnik and Edith Cohen. Bootstrapped Graph Diffusions: Exposing the Power of Nonlinearity. Proceedings of the ACM on Measurement and Analysis of Computing Systems (POMACS), 2 (1):1–19, April 2018.   \nJie Chen, Tengfei Ma, and Cao Xiao. FastGCN: Fast Learning with Graph Convolutional Networks via Importance Sampling. ICLR, 2018.   \nZhengdao Chen, Lisha Li, and Joan Bruna. Supervised Community Detection with Line Graph Neural Networks. In ICLR, 2019.   \nHanjun Dai, Zornitsa Kozareva, Bo Dai, Alexander J. Smola, and Le Song. Learning Steady-States of Iterative Algorithms over Graphs. In ICML, 2018.   \nMichael Defferrard, Xavier Bresson, and Pierre Vandergheynst. Convolutional Neural Networks on ¨ Graphs with Fast Localized Spectral Filtering. In NIPS, 2016.   \nDavid K. Duvenaud, Dougal Maclaurin, Jorge Aguilera-Iparraguirre, Rafael Gomez-Bombarelli, ´ Timothy Hirzel, Alan Aspuru-Guzik, and Ryan P. Adams. Convolutional Networks on Graphs for ´ Learning Molecular Fingerprints. In NIPS, 2015. ",
+        "bbox": [
+            171,
+            404,
+            826,
+            924
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Justin Gilmer, Samuel S. Schoenholz, Patrick F. Riley, Oriol Vinyals, and George E. Dahl. Neural Message Passing for Quantum Chemistry. In ICML, 2017. ",
+        "bbox": [
+            171,
+            103,
+            825,
+            132
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Xavier Glorot and Yoshua Bengio. Understanding the difficulty of training deep feedforward neural networks. In AISTATS, 2010. ",
+        "bbox": [
+            171,
+            140,
+            823,
+            170
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Aditya Grover and Jure Leskovec. node2vec: Scalable Feature Learning for Networks. In KDD, 2016. ",
+        "bbox": [
+            174,
+            179,
+            823,
+            208
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "William L. Hamilton, Zhitao Ying, and Jure Leskovec. Inductive Representation Learning on Large Graphs. In NIPS, 2017. ",
+        "bbox": [
+            176,
+            217,
+            823,
+            246
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Taher H. Haveliwala. Topic-sensitive PageRank. In WWW, 2002. ",
+        "bbox": [
+            174,
+            253,
+            602,
+            270
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Tatsuro Kawamoto, Masashi Tsubaki, and Tomoyuki Obuchi. Mean-field theory of graph neural networks in graph partitioning. In NeurIPS, 2018. ",
+        "bbox": [
+            173,
+            279,
+            825,
+            308
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Steven M. Kearnes, Kevin McCloskey, Marc Berndl, Vijay S. Pande, and Patrick Riley. Molecular graph convolutions: moving beyond fingerprints. Journal of Computer-Aided Molecular Design, 30(8):595–608, 2016. ",
+        "bbox": [
+            173,
+            315,
+            823,
+            359
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Diederik P. Kingma and Jimmy Ba. Adam: A Method for Stochastic Optimization. ICLR, 2015. ",
+        "bbox": [
+            173,
+            367,
+            803,
+            383
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Thomas N. Kipf and Max Welling. Semi-Supervised Classification with Graph Convolutional Networks. ICLR, 2017. ",
+        "bbox": [
+            173,
+            392,
+            823,
+            421
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jure Leskovec, Jon Kleinberg, and Christos Faloutsos. Graphs over Time: Densification Laws, Shrinking Diameters and Possible Explanations. In KDD, 2005. ",
+        "bbox": [
+            173,
+            430,
+            823,
+            459
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Qimai Li, Zhichao Han, and Xiao-Ming Wu. Deeper Insights Into Graph Convolutional Networks for Semi-Supervised Learning. In AAAI, 2018. ",
+        "bbox": [
+            171,
+            468,
+            823,
+            497
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Yujia Li, Daniel Tarlow, Marc Brockschmidt, and Richard S. Zemel. Gated Graph Sequence Neural Networks. In ICLR, 2016. ",
+        "bbox": [
+            173,
+            505,
+            823,
+            535
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Peter Lofgren, Siddhartha Banerjee, Ashish Goel, and Seshadhri Comandur. FAST-PPR: scaling personalized pagerank estimation for large graphs. In KDD, 2014. ",
+        "bbox": [
+            169,
+            544,
+            825,
+            573
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Mart´ın Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S. Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Ian Goodfellow, Andrew Harp, Geoffrey Irving, Michael Isard, Yangqing Jia, Rafal Jozefowicz, Lukasz Kaiser, Manjunath Kudlur, Josh Levenberg, Dandelion Mane, Rajat Monga, Sherry Moore, Derek Murray, ´ Chris Olah, Mike Schuster, Jonathon Shlens, Benoit Steiner, Ilya Sutskever, Kunal Talwar, Paul Tucker, Vincent Vanhoucke, Vijay Vasudevan, Fernanda Viegas, Oriol Vinyals, Pete Warden, ´ Martin Wattenberg, Martin Wicke, Yuan Yu, and Xiaoqiang Zheng. TensorFlow: Large-Scale Machine Learning on Heterogeneous Systems, 2015. ",
+        "bbox": [
+            173,
+            580,
+            825,
+            694
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Andrew Kachites McCallum, Kamal Nigam, Jason Rennie, and Kristie Seymore. Automating the construction of internet portals with machine learning. Information Retrieval, 3(2):127–163, 2000. ",
+        "bbox": [
+            176,
+            702,
+            823,
+            744
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Federico Monti, Davide Boscaini, Jonathan Masci, Emanuele Rodola, Jan Svoboda, and Michael M. Bronstein. Geometric Deep Learning on Graphs and Manifolds Using Mixture Model CNNs. In CVPR, 2017. ",
+        "bbox": [
+            173,
+            753,
+            823,
+            796
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Galileo Namata, Ben London, Lise Getoor, and Bert Huang. Query-driven Active Surveying for Collective Classification. In International Workshop on Mining and Learning with Graphs (MLG), 2012. ",
+        "bbox": [
+            173,
+            805,
+            825,
+            848
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Sharad Nandanwar and M. N. Murty. Structural Neighborhood Based Classification of Nodes in a Network. In KDD, 2016. ",
+        "bbox": [
+            173,
+            857,
+            821,
+            886
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Mathias Niepert, Mohamed Ahmed, and Konstantin Kutzkov. Learning Convolutional Neural Networks for Graphs. In ICML, 2016. ",
+        "bbox": [
+            173,
+            895,
+            820,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Lawrence Page, Sergey Brin, Rajeev Motwani, and Terry Winograd. The pagerank citation ranking: Bringing order to the web. Technical report, Stanford InfoLab, 1998. ",
+        "bbox": [
+            171,
+            103,
+            823,
+            133
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Bryan Perozzi, Rami Al-Rfou, and Steven Skiena. DeepWalk: online learning of social representations. In KDD, 2014. ",
+        "bbox": [
+            173,
+            142,
+            823,
+            171
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Trang Pham, Truyen Tran, Dinh Q. Phung, and Svetha Venkatesh. Column Networks for Collective Classification. In AAAI, 2017. ",
+        "bbox": [
+            174,
+            181,
+            823,
+            212
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Jiezhong Qiu, Yuxiao Dong, Hao Ma, Jian Li, Kuansan Wang, and Jie Tang. Network Embedding as Matrix Factorization: Unifying DeepWalk, LINE, PTE, and node2vec. In ACM International Conference on Web Search and Data Mining (WSDM), 2018. ",
+        "bbox": [
+            173,
+            222,
+            823,
+            265
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "F. Scarselli, M. Gori, Ah Chung Tsoi, M. Hagenbuchner, and G. Monfardini. The Graph Neural Network Model. IEEE Transactions on Neural Networks, 20(1):61–80, January 2009. ",
+        "bbox": [
+            173,
+            275,
+            825,
+            305
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Michael Sejr Schlichtkrull, Thomas N. Kipf, Peter Bloem, Rianne van den Berg, Ivan Titov, and Max Welling. Modeling Relational Data with Graph Convolutional Networks. In Extended Semantic Web Conference (ESWC), 2018. ",
+        "bbox": [
+            174,
+            314,
+            823,
+            358
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Prithviraj Sen, Galileo Namata, Mustafa Bilgic, Lise Getoor, Brian Gallagher, and Tina Eliassi-Rad. Collective Classification in Network Data. AI Magazine, 29(3):93–106, 2008. ",
+        "bbox": [
+            169,
+            367,
+            823,
+            397
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Oleksandr Shchur, Maximilian Mumme, Aleksandar Bojchevski, and Stephan Gunnemann. Pitfalls¨ of Graph Neural Network Evaluation. In Relational Representation Learning Workshop (R2L 2018), NeurIPS, 2018. ",
+        "bbox": [
+            173,
+            407,
+            823,
+            452
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Jian Tang, Meng Qu, Mingzhe Wang, Ming Zhang, Jun Yan, and Qiaozhu Mei. LINE: Large-scale Information Network Embedding. In WWW, 2015. ",
+        "bbox": [
+            171,
+            462,
+            825,
+            491
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Petar Velickovi ˇ c, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Li ´ o, and Yoshua \\` Bengio. Graph Attention Networks. ICLR, 2018. ",
+        "bbox": [
+            171,
+            501,
+            825,
+            530
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Keyulu Xu, Chengtao Li, Yonglong Tian, Tomohiro Sonobe, Ken-ichi Kawarabayashi, and Stefanie Jegelka. Representation Learning on Graphs with Jumping Knowledge Networks. In ICML, 2018. ",
+        "bbox": [
+            171,
+            540,
+            823,
+            570
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Zhilin Yang, William W. Cohen, and Ruslan Salakhutdinov. Revisiting Semi-Supervised Learning with Graph Embeddings. In ICML, 2016. ",
+        "bbox": [
+            173,
+            580,
+            823,
+            609
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Rex Ying, Ruining He, Kaifeng Chen, Pong Eksombatchai, William L. Hamilton, and Jure Leskovec. Graph Convolutional Neural Networks for Web-Scale Recommender Systems. KDD, 2018. ",
+        "bbox": [
+            174,
+            619,
+            826,
+            662
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A EXISTENCE OF $\\Pi _ { \\mathrm { P P R } }$ ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            690,
+            377,
+            708
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The matrix ",
+        "bbox": [
+            174,
+            723,
+            248,
+            737
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2a55d232ec3801c1dfe24455d46f4f3b2ba451800743466815f2ed67c6798bda.jpg",
+        "text": "$$\n\\Pi _ { \\mathrm { p p r } } = \\alpha \\left( I _ { n } - ( 1 - \\alpha ) \\hat { \\tilde { A } } \\right) ^ { - 1 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            393,
+            734,
+            602,
+            765
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "exists iff the determinant $\\operatorname* { d e t } ( I _ { n } - ( 1 - \\alpha ) { \\hat { \\tilde { A } } } ) \\neq 0$ , which is the case iff $\\operatorname* { d e t } ( \\hat { \\tilde { A } } - \\frac { 1 } { 1 - \\alpha } I _ { n } ) \\neq 0$ , i.e. iff $\\frac { 1 } { 1 - \\alpha }$ is not an eigenvalue of $\\hat { \\tilde { A } }$ . This value is always larger than 1 since the teleport probability $\\alpha \\in \\ ( 0 , 1 ]$ . Furthermore, the symmetrically normalized matrix $\\hat { \\tilde { A } }$ has the same eigenvalues as the row-stochastic matrix $\\tilde { A } _ { \\mathrm { r w } }$ . This can be shown by multiplying the eigenvalue equation $\\hat { \\tilde { A } } v = \\lambda v$ with $\\tilde { D } ^ { - 1 / 2 }$ from left and substituting $\\pmb { w } = \\tilde { \\pmb { D } } ^ { - 1 / 2 } \\pmb { v }$ . This also shows that the eigenvectors of $\\hat { \\tilde { A } }$ are the eigenvectors of $\\tilde { A } _ { \\mathrm { r w } }$ scaled by $\\tilde { D } ^ { 1 / 2 }$ . The largest eigenvalue of a row-stochastic matrix is 1, as can be proven using the Gershgorin circle theorem. Hence, $\\frac { 1 } { 1 - \\alpha }$ cannot be an eigenvalue and $\\Pi _ { \\mathrm { p p r } }$ always exists. ",
+        "bbox": [
+            173,
+            776,
+            826,
+            925
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "B CONVERGENCE OF APPNP ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            437,
+            118
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "APPNP uses the iterative equation ",
+        "bbox": [
+            174,
+            133,
+            401,
+            148
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ad2ef76cbe4be7eb4a98e00e3da1d4e68f1dc6951624b6ed998fb0703566257d.jpg",
+        "text": "$$\n{ \\cal Z } ^ { ( k + 1 ) } = ( 1 - \\alpha ) \\hat { \\tilde { \\cal A } } { \\cal Z } ^ { ( k ) } + \\alpha { \\cal H } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            387,
+            155,
+            611,
+            178
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "After the $k$ -th propagation step, the resulting predictions are ",
+        "bbox": [
+            173,
+            190,
+            568,
+            207
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/327146d4df4ef5cedcdb6de53aa0d9913e71f69e52f9ccc527ce49c6ad6e0926.jpg",
+        "text": "$$\n\\pmb { Z } ^ { ( k ) } = \\left( ( 1 - \\alpha ) ^ { k } \\hat { \\tilde { \\mathbf { A } } } ^ { k } + \\alpha \\sum _ { i = 0 } ^ { k - 1 } ( 1 - \\alpha ) ^ { i } \\hat { \\tilde { \\mathbf { A } } } ^ { i } \\right) \\pmb { H } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            338,
+            212,
+            661,
+            256
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "If we take the limit $k  \\infty$ the left term tends to 0 and the right term becomes a geometric series. The series converges since $\\alpha \\in ( 0 , 1 ]$ and $\\hat { \\tilde { A } }$ is symmetrically normalized and therefore $\\operatorname* { d e t } ( \\hat { \\tilde { A } } ) \\leq 1$ , resulting in ",
+        "bbox": [
+            173,
+            268,
+            825,
+            315
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/36877412057a79dd8fa8927a78970b2eb40551f0a0805c0a6c8b6c39dc231e82.jpg",
+        "text": "$$\n\\pmb { Z } ^ { ( \\infty ) } = \\alpha \\left( \\pmb { I _ { n } } - ( 1 - \\alpha ) \\hat { \\tilde { A } } \\right) ^ { - 1 } \\pmb { H } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            379,
+            313,
+            619,
+            343
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "which is the equation for calculating (exact) PPNP. ",
+        "bbox": [
+            173,
+            345,
+            508,
+            361
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "C EXPERIMENTAL DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            380,
+            416,
+            396
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/922f154318507fafbeb2f3ed5f38a53906d5fadd66a576d6fed279871fc59744.jpg",
+        "image_caption": [
+            "Figure 7: Illustration of the node sampling procedure. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            387,
+            417,
+            606,
+            507
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The sampling procedure is illustrated in Figure 7. The data is first split into a visible and a test set. For the visible set 1500 nodes were sampled for the citation graphs and 5000 for MICROSOFT ACADEMIC. The test set contains all remaining nodes. We use three different label sets in each experiment: A training set of 20 nodes per class, an early stopping set of 500 nodes and either a validation or test set. The validation set contains the remaining nodes of the visible set. We use 20 random seeds for determining the splits. These seeds are drawn once and fixed across runs to facilitate comparisons. We use one set of seeds for the validation splits and a different set for the test splits. Each experiment is run with 5 random initializations on each data split, leading to a total of 100 runs per experiment. ",
+        "bbox": [
+            173,
+            550,
+            825,
+            676
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The early stopping criterion uses a patience of $p = 1 0 0$ and an (unreachably high) maximum of $n = 1 0 0 0 0$ epochs. The patience is reset whenever the accuracy increases or the loss decreases on the early stopping set. We choose the parameter set achieving the highest accuracy and break ties by selecting the lowest loss on this set. This criterion was inspired by GAT (Velickovi ˇ c et al., 2018). ´ ",
+        "bbox": [
+            174,
+            683,
+            825,
+            739
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We used TensorFlow (Mart´ın Abadi et al., 2015) for all experiments except bootstrapped feature propagation. All uncertainties and confidence intervals correspond to a confidence level of $9 5 \\%$ and were calculated by bootstrapping with 1000 samples. ",
+        "bbox": [
+            174,
+            746,
+            825,
+            787
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We use the Adam optimizer with a learning rate of $l = 0 . 0 1$ and cross-entropy loss for all models (Kingma & Ba, 2015). Weights are initialized as described in Glorot & Bengio (2010). The feature matrix is $L _ { 1 }$ normalized per row. ",
+        "bbox": [
+            174,
+            795,
+            825,
+            838
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "D BASELINE HYPERPARAMETERS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            467,
+            117
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Vanilla GCN uses the original settings of two layers with $h = 1 6$ hidden units, no dropout on the adjacency matrix, $L _ { 2 }$ regularization parameter $\\lambda { \\stackrel { \\cdot } { = } } 5 \\times 1 0 ^ { - 4 }$ and the original early stopping with a maximum of 200 steps and a patience of 10 steps based on the loss. ",
+        "bbox": [
+            174,
+            133,
+            825,
+            175
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The optimized GCN uses two layers with $h = 6 4$ hidden units, dropout on the adjacency matrix with $d = 0 . 5$ and $L _ { 2 }$ regularization parameter $\\lambda = 0 . 0 2$ . ",
+        "bbox": [
+            176,
+            181,
+            820,
+            210
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "N-GCN uses $h = 1 6$ hidden units, $R = 4$ heads per random walk length and random walks of up to $K - 1 = 4$ steps. It uses $L _ { 2 }$ regularization on all layers with $\\lambda = \\overline { { 1 } } \\times 1 0 ^ { - 5 }$ and the attention variant for merging the predictions (Abu-El-Haija et al., 2018a). Note that this model effectively uses $R K h = 3 2 0$ hidden units, which is 5 times as many units compared to GCN, GAT, and PPNP. ",
+        "bbox": [
+            174,
+            217,
+            825,
+            273
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For GAT we use the (well optimized) original hyperparameters, except the $L _ { 2 }$ regularization parameter $\\lambda = 0 . 0 0 1$ and learning rate $l = 0 . 0 1$ . As opposed to the original paper, we do not use different hyperparameters on PUBMED, as described in our experimental setup. ",
+        "bbox": [
+            176,
+            280,
+            820,
+            323
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Bootstrapped feature propagation uses a return probability of $\\alpha = 0 . 2$ , 10 propagation steps, 10 bootstrapping (self-training) steps with $r = 0 . 1 n$ training nodes added per step. We add the training nodes with the lowest entropy on the predictions. The number of nodes added per class is based on the class proportions estimated using the predictions. Note that this model does not include any stochasticity in its initialization. We therefore only run it once per train/early stopping/test split. ",
+        "bbox": [
+            174,
+            329,
+            825,
+            400
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For the jumping knowledge networks we use the concatenation variant with three layers and $h = 6 4$ hidden units per layer. We apply $L _ { 2 }$ regularization with $\\lambda = 0 . 0 0 1$ on all layers and perform dropout with $d = 0 . 5$ on all layers but not on the adjacency matrix. ",
+        "bbox": [
+            174,
+            406,
+            825,
+            449
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "E F1 SCORE ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            468,
+            290,
+            484
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/50c3c5cb813b03bec49f4a66841a48a4d8f0c6bda9aad033c6007597101fe7b0.jpg",
+        "table_caption": [
+            "Table 4: Average macro F1 score with uncertainties showing the $9 5 \\%$ confidence level calculated by bootstrapping. PPNP achieves the highest F1 score on all datasets investigated. "
+        ],
+        "table_footnote": [
+            "∗out of memory on PUBMED, MS ACADEMIC (see efficiency analysis in Section 3) "
+        ],
+        "table_body": "<table><tr><td>Model</td><td>CITESEER</td><td>CORA-ML</td><td>PUBMED</td><td>MS ACADEMIC</td></tr><tr><td>V. GCN</td><td>0.7002±0.0043</td><td>0.8205±0.0027</td><td>0.7801 ± 0.0038</td><td>0.9000±0.0008</td></tr><tr><td>GCN</td><td>0.7065 ± 0.0037</td><td>0.8289 ± 0.0030</td><td>0.7883 ± 0.0032</td><td>0.9045 ± 0.0008</td></tr><tr><td>N-GCN</td><td>0.7021 ± 0.0035</td><td>0.8183 ± 0.0024</td><td>0.7773 ± 0.0040</td><td>0.9144 ± 0.0012</td></tr><tr><td>GAT</td><td>0.7062 ± 0.0029</td><td>0.8359 ± 0.0025</td><td>0.7777 ± 0.0040</td><td>0.8917 ± 0.0007</td></tr><tr><td>JK</td><td>0.6914 ± 0.0043</td><td>0.8202 ± 0.0026</td><td>0.7799 ± 0.0039</td><td>0.8985 ±0.0012</td></tr><tr><td>Bt.FP</td><td>0.6789 ± 0.0055</td><td>0.8026 ± 0.0082</td><td>0.7448 ± 0.0079</td><td>0.8997 ± 0.0018</td></tr><tr><td>PPNP*</td><td>0.7102 ± 0.0041</td><td>0.8454 ± 0.0021</td><td></td><td></td></tr><tr><td>APPNP</td><td>0.7105 ± 0.0038</td><td>0.8429 ±0.0022</td><td>0.7966 ± 0.0031</td><td>0.9184 ± 0.0009</td></tr></table>",
+        "bbox": [
+            174,
+            541,
+            831,
+            671
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "F PAIRED $t$ -TEST ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            715,
+            330,
+            732
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/0c1ef6e53578861dd5b1ffecd40f8b69aeab346389f9bbf3fae28a8a3613e8be.jpg",
+        "table_caption": [
+            "Table 5: p-value of the paired $t$ -test with respect to accuracy. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>CITESEER</td><td>CORA-ML</td><td>PUBMED</td><td>MS ACADEMIC</td></tr><tr><td>PPNP</td><td>1.02 ×10-3</td><td>5.96×10-14</td><td></td><td></td></tr><tr><td>APPNP</td><td>1.77 × 10-2</td><td>4.27 ×10-9</td><td>2.19 ×10-15</td><td>5.93 × 10-13</td></tr></table>",
+        "bbox": [
+            233,
+            775,
+            751,
+            823
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/921492bd3a768e4264889ce5758b459c61a0affc926c014b2bc32d88ec2e5e4c.jpg",
+        "table_caption": [
+            "Table 6: p-value of the paired $t$ -test with respect to F1 score. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>CITESEER</td><td>CORA-ML</td><td>PUBMED</td><td>MS ACADEMIC</td></tr><tr><td>PPNP</td><td>4.49×10-2</td><td>4.50×10-14</td><td></td><td></td></tr><tr><td>APPNP</td><td>2.32 ×10-2</td><td>1.07 × 10-8</td><td>8.70 × 10-14</td><td>1.99 × 10-8</td></tr></table>",
+        "bbox": [
+            236,
+            866,
+            753,
+            914
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/f5436eb85eb29bfa0226b787ea9e7661e7b18c50f0879781bcd8952e22d5f919.jpg",
+        "image_caption": [
+            "Figure 8: Validation accuracy of APPNP for varying numbers of neural network (NN) layers. Deep NNs do not improve the accuracy, which is probably due to the simple bag-of-words features and the small training set size. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
+            135,
+            803,
+            246
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "H TRAINING SET SIZE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            330,
+            374,
+            347
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/1df3e15b9e80568ac6bbedbc6c5497b5b9982cec9d08137aa5b32a1273986c1f.jpg",
+        "image_caption": [
+            "Figure 9: Accuracy for different training set sizes on CITESEER. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            189,
+            364,
+            807,
+            467
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/c9a16536a4e516b2d58581e5824a4760c7222b772af1594436a217beb5a80ac1.jpg",
+        "image_caption": [
+            "Figure 10: Accuracy for different training set sizes on PUBMED. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            191,
+            511,
+            807,
+            613
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/5389f5b7b8ef0d13b1c87be6e2841f435778ee204063bc1b91ca6aa3f22410a4.jpg",
+        "image_caption": [
+            "Figure 11: Accuracy for different training set sizes on MICROSOFT ACADEMIC. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            192,
+            657,
+            807,
+            758
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/0072cf936b72a26218ba2729a695270a0013a7756cf97608c3cdfb41adef98a8.jpg",
+        "image_caption": [
+            "Figure 12: $\\Delta$ Accuracy $( \\% )$ denotes the average improvement in percentage points of APPNP over GCN depending on the distance (number of hops) from the training nodes on CORA-ML. $\\bar { n }$ denotes the average number of nodes at each distance. The improvement increases with distance. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            189,
+            136,
+            803,
+            296
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/2fd5099893deec0cdde802e9feb63de47fc5c089a3e5393abf3aaeca4685822b.jpg",
+        "image_caption": [
+            "Figure 13: $\\Delta$ Accuracy $( \\% )$ denotes the average improvement in percentage points of APPNP over GCN depending on the distance (number of hops) from the training nodes on different graphs. $\\bar { n }$ denotes the average number of nodes at each distance over different splits. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
+            369,
+            795,
+            527
+        ],
+        "page_idx": 14
+    }
+]

parse/train/H1vEXaxA-/H1vEXaxA-.md

@@ -0,0 +1,411 @@
+# EMERGENT TRANSLATION IN MULTI-AGENT COMMUNICATION
+
+Jason Lee∗ New York University jason@cs.nyu.edu
+
+Kyunghyun Cho New York University Facebook AI Research kyunghyun.cho@nyu.edu
+
+Jason Weston Facebook AI Research jase@fb.com
+
+Douwe Kiela Facebook AI Research dkiela@fb.com
+
+# ABSTRACT
+
+While most machine translation systems to date are trained on large parallel corpora, humans learn language in a different way: by being grounded in an environment and interacting with other humans. In this work, we propose a communication game where two agents, native speakers of their own respective languages, jointly learn to solve a visual referential task. We find that the ability to understand and translate a foreign language emerges as a means to achieve shared goals. The emergent translation is interactive and multimodal, and crucially does not require parallel corpora, but only monolingual, independent text and corresponding images. Our proposed translation model achieves this by grounding the source and target languages into a shared visual modality, and outperforms several baselines on both word-level and sentence-level translation tasks. Furthermore, we show that agents in a multilingual community learn to translate better and faster than in a bilingual communication setting.
+
+# 1 INTRODUCTION
+
+Building intelligent machines that can converse with humans is a longstanding challenge in artificial intelligence. Remarkable successes have been achieved in natural language processing (NLP) via the use of supervised learning approaches on large-scale datasets (Bahdanau et al., 2015; Wu et al., 2016; Gehring et al., 2017; Sennrich et al., 2017). Machine translation is no exception: most translation systems are trained to derive statistical patterns from huge parallel corpora. Parallel corpora, however, are expensive and difficult to obtain for many language pairs. This is especially the case for low resource languages, where parallel texts are often small or nonexistent. We address these issues by designing a multi-agent communication task, where agents interact with each other in their own native languages and try to work out what the other agent meant to communicate. We find that the ability to translate foreign languages emerges as a means to achieve a common goal.
+
+Aside from the benefit of not requiring parallel data, we argue that our approach to learning to translate is also more natural than learning from large corpora. Humans learn languages by interacting with other humans and referring to their shared environment, i.e., by being grounded in physical reality. More abstract knowledge is built on top of this concrete foundation. It is natural to use vision as an intermediary: when communicating with someone who does not speak our language, we often directly refer to our surroundings. Even linguistically distant languages will, by physical and cognitive necessity, still refer to scenes and objects in the same visual space.
+
+We compare our model against a number of baselines, including a nearest neighbor method and a recently proposed model (Nakayama & Nishida, 2017) that maps languages and images to a shared space, but lacks communication. We evaluate performance on both word- and sentence-level translation, and show that our model outperforms the baselines in both settings. Additionally, we show that multilingual communities of agents, comprised of native speakers of different languages, learn faster and ultimately become better translators.
+
+# 2 PRIOR WORK
+
+Recent work has used neural networks and reinforcement learning in multi-agent settings to solve a variety of tasks with communication, including simple coordination (Sukhbaatar et al., 2016), logic riddles (Foerster et al., 2016), complex coordination with verbal and physical interaction (Lowe et al., 2017), cooperative dialogue (Das et al., 2017) and negotiation (Lewis et al., 2017).
+
+At the same time, there has been a surge of interest in communication protocols or languages that emerge from multi-agent communication in solving these various tasks. Lazaridou et al. (2017) first showed that simple neural network agents can learn to coordinate in an image referential game with single-symbol bandwidth. This work has been extended to induce communication protocols that are more similar to human language, allowing multi-turn communication (Jorge et al., 2016), adaptive communication bandwidth (Havrylov & Titov, 2017) and multi-turn communication with a variable-length conversation (Evtimova et al., 2017), and simple compositionality (Kottur et al., 2017; Mordatch & Abbeel, 2017). Meanwhile, Andreas et al. (2017) proposed a model to interpret continuous message vectors by “translating” them.
+
+Our work is related to a long line of work on learning multimodal representations. Several approaches proposed to learn a joint space for images and text using Canonical Correlation Analysis (CCA) or its variants (Hodosh et al., 2013; Andrew et al., 2013; Chandar et al., 2016). Other works minimize pairwise ranking loss to learn multimodal embeddings (Socher et al., 2014; Kiros et al., 2014; Ma et al., 2015; Vendrov et al., 2015; Kiela et al., 2017). Most recently, others extended this work to learn joint representations between images and multiple languages (Gella et al., 2017; Calixto et al., 2017b; Rajendran et al., 2016).
+
+In machine translation, our work is related to image-guided (Calixto et al., 2017a; Elliott & Kad´ ar, ´ 2017; Caglayan et al., 2016) and pivot-based (Firat et al., 2016; Hitschler et al., 2016) approaches. It is also related to previous work on multiagent translation for low-resource language pairs (without grounding) (He et al., 2016a). At word-level, there has been work on translation via a visual intermediate (Bergsma & Van Durme, 2011), including with convolutional neural network features (Kiela et al., 2015; Joulin et al., 2016).
+
+It was recently shown that zero-resource translation is possible by separately learning an image encoder and a language decoder (Nakayama & Nishida, 2017). The main difference to our work is that their models do not perform communication.
+
+# 3 TASK AND MODELS
+
+# 3.1 COMMUNICATION TASK
+
+We let two agents communicate with each other in their own respective languages to solve a visual referential task. One agent sees an image and describes it in its native language to the other agent. The other agent is given several images, one of which is the same image shown to the first agent, and has to choose the correct image using the description. The game is played in both directions simultaneously, and the agents are jointly trained to solve this task. We only allow agents to send a sequence of discrete symbols to each other, and never a continuous vector.
+
+Our task is similar to Lazaridou et al. (2017), but with the following differences: communication (1) is bidirectional and (2) of variable length; (3) the speaker is trained on both the listener’s feedback and ground-truth annotations; and (4) the speaker only observes the target image and no distractors.
+
+Let $P _ { A }$ and $P _ { B }$ be our agents, who speak the languages $L _ { A }$ and $L _ { B }$ respectively. We have two disjoint sets of image-annotation pairs: $( I _ { A } , M _ { A } )$ in language $L _ { A }$ and $( I _ { B } , M _ { B } )$ in language $L _ { B }$ .
+
+Task in language $L _ { A }$ : $P _ { A }$ is the speaker and $P _ { B }$ is the listener.
+
+1. A target image and annotation $( i , m ) \in \{ I _ { A } , M _ { A } \}$ is drawn from the training set in $L _ { A }$
+
+2. Given $i$ , the speaker $( P _ { A } )$ produces a sequence of symbols $\hat { m }$ in language $L _ { A }$ to describe the image and sends it to the listener. The speaker’s goal is to produce a message that is both an accurate prediction of the ground-truth annotation $m$ , and helps the listener $( P _ { B } )$ identify the target image.
+
+3. $K - 1$ distracting images are drawn from $I _ { A }$ at random. The target image $i$ is added to this set and all $K$ images are shuffled.
+
+4. Given the message $\hat { m }$ and the $K$ images, the listener’s goal is to identify the target image.
+
+Task with language $L _ { B }$ : The agents exchange the roles and play similarly.
+
+We explore two different settings: (1) a word-level task where the agents communicate with a single word, and (2) a sentence-level task where agents can transmit a sequence of symbols.
+
+# 3.2 MODEL ARCHITECTURE AND TRAINING
+
+Each agent has an image encoder, a native speaker module and a foreign language encoder. In English-Japanese communication, for instance, the English-speaking agent $P _ { A }$ consists of an image encoder $\dot { E _ { \mathrm { I M G } } ^ { A } }$ , a native English speaker module $S _ { \mathrm { E N } } ^ { A }$ , and a Japanese encoder $E _ { \mathrm { J A } } ^ { A }$ . Similarly, the Japanese-speaking agent $P _ { B } = ( E _ { \mathrm { I M G } } ^ { B } , S _ { \mathrm { J A } } ^ { B } , E _ { \mathrm { E N } } ^ { B } )$ .
+
+![](images/f55d70a48a7c0202874339c4988fe3a345248b04b4950a134970866a4a077b82.jpg)  
+Figure 1: Sentence-level communication task and translation between English and Japanese. (a) The red dotted line delimits the agents and the gray dotted line delimits the communication tasks for different languages. Representations residing in the multimodal space of Agent A and B are shown in green and yellow, respectively. (b) An illustration of how the Japanese agent might translate an unseen English sentence to Japanese.
+
+We now illustrate the architecture of our model using the English part of the communication task as an example (upper half of Figure 1a). We first describe the sentence-level model.
+
+Speaker $( P _ { A } )$ Given an image-annotation pair $( i , m ) \in \{ I _ { \mathrm { E N } } , M _ { \mathrm { E N } } \}$ sampled from the English training set, let $i$ be represented as a $D _ { \mathrm { i m g } }$ -dimensional vector. $P _ { A }$ ’s speaker encodes $i$ into a $D _ { \mathrm { h i d } }$ - dimensional vector with a feedforward image encoder: $h _ { 0 } = E _ { \mathrm { I M G } } ^ { A } ( i )$ .
+
+Our speaker module $S _ { \mathrm { E N } } ^ { A }$ is a recurrent neural network (RNN) with gated recurrent units (GRU, (Cho et al., 2014)). Our RNN takes the image representation $h _ { 0 }$ as initial hidden state and updates its state as $h _ { t + 1 } = \mathtt { G R U } ( h _ { t } , m _ { t } )$ where $m _ { t }$ is the $t$ -th token in $m$ . The output layer projects each hidden state $h _ { t }$ over the English vocabulary $\mathbb { V } _ { \mathrm { E N } }$ , followed by a softmax to predict the next token: $p _ { t } = { \tt s o f t m a x } ( W _ { o } h _ { t } + b _ { o } )$ . The speaker’s predictions are trained on the ground truth English
+
+annotation $m$ using the cross entropy loss:
+
+$$
+\mathcal { I } _ { \mathrm { s p k } } ^ { \mathrm { E N } } = - \frac { 1 } { N _ { \mathrm { E N } } } \sum _ { ( i , m ) } ^ { \left\{ I _ { \mathrm { E N } } , M _ { \mathrm { E N } } \right\} } \sum _ { t = 1 } ^ { T _ { m } } \log p ( m _ { t } | m _ { ( < t ) } , i ) ,
+$$
+
+where $N _ { \mathrm { E N } }$ is the size of the English training set, and $T _ { m }$ is the length of $m$
+
+To generate a sequence of tokens, we sample from the categorical distribution $\mathtt { C a t } ( p _ { t } )$ . However, sampling is a non-differentiable computation. To allow our model to be end-to-end differentiable, we use the straight-through Gumbel-softmax estimator (Jang et al., 2017; Maddison et al., 2017) to sample from $\mathtt { C a t } ( p _ { t } )$ and let the gradient flow, while the speaker sends a sequence of discrete symbols.1 The message $\hat { m }$ is a sequence of one-hot vectors: $\hat { m } = \{ y _ { t } \} _ { t = 1 } ^ { T _ { m } }$ , where $y _ { t } = \mathtt { G u m b e l { - } S T } ( p _ { t } )$ is discretized in the forward pass.
+
+Listener $( P _ { B } )$ The Japanese-speaking agent $P _ { B }$ encodes the $K$ images into $D _ { \mathrm { h i d } }$ -dimensional multimodal space with its own feedforward image encoder: $\{ E _ { \mathrm { I M G } } ^ { B } ( i _ { k } ) \} _ { k = 1 } ^ { \tilde { K } }$ . It also feeds each token from Takin $\hat { m }$ into its English encoder RNN with he last hidden state, the representati $D _ { \mathrm { h i d } }$ - menis a l hidden states: -dimensional ve $\boldsymbol { s } _ { t + 1 } = \operatorname { G R U } ( \boldsymbol { s } _ { t } , \hat { m } _ { t } )$ .. $\hat { m }$ $D _ { \mathrm { h i d } }$ $E _ { \mathrm { E N } } ^ { B } ( \hat { m } ) = s _ { T _ { m } }$ Note, that encodings of the images and the message have the same dimensionality.
+
+To encourage the listener to align the message representation closest to the target image, it is trained using a cross entropy loss where the logits are given by the reciprocal of the mean squared error (MSE) between the target image and the message representation: $\bar { \{ 1 / \big ( E _ { \mathrm { E N } } ^ { B } ( \hat { m } ) - E _ { \mathrm { I M G } } ^ { B } ( { i } _ { k } ) \big ) ^ { 2 } \} _ { k = 1 } ^ { K } }$
+
+$$
+\mathcal { I } _ { \mathrm { l s n } } ^ { \mathrm { E N } } = - \frac { 1 } { N _ { \mathrm { E N } } } \sum _ { i \in I _ { \mathrm { F N } } } \sum _ { \hat { m } } \log \left( \mathrm { s o f t m a x } \Big ( 1 / \big ( E _ { \mathrm { E N } } ^ { B } ( \hat { m } ) - E _ { \mathrm { I M G } } ^ { B } ( i ) \big ) ^ { 2 } \Big ) \right)
+$$
+
+where the softmax operation is performed over $K$ images. We observed that optimization significantly slows down after the initial stage of learning when training with the standard MSE loss. In order to ensure fast convergence throughout training, we use this modified form of MSE as a loss function whose slope gets steeper as the loss is minimized. See Appendix A for a discussion and a more thorough comparison and analysis.
+
+Training These two agents are jointly trained by minimizing the sum of speaker and listener loss:
+
+$$
+\mathcal { T } = \sum _ { x \in \{ \mathrm { E N } , \mathrm { J A } \} } ( \mathcal { T } _ { \mathrm { s p k } } ^ { x } + \mathcal { T } _ { \mathrm { l s n } } ^ { x } ) .
+$$
+
+Note that the listener is only trained on $\mathcal { I } _ { \mathrm { l s n } }$ , while the speaker is trained on both $\mathcal { T } _ { \mathrm { l s n } }$ and $\mathcal { I } _ { \mathrm { s p k } }$
+
+Word-level model The word-level model has a similar architecture to the sentence-level one: instead of an RNN, the speaker module $S _ { \mathrm { E N } } ^ { A }$ is a feedforward layer that projects $h _ { 0 }$ over the native vocabulary. We again use a straight-through Gumbel-softmax to sample a one-hot vector. Similarly, the foreign language encoder consists simply of the $D _ { \mathrm { h i d } }$ -dimensional foreign word embeddings.
+
+General training details In both word- and sentence-level experiments, we use 2048-dimensional pre-softmax features from a pre-trained ResNet with 50 layers (He et al., 2016b), instead of raw images. Our models are trained using stochastic gradient descent with the Adam optimizer (Kingma & Ba, 2014). The norm of the gradient is clipped with a threshold of 1 (Pascanu et al., 2013). Gumbelsoftmax temperature is tuned on the validation set, but fixed throughout training, not annealed or learned.
+
+# 3.3 HOW TRANSLATION ARISES
+
+To translate an English sentence $m _ { \mathrm { s r c } }$ to Japanese, we let the Japanese-speaking agent $P _ { B }$ encode $m _ { \mathrm { s r c } }$ with its English encoder, and decode this representation using its Japanese speaker module:
+
+$m _ { \mathrm { h y p } } = S _ { \mathrm { J A } } ^ { B } ( E _ { \mathrm { E N } } ^ { B } ( m _ { \mathrm { s r c } } ) )$ (see Figure 1b). Solving the image referential task requires aligning the foreign (source) sentence representation with the representation of the correct image, which will allow the speaker module to describe the source sentence in its native (target) language, as though it were an image.
+
+# 4 WORD-LEVEL EXPERIMENTS
+
+Task and dataset We train our model on a word-level communication task, where the agent $P _ { A }$ is given an image and needs to find the right word to communicate it so that the agent $P _ { B }$ can pick the right image from a set of distractors. We use the Bergsma500 dataset (Bergsma & Van Durme, 2011), a collection of up to 20 image search results per concept and language, for 500 common concepts across 6 languages: English, Spanish, German, French, Italian and Dutch. We train on $80 \%$ of the images, and choose the model with the best communication accuracy on the $20 \%$ validation set when reporting translation performance. As the Bergsma500 is an extremely small dataset, we do not have a separate test set to report the communication accuracy on. We only report the translation performance instead. Note that the translation task involves translating 500 words from the vocabulary, therefore the data split of images is not relevant for this task.
+
+Baselines For our baselines, we use a variety of nearest neighbor methods based on similarity metrics in the ConvNet feature space (Kiela et al., 2015). Given a set of 20 ResNet image vectors per concept and language, we can either average them (CNN-Mean) or take the dimension-wise maximum (CNN-Max) to derive a single aggregated image vector. To find the German word for dog, for instance, we rank all German words based on cosine similarity between the image vector of dog and their image vectors. We then examine precision in retrieving the correct German word, Hund. Alternatively, we also consider the similarities between individual image vectors instead of their aggregation: Bergsma & Van Durme (2011) propose taking the average of the maximum similarity scores (CNN-AvgMax) and the maximum of the maximum similarity scores (CNN-MaxMax).
+
+Experimental settings We train with 1 distractor $( K = 2 ) ^ { 2 }$ , learning rate 3e−4, and minibatch size 128. The embedding and hidden state dimensionalities are set to 400. When the validation accuracies of both the speaker and the listener stop improving, training terminates and we evaluate the performance on the word-level translation task, as described in $\ S 3 . 3$ . We consider all 15 language pairs in both directions, reporting results averaged across 30 translation cases.
+
+Results In all 15 language pairs, we observe that translation performance improves with communication performance (Figure 3). In translation, our model outperforms all the nearest neighbor baselines (Table 2). This shows that our agents can learn foreign word representations that are not only effective in solving referential tasks, but also more meaningful than raw image features in translation. It also demonstrates that communication helps to identify and learn correspondences between concepts in different languages.
+
+Figure 2: Word-level translation results, in precision at $k$ . Results are averaged over 30 translation cases (15 two-way pairs).   
+
+<table><tr><td>Model</td><td>P@1</td><td>P@5</td><td>P@20</td></tr><tr><td>CNN-AvgMax</td><td>53.00</td><td>68.30</td><td>78.36</td></tr><tr><td>CNN-MaxMax</td><td>49.85</td><td>65.91</td><td>77.17</td></tr><tr><td>CNN-Mean</td><td>51.89</td><td>66.47</td><td>77.14</td></tr><tr><td>CNN-Max</td><td>33.81</td><td>50.62</td><td>65.81</td></tr><tr><td>Our model</td><td>56.39</td><td>70.43</td><td>79.19</td></tr></table>
+
+![](images/e0d9502a310d495b8e4422e5e7079fbb48c6ab74fd5417d3d56bce5792b18f33.jpg)  
+Figure 3: Learning curve for the EN-DE word-level model.
+
+Qualitative analysis As our agents learn foreign words by grounding them in visual space, we expect the learned foreign word embeddings to be semantically similar to corresponding images. We inspect the nearest neighbors of foreign word embeddings in each language, and find that concepts with similar images indeed have close word embeddings. See Appendix B for a discussion and relevant examples.
+
+# 5 SENTENCE-LEVEL EXPERIMENTS
+
+Task We next train our models on a sentence-level communication task where agent $P _ { A }$ is given an image, and needs to communicate its content in a sentence in its language $L _ { A }$ to allow agent $P _ { B }$ to identify the right image from a set of distractors (see §3.1).
+
+Datasets and preprocessing We use three datasets of images with annotations in multiple languages. The Multi30k (Elliott et al., 2016) dataset contains $3 0 \mathrm { k }$ images and two types of bilingual annotations for two different tasks:
+
+• (Task 1) English-German translation task, where this can be aided by images; and • (Task 2) German image captioning task, where this can be helped with English captions.
+
+Training data for Task 1 consists of 1 English caption and its German translation for every image, translated by a professional translator. For Task 2, five English and five German captions are collected independently for every image. We use the original data split: $2 9 \mathrm { k }$ training, 1k validation and 1k test images.
+
+We experiment with another language pair: English-Japanese. We use MS COCO (Lin et al., 2014; Chen et al., 2015), which contains $1 2 0 \mathrm { k }$ images and 5 English captions per image, and STAIR (Yoshikawa et al., 2017), a collection of Japanese annotations of the same dataset (also 5 per image). Following Karpathy & Li (2015), we use $1 1 0 \mathrm { k }$ training, 5k validation and $5 \mathrm { k }$ test images.
+
+To ensure no parallel corpus is used to train our models, we partition the images in the training set into two parts (one for each langauge) and only use captions in one language for each half and not the other. With Multi30k, for instance, we have $1 4 . 5 \mathrm { k }$ English training images (whose German captions we discard) and 14.5k German training images.
+
+We use tokenized Japanese captions in STAIR.3 We lowercase, normalize and tokenize English and German captions using preprocessing scripts from Moses.4 In addition, we tokenize German captions into subword symbols using the byte pair encoding (BPE) algorithm with 10k merge operations (Sennrich et al., 2015).
+
+Baselines We compare against several baselines that similarly only make use of disjoint imagedescription data. In increasing order of sophistication:
+
+Nearest neighbor To translate an English sentence into German, we use its corresponding image to find the closest image in our German training set. We then retrieve all corresponding German captions and compute BLEU score against the ground truth German test captions. This model is similar to our word-level nearest neighbor baselines.
+
+NMT with neighboring pairs Given our non-aligned training set of English and German image captions, without any overlapping images, we can form new EN-DE sentence pairs by finding the closest German training image for every English training image. We then pair every corresponding German caption with every corresponding English caption, and train a standard NMT model without attention (Cho et al., 2014; Sutskever et al., 2014) on these pairs. We do not compare against an NMT model with attention because our models do not use attention (since incorporating attention would mean that agents have access to each other’s hidden states, which is no longer a multi-agent setting).
+
+N&N We implement and train end-to-end models from (Nakayama & Nishida, 2017). Their twoway model learns separate encoders to align the source language and images in a multimodal space. Then, a captioning model in the target language is trained on image representations, and is used to decode source representations to translate them. Their three-way models align both source and target languages with images using a target language encoder. Their models are similar to our models, with two key differences: (1) they are trained on a fixed corpus, without interaction between agents or learned communication, and (2) their model unit-normalizes the output of every encoder and is trained on pairwise ranking loss. In order to specifically examine the effectiveness of communication in learning to translate, we train these baselines using both their original loss function and our own loss function (see Appendix A).
+
+Models In our base model, the agents learn to speak their native languages simultaneously as they learn to communicate with each other (not pretrained). In a sense, this can be seen as a tabula rasa situation where both agents start from a blank slate. However, we also experiment with agents who already speak their languages, by using the weights from pretrained image captioning models in both languages to initialize our speaker modules and image encoders. Furthermore, we can freeze the parameters of image encoders or speaker modules to investigate their impact on communication and translation performance. In the most extreme case, where we pretrain and fix the speaker modules and image encoders (pretrained, spk $\pmb { \& }$ enc fixed), we only train the foreign language encoder, using only the listener loss. All other models are trained on both the speaker and listener loss.
+
+Experimental settings We train with 1 distractor $( K = 2 ) ^ { 5 }$ and minibatch size 64. The hidden state size and embedding dimensionalities are 1024 and 512, respectively. The learning rate and dropout rate are tuned on the validation set for each task. The vocabulary sizes of each language used in our experiments are: EN (4k) and DE (5K) for Multi30k Task 1, EN (8k) and DE (13k) for Multi30k Task2 and EN (10k) and JP (13k) for MS COCO.
+
+We train the model on the communication task, and early stop when the validation translation BLEU score stops improving. We use beam search at inference time, with beam width tuned on the validation set.
+
+Table 1: Test BLEU scores for each model and dataset. The best performing (unaligned) model for each dataset is shown in bold. We show the results of baselines from (Nakayama & Nishida, 2017) using two different loss functions (Appendix A). Pretrained denotes initializing the speaker modules and image encoders with pretrained image captioning models. Fixed denotes fixing the parameters of either the speaker module or the image encoder.   
+
+<table><tr><td></td><td></td><td colspan="2">Multi30k Task 1 EN-DE DE-EN</td><td colspan="2">Multi30k Task 2 EN-DE DE-EN</td><td colspan="2">COCO &amp; STAIR EN-JA JA-EN</td></tr><tr><td>NMT with neighboring pairs</td><td>Nearest neighbor</td><td>1.41</td><td>1.77</td><td>3.75</td><td>5.87</td><td>15.88</td><td>10.94</td></tr><tr><td></td><td></td><td>3.07</td><td>3.41</td><td>6.83</td><td>14.78</td><td>32.17</td><td>22.39</td></tr><tr><td></td><td>N&amp;N, 2-way, img</td><td>2.57</td><td>2.69</td><td>5.22</td><td>12.78</td><td>28.68</td><td>20.61</td></tr><tr><td></td><td>N&amp;N, 3-way, img</td><td>2.01</td><td>3.51</td><td>6.19</td><td>14.60</td><td>29.81</td><td>21.25</td></tr><tr><td>Triiss</td><td>N&amp;N,3-way, desc</td><td>3.34</td><td>3.87</td><td>9.66</td><td>15.96</td><td>27.53</td><td>17.51</td></tr><tr><td></td><td>N&amp;N, 3-way, both</td><td>1.50</td><td>3.62</td><td>9.89</td><td>15.50</td><td>31.01</td><td>20.59</td></tr><tr><td></td><td>N&amp;N,2-way, img</td><td>4.20</td><td>6.04</td><td>11.95</td><td>17.22</td><td>33.10</td><td>23.43</td></tr><tr><td>ssor Jno</td><td>N&amp;N, 3-way, img</td><td>2.32</td><td>5.91</td><td>11.62</td><td>17.84</td><td>32.11</td><td>23.61</td></tr><tr><td></td><td>N&amp;N, 3-way, desc</td><td>5.13</td><td>6.02</td><td>11.07</td><td>17.01</td><td>26.65</td><td>17.82</td></tr><tr><td></td><td>N&amp;N,3-way, both</td><td>4.89</td><td>6.59</td><td>13.53</td><td>18.48</td><td>32.84</td><td>23.28</td></tr><tr><td>not pretrained</td><td></td><td>5.80</td><td>7.20</td><td>14.81</td><td>17.70</td><td>33.26</td><td>23.66</td></tr><tr><td></td><td> pretrained, spk &amp; enc fixed</td><td>5.81</td><td>7.36</td><td>13.87</td><td>18.68</td><td>35.25</td><td>24.61</td></tr><tr><td></td><td>pretrained, spk fixed</td><td>6.49</td><td>7.42</td><td>14.93</td><td>19.81</td><td>33.01</td><td>23.59</td></tr><tr><td>sopou gni</td><td>pretrained, not fixed</td><td>5.02</td><td>6.06</td><td>13.44</td><td>17.41</td><td>33.58</td><td>23.19</td></tr><tr><td colspan="2">Aligned NMT</td><td>17.21</td><td>16.65</td><td>19.99</td><td>21.44</td><td>38.55</td><td>28.36</td></tr></table>
+
+Results We find that na¨ıvely looking up the nearest training image and retrieving its captions gives relatively poor BLEU scores (Table 1, Nearest neighbor). On the other hand, training an NMT model on these visually closest neighbor pairs gives much better translation performance.
+
+From the results of baselines from (Nakayama & Nishida, 2017), it is clear that our loss function gives better performance than the pairwise ranking loss with unit-normalized encoder outputs. We note that these baselines perform worse than our models even when our loss function is used, an indication that communication helps in learning to translate. We conjecture that our listeners become better at aligning multimodal representations compared to these baselines, as our listeners are trained on speaker’s output, and hence are exposed to a bigger and more diverse set of image descriptions. In contrast, the N&N models only make use of the ground truth captions. We also note that their 3-way models have an additional encoder for the target language, which our models lack. Although this is not used at test time, their 3-way models have $33 \%$ more parameters to train $( 9 7 \mathrm { m } )$ than our models $( 7 3 \mathrm { m } )$ .
+
+In contrast to (Nakayama & Nishida, 2017), where the best performance was obtained with endto-end trained models, we find that our models benefit from initializing weights with pretrained captioning models. The model with fixed speaker modules and non-fixed image encoders gave best results in two out of three datasets, even outperforming the (spk & enc fixed) model, which only produces messages that are trained to predict ground truth captions. This shows that learning to send messages differently from the pretrained image captioning models achieves better translation performance than learning to send ground truth captions. We provide sample translations for our baselines and models in Appendix C. We also qualitatively examine completely zero-resource German-Japanese translation in Appendix F.
+
+We compare our results with a standard non-attentional NMT model trained on parallel data (Cho et al., 2014; Sutskever et al., 2014). On COCO & STAIR, we observe a gap of approximately 4 BLEU compared to our best models On Multi30k Task 2, which is slightly smaller, the gap grows up to 5 BLEU scores. On Multi30k Task 1, where the dataset is the smallest and also of the highest quality (annotated by professional translators), the gap is around 11 BLEU. In other words, we find that our approach performs closer to supervised NMT as more training data is available, but that there still is a gap, which is unsurprising given the lack of parallel data.
+
+Qualitative analysis We conjecture that our models learn to translate by having a shared visual space to ground source and target languages onto. Indeed, we show that our translation system fails without a common visual modality (see Appendix D). We note that using a larger number of distractors helps the model learn faster initially, but does not affect translation performance (see also Appendix D).
+
+As expected, our models struggle with translating abstract sentences, although we observe that they can capture some visual elements in the source sentence (see Appendix E). This observation applies to most current grounded NMT systems, and it is an avenue worth exploring in future work but beyond the scope of the current work.
+
+Inspired by the movie Arrival (2016), we show that our agents can learn to play the referential game, and learn to translate, using an alien language (Klingon) with only a small number of captions (see Appendix G). This example is meant to illustrate the point that our models can learn to translate even in situations where there is no knowledge whatsoever of the other language, and where training a professional translator would potentially take a long time.
+
+# 6 MULTILINGUAL COMMUNITY OF AGENTS
+
+Task and dataset Humans learn to speak languages within communities. We next investigate whether we can learn to translate better in a community of different language speakers, where every agent interacts with every other agent. We use the recently released multilingual Multi30k Task 1, which contains annotations in English, German and French for $3 0 \mathrm { k }$ images (Elliott et al., 2017). We train a community of three agents (each speaking one language) and let each agent learn the other two languages simultaneously.
+
+We again partition the set of images into two halves (M1 and M2 in Figure 4), and ensure that the speaker and the listener do not see the same set of images (see Figure 4a for example). We experiment with two different settings: 1) a full community model, where having a multilingual community allows us to expose agents to more data than in the single-pair model while maintaining disjoint sets of images between agents (Figure 4c); and 2) a fair community model where the agents are trained on exactly the same number of training examples (Figure 4b). We point out that the difference in training data should mainly affect the speaker module; the image encoder and foreign language encoder are trained on the same number of examples in all the models.
+
+![](images/e8067262d6d5c5d926de61450f3d79e027eca8e2ee21704e04ba4a4859c62dd4.jpg)  
+Figure 4: Training data in single-pair and community models. M1 EN denotes the English annotations for the first half images in Multi30k. Red and blue indicate training data for the English and the German agents’ speaker modules, respectively. Note that compared to the single pair model, English and German speakers see twice the amount of training data in the full model, but see the same number of examples in the fair model.
+
+Experimental settings We train our base model (not pretrained) on a three-way sentence-level communication task. We sample a language pair with an equal probability every minibatch and let the two agents communicate. Every agent has one image encoder, one native speaker module and two foreign language encoders. We tokenize every corpus using BPE with 10k merge operations. We use the same architecture for the three models: $D _ { \mathrm { e m b } } = 1 2 8 , D _ { \mathrm { h i d } } = 2 5 6$ and we use a learning rate of 3e-4 and batch size of 128.
+
+![](images/42ac9b9e0a4d0838974aa89bfe3f2f5ae9cd65898675b4141291040beafeed4f.jpg)  
+Figure 5: DE-EN learning curve for different models.
+
+<table><tr><td>Model</td><td>EN-DE</td><td>DE-EN</td><td>EN-FR</td><td>FR-EN</td><td>DE-FR</td><td>FR-DE</td></tr><tr><td>Single</td><td>3.85</td><td>5.36</td><td>5.20</td><td>5.87</td><td>4.31</td><td>3.92</td></tr><tr><td>Fair</td><td>3.73</td><td>5.56</td><td>4.81</td><td>5.96</td><td>5.08</td><td>4.00</td></tr><tr><td>Full</td><td>4.83</td><td>7.21</td><td>7.09</td><td>8.10</td><td>6.55</td><td>5.15</td></tr></table>
+
+Table 2: Multi30k Task 1 Test BLEU scores. Results should be compared with the first two columns in Table 1.
+
+Results We observe that multilingual communities learn better translations. By having access to more target-side data, the full community model achieves the best translation performance in every language pair (Table 2). The fair community model achieves comparable performance to the single pair model.
+
+We show the learning curves of different models in Figure 5. The full community model clearly learns much faster than the other two. The fair community also learns faster than the single-pair model, as it learns with equivalent speed but with less exposure to individual language pairs, since we sample a language pair for each batch, rather than always having the same one.
+
+# 7 CONCLUSIONS AND FUTURE WORK
+
+In this paper, we have shown that the ability to understand a foreign language, and to translate it in the agent’s native language, can emerge from a communication task. We argue that this setting is natural, since humans learn language in a similar way: by trying to understand other humans while being grounded in a shared environment.
+
+We empirically confirm that the capability to translate is facilitated by the fact that agents have a shared visual modality they can refer to in their respective languages. Our experiments show that our model outperforms recently proposed baselines where agents do not communicate, as well as several nearest neighbor based baselines, in both sentence- and word-level scenarios.
+
+In future work, we plan to examine how we can enrich our agents with the ability to understand and translate abstract language, possibly through multi-task learning.
+
+# ACKNOWLEDGEMENT
+
+KC thanks support by eBay, TenCent, Facebook, Google and NVIDIA. We thank our colleagues from FAIR for helpful discussions.
+
+# REFERENCES
+
+Jacob Andreas, Anca D. Dragan, and Dan Klein. Translating neuralese. In Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics, pp. 232–242, 2017.
+
+Galen Andrew, Raman Arora, Jeff A. Bilmes, and Karen Livescu. Deep canonical correlation analysis. In Proceedings of the 30th International Conference on Machine Learning, ICML 2013, Atlanta, GA, USA, 16-21 June 2013, pp. 1247–1255, 2013.
+
+Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. In Proceedings of the International Conference on Learning Representations, 2015.
+
+Shane Bergsma and Benjamin Van Durme. Learning bilingual lexicons using the visual similarity of labeled web images. In Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence, pp. 1764–1769, 2011.
+
+Ozan Caglayan, Lo¨ıc Barrault, and Fethi Bougares. Multimodal attention for neural machine translation. arXiv preprint arXiv:1609.03976, 2016.
+
+Iacer Calixto, Qun Liu, and Nick Campbell. Doubly-attentive decoder for multi-modal neural machine translation. In Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics, pp. 1913–1924, 2017a.
+
+Iacer Calixto, Qun Liu, and Nick Campbell. Multilingual multi-modal embeddings for natural language processing. arXiv preprint arXiv:1702.01101, 2017b.
+
+Sarath Chandar, Mitesh M. Khapra, Hugo Larochelle, and Balaraman Ravindran. Correlational neural networks. Neural Computation, 28(2):257–285, 2016.
+
+Xinlei Chen, Hao Fang, Tsung-Yi Lin, Ramakrishna Vedantam, Saurabh Gupta, Piotr Dollar, and ´ C. Lawrence Zitnick. Microsoft COCO captions: Data collection and evaluation server. arXiv preprint arXiv:1504.00325, 2015.
+
+Kyunghyun Cho, Bart van Merrienboer, Dzmitry Bahdanau, and Yoshua Bengio. On the properties ¨ of neural machine translation: Encoder-decoder approaches. In Proceedings of the 8th Workshop on Syntax, Semantics, and Structure in Statistical Translation, pp. 103–111, 2014.
+
+Grzegorz Chrupala, Akos K ´ ad´ ar, and Afra Alishahi. Learning language through pictures. In ´ Proceedings of the 53rd Annual Meeting of the Association for Computational Linguistics, pp. 112– 118, 2015.
+
+Abhishek Das, Satwik Kottur, Jose M. F. Moura, Stefan Lee, and Dhruv Batra. Learning cooperative ´ visual dialog agents with deep reinforcement learning. arXiv preprint arXiv:1703.06585, 2017.
+
+Desmond Elliott and Akos K ´ ad´ ar. Imagination improves multimodal translation. ´ arXiv preprint arXiv:1705.04350, 2017.
+
+Desmond Elliott, Stella Frank, Khalil Sima’an, and Lucia Specia. Multi30k: Multilingual englishgerman image descriptions. In Proceedings of the 5th Workshop on Vision and Language, pp. 70–74, 2016.
+
+Desmond Elliott, Stella Frank, Lo¨ıc Barrault, Fethi Bougares, and Lucia Specia. Findings of the second shared task on multimodal machine translation and multilingual image description. In Proceedings of the Second Conference on Machine Translation, pp. 215–233, 2017.
+
+Katrina Evtimova, Andrew Drozdov, Douwe Kiela, and Kyunghyun Cho. Emergent language in a multi-modal, multi-step referential game. arXiv preprint arXiv:1705.10369, 2017.
+
+Orhan Firat, Baskaran Sankaran, Yaser Al-Onaizan, Fatos T. Yarman-Vural, and Kyunghyun Cho. Zero-resource translation with multi-lingual neural machine translation. In Proceedings of the 2016 Conference on Empirical Methods in Natural Language Processing, EMNLP 2016, Austin, Texas, USA, November 1-4, 2016, pp. 268–277, 2016.
+
+Jakob N. Foerster, Yannis M. Assael, Nando de Freitas, and Shimon Whiteson. Learning to communicate with deep multi-agent reinforcement learning. In Advances in Neural Information Processing Systems, pp. 2137–2145, 2016.
+
+Jonas Gehring, Michael Auli, David Grangier, Denis Yarats, and Yann N. Dauphin. Convolutional sequence to sequence learning. In Proceedings of the 34th International Conference on Machine Learning, pp. 1243–1252, 2017.
+
+Spandana Gella, Rico Sennrich, Frank Keller, and Mirella Lapata. Image Pivoting for Learning Multilingual Multimodal Representations. In Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing, pp. 2829–2835, 2017.
+
+Serhii Havrylov and Ivan Titov. Emergence of language with multi-agent games: Learning to communicate with sequences of symbols. In Proceedings of the International Conference on Learning Representations Workshop Track, 2017.
+
+Di He, Yingce Xia, Tao Qin, Liwei Wang, Nenghai Yu, Tieyan Liu, and Wei-Ying Ma. Dual learning for machine translation. In Advances in Neural Information Processing Systems, pp. 820–828, 2016a.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 770–778, 2016b.
+
+Julian Hitschler, Shigehiko Schamoni, and Stefan Riezler. Multimodal pivots for image caption translation. In Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics, 2016.
+
+Micah Hodosh, Peter Young, and Julia Hockenmaier. Framing image description as a ranking task: Data, models and evaluation metrics. J. Artif. Intell. Res., 47:853–899, 2013.
+
+Eric Jang, Shixiang Gu, and Ben Poole. Categorical Reparameterization with Gumbel-Softmax. In Proceedings of the International Conference on Learning Representations, 2017.
+
+Emilio Jorge, Mikael Kageb ˚ ack, and Emil Gustavsson. Learning to play guess who? and inventing ¨ a grounded language as a consequence. arXiv preprint arXiv:1611.03218, 2016.
+
+Armand Joulin, Laurens van der Maaten, Allan Jabri, and Nicolas Vasilache. Learning visual features from large weakly supervised data. In Computer Vision - ECCV 2016 - 14th European Conference, pp. 67–84, 2016.
+
+Andrej Karpathy and Fei-Fei Li. Deep visual-semantic alignments for generating image descriptions. In IEEE Conference on Computer Vision and Pattern Recognition, pp. 3128–3137, 2015.
+
+Douwe Kiela, Ivan Vulic, and Stephen Clark. Visual bilingual lexicon induction with transferred ´ convnet features. In Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing, pp. 148–158, 2015.
+
+Douwe Kiela, Alexis Conneau, Allan Jabri, and Maximilian Nickel. Learning visually grounded sentence representations. arXiv:1707.06320, 2017.
+
+Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In Proceedings of the 3rd International Conference for Learning Representations, 2014.
+
+Ryan Kiros, Ruslan Salakhutdinov, and Richard S. Zemel. Unifying visual-semantic embeddings with multimodal neural language models. arXiv:1411.2539, 2014.
+
+Satwik Kottur, Jose M. F. Moura, Stefan Lee, and Dhruv Batra. Natural language does not emerge ´ ’naturally’ in multi-agent dialog. In Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing, pp. 2952–2957, 2017.
+
+Angeliki Lazaridou, Alexander Peysakhovich, and Marco Baroni. Multi-agent cooperation and the emergence of (natural) language. In Proceedings of the International Conference on Learning Representations, 2017.
+
+Mike Lewis, Denis Yarats, Yann N. Dauphin, Devi Parikh, and Dhruv Batra. Deal or no deal? endto-end learning for negotiation dialogues. In Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing, pp. 2433–2443, 2017.
+
+Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dollar, and C. Lawrence Zitnick. Microsoft coco: Common objects in context. In ´ Computer Vision - ECCV 2014 - 13th European Conference, pp. 740–755, 2014.
+
+Ryan Lowe, Yi Wu, Aviv Tamar, Jean Harb, Pieter Abbeel, and Igor Mordatch. Multi-agent actorcritic for mixed cooperative-competitive environments. arXiv preprint arXiv:1706.02275, 2017.
+
+Lin Ma, Zhengdong Lu, Lifeng Shang, and Hang Li. Multimodal convolutional neural networks for matching image and sentence. In 2015 IEEE International Conference on Computer Vision, ICCV 2015, Santiago, Chile, December 7-13, 2015, pp. 2623–2631, 2015.
+
+Chris J. Maddison, Andriy Mnih, and Yee Whye Teh. The concrete distribution: A continuous relaxation of discrete random variables. In Proceedings of the International Conference on Learning Representations, 2017.
+
+Igor Mordatch and Pieter Abbeel. Emergence of grounded compositional language in multi-agent populations. arXiv preprint arXiv:1705.10369, 2017.
+
+Hideki Nakayama and Noriki Nishida. Zero-resource machine translation by multimodal encoder– decoder network with multimedia pivot. Machine Translation, 31(1):49–64, 2017.
+
+Razvan Pascanu, Tomas Mikolov, and Yoshua Bengio. On the difficulty of training recurrent neural networks. In Proceedings of the 30th International Conference on Machine Learning, pp. 1310– 1318, 2013.
+
+Janarthanan Rajendran, Mitesh M. Khapra, Sarath Chandar, and Balaraman Ravindran. Bridge correlational neural networks for multilingual multimodal representation learning. In NAACL HLT 2016, The 2016 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, San Diego California, USA, June 12-17, 2016, pp. 171–181, 2016.
+
+Rico Sennrich, Barry Haddow, and Alexandra Birch. Neural machine translation of rare words with subword units. In Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics, pp. 1715–1725, 2015.
+
+Rico Sennrich, Alexandra Birch, Anna Currey, Ulrich Germann, Barry Haddow, Kenneth Heafield, Antonio Valerio Miceli Barone, and Philip Williams. The university of edinburgh’s neural MT systems for WMT17. In Proceedings of the Second Conference on Machine Translation, pp. 389–399, 2017.
+
+Richard Socher, Andrej Karpathy, Quoc V. Le, Christopher D. Manning, and Andrew Y. Ng. Grounded compositional semantics for finding and describing images with sentences. TACL, 2:207–218, 2014.
+
+Sainbayar Sukhbaatar, Arthur Szlam, and Rob Fergus. Learning multiagent communication with backpropagation. In Advances in Neural Information Processing Systems, pp. 2244–2252, 2016.
+
+Ilya Sutskever, Oriol Vinyals, and Quoc V Le. Sequence to sequence learning with neural networks. In Advances in Neural Information Processing Systems, volume 27, pp. 3104–3112, 2014.
+
+Ivan Vendrov, Ryan Kiros, Sanja Fidler, and Raquel Urtasun. Order-embeddings of images and language. arXiv:1511.06361, 2015.
+
+Ronald J Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine learning, 8(3-4):229–256, 1992.
+
+Yonghui Wu, Mike Schuster, Zhifeng Chen, Quoc V. Le, Mohammad Norouzi, Wolfgang Macherey, Maxim Krikun, Yuan Cao, Qin Gao, Klaus Macherey, Jeff Klingner, Apurva Shah, Melvin Johnson, Xiaobing Liu, Łukasz Kaiser, Stephan Gouws, Yoshikiyo Kato, Taku Kudo, Hideto Kazawa, Keith Stevens, George Kurian, Nishant Patil, Wei Wang, Cliff Young, Jason Smith, Jason Riesa, Alex Rudnick, Oriol Vinyals, Greg Corrado, Macduff Hughes, and Jeffrey Dean. Google’s neural machine translation system: Bridging the gap between human and machine translation. arXiv:1609.08144, 2016.
+
+Yuya Yoshikawa, Yutaro Shigeto, and Akikazu Takeuchi. STAIR captions: Constructing a largescale japanese image caption dataset. In Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics, pp. 417–421, 2017.
+
+# A CHOICE OF LISTENER LOSS FUNCTION
+
+We compare two alternatives for training the listener: (1) a pairwise ranking loss, as used in Havrylov & Titov (2017) and Nakayama & Nishida (2017), and (2) the MSE loss, in addition to the one (Eq. (1)) used in this paper.
+
+Pairwise ranking loss Denoting the message vector, the target image and the $k$ -th distractor image as $\hat { m } , i$ and $i _ { k }$ , respectively, this cost function is expressed as:
+
+$$
+\mathcal { I } _ { ( \mathrm { l s n } , \mathrm { r a n k } ) } = \sum _ { k = 1 } ^ { K } \operatorname* { m a x } \Big ( 0 , \alpha - \sin \big ( E _ { \mathrm { E N } } ^ { B } ( \hat { m } ) , E _ { \mathrm { I M G } } ^ { B } ( i ) \big ) + \sin \big ( E _ { \mathrm { E N } } ^ { B } ( \hat { m } ) , E _ { \mathrm { I M G } } ^ { B } ( i _ { k } ) \big ) \Big ) ,
+$$
+
+where $\alpha$ is a margin hyperparameter and $\mathrm { s i m } ( \cdot )$ is a function that computes the similarity between two vectors. The most common choice for $\mathrm { s i m } ( \cdot )$ in practice is cosine similarity (Nakayama & Nishida, 2017). Note, however, that this only aligns the direction of the message and the target image vectors, not the magnitude. To facilitate translation, the speaker module should take as input a normalized image vector. We found this use of normalized image vectors to consistently hurt performance in all our experiments.
+
+MSE loss It has been found that minimizing the MSE loss can be effective in learning visually grounded representations of language (see, e.g., Chrupala et al., 2015). While our loss function is very similar to minimizing the MSE loss, there is an important distinction. Letting $x = E _ { \mathrm { E N } } ^ { B } ( \hat { m } ) \stackrel { \cdot } { - } E _ { \mathrm { I M G } } ^ { B } ( i )$ , the MSE loss function is given by $x ^ { 2 }$ , with a derivative of $2 x$ . Our loss function is $- \log ( 1 / x ^ { 2 } )$ , of which the derivative is $\bar { 2 } / x$ . Note that the gradient is initially small, when $x$ is large, but grows larger in magnitude as the listener loss is minimized.
+
+![](images/681ac53f3cb3e67802920f7035cd3f95b359a6e27f134b18d3a7891c3b9ffa3d.jpg)  
+Figure 6: Comparison of our listener loss function with MSE loss.
+
+In Figure 6, we show the learning curve of our model on Multi30k Task 2 EN-DE trained using either our listener loss function or the MSE loss. At the initial stage of learning, the listener trained using our loss function learns very slowly due to a small gradient. As the listener becomes better at aligning target images and sentences, it begins to learn much more quickly. On the other hand, we observe that the listener trained using the MSE loss immediately starts to learn, but learning slows down quickly.
+
+Our loss term $- \log ( 1 / x ^ { 2 } )$ is not defined at 0. In practice, $x$ is almost never 0, and we further add small noise to $x$ both to avoid $x = 0$ and to regularize learning. Empirically, we observe our formulation gives much better translation performance than both the pairwise ranking loss and the MSE loss.
+
+# B WORD-LEVEL NEAREST NEIGHBOR ANALYSIS
+
+Table 3 showcases three concepts in English and German, where for each concept, the most representative image is shown, as well as the five nearest neighboring words (by cosine similarity).
+
+We note that nearest neighbors for most words correspond to our semantic judgments: galaxy is closest to universe in English and Galaxie to Universum in German. Similarly, plant or Pflanze is closest to leaf or Blatt. For concepts that evoke different senses across languages, however, we observe different nearest neighbors. For example, most images for Fox in the dataset contain a person, whereas images for Fuchs contain an animal. This encourages the German agent to associate Fox closely with Celebrity and Girl, whereas the English agent learns that Fuchs is a furry animal, similar to Kanguru ¨ or Lowe ¨ .
+
+![](images/71b253b646af357fdfe40e3d015b506be039eba8243bdf68a5631a8f5285cf45.jpg)
+
+# C SENTENCE-LEVEL SAMPLE TRANSLATIONS
+
+![](images/1190db0935d3a19d3919f21c2b0b94ab613b1994684920eaa56b9ef049c0196c.jpg)
+
+![](images/5b8dbdfac70a9609f20a5433f4d79caf326f67f077c280000101fced18647f12.jpg)
+
+![](images/65d023ebe07a8a0fae680e5059a4cdb485f94a706a6ce08bcf42efc4ae0710e7.jpg)
+
+Table 3: Nearest neighbors of foreign word embeddings learned from communication, along with a sample image for each concept in the dataset. The English word embeddings were learned by the German agent and vice versa.   
+Table 4: Sample DE-EN translations from Multi30k Task 2 test set. Images were not used to aid translation and are only shown for references. We show the source sentence as $S r c$ and one of the five target sentences as Ref. The outputs from the nearest neighbor baseline are shown as NN, NMT baseline with neighbor pairs as NMT, the 3-way, both decoder model from (Nakayama & Nishida, 2017) as N&N, and our (pretrained, spk fixed) model as Model.   
+
+<table><tr><td>Src</td><td>einhund springt auf einer wiese vor einem weiBen zaun in die luft</td><td>ein mann hängt an einem seil mit rollen das iber ein wasser gespannt ist .</td><td>ein skateboarder an einer boschung zu einem parkplatz.</td></tr><tr><td>Ref</td><td>a dogruns on the green grass near a wooden fence.</td><td>aman wearing bathing trunks is para- sailing in the water .</td><td>a skateboard is grinding on a curb with his skateboard</td></tr><tr><td>NN</td><td>a brunettephotographer is kneeling down to take a photo .</td><td>police watch some punk rock types at a protest.</td><td>a man is standing on the streets taking photographs.</td></tr><tr><td>NMT</td><td>a dog is jumping overa fence</td><td>a man ina blue shirt is ridinga bike.</td><td>aman in a blue shirt is ridinga bike</td></tr><tr><td>N&amp;N</td><td>a brown dog is running on the grass.</td><td>a man ina wetsuit is surfinga large wave.</td><td>aman ina blue shirt iswalking down the sidewalk .</td></tr><tr><td>Model</td><td>two dogs playing with a ballin the grass</td><td>a man is parasailing in the ocean.</td><td>a man in a blue shirt and black pants is skateboarding.</td></tr></table>
+
+In Table 4, we compare sample translations from our nearest neighbor and NMT baselines, as well as the best model from (Nakayama & Nishida, 2017) and our best model. We observe that the nearest neighbor baseline generates translations that are mostly unrelated to the reference sentence. The NMT baseline, on the other hand, often generates the correct subject of the sentence, and seems capable of capturing the main actor in the scene. The 3-way, both decoder model from (Nakayama & Nishida, 2017) captures the main actors and the environment in the scene. Our model appears to capture even the minor details, such as quantity (number of dogs) and specific activity (parasailing, skateboarding).
+
+D SENTENCE-LEVEL QUALITATIVE ANALYSIS
+
+![](images/03174d2e367e3ab1c1917d81e07eb837ce5cc419e08b2949f4f8fb86b74ec01c.jpg)  
+Figure 7: Sharing $E ^ { I }$
+
+![](images/89db8c45d18bf943a570b3a7d2c5c54717921914885c3ce24390b6f5ff42c89b.jpg)  
+Figure 8: Number of distractors.
+
+We conjectured that grounding the native and foreign languages into a shared visual space allows agents to understand foreign languages. To verify this, we trained our base model (not pretrained) on COCO & STAIR, where agents have language-specific image encoders, e.g. the English agent has two separate image encoders: one for the English speaking module and another for the Japanese encoder. We compare this with our original model with the same architecture. We plot the communication accuracy and JA-EN validation BLEU scores in Figure 7. Red curves indicate validation communication accuracy (left axis), and green curves indicate BLEU score (right axis). Solid lines denote the standard model and dotted lines denote the model without sharing the image encoder.
+
+We observe no significant difference in the communication performance: the model that does not share the image encoder performs just as well in the communication task. However, translation performance greatly suffers without access to the shared visual modality.
+
+In Figure 8, we show the learning curve of four base models, with different number of distractors $( K )$ . We observe that a larger number of distractors helps the model learn faster initially, but otherwise gives no performance benefit.
+
+# E TRANSLATION IN NON-CONCRETE DOMAINS
+
+As our agents understand foreign languages through grounding in the visual modality, we investigate their ability to generalize to non-visual, abstract domains. We train our (pretrained, spk fixed) model on Multi30k Task 2, and let it translate German sentences from WMT’15 DE-EN validation and test set. See Table 5.
+
+We note that our model is able to capture some visual elements in a sentence, such as snow or mountain, but generally produces poor quality translations. We observe that most words in the source sentence from WMT’15 do not occur in Multi30k’s training set, hence our model mostly receives ${ < } U N K { > }$ vectors.
+
+# F ZERO-RESOURCE TRANSLATION
+
+To showcase our models’ ability to learn to translate without parallel corpora, we train our base model on a communication task between a low-resource language pair: German and Japanese. We take the German corpus from Multi30k Task2, the Japanese corpus from STAIR, and train two models on a sentence-level communication task between the two languages. In Tables 6 and 7, we show the Japanese source sentence (src), the model output in German (hyp), and their translation to English using Google Translate. We observe that our model mostly generates reasonable sentences, and captures properties such as color and action in the scene.
+
+Table 5: Sample translations from WMT’15 DE-EN validation and test set.   
+
+<table><tr><td rowspan=1 colspan=1>Src</td><td rowspan=1 colspan=1>Schnee liegt insbesondere auf den StraBen im Riesengebirge und im Gebirge Orlické hory.</td></tr><tr><td rowspan=1 colspan=1>Ref</td><td rowspan=1 colspan=1>Snow is particularly affecting the roads in the Krkonose and Orlicke mountains.</td></tr><tr><td rowspan=1 colspan=1>Hyp</td><td rowspan=1 colspan=1>a man is standing on a snow covered mountain.</td></tr><tr><td></td><td></td></tr><tr><td rowspan=1 colspan=1>Src</td><td rowspan=1 colspan=1>Das Tote Meer ist sogar noch wärmer und dort wird das ganze Jahr über gebadet.</td></tr><tr><td rowspan=1 colspan=1>Ref</td><td rowspan=1 colspan=1>The Dead Sea is even warmer,and people swim in it all year round.</td></tr><tr><td rowspan=1 colspan=1>Hyp</td><td rowspan=1 colspan=1>a man is surfing in the ocean.</td></tr><tr><td></td><td></td></tr><tr><td rowspan=1 colspan=1>Src</td><td rowspan=1 colspan=1>Esfolgten die erstenRadfahrerundLäuferum1O Uhr.</td></tr><tr><td rowspan=1 colspan=1>Ref</td><td rowspan=1 colspan=1>Then it was the turn of the cyclists and runners,who began at 1O am.</td></tr><tr><td rowspan=1 colspan=1>Hyp</td><td rowspan=1 colspan=1>a man ina red and white uniform is riding a dirtbike in front of a crowd.</td></tr><tr><td></td><td></td></tr><tr><td rowspan=1 colspan=1>Src</td><td rowspan=1 colspan=1>Das Baby,das so ergreifend von einem Spezialeinsatzkommando in Sicherheit getragen wurde</td></tr><tr><td rowspan=1 colspan=1>Ref</td><td rowspan=1 colspan=1>The baby who was carried poignantly to safety by a special forces commando</td></tr><tr><td rowspan=1 colspan=1>Hyp</td><td rowspan=1 colspan=1>a baby ina yellow shirt is sleeping.</td></tr></table>
+
+![](images/99e30d4727a80dd0582917bee7d5300486f8fc7c1d188cc5c81514885d3723c5.jpg)
+
+Table 6: Sample JA-DE translations   
+
+<table><tr><td rowspan=1 colspan=1>Src</td><td rowspan=1 colspan=1>キよ構之る犬</td><td rowspan=1 colspan=1>赤と白の野球の二フオー着た少年ト構之い</td><td rowspan=1 colspan=1>黄色い機離瞬間</td></tr><tr><td rowspan=1 colspan=1>Hyp</td><td rowspan=1 colspan=1>ein hund fangt einen frisbee.</td><td rowspan=1 colspan=1>ein baseballspieler mit rotem helmund weibem trikot holt mit demschläger aus,um den heranfliegendenball zu treffen</td><td rowspan=1 colspan=1>ein flugzeug fliegt üiber eine flugzeug-piste.</td></tr><tr><td rowspan=1 colspan=1>Src (en)</td><td rowspan=1 colspan=1>Dogkeeps trying to catch Frisbee</td><td rowspan=1 colspan=1>A boy in a red and white baseball uni-form wears a bat</td><td rowspan=1 colspan=1>The moment when a yellow propellerplane takes off</td></tr><tr><td rowspan=1 colspan=1>Hyp (en)</td><td rowspan=1 colspan=1>a dog catchesa frisbee.</td><td rowspan=1 colspan=1>abaseball playerwitharedhelmetand a white tricot takes the bat out tohit the ball.</td><td rowspan=1 colspan=1>Anairplane flies over an airplane run-way.</td></tr></table>
+
+![](images/9b70299e70c4f91b4931e1cb4dfed6e0bf7ec8a3a4b4482fafcab38e40ca1768.jpg)
+
+Table 7: Sample JA-DE translations   
+
+<table><tr><td rowspan=1 colspan=1>Src</td><td rowspan=1 colspan=1>黄色い打返女性</td><td rowspan=1 colspan=1>黄色いのト部分取付たパクに黄色い映い</td><td rowspan=1 colspan=1>木の枝に正目の周足赤い鳥</td></tr><tr><td rowspan=1 colspan=1>Hyp</td><td rowspan=1 colspan=1>eine tennisspielerin in weiBundgruinem oberteil spielt tennis</td><td rowspan=1 colspan=1>ein gelber bus steht geparkt vor einemnicht fertigen gebäude.</td><td rowspan=1 colspan=1>ein vogel sitzt auf einem ast .</td></tr><tr><td rowspan=1 colspan=1>Src (en)</td><td rowspan=1 colspan=1>A woman striking a yellow tennis ball</td><td rowspan=1 colspan=1>A yellow busis reflected on therearview mirror attached to the frontpart of the yellow bus.</td><td rowspan=1 colspan=1>Around the eyes stopping at thebranches of the tree and a red bird</td></tr><tr><td rowspan=1 colspan=1>Hyp (en)</td><td rowspan=1 colspan=1>atennis player in white and greenshell playing tennis</td><td rowspan=1 colspan=1>A yellow bus is parked in front of anon-finished building.</td><td rowspan=1 colspan=1>A bird sitting on a branch.</td></tr></table>
+
+# G ALIEN LANGUAGE TRANSLATION
+
+To demonstrate our models’ ability to learn to translate only with monolingual captions, we experiment with a language for which no parallel corpus exists, nor the knowledge of the language itself: Klingon. As no image captions are available in Klingon, we translate $1 5 \mathrm { k }$ English captions in Multi30k Task 1 into Klingon pIqaD6 using Bing Translator.7 We tokenize the Klingon captions and discard words occurring less than 5 times in the training data. We then train our base model (no pretraining) on English and Klingon communication. In Tables 9 and 10, the source sentence in English is shown as src, the Klingon model output in hyp, and the English translation of the output in hyp (en) (using Bing Translator). Although the Klingon training data is noisy and imperfect, we observe that our model learns to translate only with 15k Klingon captions. This example illustrates how we can learn to translate even if there is no knowledge of the other language, and where a professional translator would take a long time to first acquire the other language.
+
+![](images/11608736fe8b18530ce46fb801b14d5b5fe0db16985ea70fc8b5e5ad98009a68.jpg)  
+Figure 9: Sample English-Klingon translations
+
+![](images/6740918e4399b8d8a3eabd0458314605505e01fd42c730163c8995b214cd3a58.jpg)  
+Figure 10: Sample English-Klingon translations

parse/train/HygjqjR9Km/HygjqjR9Km.md

@@ -0,0 +1,512 @@
+# IMPROVING MMD-GAN TRAINING WITH REPULSIVE LOSS FUNCTION
+
+Wei Wang∗ University of Melbourne
+
+Yuan Sun RMIT University
+
+Saman Halgamuge University of Melbourne
+
+# ABSTRACT
+
+Generative adversarial nets (GANs) are widely used to learn the data sampling process and their performance may heavily depend on the loss functions, given a limited computational budget. This study revisits MMD-GAN that uses the maximum mean discrepancy (MMD) as the loss function for GAN and makes two contributions. First, we argue that the existing MMD loss function may discourage the learning of fine details in data as it attempts to contract the discriminator outputs of real data. To address this issue, we propose a repulsive loss function to actively learn the difference among the real data by simply rearranging the terms in MMD. Second, inspired by the hinge loss, we propose a bounded Gaussian kernel to stabilize the training of MMD-GAN with the repulsive loss function. The proposed methods are applied to the unsupervised image generation tasks on CIFAR-10, STL-10, CelebA, and LSUN bedroom datasets. Results show that the repulsive loss function significantly improves over the MMD loss at no additional computational cost and outperforms other representative loss functions. The proposed methods achieve an FID score of 16.21 on the CIFAR-10 dataset using a single DCGAN network and spectral normalization. 1
+
+# 1 INTRODUCTION
+
+Generative adversarial nets (GANs) (Goodfellow et al. (2014)) are a branch of generative models that learns to mimic the real data generating process. GANs have been intensively studied in recent years, with a variety of successful applications (Karras et al. (2018); Li et al. (2017b); Lai et al. (2017); Zhu et al. (2017); Ho & Ermon (2016)). The idea of GANs is to jointly train a generator network that attempts to produce artificial samples, and a discriminator network or critic that distinguishes the generated samples from the real ones. Compared to maximum likelihood based methods, GANs tend to produce samples with sharper and more vivid details but require more efforts to train.
+
+Recent studies on improving GAN training have mainly focused on designing loss functions, network architectures and training procedures. The loss function, or simply loss, defines quantitatively the difference of discriminator outputs between real and generated samples. The gradients of loss functions are used to train the generator and discriminator. This study focuses on a loss function called maximum mean discrepancy (MMD), which is well known as the distance metric between two probability distributions and widely applied in kernel two-sample test (Gretton et al. (2012)). Theoretically, MMD reaches its global minimum zero if and only if the two distributions are equal. Thus, MMD has been applied to compare the generated samples to real ones directly (Li et al. (2015); Dziugaite et al. (2015)) and extended as the loss function to the GAN framework recently (Unterthiner et al. (2018); Li et al. (2017a); Binkowski et al. (2018)). ´
+
+In this paper, we interpret the optimization of MMD loss by the discriminator as a combination of attraction and repulsion processes, similar to that of linear discriminant analysis. We argue that the existing MMD loss may discourage the learning of fine details in data, as the discriminator attempts to minimize the within-group variance of its outputs for the real data. To address this issue, we propose a repulsive loss for the discriminator that explicitly explores the differences among real data. The proposed loss achieved significant improvements over the MMD loss on image generation tasks of four benchmark datasets, without incurring any additional computational cost. Furthermore, a bounded Gaussian kernel is proposed to stabilize the training of discriminator. As such, using a single kernel in MMD-GAN is sufficient, in contrast to a linear combination of kernels used in Li et al. (2017a) and Binkowski et al. (2018). By using a single kernel, the computational cost of the ´ MMD loss can potentially be reduced in a variety of applications.
+
+The paper is organized as follows. Section 2 reviews the GANs trained using the MMD loss (MMDGAN). We propose the repulsive loss for discriminator in Section 3, introduce two practical techniques to stabilize the training process in Section 4, and present the results of extensive experiments in Section 5. In the last section, we discuss the connections between our model and existing work.
+
+# 2 MMD-GAN
+
+In this section, we introduce the GAN model and MMD loss. Consider a random variable $\mathbf x \in \mathcal X$ with an empirical data distribution $P _ { \mathbf { X } }$ to be learned. A typical GAN model consists of two neural networks: a generator $G$ and a discriminator $D$ . The generator $G$ maps a latent code $_ z$ with a fixed distribution $P _ { \mathbf { Z } }$ (e.g., Gaussian) to the data space $\mathcal { X }$ : $\pmb { y } = G ( \pmb { z } ) \in \mathcal { X }$ , where $\textbf {  { y } }$ represents the generated samples with distribution $P _ { G }$ . The discriminator $D$ evaluates the scores $\mathbf { \bar { \Gamma } } \mathbf { \bar { \Gamma } } D ( \mathbf { \bar { a } } ) \in \mathbb { R } ^ { d }$ of a real or generated sample $\textbf { \em a }$ . This study focuses on image generation tasks using convolutional neural networks (CNN) for both $G$ and $D$ .
+
+Several loss functions have been proposed to quantify the difference of the scores between real and generated samples: $\{ D ( { \pmb x } ) \}$ and $\{ D ( \pmb { y } ) \}$ , including the minimax loss and non-saturating loss (Goodfellow et al. (2014)), hinge loss (Tran et al. (2017)), Wasserstein loss (Arjovsky et al. (2017); Gulrajani et al. (2017)) and maximum mean discrepancy (MMD) (Li et al. (2017a); Binkowski et al. (2018)) (see Appendix B.1 for more details). Among them, MMD uses kernel em- ´ bedding $\boldsymbol { \phi } ( \boldsymbol { \mathbf { \em a } } ) = \boldsymbol { k } ( \cdot , \boldsymbol { \mathbf { \em a } } )$ associated with a characteristic kernel $k$ such that $\phi$ is infinite-dimensional and $\langle \phi ( { \pmb a } ) , \phi ( { \pmb b } ) \rangle _ { \mathcal { H } } = k ( { \pmb a } , { \pmb b } )$ . The squared MMD distance between two distributions $P$ and $Q$ is
+
+$$
+M _ { k } ^ { 2 } ( P , Q ) = \left\| \mu _ { P } - \mu _ { Q } \right\| _ { \mathcal { H } } ^ { 2 } = \mathbb { E } _ { a , a ^ { \prime } \sim P } [ k ( a , a ^ { \prime } ) ] + \mathbb { E } _ { b , b ^ { \prime } \sim Q } [ k ( b , b ^ { \prime } ) ] - 2 \mathbb { E } _ { a \sim P , b \sim Q } [ k ( a , b ) ]
+$$
+
+The kernel $k ( a , b )$ measures the similarity between two samples $\textbf { \em a }$ and $^ { b }$ . Gretton et al. (2012) proved that, using a characteristic kernel $k$ , $M _ { k } ^ { 2 } ( P , Q ) \ge 0$ with equality applies if and only if $P = Q$ .
+
+In MMD-GAN, the discriminator $D$ can be interpreted as forming a new kernel with $k$ : $k \circ D ( { \pmb a } , { \pmb b } ) =$ $k ( D ( { \pmb a } ) , D ( { \pmb b } ) ) = k _ { D } ( { \pmb a } , { \pmb b } )$ . If $D$ is injective, $k \circ D$ is characteristic and $M _ { k \circ D } ^ { 2 } ( P _ { { \bf X } } , P _ { G } )$ reaches its minimum if and only if $P _ { \mathbf { X } } = P _ { G }$ (Li et al. (2017a)). Thus, the objective functions for $G$ and $D$ could be (Li et al. (2017a); Binkowski et al. (2018)): ´
+
+$$
+\begin{array} { r l } & { \underset { G } { \mathrm { m i n } } L _ { G } ^ { \mathrm { m n d } } = M _ { k o D } ^ { 2 } ( P _ { \mathbf { X } } , P _ { G } ) = \mathbb { E } _ { P _ { G } } [ k _ { D } ( \pmb { y } , \pmb { y } ^ { \prime } ) ] - 2 \mathbb { E } _ { P _ { \mathbf { X } } , P _ { G } } [ k _ { D } ( \pmb { x } , \pmb { y } ) ] + \mathbb { E } _ { P _ { \mathbf { X } } } [ k _ { D } ( \pmb { x } , \pmb { x } ^ { \prime } ) ] } \\ & { \underset { D } { \mathrm { m i n } } L _ { D } ^ { \mathrm { a t t } } = - M _ { k o D } ^ { 2 } ( P _ { \mathbf { X } } , P _ { G } ) = 2 \mathbb { E } _ { P _ { \mathbf { X } } , P _ { G } } [ k _ { D } ( \pmb { x } , \pmb { y } ) ] - \mathbb { E } _ { P _ { \mathbf { X } } } [ k _ { D } ( \pmb { x } , \pmb { x } ^ { \prime } ) ] - \mathbb { E } _ { P _ { G } } [ k _ { D } ( \pmb { y } , \pmb { y } ^ { \prime } ) ] } \end{array}
+$$
+
+MMD-GAN has been shown to be more effective than the model that directly uses MMD as the loss function for the generator $G$ (Li et al. (2017a)).
+
+Liu et al. (2017) showed that MMD and Wasserstein metric are weaker objective functions for GAN than the Jensen–Shannon (JS) divergence (related to minimax loss) and total variation (TV) distance (related to hinge loss). The reason is that convergence of $P _ { G }$ to $P _ { \mathbf { X } }$ in JS-divergence and TV distance also implies convergence in MMD and Wasserstein metric. Weak metrics are desirable as they provide more information on adjusting the model to fit the data distribution (Liu et al. (2017)). Nagarajan & Kolter (2017) proved that the GAN trained using the minimax loss and gradient updates on model parameters is locally exponentially stable near equilibrium, while the GAN using Wasserstein loss is not. In Appendix A, we demonstrate that the MMD-GAN trained by gradient descent is locally exponentially stable near equilibrium.
+
+# 3 REPULSIVE LOSS FUNCTION
+
+In this section, we interpret the training of MMD-GAN (using $L _ { D } ^ { \mathrm { a t t } }$ and $L _ { G } ^ { \mathrm { m m d } }$ ) as a combination of attraction and repulsion processes, and propose a novel repulsive loss function for the discriminator by rearranging the components in $L _ { D } ^ { \mathrm { a t t } }$ .
+
+$$
+\begin{array} { r l r l r l r l } & { \underset { \stackrel { \mathrm { \tiny ~ \tiny ~ \overrightarrow ~ { \textrm { \tiny ~ C } ~ } } } { \tiny \textrm { \tiny { \textrm { \tiny ~ C } ~ } } } } { \neg } } & & { \bar { \mathbb { C } } } & & { \bar { \mathbb { C } } } & & { \bar { \mathbb { C } } } \\ & { \longrightarrow \bar { \mathbb { C } } } & & { \bar { \mathbb { C } } } & & { \underline { { \mathbb { C } } } } & & { \underset { \stackrel { \mathrm { \tiny ~ \overrightarrow ~ { \wedge ~ } } } { \tiny \textrm { \textrm { \tiny ~ C } } } } { \neg } } & & { \bar { \mathbb { C } } } \\ & { \underset { \stackrel { \mathrm { \tiny ~ \geq ~ \alpha ~ } } { \geq } } { \geq } \left. \begin{array} { l l l l l l l l l } { \bar { \Gamma } } & { \bar { \Gamma } } & & { } & & { \bar { \mathbb { C } } } & & { } & & { \bar { \mathbb { C } } } & { \bar { \mathbb { C } } } \\ { \bar { \Gamma } } & { \bar { \Gamma } } & { } & & { } & & { \bar { \mathbb { C } } } & { \underset { \stackrel { \mathrm { \tiny ~ \geq ~ \alpha ~ } } { \geq } } { \overset { \mathrm { \tiny ~ \wedge ~ } } { \geq } } } & & { \bar { \mathbb { C } } } & { \bar { \mathbb { C } } } \end{array} \right. } \\ & { \underset { \stackrel { \mathrm { \tiny ~ \geq ~ \alpha ~ } } { \to } } { \overset { \mathbb { C } } } { \geq } } & { \underline { { \mathbb { C } } } } & { \qquad \quad \mathrm { ( \mathbb { C } ) } } & { \underset { \bar { \mathbb { C } } } { \geq } } & { \qquad \bar { \mathbb { C } } } & { \qquad \bar { \mathbb { C } } } & { \bar { \mathbb { C } } } \\ & { \underset { \bar { \mathcal { F } } } { \geq } } & { \qquad \mathrm { ( \mathbb { C } ) } } & { \quad \mathrm { ( \mathbb { C } ) } } & & { } & { \qquad \bar { \mathbb { C } } } & { \qquad \bar { \mathbb { C } } } \end{array}
+$$
+
+Figure 1: Illustration of the gradient directions of each loss on the real sample scores $\{ D ( { \pmb x } ) \}$ (“r” nodes) and generated sample scores $\{ D ( \pmb { y } ) \}$ (“g” nodes). The blue arrows stand for attraction and the orange arrows for repulsion. When $L _ { G } ^ { \mathrm { m m d } }$ is paired with $L _ { D } ^ { \mathrm { a t t } }$ , the gradient directions of $L _ { G } ^ { \mathrm { m m d } }$ on $\{ D ( \pmb { y } ) \}$ can be obtained by reversing the arrows in (a), thus are omitted.
+
+First, consider a linear discriminant analysis (LDA) model as the discriminator. The task is to find a projection $\textbf { \em w }$ to maximize the between-group variance $\left\| \pmb { w } ^ { T } \pmb { \mu } _ { x } - \pmb { w } ^ { T } \pmb { \mu } _ { y } \right\|$ and minimize the withingroup variance $\pmb { w } ^ { T } ( \pmb { \Sigma } _ { x } + \pmb { \Sigma } _ { y } ) \pmb { w }$ , where $\pmb { \mu }$ and $\pmb { \Sigma }$ are group mean and covariance.
+
+In MMD-GAN, the neural-network discriminator works in a similar way as LDA. By minimizing $L _ { D } ^ { \mathrm { a t t } }$ , the discriminator $D$ tackles two tasks: 1) $D$ reduces $\mathbb { E } _ { P _ { \mathbf { X } } , P _ { G } } [ k _ { D } ( \pmb { x } , \pmb { y } ) ]$ , i.e., causes the two groups $\{ D ( { \pmb x } ) \}$ and $\{ D ( \pmb { y } ) \}$ to repel each other (see Fig. 1a orange arrows), or maximize between– group variance; and 2) $D$ increases $\mathbb { E } _ { P _ { \mathbf { X } } } [ k _ { D } ( { \pmb x } , { \pmb x } ^ { \prime } ) ]$ and $\mathbb { E } _ { P _ { G } } [ k ( { \pmb y } , { \pmb y } ^ { \prime } ) ]$ , i.e. contracts $\{ D ( { \pmb x } ) \}$ and $\bar { \{ D ( \pmb { y } ) } \}$ within each group (see Fig. 1a blue arrows), or minimize the within-group variance. We refer to loss functions that contract real data scores as attractive losses.
+
+We argue that the attractive loss $L _ { D } ^ { \mathrm { a t t } }$ (Eq. 3) has two issues that may slow down the GAN training:
+
+1. The discriminator $D$ may focus more on the similarities among real samples (in order to contract $\{ D ( { \pmb x } ) \} )$ than the fine details that separate them. Initially, $G$ produces low-quality samples and it may be adequate for $D$ to learn the common features of $\{ { \pmb x } \}$ in order to distinguish between $\{ x \}$ and $\{ y \}$ . Only when $\{ D ( \pmb { y } ) \}$ is sufficiently close to $\{ D ( { \pmb x } ) \}$ will $D$ learn the fine details of $\{ x \}$ to be able to separate $\{ D ( { \pmb x } ) \}$ from $\{ D ( \pmb { y } ) \}$ . Consequently, $D$ may leave out some fine details in real samples, thus $G$ has no access to them during training.
+
+2. As shown in Fig. 1a, the gradients on $D ( \pmb { y } )$ from the attraction (blue arrows) and repulsion (orange arrows) terms in $L _ { D } ^ { \mathrm { a t t } }$ (and thus $L _ { G } ^ { \mathrm { m m d } }$ ) may have opposite directions during training. Their summation may be small in magnitude even when $D ( \pmb { y } )$ is far away from $D ( \pmb { x } )$ , which may cause $G$ to stagnate locally.
+
+Therefore, we propose a repulsive loss for $D$ to encourage repulsion of the real data scores $\{ D ( { \pmb x } ) \}$
+
+$$
+L _ { D } ^ { \mathrm { r e p } } = \mathbb { E } _ { P _ { \mathbf { X } } } [ k _ { D } ( { \pmb x } , { \pmb x } ^ { \prime } ) ] - \mathbb { E } _ { P _ { G } } [ k _ { D } ( { \pmb y } , { \pmb y } ^ { \prime } ) ]
+$$
+
+The generator $G$ uses the same MMD loss $L _ { G } ^ { \mathrm { m m d } }$ as before (see Eq. 2). Thus, the adversary lies in the fact that $D$ contracts $\{ D ( \pmb { y } ) \}$ via maximizing $\mathbb { E } _ { P _ { G } } [ k _ { D } ( { \pmb y } , { \pmb y } ^ { \prime } ) ]$ (see Fig. 1b) while $G$ expands $\{ D ( \pmb { y } ) \}$ (see Fig. 1c). Additionally, $D$ also learns to separate the real data by minimizing $\mathbb { E } _ { P _ { \mathbf { X } } } [ k _ { D } ( { \pmb x } , { \pmb x } ^ { \prime } ) ]$ , which actively explores the fine details in real samples and may result in more meaningful gradients for $G$ . Note that in Eq. 4, $D$ does not explicitly push the average score of $\{ D ( \pmb { y } ) \}$ away from that of $\{ D ( { \pmb x } ) \}$ because it may have no effect on the pair-wise sample distances. But $G$ aims to match the average scores of both groups. Thus, we believe, compared to the model using $L _ { G } ^ { \mathrm { m m d } }$ and $L _ { D } ^ { \mathrm { a t t } }$ , our model of $L _ { G } ^ { \mathrm { m m d } }$ and $L _ { D } ^ { \mathrm { i e p } }$ is less likely to yield opposite gradients when $\{ D ( \pmb { y } ) \}$ and $\{ D ( { \pmb x } ) \}$ are distinct (see Fig. 1c). In Appendix A, we demonstrate that GAN trained using gradient descent and the repulsive MMD loss $( \dot { L } _ { D } ^ { \mathrm { r e p } } , L _ { G } ^ { \mathrm { m m d } } )$ is locally exponentially stable near equilibrium.
+
+At last, we identify a general form of loss function for the discriminator $D$ :
+
+$$
+L _ { D , \lambda } = \lambda \mathbb { E } _ { P _ { \mathbf { X } } } [ k _ { D } ( \pmb { x } , \pmb { x } ^ { \prime } ) ] - ( \lambda - 1 ) \mathbb { E } _ { P _ { \mathbf { X } } , P _ { G } } [ k _ { D } ( \pmb { x } , \pmb { y } ) ] - \mathbb { E } _ { P _ { G } } [ k _ { D } ( \pmb { y } , \pmb { y } ^ { \prime } ) ]
+$$
+
+![](images/2af5c2c1ae392c7326b2127e5634938432888d2a89009c214b09d4d785bdb7c4.jpg)  
+Figure 2: (a) Gaussian kernels $\{ k _ { \sigma _ { i } } ^ { \mathrm { r b f } } ( \pmb { a } , \pmb { b } ) \}$ and their mean as a function of $e = \| a - b \|$ , where $\sigma _ { i } \in \{ 1 , \sqrt { 2 } , 2 , 2 \sqrt { 2 } , 4 \}$ were used in our experiments; (b) derivatives of $\{ k _ { \sigma _ { i } } ^ { \mathrm { r b f } } ( \pmb { a } , \pmb { b } ) \}$ in (a); (c) rational quadratic kernel $\{ k _ { \alpha _ { i } } ^ { \mathrm { r q } } ( { \pmb a } , { \pmb b } ) \}$ and their mean, where $\alpha _ { i } \in \{ 0 . 2 , 0 . 5 , 1 , 2 , 5 \}$ ; (d) derivatives of $\{ k _ { \alpha _ { i } } ^ { \mathrm { r q } } ( { \bar { \mathbf { a } } } , { \mathbf { } } b ) \}$ in (c).
+
+where $\lambda$ is a hyper-parameter2. When $\lambda < 0$ , the discriminator loss $L _ { D , \lambda }$ is attractive, with $\lambda =$ $- 1$ corresponding to the original MMD loss $L _ { D } ^ { \mathrm { a t t } }$ in Eq. 3; when $\lambda > 0$ , $L _ { D , \lambda }$ is repulsive and $\lambda = 1$ corresponds to $L _ { D } ^ { \mathrm { r e p } }$ in Eq. 4. It is interesting that when $\lambda > 1$ , the discriminator explicitly contracts $\{ D ( { \pmb x } ) \}$ and $\{ \bar { D } ( \pmb y ) \}$ via maximizing $\mathbb { E } _ { P _ { \mathbf { X } } , P _ { G } } [ k _ { D } ( \pmb { x } , \pmb { y } ) ]$ , which may work as a penalty that prevents the pairwise distances of $\{ D ( { \pmb x } ) \bar  \}$ from increasing too fast. Note that $L _ { D , \lambda }$ has the same computational cost as $L _ { D } ^ { \mathrm { a t t } }$ (Eq. 3) as we only rearranged the terms in $L _ { D } ^ { \mathrm { a t t } }$ .
+
+# 4 REGULARIZATION ON MMD AND DISCRIMINATOR
+
+In this section, we propose two approaches to stabilize the training of MMD-GAN: 1) a bounded kernel to avoid the saturation issue caused by an over-confident discriminator; and 2) a generalized power iteration method to estimate the spectral norm of a convolutional kernel, which was used in spectral normalization on the discriminator in all experiments in this study unless specified otherwise.
+
+# 4.1 KERNEL IN MMD
+
+For MMD-GAN, the following two kernels have been used:
+
+• Gaussian radial basis function (RBF), or Gaussian kernel (Li et al. (2017a)), $k _ { \sigma } ^ { \mathrm { r b f } } ( a , b ) ~ =$ $\exp ( - \frac { 1 } { 2 \sigma ^ { 2 } } \left\| \pmb { a } - \pmb { b } \right\| ^ { 2 } )$ where $\sigma > 0$ is the kernel scale or bandwidth.   
+• Rational quadratic kernel (Binkowski et al. (2018)), ´ $\begin{array} { r l r } & { } & { k _ { \alpha } ^ { \mathrm { r q } } ( { \pmb a } , { \pmb b } ) = ( 1 + \frac { 1 } { 2 \alpha } \left\| { \pmb a } - { \pmb b } \right\| ^ { 2 } ) ^ { - \alpha } } \end{array}$ , where the kernel scale $\alpha > 0$ corresponds to a mixture of Gaussian kernels with a $\mathrm { G a m m a } ( \alpha , 1 )$ prior on the inverse kernel scales $\sigma ^ { - 1 }$ .
+
+It is interesting that both studies used a linear combination of kernels with five different kernel scales, i.e., krbf = P5i=1 and = P5i=1 k , where $\sigma _ { i } \in \{ 1 , 2 , 4 , 8 , 1 6 \}$ , $\alpha _ { i } \in \{ 0 . 2 , 0 . 5 , 1 , 2 , 5 \}$ (see Fig. 2a and 2c for illustration). We suspect the reason is that a single kernel $k ( \boldsymbol { a } , \boldsymbol { b } )$ is saturated when the distance $\| { \pmb a } - { \pmb b } \|$ is either too large or too small compared to the kernel scale (see Fig. 2b and 2d), which may cause diminishing gradients during training. Both Li et al. (2017a) and Binkowski ´ et al. (2018) applied penalties on the discriminator parameters but not to the MMD loss itself. Thus the saturation issue may still exist. Using a linear combination of kernels with different kernel scales may alleviate this issue but not eradicate it.
+
+Inspired by the hinge loss (see Appendix B.1), we propose a bounded RBF (RBF-B) kernel for the discriminator. The idea is to prevent $D$ from pushing $\{ D ( { \pmb x } ) \}$ too far away from $\{ D ( \pmb { y } ) \}$ and causing saturation. For $L _ { D } ^ { \mathrm { a t t } }$ in Eq. 3, the RBF-B kernel is:
+
+$$
+\begin{array} { r } { k _ { \sigma } ^ { \mathrm { { t h f f b } } } ( a , b ) = \left\{ \begin{array} { l l } { \exp \bigl ( - \frac { 1 } { 2 \sigma ^ { 2 } } \operatorname* { m a x } \bigl ( \left\| a - b \right\| ^ { 2 } , b _ { l } \bigr ) \bigr ) } & { \mathrm { i f ~ } a , b \in \left\{ D ( x ) \right\} \mathrm { ~ o r ~ } a , b \in \left\{ D ( y ) \right\} } \\ { \exp \bigl ( - \frac { 1 } { 2 \sigma ^ { 2 } } \operatorname* { m i n } \bigl ( \left\| a - b \right\| ^ { 2 } , b _ { u } \bigr ) \bigr ) } & { \mathrm { i f ~ } a \in \left\{ D ( x ) \right\} \mathrm { ~ a n d ~ } b \in \left\{ D ( y ) \right\} } \end{array} \right. } \end{array}
+$$
+
+For $L _ { D } ^ { \mathrm { r e p } }$ in Eq. 4, the RBF-B kernel is:
+
+$$
+k _ { \sigma } ^ { \mathrm { r b f - b } } ( a , b ) = \left\{ \begin{array} { l l } { \exp \bigl ( - \frac { 1 } { 2 \sigma ^ { 2 } } \operatorname* { m a x } \bigl ( \left\| a - b \right\| ^ { 2 } , b _ { l } \bigr ) \bigr ) } & { \mathrm { i f } a , b \in \{ D ( y ) \} } \\ { \exp \bigl ( - \frac { 1 } { 2 \sigma ^ { 2 } } \operatorname* { m i n } \bigl ( \left\| a - b \right\| ^ { 2 } , b _ { u } \bigr ) \bigr ) } & { \mathrm { i f } a , b \in \{ D ( x ) \} } \end{array} \right.
+$$
+
+where $b _ { l }$ and $b _ { u }$ are the lower and upper bounds. As such, a single kernel is sufficient and we set $\sigma = 1$ , $b _ { l } = 0 . 2 5$ and $b _ { u } = 4$ in all experiments for simplicity and leave their tuning for future work. It should be noted that, like the case of hinge loss, the RBF-B kernel is used only for the discriminator to prevent it from being over-confident. The generator is always trained using the original RBF kernel, thus we retain the interpretation of MMD loss $L _ { G } ^ { \mathrm { m m d } }$ as a metric.
+
+RBF-B kernel is among many methods to address the saturation issue and stabilize MMD-GAN training. We found random sampling kernel scale, instance noise (Sønderby et al. (2017)) and label smoothing (Szegedy et al. (2016); Salimans et al. (2016)) may also improve the model performance and stability. However, the computational cost of RBF-B kernel is relatively low.
+
+# 4.2 SPECTRAL NORMALIZATION IN DISCRIMINATOR
+
+Without any Lipschitz constraints, the discriminator $D$ may simply increase the magnitude of its outputs to minimize the discriminator loss, causing unstable training3. Spectral normalization divides the weight matrix of each layer by its spectral norm, which imposes an upper bound on the magnitudes of outputs and gradients at each layer of $D$ (Miyato et al. (2018)). However, to estimate the spectral norm of a convolution kernel, Miyato et al. (2018) reshaped the kernel into a matrix. We propose a generalized power iteration method to directly estimate the spectral norm of a convolution kernel (see Appendix C for details) and applied spectral normalization to the discriminator in all experiments. In Appendix D.1, we explore using gradient penalty to impose the Lipschitz constraint (Gulrajani et al. (2017); Binkowski et al. (2018); Arbel et al. (2018)) for the proposed repulsive loss. ´
+
+# 5 EXPERIMENTS
+
+In this section, we empirically evaluate the proposed 1) repulsive loss $L _ { D } ^ { \mathrm { r e p } }$ (Eq. 4) on unsupervised training of GAN for image generation tasks; and 2) RBF-B kernel to stabilize MMD-GAN training. The generalized power iteration method is evaluated in Appendix C.3. To show the efficacy of $L _ { D } ^ { \mathrm { r e p } }$ , we compared the loss functions $( L _ { D } ^ { \mathrm { r e p } } , L _ { G } ^ { \mathrm { m m d } } )$ using Gaussian kernel (MMD-rep) with $( L _ { D } ^ { \mathrm { a t t } } , L _ { G } ^ { \mathrm { m m d } } )$ using Gaussian kernel (MMD-rbf) (Li et al. (2017a)) and rational quadratic kernel (MMD-rq) (Binkowski et al. (2018)), as well as non-saturating loss (Goodfellow et al. (2014)) and ´ e loss (Tran et al. (2017)). To show the efficacy of RBF-B kernel, we applied it to both , resulting in two methods MMD-rbf-b and MMD-rep-b. The Wasserstein loss was excl $L _ { D } ^ { \mathrm { a t t } }$ andd for $L _ { D } ^ { \mathrm { r e p } }$
+
+# 5.1 EXPERIMENT SETUP
+
+Dataset: The loss functions were evaluated on four datasets: 1) CIFAR-10 $5 0 K$ images, $3 2 \times 3 2$ pixels) (Krizhevsky & Hinton (2009)); 2) STL-10 $1 0 0 K$ images, $4 8 \times 4 8$ pixels) (Coates et al. (2011)); 3) CelebA (about $2 0 3 K$ images, $6 4 \times 6 4$ pixels) (Liu et al. (2015)); and 4) LSUN bedrooms (around 3 million images, $6 4 \times 6 4$ pixels) (Yu et al. (2015)). The images were scaled to range $[ - 1 , 1 ]$ to avoid numeric issues.
+
+Network architecture: The DCGAN (Radford et al. (2016)) architecture was used with hyperparameters from Miyato et al. (2018) (see Appendix B.2 for details). In all experiments, batch normalization (BN) (Ioffe & Szegedy (2015)) was used in the generator, and spectral normalization with the generalized power iteration (see Appendix C) in the discriminator. For MMD related losses, the dimension of discriminator output layer was set to 16; for non-saturating loss and hinge loss, it was 1. In Appendix D.2, we investigate the impact of discriminator output dimension on the performance of repulsive loss.
+
+Table 1: Inception score (IS), Frechet Inception distance (FID) and multi-scale structural similarity ´ (MS-SSIM) on image generation tasks using different loss functions   
+
+<table><tr><td rowspan="2">Methods1</td><td colspan="2">CIFAR-10</td><td colspan="2">STL-10</td><td colspan="2">CelebA²</td><td colspan="2">LSUN-bedrom²</td></tr><tr><td>IS</td><td>FID</td><td>IS</td><td>FID</td><td>FID</td><td>MS-SSIM</td><td>FID</td><td>MS-SSIM</td></tr><tr><td>Real data</td><td>11.31</td><td>2.09</td><td>26.37</td><td>2.10</td><td>1.09</td><td>0.2678</td><td>1.24</td><td>0.0915</td></tr><tr><td>Non-saturating</td><td>7.39</td><td>23.23</td><td>8.25</td><td>48.53</td><td>10.64</td><td>0.2895</td><td>23.66</td><td>0.1027</td></tr><tr><td>Hinge</td><td>7.33</td><td>23.46</td><td>8.24</td><td>49.44</td><td>8.60</td><td>0.2894</td><td>16.73</td><td>0.0946</td></tr><tr><td>MMD-rbf3</td><td>7.05</td><td>28.38</td><td>8.13</td><td>57.52</td><td>13.03</td><td>0.2937</td><td></td><td></td></tr><tr><td>MMD-rq³</td><td>7.22</td><td>27.00</td><td>8.11</td><td>54.05</td><td>12.74</td><td>0.2935</td><td></td><td></td></tr><tr><td>MMD-rbf-b</td><td>7.18</td><td>25.25</td><td>8.07</td><td>51.86</td><td>10.09</td><td>0.3090</td><td>32.29</td><td>0.1001</td></tr><tr><td>MMD-rep</td><td>7.99</td><td>16.65</td><td>9.36</td><td>36.67</td><td>7.20</td><td>0.2761</td><td>16.91</td><td>0.0901</td></tr><tr><td>MMD-rep-b</td><td>8.29</td><td>16.21</td><td>9.34</td><td>37.63</td><td>6.79</td><td>0.2659</td><td>12.52</td><td>0.0908</td></tr></table>
+
+1 The models here differ only by the loss functions and dimension of discriminator outputs. See Section 5.1. 2 For CelebA and LSUN-bedroom, IS is not meaningful (Binkowski et al. (2018)) and thus omitted. ´ 3 On LSUN-bedroom, MMD-rbf and MMD-rq did not achieve reasonable results and thus are omitted.
+
+Hyper-parameters: We used Adam optimizer (Kingma & Ba (2015)) with momentum parameters $\beta _ { 1 } = 0 . 5$ , $\beta _ { 2 } = 0 . 9 9 9$ ; two-timescale update rule (TTUR) (Heusel et al. (2017)) with two learning rates $\left( \rho _ { D } , \rho _ { G } \right)$ chosen from $\{ 1 e \mathrm { - } 4 , 2 e \mathrm { - } 4 , 5 e \mathrm { - } 4 , 1 e \mathrm { - } 3 \}$ (16 combinations in total); and batch size 64. Fine-tuning on learning rates may improve the model performance, but constant learning rates were used for simplicity. All models were trained for $1 0 0 K$ iterations on CIFAR-10, STL-10, CelebA and LSUN bedroom datasets, with $n _ { d i s } = 1$ , i.e., one discriminator update per generator update4.√ For MMD-rbf, the kernel scales $\sigma _ { i } \in \{ 1 , \sqrt { 2 } , 2 , 2 \sqrt { 2 } , 4 \}$ were used due to a better performance than the original values used in Li et al. (2017a). For MMD-rq, $\alpha _ { i } \in \{ 0 . 2 , 0 . 5 , 1 , 2 , 5 \}$ . For MMD-rbf- $\mathbf { b }$ , MMD-rep, MMD-rep-b, a single Gaussian kernel with $\sigma = 1$ was used.
+
+Evaluation metrics: Inception score (IS) (Salimans et al. (2016)), Frechet Inception distance (FID) ´ (Heusel et al. (2017)) and multi-scale structural similarity (MS-SSIM) (Wang et al. (2003)) were used for quantitative evaluation. Both IS and FID are calculated using a pre-trained Inception model (Szegedy et al. (2016)). Higher IS and lower FID scores indicate better image quality. MS-SSIM calculates the pair-wise image similarity and is used to detect mode collapses among images of the same class (Odena et al. (2017)). Lower MS-SSIM values indicate perceptually more diverse images. For each model, $5 0 K$ randomly generated samples and $5 0 K$ real samples were used to calculate IS, FID and MS-SSIM.
+
+# 5.2 QUANTITATIVE ANALYSIS
+
+Table 1 shows the Inception score, FID and MS-SSIM of applying different loss functions on the benchmark datasets with the optimal learning rate combinations tested experimentally. Note that the same training setup (i.e., $\mathrm { D C G A N } + \mathrm { B N } + \mathrm { S N } + \mathrm { T T U R } )$ was applied for each loss function. We observed that: 1) MMD-rep and MMD-rep-b performed significantly better than MMD-rbf and MMD-rbf-b respectively, showing the proposed repulsive loss $\bar { L } _ { D } ^ { \mathrm { r e p } }$ (Eq. 4) greatly improved over the attractive loss $L _ { D } ^ { \mathrm { a t t } }$ (Eq. 3); 2) Using a single kernel, MMD-rbf-b performed better than MMD-rbf and MMD-rq which used a linear combination of five kernels, indicating that the kernel saturation may be an issue that slows down MMD-GAN training; 3) MMD-rep-b performed comparable or better than MMD-rep on benchmark datasets where we found the RBF-B kernel managed to stabilize MMD-GAN training using repulsive loss. 4) MMD-rep and MMD-rep-b performed significantly better than the non-saturating and hinge losses, showing the efficacy of the proposed repulsive loss.
+
+Additionally, we trained MMD-GAN using the general loss (Eq. 2) for generator on the CIFAR-10 dataset. Fig. 3 show $L _ { D , \lambda }$ (Eq. 5) for d influence of criminator and on the perfor $L _ { G } ^ { \mathrm { m m d } }$ $\lambda$
+
+![](images/a8da475d50d62f2acf9bbdb01f8be57f2bdaebbd17f1d87a8dbaa5fe9db7d596.jpg)  
+Figure 3: FID scores of MMD-GAN using (a) RBF kernel and (b) RBF-B kernel in $L _ { D , \lambda }$ on CIFAR10 dataset for 16 learning rate combinations. Each color bar represents the FID score using a learning rate combination $\left( \rho _ { D } , \rho _ { G } \right)$ , in the order of $( 1 e \mathrm { - } 4 , 1 e \mathrm { - } 4 )$ , $( 1 e \mathrm { - } 4 , 2 e \mathrm { - } 4 ) , \ldots , ( 1 e \mathrm { - } 3 , 1 e \mathrm { - } 3 )$ . The discriminator was trained (Eq. 2). We use the $L _ { D , \lambda }$ (Eq. 5) with o indicate that $\lambda \in \{ - 1 , - 0 . 5 , 0 , 0 . 5 , 1 , 2 \}$ , and generator using duced poor results. $L _ { G } ^ { \mathrm { m m d } }$ $\mathrm { F I D > 3 0 }$
+
+of MMD-GAN with RBF and RBF-B kernel5. Note that when $\lambda = - 1$ , the models are essentially MMD-rbf (with a single Gaussian kernel) and MMD-rbf-b when RBF and RBF-B kernel are used respectively. We observed that: 1) the model performed well using repulsive loss (i.e., $\lambda \geq 0$ ), with $\lambda = 0 . 5 , 1$ slightly better than $\lambda = - 0 . 5 , 0 , 2 ; 2$ ) the MMD-rbf model can be significantly improved by simply increasing $\lambda$ from $- 1$ to $- 0 . 5$ , which reduces the attraction of discriminator on real sample scores; 3) larger $\lambda$ may lead to more diverged models, possibly because the discriminator focuses more on expanding the real sample scores over adversarial learning; note when $\lambda \gg 1$ , the model would simply learn to expand all real sample scores and pull the generated sample scores to real samples’, which is a divergent process; 4) the RBF-B kernel managed to stabilize MMD-rep for most diverged cases but may occasionally cause the FID score to rise up.
+
+The proposed methods were further evaluated in Appendix A, C and D. In Appendix A.2, we used a simulation study to show the local stability of MMD-rep trained by gradient descent, while its global stability is not guaranteed as bad initialization may lead to trivial solutions. The problem may be alleviated by adjusting the learning rate for generator. In Appendix C.3, we showed the proposed generalized power iteration (Section 4.2) imposes a stronger Lipschitz constraint than the method in Miyato et al. (2018), and benefited MMD-GAN training using the repulsive loss. Moreover, the RBF-B kernel managed to stabilize the MMD-GAN training for various configurations of the spectral normalization method. In Appendix D.1, we showed the gradient penalty can also be used with the repulsive loss. In Appendix D.2, we showed that it was better to use more than one neuron at the discriminator output layer for the repulsive loss.
+
+# 5.3 QUALITATIVE ANALYSIS
+
+The discriminator outputs may be interpreted as a learned representation of the input samples. Fig. 4 visualizes the discriminator outputs learned by the MMD-rbf and proposed MMD-rep methods on CIFAR-10 dataset using t-SNE (van der Maaten (2014)). MMD-rbf ignored the class structure in data (see Fig. 4a) while MMD-rep learned to concentrate the data from the same class and separate different classes to some extent (Fig. 4b). This is because the discriminator $D$ has to actively learn the data structure in order to expands the real sample scores $\{ D ( { \pmb x } ) \}$ . Thus, we speculate that techniques reinforcing the learning of cluster structures in data may further improve the training of MMD-GAN.
+
+![](images/d55272a4d87aa3b04539423276fe1a44e5e1b7b25e3fe3d572e89f7220eccf05.jpg)  
+Figure 4: t-SNE visualization of discriminator outputs $\{ D ( { \pmb x } ) \}$ learned by (a) MMD-rbf and (b) MMD-rep for 2560 real samples from the CIFAR-10 dataset, colored by their class labels.
+
+In addition, the performance gain of proposed repulsive loss (Eq. 4) over the attractive loss (Eq. 3) comes at no additional computational cost. In fact, by using a single kernel rather than a linear combination of kernels, MMD-rep and MMD-rep-b are simpler than MMD-rbf and MMD-rq. Besides, given a typically small batch size and a small number of discriminator output neurons (64 and 16 in our experiments), the cost of MMD over the non-saturating and hinge loss is marginal compared to the convolution operations.
+
+In Appendix D.3, we provide some random samples generated by the methods in our study.
+
+# 6 DISCUSSION
+
+This study extends the previous work on MMD-GAN (Li et al. (2017a)) with two contributions. First, we interpreted the optimization of MMD loss as a combination of attraction and repulsion processes, and proposed a repulsive loss for the discriminator that actively learns the difference among real data. Second, we proposed a bounded Gaussian RBF (RBF-B) kernel to address the saturation issue. Empirically, we observed that the repulsive loss may result in unstable training, due to factors including initialization (Appendix A.2), learning rate (Fig. 3b) and Lipschitz constraints on the discriminator (Appendix C.3). The RBF-B kernel managed to stabilize the MMD-GAN training in many cases. Tuning the hyper-parameters in RBF-B kernel or using other regularization methods may further improve our results.
+
+The theoretical advantages of MMD-GAN require the discriminator to be injective. The proposed repulsive loss (Eq. 4) attempts to realize this by explicitly maximizing the pair-wise distances among the real samples. Li et al. (2017a) achieved the injection property by using the discriminator as the encoder and an auxiliary network as the decoder to reconstruct the real and generated samples, which is more computationally extensive than our proposed approach. On the other hand, Binkowski et al. ´ (2018); Arbel et al. (2018) imposed a Lipschitz constraint on the discriminator in MMD-GAN via gradient penalty, which may not necessarily promote an injective discriminator.
+
+The idea of repulsion on real sample scores is in line with existing studies. It has been widely accepted that the quality of generated samples can be significantly improved by integrating labels (Odena et al. (2017); Miyato & Koyama (2018); Zhou et al. (2018)) or even pseudo-labels generated by $\mathbf { k }$ -means method (Grinblat et al. (2017)) in the training of discriminator. The reason may be that the labels help concentrate the data from the same class and separate those from different classes. Using a pre-trained classifier may also help produce vivid image samples (Huang et al. (2017)) as the learned representations of the real samples in the hidden layers of the classifier tend to be well separated/organized and may produce more meaningful gradients to the generator.
+
+At last, we note that the proposed repulsive loss is orthogonal to the GAN studies on designing network structures and training procedures, and thus may be combined with a variety of novel techniques. For example, the ResNet architecture (He et al. (2016)) has been reported to outperform the plain DCGAN used in our experiments on image generation tasks (Miyato et al. (2018); Gulrajani et al. (2017)) and self-attention module may further improve the results (Zhang et al. (2018)). On the other hand, Karras et al. (2018) proposed to progressively grows the size of both discriminator and generator and achieved the state-of-the-art performance on unsupervised training of GANs on the CIFAR-10 dataset. Future work may explore these directions.
+
+# ACKNOWLEDGMENTS
+
+Wei Wang is fully supported by the Ph.D. scholarships of The University of Melbourne. This work is partially funded by Australian Research Council grant DP150103512 and undertaken using the LIEF HPC-GPGPU Facility hosted at the University of Melbourne. The Facility was established with the assistance of LIEF Grant LE170100200.
+
+# REFERENCES
+
+Michael Arbel, Dougal J. Sutherland, Mikołaj Binkowski, and Arthur Gretton. On gradient regular- ´ izers for MMD GANs. In NIPS, 2018.   
+Martin Arjovsky, Soumith Chintala, and Leon Bottou. Wasserstein generative adversarial networks. ´ In ICML, volume 70 of PMLR, pp. 214–223, 2017.   
+Mikołaj Binkowski, Dougal J. Sutherland, Michael Arbel, and Arthur Gretton. Demystifying MMD ´ GANs. In ICLR, 2018.   
+Adam Coates, Andrew Ng, and Honglak Lee. An analysis of single-layer networks in unsupervised feature learning. In AISTATS, volume 15 of PMLR, pp. 215–223, 2011.   
+Vincent Dumoulin and Francesco Visin. A guide to convolution arithmetic for deep learning, 2016. arxiv:1603.07285.   
+Gintare Karolina Dziugaite, Daniel M. Roy, and Zoubin Ghahramani. Training generative neural networks via maximum mean discrepancy optimization. In UAI, pp. 258–267, 2015.   
+Marc G. Genton. Classes of kernels for machine learning: A statistics perspective. J. Mach. Learn. Res., 2:299–312, 2002.   
+Ian J. Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In NIPS, pp. 2672–2680, 2014.   
+Arthur Gretton, Karsten M. Borgwardt, Malte J. Rasch, Bernhard Scholkopf, and Alexander Smola.¨ A kernel two-sample test. J. Mach. Learn. Res., 13:723–773, 2012.   
+Guillermo L. Grinblat, Lucas C. Uzal, and Pablo M. Granitto. Class-splitting generative adversarial networks, 2017. arXiv:1709.07359.   
+Ishaan Gulrajani, Faruk Ahmed, Martin Arjovsky, Vincent Dumoulin, and Aaron Courville. Improved training of Wasserstein GANs. In NIPS, pp. 5767–5777, 2017.   
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In CVPR, pp. 770–778, 2016.   
+Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter. GANs trained by a two time-scale update rule converge to a local Nash equilibrium. In NIPS, pp. 6626–6637, 2017.   
+Jonathan Ho and Stefano Ermon. Generative adversarial imitation learning. In NIPS, pp. 4565–4573, 2016.   
+Xun Huang, Yixuan Li, Omid Poursaeed, John E. Hopcroft, and Serge J. Belongie. Stacked generative adversarial networks. CVPR, pp. 1866–1875, 2017.   
+Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. In ICML, pp. 448–456, 2015.   
+Tero Karras, Timo Aila, Samuli Laine, and Jaakko Lehtinen. Progressive growing of GANs for improved quality, stability, and variation. In ICLR, 2018.   
+Diederik P. Kingma and Jimmy Lei Ba. Adam: A method for stochastic optimization. In ICLR, 2015.   
+Alex Krizhevsky and Geoffrey Hinton. Learning multiple layers of features from tiny images. Master’s thesis, Department of Computer Science, University of Toronto, 2009.   
+Wei-Sheng Lai, Jia-Bin Huang, and Ming-Hsuan Yang. Semi-supervised learning for optical flow with generative adversarial networks. In NIPS, pp. 354–364, 2017.   
+Chun-Liang Li, Wei-Cheng Chang, Yu Cheng, Yiming Yang, and Barnabas Poczos. MMD GAN: Towards deeper understanding of moment matching network. In NIPS, pp. 2203–2213, 2017a.   
+Jiwei Li, Will Monroe, Tianlin Shi, Sebastien Jean, Alan Ritter, and Dan Jurafsky. Adversarial ´ learning for neural dialogue generation. In EMNLP, pp. 2157–2169, 2017b.   
+Yujia Li, Kevin Swersky, and Rich Zemel. Generative moment matching networks. In ICML, volume 37, pp. 1718–1727, 2015.   
+Shuang Liu, Olivier Bousquet, and Kamalika Chaudhuri. Approximation and convergence properties of generative adversarial learning. In NIPS, pp. 5545–5553, 2017.   
+Ziwei Liu, Ping Luo, Xiaogang Wang, and Xiaoou Tang. Deep learning face attributes in the wild. In ICCV, pp. 3730–3738, 2015.   
+Takeru Miyato and Masanori Koyama. cGANs with projection discriminator. In ICLR, 2018.   
+Takeru Miyato, Toshiki Kataoka, Masanori Koyama, and Yuichi Yoshida. Spectral normalization for generative adversarial networks. In ICLR, 2018.   
+Vaishnavh Nagarajan and J. Zico Kolter. Gradient descent GAN optimization is locally stable. In NIPS, pp. 5585–5595, 2017.   
+XuanLong Nguyen, Martin J. Wainwright, and Michael I. Jordan. On surrogate loss functions and f-divergences. Ann. Stat., 37(2):876–904, 04 2009.   
+Augustus Odena, Christopher Olah, and Jonathon Shlens. Conditional image synthesis with auxiliary classifier GANs. In ICML, pp. 2642–2651, 2017.   
+Alec Radford, Luke Metz, and Soumith Chintala. Unsupervised representation learning with deep convolutional generative adversarial networks. In ICLR, 2016.   
+Tim Salimans, Ian Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, and Xi Chen. Improved techniques for training GANs. In NIPS, pp. 2234–2242, 2016.   
+Hanie Sedghi, Vineet Gupta, and Philip M. Long. The singular values of convolutional layers. In ICLR, 2019.   
+Casper K. Sønderby, Jose Caballero, Lucas Theis, Wenzhe Shi, and Ferenc Huszar. Amortised map ´ inference for image super-resolution. In ICLR, 2017.   
+Christian Szegedy, Vincent Vanhoucke, Sergey Ioffe, Jonathon Shlens, and Zbigniew Wojna. Rethinking the inception architecture for computer vision. In CVPR, pp. 2818–2826, 2016.   
+Dustin Tran, Rajesh Ranganath, and David M. Blei. Hierarchical implicit models and likelihood-free variational inference, 2017. arXiv:1702.08896.   
+Yusuke Tsuzuku, Issei Sato, and Masashi Sugiyama. Lipschitz-margin training: scalable certification of perturbation invariance for deep neural networks. In NIPS, 2018.
+
+Thomas Unterthiner, Bernhard Nessler, Calvin Seward, Gunter Klambauer, Martin Heusel, Hubert ¨ Ramsauer, and Sepp Hochreiter. Coulomb GANs: Provably optimal Nash equilibria via potential fields. In ICLR, 2018.
+
+Laurens van der Maaten. Accelerating t-SNE using tree-based algorithms. J. Mach. Learn. Res., 15: 3221–3245, 2014.
+
+Aladin Virmaux and Kevin Scaman. Lipschitz regularity of deep neural networks: analysis and efficient estimation. In NIPS, 2018.
+
+Zhou Wang, Eero. P. Simoncelli, and Alan. C. Bovik. Multiscale structural similarity for image quality assessment. In Asilomar Conference on Signals, Systems & Computers, volume 2, pp. 1398–1402, 2003.
+
+Fisher Yu, Yinda Zhang, Shuran Song, Ari Seff, and Jianxiong Xiao. LSUN: Construction of a large-scale image dataset using deep learning with humans in the loop. 2015. arXiv:1506.03365.
+
+Han Zhang, Ian Goodfellow, Dimitris Metaxas, and Augustus Odena. Self-attention generative adversarial networks, 2018. arXiv:1805.08318.
+
+Zhiming Zhou, Han Cai, Shu Rong, Yuxuan Song, Kan Ren, Weinan Zhang, Jun Wang, and Yong Yu. Activation maximization generative adversarial nets. In ICLR, 2018.
+
+Jun-Yan. Zhu, Taesung Park, Phillip Isola, and Alexei A. Efros. Unpaired image-to-image translation using cycle-consistent adversarial networks. In ICCV, pp. 2242–2251, 2017.
+
+# Appendices
+
+# A STABILITY ANALYSIS OF MMD-GAN
+
+This section demonstrates that, under mild assumptions, MMD-GAN trained by gradient descent is locally exponentially stable at equilibrium. It is organized as follows. The main assumption and proposition are presented in Section A.1, followed by simulation study in Section A.2 and proof in Section A.3. We discuss the indications of assumptions on the discriminator of GAN in Section A.4.
+
+# A.1 MAIN PROPOSITION
+
+We consider GAN trained using the MMD loss $L _ { G } ^ { \mathrm { m m d } }$ for generator $G$ and either the attractive loss $L _ { D } ^ { \mathrm { a t t } }$ or repulsive loss $L _ { D } ^ { \mathrm { r e p } }$ for discriminator $D$ , listed below:
+
+$$
+\begin{array} { l } { { { \cal L } _ { G } ^ { \mathrm { m o d } } = M _ { k o D } ^ { 2 } ( P _ { { \bf X } } , P _ { G } ) = \mathbb { E } _ { P _ { { \bf X } } } [ k _ { D } ( { \bf x } , { \bf x } ^ { \prime } ) ] - 2 \mathbb { E } _ { P _ { { \bf X } } , P _ { G } } [ k _ { D } ( { \bf x } , { \bf y } ) ] + \mathbb { E } _ { P _ { G } } [ k _ { D } ( { \bf y } , { \bf y } ^ { \prime } ) ] } } \\ { { { \cal L } _ { D } ^ { \mathrm { a u t } } = - { \cal L } _ { G } ^ { \mathrm { m o d } } } } \\ { { { \cal L } _ { D } ^ { \mathrm { r e p } } = \mathbb { E } _ { P _ { { \bf X } } } [ k _ { D } ( { \bf x } , { \bf x } ^ { \prime } ) ] - \mathbb { E } _ { P _ { G } } [ k _ { D } ( { \bf y } , { \bf y } ^ { \prime } ) ] } } \end{array}
+$$
+
+where $k _ { D } ( { \bf a } , { \pmb b } ) = k ( D ( { \bf a } ) , D ( { \pmb b } ) )$ . Let $S ( P )$ be the support of distribution $P$ ; let $\pmb { \theta } _ { G } \in \Theta _ { G }$ , $\theta _ { D } \in \Theta _ { D }$ be the parameters of the generator $G$ and discriminator $D$ respectively. To prove that GANs trained using the minimax loss and gradient updates is locally stable at the equilibrium point $( \pmb { \theta } _ { D } ^ { * } , \pmb { \theta } _ { G } ^ { * } )$ , Nagarajan & Kolter (2017) made the following assumption:
+
+Assumption 1 (Nagarajan & Kolter (2017)). $P _ { \pmb { \theta } _ { G } ^ { * } } = P _ { \mathbf { X } } a n d \forall \pmb { x } \in S ( P _ { \mathbf { X } } ) , D _ { \pmb { \theta } _ { D } ^ { * } } ( \pmb { x } ) = 0 .$
+
+For loss functions like minimax and Wasserstein, $D _ { \pmb { \theta } _ { D } } ( \pmb { x } )$ may be interpreted as how plausible a sample is real. Thus at equilibrium, it may be reasonable to assume all real and generated samples are equally plausible. However, $D _ { \pmb { \theta } _ { D } ^ { * } } ( { \pmb x } ) \dot { = } 0$ also indicates that $D _ { \pmb { \theta } _ { D } ^ { * } }$ may have no discrimination power (see Appendix A.4 for discussion). For MMD loss in Eq. S1, $D _ { \pmb { \theta } _ { D } } ^ { - } ( \pmb { x } ) | _ { \pmb { x } \sim P }$ may be interpreted as a learned representation of the distribution $P$ . As long as two distributions $P$ and $Q$ match, $M _ { k \circ D _ { \pmb { \theta } _ { D } } } ^ { 2 } ( P , Q ) \ = 0$ . On the other hand, $D _ { \pmb \theta _ { D } } ( { \pmb x } ) = 0$ is a minima solution for $D$ but $D$ is trained to find local maxima. Thus in contrast to Assumption 1, we assume
+
+Assumption 2. For GANs using MMD loss in Eq. S1, and random initialization on parameters, at equilibrium, $D _ { \theta _ { D } ^ { * } } ( { \pmb x } )$ is injective on $S ( P _ { \mathbf { X } } ) \bigcup S ( P _ { \pmb { \theta } _ { G } ^ { * } } )$ .
+
+Assumption 2 indicates that $D _ { \pmb { \theta } _ { D } ^ { * } } ( \pmb { x } )$ is not constant almost everywhere. We use a simulation study in Section A.2 to show that $D _ { \theta _ { D } ^ { * } } ( { \pmb x } ) = 0$ does not hold in general for MMD loss. Based on Assumption 2, we propose the following proposition and prove it in Appendix A.3:
+
+Proposition 1. If there exists $\pmb { \theta } _ { G } ^ { * } \in \Theta _ { G }$ such that $P _ { \theta _ { G } ^ { * } } = P _ { \mathbf { X } }$ , then GANs with MMD loss in Eq. S1 has equilibria $( \theta _ { G } ^ { * } , \theta _ { D } )$ for any ${ \pmb \theta } _ { D } \in \Theta _ { D }$ . Moreover, the model trained using gradient descent methods is locally exponentially stable at $( \pmb { \theta } _ { G } ^ { * } , \pmb { \theta } _ { D } )$ for any $\pmb { \theta } _ { D } \in \Theta _ { D }$ .
+
+There may exist non-realizable cases where the mapping between $P z$ and $P _ { \mathbf { X } }$ cannot be represented by any generator $G _ { \pmb { \theta } _ { G } }$ with $\pmb { \theta } _ { G } \in \Theta _ { G }$ . In Section A.2, we use a simulation study to show that both the attractive MMD loss $L _ { D } ^ { \mathrm { a t t } }$ (Eq. S1b) and the proposed repulsive loss $L _ { D } ^ { \mathrm { r e p } }$ (Eq. S1c) may be locally stable and leave the proof for future work.
+
+# A.2 SIMULATION STUDY
+
+In this section, we reused the example from Nagarajan & Kolter (2017) to show that GAN trained using the MMD loss in Eq. S1 is locally stable. Consider a two-parameter MMD-GAN with uniform latent distribution $P _ { Z }$ over $[ - 1 , 1 ]$ , generator $G ( z ) = w _ { 1 } z$ , discriminator $D ( x ) = w _ { 2 } x ^ { 2 }$ , and Gaussian kernel $k _ { 0 . 5 } ^ { \mathrm { r b f } }$ . The MMD-rbf model $L _ { G } ^ { \mathrm { m m d } }$ and $L _ { D } ^ { \mathrm { a t t } }$ from Eq. S1b) and the MMD-rep model $\left( L _ { G } ^ { \mathrm { m m d } } \right.$ and $L _ { D } ^ { \mathrm { r e p } }$ from Eq. S1c) were tested. Each model was applied to two cases:
+
+(a) the data distribution $P _ { X }$ is the same as $P _ { Z }$ , i.e., uniform over $[ - 1 , 1 ]$ , thus $P _ { X }$ is realizable;
+
+![](images/4535f0c0674d79b653f7b845db691ef69b8a509a4120e7039f24c149a628df9e.jpg)  
+Figure S1: Streamline plots of MMD-GAN using the MMD-rbf and the MMD-rep model on distributions: $P _ { Z } = \mathcal { U } ( - 1 , 1 )$ , $P _ { X } = \mathcal { U } ( - 1 , 1 )$ or $\bar { P _ { X } = \mathcal { N } ( 0 , 1 ) }$ . In (a) and (b), the equilibria satisfying $P _ { G } = P _ { X }$ lie on the line $w _ { 1 } = 1$ . In (c), the equilibrium lies around point (1.55, 0.74); in (d), it is around (1.55, 0.32).
+
+(b) $P _ { X }$ is standard Gaussian, thus non-realizable for any $w _ { 1 } \in \mathbb { R }$ .
+
+Fig. S1 shows that MMD-GAN are locally stable in both cases and $D _ { \pmb { \theta } _ { D } ^ { * } } ( { \pmb x } ) = 0$ does not hold in general for MMD loss. However, MMD-rep may not be globally stable for the tested cases: initialization of $( w _ { 1 } , w _ { 2 } )$ in some regions may lead to the trivial solution $w _ { 2 } = 0$ (see Fig. S1b and S1d). We note that by decreasing the learning rate for $G$ , the area of such regions decreased. At last, it is interesting to note that both MMD-rbf and MMD-rep had the same nontrivial solution $w _ { 1 } \approx 1 . 5 5$ for generator in the non-realizable cases (see Fig. S1c and S1d).
+
+# A.3 PROOF OF PROPOSITION 1
+
+This section divides the proof for Proposition 1 into two parts. First, we show that GAN with the MMD loss in Eq. S1 has equilibria for any parameter configuration of discriminator $D$ ; second, we prove the model is locally exponentially stable. For convenience, we consider the general form of discriminator loss in Eq. 5:
+
+$$
+L _ { D , \lambda } = \lambda \mathbb { E } _ { P _ { \mathbf { X } } } [ k _ { D } ( \pmb { x } , \pmb { x } ^ { \prime } ) ] - ( \lambda - 1 ) \mathbb { E } _ { P _ { \mathbf { X } } , P _ { G } } [ k _ { D } ( \pmb { x } , \pmb { y } ) ] - \mathbb { E } _ { P _ { G } } [ k _ { D } ( \pmb { y } , \pmb { y } ^ { \prime } ) ]
+$$
+
+which has $L _ { D } ^ { \mathrm { a t t } }$ and $L _ { D } ^ { \mathrm { r e p } }$ as the special cases when $\lambda$ equals $- 1$ and 1 respectively. Consider real data $\mathbf { X } _ { r } \sim \breve { P } _ { \mathbf { X } }$ , latent variable $\mathbf { Z } \sim P _ { \mathbf { Z } }$ and generated variable $\mathbf { Y } _ { g } = G _ { \pmb { \theta } _ { G } } ( \mathbf { Z } )$ . Let ${ \pmb x } _ { r } , z , { \pmb y } _ { g }$ be their s les. note $\begin{array} { r } { \nabla _ { b } ^ { a } = \frac { \partial a } { \partial b } } \end{array}$ ; $\dot { \pmb { \theta } } _ { D } = - \nabla _ { \pmb { \theta } _ { D } } ^ { L _ { D } }$ , $\dot { \pmb { \theta } } _ { G } = - \nabla _ { \pmb { \theta } _ { G } } ^ { L G }$ ; $\begin{array} { r } { \pmb { d } _ { g } = \pmb { D } ( G ( \pmb { z } ) ) } \end{array}$ , $\pmb { d } _ { r } = D ( \pmb { x } _ { r } )$ $L _ { D }$ $L _ { G }$ are the losses for $D$ and $G$ respectively. Assume an isotropic stationary kernel $k ( { \pmb a } , { \pmb b } ) = k _ { I } ( | | { \pmb a } - { \pmb b } | | )$ (Genton (2002)) is used in MMD. We first show:
+
+Proposition 1 (Part 1). If there exists $\pmb { \theta } _ { G } ^ { * } \in \Theta _ { G }$ such that $P _ { \theta _ { G } ^ { * } } = P _ { \mathbf { X } } ,$ , the GAN with the MMD loss in Eq. S1a and Eq. S2 has equilibria $( \theta _ { G } ^ { * } , \theta _ { D } )$ for any $\pmb { \theta } _ { D } \in \Theta _ { D }$ .
+
+Proof. Denote $\mathbf { e } _ { i , j } = \mathbf { a } _ { i } - b _ { j }$ and $\begin{array} { r } { \nabla _ { e _ { i , j } } ^ { k } = \frac { \partial k ( a _ { i } , b _ { j } ) } { \partial e } } \end{array}$ ∂k(ai,bj ) where k is the kernel of MMD. The gradients of MMD loss are
+
+$$
+\begin{array} { r l } & { \dot { \theta } _ { D } = ( \lambda - 1 ) \mathbb { E } _ { P _ { \mathbf { x } } , P _ { \theta _ { G } } } [ \nabla _ { e _ { r , g } } ^ { k } \nabla _ { \theta _ { D } } ^ { e _ { r , g } } ] - \lambda \mathbb { E } _ { P _ { \mathbf { x } } } [ \nabla _ { e _ { r 1 , r 2 } } ^ { k } \nabla _ { \theta _ { D } } ^ { e _ { r 1 , r 2 } } ] + \mathbb { E } _ { P _ { \theta _ { G } } } [ \nabla _ { e _ { g 1 , g 2 } } ^ { k } \nabla _ { \theta _ { D } } ^ { e _ { g 1 , g 2 } } ] } \\ & { \dot { \theta } _ { G } = 2 \mathbb { E } _ { P _ { \theta _ { G } } , P _ { \mathbf { x } } } [ \nabla _ { e _ { g , r } } ^ { k } \nabla _ { x _ { g } } ^ { d _ { g } } \nabla _ { \theta _ { G } } ^ { \alpha _ { g } } ] - \mathbb { E } _ { P _ { \theta _ { G } } } [ \nabla _ { e _ { g 1 , g 2 } } ^ { k } ( \nabla _ { x _ { g 1 } } ^ { d _ { g 1 } } \nabla _ { \theta _ { G } } ^ { \alpha _ { g 1 } } - \nabla _ { x _ { g 2 } } ^ { d _ { g 2 } } \nabla _ { \theta _ { G } } ^ { \alpha _ { g 2 } } ) ] } \end{array}
+$$
+
+Note that, given i.i.d. drawn samples $\pmb { X } = \{ \pmb { x } _ { i } \} _ { i = 1 } ^ { n } \sim P _ { \mathbf { X } }$ and $Y = \{ y _ { i } \} _ { i = 1 } ^ { n } \sim P _ { G }$ , an unbiased estimator of the squared MMD is (Gretton et al. (2012))
+
+$$
+{ \hat { M } } _ { k } ^ { 2 } ( P _ { \mathbf { X } } , P _ { G } ) = { \frac { 1 } { n ( n - 1 ) } } \sum _ { i \neq j } ^ { n } k ( { \pmb x } _ { i } , { \pmb x } _ { j } ) + { \frac { 1 } { n ( n - 1 ) } } \sum _ { i \neq j } ^ { n } k ( { \pmb y } _ { i } , { \pmb y } _ { j } ) - { \frac { 2 } { n ( n - 1 ) } } \sum _ { i \neq j } ^ { n } k ( { \pmb x } _ { i } , { \pmb y } _ { j } )
+$$
+
+At equilibrium, consider a sequence of $N$ samples $\pmb { d } _ { r i } = \pmb { d } _ { g i } = \pmb { d } _ { i }$ with $N \to \infty$ , we have
+
+$$
+\begin{array} { r l } & { \dot { \pmb \theta } _ { D } \propto ( \lambda - 1 ) \displaystyle \sum _ { i \neq j } \nabla _ { e _ { i , j } } ^ { k } \nabla _ { { \pmb \theta } _ { D } } ^ { e _ { i , j } } - \lambda \displaystyle \sum _ { i \neq j } \nabla _ { { \pmb e } _ { i , j } } ^ { k } \nabla _ { { \pmb \theta } _ { D } } ^ { e _ { i , j } } + \sum _ { i \neq j } \nabla _ { { \pmb e } _ { i , j } } ^ { k } \nabla _ { { \pmb \theta } _ { D } } ^ { e _ { i , j } } = { \bf 0 } } \\ & { \dot { \pmb \theta } _ { G } ^ { * } \propto - \displaystyle \sum _ { i \neq j } \nabla _ { e _ { i , j } } ^ { k } ( \nabla _ { { \pmb x } _ { i } } ^ { d _ { i } } \nabla _ { { \pmb \theta } _ { G } } ^ { { \pmb x } _ { i } } - \nabla _ { { \pmb x } _ { j } } ^ { d _ { j } } \nabla _ { { \pmb \theta } _ { G } } ^ { { \pmb x } _ { j } } ) + 2 \displaystyle \sum _ { i \neq j } \nabla _ { e _ { i , j } } ^ { k } \nabla _ { { \pmb x } _ { i } } ^ { d _ { i } } \nabla _ { { \pmb \theta } _ { G } } ^ { { \pmb x } _ { i } } } \\ & { \qquad = \displaystyle \sum _ { i \neq j } \nabla _ { e _ { i , j } } ^ { k } ( \nabla _ { { \pmb x } _ { i } } ^ { d _ { i } } \nabla _ { { \pmb \theta } _ { G } } ^ { { \pmb x } _ { i } } + \nabla _ { { \pmb x } _ { j } } ^ { d _ { j } } \nabla _ { { \pmb \theta } _ { G } } ^ { { \pmb x } _ { j } } ) = { \bf 0 } } \end{array}
+$$
+
+$i , j$ ere forrevers. $\dot { \theta } _ { G } ^ { * }$ wnd $\nabla _ { e _ { i , j } } ^ { k } = - \nabla _ { e _ { j , i } } ^ { k }$ act that for each term in the summation, there exists an term with thus the summation is zero. Since we have not assumed the status of $\theta _ { D }$ , $\dot { \pmb \theta } _ { D } = { \bf 0 }$ for any ${ \pmb \theta } _ { D } \in \Theta _ { D }$ . □
+
+We proceed to prove the model stability. First, following Theorem 5 in Gretton et al. (2012) and Theorem 4 in Li et al. (2017a), it is straightforward to see:
+
+Lemma A.1. Under Assumption 2, $M _ { k \circ D _ { \pmb \theta _ { D } } } ^ { 2 } ( P _ { \mathbf { X } } , P _ { \pmb \theta _ { G } } ) \geq 0$ with the equality if and only if $P _ { \mathbf { X } } =$ PθG .
+
+Lemma A.1 and Proposition 1 (Part 1) state that at equilibrium $P _ { \pmb { \theta } _ { G } ^ { * } } = P _ { \mathbf { X } }$ , every discriminator $D _ { \theta _ { D } }$ and kernel $k$ will give $M _ { k \circ D _ { \pmb \theta _ { D } } } ^ { 2 } ( P _ { \pmb \theta _ { G } ^ { * } } , P _ { \mathbf { X } } ) = 0$ , thus no discriminator can distinguish the two distributions. On the other hand, we cite Theorem A.4 from Nagarajan & Kolter (2017):
+
+Lemma A.2 (Nagarajan & Kolter (2017)). Consider a non-linear system of parameters $( \theta , \gamma )$ : $\dot { \pmb { \theta } } = h _ { 1 } ( \pmb { \theta } , \gamma )$ , $\dot { \gamma } = h _ { 2 } ( \theta , \gamma )$ with an equilibrium point at $( \mathbf { 0 } , \mathbf { 0 } )$ . Let there exist  such that $\forall \gamma \in$ $\mathbb { B } _ { \epsilon } ( \mathbf { 0 } ) , ( \mathbf { 0 } , \gamma )$ is an equilibrium. If $\begin{array} { r } { J = \frac { \partial h _ { 1 } ( \pmb { \theta } , \pmb { \gamma } ) } { \partial \pmb { \theta } } \Big | _ { ( \mathbf { 0 } , \mathbf { 0 } ) } } \end{array}$ is a Hurwitz matrix, the non-linear system is exponentially stable.
+
+Now we can prove:
+
+Proposition 1 (Part 2). At equilibrium $P _ { \theta _ { G } ^ { * } } = P _ { \mathbf { X } }$ , the GAN trained using MMD loss and gradient descent methods is locally exponentially stable at $( \pmb { \theta } _ { G } ^ { * } , \pmb { \theta } _ { D } )$ for any $\pmb { \theta } _ { D } \in \Theta _ { D }$ .
+
+Proof. Inspired by Nagarajan & Kolter (2017), we first derive the Jacobian of the system
+
+$$
+{ \pmb J } = \left[ { \pmb J } _ { D D } \quad { \pmb J } _ { D G } \right] \triangleq \left[ \partial \dot { \pmb \theta } _ { D } ^ { T } / \partial { \pmb \theta } _ { D } \quad \partial \dot { \pmb \theta } _ { D } ^ { T } / \partial { \pmb \theta } _ { G } \right]
+$$
+
+Denote $\begin{array} { r } { \Delta _ { b } ^ { a } = \frac { \partial ^ { 2 } a } { \partial b ^ { 2 } } } \end{array}$ and $\begin{array} { r } { \Delta _ { b c } ^ { a } = \frac { \partial ^ { 2 } a } { \partial b \partial c } } \end{array}$ ∂2a∂b∂c . Based on Eq. S3, we have
+
+$$
+\begin{array} { r l } { J _ { D D } = ( \lambda - 1 ) \mathbb { E } _ { \mathbb { P } _ { \alpha , 0 } } [ ( \Delta _ { \theta , \pi } ^ { \boldsymbol { e } , \boldsymbol { n } } ) ^ { T } \otimes ( \nabla _ { \theta , \pi } ^ { \boldsymbol { e } } ) ^ { T } + ( \nabla _ { \theta , \pi } ^ { \boldsymbol { e } , \boldsymbol { n } } ) ^ { T } \Delta _ { \theta , \pi , \eta } ^ { \boldsymbol { e } } \nabla _ { \theta , \pi } ^ { \boldsymbol { e } , \boldsymbol { n } } ] } & { } \\ { - \lambda \mathbb { E } _ { \mathbb { P } _ { \alpha , 0 } } [ ( \Delta _ { \theta , \pi } ^ { \boldsymbol { e } , 1 , 1 , 2 } ) ^ { T } \otimes ( \nabla _ { \theta , 1 , 1 , 2 } ^ { k } ) ^ { T } + ( \nabla _ { \theta , \pi } ^ { \boldsymbol { e } , 1 , 1 , 2 } ) ^ { T } \Delta _ { \theta , \pi , 1 , 2 } ^ { k } \nabla _ { \theta , \pi } ^ { \boldsymbol { e } , 1 , 1 , 2 } ] } & { } \\ { + \mathbb { E } _ { \mathbb { P } _ { \alpha , 0 } } [ ( \Delta _ { \theta , \pi } ^ { \theta , 1 , 2 } ) ^ { T } \otimes ( \nabla _ { \theta , 1 , 2 } ^ { k } ) ^ { T } + ( \nabla _ { \theta , 1 , 2 } ^ { \theta , 1 , 2 } ) ^ { T } \Delta _ { \theta , \pi , 1 , 2 } ^ { k } \nabla _ { \theta , 1 , 2 } ^ { \theta , 1 , 2 } ] } & { } \\ { J _ { D G } = ( \lambda - 1 ) \mathbb { E } _ { \mathbb { P } _ { \alpha , 0 } , P _ { G } } [ ( \Delta _ { \theta , \pi , \theta } ^ { \boldsymbol { e } , \boldsymbol { n } } ) ^ { T } \otimes ( \nabla _ { \theta , \pi , \theta } ^ { \boldsymbol { e } , 1 , 1 } ) ^ { T } - ( \nabla _ { \theta , \pi } ^ { \theta , 1 , 2 } ) ^ { T } \Delta _ { \theta , \pi , 0 } ^ { k } \nabla _ { \theta , \pi } ^ { \theta , 1 } ] } & { } \\  + \mathbb { E } _ { \mathbb { P } _ { \alpha , 0 } } [ ( \Delta _ { \theta , \pi , \theta } ^ { \theta , 1 , 2 } ) ^ { T } \otimes ( \nabla _ { \theta , \pi , 1 , 2 } ^ { k } ) ^ { T } + ( \nabla _ { \theta , \pi } ^ { \theta , 1 , 2 } ) ^ { T } \end{array}
+$$
+
+where $\otimes$ is the kronecker product. At equilibrium, consider a sequence of $N$ samples $d _ { r i } = d _ { g i } =$ $\mathbf { } d _ { i }$ with $N \to \infty$ , we have ${ \pmb J } _ { D D } = { \bf 0 }$ , ${ \pmb J } _ { G D } = { \bf 0 }$ and
+
+$$
+\begin{array} { r l } & { J _ { D G } \propto ( \lambda + 1 ) \displaystyle \sum _ { i < j } [ ( \Delta _ { \theta _ { D } \theta _ { G } } ^ { e _ { i , j } } ) ^ { T } \otimes ( \nabla _ { e _ { i , j } } ^ { k } ) ^ { T } + ( \nabla _ { \theta _ { D } } ^ { e _ { i , j } } ) ^ { T } \Delta _ { e _ { i , j } } ^ { k } \nabla _ { \theta _ { G } } ^ { e _ { i , j } } ] } \\ & { J _ { G G } = \mathbb { E } _ { P _ { \theta _ { G } } } [ ( \nabla _ { \theta _ { G } } ^ { d _ { g 1 } } ) ^ { T } \Delta _ { e _ { g 1 , g 2 } } ^ { k } \nabla _ { \theta _ { G } } ^ { d _ { g 1 } } + ( \nabla _ { \theta _ { G } } ^ { d _ { g 2 } } ) ^ { T } \Delta _ { e _ { g 1 , g 2 } } ^ { k } \nabla _ { \theta _ { G } } ^ { d _ { g 2 } } - ( \nabla _ { \theta _ { G } } ^ { e _ { g 1 , g 2 } } ) ^ { T } \Delta _ { e _ { g 1 , g 2 } } ^ { k } \nabla _ { \theta _ { G } } ^ { e _ { g 1 , g 2 } } ] } \\ & { \quad \quad \quad = \mathbb { E } _ { P _ { \theta _ { G } } } [ ( \nabla _ { \theta _ { G } } ^ { d _ { g 1 } } ) ^ { T } \Delta _ { e _ { g 1 , g 2 } } ^ { k } \nabla _ { \theta _ { G } } ^ { d _ { g 2 } } + ( \nabla _ { \theta _ { G } } ^ { d _ { g 2 } } ) ^ { T } \Delta _ { e _ { g 1 , g 2 } } ^ { k } \nabla _ { \theta _ { G } } ^ { d _ { g 1 } } ] } \end{array}
+$$
+
+Given Lemma A.1 and fact that $J _ { G G }$ is the Hessian matrix of $M _ { k \circ D _ { \pmb { \theta } _ { D } } } ^ { 2 } ( P _ { \mathbf { X } } , P _ { \pmb { \theta } _ { G } } )$ , $J _ { G G }$ is negative semidefinite. The eigenvectors of $J _ { G G }$ corresponding to zero eigenvalues form $\operatorname { n u l l } ( J _ { G G } )$ . There may exist small distortion $\delta \pmb { \theta } _ { G } \in \mathrm { \ n u l l } ( \mathbf { \cal J } _ { G G } )$ such that $P _ { \pmb { \theta } _ { G } ^ { \ast } + \delta \pmb { \theta } _ { G } } = P _ { \pmb { \theta } _ { G } ^ { \ast } }$ . That is, $P _ { \theta _ { G } ^ { * } }$ is locally constant along some directions in the parameter space of $G$ . As a result, $\operatorname { n u l l } ( J _ { G G } ) \subseteq \operatorname { n u l l } ( J _ { D G } )$ because varying $\pmb { \theta } _ { G } ^ { * }$ along these directions has no effect on $D$ .
+
+Following Lemma C.3 of Nagarajan & Kolter (2017), we consider eigenvalue decomposition $J _ { G G } = \breve { U } _ { G } \mathbf { { A } } _ { G } U _ { G } ^ { T }$ and $\begin{array} { r } { J _ { D G } \breve { J } _ { D G } ^ { T } = U _ { D } \Lambda _ { D } U _ { D } ^ { T } } \end{array}$ . Let $U _ { G } = [ \pmb { T } _ { G } ^ { \prime T } , \pmb { T } _ { G } ^ { T } ]$ , $U _ { D } = [ \pmb { T } _ { D } ^ { \prime T } , \pmb { T } _ { D } ^ { T } ]$ such that $\mathrm { C o l } ( { \cal T } _ { G } ^ { \prime T } ) = \mathrm { n u l l } ( J _ { G G } )$ , $\mathrm { C o l } ( { \mathbf { { \cal T } } _ { D } ^ { \prime T } } ) \ = \ \mathrm { n u l l } ( { \mathbf { { \cal J } } _ { D G } ^ { T } } )$ . Thus, the projections $\gamma _ { G } = T _ { G } \pmb { \theta } _ { G }$ are orthogonal to $\mathrm { \Pi } \mathrm { n u l l } ( J _ { G G } )$ . Then, the Jacobian corresponding to the projected system has the form $J ^ { \prime } = [ \mathbf { 0 } , J _ { D G } ^ { \prime } ; \mathbf { 0 } , J _ { G G } ^ { \prime } ]$ with block $\begin{array} { r } { J _ { D G } ^ { \prime } = \pmb { T } _ { D } \pmb { J } _ { D G } \pmb { T } _ { G } ^ { T } } \end{array}$ and ${ \bf J } _ { G G } ^ { \prime } \bar { = } \bar { \bf T } _ { G } { \bf J } _ { G G } \bar { \bf T } _ { G } ^ { T }$ , where $J _ { G G } ^ { \prime }$ is negative definite. Moreover, on all directions exclude those described by $J _ { G G } ^ { \prime }$ , the system is surrounded by a neighborhood of equilibia at least locally. According to Lemma A.2, the system is exponentially stable. □
+
+# A.4 DISCUSSION ON ASSUMPTION 1
+
+This section shows that constant discriminator output $D _ { \theta _ { D } ^ { * } } ( { \pmb x } ) = { \pmb c }$ for $\pmb { x } \in \mathcal { S } ( P _ { \mathbf { X } } ) \bigcup \mathcal { S } ( P _ { \pmb { \theta } _ { G } ^ { \ast } } )$ indicates that $D _ { \theta _ { D } ^ { * } }$ may have no discrimination power. First, we make the following assumptions:
+
+Assumption 3. 1. $D$ is a multilayer perceptron where each layer $l$ can be factorized into an affine transform and an element-wise activation function $f _ { l }$ . 2. Each activation function $f _ { l } \in C ^ { 0 }$ ; furthermore, $f _ { l } ^ { \prime }$ has a finite number of discontinuities and $f _ { l } ^ { \prime \prime } \in C ^ { 0 6 }$ . 3. Input data to $D$ is continuous and its support $s$ is compact in $\mathbb { R } ^ { d }$ with non-zero measure in each dimension and $d > 1 ^ { 7 }$ .
+
+Based on Assumption 3, we have the following proposition:
+
+Proposition 2. If $\forall x \in S$ , $D ( { \pmb x } ) = { \pmb c }$ , where $^ c$ is constant, then there always exists distortion $\delta \mathbfit { x }$ such that ${ \pmb x } + \delta { \pmb x } \notin { \pmb S }$ and $D ( { \pmb x } + \delta { \pmb x } ) = { \pmb c }$ .
+
+Proof. Without loss of generality, we consider ${ \cal D } ( { \pmb x } ) = { \cal W } _ { 2 } h ( { \pmb x } ) + b _ { 2 }$ and $h ( { \pmb x } ) = f ( { \pmb W } _ { 1 } { \pmb x } + { \pmb b } _ { 1 } )$ , where $W _ { 1 } \in \mathbb { R } ^ { d _ { h } \times d }$ , $\bar { W _ { 2 } } , b _ { 1 } ,$ $b _ { 2 }$ are model weights and biases, $f$ is an activation function satisfying Assumption 3. For ${ \pmb x } \in { \pmb S }$ , since $D ( { \pmb x } ) = c$ , we have $h ( \pmb { x } ) \in \mathrm { n u l l } ( \pmb { W } _ { 2 } )$ . Furthermore:
+
+(a) I ${ \mathrm { f ~ r a n k } } ( W _ { 1 } ) < d , { \mathrm { f o r ~ a n y ~ } } \delta x \in { \mathrm { n u l l } } ( W _ { 1 } ) , h ( { \pmb x } + \delta { \pmb x } ) \in { \mathrm { n u l l } } ( W _ { 2 } ) .$   
+(b) If $\operatorname { r a n k } ( \pmb { W } _ { 1 } ) = d = d _ { h }$ , the problem $h ( { \pmb x } + \delta { \pmb x } ) = k \cdot h ( { \pmb x } )$ has unique solution for any $k \in \mathbb { R }$ as long as $k \cdot h ( { \pmb x } )$ is within the output range of $f$ .   
+(c) If $\operatorname { r a n k } ( { \boldsymbol { W } } _ { 1 } ) ~ = ~ d ~ < ~ d _ { h }$ , let $U$ and $V$ be two basis matrices of $R ^ { d _ { h } }$ such that $W _ { 1 } { \pmb x } =$ $\begin{array} { r l } { U \left[ \hat { \pmb { x } } ^ { T } } & { { } \mathbf { 0 } ^ { T } \right] ^ { T } } \end{array}$ and any vector in $\mathrm { { n u l l } } ( W _ { 2 } )$ can be represented as $V \left[ z ^ { T } \quad \mathbf { 0 } ^ { T } \right] ^ { T }$ , where $\hat { \pmb x } \in \mathbb { R } ^ { d _ { h } \times d }$ , $\boldsymbol { z } \in \mathbb { R } ^ { d _ { h } \times n }$ and $n$ is the nullity of $W _ { 2 }$ . Let the projected support be $\hat { S }$ . Thus, $\forall \hat { \pmb x } \in \hat { S }$ , there exists $_ z$ such that $f ( U \left[ \hat { \pmb x } ^ { T } \quad { \bf 0 } ^ { T } \right] ^ { T } + { \pmb b } _ { 1 } ) = V \left[ z ^ { T } \quad z _ { c } ^ { T } \right] ^ { T }$ with $z _ { c } = 0$ . Consider the Jacobian:
+
+$$
+\pmb { J } = \frac { \partial \left[ z ^ { T } \quad z _ { c } ^ { T } \right] ^ { T } } { \partial \left[ \hat { \pmb { x } } ^ { T } \quad \mathbf { 0 } ^ { T } \right] ^ { T } } = V ^ { - 1 } \nabla \pmb { \Sigma } \pmb { U }
+$$
+
+where $\begin{array} { r } { \nabla \Sigma = \operatorname { d i a g } ( \frac { \textrm { d } f } { \textrm { d } a _ { i } } ) } \end{array}$ and $\pmb { a } = [ a _ { i } ] _ { i = 1 } ^ { d _ { h } }$ is the input to activation, or pre-activations. Since $\hat { S }$ is continuous and compact, it has infinite number of boundary points $\{ \hat { \pmb x } _ { b } \}$ for $d > 1$ . Consider one boundary point $\hat { \mathbf { \mathscr { x } } } _ { b }$ and its normal line $\delta \hat { \mathbfit { x } } _ { b }$ . Let $\epsilon > 0$ be a small scalar such that $\hat { \pmb x } _ { b } - \epsilon \delta \hat { \pmb x } _ { b } \in$ $\hat { S }$ and $\hat { \mathbf { x } } _ { b } + \epsilon \delta \hat { \mathbf { x } } _ { b } \notin \hat { S }$ .
+
+• For linear activation, $\nabla \Sigma = \pmb { I }$ and $\textbf {  { J } }$ is constant. Then $z _ { c }$ remains 0 for $\hat { \pmb x } _ { b } + \epsilon \delta \hat { \pmb x } _ { b }$ , i.e., there exists $_ z$ such that $h ( \hat { \pmb x } + \epsilon \delta \hat { \pmb x } ) \in \mathrm { n u l l } ( { \pmb W } _ { 2 } )$ .   
+For nonlinear activations, assume $f ^ { \prime }$ has $N$ discontinuities. Since $\begin{array} { r l } { U \left[ \hat { \pmb x } ^ { T } } & { { } \mathbf { 0 } ^ { T } \right] ^ { T } + b _ { 1 } = c } \end{array}$ has unique solution for any vector $^ c$ , the boundary points $\{ \hat { \pmb x } _ { b } \}$ cannot yield pre-activations $\left\{ a _ { b } \right\}$ that all lie on the discontinuities in any of the $d _ { h }$ directions. Though we might need   
+to sample $d _ { h } ^ { N + 1 }$ points in the worst case to find an exception, there are infinite number of exceptions. Let $\hat { \mathbf { x } } _ { b }$ be a sample where $\left\{ { { a } _ { b } } \right\}$ does not lie on the discontinuities in any direction. Because $f ^ { \prime \prime }$ is continuous, $z _ { c }$ remains 0 for $\hat { \pmb x } _ { b } + \epsilon \delta \hat { \pmb x } _ { b }$ , i.e., there exists $_ z$ such that $h ( \hat { \pmb x } + \epsilon \delta \hat { \pmb x } ) \in \mathrm { n u l l } ( { \pmb W } _ { 2 } )$ .
+
+In conclusion, we can always find $\delta \mathbfit { x }$ such that ${ \pmb x } + \delta { \pmb x } \notin \mathcal { S }$ and $D ( { \pmb x } + \delta { \pmb x } ) = { \pmb c }$
+
+Proposition 2 indicates that if $D _ { \pmb { \theta } _ { D } ^ { * } } ( { \pmb x } ) = { \bf 0 }$ for $\pmb { x } \in \mathcal { S } ( P _ { \mathbf { X } } ) \bigcup \mathcal { S } ( P _ { \pmb { \theta } _ { G } ^ { \ast } } ) , D _ { \pmb { \theta } _ { D } ^ { \ast } }$ cannot discriminate against fake samples with distortions to the original data. In contrast, Assumption 2 and Lemma A.1 guarantee that, at equilibrium, the discriminator trained using MMD loss function is effective against such fake samples given a large number of i.i.d. test samples (Gretton et al. (2012)).
+
+# B SUPPLEMENTARY METHODOLOGY
+
+# B.1 REPRESENTATIVE LOSS FUNCTIONS IN LITERATURE
+
+Several loss functions have been proposed to quantify the difference between real and generated sample scores, including: (assume linear activation is used at the last layer of $D$ )
+
+• The Minimax loss (Goodfellow et al. (2014)): $\begin{array} { r l r } { L _ { D } } & { { } = } & { \mathbb { E } _ { P _ { \mathbf { X } } } [ \mathrm { S o f t p l u s } ( - D ( \mathbf { x } ) ) ] \ + } \end{array}$ $\mathbb { E } _ { P _ { \mathbf { Z } } } [ \mathrm { S o f t p l u s } ( D ( G ( z ) ) ) ]$ and $L _ { G } = - L _ { D }$ , which can be derived from the Jensen–Shannon (JS) divergence between $P _ { \mathbf { X } }$ and the model distribution $P _ { G }$ .   
+• The non-saturating loss (Goodfellow et al. (2014)), which is a variant of the minimax loss with the same $L _ { D }$ and $L _ { G } = \mathbb { E } _ { P _ { \mathbf { Z } } } [ \mathrm { S o f t p l u s } ( - D ( G ( z ) ) ) ]$ .   
+• The Hinge loss (Tran et al. (2017)): $L _ { D } = \mathbb { E } _ { P _ { \mathbf { X } } } [ \mathrm { R e L U } ( 1 - D ( \pmb { x } ) ) ] + \mathbb { E } _ { P _ { \mathbf { Z } } } [ \mathrm { R e L U } ( 1 + D ( G ( \pmb { z } ) ) ) ] .$ , $L _ { G } = \bar { \mathbb { E } } _ { P _ { \mathbf { Z } } } [ - D ( G ( z ) ) ]$ , which is notably known for usage in support vector machines and is related to the total variation (TV) distance (Nguyen et al. (2009)).
+
+• The Wasserstein loss (Arjovsky et al. (2017); Gulrajani et al. (2017)), which is derived from the Wasserstein distance between $P _ { \mathbf { X } }$ and $P _ { G }$ : ${ \cal L } _ { \cal G } = - \mathbb { E } _ { P _ { \bf Z } } [ D ( G ( z ) ) ]$ , $L _ { D } = \mathbb { E } _ { P _ { \mathbf { Z } } } [ D ( G ( z ) ) ] -$ $\mathbb { E } _ { P _ { \mathbf { X } } } [ D ( \pmb { x } ) ]$ , where $D$ is subject to some Lipschitz constraint.
+
+• The maximum mean discrepancy (MMD) (Li et al. (2017a); Binkowski et al. (2018)), as described ´ in Section 2.
+
+# B.2 NETWORK ARCHITECTURE
+
+For unsupervised image generation tasks on CIFAR-10 and STL-10 datasets, the DCGAN architecture from Miyato et al. (2018) was used. For CelebA and LSUN bedroom datasets, we added more layers to the generator and discriminator accordingly. See Table S1 and S2 for details.
+
+Table S1: DCGAN models for image generation on CIFAR-10 $h = w = 4$ , $H = W = 3 2$ ) and STL-10 $h = w = 6$ , $H = W = 4 8$ ) datasets. For non-saturating loss and hinge loss, $s = 1$ ; for MMD-rand, MMD-rbf, MMD-rq, $s = 1 6$ .
+
+(a) Generator   
+
+<table><tr><td rowspan=1 colspan=1>z ∈ R128 ~ N(0,I)</td></tr><tr><td rowspan=1 colspan=1>128 → h × w × 512,dense,linear</td></tr><tr><td rowspan=1 colspan=1>4 × 4,stride 2 deconv, 256,BN,ReLU</td></tr><tr><td rowspan=1 colspan=1>4 × 4, stride 2 deconv,128,BN,ReLU</td></tr><tr><td rowspan=1 colspan=1>4 × 4,stride 2 deconv, 64,BN,ReLU</td></tr><tr><td rowspan=1 colspan=1>3 × 3, stride 1 conv, 3, Tanh</td></tr></table>
+
+(b) Discriminator   
+
+<table><tr><td rowspan=1 colspan=1>RGB image x ∈[-1,1]H×W×3</td></tr><tr><td rowspan=1 colspan=1>3 × 3, stride 1 conv, 64, LReLU</td></tr><tr><td rowspan=1 colspan=1>4 × 4, stride 2 conv, 128,LReLU3 × 3, stride 1 conv, 128,LReLU</td></tr><tr><td rowspan=1 colspan=1>4 × 4, stride 2 conv, 256,LReLU3 × 3, stride 1 conv, 256,LReLU</td></tr><tr><td rowspan=1 colspan=1> 4 × 4, stride 2 conv, 512,LReLU3 × 3, stride 1 conv, 512, LReLU</td></tr><tr><td rowspan=1 colspan=1>h × w × 512 -→ s,dense,linear</td></tr></table>
+
+Table S2: DCGAN models for image generation on CelebA and LSUN-bedroom datasets. For non-saturating loss and hinge loss, $s = 1$ ; for MMD-rand, MMD-rbf, MMD-rq, $s = 1 6$ .   
+(b) Discriminator   
+
+<table><tr><td rowspan=1 colspan=1>RGB image x ∈[-1,1]64×64×3</td></tr><tr><td rowspan=1 colspan=1>3 × 3, stride 1 conv, 64, LReLU</td></tr><tr><td rowspan=1 colspan=1>4 × 4, stride 2 conv, 128,LReLU3 × 3, stride 1 conv, 128,LReLU</td></tr><tr><td rowspan=1 colspan=1>4 × 4, stride 2 conv, 256,LReLU3 × 3, stride 1 conv, 256,LReLU</td></tr><tr><td rowspan=1 colspan=1>4 × 4,stride 2 conv, 512,LReLU3 × 3, stride 1 conv, 512,LReLU</td></tr><tr><td rowspan=1 colspan=1>4 × 4, stride 2 conv,1024,LReLU3 × 3, stride 1 conv, 1024,LReLU</td></tr><tr><td rowspan=1 colspan=1>4 × 4× 512 →s,dense,linear</td></tr></table>
+
+(a) Generator   
+
+<table><tr><td rowspan=1 colspan=1>z ∈ R128 ~ N(0,I)</td></tr><tr><td rowspan=1 colspan=1>128 → 4 × 4 × 1024,dense,linear</td></tr><tr><td rowspan=1 colspan=1>4 × 4, stride 2 deconv, 512,BN, ReLU</td></tr><tr><td rowspan=1 colspan=1>4 × 4, stride 2 deconv, 256,BN, ReLU</td></tr><tr><td rowspan=1 colspan=1>4 × 4, stride 2 deconv, 128, BN, ReLU</td></tr><tr><td rowspan=1 colspan=1>4 × 4, stride 2 deconv, 64, BN,ReLU</td></tr><tr><td rowspan=1 colspan=1>3 × 3, stride 1 conv, 3, Tanh</td></tr></table>
+
+# C POWER ITERATION FOR CONVOLUTION OPERATION
+
+This section introduces the power iteration for convolution operation (PICO) method to estimate the spectral norm of a convolution kernel, and compare PICO with the power iteration for matrix (PIM) method used in Miyato et al. (2018).
+
+# C.1 METHOD FORMATION
+
+For a weight matrix $W$ , the spectral norm is defined as $\begin{array} { r } { \sigma ( \pmb { W } ) = \operatorname* { m a x } _ { \| \pmb { v } \| _ { 2 } \leq 1 } \| \pmb { W } \pmb { v } \| _ { 2 } } \end{array}$ . The PIM is used to estimate $\sigma ( W )$ (Miyato et al. (2018)), which iterates between two steps:
+
+1. Update $\pmb { u } = \pmb { W v } / \| \pmb { W v } \| _ { 2 }$ ;   
+2. Update $\pmb { v } = \pmb { W } ^ { T } \pmb { u } / \left| \left| \pmb { W } ^ { T } \pmb { u } \right| \right| _ { 2 }$ .
+
+The convolutional kernel $W _ { c }$ is a tensor of shape $h \times w \times c _ { i n } \times c _ { o u t }$ with $h , w$ the receptive field size and $c _ { i n } , c _ { o u t }$ the number of input/output channels. To estimate $\sigma ( { \pmb W } _ { c } )$ , Miyato et al. (2018) reshaped it into a matrix $W _ { r s }$ of shape $( h w c _ { i n } ) \times c _ { o u t }$ and estimated $\sigma ( W _ { r s } )$ .
+
+We propose a simple method to calculate $W _ { c }$ directly based on the fact that convolution operation is linear. For any linear map $T : \mathbb { R } ^ { m }  \mathbb { R } ^ { n }$ , there exists matrix $W _ { L } \in \mathbb { R } ^ { n \times m }$ such that ${ \pmb y } = T ( { \pmb x } )$ can be represented as $\pmb { y } = \pmb { W } _ { L } \pmb { x }$ . Thus, we may simply substitute $\begin{array} { r } { W _ { L } = \frac { \partial \mathbf { y } } { \partial \mathbf { x } } } \end{array}$ ∂y∂x in the PIM method to estimate the spectral norm of any linear operation. In the case of convolution operation $^ *$ , there exist doubly block circulant matrix $W _ { d b c }$ such that $\pmb { u } = \pmb { W _ { c } } * \pmb { v } = \pmb { W _ { d b c } } \pmb { v }$ . Consider ${ \pmb v } ^ { \prime } = { \pmb W } _ { d b c } ^ { T } { \pmb u } =$ $[ \frac { \partial \pmb { u } } { \partial \pmb { v } } ] ^ { T } \pmb { u }$ which is essentially the transpose convolution of $W _ { c }$ on $\textbf { \em u }$ (Dumoulin & Visin (2016)). Thus, similar to PIM, PICO iterates between the following two steps:
+
+1. Update $\pmb { u } = \pmb { W _ { c } } * \pmb { v } / \left| \left| \pmb { W _ { c } } * \pmb { v } \right| \right| _ { 2 }$ ;   
+2. Do transpose convolution of $W _ { c }$ on $\textbf { \em u }$ to get $\hat { v }$ ; update $\pmb { v } = \hat { \pmb { v } } / \| \hat { \pmb { v } } \| _ { 2 }$
+
+Similar approaches have been proposed in Tsuzuku et al. (2018) and Virmaux & Scaman (2018) from different angles, which we were not aware during this study. In addition, Sedghi et al. (2019) proposes to compute the exact singular values of convolution kernels using FFT and SVD. In spectral normalization, only the first singular value is concerned, making the power iteration methods PIM and PICO more efficient than FFT and thus preferred in our study. However, we believe the exact method F $\mathrm { F T } { + } \mathrm { S V D }$ (Sedghi et al. (2019)) may eventually inspire more rigorous regularization methods for GAN.
+
+The proposed PICO method estimates the real spectral norm of a convolution kernel at each layer, thus enforces an upper bound on the Lipschitz constant of the discriminator $D$ . Denote the upper bound as LIPPICO. In this study, Leaky ReLU (LReLU) was used at each layer of $D$ , thus $\mathrm { L I P _ { P I C O } \approx }$ 1 (Virmaux & Scaman (2018)). In practice, however, PICO would often cause the norm of the signal passing through $D$ to decrease to zero, because at each layer,
+
+• the signal hardly coincides with the first singular-vector of the convolution kernel; and
+
+Consequently, the discriminator outputs tend to be similar for all the inputs. To compensate the loss of norm at each layer, the signal is multiplied by a constant $C$ after each spectral normalization. This essentially enlarges $\mathrm { L I P _ { P I C O } }$ by $C ^ { K }$ where $K$ is the number of layers in the DCGAN discriminator. For all experiments in Section 5, we fixed $\begin{array} { r } { C = \frac { 1 } { 0 . 5 5 } \approx 1 . 8 2 } \end{array}$ as all loss functions performed relatively well empirically. In Appendix Section C.3, we tested the effects of coefficient $C ^ { K }$ on the performance of several loss functions.
+
+# C.2 COMPARISON TO PIM
+
+PIM (Miyato et al. (2018)) also enforces an upper bound $\mathrm { L I P } _ { \mathrm { P I M } }$ on the Lipschitz constant of the discriminator $D$ . Consider a convolution kernel $W _ { c }$ with receptive field size $h \times w$ and stride $s$ Let $\sigma _ { \mathrm { P I C O } }$ and $\sigma _ { \mathrm { P I M } }$ be the spectral norm estimated by PICO and PIM respectively. We empirically found8 that $\sigma _ { \mathrm { P I M } } ^ { - 1 }$ varies in the range $\big [ \sigma _ { \mathrm { P I C O } } ^ { - 1 }$ , $\begin{array} { r l r } {  { \frac { \sqrt { h w } } { s } \sigma _ { \mathrm { P I C O } } ^ { - 1 } } } \end{array}$ , depending on the kernel $W _ { c }$ . For a typical kernel of size $3 \times 3$ and stride 1, $\sigma _ { \mathrm { P I M } } ^ { - 1 }$ may vary from $\sigma _ { \mathrm { P I C O } } ^ { - 1 }$ to $3 \sigma _ { \mathrm { P I C O } } ^ { - 1 }$ . Thus, $\mathrm { L I P } _ { \mathrm { P I M } }$ is indefinite and may vary during training. In deep convolutional networks, PIM could potentially result in a very loose constraint on the Lipschitz constant of the network. In Appendix Section C.3, we experimentally compare the performance of PICO and PIM with several loss functions.
+
+Table S3: Frechet Inception distance (FID) on image generation tasks using spectral normalization ´ with two power iteration methods PICO and PIM   
+
+<table><tr><td rowspan="2">Methods</td><td rowspan="2">CK in PICO</td><td colspan="2">CIFAR-10</td><td colspan="2">STL-10</td><td colspan="2">CelebA</td><td colspan="2">LSUN-bedrom</td></tr><tr><td>PIM</td><td>PICO</td><td>PIM</td><td>PICO</td><td>PIM</td><td>PICO</td><td>PIM</td><td>PICO</td></tr><tr><td>Hinge</td><td>128</td><td>23.60</td><td>22.89</td><td>47.10</td><td>47.24</td><td>10.02</td><td>9.08</td><td>27.38</td><td>17.20</td></tr><tr><td>MMD-rbf</td><td>128</td><td>26.56</td><td>26.50</td><td>53.17</td><td>54.23</td><td>13.06</td><td>12.81</td><td></td><td></td></tr><tr><td>MMD-rep</td><td>64</td><td>19.98</td><td>17.00</td><td>40.40</td><td>37.15</td><td>8.51</td><td>6.81</td><td>74.03</td><td>16.01</td></tr><tr><td>MMD-rep-b</td><td>64</td><td>18.24</td><td>16.65</td><td>39.78</td><td>37.31</td><td>7.09</td><td>6.42</td><td>20.12</td><td>11.22</td></tr></table>
+
+# C.3 EXPERIMENTS
+
+In this section, we empirically evaluate the effects of coefficient $C ^ { K }$ on the performance of PICO and compare PICO against PIM using several loss functions.
+
+Experiment setup: We used a similar setup as Section 5.1 with the following adjustments. Four loss functions were tested: hinge, MMD-rbf, MMD-rep and MMD-rep-b. Either PICO or PIM was used at each layer of the discriminator. For PICO, five coefficients $C ^ { \dot { K } }$ were tested: 16, 32, 64, 128 and 256 (note this is the overall coefficient for $K$ layers; $K = 8$ for CIFAR-10 and STL-10; $K = 1 0$ for CelebA and LSUN-bedroom; see Appendix B.2). FID was used to evaluate the performance of each combination of loss function and power iteration method, e.g., hinge $+ \mathrm { P I C O }$ with $C ^ { K } = 1 6$ .
+
+Results: For each combination of loss function and power iteration method, the distribution of FID scores over 16 learning rate combinations is shown in Fig. S2. We separated well-performed learning rate combinations from diverged or poorly-performed ones using a threshold $\tau$ as the diverged cases often had non-meaningful FID scores. The boxplot shows the distribution of FID scores for goodperformed cases while the number of diverged or poorly-performed cases was shown above each box if it is non-zero.
+
+Fig. S2 shows that:
+
+1) When PICO was used, the hinge, MMD-rbf and MMD-rep methods were sensitive to the choices of $C ^ { K }$ while MMD-rep-b was robust. For hinge and MMD-rbf, higher $C ^ { K }$ may result in better FID scores and less diverged cases over 16 learning rate combinations. For MMD-rep, higher $C ^ { K }$ may cause more diverged cases; however, the best FID scores were often achieved with $C ^ { K } = { \dot { 6 } } 4$ or 128.
+
+2) For CIFAR-10, STL-10 and CelebA datasets, PIM performed comparable to PICO with $C ^ { K } =$ 128 or 256 on four loss functions. For LSUN bedroom dataset, it is likely that the performance of PIM corresponded to that of PICO with $C ^ { K } > 2 5 6$ . This implies that PIM may result in a relatively loose Lipschitz constraint on deep convolutional networks.
+
+3) MMD-rep-b performed generally better than hinge and MMD-rbf with tested power iteration methods and hyper-parameter configurations. Using PICO, MMD-rep also achieved generally better FID scores than hinge and MMD-rbf. This implies that, given a limited computational budget, the proposed repulsive loss may be a better choice than the hinge and MMD loss for the discriminator.
+
+Table S4: Inception score (IS) and Frechet Inception distance (FID) on ´ CIFAR-10 dataset using gradient penalty and different loss functions   
+
+<table><tr><td>Methods1</td><td>IS</td><td>FID</td></tr><tr><td>Real data</td><td>11.31</td><td>2.09</td></tr><tr><td>SMMDGAN2</td><td>7.0</td><td>31.5</td></tr><tr><td>SN-SMMDGAN2</td><td>7.3</td><td>25.0</td></tr><tr><td>MMD-rep-gp</td><td>7.26</td><td>23.01</td></tr></table>
+
+1 All methods used the same DCGAN architecture. 2 Results from Arbel et al. (2018) Table 1.
+
+Table S5: Inception score (IS), Frechet ´ Inception distance (FID) on CIFAR-10 dataset using MMD-rep and different dimensions of discriminator outputs   
+
+<table><tr><td>Methods</td><td>cK</td><td>IS</td><td>FID</td></tr><tr><td>Real data</td><td></td><td>11.31</td><td>2.09</td></tr><tr><td>MMD-rep-1</td><td>64</td><td>7.43</td><td>22.43</td></tr><tr><td>MMD-rep-4</td><td></td><td>7.81</td><td>17.87</td></tr><tr><td>MMD-rep-16</td><td>32</td><td>8.20</td><td>16.99</td></tr><tr><td>MMD-rep-64</td><td>32</td><td>8.08</td><td>15.65</td></tr><tr><td>MMD-rep-256</td><td>32</td><td>7.96</td><td>16.61</td></tr></table>
+
+Table S3 shows the best FID scores obtained by PICO and PIM where $C ^ { K }$ was fixed at 128 for hinge and MMD-rbf, and 64 for MMD-rep and MMD-rep-b. For hinge and MMD-rbf, PICO performed significantly better than PIM on the LSUN-bedroom dataset and comparably on the rest datasets. For MMD-rep and MMD-rep-b, PICO achieved consistently better FID scores than PIM.
+
+However, compared to PIM, PICO has a higher computational cost which roughly equals the additional cost incurred by increasing the batch size by two (Tsuzuku et al. (2018)). This may be problematic when a small batch has to be used due to memory constraints, e.g., when handling high resolution images on a single GPU. Thus, we recommend using PICO when the computational cost is less of a concern.
+
+# D SUPPLEMENTARY EXPERIMENTS
+
+# D.1 LIPSCHITZ CONSTRAINT VIA GRADIENT PENALTY
+
+Gradient penalty has been widely used to impose the Lipschitz constraint on the discriminator arguably since Wasserstein GAN (Gulrajani et al. (2017)). This section explores whether the proposed repulsive loss can be applied with gradient penalty.
+
+Several gradient penalty methods have been proposed for MMD-GAN. Binkowski et al. (2018) pe- ´ nalized the gradient norm of witness function $\bar { f _ { w } } ( z ) = \mathbb { E } _ { P _ { \mathbf { X } } } [ k _ { D } ( z , \pmb { x } ) ] - \mathbb { E } _ { P _ { G } } [ k _ { D } ( z , \pmb { y } ) ]$ w.r.t. the interpolated sample $\begin{array} { r } { z = u \mathbf { x } + ( 1 - u ) \mathbf { y } } \end{array}$ to one, where $u \sim \mathcal { U } ( 0 , 1 ) ^ { 9 }$ . More recently, Arbel et al. (2018) proposed to impose the Lipschitz constraint on the mapping $\phi \circ D$ directly and derived the Scaled MMD (SMMD) as $S M _ { k } ( P , Q ) = \sigma _ { \mu , k , \lambda } M _ { k } ( P , Q )$ , where the scale $\sigma _ { \mu , k , \lambda }$ incorporates gradient and smooth penalties. Using the Gaussian kernel and measure $\mu = P _ { \mathbf { X } } $ leads to the discriminator loss:
+
+$$
+L _ { D } ^ { \mathrm { S M M D } } = \frac { L _ { D } ^ { \mathrm { a t t } } } { 1 + \lambda \mathbb { E } _ { P _ { \mathbf { X } } } [ \left. \nabla D ( \pmb { x } ) \right. _ { F } ^ { 2 } ] }
+$$
+
+We apply the same formation of gradient penalty to the repulsive loss:
+
+$$
+L _ { D } ^ { \mathrm { r e p - g p } } = \frac { L _ { D } ^ { \mathrm { r e p } } - 1 } { 1 + \lambda \mathbb { E } _ { P _ { \mathbf { X } } } [ \left\| \nabla D ( \pmb { x } ) \right\| _ { F } ^ { 2 } ] }
+$$
+
+where the numerator $L _ { D } ^ { \mathrm { r e p } } - 1 \leq 0$ so that the discriminator will always attempt to minimize both $L _ { D } ^ { \mathrm { r e p } }$ and the Frobenius norm of gradients $\nabla D ( { \pmb x } )$ w.r.t. real samples. Meanwhile, the generator is trained using the MMD loss $L _ { G } ^ { \mathrm { m m d } }$ (Eq. 2).
+
+Experiment setup: The gradient-penalized repulsive loss $L _ { D } ^ { \mathrm { r e p - g p } }$ (Eq. S8, referred to as MMD-repgp) was evaluated on the CIFAR-10 dataset. We found $\lambda = 1 0$ in Arbel et al. (2018) too restrictive and used $\lambda = 0 . 1$ instead. Same as Arbel et al. (2018), the output dimension of discriminator was set to one. Since we entrusted the Lipschitz constraint to the gradient penalty, spectral normalization was not used. The rest experiment setup can be found in Section 5.1.
+
+Results: Table S4 shows that the proposed repulsive loss can be used with gradient penalty to achieve reasonable results on CIFAR-10 dataset. For comparison, we cited the Inception score and FID for Scaled MMD-GAN (SMMDGAN) and Scaled MMD-GAN with spectral normalization (SN-SMMDGAN) from Arbel et al. (2018). Note that SMMDGAN and SN-SMMDGAN used the same DCGAN architecture as MMD-rep-gp, but were trained for $1 5 0 \mathrm { k }$ generator updates and $7 5 0 \mathrm { k }$ discriminator updates, much more than that of MMD-rep-gp (100k for both $G$ and $D$ ). Thus, the repulsive loss significantly improved over the attractive MMD loss for discriminator.
+
+# D.2 OUTPUT DIMENSION OF DISCRIMINATOR
+
+In this section, we investigate the impact of the output dimension of discriminator on the performance of repulsive loss.
+
+Experiment setup: We used a similar setup as Section 5.1 with the following adjustments. The repulsive loss was tested on the CIFAR-10 dataset with a variety of discriminator output dimensions: $d \in \{ 1 , 4 , 1 6 , 6 4 , 2 5 6 \}$ . Spectral normalization was applied to discriminator with the proposed PICO method (see Appendix C) and the coefficients $C ^ { K }$ selected from $\{ 1 6 , 3 2 , 6 4 , 1 2 8 , 2 5 6 \}$ .
+
+Results: Table S5 shows that using more than one output neuron in the discriminator $D$ significantly improved the performance of repulsive loss over the one-neuron case on CIFAR-10 dataset. The reason may be that using insufficient output neurons makes it harder for the discriminator to learn an injective and discriminative representation of the data (see Fig. 4b). However, the performance gain diminished when more neurons were used, perhaps because it becomes easier for $D$ to surpass the generator $G$ and trap it around saddle solutions. The computation cost also slightly increased due to more output neurons.
+
+# D.3 SAMPLES OF UNSUPERVISED IMAGE GENERATION
+
+Generated samples on CelebA dataset are given in Fig. S3 and LSUN bedrooms in Fig. S4.
+
+![](images/d2b78adf415d4edb093a9916c69964cd1637714f8f1e07c356f82b1cf970a7f8.jpg)  
+Figure S2: Boxplot of the FID scores for 16 learning rate combinations on four datasets: (a) CIFAR10, (b) STL-10, (c) CelebA, (d) LSUN-bedroom, using four loss functions, Hinge, MMD-rbf, MMD-rep and MMD-rmb. Spectral normalization was applied to discriminator with two power iteration methods: PICO and PIM. For PICO, five coefficients $C ^ { K }$ were tested: 16, 32, 64, 128, and 256. A learning rate combination was considered diverged or poorly-performed if the FID score exceeded a threshold $\tau$ , which is 50, 80, 50, 90 for CIFAR-10, STL-10, CelebA and LSUN-bedroom respectively. The box quartiles were plotted based on the cases with $ { \mathrm { F I D } } < \tau$ while the number of diverged or poorly-performed cases (out of 16 learning rate combinations) was shown above each box if it is non-zero. We introduced $\tau$ because the diverged cases often had arbitrarily large and non-meaningful FID scores.
+
+![](images/915d92f786260c4c888839c5f8a52011335c3e2aa3037cca5b1cd6be17ca0170.jpg)  
+Figure S3: Image generation using different loss functions on $6 4 \times 6 4$ CelebA dataset.
+
+![](images/651d50f89ae3398526497e9f5bd123611f6a4068027b76467585714286be96b4.jpg)  
+Figure S4: Image generation using different loss functions on $6 4 \times 6 4$ LSUN bedroom dataset.

parse/train/MDsQkFP1Aw/MDsQkFP1Aw.md

@@ -0,0 +1,497 @@
+# INTO THE WILD WITH AUDIOSCOPE: UNSUPERVISED AUDIO-VISUAL SEPARATION OF ON-SCREEN SOUNDS
+
+Efthymios Tzinis∗1, Scott Wisdom2, Aren Jansen2, Shawn Hershey2, Tal Remez2, Daniel P.W. Ellis2, John R. Hershey2 1University of Illinois at Urbana-Champaign 2Google Research etzinis2@illinois.edu scottwisdom@google.com
+
+# ABSTRACT
+
+Recent progress in deep learning has enabled many advances in sound separation and visual scene understanding. However, extracting sound sources which are apparent in natural videos remains an open problem. In this work, we present AudioScope, a novel audio-visual sound separation framework that can be trained without supervision to isolate on-screen sound sources from real in-the-wild videos. Prior audio-visual separation work assumed artificial limitations on the domain of sound classes (e.g., to speech or music), constrained the number of sources, and required strong sound separation or visual segmentation labels. AudioScope overcomes these limitations, operating on an open domain of sounds, with variable numbers of sources, and without labels or prior visual segmentation. The training procedure for AudioScope uses mixture invariant training (MixIT) to separate synthetic mixtures of mixtures (MoMs) into individual sources, where noisy labels for mixtures are provided by an unsupervised audio-visual coincidence model. Using the noisy labels, along with attention between video and audio features, AudioScope learns to identify audio-visual similarity and to suppress off-screen sounds. We demonstrate the effectiveness of our approach using a dataset of video clips extracted from open-domain YFCC100m video data. This dataset contains a wide diversity of sound classes recorded in unconstrained conditions, making the application of previous methods unsuitable. For evaluation and semi-supervised experiments, we collected human labels for presence of on-screen and off-screen sounds on a small subset of clips.
+
+# 1 INTRODUCTION
+
+![](images/e7a74024546f98607bce55ae4808b7762b633c8156274fe64bb5bb7ef9ba6a8c.jpg)  
+Figure 1: AudioScope separating on-screen bird chirping from wind noise and off-screen sounds from fireworks and human laugh. More demos online at https://audioscope.github.io.
+
+Audio-visual machine perception has been undergoing a renaissance in recent years driven by advances in large-scale deep learning. A motivating observation is the interplay in human perception between auditory and visual perception. We understand the world by parsing it into the objects that are the sources of the audio and visual signals we can perceive. However, the sounds and sights produced by these sources have rather different and complementary properties. Objects may make sounds intermittently, whereas their visual appearance is typically persistent. The visual percepts of different objects tend to be spatially distinct, whereas sounds from different sources can blend together and overlap in a single signal, making it difficult to separately perceive the individual sources.
+
+This suggests that there is something to be gained by aligning our audio and visual percepts: if we can identify which audio signals correspond to which visual objects, we can selectively attend to an object’s audio signal by visually selecting the object.
+
+This intuition motivates using vision as an interface for audio processing, where a primary problem is to selectively preserve desired sounds, while removing unwanted sounds. In some tasks, such as speech enhancement, the desired sounds can be selected by their class: speech versus non-speech in this case. In an open-domain setting, the selection of desired sounds is at the user’s discretion. This presents a user-interface problem: it is challenging to select sources in an efficient way using audio. This problem can be greatly simplified in the audio-visual case if we use video selection as a proxy for audio selection, for example, by selecting sounds from on-screen objects, and removing off-screen sounds. Recent work has used video for selection and separation of speech (Ephrat et al., 2018; Afouras et al., 2020) or music (Zhao et al., 2018; Gao & Grauman, 2019; Gan et al., 2020). However, systems that address this for arbitrary sounds (Gao et al., 2018; Rouditchenko et al., 2019; Owens & Efros, 2018) may be useful in more general cases, such as video recording, where the sounds of interest cannot be defined in advance.
+
+The problem of associating arbitrary sounds with their visual objects is challenging in an open domain. Several complications arise that have not been fully addressed by previous work. First, a large amount of training data is needed in order to cover the space of possible sound. Supervised methods require labeled examples where isolated on-screen sounds are known. The resulting data collection and labeling burden limits the amount and quality of available data. To overcome this, we propose an unsupervised approach using mixture invariant training (MixIT) (Wisdom et al., 2020), that can learn to separate individual sources from in-the-wild videos, where the on-screen and off-screen sounds are unknown. Another problem is that different audio sources may correspond to a dynamic set of on-screen objects in arbitrary spatial locations. We accommodate this by using attention mechanisms that align each hypothesized audio source with the different spatial and temporal positions of the corresponding objects in the video. Finally we need to determine which audio sources appear on screen, in the absence of strong labels. This is handled using a weakly trained classifier for sources based on audio and video embeddings produced by the attention mechanism.
+
+# 2 RELATION TO PREVIOUS WORK
+
+Separation of arbitrary sounds from a mixture, known as “universal sound separation,” was recently shown to be possible with a fixed number of sounds (Kavalerov et al., 2019). Conditional information about which sound classes are present can improve separation performance (Tzinis et al., 2020). The FUSS dataset (Wisdom et al., 2021) expanded the scope to separate a variable number of sounds, in order to handle more realistic data. A framework has also been proposed where specific sound classes can be extracted from input sound mixtures (Ochiai et al., 2020). These approaches require curated data containing isolated sounds for training, which prevents their application to truly open-domain data and introduces difficulties such as annotation cost, accurate simulation of realistic acoustic mixtures, and biased datasets.
+
+To avoid these issues, a number of recent works have proposed replacing the strong supervision of reference source signals with weak supervision labels from related modalities such as sound class (Pishdadian et al., 2020; Kong et al., 2020), visual input (Gao & Grauman, 2019), or spatial location from multi-microphone recordings (Tzinis et al., 2019; Seetharaman et al., 2019; Drude et al., 2019). Most recently, Wisdom et al. (2020) proposed mixture invariant training (MixIT), which provides a purely unsupervised source separation framework for a variable number of latent sources.
+
+A variety of research has laid the groundwork towards solving audio-visual on-screen source separation (Michelsanti et al., 2020). Generally, the two main approaches are to use audio-visual localization (Hershey & Movellan, 2000; Senocak et al., 2018; Wu et al., 2019; Afouras et al., 2020), or object detection networks, either supervised (Ephrat et al., 2018; Gao & Grauman, 2019; Gan et al., 2020) or unsupervised (Zhao et al., 2018), to predict visual conditioning information. However, these works only consider restricted domains such as speech (Hershey & Casey, 2002; Ephrat et al., 2018; Afouras et al., 2020) or music (Zhao et al., 2018; Gao & Grauman, 2019; Gan et al., 2020). Gao et al. (2018) reported results with videos from a wide domain, but relied on supervised visual object detectors, which precludes learning about the appearance of sound sources outside of a closed set of classes defined by the detectors. Rouditchenko et al. (2019) proposed a system for a wide domain of sounds, but required sound class labels as well as isolated sounds from these classes. Our approach avoids the supervision of class labels and isolated sources in order to handle unknown visual and sound classes occurring in multi-source data.
+
+Towards learning directly from a less restrictive open domain of in-the-wild video data, Tian et al. (2018) learned to localize audio-visual events in unconstrained videos and presented an ad hoc dataset. Korbar et al. (2018) pretrained models to discern temporal synchronization of audio-video pairs, and demonstrated promising results on action recognition and audio classification. Arandjelovic & Zisserman (2017) took a similar approach by classifying audio-visual correspondences of pairs of one video frame and one second of audio. Hu et al. (2020) proposed a curriculum learning approach where the model gradually learns harder examples to separate.
+
+Closest to our work is the approach of Owens & Efros (2018), a self-supervised audio-visual onscreen speech separation system based on temporal audio-visual alignment. However, Owens & Efros (2018) assumes training videos containing only on-screen sources, and it is unclear how to adapt it to the case where training videos include off-screen sources.
+
+Our approach significantly differs from these prior works in that we do not restrict our domain to musical instruments or human speakers, and we train and test with real in-the-wild videos containing an arbitrary number of objects with no object class restrictions. Our proposed framework can deal with noisy labels (e.g. videos with no on-screen sounds), operate on a completely open-domain of in-the-wild videos, and effectively isolate sounds coming from on-screen objects.
+
+We address the following task, which extends the formulation of the on-screen speech separation problem (Owens & Efros, 2018). Given an input video, the goal is to separate all sources that constitute the input mixture, and then estimate an audio-visual correspondence score for each separated source. These probability scores should be high for separated sources which are apparent on-screen, and low otherwise. The separated audio sources, weighted by their estimated on-screen probabilities, can be summed together to reconstruct the on-screen mixture. We emphasize that our approach is more generally applicable than previous proposals, because real-world videos may contain an unknown number of both on-screen and off-screen sources belonging to an undefined ontology of classes.
+
+We make the following contributions in this paper:
+
+1. We provide the first solution for training an unsupervised, open-domain, audio-visual onscreen separation system from scratch on real in-the-wild video data, with no requirement on modules such as object detectors that require supervised data.   
+2. We develop a new dataset for the on-screen audio-visual separation task, drawn from 2,500 hours of unlabeled videos from YFCC100m, and 55 hours of videos that are human-labeled for presence of on-screen and off-screen sounds.
+
+# 3 MODEL ARCHITECTURE
+
+The overall architecture of AudioScope is built from the following blocks: an image embedding network, an audio separation network, an audio embedding network, an audio-visual attention mechanism, and an on-screen classifier (see Figure 2). The separation and embedding networks are based on prior work and are described in the following subsections. However, the main focus of this work is the overall architecture, as well as the training framework and loss functions.
+
+The video is analyzed with the image embedding network, which generates local embeddings for each of 64 locations within each frame, as well as an embedding of the whole frame. These embeddings are used both as a conditioning input to an audio separation network, as well as an input for classification of the on-screen sounds. The audio separation network takes the mixed input waveform as input, and generates a fixed number of output waveforms, a variable number of which are non-zero depending on the estimated number of sources in the mixture. Conditioning on the video enables the separation to take advantage of cues about the sources present when performing separation. The audio embedding network is applied to each estimated source to obtain one embedding per frame for each source. These audio embeddings are then pooled over time and used in the audio-visual spatio-temporal attention network to retrieve, for each source, a representation of the visual activity that best matches the audio, similar to the associative maps extracted from the internal network representations proposed by Harwath et al. (2018).
+
+The architecture is designed to address the problem of unsupervised learning on in-the-wild opendomain data. First, because the target training videos can contain both on-screen and off-screen sounds, training a system to directly produce the audio of the target video would encourage inclusion of off-screen sounds as well as on-screen ones1. Our proposed multi-source separation network instead produces latent source estimates using an unsupervised MixIT objective, which has been shown to perform well at general sound separation (Wisdom et al., 2020). By decoupling separation from on-screen classification, our architecture facilitates the use of robust objectives that allow some of the sources to be considered off-screen, even if they appear in the soundtrack of the target videos.
+
+The audio-visual attention architecture is motivated by the alignment problem between audio and video: sound source objects in video may be localized, may move over time, and may be present before and after the corresponding audio activity. Because of the open domain we cannot rely on a pre-defined set of object detectors to anchor the video representations of on-screen sources, as is done in some prior works (Ephrat et al., 2018; Gao & Grauman, 2019; Gan et al., 2020). Instead we propose attention to find the video representations that correspond to a source in a more flexible way.
+
+The proposed strategy of temporal pooling of the audio embeddings, before using them in the spatiotemporal attention, allows the network to derive embeddings that represent the active segments of the source audio, and ignore the ambiguous silent regions. In the present model, video is analyzed at a low frame rate, and so the audio-visual correspondence is likely based on relatively static properties of the objects, rather than the synchrony of their motion with the audio. In this case, a single time-invariant representation of the audio may be sufficient as a proof of concept. However, in future work, with higher video frame rates, it may be worthwhile to consider using attention to align sequences of audio and video embeddings in order to detect synchrony in their activity patterns.
+
+The on-screen classifier operates on an audio embedding for one estimated source, as well as the video embedding retrieved by the spatio-temporal attention mechanism, using a dense network. This presumably allows detection of the congruence between the embeddings. To provide additional context for this decision, a global video embedding, produced by temporal pooling, is provided as an additional input. Many alternative choices are possible for this classifier design, which we leave for future work, such as using a more complex classification architecture, or providing additional audio embeddings as input.
+
+![](images/8266a2fa4ec53d827ec4ed06788cf3dfc28a5f6d599ed73e0dc00288ff358b45.jpg)  
+Figure 2: AudioScope system diagram with 2 input mixtures and 4 output sources.
+
+# 3.1 AUDIO SEPARATION NETWORK
+
+The separation network $\mathcal { M } ^ { \mathrm { s } }$ architecture consists of learnable convolutional encoder and decoder layers with an improved time-domain convolutional network $\mathrm { ( T D C N + + ) }$ ) masking network (Wisdom et al., 2020). A mixture consistency projection (Wisdom et al., 2019) is applied to constrain separated sources to add up to the input mixture. The separation network processes a $T$ -sample input mixture waveform and outputs $M$ estimated sources $\hat { s } \in \mathbb { R } ^ { M \times T }$ . Internally, the network estimates $M$ masks which are multiplied with the activations of the encoded input mixture. The time-domain signals $\hat { s }$ are computed by applying the decoder, a transposed convolutional layer, to the masked coefficients.
+
+# 3.2 AUDIO EMBEDDING NETWORK
+
+For each separated source $\hat { s } _ { m }$ , we extract a corresponding global audio embedding using the MobileNet v1 architecture (Howard et al., 2017) which consists of stacked 2D separable dilated convolutional blocks with a dense layer at the end. This network $\mathcal { M } ^ { \mathrm { a } }$ first computes log Mel-scale spectrograms with $F _ { \mathrm { a } }$ audio frames from the time-domain separated sources, and then applies stacks of depthwise separable convolutions to produce the $F _ { \mathrm { a } } \times N$ embedding matrix $Z _ { m } ^ { \mathrm { a } }$ , which contains an $N$ -dimensional row embedding for each frame. An attentional pooling operation (Girdhar & Ramanan, 2017) is used, for each source, $m$ , to form a static audio embedding vector $z _ { m } ^ { \mathrm { a } } = \mathrm { a t t e n d } ( \bar { Z } _ { m } ^ { \mathrm { a } } , Z _ { m } ^ { \mathrm { a } } , Z _ { m } ^ { \mathrm { a } } )$ , where the average embedding $\begin{array} { r } { \bar { Z } _ { m } ^ { \mathrm { a } } = \frac { 1 } { F _ { \mathrm { a } } } \sum _ { i } Z _ { m , i } ^ { \mathrm { a } } } \end{array}$ is the query vector for source $m$ . The attention mechanism (Bahdanau et al., 2015) is defined as follows:
+
+$$
+\mathrm { a t t e n d } ( q , K , V ) = \alpha ^ { T } f _ { \mathrm { V } } ( V ) , \ \alpha = \mathrm { s o f t m a x } ( \mathrm { t a n h } \left( f _ { \mathrm { K } } ( K ) \right) \mathrm { t a n h } \left( f _ { \mathrm { q } } ( q ) \right) ^ { T } ) ,
+$$
+
+with query row vector $q$ , the attention weight distribution column vector $\alpha$ , key matrix $K$ , value matrix $V$ , and trainable row-wise dense layers $f _ { \mathrm { q } } , f _ { \mathrm { V } } , f _ { \mathrm { K } }$ , all having conforming dimensions.
+
+# 3.3 IMAGE EMBEDDING NETWORK
+
+To extract visual features from video frames, we again use a MobileNet v1 architecture. This visual embedding model $\mathcal { M } ^ { \mathrm { v } }$ is applied independently to each one of the $F _ { \mathrm { v } }$ input video frames and a static-length embedding is extracted for each image $Z _ { j } ^ { \mathrm { v } } , ~ j \in \{ 1 , \ldots , F _ { \mathrm { v } } \} .$ .
+
+Conditioning separation network with the temporal video embedding: The embeddings of the video input $Z _ { j } ^ { \mathrm { v } }$ can be used to condition the separation network (Tzinis et al., 2020). Specifically, the image embeddings are fed through a dense layer, and a simple nearest neighbor upsampling matches the time dimension to the time dimension of the intermediate separation network activations. These upsampled and transformed image embeddings are concatenated with the intermediate $\mathrm { T D C N + + }$ activations and fed as input to the separation network layers.
+
+Global video embedding: A global embedding of the video input is extracted using attentional pooling over all video frames, given by $z ^ { \mathrm { v g } } = \bar { \mathrm { a t t e n d } } ( \bar { Z } ^ { \mathrm { v } } , Z ^ { \mathrm { v } } , \bar { Z } ^ { \mathrm { \bar { v } } } )$ , where the average embedding $\begin{array} { r } { \bar { Z } ^ { \mathrm { v } } = \frac { 1 } { F _ { \mathrm { v } } } \sum _ { j } Z _ { j } ^ { \mathrm { v } } } \end{array}$ is the query vector.
+
+Local spatio-temporal video embedding: We also use local features extracted from an intermediate level in the visual convolutional network, that has $8 \times 8$ spatial locations. These are denoted $Z _ { k } ^ { \mathrm { v l } }$ where $k = ( j , n )$ indexes video frame $j$ and spatial location index $n$ . These provide spatial features for identification of sources with visual objects to be used with audio-visual spatio-temporal attention.
+
+# 3.4 AUDIO-VISUAL SPATIO-TEMPORAL ATTENTION
+
+An important aspect of this work is to combine audio and visual information in order to infer correspondence between each separated source and the relevant objects in video. This in turn will be used to identify which sources are visible on-screen. To this end, we employ an audio-visual spatiotemporal attention scheme by letting the network attend to the local features of the visual embeddings for each separated source. In this mechanism, we use the audio embedding $z _ { m } ^ { \mathrm { a } }$ as the query input for source $m$ , and the key and value inputs are given by the spatio-temporal video embeddings, $\dot { Z } ^ { \mathrm { v l } }$ . As a result, the flattened version of the output spatio-temporal embedding, corresponding to the $m \cdot$ -th source, is $z _ { m } ^ { \mathrm { a v } } = \mathrm { a t t e n d } ( z _ { m } ^ { \mathrm { a } } , Z ^ { \mathrm { v l } } , Z ^ { \mathrm { v l } } )$ .
+
+# 3.5 ON-SCREEN CLASSIFIER
+
+To infer the visual presence each separated source, we concatenate the global video embedding $z ^ { \mathrm { v g } }$ , the global audio embedding for each source $z _ { m } ^ { \mathrm { a } }$ , and the corresponding local spatio-temporal audio-visual embedding $z _ { m } ^ { \mathrm { a v } }$ . The concatenated vector is fed through a dense layer $f _ { \mathrm { C } }$ with a logistic activation: $\hat { y } _ { m } =$ logistic $\bar { \mathcal { f } } _ { \mathrm { C } } \left( \left[ z ^ { \mathrm { v g } } , z _ { m } ^ { \mathrm { a } } , z _ { m } ^ { \mathrm { a v } } \right] \right) )$ .
+
+# 3.6 SEPARATION LOSS
+
+We use a MixIT separation loss (Wisdom et al., 2020), which optimizes the assignment of $M$ estimated sources $\hat { s } = \mathcal { M } ^ { \mathrm { s } } \left( x _ { 1 } + x _ { 2 } \right)$ to two reference mixtures $x _ { 1 } , x _ { 2 }$ as follows:
+
+$$
+\mathcal { L } _ { \mathrm { s e p } } \left( x _ { 1 } , x _ { 2 } , \hat { s } \right) = \operatorname* { m i n } _ { A } \Big ( \mathcal { L } _ { \mathrm { S N R } } \left( x _ { 1 } , \left[ A \hat { s } \right] _ { 1 } \right) + \mathcal { L } _ { \mathrm { S N R } } \left( x _ { 2 } , \left[ A \hat { s } \right] _ { 2 } \right) \Big ) ,
+$$
+
+where the mixing matrix $A \in \mathbb { B } ^ { 2 \times M }$ is constrained to the set of $2 \times M$ binary matrices where each column sums to 1. Due to the constraints on $A$ , each source $\hat { s } _ { m }$ can only be assigned to one reference mixture. The SNR loss for an estimated signal $\boldsymbol { \hat { t } } \in \mathbb { R } ^ { T }$ and a target signal $t \in { \check { \mathbb { R } } } ^ { T }$ is defined as:
+
+$$
+\mathcal { L } _ { \mathrm { S N R } } ( t , \hat { t } ) = 1 0 \log _ { 1 0 } \left( \Vert t - \hat { t } \Vert ^ { 2 } + 1 0 ^ { - 3 } \Vert t \Vert ^ { 2 } \right) .
+$$
+
+# 3.7 CLASSIFICATION LOSS
+
+To train the on-screen classifier, we consider the following classification losses. These losses use the binary labels $y _ { m }$ , where are given for supervised examples, and in the unsupervised case $y _ { m } = A _ { 1 , m } ^ { * }$ for each source $m$ , where $A ^ { * }$ is the optimial mixing matrix found by the minimization in (2). We also use the notation $\mathcal { R } = \{ m | y _ { m } = \bar { 1 } , \ m \in \{ 1 , \ldots , M \} \}$ to denote the set of positive labels.
+
+# Exact binary cross entropy:
+
+$$
+\mathcal { L } _ { \mathrm { e x a c t } } \left( y , \hat { y } \right) = \sum _ { m = 1 } ^ { M } \Big ( - y _ { m } \log \left( \hat { y } _ { m } \right) + \left( y _ { m } - 1 \right) \log \left( 1 - \hat { y } _ { m } \right) \Big ) .
+$$
+
+Multiple-instance cross entropy: Since some separated sources assigned to the on-screen mixture are not on-screen, a multiple-instance (MI) (Maron & Lozano-Pérez, 1998) loss, which minimizes over the set of positive labels $\mathcal { R }$ may be more robust:
+
+$$
+\mathcal { L } _ { \mathrm { M I } } \left( y , \hat { y } \right) = \operatorname* { m i n } _ { m \in \mathcal { R } } \Big ( - \log \left( \hat { y } _ { m } \right) - \sum _ { m ^ { \prime } \notin \mathcal { R } } \log \left( 1 - \hat { y } _ { m ^ { \prime } } \right) \Big ) .
+$$
+
+Active combinations cross entropy: An alternative to the MI loss, active combinations (AC), corresponds to the minimum loss over all settings $\wp _ { \geq 1 } ( \mathcal { R } )$ of the labels s.t. at least one label is 1:
+
+$$
+\mathcal { L } _ { \mathrm { A C } } \left( y , \hat { y } \right) = \operatorname* { m i n } _ { s \in \bigcup _ { \geq 1 } ( \mathcal { R } ) } \Big ( - \sum _ { m \in S } \log \left( \hat { y } _ { m } \right) - \sum _ { m ^ { \prime } \notin S } \log \left( 1 - \hat { y } _ { m ^ { \prime } } \right) \Big ) .
+$$
+
+where $\left\{ { \mathcal { O } } _ { \geq 1 } \left( { \mathcal { R } } \right) \right.$ denotes the power set of indices with label of 1.
+
+# 4 EXPERIMENTAL FRAMEWORK
+
+# 4.1 DATA PREPARATION
+
+In order to train on real-world audio-visual recording environments for our open-domain system, we use the Yahoo Flickr Creative Commons 100 Million Dataset $( \mathrm { Y F C C l 0 0 m } )$ (Thomee et al., 2016). The dataset is drawn from about 200,000 videos (2,500 total hours) of various lengths and covering a diverse range of semantic sound categories. By splitting on video uploader, we select 1,600 videos for training, and use the remaining videos for validation and test. We extract 5-second clips with a hop size of 1 second, resulting in around 7.2 million clips. Clips consist of a 5-second audio waveform sampled at $1 6 \mathrm { k H z }$ and 5 video frames $x ^ { ( f ) }$ , where each frame is a $1 2 8 \times 1 2 8 \times 3$ RGB image.
+
+Our goal is to train our system completely unsupervised, but we sought to reduce the proportion of videos with no on-screen sounds. We thus created a filtered subset $\mathcal { D } _ { f }$ of $\mathtt { Y F C C l 0 0 m }$ of clips with a high audio-visual coincidence probability predicted by an unsupervised audio-visual coincidence prediction model (Jansen et al., 2020) trained on sounds from AudioSet (Gemmeke et al., 2017). The resulting selection is noisy, because the coincidence model is not perfect, and clips that have high audio-visual coincidence may contain both on-screen and off-screen sounds, or even no on-screen sounds. However, this selection does increase the occurrence of on-screen sounds, as shown below. The final filtered dataset consists of all clips (about 336,000) extracted from the 36,000 highest audio-visual coincidence scoring videos. The threshold for filtering was empirically set to keep a fair amount of diverse videos while ensuring that not too many off-screen-only clips were accepted.
+
+To evaluate the performance of the unsupervised filtering and our proposed models, and to experiment with a small amount of supervised training data, we obtained human annotations for 10,000 unfiltered training clips, 10,000 filtered training clips, and 10,000 filtered validation/test clips. In the annotation process, the raters indicated “present” or “not present” for on-screen and off-screen sounds. Each clip is labeled by 3 individual raters, and is only considered on-screen-only or off-screen-only if raters are unanimous. We constructed an on-screen-only subset with 836 training, 735 validation, and 295 test clips, and an off-screen-only subset with 3,681 training, 836 validation, and 370 test clips.
+
+Based on human annotations, we estimate that for unfiltered data $7 1 . 3 \%$ of clips contain both on-andoff-screen sounds, $2 . 8 \%$ contain on-screen-only sounds, and $2 5 . 9 \%$ only off-screen sounds. For the filtered data, $8 3 . 5 \%$ of clips contain on-screen and off-screen sounds, $5 . 6 \%$ of clips are on-screenonly, and $1 0 . 9 \%$ are off-screen-only. Thus, the unsupervised filtering reduced the proportion of off-screen-only clips and increased the proportion of clips with on-screen sounds.
+
+# 4.2 TRAINING
+
+Both audio and visual embedding networks were pre-trained on AudioSet (Gemmeke et al., 2017) for unsupervised coincidence prediction (Jansen et al., 2020) and fine-tuned on our data (see Appendix A.3.1 for ablation), whereas the separation network is trained from scratch using MixIT (2) on mixtures of mixtures (MoMs) from the audio of our data. All models are trained on 4 Google Cloud TPUs (16 chips) with Adam (Kingma & Ba, 2015), batch size 256, and learning rate $1 0 ^ { - 4 }$ .
+
+To train the overall network, we construct minibatches of video clips, where the clip’s audio is either a single video’s soundtrack (“single mixture” example), or a mixture of two videos’ soundtracks (“MoM” example): NOn (noisy-labeled on-screen), SOff (synthetic off-screen-only), LOn (humanlabeled on-screen-only), and LOff (human-labeled off-screen-only). For all MoM examples, the second audio mixture is drawn from a different random video in the filtered data. Unsupervised minibatches consist of either $0 \%$ or $2 5 \%$ SOff examples, with the remainder as NOn. NOn examples are always MoMs, and SOff examples are evenly split between single mixtures and MoMs. A NOn MoM uses video clip frames and audio from the filtered high-coincidence subset of our data, $\mathcal { D } _ { f }$ , and SOff MoMs combine video frames of a filtered clip with random audio drawn from the dataset $\mathcal { D } _ { f }$ .
+
+Semi-supervised minibatches additionally include LOn and LOff examples. Half of these examples in the minibatch are single-mixture examples, and the other half are MoM examples. LOn and LOff examples are constructed in the manner as NOn, except that the corresponding video clip is drawn from unanimously human-labeled on-screen-only videos and unanimously human-labeled offscreen-only videos, respectively. We experiment with using $0 \%$ or $2 5 \%$ SOff examples: (NOn, SOff) proportions of $( 5 0 \% , 0 \% )$ or $( \dot { 2 } 5 \%$ , $2 5 \%$ ), respectively, with the remainder of the minibatch evenly split between LOn single-mixture, LOn MoM, LOff single-mixture, and LOff MoM.
+
+Classification labels $y _ { m }$ for all separated sources $\hat { s } _ { m }$ in SOff and LOff examples are set to 0. For NOn and LOn examples, we set the label for each separated source as the first row of the MixIT mixing matrix (2): $y _ { m } = A _ { 1 , m }$ . The MixIT separation loss (2) is used for all MoM example types.
+
+# 4.3 EVALUATION
+
+All evaluations use human-labeled test videos, which have been unanimously labeled as containing either only on-screen or only off-screen sounds. Using this data, we construct four evaluation sets: on-screen single mixtures, off-screen single mixtures, on-screen MoMs, and off-screen MoMs. The single-mixture evaluations consist of only data drawn from the particular label, either on-screen or off-screen. Each on-screen (off-screen) MoM consists of an on-screen-only (off-screen-only) video clip, mixed with the audio from another random clip, drawn from the off-screen-only examples.
+
+# 4.3.1 ON-SCREEN DETECTION
+
+Detection performance for the on-screen classifier is measured using the area under the curve of the weighted receiver operator characteristic (AUC-ROC). Specifically, we set the weight for each source’s prediction equal to the linear ratio of source power to input power, which helps avoid ambiguous classification decisions for inactive or very quiet sources. For single-mixture evaluations, positive labels are assigned for all separated sources from on-screen-only mixtures, and negative labels for all separated sources from off-screen-only mixtures. For on-screen MoM evaluations, labels for separated sources from on-screen MoMs are assigned using the first row of the oracle MixIT mixing matrix, and negative labels are assigned to sources separated from off-screen MoMs.
+
+# 4.3.2 SEPARATION
+
+Since we do not have access to individual ground-truth reference sources for our in-the-wild data, we cannot evaluate the per-source separation performance. The only references we have are mixtures. Thus, we compute an estimate of the on-screen audio by combining the separated sources using classifier predictions: $\begin{array} { r } { \hat { x } ^ { \mathrm { { o n } } } = \sum _ { m = 1 } ^ { M } p _ { m } \hat { s } _ { m } } \end{array}$ . For on-screen single mixture and MoM evaluations, we measure scale-invariant SNR (SI-SNR) (Le Roux et al., 2019), between ${ \hat { x } } ^ { \mathrm { { o n } } }$ and the reference on-screen-only mixture $x ^ { ( o n ) }$ . SI-SNR measures the fidelity between a target $t \in \mathbb { R } ^ { T }$ and an estimate $\hat { t } \in \mathbb { R } ^ { T }$ within an arbitrary scale factor in units of decibels:
+
+$$
+\mathrm { S I - S N R } ( t , { \hat { t } } ) = 1 0 \log _ { 1 0 } { \frac { \| \alpha t \| ^ { 2 } } { \| \alpha t - { \hat { t } } \| ^ { 2 } } } , \quad \alpha = \mathrm { a r g m i n } _ { a } \| a t - { \hat { t } } \| ^ { 2 } = { \frac { t ^ { T } { \hat { t } } } { \| t \| ^ { 2 } } } .
+$$
+
+To measure the degree to which AudioScope rejects off-screen audio, we define the off-screen suppression ratio (OSR), which is the ratio in decibels of the power of the input mixture to the power of the on-screen estimate ${ \hat { x } } ^ { \mathrm { { o n } } }$ . We only compute OSR for off-screen evaluation examples where the input mixture only contains off-screen audio. Thus, higher OSR implies greater suppression of off-screen sounds. The minimum value of OSR is $0 \mathrm { d B }$ , which means that ${ \hat { x } } ^ { \mathrm { o n } }$ is equal to the input mixture, which corresponds to all on-screen classifier probabilities being equal to 1.
+
+In some cases, SI-SNR and OSR might yield infinite values. For example, the estimate $\hat { y }$ may be zero, in which case SI-SNR (7) is $- \infty$ dB. This can occur when the input SNR of an on-screen mixture in a MoM is very low and none of the separated sources are assigned to it by MixIT. Conversely, if the estimate perfectly matches the target, SI-SNR can yield a value of $\infty$ dB, which occurs for on-screen single mixture evaluation cases when the separated sources trivially add up to the on-screen input due to mixture consistency of the separation model. For off-screen examples, OSR can also be infinite if the separation model achieves perfect off-screen suppression by predicting zero for ${ \hat { x } } ^ { \mathrm { { o n } } }$ . To avoid including these infinite values, we elect to measure median SI-SNR and OSR.
+
+# 5 RESULTS
+
+Results are shown in Table 1. Note that there is a trade-off between preservation of on-screen sounds, as measured by SI-SNR, and suppression of off-screen sounds, as measured by OSR: higher on-screen SI-SNR on on-screen examples generally means lower OSR on off-screen examples. Different classification losses have different operating points: for MoMs, compared to using the exact cross-entropy loss, models trained with active combinations or multiple instance loss achieve lower on-screen SI-SNR, while achieving more suppression (higher OSR) of off-screen sounds. Exact cross-entropy models achieve higher AUC-ROC for single mixtures and MoMs, and achieve better reconstruction of on-screen single mixtures at the expense of less rejection of off-screen mixtures.
+
+Training only with the noisy labels provided by the unsupervised coincidence model (Jansen et al., 2020) achieves lower AUC-ROC compared to the semi-supervised condition that adds a small amount of human-labeled examples. Semi-supervised and unsupervised models achieve comparable onscreen SI-SNR, but semi-supervised models achieve better off-screen suppression. For example, the best on-screen SI-SNR for unsupervised and semi-supervised is 8.0 dB and $7 . 3 \ \mathrm { d B }$ , respectively, while OSR is $5 . 3 \ \mathrm { d B }$ and $1 0 . 7 \mathrm { d B }$ . Using $2 5 \%$ synthetic off-screen particularly shifts the behavior of semi-supervised models by biasing them towards predicting lower probabilities of on-screen. This bias results in lower on-screen SI-SNR, yet very strong off-screen rejection (i.e. very large OSRs).
+
+Table 1: Evaluation results for unanimously-annotated on-screen and off-screen mixtures. Training uses unsupervised or semi-supervised examples, with either $0 \%$ or $2 5 \%$ synthetic off-screen (SOff) examples (Section 4.2). Cross-entropy (CE) losses include active combinations (AC), multiple instance (MI), and exact. Note that “MixIT\*” indicates SI-SNR of an oracle estimate derived using MixIT with reference mixtures, and ${ \hat { x } } ^ { \mathrm { { o n } } }$ is the on-screen estimate produced by mixing separated sources with classifier probabilities. On-screen MoMs have a median input SI-SNR of $4 . 4 \ : \mathrm { d B }$ .   
+
+<table><tr><td colspan="3"></td><td colspan="4">Single mixture</td><td colspan="4">Mixture of mixtures</td></tr><tr><td></td><td></td><td></td><td></td><td colspan="2">On: SI-SNR (dB)</td><td>Off: OSR (dB)</td><td></td><td colspan="2">On: SI-SNR (dB)</td><td>Off: OSR (dB)</td></tr><tr><td>Training</td><td>SOff</td><td>CE loss</td><td>AUC</td><td>MixIT*</td><td>xon</td><td>xon</td><td>AUC</td><td>MixIT*</td><td>xon</td><td>xon</td></tr><tr><td>Unsup</td><td>0%</td><td>AC</td><td>0.58</td><td>8</td><td>13.5</td><td>2.5</td><td>0.77</td><td>10.5</td><td>6.3</td><td>9.4</td></tr><tr><td>Unsup</td><td>0%</td><td>MI</td><td>0.55</td><td>8</td><td>11.9</td><td>3.6</td><td>0.75</td><td>10.1</td><td>5.3</td><td>10.8</td></tr><tr><td>Unsup</td><td>0%</td><td>Exact</td><td>0.62</td><td>8</td><td>36.6</td><td>0.5</td><td>0.81</td><td>10.6</td><td>8.0</td><td>5.3</td></tr><tr><td>Unsup</td><td>25%</td><td>AC</td><td>0.57</td><td>8</td><td>12.1</td><td>4.3</td><td>0.76</td><td>10.5</td><td>6.4</td><td>9.3</td></tr><tr><td>Unsup</td><td>25%</td><td>MI</td><td>0.62</td><td>8</td><td>13.6</td><td>4.0</td><td>0.78</td><td>9.6</td><td>6.1</td><td>9.6</td></tr><tr><td>Unsup</td><td>25%</td><td>Exact</td><td>0.64</td><td>8</td><td>41.0</td><td>0.6</td><td>0.81</td><td>10.6</td><td>7.5</td><td>4.5</td></tr><tr><td>Semi</td><td>0%</td><td>AC</td><td>0.71</td><td>8</td><td>14.8</td><td>6.6</td><td>0.82</td><td>10.4</td><td>6.1</td><td>14.1</td></tr><tr><td>Semi</td><td>0%</td><td>MI</td><td>0.68</td><td>8</td><td>12.3</td><td>11.3</td><td>0.79</td><td>9.6</td><td>4.7</td><td>21.0</td></tr><tr><td>Semi</td><td>0%</td><td>Exact</td><td>0.73</td><td>8</td><td>32.8</td><td>4.5</td><td>0.81</td><td>10.1</td><td>7.3</td><td>10.7</td></tr><tr><td>Semi</td><td>25%</td><td>AC</td><td>0.79</td><td>8</td><td>6.7</td><td>54.3</td><td>0.78</td><td>10.0</td><td>3.4</td><td>61.8</td></tr><tr><td>Semi</td><td>25%</td><td>MI</td><td>0.82</td><td>8</td><td>6.6</td><td>52.9</td><td>0.78</td><td>9.4</td><td>2.0</td><td>60.1</td></tr><tr><td>Semi</td><td>25%</td><td>Exact</td><td>0.83</td><td>8</td><td>6.6</td><td>53.9</td><td>0.81</td><td>10.0</td><td>2.4</td><td>61.5</td></tr></table>
+
+![](images/a167c9cd5ce8b3013f075b31603a5cc7e62a66623fb5faaac95be54b50f9908a.jpg)  
+Figure 3: Scatter plots of input SI-SNR versus on-screen SI-SNR for on-screen evaluation examples, and mixture power in dB versus OSR. The settings for the models are SOff $0 \%$ and exact CE loss, and colormap indicates density of points.
+
+Figure 3 shows scatter plots of input SI-SNR versus SI-SNR of MixIT or ${ \hat { x } } ^ { \mathrm { { o n } } }$ on-screen estimates. From these plots, it is clear that the models tend to improve on-screen SI-SNR more often than not, and that these improvements are most significant around $\pm 1 0$ dB input SI-SNR. Note that for MixIT, a number of points have a SI-SNR of $- \infty$ , which happens when MixIT assigns all separated sources to the off-screen mixture. OSR is sometimes $\infty$ when AudioScope achieves excellent off-screen suppression by predicting nearly 0 for the on-screen audio from off-screen-only input. To provide a sense of the qualitative performance of AudioScope, we include visualizations of best, worst, and typical predictions in the appendix, and the supplementary material contains audio-visual demos.
+
+To benchmark AudioScope against other audio-visual separation approaches and measure performance on mismatched data, we evaluate on existing audio-visual separation test sets in Appendix A.2. We also performed a number of ablations for AudioScope, described in Appendix A.3.
+
+# 6 CONCLUSION
+
+In this paper we have proposed the first solution for training an unsupervised, open-domain, audiovisual on-screen separation system, without reliance on prior class labels or classifiers. We demonstrated the effectiveness of our system using a small amount of human-labeled, in-the-wild videos. A recipe for these will be available on the project webpage: https://audioscope.github.io. In future work, we will explore more fine-grained visual features, especially synchrony, which we expect will be especially helpful when multiple instances of the same object are present in the video. We also plan to use our trained classifier to refilter YFCC100m to get better noisy labels for the presence of on-screen sounds, which should further improve the performance of the system.
+
+# REFERENCES
+
+Triantafyllos Afouras, Andrew Owens, Joon Son Chung, and Andrew Zisserman. Self-supervised learning of audio-visual objects from video. In Proc. European Conference on Computer Vision (ECCV), 2020.
+
+Relja Arandjelovic and Andrew Zisserman. Look, listen and learn. In Proc. IEEE International Conference on Computer Vision (ICCV), pp. 609–617, 2017.
+
+Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. In Proc. International Conference on Learning Representations (ICLR), 2015.
+
+Lukas Drude, Daniel Hasenklever, and Reinhold Haeb-Umbach. Unsupervised training of a deep clustering model for multichannel blind source separation. In Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), pp. 695–699, 2019.
+
+Ariel Ephrat, Inbar Mosseri, Oran Lang, Tali Dekel, Kevin Wilson, Avinatan Hassidim, William T Freeman, and Michael Rubinstein. Looking to listen at the cocktail party: a speaker-independent audio-visual model for speech separation. ACM Transactions on Graphics (TOG), 37(4):1–11, 2018.
+
+Chuang Gan, Deng Huang, Hang Zhao, Joshua B Tenenbaum, and Antonio Torralba. Music gesture for visual sound separation. In Proc. IEEE International Conference on Computer Vision (CVPR), pp. 10478–10487, 2020.
+
+Ruohan Gao and Kristen Grauman. Co-separating sounds of visual objects. In Proc. IEEE International Conference on Computer Vision (CVPR), pp. 3879–3888, 2019.
+
+Ruohan Gao, Rogerio Feris, and Kristen Grauman. Learning to separate object sounds by watching unlabeled video. In Proc. European Conference on Computer Vision (ECCV), pp. 35–53, 2018.
+
+Jort F Gemmeke, Daniel P. W. Ellis, Dylan Freedman, Aren Jansen, Wade Lawrence, R Channing Moore, Manoj Plakal, and Marvin Ritter. Audio set: An ontology and human-labeled dataset for audio events. In Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), pp. 776–780, 2017.
+
+Rohit Girdhar and Deva Ramanan. Attentional pooling for action recognition. In Advances in Neural Information Processing Systems, pp. 34–45, 2017.
+
+David Harwath, Adria Recasens, Dídac Surís, Galen Chuang, Antonio Torralba, and James Glass. Jointly discovering visual objects and spoken words from raw sensory input. In Proc. European Conference on Computer Vision (ECCV), pp. 649–665, 2018.
+
+John R Hershey and Michael Casey. Audio-visual sound separation via hidden Markov models. In Advances in Neural Information Processing Systems, pp. 1173–1180, 2002.
+
+John R Hershey and Javier R Movellan. Audio vision: Using audio-visual synchrony to locate sounds. In Advances in Neural Information Processing Systems, pp. 813–819, 2000.
+
+Jen-Cheng Hou, Syu-Siang Wang, Ying-Hui Lai, Yu Tsao, Hsiu-Wen Chang, and Hsin-Min Wang. Audio-visual speech enhancement using multimodal deep convolutional neural networks. IEEE Transactions on Emerging Topics in Computational Intelligence, 2(2):117–128, 2018.
+
+Andrew G Howard, Menglong Zhu, Bo Chen, Dmitry Kalenichenko, Weijun Wang, Tobias Weyand, Marco Andreetto, and Hartwig Adam. Mobilenets: Efficient convolutional neural networks for mobile vision applications. arXiv preprint arXiv:1704.04861, 2017.
+
+Di Hu, Zheng Wang, Haoyi Xiong, Dong Wang, Feiping Nie, and Dejing Dou. Curriculum audiovisual learning. arXiv preprint arXiv:2001.09414, 2020.
+
+Aren Jansen, Daniel PW Ellis, Shawn Hershey, R Channing Moore, Manoj Plakal, Ashok C Popat, and Rif A Saurous. Coincidence, categorization, and consolidation: Learning to recognize sounds with minimal supervision. In Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), pp. 121–125, 2020.
+
+Ilya Kavalerov, Scott Wisdom, Hakan Erdogan, Brian Patton, Kevin Wilson, Jonathan Le Roux, and John R. Hershey. Universal sound separation. In Proc. IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (WASPAA), 2019.
+
+Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In Proc. International Conference on Learning Representations (ICLR), 2015.
+
+Qiuqiang Kong, Yuxuan Wang, Xuchen Song, Yin Cao, Wenwu Wang, and Mark D Plumbley. Source separation with weakly labelled data: An approach to computational auditory scene analysis. In Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), pp. 101–105, 2020.
+
+Bruno Korbar, Du Tran, and Lorenzo Torresani. Cooperative learning of audio and video models from self-supervised synchronization. In Advances in Neural Information Processing Systems, pp. 7763–7774, 2018.
+
+Jonathan Le Roux, Scott Wisdom, Hakan Erdogan, and John R. Hershey. SDR–half-baked or well done? In Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), pp. 626–630, 2019.
+
+Oded Maron and Tomás Lozano-Pérez. A framework for multiple-instance learning. In Advances in Neural Information Processing Systems, pp. 570–576, 1998.
+
+Daniel Michelsanti, Zheng-Hua Tan, Shi-Xiong Zhang, Yong Xu, Meng Yu, Dong Yu, and Jesper Jensen. An overview of deep-learning-based audio-visual speech enhancement and separation. arXiv preprint arXiv:2008.09586, 2020.
+
+Tsubasa Ochiai, Marc Delcroix, Yuma Koizumi, Hiroaki Ito, Keisuke Kinoshita, and Shoko Araki. Listen to what you want: Neural network-based universal sound selector. In Proc. Interspeech, 2020.
+
+Andrew Owens and Alexei A Efros. Audio-visual scene analysis with self-supervised multisensory features. In Proc. European Conference on Computer Vision (ECCV), pp. 631–648, 2018.
+
+F. Pishdadian, G. Wichern, and J. Le Roux. Finding strength in weakness: Learning to separate sounds with weak supervision. IEEE/ACM Transactions on Audio, Speech, and Language Processing, 28: 2386–2399, 2020.
+
+Andrew Rouditchenko, Hang Zhao, Chuang Gan, Josh McDermott, and Antonio Torralba. Selfsupervised audio-visual co-segmentation. In Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), pp. 2357–2361. IEEE, 2019.
+
+Prem Seetharaman, Gordon Wichern, Jonathan Le Roux, and Bryan Pardo. Bootstrapping singlechannel source separation via unsupervised spatial clustering on stereo mixtures. In Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), pp. 356–360, 2019.
+
+Arda Senocak, Tae-Hyun Oh, Junsik Kim, Ming-Hsuan Yang, and In So Kweon. Learning to localize sound source in visual scenes. In Proc. IEEE International Conference on Computer Vision (CVPR), pp. 4358–4366, 2018.
+
+Bart Thomee, David A Shamma, Gerald Friedland, Benjamin Elizalde, Karl Ni, Douglas Poland, Damian Borth, and Li-Jia Li. Yfcc100m: The new data in multimedia research. Communications of the ACM, 59(2):64–73, 2016.
+
+Yapeng Tian, Jing Shi, Bochen Li, Zhiyao Duan, and Chenliang Xu. Audio-visual event localization in unconstrained videos. In Proc. European Conference on Computer Vision (ECCV), pp. 247–263, 2018.
+
+Efthymios Tzinis, Shrikant Venkataramani, and Paris Smaragdis. Unsupervised deep clustering for source separation: Direct learning from mixtures using spatial information. In Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), pp. 81–85, 2019.
+
+Efthymios Tzinis, Scott Wisdom, John R. Hershey, Aren Jansen, and Daniel P. W. Ellis. Improving universal sound separation using sound classification. In Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), pp. 96–100, 2020.
+
+Emmanuel Vincent, Rémi Gribonval, and Cédric Févotte. Performance measurement in blind audio source separation. IEEE/ACM Transactions on Audio, Speech, and Language Processing, 14(4): 1462–1469, 2006.
+
+Scott Wisdom, John R. Hershey, Kevin Wilson, Jeremy Thorpe, Michael Chinen, Brian Patton, and Rif A. Saurous. Differentiable consistency constraints for improved deep speech enhancement. In Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), pp. 900–904, 2019.
+
+Scott Wisdom, Efthymios Tzinis, Hakan Erdogan, Ron J. Weiss, Kevin Wilson, and John R. Hershey. Unsupervised sound separation using mixture invariant training. In Advances in Neural Information Processing Systems, 2020.
+
+Scott Wisdom, Hakan Erdogan, Daniel Ellis, Romain Serizel, Nicolas Turpault, Eduardo Fonseca, Justin Salamon, Prem Seetharaman, and John Hershey. What’s all the fuss about free universal sound separation data? In Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), 2021.
+
+Yu Wu, Linchao Zhu, Yan Yan, and Yi Yang. Dual attention matching for audio-visual event localization. In Proc. IEEE International Conference on Computer Vision (CVPR), pp. 6292–6300, 2019.
+
+Xudong Xu, Bo Dai, and Dahua Lin. Recursive visual sound separation using minus-plus net. In Proc. IEEE International Conference on Computer Vision (CVPR), pp. 882–891, 2019.
+
+Hang Zhao, Chuang Gan, Andrew Rouditchenko, Carl Vondrick, Josh McDermott, and Antonio Torralba. The sound of pixels. In Proc. European Conference on Computer Vision (ECCV), pp. 570–586, 2018.
+
+Hang Zhao, Chuang Gan, Wei-Chiu Ma, and Antonio Torralba. The sound of motions. In Proc. IEEE International Conference on Computer Vision (CVPR), pp. 1735–1744, 2019.
+
+Lingyu Zhu and Esa Rahtu. Separating sounds from a single image. arXiv preprint arXiv:2007.07984, 2020a.
+
+Lingyu Zhu and Esa Rahtu. Visually guided sound source separation using cascaded opponent filter network. In Proc. Asian Conference on Computer Vision, 2020b.
+
+# A APPENDIX
+
+# A.1 QUALITATIVE EXAMPLES
+
+For a range of input SNRs, Figure 4 shows best-case examples of separating on-screen sounds with AudioScope, while Figure 5 shows failure cases. Figures 6 and 7 show random examples at various SNRs, comparing the outputs of semi-supervised SOff $0 \%$ models trained with either exact cross entropy (4) or active combinations cross entropy (6). Figure 6 shows the outputs of the two models on 7 random examples, and Figure 7 shows the outputs of the two models on 5 examples that have maximum absolute difference in terms of SI-SNR of the on-screen estimate.
+
+The supplementary material includes audio-visual demos of AudioScope on single mixtures and MoMs with visualizations of MixIT assignments and predicted on-screen probabilities. For more examples please see: https://audioscope.github.io/.
+
+![](images/faf361b8860389d03257519e5ac3ba50205d5604b24ce703792fdd20da7220c9.jpg)  
+Figure 4: Best cases for AudioScope separation of on-screen sounds under various SNR conditions.
+
+![](images/6b87ce102dd356d8b5e0541f4fff56cbb4857ca4087b1912e78709ed3a9650c1.jpg)  
+Figure 5: Failure cases for AudioScope separation of on-screen sounds under various SNR conditions.
+
+![](images/17e3062d21a74fb4dada67cde08a2b53d0647fc4e4a53b6338af695468af8be2.jpg)  
+Figure 6: Comparison of random examples of separating on-screen sounds under various SNR conditions using either exact cross entropy (4) or active combinations cross entropy (6) as the classification loss.
+
+![](images/6ef06085b3b6e30e7daaa1b7ef1d95d27fff7884296ce91f9e736d7c6a9b7d7f.jpg)  
+Video frame On-screen audio Off-screen audio Input audio mixture Attention map On-screen estimate Attention map On-screen estimate   
+Figure 7: Comparison of examples with maximum absolute performance difference under various SNR conditions using either exact cross entropy (4) or active combinations cross entropy (6) as the classification loss.
+
+# A.2 EVALUATION ON MISMATCHED DATA
+
+To evaluate the generalization capability of AudioScope and facilitate a comparison to prior works, we evaluated our model using test data from an audio-visual speech enhancement task (Hou et al., 2018) as well as an audio-visual task derived from a single-source subset of AudioSet (Gao et al., 2018). In both cases, the evaluation is on a restricted domain, and the prior methods used both matched and supervised training on that domain. In contrast, the AudioScope model is trained on open-domain YFCC100m videos using unsupervised training. For all evaluations we use the unsupervised AudioScope model using $0 \%$ SOff and active combinations loss.
+
+# A.2.1 EVALUATION ON AUDIO-VISUAL SPEECH ENHANCEMENT
+
+Since our method can be used to separate on-screen sounds for arbitrary classes of sound, to compare to existing approaches we evaluate the trained AudioScope model on the more restricted domain of audio-visual speech enhancement. To that end, we used the Mandarin sentences dataset, introduced by Hou et al. (2018). The dataset contains video utterances of Mandarin sentences spoken by a native speaker. Each sentence is unique and contains 10 Chinese characters. The length of each utterance is approximately 3 to 4 seconds. Synthetic noise is added to each ground truth audio. Forty such videos are used as the official testing set. For our evaluation we regard the speech of the filmed speaker as the on-screen sounds and the interference as off-screen sounds. Thus, we can compute quality metrics for the on-screen estimate while comparing to speech enhancement methods. To compare to previously-published numbers, we use signal-to-distortion ratio (SDR) (Vincent et al., 2006), which measures signal-to-noise ratio within a linear filtering of the reference signal.
+
+Table 2 shows the comparison between Hou et al. (2018), Ephrat et al. (2018), AudioScope ${ \hat { x } } ^ { \mathrm { { o n } } }$ (on-screen estimate using predicted on-screen probabilities), AudioScope source with max $\hat { y } _ { m }$ (use separated source with highest predicted on-screen probability), AudioScope best source (oracle selection of the separated source with the highest SDR with on-screen reference), and AudioScope MixIT\* (on-screen estimate using oracle binary on-screen weights using references). Note that the AudioScope models are trained with mismatched open-domain training data, whereas the others were trained on matched speech enhancement data. It can be seen that although non-oracle AudioScope estimates do not advance on state-of-the-art performance of speech enhancement-specific methods, the oracle AudioScope estimates improve over (Hou et al., 2018). Thus AudioScope show promising results on this challenging data which is not explicitly represented in its open-domain training set. We believe that by adding such data to our training set, perhaps by fine-tuning, AudioScope could improve its performance significantly on this more specific task, which we leave for future work.
+
+Table 2: Audio-visual enhancement results on the Mandarin test set (Hou et al., 2018).   
+
+<table><tr><td>Method</td><td>In-Domain</td><td>Supervised</td><td>SDR</td><td>STOI</td></tr><tr><td>Hou et al. (2018)</td><td>、</td><td></td><td>2.8</td><td>0.66</td></tr><tr><td>Ephrat et al. (2018)</td><td></td><td>·</td><td>6.1</td><td>0.71</td></tr><tr><td>AudioScope xon</td><td>×</td><td>X</td><td>2.5</td><td>0.59</td></tr><tr><td>AudioScope source with max 9m</td><td>×</td><td>X</td><td>2.3</td><td>0.58</td></tr><tr><td>AudioScope best source (oracle)</td><td>X</td><td>X</td><td>3.2</td><td>0.60</td></tr><tr><td>AudioScope MixIT* (oracle)</td><td>X</td><td>X</td><td>3.4</td><td>0.61</td></tr></table>
+
+# A.2.2 EVALUATION ON AUDIOSET-SINGLESOURCE
+
+We evaluated AudioScope on the musical instrument portion of the AudioSet-SingleSource dataset (Gao et al., 2018), which is a small number of clips from AudioSet (Gemmeke et al., 2017) that have been verified by humans to contain single sources. We use the same procedure as Gao & Grauman (2019) to construct a MoM test set, which creates 105 synthetic mixtures from all pairs of 15 musical instrument classes. For each pair, audio tracks are mixed together, and we perform separation twice for each pair, conditioning on the video for each source. The results are shown in table 3.
+
+The non-oracle AudioScope methods perform rather poorly, but the oracle methods, especially MixIT\* (which matches the MixIT training loss), achieve state-of-the-art performance compared to methods form the literature. This suggests that the on-screen classifier is less accurate on this data. Also, mixing the predicted AudioScope sources using the probabilities of the on-screen classifier may be suboptimal, and exploring alternative mixing methods to estimate on-screen audio is an avenue for future work. Fine-tuning on data for this specific task could also improve performance, which we also leave for future work.
+
+Table 3: Audio-visual separation results on AudioSet-SingleSource musical instruments test set (Gao & Grauman, 2019).   
+
+<table><tr><td>Method</td><td>In-Domain</td><td> Supervised</td><td>SDR</td><td>SIR</td><td>SAR</td></tr><tr><td>Sound-of-Pixels (Zhao et al.,2018)</td><td>·</td><td>1</td><td>1.7</td><td>3.6</td><td>11.5</td></tr><tr><td>AV-MIML (Gao et al.,2018)</td><td></td><td></td><td>1.8</td><td>1</td><td>1</td></tr><tr><td>Co-Separation (Gao &amp; Grauman,2019)</td><td>√</td><td>√</td><td>4.3</td><td>7.1</td><td>13.0</td></tr><tr><td>AudioScope xon</td><td>X</td><td>X</td><td>0.4</td><td>2.7</td><td>11.4</td></tr><tr><td>AudioScope source with max ym</td><td>X</td><td>×</td><td>-0.9</td><td>2.8</td><td>7.9</td></tr><tr><td>AudioScope best source (oracle)</td><td>X</td><td>×</td><td>4.6</td><td>9.9</td><td>12.1</td></tr><tr><td>AudioScope MixIT* (oracle)</td><td>X</td><td>×</td><td>5.7</td><td>8.4</td><td>12.5</td></tr></table>
+
+# A.2.3 EVALUATION ON MUSIC
+
+We also evaluated AudioScope on MUSIC (Zhao et al., 2018), which includes video clips of solo musical performances that have been verified by humans to contain single sources. We use the same procedure as Gao & Grauman (2019) to construct a MoM test set, which creates 550 synthetic mixtures from all 55 pairs of 11 musical instrument classes, with 10 random 10 second clips per pair2. For each pair, the two audio clips are mixed together, and we perform separation twice for each pair, conditioning on the video for each source. The results are shown in table 4.
+
+Table 4: Audio-visual separation results on MUSIC test set (Zhao et al., 2018)   
+
+<table><tr><td>Method</td><td>In-Domain</td><td>Supervised</td><td>SDR</td><td>SIR</td><td>SAR</td></tr><tr><td>Sound-of-Pixels (Zhao et al., 2018)</td><td></td><td></td><td>5.4</td><td>11.0</td><td>9.8</td></tr><tr><td>Sound-of-Motions (Zhao et al.,2019)</td><td>?</td><td>专</td><td>4.8</td><td>11.0</td><td>8.7</td></tr><tr><td>MP-Net Xu et al. (2019)</td><td></td><td></td><td>5.7</td><td>11.4</td><td>10.4</td></tr><tr><td>Co-Separation (Gao &amp; Grauman, 2019)</td><td></td><td></td><td>7.4</td><td>13.7</td><td>10.8</td></tr><tr><td>Cascaded Opponent Filter (Zhu &amp; Rahtu, 2020b)</td><td></td><td></td><td>10.1</td><td>16.7</td><td>13.0</td></tr><tr><td>A(Res-50,att) + S(DV3P) (Zhu &amp; Rahtu,2020a)</td><td></td><td></td><td>9.4</td><td>15.6</td><td>12.7</td></tr><tr><td>A(Res-50,class.) + S(DV3P) (Zhu &amp; Rahtu,2020a)</td><td>！</td><td>√</td><td>10.6</td><td>17.2</td><td>12.8</td></tr><tr><td>AudioScopexon</td><td>X</td><td>X</td><td>-0.5</td><td>2.8</td><td>11.2</td></tr><tr><td>AudioScope source with max ym</td><td>X</td><td>X</td><td>-2.0</td><td>3.3</td><td>7.6</td></tr><tr><td>AudioScope best source (oracle)</td><td>X</td><td>X</td><td>7.1</td><td>14.9</td><td>12.5</td></tr><tr><td>AudioScope MixIT* (oracle)</td><td>X</td><td>X</td><td>8.8</td><td>13.0</td><td>13.1</td></tr></table>
+
+We see a similar pattern compared to the results for AudioSet-SingleSource in Table 3: non-oracle methods that use the predicted on-screen probability $\hat { y } _ { m }$ do not perform very well. However, oracle selection of the best source, or oracle remixing of the sources, both achieve better performance than a number of recent specialized supervised in-domain systems from the literature, though they do not achieve state-of-the-art performance. These results seem to suggest that the predictions $\hat { y } _ { m }$ are less accurate for this restricted-domain task, but the excellent oracle results suggest potential. In particular, non-oracle performance could improve if the classifier were more accurate, perhaps by fine-tuning. Also, there may be better ways of combining separated sources together to reconstruct on-screen sounds.
+
+# A.3 ABLATIONS
+
+We performed a number of ablations on AudioScope. The following subsections show the results of a number of ablations using either unsupervised or semi-supervised training. All models for these ablation use $0 \%$ SOff examples and the active combinations loss (6).
+
+# A.3.1 AUDIO AND VIDEO EMBEDDINGS
+
+Table 5 shows the results of various ablations involving audio and video embeddings in the model.
+
+Table 5: Ablations related to audio and video embeddings.   
+
+<table><tr><td rowspan="3"></td><td rowspan="3"></td><td colspan="3">Single mixture</td><td colspan="4">Mixture of mixtures</td></tr><tr><td rowspan="2"></td><td colspan="2">On: SI-SNR</td><td rowspan="2">Off: OSR AUC</td><td colspan="2">On: SI-SNR</td><td rowspan="2">Off: OSR con</td></tr><tr><td>AUC xon</td><td>xon</td><td>MixIT*</td><td>gon</td></tr><tr><td>Training Unsup</td><td></td><td>0.58</td><td>13.5</td><td>2.5</td><td>0.77</td><td>10.5</td><td>6.3</td><td>9.4</td></tr><tr><td>Unsup</td><td>No video conditioning for separation</td><td>0.54</td><td>11.5</td><td>2.7</td><td>0.75</td><td>10.6</td><td>5.4</td><td>11.2</td></tr><tr><td>Unsup</td><td>Emb. networks from scratch</td><td>0.45</td><td>8.9</td><td>7.0</td><td>0.61</td><td>10.6</td><td>2.4</td><td>13.4</td></tr><tr><td>Unsup</td><td>No global video and audio emb.to classifier</td><td>0.55</td><td>14.0</td><td>1.1</td><td>0.77</td><td>10.3</td><td>6.9</td><td>5.9</td></tr><tr><td>Unsup</td><td>No global video emb.to classifier</td><td>0.57</td><td>13.3</td><td>2.3</td><td>0.77</td><td>10.6</td><td>6.3</td><td>9.8</td></tr><tr><td>Semi</td><td></td><td>0.71</td><td>14.8</td><td>6.6</td><td>0.82</td><td>10.4</td><td>6.1</td><td>14.1</td></tr><tr><td>Semi</td><td>No video conditioning for separation</td><td>0.65</td><td>12.8</td><td>12.3</td><td>0.77</td><td>10.4</td><td>5.3</td><td>19.7</td></tr><tr><td>Semi</td><td>Emb. networks from scratch</td><td>0.48</td><td>6.3</td><td>12.7</td><td>0.59</td><td>9.7</td><td>2.0</td><td>16.4</td></tr><tr><td>Semi</td><td>No global video and audio emb.to classifier</td><td>0.69</td><td>11.1</td><td>14.7</td><td>0.78</td><td>10.1</td><td>4.5</td><td>21.4</td></tr><tr><td>Semi</td><td>No global video emb. to classifier</td><td>0.66</td><td>11.7</td><td>10.0</td><td>0.77</td><td>10.2</td><td>5.6</td><td>16.3</td></tr></table>
+
+First, notice that removing video conditioning for the separation model reduces on-screen SI-SNR by $2 \mathrm { d B }$ on single mixtures and $0 . 9 \mathrm { d B }$ on MoMs, with negligible or slight improvement in OSR. Thus, we can conclude that visual conditioning does have some benefit for the model.
+
+Next, we consider training the audio and video embedding networks from scratch, instead of using the coincidence model weights pretrained using AudioSet (Jansen et al., 2020). Training from scratch is quite detrimental, as AUC-ROC decreases by a minimum of 0.13 and maximum of 0.23 across single-mixtures/MoMs and unsupervised/semi-supervised conditions. Furthermore, separation performance suffers, with on-screen SI-SNR dropping by multiple for all conditions.
+
+Finally, we consider removing the global video embedding, or both the global video embedding and audio embeddings, from the input of the on-screen classifier. This results in equivalent or slightly worse AUC-ROC, with equivalent or worse on-screen SI-SNR. For unsupervised training, removing both embeddings at the classifier input improves on-screen SI-SNR a bit $0 . 5 \mathrm { d B }$ for single mixtures, $0 . 6 \mathrm { d B }$ for MoMs) with a slight drop in OSR, though for semi-supervised on-screen SI-SNR drops by $3 . 7 \ \mathrm { d B }$ for single mixtures and $0 . 5 \ : \mathrm { d B }$ for MoMs. Overall, the best result is achieved by including these embeddings at the classifier input.
+
+# A.3.2 ATTENTIONAL POOLING
+
+We tried decreasing the embedding dimension from 256 to 128, as well as replacing the attentional pooling with mean pooling for audio sources, video frames, or both. The results are shown in Table 6.
+
+Table 6: Ablations related to attentional pooling run for the unsupervised setting.   
+
+<table><tr><td></td><td></td><td colspan="3">Single mixture</td><td colspan="4">Mixture of mixtures</td></tr><tr><td>Training</td><td></td><td></td><td>On: SI-SNR</td><td>Off: OSR</td><td></td><td colspan="2">On: SI-SNR</td><td>Off: OSR</td></tr><tr><td></td><td>Ablation</td><td>AUC</td><td>gon</td><td>xon</td><td>AUC</td><td>MixIT*</td><td>on</td><td>xon</td></tr><tr><td>Unsup</td><td></td><td>0.58</td><td>13.5</td><td>2.5</td><td>0.77</td><td>10.5</td><td>6.3</td><td>9.4</td></tr><tr><td>Unsup</td><td>Attention embedding dimension 128</td><td>0.56</td><td>12.1</td><td>1.9</td><td>0.76</td><td>10.1</td><td>5.7</td><td>7.5</td></tr><tr><td>Unsup</td><td>No attentional pooling for audio sources</td><td>0.59</td><td>14.3</td><td>1.9</td><td>0.78</td><td>10.3</td><td>6.4</td><td>7.5</td></tr><tr><td>Unsup</td><td>No attentional pooling for video</td><td>0.57</td><td>16.2</td><td>1.4</td><td>0.77</td><td>10.5</td><td>6.8</td><td>6.2</td></tr><tr><td>Unsup</td><td>No attentional pooling for audio and video</td><td>0.56</td><td>12.8</td><td>1.8</td><td>0.76</td><td>10.5</td><td>6.3</td><td>7.7</td></tr></table>
+
+Decreasing the embedding dimension reduces performance, dropping on-screen SI-SNR by $1 . 4 \mathrm { d B }$ on single mixtures and $0 . 6 \mathrm { d B }$ on MoMs, also with reduction in OSR. Replacing attentional pooling with mean pooling generally does not change AUC-ROC or on-screen SI-SNR that much, but does result
+
+in a OSR reduction of at least 0.6 dB for single mixtures and $1 . 7 \ \mathrm { d B }$ for MoMs. Thus, attentional pooling seems to have a beneficial effect in that it improves off-screen suppression, with equivalent classification and on-screen separation performance.
+
+# A.3.3 DATA FILTERING
+
+As described in section 4.1, we use an unsupervised audio-visual coincidence model (Jansen et al., 2020) to filter training videos for on-screen sounds. To ablate the benefit of this filtering, we tried using different combinations of filtered and unfiltered data for NOn examples, as described in section 4.2, which uses filterd data for both on-screen and off-screen mixtures. Filtered data has the advantage of less noisy on-screen labels, but the disadvantage that it lacks the variety of unfiltered data, being only $4 . 7 \%$ of the unfiltered data.
+
+Table 7: Ablations for different data configurations.   
+
+<table><tr><td></td><td></td><td colspan="3">Single mixture</td><td colspan="4">Mixture of mixtures</td></tr><tr><td></td><td></td><td></td><td>On: SI-SNR</td><td>Off: OSR</td><td></td><td colspan="2">On: SI-SNR</td><td>Off: OSR</td></tr><tr><td>Training</td><td>Ablation</td><td>AUC</td><td>gon</td><td>xon</td><td>AUC</td><td>MixIT*</td><td>mon</td><td>xon</td></tr><tr><td>Unsup</td><td></td><td>0.58</td><td>13.5</td><td>2.5</td><td>0.77</td><td>10.5</td><td>6.3</td><td>9.4</td></tr><tr><td>Unsup</td><td>Filtered on-screen,unfiltered off-screen</td><td>0.60</td><td>11.5</td><td>3.4</td><td>0.78</td><td>10.7</td><td>5.7</td><td>11.0</td></tr><tr><td>Unsup</td><td>Unfiltered on-screen,unfiltered off-screen</td><td>0.60</td><td>12.6</td><td>2.5</td><td>0.77</td><td>10.5</td><td>6.4</td><td>6.7</td></tr><tr><td>Unsup</td><td>Unfiltered on-screen,filtered off-screen</td><td>0.63</td><td>15.3</td><td>2.3</td><td>0.79</td><td>10.7</td><td>7.0</td><td>5.6</td></tr><tr><td>Semi</td><td></td><td>0.71</td><td>14.8</td><td>6.6</td><td>0.82</td><td>10.4</td><td>6.1</td><td>14.1</td></tr><tr><td>Semi</td><td>Filtered on-screen,unfiltered off-screen</td><td>0.68</td><td>11.1</td><td>7.1</td><td>0.78</td><td>10.2</td><td>5.3</td><td>13.1</td></tr><tr><td>Semi</td><td>Unfiltered on-screen,unfiltered off-screen</td><td>0.66</td><td>13.5</td><td>6.0</td><td>0.76</td><td>10.4</td><td>6.2</td><td>9.6</td></tr><tr><td>Semi</td><td>Unfiltered on-screen,filtered off-screen</td><td>0.65</td><td>14.8</td><td>7.8</td><td>0.74</td><td>10.4</td><td>5.6</td><td>8.9</td></tr></table>
+
+The results are shown in Table 7. For unsupervised training, unfiltered on-screen with filtered off-screen achieves improved performance in terms of AUC-ROC and on-screen SI-SNR, yet OSR decreases for MoMs. This suggests that in the absence of cleanly-labeled on-screen videos, a larger amount of data with noisier labels is better compares to a smaller amount of data with less noisy labels. However, for semi-supervised training that includes a small amount of cleanly-labeled on-screen examples, AUC-ROC is consistently worse for all ablations, and on-screen SI-SNR and OSR are generally equivalent or worse for all ablations. Thus, these ablations validate that using filtered data for both on-screen and off-screen components of NOn examples with semi-supervised training achieves the best results overall.
+
+# A.3.4 NUMBER OF OUTPUT SOURCES
+
+For all experiments in this paper, we generally used $M = 4$ output sources for the separation model, which is the maximum number of sources that it can predict. Here we see if increasing the number of output sources can improve performance. More output source slots provides a separation model with more flexibility in decomposing the input waveform, yet the drawback is that the model may over-separate (i.e. split sources into multiple components), and there is more pressure on the classifier to correctly group components of the on-screen sound together. The results are shown in Table 8.
+
+Table 8: Ablations for number of max output sources.   
+
+<table><tr><td rowspan="3"></td><td rowspan="3"></td><td colspan="3">Single mixture</td><td colspan="4">Mixture of mixtures</td></tr><tr><td colspan="2">On: SI-SNR</td><td>Off: OSR</td><td rowspan="2"></td><td colspan="2">On: SI-SNR</td><td>Off: OSR</td></tr><tr><td>AUC</td><td>xon</td><td>xon</td><td>AUC MixIT*</td><td>xon</td><td>xon</td></tr><tr><td>Training Unsup</td><td></td><td>一</td><td></td><td>2.5</td><td>0.77</td><td>10.5</td><td>6.3</td><td>9.4</td></tr><tr><td>Unsup</td><td>6 output sources</td><td>0.58 0.54</td><td>13.5 9.6</td><td>2.1</td><td>0.71</td><td>10.3</td><td>5.0</td><td>6.7</td></tr><tr><td>Unsup</td><td>8 output sources</td><td>0.52</td><td>6.9</td><td>4.5</td><td>0.69</td><td>11.1</td><td>3.6</td><td>10.1</td></tr><tr><td>Semi</td><td></td><td>1 0.71</td><td>14.8</td><td>6.6</td><td>0.82</td><td>10.4</td><td>6.1</td><td>14.1</td></tr><tr><td>Semi</td><td>6 output sources</td><td>0.67</td><td>7.9</td><td>12.4</td><td>0.76</td><td>10.8</td><td>4.0</td><td>20.4</td></tr><tr><td>Semi</td><td>8 output sources</td><td>0.67</td><td>6.5</td><td>18.8</td><td>0.77</td><td>10.9</td><td>2.6</td><td>24.6</td></tr></table>
+
+For unsupervised training, increasing the number of output sources generally degrades AUC-ROC and on-screen SI-SNR, while boosting OSR a bit. Note that the MixIT\* improves for MoMs with 8 output sources $1 0 . 5 \ : \mathrm { d B }  1 1 . 1 \ : \cdot$ dB), which suggests the greater flexibility of the model, yet the on-screen estimate ${ \hat { x } } ^ { \mathrm { { o n } } }$ is quite a bit worse (3.6 dB), also compared to on-screen SI-SNR for 4 output sources (6.3 dB).
+
+For semi-supervised training, MixIT\* performance also improves with more output sources, but AUC-ROC and on-screen SI-SNR decrease, suggesting the increased pressure on the classifier to make correct predictions for more, and potentially partial, sources. OSR increases with more output sources, which suggests the classifier biases towards predicting 0s more often. Thus, increasing the number of sources shifts the operating point of the model away from separating on-screen sounds and towards suppressing off-screen sounds.
+
+# A.3.5 BASELINE SEPARATION MODEL
+
+We also trained two-output baseline separation models without the on-screen classifier, where the first estimated source is the on-screen estimate ${ \hat { x } } ^ { \mathrm { { o n } } }$ with training target of on-screen audio, and the second estimated source is the off-screen estimate $\hat { x } ^ { \mathrm { o f f } }$ with training target of off-screen audio. These models were trained with or without video conditioning, using the negative SNR loss (3). The training data is exactly the same as in Table 1, with $0 \%$ SOff.
+
+Table 9: Results for baseline two-output separation model without on-screen classifier.   
+
+<table><tr><td rowspan="3"></td><td rowspan="3"></td><td colspan="5">Single mixture</td></tr><tr><td colspan="3">On-screen</td><td colspan="2">Off-screen</td></tr><tr><td>SI-SNR xon</td><td>ISR xon</td><td>ISR xoff</td><td>ISRxon</td><td>ISR off</td></tr><tr><td>Training Predict input for xon</td><td>and O forxoff</td><td>8</td><td>0.0</td><td>8</td><td>0.0</td><td>8</td></tr><tr><td>Predict 1/2 input for xon </td><td>andxoff</td><td>8</td><td>3.0</td><td>3.0</td><td>3.0</td><td>3.0</td></tr><tr><td>Predict O for xon</td><td> and input forxoff</td><td>NaN</td><td>8</td><td>0.0</td><td>8</td><td>0.0</td></tr><tr><td>Unsup</td><td></td><td>66.2</td><td>6.0</td><td>6.0</td><td>6.0</td><td>6.0</td></tr><tr><td>Unsup</td><td>X</td><td>29.7</td><td>0.2</td><td>27.3</td><td>2.4</td><td>6.7</td></tr><tr><td>Semi</td><td>×</td><td>-18.0</td><td>51.1</td><td>0.0</td><td>51.2</td><td>0.0</td></tr><tr><td>Semi</td><td>√</td><td>18.8</td><td>10.5</td><td>3.0</td><td>48.5</td><td>0.0</td></tr><tr><td rowspan="2"></td><td colspan="5"></td><td rowspan="2"></td></tr><tr><td></td><td></td><td>On-screen</td><td></td><td>Off-screen</td></tr><tr><td rowspan="2">Training</td><td>Video cond.</td><td>SI-SNR xon</td><td>ISR xon</td><td>ISR xoff</td><td>ISRxon</td><td>ISR xoff</td></tr><tr><td>Predict input for xon : and O forxoff</td><td>4.4</td><td>0.0</td><td>8</td><td>0.0</td><td>8</td></tr><tr><td colspan="2">Predict 1/2 input for xon andxoff and input forxoff</td><td>4.4</td><td>3.0</td><td>3.0</td><td>3.0</td><td>3.0</td></tr><tr><td colspan="2">Predict O forxon ×</td><td>NaN</td><td>8</td><td>0.0</td><td>8</td><td>0.0</td></tr><tr><td colspan="2">Unsup</td><td>4.4</td><td>6.0</td><td>6.0</td><td>6.0</td><td>6.0</td></tr><tr><td colspan="2">Unsup</td><td>4.1</td><td>5.8</td><td>3.7</td><td>9.5</td><td>1.7</td></tr><tr><td colspan="2">Semi</td><td>-19.7</td><td>53.3</td><td>0.0</td><td>54.2</td><td>0.0</td></tr><tr><td colspan="2">Semi</td><td>-5.3</td><td>29.5</td><td>0.3</td><td>53.3</td><td>0.0</td></tr></table>
+
+Table 9 shows the results in terms of the same metrics used in Tables 1, except that instead of “off-screen rejection ratio (OSR)”, we report “input-to-source ratio (ISR)” (i.e. $1 0 \log _ { 1 0 }$ of the ratio of input power to estimated source power) for each of the two output sources. High ISR means that the source power is lower compared to the input power. Note that ISR ${ \hat { x } } ^ { \mathrm { { o n } } }$ for off-screen single-mixtures and MoMs is equivalent to OSR. Table 9 also includes several trivial baselines with expected scores.
+
+First, notice that none of these models approach the performance of separation models that include the on-screen classifier, as shown in Table 1. Second, the unsupervised and semi-supervised models here achieve distinctly different operating points. Without video conditioning, the unsupervised model achieves a trivial solution, nearly equivalent to just outputting $1 / 2$ the input mixture for each estimated source. Adding video conditioning for the unsupervised model actually reduces single-mixture performance a bit (66.2 dB to 29.7 dB).
+
+The semi-supervised model without video conditioning is very poor at single-mixture on-screen SI-SNR (-18.0 dB), yet achieves quite high single-mixture OSR (51.1 dB). As indicated by the ISRs, the model tends to prefer nearly-zero on-screen estimates, which may be due to the additional cleanlylabeled off-screen examples provided during training. For the video-conditioned semi-supervised model, single-mixture on-screen SI-SNR improves by quite a lot (- $- 1 8 . 0 \mathrm { d B }$ to $1 8 . 8 \mathrm { d B }$ ), but on-screen SI-SNR performance for on-screen MoMs is abysmal (-19.7 dB without visual conditioning, -5.3 dB with visual conditioning).
+
+Overall, we can conclude from these baselines that simply training a two-output separation model with on-screen and off-screen targets, even with visual conditioning, is not a feasible approach for our open-domain and noisily-labeled data.
+
+# A.4 NEURAL NETWORK ARCHITECTURES
+
+We briefly present the architectures used in this work for the separation network $\mathcal { M } ^ { \mathrm { s } }$ , the audio embedding network $\mathcal { M } ^ { \mathrm { a } }$ , and the image embedding network $\mathcal { M } ^ { \mathrm { v } }$ , and referred in Sections 3.1, 3.2 and 3.3, respectively.
+
+We present the architecture of the $\mathrm { T D C N + + }$ separation network in Table 10. The input to the separation network is a mixture waveform with $T$ time samples and the output is a tensor containing the $M$ estimated source waveforms $\hat { s } \in \mathbb { R } ^ { M \times T }$ . The input for the ith depth-wise (DW) separable convolutional block is the summation of all skip-residual connections and the output of the previous block. Specifically, there are the following skip connections defined w.r.t. their index $i = 0 , \ldots , 3 1$ : $0  8$ , $0  1 6$ , $0  2 4$ , $8  1 6$ , $8  2 4$ and $1 6  2 4$ .
+
+Table 10: $\mathrm { T D C N + + }$ separation network architecture for an input mixture waveform corresponding to a time-length of 5 seconds, sampled at $1 6 \mathrm { k H z }$ . The output of the separation network are $M = 4$ separated sources. The dilation factor for each block is defined as $D _ { i } = 2 ^ { \mathrm { m o d } ( i , 8 ) } , i = 0 , \dots , 3 1$ .   
+
+<table><tr><td>Multiples</td><td>Layer operation</td><td>Filter size</td><td>Stride</td><td>Dilation</td><td> Input shape</td></tr><tr><td rowspan="2">×1</td><td>Conv1D</td><td>40×1×256</td><td>20</td><td>1</td><td>1 × 80,000</td></tr><tr><td>Dense</td><td>1 × 256× 256</td><td>1</td><td>1</td><td>256×4,000</td></tr><tr><td rowspan="7">×32</td><td>Dense</td><td>1× 256×512</td><td>1</td><td>1</td><td>256 ×4,000</td></tr><tr><td>PReLU</td><td>1×1×1</td><td></td><td>1</td><td>512 × 4,000</td></tr><tr><td>Instance Norm</td><td>1 × 512 Separable</td><td></td><td></td><td>512 × 4,000</td></tr><tr><td>DW Conv1D</td><td>3 × 512 Separable</td><td>1</td><td>Di</td><td>512 × 4,000</td></tr><tr><td>PReLU</td><td>1×1×1</td><td></td><td>1</td><td>512 × 4,000</td></tr><tr><td>Instance Norm</td><td>1 × 512 Separable</td><td></td><td></td><td>512 × 4,000</td></tr><tr><td>Dense</td><td>1 × 512 × 256</td><td>1</td><td>1</td><td>512 × 4,000</td></tr><tr><td rowspan="3">×1</td><td>Dense</td><td>1× 256×M·256</td><td>1</td><td>1</td><td>256×4,000</td></tr><tr><td>Reshape</td><td></td><td></td><td></td><td>M· 256 × 4,000</td></tr><tr><td>ConvTranspose1D</td><td>40 × 256×1</td><td>20</td><td>1</td><td>M × 256 × 4,000</td></tr></table>
+
+In a similar way, in Table 11 we define the image and audio embedding networks, which use the same MobileNet v1 architecture (Howard et al., 2017) with different input tensors.
+
+The extraction of each image embedding $Z _ { j } ^ { \mathrm { v } } , ~ j = 1 , \ldots , 5$ relies on the application of the image embedding network $\mathcal { M } ^ { \mathrm { v } }$ on top of each input video frame individually. Moreover, in order to extract the local video spatio-temporal embedding, we extract the output of the $8 \times 8$ convolutional map (denoted with a \* in Table 11) for each input video frame and feed it through a dense layer in order to reduce its channel dimensions to 1. By concatenating all these intermediate convolutional maps we form the local spatio-temporal video embedding $Z ^ { \mathrm { v l } }$ as specificed in Section 3.3.
+
+On the other hand, we extract a time-varying embedding $Z _ { m } ^ { \mathrm { a } }$ for the mth separated source waveform by applying the audio embedding network $\mathcal { M } ^ { \mathrm { a } }$ on overlapping audio segments and concatenating those outputs. The audio segments are extracted with an overlap of 86 windows or equivalently 0.86 seconds. Specifically, for each segment, we extract the mel-spectrogram representation from 96 windows with a length of $2 5 \mathrm { m s }$ and a hop size of $1 0 \mathrm { m s }$ forming the input for the audio embedding network as a matrix with size $9 6 \times 6 4$ , where 64 is the number of mel-features. After feeding this mel-spectrogram as an input to our audio embedding network $\mathcal { M } ^ { \mathrm { a } }$ , we extract the corresponding static length representation for this segment $Z _ { j } ^ { \mathrm { a } }$ , where $j$ denotes the segment index.
+
+Table 11: Audio and image embedding network architectures for an input segment log-mel spectrogram corresponding to a 0.96 seconds, sampled at $1 6 \mathrm { k H z }$ and an input image represented as an RGB tensor with shape $1 2 8 \times 1 2 8 \times 3$ , respectively. The log-mel spectrogram input to the audio embedding network has a number of input channels $C _ { i n } = 1$ and the input video frame to the image embedding network has a number of input channels $C _ { i n } = 3$ . Each depth-wise (DW) convolution or regular convolution is followed by a batch normalization layer and a ReLU activation. \* denotes the layer from which the local $8 \times 8$ spatial feature map is extracted, as described in Section 3.3.   
+
+<table><tr><td rowspan=1 colspan=5>Layer operation   Filter size                  Stride</td><td rowspan=1 colspan=2>Input shapeAudio network  Image network</td></tr><tr><td rowspan=1 colspan=5>Conv              3 ×3× Cin × 32        2</td><td rowspan=1 colspan=1>96×64×1</td><td rowspan=1 colspan=1>128 × 128×3</td></tr><tr><td rowspan=1 colspan=5>DW Conv         3 × 3 × 32 Separable     1</td><td rowspan=1 colspan=1>48× 32× 32</td><td rowspan=1 colspan=1>64 × 64 × 32</td></tr><tr><td rowspan=1 colspan=5>Conv              1×1× 32×64          1</td><td rowspan=1 colspan=1>48 ×32× 32</td><td rowspan=1 colspan=1>64 ×64× 32</td></tr><tr><td rowspan=1 colspan=5>DW Conv         3 ×3 × 64 Separable     2</td><td rowspan=1 colspan=1>48× 32×64</td><td rowspan=1 colspan=1>64 × 64×64</td></tr><tr><td rowspan=1 colspan=5>Conv               1×1×64×128         1</td><td rowspan=1 colspan=1>24×16×64</td><td rowspan=1 colspan=1>32 × 32 × 64</td></tr><tr><td rowspan=1 colspan=5>DW Conv         3 ×3 × 128 Separable    1</td><td rowspan=1 colspan=1>24 × 16 ×128</td><td rowspan=1 colspan=1>32 × 32 × 128</td></tr><tr><td rowspan=1 colspan=3>Conv              1 ×1×128×128</td><td rowspan=1 colspan=2>1</td><td rowspan=1 colspan=1>24×16×128</td><td rowspan=1 colspan=1>32 × 32 × 128</td></tr><tr><td rowspan=1 colspan=3>DW Conv         3 × 3 × 128 Separable</td><td rowspan=1 colspan=2>2</td><td rowspan=1 colspan=1>24 × 16 ×128</td><td rowspan=1 colspan=1>32 × 32 × 128</td></tr><tr><td rowspan=1 colspan=3>Conv              1 ×1× 128× 256</td><td rowspan=1 colspan=2>1</td><td rowspan=1 colspan=1>12 ×8×128</td><td rowspan=1 colspan=1>16 × 16 ×128</td></tr><tr><td rowspan=1 colspan=3>DW Conv</td><td rowspan=1 colspan=2>1</td><td rowspan=1 colspan=1>12×8× 256</td><td rowspan=1 colspan=1>16 ×16× 256</td></tr><tr><td rowspan=1 colspan=1>Conv</td><td rowspan=1 colspan=2>1×1× 256 × 256</td><td rowspan=1 colspan=2>1</td><td rowspan=1 colspan=1>12 ×8×256</td><td rowspan=1 colspan=1>16 ×16× 256</td></tr><tr><td rowspan=1 colspan=1>DW Conv</td><td rowspan=1 colspan=2>3×3×256</td><td rowspan=1 colspan=2>2</td><td rowspan=1 colspan=1>12×8×256</td><td rowspan=1 colspan=1>16 × 16× 256</td></tr><tr><td rowspan=1 colspan=1>Conv</td><td rowspan=1 colspan=2>1 ×1× 256×512</td><td rowspan=1 colspan=2>1</td><td rowspan=1 colspan=1>6×4×256</td><td rowspan=1 colspan=1>8×8×256</td></tr><tr><td rowspan=1 colspan=1>DW Conv</td><td rowspan=1 colspan=2>3×3×512</td><td rowspan=1 colspan=2>1</td><td rowspan=1 colspan=1>6×4×512</td><td rowspan=1 colspan=1>8×8×512</td></tr><tr><td rowspan=1 colspan=1>Conv *</td><td rowspan=1 colspan=2>1 ×1× 512×512</td><td rowspan=1 colspan=2>1</td><td rowspan=1 colspan=1>6×4×512</td><td rowspan=1 colspan=1>8×8×512</td></tr><tr><td rowspan=1 colspan=1>DW Conv</td><td rowspan=1 colspan=1>3×3×</td><td rowspan=1 colspan=1>512</td><td rowspan=1 colspan=1>Separable</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>6×4×512</td><td rowspan=1 colspan=1>8×8×512</td></tr><tr><td rowspan=1 colspan=1>Conv</td><td rowspan=1 colspan=1>1×1×</td><td rowspan=1 colspan=1>512</td><td rowspan=1 colspan=2>1</td><td rowspan=1 colspan=1>6×4×512</td><td rowspan=1 colspan=1>8×8×512</td></tr><tr><td rowspan=1 colspan=1>DW Conv</td><td rowspan=1 colspan=1>3×3x</td><td rowspan=1 colspan=1>512</td><td rowspan=1 colspan=2>1</td><td rowspan=1 colspan=1>6×4×512</td><td rowspan=1 colspan=1>8×8×512</td></tr><tr><td rowspan=1 colspan=1>Conv</td><td rowspan=1 colspan=1>1×1×</td><td rowspan=1 colspan=1>512</td><td rowspan=1 colspan=1>× 512</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>6×4×512</td><td rowspan=1 colspan=1>8×8×512</td></tr><tr><td rowspan=1 colspan=1>DW Conv</td><td rowspan=1 colspan=1>3×3×</td><td rowspan=1 colspan=1>512</td><td rowspan=1 colspan=2>1</td><td rowspan=1 colspan=1>6×4×512</td><td rowspan=1 colspan=1>8×8×512</td></tr><tr><td rowspan=1 colspan=1>Conv</td><td rowspan=1 colspan=1>1×1×</td><td rowspan=1 colspan=1>512</td><td rowspan=1 colspan=2>1</td><td rowspan=1 colspan=1>6×4×512</td><td rowspan=1 colspan=1>8×8×512</td></tr><tr><td rowspan=1 colspan=1>DW Conv</td><td rowspan=1 colspan=1>3×3×</td><td rowspan=1 colspan=1>512</td><td rowspan=1 colspan=2>1</td><td rowspan=1 colspan=1>6×4×512</td><td rowspan=1 colspan=1>8×8×512</td></tr><tr><td rowspan=1 colspan=1>Conv</td><td rowspan=1 colspan=2>1 ×1× 512× 512</td><td rowspan=1 colspan=2>1</td><td rowspan=1 colspan=1>6×4×512</td><td rowspan=1 colspan=1>8×8×512</td></tr><tr><td rowspan=1 colspan=1>DW Conv</td><td rowspan=1 colspan=2>3×3×512</td><td rowspan=1 colspan=2>2</td><td rowspan=1 colspan=1>6×4×512</td><td rowspan=1 colspan=1>8×8×512</td></tr><tr><td rowspan=1 colspan=1>Conv</td><td rowspan=1 colspan=2>1 ×1 × 512 ×1024</td><td rowspan=1 colspan=2>1</td><td rowspan=1 colspan=1>3×2×512</td><td rowspan=1 colspan=1>4×4×512</td></tr><tr><td rowspan=1 colspan=1>DW Conv</td><td rowspan=1 colspan=1>3×3x</td><td rowspan=1 colspan=1>102</td><td rowspan=1 colspan=2>1</td><td rowspan=1 colspan=1>3×2×1024</td><td rowspan=1 colspan=1>4×4×1024</td></tr><tr><td rowspan=1 colspan=1>Conv</td><td rowspan=1 colspan=4>1 ×1× 1024 ×1024     1</td><td rowspan=1 colspan=1>3×2×1024</td><td rowspan=1 colspan=1>4×4×1024</td></tr><tr><td rowspan=2 colspan=5>Average PoolingDense              1 ×1 × 1024 ×128      1</td><td rowspan=1 colspan=1>3×2×1024</td><td rowspan=2 colspan=1>4×4×10241×1×1024</td></tr><tr><td rowspan=1 colspan=1>1×1×1024</td></tr></table>
+
+# A.5 HUMAN EVALUATION
+
+To determine the subjective quality of AudioScope predictions, we performed another round of human annotation on on-screen test MoM videos. The rating task is the same as the one used to annotate data, as described in Section 4.1, where raters were asked to mark the presence of on-screen sounds and off-screen sounds. All models for these evaluations are the same as the base model used in Appendix A.3: $0 \%$ SOff examples with active combinations loss (6). Each example was annotated by 3 raters, and the ultimate binary rating for each example is determined by majority.
+
+Table 12: Results of human annotation task on on-screen MoM test set.   
+
+<table><tr><td>Method</td><td>% on-screen only</td><td>% off-screen only</td><td>% on-and-off-screen</td><td>% unsure</td></tr><tr><td>Unprocessed</td><td>24.1</td><td>2.7</td><td>67.4</td><td>5.9</td></tr><tr><td>Unsup xon</td><td>38.0</td><td>2.5</td><td>53.8</td><td>5.7</td></tr><tr><td>Unsup MixIT*</td><td>37.1</td><td>1.1</td><td>53.7</td><td>8.0</td></tr><tr><td>Semisup xon</td><td>37.6</td><td>2.3</td><td>54.0</td><td>6.1</td></tr><tr><td>Semisup MixIT*</td><td>36.9</td><td>1.2</td><td>52.8</td><td>9.1</td></tr></table>
+
+The results for the on-screen MoM test set are shown in Table 12. We evaluated both the estimate ${ \hat { x } } ^ { \mathrm { { o n } } }$ computed by a weighted sum of the separated sources $\hat { s } _ { m }$ with the predicted probabilities $\hat { y } _ { m }$ , as well as the oracle remixture of separated sources to match the on-screen and off-screen reference audios (denoted by MixIT\*). In this case, notice that all methods improve the percentage of videos rated as on-screen-only from $2 5 . 7 \%$ to about $37 \%$ or $38 \%$ for all methods.
+
+Overall, these human evaluation results suggest lower performance than the objective metrics in Table 1. One reason for this is that the binary rating task is ill-suited towards measuring variable levels of off-screen sounds. That is, a video will be rated as on-screen only if there is absolutely no off-screen sound. However, even if there is quiet off-screen sound present, or artifacts from the separation, a video will be rated as having off-screen sound. Thus, the proportion of human-rated on-screen-only videos can be interpreted as the number of cases where the model did a perfect job at removing off-screen sounds.
+
+We plan to run new human evaluation tasks with better-matched questions. For example, we could ask raters to use a categorical scale, e.g. mean opinion score from 1 to 5. Another idea is to ask raters to score the loudness of on-screen sounds with respect to off-screen sounds on a sliding scale, where the bottom of the scale means on-screen sound is much quieter than off-screen sound, middle of the scale means on-screen sound is equal in loudness to off-screen sound, and top of the scale means on-screen sound is much louder than off-screen sound.
+
+In Figure 8, we show the distributions of overall SI-SNR and SI-SNR improvement, as well as OSR for the best unsupervised and semi-supervised models. We have neglected outliers (including infinite values) in both axes in order to focus on the most common samples.
+
+![](images/6deefd5e337e4f772c5def5b37143af2cd5ae84f882f04d5756569dcef90964b.jpg)  
+Figure 8: Distribution plots for the performance obtained by the best model in terms of on-screen SI-SNR (Figure 8a) and SI-SNRi (Figure 8b) reconstruction and off-screen power suppression (Figure 8c). The settings for the models are SOff $0 \%$ and active combinations (AC) cross-entropy loss.
+
+In Figure 9, for on-screen MoMs we show the distribution of each performance metric for these models versus different ranges of input SI-SNRs lying between $[ - 3 0 , 3 0 ] \mathrm { d B }$ , both for absolute onscreen SI-SNR (Figure 9a) and on-screen SI-SNR improvement (Figure 9b). For off-screen test MoM videos, we plot the distribution of OSR for different ranges of input mixture power lying between $[ - 4 0 , 0 ] \mathrm { d B }$ (Figure 9c).
+
+For on-screen SI-SNR and SI-SNRi, notice that the performance of the unsupervised and semisupervised models is similar except for the $[ - 3 0 , - 2 0 ]$ dB range of input SI-SNR. In Figure ${ 9 \mathrm { c } }$ , note that both models achieve OSR of at least $0 \mathrm { d B }$ for $7 5 \%$ of examples, and thus suppress off-screen sounds for at least $7 5 \%$ of the test data.
+
+![](images/2e232021f75fb318bb22d55b5fa2a7d0954d48a1cc6f1bb8a3f9b834cb866e33.jpg)
+
+(a) On-screen reconstruction performance in terms of SI-SNR for on-screen MoMs, for each input SI-SNR bucket.
+
+![](images/7ea8adf48b1d71cc0960212c91f8708f84d6d5cb76c7ec73074411ce726f1540.jpg)
+
+(b) On-screen reconstruction performance in terms of SI-SNR improvement (SI-SNRi) for on-screen MoMs, for each input SI-SNR bucket.
+
+![](images/a6ec79c34ccd0266a9b529a1560756c62ae607dfec3c7e66b625b6440c579d2f.jpg)  
+(c) Off-screen power suppression (OSR) distribution for off-screen MoMs, for each input mixture power bucket in dB.
+
+Figure 9: Distribution plots for the performance obtained by the best model under different ranges of input SI-SNR and input mixture powers, for both unsupervised and semi-supervised settings. For each distribution plot, we depict the 25th, 50th and 75th percentiles with dashed lines. The settings for the models are SOff $0 \%$ and active combinations (AC) cross-entropy loss.
+
+# A.7 ATTRIBUTIONS
+
+Images in figures are resized stills with or without overlaid attention maps from the following videos.
+
+“Whitethroat” by S. Rae CC-BY 2.0   
+Figure 2   
+“Luchador and Yellow Jumpsuit” by tenaciousme CC-BY 2.0   
+Figure 4   
+“Video of six hands on the piano” by superhua CC-BY 2.0   
+“Small waterfall” by Jay Tamboli CC-BY 2.0   
+“Roval lap” by mcipseric CC-BY 2.0   
+“photo” by Lentini CC-BY 2.0   
+“IMG_0202 (2 of 14)” by Kerry Goodwin Photography CC-BY 2.0   
+“Archive Video 7: Party in my tummy.” by danoxster CC-BY-SA 2.0   
+“Untitled” by Jacob Davies CC-BY-SA 2.0   
+Figure 5   
+“Steve’s Bobber finished , test ride Video !” by REDMAXSPEEDSHOP.COM CC-BY-SA 2.0   
+“Natural Resources Program” by Kentuckyguard CC-BY 2.0   
+“Ray and Nana” by spilltojill CC-BY-SA 2.0   
+“HDV_0083” by winsors CC-BY 2.0   
+“Somewhere Over the Rainbow” by Mikol CC-BY-SA 2.0   
+“IMG_2797” by Lentini CC-BY 2.0   
+Figure 6   
+“MOV04180” by mike_troiano CC-BY 2.0   
+“Video of him playing drums” superhua CC-BY 2.0   
+“Day 08 - Killarney” by brandonzeman CC-BY-SA 2.0   
+“Jenner” by Mr. Gunn CC-BY-SA 2.0   
+“Ray - 6 months” by spilltojill CC-BY-SA 2.0   
+“Hedwig’s Theme” by Mikol CC-BY-SA 2.0   
+“5-20 (6)” by nmuster1 CC-BY 2.0   
+Figure 7   
+“Bretagne” by MadMonday CC-BY 2.0   
+“Untitled” by Sack-Sama CC-BY-SA 2.0   
+“Natural Resources Program” by Kentuckyguard CC-BY 2.0   
+“Archive Video 7: Party in my tummy.” by danoxster CC-BY-SA 2.0   
+“Video of Micah singing "Old MacDonald"” by superhua CC-BY 2.0

parse/train/SJeD3CEFPH/SJeD3CEFPH.md

@@ -0,0 +1,381 @@
+# M E T A - Q - L E A R N I N G
+
+Rasool Fakoor1, Pratik Chaudhari2∗, Stefano Soatto1, Alexander Smola1
+
+1 Amazon Web Services   
+2 University of Pennsylvania   
+Email: {fakoor, soattos, smola}@amazon.com, pratikac@seas.upenn.edu
+
+# A B S T R A C T
+
+This paper introduces Meta-Q-Learning (MQL), a new off-policy algorithm for meta-Reinforcement Learning (meta-RL). MQL builds upon three simple ideas. First, we show that Q-learning is competitive with state-of-the-art meta-RL algorithms if given access to a context variable that is a representation of the past trajectory. Second, a multi-task objective to maximize the average reward across the training tasks is an effective method to meta-train RL policies. Third, past data from the meta-training replay buffer can be recycled to adapt the policy on a new task using off-policy updates. MQL draws upon ideas in propensity estimation to do so and thereby amplifies the amount of available data for adaptation. Experiments on standard continuous-control benchmarks suggest that MQL compares favorably with the state of the art in meta-RL.
+
+# 1 I N T R O D U C T I O N
+
+Reinforcement Learning (RL) algorithms have demonstrated good performance on simulated data. There are however two main challenges in translating this performance to real robots: (i) robots are complex and fragile which precludes extensive data collection, and (ii) a real robot may face an environment that is different than the simulated environment it was trained in. This has fueled research into MetaReinforcement Learning (meta-RL) which develops algorithms that “meta-train” on a large number of different environments, e.g., simulated ones, and aim to adapt to a new environment with few data.
+
+How well does meta-RL work today? Fig. 1 shows the performance of two prototypical meta-RL algorithms on four standard continuous-control benchmarks.1 We compared them to the following simple baseline: an off-policy RL algorithm (TD3 by Fujimoto et al. (2018b)) and which was trained to maximize the average reward over all training tasks and modified to use a “context variable” that represents the trajectory. All algorithms in this figure use the same evaluation protocol. It is surprising that this simple non-meta-learning-based method is competitive with state-of-the-art meta-RL algorithms. This is the first contribution of our paper: we demonstrate that it is not necessary to meta-train policies to do well on existing benchmarks.
+
+![](images/b1a6beb24d0dbc9d0d33b828a72a6b9c62826ee97037a856fc0ced54b492f7ae.jpg)  
+Figure 1: How well does meta-RL work? Average returns on validation tasks compared for two prototypical meta-RL algorithms, MAML (Finn et al., 2017) and PEARL (Rakelly et al., 2019), with those of a vanilla Q-learning algorithm named TD3 (Fujimoto et al., 2018b) that was modified to incorporate a context variable that is a representation of the trajectory from a task (TD3-context). Even without any meta-training and adaptation on a new task, TD3-context is competitive with these sophisticated algorithms.
+
+Our second contribution is an off-policy meta-RL algorithm named Meta-Q-Learning (MQL) that builds upon the above result. MQL uses a simple meta-training procedure: it maximizes the average rewards across all meta-training tasks using off-policy updates to obtain
+
+$$
+\widehat { \theta } _ { \mathrm { m e t a } } = \arg \operatorname* { m a x } _ { \theta } \frac { 1 } { n } \sum _ { k = 1 } ^ { n } \mathbb { E } _ { \sim D ^ { k } } \left[ \ell ^ { k } ( \theta ) \right]
+$$
+
+where $\ell ^ { k } ( \theta )$ is the objective evaluated on the transition $\tau$ obtained from the task $D ^ { k } ( \theta )$ , e.g., 1-step temporal-difference (TD) error would set $\ell ^ { k } ( \theta ) \ : = \ : \mathrm { T D ^ { 2 } } ( \theta ; \tau )$ . This objective, which we call the multi-task objective, is the simplest form of meta-training.
+
+For adapting the policy to a new task, MQL samples transitions from the meta-training replay buffer that are similar to those from the new task. This amplifies the amount of data available for adaptation but it is difficult to do because of the large potential bias. We use techniques from the propensity estimation literature for performing this adaptation and the off-policy updates of MQL are crucial to doing so. The adaptation phase of MQL solves
+
+$$
+\arg \operatorname* { m a x } _ { \theta } \left\{ \underset { - \sqrt { S ^ { \mathrm { a e s } } } } { \mathbb { E } } \left[ \ell ^ { \mathrm { n e s } } ( \theta ) \right] + \underset { \tau \sim \mathcal { D } _ { \mathrm { m e s } } } { \mathbb { E } } \left[ \beta ( \tau ; D ^ { \mathrm { n e s } } , \mathcal { D } _ { \mathrm { m e t a } } ) \ell ^ { \mathrm { n e s } } ( \theta ) \right] - \left( 1 - \widehat { \mathrm { E S S } } \right) \Vert \theta - \widehat { \theta } _ { \mathrm { m e t a } } \Vert _ { 2 } ^ { 2 } \right\}
+$$
+
+where $\mathcal { D } _ { \mathrm { m e t a } }$ is the meta-training replay buffer, the propensity score $\beta ( \tau ; D ^ { \mathrm { n e w } } , \mathcal { D } _ { \mathrm { m e t a } } )$ is the odds of a transition $\tau$ belonging to $D ^ { \mathrm { n e w } }$ versus $\mathcal { D } _ { \mathrm { m e t a } }$ , and $\widehat { \mathrm { E S S } }$ is the Effective Sample Size between $D ^ { \mathrm { n e w } }$ and $\mathcal { D } _ { \mathrm { m e t a } }$ that is a measure of the similarly of the new task with the meta-training tasks. The first term computes off-policy updates on the new task, the second term performs $\beta ( \cdot )$ -weighted off-policy updates on old data, while the third term is an automatically adapting proximal term that prevents degradation of the policy during adaptation.
+
+We perform extensive experiments in Sec. 4.2 including ablation studies using standard meta-RL benchmarks that demonstrate that MQL policies obtain higher average returns on new tasks even if they are meta-trained for fewer time-steps than state-of-the-art algorithms.
+
+# 2 B A C K G R O U N D
+
+This section introduces notation and formalizes the meta-RL problem. We discuss techniques for estimating the importance ratio between two probability distributions in Sec. 2.2.
+
+Consider a Markov Decision Processes (MDP) denoted by
+
+$$
+x _ { t + 1 } = f ^ { k } ( x _ { t } , u _ { t } , \xi _ { t } ) \quad x _ { 0 } \sim p _ { 0 } ^ { k } ,
+$$
+
+where $x _ { t } ~ \in ~ X ~ \subset ~ \mathbb { R } ^ { d }$ are the states and $u _ { t } ~ \in ~ U ~ \subset ~ \mathbb { R } ^ { p }$ are the actions. The dynamics $f ^ { k }$ is parameterized by $k \in \{ 1 , \ldots , n \}$ where each $k$ corresponds to a different task. The domain of all these tasks, $X$ for the states and $U$ for the actions, is the same. The distribution $p _ { 0 } ^ { k }$ denotes the initial state distribution and $\xi _ { t }$ is the noise in the dynamics. Given a deterministic policy $u _ { \theta } ( x _ { t } )$ , the actionvalue function for $\gamma$ -discounted future rewards $r _ { t } ^ { k } : = r ^ { k } ( x _ { t } , u _ { \theta } ( x _ { t } ) )$ over an infinite time-horizon is
+
+$$
+q ^ { k } ( x , u ) = \operatorname * { \mathbb { E } } _ { \xi _ { ( \cdot ) } } \Big [ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } r _ { t } ^ { k } \ | \ x _ { 0 } = x , u _ { 0 } = u , u _ { t } = u _ { \theta } ( x _ { t } ) \Big ] .
+$$
+
+Note that we have assumed that different tasks have the same state and action space and may only differ in their dynamics $f ^ { k }$ and reward function $r ^ { k }$ . Given one task $k \in \{ 1 , \ldots , n \}$ , the standard Reinforcement Learning (RL) formalism solves for
+
+$$
+\widehat { \theta ^ { k } } = \arg \operatorname* { m a x } _ { \theta } \ell ^ { k } ( \theta ) \quad \mathrm { w h e r e } \ \ell ^ { k } ( \theta ) = \underset { x \sim p _ { 0 } } { \mathbb { E } } \left[ q ^ { k } ( x , u _ { \theta } ( x ) ) \right] .
+$$
+
+Let us denote the dataset of all states, actions and rewards pertaining to a task $k$ and policy $u _ { \theta } ( x )$ by
+
+$$
+\begin{array} { r } { D ^ { k } ( \theta ) = \left\{ x _ { t } , u _ { \theta } ( x _ { t } ) , r ^ { k } , x _ { t + 1 } = f ^ { k } ( x _ { t } , u _ { \theta } ( x _ { t } ) , \xi _ { t } ) \right\} _ { t \geq 0 , \ x ( 0 ) \sim p _ { 0 } ^ { k } , \ \xi ( \cdot ) } ; } \end{array}
+$$
+
+we will often refer to $D ^ { k }$ as the “task” itself. The Deterministic Policy Gradient (DPG) algorithm (Silver et al., 2014) for solving (5) learns a $\varphi$ -parameterized approximation $q _ { \varphi }$ to the optimal value func
+
+tion $q ^ { k }$ by minimizing the Bellman error and the optimal policy $u _ { \theta }$ that maximizes this approximation by solving the coupled optimization problem
+
+$$
+\begin{array} { r l } & { \widehat { \varphi ^ { k } } = \arg \operatorname* { m i n } _ { \varphi } \underset { \tau \sim D ^ { k } } { \mathbb { E } } \Big [ \Big ( q _ { \varphi } ( x , u ) - r ^ { k } - \gamma q _ { \varphi } ( x ^ { \prime } , u _ { \widehat { \theta ^ { k } } } ( x ^ { \prime } ) ) \Big ) ^ { 2 } \Big ] , } \\ & { \widehat { \theta ^ { k } } = \arg \underset { \theta } { \operatorname* { m a x } } \underset { \tau \sim D ^ { k } } { \mathbb { E } } \Big [ q _ { \widehat { \varphi ^ { k } } } ( x , u _ { \theta } ( x ) ) \Big ] . } \end{array}
+$$
+
+The 1-step temporal difference error (TD error) is defined as
+
+$$
+\mathrm { T D } ^ { 2 } ( \theta ) = \bigl ( q _ { \varphi } ( x , u ) - r ^ { k } - \gamma q _ { \varphi } ( x ^ { \prime } , u _ { \theta } ( x ^ { \prime } ) ) \bigr ) ^ { 2 }
+$$
+
+where we keep the dependence of $\operatorname { T D } ( \cdot )$ on $\varphi$ implicit. DPG, or its deep network-based variant DDPG (Lillicrap et al., 2015), is an off-policy algorithm. This means that the expectations in (6) are computed using data that need not be generated by the policy being optimized $( u _ { \theta } )$ , this data can come from some other policy.
+
+In the sequel, we will focus on the parameters $\theta$ parameterizing the policy. The parameters $\varphi$ of the value function are always updated to minimize the TD-error and are omitted for clarity.
+
+# 2 . 1 M E T A - R E I N F O R C E M E N T L E A R N I N G ( M E T A - R L )
+
+Meta-RL is a technique to learn an inductive bias that accelerates the learning of a new task by training on a large of number of training tasks. Formally, meta-training on tasks from the meta-training set $\mathcal { D } _ { \mathrm { m e t a } } = \ \backslash \ \{ D ^ { k } \} _ { k = 1 , \ldots , n }$ involves learning a policy
+
+$$
+\widehat { \theta } _ { \mathrm { m e t a } } = \arg \operatorname* { m a x } _ { \theta } \frac { 1 } { n } \sum _ { k = 1 } ^ { n } \ell _ { \mathrm { m e t a } } ^ { k } ( \theta )
+$$
+
+where let us $\ell _ { \mathrm { m e t a } } ^ { k } ( \theta )$ is a meta-training loss that depends on the particular method. Gradient-based meta-RL,ML by Finn et al. (2017) as a concrete example, sets
+
+$$
+\ell _ { \mathrm { m e t a } } ^ { k } ( \theta ) = \ell ^ { k } ( \theta + \alpha \nabla _ { \theta } \ell ^ { k } ( \theta ) )
+$$
+
+for a step-size $\alpha > 0 ; \ell ^ { k } ( \theta )$ is the objective of non-meta-RL (5). In this case $\ell _ { \mathrm { m e t a } } ^ { k }$ is the objective obtained on the task $D ^ { k }$ after one (or in general, more) updates of the policy on the task. The idea behind this is that even if the policy $\widehat { \theta } _ { \mathrm { { m e t a } } }$ does not perform well on all tasks in $\mathcal { D } _ { \mathrm { m e t a } }$ it may be updated quickly on a new task $D ^ { \mathrm { n e w } }$ to obtain a well-performing policy. This can either be done using the same procedure as that of meta-training time, i.e., by maximizing $\ell _ { \mathrm { m e t a } } ^ { \mathrm { n e w } } ( \theta )$ with the policy $\widehat { \theta } _ { \mathrm { { m e t a } } }$ as the initialization, or by some other adaptation procedure. The meta-training method and the adaptation method in meta-RL, and meta-learning in general, can be different from each other.
+
+# 2 . 2 L O G I S T I C R E G R E S S I O N F O R E S T I M A T I N G T H E P R O P E N S I T Y S C O R E
+
+Consider standard supervised learning: given two distributions $q ( x )$ (say, train) and $p ( x )$ (say, test), we would like to estimate how a model’s predictions ${ \hat { y } } ( x )$ change across them. This is formally done using importance sampling:
+
+$$
+{ \underset { x \sim p ( x ) } { \mathbb { E } } } { \underset { y \mid x } { \mathbb { E } } } \left[ \ell ( y , { \hat { y } } ( x ) ) \right] = { \underset { x \sim q ( x ) } { \mathbb { E } } } { \underset { y \mid x } { \mathbb { E } } } \left[ \beta ( x ) \ell ( y , { \hat { y } } ( x ) ) \right] ;
+$$
+
+where $y | x$ are the true labels of data, the predictions of the model are ${ \hat { y } } ( x )$ and $\ell ( y , \hat { y } ( x ) )$ is the loss for each datum $( x , y )$ . The importance ratio $\begin{array} { r } { \beta ( x ) \ = \ \frac { \mathrm { d } p } { \mathrm { d } q } ( x ) \ } \end{array}$ , also known as the propensity score, is the Radon-Nikodym derivative (Resnick, 2013) of the two data densities and measures the odds of a sample $x$ coming from the distribution $p$ versus the distribution $q$ . In practice, we do not know the densities $q ( x )$ and $p ( x )$ and therefore need to estimate $\beta ( x )$ using some finite data $X _ { q } = \{ x _ { 1 } , \dots , x _ { m } \}$ drawn from $q$ and $X _ { p } = \{ x _ { 1 } ^ { \prime } , \ldots , x _ { m } ^ { \prime } \}$ drawn from $p$ . As Agarwal et al. (2011) show, this is easy to do using logistic regression. Set $z _ { k } = 1$ to be the labels for the data in $X _ { q }$ and $z _ { k } = - 1$ to be the labels of the data in $X _ { p }$ for $k \leq m$ and fit a logistic classifier on the combined $2 m$
+
+samples by solving
+
+$$
+w ^ { * } = \operatorname* { m i n } _ { w } \ \frac { 1 } { 2 m } \sum _ { ( x , z ) } \log \left( 1 + e ^ { - z w ^ { \top } x } \right) + c \left\| w \right\| ^ { 2 } .
+$$
+
+This gives
+
+$$
+\beta ( x ) = \frac { \mathbb { P } ( z = - 1 | x ) } { \mathbb { P } ( z = 1 | x ) } = e ^ { - w ^ { * } ^ { \top } x } .
+$$
+
+Normalized Effective Sample Size $\widehat { ( \mathbf { E S S } ) }$ : A related quantity to $\beta ( x )$ is the normalized Effective Sample Size (ESS ) which we define as the relative number of samples from the target distribution $p ( x )$ required to obtain an estimator with performance (say, variance) equal to that of the importance sampling estimator (10). It is not possible to compute the $\widehat { \mathrm { E S S } }$ without knowing both densities $q ( x )$ and $p ( x )$ but there are many heuristics for estimating it. A popular one in the Monte Carlo literature (Kong, 1992; Smith, 2013; Elvira et al., 2018) is
+
+$$
+\widehat { \mathrm { E S S } } = \frac { 1 } { m } \frac { \left( \sum _ { k = 1 } ^ { m } \beta ( x _ { k } ) \right) ^ { 2 } } { \sum _ { k = 1 } ^ { m } \beta ( x _ { k } ) ^ { 2 } } \in [ 0 , 1 ]
+$$
+
+where $X = \{ x _ { 1 } , \ldots , x _ { m } \}$ is some finite batch of data. Observe that if two distributions $q$ and $p$ are close then the $\widehat { \mathrm { E S S } }$ is close to one; if they are far apart the $\widehat { \mathrm { E S S } }$ is close to zero.
+
+#
+
+This section describes the MQL algorithm. We begin by describing the meta-training procedure of MQL including a discussion of multi-task training in Sec. 3.1. The adaptation procedure is described in Sec. 3.2.
+
+# 3 . 1 M E T A - T R A I N I N G
+
+MQL performs meta-training using the multi-task objective. Note that if one sets
+
+$$
+\ell _ { \mathrm { m e t a } } ^ { k } ( \theta ) \triangleq \ell ^ { k } ( \theta ) = \underset { x \sim p _ { 0 } ^ { k } } { \mathbb { E } } \left[ q ^ { k } ( x , u _ { \theta } ( x ) ) \right]
+$$
+
+in (8) then the parameters $\widehat { \theta } _ { \mathrm { { m e t a } } }$ are such that they maximize the average returns over all tasks from the meta-training set. We use an off-policy algorithm named TD3 (Fujimoto et al., 2018b) as the building block and solve for
+
+$$
+\widehat { \theta } _ { \mathrm { m e t a } } = \arg \operatorname* { m i n } _ { \theta } \frac { 1 } { n } \sum _ { k = 1 } ^ { n } \mathbb { E } _ { \sim D ^ { k } } \left[ \mathbf { T D } ^ { 2 } ( \theta ) \right] ;
+$$
+
+where $\operatorname { T D } ( \cdot )$ is defined in (7). As is standard in TD3, we use two action-value functions parameterized by $\varphi _ { 1 }$ and $\varphi _ { 2 }$ and take their minimum to compute the target in (7). This trick known as “doubleQ-learning” reduces the over-estimation bias. Let us emphasize that (14) is a special case of the procedure outlined in (8). The following remark explains why MQL uses the multi-task objective as opposed to the meta-training objective used, for instance, in existing gradient-based meta-RL algorithms.
+
+Remark 1. Let us compare the critical points of the $m$ -step MAML objective (9) to those of the multi-task objective which uses (14). As is done by the authors in Nichol et al. (2018), we can perform a Taylor series expansion around the parameters $\theta$ to obtain
+
+$$
+\nabla \ell _ { \mathrm { m e t a } } ^ { k } ( \theta ) = \nabla \ell ^ { k } ( \theta ) + 2 \alpha ( m - 1 ) \left( \nabla ^ { 2 } \ell ^ { k } ( \theta ) \right) \nabla \ell ^ { k } ( \theta ) + \mathcal { O } ( \alpha ^ { 2 } ) .
+$$
+
+Further, note that $\nabla \ell _ { \mathrm { m e t a } } ^ { k }$ in (16) is also the gradient of the loss
+
+$$
+\ell ^ { k } ( \theta ) + \alpha ( m - 1 ) \| \nabla \ell ^ { k } ( \theta ) \| _ { 2 } ^ { 2 }
+$$
+
+up to first order. This lends a new interpretation that MAML is attracted towards regions in the loss landscape that under-fit on individual tasks: parameters with large $\| \nabla \ell ^ { k } \| _ { 2 }$ will be far from the local maxima of $\ell ^ { k } ( \theta )$ . The parameters $\alpha$ and $m$ control this under-fitting. Larger the number of gradient steps, larger the under-fitting effect. This remark suggests that the adaptation speed of gradient-based meta-learning comes at the cost of under-fitting on the tasks.
+
+# 3 . 1 . 1 D E S I G N I N G C O N T E X T
+
+As discussed in Sec. 1 and 4.4, the identity of the task in meta-RL can be thought of as the hidden variable of an underlying partially-observable MDP. The optimal policy on the entire trajectory of the states, actions and the rewards. We therefore design a recurrent context variable $z _ { t }$ that depends on $\{ ( x _ { i } , u _ { i } , r _ { i } ) \} _ { i \leq t }$ . We set $z _ { t }$ to the hidden state at time $t$ of a Gated Recurrent Unit (GRU by Cho et al. (2014)) model. All the policies $u _ { \theta } ( x )$ and value functions $q _ { \varphi } ( x , u )$ in MQL are conditioned on the context and implemented as $u _ { \theta } ( x , z )$ and $q _ { \varphi } ( x , u , z )$ . Any other recurrent model can be used to design the context; we used a GRU because it offers a good trade-off between a rich representation and computational complexity.
+
+Remark 2 (MQL uses a deterministic context that is not permutation invariant). We have aimed for simplicity while designing the context. The context in MQL is built using an off-the-shelf model like GRU and is not permutation invariant. Indeed, the direction of time affords crucial information about the dynamics of a task to the agent, e.g., a Half-Cheetah running forward versus backward has arguably the same state trajectory but in a different order. Further, the context in MQL is a deterministic function of the trajectory. Both these aspects are different than the context used by Rakelly et al. (2019) who design an inference network and sample a probabilistic context conditioned on a moving window. RL algorithms are quite complex and challenging to reproduce. Current meta-RL techniques which build upon them further exacerbate this complexity. Our demonstration that a simple context variable is enough is an important contribution.
+
+# 3 . 2 A D A P T AT I O N T O A N E W T A S K
+
+We next discuss the adaptation procedure which adapts the meta-trained policy $\widehat { \theta } _ { \mathrm { { m e t a } } }$ to a new task $D ^ { \mathrm { n e w } }$ with few data. MQL optimizes the adaptation objective introduced in (2) into two steps.
+
+1. Vanilla off-policy adaptation: The first step is to update the policy using the new data as
+
+$$
+\underset { \theta } { \arg \operatorname* { m a x } } \left\{ \underset { \tau \sim D ^ { \mathrm { n e w } } } { \mathbb { E } } \left[ \ell ^ { \mathrm { n e w } } ( \theta ) \right] - \frac { \lambda } { 2 } \| \theta - \widehat { \theta } _ { \mathrm { m e t a } } \| _ { 2 } ^ { 2 } \right\} .
+$$
+
+The quadratic penalty $\lVert \boldsymbol { \theta } - \widehat { \boldsymbol { \theta } } _ { \mathrm { { m e t a } } } \rVert ^ { 2 }$ keeps the parameters close to $\widehat { \theta } _ { \mathrm { { m e t a } } }$ . This is crucial to reducing the variance of the model that is adapted using few data from the new task (Reddi et al., 2015). Off-policy learning is critical in this step because of its sample efficiency. We initialize $\theta$ to $\widehat { \theta } _ { \mathrm { { m e t a } } }$ while solving (18).
+
+2. Importance-ratio corrected off-policy updates: The second step of MQL exploits the metatraining replay buffer. Meta-training tasks $\mathcal { D } _ { \mathrm { m e t a } }$ are disjoint from $D ^ { \mathrm { n e w } }$ but because they are expected to come from the same task distribution, transitions collected during meta-training can potentially be exploited to adapt the policy. This is difficult to do on two counts. First, the meta-training transitions do not come from $D ^ { \mathrm { n e w } }$ . Second, even for transitions from the same task, it is non-trivial to update the policy because of extrapolation error (Fujimoto et al., 2018a): the value function has high error on states it has not seen before. Our use of the propensity score to reweigh transitions is a simpler version of the conditional generative model used by Fujimoto et al. (2018a) in this context.
+
+MQL fits a logistic classifier on a mini-batch of transitions from the meta-training replay buffer and the transitions collected from the new task in step 1. The context variable $z _ { t }$ is the feature for this classifier. The logistic classifier estimates the importance ratio $\beta ( \tau ; D ^ { \mathrm { n e w } } , \mathcal { D } _ { \mathrm { m e t a } } )$ and can be used to reweigh data from the meta-training replay buffer for taking updates as
+
+$$
+\arg \operatorname* { m a x } _ { \theta } \left\{ { \underset { \tau \sim \mathcal { D } _ { \mathrm { m e t a } } } { \mathbb { E } } } \left[ \beta ( \tau ; D ^ { \mathrm { n e w } } , \mathcal { D } _ { \mathrm { m e t a } } ) \ \ell ^ { \mathrm { n e w } } ( \theta ) \right] - \frac { \lambda } { 2 } \Vert \theta - \widehat { \theta } _ { \mathrm { m e t a } } \Vert _ { 2 } ^ { 2 } \right\} .
+$$
+
+We have again included a quadratic penalty $\lVert \boldsymbol { \theta } - \widehat { \boldsymbol { \theta } } _ { \mathrm { { m e t a } } } \rVert ^ { 2 }$ that keeps the new parameters close to $\widehat { \theta } _ { \mathrm { { m e t a } } }$ . Estimating the importance ratio involves solving a convex optimization problem on few samples (typically, 200 from the new task and 200-400 from the meta-training tasks). This classifier allows MQL to exploit the large amount of past data. In practice, we perform as many as $1 0 0 \times$ more weight updates using (19) than (18).
+
+Remark 3 (Picking the coefficient $\boldsymbol { \lambda }$ ). Following Fakoor et al. (2019), we pick
+
+$$
+\lambda = 1 - { \widehat { \mathrm { E S S } } }
+$$
+
+for both the steps (18–19). This relaxes the quadratic penalty if the new task is similar to the metatraining tasks (ESS  is large) and vice-versa. While $\lambda$ could be tuned as a hyper-parameter, our empirical results show that adapting it using $\widehat { \mathrm { E S S } }$ is a simple and effective heuristic.
+
+Remark 4 (Details of estimating the importance ratio). It is crucial to ensure that the logistic classifier for estimating $\beta$ generalizes well if we are to reweigh transitions in the meta-training replay buffer that are different than the ones the logistic was fitted upon. We do so in two ways: (i) the regularization co-efficient in (11) is chosen to be relatively large, that way we prefer false negatives than risk false positives; (ii) transitions with very high $\beta$ are valuable for updating (19) but cause a large variance in stochastic gradient descent-based updates, we clip $\beta$ before taking the update in (19). The clipping constant is a hyper-parameter and is given in Sec. 4.
+
+MQL requires having access to the meta-training replay buffer during adaptation. This is not a debilitating requirement and there are a number of clustering techniques that can pick important transitions from the replay-buffer if a robotic agent is limited by available hard-disk space. The meta-training replay buffer is at most 3 GB for the experiments in Sec. 4.
+
+# 4 E X P E R I M E N T S
+
+This section presents the experimental results of MQL. We first discuss the setup and provide details the benchmark in Sec. 4.1. This is followed by empirical results and ablation experiments in Sec. 4.2.
+
+# 4 . 1 S E T U P
+
+Tasks and algorithms: We use the MuJoCo (Todorov et al., 2012) simulator with OpenAI Gym (Brockman et al., 2016) on continuous-control meta-RL benchmark tasks. These tasks have different rewards, randomized system parameters (Walker-2D-Params) and have been used in previous papers such as Finn et al. (2017); Rothfuss et al. (2018); Rakelly et al. (2019). We compare against standard baseline algorithms, namely MAML (TRPO (Schulman et al., 2015) variant) (Finn et al., 2017), RL2 (Duan et al., 2016), ProMP (Rothfuss et al., 2018) and PEARL (Rakelly et al., 2019). We obtained the training curves and hyper-parameters for all the three algorithms from the published code by Rakelly et al. (2019).
+
+![](images/eafe95396a5aef45830b380729fae5360000332773a15ea5df288a806f78b861.jpg)  
+Figure 2: Average undiscounted return of TD3 and TD3-context compared with PEARL for validation tasks from four meta-RL environments. The agent fails to learn if the policy is conditioned only on the state. In contrast, everything else remaining same, if TD3 is provided access to context, the rewards are much higher. In spite of not adaptating on the validation tasks, TD3- context is comparable to PEARL.
+
+We will compare the above algorithms against: (i) vanilla TD3 (Fujimoto et al., 2018a) without any adaptation on new tasks, (ii) TD3-context: TD3 with GRU-based context Sec. 3.1.1 without any adaptation, and (iii) MQL: TD3 with context and adaptation on new task using the procedure in Sec. 3.2. All the three variants use the multi-task objective for meta-training (15). We use Adam (Kingma & Ba, 2014) for optimizing all the loss functions in this paper.
+
+Evaluation: Current meta-RL benchmarks lack a systematic evaluation procedure. 2 For each environment, Rakelly et al. (2019) constructed a fixed set of meta-training tasks $\left( \mathcal { D } _ { \mathrm { m e t a } } \right)$ and a validation set of tasks $D ^ { \mathrm { n e w } }$ that are disjoint from the meta-training set. To enable direct comparison with published empirical results, we closely followed the evaluation code of Rakelly et al. (2019) to create these tasks. We also use the exact same evaluation protocol as that of these authors, e.g., 200 timesteps of data from the new task, or the number of evaluation episodes. We report the undiscounted return on the validation tasks with statistics computed across 5 random seeds.
+
+# 4 . 2 R E S U L T S
+
+![](images/7543b1ffa2f4cafc441b9270fd2319318343cb8edcebc6d157ae52d1a9808a03.jpg)  
+Figure 3: Comparison of the average undiscounted return of MQL (orange) against existing meta-RL algorithms on continuous-control environments. We compare against four existing algorithms, namely MAML (green), RL2 (red), PROMP (purple) and PEARL (blue). In all environments except Walker-2D-Params and Ant-Goal-2D, MQL is better or comparable to existing algorithms in terms of both sample complexity and final returns.
+
+Our first result, in Fig. 2, is to show that vanilla off-policy learning with context, without any adaptation is competitive with state of the art meta-RL algorithms. We used a standard implementation of TD3 and train on the meta-training tasks using the multi-task objective (15). Hyper-parameters for these tasks are provided in Appendix D. This result is surprising and had gone unnoticed in the current literature. Policies that have access to the context can easily generalize to the validation tasks and achieve performance that is comparable to more sophisticated meta-RL algorithms.
+
+We next evaluate MQL against existing meta-RL benchmarks on all environments. The results are shown in Fig. 3. We see that for all environments except Walker-2D-Params and Ant-Goal-2D, MQL obtains comparable or better returns on the validation tasks. In most cases, in particular for the challenging Humanoid-Direc-2D environment, MQL converges faster than existing algorithms. MAML and ProMP require about 100M time-steps to converge to returns that are significantly worse than the returns of off-policy algorithms like MQL and PEARL. Compare the training curve for TD3-context for the Ant-Goal-2D environment in Fig. 2 with that of the same environment in Fig. 3: the former shows a prominent dip in performance as meta-training progresses; this dip is absent in Fig. 3 and can be attributed to the adaptation phase of MQL.
+
+# 4 . 3 A B L A T I O N E X P E R I M E N T S
+
+We conduct a series of ablation studies to analyze the different components of the MQL algorithm. We use two environments for this purpose, namely Half-Cheetah-Fwd-Back and Ant-Fwd-Back. Fig. 4a shows that the adaptation in MQL in (18) and (19) improves performance. Also observe that MQL has a smaller standard deviation in the returns as compared to TD3-context which does not perform any adaptation; this can be seen as the adaptation phase making up for the lost performance of the meta-trained policy on a difficult task. Next, we evaluate the importance of the additional data from the replay buffer in MQL. Fig. 4b compares the performance of MQL with and without updates in (19). We see that the old data, even if it comes from different tasks, is useful to improve the performance on top of (18). Fig. 4c shows the effectiveness of setting $\lambda = 1 - { \widehat { \mathrm { E S S } } }$ as compared to a fixed value of $\lambda = 0 . 5$ . We see that modulating the quadratic penalty with $\widehat { \mathrm { E S S } }$ helps, the effect is minor for Sec. 4.3. The ideal value of $\lambda$ depends on a given task and using $1 - { \widehat { \mathrm { E S S } } }$ can help to adjust to different tasks without the need to do hyper-parameter search per task. Finally, Fig. 5 shows the evolution of $\lambda$ and $\beta ( z )$ during meta-training. The coefficient $\lambda$ is about 0.55 and $\beta ( z )$ is 0.8 for a large fraction of the time. The latter indicates that propensity score estimation is successful in sampling transitions from the meta-training replay buffer that are similar to the validation tasks. The value of $\lambda$ remains relatively unchanged during training. This value indicates the fraction of transitions in the old data that are similar to those from the new tasks; since there are two distinct tasks in Ant-Fwd-Back, the value $\lambda = 0 . 5 5$ is appropriate.
+
+![](images/50ecd84362774187e893ccd7bd046411a834c2b7f94e64540448031304c5a17c.jpg)  
+Figure 4: Ablation studies to examine various components of MQL.
+
+# 4 . 4 R E L AT E D W O R K
+
+Learning to learn: The idea of building an inductive bias for learning a new task by training on a large number of related tasks was established in a series of works (Utgoff, 1986; Schmidhuber, 1987; Baxter, 1995; Thrun, 1996; Thrun & Pratt, 2012). These papers propose building a base learner that fits on each task and a meta-learner that learns properties of the base learners to output a new base learner for a new task. The recent literature instantiates this idea in two forms: (i) the meta-learner directly predicts the base-learner (Wang et al., 2016; Snell et al., 2017) and (ii) the meta-learner learns the updates of the base-learner (Bengio et al., 1992; Hochreiter et al., 2001; Finn et al., 2017).
+
+Meta-training versus multi-task training: Metatraining aims to train a policy that can be adapted efficiently on a new task. Conceptually, the improved efficiency of a meta-learner comes from two things: (i) building a better inductive bias to initialize the learning (Schmidhuber et al., 1997; Baxter, 1995; 2000; Mitchell, 1980), or (ii) learning a better learning procedure (Bengio et al., 1997; Lee et al., 2019). The two notions of meta-learning above are complementary to each other and in fact, most recent literature using deep neural networks, e.g., MAML (Finn et al., 2017) and Prototypical Networks (Snell et al., 2017) confirms to the first notion of building a better inductive bias.
+
+The multi-task training objective in MQL is the simplest possible instantiation of this idea: it maximizes the average reward on all tasks and learns a better prior without explicitly training for improving adaptation. This aspect of MQL coincides with a recent trend in meta-learning for image classification where it has been observed that modifications to episodic meta-training (Snell et al., 2017; Gidaris & Komodakis, 2018; Chen et al., 2018), or even foregoing meta-training completely (Dhillon et al., 2019) performs better. We speculate two reasons for this phenomenon: (i) meta-training methods are complex to implement and tune, and (ii) powerful function classes such as deep neural networks may have leftover capacity to adapt to a new task even if they are not explicitly trained for adaptation.
+
+![](images/36c01716fcb64d79d370b897f1c29ebb601f84dbce2eae64ed26cdd023f1eddf.jpg)  
+Figure 5: Evolution of $\lambda$ and $\beta ( z )$ during meta-training.
+
+Context-based approaches: Both forms of meta-learning above have been employed relatively successfully for image classification (Snell et al., 2017; Ravi & Larochelle, 2016; Finn et al., 2017). It has however been difficult to replicate that empirical performance in RL: sensitivity to hyperparameters (Henderson et al., 2018) precludes directly predicting the base-learner while long-range temporal dependencies make it difficult to learn the updates of the base learner (Nichol et al., 2018). Recent methods for meta-RL instead leverage context and learn a policy that depends on just on the current state $x _ { t }$ but on the previous history. This may be done in a recurrent fashion (Heess et al., 2015; Hausknecht & Stone, 2015) or by learning a latent representation of the task (Rakelly et al., 2019). Context is a powerful construct: as Fig. 1 shows, even a simple vanilla RL algorithm (TD3) when combined with context performs comparably to state-of-the-art meta-RL algorithms. However, context is a meta-training technique, it does not suggest a way to adapt a policy to a new task. For instance, Rakelly et al. (2019) do not update parameters of the policy on a new task. They rely on the latent representation of the context variable generalizing to new tasks. This is difficult if the new task is different from the training tasks; we discuss this further in Sec. 3.1.1.
+
+Policy-gradient-based algorithms versus off-policy methods: Policy-gradient-based methods have high sample complexity (Ilyas et al., 2018). This is particularly limiting for meta-RL (Finn et al., 2017; Rothfuss et al., 2018; Houthooft et al., 2018) where one (i) trains on a large number of tasks and, (ii) aims to adapt to a new task with few data. Off-policy methods offer substantial gains in sample complexity. This motivates our use of off-policy updates for both meta-training and adaptation. Off-policy updates allow using past data from other policies. MQL exploits this substantially, it takes up to $1 0 0 \times$ more updates using old data than new data during adaptation. Off-policy algorithms are typically very sensitive to hyper-parameters (Fujimoto et al., 2018a) but we show that MQL is robust to such sensitivity because it adapts automatically to the distribution shift using the Effective Sample Size (ESS).
+
+Propensity score estimation has been extensively studied in both statistics (Robert & Casella, 2013; Quionero-Candela et al., 2009) and RL (Dud´ık et al., 2011; Jiang & Li, 2015; Kang et al., 2007; Bang & Robins, 2005). It is typically used to reweigh data from the proposal distribution to compute estimators on the target distribution. MQL uses propensity scores in a novel way: we fit a propensity score estimator on a subset of the meta-training replay buffer and use this model to sample transitions from the replay buffer that are similar to the new task. The off-policy updates in MQL are essential to exploiting this data. The coefficient of the proximal term in the adaptation-phase objective (18–19) using the effective sample size (ESS) is inspired from the recent work of Fakoor et al. (2019).
+
+# 5 D I S C U S S I O N
+
+The algorithm proposed in this paper, namely MQL, builds upon on three simple ideas. First, Qlearning with context is sufficient to be competitive on current meta-RL benchmarks. Second, maximizing the average reward of training tasks is an effective meta-learning technique. The meta-training phase of MQL is significantly simpler than that of existing algorithms and yet it achieves comparable performance to the state of the art. This suggests that we need to re-think meta-learning in the context of rich function approximators such as deep networks. Third, if one is to adapt to new tasks with few data, it is essential to exploit every available avenue. MQL recycles data from the meta-training replay buffer using propensity estimation techniques. This data is essentially free and is completely neglected by other algorithms. This idea can potentially be used in problems outside RL such as few-shot and zero-shot image classification.
+
+Finally, this paper sheds light on the nature of benchmark environments in meta-RL. The fact that even vanilla Q-learning with a context variable—without meta-training and without any adaptation— is competitive with state of the art algorithms indicates that (i) training and validation tasks in the current meta-RL benchmarks are quite similar to each other and (ii) current benchmarks may be insufficient to evaluate meta-RL algorithms. Both of these are a call to action and point to the need to invest resources towards creating better benchmark problems for meta-RL that drive the innovation of new algorithms.
+
+# R E F E R E N C E S
+
+Deepak Agarwal, Lihong Li, and Alexander Smola. Linear-time estimators for propensity scores. In Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics, pp. 93–100, 2011.   
+Heejung Bang and James M Robins. Doubly robust estimation in missing data and causal inference models. Biometrics, 61(4):962–973, 2005.   
+Jonathan Baxter. Learning internal representations. Flinders University of S. Aust., 1995.   
+Jonathan Baxter. A model of inductive bias learning. Journal of artificial intelligence research, 12:149–198, 2000.   
+Samy Bengio, Yoshua Bengio, Jocelyn Cloutier, and Jan Gecsei. On the optimization of a synaptic learning rule. In Preprints Conf. Optimality in Artificial and Biological Neural Networks, pp. 6–8. Univ. of Texas, 1992.   
+Samy Bengio, Yoshua Bengio, Jocelyn Cloutier, and Jan Gecsei. On the optimization of a synaptic learning rule, 1997.   
+Greg Brockman, Vicki Cheung, Ludwig Pettersson, Jonas Schneider, John Schulman, Jie Tang, and Wojciech Zaremba. OpenAI Gym. arXiv:1606.01540, 2016.   
+Wei-Yu Chen, Yen-Cheng Liu, Zsolt Kira, Yu-Chiang Frank Wang, and Jia-Bin Huang. A closer look at few-shot classification. 2018.   
+Kyunghyun Cho, Bart Van Merrienboer, Caglar Gulcehre, Dzmitry Bahdanau, Fethi Bougares, Holger Schwenk, ¨ and Yoshua Bengio. Learning phrase representations using rnn encoder-decoder for statistical machine translation. arXiv:1406.1078, 2014.   
+Guneet S Dhillon, Pratik Chaudhari, Avinash Ravichandran, and Stefano Soatto. A baseline for few-shot image classification. arXiv:1909.02729, 2019.   
+Yan Duan, John Schulman, Xi Chen, Peter L Bartlett, Ilya Sutskever, and Pieter Abbeel. Rl2: Fast reinforcement learning via slow reinforcement learning. arXiv:1611.02779, 2016.   
+Miroslav Dud´ık, John Langford, and Lihong Li. Doubly robust policy evaluation and learning. arXiv:1103.4601, 2011.   
+V´ıctor Elvira, Luca Martino, and Christian P Robert. Rethinking the effective sample size. arXiv:1809.04129, 2018.   
+Rasool Fakoor, Pratik Chaudhari, and Alexander J Smola. P3o: Policy-on policy-off policy optimization. arXiv:1905.01756, 2019.   
+Chelsea Finn, Pieter Abbeel, and Sergey Levine. Model-agnostic meta-learning for fast adaptation of deep networks. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pp. 1126– 1135. JMLR. org, 2017.   
+Scott Fujimoto, David Meger, and Doina Precup. Off-policy deep reinforcement learning without exploration. arXiv:1812.02900, 2018a.   
+Scott Fujimoto, Herke van Hoof, and Dave Meger. Addressing function approximation error in actor-critic methods. arXiv:1802.09477, 2018b.   
+Spyros Gidaris and Nikos Komodakis. Dynamic few-shot visual learning without forgetting. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 4367–4375, 2018.   
+Matthew Hausknecht and Peter Stone. Deep recurrent q-learning for partially observable mdps. In 2015 AAAI Fall Symposium Series, 2015.   
+Nicolas Heess, Jonathan J Hunt, Timothy P Lillicrap, and David Silver. Memory-based control with recurrent neural networks. arXiv:1512.04455, 2015.   
+Peter Henderson, Riashat Islam, Philip Bachman, Joelle Pineau, Doina Precup, and David Meger. Deep reinforcement learning that matters. In Thirty-Second AAAI Conference on Artificial Intelligence, 2018.   
+Sepp Hochreiter, A Steven Younger, and Peter R Conwell. Learning to learn using gradient descent. In International Conference on Artificial Neural Networks, pp. 87–94. Springer, 2001.   
+Rein Houthooft, Yuhua Chen, Phillip Isola, Bradly Stadie, Filip Wolski, OpenAI Jonathan Ho, and Pieter Abbeel. Evolved policy gradients. In Advances in Neural Information Processing Systems, pp. 5400–5409, 2018.   
+Andrew Ilyas, Logan Engstrom, Shibani Santurkar, Dimitris Tsipras, Firdaus Janoos, Larry Rudolph, and Aleksander Madry. Are deep policy gradient algorithms truly policy gradient algorithms? arXiv:1811.02553, 2018.   
+Nan Jiang and Lihong Li. Doubly robust off-policy value evaluation for reinforcement learning. arXiv:1511.03722, 2015.   
+Joseph DY Kang, Joseph L Schafer, et al. Demystifying double robustness: A comparison of alternative strategies for estimating a population mean from incomplete data. Statistical science, 22(4):523–539, 2007.   
+Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv:1412.6980, 2014.   
+Augustine Kong. A note on importance sampling using standardized weights. University of Chicago, Dept. of Statistics, Tech. Rep, 348, 1992.   
+Kwonjoon Lee, Subhransu Maji, Avinash Ravichandran, and Stefano Soatto. Meta-learning with differentiable convex optimization. arXiv:1904.03758, 2019.   
+Timothy P Lillicrap, Jonathan J Hunt, Alexander Pritzel, Nicolas Heess, Tom Erez, Yuval Tassa, David Silver, and Daan Wierstra. Continuous control with deep reinforcement learning. arXiv:1509.02971, 2015.   
+Tom M Mitchell. The need for biases in learning generalizations. Department of Computer Science, Laboratory for Computer Science Research . . . , 1980.   
+Alex Nichol, Joshua Achiam, and John Schulman. On first-order meta-learning algorithms. arXiv:1803.02999, 2018.   
+Joaquin Quionero-Candela, Masashi Sugiyama, Anton Schwaighofer, and Neil D Lawrence. Dataset shift in machine learning. The MIT Press, 2009.   
+Kate Rakelly, Aurick Zhou, Deirdre Quillen, Chelsea Finn, and Sergey Levine. Efficient off-policy metareinforcement learning via probabilistic context variables. arXiv:1903.08254, 2019.   
+Sachin Ravi and Hugo Larochelle. Optimization as a model for few-shot learning. 2016.   
+Sashank J. Reddi, Barnabas P ´ oczos, and Alexander J. Smola. Doubly robust covariate shift correction. In ´ AAAI, 2015.   
+Sidney I Resnick. A probability path. Springer Science & Business Media, 2013.   
+Christian Robert and George Casella. Monte Carlo statistical methods. Springer Science & Business Media, 2013.   
+Jonas Rothfuss, Dennis Lee, Ignasi Clavera, Tamim Asfour, and Pieter Abbeel. Promp: Proximal meta-policy search. arXiv:1810.06784, 2018.   
+Jurgen Schmidhuber. Evolutionary principles in self-referential learning. On learning how to learn: The meta-meta-... hook.) Diploma thesis, Institut f. Informatik, Tech. Univ. Munich, 1987.   
+Jurgen Schmidhuber, Jieyu Zhao, and Marco Wiering. Shifting inductive bias with success-story algorithm,¨ adaptive levin search, and incremental self-improvement. Machine Learning, 28(1):105–130, Jul 1997. ISSN 1573-0565.   
+John Schulman, Sergey Levine, Pieter Abbeel, Michael I Jordan, and Philipp Moritz. Trust region policy optimization. In International Conference on Machine Learning, volume 37, pp. 1889–1897, 2015.   
+David Silver, Guy Lever, Nicolas Heess, Thomas Degris, Daan Wierstra, and Martin Riedmiller. Deterministic policy gradient algorithms. In International Conference on Machine Learning, 2014.   
+Adrian Smith. Sequential Monte Carlo methods in practice. Springer Science & Business Media, 2013.   
+Jake Snell, Kevin Swersky, and Richard Zemel. Prototypical networks for few-shot learning. In Advances in Neural Information Processing Systems, pp. 4077–4087, 2017.   
+Sebastian Thrun. Is learning the n-th thing any easier than learning the first? In Advances in neural information processing systems, pp. 640–646, 1996.   
+Sebastian Thrun and Lorien Pratt. Learning to learn. Springer Science & Business Media, 2012.   
+Emanuel Todorov, Tom Erez, and Yuval Tassa. Mujoco: A physics engine for model-based control. In 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 5026–5033. IEEE, 2012.   
+Paul E Utgoff. Shift of bias for inductive concept learning. Machine learning: An artificial intelligence approach, 2:107–148, 1986.   
+Jane X. Wang, Zeb Kurth-Nelson, Dhruva Tirumala, Hubert Soyer, Joel Z. Leibo, Remi Munos, Charles Blundell, ´ Dharshan Kumaran, and Matthew Botvinick. Learning to reinforcement learn. CoRR, abs/1611.05763, 2016. URL http://arxiv.org/abs/1611.05763.
+
+# A P S E U D O - C O D E
+
+The pseudo-code for MQL during training and adaption are given in Algorithm 1 and Algorithm 2. After MQL is trained for a given environment as described in Algorithm 1, it returns the meta-trained policy $\theta$ and replay buffer containing train tasks.
+
+Next, Algorithm 2 runs the adaptation procedure which adapts the meta-trained policy to a test task $D$ with few data. To do so, MQL optimizes the adaptation objective into two steps. After gathering data from a test task $D$ , MQL first updates the policy using the new data (line 4). MQL then fits a logistic classifier on a mini-batch of transitions from the meta-training replay buffer and the transitions collected from the test task and then estimates $\widehat { \mathrm { E S S } }$ (lines 5-6). Finally, the adaptation step runs for $n$ iterations (lines $7 \textrm { - } 1 0 $ ) in which MQL can exploit past data in which it uses propensity score to decide whether or not a given sample is related to the current test task.
+
+# Algorithm 1: MQL - Meta-training
+
+<table><tr><td colspan="2">Algorithm1:MQL -Meta-training</td></tr><tr><td colspan="2">Input: Set of training tasks Dmeta</td></tr><tr><td colspan="2">1 Initialize the replay buffer</td></tr><tr><td colspan="2">2 Initialize parameters θ of an off-policy method,e.g., TD3</td></tr><tr><td>3 while not done do</td><td></td></tr><tr><td>4</td><td>// Rollout and update policy</td></tr><tr><td>5 6</td><td>Sample a task D ~ Dmeta Gather data from task D using policy Te while feeding transitions through context GRU.Add</td></tr><tr><td></td><td>trajectory to the replay buffer.</td></tr><tr><td>7 8</td><td>&amp;←Sample mini-batch from buffer Update parameters θ using mini-batch t and Eqn. (15)</td></tr><tr><td colspan="2">90meta←0</td></tr><tr><td colspan="2">10 return θmeta ,replay buffer</td></tr></table>
+
+# Algorithm 2: MQL - Adaptation
+
+Input: Test task $D$ , meta-training replay buffer, meta-trained policy $\theta _ { \mathrm { { m e t a } } }$
+
+1 Initialize temporary buffer buf   
+2 $\theta \gets \theta _ { m e t a }$   
+3 buf ← Gather data from D using πθmeta   
+4 Update Eqn. (18) using buf   
+5 Fit $\beta ( D )$ using buf and meta-training replay buffer using Eqn. (12)   
+6 Estimate $\widehat { \mathrm { E S S } }$ using $\beta ( D )$ using Eqn. (13)   
+7 for $i \leq n$ do   
+8 $\ell ~ \gets$ sample mini-batch from meta-training replay buffer   
+9 Calculate $\beta$ for $\mathcal { \ell }$   
+10 Update $\theta$ using Eqn. (19)   
+11 Evaluate $\theta$ on a new rollout from task $D$   
+12 return $\theta$
+
+# B O U T - O F - D I S T R I B U T I O N T A S K S
+
+MQL is designed for explicitly using data from the new task along with off-policy data from old, possibly very different tasks. This is on account of two things: (i) the loss function of MQL does not use the old data if it is very different from the new task, $\beta$ is close to zero for all samples, and (ii) the first term in (18) makes multiple updates using data from the new task. To explore this aspect, we create an out-of-distribution task using the “Half-Cheetah-Vel” environment wherein we use disjoint sets of velocities for meta-training and testing. The setup is as follows:
+
+• Half-Cheetah-Vel-OOD-Medium: target velocity for a training task is sampled uniformly randomly from [0, 2.5] while that for test task is sampled uniformly randomly from [2.5, 3.0].
+
+This is what we call “medium” hardness task because although the distributions of train and test velocities is disjoint, they are close to each other.
+
+• Half-Cheetah-Vel-OOD-Hard: target velocity for a training task is sampled uniformly randomly from [0, 1.5] while that for test task is sampled uniformly randomly from [2.5, 3.0]. This is a “hard” task because the distributions of train and test velocities are far away from each other.
+
+Fig. 6a shows that MQL significantly outperforms PEARL when the train and test target velocities come from disjoint sets. We used the published code of PEARL (Rakelly et al., 2019) for this experiment. This shows that the adaptation in MQL is crucial to generalizing to new situations which are not a part of the meta-training process. Fig. 6b shows the evolution of the proximal penalty coefficient $\lambda$ and the propensity score $\beta ( z )$ during meta-training for the medium-hard task. We see that $\lambda \approx 0 . 8$ while $\beta ( z ) \approx 0 . 2$ throughout training. This indicates that MQL automatically adjusts its test-time adaptation to use only few samples in (19) if the test task provides transitions quite different than those in the replay buffer.
+
+We next discuss results on the harder task Half-Cheetah-Vel-OOD-Hard. There is a very large gap between training and test target velocities in this case. Fig. 7a shows the comparison with the same test protocol as the other experiments in this paper. In particular, we collect 200 time-steps from the new task and use it for adaptation in both MQL and TD3-context. Since this task is particularly hard, we also ran an experiment where 1200 time-steps (6 episodes) are given to the two algorithms for adaptation. The results are shown in Fig. 7b. In both cases, we see that MQL is better than TD3-context by a large margin (the standard deviation on these plots is high because the environment is hard). Note that since we re-initialize the hidden state of the context network at the beginning of each episode, TD3-context cannot take advantage of the extra time-steps. MQL on the other hand updates the policy explicitly and can take advantage of this extra data.
+
+For sake of being thorough, we collected 800 time-steps from the new task from the same episode, the results are shown in Fig. 8a. We again notice that MQL results in slightly higher rewards than TD3-context in spite of the fact that both the algorithms suffer large degradation in performance as compared to Figs. 7a and 7b.
+
+Figs. 7c, 7d and 8b show that the proximal penalty coefficient $\lambda \approx 1$ and the propensity score $\beta ( z ) \approx 0$ for a large fraction of training. This proves that MQL is able to automatically discard samples unrelated to the new test during the adaptation phase.
+
+![](images/a5e4ffc4e9104fac466ee09ac523a3bcc4115a1f93e779b146394ffd7f26c1b9.jpg)  
+Figure 6: Comparison of the average return of MQL (orange) against existing PEARL algorithms (blue). Fig. 6a shows that MQL significantly outperforms PEARL when the train and test target velocities come from disjoint sets. Fig. 6b shows the evolution of the proximal penalty coefficient $\lambda$ and the propensity score $\beta ( z )$ . We see that $\beta ( z )$ is always small which demonstrates that MQL automatically adjusts the adaptation in (19) if the test task is different from the training tasks.
+
+![](images/5128fdf6d6ee3f9aa4b187936e2fbdbe73f675fdf553c3b2f8cd2fbdf12229c8.jpg)  
+Figure 7: (a,b) Comparison of the average return of MQL (orange) against TD3-context (blue). Fig. 7a shows the comparison with the same test protocol as the other experiments in this paper. In particular, we collect 200 time-steps from the new task and use it for adaptation in both MQL and TD3-context. Since this task is particularly hard, we also ran an experiment where 1200 time-steps (6 episodes) where results are shown in Fig. 7b. In both cases, we see that MQL is better than TD3-context by a large margin (the standard deviation on these plots is high because the environment is hard). (c,d) Evolution of $\lambda$ and $\beta ( z )$ during meta-training. shows the evolution of the proximal penalty coefficient $\lambda$ and the propensity score $\beta ( z )$ . We see in Fig. 7c and Fig. 7d that $\beta ( z )$ is always small which demonstrates that MQL automatically adjusts the adaptation if the test task is different from the training tasks.
+
+![](images/536e4d33a9916c5fa1ea32db4a5093aebc3344fb15594b6f73ee82052a68d4dc.jpg)  
+Figure 8: (a) Comparison of the average return of MQL (orange) against TD3-context (blue). For these experiments, we collected 800 time-steps from the new task from the same episode, the results are shown in Fig. 8a. We again notice that MQL results in slightly higher rewards than TD3-context in spite of the fact that both the algorithms suffer large degradation in performance as compared to Figs. 7a and 7b. (b) Evolution of $\lambda$ and $\beta ( z )$ during meta-training shows the evolution of the proximal penalty coefficient $\lambda$ and the propensity score $\beta ( z )$ .
+
+![](images/07d7de0e698367a58640db2cdbdd886a8b673b1442d6db6f5dfc3bdaf74030b0.jpg)  
+Figure 9: Ablation studies to examine various components of MQL. Fig. 9a shows that the adaptation in MQL in (18) and (19) improves performance. One reason for that is because test and training tasks in Walker2D-Params are very similar as it shown in Fig. 10b. Next, we evaluate the importance of the additional data from the replay buffer in MQL. Fig. 9b compares the performance of MQL with and without updates in (19). We see that the old data, even if it comes from different tasks, is useful to improve the performance on top of (18). Fig. 9c and Fig. 9f show the effectiveness of setting $\lambda = 1 - { \widehat { \mathrm { E S S } } }$ as compared to a fixed value of $\lambda = 0 . 5$ . We see that modulating the quadratic penalty with ESS  helps, the effect is minor for Sec. 4.3. The ideal value of $\lambda$ depends on a given task and using $1 - { \widehat { \mathrm { E S S } } }$ can help to adjust to different tasks without the need to do hyper-parameter search per task.
+
+![](images/e58b0c971d517dac40d33263b6e8ced7c530b4827ba0c08c85a99984da9bc9ae.jpg)  
+Figure 10: Evolution of $\lambda$ and $\beta ( z )$ during meta-training shows the evolution of the proximal penalty coefficient $\lambda$ and the propensity score $\beta ( z )$ . We see in Fig. 10a that $\beta ( z )$ stays around 0.4 which demonstrates MQL automatically adjusts the adaptation if the test task is different from the training tasks. Fig. 10b shows that test and training tasks are very similar as $\beta ( z )$ is around 0.6.
+
+# C M O R E A B L A T I O N S T U D I E S
+
+We conduct a series of additional ablation studies to analyze the different components of the MQL algorithm. We use two environments for this purpose, namely Half-Cheetah-Vel and Walker-2DParams. Fig. 9 and Fig. 10 show the result of these experiments. These experiments show that adaptation phase is more useful for Half-Cheetah-Vel than Walker-2D-Params as test and training tasks are very similar in Walker-2D-Params which helps TD3-context achieves strong performance that leaves no window for improvement with adaptation.
+
+D H Y P E R - P A R A M E T E R S A N D M O R E D E T A I L S O F T H E E M P I R I C A L R E S U LT S
+
+Table 1: Hyper-parameters for MQL and TD3 for continuous-control meta-RL benchmark tasks. We use a network with two full-connected layers for all environments. The batch-size in Adam is fixed to 256 for all environments. The abbreviation HC stands for Half-Cheetah. These hyper-parameters were tuned by grid-search.   
+
+<table><tr><td></td><td>Humanoid</td><td>HC-Vel</td><td> Ant-FB</td><td> Ant-Goal</td><td> Walker</td><td>HC-FB</td></tr><tr><td>β clipping</td><td>1</td><td>1.1</td><td>1</td><td>1.2</td><td>2</td><td>0.8</td></tr><tr><td>TD3 exploration noise</td><td>0.3</td><td>0.3</td><td>0.3</td><td>0.2</td><td>0.3</td><td>0.2</td></tr><tr><td>TD3 policy noise</td><td>0.2</td><td>0.3</td><td>0.3</td><td>0.4</td><td>0.3</td><td>0.2</td></tr><tr><td>TD3 policy update frequency</td><td>2</td><td>2</td><td>2</td><td>4</td><td>4</td><td>3</td></tr><tr><td>Parameter updates per iteration (meta-training)</td><td>500</td><td>1000</td><td>200</td><td>1000</td><td>200</td><td>200</td></tr><tr><td>Adaptation parameter updates per episode (eq. 18)</td><td>10</td><td>5</td><td>10</td><td>5</td><td>10</td><td>10</td></tr><tr><td>Adaptation parameter updates per episode (eq. 19)</td><td>200</td><td>400</td><td>100</td><td>400</td><td>400</td><td>300</td></tr><tr><td>GRU sequence length</td><td>20</td><td>20</td><td>20</td><td>25</td><td>10</td><td>10</td></tr><tr><td>Context dimension</td><td>20</td><td>20</td><td>15</td><td>30</td><td>30</td><td>30</td></tr><tr><td>Adam learning rate</td><td>0.0003</td><td>0.001</td><td>0.0003</td><td>0.0004</td><td>0.0008</td><td>0.0003</td></tr></table>

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+        "type": "text",
+        "text": "M E T A - Q - L E A R N I N G ",
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+        "text": "Rasool Fakoor1, Pratik Chaudhari2∗, Stefano Soatto1, Alexander Smola1 ",
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+        "text": "1 Amazon Web Services   \n2 University of Pennsylvania   \nEmail: {fakoor, soattos, smola}@amazon.com, pratikac@seas.upenn.edu ",
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+        "type": "text",
+        "text": "A B S T R A C T ",
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+        "text": "This paper introduces Meta-Q-Learning (MQL), a new off-policy algorithm for meta-Reinforcement Learning (meta-RL). MQL builds upon three simple ideas. First, we show that Q-learning is competitive with state-of-the-art meta-RL algorithms if given access to a context variable that is a representation of the past trajectory. Second, a multi-task objective to maximize the average reward across the training tasks is an effective method to meta-train RL policies. Third, past data from the meta-training replay buffer can be recycled to adapt the policy on a new task using off-policy updates. MQL draws upon ideas in propensity estimation to do so and thereby amplifies the amount of available data for adaptation. Experiments on standard continuous-control benchmarks suggest that MQL compares favorably with the state of the art in meta-RL. ",
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+        "type": "text",
+        "text": "1 I N T R O D U C T I O N ",
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+        "text": "Reinforcement Learning (RL) algorithms have demonstrated good performance on simulated data. There are however two main challenges in translating this performance to real robots: (i) robots are complex and fragile which precludes extensive data collection, and (ii) a real robot may face an environment that is different than the simulated environment it was trained in. This has fueled research into MetaReinforcement Learning (meta-RL) which develops algorithms that “meta-train” on a large number of different environments, e.g., simulated ones, and aim to adapt to a new environment with few data. ",
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+        "text": "How well does meta-RL work today? Fig. 1 shows the performance of two prototypical meta-RL algorithms on four standard continuous-control benchmarks.1 We compared them to the following simple baseline: an off-policy RL algorithm (TD3 by Fujimoto et al. (2018b)) and which was trained to maximize the average reward over all training tasks and modified to use a “context variable” that represents the trajectory. All algorithms in this figure use the same evaluation protocol. It is surprising that this simple non-meta-learning-based method is competitive with state-of-the-art meta-RL algorithms. This is the first contribution of our paper: we demonstrate that it is not necessary to meta-train policies to do well on existing benchmarks. ",
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+            "Figure 1: How well does meta-RL work? Average returns on validation tasks compared for two prototypical meta-RL algorithms, MAML (Finn et al., 2017) and PEARL (Rakelly et al., 2019), with those of a vanilla Q-learning algorithm named TD3 (Fujimoto et al., 2018b) that was modified to incorporate a context variable that is a representation of the trajectory from a task (TD3-context). Even without any meta-training and adaptation on a new task, TD3-context is competitive with these sophisticated algorithms. "
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+        "text": "Our second contribution is an off-policy meta-RL algorithm named Meta-Q-Learning (MQL) that builds upon the above result. MQL uses a simple meta-training procedure: it maximizes the average rewards across all meta-training tasks using off-policy updates to obtain ",
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+        "text": "$$\n\\widehat { \\theta } _ { \\mathrm { m e t a } } = \\arg \\operatorname* { m a x } _ { \\theta } \\frac { 1 } { n } \\sum _ { k = 1 } ^ { n } \\mathbb { E } _ { \\sim D ^ { k } } \\left[ \\ell ^ { k } ( \\theta ) \\right]\n$$",
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+        "text": "where $\\ell ^ { k } ( \\theta )$ is the objective evaluated on the transition $\\tau$ obtained from the task $D ^ { k } ( \\theta )$ , e.g., 1-step temporal-difference (TD) error would set $\\ell ^ { k } ( \\theta ) \\ : = \\ : \\mathrm { T D ^ { 2 } } ( \\theta ; \\tau )$ . This objective, which we call the multi-task objective, is the simplest form of meta-training. ",
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+        "text": "For adapting the policy to a new task, MQL samples transitions from the meta-training replay buffer that are similar to those from the new task. This amplifies the amount of data available for adaptation but it is difficult to do because of the large potential bias. We use techniques from the propensity estimation literature for performing this adaptation and the off-policy updates of MQL are crucial to doing so. The adaptation phase of MQL solves ",
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+        "text": "$$\n\\arg \\operatorname* { m a x } _ { \\theta } \\left\\{ \\underset { - \\sqrt { S ^ { \\mathrm { a e s } } } } { \\mathbb { E } } \\left[ \\ell ^ { \\mathrm { n e s } } ( \\theta ) \\right] + \\underset { \\tau \\sim \\mathcal { D } _ { \\mathrm { m e s } } } { \\mathbb { E } } \\left[ \\beta ( \\tau ; D ^ { \\mathrm { n e s } } , \\mathcal { D } _ { \\mathrm { m e t a } } ) \\ell ^ { \\mathrm { n e s } } ( \\theta ) \\right] - \\left( 1 - \\widehat { \\mathrm { E S S } } \\right) \\Vert \\theta - \\widehat { \\theta } _ { \\mathrm { m e t a } } \\Vert _ { 2 } ^ { 2 } \\right\\}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\mathcal { D } _ { \\mathrm { m e t a } }$ is the meta-training replay buffer, the propensity score $\\beta ( \\tau ; D ^ { \\mathrm { n e w } } , \\mathcal { D } _ { \\mathrm { m e t a } } )$ is the odds of a transition $\\tau$ belonging to $D ^ { \\mathrm { n e w } }$ versus $\\mathcal { D } _ { \\mathrm { m e t a } }$ , and $\\widehat { \\mathrm { E S S } }$ is the Effective Sample Size between $D ^ { \\mathrm { n e w } }$ and $\\mathcal { D } _ { \\mathrm { m e t a } }$ that is a measure of the similarly of the new task with the meta-training tasks. The first term computes off-policy updates on the new task, the second term performs $\\beta ( \\cdot )$ -weighted off-policy updates on old data, while the third term is an automatically adapting proximal term that prevents degradation of the policy during adaptation. ",
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+    {
+        "type": "text",
+        "text": "We perform extensive experiments in Sec. 4.2 including ablation studies using standard meta-RL benchmarks that demonstrate that MQL policies obtain higher average returns on new tasks even if they are meta-trained for fewer time-steps than state-of-the-art algorithms. ",
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+    {
+        "type": "text",
+        "text": "2 B A C K G R O U N D ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "This section introduces notation and formalizes the meta-RL problem. We discuss techniques for estimating the importance ratio between two probability distributions in Sec. 2.2. ",
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+    {
+        "type": "text",
+        "text": "Consider a Markov Decision Processes (MDP) denoted by ",
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+        "type": "equation",
+        "img_path": "images/48cc9e8e9bab6e0245b2760914f5730a5e6f6bd3a26f70ad7ff826c888677478.jpg",
+        "text": "$$\nx _ { t + 1 } = f ^ { k } ( x _ { t } , u _ { t } , \\xi _ { t } ) \\quad x _ { 0 } \\sim p _ { 0 } ^ { k } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $x _ { t } ~ \\in ~ X ~ \\subset ~ \\mathbb { R } ^ { d }$ are the states and $u _ { t } ~ \\in ~ U ~ \\subset ~ \\mathbb { R } ^ { p }$ are the actions. The dynamics $f ^ { k }$ is parameterized by $k \\in \\{ 1 , \\ldots , n \\}$ where each $k$ corresponds to a different task. The domain of all these tasks, $X$ for the states and $U$ for the actions, is the same. The distribution $p _ { 0 } ^ { k }$ denotes the initial state distribution and $\\xi _ { t }$ is the noise in the dynamics. Given a deterministic policy $u _ { \\theta } ( x _ { t } )$ , the actionvalue function for $\\gamma$ -discounted future rewards $r _ { t } ^ { k } : = r ^ { k } ( x _ { t } , u _ { \\theta } ( x _ { t } ) )$ over an infinite time-horizon is ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/1652e5e095c5753afe456895c6c87c36ff3305a2d59f05d43c287b871dfb6bee.jpg",
+        "text": "$$\nq ^ { k } ( x , u ) = \\operatorname * { \\mathbb { E } } _ { \\xi _ { ( \\cdot ) } } \\Big [ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } r _ { t } ^ { k } \\ | \\ x _ { 0 } = x , u _ { 0 } = u , u _ { t } = u _ { \\theta } ( x _ { t } ) \\Big ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Note that we have assumed that different tasks have the same state and action space and may only differ in their dynamics $f ^ { k }$ and reward function $r ^ { k }$ . Given one task $k \\in \\{ 1 , \\ldots , n \\}$ , the standard Reinforcement Learning (RL) formalism solves for ",
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+    {
+        "type": "equation",
+        "img_path": "images/63ab6ddf3cf26ff282a8e264b7e67b84c3b016a2a7b2df8addccc866e1a533d3.jpg",
+        "text": "$$\n\\widehat { \\theta ^ { k } } = \\arg \\operatorname* { m a x } _ { \\theta } \\ell ^ { k } ( \\theta ) \\quad \\mathrm { w h e r e } \\ \\ell ^ { k } ( \\theta ) = \\underset { x \\sim p _ { 0 } } { \\mathbb { E } } \\left[ q ^ { k } ( x , u _ { \\theta } ( x ) ) \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Let us denote the dataset of all states, actions and rewards pertaining to a task $k$ and policy $u _ { \\theta } ( x )$ by ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/01c0ca56174998ce2e54efe808ff05b40a6a8aa5d34fdc1097886409ec4f7bad.jpg",
+        "text": "$$\n\\begin{array} { r } { D ^ { k } ( \\theta ) = \\left\\{ x _ { t } , u _ { \\theta } ( x _ { t } ) , r ^ { k } , x _ { t + 1 } = f ^ { k } ( x _ { t } , u _ { \\theta } ( x _ { t } ) , \\xi _ { t } ) \\right\\} _ { t \\geq 0 , \\ x ( 0 ) \\sim p _ { 0 } ^ { k } , \\ \\xi ( \\cdot ) } ; } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "we will often refer to $D ^ { k }$ as the “task” itself. The Deterministic Policy Gradient (DPG) algorithm (Silver et al., 2014) for solving (5) learns a $\\varphi$ -parameterized approximation $q _ { \\varphi }$ to the optimal value func",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "tion $q ^ { k }$ by minimizing the Bellman error and the optimal policy $u _ { \\theta }$ that maximizes this approximation by solving the coupled optimization problem ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/932b9eab9dbfafb3245c4048f960d6e4e67cfa524b85fbc6449b50aeaf6a9e52.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\widehat { \\varphi ^ { k } } = \\arg \\operatorname* { m i n } _ { \\varphi } \\underset { \\tau \\sim D ^ { k } } { \\mathbb { E } } \\Big [ \\Big ( q _ { \\varphi } ( x , u ) - r ^ { k } - \\gamma q _ { \\varphi } ( x ^ { \\prime } , u _ { \\widehat { \\theta ^ { k } } } ( x ^ { \\prime } ) ) \\Big ) ^ { 2 } \\Big ] , } \\\\ & { \\widehat { \\theta ^ { k } } = \\arg \\underset { \\theta } { \\operatorname* { m a x } } \\underset { \\tau \\sim D ^ { k } } { \\mathbb { E } } \\Big [ q _ { \\widehat { \\varphi ^ { k } } } ( x , u _ { \\theta } ( x ) ) \\Big ] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The 1-step temporal difference error (TD error) is defined as ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/b912e63c4cff921a3d5a491cd7b951d35d047954d0f3100936e95f9aa31b0305.jpg",
+        "text": "$$\n\\mathrm { T D } ^ { 2 } ( \\theta ) = \\bigl ( q _ { \\varphi } ( x , u ) - r ^ { k } - \\gamma q _ { \\varphi } ( x ^ { \\prime } , u _ { \\theta } ( x ^ { \\prime } ) ) \\bigr ) ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where we keep the dependence of $\\operatorname { T D } ( \\cdot )$ on $\\varphi$ implicit. DPG, or its deep network-based variant DDPG (Lillicrap et al., 2015), is an off-policy algorithm. This means that the expectations in (6) are computed using data that need not be generated by the policy being optimized $( u _ { \\theta } )$ , this data can come from some other policy. ",
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+    {
+        "type": "text",
+        "text": "In the sequel, we will focus on the parameters $\\theta$ parameterizing the policy. The parameters $\\varphi$ of the value function are always updated to minimize the TD-error and are omitted for clarity. ",
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+    {
+        "type": "text",
+        "text": "2 . 1 M E T A - R E I N F O R C E M E N T L E A R N I N G ( M E T A - R L ) ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Meta-RL is a technique to learn an inductive bias that accelerates the learning of a new task by training on a large of number of training tasks. Formally, meta-training on tasks from the meta-training set $\\mathcal { D } _ { \\mathrm { m e t a } } = \\ \\backslash \\ \\{ D ^ { k } \\} _ { k = 1 , \\ldots , n }$ involves learning a policy ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/b09da969763b9d654647f221b89a14b0bd0b0a2b942f525215c68114f3d0fc14.jpg",
+        "text": "$$\n\\widehat { \\theta } _ { \\mathrm { m e t a } } = \\arg \\operatorname* { m a x } _ { \\theta } \\frac { 1 } { n } \\sum _ { k = 1 } ^ { n } \\ell _ { \\mathrm { m e t a } } ^ { k } ( \\theta )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where let us $\\ell _ { \\mathrm { m e t a } } ^ { k } ( \\theta )$ is a meta-training loss that depends on the particular method. Gradient-based meta-RL,ML by Finn et al. (2017) as a concrete example, sets ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/457771362627efc97c62fff5a17acc1db5001492655a1f785b12cc36b6d6f6e2.jpg",
+        "text": "$$\n\\ell _ { \\mathrm { m e t a } } ^ { k } ( \\theta ) = \\ell ^ { k } ( \\theta + \\alpha \\nabla _ { \\theta } \\ell ^ { k } ( \\theta ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "for a step-size $\\alpha > 0 ; \\ell ^ { k } ( \\theta )$ is the objective of non-meta-RL (5). In this case $\\ell _ { \\mathrm { m e t a } } ^ { k }$ is the objective obtained on the task $D ^ { k }$ after one (or in general, more) updates of the policy on the task. The idea behind this is that even if the policy $\\widehat { \\theta } _ { \\mathrm { { m e t a } } }$ does not perform well on all tasks in $\\mathcal { D } _ { \\mathrm { m e t a } }$ it may be updated quickly on a new task $D ^ { \\mathrm { n e w } }$ to obtain a well-performing policy. This can either be done using the same procedure as that of meta-training time, i.e., by maximizing $\\ell _ { \\mathrm { m e t a } } ^ { \\mathrm { n e w } } ( \\theta )$ with the policy $\\widehat { \\theta } _ { \\mathrm { { m e t a } } }$ as the initialization, or by some other adaptation procedure. The meta-training method and the adaptation method in meta-RL, and meta-learning in general, can be different from each other. ",
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+    {
+        "type": "text",
+        "text": "2 . 2 L O G I S T I C R E G R E S S I O N F O R E S T I M A T I N G T H E P R O P E N S I T Y S C O R E ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Consider standard supervised learning: given two distributions $q ( x )$ (say, train) and $p ( x )$ (say, test), we would like to estimate how a model’s predictions ${ \\hat { y } } ( x )$ change across them. This is formally done using importance sampling: ",
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+        "img_path": "images/a0c2e2f9b7635714b71dbadca8306be315ae3988d9318172473ef77cca68ed51.jpg",
+        "text": "$$\n{ \\underset { x \\sim p ( x ) } { \\mathbb { E } } } { \\underset { y \\mid x } { \\mathbb { E } } } \\left[ \\ell ( y , { \\hat { y } } ( x ) ) \\right] = { \\underset { x \\sim q ( x ) } { \\mathbb { E } } } { \\underset { y \\mid x } { \\mathbb { E } } } \\left[ \\beta ( x ) \\ell ( y , { \\hat { y } } ( x ) ) \\right] ;\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $y | x$ are the true labels of data, the predictions of the model are ${ \\hat { y } } ( x )$ and $\\ell ( y , \\hat { y } ( x ) )$ is the loss for each datum $( x , y )$ . The importance ratio $\\begin{array} { r } { \\beta ( x ) \\ = \\ \\frac { \\mathrm { d } p } { \\mathrm { d } q } ( x ) \\ } \\end{array}$ , also known as the propensity score, is the Radon-Nikodym derivative (Resnick, 2013) of the two data densities and measures the odds of a sample $x$ coming from the distribution $p$ versus the distribution $q$ . In practice, we do not know the densities $q ( x )$ and $p ( x )$ and therefore need to estimate $\\beta ( x )$ using some finite data $X _ { q } = \\{ x _ { 1 } , \\dots , x _ { m } \\}$ drawn from $q$ and $X _ { p } = \\{ x _ { 1 } ^ { \\prime } , \\ldots , x _ { m } ^ { \\prime } \\}$ drawn from $p$ . As Agarwal et al. (2011) show, this is easy to do using logistic regression. Set $z _ { k } = 1$ to be the labels for the data in $X _ { q }$ and $z _ { k } = - 1$ to be the labels of the data in $X _ { p }$ for $k \\leq m$ and fit a logistic classifier on the combined $2 m$ ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "samples by solving ",
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/361441edf12186c8a83cb79ee8ab40a681331f05896b6d4d1a9a5be4123072eb.jpg",
+        "text": "$$\nw ^ { * } = \\operatorname* { m i n } _ { w } \\ \\frac { 1 } { 2 m } \\sum _ { ( x , z ) } \\log \\left( 1 + e ^ { - z w ^ { \\top } x } \\right) + c \\left\\| w \\right\\| ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "This gives ",
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ef1769398b5e5f06c0c58cc77eff451ddb7eb9462bcad4d93bcfa75b4d080142.jpg",
+        "text": "$$\n\\beta ( x ) = \\frac { \\mathbb { P } ( z = - 1 | x ) } { \\mathbb { P } ( z = 1 | x ) } = e ^ { - w ^ { * } ^ { \\top } x } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Normalized Effective Sample Size $\\widehat { ( \\mathbf { E S S } ) }$ : A related quantity to $\\beta ( x )$ is the normalized Effective Sample Size (ESS ) which we define as the relative number of samples from the target distribution $p ( x )$ required to obtain an estimator with performance (say, variance) equal to that of the importance sampling estimator (10). It is not possible to compute the $\\widehat { \\mathrm { E S S } }$ without knowing both densities $q ( x )$ and $p ( x )$ but there are many heuristics for estimating it. A popular one in the Monte Carlo literature (Kong, 1992; Smith, 2013; Elvira et al., 2018) is ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/14bcb60c20fffa3243816502ba2ff6f0942f6c6802dd0a39412c0cc0296c03be.jpg",
+        "text": "$$\n\\widehat { \\mathrm { E S S } } = \\frac { 1 } { m } \\frac { \\left( \\sum _ { k = 1 } ^ { m } \\beta ( x _ { k } ) \\right) ^ { 2 } } { \\sum _ { k = 1 } ^ { m } \\beta ( x _ { k } ) ^ { 2 } } \\in [ 0 , 1 ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $X = \\{ x _ { 1 } , \\ldots , x _ { m } \\}$ is some finite batch of data. Observe that if two distributions $q$ and $p$ are close then the $\\widehat { \\mathrm { E S S } }$ is close to one; if they are far apart the $\\widehat { \\mathrm { E S S } }$ is close to zero. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This section describes the MQL algorithm. We begin by describing the meta-training procedure of MQL including a discussion of multi-task training in Sec. 3.1. The adaptation procedure is described in Sec. 3.2. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "3 . 1 M E T A - T R A I N I N G ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "MQL performs meta-training using the multi-task objective. Note that if one sets ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3f32fe7fb42551220d0b757dfa0c5c339bcd45468d3c4588cc7cdf3713411e18.jpg",
+        "text": "$$\n\\ell _ { \\mathrm { m e t a } } ^ { k } ( \\theta ) \\triangleq \\ell ^ { k } ( \\theta ) = \\underset { x \\sim p _ { 0 } ^ { k } } { \\mathbb { E } } \\left[ q ^ { k } ( x , u _ { \\theta } ( x ) ) \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "in (8) then the parameters $\\widehat { \\theta } _ { \\mathrm { { m e t a } } }$ are such that they maximize the average returns over all tasks from the meta-training set. We use an off-policy algorithm named TD3 (Fujimoto et al., 2018b) as the building block and solve for ",
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+        "type": "equation",
+        "img_path": "images/5e2a29f38c4d00d398359a0168f1694ffaf6f52ebd02316f1f203d2d0253d1b8.jpg",
+        "text": "$$\n\\widehat { \\theta } _ { \\mathrm { m e t a } } = \\arg \\operatorname* { m i n } _ { \\theta } \\frac { 1 } { n } \\sum _ { k = 1 } ^ { n } \\mathbb { E } _ { \\sim D ^ { k } } \\left[ \\mathbf { T D } ^ { 2 } ( \\theta ) \\right] ;\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            665,
+            630,
+            707
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\operatorname { T D } ( \\cdot )$ is defined in (7). As is standard in TD3, we use two action-value functions parameterized by $\\varphi _ { 1 }$ and $\\varphi _ { 2 }$ and take their minimum to compute the target in (7). This trick known as “doubleQ-learning” reduces the over-estimation bias. Let us emphasize that (14) is a special case of the procedure outlined in (8). The following remark explains why MQL uses the multi-task objective as opposed to the meta-training objective used, for instance, in existing gradient-based meta-RL algorithms. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Remark 1. Let us compare the critical points of the $m$ -step MAML objective (9) to those of the multi-task objective which uses (14). As is done by the authors in Nichol et al. (2018), we can perform a Taylor series expansion around the parameters $\\theta$ to obtain ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4a7a8dbe88887128a70b26a53895cccf9196aaeb328b0b066c9971ac3e39e529.jpg",
+        "text": "$$\n\\nabla \\ell _ { \\mathrm { m e t a } } ^ { k } ( \\theta ) = \\nabla \\ell ^ { k } ( \\theta ) + 2 \\alpha ( m - 1 ) \\left( \\nabla ^ { 2 } \\ell ^ { k } ( \\theta ) \\right) \\nabla \\ell ^ { k } ( \\theta ) + \\mathcal { O } ( \\alpha ^ { 2 } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            877
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Further, note that $\\nabla \\ell _ { \\mathrm { m e t a } } ^ { k }$ in (16) is also the gradient of the loss ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3a75c7831780c297839a8c5ceabe8a3a732ea8db34d8542fae8fd2ae7ad5af06.jpg",
+        "text": "$$\n\\ell ^ { k } ( \\theta ) + \\alpha ( m - 1 ) \\| \\nabla \\ell ^ { k } ( \\theta ) \\| _ { 2 } ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "up to first order. This lends a new interpretation that MAML is attracted towards regions in the loss landscape that under-fit on individual tasks: parameters with large $\\| \\nabla \\ell ^ { k } \\| _ { 2 }$ will be far from the local maxima of $\\ell ^ { k } ( \\theta )$ . The parameters $\\alpha$ and $m$ control this under-fitting. Larger the number of gradient steps, larger the under-fitting effect. This remark suggests that the adaptation speed of gradient-based meta-learning comes at the cost of under-fitting on the tasks. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 . 1 . 1 D E S I G N I N G C O N T E X T ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "As discussed in Sec. 1 and 4.4, the identity of the task in meta-RL can be thought of as the hidden variable of an underlying partially-observable MDP. The optimal policy on the entire trajectory of the states, actions and the rewards. We therefore design a recurrent context variable $z _ { t }$ that depends on $\\{ ( x _ { i } , u _ { i } , r _ { i } ) \\} _ { i \\leq t }$ . We set $z _ { t }$ to the hidden state at time $t$ of a Gated Recurrent Unit (GRU by Cho et al. (2014)) model. All the policies $u _ { \\theta } ( x )$ and value functions $q _ { \\varphi } ( x , u )$ in MQL are conditioned on the context and implemented as $u _ { \\theta } ( x , z )$ and $q _ { \\varphi } ( x , u , z )$ . Any other recurrent model can be used to design the context; we used a GRU because it offers a good trade-off between a rich representation and computational complexity. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Remark 2 (MQL uses a deterministic context that is not permutation invariant). We have aimed for simplicity while designing the context. The context in MQL is built using an off-the-shelf model like GRU and is not permutation invariant. Indeed, the direction of time affords crucial information about the dynamics of a task to the agent, e.g., a Half-Cheetah running forward versus backward has arguably the same state trajectory but in a different order. Further, the context in MQL is a deterministic function of the trajectory. Both these aspects are different than the context used by Rakelly et al. (2019) who design an inference network and sample a probabilistic context conditioned on a moving window. RL algorithms are quite complex and challenging to reproduce. Current meta-RL techniques which build upon them further exacerbate this complexity. Our demonstration that a simple context variable is enough is an important contribution. ",
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+    },
+    {
+        "type": "text",
+        "text": "3 . 2 A D A P T AT I O N T O A N E W T A S K ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We next discuss the adaptation procedure which adapts the meta-trained policy $\\widehat { \\theta } _ { \\mathrm { { m e t a } } }$ to a new task $D ^ { \\mathrm { n e w } }$ with few data. MQL optimizes the adaptation objective introduced in (2) into two steps. ",
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+    {
+        "type": "text",
+        "text": "1. Vanilla off-policy adaptation: The first step is to update the policy using the new data as ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
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+        "img_path": "images/5f1edbdb2dab047f17f3d1ed8007cfda7e18d8f9e38b916507c9c15a7b2e32c4.jpg",
+        "text": "$$\n\\underset { \\theta } { \\arg \\operatorname* { m a x } } \\left\\{ \\underset { \\tau \\sim D ^ { \\mathrm { n e w } } } { \\mathbb { E } } \\left[ \\ell ^ { \\mathrm { n e w } } ( \\theta ) \\right] - \\frac { \\lambda } { 2 } \\| \\theta - \\widehat { \\theta } _ { \\mathrm { m e t a } } \\| _ { 2 } ^ { 2 } \\right\\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The quadratic penalty $\\lVert \\boldsymbol { \\theta } - \\widehat { \\boldsymbol { \\theta } } _ { \\mathrm { { m e t a } } } \\rVert ^ { 2 }$ keeps the parameters close to $\\widehat { \\theta } _ { \\mathrm { { m e t a } } }$ . This is crucial to reducing the variance of the model that is adapted using few data from the new task (Reddi et al., 2015). Off-policy learning is critical in this step because of its sample efficiency. We initialize $\\theta$ to $\\widehat { \\theta } _ { \\mathrm { { m e t a } } }$ while solving (18). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "2. Importance-ratio corrected off-policy updates: The second step of MQL exploits the metatraining replay buffer. Meta-training tasks $\\mathcal { D } _ { \\mathrm { m e t a } }$ are disjoint from $D ^ { \\mathrm { n e w } }$ but because they are expected to come from the same task distribution, transitions collected during meta-training can potentially be exploited to adapt the policy. This is difficult to do on two counts. First, the meta-training transitions do not come from $D ^ { \\mathrm { n e w } }$ . Second, even for transitions from the same task, it is non-trivial to update the policy because of extrapolation error (Fujimoto et al., 2018a): the value function has high error on states it has not seen before. Our use of the propensity score to reweigh transitions is a simpler version of the conditional generative model used by Fujimoto et al. (2018a) in this context. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "MQL fits a logistic classifier on a mini-batch of transitions from the meta-training replay buffer and the transitions collected from the new task in step 1. The context variable $z _ { t }$ is the feature for this classifier. The logistic classifier estimates the importance ratio $\\beta ( \\tau ; D ^ { \\mathrm { n e w } } , \\mathcal { D } _ { \\mathrm { m e t a } } )$ and can be used to reweigh data from the meta-training replay buffer for taking updates as ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e9e0a0a4733a4446326e8ea2b6fde6239f47b6cea29d84e8358414759da5c884.jpg",
+        "text": "$$\n\\arg \\operatorname* { m a x } _ { \\theta } \\left\\{ { \\underset { \\tau \\sim \\mathcal { D } _ { \\mathrm { m e t a } } } { \\mathbb { E } } } \\left[ \\beta ( \\tau ; D ^ { \\mathrm { n e w } } , \\mathcal { D } _ { \\mathrm { m e t a } } ) \\ \\ell ^ { \\mathrm { n e w } } ( \\theta ) \\right] - \\frac { \\lambda } { 2 } \\Vert \\theta - \\widehat { \\theta } _ { \\mathrm { m e t a } } \\Vert _ { 2 } ^ { 2 } \\right\\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We have again included a quadratic penalty $\\lVert \\boldsymbol { \\theta } - \\widehat { \\boldsymbol { \\theta } } _ { \\mathrm { { m e t a } } } \\rVert ^ { 2 }$ that keeps the new parameters close to $\\widehat { \\theta } _ { \\mathrm { { m e t a } } }$ . Estimating the importance ratio involves solving a convex optimization problem on few samples (typically, 200 from the new task and 200-400 from the meta-training tasks). This classifier allows MQL to exploit the large amount of past data. In practice, we perform as many as $1 0 0 \\times$ more weight updates using (19) than (18). ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "Remark 3 (Picking the coefficient $\\boldsymbol { \\lambda }$ ). Following Fakoor et al. (2019), we pick ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/342668fdc7a46d406ac42793870c44087cdad0e81ebfd0dfc4660cb5585a9e1e.jpg",
+        "text": "$$\n\\lambda = 1 - { \\widehat { \\mathrm { E S S } } }\n$$",
+        "text_format": "latex",
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "for both the steps (18–19). This relaxes the quadratic penalty if the new task is similar to the metatraining tasks (ESS  is large) and vice-versa. While $\\lambda$ could be tuned as a hyper-parameter, our empirical results show that adapting it using $\\widehat { \\mathrm { E S S } }$ is a simple and effective heuristic. ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "Remark 4 (Details of estimating the importance ratio). It is crucial to ensure that the logistic classifier for estimating $\\beta$ generalizes well if we are to reweigh transitions in the meta-training replay buffer that are different than the ones the logistic was fitted upon. We do so in two ways: (i) the regularization co-efficient in (11) is chosen to be relatively large, that way we prefer false negatives than risk false positives; (ii) transitions with very high $\\beta$ are valuable for updating (19) but cause a large variance in stochastic gradient descent-based updates, we clip $\\beta$ before taking the update in (19). The clipping constant is a hyper-parameter and is given in Sec. 4. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "MQL requires having access to the meta-training replay buffer during adaptation. This is not a debilitating requirement and there are a number of clustering techniques that can pick important transitions from the replay-buffer if a robotic agent is limited by available hard-disk space. The meta-training replay buffer is at most 3 GB for the experiments in Sec. 4. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 E X P E R I M E N T S ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "This section presents the experimental results of MQL. We first discuss the setup and provide details the benchmark in Sec. 4.1. This is followed by empirical results and ablation experiments in Sec. 4.2. ",
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+    {
+        "type": "text",
+        "text": "4 . 1 S E T U P ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Tasks and algorithms: We use the MuJoCo (Todorov et al., 2012) simulator with OpenAI Gym (Brockman et al., 2016) on continuous-control meta-RL benchmark tasks. These tasks have different rewards, randomized system parameters (Walker-2D-Params) and have been used in previous papers such as Finn et al. (2017); Rothfuss et al. (2018); Rakelly et al. (2019). We compare against standard baseline algorithms, namely MAML (TRPO (Schulman et al., 2015) variant) (Finn et al., 2017), RL2 (Duan et al., 2016), ProMP (Rothfuss et al., 2018) and PEARL (Rakelly et al., 2019). We obtained the training curves and hyper-parameters for all the three algorithms from the published code by Rakelly et al. (2019). ",
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+    {
+        "type": "image",
+        "img_path": "images/eafe95396a5aef45830b380729fae5360000332773a15ea5df288a806f78b861.jpg",
+        "image_caption": [
+            "Figure 2: Average undiscounted return of TD3 and TD3-context compared with PEARL for validation tasks from four meta-RL environments. The agent fails to learn if the policy is conditioned only on the state. In contrast, everything else remaining same, if TD3 is provided access to context, the rewards are much higher. In spite of not adaptating on the validation tasks, TD3- context is comparable to PEARL. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We will compare the above algorithms against: (i) vanilla TD3 (Fujimoto et al., 2018a) without any adaptation on new tasks, (ii) TD3-context: TD3 with GRU-based context Sec. 3.1.1 without any adaptation, and (iii) MQL: TD3 with context and adaptation on new task using the procedure in Sec. 3.2. All the three variants use the multi-task objective for meta-training (15). We use Adam (Kingma & Ba, 2014) for optimizing all the loss functions in this paper. ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Evaluation: Current meta-RL benchmarks lack a systematic evaluation procedure. 2 For each environment, Rakelly et al. (2019) constructed a fixed set of meta-training tasks $\\left( \\mathcal { D } _ { \\mathrm { m e t a } } \\right)$ and a validation set of tasks $D ^ { \\mathrm { n e w } }$ that are disjoint from the meta-training set. To enable direct comparison with published empirical results, we closely followed the evaluation code of Rakelly et al. (2019) to create these tasks. We also use the exact same evaluation protocol as that of these authors, e.g., 200 timesteps of data from the new task, or the number of evaluation episodes. We report the undiscounted return on the validation tasks with statistics computed across 5 random seeds. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4 . 2 R E S U L T S ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
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+        "img_path": "images/7543b1ffa2f4cafc441b9270fd2319318343cb8edcebc6d157ae52d1a9808a03.jpg",
+        "image_caption": [
+            "Figure 3: Comparison of the average undiscounted return of MQL (orange) against existing meta-RL algorithms on continuous-control environments. We compare against four existing algorithms, namely MAML (green), RL2 (red), PROMP (purple) and PEARL (blue). In all environments except Walker-2D-Params and Ant-Goal-2D, MQL is better or comparable to existing algorithms in terms of both sample complexity and final returns. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Our first result, in Fig. 2, is to show that vanilla off-policy learning with context, without any adaptation is competitive with state of the art meta-RL algorithms. We used a standard implementation of TD3 and train on the meta-training tasks using the multi-task objective (15). Hyper-parameters for these tasks are provided in Appendix D. This result is surprising and had gone unnoticed in the current literature. Policies that have access to the context can easily generalize to the validation tasks and achieve performance that is comparable to more sophisticated meta-RL algorithms. ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "We next evaluate MQL against existing meta-RL benchmarks on all environments. The results are shown in Fig. 3. We see that for all environments except Walker-2D-Params and Ant-Goal-2D, MQL obtains comparable or better returns on the validation tasks. In most cases, in particular for the challenging Humanoid-Direc-2D environment, MQL converges faster than existing algorithms. MAML and ProMP require about 100M time-steps to converge to returns that are significantly worse than the returns of off-policy algorithms like MQL and PEARL. Compare the training curve for TD3-context for the Ant-Goal-2D environment in Fig. 2 with that of the same environment in Fig. 3: the former shows a prominent dip in performance as meta-training progresses; this dip is absent in Fig. 3 and can be attributed to the adaptation phase of MQL. ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4 . 3 A B L A T I O N E X P E R I M E N T S ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We conduct a series of ablation studies to analyze the different components of the MQL algorithm. We use two environments for this purpose, namely Half-Cheetah-Fwd-Back and Ant-Fwd-Back. Fig. 4a shows that the adaptation in MQL in (18) and (19) improves performance. Also observe that MQL has a smaller standard deviation in the returns as compared to TD3-context which does not perform any adaptation; this can be seen as the adaptation phase making up for the lost performance of the meta-trained policy on a difficult task. Next, we evaluate the importance of the additional data from the replay buffer in MQL. Fig. 4b compares the performance of MQL with and without updates in (19). We see that the old data, even if it comes from different tasks, is useful to improve the performance on top of (18). Fig. 4c shows the effectiveness of setting $\\lambda = 1 - { \\widehat { \\mathrm { E S S } } }$ as compared to a fixed value of $\\lambda = 0 . 5$ . We see that modulating the quadratic penalty with $\\widehat { \\mathrm { E S S } }$ helps, the effect is minor for Sec. 4.3. The ideal value of $\\lambda$ depends on a given task and using $1 - { \\widehat { \\mathrm { E S S } } }$ can help to adjust to different tasks without the need to do hyper-parameter search per task. Finally, Fig. 5 shows the evolution of $\\lambda$ and $\\beta ( z )$ during meta-training. The coefficient $\\lambda$ is about 0.55 and $\\beta ( z )$ is 0.8 for a large fraction of the time. The latter indicates that propensity score estimation is successful in sampling transitions from the meta-training replay buffer that are similar to the validation tasks. The value of $\\lambda$ remains relatively unchanged during training. This value indicates the fraction of transitions in the old data that are similar to those from the new tasks; since there are two distinct tasks in Ant-Fwd-Back, the value $\\lambda = 0 . 5 5$ is appropriate. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/50ecd84362774187e893ccd7bd046411a834c2b7f94e64540448031304c5a17c.jpg",
+        "image_caption": [
+            "Figure 4: Ablation studies to examine various components of MQL. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "4 . 4 R E L AT E D W O R K ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "Learning to learn: The idea of building an inductive bias for learning a new task by training on a large number of related tasks was established in a series of works (Utgoff, 1986; Schmidhuber, 1987; Baxter, 1995; Thrun, 1996; Thrun & Pratt, 2012). These papers propose building a base learner that fits on each task and a meta-learner that learns properties of the base learners to output a new base learner for a new task. The recent literature instantiates this idea in two forms: (i) the meta-learner directly predicts the base-learner (Wang et al., 2016; Snell et al., 2017) and (ii) the meta-learner learns the updates of the base-learner (Bengio et al., 1992; Hochreiter et al., 2001; Finn et al., 2017). ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Meta-training versus multi-task training: Metatraining aims to train a policy that can be adapted efficiently on a new task. Conceptually, the improved efficiency of a meta-learner comes from two things: (i) building a better inductive bias to initialize the learning (Schmidhuber et al., 1997; Baxter, 1995; 2000; Mitchell, 1980), or (ii) learning a better learning procedure (Bengio et al., 1997; Lee et al., 2019). The two notions of meta-learning above are complementary to each other and in fact, most recent literature using deep neural networks, e.g., MAML (Finn et al., 2017) and Prototypical Networks (Snell et al., 2017) confirms to the first notion of building a better inductive bias. ",
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+        "text": "The multi-task training objective in MQL is the simplest possible instantiation of this idea: it maximizes the average reward on all tasks and learns a better prior without explicitly training for improving adaptation. This aspect of MQL coincides with a recent trend in meta-learning for image classification where it has been observed that modifications to episodic meta-training (Snell et al., 2017; Gidaris & Komodakis, 2018; Chen et al., 2018), or even foregoing meta-training completely (Dhillon et al., 2019) performs better. We speculate two reasons for this phenomenon: (i) meta-training methods are complex to implement and tune, and (ii) powerful function classes such as deep neural networks may have leftover capacity to adapt to a new task even if they are not explicitly trained for adaptation. ",
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+            "Figure 5: Evolution of $\\lambda$ and $\\beta ( z )$ during meta-training. "
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+        "text": "Context-based approaches: Both forms of meta-learning above have been employed relatively successfully for image classification (Snell et al., 2017; Ravi & Larochelle, 2016; Finn et al., 2017). It has however been difficult to replicate that empirical performance in RL: sensitivity to hyperparameters (Henderson et al., 2018) precludes directly predicting the base-learner while long-range temporal dependencies make it difficult to learn the updates of the base learner (Nichol et al., 2018). Recent methods for meta-RL instead leverage context and learn a policy that depends on just on the current state $x _ { t }$ but on the previous history. This may be done in a recurrent fashion (Heess et al., 2015; Hausknecht & Stone, 2015) or by learning a latent representation of the task (Rakelly et al., 2019). Context is a powerful construct: as Fig. 1 shows, even a simple vanilla RL algorithm (TD3) when combined with context performs comparably to state-of-the-art meta-RL algorithms. However, context is a meta-training technique, it does not suggest a way to adapt a policy to a new task. For instance, Rakelly et al. (2019) do not update parameters of the policy on a new task. They rely on the latent representation of the context variable generalizing to new tasks. This is difficult if the new task is different from the training tasks; we discuss this further in Sec. 3.1.1. ",
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+        "text": "Policy-gradient-based algorithms versus off-policy methods: Policy-gradient-based methods have high sample complexity (Ilyas et al., 2018). This is particularly limiting for meta-RL (Finn et al., 2017; Rothfuss et al., 2018; Houthooft et al., 2018) where one (i) trains on a large number of tasks and, (ii) aims to adapt to a new task with few data. Off-policy methods offer substantial gains in sample complexity. This motivates our use of off-policy updates for both meta-training and adaptation. Off-policy updates allow using past data from other policies. MQL exploits this substantially, it takes up to $1 0 0 \\times$ more updates using old data than new data during adaptation. Off-policy algorithms are typically very sensitive to hyper-parameters (Fujimoto et al., 2018a) but we show that MQL is robust to such sensitivity because it adapts automatically to the distribution shift using the Effective Sample Size (ESS). ",
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+        "text": "Propensity score estimation has been extensively studied in both statistics (Robert & Casella, 2013; Quionero-Candela et al., 2009) and RL (Dud´ık et al., 2011; Jiang & Li, 2015; Kang et al., 2007; Bang & Robins, 2005). It is typically used to reweigh data from the proposal distribution to compute estimators on the target distribution. MQL uses propensity scores in a novel way: we fit a propensity score estimator on a subset of the meta-training replay buffer and use this model to sample transitions from the replay buffer that are similar to the new task. The off-policy updates in MQL are essential to exploiting this data. The coefficient of the proximal term in the adaptation-phase objective (18–19) using the effective sample size (ESS) is inspired from the recent work of Fakoor et al. (2019). ",
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+    {
+        "type": "text",
+        "text": "5 D I S C U S S I O N ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The algorithm proposed in this paper, namely MQL, builds upon on three simple ideas. First, Qlearning with context is sufficient to be competitive on current meta-RL benchmarks. Second, maximizing the average reward of training tasks is an effective meta-learning technique. The meta-training phase of MQL is significantly simpler than that of existing algorithms and yet it achieves comparable performance to the state of the art. This suggests that we need to re-think meta-learning in the context of rich function approximators such as deep networks. Third, if one is to adapt to new tasks with few data, it is essential to exploit every available avenue. MQL recycles data from the meta-training replay buffer using propensity estimation techniques. This data is essentially free and is completely neglected by other algorithms. This idea can potentially be used in problems outside RL such as few-shot and zero-shot image classification. ",
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+        "type": "text",
+        "text": "Finally, this paper sheds light on the nature of benchmark environments in meta-RL. The fact that even vanilla Q-learning with a context variable—without meta-training and without any adaptation— is competitive with state of the art algorithms indicates that (i) training and validation tasks in the current meta-RL benchmarks are quite similar to each other and (ii) current benchmarks may be insufficient to evaluate meta-RL algorithms. Both of these are a call to action and point to the need to invest resources towards creating better benchmark problems for meta-RL that drive the innovation of new algorithms. ",
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+    {
+        "type": "text",
+        "text": "R E F E R E N C E S ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Deepak Agarwal, Lihong Li, and Alexander Smola. Linear-time estimators for propensity scores. In Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics, pp. 93–100, 2011.   \nHeejung Bang and James M Robins. Doubly robust estimation in missing data and causal inference models. Biometrics, 61(4):962–973, 2005.   \nJonathan Baxter. Learning internal representations. Flinders University of S. Aust., 1995.   \nJonathan Baxter. A model of inductive bias learning. Journal of artificial intelligence research, 12:149–198, 2000.   \nSamy Bengio, Yoshua Bengio, Jocelyn Cloutier, and Jan Gecsei. On the optimization of a synaptic learning rule. In Preprints Conf. Optimality in Artificial and Biological Neural Networks, pp. 6–8. Univ. of Texas, 1992.   \nSamy Bengio, Yoshua Bengio, Jocelyn Cloutier, and Jan Gecsei. On the optimization of a synaptic learning rule, 1997.   \nGreg Brockman, Vicki Cheung, Ludwig Pettersson, Jonas Schneider, John Schulman, Jie Tang, and Wojciech Zaremba. OpenAI Gym. arXiv:1606.01540, 2016.   \nWei-Yu Chen, Yen-Cheng Liu, Zsolt Kira, Yu-Chiang Frank Wang, and Jia-Bin Huang. A closer look at few-shot classification. 2018.   \nKyunghyun Cho, Bart Van Merrienboer, Caglar Gulcehre, Dzmitry Bahdanau, Fethi Bougares, Holger Schwenk, ¨ and Yoshua Bengio. Learning phrase representations using rnn encoder-decoder for statistical machine translation. arXiv:1406.1078, 2014.   \nGuneet S Dhillon, Pratik Chaudhari, Avinash Ravichandran, and Stefano Soatto. A baseline for few-shot image classification. arXiv:1909.02729, 2019.   \nYan Duan, John Schulman, Xi Chen, Peter L Bartlett, Ilya Sutskever, and Pieter Abbeel. Rl2: Fast reinforcement learning via slow reinforcement learning. arXiv:1611.02779, 2016.   \nMiroslav Dud´ık, John Langford, and Lihong Li. Doubly robust policy evaluation and learning. arXiv:1103.4601, 2011.   \nV´ıctor Elvira, Luca Martino, and Christian P Robert. Rethinking the effective sample size. arXiv:1809.04129, 2018.   \nRasool Fakoor, Pratik Chaudhari, and Alexander J Smola. P3o: Policy-on policy-off policy optimization. arXiv:1905.01756, 2019.   \nChelsea Finn, Pieter Abbeel, and Sergey Levine. Model-agnostic meta-learning for fast adaptation of deep networks. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pp. 1126– 1135. JMLR. org, 2017.   \nScott Fujimoto, David Meger, and Doina Precup. Off-policy deep reinforcement learning without exploration. arXiv:1812.02900, 2018a.   \nScott Fujimoto, Herke van Hoof, and Dave Meger. Addressing function approximation error in actor-critic methods. arXiv:1802.09477, 2018b.   \nSpyros Gidaris and Nikos Komodakis. Dynamic few-shot visual learning without forgetting. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 4367–4375, 2018.   \nMatthew Hausknecht and Peter Stone. Deep recurrent q-learning for partially observable mdps. In 2015 AAAI Fall Symposium Series, 2015.   \nNicolas Heess, Jonathan J Hunt, Timothy P Lillicrap, and David Silver. Memory-based control with recurrent neural networks. arXiv:1512.04455, 2015.   \nPeter Henderson, Riashat Islam, Philip Bachman, Joelle Pineau, Doina Precup, and David Meger. Deep reinforcement learning that matters. In Thirty-Second AAAI Conference on Artificial Intelligence, 2018.   \nSepp Hochreiter, A Steven Younger, and Peter R Conwell. Learning to learn using gradient descent. In International Conference on Artificial Neural Networks, pp. 87–94. Springer, 2001.   \nRein Houthooft, Yuhua Chen, Phillip Isola, Bradly Stadie, Filip Wolski, OpenAI Jonathan Ho, and Pieter Abbeel. Evolved policy gradients. In Advances in Neural Information Processing Systems, pp. 5400–5409, 2018.   \nAndrew Ilyas, Logan Engstrom, Shibani Santurkar, Dimitris Tsipras, Firdaus Janoos, Larry Rudolph, and Aleksander Madry. Are deep policy gradient algorithms truly policy gradient algorithms? arXiv:1811.02553, 2018.   \nNan Jiang and Lihong Li. Doubly robust off-policy value evaluation for reinforcement learning. arXiv:1511.03722, 2015.   \nJoseph DY Kang, Joseph L Schafer, et al. Demystifying double robustness: A comparison of alternative strategies for estimating a population mean from incomplete data. Statistical science, 22(4):523–539, 2007.   \nDiederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv:1412.6980, 2014.   \nAugustine Kong. A note on importance sampling using standardized weights. University of Chicago, Dept. of Statistics, Tech. Rep, 348, 1992.   \nKwonjoon Lee, Subhransu Maji, Avinash Ravichandran, and Stefano Soatto. Meta-learning with differentiable convex optimization. arXiv:1904.03758, 2019.   \nTimothy P Lillicrap, Jonathan J Hunt, Alexander Pritzel, Nicolas Heess, Tom Erez, Yuval Tassa, David Silver, and Daan Wierstra. Continuous control with deep reinforcement learning. arXiv:1509.02971, 2015.   \nTom M Mitchell. The need for biases in learning generalizations. Department of Computer Science, Laboratory for Computer Science Research . . . , 1980.   \nAlex Nichol, Joshua Achiam, and John Schulman. On first-order meta-learning algorithms. arXiv:1803.02999, 2018.   \nJoaquin Quionero-Candela, Masashi Sugiyama, Anton Schwaighofer, and Neil D Lawrence. Dataset shift in machine learning. The MIT Press, 2009.   \nKate Rakelly, Aurick Zhou, Deirdre Quillen, Chelsea Finn, and Sergey Levine. Efficient off-policy metareinforcement learning via probabilistic context variables. arXiv:1903.08254, 2019.   \nSachin Ravi and Hugo Larochelle. Optimization as a model for few-shot learning. 2016.   \nSashank J. Reddi, Barnabas P ´ oczos, and Alexander J. Smola. Doubly robust covariate shift correction. In ´ AAAI, 2015.   \nSidney I Resnick. A probability path. Springer Science & Business Media, 2013.   \nChristian Robert and George Casella. Monte Carlo statistical methods. Springer Science & Business Media, 2013.   \nJonas Rothfuss, Dennis Lee, Ignasi Clavera, Tamim Asfour, and Pieter Abbeel. Promp: Proximal meta-policy search. arXiv:1810.06784, 2018.   \nJurgen Schmidhuber. Evolutionary principles in self-referential learning. On learning how to learn: The meta-meta-... hook.) Diploma thesis, Institut f. Informatik, Tech. Univ. Munich, 1987.   \nJurgen Schmidhuber, Jieyu Zhao, and Marco Wiering. Shifting inductive bias with success-story algorithm,¨ adaptive levin search, and incremental self-improvement. Machine Learning, 28(1):105–130, Jul 1997. ISSN 1573-0565.   \nJohn Schulman, Sergey Levine, Pieter Abbeel, Michael I Jordan, and Philipp Moritz. Trust region policy optimization. In International Conference on Machine Learning, volume 37, pp. 1889–1897, 2015.   \nDavid Silver, Guy Lever, Nicolas Heess, Thomas Degris, Daan Wierstra, and Martin Riedmiller. Deterministic policy gradient algorithms. In International Conference on Machine Learning, 2014.   \nAdrian Smith. Sequential Monte Carlo methods in practice. Springer Science & Business Media, 2013.   \nJake Snell, Kevin Swersky, and Richard Zemel. Prototypical networks for few-shot learning. In Advances in Neural Information Processing Systems, pp. 4077–4087, 2017.   \nSebastian Thrun. Is learning the n-th thing any easier than learning the first? In Advances in neural information processing systems, pp. 640–646, 1996.   \nSebastian Thrun and Lorien Pratt. Learning to learn. Springer Science & Business Media, 2012.   \nEmanuel Todorov, Tom Erez, and Yuval Tassa. Mujoco: A physics engine for model-based control. In 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 5026–5033. IEEE, 2012.   \nPaul E Utgoff. Shift of bias for inductive concept learning. Machine learning: An artificial intelligence approach, 2:107–148, 1986.   \nJane X. Wang, Zeb Kurth-Nelson, Dhruva Tirumala, Hubert Soyer, Joel Z. Leibo, Remi Munos, Charles Blundell, ´ Dharshan Kumaran, and Matthew Botvinick. Learning to reinforcement learn. CoRR, abs/1611.05763, 2016. URL http://arxiv.org/abs/1611.05763. ",
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+    {
+        "type": "text",
+        "text": "A P S E U D O - C O D E ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The pseudo-code for MQL during training and adaption are given in Algorithm 1 and Algorithm 2. After MQL is trained for a given environment as described in Algorithm 1, it returns the meta-trained policy $\\theta$ and replay buffer containing train tasks. ",
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+        "text": "Next, Algorithm 2 runs the adaptation procedure which adapts the meta-trained policy to a test task $D$ with few data. To do so, MQL optimizes the adaptation objective into two steps. After gathering data from a test task $D$ , MQL first updates the policy using the new data (line 4). MQL then fits a logistic classifier on a mini-batch of transitions from the meta-training replay buffer and the transitions collected from the test task and then estimates $\\widehat { \\mathrm { E S S } }$ (lines 5-6). Finally, the adaptation step runs for $n$ iterations (lines $7 \\textrm { - } 1 0 $ ) in which MQL can exploit past data in which it uses propensity score to decide whether or not a given sample is related to the current test task. ",
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+        "type": "text",
+        "text": "Algorithm 1: MQL - Meta-training ",
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+        "table_body": "<table><tr><td colspan=\"2\">Algorithm1:MQL -Meta-training</td></tr><tr><td colspan=\"2\">Input: Set of training tasks Dmeta</td></tr><tr><td colspan=\"2\">1 Initialize the replay buffer</td></tr><tr><td colspan=\"2\">2 Initialize parameters θ of an off-policy method,e.g., TD3</td></tr><tr><td>3 while not done do</td><td></td></tr><tr><td>4</td><td>// Rollout and update policy</td></tr><tr><td>5 6</td><td>Sample a task D ~ Dmeta Gather data from task D using policy Te while feeding transitions through context GRU.Add</td></tr><tr><td></td><td>trajectory to the replay buffer.</td></tr><tr><td>7 8</td><td>&amp;←Sample mini-batch from buffer Update parameters θ using mini-batch t and Eqn. (15)</td></tr><tr><td colspan=\"2\">90meta←0</td></tr><tr><td colspan=\"2\">10 return θmeta ,replay buffer</td></tr></table>",
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+        "type": "text",
+        "text": "Algorithm 2: MQL - Adaptation ",
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+        "type": "text",
+        "text": "Input: Test task $D$ , meta-training replay buffer, meta-trained policy $\\theta _ { \\mathrm { { m e t a } } }$ ",
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+        "text": "1 Initialize temporary buffer buf   \n2 $\\theta \\gets \\theta _ { m e t a }$   \n3 buf ← Gather data from D using πθmeta   \n4 Update Eqn. (18) using buf   \n5 Fit $\\beta ( D )$ using buf and meta-training replay buffer using Eqn. (12)   \n6 Estimate $\\widehat { \\mathrm { E S S } }$ using $\\beta ( D )$ using Eqn. (13)   \n7 for $i \\leq n$ do   \n8 $\\ell ~ \\gets$ sample mini-batch from meta-training replay buffer   \n9 Calculate $\\beta$ for $\\mathcal { \\ell }$   \n10 Update $\\theta$ using Eqn. (19)   \n11 Evaluate $\\theta$ on a new rollout from task $D$   \n12 return $\\theta$ ",
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+    {
+        "type": "text",
+        "text": "B O U T - O F - D I S T R I B U T I O N T A S K S ",
+        "text_level": 1,
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+        "text": "MQL is designed for explicitly using data from the new task along with off-policy data from old, possibly very different tasks. This is on account of two things: (i) the loss function of MQL does not use the old data if it is very different from the new task, $\\beta$ is close to zero for all samples, and (ii) the first term in (18) makes multiple updates using data from the new task. To explore this aspect, we create an out-of-distribution task using the “Half-Cheetah-Vel” environment wherein we use disjoint sets of velocities for meta-training and testing. The setup is as follows: ",
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+        "text": "• Half-Cheetah-Vel-OOD-Medium: target velocity for a training task is sampled uniformly randomly from [0, 2.5] while that for test task is sampled uniformly randomly from [2.5, 3.0]. ",
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+        "text": "This is what we call “medium” hardness task because although the distributions of train and test velocities is disjoint, they are close to each other. ",
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+        "text": "• Half-Cheetah-Vel-OOD-Hard: target velocity for a training task is sampled uniformly randomly from [0, 1.5] while that for test task is sampled uniformly randomly from [2.5, 3.0]. This is a “hard” task because the distributions of train and test velocities are far away from each other. ",
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+        "text": "Fig. 6a shows that MQL significantly outperforms PEARL when the train and test target velocities come from disjoint sets. We used the published code of PEARL (Rakelly et al., 2019) for this experiment. This shows that the adaptation in MQL is crucial to generalizing to new situations which are not a part of the meta-training process. Fig. 6b shows the evolution of the proximal penalty coefficient $\\lambda$ and the propensity score $\\beta ( z )$ during meta-training for the medium-hard task. We see that $\\lambda \\approx 0 . 8$ while $\\beta ( z ) \\approx 0 . 2$ throughout training. This indicates that MQL automatically adjusts its test-time adaptation to use only few samples in (19) if the test task provides transitions quite different than those in the replay buffer. ",
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+        "text": "We next discuss results on the harder task Half-Cheetah-Vel-OOD-Hard. There is a very large gap between training and test target velocities in this case. Fig. 7a shows the comparison with the same test protocol as the other experiments in this paper. In particular, we collect 200 time-steps from the new task and use it for adaptation in both MQL and TD3-context. Since this task is particularly hard, we also ran an experiment where 1200 time-steps (6 episodes) are given to the two algorithms for adaptation. The results are shown in Fig. 7b. In both cases, we see that MQL is better than TD3-context by a large margin (the standard deviation on these plots is high because the environment is hard). Note that since we re-initialize the hidden state of the context network at the beginning of each episode, TD3-context cannot take advantage of the extra time-steps. MQL on the other hand updates the policy explicitly and can take advantage of this extra data. ",
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+        "type": "text",
+        "text": "For sake of being thorough, we collected 800 time-steps from the new task from the same episode, the results are shown in Fig. 8a. We again notice that MQL results in slightly higher rewards than TD3-context in spite of the fact that both the algorithms suffer large degradation in performance as compared to Figs. 7a and 7b. ",
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+        "text": "Figs. 7c, 7d and 8b show that the proximal penalty coefficient $\\lambda \\approx 1$ and the propensity score $\\beta ( z ) \\approx 0$ for a large fraction of training. This proves that MQL is able to automatically discard samples unrelated to the new test during the adaptation phase. ",
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+            "Figure 6: Comparison of the average return of MQL (orange) against existing PEARL algorithms (blue). Fig. 6a shows that MQL significantly outperforms PEARL when the train and test target velocities come from disjoint sets. Fig. 6b shows the evolution of the proximal penalty coefficient $\\lambda$ and the propensity score $\\beta ( z )$ . We see that $\\beta ( z )$ is always small which demonstrates that MQL automatically adjusts the adaptation in (19) if the test task is different from the training tasks. "
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+    {
+        "type": "image",
+        "img_path": "images/5128fdf6d6ee3f9aa4b187936e2fbdbe73f675fdf553c3b2f8cd2fbdf12229c8.jpg",
+        "image_caption": [
+            "Figure 7: (a,b) Comparison of the average return of MQL (orange) against TD3-context (blue). Fig. 7a shows the comparison with the same test protocol as the other experiments in this paper. In particular, we collect 200 time-steps from the new task and use it for adaptation in both MQL and TD3-context. Since this task is particularly hard, we also ran an experiment where 1200 time-steps (6 episodes) where results are shown in Fig. 7b. In both cases, we see that MQL is better than TD3-context by a large margin (the standard deviation on these plots is high because the environment is hard). (c,d) Evolution of $\\lambda$ and $\\beta ( z )$ during meta-training. shows the evolution of the proximal penalty coefficient $\\lambda$ and the propensity score $\\beta ( z )$ . We see in Fig. 7c and Fig. 7d that $\\beta ( z )$ is always small which demonstrates that MQL automatically adjusts the adaptation if the test task is different from the training tasks. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/536e4d33a9916c5fa1ea32db4a5093aebc3344fb15594b6f73ee82052a68d4dc.jpg",
+        "image_caption": [
+            "Figure 8: (a) Comparison of the average return of MQL (orange) against TD3-context (blue). For these experiments, we collected 800 time-steps from the new task from the same episode, the results are shown in Fig. 8a. We again notice that MQL results in slightly higher rewards than TD3-context in spite of the fact that both the algorithms suffer large degradation in performance as compared to Figs. 7a and 7b. (b) Evolution of $\\lambda$ and $\\beta ( z )$ during meta-training shows the evolution of the proximal penalty coefficient $\\lambda$ and the propensity score $\\beta ( z )$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/07d7de0e698367a58640db2cdbdd886a8b673b1442d6db6f5dfc3bdaf74030b0.jpg",
+        "image_caption": [
+            "Figure 9: Ablation studies to examine various components of MQL. Fig. 9a shows that the adaptation in MQL in (18) and (19) improves performance. One reason for that is because test and training tasks in Walker2D-Params are very similar as it shown in Fig. 10b. Next, we evaluate the importance of the additional data from the replay buffer in MQL. Fig. 9b compares the performance of MQL with and without updates in (19). We see that the old data, even if it comes from different tasks, is useful to improve the performance on top of (18). Fig. 9c and Fig. 9f show the effectiveness of setting $\\lambda = 1 - { \\widehat { \\mathrm { E S S } } }$ as compared to a fixed value of $\\lambda = 0 . 5$ . We see that modulating the quadratic penalty with ESS  helps, the effect is minor for Sec. 4.3. The ideal value of $\\lambda$ depends on a given task and using $1 - { \\widehat { \\mathrm { E S S } } }$ can help to adjust to different tasks without the need to do hyper-parameter search per task. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/e58b0c971d517dac40d33263b6e8ced7c530b4827ba0c08c85a99984da9bc9ae.jpg",
+        "image_caption": [
+            "Figure 10: Evolution of $\\lambda$ and $\\beta ( z )$ during meta-training shows the evolution of the proximal penalty coefficient $\\lambda$ and the propensity score $\\beta ( z )$ . We see in Fig. 10a that $\\beta ( z )$ stays around 0.4 which demonstrates MQL automatically adjusts the adaptation if the test task is different from the training tasks. Fig. 10b shows that test and training tasks are very similar as $\\beta ( z )$ is around 0.6. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "C M O R E A B L A T I O N S T U D I E S ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "We conduct a series of additional ablation studies to analyze the different components of the MQL algorithm. We use two environments for this purpose, namely Half-Cheetah-Vel and Walker-2DParams. Fig. 9 and Fig. 10 show the result of these experiments. These experiments show that adaptation phase is more useful for Half-Cheetah-Vel than Walker-2D-Params as test and training tasks are very similar in Walker-2D-Params which helps TD3-context achieves strong performance that leaves no window for improvement with adaptation. ",
+        "bbox": [
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+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "D H Y P E R - P A R A M E T E R S A N D M O R E D E T A I L S O F T H E E M P I R I C A L R E S U LT S ",
+        "bbox": [
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+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "table",
+        "img_path": "images/584357b4197c4998aa67100ec2a798971d756f2b98ea21a498b023a2938eacb5.jpg",
+        "table_caption": [
+            "Table 1: Hyper-parameters for MQL and TD3 for continuous-control meta-RL benchmark tasks. We use a network with two full-connected layers for all environments. The batch-size in Adam is fixed to 256 for all environments. The abbreviation HC stands for Half-Cheetah. These hyper-parameters were tuned by grid-search. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Humanoid</td><td>HC-Vel</td><td> Ant-FB</td><td> Ant-Goal</td><td> Walker</td><td>HC-FB</td></tr><tr><td>β clipping</td><td>1</td><td>1.1</td><td>1</td><td>1.2</td><td>2</td><td>0.8</td></tr><tr><td>TD3 exploration noise</td><td>0.3</td><td>0.3</td><td>0.3</td><td>0.2</td><td>0.3</td><td>0.2</td></tr><tr><td>TD3 policy noise</td><td>0.2</td><td>0.3</td><td>0.3</td><td>0.4</td><td>0.3</td><td>0.2</td></tr><tr><td>TD3 policy update frequency</td><td>2</td><td>2</td><td>2</td><td>4</td><td>4</td><td>3</td></tr><tr><td>Parameter updates per iteration (meta-training)</td><td>500</td><td>1000</td><td>200</td><td>1000</td><td>200</td><td>200</td></tr><tr><td>Adaptation parameter updates per episode (eq. 18)</td><td>10</td><td>5</td><td>10</td><td>5</td><td>10</td><td>10</td></tr><tr><td>Adaptation parameter updates per episode (eq. 19)</td><td>200</td><td>400</td><td>100</td><td>400</td><td>400</td><td>300</td></tr><tr><td>GRU sequence length</td><td>20</td><td>20</td><td>20</td><td>25</td><td>10</td><td>10</td></tr><tr><td>Context dimension</td><td>20</td><td>20</td><td>15</td><td>30</td><td>30</td><td>30</td></tr><tr><td>Adam learning rate</td><td>0.0003</td><td>0.001</td><td>0.0003</td><td>0.0004</td><td>0.0008</td><td>0.0003</td></tr></table>",
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+    }
+]