Add files using upload-large-folder tool

parse/dev/3FvF1db-bKT/3FvF1db-bKT.md

@@ -0,0 +1,385 @@
+# LOCAL AUGMENTATION FOR GRAPH NEURAL NETWORKS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Data augmentation has been widely used in image data and linguistic data but remains under-explored for Graph Neural Networks (GNNs). Existing methods focus on augmenting the graph data from a global perspective and largely fall into two genres: structural manipulation and adversarial training with feature noise injection. However, recent graph data augmentation methods ignore the importance of local information for the GNNs’ message passing mechanism. In this work, we introduce the local augmentation, which enhances the locality of node representations by their subgraph structures. Specifically, we model the data augmentation as a feature generation process. Given a node’s features, our local augmentation approach learns the conditional distribution of its neighbors’ features and generates more neighbors’ features to boost the performance of downstream tasks. Based on the local augmentation, we further design a novel framework: LA-GNN, which can apply to any GNN models in a plug-and-play manner. Extensive experiments and analyses show that local augmentation consistently yields performance improvement for various GNN architectures across a diverse set of benchmarks.
+
+# 1 INTRODUCTION
+
+Graph Neural Networks (GNNs) and their variants (Abu-El-Haija et al., 2019; Kipf & Welling, 2017; Velickovi ˇ c et al., 2018) have achieved state-of-the-art performance for many tasks on graphs such as ´ recommendation system (Ying et al., 2018) and traffic prediction (Guo et al., 2019). However, most of the GNN models, such as GCN (Kipf & Welling, 2017) and GAT (Velickovi ˇ c et al., 2018), learn ´ the node representations by aggregating information over only the 2-hop neighborhood. Such shallow architectures limit their ability to extract information from higher-layer neighborhoods (Wang & Derr, 2021). But deep GNNs are prone to over-smoothing (Li et al., 2018), which suggests the node representations tend to converge to a certain vector and thus become indistinguishable. One solution to address this problem is to preserve the locality of node representations when increasing the number of layers. For example, JKNet (Xu et al., 2018) densely connects (Huang et al., 2017) each hidden layer to the final layer. GCNII (Chen et al., 2020) employs an initial residual to construct a skip connection from the input layer. Besides, Zeng et al. (2021) pointed out that the key for GNN is to smooth the local neighborhood into informative representation, no matter how deep it is. And they decouple the depth and scope of GNNs to help capture local graph structure. Prior works have emphasized the importance of local information, but one property of the graph is that the number of nodes in the local neighborhood is far fewer than higher-order neighbors. And this property limits the expressive power of GNNs due to the limited neighbors in the local structure. A very intuitive idea is to use data augmentation to increase the number of nodes in the local substructure.
+
+However, existing graph data augmentation methods ignore the importance of local information and only perturb at the topology-level and feature-level from a global perspective, which can be divided into two categories: topology-level augmentation (Rong et al., 2020; Wang et al., 2020b; Zhao et al., 2021) and feature-level augmentation (Deng et al., 2019; Feng et al., 2019; Kong et al., 2020). Topology-level augmentation perturbs the adjacency matrix, yielding different graph structures. On the other hand, existing feature-level augmentation mainly exploits perturbation of node attributes guided by adversarial training (Deng et al., 2019; Feng et al., 2019; Kong et al., 2020). These augmentation techniques have two drawbacks. 1) Some of they employ full-batch training for augmentation, which is computationally expensive, and introduce some additional side effects such as over-smoothing. 2) The type of feature-level augmentation is coarse-grained, which focuses on global augmentation and overlooks the local information of the neighborhood. Moreover, to our best knowledge, none of the existing approaches combines both the feature representations and the graph topology, especially the local subgraph structures, for graph-level data augmentation.
+
+In this work, we propose a framework: Local Augmentation for Graph Neural Networks (LA-GNNs), to further enhance the locality of node representations based on both the topology-level and featurelevel information in the substructure. The term "local augmentation" refers to the generation of neighborhood features via a generative model conditioned on local structures and node features. Specifically, our proposed framework learns the conditional distribution of the connected neighbors’ representations given the representation of the central node, bearing some similarities with the Skipgram (Mikolov et al., 2013) and Deepwalk Perozzi et al. (2014), with the difference that our method does not base on word or graph embedding.
+
+The motivation behind this work concludes three-fold. 1) Existing feature-level augmentation works primarily pay attention to global augmentation without considering the informative neighborhood. 2) The distributions of the representations of the neighbors are closely connected to the central node, making ample room for feature augmentation. 3) Preserving the locality of node representations is key to avoiding over-smoothing $\mathrm { { X u } }$ et al., 2018; Klicpera et al., 2019; Chen et al., 2020). And there are several benefits in applying local augmentation for the GNN training. First, local augmentation is essentially a data augmentation technique that can improve the generalization of the GNN models and prevent over-fitting. Second, we can recover some missing contextual information of the local neighborhood in an attributed graph via the generative model (Jia & Benson, 2020). Third, our proposed framework is flexible and can be applied to various popular backbone networks such as GCN (Kipf & Welling, 2017), GAT (Velickovi ˇ c et al., 2018), GCNII (Chen et al., 2020), and ´ GRAND (Feng et al., 2020) to enhance their performance. Extensive experimental results demonstrate that our proposed framework could improve the performance of GNN variants on 7 benchmark datasets.
+
+# 2 BACKGROUND
+
+Notations. Let $G = ( V , E )$ represent the graph, where $V$ is the set of vertices $\{ v _ { 1 } , \cdots , v _ { N } \}$ with $| V | = N$ and $E$ is the set of edges. The adjacency matrix is defined as $\mathbf { A } \in \{ 0 , 1 \} ^ { N \times N }$ , and nod $\mathbf { A } _ { i j } = 1$ f and only if denote the d $( v _ { i } , v _ { j } ) \in E$ . Let ee ma $\mathcal { N } _ { i } \overset { \cdot } { = } \{ v _ { j } \vert  \mathbf { A } _ { i j } = 1 \}$ the neighborhood of. The feature matrix $v _ { i }$ $\mathbf { D }$ $\begin{array} { r } { \dot { \bf D } _ { i i } = \dot { \sum } _ { j = 1 } ^ { n } { \bf A } _ { i j } } \end{array}$ is denoted as $\mathbf { X } \in \mathbb { R } ^ { N \times F }$ where each node $v$ is associated with a $F$ -dimensional feature vector $\mathbf { X } _ { v }$ . $\mathbf { Y } \in \{ 0 , 1 \} ^ { N \times C }$ denote the one-hot label matrix, where $\mathbf { Y } _ { i } \in \{ 0 , 1 \} ^ { C }$ is a one-hot vector and $\begin{array} { r } { \sum _ { j = 1 } ^ { C } \mathbf { Y } _ { i j } = 1 } \end{array}$ for any $v _ { i } \in V$ .
+
+GNN. Graph Neural Network (GNN) is a type of neural network that directly operates on the graph structure, such as GCN and GAT (Kipf & Welling, 2017; Velickovi ˇ c et al., 2018), that capture the ´ dependence of graphs via message passing between the nodes of a graph as
+
+$$
+\mathbf { H } ^ { ( \ell ) } = f ( \mathbf { A } , \mathbf { H } ^ { ( \ell - 1 ) } ) ,
+$$
+
+where $f$ denotes the specific GNN layer for different models, $\mathbf { H } ^ { ( \ell ) }$ are the hidden vectors of the $\ell$ -th layer and $\mathbf { H } ^ { ( 0 ) } = \mathbf { X }$ . For example, $\dot { f ( \mathbf { A } , \mathbf { H } ) } = \sigma ( \hat { \mathbf { A } } \mathbf { H } \mathbf { W } )$ for GCN, where $\hat { \mathbf { A } } = \tilde { \mathbf { D } } ^ { - \frac { 1 } { 2 } } \tilde { \mathbf { A } } \tilde { \mathbf { D } } ^ { - \frac { 1 } { 2 } } ,$ $\tilde { \bf D }$ is the degree matrix of $\tilde { \mathbf { A } }$ , i.e., $\begin{array} { r } { \tilde { \bf D } _ { i i } = \sum _ { j } \tilde { \bf A } _ { i j } } \end{array}$ , and $\tilde { \mathbf { A } } = \mathbf { A } + \mathbf { I }$ .
+
+Topology-level Augmentation. Topology-level augmentation usually perturbs $\mathbf { A }$ to generate different graph structures, which can be formulated as $\mathbf { A } ^ { \prime } = \mathcal { F } ( \mathbf { A } , \mathbf { X } )$ , where $\mathcal F ( \cdot )$ is a structure perturbation function. For example, DropEdge (Rong et al., 2020) considers $\mathcal { F } ( \mathbf { A } , \mathbf { X } ) = \mathbf { A } - \mathbf { A _ { s } }$ which is independent of $\mathbf { X }$ , where $\mathbf { A _ { s } }$ is a sparse matrix consists of a subset of the original edges $E$ . GAUG-O (Zhao et al., 2021) leverages their proposed neural edge predictors to produce a different structure $\mathbf { A } ^ { \prime }$ where $\begin{array} { r } { \mathbf { A } _ { i j } ^ { \prime } = \left\lfloor \frac { 1 } { 1 + e ^ { - \left( \log \mathbf { P } _ { i j } + G \right) / \tau } } + \frac { 1 } { 2 } \right\rfloor } \end{array}$ , $\mathbf { P } _ { i j } = \alpha \mathbf { M } _ { i j } + ( 1 - \alpha ) \mathbf { A } _ { i j }$ , $\mathbf { M } = { \boldsymbol { \sigma } } \left( \mathbf { Z } \mathbf { Z } ^ { T } \right)$ , $\mathbf { Z } = f \left( \mathbf { A } , f ( \mathbf { A } , \mathbf { X } ) \right)$ , $\tau$ is the temperature of Gumbel-Softmax distribution, $G \sim { \mathrm { G u m b e l } } ( 0 , 1 )$ is a Gumbel random variate, and $\alpha$ is a hyperparameter mediating the influence of edge predictor on the original graph.
+
+# Feature-level Augmentation.
+
+Besides, feature-level augmentation function can be defines as $\mathbf { X } ^ { \prime } \ = \ \mathcal { H } ( \mathbf { A } , \mathbf { X } )$ , where $\mathcal { H } ( \cdot )$ is a feature perturbation function. FLAG (Kong et al., 2020) defines the perturbation function as $\begin{array} { r } { \mathcal { H } ( \mathbf { A } , \mathbf { X } ) = \textbf { X } + \boldsymbol { \delta } } \end{array}$ where
+
+Table 1: Comparison of existing graph data augmentation.   
+
+<table><tr><td colspan="4">GraphData Augmentation</td></tr><tr><td>Method</td><td>ConsideredPart</td><td>Type</td><td>Perturbed Part</td></tr><tr><td>DropEdge</td><td>A</td><td>Sampling</td><td>A</td></tr><tr><td>GAUG-O</td><td>A&amp;X</td><td>Reconstruction</td><td></td></tr><tr><td>FLAG</td><td>X</td><td>Noise Injection</td><td></td></tr><tr><td>G-GCN</td><td>A&amp;X</td><td>Reconstruction</td><td>AXX</td></tr><tr><td>Local Augmentation</td><td>A&amp;X</td><td>Generation</td><td>X</td></tr></table>
+
+perturbation $\pmb { \delta }$ is updated iteratively during the adversarial training phase. G-GCN (plain) (Zhu et al., 2020) obtains the global attribute feature matrix $\mathbf { X } ^ { ( a ) } \in \bar { \mathbb { R } ^ { N \times d _ { a } } }$ through minimizing the objective Qv∈V Qa∈CA(v) v a Pk∈U expX(a)v ·Vk where $U$ is the set of all attributes, $C A ( v )$ is the sampled context attributes of $v$ , and $\mathbf { V } \in \mathbb { R } ^ { d _ { a } \times F }$ denotes the parameters. Obviously, the perturbation function of G-GCN has no close-form solution. In this work, we propose a novel feature-level augmentation method, named local augmentation. And the comparison of the details of various graph data augmentation techniques can be found in Table 1.
+
+# 3 LOCAL AUGMENTATION
+
+In this section, we describe details of the proposed method. The local augmentation framework consists of three modules: learning the conditional distribution via a generative model, the active learning trick, and the downstream GNN models, as illustrated in Figure 1. Note that the proposed algorithm enhances the locality of node representations through augmenting 1-hop neighbors in a generative way. Specifically, we exploit a generative model to learn the conditional distribution of the connected neighbors’ representations given the representation of a node. We describe the details of learning the conditional distribution and the motivation for why local augmentation is able to improve the performance in a probabilistic view in Sec. 3.1, detail the architecture of downstream GNN models in Sec. 3.2. We finally elaborate the training procedure of both the generative model and the downstream GNN models with the active learning trick in Sec. 3.3.
+
+![](images/a25e0a452d09cb122f2cc05b960b1e984bdab4b6769823194b1ac823b6b03d23.jpg)  
+Figure 1: A schematic depiction of our local augmentation. The purple and yellow circles on the graph correspond to the central node and its augmented neighbors respectively. After augmenting the neighborhood, we exploit the initial and the generated feature matrix as input for downstream GNNs.
+
+# 3.1 LEARNING THE CONDITIONAL DISTRIBUTION
+
+We start by reviewing the semi-supervised learning of GNNs in a probabilistic view. Most existing GNN models (Kipf & Welling, 2017; Velickovi ˇ c et al., 2018) are viewed as a classification function ´ to predict the class labels of the graph nodes. In this work, we use a GNN classification estimator $P _ { \theta } ( \mathbf { Y } | \mathbf { A } , \mathbf { X } )$ $\theta$ is the parameter) to model the conditional distribution of label $\mathbf { Y }$ with respect to the graph structure A and feature matrix X. Given training samples $\{ \mathbf { A } , \mathbf { X } , \mathbf { Y } \}$ , the parameter $\theta$ can be estimated using Maximum Likelihood Estimation (MLE), by optimizing the following likelihood function:
+
+$$
+\operatorname* { m a x } \prod _ { k \in \mathbf { K } } P _ { \theta } \left( \mathbf { Y } _ { k } | \mathbf { A } , \mathbf { X } \right) ,
+$$
+
+where $\mathbf { K }$ is the set of node indices of the training dataset whose labels are visible during the semi-supervised training. To further boost the performance of GNN, we introduce a new model $P _ { \theta } ( { \bf Y } , \overline { { { \bf X } } } | { \bf A } , { \bf X } )$ , where $\overline { { \mathbf { X } } }$ is generated features by feature-level augmentation. For this model, the MLE method needs to optimize a marginalized probability $P _ { \theta }$ over the generated feature matrix $\overline { { \mathbf { X } } }$ :
+
+$$
+\operatorname* { m a x } \prod _ { k \in \mathbf { K } } \int _ { \overline { { \mathbf { X } } } } P _ { \theta } \left( \mathbf { Y } _ { k } , \overline { { \mathbf { X } } } | \mathbf { A } , \mathbf { X } \right) .
+$$
+
+For Bayesian tractability, we decompose $P _ { \theta }$ in Eq.(3) as a product of two posterior probabilities:
+
+$$
+\begin{array} { r } { P _ { \theta , \phi } ( \mathbf { Y } _ { k } , \overline { { \mathbf { X } } } | \mathbf { A } , \mathbf { X } ) : = P _ { \theta } ( \mathbf { Y } _ { k } | \mathbf { A } , \mathbf { X } , \overline { { \mathbf { X } } } ) Q _ { \phi } ( \overline { { \mathbf { X } } } | \mathbf { A } , \mathbf { X } ) , } \end{array}
+$$
+
+where $P _ { \theta } ( \mathbf { Y } _ { k } | \mathbf { A } , \mathbf { X } , \mathbf { \overline { { X } } } )$ and $Q _ { \phi } ( { \overline { { \mathbf { X } } } } | \mathbf { A } , \mathbf { X } )$ denote the probabilistic distributions approximated by the downstream GNN and the (feature-level augmentation) generator respectively, parameterized by $\theta$ and $\phi$ . There are two benefits in the decomposition above. First, it allows us to decouple the training of the downstream predictor $P _ { \theta }$ and the generator $Q _ { \phi }$ , enabling the generator to easily generalize to other downstream tasks. Moreover, inspired by the successes of data augmentation via deep-learning-based generative modeling (Antoniou et al., 2017), the representation power of Eq.(4) is superior than that of a single predictor $P _ { \theta } \left( \mathbf { Y } _ { k } | \mathbf { A } , \mathbf { X } \right)$ without data augmentation.
+
+Consequently, once a generator $Q _ { \phi }$ is trained very well, our training procedure can optimize $P _ { \theta } ( \mathbf { Y } _ { k } | \mathbf { A } , \mathbf { X } , \mathbf { \overline { { X } } } )$ with samples $\overline { { \mathbf { X } } }$ drawn from the fixed conditional distribution $Q _ { \phi }$ . Now, we show how to train the generator as follows.
+
+Generator To learn a feature augmentation generator, a naive solution is to learn one single distribution for all the neighbors using the MLE method, i.e., solving the following optimization problem
+
+$$
+\operatorname* { m a x } _ { \psi } \sum _ { j \in \mathcal { N } _ { i } } \log p _ { \psi } \left( \mathbf { X } _ { j } | \mathbf { X } _ { i } \right) = \operatorname* { m a x } _ { \psi } \log \prod _ { j \in \mathcal { N } _ { i } } p _ { \psi } \left( \mathbf { X } _ { j } | \mathbf { X } _ { i } \right) ,
+$$
+
+where $\{ \mathbf { X } _ { j \mid j \in \mathcal { N } _ { i } } , \mathbf { X } _ { i } \}$ . Then $p _ { \psi }$ can be used to augment features for all the neighbors. However, this method ignores the differences between all the neighbors, which may induce severe noise.
+
+To overcome the limitation, we assume that each neighbor satisfies a different conditional distribution. Specifically, there exists a conditional distribution $p ( \cdot | \mathbf { X } _ { i } , \mathbf { z } _ { j } )$ with latent random variable $\mathbf { z } _ { j }$ , such that we have $\mathbf { X } _ { j } \sim p ( \mathbf { X } | \mathbf { X } _ { i } , \mathbf { z } _ { j } )$ for $\mathbf { X } _ { j \mid j \in \mathcal { N } _ { i } }$ . Once we obtain $p ( \cdot | \mathbf { X } _ { i } , \mathbf { z } _ { j } )$ in some way, we can generate augmented features $\overline { { \mathbf { X } } }$ , and then we can train $P _ { \theta } ( \mathbf { Y } _ { k } | \mathbf { A } , \mathbf { X } , \mathbf { \overline { { X } } } )$ instead of $P _ { \theta } ( \mathbf { Y } _ { k } | \mathbf { A } , \mathbf { X } )$ to improve the final performance of $P _ { \theta }$ . Below, we will present how to find $p ( \cdot | \mathbf { X } _ { i } , \mathbf { z } _ { j } )$ , which will produce the generator $Q _ { \phi }$ .
+
+To achieve our purpose, a suitable method is the conditional variational auto-encoder (CVAE) (Kingma & Welling, 2013; Sohn et al., 2015), which can help learn the distribution of the latent variable $\mathbf { z } _ { j }$ , and the conditional distribution $p ( \cdot | \mathbf { X } _ { i } , \mathbf { z } _ { j } )$ . So, a CVAE model $Q _ { \phi } \left( \overline { { \mathbf { X } } } | \mathbf { A } , \mathbf { X } \right)$ is adopted as our generator, where $\phi = \{ \varphi , \psi \}$ , $\varphi$ denotes the variational parameters and $\psi$ represents the generative parameters. To derive the optimization problem for CVAE, $\log p _ { \psi } \left( \mathbf { X } _ { j } | \mathbf { X } _ { i } \right)$ can be written with latent variables $\mathbf { z }$ as follows, following previous work (Pandey & Dukkipati, 2017; Sohn et al., 2015):
+
+$$
+\begin{array} { l } { \log p _ { \psi } ( \mathbf { X } _ { j } | \mathbf { X } _ { i } ) = \displaystyle \int q _ { \varphi } ( \mathbf { z } | \mathbf { X } _ { j } , \mathbf { X } _ { i } ) \log \frac { p _ { \psi } ( \mathbf { X } _ { j } , \mathbf { z } | \mathbf { X } _ { i } ) } { q _ { \varphi } ( \mathbf { z } | \mathbf { X } _ { j } , \mathbf { X } _ { i } ) } \mathrm { d } \mathbf { z } + K L ( q _ { \varphi } ( \mathbf { z } | \mathbf { X } _ { j } , \mathbf { X } _ { i } ) \| p _ { \psi } ( \mathbf { z } | \mathbf { X } _ { j } , \mathbf { X } _ { i } ) ) } \\ { \displaystyle \qquad \geq \int q _ { \varphi } ( \mathbf { z } | \mathbf { X } _ { j } , \mathbf { X } _ { i } ) \log \frac { p _ { \psi } ( \mathbf { X } _ { j } , \mathbf { z } | \mathbf { X } _ { i } ) } { q _ { \varphi } ( \mathbf { z } | \mathbf { X } _ { j } , \mathbf { X } _ { i } ) } \mathrm { d } \mathbf { z } , } \end{array}
+$$
+
+and the evidence lower bound (ELBO) can be written as:
+
+$$
+\mathcal { L } ( \mathbf { X } _ { j } , \mathbf { X } _ { i } ; \psi , \varphi ) = - K L ( q _ { \varphi } ( \mathbf { z } | \mathbf { X } _ { j } , \mathbf { X } _ { i } ) | | p _ { \psi } ( \mathbf { z } | \mathbf { X } _ { i } ) ) + \int q _ { \varphi } ( \mathbf { z } | \mathbf { X } _ { j } , \mathbf { X } _ { i } ) \log p _ { \psi } ( \mathbf { X } _ { j } | \mathbf { X } _ { i } , \mathbf { z } ) \mathrm { d } \mathbf { z } ,
+$$
+
+where the encoder $q _ { \varphi } ( \mathbf { z } | \mathbf { X } _ { j } , \mathbf { X } _ { i } ) \ = \ { \mathcal { N } } ( f ( \mathbf { X } _ { j } , \mathbf { X } _ { i } ) , g ( \mathbf { X } _ { j } , \mathbf { X } _ { i } ) )$ and decoder $p _ { \psi } ( { \bf X } _ { j } | { \bf X } _ { i } , { \bf z } ) =$ $\mathcal { N } ( h ( \mathbf { X } _ { i } , \mathbf { z } ) , c I )$ . The encoder is a two-layer MLP. $f$ and $g$ share the first layer, and their second layers employ different parameters. The decoder $h$ is two-layer MLP. For simplicity and tractability, the implemented generator $Q \left( \overline { { \mathbf { X } } } | \mathbf { A } , \mathbf { X } \right)$ uses the same parameters across all nodes $v _ { i } \in V$ .
+
+Optimization of the MLE Now, we present how to optimize the MLE Eq.(4) using the feature matrix produced from the generator. Once the augmented feature matrix can be sampled from the generator, we can optimize the parameters of Eq.(4) in the following way. Firstly, the parameter $\bar { \phi } = \{ \psi , \varphi \}$ can be optimized by maximizing the ELBO of the generator (6), i.e., we train the generator. Secondly, the parameter $\theta$ is optimized by maximizing the MLE Eq.(4) with $\phi$ fixed, which is the conditional distribution of ${ \bf Y } _ { k }$ given A, $\mathbf { X }$ , and $\overline { { \mathbf { X } } }$ , i.e., we train the downstream GNN model.
+
+In this paper, the MLE is formulated by a downstream GNN model as follows:
+
+$$
+P _ { \theta } \left( \mathbf { Y } _ { k } \mid \mathbf { A } , \mathbf { X } , { \overline { { \mathbf { X } } } } \right) \propto - { \overline { { \mathcal { L } } } } ( \theta | \mathbf { A } , \mathbf { X } , { \overline { { \mathbf { X } } } } , \phi ) ,
+$$
+
+$$
+\begin{array} { r } { \overline { { \mathcal { L } } } ( \theta | \mathbf { A } , \mathbf { X } , \overline { { \mathbf { X } } } , \phi ) = - \sum _ { k \in \mathbf { T } } \sum _ { f = 1 } ^ { C } \mathbf { Y } _ { k f } \ln \Big ( \mathrm { s o f t m a x } \big ( \mathrm { G N N } ( \mathbf { A } , \mathbf { X } , \overline { { \mathbf { X } } } ) \big ) _ { k f } \Big ) . } \end{array}
+$$
+
+# 3.2 THE ARCHITECTURE OF LA-GNN
+
+We discuss the details of downstream GNN models. And we use GCN, GAT, GCNII, and GRAND as the backbones and test them on semi-supervised node classification tasks. We name the modified GNN architecture as LA-GNN, where LA means local augmentation.
+
+LA-GCN A 2-layer LA-GCN is defined as follows:
+
+$$
+\mathbf { H } ^ { ( 2 ) } = \sigma \left( \hat { \mathbf { A } } \left( \sigma \left( \hat { \mathbf { A } } \mathbf { X } \mathbf { W } _ { 1 } ^ { ( 1 ) } \right) \bigg | \bigg | \sigma \left( \hat { \mathbf { A } } \overline { { \mathbf { X } } } _ { 1 } \mathbf { W } _ { 2 } ^ { ( 1 ) } \right) \bigg | \bigg | \cdots \bigg | \bigg | \sigma \left( \hat { \mathbf { A } } \overline { { \mathbf { X } } } _ { n } \mathbf { W } _ { n + 1 } ^ { ( 1 ) } \right) \right) \mathbf { W } ^ { ( 2 ) } \right) ,
+$$
+
+where $\overline { { \mathbf { X } } } _ { i }$ $\bar { \mathsf { \bar { c } } } _ { i } ( i = 1 , 2 , \cdots , n )$ is the augmented feature matrix produced by the generator, $\parallel$ denotes an operator of column-wise concatenation, $\mathbf { W } _ { i } ^ { ( 1 ) } \left( i = 1 , 2 , \cdots , n \right)$ denotes the parameters of the first LA-GCN layer, and $\mathbf { W } ^ { ( 2 ) }$ denotes the parameters of the second LA-GCN layer.
+
+LA-GCNII Since GCNII (Chen et al., 2020) applies a fully-connected neural network on $\mathbf { X }$ to obtain a lower-dimensional initial representation $\mathbf { H } ^ { ( 0 ) }$ before the forward propagation, we apply a fully-connected neural network on $\mathbf { X }$ and $\overline { { \mathbf { X } } }$ to obtain $\mathbf { H } ^ { ( 0 ) }$ for LA-GCNII as follows:
+
+$$
+\mathbf { H } ^ { ( 0 ) } = \sigma ( \mathbf { X } \mathbf { W } _ { 1 } ^ { ( 0 ) } ) \| \sigma ( \overline { { \mathbf { X } } } _ { 1 } \mathbf { W } _ { 2 } ^ { ( 0 ) } ) \| \cdots \| \sigma ( \overline { { \mathbf { X } } } _ { n } \mathbf { W } _ { n + 1 } ^ { ( 0 ) } ) .
+$$
+
+$\mathbf { H } ^ { ( 0 ) }$ is fed into the next forward propagation layer. Besides, we do not modify the architecture of GAT and GRAND, and just add our generated feature matrix to the input.
+
+# 3.3 ACTIVE LEARNING
+
+In this section, we introduce a trick for the overall training framework. After the training of the generator finishes, it contains an issue of using $Q _ { \phi } ( { \overline { { \mathbf { X } } } } | \mathbf { A } , \mathbf { X } )$ of Eq.(4) for inference because $Q$ may generate some samples from the side part of the distribution. This critical question makes the inferences inefficient. Inspired by Nielsen & Okoniewski (2019), we introduce active learning to capture the suitable generated feature matrix and the corresponding generator, which improves the inference efficiency and helps the optimization of the MLE. During active learning, the probability of each feature is proportional to its uncertainty evaluated by an acquisition function. We adopt the Bayesian Active Learning by Disagreement (BALD) acquisition function (Houlsby et al., 2011) to sample the most important inferences with the approximation from the Monte Carlo (MC) dropout samples as
+
+$$
+{ \cal U } ( \overline { { \mathbf { X } } } ) \approx H \left[ \frac { 1 } { N } \sum _ { n = 1 } ^ { N } P \left( \mathbf { Y } _ { k } | \overline { { \mathbf { X } } } , \omega _ { n } \right) \right] - \frac { 1 } { N } \sum _ { n = 1 } ^ { N } H \left[ P \left( \mathbf { Y } _ { k } | \overline { { \mathbf { X } } } , \omega _ { n } \right) \right] ,
+$$
+
+where $N$ is the number of MC samples and $\omega _ { n }$ are the parameters of the network sampled for the $n$ -th MC dropout sample. A high BLAD score indicates a network with high uncertainty about the generated feature matrix. So it tends to be selected to improve the GNN model. Finally, the overall algorithm framework is summarized in Algorithm 1, which shows the optimization of Eq.(4).
+
+Algorithm 1 The framework to train the Generator $Q _ { \phi }$ and the downstream GNN $P _ { \theta }$ using the initial feature matrix $\mathbf { X }$ and the generated feature matrix $\overline { { \mathbf { X } } }$ selected by the acquisition function
+
+1: Initialize $U { = }$ -inf, $\overline { { \mathbf { X } } }$ , $Q _ { \phi }$ , $\overline { { \mathbf { X } } } ^ { \prime }$ , and $Q _ { \phi } ^ { \prime }$   
+2: for $i = 1$ to the number of generator iterations do   
+3: Train the generator $Q _ { \phi }$ using $\mathbf { A }$ and $\mathbf { X }$   
+4: Generate feature matrix $\overline { { \mathbf { X } } }$ using $Q _ { \phi }$   
+5: Compute $U ( { \overline { { \mathbf { X } } } } )$ using Eq.(10).   
+6: if $U ( { \overline { { \mathbf { X } } } } ) > U$ then   
+7: $U = U ( { \overline { { \mathbf { X } } } } )$   
+8: if $i > N _ { w a r m u p }$ then   
+9: Train GNN $P _ { \theta }$ using $\mathbf { A }$ and $\overline { { \mathbf { X } } }$ for the number of continued GNN training iterations   
+10: X 0 = X , Q 0φ = Q φ   
+11: $\overline { { \mathbf { X } } } = \overline { { \mathbf { X } } } ^ { \prime }$ , $Q _ { \phi } = Q _ { \phi } ^ { \prime }$   
+12: Train the downstream GNN $P _ { \theta }$ with the generated feature matrix $\overline { { \mathbf { X } } }$ , and generator $Q _ { \phi }$
+
+# 4 DISCUSSION
+
+In this section, we discuss the motivation of this work and provide some analysis.
+
+Connection to EP-B and GraphSAGE We discuss how our proposed model distinguishes from the classical representation learning models on graphs. Previous methods such as EP-B (GarcíaDurán & Niepert, 2017) and GraphSAGE (Hamilton et al., 2017) rely on reconstruction loss function between the central node and its neighbors’ embeddings. EP-B aims to minimize the reconstruction error by optimizing the objective $\begin{array} { r } { \operatorname* { m i n } { \sum _ { u \in V \backslash \{ v \} } } \left[ \gamma + d ( \widetilde { \mathbf { X } } _ { v } , \mathbf { X } _ { v } ) - d ( \widetilde { \mathbf { X } } _ { v } , \mathbf { X } _ { u } ) \right] } \end{array}$ where $\mathbf { X } _ { v }$ represents the target node; $\mathbf { X } _ { u }$ denotes the neighbor nodes; $\widetilde { \mathbf { X } } _ { v } = \mathrm { A G G } ( \mathbf { X } _ { l } | l \in \mathcal { N } ( v ) )$ indicates the reconstruction from neighbors; and $\gamma$ refers to the bias. Besides, GraphSAGE exploits the negative sampling to differentiate the representations of remote node-pairs. GraphSAGE enforce nearby nodes to have similar representations and to enforce disparate nodes to be distinct by minimizing the objective $\operatorname* { m i n } - E _ { u \sim \mathcal { N } ( v ) } \overset { \cdot } { \log } \left( \left( \sigma ( \mathbf { X } _ { u } ^ { T } \mathbf { X } _ { v } ) \right) \right) - \lambda E _ { v _ { n } \sim P _ { n } ( v ) } \log \left( \left( \sigma ( - \mathbf { X } _ { v _ { n } } ^ { T } \mathbf { X } _ { v } ) \right) \right)$ where $\mathbf { X } _ { v }$ denotes target node; $\mathbf { X } _ { u }$ represents the neighbor node; ${ \bf X } _ { v _ { n } }$ is disparate node; and $P _ { n } ( v )$ is the negative sampling. These approaches build upon the assumption that adjacent nodes share similar attributes. In contrast, our model does not rely on such assumption and instead generates the neighboring node features from the conditional distribution of central node representations. Given the target node, $\mathbf { X } _ { v }$ , our aim is to learn the conditional distribution of the neighbor nodes, $\mathbf { X } _ { u }$ . A comparison between the reconstruction-based representation learning on graphs and our proposed framework is illustrated in Figure 2. And our local augmentation method is the third paradigm to exploit neighbors in a generative way.
+
+![](images/02db34de261ebadeed56e54daa7a15cd3eaa86f7795ea776ca63ea498b28e4a9.jpg)  
+Figure 2: (a) The original graph. (b) EP-B exploits the neighbors to reconstruct the central node’s embedding. (c) GraphSAGE encourages nearby nodes to have similar embeddings. (d) Given the representation of the central node, our aim is to infer the representations of the connected distribution of neighbors.
+
+Local Augmentation vs. General Augmentation General image augmentation algorithms include geometric transformations, feature space augmentation, adversarial training, and generative adversarial networks (Shorten & Khoshgoftaar, 2019). It is impossible to apply geometric transformations directly to graph data augmentation since graphs are sensitive to node permutation. General adversarial training, feature space augmentation, and generative adversarial networks don’t take the graph structure into account. Graphs consist of a set of identities with certain pairs of these identities connected by edges. We need to consider node features and the graph structure when designing the graph data augmentation framework. Our proposed method of local augmentation fully considers these two points. By extracting the neighbors’ feature vectors, we have enough data points to learn the distribution. There are two benefits to designing local augmentation. First, by taking the sub-graph structure and feature representation associated with this sub-graph structure as input for the generative model, we can learn the information of the sub-graph structure. Second, the number of data points to learn the distribution depends on the node degree. This assures that we have enough data points compared with the general feature augmentation and we can learn a better distribution.
+
+Complementing missing information Jia & Benson (2020) points out that some attribute information might be missing on a subset of vertices. By learning the distribution of node representations from the observed data, we can utilize the produced node representations from the generative model to complement the information missing in the nodes’ attributes, which boosts the robustness of downstream tasks. And we show that our model still works in the scenario that nodes lose a certain percentage of attributes. In other words, we can exploit the well-learned distribution to complement the contextual information of the local neighborhood to enhance the locality of the node representations.
+
+# 5 EXPERIMENTS
+
+In this section, we evaluate the performance of our proposed model on semisupervised node classification tasks on a variety of public graph datasets and compare our model with the state-of-the-art graph neural networks. We also carry out additional experiments to showcase the necessity of our design and its robustness to missing information.
+
+Table 2: Classification results on fixed split $( \% )$   
+
+<table><tr><td>Method</td><td>Cora</td><td>Citeseer</td><td>Pubmed</td></tr><tr><td>Chebyshev (Defferrard et al.,2016)</td><td>81.2</td><td>69.8</td><td>74.4</td></tr><tr><td>APPNP (Klicpera et al.,2019)</td><td>83.8</td><td>71.6</td><td>79.7</td></tr><tr><td>MixHop (Abu-El-Haija et al.,2019)</td><td>81.9</td><td>71.4</td><td>80.8</td></tr><tr><td>Graph U-net (Gao&amp; Ji,2019)</td><td>84.4</td><td>73.2</td><td>79.6</td></tr><tr><td>GSNN-M (Wang et al.,2020a)</td><td>83.9</td><td>72.2</td><td>79.1</td></tr><tr><td>S²GC (Zhu &amp; Koniusz,2021)</td><td>83.5</td><td>73.6</td><td>80.2</td></tr><tr><td>GCN (Kipf &amp; Welling,2017)</td><td>81.6</td><td>70.3</td><td>78.9</td></tr><tr><td>G-GCN (Zhu et al.,2020)</td><td>83.7</td><td>71.3</td><td>80.9</td></tr><tr><td>DropEdge-GCN (Rong et al.,2020)</td><td>82.8</td><td>72.3</td><td>79.6</td></tr><tr><td>GAUG-O-GCN (Zhao et al.,2021)</td><td>83.6</td><td>73.3</td><td>79.3</td></tr><tr><td>LA-GCN</td><td>84.1</td><td>72.5</td><td>81.3</td></tr><tr><td>GAT (Velickovic et al., 2018)</td><td>83.0</td><td>70.4</td><td>0OM</td></tr><tr><td>LA-GAT</td><td>83.9</td><td>72.3</td><td>OOM</td></tr><tr><td>GCNII (Chen et al.,2020) LA-GCNII</td><td>85.2</td><td>73.1</td><td>80.0</td></tr><tr><td></td><td>85.2</td><td>73.7</td><td>81.6</td></tr><tr><td>GRAND (Feng et al.,2020)</td><td>85.4</td><td>75.4</td><td>82.7</td></tr><tr><td>LA-GRAND</td><td>85.8</td><td>75.8</td><td>83.3</td></tr></table>
+
+# 5.1 DATASETS
+
+We utilize seven public graph datasets (Cora, Citeseer, Pubmed, Squirrel, Actor, Chameleon, and Cornell) for semisupervised node classification tasks. The details of these datasets can be found in the appendix.
+
+# 5.2 SEMI-SUPERVISED NODE CLASSIFICATION
+
+Baselines and Experimental Setup. We apply the standard fixed splits (Yang et al., 2016) on three datasets Cora, Citeseer, and Pubmed, with 20 nodes per class for training, 500 nodes for validation, and 1,000 nodes for testing. And we consider four backbones: GCN (Kipf & Welling, 2017), GAT (Velickovi ˇ c et al., 2018), GCNII (Chen ´ et al., 2020), and GRAND (Feng et al., 2020) to evaluate our proposed framework and compare our model against state-of-the-art models including 1) backbone models: Chebyshev (Defferrard et al., 2016), GCN, GAT,
+
+Table 3: Classification results on random split $( \% )$   
+
+<table><tr><td>Method</td><td>Squirrel</td><td>Actor</td><td>Chameleon</td><td>Cornell</td></tr><tr><td>APPNP</td><td>21.6</td><td>32.1</td><td>33.0</td><td>58.7</td></tr><tr><td>S²GC</td><td>21.3</td><td>27.8</td><td>30.2</td><td>57.2</td></tr><tr><td>GCN</td><td>22.5</td><td>26.2</td><td>25.1</td><td>55.7</td></tr><tr><td>DropEdge-GCN</td><td>21.9</td><td>26.5</td><td>25.0</td><td>53.6</td></tr><tr><td>LA-GCN</td><td>23.2</td><td>27.0</td><td>28.9</td><td>56.1</td></tr><tr><td>GAT</td><td>24.2</td><td>27.2</td><td>34.8</td><td>55.8</td></tr><tr><td>LA-GAT</td><td>28.2</td><td>27.4</td><td>38.6</td><td>56.5</td></tr><tr><td>GCNII</td><td>25.3</td><td>31.9</td><td>30.2</td><td>57.3</td></tr><tr><td>LA-GCNII</td><td>28.6</td><td>32.7</td><td>32.5</td><td>56.6</td></tr></table>
+
+APPNP (Klicpera et al., 2019), Graph U-net (Gao & Ji, 2019), MixHop (Abu-El-Haija et al., 2019), GCNII, GSNN-M (Wang et al., 2020a), $\mathrm { { \cal S } ^ { 2 } { \cal G } { \cal C } }$ (Zhu & Koniusz, 2021), and GRAND and 2) featurelevel and topology-level augmentation models: G-GNNs (Zhu et al., 2020), DropEdge (Rong et al.,
+
+2020) and GAUG-O (Zhao et al., 2021). For four datasets Squirrel, Actor, Chameleon, and Cornell, we take 10 random splits (Shchur et al., 2018) where $10 \%$ , $30 \%$ , and $60 \%$ of the date for training, validation, testing; measure the performance of GCN, GAT, GCNII, and corresponding modified models.
+
+Results For three datasets Cora, Citeseer, and Pubmed, we report the mean classification accuracy on the test nodes of all our models after 100 runs and report the values after running the experiments of their models with our server under their setting hyperparameters in their original papers. The results of the evaluation experiments are summarized in Tables 2, 3, and in the appendix, which demonstrate that the backbone models equipped with our method achieve the best performance across all the datasets except the Cornell dataset. More specifically, we can improve upon GCN by a margin of $2 . 5 \%$ , $2 . 2 \%$ , and $2 . 4 \%$ on Cora, Citeseer, and Pubmed respectively. Moreover, LA-GNN outperforms other backbone models including GAT and GCNII as well as data augmentation models (Zhu et al., 2020; Rong et al., 2020; Zhao et al., 2021) on these citation network datasets. Besids, we also provide the analysis of the distribution of our generated feature matrix. And Figure 3 shows the distribution of the attributes of the original
+
+and inference neighbors, which can demonstrate our inference feature matrix follow the distribution of the initial feature matrix.
+
+![](images/410d67587b543d15cc65cba5cdc78b6ecc62b88dfffe4c1dfb7018aab80db729.jpg)  
+Figure 3: The distribution of the attribute bin of the inference neighbors vs. the distribution of the attribute bin of the original neighbors, with KL divergence $= 0 . 0 0 2 6$ . The value of each feature bin is the sum of the attribute values of multiple dimensions of the feature vector. We split the feature vector into multiple feature bins.
+
+# 5.3 ABLATION STUDY
+
+In this section, to demonstrate the effectiveness of our proposed generative framework, we conduct experiments that compare LA-GNN to several of its ablated variants without generative modeling. The results are shown in Table 4. ${ } " \mathrm { G C N } +$ width" only increases the first network layer width for GCN and GCNII to match LAGNN without giving generated samples as input. $" +$ concatenation" only replaces the generated feature matrix of LA-GNN with the original feature matrix of the central node. $" +$ plain neighborhood" replaces the generated feature matrix of LA-GNN with a neighborhood feature matrix where each row corresponds to the feature vector of the randomly sampled neighbor. The
+
+Table 4: Effects of different components of our framework evaluated on the standard split of the Cora, Citeseer and Pubmed dataset.   
+
+<table><tr><td>Method</td><td>Cora</td><td>Citeseer</td><td>Pubmed</td></tr><tr><td>GCN</td><td>81.6</td><td>70.3</td><td>78.9</td></tr><tr><td>GCNII</td><td>85.2</td><td>73.1</td><td>80.0</td></tr><tr><td>GCN + width</td><td>82.0</td><td>71.4</td><td>79.5</td></tr><tr><td>GCN + concatenation</td><td>81.8</td><td>71.6</td><td>78.8</td></tr><tr><td>GCN + plain neighborhood</td><td>80.9</td><td>68.8</td><td>75.0</td></tr><tr><td>GCNII + width</td><td>85.1</td><td>73.1</td><td>80.2</td></tr><tr><td>GCNII + concatenation</td><td>85.2</td><td>73.3</td><td>80.2</td></tr><tr><td>GCNII + plain neighborhood</td><td>83.3</td><td>71.9</td><td>78.1</td></tr><tr><td>LA-GCN</td><td>84.1</td><td>72.5</td><td>81.3</td></tr><tr><td>LA-GCNII</td><td>85.2</td><td>73.7</td><td>81.6</td></tr></table>
+
+results show that the first two variants provide no notable improvement for the backbone models, and the third variant even results in degradation. By eliminating the possibility that these confounding factors irrelevant to our core approach may contribute to the final performance, it’s evident that the performance gain in Table 2 and 3 are due to our proposed generative local augmentation framework.
+
+# 5.4 ROBUSTNESS TO MISSING INFORMATION
+
+In this section, we conduct experiments to verify that our proposed framework can robustify downstream tasks against missing information in the feature attributes. Specifically, we mask a certain percentage of the attributes of each feature vector and use the same pipeline to do augmentation for the masked feature matrix. As shown in Table 5, we can see that as the mask ratio increases, the gap of the performance between the GCN and LA-GCN enlarges in most cases in Cora and Citeseer, which corroborates our insight discussed in Section 4. Since there exists large redundancy in the features of the Pubmed dataset, the performance of GCN and LA-GCN decreases little as the mask ratio increases and the gap of the performance does not enlarge. To conclude, our model can complement the contextual information of the local neighborhood to enhance the locality of the node representations.
+
+Table 5: Summary of results on recovering study in terms of classification accuracy $( \% )$ . $\downarrow$ means a decrease compared with the accuracy if features are not masked.   
+
+<table><tr><td>Dataset</td><td colspan="4">Cora</td><td colspan="4">Citeseer</td><td colspan="4">Pubmed</td></tr><tr><td>Mask Ratio</td><td>0.1</td><td>0.2</td><td>0.4</td><td>0.8</td><td>0.1</td><td>0.2</td><td>0.4</td><td>0.8</td><td>0.1</td><td>0.2</td><td>0.4</td><td>0.8</td></tr><tr><td>GCN</td><td>81.0(↓0.6)</td><td>80.6(↓1.0)</td><td>80.1(↓1.5)</td><td>76.0 (↓5.6)</td><td>70.1(↓0.2)</td><td>69.3 (↓1.0)</td><td>67.2 (↓3.1)</td><td>61.0(↓9.3)</td><td>78.5(↓0.4)</td><td>78.5(↓0.4)</td><td>77.5 (↓1.4)</td><td>76.9 (↓2.0)</td></tr><tr><td>LA-GCN</td><td>83.5 (↓0.6)</td><td>83.1(↓1.0)</td><td>81.6(↓2.5)</td><td>81.1 (↓3.0)</td><td>72.2(↓0.3)</td><td>71.7 (↓0.8)</td><td>69.3 (↓3.2)</td><td>65.9 (↓6.6)</td><td>81.4(↓0.1)</td><td>80.9 (↓0.6)</td><td>80.5 (↓1.0)</td><td>79.4 (↓2.1)</td></tr></table>
+
+# 6 RELATED WORK
+
+Graph Neural Networks In general, convolution in the graph domain involves non-spectral (spatial) and spectral approaches. Non-spectral methods generalize convolutions operating on spatially close neighbors to the graph domain, such as Duvenaud et al. (2015); Atwood & Towsley (2016); Niepert et al. (2016); Monti et al. (2017). Spectral approaches define the convolution operations based on the spectral formulation, such as Bruna et al. (2014); Defferrard et al. (2016); Kipf & Welling (2017). Recently, several methods (Abu-El-Haija et al., 2019; Liao et al., 2019) based on GCN have been proposed to obtain the higher-order filters. Besides, GAT (Velickovi ˇ c et al., 2018), Graph ´ U-Nets (Gao & Ji, 2019) combine attention networks and pooling operation with GNN separately, which achieve state-of-the-art performance on node and link classification tasks. In this work, local augmentation can be applied on various backbone models to improve performance.
+
+Graph Generative Models Generative models (Goodfellow et al., 2014; Kingma & Welling, 2013) are powerful tools of learning data distribution through unsupervised learning, and they have achieved tremendous success in various applications. Recently, researchers have proposed several interesting generative models for graph data generation. Variational graph auto-encoder (VGAE) (Kipf & Welling, 2016) makes use of latent variables and learns interpretable latent representations for undirected graphs. Salha et al. (2019) replace the GCN encoder in VGAE with a simple linear model and emphasize the effectiveness of a simple node encoding scheme. Xu et al. (2019) propose a generative model framework to learn node representations, by sampling graph generation sequences constructed from observed graph data. ConDgen (Yang et al., 2019) exploits the GCN encoder to handle the inherent challenges of flexible context-structure conditioning and permutation-invariant generation. Besides, some methods have been proposed to apply the graph generative models in various applications such as graph matching (Simonovsky & Komodakis, 2018), molecule design (Liu et al., 2018), retrosynthesis prediction (Shi et al., 2020) and chemical design (Samanta et al., 2018). Compared with these approaches mainly focusing on structure generation, our model takes full use of the power of the generative model for feature representation generation, which can serve as an enhanced technique for the downstream backbone models.
+
+# 7 CONCLUSION
+
+We propose local augmentation, a brand-new technique that exploits the generative model to learn the conditional distribution of the central node’s neighbors’ feature representations given its representation. We can augment more 1-hop neighbors from a well-trained generative model to enhance the performance of backbone GNN models. Experiments show that our model can improve performance across various GNN architectures and benchmark datasets by enriching local information. Besides, our model achieves new state-of-the-art results on various semi-supervised node classification tasks. One limitation of our proposed framework is that we do not exploit the 2-hop neighbors or use the random walk to find more related neighbors for the central node. And one future work is that we can extract more $^ { 2 / 3 }$ -hop neighbors if the central node’s degree is small and learn the conditional distribution for random sampling nodes if the graph is large.
+
+# REFERENCES
+
+Martín Abadi, Paul Barham, Jianmin Chen, Zhifeng Chen, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Geoffrey Irving, Michael Isard, et al. Tensorflow: A system for large-scale machine learning. In 12th {USENIX} symposium on operating systems design and implementation ({OSDI} 16), pp. 265–283, 2016.
+
+Sami Abu-El-Haija, Bryan Perozzi, Amol Kapoor, Nazanin Alipourfard, Kristina Lerman, Hrayr Harutyunyan, Greg Ver Steeg, and Aram Galstyan. Mixhop: Higher-order graph convolutional architectures via sparsified neighborhood mixing. In international conference on machine learning, pp. 21–29. PMLR, 2019.
+
+Antreas Antoniou, Amos Storkey, and Harrison Edwards. Data augmentation generative adversarial networks. arXiv preprint arXiv:1711.04340, 2017.
+
+James Atwood and Don Towsley. Diffusion-convolutional neural networks. In Advances in neural information processing systems, pp. 1993–2001, 2016.
+
+Joan Bruna, Wojciech Zaremba, Arthur Szlam, and Yann LeCun. Spectral networks and locally connected networks on graphs. In International Conference on Learning Representations, 2014.
+
+Ming Chen, Zhewei Wei, Zengfeng Huang, Bolin Ding, and Yaliang Li. Simple and deep graph convolutional networks. In International Conference on Machine Learning, pp. 1725–1735. PMLR, 2020.
+
+Michaël Defferrard, Xavier Bresson, and Pierre Vandergheynst. Convolutional neural networks on graphs with fast localized spectral filtering. In Advances in Neural Information Processing Systems, 2016.
+
+Zhijie Deng, Yinpeng Dong, and Jun Zhu. Batch virtual adversarial training for graph convolutional networks. arXiv preprint arXiv:1902.09192, 2019.
+
+David Duvenaud, Dougal Maclaurin, Jorge Aguilera-Iparraguirre, Rafael Gómez-Bombarelli, Timothy Hirzel, Alán Aspuru-Guzik, and Ryan P Adams. Convolutional networks on graphs for learning molecular fingerprints. In Advances in Neural Information Processing Systems, 2015.
+
+Fuli Feng, Xiangnan He, Jie Tang, and Tat-Seng Chua. Graph adversarial training: Dynamically regularizing based on graph structure. IEEE Transactions on Knowledge and Data Engineering, 2019.
+
+Wenzheng Feng, Jie Zhang, Yuxiao Dong, Yu Han, Huanbo Luan, Qian Xu, Qiang Yang, Evgeny Kharlamov, and Jie Tang. Graph random neural network for semi-supervised learning on graphs. In NeurIPS’20, 2020.
+
+Hongyang Gao and Shuiwang Ji. Graph u-nets. In international conference on machine learning, pp. 2083–2092. PMLR, 2019.
+
+Alberto García-Durán and Mathias Niepert. Learning graph representations with embedding propagation. In Advances in Neural Information Processing Systems, 2017.
+
+Ian J Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial networks. In Advances in Neural Information Processing Systems, 2014.
+
+Shengnan Guo, Youfang Lin, Ning Feng, Chao Song, and Huaiyu Wan. Attention based spatialtemporal graph convolutional networks for traffic flow forecasting. In Proceedings of the AAAI Conference on Artificial Intelligence, pp. 922–929, 2019.
+
+William L Hamilton, Rex Ying, and Jure Leskovec. Inductive representation learning on large graphs. In Advances in Neural Information Processing Systems, 2017.
+
+Neil Houlsby, Ferenc Huszár, Zoubin Ghahramani, and Máté Lengyel. Bayesian active learning for classification and preference learning. arXiv preprint arXiv:1112.5745, 2011.
+
+Gao Huang, Zhuang Liu, Laurens Van Der Maaten, and Kilian Q Weinberger. Densely connected convolutional networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 4700–4708, 2017.
+
+Junteng Jia and Austion R Benson. Residual correlation in graph neural network regression. In Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pp. 588–598, 2020.
+
+Diederik P Kingma and Max Welling. Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114, 2013.
+
+Thomas N Kipf and Max Welling. Variational graph auto-encoders. NIPS Workshop on Bayesian Deep Learning, 2016.
+
+Thomas N. Kipf and Max Welling. Semi-supervised classification with graph convolutional networks. In International Conference on Learning Representation, 2017.
+
+Johannes Klicpera, Aleksandar Bojchevski, and Stephan Günnemann. Predict then propagate: Graph neural networks meet personalized pagerank. In International Conference on Learning Representation, 2019.
+
+Kezhi Kong, Guohao Li, Mucong Ding, Zuxuan Wu, Chen Zhu, Bernard Ghanem, Gavin Taylor, and Tom Goldstein. Flag: Adversarial data augmentation for graph neural networks. arXiv preprint arXiv:2010.09891, 2020.
+
+Qimai Li, Zhichao Han, and Xiao-Ming Wu. Deeper insights into graph convolutional networks for semi-supervised learning. In Proceedings of the AAAI Conference on Artificial Intelligence, 2018.
+
+Renjie Liao, Zhizhen Zhao, Raquel Urtasun, and Richard S Zemel. Lanczosnet: Multi-scale deep graph convolutional networks. In International Conference on Learning Representations, 2019.
+
+Qi Liu, Miltiadis Allamanis, Marc Brockschmidt, and Alexander L Gaunt. Constrained graph variational autoencoders for molecule design. In Advances in Neural Information Processing Systems, 2018.
+
+Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg Corrado, and Jeffrey Dean. Distributed representations of words and phrases and their compositionality. In Advances in Neural Information Processing Systems, 2013.
+
+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 Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 5115–5124, 2017.
+
+Christopher Nielsen and Michal M Okoniewski. Gan data augmentation through active learning inspired sample acquisition. In CVPR Workshops, pp. 109–112, 2019.
+
+Mathias Niepert, Mohamed Ahmed, and Konstantin Kutzkov. Learning convolutional neural networks for graphs. In International conference on machine learning, pp. 2014–2023. PMLR, 2016.
+
+Gaurav Pandey and Ambedkar Dukkipati. Variational methods for conditional multimodal deep learning. In 2017 International Joint Conference on Neural Networks (IJCNN), pp. 308–315. IEEE, 2017.
+
+Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, et al. Pytorch: An imperative style, high-performance deep learning library. In Advances in Neural Information Processing Systems, 2019.
+
+Hongbin Pei, Bingzhe Wei, Kevin Chen-Chuan Chang, Yu Lei, and Bo Yang. Geom-gcn: Geometric graph convolutional networks. In International Conference on Learning Representations, 2020.
+
+Bryan Perozzi, Rami Al-Rfou, and Steven Skiena. Deepwalk: Online learning of social representations. In Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 701–710, 2014.
+
+Yu Rong, Wenbing Huang, Tingyang Xu, and Junzhou Huang. Dropedge: Towards deep graph convolutional networks on node classification. In International Conference on Learning Representation, 2020.
+
+Benedek Rozemberczki, Carl Allen, and Rik Sarkar. Multi-scale attributed node embedding. Journal of Complex Networks, 9(2):cnab014, 2021.
+
+Guillaume Salha, Romain Hennequin, and Michalis Vazirgiannis. Keep it simple: Graph autoencoders without graph convolutional networks. Workshop on Graph Representation Learning, 33rd Conference on Neural Information Processing Systems (NeurIPS), 2019.
+
+Bidisha Samanta, Abir De, Niloy Ganguly, and Manuel Gomez-Rodriguez. Designing random graph models using variational autoencoders with applications to chemical design. arXiv preprint arXiv:1802.05283, 2018.
+
+Prithviraj Sen, Galileo Namata, Mustafa Bilgic, Lise Getoor, Brian Galligher, and Tina Eliassi-Rad. Collective classification in network data. AI magazine, 29(3):93–93, 2008.
+
+Oleksandr Shchur, Maximilian Mumme, Aleksandar Bojchevski, and Stephan Günnemann. Pitfalls of graph neural network evaluation. arXiv preprint arXiv:1811.05868, 2018.
+
+Chence Shi, Minkai Xu, Hongyu Guo, Ming Zhang, and Jian Tang. A graph to graphs framework for retrosynthesis prediction. In International Conference on Machine Learning, pp. 8818–8827. PMLR, 2020.
+
+Connor Shorten and Taghi M Khoshgoftaar. A survey on image data augmentation for deep learning. Journal of Big Data, 6(1):1–48, 2019.
+
+Martin Simonovsky and Nikos Komodakis. Graphvae: Towards generation of small graphs using variational autoencoders. In International Conference on Artificial Neural Networks, pp. 412–422. Springer, 2018.
+
+Kihyuk Sohn, Honglak Lee, and Xinchen Yan. Learning structured output representation using deep conditional generative models. Advances in neural information processing systems, 28:3483–3491, 2015.
+
+Jie Tang, Jimeng Sun, Chi Wang, and Zi Yang. Social influence analysis in large-scale networks. In Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 807–816, 2009.
+
+Petar Velickovi ˇ c, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Lio, and Yoshua ´ Bengio. Graph attention networks. In International Conference on Learning Representations, 2018.
+
+Haibo Wang, Chuan Zhou, Xin Chen, Jia Wu, Shirui Pan, and Jilong Wang. Graph stochastic neural networks for semi-supervised learning. In Advances in Neural Information Processing Systems, 2020a.
+
+Minjie Wang, Da Zheng, Zihao Ye, Quan Gan, Mufei Li, Xiang Song, Jinjing Zhou, Chao Ma, Lingfan Yu, Yu Gai, Tianjun Xiao, Tong He, George Karypis, Jinyang Li, and Zheng Zhang. Deep graph library: A graph-centric, highly-performant package for graph neural networks. arXiv preprint arXiv:1909.01315, 2019.
+
+Yiwei Wang, Wei Wang, Yuxuan Liang, Yujun Cai, Juncheng Liu, and Bryan Hooi. Nodeaug: Semi-supervised node classification with data augmentation. In Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pp. 207–217, 2020b.
+
+Yu Wang and Tyler Derr. Tree decomposed graph neural network. In Proceedings of the 30th ACM International Conference on Information & Knowledge Management, pp. 2040–2049, 2021.
+
+Da Xu, Chuanwei Ruan, Kamiya Motwani, Evren Korpeoglu, Sushant Kumar, and Kannan Achan. Generative graph convolutional network for growing graphs. In ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 3167–3171. IEEE, 2019.
+
+Keyulu Xu, Chengtao Li, Yonglong Tian, Tomohiro Sonobe, Ken-ichi Kawarabayashi, and Stefanie Jegelka. Representation learning on graphs with jumping knowledge networks. In International Conference on Machine Learning, pp. 5453–5462. PMLR, 2018.
+
+Carl Yang, Peiye Zhuang, Wenhan Shi, Alan Luu, and Pan Li. Conditional structure generation through graph variational generative adversarial nets. In NeurIPS, pp. 1338–1349, 2019.
+
+Zhilin Yang, William Cohen, and Ruslan Salakhudinov. Revisiting semi-supervised learning with graph embeddings. In International Conference on Machine Learning, pp. 40–48, 2016.
+
+Rex Ying, Ruining He, Kaifeng Chen, Pong Eksombatchai, William L Hamilton, and Jure Leskovec. Graph convolutional neural networks for web-scale recommender systems. In Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pp. 974–983, 2018.
+
+Hanqing Zeng, Muhan Zhang, Yinglong Xia, Ajitesh Srivastava, Andrey Malevich, Rajgopal Kannan, Viktor Prasanna, Long Jin, and Ren Chen. Decoupling the depth and scope of graph neural networks. In Thirty-Fifth Conference on Neural Information Processing Systems, 2021.
+
+Tong Zhao, Yozen Liu, Leonardo Neves, Oliver Woodford, Meng Jiang, and Neil Shah. Data augmentation for graph neural networks. In The Thirty-Fifth AAAI Conference on Artificial Intelligence, 2021.
+
+Danhao Zhu, Xin-Yu Dai, and Jiajun Chen. Pre-train and learn: Preserve global information for graph neural networks. In Proceedings of the AAAI Conference on Artificial Intelligence, 2020.
+
+Hao Zhu and Piotr Koniusz. Simple spectral graph convolution. In International Conference on Learning Representations, 2021.
+
+A PROOF OF EQ.(6)
+
+We give more details of the derivation of the generator ELBO as follows:
+
+$$
+\begin{array} { r l } { \log _ { \rho } | \mathbf { X } _ { i } | \mathbb { X } _ { j } - j } & { \neq \langle z | \mathbf { z } | \mathbf { z } | \mathbf { X } _ { j } , ~ \mathbf { X } _ { i } \rangle \log _ { \rho } \langle ~ \mathbf { X } _ { i } | \mathbf { X } _ { j } \rangle \mathrm { d } \mathbf { z } } \\ & { = \int \langle z | \mathbf { z } | \mathbf { A } _ { \mathbf { x } } \mathbf { x } , ~ \mathbf { X } _ { i } | \log _ { \rho } | \mathbf { X } _ { i } \mathbf { X } _ { j } | \log _ { \rho } \langle ~ \mathbf { X } _ { i } | \mathbf { X } _ { j } , ~ \mathbf { X } _ { i } \rangle } \\ & { \quad - \int \langle z | \mathbf { z } | \mathbf { z } | \mathbf { X } _ { i } , ~ \mathbf { X } _ { i } \rangle \log _ { \rho } | \langle ~ \mathbf { X } _ { i } | \mathbf { X } _ { j } \rangle \mathrm { d } \mathbf { z } | } \\ & { = \int \langle \exp \{ \mathbf { X } _ { i } \mathbf { X } _ { j } \} | \exp \{ \exp \{ | \mathbf { X } _ { i } \mathbf { X } _ { j } | \} \} \exp \{ | \langle \mathbf { X } _ { i } \mathbf { X } _ { j } , ~ \mathbf { X } _ { i } , ~ \mathbf { X } _ { j } | \} \mathrm { d } \mathbf { z } } \\ & { \quad - \int \langle z | \mathbf { z } | \mathbf { X } _ { i } \rangle \exp \{ | \langle \mathbf { X } _ { j } | \mathbf { X } _ { i } , ~ \mathbf { X } _ { j } \rangle | \} } \\ & { \quad - \int \langle z | \mathbf { z } | \mathbf { z } | \mathrm { X } _ { i } \rangle \exp \{ | \langle \mathbf { X } _ { i } | \mathbf { X } _ { j } \rangle | \} } \\ & { = \int \langle z | \mathbf { z } | \mathbf { X } _ { i } \mathbf { X } _ { j } \rangle \log _ { \rho } \langle \mathbf { X } _ { i } | \mathbf { X } _ { j } \rangle \exp \{ | \langle \mathbf { X } _ { i } | \mathbf { X } _ { j } \rangle | } \\ & { \quad - \int \langle z | \mathbf { z } | \mathbf { Z } _ { j } \rangle \exp \{ | \langle \mathbf { X } _ { j } | \mathbf { X } _ { j } \rangle | } \\ &  \quad - \int \langle z | \mathbf { z } | \mathbf { Z } _ { j } \rangle \exp \{  \end{array}
+$$
+
+$$
+\begin{array} { r l } { L _ { E L B O } = \displaystyle { \int } { q _ { \mathcal { G } } ( { \mathbf z } | \mathbf X _ { j } , \mathbf X _ { i } ) \log \frac { p _ { \phi } ( { \mathbf x } _ { j } , { \mathbf X _ { i } } ) \log } { q _ { \mathcal { G } } ( { \mathbf z } | \mathbf X _ { j } , { \mathbf X _ { i } } ) } } \mathrm { d } { \mathbf z } } \\ { } & { ~ = \displaystyle { \int } { q _ { \mathcal { G } } ( { \mathbf z } | \mathbf X _ { j } , \mathbf X _ { i } ) \log \frac { p _ { \psi } ( { \mathbf X _ { j } } , { \mathbf X _ { i } } , { \mathbf z } ) } { q _ { \mathcal { G } } ( { \mathbf z } | \mathbf X _ { j } , { \mathbf X _ { i } } ) p _ { \psi } ( { \mathbf X _ { i } } ) } } \mathrm { d } { \mathbf z } } \\ { } & { ~ = \displaystyle { \int } { q _ { \mathcal { G } } ( { \mathbf z } | \mathbf X _ { j } , \mathbf X _ { i } ) \log \frac { p _ { \phi } ( { \mathbf X _ { j } } | \mathbf X _ { i } , { \mathbf z } ) p _ { \psi } ( { \mathbf X _ { i } } , { \mathbf z } ) } { q _ { \mathcal { G } } ( { \mathbf z } | \mathbf X _ { j } , { \mathbf X _ { i } } ) p _ { \psi } ( { \mathbf X _ { i } } ) } } \mathrm { d } { \mathbf z } } \\ { } & { ~ = \displaystyle { \int } { q _ { \mathcal { G } } ( { \mathbf z } | \mathbf X _ { j } , \mathbf X _ { i } ) \log \frac { p _ { \phi } ( { \mathbf X _ { j } } | \mathbf X _ { i } , { \mathbf z } ) p _ { \psi } ( { \mathbf z } | \mathbf X _ { i } ) } { q _ { \mathcal { G } } ( { \mathbf z } | \mathbf X _ { j } , { \mathbf X _ { i } } ) } } \mathrm { d } { \mathbf z } } \\ { } &  ~ = \displaystyle { \int } { q _ { \mathcal { G } } ( { \mathbf z } | \mathbf X _ { j } , \mathbf X _ { i } ) \log \frac { p _ { \phi } ( { \mathbf z } | \mathbf X _ { j } , { \mathbf z } ) p _ { \phi } ( { \mathbf Z } | \mathbf X _ { i } ) } { q _ { \mathcal { G } } ( { \mathbf z } | \mathbf X _ { j } , { \mathbf X _ { i } } ) } } \\ { } &  ~ = \displaystyle { \int }  q _ { \mathcal { G } } ( { \mathbf z } | \mathbf X _ { j } , \mathbf X _ { i }  \end{array}
+$$
+
+# B REPRODUCIBILITY
+
+# B.1 DATASETS DETAILS
+
+Cora, Citeseer, and Pubmed are standard citation network benchmark datasets Sen et al. (2008). In these datasets, nodes represent documents, and edges denote citations; node feature corresponds to elements of a bag-of-words representation of a document, and node label corresponds to one of the academic topics. Besides, we utilize four datasets used in Pei et al. (2020) for evaluation. Chameleon and squirrel are two page-page networks on specific topics in Wikipedia Rozemberczki et al. (2021). In these datasets, nodes represent web pages, and edges denote mutual links between pages; node features correspond to several informative nouns in the Wikipedia pages and labels correspond to the number of the average monthly traffic of the web page. WebKB1 is a webpage dataset collected from various universities. We use the one subdataset of it, Cornell. In this dataset, nodes represent web pages, and edges are hyperlinks between them; node features correspond to the bag-of-words representation of web pages and labels correspond to five categories, student, project, course, staff, and faculty. Film dataset is the actor-only induced subgraph of the film-directoractor-writer network Tang et al. (2009). In this dataset, Nodes represent actors, and edges denote co-occurrence on the same Wikipedia page; node features correspond to some keywords in the Wikipedia pages and labels correspond to five categories in terms of words of actor’s Wikipedia. All the dataset statistics are summarized in Table 6.
+
+Table 6: Datasets statistics   
+
+<table><tr><td>Dataset</td><td>Cora</td><td>Cite.</td><td>Pubm.</td><td>Cham.</td><td>Squi.</td><td>Actor</td><td>Corn.</td></tr><tr><td>#Nodes</td><td>2708</td><td>3327</td><td>19717</td><td>2277</td><td>5201</td><td>7600</td><td>183</td></tr><tr><td>#Edges</td><td>5429</td><td>4732</td><td>44338</td><td>36101</td><td>217073</td><td>33544</td><td>295</td></tr><tr><td>#Features</td><td>1433</td><td>3703</td><td>500</td><td>2325</td><td>2089</td><td>931</td><td>1703</td></tr><tr><td># Classes</td><td>7</td><td>6</td><td>3</td><td>5</td><td>5</td><td>5</td><td>5</td></tr></table>
+
+# B.2 IMPLEMENTATION DETAILS
+
+We use Pytorch (Paszke et al., 2019) to implement LA-GNNs. The codes of $S ^ { 2 } G C$ (Zhu & Koniusz, 2021), LA-GCN, LA-GAT, LA-GCNII, LA-GRAND, and DropEdge-GCN are implemented referring to Pytorch implementation of $\mathrm { S } ^ { 2 } \mathrm { G } \mathrm { C } ^ { 2 }$ , $\mathrm { G C N } ^ { 3 }$ (Kipf & Welling, 2017), $\mathrm { G A T ^ { 4 } }$ (Velickovi ˇ c et al., 2018), ´ $\mathrm { G C N I I } ^ { 5 }$ (Chen et al., 2020) $\mathrm { G R A N D } ^ { 6 }$ (Feng et al., 2020), and DropEdge- $\mathbf { \Delta } G \mathbf { C N } ^ { 7 }$ (Rong et al., 2020). Besides, we implement APPNP (Klicpera et al., 2019) with DGL (Wang et al., 2019) version of APPNP8. The datasets Cora, Citeseer, Pubmed are downloaded from TensorFlow (Abadi et al., 2016) implementation of $\mathrm { G C N ^ { 9 } }$ , and the datasets Chameleon, Squirrel, Actor, and Cornell are downloaded from the implementation of Geom- $\mathrm { G C N ^ { 1 0 } }$ (Pei et al., 2020). All the experiments in this work are conducted on a single NVIDIA Tesla V100 with 32GB memory size. The operating system behind the Docker where the experiments are running is Red Hat 4.8.2-16. And the software that we use for experiments are Python 3.6.8, numpy 1.19.2, sklearn 0.0, scipy 1.5.4, networkx 2.5.1, torch 1.6.0, torchvision 0.7.0, CUDA 10.2.89, and CUDNN 8.0.2.
+
+# B.3 HYPERPARAMETER DETAILS
+
+LA-GNNs introduce an additional parameter, that is the hidden layer for generated feature matrix $\overline { { \mathbf { X } } }$ before concatenation. The difference of architectures between GCN and LA-GCN can be found in Figure 4, and the LA-GCNII architecture can be found in Figure 5.
+
+![](images/d7816fad620dd0825bfeea46796303662cc8614bc99cb04d713507ea99f5f1a5.jpg)  
+Figure 4: GCN and LA-GCN architectures. The difference between GCN and LA-GCN architectures is that the LA-GCN has an additional convolutional layer for $\overline { { \mathbf { X } } }$ and it uses a concatenation operation to mix the hidden representations.
+
+![](images/5443477c1a5cf6f340de4b0323cbbb8400ffee35c029505e3e5d44d84aa3e826.jpg)  
+Figure 5: LA-GCNII architecture. The difference between GCNII and LA-GCNII is that the LAGCNII has an additional MLP layer for $\overline { { \mathbf { X } } }$ and it uses a concatenation operation to mix the hidden representations.
+
+The difference of hyperparameters between the GCN and LA-GCN is only the hidden layer size before concatenation. For the LA-GCNII, LA-GAT, LA-GRAND, we tune the hyperparameters in the same way as described in their original papers with validation set.

parse/dev/AXDNM76T1nc/AXDNM76T1nc.md

@@ -0,0 +1,324 @@
+# Video PreTraining (VPT): Learning to Act by Watching Unlabeled Online Videos
+
+Bowen Baker⇤† bowen@openai.com
+
+Ilge Akkaya⇤† ilge@openai.com
+
+Peter Zhokhov⇤† peterz@openai.com
+
+Joost Huizinga⇤† joost@openai.com
+
+Jie Tang⇤† jietang@openai.com
+
+Adrien Ecoffet⇤† adrien@openai.com
+
+Brandon Houghton⇤† brandon@openai.com
+
+Raul Sampedro⇤† raulsamg@gmail.com
+
+Jeff Clune⇤†‡ jclune@gmail.com
+
+# Abstract
+
+Pretraining on noisy, internet-scale datasets has been heavily studied as a technique for training models with broad, general capabilities for text, images, and other modalities. 1–6 However, for many sequential decision domains such as robotics, video games, and computer use, publicly available data does not contain the labels required to train behavioral priors in the same way. We extend the internet-scale pretraining paradigm to sequential decision domains through semi-supervised imitation learning wherein agents learn to act by watching online unlabeled videos. Specifically, we show that with a small amount of labeled data we can train an inverse dynamics model accurate enough to label a huge unlabeled source of online data – here, online videos of people playing Minecraft – from which we can then train a general behavioral prior. Despite using the native human interface (mouse and keyboard at $2 0 \mathrm { H z }$ ), we show that this behavioral prior has nontrivial zeroshot capabilities and that it can be fine-tuned, with both imitation learning and reinforcement learning, to hard-exploration tasks that are impossible to learn from scratch via reinforcement learning. For many tasks our models exhibit humanlevel performance, and we are the first to report computer agents that can craft diamond tools, which can take proficient humans upwards of 20 minutes (24,000 environment actions) of gameplay to accomplish.
+
+# 1 Introduction
+
+Work in recent years has demonstrated the efficacy of pretraining large and general foundation models 7 on noisy internet-scale datasets for use in downstream tasks in natural language 1–4, computer vision, 5,6,8 and multi-task models. 9 For sequential decision domains (e.g. robotics, game playing, and computer usage) where agents must repeatedly act within an environment, a wealth of data also exists on the web; however, most of this data is in the form of unlabeled video (i.e. without the actions taken at each frame), making it much less straightforward to train a behavioral prior in these domains than it is in e.g. natural language. In a few rare settings, such as Chess, Go, and StarCraft, there already exist large datasets with action labels from various online platforms that researchers have used for imitation learning. 10,11 When large labeled datasets do not exist, the canonical strategy for training capable agents is reinforcement learning (RL), 12 which can be sample inefficient and expensive for hard-exploration problems. 13–19 Many virtual tasks, e.g. navigating websites, using Photoshop, booking flights, etc., can be very hard to learn with RL and do not have large, commonly available sources of labeled data. 20,21 In this paper, we seek to extend the paradigm of training large, general-purpose foundation models to sequential decision domains by utilizing freely available internet-scale unlabeled video datasets with a simple semi-supervised imitation learning method. We call this method Video PreTraining (VPT) and demonstrate its efficacy in the domain of Minecraft.
+
+Existing semi-supervised imitation learning methods aim to learn with few or no explicit action labels; however, they generally rely on the policy’s ability to explore the environment throughout training, making them susceptible to exploration bottlenecks. 22–26 Furthermore, most prior semi-supervised imitation learning work was tested in the relatively low data regime; because we experiment with far more data ( $\mathord { \sim } 7 0 \mathrm { k }$ hours of unlabeled video), we hypothesize that we can achieve good performance with a much simpler method, a trend that has proven true for pretraining in other modalities such as text. 1 In particular, given a large but unlabeled dataset, we propose generating pseudo-labels by gathering a small amount of labeled data to train an inverse dynamics model (IDM) that predicts the action taken at each timestep in a video. Behavioral cloning (BC) can require a large amount of data because the model must learn to infer intent and the distribution over future behaviors from only past observations. In contrast, the inverse dynamics modeling task is simpler because it is non-causal, meaning it can look at both past and future frames to infer actions. In most settings, environment mechanics are far simpler than the breadth of human behavior that can take place within the environment, suggesting that non-causal IDMs could require far less data to train than causal BC models. Using pseudo-labels generated from the IDM, we then train a model to mimic the distribution of behavior in the previously unlabeled dataset with standard behavioral cloning at scale, which does not require any model rollouts and thus does not suffer from any potential exploration bottlenecks in the environment. Finally, we show we can fine-tune this model to downstream tasks with either behavioral cloning or reinforcement learning.
+
+We chose to test our method in Minecraft because it (a) is one of the most actively played games in the world27 and thus has a wealth of online video data, (b) is an open-ended sandbox game with an extremely wide variety of potential things to do, build, and collect, making our results more applicable to real-world applications such as computer usage, which also tends to be varied and open-ended, and (c) has already garnered interest by the RL community as a research domain due to its complexity and correspondingly difficult exploration challenges. 28–32 In this work we use the native human interface for Minecraft so that we can (1) most accurately model the human behavior distribution and reduce domain shift between video data and the environment, (2) make data collection easier by allowing our human contractors to play the game without modification, and (3) eliminate the need to hand-engineer a custom interface for models to interact with the environment. This choice means that our models play at 20 frames per second and must use a mouse and keyboard interface to interact with human GUIs for crafting, smelting, trading, etc., including dragging items to specific slots or navigating the recipe book with the mouse cursor (Fig. 1). Compared to prior work in Minecraft that uses a lower frame rate and constructs crafting and attacking macros, 31,33–35 using the native human interface drastically increases the environment’s exploration difficulty, making most simple tasks near impossible with RL from scratch. Even the simple task of gathering a single wooden log while already facing a tree takes 60 consecutive attack actions with the human interface, meaning the chance for a naive random policy to succeed is $1 / 2 ^ { 6 0 }$ . While this paper shows results in Minecraft only, the VPT method is general and could be applied to any domain.
+
+![](images/9ecaae8777654a20f20ae479e79fd1eb563c824f512fc612e394f000c310a23c.jpg)  
+Figure 1: Example Minecraft crafting GUI. Agents use the mouse and keyboard to navigate menus and drag and drop items.
+
+In Section 4 we show that the VPT foundation model has nontrivial zero-shot performance, accomplishing tasks impossible to learn with RL alone, such as crafting planks and crafting tables (tasks requiring a human proficient in Minecraft a median of 50 seconds or ${ \sim } 9 7 0 $ consecutive actions). Through fine-tuning with behavioral cloning to smaller datasets that target more specific behavior distributions, our agent is able to push even further into the technology tree, crafting stone tools (taking a human a median of 2.3 minutes or ${ \sim } 2 7 9 0$ actions). Finally, fine-tuning via RL produces the most dramatic improvements: our agent is able to craft diamond tools, an unprecedented result in Minecraft made even more challenging by using the native human interface. This task requires a proficient human a median upwards of 20 minutes or ${ \sim } 2 4 0 0 0$ actions. The main contributions of this work are (1) we are the first to show promising results applying semi-supervised imitation learning to extremely large, noisy, and freely available video datasets for sequential decision domains, (2) we show that such pretraining plus fine-tuning enables agents to solve tasks that were otherwise impossible to learn, (3) we show that labeled contractor data is far more efficiently used within the VPT method than it would be by directly training a foundation model from it and (4) we open source our contractor data, trained model weights, and Minecraft environment for future research into learning to act via semi-supervised imitation learning at scale.
+
+# 2 Preliminaries and Related Work
+
+Imitation learning methods 36–39 seek to construct a policy that accurately models the distribution of behavior in some dataset $D = \{ ( o _ { i } , a _ { i } ) \}$ , $i \in \{ 1 . . . N \}$ of action-observation pairs. In order to roll out these policies in an environment, they must be causal, meaning they condition on observations from the current timestep $t$ and past timesteps only, i.e. $\pi \sim p ( \bar { a } _ { t } | o _ { 1 } . . . o _ { t } )$ . Imitation learning is simplest when demonstrations are labeled with corresponding actions. Imitating labeled trajectories has seen success in aerial vehicles, 40,41 self-driving cars, 42,43 board games, 10,44 and video games. 11,45
+
+When labeled demonstrations are not available, standard behavioral cloning will not work; however, there is a large body of work in imitating behavior from unlabeled demonstrations. 23 For instance, GAIL24 constructs an adversarial objective incentivizing the trained policy to exhibit behaviors indistinguishable from those in the target dataset. Edwards et al. 46 propose to first learn a latent policy using unlabeled demonstrations and then map the learned latent actions to real actions using environment interaction. Peng et al. 47 use motion-capture methods to track agent positions in videos and then train RL agents to match these waypoints. Similarly, Behbahani et al. 48 and Aytar et al. 49 task a RL agent to match waypoints; however, their waypoints are embeddings from unsupervised feature learning models. Pathak et al. 50 and Nair et al. 51 train goal conditioned policies to take actions that move towards expert-provided goal states expressed as high dimensional visual waypoints. Most similar to our own work, Torabi et al. 25 simultaneously train (1) an inverse dynamics model (IDM), 52 which aims to uncover the underlying action between timesteps given observations of past and future timesteps, e.g. $p _ { \mathrm { I D M } } ( a _ { t } | o _ { t } , o _ { t + 1 } )$ , and (2) a behavioral cloning (BC) model on trajectories of observations labeled with the IDM. Data to train the IDM is collected by rolling out the BC model in the target environment such that both models improve in tandem. However, at any point in training if there are sequences in the dataset that the IDM performs poorly on, it requires that the BC model perform those or similar sequences in order for the IDM to improve and correctly label them. Therefore, if the BC model does not explore efficiently, it could severely slow down learning. In order to avoid this potential issue we opted for a simpler two-stage approach: we first train an IDM on a small number of labeled trajectories collected from human contractors (they play the game as would normally as we record their keypresses and mouse movements). Because human contractors reach most relevant parts of the state space, we can hold the IDM fixed throughout BC training.
+
+Compared to most previous work in semi-supervised imitation learning, we experiment in the much more complex and open-ended environment of Minecraft. Minecraft is a voxel-based 3D video game that, due its popularity and wide variety of mechanics, has attracted a vast amount of RL research. 28,29,31–35,53–61 A large body of work focuses on small, custom-made Minecraft worlds with tasks such as navigation, 54,61 block placing, 55,56 instruction following, 59,60 combat, 57 and others. 29,32,58 Work operating in the massive, randomly generated environments of Minecraft itself has included hill climbing, 53 automated curriculum learning31 and, most closely related to the RL experiments presented in Sec. 4.4, diamond mining. 28,33–35 However, to the best of our knowledge, there is no published work that operates in the full, unmodified human action space, which includes drag-and-drop inventory management and item crafting.
+
+# 3 Methods
+
+Inverse Dynamics Models (IDM) VPT, illustrated in Figure 2, requires we first collect a small amount of labeled contractor data with which to train an inverse dynamics model $p _ { \mathrm { I D M } } ( a _ { t } | o _ { 1 \ldots T } )$
+
+![](images/064a1527f9fa1ec271241672e70e73bab60b52c51f73a35ec94e5f1f718ad9c6.jpg)  
+Figure 2: Video Pretraining (VPT) Method Overview.
+
+which seeks to minimize the negative log-likelihood of an action at timestep $t$ given a trajectory of $T$ observations $o _ { t } ~ : ~ t \in [ 1 . . . T ]$ . In contrast to an imitation learning policy, the IDM can be non-causal, meaning its prediction for $a _ { t }$ can be a function of both past and future events, i.e. $o _ { t ^ { \prime } > t }$ . Compared to the behavioral cloning objective of modeling the distribution of human intent given past frames only, we hypothesize that inverting environment dynamics is easier and more data efficient to learn. Indeed, Sec. 4.1 will show that the IDM objective is much easier to learn, and furthermore Sec. 4.6 will show that with very little labeled data (as few as 100 hours) we can train a fairly accurate IDM. This IDM can be used to label online videos, providing the large amount of data required for the harder task of behavioral cloning. See appendices D and B for IDM training and data collection details.
+
+Data Filtering We gather a large dataset of Minecraft videos by searching the web for related keywords (Appendix A). Online videos often (1) include overlaid artifacts, such as a video feed of the player’s face, channel logos, watermarks, etc., (2) are collected from platforms other than a computer with different gameplay, or (3) are from different game modes, e.g. in Minecraft we only want "survival mode" where players start from scratch and must gather or craft all their items. We call data “clean” if it does not contain visual artifacts and is from survival mode, and call all other data “unclean.” With enough data, a large enough model, and enough training compute, a BC model trained on both unclean and clean videos would likely still perform well in a clean Minecraft environment. However, for simplicity and training compute efficiency, we choose to filter out unclean segments of video (note that a video may contain both clean and unclean segments). We do this by training a model to filter out unclean segments using a small dataset (8800) of images sampled from online videos labeled by contractors as clean or unclean. We did not tune this process as it is fairly standard; see Appendix A.2 for more details and ablations showing data cleaning is beneficial.
+
+VPT Foundation Model We train a foundation model with standard behavioral cloning, i.e. minimizing the negative log-likelihood of actions predicted by the IDM on clean data. For a particular trajectory of length $T$ we minimize
+
+$$
+\operatorname* { m i n } _ { \theta } \sum _ { t \in [ 1 \ldots T ] } - \log \pi _ { \theta } ( a _ { t } | o _ { 1 } , \ldots , o _ { t } ) , { \mathrm { w h e r e ~ } } a _ { t } \sim p _ { \mathrm { I D M } } ( a _ { t } | o _ { 1 } , \ldots , o _ { t } , \ldots , o _ { T } )
+$$
+
+As we will see in the following sections, this model exhibits nontrivial zero-shot behavior and can be fine-tuned with both imitation learning and RL to perform even more complex skills.
+
+# 4 Results
+
+# 4.1 Performance of the Inverse Dynamics Model
+
+The IDM architecture is comprised primarily of a temporal convolution layer, a ResNet 63 image processing stack, and residual unmasked attention layers, from which the IDM simultaneously predicts keypresses and mouse movements (see Appendix D for IDM architecture and training details). A key hypothesis behind our work is that IDMs can be trained with a relatively small amount of labeled data. While more data improves both mouse movement and keypress predictions, our best IDM trains on only 1962 hours of data (compared to the $\mathrm { \sim } 7 0 \mathrm { k }$ hours of clean data we collected from the internet) and achieves $9 0 . 6 \%$ keypress accuracy and a $0 . 9 7 ~ R ^ { 2 }$ for mouse movements evaluated on a held-out validation set of contractor-labeled data (Figure 3 left).
+
+![](images/4b3ea22386575829558b307187db5f20ebab1ca9a8165026d52849f357087422.jpg)  
+Figure 3: (Left) IDM keypress accuracy and mouse movement $R ^ { 2 }$ (explained variance 62) as a function of dataset size. (Right) IDM vs. behavioral cloning data efficiency.
+
+Figure 3 (right) validates our hypothesis that IDMs are far more data efficient than BC models, likely because inverting environment mechanics is far easier than modeling the entire distribution of human behavior. The IDM is two orders of magnitude more data efficient than a BC model trained on the same data and improves more quickly with more data. This evidence supports our hypothesis that it is more effective to use contractor data within the VPT pipeline by training an IDM than it is to train a foundation model from contractor data directly (Sections 4.5 and 4.6 provide additional evidence). Due to their data efficiency, training an IDM uses a negligible fraction of the overall compute needed to train a VPT model.
+
+# 4.2 VPT Foundation Model Training and Zero-Shot Performance
+
+![](images/34a95ee247846f0d6edcf6bdccdb1ad4fe749757d92015a2c7b77655f520b49f.jpg)  
+Figure 4: (Left) Training and validation loss on the web_clean internet dataset with IDM pseudolabels, and loss on the main IDM contractor dataset, which has ground-truth labels but is out-ofdistribution (see text). (Right) Amount a given item was collected per episode averaged over 2500 60-minute survival episodes as a function of training epoch, shaded with the standard error of the mean. Basic mining refers to collection of dirt, gravel, or sand (all materials that can be gathered without tools). Logs are obtained by repeatedly hitting trees for three seconds, a difficult feat for an RL agent to achieve as we show in Sec. 4.4. Planks can be crafted from logs, and crafting tables crafted from planks. Crafting requires using in-game crafting GUIs, and proficient humans take a median of 50 seconds (970 consecutive actions) to make a crafting table.
+
+We now explore the emergent behavior learned by a behavioral cloning policy trained on an extremely large, but noisy, internet dataset labeled with our IDM. To collect the unlabeled internet dataset, we searched for publicly available videos of Minecraft play with search terms such as “minecraft survival for beginners.” These searches resulted in ${ \sim } 2 7 0 \mathrm { k }$ hours of video, which we filtered down to “clean” video segments yielding an unlabeled dataset of $\mathrm { \sim } 7 0 \mathrm { k }$ hours, which we refer to as web_clean (Appendix A has further details on data scraping and filtering). We then generated pseudo-labels for web_clean with our best IDM (Section 3) and then trained the VPT foundation model with behavioral cloning. Preliminary model scaling experiments suggested that our model could benefit from 30 epochs of training and that a 0.5 billion parameter model was required to stay in the efficient learning regime 64 for that training duration (Appendix H shows results comparing model size and the benefit of scaling to 0.5B parameters), which took ${ \sim } 9$ days on 720 V100 GPUs.
+
+We evaluate our models by measuring validation loss (Fig. 4, left) and rolling them out in the Minecraft environment. Unless otherwise noted, in all environment evaluations we spawn agents in a standard survival mode game where they play for 60 minutes, i.e. 72000 consecutive actions, and we plot the mean and shade the standard error of the mean for various game statistics such as crafting and collection rates (Fig. 4, right). The VPT foundation model quickly learns to chop down trees to collect logs, a task we found near impossible for an RL agent to achieve with the native human interface (Sec. 4.4). It also learns to craft those logs into wooden planks and then use those planks to craft a crafting table, which are required to unlock most other technology in the game and take a human proficient in Minecraft approximately 50 seconds (970 consecutive actions) to collect. While these behaviors are fairly complex in the native human action space, the VPT foundation model crafts these items at a rate far below that of our proficient contractors, e.g. on average our contractors craft 5.44 crafting tables in 60 minutes of play versus 0.19 for the foundation model. The model also crafts a non-negligible amount of wooden sticks, which are required to make wooden tools; collects various flowers and crafts dyes from them; kills zombies that appear during the night; hunts wild animals; collects various berries and mushrooms and eats them; and finds game-generated villages from which to collect various rare items from chests. The model also learned to navigate uneven terrain, swim, and pillar jump, which involves the agent repeatedly jumping and quickly placing a block below itself such that it climbs upward by making a pillar.(iv)
+
+While training and validation loss decrease healthily over training (Fig. 4, left), loss on our contractor dataset (which the VPT model does not train on) begins increasing after 7 epochs. Contractor data could be out-of-distribution because our contractors may have a different distribution of play or because there is some impactful visual domain shift compared to videos from the web, and we provide some evidence for this phenomenon in Appendix H. While one could have expected this would be predictive of declining evaluation performance, we do not see notable game statistics from the VPT foundation model rollouts (Figure 4, right) decrease over training, and in the next section we show that BC fine-tuning performance continually improves as the VPT foundation model trains.
+
+# 4.3 Fine-Tuning with Behavioral Cloning
+
+Foundation models are designed to have a broad behavior profile and be generally capable across a wide variety of tasks. To incorporate new knowledge or allow them to specialize on a narrower task distribution, it is common practice to fine-tune these models to smaller, more specific datasets. 1 The VPT foundation model trained on the broad web_clean dataset had nontrivial zero-shot performance; it was able to craft a crafting table yet unable to go past this in the technology tree. As a case study into BC fine-tuning, we attempt to improve the VPT foundation model’s ability to collect and craft these “early game” items by fine-tuning to two narrower datasets targeted at Minecraft behavior within the first few minutes of players starting in a fresh world. In the first dataset, contractor_house, contractors have 10 minutes to build a basic house from scratch using primarily wood, sand, and dirt. Collecting contractor data can be difficult and expensive, so we also construct a dataset earlygame_keyword by searching for videos online with descriptions that match keywords such as “new world”, “let’s play episode 1”, etc.; this is a subset of web_clean and is labeled with the IDM. See Appendix B.4 and A.3 for full descriptions of both datasets.
+
+![](images/6de54e50401bab8c4182be82806eb8a59bcf1d9c74a1f831f1c8e532f5d5450c.jpg)  
+Figure 5: (Left) Collection and crafting rates for three policies: the zero-shot VPT foundation model, and the VPT foundation model BC fine-tuned to the earlygame_keyword or contractor_house datasets. BC fine-tuning to either dataset improves performance, including (for the contractor_house dataset) yielding wooden and stone tools. Proficient Minecraft players take a median of 1.2 minutes (1390 actions) to construct wooden tools and 2.3 minutes (2790 actions) to construct stone tools. (Right) Collection and crafting rates for VPT foundation model snapshots throughout training after they are BC fine-tuned to the contractor_house dataset. In general, crafting-related behaviors increase throughout foundation model training. Fig. 4 defines the other task terms (logs, planks, crafting tables, and total crafting).
+
+Fine-tuning to earlygame_keyword results in a large boost compared to the zero-shot foundation model: $2 . 5 \mathrm { x }$ more crafting tables, 6.1x more planks, $4 . 3 \mathbf { x }$ more logs, and $5 . 5 \mathrm { x }$ more crafting overall (Fig. 5). However, when fine-tuning to this dataset we did not see any new behaviors emerge, only a refinement of existing skills. We saw an even bigger improvement when fine-tuning to the contractor_house dataset: $2 1 3 \mathrm { x }$ more crafting tables, $5 9 \mathrm { x }$ more wooden planks, $7 \mathbf { x }$ more logs, and $5 9 \mathrm { x }$ more crafting over all. In addition, we saw the emergence of crafting wooden tools, which requires placing a crafting table on the ground, opening it to reveal a new crafting interface, and then using it to craft wooden tools. This entire sequence takes a proficient human player a median of 1.2 minutes (1390 consecutive actions) to accomplish. The model goes further and collects cobblestone, which requires a wooden pickaxe to mine, and crafts stone tools, requiring it to again use a crafting table; this takes a proficient human player a median of 2.3 minutes (2790 consecutive actions). We also saw this model more frequently raiding villages that randomly spawn in the game, hunting animals for food, in addition to many behaviors we saw performed by the foundation model.(v)
+
+Despite the foundation model’s zero-shot rollout performance plateauing 1/3 into training (Fig. 4, right), fine-tuning performance does continue to increase throughout foundation model training (Fig. 5, right). Additionally, there is a stark difference in performance when training from scratch vs. fine-tuning from the VPT foundation model (Fig. 5 right, comparing the left and rightmost points).
+
+# 4.4 Fine-Tuning with Reinforcement Learning
+
+![](images/cba0e6b9fd67795bb3b4d12861ce77a9904cf7d4c4cee89a406ab396ae3a0429.jpg)  
+Figure 6: Typical sequence of items for obtaining a diamond pickaxe. Below each item is the median time and number of actions contractors required to obtain that item and the percentage of contractors that got the item within 10 minutes. The median time to obtain a diamond pickaxe is unknown (except that it is $> 2 0 \mathrm { m } ,$ ) because contractors obtained this item in less than $5 0 \%$ of 20-minute episodes.
+
+To demonstrate the efficacy of RL fine-tuning, we chose the challenging goal of obtaining a diamond pickaxe within 10 minutes starting from a fresh Minecraft survival world. Doing so involves acquiring a sequence of difficult-to-obtain items that require complex skills like mining, inventory management, crafting with and without a crafting table, tool use, operating a furnace, and mining at the lowest depths, where many hazards like enemies and lava exist (Fig. 6). Adding to the difficulty, progress can be easily lost by dropping items, destroying items, or dying. Obtaining a diamond pickaxe more often than not takes a proficient human over 20 minutes (24,000 actions).
+
+Agents are rewarded for each item obtained in the sequence, with lower rewards for items that have to be collected in bulk and higher rewards for items near the end of the sequence. Agents are optimized with the phasic policy gradient 65 RL algorithm for ${ \sim } 1 . 3$ million episodes (roughly $1 . 4 \times 1 0 ^ { 1 0 }$ frames). Episodes last for 10 minutes. See Appendix G.1 for reward function and RL training details. Due to computational constraints, RL experiments use a $\sim 2 4 8$ million parameter VPT model (Appendix H).
+
+A major problem when fine-tuning with RL is catastrophic forgetting66,67 because previously learned skills can be lost before their value is realized. For instance, while our VPT foundation model never exhibits the entire sequence of behaviors required to smelt iron zero-shot, it did train on examples of players smelting with furnaces. It therefore may have some latent ability to smelt iron once the many prerequisites to do so have been performed. To combat the catastrophic forgetting of latent skills such that they can continually improve exploration throughout RL fine-tuning, we add an auxiliary Kullback-Leibler (KL) divergence loss between the RL model and the frozen pretrained policy. 11
+
+Training from a randomly initialized policy fails to achieve almost any reward, underscoring how hard an exploration challenge the diamond pickaxe task is for RL in the native human action space (Fig. 7a). The model never learns to reliably collect logs, typically the first of many steps to obtaining a diamond pickaxe (Fig. 7b). RL fine-tuning from the VPT foundation model does substantially better (Fig. 7a), learning everything up to mining iron ore and crafting furnaces. (Fig. 7c). However, this agent fails at smelting an iron ingot, the next item required to get further into the tech tree, likely because the zero-shot probability that the VPT foundation model smelts an iron ingot is too low, even when given the prerequisite materials.
+
+![](images/3465129888555f408ab610b7e6a3dbdf535575989fd849fe8c5dda363f075cfd.jpg)  
+Figure 7: RL Fine-tuning results. (a) RL from a randomly initialized model fails to get almost any reward, RL fine-tuning from the VPT foundation model performs substantially better with a reward near 13, and RL fine-tuning from the early-game model performs best with a reward of 25. When training the early-game model without a KL loss to the original policy (No KL-loss) progress stalls after 100,000 episodes, suggesting that the skills necessary to make further progress have been catastrophically forgotten. (b) RL from a randomly initialized model occasionally collects sticks by breaking leaves (an easy but inefficient method of getting sticks that does not require logs or planks) and never learns to reliably collect logs. (c) RL fine-tuning from the VPT Foundation model learns everything in the curriculum up to iron ore and making furnaces, but fails to learn to use the furnace to smelt iron ingots. (d) RL fine-tuning from the early-game model learns to obtain (at human-level) all items in the sequence towards a diamond pickaxe and crafts a diamond pickaxe in $2 . 5 \%$ of episodes.
+
+Results further improve by first BC fine-tuning the VPT Foundation Model to the earlygame_keyword dataset (the early-game model, Sec. 4.3) and then fine-tuning with RL (Fig. 7a), which in preliminary experiments we found to perform better than first fine-tuning to contractor_house followed by fine-tuning with RL (Appendix G.2). The three-phase training (pretraining, BC fine-tuning, and then RL fine-tuning) succeeds in learning extremely difficult tasks: it achieves over $8 0 \%$ reliability on iron pickaxes, almost $2 0 \%$ reliability on collecting diamonds, and $2 . 5 \%$ reliability on obtaining a diamond pickaxe (Fig. 7d). For comparison, human players given the objective of obtaining a diamond pickaxe collect these items in $5 7 \%$ , $1 5 \%$ , and $1 2 \%$ of episodes, respectively, meaning our model is human-level for crafting iron pickaxes and mining diamonds. Others have managed to obtain diamonds with $\sim 0 . 1 \%$ reliability in 15 minutes 33,34 but always with a simplified action space designed to ease exploration. To the best of our knowledge, we are the first to report non-zero success rates on crafting a diamond pickaxe. Qualitatively, the model developed useful skills for diamond mining, such as efficient mining patterns, cave exploration, returning to previously placed objects like crafting tables, and advanced techniques like using wooden pickaxes as fuel when moving on to iron tools.(vi)
+
+Finally, we validated the importance of the KL loss to the pretrained model during RL fine-tuning. The treatment without a KL loss obtains only items early in the sequence (logs, planks, sticks, and crafting tables) limiting its reward (Fig. 7a). This failure to progress further into the sequence is likely because, while the initial skills of chopping logs and crafting planks are being learned with RL, subsequent skills like crafting a wooden pickaxe are lost due to catastrophic forgetting.
+
+# 4.5 Data Scaling Properties of the Foundation Model
+
+In this section we validate a core hypothesis behind this work: that it is far more effective to use labeled contractor data to train an IDM within the VPT method than it is to directly train a BC foundation model from that same small contractor dataset. If we could cheaply collect a labeled contractor dataset of a similar order of magnitude as web_clean, then this would not be important; however, collecting that scale of data would have cost millions of dollars. Figure 8 compares foundation models trained on increasing orders of magnitude of data from 1 hour up to the full ${ \sim } 7 0 \mathrm { k }$ web_clean dataset. Foundation models trained up to and including 1k hours are trained on the IDM contractor data, and those trained on 5k hours and above are trained on subsets of web_clean, which does not contain any IDM contractor data. Scaling training data increases log collection, mining, and crafting capabilities. The zero-shot model only begins to start crafting crafting tables at over 5000 hours of training data. When fine-tuning each foundation model to contractor_house, we see that crafting rates for crafting tables and wooden tools increase by orders of magnitude when using the entire ${ \sim } 7 0 \mathrm { k }$ hour web_clean dataset. We furthermore only see the emergence of crafting stone tools at the largest data scale.
+
+![](images/a4e776f01b9e9a1ad25c549eef4298ddfee753b2cc183af6b0d9ea8fde1fc6bf.jpg)  
+Figure 8: (Left) Zero-shot rollout performance of foundation models trained on varying amounts of data. Models to the left of the dashed black line (points $\leq 1 \mathrm { k }$ hours) were trained on contractor data (ground-truth labels), and models to the right were trained on IDM pseudo-labeled subsets of web_clean. Due to compute limitations, this analysis was performed with smaller (71 million parameter) models except for the final point, which is the 0.5 billion parameter VPT foundation model. (Right) The corresponding performance of each model after BC fine-tuning each model to the contractor_house dataset.
+
+# 4.6 Effect of Inverse Dynamics Model Quality on Behavioral Cloning
+
+This section investigates how downstream BC performance is affected by IDM quality. We train IDMs on increasingly larger datasets and use each to independently label the earlygame_keyword dataset (this smaller dataset was chosen due to a limited compute budget). We then train a BC model from scratch on each dataset and report game statistics for each model as a function of IDM contractor dataset size (Fig. 9).
+
+IDMs trained on at least 10 hours of data are required for any crafting, and the crafting rate increases quickly up until 100 hours of data,
+
+![](images/555a4d47272fd3d0f24896ab4bc7825e8002ccc73861b6de1f09a96cbfe40ed5.jpg)  
+Figure 9: Zero-shot performance of BC models trained from scratch on the earlygame_keyword dataset labeled with IDMs that were trained on increasing amounts of contractor data.
+
+after which there are few to no gains and differences are likely due to noise. Similarly, crafting tables are only crafted after 50 or more hours of IDM data, and again gains plateau after 100 hours. While in all previous experiments we use our best IDM trained on 1962 hours of data, these results suggest we could reduce that number to as low as 100 hours.
+
+# 5 Discussion and Conclusion
+
+The results presented in this paper help pave the path to utilizing the wealth of unlabeled data on the web for many sequential decision domains. Compared to representation learning methods, e.g. generative video modeling, VPT offers the exciting possibility of directly learning to act during pretraining and using these learned behavioral priors as extremely effective exploration priors for RL. VPT could even be an effective representation learning method for downstream tasks that do not require acting, e.g. video captioning, because arguably the most important information in any given scene would be present in features trained to correctly predict the distribution over future human actions. We leave this intriguing direction to future work.
+
+Future work could improve results with more data (we estimate we could collect ${ \bf \Lambda } > 1 { \bf M }$ hours) and larger, better-tuned models. Our internet data was fairly noisy and varied (players choose their own graphics settings); we hope future work will investigate even noisier sources of data, as well as how to use both first and third person demonstrations. Furthermore, all models in this work condition on past observations only; we cannot ask the model to perform specific tasks. Appendix I presents preliminary experiments on conditioning our models on closed captions (text transcripts of speech in videos), showing they become weakly steerable; we believe this a rich direction for future research. By definition behavioral priors must predict actions, and in this work we found this objective sufficient to train capable agents; however, a fruitful direction could be incorporating auxiliary representation learning objectives (e.g. contrastive losses, environment dynamics modeling, etc.) to reduce the sample complexity of both the IDM and foundation models. Similarly, it would be interesting to see if VPT could benefit from pretraining its attention layers with a language modeling task as in Li et al. 68 and Reid et al. 69 Loss was not consistently correlated with downstream evaluation metrics (Sec. 4.2), which often made progress slow. Another worthwhile future direction would be to investigate the correlation between various training metrics and downstream evaluations.
+
+For RL fine-tuning we only experimented with a standard policy gradient based RL algorithm (PPG); an interesting future direction would be to investigate how well VPT can be combined with other RL algorithms, e.g. off-policy or model based. Furthermore, we showed the efficacy of fine-tuning VPT with RL using a very difficult, albeit handcrafted, reward function aimed at crafting diamond tools. We hope future work will combine VPT with methods that can generate more generic reward functions, e.g. natural language based reward functions as proposed in MineDojo70 (released after this paper). Finally, while we do not anticipate any direct negative societal impacts from the models trained in this work, as VPT improves and expands to other domains it will be important to assess and mitigate harms that emerge with other forms of pretraining on internet datasets, such as emulating inappropriate behavior. 71
+
+In conclusion, VPT extends the paradigm of training large and general purpose behavioral priors to sequential decision domains that have commonly available unlabeled internet data. Our models exhibited impressive zero-shot behavior and, when fine-tuned with RL, achieved an unprecedented result of crafting a diamond pickaxe in Minecraft (all the more difficult given the human interface). We further showed that contractor data is far better used within the VPT pipeline than to train a foundation model directly and that only a small amount of contractor data (about $\$ 2000$ USD) was required to unlock massive amounts of unlabeled online data for use in BC. Finally, learning with the human keyboard and mouse interface is highly general and allows losslessly modeling the entire distribution of human behavior. While we only experiment in Minecraft, we believe that VPT provides a general recipe for training behavioral priors in hard, yet generic, action spaces in any domain that has a large amount of freely available unlabeled data, such as computer usage.
+
+# Acknowledgements
+
+We thank the following people for helpful discussions and support: Bob McGrew, Ken Stanley, Joel Lehman, Ilya Sutskever, Wojciech Zaremba, Ingmar Kanitscheider, David Farhi, Glenn Powell, Jonathan Gordon, and the OpenAI supercomputing team, especially Christian Gibson, Ben Chess, and Christopher Berner.
+
+# References
+
+[1] Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. Language models are few-shot learners. Advances in neural information processing systems, 33:1877–1901, 2020.   
+[2] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. Bert: Pre-training of deep bidirectional transformers for language understanding. arXiv preprint arXiv:1810.04805, 2018.   
+[3] Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov. Roberta: A robustly optimized bert pretraining approach. arXiv preprint arXiv:1907.11692, 2019.
+
+[4] Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J Liu. Exploring the limits of transfer learning with a unified text-to-text transformer. arXiv preprint arXiv:1910.10683, 2019.
+
+[5] Dhruv Mahajan, Ross Girshick, Vignesh Ramanathan, Kaiming He, Manohar Paluri, Yixuan Li, Ashwin Bharambe, and Laurens Van Der Maaten. Exploring the limits of weakly supervised pretraining. In Proceedings of the European conference on computer vision (ECCV), pages 181–196, 2018.
+
+[6] Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, et al. Learning transferable visual models from natural language supervision. In International Conference on Machine Learning, pages 8748–8763. PMLR, 2021.
+
+[7] Rishi Bommasani, Drew A Hudson, Ehsan Adeli, Russ Altman, Simran Arora, Sydney von Arx, Michael S Bernstein, Jeannette Bohg, Antoine Bosselut, Emma Brunskill, et al. On the opportunities and risks of foundation models. arXiv preprint arXiv:2108.07258, 2021.
+
+[8] Xiaohua Zhai, Alexander Kolesnikov, Neil Houlsby, and Lucas Beyer. Scaling vision transformers. CoRR, abs/2106.04560, 2021. URL https://arxiv.org/abs/2106.04560.
+
+[9] Scott Reed, Konrad Zolna, Emilio Parisotto, Sergio Gomez Colmenarejo, Alexander Novikov, Gabriel Barth-Maron, Mai Gimenez, Yury Sulsky, Jackie Kay, Jost Tobias Springenberg, et al. A generalist agent. arXiv preprint arXiv:2205.06175, 2022.
+
+[10] David Silver, Aja Huang, Chris J Maddison, Arthur Guez, Laurent Sifre, George Van Den Driessche, Julian Schrittwieser, Ioannis Antonoglou, Veda Panneershelvam, Marc Lanctot, et al. Mastering the game of go with deep neural networks and tree search. Nature, 529(7587):484–489, 2016.
+
+[11] Oriol Vinyals, Igor Babuschkin, Wojciech M Czarnecki, Michaël Mathieu, Andrew Dudzik, Junyoung Chung, David H Choi, Richard Powell, Timo Ewalds, Petko Georgiev, et al. Grandmaster level in starcraft ii using multi-agent reinforcement learning. Nature, 575(7782):350–354, 2019.
+
+[12] Richard S Sutton and Andrew G Barto. Reinforcement learning: An introduction. MIT press, 2018.
+
+[13] Christopher Berner, Greg Brockman, Brooke Chan, Vicki Cheung, Przemysław D˛ebiak, Christy Dennison, David Farhi, Quirin Fischer, Shariq Hashme, Chris Hesse, et al. Dota 2 with large scale deep reinforcement learning. arXiv preprint arXiv:1912.06680, 2019.
+
+[14] Bowen Baker, Ingmar Kanitscheider, Todor Markov, Yi Wu, Glenn Powell, Bob McGrew, and Igor Mordatch. Emergent tool use from multi-agent autocurricula. arXiv preprint arXiv:1909.07528, 2019.
+
+[15] Max Jaderberg, Wojciech M Czarnecki, Iain Dunning, Luke Marris, Guy Lever, Antonio Garcia Castaneda, Charles Beattie, Neil C Rabinowitz, Ari S Morcos, Avraham Ruderman, et al. Human-level performance in 3d multiplayer games with population-based reinforcement learning. Science, 364(6443):859–865, 2019.
+
+[16] Adrià Puigdomènech Badia, Bilal Piot, Steven Kapturowski, Pablo Sprechmann, Alex Vitvitskyi, Zhaohan Daniel Guo, and Charles Blundell. Agent57: Outperforming the atari human benchmark. In International Conference on Machine Learning, pages 507–517. PMLR, 2020.
+
+[17] Marc Bellemare, Sriram Srinivasan, Georg Ostrovski, Tom Schaul, David Saxton, and Remi Munos. Unifying count-based exploration and intrinsic motivation. Advances in neural information processing systems, 29, 2016.
+
+[18] Yuri Burda, Harrison Edwards, Amos Storkey, and Oleg Klimov. Exploration by random network distillation. arXiv preprint arXiv:1810.12894, 2018.
+
+[19] Adrien Ecoffet, Joost Huizinga, Joel Lehman, Kenneth O Stanley, and Jeff Clune. First return, then explore. Nature, 590(7847):580–586, 2021.
+
+[20] Peter C Humphreys, David Raposo, Toby Pohlen, Gregory Thornton, Rachita Chhaparia, Alistair Muldal, Josh Abramson, Petko Georgiev, Alex Goldin, Adam Santoro, et al. A data-driven approach for learning to control computers. arXiv preprint arXiv:2202.08137, 2022.
+
+[21] Tianlin Shi, Andrej Karpathy, Linxi Fan, Jonathan Hernandez, and Percy Liang. World of bits: An open-domain platform for web-based agents. In Doina Precup and Yee Whye Teh, editors, Proceedings of the 34th International Conference on Machine Learning, volume 70 of Proceedings of Machine Learning Research, pages 3135–3144. PMLR, 06–11 Aug 2017. URL https://proceedings.mlr.press/v70/shi17a.html.
+
+[22] Andrew Y Ng, Stuart J Russell, et al. Algorithms for inverse reinforcement learning. In Icml, volume 1, page 2, 2000.
+
+[23] Faraz Torabi, Garrett Warnell, and Peter Stone. Recent advances in imitation learning from observation. arXiv preprint arXiv:1905.13566, 2019.
+
+[24] Jonathan Ho and Stefano Ermon. Generative adversarial imitation learning. Advances in neural information processing systems, 29, 2016.
+
+[25] Faraz Torabi, Garrett Warnell, and Peter Stone. Behavioral cloning from observation. arXiv preprint arXiv:1805.01954, 2018.
+
+[26] YuXuan Liu, Abhishek Gupta, Pieter Abbeel, and Sergey Levine. Imitation from observation: Learning to imitate behaviors from raw video via context translation. In 2018 IEEE International Conference on Robotics and Automation (ICRA), pages 1118–1125. IEEE, 2018.
+
+[27] Twinfinite Staff. Most played games in 2021, ranked by peak concurrent players. Twinfinite. URL https://twinfinite.net/2021/12/ most-played-games-in-2020-ranked-by-peak-concurrent-players/.
+
+[28] William H Guss, Brandon Houghton, Nicholay Topin, Phillip Wang, Cayden Codel, Manuela Veloso, and Ruslan Salakhutdinov. Minerl: A large-scale dataset of minecraft demonstrations. arXiv preprint arXiv:1907.13440, 2019.
+
+[29] Chen Tessler, Shahar Givony, Tom Zahavy, Daniel Mankowitz, and Shie Mannor. A deep hierarchical approach to lifelong learning in minecraft. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 31, 2017.
+
+[30] Christian Scheller, Yanick Schraner, and Manfred Vogel. Sample efficient reinforcement learning through learning from demonstrations in minecraft. In NeurIPS 2019 Competition and Demonstration Track, pages 67–76. PMLR, 2020.
+
+[31] Ingmar Kanitscheider, Joost Huizinga, David Farhi, William Hebgen Guss, Brandon Houghton, Raul Sampedro, Peter Zhokhov, Bowen Baker, Adrien Ecoffet, Jie Tang, et al. Multi-task curriculum learning in a complex, visual, hard-exploration domain: Minecraft. arXiv preprint arXiv:2106.14876, 2021.
+
+[32] Junhyuk Oh, Valliappa Chockalingam, Honglak Lee, et al. Control of memory, active perception, and action in minecraft. In International Conference on Machine Learning, pages 2790–2799. PMLR, 2016.
+
+[33] Vihang P Patil, Markus Hofmarcher, Marius-Constantin Dinu, Matthias Dorfer, Patrick M Blies, Johannes Brandstetter, Jose A Arjona-Medina, and Sepp Hochreiter. Align-rudder: Learning from few demonstrations by reward redistribution. arXiv preprint arXiv:2009.14108, 2020.
+
+[34] Alexey Skrynnik, Aleksey Staroverov, Ermek Aitygulov, Kirill Aksenov, Vasilii Davydov, and Aleksandr I Panov. Forgetful experience replay in hierarchical reinforcement learning from demonstrations. arXiv preprint arXiv:2006.09939, 2020.
+
+[35] Zichuan Lin, Junyou Li, Jianing Shi, Deheng Ye, Qiang Fu, and Wei Yang. Juewu-mc: Playing minecraft with sample-efficient hierarchical reinforcement learning. arXiv preprint arXiv:2112.04907, 2021.
+
+[36] Dean A Pomerleau. Alvinn: An autonomous land vehicle in a neural network. Advances in neural information processing systems, 1, 1988.
+
+[37] Stefan Schaal. Is imitation learning the route to humanoid robots? Trends in cognitive sciences, 3(6):233–242, 1999.
+
+[38] Brenna D Argall, Sonia Chernova, Manuela Veloso, and Brett Browning. A survey of robot learning from demonstration. Robotics and autonomous systems, 57(5):469–483, 2009.
+
+[39] Ahmed Hussein, Mohamed Medhat Gaber, Eyad Elyan, and Chrisina Jayne. Imitation learning: A survey of learning methods. ACM Computing Surveys (CSUR), 50(2):1–35, 2017.
+
+[40] Claude Sammut, Scott Hurst, Dana Kedzier, and Donald Michie. Learning to fly. In Machine Learning Proceedings 1992, pages 385–393. Elsevier, 1992.
+
+[42] Mariusz Bojarski, Davide Del Testa, Daniel Dworakowski, Bernhard Firner, Beat Flepp, Prasoon Goyal, Lawrence D Jackel, Mathew Monfort, Urs Muller, Jiakai Zhang, et al. End to end learning for self-driving cars. arXiv preprint arXiv:1604.07316, 2016.
+
+[43] Felipe Codevilla, Matthias Müller, Antonio López, Vladlen Koltun, and Alexey Dosovitskiy. End-to-end driving via conditional imitation learning. In 2018 IEEE international conference on robotics and automation (ICRA), pages 4693–4700. IEEE, 2018.
+
+[44] Rémi Coulom. Computing “elo ratings” of move patterns in the game of go. ICGA journal, 30 (4):198–208, 2007.
+
+[45] Todd Hester, Matej Vecerik, Olivier Pietquin, Marc Lanctot, Tom Schaul, Bilal Piot, Dan Horgan, John Quan, Andrew Sendonaris, Ian Osband, et al. Deep q-learning from demonstrations. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 32, 2018.
+
+[46] Ashley Edwards, Himanshu Sahni, Yannick Schroecker, and Charles Isbell. Imitating latent policies from observation. In International conference on machine learning, pages 1755–1763. PMLR, 2019.
+
+[47] Xue Bin Peng, Angjoo Kanazawa, Jitendra Malik, Pieter Abbeel, and Sergey Levine. Sfv: Reinforcement learning of physical skills from videos. ACM Transactions On Graphics (TOG), 37(6):1–14, 2018.
+
+[48] Feryal Behbahani, Kyriacos Shiarlis, Xi Chen, Vitaly Kurin, Sudhanshu Kasewa, Ciprian Stirbu, Joao Gomes, Supratik Paul, Frans A Oliehoek, Joao Messias, et al. Learning from demonstration in the wild. In 2019 International Conference on Robotics and Automation (ICRA), pages 775–781. IEEE, 2019.
+
+[49] Yusuf Aytar, Tobias Pfaff, David Budden, Thomas Paine, Ziyu Wang, and Nando De Freitas. Playing hard exploration games by watching youtube. Advances in neural information processing systems, 31, 2018.
+
+[50] Deepak Pathak, Parsa Mahmoudieh, Guanghao Luo, Pulkit Agrawal, Dian Chen, Yide Shentu, Evan Shelhamer, Jitendra Malik, Alexei A. Efros, and Trevor Darrell. Zero-shot visual imitation. In ICLR, 2018.
+
+[51] Ashvin Nair, Dian Chen, Pulkit Agrawal, Phillip Isola, Pieter Abbeel, Jitendra Malik, and Sergey Levine. Combining self-supervised learning and imitation for vision-based rope manipulation. pages 2146–2153, 05 2017. doi: 10.1109/ICRA.2017.7989247.
+
+[52] Duy Nguyen-Tuong, Jan Peters, Matthias Seeger, and Bernhard Schölkopf. Learning inverse dynamics: a comparison. In European symposium on artificial neural networks, number CONF, 2008.
+
+[53] David Abel, Alekh Agarwal, Fernando Diaz, Akshay Krishnamurthy, and Robert E Schapire. Exploratory gradient boosting for reinforcement learning in complex domains. arXiv preprint arXiv:1603.04119, 2016.
+
+[54] Dilip Arumugam, Jun Ki Lee, Sophie Saskin, and Michael L Littman. Deep reinforcement learning from policy-dependent human feedback. arXiv preprint arXiv:1902.04257, 2019.
+
+[55] Alexander Trott, Stephan Zheng, Caiming Xiong, and Richard Socher. Keeping your distance: Solving sparse reward tasks using self-balancing shaped rewards. Advances in Neural Information Processing Systems, 32, 2019.
+
+[56] Stephan Alaniz. Deep reinforcement learning with model learning and monte carlo tree search in minecraft. arXiv preprint arXiv:1803.08456, 2018.
+
+[57] Hiroto Udagawa, Tarun Narasimhan, and Shim-Young Lee. Fighting zombies in minecraft with deep reinforcement learning. Technical report, Technical report, Technical report, Stanford University, 2016.
+
+[58] Tianmin Shu, Caiming Xiong, and Richard Socher. Hierarchical and interpretable skill acquisition in multi-task reinforcement learning. arXiv preprint arXiv:1712.07294, 2017.
+
+[59] Junhyuk Oh, Satinder Singh, Honglak Lee, and Pushmeet Kohli. Zero-shot task generalization with multi-task deep reinforcement learning. In International Conference on Machine Learning, pages 2661–2670. PMLR, 2017.
+
+[60] Zhengxiang Shi, Yue Feng, and Aldo Lipani. Learning to execute or ask clarification questions. arXiv preprint arXiv:2204.08373, 2022.
+
+[61] Tambet Matiisen, Avital Oliver, Taco Cohen, and John Schulman. Teacher–student curriculum learning. IEEE transactions on neural networks and learning systems, 31(9):3732–3740, 2019.
+
+[62] Robert George Douglas Steel, James Hiram Torrie, et al. Principles and procedures of statistics. Principles and procedures of statistics., 1960.
+
+[63] 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, pages 770–778, 2016.
+
+[64] Jared Kaplan, Sam McCandlish, Tom Henighan, Tom B Brown, Benjamin Chess, Rewon Child, Scott Gray, Alec Radford, Jeffrey Wu, and Dario Amodei. Scaling laws for neural language models. arXiv preprint arXiv:2001.08361, 2020.
+
+[65] Karl W Cobbe, Jacob Hilton, Oleg Klimov, and John Schulman. Phasic policy gradient. In Marina Meila and Tong Zhang, editors, Proceedings of the 38th International Conference on Machine Learning, volume 139 of Proceedings of Machine Learning Research, pages 2020– 2027. PMLR, 18–24 Jul 2021. URL https://proceedings.mlr.press/v139/cobbe21a. html.
+
+[66] Dhireesha Kudithipudi, Mario Aguilar-Simon, Jonathan Babb, Maxim Bazhenov, Douglas Blackiston, Josh Bongard, Andrew P Brna, Suraj Chakravarthi Raja, Nick Cheney, Jeff Clune, et al. Biological underpinnings for lifelong learning machines. Nature Machine Intelligence, 4 (3):196–210, 2022.
+
+[67] James Kirkpatrick, Razvan Pascanu, Neil Rabinowitz, Joel Veness, Guillaume Desjardins, Andrei A Rusu, Kieran Milan, John Quan, Tiago Ramalho, Agnieszka Grabska-Barwinska, et al. Overcoming catastrophic forgetting in neural networks. Proceedings of the national academy of sciences, 114(13):3521–3526, 2017.
+
+[68] Shuang Li, Xavier Puig, Yilun Du, Clinton Wang, Ekin Akyurek, Antonio Torralba, Jacob Andreas, and Igor Mordatch. Pre-trained language models for interactive decision-making. arXiv preprint arXiv:2202.01771, 2022.
+
+[69] Machel Reid, Yutaro Yamada, and Shixiang Shane Gu. Can wikipedia help offline reinforcement learning? arXiv preprint arXiv:2201.12122, 2022.
+
+[70] Linxi Fan, Guanzhi Wang, Yunfan Jiang, Ajay Mandlekar, Yuncong Yang, Haoyi Zhu, Andrew Tang, De-An Huang, Yuke Zhu, and Anima Anandkumar. Minedojo: Building open-ended embodied agents with internet-scale knowledge. arXiv preprint arXiv:2206.08853, 2022.
+
+[71] Emily M Bender, Timnit Gebru, Angelina McMillan-Major, and Shmargaret Shmitchell. On the dangers of stochastic parrots: Can language models be too big???. In Proceedings of the 2021 ACM Conference on Fairness, Accountability, and Transparency, pages 610–623, 2021.
+
+[72] Fabian Pedregosa, Gaël Varoquaux, Alexandre Gramfort, Vincent Michel, Bertrand Thirion, Olivier Grisel, Mathieu Blondel, Peter Prettenhofer, Ron Weiss, Vincent Dubourg, et al. Scikitlearn: Machine learning in python. the Journal of machine Learning research, 12:2825–2830, 2011.
+
+[73] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. Advances in neural information processing systems, 30, 2017.
+
+[74] Jimmy Lei Ba, Jamie Ryan Kiros, and Geoffrey E Hinton. Layer normalization. arXiv preprint arXiv:1607.06450, 2016.
+
+[75] Yann A LeCun, Léon Bottou, Genevieve B Orr, and Klaus-Robert Müller. Efficient backprop. In Neural networks: Tricks of the trade, pages 9–48. Springer, 2012.
+
+[76] Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
+
+[77] Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Kopf, Edward Yang, Zachary DeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala. Pytorch: An imperative style, highperformance deep learning library. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché- Buc, E. Fox, and R. Garnett, editors, Advances in Neural Information Processing Systems 32, pages 8024–8035. Curran Associates, Inc., 2019. URL http://papers.neurips.cc/paper/ 9015-pytorch-an-imperative-style-high-performance-deep-learning-library. pdf.
+
+[78] Yann N Dauphin, Razvan Pascanu, Caglar Gulcehre, Kyunghyun Cho, Surya Ganguli, and Yoshua Bengio. Identifying and attacking the saddle point problem in high-dimensional nonconvex optimization. Advances in neural information processing systems, 27, 2014.
+
+[79] Jason Yosinski, Jeff Clune, Yoshua Bengio, and Hod Lipson. How transferable are features in deep neural networks? Advances in neural information processing systems, 27, 2014.
+
+[80] Zihang Dai, Zhilin Yang, Yiming Yang, Jaime Carbonell, Quoc V Le, and Ruslan Salakhutdinov. Transformer-xl: Attentive language models beyond a fixed-length context. arXiv preprint arXiv:1901.02860, 2019.
+
+[81] John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms. arXiv preprint arXiv:1707.06347, 2017.
+
+[82] John Schulman, Philipp Moritz, Sergey Levine, Michael Jordan, and Pieter Abbeel. Highdimensional continuous control using generalized advantage estimation. arXiv preprint arXiv:1506.02438, 2015.
+
+[83] Ronald J Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine learning, 8(3):229–256, 1992.
+
+[84] Volodymyr Mnih, Adria Puigdomenech Badia, Mehdi Mirza, Alex Graves, Timothy Lillicrap, Tim Harley, David Silver, and Koray Kavukcuoglu. Asynchronous methods for deep reinforcement learning. In International conference on machine learning, pages 1928–1937. PMLR, 2016.
+
+[85] Marcin Andrychowicz, Filip Wolski, Alex Ray, Jonas Schneider, Rachel Fong, Peter Welinder, Bob McGrew, Josh Tobin, OpenAI Pieter Abbeel, and Wojciech Zaremba. Hindsight experience replay. Advances in neural information processing systems, 30, 2017.   
+[86] Tom Schaul, Daniel Horgan, Karol Gregor, and David Silver. Universal value function approximators. In International conference on machine learning, pages 1312–1320. PMLR, 2015.   
+[87] Open Ended Learning Team, Adam Stooke, Anuj Mahajan, Catarina Barros, Charlie Deck, Jakob Bauer, Jakub Sygnowski, Maja Trebacz, Max Jaderberg, Michael Mathieu, et al. Openended learning leads to generally capable agents. arXiv preprint arXiv:2107.12808, 2021.   
+[88] Jelena Luketina, Nantas Nardelli, Gregory Farquhar, Jakob Foerster, Jacob Andreas, Edward Grefenstette, Shimon Whiteson, and Tim Rocktäschel. A survey of reinforcement learning informed by natural language. In Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence, IJCAI-19, pages 6309–6317. International Joint Conferences on Artificial Intelligence Organization, 7 2019. doi: 10.24963/ijcai.2019/880. URL https://doi.org/10.24963/ijcai.2019/880.   
+[89] DeepMind Interactive Agents Team, Josh Abramson, Arun Ahuja, Arthur Brussee, Federico Carnevale, Mary Cassin, Felix Fischer, Petko Georgiev, Alex Goldin, Tim Harley, et al. Creating multimodal interactive agents with imitation and self-supervised learning. arXiv preprint arXiv:2112.03763, 2021.   
+[90] Aditya Ramesh, Prafulla Dhariwal, Alex Nichol, Casey Chu, and Mark Chen. Hierarchical text-conditional image generation with clip latents. arXiv preprint arXiv:2204.06125, 2022.   
+[91] Dong Yu and Li Deng. Automatic speech recognition, volume 1. Springer, 2016.   
+[92] Daulet Nurmanbetov. rpunct, May 25 2021. URL https://github.com/Felflare/rpunct. accessed 2022-04-22.   
+[93] Arvind Neelakantan, Tao Xu, Raul Puri, Alec Radford, Jesse Michael Han, Jerry Tworek, Qiming Yuan, Nikolas Tezak, Jong Wook Kim, Chris Hallacy, et al. Text and code embeddings by contrastive pre-training. arXiv preprint arXiv:2201.10005, 2022.

parse/dev/Ms6QZafNv01/Ms6QZafNv01.md

@@ -0,0 +1,493 @@
+# Optimal algorithms for group distributionally robust optimization and beyond
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 Distributionally robust optimization (DRO) can improve the robustness and fairness   
+2 of learning methods. In this paper, we devise stochastic algorithms for a class   
+3 of DRO problems including group DRO, subpopulation fairness, and empirical   
+4 conditional value at risk (CVaR) optimization. Our new algorithms achieve faster   
+5 convergence rates than existing algorithms for multiple DRO settings. We also   
+6 provide a new information-theoretic lower bound that implies our bounds are tight   
+7 for group DRO. Empirically, too, our algorithms outperform known methods.
+
+# 8 1 Introduction
+
+9 Commonly, machine learning models are trained to optimize the average performance. However,   
+10 such models may not perform equally well among all demographic subgroups due to a hidden bias in   
+11 the training set or distribution shift in training and test phases [Hovy and Søgaard, 2015; Hashimoto   
+12 et al., 2018; Martinez et al., 2021; Duchi and Namkoong, 2021]. Biases in datasets are also directly   
+13 related to fairness concerns in machine learning [Buolamwini and Gebru, 2018; Jurgens et al., 2017].   
+14 Recently, various algorithms based on distributionally robust optimization (DRO) have been proposed   
+15 to address these problems [Hovy and Søgaard, 2015; Hashimoto et al., 2018; Hu et al., 2018; Oren et   
+16 al., 2019; Williamson and Menon, 2019; Sagawa et al., 2020; Curi et al., 2020; Zhang et al., 2021;   
+17 Martinez et al., 2021; Duchi and Namkoong, 2021]. However, these algorithms are often highly   
+18 tailored to each specific DRO formulation. Furthermore, it is often unclear whether these proposed   
+19 algorithms are optimal in terms of the convergence rate. Are there a unified algorithmic methodology   
+20 and a lower bound for these problems?   
+21 Contributions. In this paper, we study a general class of DRO problems, which includes group   
+22 DRO [Hu et al., 2018; Oren et al., 2019; Sagawa et al., 2020], subpopulation fairness [Martinez et   
+23 al., 2021], conditional value at risk (CVaR) optimization [Curi et al., 2020], and many others. Let   
+24 $\boldsymbol \Theta \subseteq \mathbb { R } ^ { n }$ be a convex set of model parameters and $\ell ( \theta ; z ) : \Theta \to \mathbb { R } _ { + }$ be a convex loss of the model   
+25 with parameter $\theta$ with respect to data point $z$ . The data point $z$ may be drawn from one out of $m$   
+26 distributions $P _ { 1 } , \ldots , P _ { m }$ which are accessible via a stochastic oracle that returns an i.i.d. sample   
+27 $z \sim P _ { i }$ . Let $Q$ be a convex subset of the probability simplex in $\mathbb { R } ^ { m }$ that contains the uniform vector,   
+28 i.e., $( 1 / m , \ldots , 1 / m ) \in Q$ . Our DRO formulation is as follows:
+
+$$
+\operatorname* { m i n } _ { \theta \in \Theta } \operatorname* { m a x } _ { q \in Q } \sum _ { i = 1 } ^ { m } q _ { i } \operatorname* { \bf { E } } _ { z \sim P _ { i } } [ \ell ( \theta ; z ) ] .
+$$
+
+29 If $Q$ are the probability simplex and scaled $k$ -set polytope, we can recover group DRO [Sagawa et   
+30 al., 2020] and subpopulation fairness [Martinez et al., 2021], respectively. Moreover, we formulate   
+31 a new, more general fairness concept based on weighted rankings with $Q$ being a permutahedron,   
+32 which includes these special cases; see Section 2 for details.
+
+Submitted to 36th Conference on Neural Information Processing Systems (NeurIPS 2022). Do not distribute.
+
+Table 1: Summary of convergence results for group DRO. Here, $m$ denotes the number of groups, $n$ the dimension of $\theta$ , $G$ the Lipschitz constant of loss function $\ell$ , $D$ the diameter of feasible set $\Theta$ , $M$ the range of loss function $\ell$ , and $T$ the number of calls to stochastic oracle.   
+
+<table><tr><td>reference</td><td>convergence rate E[εr]</td><td>iteration complexity</td><td>lower bound</td><td></td></tr><tr><td>[Sagawa et al., 2020]</td><td>0（m</td><td>G2 D²+M2 log m T</td><td>O(m + n) + proj. onto Θ</td><td rowspan="3">( G²D²+M2m T (Theorem 5)</td></tr><tr><td>Ours (Theorem 2)</td><td></td><td>T</td><td>O(m + n) + proj. onto Θ</td></tr><tr><td>Ours (Theorem 3)</td><td>0（√</td><td>G²D²+M²m） T</td><td>O(m + n) + proj. onto Θ + solving scalar equation</td></tr></table>
+
+33 For our general DRO, we devise an efficient stochastic gradient algorithm. Furthermore, we show that   
+34 it achieves the information-theoretic optimal convergence rate for group DRO. Our main technical   
+35 contributions are as follows;
+
+• We provide a generic stochastic gradient algorithm for our general DRO. By specializing it in the group DRO setting, we provide two algorithms (GDRO-EXP3 and GDRO-TINF)√ that improve the rate of Sagawa et al. [2020] by a factor of $\Omega ( { \sqrt { m } } )$ with the almost same complexity per iteration; see Table 1. Furthermore, our generic algorithm can be specialized to improve the convergence rate of Curi et al. [2020] for subpopulation fairness (a.k.a. empirical CVaR optimization). Finally, we show that our algorithm runs efficiently if $Q$ is a permutahedron, which includes all aforementioned subclasses.   
+• We prove a matching information-theoretic lower bound for the convergence rate of group DRO. This implies that no algorithm can improve the convergence rate of GDRO-TINF (up to a constant factor). To the best of our knowledge, this is the first information-theoretic lower bound for group DRO.   
+• Our experiments on real-world and synthetic datasets show that our algorithms also empirically outperform the known algorithm, supporting our theoretical analysis.
+
+# 49 1.1 Our techniques
+
+50 Algorithms. The core idea of our algorithms is stochastic no-regret dynamics [Hazan, 2016]. We   
+51 regard DRO (1) as a two-player zero-sum game between a player who picks $\theta \in \Theta$ and another player   
+52 who picks $q \in Q$ . The two players iteratively update their solution using online learning algorithms;   
+53 in particular, we will use online gradient descent (OGD) [Zinkevich, 2003] and online mirror descent   
+54 (OMD) [Cesa-Bianchi and Lugosi, 2006] for the $\theta$ -player and $q$ -player, respectively. In addition, we   
+55 need to estimate gradients for both players, since the objective function of our DRO is stochastic and   
+56 we cannot obtain exact gradients.   
+57 The convergence rate of stochastic no-regret dynamics depends on the expected regret of OGD and   
+58 OMD. To obtain the optimal convergence rate, we must carefully choose the regularizer in OMD as   
+59 well as gradient estimators, exploiting the structure of our DRO. In particular, we need to balance   
+60 the variance of gradient estimators and the diameter terms in both OGD and OMD. This is the most   
+61 challenging part of the algorithm design. Inspired by adversarial multi-armed bandit algorithms,   
+62 we design gradient estimators for no-regret dynamics of OGD and OMD in our DRO. Indeed, our   
+63 algorithms for group DRO (GDRO-EXP3 and GDRO-TINF) are based on adversarial multi-armed   
+64 bandit algorithms, EXP3 [Auer et al., 2003] and Tsallis-INF [Zimmert and Seldin, 2021], respectively,   
+65 hence the name. Although each building block (OGD, OMD, and gradient estimators) is fairly known   
+66 in the literature, we need to put them together in the right combination to obtain the optimal rate.   
+67 Lower bound. For the lower bound, we carefully design a family of group DRO instances for which   
+68 any algorithm requires a certain number of queries to achieve a good objective value. To bound   
+69 the number of queries, we use information-theoretic tools such as Le Cam’s lemma and bound the   
+70 Kullback-Leibler divergence between Bernoulli distributions. Such tools are also used at the heart of   
+71 lower bounds for stochastic convex optimization [Agarwal et al., 2012] and adversarial multi-armed   
+72 bandits [Auer et al., 2003], but the connection to those settings is much more subtle here, and our   
+73 construction is specifically designed for group DRO-type problems.   
+75 DRO is a wide field ranging from robust optimization to machine learning and statistics [Goh and Sim,   
+76 2010; Bertsimas et al., 2018], whose original idea dates back to Scarf [1958]. Popular choices of the   
+77 uncertainty set in DRO include balls around an empirical distribution in Wasserstein distance [Esfahani   
+78 and Kuhn, 2018; Blanchet et al., 2019], $f$ -divergence [Namkoong and Duchi, 2016; Duchi and   
+79 Namkoong, 2021], $\chi ^ { 2 }$ -divergence [Staib et al., 2019], and maximum mean discrepancy [Staib and   
+80 Jegelka, 2019; Kirschner et al., 2020].   
+81 DRO algorithms have been mainly studied for the offline setting, i.e., algorithms can access all data   
+82 points of the empirical distribution. Note that our DRO is not offline because the group distributions   
+83 are given by the stochastic oracles. Namkoong and Duchi [2016] proposed stochastic gradient   
+84 algorithms for offline DRO with $f$ -divergence uncertainty sets. Curi et al. [2020] used no-regret   
+85 dynamics for empirical CVaR minimization. Their algorithm invokes sampling from $k$ -DPP in each   
+86 iteration, which is more computationally demanding than our algorithm. Furthermore, our algorithm   
+87 gets rid of an $O ( \log m )$ factor in the convergence rate using the Tsallis entropy regularizer; see   
+88 Theorem 4. Qi et al. [2021]; Jin et al. [2021] devised stochastic gradient algorithms for several DRO   
+89 with non-convex losses.   
+90 Agarwal et al. [2012] gave a lower bound for stochastic convex optimization, which is a special case   
+91 of our DRO with only one distribution. Recently, Carmon et al. [2021] showed a lower bound for   
+92 minimax problem $\begin{array} { r } { \operatorname* { m i n } _ { x } \operatorname* { m a x } _ { i = 1 } ^ { m } f _ { i } ( x ) } \end{array}$ for non-stochastic Lipschitz convex $f _ { i }$ . Our lower bound deals   
+93 with the stochastic functions, so this result does not apply.   
+94 In this paper, we assume that the group information is given in advance. However, the group   
+95 information might not be easy to define in practice. Bao et al. [2021] propose a simple method to   
+96 define groups for classification problems based on mistakes of models in the training phase. Their   
+97 method often generates group DRO instances with large $m$ . Our algorithms are more efficient for   
+98 such group DRO thanks to the better dependence on $m$ in the convergence rate.   
+99 No-regret dynamics is a well-studied method for solving two-player zero-sum games [Cesa-Bianchi   
+100 and Lugosi, 2006]. For non-stochastic convex-concave games, one can achieve $O ( 1 / T )$ convergence   
+101 via predictable sequences [Rakhlin and Sridharan, 2013]. This result does not apply to our setting   
+102 because our DRO is a stochastic game.   
+103 Notations. Throughout the paper, $m$ denotes the number of distributions (groups) and $n$ denotes   
+104 the dimension of a variable $\theta$ . For a positive integer $m$ , we write $[ m ] : = \{ 1 , \ldots , m \}$ . The orthogonal   
+105 projection onto set $\Theta$ is denoted by $\mathrm { p r o j } _ { \Theta }$ . The ith standard unit vector is denoted by $\mathbf { e } _ { i }$ and the   
+106 all-one vector is denoted by 1. The probability simplex in $\mathbb { R } ^ { m }$ is denoted by $\Delta _ { m }$ .
+
+# 107 2 Examples contained in our general DRO
+
+108 In this section, we show how several DRO formulations in the literature can be phrased in our general   
+109 DRO formulation (1). In addition, we propose a novel fairness constraint based on weighted rankings   
+110 using our general DRO.   
+111 Group DRO. When $Q$ equals the probablility simplex, we obtain group DRO [Hu et al., 2018;   
+112 Oren et al., 2019; Sagawa et al., 2020]:
+
+$$
+\operatorname* { m i n } _ { \theta \in \Theta } \operatorname* { m a x } _ { i = 1 } ^ { m } \ \underset { z \sim P _ { i } } { \mathbf { E } } [ \ell ( \theta ; z ) ] .
+$$
+
+113 That is, group DRO aims to minimize the expected loss in the worst group, thereby ensuring better   
+114 performance across all groups.   
+115 Empirical CVaR, Subpopulation fairness, Average top- $k$ worst group loss. Group DRO may   
+116 yield overly pessimistic solutions. For instance, the groups might be automatically generated by other   
+117 algorithms (such as one in Bao et al. [2021]) and there might exist a few “outlier” groups that make   
+118 the group DRO objective trivial.
+
+119 For such a case, we can restrict $Q$ to a small subset of the probability simplex so that the solution cannot put large weights on a few outlier groups. Especially, let 120 $Q = \left\{ q \in \Delta _ { m } : 0 \leq q _ { i } \leq { \frac { 1 } { p m } } \right\}$ for
+
+121 some parameter $p \in ( 0 , 1 )$ , i.e., $Q$ is a scaled $k$ -set polytope. The intuition behind the choice of $Q$   
+122 is that, by limiting the largest entry of $q$ to $1 / p m$ , DRO would optimize the expected loss over the   
+123 worst $p$ -fraction subgroups of $m$ groups. Therefore, if the fraction of outlier groups is sufficiently   
+124 small compared to $p$ , then $p$ -fraction subgroups must contain “inlier” groups as well. Therefore, it is   
+125 likely that DRO with $Q$ finds solutions more robust than group DRO.   
+126 When $P _ { i }$ is the Dirac measure of data $z _ { i }$ , then the resulting DRO is empirical CVaR optimization [Curi   
+127 et al., 2020]. In the fairness context, the same problem is called subpopulation fairness [Williamson   
+128 and Menon, 2019; Martinez et al., 2021; Duchi and Namkoong, 2021].   
+129 If $p \ = \ m / k$ for some positive integer $k$ , the resulting DRO is the average top- $k$ worst group   
+130 loss [Zhang et al., 2021]:
+
+$$
+\operatorname* { m i n } _ { \theta \in \Theta } \frac { 1 } { k } \sum _ { i = 1 } ^ { k } L _ { i } ^ { \downarrow } ( \theta ) ,
+$$
+
+where 131 $L _ { i } ^ { \downarrow } ( \theta )$ denotes the the $i$ th largest population group loss of $\theta$ . More precisely, let $L _ { i } ( \theta ) =$ 132 $\mathbf { E } _ { z \sim P _ { i } } [ \ell ( \theta ; z ) ]$ for $i \in [ m ]$ and sort them in the non-increasing order: $L _ { 1 } ^ { \downarrow } ( \theta ) \ge \cdots \ge L _ { m } ^ { \downarrow } ( \theta )$ .
+
+133 Weighted ranking of group losses. The aforementioned DRO formulations are special cases of the   
+134 following DRO, which we call the weighted ranking of group losses. Let $\alpha \in \Delta ^ { m }$ be a fixed vector   
+135 with non-increasing entries. Let $Q$ be the permutahedron of $\alpha$ , the convex hull of $( \alpha _ { \sigma ( 1 ) } , \ldots , \alpha _ { \sigma ( m ) } )$   
+136 for all permutations $\sigma$ of $[ m ]$ . Then, the resulting DRO is
+
+$$
+\operatorname* { m i n } _ { \theta \in \Theta } \sum _ { i = 1 } ^ { m } \alpha _ { i } L _ { i } ^ { \downarrow } ( \theta ) .
+$$
+
+137 Group DRO corresponds to $\alpha = ( 1 , 0 , \ldots , 0 )$ and the average top- $k$ worst group losses corresponds 138 to $\alpha \overset { \cdot } { = } ( 1 / k , \ldots , \bar { 1 ^ { \prime } } k , 0 , \ldots , 0 )$ . Another example that is contained in none of the above examples is {zk times
+
+139 lexicographic minimax fairness [Diana et al., 2021]. The goal of lexicographical minimax fairness   
+140 is to find $\theta \in \Theta$ such that the sequence $( L _ { 1 } ^ { \downarrow } ( \theta ) , \dots , L _ { m } ^ { \downarrow } ( \theta ) )$ is lexicographically minimum. This   
+141 corresponds to $\alpha$ with sufficiently varied entries, i.e., $\alpha _ { 1 } \gg \alpha _ { 2 } \gg \cdot \cdot \cdot \gg \alpha _ { m }$ .
+
+# 142 3 Algorithms
+
+143 In this section, we describe our algorithms. First, we present a generic algorithm for our general   
+144 DRO (1) and provide a unified convergence analysis in Section 3.1. Then, we specialize it into two   
+145 concrete algorithms for group DRO (2) in Section 3.2. We sketch algorithms for the average of top- $k$   
+146 group losses and weighted ranking of group loss in Section 3.3.
+
+# 7 3.1 Algorithm for the general case
+
+148 We present our algorithm for geranal DRO (1). At a high level, our algorithm can be regarded as   
+149 stochastic no-regret dynamics. Let us denote $\begin{array} { r } { L ( \boldsymbol { \theta } , \boldsymbol { q } ) : = \sum _ { i = 1 } ^ { m } q _ { i } \mathbf { E } _ { z \sim P _ { i } } [ \ell ( \boldsymbol { \theta } ; z ) ] } \end{array}$ . Imagine that the   
+150 $\theta$ -player and $q$ -player run online algorithms $\scriptstyle A _ { \theta }$ and $A _ { q }$ , respectively, to solve the minimax problem   
+151 $\begin{array} { r } { \operatorname* { m i n } _ { \theta \in \Theta } \operatorname* { m a x } _ { q \in Q } L ( \theta , q ) } \end{array}$ . That is, for $t = 1 , \dots , T$ ,
+
+• $\theta _ { t } \in \Theta$ and $q _ { t } \in Q$ are determined by $\scriptstyle A _ { \theta }$ and $A _ { q }$ , respectively.   
+• Both players feed gradient estimators $\hat { \nabla } _ { \theta , t }$ and $\hat { \nabla } _ { \boldsymbol { q } , t }$ to $\scriptstyle A _ { \theta }$ and $A _ { q }$ , respectively. Here, $\mathbf { E } [ \hat { \nabla } _ { \boldsymbol { \theta } , t } ] = \nabla _ { \boldsymbol { \theta } } L ( \theta _ { t } , q _ { t } )$ and $\mathbf { E } [ \hat { \nabla } _ { q , t } ] = \nabla _ { q } L ( \theta _ { t } , q _ { t } )$ .
+
+155 Let
+
+$$
+\varepsilon _ { T } : = \operatorname* { m a x } _ { q \in Q } L ( \bar { \theta } _ { 1 : T } , q ) - \operatorname* { m i n } _ { \theta \in \Theta } \operatorname* { m a x } _ { q \in Q } L ( \theta , q )
+$$
+
+be the optimality156 convergence rate of the averavia regrets iterand $\begin{array} { r } { \bar { \theta } _ { 1 : T } = \frac { 1 } { T } \sum _ { t = 1 } ^ { T } \theta _ { t } } \end{array}$ . We can bound the eorithms (see Appendix ectedfor a $\mathbf { E } [ \varepsilon _ { T } ]$ $R _ { \theta }$ $R _ { q }$ $\mathbf { A }$ 158 formal definition), i.e.,
+
+$$
+\mathbf { E } [ \varepsilon _ { T } ] \leq \frac { \mathbf { E } [ R _ { \theta } ( T ) ] + \mathbf { E } [ R _ { q } ( T ) ] } { T } .
+$$
+
+159 We can obtain hence the convergence rate of the above algorithms by investigating the expected regret   
+160 bounds of these online algorithms.   
+161 To get a concrete algorithm, we must specify the online algorithms $A _ { \theta } , A _ { q }$ as well as the gradient   
+162 estimators $\hat { \nabla } _ { \boldsymbol { \theta } , t } , \hat { \nabla } _ { \boldsymbol { q } , t }$ . We use OGD and OMD as $\scriptstyle A _ { \theta }$ and $A _ { q }$ , respectively. We construct the gradient   
+163 estimators by sampling $i _ { t } \sim q _ { t }$ and $z \sim P _ { i _ { t } }$ and setting $\hat { \nabla } _ { \boldsymbol { \theta } , t } = \nabla _ { \boldsymbol { \theta } } \ell ( \theta _ { t } ; z )$ and $\begin{array} { r } { \hat { \nabla } _ { q , t } = \frac { \ell ( \theta _ { t } ; z ) } { q _ { t , i _ { t } } } \mathbf { e } _ { i _ { t } } } \end{array}$   
+164 This leads to Algorithm 1. There, $\Psi : Q  \mathbb { R }$ denotes the regularizer of OMD and $\eta _ { \theta , t }$ and $\eta _ { q }$   
+165 denote the step sizes of OGD and OMD, respectively. 1 It turns out that this combination of online   
+166 algorithms and gradient estimators yields the best convergence rate (for group DRO) because the   
+167 expected regrets of both players are optimal.
+
+# Algorithm 1 Algorithm for general DRO (1)
+
+Require: initial solution $\theta _ { 1 } \in \Theta$ , number of iterations $T$ , step sizes $\eta _ { \theta , t } > 0 ( t \in [ T ] ) , \eta _ { q } > 0$ , and a strictly convex function $\Psi : Q  \mathbb { R }$ .   
+1: Let $q _ { 1 } \dot { = } ( 1 / m , \dots , 1 / m )$ .   
+2: for $t = 1 , \dots , T$ do   
+3: Sample $i _ { t } \sim q _ { t }$ .   
+4: Call the stochastic oracle to obtain $z \sim P _ { i _ { t } }$ .   
+5: $\theta _ { t + 1 }  \mathrm { p r o j } _ { \Theta } ( \theta _ { t } - \eta _ { \theta , t } \nabla _ { \theta } \ell ( \theta _ { t } ; z ) )$   
+6: $\begin{array} { r l r } { \nabla \Psi ( \widetilde { q } _ { t + 1 } ) } & { { }  } & { \nabla \Psi ( q _ { t } ) \ - \ \frac { \eta _ { q } } { q _ { t , i _ { t } } } \ell ( \theta _ { t } ; z ) \mathbf { e } _ { i _ { t } } } \end{array}$ ; $\begin{array} { r l r } { q _ { t + 1 } } & { { }  } & { \arg \operatorname* { m i n } _ { q \in Q } D _ { \Psi } ( q , \tilde { q } _ { t + 1 } ) } \end{array}$ , where $D _ { \Psi } ( x , y ) = \Psi ( x ) - \Psi ( y ) - \nabla \Psi ( x ) ^ { \top } ( y - x )$ is the Bregman divergence with respect to $\Psi$ .   
+7: return $\textstyle { \frac { 1 } { T } } \sum _ { t = 1 } ^ { T } \theta _ { t }$ .
+
+168 We now analyze the convergence rate of Algorithm 1. We make the following standard assumptions.
+
+69 Assumption 1. The loss function $\ell ( \theta ; z )$ is continuously differentiable and $G$ -Lipchitz in $\theta$ , and has   
+170 range $[ 0 , M ]$ for all $z$ . The Euclidean diameter of the feasible region $\Theta$ is at most $D$ .
+
+The following theorem follows from plugging regret bounds of OGD and OGD, and the construction 2 of the gradient estimators into (3).
+
+73 Theorem 1. If $\eta _ { \theta , t }$ is nonincreasing, Algorithm 1 achieves the expected convergence rate
+
+$$
+\Im [ \varepsilon _ { T } ] \leq \frac { 1 } { T } \left( \frac { G ^ { 2 } } { 2 } \sum _ { t = 1 } ^ { T } \eta _ { \theta , t } + \frac { D ^ { 2 } } { 2 \eta _ { \theta , T } } + \frac { M ^ { 2 } } { 2 } \eta _ { \theta } \sum _ { t = 1 } ^ { T } \frac { \mathbf { F } } { i _ { t } } \left[ \frac { ( \nabla ^ { 2 } \Psi ( q _ { t } ) ) _ { i _ { t } , i _ { t } } ^ { - 1 } } { q _ { t , i _ { t } } ^ { 2 } } \right] + \frac { \operatorname* { m a x } _ { q ^ { * } \in Q } D _ { \Psi } ( q ^ { * } , \mathbf { 1 } / m ) } { \eta _ { q } } \right) .
+$$
+
+174 A formal proof can be found in Appendix B. We will see how specific choices of the regularizer $\Psi$   
+175 yield various algorithms and convergence rates for group DRO and others in the next subsections. A   
+176 few remarks on the regularizers, step sizes, and projection step are in order.   
+177 Regularizer. Although Algorithm 1 works with general $\Psi$ , we can choose a specific regularizer for   
+178 $Q$ appearing in applications, e.g, the probability simplex, scaled $k$ -set polytope, or a permutahedron.   
+179 In the next subsections, we show that the entropy regularizer $\begin{array} { r } { \Psi ( x ) = \bar { \sum _ { i } } ( x _ { i } \log x _ { i } - \bar { \operatorname { x } _ { i } } ) } \end{array}$ and Tsallis   
+180 entropy regularizer $\begin{array} { r } { \Psi ( x ) = 2 ( 1 - \sum _ { i } \sqrt { x _ { i } } ) } \end{array}$ yield efficient algorithms with improved convergence   
+181 rates for these cases.   
+182 Step sizes. The theorem includes decreasing step sizes such as $\begin{array} { r } { \eta _ { \theta , t } = \frac { D } { m G \sqrt { t } } } \end{array}$ in addition to fixed   
+183 step sizes. Decreasing step sizes have the advantage that we do not require the knowledge of $T$   
+184 at the beginning of the algorithm but come at the cost of an extra constant factor in the expected   
+185 convergence rate. Since both step size policies give the asymptotically same convergence rate, we   
+186 describe only fixed step sizes in the theorems in the next subsections. In practice, decreasing step   
+187 sizes stabilize the algorithm and often outperform fixed step sizes.
+
+Projection step. In general, the Bregman projection argminq∈Q $\mathrm { a r g m i n } _ { q \in Q } D _ { \Psi } ( q , \tilde { q } _ { t + 1 } )$ is convex, but may be costly to compute. For the applications described in Section 2, $\tilde { Q }$ is a permutahedron. In this case,
+
+# Algorithm 2 GDRO-EXP3
+
+Require: initial solution $\theta _ { 1 } \in \Theta$ , number of iterations $T$ , and step sizes $\eta _ { \theta , t } > 0 \left( t \in [ T ] \right)$ , $\eta _ { q } > 0$ .   
+1: Let $q _ { t } = ( 1 / m , \dots , 1 / m )$ .   
+2: for $t = 1 , \dots , T$ do   
+3: Sample $i _ { t } \sim q _ { t }$ .   
+4: Call the stochastic oracle to obtain $z \sim P _ { i _ { t } }$ 5: θt+1 ← projΘ(θt − ηθ,t∇θ\`(θt; z))   
+6: q˜t+1 ← qt exp  ηq\`(θt;z)eitqt,i    
+7: qt+1 ← Pq˜t+1q˜t+1,i .   
+8: return 1T PTt=1 θt.
+
+# Algorithm 3 GDRO-TINF
+
+Require: initial solution $\theta _ { 1 } \in \Theta$ , number of iterations $T$ , and step sizes $\eta _ { \theta , t } > 0 \left( t \in [ T ] \right)$ , $\eta _ { q } > 0$ .   
+1: Let $q _ { t } = ( 1 / m , \dots , 1 / m )$ .   
+2: for $t = 1 , \dots , T$ do   
+3: Sample $i _ { t } \sim q _ { t }$ . 4: Call the stochastic oracle to obtain $z \sim P _ { i _ { t } }$ .   
+5: $\begin{array} { r l } & { \theta _ { t + 1 } \gets \mathrm { p r o j } _ { \Theta } ( \theta _ { t } - \eta _ { \theta , t } \nabla _ { \theta } \ell ( \theta _ { t } ; z ) ) } \\ & { \widetilde { q } _ { t + 1 } \gets q _ { t } \left( \mathbf { 1 } - \frac { \eta _ { q } \sqrt { q _ { t } } } { q _ { t , i _ { t } } } \ell ( \theta _ { t } ; z ) \mathbf { e } _ { i _ { t } } \right) ^ { - 2 } } \\ & { \mathrm { C o m p u t e ~ } \quad \alpha \qquad \in \qquad \mathbb { R } \quad \mathrm { s u c h } } \\ & { \sum _ { i = 1 } ^ { m } \left( \sqrt { \widetilde { q } _ { t + 1 , i } } - \alpha \right) ^ { - 2 } = 1 . } \\ & { q _ { t + 1 } \gets \left( \sqrt { \widetilde { q } _ { t + 1 } } - \alpha \mathbf { 1 } \right) ^ { - 2 } } \\ & { \mathrm { t u r n ~ } \frac { 1 } { T } \sum _ { t = 1 } ^ { T } \theta _ { t } . } \end{array}$   
+6: 7: that   
+8:   
+9: return 1T PTt=1 θt.
+
+190 it is known that the Bregman projection with respect to the entropy and Tsallis entropy regularizers   
+191 can be done in $O ( m \log m )$ time [Lim and Wright, 2016]. If $Q$ is the probability simplex, we even   
+192 have a closed form for the Bregman projection.
+
+# 3.2 Algorithms for Group DRO
+
+As applications of our generic algorithm, we now describe two concrete algorithms for group DRO (2) and their convergence rates.
+
+GDRO-EXP3. Let $\Psi$ be the entropy regularizer, which corresponds to the EXP3 algorithm for 7 the $q$ -player. The resulting algorithm, GDRO-EXP3, is shown in Algorithm 2. The update is in a 8 closed formula and its complexity is $O ( m + n )$ time. The convergence rate follows from Theorem 1.
+
+99 Theorem 2. If $\eta _ { \theta , t }$ is nonincreasing, GDRO-EXP3 (Algorithm 2) achieves the expected convergence   
+00 rate
+
+$$
+\mathbf { E } [ \varepsilon _ { T } ] \leq { \frac { 1 } { T } } \left( { \frac { G ^ { 2 } } { 2 } } \sum _ { t = 1 } ^ { T } \eta _ { \theta , t } + { \frac { D ^ { 2 } } { 2 \eta _ { \theta , T } } } + { \frac { m M ^ { 2 } } { 2 } } \eta _ { q } T + { \frac { \log m } { \eta _ { q } } } \right) .
+$$
+
+For 201 $\begin{array} { r } { \eta _ { \theta , t } = \frac { D } { G \sqrt { T } } } \end{array}$ and ηq = $\begin{array} { r } { \eta _ { q } = \sqrt { \frac { 2 \log m } { m M ^ { 2 } T } } } \end{array}$ , we obtain
+
+$$
+\mathbf { E } [ \varepsilon _ { T } ] \leq { \sqrt { 2 } } { \frac { { \sqrt { G ^ { 2 } D ^ { 2 } + 2 M ^ { 2 } m \log m } } } { \sqrt { T } } } .
+$$
+
+202 GRDO-TINF. We can further improve the convergence rate using the Tsallis entropy regularizer   
+203 at the cost of a slightly higher iteration complexity. The update of $q _ { t }$ is then
+
+$$
+\tilde { q } _ { t + 1 } = q _ { t } \left( \mathbf { 1 } - \frac { \eta _ { q } \sqrt { q _ { t } } } { q _ { t , i _ { t } } } \ell ( \theta _ { t } ; z ) \mathbf { e } _ { i _ { t } } \right) ^ { - 2 } , \quad q _ { t + 1 } : = \left( \sqrt { \tilde { q } _ { t + 1 } } - \alpha \mathbf { 1 } \right) ^ { - 2 } ,
+$$
+
+204 where the multiplication, square-root, and power operations are entry-wise and $\alpha \in \mathbb { R }$ is the unique   
+205 solution of equation $\begin{array} { r } { \sum _ { i = 1 } ^ { m } \overline { { \left( \sqrt { \tilde { q } _ { t + 1 , i } } - \alpha \right) ^ { - 2 } } } = 1 } \end{array}$ . The solution $\alpha$ can be computed via the Newton   
+206 method. Practically, one can use $\alpha$ in the previous iteration to warm start the Newton method. In   
+207 each iteration, the algorithm performs a single orthogonal projection onto $\Theta$ , the Newton method for   
+208 finding $\alpha$ , and $O ( m + n )$ operations to update $\theta _ { t } , q _ { t }$ . The pseudocode is given in Algorithm 3. From   
+209 Theorem 1, we obtain the following convergence rate.   
+210 Theorem 3. If $\eta _ { \theta , t }$ is nonincreasing, GDRO-TINF (Algorithm 3) achieves the expected convergence   
+211 rate
+
+$$
+\mathbf { E } [ \varepsilon _ { T } ] \leq { \frac { 1 } { T } } \left( { \frac { G ^ { 2 } } { 2 } } \sum _ { t = 1 } ^ { T } \eta _ { \theta , t } + { \frac { D ^ { 2 } } { 2 \eta _ { \theta , T } } } + { \sqrt { m } } M ^ { 2 } \eta _ { q } T + { \frac { \sqrt { m } } { \eta _ { q } } } \right) .
+$$
+
+For 212 $\begin{array} { r } { \eta _ { \theta , t } = \frac { D } { G \sqrt { T } } } \end{array}$ and $\begin{array} { r } { \eta _ { q } = \frac { 1 } { M \sqrt { T } } } \end{array}$ , we obtain
+
+$$
+\mathbf { E } [ \varepsilon _ { T } ] \leq \sqrt { 2 } \frac { \sqrt { G ^ { 2 } D ^ { 2 } + 4 M ^ { 2 } m } } { \sqrt { T } } .
+$$
+
+213 Comparison to Sagawa et al. [2020]. Our algorithms improve the convergence rate of Sagawa et√   
+214 al. [2020] by a factor of $O ( \sqrt { m } )$ ; see Table 1. The reason lies in the choice of gradient estimator. All   
+215 algorithms are stochastic no-regret dynamics. As outlined above, their convergence hence can be   
+216 bounded by the regrets of the players, which depend on the variance of the local norm of the gradient   
+217 estimators. Their strategy is based on uniform sampling that yields a variance of √ $O ( m )$ for both   
+218 players, whereas our bound is $O ( { \sqrt { m } } )$ thanks to the gradient estimators tailored to the regularizer of   
+219 OMD. More details may be found in Appendix D.
+
+# 3.3 Algorithm for weighted ranking of group losses
+
+We now consider a more general case that $Q$ is a permutahedron. Applying Algorithm 1 with the Tsallis entropy regularizer, we obtain the following result.
+
+Theorem 4. If $\eta _ { \theta , t }$ is nonincreasing and $Q$ is a permutahedron, Algorithm 1 with the Tsallis entropy regularizer achieves the same expected convergence rate as Theorem 3. Furthermore, the iteration complexity is $O ( m \log m + n )$ .
+
+This implies a convergence rate of G2D2+M2mT ) for empirical CVaR optimization, which improves q G2D2+M2m log mT ) convergence by Curi et al. [2020]. Furthermore, their iteration complexity is $O ( m ^ { 3 } )$ due to the $k$ -DPP sampling step, so our algorithm is even faster in terms of iteration complexity.
+
+# 4 Lower bound
+
+Theorem 3 states that we can find an $\varepsilon$ -optimal solution for group DRO in $ { \mathcal { O } } ( \frac { G ^ { 2 } D ^ { 2 } + M ^ { 2 } m } { \varepsilon ^ { 2 } } )$ calls to stochastic oracles. Next, we show that this query complexity is information-theoretically optimal.
+
+33 Let $\mathcal { L }$ be a class of convex $G$ -Lipschitz loss functions $\ell : \Theta \to [ 0 , M ]$ . Given a loss function $\ell \in { \mathcal { L } }$ ,   
+34 and an $m$ -set $\mathcal { P } = \{ P _ { 1 } , \ldots , P _ { m } \}$ of distributions, denote the optimality gap of $\theta \in \Theta$ by
+
+$$
+R ( \theta , \ell , \mathcal { P } ) = \operatorname* { m a x } _ { P \in \mathcal { P } } \mathbf { \Xi } _ { z \sim P } ^ { \mathbf { E } } [ \ell ( \theta ; z ) ] - \operatorname* { m i n } _ { \theta ^ { * } \in \Theta } \operatorname* { m a x } _ { P \in \mathcal { P } } \mathbf { \Xi } _ { z \sim P } ^ { \mathbf { E } } [ \ell ( \theta ^ { * } ; z ) ] .
+$$
+
+Let 235 $\boldsymbol { \mathcal { A } } _ { T }$ be the set of algorithms that outputs $\hat { \theta } \in \Theta$ making $T$ queries to the stochastic oracle.
+
+Theorem 5 (Lower Bound).
+
+$$
+\operatorname* { i n f } _ { \boldsymbol { \hat { \theta } } \in A _ { T } } \operatorname* { s u p } _ { \boldsymbol { \ell } \in \mathcal { L } , \boldsymbol { \Theta } , \mathcal { P } } \mathbf { E } [ R ( \boldsymbol { \hat { \theta } } , \boldsymbol { \ell } , \mathcal { P } ) ] \geq \Omega \left( \operatorname* { m a x } \left\{ \frac { G D } { \sqrt { T } } , M \sqrt { \frac { m } { T } } \right\} \right) ,
+$$
+
+236 where $\Theta$ runs over convex sets with diameter $D$ and $\mathcal { P }$ over $m$ -sets of distributions, and $\mathbf { E } _ { \mathcal { P } }$ denotes   
+237 the expectation over outcomes of the stochastic oracle in $\mathcal { P }$ .
+
+As ${ \sqrt { x + y } } \leq { \sqrt { x } } + { \sqrt { y } } \leq { \sqrt { 2 ( x + y ) } }$ for $x , y \geq 0$ , this theorem immediately implies that the minimax convergence rate is $\Omega \left( { \sqrt { \frac { G ^ { 2 } D ^ { 2 } + M ^ { 2 } m } { T } } } \right)$ , which equals the convergence rate achieved by Algorithm 3 up to a constant factor.
+
+241 Proof Sketch. It suffices to show two lower bounds $\textstyle { \frac { G D } { \sqrt { T } } }$ and $M _ { \sqrt { T } }$ independently. The former is   
+242 a well-known lower bound for stochastic convex optimization [Agarwal et al., 2012]. To illustrate the   
+243 latter, we take an algorithmic dependent point of view via the Le cam’s method. For any algorithm   
+244 in $\boldsymbol { \mathcal { A } } _ { T }$ , we need to construct instances $\mathcal { P } _ { 0 } , \mathcal { P } _ { 1 }$ such that the total variation distance between the   
+245 distributions over the query outcomes (they depend on both the behavior of the algorithm and the   
+246 instance) with respect to $\mathcal { P } _ { 0 }$ and $\mathcal { P } _ { 1 }$ is small. On the other hand, the objective function of the two   
+247 instances must be well-separated, i.e., any fixed $\theta$ is $\delta$ sub-optimal for either $\mathcal { P } _ { 0 }$ or $\mathcal { P } _ { 1 }$ . So, any   
+248 algorithm that solves group DRO up to error $\delta$ needs to distinguish two instances $\mathcal { P } _ { 0 }$ and $\mathcal { P } _ { 1 }$ . This   
+249 implies a query lower bound because the total variation distance of the outcome distributions of   
+250 these instances is small. The challenge is how to construct such instances for the regime of small   
+251 dimensions of $\theta$ , e.g, $n = 1$ . To this end, we carefully construct linear functions for $m$ groups using   
+252 opposite slopes. Then, based on the behavior of the algorithm, we tweak the noise bias in one of the   
+253 groups with a positive slope, in a way that any fixed $\theta$ is $\Theta ( \delta )$ sub-optimal for one of these instances.   
+254 For the detailed proof, see Appendix C.
+
+![](images/d9492d45448eb847bb159864789d12e10cf5a5dda3bde0a928bacf6167b13ceb.jpg)  
+Figure 1: Results on Adult dataset
+
+![](images/5b04a8e0db493af6fd2e274be9b0b710c499c0b8edd296bd1ca151d3d95db3d7.jpg)  
+Figure 2: Results on synthetic dataset
+
+# 5 Experiments
+
+In this section, we compare our algorithms with the known algorithm using real-world and synthetic datasets. We follow the setup in [Namkoong and Duchi, 2016].
+
+Adult dataset. For the real-world dataset, we use Adult dataset [Dua and Graff, 2017]. The dataset consists of age, gender, race, educational background, and many other attributes of 48, 842 individuals from the US census. The task is to predict whether the person’s income is greater than 50, 000 USD or not. We set up 6 groups based on the race and gender attributes: each group corresponds to a combination of {black, white, others} $\times \left\{ \begin{array} { r l } \end{array} \right.$ {female, male}. Converting the categorical features to dummy variables, we obtain a 101-dimensional feature vector $a \in \mathbb { R } ^ { n }$ ( $n = 1 0 1$ ) for each individual. We train the linear model with the logistic loss and hinge loss functions. The group-DRO objective is the worst empirical loss over the 6 groups:
+
+$$
+\operatorname* { m a x } _ { i = 1 } ^ { 6 } { \frac { 1 } { | I _ { i } | } } \sum _ { ( a , b ) \in I _ { i } } \ell ( \theta ; a , b ) ,
+$$
+
+where 266 $I _ { i }$ is the set of data points in the ith group. The feasible region is set to the Euclidean ball of 267 radius $D = 1 0$ .
+
+268 Synthetic dataset. To observe the performance of the algorithms over the regime of high-dimension   
+269 model parameters and the larger number of groups, we also conducted experiments using the following   
+270 synthetic instances. First, we set $n = 5 0 0$ and varied $m \in \{ 1 0 , 5 0 , 1 0 0 \}$ . For each group $i \in [ m ]$ , we   
+271 generated the true classifier $\theta _ { i } ^ { * } \in \mathbb { R } ^ { n }$ from the uniform distribution over the unit sphere in $\mathbb { R } ^ { n }$ . The ith   
+272 group distribution $P _ { i }$ was the empirical distribution of 1,000 data points, where each data point $( a , b )$   
+273 was drawn as $a \sim N ( 0 , I _ { n } )$ and $b = \mathrm { s i g n } ( a ^ { \top } \theta _ { i } ^ { * } )$ with probability 0.9 and $b = - \mathrm { s i g n } ( a ^ { \top } \theta _ { i } ^ { * } )$ with   
+274 probability 0.1. We trained the linear model with the hinge loss function. Finally, the group-DRO   
+275 objective is
+
+$$
+\underset { i = 1 } { \operatorname* { m a x } } \ \underset { ( a , b ) \sim P _ { i } } { \mathbf { E } } [ \ell ( \theta ; a , b ) ] .
+$$
+
+276 The feasible region is set to the Euclidean ball of radius $D = 1 0$ .
+
+# 5.1 Algorithms
+
+We implemented GDRO-EXP3, GDRO-TINF, and the algorithm in [Sagawa et al., 2020] in Python.   
+We ran our algorithms for $T = 2 , 0 0 0 , 0 0 0$ iterations.
+
+Inner online algorithms. It is known that EXP3 has a variance as large as $O ( T ^ { 2 } )$ [Lattimore and Szepesvári, 2020]. Therefore, vanilla EXP3 often fails to achieve a sublinear regret even though it achieves $O ( \sqrt { T } )$ regret in expectation. This large variance makes it difficult to reliably evaluate the performance of the algorithms. To stabilize the algorithms, we replaced EXP3 with its variation,√ EXP3P [Auer et al., 2003], which achieves $O ( \sqrt { T } )$ regret with high probability. Note that this change does not harm our expected convergence bounds.
+
+Step sizes. The choice of step sizes is crucial to the practical performance of first-order methods. We found that the decreasing step size $\eta _ { \theta , t } \sim 1 / \sqrt { t }$ for $\theta _ { t }$ and the fixed step size $\eta _ { q } \sim 1 / \sqrt { T }$ for $q _ { t }$ gave the best results. More precisely, we set $\begin{array} { r } { \eta _ { \theta , t } = \frac { C _ { \theta } D } { \sqrt { t } } } \end{array}$ $( t \in [ T ] )$ and $\begin{array} { r } { \eta _ { q } = C _ { q } \sqrt { \frac { \log m } { m T } } } \end{array}$ , where $C _ { \theta } \in [ 0 . 1 , 5 . 0 ]$ and $C _ { q } \in [ 0 . 1 , 3 . 0 ]$ are hyper-parameters tuned for each algorithm. We used the best hyper-parameter found by Optuna [Akiba et al., 2019] for the shown results.
+
+Mini-batch. The use of mini-batch often improves the stability of stochastic gradient algorithms. In our experiments, we used mini-batches of size 10 to evaluate stochastic gradients. Neither the objective values of outputs nor the stability was improved with larger mini-batch sizes. The group DRO objective is evaluated using the entire dataset.
+
+Initialization. For both datasets, we initialized the algorithms with $\theta _ { 1 } = \mathbf { 0 }$ .
+
+# 5.2 Results
+
+We show the results of our experiment in Figures 1 and 2.
+
+Adult dataset. In Figure 1, we plot the optimality gap of the averaged iterate $\textstyle { \frac { 1 } { T } } \sum _ { t = 1 } ^ { T } \theta _ { t }$ against the number of iteration . We observe that all the algorithms converge with a rate roughly $T ^ { - 0 . 5 }$ for both loss functions, consistent with our convergence bound. Furthermore, our algorithms (GDRO-EXP3 and GDRO-TINF) achieve faster convergence compared to the algorithm by Sagawa et al. [2020]. Interestingly, GDRO-TINF achieves a $1 \overline { { 0 } } ^ { - 4 }$ optimality gap in $T = 1 0 ^ { 6 }$ iterations, which is faster than the theoretical $T ^ { - 0 . 5 }$ rate in Theorem 3.
+
+Synthetic dataset. In Figure 2, we plot the objective values of the averaged iterate against the number of iterations. For all the values of $m$ , our algorithms (especially GDRO-EXP3) consistently achieve smaller loss values faster than the known algorithm. The performance gap between our algorithms and the known algorithm increased as $m$ grows, which verifies that our algorithms have better dependence on $m$ in the convergence rate.
+
+09 References   
+10 Alekh Agarwal, Peter L. Bartlett, Pradeep Ravikumar, and Martin J. Wainwright. Informationtheoretic lower bounds on the oracle complexity of stochastic convex optimization. IEEE Transactions on Information Theory, pages 3235–3249, 2012.   
+13 Takuya Akiba, Shotaro Sano, Toshihiko Yanase, Takeru Ohta, and Masanori Koyama. Optuna: A next-generation hyperparameter optimization framework. In Proceedings of the 25rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 2019.   
+316 Peter Auer, Nicolò Cesa-Bianchi, Yoav Freund, and Robert E. Schapire. The nonstochastic multiarmed bandit problem. SIAM Journal on Computing, 32(1):48–77, 2003.   
+18 Yujia Bao, Shiyu Chang, and Regina Barzilay. Predict then interpolate: A simple algorithm to learn stable classifiers. In Proceedings of the 38th International Conference on Machine Learning, volume 139, pages 640–650, 2021. Dimitris Bertsimas, Vishal Gupta, and Nathan Kallus. Data-driven robust optimization. Mathematical Programming, 167(2):235–292, 2018. Jose Blanchet, Yang Kang, and Karthyek Murthy. Robust wasserstein profile inference and applications to machine learning. Journal of Applied Probability, 56(3):830–857, 2019.   
+25 Joy Buolamwini and Timnit Gebru. Gender shades: Intersectional accuracy disparities in commercial gender classification. In Proceedings of the 1st Conference on Fairness, Accountability and Transparency, pages 77–91, 2018. Yair Carmon, Arun Jambulapati, Yujia Jin, and Aaron Sidford. Thinking inside the ball: Near-optimal minimization of the maximal loss. In Proceedings of 34th Conference on Learning Theory, volume   
+134 of Proceedings of Machine Learning Research, pages 866–882, 2021.   
+31 Nicolo Cesa-Bianchi and Gabor Lugosi. Prediction, Learning, and Games. Cambridge University Press, 2006.   
+33 Sebastian Curi, Kfir Y. Levy, Stefanie Jegelka, and Andreas Krause. Adaptive sampling for stochastic risk-averse learning. In Advances in Neural Information Processing Systems, pages 1036–1047,   
+2020.   
+36 Emily Diana, Wesley Gill, Ira Globus-Harris, Michael Kearns, Aaron Roth, and Saeed SharifiMalvajerdi. Lexicographically fair learning: Algorithms and generalization. In Proceedings of the   
+2nd Symposium on Foundations of Responsible Computing, pages 6:1–6:23, 2021.   
+39 Dheeru Dua and Casey Graff. UCI machine learning repository, 2017.   
+40 John C. Duchi and Hongseok Namkoong. Learning models with uniform performance via distributionally robust optimization. The Annals of Statistics, 49(3):1378 – 1406, 2021.   
+42 Peyman Mohajerin Esfahani and Daniel Kuhn. Data-driven distributionally robust optimization using the wasserstein metric: Performance guarantees and tractable reformulations. Mathematical Programming, 171(1):115–166, 2018.   
+45 Joel Goh and Melvyn Sim. Distributionally robust optimization and its tractable approximations. Operations Research, 58(4-part-1):902–917, 2010.   
+47 Tatsunori Hashimoto, Megha Srivastava, Hongseok Namkoong, and Percy Liang. Fairness without demographics in repeated loss minimization. In Proceedings of the 35th International Conference on Machine Learning, pages 1929–1938, 2018.   
+50 Elad Hazan. Introduction to Online Convex Optimization. 2016.   
+51 Dirk Hovy and Anders Søgaard. Tagging performance correlates with author age. In Proceedings of the 53rd Annual Meeting of the Association for Computational Linguistics and the 7th International Joint Conference on Natural Language Processing, pages 483–488, 2015.
+
+354 Weihua Hu, Gang Niu, Issei Sato, and Masashi Sugiyama. Does distributionally robust supervised learning give robust classifiers? In Proceedings of the 35th International Conference on Machine Learning, pages 2029–2037, 2018. Jikai Jin, Bohang Zhang, Haiyang Wang, and Liwei Wang. Non-convex distributionally robust optimization: Non-asymptotic analysis. In Advances in Neural Information Processing Systems, volume 34, pages 2771–2782, 2021. David Jurgens, Yulia Tsvetkov, and Dan Jurafsky. Incorporating dialectal variability for socially equitable language identification. In Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics, pages 51–57, 2017. Johannes Kirschner, Ilija Bogunovic, Stefanie Jegelka, and Andreas Krause. Distributionally robust bayesian optimization. In Proceedings of the 33rd International Conference on Artificial Intelligence and Statistics, pages 2174–2184, 2020. Tor Lattimore and Csaba Szepesvári. Bandit algorithms. Cambridge University Press, 2020. Cong Han Lim and Stephen J. Wright. Efficient bregman projections onto the permutahedron and related polytopes. In Proceedings of the 19th International Conference on Artificial Intelligence and Statistics, pages 1205–1213, 2016. Natalia L Martinez, Martin A Bertran, Afroditi Papadaki, Miguel Rodrigues, and Guillermo Sapiro. Blind pareto fairness and subgroup robustness. In Proceedings of the 38th International Conference on Machine Learning, pages 7492–7501, 2021. Hongseok Namkoong and John C Duchi. Stochastic gradient methods for distributionally robust optimization with $f$ -divergences. In Advances in Neural Information Processing Systems, 2016. Yonatan Oren, Shiori Sagawa, Tatsunori Hashimoto, and Percy Liang. Distributionally robust language modeling. In Proceedings of the Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLPIJCNLP), pages 4227–4237, 2019. Qi Qi, Zhishuai Guo, Yi Xu, Rong Jin, and Tianbao Yang. An online method for a class of distributionally robust optimization with non-convex objectives. In Advances in Neural Information Processing Systems, volume 34, pages 10067–10080, 2021. Alexander Rakhlin and Karthik Sridharan. Optimization, learning, and games with predictable sequences. In Advances in Neural Information Processing Systems, 2013. Shiori Sagawa, Pang Wei Koh, Tatsunori B. Hashimoto, and Percy Liang. Distributionally robust neural networks for group shifts: On the importance of regularization for worst-case generalization. In The 8th International Conference on Learning Representations, 2020. Herbert Scarf. A min-max solution of an inventory problem. Studies in the mathematical theory of inventory and production, 1958. Matthew Staib and Stefanie Jegelka. Distributionally robust optimization and generalization in kernel methods. In Advances in Neural Information Processing Systems, 2019. Matthew Staib, Bryan Wilder, and Stefanie Jegelka. Distributionally robust submodular maximization. In Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics, pages 506–516, 2019. Robert Williamson and Aditya Menon. Fairness risk measures. In Proceedings of the 36th International Conference on Machine Learning, pages 6786–6797, 2019. Jingzhao Zhang, Aditya Krishna Menon, Andreas Veit, Srinadh Bhojanapalli, Sanjiv Kumar, and Suvrit Sra. Coping with label shift via distributionally robust optimisation. In The 9th International Conference on Learning Representations, 2021. Julian Zimmert and Yevgeny Seldin. Tsallis-inf: An optimal algorithm for stochastic and adversarial bandits. Journal of Machine Learning Research, 22(28):1–49, 2021.
+
+Martin Zinkevich. Online convex programming and generalized infinitesimal gradient ascent. In Proceedings of the 20th International Conference on International Conference on Machine Learning, pages 928–935, 2003.
+
+# 404 Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [No]   
+(c) Did you discuss any potential negative societal impacts of your work? [No] This paper is a theoretical paper.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Assumption 1.   
+(b) Did you include complete proofs of all theoretical results? [Yes] Ommited Proof can be found in the supplemental material.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] The experiment code can be found in the supplemental material.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section 5.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No]   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes]
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes]   
+(b) Did you mention the license of the assets? [Yes]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [No]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [No] We used a public dataset.   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [No] Not applicable   
+(b) DidNo describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [No]   
+(c) DidNo include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [No]

parse/dev/Ms6QZafNv01/Ms6QZafNv01_content_list.json

@@ -0,0 +1,1505 @@
+[
+    {
+        "type": "text",
+        "text": "Optimal algorithms for group distributionally robust optimization and beyond ",
+        "text_level": 1,
+        "bbox": [
+            179,
+            122,
+            818,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
+        "bbox": [
+            423,
+            226,
+            580,
+            281
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            318,
+            535,
+            334
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Distributionally robust optimization (DRO) can improve the robustness and fairness   \n2 of learning methods. In this paper, we devise stochastic algorithms for a class   \n3 of DRO problems including group DRO, subpopulation fairness, and empirical   \n4 conditional value at risk (CVaR) optimization. Our new algorithms achieve faster   \n5 convergence rates than existing algorithms for multiple DRO settings. We also   \n6 provide a new information-theoretic lower bound that implies our bounds are tight   \n7 for group DRO. Empirically, too, our algorithms outperform known methods. ",
+        "bbox": [
+            150,
+            347,
+            766,
+            445
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "8 1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            151,
+            468,
+            312,
+            486
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "9 Commonly, machine learning models are trained to optimize the average performance. However,   \n10 such models may not perform equally well among all demographic subgroups due to a hidden bias in   \n11 the training set or distribution shift in training and test phases [Hovy and Søgaard, 2015; Hashimoto   \n12 et al., 2018; Martinez et al., 2021; Duchi and Namkoong, 2021]. Biases in datasets are also directly   \n13 related to fairness concerns in machine learning [Buolamwini and Gebru, 2018; Jurgens et al., 2017].   \n14 Recently, various algorithms based on distributionally robust optimization (DRO) have been proposed   \n15 to address these problems [Hovy and Søgaard, 2015; Hashimoto et al., 2018; Hu et al., 2018; Oren et   \n16 al., 2019; Williamson and Menon, 2019; Sagawa et al., 2020; Curi et al., 2020; Zhang et al., 2021;   \n17 Martinez et al., 2021; Duchi and Namkoong, 2021]. However, these algorithms are often highly   \n18 tailored to each specific DRO formulation. Furthermore, it is often unclear whether these proposed   \n19 algorithms are optimal in terms of the convergence rate. Are there a unified algorithmic methodology   \n20 and a lower bound for these problems?   \n21 Contributions. In this paper, we study a general class of DRO problems, which includes group   \n22 DRO [Hu et al., 2018; Oren et al., 2019; Sagawa et al., 2020], subpopulation fairness [Martinez et   \n23 al., 2021], conditional value at risk (CVaR) optimization [Curi et al., 2020], and many others. Let   \n24 $\\boldsymbol \\Theta \\subseteq \\mathbb { R } ^ { n }$ be a convex set of model parameters and $\\ell ( \\theta ; z ) : \\Theta \\to \\mathbb { R } _ { + }$ be a convex loss of the model   \n25 with parameter $\\theta$ with respect to data point $z$ . The data point $z$ may be drawn from one out of $m$   \n26 distributions $P _ { 1 } , \\ldots , P _ { m }$ which are accessible via a stochastic oracle that returns an i.i.d. sample   \n27 $z \\sim P _ { i }$ . Let $Q$ be a convex subset of the probability simplex in $\\mathbb { R } ^ { m }$ that contains the uniform vector,   \n28 i.e., $( 1 / m , \\ldots , 1 / m ) \\in Q$ . Our DRO formulation is as follows: ",
+        "bbox": [
+            148,
+            500,
+            826,
+            569
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            575,
+            825,
+            672
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            686,
+            825,
+            797
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4ac8a19812285ddf0e6d9bef2b65202c5ab1c357b9cb2a65dc49e058fb27cf94.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\theta \\in \\Theta } \\operatorname* { m a x } _ { q \\in Q } \\sum _ { i = 1 } ^ { m } q _ { i } \\operatorname* { \\bf { E } } _ { z \\sim P _ { i } } [ \\ell ( \\theta ; z ) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            400,
+            801,
+            598,
+            843
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "29 If $Q$ are the probability simplex and scaled $k$ -set polytope, we can recover group DRO [Sagawa et   \n30 al., 2020] and subpopulation fairness [Martinez et al., 2021], respectively. Moreover, we formulate   \n31 a new, more general fairness concept based on weighted rankings with $Q$ being a permutahedron,   \n32 which includes these special cases; see Section 2 for details. ",
+        "bbox": [
+            147,
+            847,
+            825,
+            902
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Submitted to 36th Conference on Neural Information Processing Systems (NeurIPS 2022). Do not distribute. ",
+        "bbox": [
+            166,
+            922,
+            815,
+            936
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "table",
+        "img_path": "images/1e10e696530b92b7f12f76f28365fe4f6cfd558647b4d433d1c309e6753bff4f.jpg",
+        "table_caption": [
+            "Table 1: Summary of convergence results for group DRO. Here, $m$ denotes the number of groups, $n$ the dimension of $\\theta$ , $G$ the Lipschitz constant of loss function $\\ell$ , $D$ the diameter of feasible set $\\Theta$ , $M$ the range of loss function $\\ell$ , and $T$ the number of calls to stochastic oracle. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>reference</td><td>convergence rate E[εr]</td><td>iteration complexity</td><td>lower bound</td><td></td></tr><tr><td>[Sagawa et al., 2020]</td><td>0（m</td><td>G2 D²+M2 log m T</td><td>O(m + n) + proj. onto Θ</td><td rowspan=\"3\">( G²D²+M2m T (Theorem 5)</td></tr><tr><td>Ours (Theorem 2)</td><td></td><td>T</td><td>O(m + n) + proj. onto Θ</td></tr><tr><td>Ours (Theorem 3)</td><td>0（√</td><td>G²D²+M²m） T</td><td>O(m + n) + proj. onto Θ + solving scalar equation</td></tr></table>",
+        "bbox": [
+            173,
+            137,
+            849,
+            250
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "33 For our general DRO, we devise an efficient stochastic gradient algorithm. Furthermore, we show that   \n34 it achieves the information-theoretic optimal convergence rate for group DRO. Our main technical   \n35 contributions are as follows; ",
+        "bbox": [
+            147,
+            273,
+            826,
+            316
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We provide a generic stochastic gradient algorithm for our general DRO. By specializing it in the group DRO setting, we provide two algorithms (GDRO-EXP3 and GDRO-TINF)√ that improve the rate of Sagawa et al. [2020] by a factor of $\\Omega ( { \\sqrt { m } } )$ with the almost same complexity per iteration; see Table 1. Furthermore, our generic algorithm can be specialized to improve the convergence rate of Curi et al. [2020] for subpopulation fairness (a.k.a. empirical CVaR optimization). Finally, we show that our algorithm runs efficiently if $Q$ is a permutahedron, which includes all aforementioned subclasses.   \n• We prove a matching information-theoretic lower bound for the convergence rate of group DRO. This implies that no algorithm can improve the convergence rate of GDRO-TINF (up to a constant factor). To the best of our knowledge, this is the first information-theoretic lower bound for group DRO.   \n• Our experiments on real-world and synthetic datasets show that our algorithms also empirically outperform the known algorithm, supporting our theoretical analysis. ",
+        "bbox": [
+            210,
+            330,
+            825,
+            523
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "49 1.1 Our techniques ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            540,
+            320,
+            555
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "50 Algorithms. The core idea of our algorithms is stochastic no-regret dynamics [Hazan, 2016]. We   \n51 regard DRO (1) as a two-player zero-sum game between a player who picks $\\theta \\in \\Theta$ and another player   \n52 who picks $q \\in Q$ . The two players iteratively update their solution using online learning algorithms;   \n53 in particular, we will use online gradient descent (OGD) [Zinkevich, 2003] and online mirror descent   \n54 (OMD) [Cesa-Bianchi and Lugosi, 2006] for the $\\theta$ -player and $q$ -player, respectively. In addition, we   \n55 need to estimate gradients for both players, since the objective function of our DRO is stochastic and   \n56 we cannot obtain exact gradients.   \n57 The convergence rate of stochastic no-regret dynamics depends on the expected regret of OGD and   \n58 OMD. To obtain the optimal convergence rate, we must carefully choose the regularizer in OMD as   \n59 well as gradient estimators, exploiting the structure of our DRO. In particular, we need to balance   \n60 the variance of gradient estimators and the diameter terms in both OGD and OMD. This is the most   \n61 challenging part of the algorithm design. Inspired by adversarial multi-armed bandit algorithms,   \n62 we design gradient estimators for no-regret dynamics of OGD and OMD in our DRO. Indeed, our   \n63 algorithms for group DRO (GDRO-EXP3 and GDRO-TINF) are based on adversarial multi-armed   \n64 bandit algorithms, EXP3 [Auer et al., 2003] and Tsallis-INF [Zimmert and Seldin, 2021], respectively,   \n65 hence the name. Although each building block (OGD, OMD, and gradient estimators) is fairly known   \n66 in the literature, we need to put them together in the right combination to obtain the optimal rate.   \n67 Lower bound. For the lower bound, we carefully design a family of group DRO instances for which   \n68 any algorithm requires a certain number of queries to achieve a good objective value. To bound   \n69 the number of queries, we use information-theoretic tools such as Le Cam’s lemma and bound the   \n70 Kullback-Leibler divergence between Bernoulli distributions. Such tools are also used at the heart of   \n71 lower bounds for stochastic convex optimization [Agarwal et al., 2012] and adversarial multi-armed   \n72 bandits [Auer et al., 2003], but the connection to those settings is much more subtle here, and our   \n73 construction is specifically designed for group DRO-type problems.   \n75 DRO is a wide field ranging from robust optimization to machine learning and statistics [Goh and Sim,   \n76 2010; Bertsimas et al., 2018], whose original idea dates back to Scarf [1958]. Popular choices of the   \n77 uncertainty set in DRO include balls around an empirical distribution in Wasserstein distance [Esfahani   \n78 and Kuhn, 2018; Blanchet et al., 2019], $f$ -divergence [Namkoong and Duchi, 2016; Duchi and   \n79 Namkoong, 2021], $\\chi ^ { 2 }$ -divergence [Staib et al., 2019], and maximum mean discrepancy [Staib and   \n80 Jegelka, 2019; Kirschner et al., 2020].   \n81 DRO algorithms have been mainly studied for the offline setting, i.e., algorithms can access all data   \n82 points of the empirical distribution. Note that our DRO is not offline because the group distributions   \n83 are given by the stochastic oracles. Namkoong and Duchi [2016] proposed stochastic gradient   \n84 algorithms for offline DRO with $f$ -divergence uncertainty sets. Curi et al. [2020] used no-regret   \n85 dynamics for empirical CVaR minimization. Their algorithm invokes sampling from $k$ -DPP in each   \n86 iteration, which is more computationally demanding than our algorithm. Furthermore, our algorithm   \n87 gets rid of an $O ( \\log m )$ factor in the convergence rate using the Tsallis entropy regularizer; see   \n88 Theorem 4. Qi et al. [2021]; Jin et al. [2021] devised stochastic gradient algorithms for several DRO   \n89 with non-convex losses.   \n90 Agarwal et al. [2012] gave a lower bound for stochastic convex optimization, which is a special case   \n91 of our DRO with only one distribution. Recently, Carmon et al. [2021] showed a lower bound for   \n92 minimax problem $\\begin{array} { r } { \\operatorname* { m i n } _ { x } \\operatorname* { m a x } _ { i = 1 } ^ { m } f _ { i } ( x ) } \\end{array}$ for non-stochastic Lipschitz convex $f _ { i }$ . Our lower bound deals   \n93 with the stochastic functions, so this result does not apply.   \n94 In this paper, we assume that the group information is given in advance. However, the group   \n95 information might not be easy to define in practice. Bao et al. [2021] propose a simple method to   \n96 define groups for classification problems based on mistakes of models in the training phase. Their   \n97 method often generates group DRO instances with large $m$ . Our algorithms are more efficient for   \n98 such group DRO thanks to the better dependence on $m$ in the convergence rate.   \n99 No-regret dynamics is a well-studied method for solving two-player zero-sum games [Cesa-Bianchi   \n100 and Lugosi, 2006]. For non-stochastic convex-concave games, one can achieve $O ( 1 / T )$ convergence   \n101 via predictable sequences [Rakhlin and Sridharan, 2013]. This result does not apply to our setting   \n102 because our DRO is a stochastic game.   \n103 Notations. Throughout the paper, $m$ denotes the number of distributions (groups) and $n$ denotes   \n104 the dimension of a variable $\\theta$ . For a positive integer $m$ , we write $[ m ] : = \\{ 1 , \\ldots , m \\}$ . The orthogonal   \n105 projection onto set $\\Theta$ is denoted by $\\mathrm { p r o j } _ { \\Theta }$ . The ith standard unit vector is denoted by $\\mathbf { e } _ { i }$ and the   \n106 all-one vector is denoted by 1. The probability simplex in $\\mathbb { R } ^ { m }$ is denoted by $\\Delta _ { m }$ . ",
+        "bbox": [
+            147,
+            566,
+            825,
+            662
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            145,
+            670,
+            825,
+            809
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            814,
+            825,
+            911
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            116,
+            825,
+            200
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            145,
+            205,
+            825,
+            330
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            337,
+            825,
+            393
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            145,
+            398,
+            825,
+            468
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            142,
+            474,
+            825,
+            530
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            545,
+            825,
+            602
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "107 2 Examples contained in our general DRO ",
+        "text_level": 1,
+        "bbox": [
+            143,
+            619,
+            544,
+            637
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "108 In this section, we show how several DRO formulations in the literature can be phrased in our general   \n109 DRO formulation (1). In addition, we propose a novel fairness constraint based on weighted rankings   \n110 using our general DRO.   \n111 Group DRO. When $Q$ equals the probablility simplex, we obtain group DRO [Hu et al., 2018;   \n112 Oren et al., 2019; Sagawa et al., 2020]: ",
+        "bbox": [
+            142,
+            650,
+            825,
+            693
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            705,
+            826,
+            736
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/432f41cd20819cb825447ee2dfe7667e9ba109d770e71f8e19c951e955993f7e.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\theta \\in \\Theta } \\operatorname* { m a x } _ { i = 1 } ^ { m } \\ \\underset { z \\sim P _ { i } } { \\mathbf { E } } [ \\ell ( \\theta ; z ) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            418,
+            741,
+            578,
+            767
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "113 That is, group DRO aims to minimize the expected loss in the worst group, thereby ensuring better   \n114 performance across all groups.   \n115 Empirical CVaR, Subpopulation fairness, Average top- $k$ worst group loss. Group DRO may   \n116 yield overly pessimistic solutions. For instance, the groups might be automatically generated by other   \n117 algorithms (such as one in Bao et al. [2021]) and there might exist a few “outlier” groups that make   \n118 the group DRO objective trivial. ",
+        "bbox": [
+            140,
+            771,
+            828,
+            801
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            142,
+            814,
+            825,
+            869
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "119 For such a case, we can restrict $Q$ to a small subset of the probability simplex so that the solution cannot put large weights on a few outlier groups. Especially, let 120 $Q = \\left\\{ q \\in \\Delta _ { m } : 0 \\leq q _ { i } \\leq { \\frac { 1 } { p m } } \\right\\}$ for ",
+        "bbox": [
+            142,
+            876,
+            825,
+            914
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "121 some parameter $p \\in ( 0 , 1 )$ , i.e., $Q$ is a scaled $k$ -set polytope. The intuition behind the choice of $Q$   \n122 is that, by limiting the largest entry of $q$ to $1 / p m$ , DRO would optimize the expected loss over the   \n123 worst $p$ -fraction subgroups of $m$ groups. Therefore, if the fraction of outlier groups is sufficiently   \n124 small compared to $p$ , then $p$ -fraction subgroups must contain “inlier” groups as well. Therefore, it is   \n125 likely that DRO with $Q$ finds solutions more robust than group DRO.   \n126 When $P _ { i }$ is the Dirac measure of data $z _ { i }$ , then the resulting DRO is empirical CVaR optimization [Curi   \n127 et al., 2020]. In the fairness context, the same problem is called subpopulation fairness [Williamson   \n128 and Menon, 2019; Martinez et al., 2021; Duchi and Namkoong, 2021].   \n129 If $p \\ = \\ m / k$ for some positive integer $k$ , the resulting DRO is the average top- $k$ worst group   \n130 loss [Zhang et al., 2021]: ",
+        "bbox": [
+            140,
+            90,
+            825,
+            161
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            145,
+            166,
+            826,
+            210
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            214,
+            825,
+            243
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0dd149f95fe20e781d725c891f16aeaa2b791d47de642552ebddeff30d5039a7.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\theta \\in \\Theta } \\frac { 1 } { k } \\sum _ { i = 1 } ^ { k } L _ { i } ^ { \\downarrow } ( \\theta ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            437,
+            241,
+            557,
+            285
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where 131 $L _ { i } ^ { \\downarrow } ( \\theta )$ denotes the the $i$ th largest population group loss of $\\theta$ . More precisely, let $L _ { i } ( \\theta ) =$ 132 $\\mathbf { E } _ { z \\sim P _ { i } } [ \\ell ( \\theta ; z ) ]$ for $i \\in [ m ]$ and sort them in the non-increasing order: $L _ { 1 } ^ { \\downarrow } ( \\theta ) \\ge \\cdots \\ge L _ { m } ^ { \\downarrow } ( \\theta )$ . ",
+        "bbox": [
+            142,
+            286,
+            825,
+            320
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "133 Weighted ranking of group losses. The aforementioned DRO formulations are special cases of the   \n134 following DRO, which we call the weighted ranking of group losses. Let $\\alpha \\in \\Delta ^ { m }$ be a fixed vector   \n135 with non-increasing entries. Let $Q$ be the permutahedron of $\\alpha$ , the convex hull of $( \\alpha _ { \\sigma ( 1 ) } , \\ldots , \\alpha _ { \\sigma ( m ) } )$   \n136 for all permutations $\\sigma$ of $[ m ]$ . Then, the resulting DRO is ",
+        "bbox": [
+            142,
+            332,
+            826,
+            390
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4a749bcd2c7c3fc817d70a8872df3c64f318e02898d7f0ca73809182f9881cbe.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\theta \\in \\Theta } \\sum _ { i = 1 } ^ { m } \\alpha _ { i } L _ { i } ^ { \\downarrow } ( \\theta ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            437,
+            392,
+            558,
+            433
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "137 Group DRO corresponds to $\\alpha = ( 1 , 0 , \\ldots , 0 )$ and the average top- $k$ worst group losses corresponds 138 to $\\alpha \\overset { \\cdot } { = } ( 1 / k , \\ldots , \\bar { 1 ^ { \\prime } } k , 0 , \\ldots , 0 )$ . Another example that is contained in none of the above examples is {zk times ",
+        "bbox": [
+            142,
+            434,
+            826,
+            479
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "139 lexicographic minimax fairness [Diana et al., 2021]. The goal of lexicographical minimax fairness   \n140 is to find $\\theta \\in \\Theta$ such that the sequence $( L _ { 1 } ^ { \\downarrow } ( \\theta ) , \\dots , L _ { m } ^ { \\downarrow } ( \\theta ) )$ is lexicographically minimum. This   \n141 corresponds to $\\alpha$ with sufficiently varied entries, i.e., $\\alpha _ { 1 } \\gg \\alpha _ { 2 } \\gg \\cdot \\cdot \\cdot \\gg \\alpha _ { m }$ . ",
+        "bbox": [
+            142,
+            481,
+            825,
+            525
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "142 3 Algorithms ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            541,
+            300,
+            559
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "143 In this section, we describe our algorithms. First, we present a generic algorithm for our general   \n144 DRO (1) and provide a unified convergence analysis in Section 3.1. Then, we specialize it into two   \n145 concrete algorithms for group DRO (2) in Section 3.2. We sketch algorithms for the average of top- $k$   \n146 group losses and weighted ranking of group loss in Section 3.3. ",
+        "bbox": [
+            142,
+            571,
+            825,
+            628
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "7 3.1 Algorithm for the general case ",
+        "text_level": 1,
+        "bbox": [
+            156,
+            643,
+            424,
+            659
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "148 We present our algorithm for geranal DRO (1). At a high level, our algorithm can be regarded as   \n149 stochastic no-regret dynamics. Let us denote $\\begin{array} { r } { L ( \\boldsymbol { \\theta } , \\boldsymbol { q } ) : = \\sum _ { i = 1 } ^ { m } q _ { i } \\mathbf { E } _ { z \\sim P _ { i } } [ \\ell ( \\boldsymbol { \\theta } ; z ) ] } \\end{array}$ . Imagine that the   \n150 $\\theta$ -player and $q$ -player run online algorithms $\\scriptstyle A _ { \\theta }$ and $A _ { q }$ , respectively, to solve the minimax problem   \n151 $\\begin{array} { r } { \\operatorname* { m i n } _ { \\theta \\in \\Theta } \\operatorname* { m a x } _ { q \\in Q } L ( \\theta , q ) } \\end{array}$ . That is, for $t = 1 , \\dots , T$ , ",
+        "bbox": [
+            143,
+            667,
+            826,
+            726
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "• $\\theta _ { t } \\in \\Theta$ and $q _ { t } \\in Q$ are determined by $\\scriptstyle A _ { \\theta }$ and $A _ { q }$ , respectively.   \n• Both players feed gradient estimators $\\hat { \\nabla } _ { \\theta , t }$ and $\\hat { \\nabla } _ { \\boldsymbol { q } , t }$ to $\\scriptstyle A _ { \\theta }$ and $A _ { q }$ , respectively. Here, $\\mathbf { E } [ \\hat { \\nabla } _ { \\boldsymbol { \\theta } , t } ] = \\nabla _ { \\boldsymbol { \\theta } } L ( \\theta _ { t } , q _ { t } )$ and $\\mathbf { E } [ \\hat { \\nabla } _ { q , t } ] = \\nabla _ { q } L ( \\theta _ { t } , q _ { t } )$ . ",
+        "bbox": [
+            215,
+            733,
+            825,
+            787
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "155 Let ",
+        "bbox": [
+            142,
+            796,
+            199,
+            809
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/02de1321aa0aa2435026ceb46cea193d96109e226340dac667c46884a0209143.jpg",
+        "text": "$$\n\\varepsilon _ { T } : = \\operatorname* { m a x } _ { q \\in Q } L ( \\bar { \\theta } _ { 1 : T } , q ) - \\operatorname* { m i n } _ { \\theta \\in \\Theta } \\operatorname* { m a x } _ { q \\in Q } L ( \\theta , q )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            359,
+            806,
+            637,
+            832
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "be the optimality156 convergence rate of the averavia regrets iterand $\\begin{array} { r } { \\bar { \\theta } _ { 1 : T } = \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\theta _ { t } } \\end{array}$ . We can bound the eorithms (see Appendix ectedfor a $\\mathbf { E } [ \\varepsilon _ { T } ]$ $R _ { \\theta }$ $R _ { q }$ $\\mathbf { A }$ 158 formal definition), i.e., ",
+        "bbox": [
+            140,
+            834,
+            825,
+            877
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bad6c8e038a2753a918f2694ce98e383faefe0b5ddc6596a324c9f7049de7990.jpg",
+        "text": "$$\n\\mathbf { E } [ \\varepsilon _ { T } ] \\leq \\frac { \\mathbf { E } [ R _ { \\theta } ( T ) ] + \\mathbf { E } [ R _ { q } ( T ) ] } { T } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            387,
+            878,
+            612,
+            910
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "159 We can obtain hence the convergence rate of the above algorithms by investigating the expected regret   \n160 bounds of these online algorithms.   \n161 To get a concrete algorithm, we must specify the online algorithms $A _ { \\theta } , A _ { q }$ as well as the gradient   \n162 estimators $\\hat { \\nabla } _ { \\boldsymbol { \\theta } , t } , \\hat { \\nabla } _ { \\boldsymbol { q } , t }$ . We use OGD and OMD as $\\scriptstyle A _ { \\theta }$ and $A _ { q }$ , respectively. We construct the gradient   \n163 estimators by sampling $i _ { t } \\sim q _ { t }$ and $z \\sim P _ { i _ { t } }$ and setting $\\hat { \\nabla } _ { \\boldsymbol { \\theta } , t } = \\nabla _ { \\boldsymbol { \\theta } } \\ell ( \\theta _ { t } ; z )$ and $\\begin{array} { r } { \\hat { \\nabla } _ { q , t } = \\frac { \\ell ( \\theta _ { t } ; z ) } { q _ { t , i _ { t } } } \\mathbf { e } _ { i _ { t } } } \\end{array}$   \n164 This leads to Algorithm 1. There, $\\Psi : Q  \\mathbb { R }$ denotes the regularizer of OMD and $\\eta _ { \\theta , t }$ and $\\eta _ { q }$   \n165 denote the step sizes of OGD and OMD, respectively. 1 It turns out that this combination of online   \n166 algorithms and gradient estimators yields the best convergence rate (for group DRO) because the   \n167 expected regrets of both players are optimal. ",
+        "bbox": [
+            147,
+            90,
+            825,
+            119
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            125,
+            825,
+            234
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 Algorithm for general DRO (1) ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            250,
+            467,
+            265
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Require: initial solution $\\theta _ { 1 } \\in \\Theta$ , number of iterations $T$ , step sizes $\\eta _ { \\theta , t } > 0 ( t \\in [ T ] ) , \\eta _ { q } > 0$ , and a strictly convex function $\\Psi : Q  \\mathbb { R }$ .   \n1: Let $q _ { 1 } \\dot { = } ( 1 / m , \\dots , 1 / m )$ .   \n2: for $t = 1 , \\dots , T$ do   \n3: Sample $i _ { t } \\sim q _ { t }$ .   \n4: Call the stochastic oracle to obtain $z \\sim P _ { i _ { t } }$ .   \n5: $\\theta _ { t + 1 }  \\mathrm { p r o j } _ { \\Theta } ( \\theta _ { t } - \\eta _ { \\theta , t } \\nabla _ { \\theta } \\ell ( \\theta _ { t } ; z ) )$   \n6: $\\begin{array} { r l r } { \\nabla \\Psi ( \\widetilde { q } _ { t + 1 } ) } & { { }  } & { \\nabla \\Psi ( q _ { t } ) \\ - \\ \\frac { \\eta _ { q } } { q _ { t , i _ { t } } } \\ell ( \\theta _ { t } ; z ) \\mathbf { e } _ { i _ { t } } } \\end{array}$ ; $\\begin{array} { r l r } { q _ { t + 1 } } & { { }  } & { \\arg \\operatorname* { m i n } _ { q \\in Q } D _ { \\Psi } ( q , \\tilde { q } _ { t + 1 } ) } \\end{array}$ , where $D _ { \\Psi } ( x , y ) = \\Psi ( x ) - \\Psi ( y ) - \\nabla \\Psi ( x ) ^ { \\top } ( y - x )$ is the Bregman divergence with respect to $\\Psi$ .   \n7: return $\\textstyle { \\frac { 1 } { T } } \\sum _ { t = 1 } ^ { T } \\theta _ { t }$ . ",
+        "bbox": [
+            176,
+            270,
+            825,
+            420
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "168 We now analyze the convergence rate of Algorithm 1. We make the following standard assumptions. ",
+        "bbox": [
+            151,
+            438,
+            823,
+            453
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "69 Assumption 1. The loss function $\\ell ( \\theta ; z )$ is continuously differentiable and $G$ -Lipchitz in $\\theta$ , and has   \n170 range $[ 0 , M ]$ for all $z$ . The Euclidean diameter of the feasible region $\\Theta$ is at most $D$ . ",
+        "bbox": [
+            150,
+            457,
+            826,
+            486
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The following theorem follows from plugging regret bounds of OGD and OGD, and the construction 2 of the gradient estimators into (3). ",
+        "bbox": [
+            156,
+            494,
+            826,
+            525
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "73 Theorem 1. If $\\eta _ { \\theta , t }$ is nonincreasing, Algorithm 1 achieves the expected convergence rate ",
+        "bbox": [
+            155,
+            527,
+            764,
+            544
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/698f30d626c7a3567fd4d8136295eb2a2956fe573275c1b4d7509e60a0a9e6f9.jpg",
+        "text": "$$\n\\Im [ \\varepsilon _ { T } ] \\leq \\frac { 1 } { T } \\left( \\frac { G ^ { 2 } } { 2 } \\sum _ { t = 1 } ^ { T } \\eta _ { \\theta , t } + \\frac { D ^ { 2 } } { 2 \\eta _ { \\theta , T } } + \\frac { M ^ { 2 } } { 2 } \\eta _ { \\theta } \\sum _ { t = 1 } ^ { T } \\frac { \\mathbf { F } } { i _ { t } } \\left[ \\frac { ( \\nabla ^ { 2 } \\Psi ( q _ { t } ) ) _ { i _ { t } , i _ { t } } ^ { - 1 } } { q _ { t , i _ { t } } ^ { 2 } } \\right] + \\frac { \\operatorname* { m a x } _ { q ^ { * } \\in Q } D _ { \\Psi } ( q ^ { * } , \\mathbf { 1 } / m ) } { \\eta _ { q } } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            549,
+            836,
+            593
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "174 A formal proof can be found in Appendix B. We will see how specific choices of the regularizer $\\Psi$   \n175 yield various algorithms and convergence rates for group DRO and others in the next subsections. A   \n176 few remarks on the regularizers, step sizes, and projection step are in order.   \n177 Regularizer. Although Algorithm 1 works with general $\\Psi$ , we can choose a specific regularizer for   \n178 $Q$ appearing in applications, e.g, the probability simplex, scaled $k$ -set polytope, or a permutahedron.   \n179 In the next subsections, we show that the entropy regularizer $\\begin{array} { r } { \\Psi ( x ) = \\bar { \\sum _ { i } } ( x _ { i } \\log x _ { i } - \\bar { \\operatorname { x } _ { i } } ) } \\end{array}$ and Tsallis   \n180 entropy regularizer $\\begin{array} { r } { \\Psi ( x ) = 2 ( 1 - \\sum _ { i } \\sqrt { x _ { i } } ) } \\end{array}$ yield efficient algorithms with improved convergence   \n181 rates for these cases.   \n182 Step sizes. The theorem includes decreasing step sizes such as $\\begin{array} { r } { \\eta _ { \\theta , t } = \\frac { D } { m G \\sqrt { t } } } \\end{array}$ in addition to fixed   \n183 step sizes. Decreasing step sizes have the advantage that we do not require the knowledge of $T$   \n184 at the beginning of the algorithm but come at the cost of an extra constant factor in the expected   \n185 convergence rate. Since both step size policies give the asymptotically same convergence rate, we   \n186 describe only fixed step sizes in the theorems in the next subsections. In practice, decreasing step   \n187 sizes stabilize the algorithm and often outperform fixed step sizes. ",
+        "bbox": [
+            140,
+            604,
+            826,
+            648
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            142,
+            661,
+            825,
+            732
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            143,
+            744,
+            825,
+            833
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Projection step. In general, the Bregman projection argminq∈Q $\\mathrm { a r g m i n } _ { q \\in Q } D _ { \\Psi } ( q , \\tilde { q } _ { t + 1 } )$ is convex, but may be costly to compute. For the applications described in Section 2, $\\tilde { Q }$ is a permutahedron. In this case, ",
+        "bbox": [
+            158,
+            847,
+            825,
+            876
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 2 GDRO-EXP3 ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            106,
+            361,
+            121
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Require: initial solution $\\theta _ { 1 } \\in \\Theta$ , number of iterations $T$ , and step sizes $\\eta _ { \\theta , t } > 0 \\left( t \\in [ T ] \\right)$ , $\\eta _ { q } > 0$ .   \n1: Let $q _ { t } = ( 1 / m , \\dots , 1 / m )$ .   \n2: for $t = 1 , \\dots , T$ do   \n3: Sample $i _ { t } \\sim q _ { t }$ .   \n4: Call the stochastic oracle to obtain $z \\sim P _ { i _ { t } }$ 5: θt+1 ← projΘ(θt − ηθ,t∇θ\\`(θt; z))   \n6: q˜t+1 ← qt exp \u0010 ηq\\`(θt;z)eitqt,i \u0011   \n7: qt+1 ← Pq˜t+1q˜t+1,i .   \n8: return 1T PTt=1 θt. ",
+        "bbox": [
+            176,
+            126,
+            495,
+            315
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 3 GDRO-TINF ",
+        "text_level": 1,
+        "bbox": [
+            504,
+            106,
+            687,
+            121
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Require: initial solution $\\theta _ { 1 } \\in \\Theta$ , number of iterations $T$ , and step sizes $\\eta _ { \\theta , t } > 0 \\left( t \\in [ T ] \\right)$ , $\\eta _ { q } > 0$ .   \n1: Let $q _ { t } = ( 1 / m , \\dots , 1 / m )$ .   \n2: for $t = 1 , \\dots , T$ do   \n3: Sample $i _ { t } \\sim q _ { t }$ . 4: Call the stochastic oracle to obtain $z \\sim P _ { i _ { t } }$ .   \n5: $\\begin{array} { r l } & { \\theta _ { t + 1 } \\gets \\mathrm { p r o j } _ { \\Theta } ( \\theta _ { t } - \\eta _ { \\theta , t } \\nabla _ { \\theta } \\ell ( \\theta _ { t } ; z ) ) } \\\\ & { \\widetilde { q } _ { t + 1 } \\gets q _ { t } \\left( \\mathbf { 1 } - \\frac { \\eta _ { q } \\sqrt { q _ { t } } } { q _ { t , i _ { t } } } \\ell ( \\theta _ { t } ; z ) \\mathbf { e } _ { i _ { t } } \\right) ^ { - 2 } } \\\\ & { \\mathrm { C o m p u t e ~ } \\quad \\alpha \\qquad \\in \\qquad \\mathbb { R } \\quad \\mathrm { s u c h } } \\\\ & { \\sum _ { i = 1 } ^ { m } \\left( \\sqrt { \\widetilde { q } _ { t + 1 , i } } - \\alpha \\right) ^ { - 2 } = 1 . } \\\\ & { q _ { t + 1 } \\gets \\left( \\sqrt { \\widetilde { q } _ { t + 1 } } - \\alpha \\mathbf { 1 } \\right) ^ { - 2 } } \\\\ & { \\mathrm { t u r n ~ } \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\theta _ { t } . } \\end{array}$   \n6: 7: that   \n8:   \n9: return 1T PTt=1 θt. ",
+        "bbox": [
+            508,
+            126,
+            826,
+            347
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "190 it is known that the Bregman projection with respect to the entropy and Tsallis entropy regularizers   \n191 can be done in $O ( m \\log m )$ time [Lim and Wright, 2016]. If $Q$ is the probability simplex, we even   \n192 have a closed form for the Bregman projection. ",
+        "bbox": [
+            143,
+            371,
+            825,
+            415
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.2 Algorithms for Group DRO ",
+        "text_level": 1,
+        "bbox": [
+            169,
+            429,
+            406,
+            444
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "As applications of our generic algorithm, we now describe two concrete algorithms for group DRO (2) and their convergence rates. ",
+        "bbox": [
+            163,
+            454,
+            825,
+            483
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "GDRO-EXP3. Let $\\Psi$ be the entropy regularizer, which corresponds to the EXP3 algorithm for 7 the $q$ -player. The resulting algorithm, GDRO-EXP3, is shown in Algorithm 2. The update is in a 8 closed formula and its complexity is $O ( m + n )$ time. The convergence rate follows from Theorem 1. ",
+        "bbox": [
+            156,
+            494,
+            826,
+            537
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "99 Theorem 2. If $\\eta _ { \\theta , t }$ is nonincreasing, GDRO-EXP3 (Algorithm 2) achieves the expected convergence   \n00 rate ",
+        "bbox": [
+            153,
+            540,
+            825,
+            568
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c6d8d8bdce7bc54a19569910ff31c1dd9441b14dbb31482abe00cbb8dc5b63e7.jpg",
+        "text": "$$\n\\mathbf { E } [ \\varepsilon _ { T } ] \\leq { \\frac { 1 } { T } } \\left( { \\frac { G ^ { 2 } } { 2 } } \\sum _ { t = 1 } ^ { T } \\eta _ { \\theta , t } + { \\frac { D ^ { 2 } } { 2 \\eta _ { \\theta , T } } } + { \\frac { m M ^ { 2 } } { 2 } } \\eta _ { q } T + { \\frac { \\log m } { \\eta _ { q } } } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            295,
+            566,
+            702,
+            611
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For 201 $\\begin{array} { r } { \\eta _ { \\theta , t } = \\frac { D } { G \\sqrt { T } } } \\end{array}$ and ηq = $\\begin{array} { r } { \\eta _ { q } = \\sqrt { \\frac { 2 \\log m } { m M ^ { 2 } T } } } \\end{array}$ , we obtain ",
+        "bbox": [
+            140,
+            613,
+            486,
+            637
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e42e5495c988f378729b090676708ac9939b90ddc1268d453baa00a703b724d9.jpg",
+        "text": "$$\n\\mathbf { E } [ \\varepsilon _ { T } ] \\leq { \\sqrt { 2 } } { \\frac { { \\sqrt { G ^ { 2 } D ^ { 2 } + 2 M ^ { 2 } m \\log m } } } { \\sqrt { T } } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            367,
+            638,
+            629,
+            675
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "202 GRDO-TINF. We can further improve the convergence rate using the Tsallis entropy regularizer   \n203 at the cost of a slightly higher iteration complexity. The update of $q _ { t }$ is then ",
+        "bbox": [
+            140,
+            683,
+            825,
+            712
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/313e12a27548be83bb3f3ae64cbf316e083e85d11b35af9a4a09de0e9953264a.jpg",
+        "text": "$$\n\\tilde { q } _ { t + 1 } = q _ { t } \\left( \\mathbf { 1 } - \\frac { \\eta _ { q } \\sqrt { q _ { t } } } { q _ { t , i _ { t } } } \\ell ( \\theta _ { t } ; z ) \\mathbf { e } _ { i _ { t } } \\right) ^ { - 2 } , \\quad q _ { t + 1 } : = \\left( \\sqrt { \\tilde { q } _ { t + 1 } } - \\alpha \\mathbf { 1 } \\right) ^ { - 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            266,
+            712,
+            728,
+            750
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "204 where the multiplication, square-root, and power operations are entry-wise and $\\alpha \\in \\mathbb { R }$ is the unique   \n205 solution of equation $\\begin{array} { r } { \\sum _ { i = 1 } ^ { m } \\overline { { \\left( \\sqrt { \\tilde { q } _ { t + 1 , i } } - \\alpha \\right) ^ { - 2 } } } = 1 } \\end{array}$ . The solution $\\alpha$ can be computed via the Newton   \n206 method. Practically, one can use $\\alpha$ in the previous iteration to warm start the Newton method. In   \n207 each iteration, the algorithm performs a single orthogonal projection onto $\\Theta$ , the Newton method for   \n208 finding $\\alpha$ , and $O ( m + n )$ operations to update $\\theta _ { t } , q _ { t }$ . The pseudocode is given in Algorithm 3. From   \n209 Theorem 1, we obtain the following convergence rate.   \n210 Theorem 3. If $\\eta _ { \\theta , t }$ is nonincreasing, GDRO-TINF (Algorithm 3) achieves the expected convergence   \n211 rate ",
+        "bbox": [
+            140,
+            750,
+            825,
+            838
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            839,
+            825,
+            867
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e0fbded07dffd9a617e9391afc04ea894ea8e9f476114d81cad31c8f14593a69.jpg",
+        "text": "$$\n\\mathbf { E } [ \\varepsilon _ { T } ] \\leq { \\frac { 1 } { T } } \\left( { \\frac { G ^ { 2 } } { 2 } } \\sum _ { t = 1 } ^ { T } \\eta _ { \\theta , t } + { \\frac { D ^ { 2 } } { 2 \\eta _ { \\theta , T } } } + { \\sqrt { m } } M ^ { 2 } \\eta _ { q } T + { \\frac { \\sqrt { m } } { \\eta _ { q } } } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            295,
+            866,
+            700,
+            910
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For 212 $\\begin{array} { r } { \\eta _ { \\theta , t } = \\frac { D } { G \\sqrt { T } } } \\end{array}$ and $\\begin{array} { r } { \\eta _ { q } = \\frac { 1 } { M \\sqrt { T } } } \\end{array}$ , we obtain ",
+        "bbox": [
+            140,
+            89,
+            464,
+            111
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/18204b6f05bc93b6eef0943a405d623b7d4fe9d39c53b547b666991f7ff3ed92.jpg",
+        "text": "$$\n\\mathbf { E } [ \\varepsilon _ { T } ] \\leq \\sqrt { 2 } \\frac { \\sqrt { G ^ { 2 } D ^ { 2 } + 4 M ^ { 2 } m } } { \\sqrt { T } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            390,
+            116,
+            607,
+            154
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "213 Comparison to Sagawa et al. [2020]. Our algorithms improve the convergence rate of Sagawa et√   \n214 al. [2020] by a factor of $O ( \\sqrt { m } )$ ; see Table 1. The reason lies in the choice of gradient estimator. All   \n215 algorithms are stochastic no-regret dynamics. As outlined above, their convergence hence can be   \n216 bounded by the regrets of the players, which depend on the variance of the local norm of the gradient   \n217 estimators. Their strategy is based on uniform sampling that yields a variance of √ $O ( m )$ for both   \n218 players, whereas our bound is $O ( { \\sqrt { m } } )$ thanks to the gradient estimators tailored to the regularizer of   \n219 OMD. More details may be found in Appendix D. ",
+        "bbox": [
+            140,
+            166,
+            826,
+            265
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "3.3 Algorithm for weighted ranking of group losses ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            280,
+            542,
+            295
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We now consider a more general case that $Q$ is a permutahedron. Applying Algorithm 1 with the Tsallis entropy regularizer, we obtain the following result. ",
+        "bbox": [
+            163,
+            304,
+            825,
+            334
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 4. If $\\eta _ { \\theta , t }$ is nonincreasing and $Q$ is a permutahedron, Algorithm 1 with the Tsallis entropy regularizer achieves the same expected convergence rate as Theorem 3. Furthermore, the iteration complexity is $O ( m \\log m + n )$ . ",
+        "bbox": [
+            173,
+            338,
+            825,
+            381
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "This implies a convergence rate of G2D2+M2mT ) for empirical CVaR optimization, which improves q G2D2+M2m log mT ) convergence by Curi et al. [2020]. Furthermore, their iteration complexity is $O ( m ^ { 3 } )$ due to the $k$ -DPP sampling step, so our algorithm is even faster in terms of iteration complexity. ",
+        "bbox": [
+            171,
+            392,
+            826,
+            470
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4 Lower bound ",
+        "text_level": 1,
+        "bbox": [
+            169,
+            489,
+            318,
+            506
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3 states that we can find an $\\varepsilon$ -optimal solution for group DRO in $ { \\mathcal { O } } ( \\frac { G ^ { 2 } D ^ { 2 } + M ^ { 2 } m } { \\varepsilon ^ { 2 } } )$ calls to stochastic oracles. Next, we show that this query complexity is information-theoretically optimal. ",
+        "bbox": [
+            173,
+            517,
+            823,
+            550
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "33 Let $\\mathcal { L }$ be a class of convex $G$ -Lipschitz loss functions $\\ell : \\Theta \\to [ 0 , M ]$ . Given a loss function $\\ell \\in { \\mathcal { L } }$ ,   \n34 and an $m$ -set $\\mathcal { P } = \\{ P _ { 1 } , \\ldots , P _ { m } \\}$ of distributions, denote the optimality gap of $\\theta \\in \\Theta$ by ",
+        "bbox": [
+            156,
+            554,
+            826,
+            584
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/83685d1c8262f85f16a7577c691f8a4db2ce4ba5ad43c336bdf62a3bb5ac94d9.jpg",
+        "text": "$$\nR ( \\theta , \\ell , \\mathcal { P } ) = \\operatorname* { m a x } _ { P \\in \\mathcal { P } } \\mathbf { \\Xi } _ { z \\sim P } ^ { \\mathbf { E } } [ \\ell ( \\theta ; z ) ] - \\operatorname* { m i n } _ { \\theta ^ { * } \\in \\Theta } \\operatorname* { m a x } _ { P \\in \\mathcal { P } } \\mathbf { \\Xi } _ { z \\sim P } ^ { \\mathbf { E } } [ \\ell ( \\theta ^ { * } ; z ) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            307,
+            590,
+            691,
+            614
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Let 235 $\\boldsymbol { \\mathcal { A } } _ { T }$ be the set of algorithms that outputs $\\hat { \\theta } \\in \\Theta$ making $T$ queries to the stochastic oracle. ",
+        "bbox": [
+            142,
+            623,
+            784,
+            640
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 5 (Lower Bound). ",
+        "bbox": [
+            174,
+            643,
+            361,
+            659
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1ff4a493cd44b8276c437b8dfb5fce18bdf1564bd2a7f4403f49257b2bb24826.jpg",
+        "text": "$$\n\\operatorname* { i n f } _ { \\boldsymbol { \\hat { \\theta } } \\in A _ { T } } \\operatorname* { s u p } _ { \\boldsymbol { \\ell } \\in \\mathcal { L } , \\boldsymbol { \\Theta } , \\mathcal { P } } \\mathbf { E } [ R ( \\boldsymbol { \\hat { \\theta } } , \\boldsymbol { \\ell } , \\mathcal { P } ) ] \\geq \\Omega \\left( \\operatorname* { m a x } \\left\\{ \\frac { G D } { \\sqrt { T } } , M \\sqrt { \\frac { m } { T } } \\right\\} \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            299,
+            665,
+            696,
+            702
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "236 where $\\Theta$ runs over convex sets with diameter $D$ and $\\mathcal { P }$ over $m$ -sets of distributions, and $\\mathbf { E } _ { \\mathcal { P } }$ denotes   \n237 the expectation over outcomes of the stochastic oracle in $\\mathcal { P }$ . ",
+        "bbox": [
+            147,
+            708,
+            823,
+            738
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "As ${ \\sqrt { x + y } } \\leq { \\sqrt { x } } + { \\sqrt { y } } \\leq { \\sqrt { 2 ( x + y ) } }$ for $x , y \\geq 0$ , this theorem immediately implies that the minimax convergence rate is $\\Omega \\left( { \\sqrt { \\frac { G ^ { 2 } D ^ { 2 } + M ^ { 2 } m } { T } } } \\right)$ , which equals the convergence rate achieved by Algorithm 3 up to a constant factor. ",
+        "bbox": [
+            166,
+            747,
+            825,
+            810
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "241 Proof Sketch. It suffices to show two lower bounds $\\textstyle { \\frac { G D } { \\sqrt { T } } }$ and $M _ { \\sqrt { T } }$ independently. The former is   \n242 a well-known lower bound for stochastic convex optimization [Agarwal et al., 2012]. To illustrate the   \n243 latter, we take an algorithmic dependent point of view via the Le cam’s method. For any algorithm   \n244 in $\\boldsymbol { \\mathcal { A } } _ { T }$ , we need to construct instances $\\mathcal { P } _ { 0 } , \\mathcal { P } _ { 1 }$ such that the total variation distance between the   \n245 distributions over the query outcomes (they depend on both the behavior of the algorithm and the   \n246 instance) with respect to $\\mathcal { P } _ { 0 }$ and $\\mathcal { P } _ { 1 }$ is small. On the other hand, the objective function of the two   \n247 instances must be well-separated, i.e., any fixed $\\theta$ is $\\delta$ sub-optimal for either $\\mathcal { P } _ { 0 }$ or $\\mathcal { P } _ { 1 }$ . So, any   \n248 algorithm that solves group DRO up to error $\\delta$ needs to distinguish two instances $\\mathcal { P } _ { 0 }$ and $\\mathcal { P } _ { 1 }$ . This   \n249 implies a query lower bound because the total variation distance of the outcome distributions of   \n250 these instances is small. The challenge is how to construct such instances for the regime of small   \n251 dimensions of $\\theta$ , e.g, $n = 1$ . To this end, we carefully construct linear functions for $m$ groups using   \n252 opposite slopes. Then, based on the behavior of the algorithm, we tweak the noise bias in one of the   \n253 groups with a positive slope, in a way that any fixed $\\theta$ is $\\Theta ( \\delta )$ sub-optimal for one of these instances.   \n254 For the detailed proof, see Appendix C. ",
+        "bbox": [
+            143,
+            821,
+            825,
+            912
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/d9492d45448eb847bb159864789d12e10cf5a5dda3bde0a928bacf6167b13ceb.jpg",
+        "image_caption": [
+            "Figure 1: Results on Adult dataset "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            245,
+            87,
+            756,
+            263
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/5b04a8e0db493af6fd2e274be9b0b710c499c0b8edd296bd1ca151d3d95db3d7.jpg",
+        "image_caption": [
+            "Figure 2: Results on synthetic dataset "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            299,
+            838,
+            472
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            518,
+            826,
+            631
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 Experiments ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            648,
+            312,
+            666
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this section, we compare our algorithms with the known algorithm using real-world and synthetic datasets. We follow the setup in [Namkoong and Duchi, 2016]. ",
+        "bbox": [
+            168,
+            680,
+            823,
+            708
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Adult dataset. For the real-world dataset, we use Adult dataset [Dua and Graff, 2017]. The dataset consists of age, gender, race, educational background, and many other attributes of 48, 842 individuals from the US census. The task is to predict whether the person’s income is greater than 50, 000 USD or not. We set up 6 groups based on the race and gender attributes: each group corresponds to a combination of {black, white, others} $\\times \\left\\{ \\begin{array} { r l } \\end{array} \\right.$ {female, male}. Converting the categorical features to dummy variables, we obtain a 101-dimensional feature vector $a \\in \\mathbb { R } ^ { n }$ ( $n = 1 0 1$ ) for each individual. We train the linear model with the logistic loss and hinge loss functions. The group-DRO objective is the worst empirical loss over the 6 groups: ",
+        "bbox": [
+            169,
+            722,
+            825,
+            834
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4c03c695791e172633b3694d44dd63eaf9ed57289a4df3544c58f47d7db01eaa.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { i = 1 } ^ { 6 } { \\frac { 1 } { | I _ { i } | } } \\sum _ { ( a , b ) \\in I _ { i } } \\ell ( \\theta ; a , b ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            408,
+            838,
+            584,
+            878
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "where 266 $I _ { i }$ is the set of data points in the ith group. The feasible region is set to the Euclidean ball of 267 radius $D = 1 0$ . ",
+        "bbox": [
+            140,
+            882,
+            823,
+            911
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "268 Synthetic dataset. To observe the performance of the algorithms over the regime of high-dimension   \n269 model parameters and the larger number of groups, we also conducted experiments using the following   \n270 synthetic instances. First, we set $n = 5 0 0$ and varied $m \\in \\{ 1 0 , 5 0 , 1 0 0 \\}$ . For each group $i \\in [ m ]$ , we   \n271 generated the true classifier $\\theta _ { i } ^ { * } \\in \\mathbb { R } ^ { n }$ from the uniform distribution over the unit sphere in $\\mathbb { R } ^ { n }$ . The ith   \n272 group distribution $P _ { i }$ was the empirical distribution of 1,000 data points, where each data point $( a , b )$   \n273 was drawn as $a \\sim N ( 0 , I _ { n } )$ and $b = \\mathrm { s i g n } ( a ^ { \\top } \\theta _ { i } ^ { * } )$ with probability 0.9 and $b = - \\mathrm { s i g n } ( a ^ { \\top } \\theta _ { i } ^ { * } )$ with   \n274 probability 0.1. We trained the linear model with the hinge loss function. Finally, the group-DRO   \n275 objective is ",
+        "bbox": [
+            140,
+            90,
+            826,
+            204
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0fca465f34037987076d44579a79c52496108a70c62fddefaa5b1846e7caf51f.jpg",
+        "text": "$$\n\\underset { i = 1 } { \\operatorname* { m a x } } \\ \\underset { ( a , b ) \\sim P _ { i } } { \\mathbf { E } } [ \\ell ( \\theta ; a , b ) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            416,
+            208,
+            576,
+            236
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "276 The feasible region is set to the Euclidean ball of radius $D = 1 0$ . ",
+        "bbox": [
+            145,
+            244,
+            598,
+            260
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.1 Algorithms ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            280,
+            292,
+            295
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We implemented GDRO-EXP3, GDRO-TINF, and the algorithm in [Sagawa et al., 2020] in Python.   \nWe ran our algorithms for $T = 2 , 0 0 0 , 0 0 0$ iterations. ",
+        "bbox": [
+            168,
+            308,
+            823,
+            335
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Inner online algorithms. It is known that EXP3 has a variance as large as $O ( T ^ { 2 } )$ [Lattimore and Szepesvári, 2020]. Therefore, vanilla EXP3 often fails to achieve a sublinear regret even though it achieves $O ( \\sqrt { T } )$ regret in expectation. This large variance makes it difficult to reliably evaluate the performance of the algorithms. To stabilize the algorithms, we replaced EXP3 with its variation,√ EXP3P [Auer et al., 2003], which achieves $O ( \\sqrt { T } )$ regret with high probability. Note that this change does not harm our expected convergence bounds. ",
+        "bbox": [
+            171,
+            354,
+            825,
+            443
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Step sizes. The choice of step sizes is crucial to the practical performance of first-order methods. We found that the decreasing step size $\\eta _ { \\theta , t } \\sim 1 / \\sqrt { t }$ for $\\theta _ { t }$ and the fixed step size $\\eta _ { q } \\sim 1 / \\sqrt { T }$ for $q _ { t }$ gave the best results. More precisely, we set $\\begin{array} { r } { \\eta _ { \\theta , t } = \\frac { C _ { \\theta } D } { \\sqrt { t } } } \\end{array}$ $( t \\in [ T ] )$ and $\\begin{array} { r } { \\eta _ { q } = C _ { q } \\sqrt { \\frac { \\log m } { m T } } } \\end{array}$ , where $C _ { \\theta } \\in [ 0 . 1 , 5 . 0 ]$ and $C _ { q } \\in [ 0 . 1 , 3 . 0 ]$ are hyper-parameters tuned for each algorithm. We used the best hyper-parameter found by Optuna [Akiba et al., 2019] for the shown results. ",
+        "bbox": [
+            173,
+            460,
+            825,
+            545
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Mini-batch. The use of mini-batch often improves the stability of stochastic gradient algorithms. In our experiments, we used mini-batches of size 10 to evaluate stochastic gradients. Neither the objective values of outputs nor the stability was improved with larger mini-batch sizes. The group DRO objective is evaluated using the entire dataset. ",
+        "bbox": [
+            174,
+            563,
+            825,
+            619
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Initialization. For both datasets, we initialized the algorithms with $\\theta _ { 1 } = \\mathbf { 0 }$ . ",
+        "bbox": [
+            169,
+            637,
+            678,
+            652
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.2 Results ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            674,
+            264,
+            688
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We show the results of our experiment in Figures 1 and 2. ",
+        "bbox": [
+            165,
+            700,
+            550,
+            715
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Adult dataset. In Figure 1, we plot the optimality gap of the averaged iterate $\\textstyle { \\frac { 1 } { T } } \\sum _ { t = 1 } ^ { T } \\theta _ { t }$ against the number of iteration . We observe that all the algorithms converge with a rate roughly $T ^ { - 0 . 5 }$ for both loss functions, consistent with our convergence bound. Furthermore, our algorithms (GDRO-EXP3 and GDRO-TINF) achieve faster convergence compared to the algorithm by Sagawa et al. [2020]. Interestingly, GDRO-TINF achieves a $1 \\overline { { 0 } } ^ { - 4 }$ optimality gap in $T = 1 0 ^ { 6 }$ iterations, which is faster than the theoretical $T ^ { - 0 . 5 }$ rate in Theorem 3. ",
+        "bbox": [
+            173,
+            734,
+            825,
+            823
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Synthetic dataset. In Figure 2, we plot the objective values of the averaged iterate against the number of iterations. For all the values of $m$ , our algorithms (especially GDRO-EXP3) consistently achieve smaller loss values faster than the known algorithm. The performance gap between our algorithms and the known algorithm increased as $m$ grows, which verifies that our algorithms have better dependence on $m$ in the convergence rate. ",
+        "bbox": [
+            169,
+            842,
+            825,
+            911
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "09 References   \n10 Alekh Agarwal, Peter L. Bartlett, Pradeep Ravikumar, and Martin J. Wainwright. Informationtheoretic lower bounds on the oracle complexity of stochastic convex optimization. IEEE Transactions on Information Theory, pages 3235–3249, 2012.   \n13 Takuya Akiba, Shotaro Sano, Toshihiko Yanase, Takeru Ohta, and Masanori Koyama. Optuna: A next-generation hyperparameter optimization framework. In Proceedings of the 25rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 2019.   \n316 Peter Auer, Nicolò Cesa-Bianchi, Yoav Freund, and Robert E. Schapire. The nonstochastic multiarmed bandit problem. SIAM Journal on Computing, 32(1):48–77, 2003.   \n18 Yujia Bao, Shiyu Chang, and Regina Barzilay. Predict then interpolate: A simple algorithm to learn stable classifiers. In Proceedings of the 38th International Conference on Machine Learning, volume 139, pages 640–650, 2021. Dimitris Bertsimas, Vishal Gupta, and Nathan Kallus. Data-driven robust optimization. Mathematical Programming, 167(2):235–292, 2018. Jose Blanchet, Yang Kang, and Karthyek Murthy. Robust wasserstein profile inference and applications to machine learning. Journal of Applied Probability, 56(3):830–857, 2019.   \n25 Joy Buolamwini and Timnit Gebru. Gender shades: Intersectional accuracy disparities in commercial gender classification. In Proceedings of the 1st Conference on Fairness, Accountability and Transparency, pages 77–91, 2018. Yair Carmon, Arun Jambulapati, Yujia Jin, and Aaron Sidford. Thinking inside the ball: Near-optimal minimization of the maximal loss. In Proceedings of 34th Conference on Learning Theory, volume   \n134 of Proceedings of Machine Learning Research, pages 866–882, 2021.   \n31 Nicolo Cesa-Bianchi and Gabor Lugosi. Prediction, Learning, and Games. Cambridge University Press, 2006.   \n33 Sebastian Curi, Kfir Y. Levy, Stefanie Jegelka, and Andreas Krause. Adaptive sampling for stochastic risk-averse learning. In Advances in Neural Information Processing Systems, pages 1036–1047,   \n2020.   \n36 Emily Diana, Wesley Gill, Ira Globus-Harris, Michael Kearns, Aaron Roth, and Saeed SharifiMalvajerdi. Lexicographically fair learning: Algorithms and generalization. In Proceedings of the   \n2nd Symposium on Foundations of Responsible Computing, pages 6:1–6:23, 2021.   \n39 Dheeru Dua and Casey Graff. UCI machine learning repository, 2017.   \n40 John C. Duchi and Hongseok Namkoong. Learning models with uniform performance via distributionally robust optimization. The Annals of Statistics, 49(3):1378 – 1406, 2021.   \n42 Peyman Mohajerin Esfahani and Daniel Kuhn. Data-driven distributionally robust optimization using the wasserstein metric: Performance guarantees and tractable reformulations. Mathematical Programming, 171(1):115–166, 2018.   \n45 Joel Goh and Melvyn Sim. Distributionally robust optimization and its tractable approximations. Operations Research, 58(4-part-1):902–917, 2010.   \n47 Tatsunori Hashimoto, Megha Srivastava, Hongseok Namkoong, and Percy Liang. Fairness without demographics in repeated loss minimization. In Proceedings of the 35th International Conference on Machine Learning, pages 1929–1938, 2018.   \n50 Elad Hazan. Introduction to Online Convex Optimization. 2016.   \n51 Dirk Hovy and Anders Søgaard. Tagging performance correlates with author age. In Proceedings of the 53rd Annual Meeting of the Association for Computational Linguistics and the 7th International Joint Conference on Natural Language Processing, pages 483–488, 2015. ",
+        "bbox": [
+            153,
+            75,
+            828,
+            922
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "354 Weihua Hu, Gang Niu, Issei Sato, and Masashi Sugiyama. Does distributionally robust supervised learning give robust classifiers? In Proceedings of the 35th International Conference on Machine Learning, pages 2029–2037, 2018. Jikai Jin, Bohang Zhang, Haiyang Wang, and Liwei Wang. Non-convex distributionally robust optimization: Non-asymptotic analysis. In Advances in Neural Information Processing Systems, volume 34, pages 2771–2782, 2021. David Jurgens, Yulia Tsvetkov, and Dan Jurafsky. Incorporating dialectal variability for socially equitable language identification. In Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics, pages 51–57, 2017. Johannes Kirschner, Ilija Bogunovic, Stefanie Jegelka, and Andreas Krause. Distributionally robust bayesian optimization. In Proceedings of the 33rd International Conference on Artificial Intelligence and Statistics, pages 2174–2184, 2020. Tor Lattimore and Csaba Szepesvári. Bandit algorithms. Cambridge University Press, 2020. Cong Han Lim and Stephen J. Wright. Efficient bregman projections onto the permutahedron and related polytopes. In Proceedings of the 19th International Conference on Artificial Intelligence and Statistics, pages 1205–1213, 2016. Natalia L Martinez, Martin A Bertran, Afroditi Papadaki, Miguel Rodrigues, and Guillermo Sapiro. Blind pareto fairness and subgroup robustness. In Proceedings of the 38th International Conference on Machine Learning, pages 7492–7501, 2021. Hongseok Namkoong and John C Duchi. Stochastic gradient methods for distributionally robust optimization with $f$ -divergences. In Advances in Neural Information Processing Systems, 2016. Yonatan Oren, Shiori Sagawa, Tatsunori Hashimoto, and Percy Liang. Distributionally robust language modeling. In Proceedings of the Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLPIJCNLP), pages 4227–4237, 2019. Qi Qi, Zhishuai Guo, Yi Xu, Rong Jin, and Tianbao Yang. An online method for a class of distributionally robust optimization with non-convex objectives. In Advances in Neural Information Processing Systems, volume 34, pages 10067–10080, 2021. Alexander Rakhlin and Karthik Sridharan. Optimization, learning, and games with predictable sequences. In Advances in Neural Information Processing Systems, 2013. Shiori Sagawa, Pang Wei Koh, Tatsunori B. Hashimoto, and Percy Liang. Distributionally robust neural networks for group shifts: On the importance of regularization for worst-case generalization. In The 8th International Conference on Learning Representations, 2020. Herbert Scarf. A min-max solution of an inventory problem. Studies in the mathematical theory of inventory and production, 1958. Matthew Staib and Stefanie Jegelka. Distributionally robust optimization and generalization in kernel methods. In Advances in Neural Information Processing Systems, 2019. Matthew Staib, Bryan Wilder, and Stefanie Jegelka. Distributionally robust submodular maximization. In Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics, pages 506–516, 2019. Robert Williamson and Aditya Menon. Fairness risk measures. In Proceedings of the 36th International Conference on Machine Learning, pages 6786–6797, 2019. Jingzhao Zhang, Aditya Krishna Menon, Andreas Veit, Srinadh Bhojanapalli, Sanjiv Kumar, and Suvrit Sra. Coping with label shift via distributionally robust optimisation. In The 9th International Conference on Learning Representations, 2021. Julian Zimmert and Yevgeny Seldin. Tsallis-inf: An optimal algorithm for stochastic and adversarial bandits. Journal of Machine Learning Research, 22(28):1–49, 2021. ",
+        "bbox": [
+            145,
+            54,
+            828,
+            920
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Martin Zinkevich. Online convex programming and generalized infinitesimal gradient ascent. In Proceedings of the 20th International Conference on International Conference on Machine Learning, pages 928–935, 2003. ",
+        "bbox": [
+            143,
+            90,
+            828,
+            133
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "404 Checklist ",
+        "text_level": 1,
+        "bbox": [
+            143,
+            159,
+            254,
+            175
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "1. For all authors... ",
+        "bbox": [
+            214,
+            186,
+            339,
+            200
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   \n(b) Did you describe the limitations of your work? [No]   \n(c) Did you discuss any potential negative societal impacts of your work? [No] This paper is a theoretical paper.   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
+        "bbox": [
+            238,
+            205,
+            825,
+            309
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "2. If you are including theoretical results... ",
+        "bbox": [
+            215,
+            314,
+            493,
+            328
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Assumption 1.   \n(b) Did you include complete proofs of all theoretical results? [Yes] Ommited Proof can be found in the supplemental material. ",
+        "bbox": [
+            238,
+            332,
+            825,
+            391
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "3. If you ran experiments... ",
+        "bbox": [
+            214,
+            395,
+            393,
+            410
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] The experiment code can be found in the supplemental material.   \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section 5.   \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No]   \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] ",
+        "bbox": [
+            238,
+            414,
+            825,
+            546
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ",
+        "bbox": [
+            215,
+            551,
+            823,
+            565
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "(a) If your work uses existing assets, did you cite the creators? [Yes]   \n(b) Did you mention the license of the assets? [Yes]   \n(c) Did you include any new assets either in the supplemental material or as a URL? [No]   \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [No] We used a public dataset.   \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No] ",
+        "bbox": [
+            238,
+            569,
+            823,
+            676
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "5. If you used crowdsourcing or conducted research with human subjects... ",
+        "bbox": [
+            214,
+            681,
+            705,
+            695
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [No] Not applicable   \n(b) DidNo describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [No]   \n(c) DidNo include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [No] ",
+        "bbox": [
+            238,
+            699,
+            825,
+            789
+        ],
+        "page_idx": 11
+    }
+]

parse/dev/VYYf6S67pQc/VYYf6S67pQc.md

@@ -0,0 +1,335 @@
+# Mildly Conservative $Q$ -Learning for Offline Reinforcement Learning
+
+Jiafei Lyu1∗, Xiaoteng $\mathbf { M } \mathbf { a } ^ { 2 * }$ , Xiu Li1†, Zongqing $\mathbf { L u ^ { 3 \dag } }$ 1Tsinghua Shenzhen International Graduate School, Tsinghua University 2Department of Automation, Tsinghua Unversity 3School of Computer Science, Peking University {lvjf20,ma-xt17}@mails.tsinghua.edu.cn, li.xiu@sz.tsinghua.edu.cn, zongqing.lu@pku.edu.cn
+
+# Abstract
+
+Offline reinforcement learning (RL) defines the task of learning from a static logged dataset without continually interacting with the environment. The distribution shift between the learned policy and the behavior policy makes it necessary for the value function to stay conservative such that out-of-distribution (OOD) actions will not be severely overestimated. However, existing approaches, penalizing the unseen actions or regularizing with the behavior policy, are too pessimistic, which suppresses the generalization of the value function and hinders the performance improvement. This paper explores mild but enough conservatism for offline learning while not harming generalization. We propose Mildly Conservative $Q$ -learning (MCQ), where OOD actions are actively trained by assigning them proper pseudo $Q$ values. We theoretically show that MCQ induces a policy that behaves at least as well as the behavior policy and no erroneous overestimation will occur for OOD actions. Experimental results on the D4RL benchmarks demonstrate that MCQ achieves remarkable performance compared with prior work. Furthermore, MCQ shows superior generalization ability when transferring from offline to online, and significantly outperforms baselines. Our code is publicly available at https://github.com/dmksjfl/MCQ.
+
+# 1 Introduction
+
+Continually interacting with the environment of online reinforcement learning (RL) is often infeasible and unrealistic, since the data collection process of the agent may be expensive, difficult, or even dangerous, especially in real-world applications. Offline RL, instead, aims at learning from a static dataset that was previously collected by some unknown process [36], hence eliminating the need for environmental interactions during training.
+
+The main challenge of offline RL is the distribution shift of state-action visitation frequency between the learned policy and the behavior policy. The evaluation of out-of-distribution (OOD) actions causes extrapolation error [14], which can be exacerbated through bootstrapping [34] and result in severe overestimation errors. Thus, keeping conservatism in value estimation is necessary in offline RL [24, 50, 60]. Previous methods achieve the conservatism by compelling the learned policy to be close to the behavior policy [14, 58, 34, 13, 57], by penalizing the learned value functions from being over-optimistic upon out-of-distribution (OOD) actions [35, 33, 59], or by learning without querying OOD samples [56, 8, 62, 32, 40].
+
+![](images/ad202e97216c552f7b5a99b52b8c911d09909fa8131df452eeb7c7da141e8778.jpg)  
+Figure 1: Comparison of prior methods against mild conservatism. The red spots represent the dataset samples. The left figure shows that penalizing OOD actions makes the value function drop sharply at the boundary of the dataset’s support, which barriers policy learning. The central figure depicts that policy regularization keeps the policy near behavior policy, leading to undesired performance if the behavior policy is unsatisfying. On the right side, we illustrate the basic idea of mild conservatism. The estimated values for OOD actions are allowed to be high as long as it does not affect the learning for the optimal policy supported by the dataset, i.e., $Q ( s , \bar { a } ^ { \mathrm { o o d } } ) < \bar { \operatorname * { m a x } } _ { a \in \mathrm { S u p p o r t } ( \mu ) } Q ( s , a ) .$
+
+In practice, we rely on neural networks to extract knowledge from the dataset and generalize it to the nearby unseen states and actions when facing continuous state and action spaces. In other words, we need the networks to “stitch” the suboptimal trajectories to generate the best possible trajectory supported by the dataset. Unfortunately, there is no free lunch. Conservatism, which offline RL celebrates, often limits the generalization and impedes the performance of the agent. Existing approaches are still inadequate in balancing conservatism and generalization. As illustrated in Figure 1, policy regularization is unreliable for offline RL when the data-collecting policy is poor, and value penalization methods often induce unnecessary pessimism in both the in-dataset region and OOD region. We argue that the proper conservatism should be as mild as possible. As depicted in Figure 1, we aim at well estimating the value function in the support of the dataset, and allowing value estimates upon OOD actions to be high (even higher than their optimal values) as long as $Q ( s , a ^ { \mathrm { o o d } } ) < \operatorname* { m a x } _ { a \in \mathrm { S u p p o r t } ( \mu ) } Q ( s , a )$ is satisfied. The mild conservatism benefits generalization since value estimates upon OOD actions are slightly optimistic instead of being overly conservative.
+
+To fulfill that, we propose a novel Mildly Conservative Bellman (MCB) operator for offline RL, where we actively train OOD actions and query their $Q$ values. We theoretically analyze the convergence property of the MCB operator under the tabular MDP setting. We show that the policy induced by the MCB operator is guaranteed to behave better than the behavior policy, and can consistently improve the policy with a tighter lower bound compared with policy constraint methods or value penalization methods like CQL [35]. For practical usage, we propose the practical MCB operator and illustrate its advantages by theoretically showing that erroneous overestimation error will not occur with it. We then estimate the behavior policy with a conditional variational autoencoder (CVAE) [29, 51], and integrate the practical MCB operator with the Soft Actor-Critic (SAC) [20] algorithm. To this end, we propose our novel offline RL algorithm, Mildly Conservative $Q$ -learning (MCQ).
+
+Experimental results on the D4RL MuJoCo locomotion tasks demonstrate that MCQ surpasses recent strong baseline methods on most of the tasks, especially on non-expert datasets. Meanwhile, MCQ shows superior generalization capability when transferring from offline to online, validating our claims that mild pessimism is of importance to offline learning.
+
+# 2 Preliminaries
+
+We consider a Markov Decision Process (MDP) specified by a tuple $\langle S , \mathcal { A } , r , \rho _ { 0 } , p , \gamma \rangle$ , where $s$ is the state space, $\mathcal { A }$ is the action space, $r ( s , a ) : S \times \mathcal { A } \mapsto \mathbb { R }$ is the reward function, $\rho _ { 0 } ( s )$ is the initial state distribution, $p ( s ^ { \prime } | s , a ) : \bar { \mathcal { S } } \check { \times } \bar { \mathcal { A } } \times \bar { \mathcal { S } } \mapsto [ 0 , 1 ]$ is the transition probability, $\gamma \in [ 0 , 1 )$ is the discount factor. Reinforcement learning (RL) aims at finding a policy $\pi ( \cdot | s )$ such that the expected cumulative long-term rewards $J ( \pi ) = \mathbb { E } _ { s _ { 0 } \sim \rho _ { 0 } ( \cdot ) , a _ { t } \sim \pi ( \cdot \vert s _ { t } ) , s _ { t + 1 } \sim p ( \cdot \vert s _ { t } , a _ { t } ) } [ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } r ( s _ { t } , a _ { t } ) ]$ are maximized. The state-action function $Q ( s , a )$ measures the discounted return starting from state $s$ and action $a$ , and following the policy $\pi$ . We assume that the reward function $r ( s , a )$ is bounded, i.e., $| r ( s , a ) | \leq r _ { \operatorname* { m a x } }$ . Given a policy $\pi ( \cdot | s )$ , the Bellman backup for obtaining the corresponding $Q$ function gives:
+
+$$
+\begin{array} { r } { \mathcal T ^ { \pi } Q ( s , a ) : = r ( s , a ) + \gamma \mathbb E _ { s ^ { \prime } } \mathbb E _ { a ^ { \prime } \sim \pi ( \cdot \vert s ^ { \prime } ) } [ Q ( s ^ { \prime } , a ^ { \prime } ) ] . } \end{array}
+$$
+
+The $Q$ function of the optimal policy satisfies the following Bellman optimal operator:
+
+$$
+\mathcal { T } Q ( s , a ) : = r ( s , a ) + \gamma \mathbb { E } _ { s ^ { \prime } } \left[ \operatorname* { m a x } _ { a ^ { \prime } \in \mathcal { A } } Q ( s ^ { \prime } , a ^ { \prime } ) \right] .
+$$
+
+In offline RL setting, the online interaction is infeasible, and we can only have access to previously collected datasets $\bar { \mathcal { D } } = \{ ( s _ { i } , a _ { i } , r _ { i } , s _ { i + 1 } ^ { \prime } , d _ { i } ) \} _ { i = 1 } ^ { N }$ , where $d$ is the done flag. We denote the behavior policy as $\mu ( \cdot | s )$ . The Bellman backup relies on actions sampled from the learned policy, $a ^ { \prime } \sim \pi ( \cdot | s ^ { \prime } )$ . However, $a ^ { \prime }$ can lie outside of the support of $\mu$ due to the distribution shift between $\pi$ and $\mu$ . The value estimates upon $a ^ { \prime }$ can then be arbitrarily wrong, resulting in bad policy training. Unlike prior work, we actively train OOD actions by constructing them pseudo target values. In this way, we retain pessimism while enjoying better generalization.
+
+# 3 Mildly Conservative $Q$ -Learning
+
+In this section, we first formally define the MCB operator and characterize its dynamic programming properties in the tabular MDP setting. We further give a practical version of the MCB operator. We show that no erroneous overestimation will occur with the MCB operator. Finally, we incorporate the MCB operator with SAC [20] and present our novel offline RL algorithm.
+
+# 3.1 Mildly Conservative Bellman (MCB) Operator
+
+Definition 1. The Mildly Conservative Bellman (MCB) operator is defined as
+
+$$
+\begin{array} { r } { \mathcal { T } _ { \mathrm { M C B } } Q ( s , a ) = ( \mathcal { T } _ { 1 } \mathcal { T } _ { 2 } ) Q ( s , a ) , } \end{array}
+$$
+
+where
+
+$$
+\begin{array} { r l } & { { \mathcal T } _ { 1 } Q ( s , a ) = \left\{ \begin{array} { l l } { Q ( s , a ) , } & { \mu ( a | s ) > 0 . } \\ { \operatorname* { m a x } _ { a ^ { \prime } \sim \mathrm { S u p p o r t } ( \mu ( \cdot | s ) ) } Q ( s , a ^ { \prime } ) - \delta , } & { e l s e . } \end{array} \right. } \\ & { { \mathcal T } _ { 2 } Q ( s , a ) = \left\{ \begin{array} { l l } { r ( s , a ) + \gamma \mathbb { E } _ { s ^ { \prime } } \left[ \operatorname* { m a x } _ { a ^ { \prime } \in A } Q ( s ^ { \prime } , a ^ { \prime } ) \right] , } & { \mu ( a | s ) > 0 , } \\ { Q ( s , a ) , } & { e l s e . } \end{array} \right. } \end{array}
+$$
+
+The basic idea behind this novel operator is that if the learned policy outputs actions that lie in the support region of $\mu$ , then we go for backup; while if OOD actions are generated, we deliberately replace their value estimates with $\begin{array} { r } { \operatorname* { m a x } _ { a ^ { \prime } \sim \operatorname { S u p p o r t } ( \mu ( \cdot | s ) ) } Q ( s , a ^ { \prime } ) - \delta } \end{array}$ , where $\delta > 0$ can be arbitrarily small. That is, different from standard Bellman backup, we set up a checking procedure (i.e., $\mathcal { T } _ { 1 , }$ ) of whether the previous backup (i.e., $\mathcal { T } _ { 2 }$ ) involves OOD actions for the update. Intrinsically, we construct pseudo target values for OOD actions. We subtract a small positive $\delta$ such that OOD actions will not be chosen when executing policy via arg $\operatorname* { m a x } _ { a \in \mathcal { A } } Q ( s , a )$ .
+
+For a better understanding of the MCB operator, we theoretically analyze its dynamic programming properties in the tabular MDP setting. All proofs are deferred to Appendix A.
+
+Proposition 1. In the support region of the behavior policy, i.e., Support $( \mu )$ , the MCB operator is a $\gamma$ -contraction operator in the $\mathcal { L } _ { \infty }$ norm, and any initial $Q$ function can converge to a unique fixed point by repeatedly applying $\mathcal { T } _ { \mathrm { M C B } }$ .
+
+Proposition 2 (Behave at least as well as behavior policy). Denote $Q _ { \mathrm { M C B } }$ as the unique fixed point acquired by the MCB operator, then in $\operatorname { S u p p o r t } ( \mu )$ we have: $Q _ { \mu } \leq Q _ { \mathrm { M C B } } \leq Q _ { \mu ^ { * } }$ , where $Q _ { \mu }$ is the $Q$ function of the behavior policy and $Q _ { \mu ^ { * } }$ is the $Q$ function of the optimal policy in the batch.
+
+Proposition 2 indicates that the policy induced by the MCB operator can behave at least as well as the behavior policy, and can approximate the optimal batch-constraint policy. Apart from this advantage, we further show that the MCB operator results in milder conservatism. We start by observing that value penalization method, like CQL [35], guarantees that the learned value function $\hat { Q } ^ { \pi } ( s , a )$ is a lower bound of its true value $Q ^ { \pi } ( s , a )$ . It is also ensured that following such conservative update leads to a safe policy improvement, i.e., $\begin{array} { r } { J ( \pi _ { \mathrm { C Q L } } ) \ge J ( \mu ) - \mathcal { O } ( \frac { 1 } { ( 1 - \gamma ) ^ { 2 } } ) } \end{array}$ (Theorem 3.6 in [35]). For explicit policy constraint methods, e.g., $\mathrm { T D } 3 { + } \mathrm { B C }$ [13], the learned policy $\pi _ { p }$ mimics the behavior policy $\mu$ , and can hardly behave significantly better than $\mu$ . We show in Proposition 3 that explicit policy constraint methods also exhibit a safe policy improvement, $\begin{array} { r } { J ( \pi _ { p } ) \geq J ( \dot { \mu } ) - \mathcal { O } ( \frac { 1 } { ( 1 - \gamma ) ^ { 2 } } ) } \end{array}$ ( 1(1−γ)2 ), while the MCB operator can consistently improve the policy with a tighter lower bound.
+
+Proposition 3 (Milder Pessimism). Suppose there exists an explicit policy constraint offline reinforcement learning algorithm such that the $K L$ -divergence of the learned policy $\pi _ { p } ( \cdot | s )$ and the behavior policy $\mu ( \cdot | s )$ is optimized to guarantee max $( \mathrm { K L } ( \mu , \pi _ { p } ) , \mathrm { K L } ( \pi _ { p } , \mu ) ) \leq \dot { \epsilon }$ , ∀s. Denote $\begin{array} { r } { \epsilon _ { \mu } ^ { \pi _ { p } } = \operatorname* { m a x } _ { s } | \mathbb { E } _ { a \sim \pi _ { p } } A ^ { \mu } ( s , a ) | . } \end{array}$ , where $A ^ { \mu } ( s , a )$ is the advantage function. Then
+
+$$
+J ( \pi _ { p } ) \geq J ( \mu ) - \frac { \sqrt { 2 } \gamma \epsilon _ { \mu } ^ { \pi _ { p } } } { ( 1 - \gamma ) ^ { 2 } } \sqrt { \epsilon } ,
+$$
+
+while for the policy $\pi _ { \mathrm { M C B } }$ learned by applying the MCB operator, we have
+
+$$
+J ( \pi _ { \mathrm { M C B } } ) \geq J ( \mu ) .
+$$
+
+In summary, the MCB operator benefits the offline learning in two aspects: (1) the operator is a contraction, and any initial $Q$ functions are guaranteed to converge to a unique fixed point; (2) the learned policy of the MCB operator is ensured to be better than the behavior policy, and reserve milder pessimism compared with policy constraint methods or CQL.
+
+# 3.2 Practical MCB Operator
+
+In practice, it is intractable to acquire $\begin{array} { r } { \operatorname* { m a x } _ { a ^ { \prime } \sim \operatorname { S u p p o r t } ( \mu ( \cdot | s ) ) } Q ( s , a ^ { \prime } ) } \end{array}$ in $\mathcal { T } _ { 1 }$ of Eq. (4) in continuous control domains, and the behavior policy is often unknown. Thus, we fit an empirical behavior policy $\hat { \mu }$ with supervised learning based on the static dataset. The pseudo target values for the OOD actions are then computed by sampling $N$ actions from $\hat { \mu }$ , and taking maximum over their value evaluation. Formally, we define the practical MCB operator below, accompanied by the theoretical analysis.
+
+Definition 2. The practical Mildly Conservative Bellman (MCB) operator is defined as
+
+$$
+\hat { \mathcal { T } } _ { \mathrm { M C B } } Q ( s , a ) = ( \hat { \mathcal { T } } _ { 1 } \mathcal { T } _ { 2 } ) Q ( s , a ) ,
+$$
+
+where
+
+$$
+\begin{array} { r } { \hat { \mathcal { T } } _ { 1 } Q ( s , a ) = \left\{ \begin{array} { l l } { Q ( s , a ) , \qquad ~ } & { \mu ( a | s ) > 0 . } \\ { \mathbb { E } _ { \{ a _ { i } ^ { \prime } \} ^ { N } \sim \hat { \mu } ( \cdot | s ) } \left[ \operatorname* { m a x } _ { a ^ { \prime } \sim \{ a _ { i } ^ { \prime } \} ^ { N } } Q ( s , a ^ { \prime } ) \right] , } & { e l s e . } \end{array} \right. } \end{array}
+$$
+
+Compared with Eq. (4), we make a small modification of $\mathcal { T } _ { 1 }$ , and keep $\mathcal { T } _ { 2 }$ unchanged. There is no need to subtract $\delta$ here as generally $\begin{array} { r } { { \mathbb E } _ { \{ a _ { i } ^ { \prime } \} ^ { N } \sim \hat { \mu } ( \cdot | s ) } \left[ \operatorname* { m a x } _ { a ^ { \prime } \sim \{ a _ { i } ^ { \prime } \} ^ { N } } Q ( s , a ^ { \prime } ) \right] \leq \operatorname* { m a x } _ { a ^ { \prime } \sim \mathrm { S u p p o r t } ( \mu ) } Q ( s , a ^ { \prime } ) . } \end{array}$ . The practical MCB operator is much easier to implement in practice. We show that the practical MCB operator is still a $\gamma$ -contraction in the support region of the behavior policy $\mu$ .
+
+Proposition 4. Proposition 1 still holds for the practical MCB operator.
+
+Since we fit the empirical distribution $\hat { \mu }$ of the behavior policy $\mu$ , there may exist a shift between $\hat { \mu }$ and $\mu$ , especially when we represent the policy via neural networks. That suggests that OOD actions $a ^ { \prime }$ can still be sampled from $\hat { \mu }$ such that $a ^ { \prime } \overset { \cdot } { \notin } \operatorname { S u p p o r t } ( \mu ( \cdot | s ) )$ . Our last main result reveals that erroneous overestimation issue will not occur with the aid of the practical MCB operator.
+
+Proposition 5 (No erroneous overestimation will occur). Assuming that $\operatorname* { s u p } _ { s } D _ { \mathrm { T V } } ( \hat { \mu } ( \cdot | s ) \quad | |$ $\begin{array} { r } { \mu ( \cdot | \bar { s } ) ) \leq \epsilon < \frac { 1 } { 2 } } \end{array}$ , we have
+
+$$
+\mathbb { E } _ { \{ a _ { i } ^ { \prime } \} ^ { N } \sim \hat { \mu } ( \cdot | s ) } \left[ \operatorname* { m a x } _ { a ^ { \prime } \in \{ a _ { i } ^ { \prime } \} ^ { N } } Q ( s , a ^ { \prime } ) \right] \leq \operatorname* { m a x } _ { a ^ { \prime } \in \mathrm { S u p p o r t } ( \mu ( \cdot | s ) ) } Q ( s , a ^ { \prime } ) + ( 1 - ( 1 - 2 \epsilon ) ^ { N } ) \frac { r _ { \operatorname* { m a x } } } { 1 - \gamma } .
+$$
+
+Remark: This proposition generally requires a comparatively well-fitted empirical behavior policy $\hat { \mu }$ . In practice, we model $\hat { \mu }$ with a CVAE. In most cases, CVAE can already fit the dataset well and guarantee a good performance. Whereas there may exist some situations, e.g., the dataset is highly multi-modal, then one can replace the CVAE as the conditional GAN (CGAN) to better capture the different modes in the dataset as depicted in [61]. We believe generative models like CGAN will be a good choice by then.
+
+Intuitively, the above conclusion says that if the empirical behavior policy $\hat { \mu }$ well fits $\mu$ , i.e., $\epsilon$ is small enough, then regardless of how $\{ a _ { i } ^ { \prime } \} ^ { N }$ are sampled, the pseudo target value will approximate the maximum $Q$ -value within the dataset’s support with high probability. The extrapolation error is under the scale of $\begin{array} { r } { ( 1 - ( 1 - 2 \epsilon ) ^ { N } ) \frac { r _ { \operatorname* { m a x } } } { 1 - \gamma } } \end{array}$ . We expect a good empirical behavior policy such that most of the actions sampled from it will be in-distribution. However, if $\epsilon$ is large, $N$ can act as a trade-off parameter. The smaller $N$ we use, the more conservative we are. Fortunately, we find empirically that our method performs well in a large interval of $N$ over different tasks (see Section 4.2). Hence, it is safe to fix a $N$ in practice.
+
+# 3.3 Algorithm
+
+As aforementioned, we often cannot get prior information about the behavior policy $\mu$ . Thus, we need to empirically fit a behavior policy $\hat { \mu }$ with supervised learning for applying the practical MCB operator. Our algorithm, Mildly Conservative $Q$ -learning (MCQ), trains an additional generative model, which is also adopted by many prior work [14, 17, 33, 67]. We build our novel offline algorithm upon an off-the-shelf off-policy online RL algorithm, Soft Actor-Critic (SAC) [20].
+
+Modelling the behavior policy with the CVAE. We utilize a conditional variational autoencoder (CVAE) [29, 51, 14] to model the behavior policy $\mu$ . Given a fixed logged dataset, the goal of the CVAE is to reconstruct actions conditioned on the states such that the reconstructed actions come from the same distribution as the actions in the dataset, i.e., $\mu ( \cdot | s )$ . That generally satisfies the assumption we make in Proposition 5. As concerned by [32], training a generative model like CVAE still may produce out-of-dataset actions, which leads to extrapolation error since undefined $Q$ values can be possibly queried. Prior methods, like BCQ [14], do not well address such issue. While for our algorithm, such concern is mitigated because overestimation error is actually under control as is guaranteed by Proposition 5.
+
+The CVAE $G _ { \omega } ( s )$ parameterized by $\omega$ is made up of an encoder $E _ { \xi } ( s , a )$ and a decoder $D _ { \psi } ( s , z )$ parameterized by $\xi$ , $\psi$ respectively, $\omega = \{ \xi , \psi \}$ . The CVAE is optimized by maximizing its variational lower bound, which is equivalent to minimizing the following objective function.
+
+$$
+\mathcal { L } _ { \mathrm { C V A E } } = \mathbb { E } _ { ( s , a ) \sim \mathcal { D } , z \sim E _ { \xi } ( s , a ) } \left[ ( a - D _ { \psi } ( s , z ) ) ^ { 2 } + \mathrm { K L } \left( E _ { \xi } ( s , a ) , \mathcal { N } ( 0 , { \bf I } ) \right) \right] ,
+$$
+
+where $\operatorname { K L } ( p , q )$ denotes the KL-divergence between probability distribution $p ( \cdot )$ and $q ( \cdot )$ , and $\mathbf { I }$ is the identity matrix. When sampling actions from the CVAE, we first sample a latent variable $z$ from the prior distribution, which is set to be multivariate normal distribution $\mathcal { N } ( 0 , \bf { I } )$ , and then pass it in conjunction with the state $s$ into the decoder $D _ { \psi } ( s , z )$ to get the desired decoded action.
+
+It is also worth noting that we do not choose GAN [19] as the generative model because it is known to suffer from training instability and mode collapse [52, 6, 5]. Also, GAN consumes much more time and memories to train compared with the CVAE.
+
+Loss functions. In deep RL, the $Q$ function is represented with a neural network parameterized by $\theta$ and is updated via minimizing the temporal difference (TD) loss $\mathbb { E } _ { s , a , r , s ^ { \prime } } [ ( Q _ { \theta } ( s , a ) - \mathcal { T } Q ( s , a ) ) ^ { 2 } ]$ . We actually are performing the regression task $( s , a ) \mapsto \mathcal { T } Q ( s , a )$ to train the $Q$ function. The target value ${ \mathcal { T } } Q ( s , a )$ is usually computed by utilizing a lagging target network parameterized by $\theta ^ { \prime }$ without gradient backpropagation. As a typical actor-critic [30, 31, 53] algorithm, SAC uses its critic networks to perform value estimation and uses a separate actor network for policy improvement. In order to incorporate the MCB operator with the off-the-shelf SAC algorithm, we need to check whether the sampled action $a ^ { \prime } \sim \bar { \pi } ( \cdot | s )$ lies outside of the behavior policy’s support, i.e., whether $\mu ( a ^ { \prime } | s ) > 0$ . However, such a criterion is not reliable, because the true behavior policy $\mu$ is unknown and it is difficult to examine whether $\mu ( a ^ { \prime } | s ) > 0$ in practice. It is also problematic if we rely on the empirical behavior policy $\hat { \mu }$ to check whether $a ^ { \prime }$ is OOD as $\hat { \mu }$ itself can produce OOD actions.
+
+We then resort to constructing an auxiliary loss for OOD actions and integrating it with the standard Bellman error. Specifically, we sample $a ^ { \mathrm { o o d } }$ from the learned policy $\pi ( \cdot | _ { s } \mathrm { i n } )$ based on the sampled state $s ^ { \mathrm { i n } } \sim \mathcal { D }$ from the dataset and assign them pseudo target values based on the practical MCB operator. Note that the superscript ood is used to distinguish from the in-dataset real actions, and $a ^ { \mathrm { { \bar { o } o d } } }$ is not necessarily an OOD action. We remark that if $a ^ { \mathrm { o o d } } \in \operatorname { S u p p o r t } ( \mu ( \cdot | s ) )$ , the pseudo $Q$
+
+# Algorithm 1 Mildly Conservative $Q$ -learning (MCQ)
+
+1: Initialize CVAE $G _ { \omega }$ , critic networks $Q _ { \theta _ { 1 } } , Q _ { \theta _ { 2 } }$ and actor network $\pi _ { \phi }$ with random parameters   
+2: Initialize target networks $\theta _ { 1 } ^ { \prime }  \theta _ { 1 } , \theta _ { 2 } ^ { \prime }  \theta _ { 2 }$ and offline replay buffer $\mathcal { D }$ .   
+3: for $t = 1$ to $T$ do   
+4: Sample a mini-batch $B = \{ ( s , a , r , s ^ { \prime } , d ) \}$ from $\mathcal { D }$ , where $d$ is the done flag   
+5: Train CVAE via minimizing Eq. (10)   
+6: Get target value: $\begin{array} { r } { y = r ( s , \bar { a } ) + \gamma \left[ \operatorname* { m i n } _ { i = 1 , 2 } Q _ { \theta _ { i } ^ { \prime } } ( s ^ { \prime } , a ^ { \prime } ) - \alpha \log \pi _ { \phi } ( a ^ { \prime } | s ^ { \prime } ) \right] , a ^ { \prime } \sim \pi _ { \phi } ( \cdot | s ^ { \prime } ) } \end{array}$   
+7: Sample $N$ actions from $\pi$ based on each $s$ and $s ^ { \prime }$ , set $s ^ { \mathrm { i n } } = \{ s , s ^ { \prime } \}$   
+8: Compute the target value for the OOD actions via Eq. (13)   
+9: Update critic $\theta _ { i }$ with gradient descent via minimizing Eq. (11)   
+10: Update actor $\phi$ with gradient ascent via Eq. (14)   
+11: Update target networks: $\theta _ { i } ^ { \prime }  \tau \theta _ { i } + ( 1 - \tau ) \theta _ { i } ^ { \prime } , i = 1 , 2$   
+12: end for
+
+value will not negatively affect the evaluation upon it, because in-distribution actions are still trained to approximate the optimal batch-constraint $Q$ value. In this way, we actively train both possible OOD actions and in-distribution actions simultaneously via $o o D$ sampling. The resulting objective function for the critic networks is presented in Eq. (11).
+
+$$
+\mathcal { L } _ { \mathrm { c r i t i c } } = \lambda \mathbb { E } _ { ( s , a , r , s ^ { \prime } ) \sim \mathcal { D } } \left[ ( Q _ { \theta _ { i } } ( s , a ) - y ) ^ { 2 } \right] + ( 1 - \lambda ) \mathbb { E } _ { s ^ { \mathrm { i n } } \sim \mathcal { D } , a ^ { \mathrm { o o d } } \sim \pi } \left[ ( Q _ { \theta _ { i } } ( s ^ { \mathrm { i n } } , a ^ { \mathrm { o o d } } ) - y ^ { \prime } ) ^ { 2 } \right] ,
+$$
+
+where the target value for the in-distribution actions gives
+
+$$
+y = r ( s , a ) + \gamma \left[ \operatorname* { m i n } _ { i = 1 , 2 } Q _ { \theta _ { i } ^ { \prime } } ( s ^ { \prime } , a ^ { \prime } ) - \alpha \log \pi _ { \phi } ( a ^ { \prime } | s ^ { \prime } ) \right] , \alpha \in \mathbb { R } _ { + } ,
+$$
+
+which follows the standard target value of vanilla SAC. The hyperparameter $\lambda$ balances the indistribution data training and OOD action training. Following the formulas of the practical MCB operator in Eq. (9), the pseudo target value for the OOD action is computed by:
+
+$$
+y ^ { \prime } = \operatorname* { m i n } _ { j = 1 , 2 } \mathbb { E } _ { \{ a _ { i } ^ { \prime } \} ^ { N } \sim \hat { \mu } } \left[ \operatorname* { m a x } _ { a ^ { \prime } \sim \{ a _ { i } ^ { \prime } \} ^ { N } } Q _ { \theta _ { j } } ( s ^ { \mathrm { i n } } , a ^ { \prime } ) \right] .
+$$
+
+Note that we experimentally find that replacing the min operator with a mean operator does not raise much difference in performance. We hence take advantage of the min operator to fulfill the pseudo clipped double $Q$ -learning for OOD actions.
+
+The policy is then optimized by solving the following optimization problem:
+
+$$
+\pi _ { \phi } : = \operatorname* { m a x } _ { \phi } \mathbb { E } _ { s \sim \mathcal { D } , a \sim \pi _ { \phi } ( \cdot | s ) } \left[ \operatorname* { m i n } _ { i = 1 , 2 } Q _ { \theta _ { i } } ( s , a ) - \alpha \log \pi _ { \phi } ( \cdot | s ) \right] .
+$$
+
+We detail the learning procedure of our MCQ in Algorithm 1. Different from [13], our method does not require normalization over states or value functions. The only change we make to the vanilla SAC algorithm is an extra auxiliary loss term (blue term in Eq. (11)) such that OOD actions are actively and properly trained. The additional critic loss term can also be plugged into other off-policy online RL algorithms directly. As an evidence, we combine the MCB operator with TD3 [15], yielding a deterministic version of MCQ. Please refer to Appendix B for more details.
+
+# 4 Experiments
+
+In this section, we first empirically demonstrate the effectiveness and advantages of our proposed MCQ algorithm on D4RL benchmarks [12]. We then conduct a detailed parameter study to show the hyperparameter sensitivity of MCQ. We also experimentally illustrate that the value estimation of MCQ will not incur severe overestimation and pessimistic value estimates are witnessed in practice. Finally, we show the superior offline-to-online fine-tuning benefits of MCQ on some MuJoCo datasets.
+
+# 4.1 Results on MuJoCo Datasets
+
+We experimentally compare our MCQ against behavior cloning (BC), SAC, and several recent strong baseline methods, CQL [35], UWAC [59], TD3+BC [13], and IQL [32], on D4RL [12] benchmarks.
+
+We choose these methods as they typically represent different categories of model-free offline RL, i.e., CQL is a value penalization method, $\mathrm { T D } 3 { + } \mathrm { B C }$ involves explicit policy constraint (BC loss), UWAC relies on uncertainty estimation for training, and IQL learns without querying OOD samples.
+
+We conduct experiments on MuJoCo locomotion tasks, which are made up of five types of datasets (random, medium, medium-replay, medium-expert, and expert), yielding a total of 15 datasets. We use the most recently released "-v2" datasets for performance evaluation. The results of BC and SAC are acquired by using our implemented code. The results of CQL and UWAC are obtained by running their official codes, because the reported scores in their papers are not obtained on MuJoCo "-v2" datasets. We take the results of $\mathrm { T D } 3 { + } \mathrm { B C }$ from its original paper (Table 7 in [13]). Since the IQL paper does not report its performance on MuJoCo random and expert datasets, we run IQL using the official codebase on them and take the results on medium, medium-replay, medium-expert datasets from its original paper directly. All methods are run for 1M gradient steps.
+
+Table 1: Normalized average score comparison of MCQ against baseline methods on D4RL benchmarks over the final 10 evaluations. 0 corresponds to a random policy and 100 corresponds to an expert policy. The experiments are run on MuJoCo "-v2" datasets over 4 random seeds. $\mathbf { r } =$ random, $\mathbf { m } =$ medium, $\mathrm { m - r = }$ medium-replay, $\mathbf { m } { - } \mathbf { e } =$ medium-expert, ${ \bf e } =$ expert. We bold the highest mean.   
+
+<table><tr><td>Task Name</td><td>BC</td><td>SAC</td><td>CQL</td><td>UWAC</td><td>TD3+BC</td><td>IQL</td><td>MCQ (ours)</td></tr><tr><td>halfcheetah-r</td><td>2.2±0.0</td><td>29.7±1.4</td><td>17.5±1.5</td><td>2.3±0.0</td><td>11.0±1.1</td><td>13.1±1.3</td><td>28.5±0.6</td></tr><tr><td>hopper-r</td><td>3.7±0.6</td><td>9.9±1.5</td><td>7.9±0.4</td><td>2.7±0.3</td><td>8.5±0.6</td><td>7.9±0.2</td><td>31.8±0.5</td></tr><tr><td>walker2d-r</td><td>1.3±0.1</td><td>0.9±0.8</td><td>5.1±1.3</td><td>2.0±0.4</td><td>1.6±1.7</td><td>5.4±1.2</td><td>17.0±3.0</td></tr><tr><td>halfcheetah-m</td><td>43.2±0.6</td><td>55.2±27.8</td><td>47.0±0.5</td><td>42.2±0.4</td><td>48.3±0.3</td><td>47.4±0.2</td><td>64.3±0.2</td></tr><tr><td>hopper-m</td><td>54.1±3.8</td><td>0.8±0.0</td><td>53.0±28.5</td><td>50.9±4.4</td><td>59.3±4.2</td><td>66.2±5.7</td><td>78.4±4.3</td></tr><tr><td>walker2d-m</td><td>70.9±11.0</td><td>-0.3±0.2</td><td>73.3±17.7</td><td>75.4±3.0</td><td>83.7±2.1</td><td>78.3±8.7</td><td>91.0±0.4</td></tr><tr><td>halfcheetah-m-r</td><td>37.6±2.1</td><td>0.8±1.0</td><td>45.5±0.7</td><td>35.9±3.7</td><td>44.6±0.5</td><td>44.2±1.2</td><td>56.8±0.6</td></tr><tr><td>hopper-m-r</td><td>16.6±4.8</td><td>7.4±0.5</td><td>88.7±12.9</td><td>25.3±1.7</td><td>60.9±18.8</td><td>94.7±8.6</td><td>101.6±0.8</td></tr><tr><td>walker2d-m-r halfcheetah-m-e</td><td>20.3±9.8</td><td>-0.4±0.3</td><td>81.8±2.7</td><td>23.6±6.9</td><td>81.8±5.5</td><td>73.8±7.1</td><td>91.3±5.7</td></tr><tr><td></td><td>44.0±1.6</td><td>28.4±19.4</td><td>75.6±25.7</td><td>42.7±0.3</td><td>90.7±4.3</td><td>86.7±5.3</td><td>87.5±1.3</td></tr><tr><td>hopper-m-e</td><td>53.9±4.7</td><td>0.7±0.0</td><td>105.6±12.9</td><td>44.9±8.1</td><td>98.0±9.4</td><td>91.5±14.3</td><td>111.2±0.1</td></tr><tr><td>walker2d-m-e</td><td>90.1±13.2</td><td>1.9±3.9</td><td>107.9±1.6</td><td>96.5±9.1</td><td>110.1±0.5</td><td>109.6±1.0</td><td>114.2±0.7</td></tr><tr><td>Average Above</td><td>36.5</td><td>11.3</td><td>59.1</td><td>37.0</td><td>58.2</td><td>59.9</td><td>72.8</td></tr><tr><td>halfcheetah-e</td><td>91.8±1.5</td><td>-0.8±1.8</td><td>96.3±1.3</td><td>92.9±0.6</td><td>96.7±1.1</td><td>95.0±0.5</td><td>96.2±0.4</td></tr><tr><td>hopper-e walker2d-e</td><td>107.7±0.7</td><td>0.7±0.0</td><td>96.5±28.0</td><td>110.5±0.5</td><td>107.8±7</td><td>109.4±0.5</td><td>111.4±0.4</td></tr><tr><td></td><td>106.7±0.2</td><td>0.7±0.3</td><td>108.5±0.5</td><td>108.4±0.4</td><td>110.2±0.3</td><td>109.9±1.2</td><td>107.2±1.1</td></tr><tr><td>Total Average</td><td>49.6</td><td>9.0</td><td>67.3</td><td>50.4</td><td>67.6</td><td>68.9</td><td>79.2</td></tr></table>
+
+In our experiments, we set the number of sampled actions $N = 1 0$ by default and tune the weighting coefficient $\lambda$ . We report the $\lambda$ used for all tasks in Appendix C, along with details on the experiments and implementation. We summarize the normalized average score comparison of MCQ against recent baselines in Table 1. Unsurprisingly, we observe that MCQ behaves better than BC on all of the tasks, which is consistent with our theoretical analysis in Proposition 2 and 3. MCQ also significantly outperforms the base SAC algorithm. Prior offline RL methods struggle for good performance on non-expert datasets like random and medium-replay, while MCQ surpasses them with a remarkable margin on many non-expert datasets. We attribute the less satisfying performance of prior offline RL methods to their strict conservatism, which restricts their generalization beyond the support of the dataset and leads to limited performance. The results, therefore, validate our claim that milder pessimism is more we need for offline learning. Furthermore, MCQ is also competitive to baselines on expert datasets. MCQ achieves the best performance on 11 out of 15 datasets, yielding a total average score of 72.8 on non-expert datasets, and an average score of 79.2 on all 15 datasets. Whereas the second best method, IQL, has an average score of 59.9 on non-expert datasets and a total average score of 68.9 across all tasks.
+
+# 4.2 Parameter Study
+
+In this subsection, we conduct a detailed parameter study on MCQ. MCQ generally contains two hyperparameters, weighting coefficient $\lambda$ and number of sampled actions $N$ . To demonstrate the parameter sensitivity of MCQ, we choose two datasets from MuJoCo locomotion tasks and conduct experiments on them, halfcheetah-medium-v2, and hopper-medium-replay-v2. The experiments are run for 1M gradient steps over 4 different random seeds.
+
+![](images/19fe1db412940950351ea94a4e995d8979f361b533ea62a8c82c16ec795be03e.jpg)  
+Figure 2: Parameter study and $Q$ function estimation on halfcheetah-medium-v2 and hopper-mediumreplay-v2. The shaded region captures the standard deviation.
+
+Weighting coefficient $\lambda$ . The weighting coefficient $\lambda$ is a critical hyperparameter for MCQ, which directly controls the balance between in-distribution actions training and OOD actions training. If we set $\lambda = 1$ , then MCQ degenerates into the base SAC algorithm. If $\lambda$ leans towards 0, the critics will be overwhelmed by OOD actions. Intuitively, one ought not to use small $\lambda$ , because more weights are desired for standard Bellman error such that in-distribution state-action pairs can be well-trained. We observe significant performance drop with smaller $\lambda$ in Figure 2(a) and 2(b). Also, we find that choosing $0 . 7 \leq \lambda < 1$ generally induces good performance.
+
+Number of sampled actions $N$ . $N$ works as a regularizer to control the potential extrapolation error. In case the behavior policy $\mu$ is known, we require $N$ to be as large as possible to better estimate the maximum $Q$ value. While in practice, we leverage the CVAE to approximate $\mu$ , from which OOD actions can be sampled. $N$ then plays a role to balance pessimism and generalization. To see the influence of $N$ , we fix $\lambda = 0 . 9 5$ for the two datasets. Experimental results in Figure 2(d) and 2(e) indicate that MCQ is insensitive to $N$ for a wide range of $N$ . We therefore set $N = 1 0$ by default.
+
+Value estimation. We present the $Q$ value estimates with respect to (w.r.t.) $\lambda$ and $N$ in Figure 2(c) and 2(f). The $Q$ estimation is calculated via $\mathbb { E } _ { i = 1 , 2 } \mathbb { E } _ { ( s , a ) \sim \mathcal { D } } \big [ Q _ { \theta _ { i } } ( s , a ) \big ]$ . The results illustrate that (1) smaller $\lambda$ will incur severe underestimation issue (as depicted by Figure 2(c), $Q$ values collapse with $\lambda = 0 . 5$ or $\lambda = 0 . 3$ ); (2) no overestimation is observed, even with a large $\lambda = 0 . 9 5$ , which validates the theoretical result in Proposition 5; (3) the $Q$ estimates resemble each other under different $N$ . We conclude that MCQ ensures a stable and good value estimation with a proper $\lambda$ .
+
+# 4.3 Offline-to-online Fine-tuning
+
+We examine the offline-to-online fine-tuning capability of MCQ against some prior strong offline RL baselines, CQL [35], $\mathrm { T D } 3 { + } \mathrm { B C }$ [13], IQL [32]. We additionally compare against AWAC [45], which is designed intrinsically for offline-to-online adaptation. We conduct experiments on MuJoCo random and medium-replay datasets. It is challenging to train on these datasets for both offline and offline-to-online fine-tuning as they are non-expert, or even contain many bad transitions. We first train baselines and MCQ for 1M gradient steps offline and then perform online fine-tuning for another 100K gradient steps. Note that IQL paper [32] adopts 1M steps for online fine-tuning. However, we argue that 1M steps of online interactions are even enough to train off-policy online RL algorithms from scratch to perform very well. We thus believe 100K steps is more reasonable for the online interaction. All methods are run over 4 random seeds. The results are shown in Figure 3, where the shaded region denotes the standard deviation. As expected, we observe that MCQ consistently outperforms prior offline RL methods as well as AWAC on all of the datasets, often surpassing all of them with a large margin. The mild pessimism of MCQ makes it adapt faster, or keep the offline good performance during online interactions. Other prior offline RL methods, unfortunately, fail in achieving satisfying performance during online interaction due to strict conservatism and lack of generalization ability.
+
+![](images/310b7d615b5c1d531901da1dfa0eedaffd43f5b6fe2a4197607012c874ce945f.jpg)  
+Figure 3: Offline-to-online fine-tuning results on 6 D4RL MuJoCo locomotion tasks.
+
+# 5 Related Work
+
+Model-free offline RL. Prior model-free offline RL methods are typically designed to restrict the learned policy from producing OOD actions. They usually achieve this by leveraging importance sampling [47, 54, 39, 44, 16], incorporating explicit policy constraints [34, 58, 17, 13, 11], learning latent actions [67, 3], penalizing learned value functions such that low values are assigned to unseen actions [35, 33, 41], using adaptive methods [18], and uncertainty quantification [59, 65, 4]. Another line of the methods, instead, resorts to learning without querying OOD actions [56, 8, 32]. By doing so, they constrain the learning process within the support of the dataset. Nevertheless, existing methods may induce unnecessarily over-pessimistic value functions, and their performance is largely confined by how well the behavior policy is [45, 38, 4]. That partly explains why these methods are not satisfiable when trained on non-expert datasets (e.g., random datasets). MCQ keeps milder conservatism and better generalization ability as OOD actions are actively trained with proper targets.
+
+Model-based offline RL. Model-based offline RL methods, in contrast, learn the dynamics model in a supervised manner, and leverage the learned dynamics for policy optimization. Advances in this field include uncertainty quantification [46, 64, 27, 10], learning conservative value functions [63], representation learning [37, 48], constraining the learned policy with a behavior cloning loss [42], and sequential modelling [7, 23, 43]. However, there is no guarantee that the trained dynamics models are reliable, e.g., poor transitions can be generated, especially in complex high-dimensional environments [22]. Meanwhile, training dynamics models raises extra computation costs.
+
+Offline-to-online RL. There are some efforts on accelerating online interactions with the aid of offline logged data, which is also referred to as learning from demonstration [21, 26, 49]. Offline-to-online RL, instead, aims at enhancing the well-trained offline policy via online interactions. To ensure a fast adaptation and stable policy improvement, many techniques are adopted, such as model ensemble [38], explicit policy constraints [45, 66]. Offline-to-online fine-tuning will be difficult if the trained value function or policy is overly pessimistic, which may lead to a suboptimal policy.
+
+# 6 Conclusion
+
+In this paper, we propose Mildly Conservative $Q$ -learning (MCQ) to alleviate the over pessimism in existing offline RL methods. MCQ actively train OOD actions by constructing them proper pseudo target values following the guidance of the practical Mildly Conservative Bellman (MCB) operator. We theoretically illustrate that the policy induced by the MCB operator behaves at least as well as the behavior policy, and no erroneous overestimation will occur for the practical MCB operator. Furthermore, we extensively compare MCQ against recent strong baselines on MuJoCo locomotion tasks. Experimental results show that MCQ surpasses these baselines with a large margin on many non-expert datasets, and is also competitive with baselines on expert datasets. Moreover, we demonstrate the superior generalization capability of MCQ when transferring from offline to online. These altogether reveal that mild conservatism is critical for offline learning. We hope this work can promote the offline RL towards mild pessimism, and bring new insights into the community.
+
+One drawback of our current algorithm lies in the need of tuning the weighting coefficient $\lambda$ . However, we empirically find that $0 . 7 \leq \lambda < 1$ can usually induce satisfying performance. We leave the automatic tuning of $\lambda$ as future work.
+
+# Acknowledgments and Disclosure of Funding
+
+This work was supported in part by the Science and Technology Innovation 2030-Key Project under Grant 2021ZD0201404, in part by the NSF China under Grant 61872009. The authors would like to thank the anonymous reviewers for their valuable comments and advice.
+
+# References
+
+[1] J. Achiam, D. Held, A. Tamar, and P. Abbeel. Constrained Policy Optimization. In International Conference on Machine Learning, 2017.   
+[2] R. Agarwal, M. Schwarzer, P. S. Castro, A. C. Courville, and M. G. Bellemare. Deep Reinforcement Learning at the Edge of the Statistical Precipice. In Advances in Neural Information Processing Systems, 2021.   
+[3] A. Ajay, A. Kumar, P. Agrawal, S. Levine, and O. Nachum. OPAL: Offline Primitive Discovery for Accelerating Offline Reinforcement Learning. In International Conference on Learning Representations, 2021.   
+[4] C. Bai, L. Wang, Z. Yang, Z.-H. Deng, A. Garg, P. Liu, and Z. Wang. Pessimistic Bootstrapping for Uncertainty-Driven Offline Reinforcement Learning. In International Conference on Learning Representations, 2022.   
+[5] D. Bang and H. Shim. MGGAN: Solving Mode Collapse Using Manifold-Guided Training. IEEE/CVF International Conference on Computer Vision Workshops, pages 2347–2356, 2021.   
+[6] D. Bau, J.-Y. Zhu, J. Wulff, W. S. Peebles, H. Strobelt, B. Zhou, and A. Torralba. Seeing What a GAN Cannot Generate. In IEEE/CVF International Conference on Computer Vision, pages 4501–4510, 2019.   
+[7] L. Chen, K. Lu, A. Rajeswaran, K. Lee, A. Grover, M. Laskin, P. Abbeel, A. Srinivas, and I. Mordatch. Decision Transformer: Reinforcement Learning via Sequence Modeling. ArXiv, abs/2106.01345, 2021.   
+[8] X. Chen, Z. Zhou, Z. Wang, C. Wang, Y. Wu, Q. Deng, and K. W. Ross. BAIL: Best-Action Imitation Learning for Batch Deep Reinforcement Learning. In Advances in Neural Information Processing Systems, 2020.   
+[9] I. Csiszár and J. Körner. Information Theory - Coding Theorems for Discrete Memoryless Systems, Second Edition. Cambridge University Press, 2011.   
+[10] C. P. Diehl, T. Sievernich, M. Krüger, F. Hoffmann, and T. Bertram. UMBRELLA: Uncertainty-Aware Model-Based Offline Reinforcement Learning Leveraging Planning. ArXiv, abs/2111.11097, 2021.   
+[11] R. Fakoor, J. Mueller, P. Chaudhari, and A. Smola. Continuous Doubly Constrained Batch Reinforcement Learning. In Advances in Neural Information Processing Systems, 2021.   
+[12] J. Fu, A. Kumar, O. Nachum, G. Tucker, and S. Levine. D4RL: Datasets for Deep Data-Driven Reinforcement Learning. ArXiv, abs/2004.07219, 2020.   
+[13] S. Fujimoto and S. S. Gu. A Minimalist Approach to Offline Reinforcement Learning. In Advances in Neural Information Processing Systems, 2021.   
+[14] S. Fujimoto, D. Meger, and D. Precup. Off-Policy Deep Reinforcement Learning without Exploration. In International Conference on Machine Learning, 2019.   
+[15] S. Fujimoto, H. van Hoof, and D. Meger. Addressing Function Approximation Error in Actor-Critic Methods. In International Conference on Machine Learning, 2018.   
+[16] C. Gelada and M. G. Bellemare. Off-Policy Deep Reinforcement Learning by Bootstrapping the Covariate Shift. In AAAI Conference on Artificial Intelligence, volume 33, 2019.   
+[17] S. K. S. Ghasemipour, D. Schuurmans, and S. S. Gu. EMaQ: Expected-Max Q-Learning Operator for Simple Yet Effective Offline and Online RL. In International Conference on Machine Learning, 2021.   
+[18] D. Ghosh, A. Ajay, P. Agrawal, and S. Levine. Offline RL Policies Should be Trained to be Adaptive. ArXiv, abs/2207.02200, 2022.   
+[19] I. J. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. C. Courville, and Y. Bengio. Generative Adversarial Nets. In Advances in Neural Information Processing Systems, 2014.   
+[20] T. Haarnoja, A. Zhou, P. Abbeel, and S. Levine. Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor. In International Conference on Machine Learning, 2018.   
+[21] T. Hester, M. Vecerík, O. Pietquin, M. Lanctot, T. Schaul, B. Piot, D. Horgan, J. Quan, A. Sendonaris, I. Osband, G. Dulac-Arnold, J. P. Agapiou, J. Z. Leibo, and A. Gruslys. Deep Q-learning From Demonstrations. In AAAI Conference on Artificial Intelligence, 2018.   
+[22] M. Janner, J. Fu, M. Zhang, and S. Levine. When to Trust Your Model: Model-Based Policy Optimization. In Advances in Neural Information Processing Systems, 2019.   
+[23] M. Janner, Q. Li, and S. Levine. Offline Reinforcement Learning as One Big Sequence Modeling Problem. In Advances in Neural Information Processing Systems, 2021.   
+[24] Y. Jin, Z. Yang, and Z. Wang. Is Pessimism Provably Efficient for Offline RL? In International Conference on Machine Learning, pages 5084–5096. PMLR, 2021.   
+[25] S. M. Kakade and J. Langford. Approximately Optimal Approximate Reinforcement Learning. In International Conference on Machine Learning, 2002.   
+[26] B. Kang, Z. Jie, and J. Feng. Policy Optimization with Demonstrations. In International Conference on Machine Learning, 2018.   
+[27] R. Kidambi, A. Rajeswaran, P. Netrapalli, and T. Joachims. MOReL: Model-Based Offline Reinforcement Learning. In Advances in Neural Information Processing Systems, 2020.   
+[28] D. P. Kingma and J. Ba. Adam: A Method for Stochastic Optimization. In International Conference on Learning Representation, 2015.   
+[29] D. P. Kingma and M. Welling. Auto-Encoding Variational Bayes. ArXiv, abs/1312.6114, 2013.   
+[30] V. R. Konda and V. S. Borkar. Actor-Critic - Type Learning Algorithms for Markov Decision Processes. SIAM J. Control. Optim., 38:94–123, 1999.   
+[31] V. R. Konda and J. N. Tsitsiklis. Actor-Critic Algorithms. In Advances in Neural Information Processing Systems, 1999.   
+[32] I. Kostrikov, A. Nair, and S. Levine. Offline Reinforcement Learning with Implicit Q-Learning. In International Conference on Learning Representations, 2022.   
+[33] I. Kostrikov, J. Tompson, R. Fergus, and O. Nachum. Offline Reinforcement Learning with Fisher Divergence Critic Regularization. In International Conference on Machine Learning, 2021.   
+[34] A. Kumar, J. Fu, G. Tucker, and S. Levine. Stabilizing Off-Policy Q-Learning via Bootstrapping Error Reduction. In Advances in Neural Information Processing Systems, 2019.   
+[35] A. Kumar, A. Zhou, G. Tucker, and S. Levine. Conservative Q-Learning for Offline Reinforcement Learning. In Advances in Neural Information Processing Systems, 2020.   
+[36] S. Lange, T. Gabel, and M. A. Riedmiller. Batch Reinforcement Learning. In Reinforcement Learning, 2012.   
+[37] B.-J. Lee, J. Lee, and K.-E. Kim. Representation Balancing Offline Model-based Reinforcement Learning. In International Conference on Learning Representations, 2021.   
+[38] S. Lee, Y. Seo, K. Lee, P. Abbeel, and J. Shin. Offline-to-Online Reinforcement Learning via Balanced Replay and Pessimistic Q-Ensemble. In Conference on Robot Learning, 2021.   
+[39] Y. Liu, A. Swaminathan, A. Agarwal, and E. Brunskill. Off-Policy Policy Gradient with State Distribution Correction. ArXiv, abs/1904.08473, 2019.   
+[40] X. Ma, Y. Yang, H. Hu, J. Yang, C. Zhang, Q. Zhao, B. Liang, and Q. Liu. Offline Reinforcement Learning with Value-based Episodic Memory. In International Conference on Learning Representations, 2022.   
+[41] Y. J. Ma, D. Jayaraman, and O. Bastani. Conservative Offline Distributional Reinforcement Learning. In Advances in Neural Information Processing Systems, volume 34, 2021.   
+[42] T. Matsushima, H. Furuta, Y. Matsuo, O. Nachum, and S. Gu. Deployment-Efficient Reinforcement Learning via Model-Based Offline Optimization. In International Conference on Learning Representations, 2021.   
+[43] L. Meng, M. Wen, Y. Yang, C. Le, X. Li, W. Zhang, Y. Wen, H. Zhang, J. Wang, and B. Xu. Offline Pre-trained Multi-Agent Decision Transformer: One Big Sequence Model Tackles All SMAC Tasks. ArXiv, abs/2112.02845, 2021.   
+[44] O. Nachum, Y. Chow, B. Dai, and L. Li. DualDICE: Behavior-Agnostic Estimation of Discounted Stationary Distribution Corrections. In Advances in Neural Information Processing Systems, 2019.   
+[45] A. Nair, M. Dalal, A. Gupta, and S. Levine. Accelerating Online Reinforcement Learning with Offline Datasets. ArXiv, abs/2006.09359, 2020.   
+[46] Y. Ovadia, E. Fertig, J. Ren, Z. Nado, D. Sculley, S. Nowozin, J. V. Dillon, B. Lakshminarayanan, and J. Snoek. Can You Trust Your Model’s Uncertainty? Evaluating Predictive Uncertainty Under Dataset Shift. In Advances in Neural Information Processing Systems, 2019.   
+[47] D. Precup, R. S. Sutton, and S. Dasgupta. Off-Policy Temporal Difference Learning with Function Approximation. In International Conference on Machine Learning, 2001.   
+[48] R. Rafailov, T. Yu, A. Rajeswaran, and C. Finn. Offline Reinforcement Learning from Images with Latent Space Models. In Learning for Dynamics and Control (L4DC), 2021.   
+[49] A. Rajeswaran, V. Kumar, A. Gupta, J. Schulman, E. Todorov, and S. Levine. Learning Complex Dexterous Manipulation with Deep Reinforcement Learning and Demonstrations. ArXiv, abs/1709.10087, 2018.   
+[50] P. Rashidinejad, B. Zhu, C. Ma, J. Jiao, and S. Russell. Bridging Offline reinforcement Learning and Imitation Learning: A Tale of Pessimism. Advances in Neural Information Processing Systems, 34, 2021.   
+[51] K. Sohn, H. Lee, and X. Yan. Learning Structured Output Representation using Deep Conditional Generative Models. In Advances in Neural Information Processing Systems, 2015.   
+[52] A. Srivastava, L. Valkov, C. Russell, M. U. Gutmann, and C. Sutton. VEEGAN: Reducing Mode Collapse in GANs using Implicit Variational Learning. In Advances in Neural Information Processing Systems, 2017.   
+[53] R. S. Sutton and A. G. Barto. Reinforcement learning: An introduction. MIT press, 2018.   
+[54] R. S. Sutton, A. R. Mahmood, and M. White. An Emphatic Approach to the Problem of Off-policy Temporal-Difference Learning. Journal of Machine Learning Research, 17:2603–2631, 2016.   
+[55] J. Wang, W. Li, H. Jiang, G. Zhu, S. Li, and C. Zhang. Offline Reinforcement Learning with Reverse Model-based Imagination. In Advances in Neural Information Processing Systems, 2021.   
+[56] Q. Wang, J. Xiong, L. Han, P. Sun, H. Liu, and T. Zhang. Exponentially Weighted Imitation Learning for Batched Historical Data. In Advances in Neural Information Processing Systems, 2018.   
+[57] Z. Wang, A. Novikov, K. Zolna, J. T. Springenberg, S. E. Reed, B. Shahriari, N. Siegel, J. Merel, C. Gulcehre, N. M. O. Heess, and N. de Freitas. Critic Regularized Regression. In Advances in Neural Information Processing Systems, 2020.   
+[58] Y. Wu, G. Tucker, and O. Nachum. Behavior Regularized Offline Reinforcement Learning. ArXiv, abs/1911.11361, 2019.   
+[59] Y. Wu, S. Zhai, N. Srivastava, J. M. Susskind, J. Zhang, R. Salakhutdinov, and H. Goh. Uncertainty Weighted Actor-Critic for Offline Reinforcement Learning. In International Conference on Machine Learning, 2021.   
+[60] T. Xie, C.-A. Cheng, N. Jiang, P. Mineiro, and A. Agarwal. Bellman-Consistent Pessimism for Offline Reinforcement Learning. Advances in Neural Information Processing Systems, 34, 2021.   
+[61] S. Yang, Z. Wang, H. Zheng, Y. Feng, and M. Zhou. A Regularized Implicit Policy for Offline Reinforcement Learning. ArXiv, abs/2202.09673, 2022.   
+[62] Y. Yang, X. Ma, L. Chenghao, Z. Zheng, Q. Zhang, G. Huang, J. Yang, and Q. Zhao. Believe what you see: Implicit constraint approach for offline multi-agent reinforcement learning. Advances in Neural Information Processing Systems, 34, 2021.   
+[63] T. Yu, A. Kumar, R. Rafailov, A. Rajeswaran, S. Levine, and C. Finn. COMBO: Conservative Offline Model-Based Policy Optimization. In Advances in Neural Information Processing Systems, 2021.   
+[64] T. Yu, G. Thomas, L. Yu, S. Ermon, J. Y. Zou, S. Levine, C. Finn, and T. Ma. MOPO: Model-based Offline Policy Optimization. In Advances in Neural Information Processing Systems, 2020.   
+[65] A. Zanette, M. J. Wainwright, and E. Brunskill. Provable Benefits of Actor-Critic Methods for Offline Reinforcement Learning. In Advances in Neural Information Processing Systems, 2021.   
+[66] Y. Zhao, R. Boney, A. Ilin, J. Kannala, and J. Pajarinen. Adaptive Behavior Cloning Regularization for Stable Offline-to-Online Reinforcement Learning, 2022.   
+[67] W. Zhou, S. Bajracharya, and D. Held. PLAS: Latent Action Space for Offline Reinforcement Learning. In Conference on Robot Learning, 2020.
+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes]   
+(c) Did you discuss any potential negative societal impacts of your work? [N/A]   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes]   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes]   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes]
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes]   
+(b) Did you mention the license of the assets? [Yes]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [No]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [No]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/dev/VYYf6S67pQc/VYYf6S67pQc_content_list.json

@@ -0,0 +1,1330 @@
+[
+    {
+        "type": "text",
+        "text": "Mildly Conservative $Q$ -Learning for Offline Reinforcement Learning ",
+        "text_level": 1,
+        "bbox": [
+            233,
+            122,
+            766,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Jiafei Lyu1∗, Xiaoteng $\\mathbf { M } \\mathbf { a } ^ { 2 * }$ , Xiu Li1†, Zongqing $\\mathbf { L u ^ { 3 \\dag } }$ 1Tsinghua Shenzhen International Graduate School, Tsinghua University 2Department of Automation, Tsinghua Unversity 3School of Computer Science, Peking University {lvjf20,ma-xt17}@mails.tsinghua.edu.cn, li.xiu@sz.tsinghua.edu.cn, zongqing.lu@pku.edu.cn ",
+        "bbox": [
+            259,
+            224,
+            738,
+            311
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            347,
+            535,
+            363
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Offline reinforcement learning (RL) defines the task of learning from a static logged dataset without continually interacting with the environment. The distribution shift between the learned policy and the behavior policy makes it necessary for the value function to stay conservative such that out-of-distribution (OOD) actions will not be severely overestimated. However, existing approaches, penalizing the unseen actions or regularizing with the behavior policy, are too pessimistic, which suppresses the generalization of the value function and hinders the performance improvement. This paper explores mild but enough conservatism for offline learning while not harming generalization. We propose Mildly Conservative $Q$ -learning (MCQ), where OOD actions are actively trained by assigning them proper pseudo $Q$ values. We theoretically show that MCQ induces a policy that behaves at least as well as the behavior policy and no erroneous overestimation will occur for OOD actions. Experimental results on the D4RL benchmarks demonstrate that MCQ achieves remarkable performance compared with prior work. Furthermore, MCQ shows superior generalization ability when transferring from offline to online, and significantly outperforms baselines. Our code is publicly available at https://github.com/dmksjfl/MCQ. ",
+        "bbox": [
+            233,
+            380,
+            766,
+            613
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            641,
+            310,
+            659
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Continually interacting with the environment of online reinforcement learning (RL) is often infeasible and unrealistic, since the data collection process of the agent may be expensive, difficult, or even dangerous, especially in real-world applications. Offline RL, instead, aims at learning from a static dataset that was previously collected by some unknown process [36], hence eliminating the need for environmental interactions during training. ",
+        "bbox": [
+            174,
+            674,
+            825,
+            743
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The main challenge of offline RL is the distribution shift of state-action visitation frequency between the learned policy and the behavior policy. The evaluation of out-of-distribution (OOD) actions causes extrapolation error [14], which can be exacerbated through bootstrapping [34] and result in severe overestimation errors. Thus, keeping conservatism in value estimation is necessary in offline RL [24, 50, 60]. Previous methods achieve the conservatism by compelling the learned policy to be close to the behavior policy [14, 58, 34, 13, 57], by penalizing the learned value functions from being over-optimistic upon out-of-distribution (OOD) actions [35, 33, 59], or by learning without querying OOD samples [56, 8, 62, 32, 40]. ",
+        "bbox": [
+            174,
+            750,
+            825,
+            861
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/ad202e97216c552f7b5a99b52b8c911d09909fa8131df452eeb7c7da141e8778.jpg",
+        "image_caption": [
+            "Figure 1: Comparison of prior methods against mild conservatism. The red spots represent the dataset samples. The left figure shows that penalizing OOD actions makes the value function drop sharply at the boundary of the dataset’s support, which barriers policy learning. The central figure depicts that policy regularization keeps the policy near behavior policy, leading to undesired performance if the behavior policy is unsatisfying. On the right side, we illustrate the basic idea of mild conservatism. The estimated values for OOD actions are allowed to be high as long as it does not affect the learning for the optimal policy supported by the dataset, i.e., $Q ( s , \\bar { a } ^ { \\mathrm { o o d } } ) < \\bar { \\operatorname * { m a x } } _ { a \\in \\mathrm { S u p p o r t } ( \\mu ) } Q ( s , a ) .$ "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            169,
+            84,
+            828,
+            273
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In practice, we rely on neural networks to extract knowledge from the dataset and generalize it to the nearby unseen states and actions when facing continuous state and action spaces. In other words, we need the networks to “stitch” the suboptimal trajectories to generate the best possible trajectory supported by the dataset. Unfortunately, there is no free lunch. Conservatism, which offline RL celebrates, often limits the generalization and impedes the performance of the agent. Existing approaches are still inadequate in balancing conservatism and generalization. As illustrated in Figure 1, policy regularization is unreliable for offline RL when the data-collecting policy is poor, and value penalization methods often induce unnecessary pessimism in both the in-dataset region and OOD region. We argue that the proper conservatism should be as mild as possible. As depicted in Figure 1, we aim at well estimating the value function in the support of the dataset, and allowing value estimates upon OOD actions to be high (even higher than their optimal values) as long as $Q ( s , a ^ { \\mathrm { o o d } } ) < \\operatorname* { m a x } _ { a \\in \\mathrm { S u p p o r t } ( \\mu ) } Q ( s , a )$ is satisfied. The mild conservatism benefits generalization since value estimates upon OOD actions are slightly optimistic instead of being overly conservative. ",
+        "bbox": [
+            173,
+            425,
+            826,
+            604
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To fulfill that, we propose a novel Mildly Conservative Bellman (MCB) operator for offline RL, where we actively train OOD actions and query their $Q$ values. We theoretically analyze the convergence property of the MCB operator under the tabular MDP setting. We show that the policy induced by the MCB operator is guaranteed to behave better than the behavior policy, and can consistently improve the policy with a tighter lower bound compared with policy constraint methods or value penalization methods like CQL [35]. For practical usage, we propose the practical MCB operator and illustrate its advantages by theoretically showing that erroneous overestimation error will not occur with it. We then estimate the behavior policy with a conditional variational autoencoder (CVAE) [29, 51], and integrate the practical MCB operator with the Soft Actor-Critic (SAC) [20] algorithm. To this end, we propose our novel offline RL algorithm, Mildly Conservative $Q$ -learning (MCQ). ",
+        "bbox": [
+            174,
+            611,
+            825,
+            750
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Experimental results on the D4RL MuJoCo locomotion tasks demonstrate that MCQ surpasses recent strong baseline methods on most of the tasks, especially on non-expert datasets. Meanwhile, MCQ shows superior generalization capability when transferring from offline to online, validating our claims that mild pessimism is of importance to offline learning. ",
+        "bbox": [
+            176,
+            756,
+            825,
+            811
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Preliminaries ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            835,
+            316,
+            852
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We consider a Markov Decision Process (MDP) specified by a tuple $\\langle S , \\mathcal { A } , r , \\rho _ { 0 } , p , \\gamma \\rangle$ , where $s$ is the state space, $\\mathcal { A }$ is the action space, $r ( s , a ) : S \\times \\mathcal { A } \\mapsto \\mathbb { R }$ is the reward function, $\\rho _ { 0 } ( s )$ is the initial state distribution, $p ( s ^ { \\prime } | s , a ) : \\bar { \\mathcal { S } } \\check { \\times } \\bar { \\mathcal { A } } \\times \\bar { \\mathcal { S } } \\mapsto [ 0 , 1 ]$ is the transition probability, $\\gamma \\in [ 0 , 1 )$ is the discount factor. Reinforcement learning (RL) aims at finding a policy $\\pi ( \\cdot | s )$ such that the expected cumulative long-term rewards $J ( \\pi ) = \\mathbb { E } _ { s _ { 0 } \\sim \\rho _ { 0 } ( \\cdot ) , a _ { t } \\sim \\pi ( \\cdot \\vert s _ { t } ) , s _ { t + 1 } \\sim p ( \\cdot \\vert s _ { t } , a _ { t } ) } [ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } r ( s _ { t } , a _ { t } ) ]$ are maximized. The state-action function $Q ( s , a )$ measures the discounted return starting from state $s$ and action $a$ , and following the policy $\\pi$ . We assume that the reward function $r ( s , a )$ is bounded, i.e., $| r ( s , a ) | \\leq r _ { \\operatorname* { m a x } }$ . Given a policy $\\pi ( \\cdot | s )$ , the Bellman backup for obtaining the corresponding $Q$ function gives: ",
+        "bbox": [
+            176,
+            868,
+            823,
+            912
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            90,
+            826,
+            162
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6792174c10d69443af97b57fa4ac6728c510d29364b660c9a06a149421176627.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal T ^ { \\pi } Q ( s , a ) : = r ( s , a ) + \\gamma \\mathbb E _ { s ^ { \\prime } } \\mathbb E _ { a ^ { \\prime } \\sim \\pi ( \\cdot \\vert s ^ { \\prime } ) } [ Q ( s ^ { \\prime } , a ^ { \\prime } ) ] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            330,
+            167,
+            668,
+            188
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The $Q$ function of the optimal policy satisfies the following Bellman optimal operator: ",
+        "bbox": [
+            171,
+            191,
+            738,
+            208
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/71c9c56580527b98417d27a8c3a4ea0b5594dbd49fb28b542f53c73ea2714329.jpg",
+        "text": "$$\n\\mathcal { T } Q ( s , a ) : = r ( s , a ) + \\gamma \\mathbb { E } _ { s ^ { \\prime } } \\left[ \\operatorname* { m a x } _ { a ^ { \\prime } \\in \\mathcal { A } } Q ( s ^ { \\prime } , a ^ { \\prime } ) \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            346,
+            213,
+            651,
+            247
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In offline RL setting, the online interaction is infeasible, and we can only have access to previously collected datasets $\\bar { \\mathcal { D } } = \\{ ( s _ { i } , a _ { i } , r _ { i } , s _ { i + 1 } ^ { \\prime } , d _ { i } ) \\} _ { i = 1 } ^ { N }$ , where $d$ is the done flag. We denote the behavior policy as $\\mu ( \\cdot | s )$ . The Bellman backup relies on actions sampled from the learned policy, $a ^ { \\prime } \\sim \\pi ( \\cdot | s ^ { \\prime } )$ . However, $a ^ { \\prime }$ can lie outside of the support of $\\mu$ due to the distribution shift between $\\pi$ and $\\mu$ . The value estimates upon $a ^ { \\prime }$ can then be arbitrarily wrong, resulting in bad policy training. Unlike prior work, we actively train OOD actions by constructing them pseudo target values. In this way, we retain pessimism while enjoying better generalization. ",
+        "bbox": [
+            173,
+            252,
+            825,
+            352
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Mildly Conservative $Q$ -Learning ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            369,
+            478,
+            387
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section, we first formally define the MCB operator and characterize its dynamic programming properties in the tabular MDP setting. We further give a practical version of the MCB operator. We show that no erroneous overestimation will occur with the MCB operator. Finally, we incorporate the MCB operator with SAC [20] and present our novel offline RL algorithm. ",
+        "bbox": [
+            174,
+            401,
+            825,
+            458
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 Mildly Conservative Bellman (MCB) Operator ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            473,
+            539,
+            488
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Definition 1. The Mildly Conservative Bellman (MCB) operator is defined as ",
+        "bbox": [
+            173,
+            498,
+            684,
+            513
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1721c867296eb610d74dfd34366cc5718808b6194a0a79d0c322be7515c92272.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { T } _ { \\mathrm { M C B } } Q ( s , a ) = ( \\mathcal { T } _ { 1 } \\mathcal { T } _ { 2 } ) Q ( s , a ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            392,
+            520,
+            604,
+            536
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where ",
+        "bbox": [
+            173,
+            544,
+            215,
+            558
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/88eae016a6944540e173c86da8918c5503f2919506c70cc4aee6304269296536.jpg",
+        "text": "$$\n\\begin{array} { r l } & { { \\mathcal T } _ { 1 } Q ( s , a ) = \\left\\{ \\begin{array} { l l } { Q ( s , a ) , } & { \\mu ( a | s ) > 0 . } \\\\ { \\operatorname* { m a x } _ { a ^ { \\prime } \\sim \\mathrm { S u p p o r t } ( \\mu ( \\cdot | s ) ) } Q ( s , a ^ { \\prime } ) - \\delta , } & { e l s e . } \\end{array} \\right. } \\\\ & { { \\mathcal T } _ { 2 } Q ( s , a ) = \\left\\{ \\begin{array} { l l } { r ( s , a ) + \\gamma \\mathbb { E } _ { s ^ { \\prime } } \\left[ \\operatorname* { m a x } _ { a ^ { \\prime } \\in A } Q ( s ^ { \\prime } , a ^ { \\prime } ) \\right] , } & { \\mu ( a | s ) > 0 , } \\\\ { Q ( s , a ) , } & { e l s e . } \\end{array} \\right. } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            272,
+            553,
+            715,
+            627
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The basic idea behind this novel operator is that if the learned policy outputs actions that lie in the support region of $\\mu$ , then we go for backup; while if OOD actions are generated, we deliberately replace their value estimates with $\\begin{array} { r } { \\operatorname* { m a x } _ { a ^ { \\prime } \\sim \\operatorname { S u p p o r t } ( \\mu ( \\cdot | s ) ) } Q ( s , a ^ { \\prime } ) - \\delta } \\end{array}$ , where $\\delta > 0$ can be arbitrarily small. That is, different from standard Bellman backup, we set up a checking procedure (i.e., $\\mathcal { T } _ { 1 , }$ ) of whether the previous backup (i.e., $\\mathcal { T } _ { 2 }$ ) involves OOD actions for the update. Intrinsically, we construct pseudo target values for OOD actions. We subtract a small positive $\\delta$ such that OOD actions will not be chosen when executing policy via arg $\\operatorname* { m a x } _ { a \\in \\mathcal { A } } Q ( s , a )$ . ",
+        "bbox": [
+            173,
+            633,
+            826,
+            732
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "For a better understanding of the MCB operator, we theoretically analyze its dynamic programming properties in the tabular MDP setting. All proofs are deferred to Appendix A. ",
+        "bbox": [
+            174,
+            737,
+            823,
+            766
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Proposition 1. In the support region of the behavior policy, i.e., Support $( \\mu )$ , the MCB operator is a $\\gamma$ -contraction operator in the $\\mathcal { L } _ { \\infty }$ norm, and any initial $Q$ function can converge to a unique fixed point by repeatedly applying $\\mathcal { T } _ { \\mathrm { M C B } }$ . ",
+        "bbox": [
+            173,
+            770,
+            825,
+            813
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Proposition 2 (Behave at least as well as behavior policy). Denote $Q _ { \\mathrm { M C B } }$ as the unique fixed point acquired by the MCB operator, then in $\\operatorname { S u p p o r t } ( \\mu )$ we have: $Q _ { \\mu } \\leq Q _ { \\mathrm { M C B } } \\leq Q _ { \\mu ^ { * } }$ , where $Q _ { \\mu }$ is the $Q$ function of the behavior policy and $Q _ { \\mu ^ { * } }$ is the $Q$ function of the optimal policy in the batch. ",
+        "bbox": [
+            173,
+            815,
+            825,
+            859
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Proposition 2 indicates that the policy induced by the MCB operator can behave at least as well as the behavior policy, and can approximate the optimal batch-constraint policy. Apart from this advantage, we further show that the MCB operator results in milder conservatism. We start by observing that value penalization method, like CQL [35], guarantees that the learned value function $\\hat { Q } ^ { \\pi } ( s , a )$ is a lower bound of its true value $Q ^ { \\pi } ( s , a )$ . It is also ensured that following such conservative update leads to a safe policy improvement, i.e., $\\begin{array} { r } { J ( \\pi _ { \\mathrm { C Q L } } ) \\ge J ( \\mu ) - \\mathcal { O } ( \\frac { 1 } { ( 1 - \\gamma ) ^ { 2 } } ) } \\end{array}$ (Theorem 3.6 in [35]). For explicit policy constraint methods, e.g., $\\mathrm { T D } 3 { + } \\mathrm { B C }$ [13], the learned policy $\\pi _ { p }$ mimics the behavior policy $\\mu$ , and can hardly behave significantly better than $\\mu$ . We show in Proposition 3 that explicit policy constraint methods also exhibit a safe policy improvement, $\\begin{array} { r } { J ( \\pi _ { p } ) \\geq J ( \\dot { \\mu } ) - \\mathcal { O } ( \\frac { 1 } { ( 1 - \\gamma ) ^ { 2 } } ) } \\end{array}$ ( 1(1−γ)2 ), while the MCB operator can consistently improve the policy with a tighter lower bound. ",
+        "bbox": [
+            174,
+            868,
+            825,
+            911
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            90,
+            826,
+            194
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 3 (Milder Pessimism). Suppose there exists an explicit policy constraint offline reinforcement learning algorithm such that the $K L$ -divergence of the learned policy $\\pi _ { p } ( \\cdot | s )$ and the behavior policy $\\mu ( \\cdot | s )$ is optimized to guarantee max $( \\mathrm { K L } ( \\mu , \\pi _ { p } ) , \\mathrm { K L } ( \\pi _ { p } , \\mu ) ) \\leq \\dot { \\epsilon }$ , ∀s. Denote $\\begin{array} { r } { \\epsilon _ { \\mu } ^ { \\pi _ { p } } = \\operatorname* { m a x } _ { s } | \\mathbb { E } _ { a \\sim \\pi _ { p } } A ^ { \\mu } ( s , a ) | . } \\end{array}$ , where $A ^ { \\mu } ( s , a )$ is the advantage function. Then ",
+        "bbox": [
+            173,
+            196,
+            825,
+            256
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3821760f8618dcd8a843ed759ea3e4a8c868b3b2c61c34c6c5c950b22abd3bc1.jpg",
+        "text": "$$\nJ ( \\pi _ { p } ) \\geq J ( \\mu ) - \\frac { \\sqrt { 2 } \\gamma \\epsilon _ { \\mu } ^ { \\pi _ { p } } } { ( 1 - \\gamma ) ^ { 2 } } \\sqrt { \\epsilon } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            397,
+            262,
+            601,
+            299
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "while for the policy $\\pi _ { \\mathrm { M C B } }$ learned by applying the MCB operator, we have ",
+        "bbox": [
+            176,
+            303,
+            665,
+            319
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/95eb0d16de4d35931aa082508432729e2a2631ff85cde32040e7624425b59cc3.jpg",
+        "text": "$$\nJ ( \\pi _ { \\mathrm { M C B } } ) \\geq J ( \\mu ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            436,
+            324,
+            562,
+            342
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In summary, the MCB operator benefits the offline learning in two aspects: (1) the operator is a contraction, and any initial $Q$ functions are guaranteed to converge to a unique fixed point; (2) the learned policy of the MCB operator is ensured to be better than the behavior policy, and reserve milder pessimism compared with policy constraint methods or CQL. ",
+        "bbox": [
+            174,
+            353,
+            825,
+            411
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 Practical MCB Operator ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            426,
+            387,
+            441
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In practice, it is intractable to acquire $\\begin{array} { r } { \\operatorname* { m a x } _ { a ^ { \\prime } \\sim \\operatorname { S u p p o r t } ( \\mu ( \\cdot | s ) ) } Q ( s , a ^ { \\prime } ) } \\end{array}$ in $\\mathcal { T } _ { 1 }$ of Eq. (4) in continuous control domains, and the behavior policy is often unknown. Thus, we fit an empirical behavior policy $\\hat { \\mu }$ with supervised learning based on the static dataset. The pseudo target values for the OOD actions are then computed by sampling $N$ actions from $\\hat { \\mu }$ , and taking maximum over their value evaluation. Formally, we define the practical MCB operator below, accompanied by the theoretical analysis. ",
+        "bbox": [
+            173,
+            450,
+            826,
+            521
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Definition 2. The practical Mildly Conservative Bellman (MCB) operator is defined as ",
+        "bbox": [
+            169,
+            525,
+            748,
+            540
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/03f691723e8c8f0f3a9a123c179ecccb0652c0b5bd4088beff3c53fbe2257f76.jpg",
+        "text": "$$\n\\hat { \\mathcal { T } } _ { \\mathrm { M C B } } Q ( s , a ) = ( \\hat { \\mathcal { T } } _ { 1 } \\mathcal { T } _ { 2 } ) Q ( s , a ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            392,
+            545,
+            604,
+            565
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where ",
+        "bbox": [
+            173,
+            571,
+            215,
+            585
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/19a6d8ab7c6e4cd3f282e0a908e50c00ce26262e3b854eb379e979e99e84e897.jpg",
+        "text": "$$\n\\begin{array} { r } { \\hat { \\mathcal { T } } _ { 1 } Q ( s , a ) = \\left\\{ \\begin{array} { l l } { Q ( s , a ) , \\qquad ~ } & { \\mu ( a | s ) > 0 . } \\\\ { \\mathbb { E } _ { \\{ a _ { i } ^ { \\prime } \\} ^ { N } \\sim \\hat { \\mu } ( \\cdot | s ) } \\left[ \\operatorname* { m a x } _ { a ^ { \\prime } \\sim \\{ a _ { i } ^ { \\prime } \\} ^ { N } } Q ( s , a ^ { \\prime } ) \\right] , } & { e l s e . } \\end{array} \\right. } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            259,
+            580,
+            728,
+            623
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Compared with Eq. (4), we make a small modification of $\\mathcal { T } _ { 1 }$ , and keep $\\mathcal { T } _ { 2 }$ unchanged. There is no need to subtract $\\delta$ here as generally $\\begin{array} { r } { { \\mathbb E } _ { \\{ a _ { i } ^ { \\prime } \\} ^ { N } \\sim \\hat { \\mu } ( \\cdot | s ) } \\left[ \\operatorname* { m a x } _ { a ^ { \\prime } \\sim \\{ a _ { i } ^ { \\prime } \\} ^ { N } } Q ( s , a ^ { \\prime } ) \\right] \\leq \\operatorname* { m a x } _ { a ^ { \\prime } \\sim \\mathrm { S u p p o r t } ( \\mu ) } Q ( s , a ^ { \\prime } ) . } \\end{array}$ . The practical MCB operator is much easier to implement in practice. We show that the practical MCB operator is still a $\\gamma$ -contraction in the support region of the behavior policy $\\mu$ . ",
+        "bbox": [
+            173,
+            631,
+            826,
+            696
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 4. Proposition 1 still holds for the practical MCB operator. ",
+        "bbox": [
+            173,
+            700,
+            648,
+            715
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Since we fit the empirical distribution $\\hat { \\mu }$ of the behavior policy $\\mu$ , there may exist a shift between $\\hat { \\mu }$ and $\\mu$ , especially when we represent the policy via neural networks. That suggests that OOD actions $a ^ { \\prime }$ can still be sampled from $\\hat { \\mu }$ such that $a ^ { \\prime } \\overset { \\cdot } { \\notin } \\operatorname { S u p p o r t } ( \\mu ( \\cdot | s ) )$ . Our last main result reveals that erroneous overestimation issue will not occur with the aid of the practical MCB operator. ",
+        "bbox": [
+            173,
+            726,
+            826,
+            781
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 5 (No erroneous overestimation will occur). Assuming that $\\operatorname* { s u p } _ { s } D _ { \\mathrm { T V } } ( \\hat { \\mu } ( \\cdot | s ) \\quad | |$ $\\begin{array} { r } { \\mu ( \\cdot | \\bar { s } ) ) \\leq \\epsilon < \\frac { 1 } { 2 } } \\end{array}$ , we have ",
+        "bbox": [
+            174,
+            785,
+            823,
+            815
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8b8f24c603d5f6227a97ac00f85a8b87b0da37edcf03374eee68bdca2cc080a5.jpg",
+        "text": "$$\n\\mathbb { E } _ { \\{ a _ { i } ^ { \\prime } \\} ^ { N } \\sim \\hat { \\mu } ( \\cdot | s ) } \\left[ \\operatorname* { m a x } _ { a ^ { \\prime } \\in \\{ a _ { i } ^ { \\prime } \\} ^ { N } } Q ( s , a ^ { \\prime } ) \\right] \\leq \\operatorname* { m a x } _ { a ^ { \\prime } \\in \\mathrm { S u p p o r t } ( \\mu ( \\cdot | s ) ) } Q ( s , a ^ { \\prime } ) + ( 1 - ( 1 - 2 \\epsilon ) ^ { N } ) \\frac { r _ { \\operatorname* { m a x } } } { 1 - \\gamma } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            204,
+            820,
+            792,
+            857
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Remark: This proposition generally requires a comparatively well-fitted empirical behavior policy $\\hat { \\mu }$ . In practice, we model $\\hat { \\mu }$ with a CVAE. In most cases, CVAE can already fit the dataset well and guarantee a good performance. Whereas there may exist some situations, e.g., the dataset is highly multi-modal, then one can replace the CVAE as the conditional GAN (CGAN) to better capture the different modes in the dataset as depicted in [61]. We believe generative models like CGAN will be a good choice by then. ",
+        "bbox": [
+            174,
+            868,
+            825,
+            911
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            176,
+            90,
+            823,
+            132
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Intuitively, the above conclusion says that if the empirical behavior policy $\\hat { \\mu }$ well fits $\\mu$ , i.e., $\\epsilon$ is small enough, then regardless of how $\\{ a _ { i } ^ { \\prime } \\} ^ { N }$ are sampled, the pseudo target value will approximate the maximum $Q$ -value within the dataset’s support with high probability. The extrapolation error is under the scale of $\\begin{array} { r } { ( 1 - ( 1 - 2 \\epsilon ) ^ { N } ) \\frac { r _ { \\operatorname* { m a x } } } { 1 - \\gamma } } \\end{array}$ . We expect a good empirical behavior policy such that most of the actions sampled from it will be in-distribution. However, if $\\epsilon$ is large, $N$ can act as a trade-off parameter. The smaller $N$ we use, the more conservative we are. Fortunately, we find empirically that our method performs well in a large interval of $N$ over different tasks (see Section 4.2). Hence, it is safe to fix a $N$ in practice. ",
+        "bbox": [
+            174,
+            138,
+            825,
+            253
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 Algorithm ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            268,
+            284,
+            284
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As aforementioned, we often cannot get prior information about the behavior policy $\\mu$ . Thus, we need to empirically fit a behavior policy $\\hat { \\mu }$ with supervised learning for applying the practical MCB operator. Our algorithm, Mildly Conservative $Q$ -learning (MCQ), trains an additional generative model, which is also adopted by many prior work [14, 17, 33, 67]. We build our novel offline algorithm upon an off-the-shelf off-policy online RL algorithm, Soft Actor-Critic (SAC) [20]. ",
+        "bbox": [
+            174,
+            294,
+            825,
+            363
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Modelling the behavior policy with the CVAE. We utilize a conditional variational autoencoder (CVAE) [29, 51, 14] to model the behavior policy $\\mu$ . Given a fixed logged dataset, the goal of the CVAE is to reconstruct actions conditioned on the states such that the reconstructed actions come from the same distribution as the actions in the dataset, i.e., $\\mu ( \\cdot | s )$ . That generally satisfies the assumption we make in Proposition 5. As concerned by [32], training a generative model like CVAE still may produce out-of-dataset actions, which leads to extrapolation error since undefined $Q$ values can be possibly queried. Prior methods, like BCQ [14], do not well address such issue. While for our algorithm, such concern is mitigated because overestimation error is actually under control as is guaranteed by Proposition 5. ",
+        "bbox": [
+            173,
+            369,
+            825,
+            494
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The CVAE $G _ { \\omega } ( s )$ parameterized by $\\omega$ is made up of an encoder $E _ { \\xi } ( s , a )$ and a decoder $D _ { \\psi } ( s , z )$ parameterized by $\\xi$ , $\\psi$ respectively, $\\omega = \\{ \\xi , \\psi \\}$ . The CVAE is optimized by maximizing its variational lower bound, which is equivalent to minimizing the following objective function. ",
+        "bbox": [
+            173,
+            501,
+            825,
+            542
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/106eb7e9c73543ed6c2db7cd6d06bc028aa4998c4eb2e301baae28daeda5c1c9.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { C V A E } } = \\mathbb { E } _ { ( s , a ) \\sim \\mathcal { D } , z \\sim E _ { \\xi } ( s , a ) } \\left[ ( a - D _ { \\psi } ( s , z ) ) ^ { 2 } + \\mathrm { K L } \\left( E _ { \\xi } ( s , a ) , \\mathcal { N } ( 0 , { \\bf I } ) \\right) \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            248,
+            547,
+            746,
+            566
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\operatorname { K L } ( p , q )$ denotes the KL-divergence between probability distribution $p ( \\cdot )$ and $q ( \\cdot )$ , and $\\mathbf { I }$ is the identity matrix. When sampling actions from the CVAE, we first sample a latent variable $z$ from the prior distribution, which is set to be multivariate normal distribution $\\mathcal { N } ( 0 , \\bf { I } )$ , and then pass it in conjunction with the state $s$ into the decoder $D _ { \\psi } ( s , z )$ to get the desired decoded action. ",
+        "bbox": [
+            173,
+            573,
+            825,
+            630
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "It is also worth noting that we do not choose GAN [19] as the generative model because it is known to suffer from training instability and mode collapse [52, 6, 5]. Also, GAN consumes much more time and memories to train compared with the CVAE. ",
+        "bbox": [
+            174,
+            635,
+            825,
+            676
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Loss functions. In deep RL, the $Q$ function is represented with a neural network parameterized by $\\theta$ and is updated via minimizing the temporal difference (TD) loss $\\mathbb { E } _ { s , a , r , s ^ { \\prime } } [ ( Q _ { \\theta } ( s , a ) - \\mathcal { T } Q ( s , a ) ) ^ { 2 } ]$ . We actually are performing the regression task $( s , a ) \\mapsto \\mathcal { T } Q ( s , a )$ to train the $Q$ function. The target value ${ \\mathcal { T } } Q ( s , a )$ is usually computed by utilizing a lagging target network parameterized by $\\theta ^ { \\prime }$ without gradient backpropagation. As a typical actor-critic [30, 31, 53] algorithm, SAC uses its critic networks to perform value estimation and uses a separate actor network for policy improvement. In order to incorporate the MCB operator with the off-the-shelf SAC algorithm, we need to check whether the sampled action $a ^ { \\prime } \\sim \\bar { \\pi } ( \\cdot | s )$ lies outside of the behavior policy’s support, i.e., whether $\\mu ( a ^ { \\prime } | s ) > 0$ . However, such a criterion is not reliable, because the true behavior policy $\\mu$ is unknown and it is difficult to examine whether $\\mu ( a ^ { \\prime } | s ) > 0$ in practice. It is also problematic if we rely on the empirical behavior policy $\\hat { \\mu }$ to check whether $a ^ { \\prime }$ is OOD as $\\hat { \\mu }$ itself can produce OOD actions. ",
+        "bbox": [
+            173,
+            683,
+            825,
+            835
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We then resort to constructing an auxiliary loss for OOD actions and integrating it with the standard Bellman error. Specifically, we sample $a ^ { \\mathrm { o o d } }$ from the learned policy $\\pi ( \\cdot | _ { s } \\mathrm { i n } )$ based on the sampled state $s ^ { \\mathrm { i n } } \\sim \\mathcal { D }$ from the dataset and assign them pseudo target values based on the practical MCB operator. Note that the superscript ood is used to distinguish from the in-dataset real actions, and $a ^ { \\mathrm { { \\bar { o } o d } } }$ is not necessarily an OOD action. We remark that if $a ^ { \\mathrm { o o d } } \\in \\operatorname { S u p p o r t } ( \\mu ( \\cdot | s ) )$ , the pseudo $Q$ ",
+        "bbox": [
+            174,
+            842,
+            823,
+            911
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 Mildly Conservative $Q$ -learning (MCQ) ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            90,
+            526,
+            106
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "1: Initialize CVAE $G _ { \\omega }$ , critic networks $Q _ { \\theta _ { 1 } } , Q _ { \\theta _ { 2 } }$ and actor network $\\pi _ { \\phi }$ with random parameters   \n2: Initialize target networks $\\theta _ { 1 } ^ { \\prime }  \\theta _ { 1 } , \\theta _ { 2 } ^ { \\prime }  \\theta _ { 2 }$ and offline replay buffer $\\mathcal { D }$ .   \n3: for $t = 1$ to $T$ do   \n4: Sample a mini-batch $B = \\{ ( s , a , r , s ^ { \\prime } , d ) \\}$ from $\\mathcal { D }$ , where $d$ is the done flag   \n5: Train CVAE via minimizing Eq. (10)   \n6: Get target value: $\\begin{array} { r } { y = r ( s , \\bar { a } ) + \\gamma \\left[ \\operatorname* { m i n } _ { i = 1 , 2 } Q _ { \\theta _ { i } ^ { \\prime } } ( s ^ { \\prime } , a ^ { \\prime } ) - \\alpha \\log \\pi _ { \\phi } ( a ^ { \\prime } | s ^ { \\prime } ) \\right] , a ^ { \\prime } \\sim \\pi _ { \\phi } ( \\cdot | s ^ { \\prime } ) } \\end{array}$   \n7: Sample $N$ actions from $\\pi$ based on each $s$ and $s ^ { \\prime }$ , set $s ^ { \\mathrm { i n } } = \\{ s , s ^ { \\prime } \\}$   \n8: Compute the target value for the OOD actions via Eq. (13)   \n9: Update critic $\\theta _ { i }$ with gradient descent via minimizing Eq. (11)   \n10: Update actor $\\phi$ with gradient ascent via Eq. (14)   \n11: Update target networks: $\\theta _ { i } ^ { \\prime }  \\tau \\theta _ { i } + ( 1 - \\tau ) \\theta _ { i } ^ { \\prime } , i = 1 , 2$   \n12: end for ",
+        "bbox": [
+            178,
+            108,
+            810,
+            279
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "value will not negatively affect the evaluation upon it, because in-distribution actions are still trained to approximate the optimal batch-constraint $Q$ value. In this way, we actively train both possible OOD actions and in-distribution actions simultaneously via $o o D$ sampling. The resulting objective function for the critic networks is presented in Eq. (11). ",
+        "bbox": [
+            173,
+            304,
+            823,
+            361
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3eb4aaa17bda69468e46e1a79ccce3076522dd064e68148480b7ef7fc8698573.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { c r i t i c } } = \\lambda \\mathbb { E } _ { ( s , a , r , s ^ { \\prime } ) \\sim \\mathcal { D } } \\left[ ( Q _ { \\theta _ { i } } ( s , a ) - y ) ^ { 2 } \\right] + ( 1 - \\lambda ) \\mathbb { E } _ { s ^ { \\mathrm { i n } } \\sim \\mathcal { D } , a ^ { \\mathrm { o o d } } \\sim \\pi } \\left[ ( Q _ { \\theta _ { i } } ( s ^ { \\mathrm { i n } } , a ^ { \\mathrm { o o d } } ) - y ^ { \\prime } ) ^ { 2 } \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            179,
+            366,
+            789,
+            386
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where the target value for the in-distribution actions gives ",
+        "bbox": [
+            178,
+            388,
+            558,
+            404
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a00c6643dfba766258da96fa5b8ce403492873e0ea7cc2fbb683b645290412d6.jpg",
+        "text": "$$\ny = r ( s , a ) + \\gamma \\left[ \\operatorname* { m i n } _ { i = 1 , 2 } Q _ { \\theta _ { i } ^ { \\prime } } ( s ^ { \\prime } , a ^ { \\prime } ) - \\alpha \\log \\pi _ { \\phi } ( a ^ { \\prime } | s ^ { \\prime } ) \\right] , \\alpha \\in \\mathbb { R } _ { + } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            290,
+            407,
+            705,
+            443
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "which follows the standard target value of vanilla SAC. The hyperparameter $\\lambda$ balances the indistribution data training and OOD action training. Following the formulas of the practical MCB operator in Eq. (9), the pseudo target value for the OOD action is computed by: ",
+        "bbox": [
+            174,
+            446,
+            825,
+            488
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/dbaf72ac100486c592298433d8c8daaaecdcbc66ed2de7f8392d5fab8623092b.jpg",
+        "text": "$$\ny ^ { \\prime } = \\operatorname* { m i n } _ { j = 1 , 2 } \\mathbb { E } _ { \\{ a _ { i } ^ { \\prime } \\} ^ { N } \\sim \\hat { \\mu } } \\left[ \\operatorname* { m a x } _ { a ^ { \\prime } \\sim \\{ a _ { i } ^ { \\prime } \\} ^ { N } } Q _ { \\theta _ { j } } ( s ^ { \\mathrm { i n } } , a ^ { \\prime } ) \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            346,
+            493,
+            651,
+            529
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Note that we experimentally find that replacing the min operator with a mean operator does not raise much difference in performance. We hence take advantage of the min operator to fulfill the pseudo clipped double $Q$ -learning for OOD actions. ",
+        "bbox": [
+            174,
+            532,
+            825,
+            575
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The policy is then optimized by solving the following optimization problem: ",
+        "bbox": [
+            174,
+            580,
+            676,
+            595
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7e4cbad32eb9c7aac5055eb1522c6ba7c4473a41fa06db477b0177987ee2ccb2.jpg",
+        "text": "$$\n\\pi _ { \\phi } : = \\operatorname* { m a x } _ { \\phi } \\mathbb { E } _ { s \\sim \\mathcal { D } , a \\sim \\pi _ { \\phi } ( \\cdot | s ) } \\left[ \\operatorname* { m i n } _ { i = 1 , 2 } Q _ { \\theta _ { i } } ( s , a ) - \\alpha \\log \\pi _ { \\phi } ( \\cdot | s ) \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            295,
+            601,
+            702,
+            635
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We detail the learning procedure of our MCQ in Algorithm 1. Different from [13], our method does not require normalization over states or value functions. The only change we make to the vanilla SAC algorithm is an extra auxiliary loss term (blue term in Eq. (11)) such that OOD actions are actively and properly trained. The additional critic loss term can also be plugged into other off-policy online RL algorithms directly. As an evidence, we combine the MCB operator with TD3 [15], yielding a deterministic version of MCQ. Please refer to Appendix B for more details. ",
+        "bbox": [
+            173,
+            638,
+            825,
+            723
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            741,
+            312,
+            758
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this section, we first empirically demonstrate the effectiveness and advantages of our proposed MCQ algorithm on D4RL benchmarks [12]. We then conduct a detailed parameter study to show the hyperparameter sensitivity of MCQ. We also experimentally illustrate that the value estimation of MCQ will not incur severe overestimation and pessimistic value estimates are witnessed in practice. Finally, we show the superior offline-to-online fine-tuning benefits of MCQ on some MuJoCo datasets. ",
+        "bbox": [
+            174,
+            772,
+            826,
+            842
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1 Results on MuJoCo Datasets ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            857,
+            411,
+            872
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We experimentally compare our MCQ against behavior cloning (BC), SAC, and several recent strong baseline methods, CQL [35], UWAC [59], TD3+BC [13], and IQL [32], on D4RL [12] benchmarks. ",
+        "bbox": [
+            173,
+            882,
+            823,
+            912
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We choose these methods as they typically represent different categories of model-free offline RL, i.e., CQL is a value penalization method, $\\mathrm { T D } 3 { + } \\mathrm { B C }$ involves explicit policy constraint (BC loss), UWAC relies on uncertainty estimation for training, and IQL learns without querying OOD samples. ",
+        "bbox": [
+            174,
+            92,
+            823,
+            133
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We conduct experiments on MuJoCo locomotion tasks, which are made up of five types of datasets (random, medium, medium-replay, medium-expert, and expert), yielding a total of 15 datasets. We use the most recently released \"-v2\" datasets for performance evaluation. The results of BC and SAC are acquired by using our implemented code. The results of CQL and UWAC are obtained by running their official codes, because the reported scores in their papers are not obtained on MuJoCo \"-v2\" datasets. We take the results of $\\mathrm { T D } 3 { + } \\mathrm { B C }$ from its original paper (Table 7 in [13]). Since the IQL paper does not report its performance on MuJoCo random and expert datasets, we run IQL using the official codebase on them and take the results on medium, medium-replay, medium-expert datasets from its original paper directly. All methods are run for 1M gradient steps. ",
+        "bbox": [
+            173,
+            138,
+            825,
+            265
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/4780d2533eed768c83a0f8be4c51b29c5dbb1eb8c822a946561b324908c65d7d.jpg",
+        "table_caption": [
+            "Table 1: Normalized average score comparison of MCQ against baseline methods on D4RL benchmarks over the final 10 evaluations. 0 corresponds to a random policy and 100 corresponds to an expert policy. The experiments are run on MuJoCo \"-v2\" datasets over 4 random seeds. $\\mathbf { r } =$ random, $\\mathbf { m } =$ medium, $\\mathrm { m - r = }$ medium-replay, $\\mathbf { m } { - } \\mathbf { e } =$ medium-expert, ${ \\bf e } =$ expert. We bold the highest mean. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Task Name</td><td>BC</td><td>SAC</td><td>CQL</td><td>UWAC</td><td>TD3+BC</td><td>IQL</td><td>MCQ (ours)</td></tr><tr><td>halfcheetah-r</td><td>2.2±0.0</td><td>29.7±1.4</td><td>17.5±1.5</td><td>2.3±0.0</td><td>11.0±1.1</td><td>13.1±1.3</td><td>28.5±0.6</td></tr><tr><td>hopper-r</td><td>3.7±0.6</td><td>9.9±1.5</td><td>7.9±0.4</td><td>2.7±0.3</td><td>8.5±0.6</td><td>7.9±0.2</td><td>31.8±0.5</td></tr><tr><td>walker2d-r</td><td>1.3±0.1</td><td>0.9±0.8</td><td>5.1±1.3</td><td>2.0±0.4</td><td>1.6±1.7</td><td>5.4±1.2</td><td>17.0±3.0</td></tr><tr><td>halfcheetah-m</td><td>43.2±0.6</td><td>55.2±27.8</td><td>47.0±0.5</td><td>42.2±0.4</td><td>48.3±0.3</td><td>47.4±0.2</td><td>64.3±0.2</td></tr><tr><td>hopper-m</td><td>54.1±3.8</td><td>0.8±0.0</td><td>53.0±28.5</td><td>50.9±4.4</td><td>59.3±4.2</td><td>66.2±5.7</td><td>78.4±4.3</td></tr><tr><td>walker2d-m</td><td>70.9±11.0</td><td>-0.3±0.2</td><td>73.3±17.7</td><td>75.4±3.0</td><td>83.7±2.1</td><td>78.3±8.7</td><td>91.0±0.4</td></tr><tr><td>halfcheetah-m-r</td><td>37.6±2.1</td><td>0.8±1.0</td><td>45.5±0.7</td><td>35.9±3.7</td><td>44.6±0.5</td><td>44.2±1.2</td><td>56.8±0.6</td></tr><tr><td>hopper-m-r</td><td>16.6±4.8</td><td>7.4±0.5</td><td>88.7±12.9</td><td>25.3±1.7</td><td>60.9±18.8</td><td>94.7±8.6</td><td>101.6±0.8</td></tr><tr><td>walker2d-m-r halfcheetah-m-e</td><td>20.3±9.8</td><td>-0.4±0.3</td><td>81.8±2.7</td><td>23.6±6.9</td><td>81.8±5.5</td><td>73.8±7.1</td><td>91.3±5.7</td></tr><tr><td></td><td>44.0±1.6</td><td>28.4±19.4</td><td>75.6±25.7</td><td>42.7±0.3</td><td>90.7±4.3</td><td>86.7±5.3</td><td>87.5±1.3</td></tr><tr><td>hopper-m-e</td><td>53.9±4.7</td><td>0.7±0.0</td><td>105.6±12.9</td><td>44.9±8.1</td><td>98.0±9.4</td><td>91.5±14.3</td><td>111.2±0.1</td></tr><tr><td>walker2d-m-e</td><td>90.1±13.2</td><td>1.9±3.9</td><td>107.9±1.6</td><td>96.5±9.1</td><td>110.1±0.5</td><td>109.6±1.0</td><td>114.2±0.7</td></tr><tr><td>Average Above</td><td>36.5</td><td>11.3</td><td>59.1</td><td>37.0</td><td>58.2</td><td>59.9</td><td>72.8</td></tr><tr><td>halfcheetah-e</td><td>91.8±1.5</td><td>-0.8±1.8</td><td>96.3±1.3</td><td>92.9±0.6</td><td>96.7±1.1</td><td>95.0±0.5</td><td>96.2±0.4</td></tr><tr><td>hopper-e walker2d-e</td><td>107.7±0.7</td><td>0.7±0.0</td><td>96.5±28.0</td><td>110.5±0.5</td><td>107.8±7</td><td>109.4±0.5</td><td>111.4±0.4</td></tr><tr><td></td><td>106.7±0.2</td><td>0.7±0.3</td><td>108.5±0.5</td><td>108.4±0.4</td><td>110.2±0.3</td><td>109.9±1.2</td><td>107.2±1.1</td></tr><tr><td>Total Average</td><td>49.6</td><td>9.0</td><td>67.3</td><td>50.4</td><td>67.6</td><td>68.9</td><td>79.2</td></tr></table>",
+        "bbox": [
+            176,
+            339,
+            821,
+            603
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In our experiments, we set the number of sampled actions $N = 1 0$ by default and tune the weighting coefficient $\\lambda$ . We report the $\\lambda$ used for all tasks in Appendix C, along with details on the experiments and implementation. We summarize the normalized average score comparison of MCQ against recent baselines in Table 1. Unsurprisingly, we observe that MCQ behaves better than BC on all of the tasks, which is consistent with our theoretical analysis in Proposition 2 and 3. MCQ also significantly outperforms the base SAC algorithm. Prior offline RL methods struggle for good performance on non-expert datasets like random and medium-replay, while MCQ surpasses them with a remarkable margin on many non-expert datasets. We attribute the less satisfying performance of prior offline RL methods to their strict conservatism, which restricts their generalization beyond the support of the dataset and leads to limited performance. The results, therefore, validate our claim that milder pessimism is more we need for offline learning. Furthermore, MCQ is also competitive to baselines on expert datasets. MCQ achieves the best performance on 11 out of 15 datasets, yielding a total average score of 72.8 on non-expert datasets, and an average score of 79.2 on all 15 datasets. Whereas the second best method, IQL, has an average score of 59.9 on non-expert datasets and a total average score of 68.9 across all tasks. ",
+        "bbox": [
+            174,
+            618,
+            825,
+            825
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2 Parameter Study ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            843,
+            331,
+            858
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this subsection, we conduct a detailed parameter study on MCQ. MCQ generally contains two hyperparameters, weighting coefficient $\\lambda$ and number of sampled actions $N$ . To demonstrate the parameter sensitivity of MCQ, we choose two datasets from MuJoCo locomotion tasks and conduct experiments on them, halfcheetah-medium-v2, and hopper-medium-replay-v2. The experiments are run for 1M gradient steps over 4 different random seeds. ",
+        "bbox": [
+            176,
+            869,
+            823,
+            911
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/19fe1db412940950351ea94a4e995d8979f361b533ea62a8c82c16ec795be03e.jpg",
+        "image_caption": [
+            "Figure 2: Parameter study and $Q$ function estimation on halfcheetah-medium-v2 and hopper-mediumreplay-v2. The shaded region captures the standard deviation. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
+            98,
+            813,
+            380
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            443,
+            823,
+            470
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Weighting coefficient $\\lambda$ . The weighting coefficient $\\lambda$ is a critical hyperparameter for MCQ, which directly controls the balance between in-distribution actions training and OOD actions training. If we set $\\lambda = 1$ , then MCQ degenerates into the base SAC algorithm. If $\\lambda$ leans towards 0, the critics will be overwhelmed by OOD actions. Intuitively, one ought not to use small $\\lambda$ , because more weights are desired for standard Bellman error such that in-distribution state-action pairs can be well-trained. We observe significant performance drop with smaller $\\lambda$ in Figure 2(a) and 2(b). Also, we find that choosing $0 . 7 \\leq \\lambda < 1$ generally induces good performance. ",
+        "bbox": [
+            173,
+            477,
+            825,
+            574
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Number of sampled actions $N$ . $N$ works as a regularizer to control the potential extrapolation error. In case the behavior policy $\\mu$ is known, we require $N$ to be as large as possible to better estimate the maximum $Q$ value. While in practice, we leverage the CVAE to approximate $\\mu$ , from which OOD actions can be sampled. $N$ then plays a role to balance pessimism and generalization. To see the influence of $N$ , we fix $\\lambda = 0 . 9 5$ for the two datasets. Experimental results in Figure 2(d) and 2(e) indicate that MCQ is insensitive to $N$ for a wide range of $N$ . We therefore set $N = 1 0$ by default. ",
+        "bbox": [
+            174,
+            580,
+            825,
+            664
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Value estimation. We present the $Q$ value estimates with respect to (w.r.t.) $\\lambda$ and $N$ in Figure 2(c) and 2(f). The $Q$ estimation is calculated via $\\mathbb { E } _ { i = 1 , 2 } \\mathbb { E } _ { ( s , a ) \\sim \\mathcal { D } } \\big [ Q _ { \\theta _ { i } } ( s , a ) \\big ]$ . The results illustrate that (1) smaller $\\lambda$ will incur severe underestimation issue (as depicted by Figure 2(c), $Q$ values collapse with $\\lambda = 0 . 5$ or $\\lambda = 0 . 3$ ); (2) no overestimation is observed, even with a large $\\lambda = 0 . 9 5$ , which validates the theoretical result in Proposition 5; (3) the $Q$ estimates resemble each other under different $N$ . We conclude that MCQ ensures a stable and good value estimation with a proper $\\lambda$ . ",
+        "bbox": [
+            173,
+            670,
+            825,
+            753
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.3 Offline-to-online Fine-tuning ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            773,
+            413,
+            787
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We examine the offline-to-online fine-tuning capability of MCQ against some prior strong offline RL baselines, CQL [35], $\\mathrm { T D } 3 { + } \\mathrm { B C }$ [13], IQL [32]. We additionally compare against AWAC [45], which is designed intrinsically for offline-to-online adaptation. We conduct experiments on MuJoCo random and medium-replay datasets. It is challenging to train on these datasets for both offline and offline-to-online fine-tuning as they are non-expert, or even contain many bad transitions. We first train baselines and MCQ for 1M gradient steps offline and then perform online fine-tuning for another 100K gradient steps. Note that IQL paper [32] adopts 1M steps for online fine-tuning. However, we argue that 1M steps of online interactions are even enough to train off-policy online RL algorithms from scratch to perform very well. We thus believe 100K steps is more reasonable for the online interaction. All methods are run over 4 random seeds. The results are shown in Figure 3, where the shaded region denotes the standard deviation. As expected, we observe that MCQ consistently outperforms prior offline RL methods as well as AWAC on all of the datasets, often surpassing all of them with a large margin. The mild pessimism of MCQ makes it adapt faster, or keep the offline good performance during online interactions. Other prior offline RL methods, unfortunately, fail in achieving satisfying performance during online interaction due to strict conservatism and lack of generalization ability. ",
+        "bbox": [
+            174,
+            800,
+            825,
+            911
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/310b7d615b5c1d531901da1dfa0eedaffd43f5b6fe2a4197607012c874ce945f.jpg",
+        "image_caption": [
+            "Figure 3: Offline-to-online fine-tuning results on 6 D4RL MuJoCo locomotion tasks. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            92,
+            823,
+            330
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            387,
+            825,
+            498
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 Related Work ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            527,
+            321,
+            545
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Model-free offline RL. Prior model-free offline RL methods are typically designed to restrict the learned policy from producing OOD actions. They usually achieve this by leveraging importance sampling [47, 54, 39, 44, 16], incorporating explicit policy constraints [34, 58, 17, 13, 11], learning latent actions [67, 3], penalizing learned value functions such that low values are assigned to unseen actions [35, 33, 41], using adaptive methods [18], and uncertainty quantification [59, 65, 4]. Another line of the methods, instead, resorts to learning without querying OOD actions [56, 8, 32]. By doing so, they constrain the learning process within the support of the dataset. Nevertheless, existing methods may induce unnecessarily over-pessimistic value functions, and their performance is largely confined by how well the behavior policy is [45, 38, 4]. That partly explains why these methods are not satisfiable when trained on non-expert datasets (e.g., random datasets). MCQ keeps milder conservatism and better generalization ability as OOD actions are actively trained with proper targets. ",
+        "bbox": [
+            174,
+            565,
+            825,
+            718
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Model-based offline RL. Model-based offline RL methods, in contrast, learn the dynamics model in a supervised manner, and leverage the learned dynamics for policy optimization. Advances in this field include uncertainty quantification [46, 64, 27, 10], learning conservative value functions [63], representation learning [37, 48], constraining the learned policy with a behavior cloning loss [42], and sequential modelling [7, 23, 43]. However, there is no guarantee that the trained dynamics models are reliable, e.g., poor transitions can be generated, especially in complex high-dimensional environments [22]. Meanwhile, training dynamics models raises extra computation costs. ",
+        "bbox": [
+            173,
+            724,
+            825,
+            821
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Offline-to-online RL. There are some efforts on accelerating online interactions with the aid of offline logged data, which is also referred to as learning from demonstration [21, 26, 49]. Offline-to-online RL, instead, aims at enhancing the well-trained offline policy via online interactions. To ensure a fast adaptation and stable policy improvement, many techniques are adopted, such as model ensemble [38], explicit policy constraints [45, 66]. Offline-to-online fine-tuning will be difficult if the trained value function or policy is overly pessimistic, which may lead to a suboptimal policy. ",
+        "bbox": [
+            174,
+            828,
+            823,
+            911
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 Conclusion ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            89,
+            299,
+            106
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose Mildly Conservative $Q$ -learning (MCQ) to alleviate the over pessimism in existing offline RL methods. MCQ actively train OOD actions by constructing them proper pseudo target values following the guidance of the practical Mildly Conservative Bellman (MCB) operator. We theoretically illustrate that the policy induced by the MCB operator behaves at least as well as the behavior policy, and no erroneous overestimation will occur for the practical MCB operator. Furthermore, we extensively compare MCQ against recent strong baselines on MuJoCo locomotion tasks. Experimental results show that MCQ surpasses these baselines with a large margin on many non-expert datasets, and is also competitive with baselines on expert datasets. Moreover, we demonstrate the superior generalization capability of MCQ when transferring from offline to online. These altogether reveal that mild conservatism is critical for offline learning. We hope this work can promote the offline RL towards mild pessimism, and bring new insights into the community. ",
+        "bbox": [
+            174,
+            119,
+            825,
+            272
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "One drawback of our current algorithm lies in the need of tuning the weighting coefficient $\\lambda$ . However, we empirically find that $0 . 7 \\leq \\lambda < 1$ can usually induce satisfying performance. We leave the automatic tuning of $\\lambda$ as future work. ",
+        "bbox": [
+            176,
+            279,
+            823,
+            320
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            339,
+            553,
+            357
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This work was supported in part by the Science and Technology Innovation 2030-Key Project under Grant 2021ZD0201404, in part by the NSF China under Grant 61872009. The authors would like to thank the anonymous reviewers for their valuable comments and advice. ",
+        "bbox": [
+            174,
+            371,
+            825,
+            412
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            431,
+            266,
+            446
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "[1] J. Achiam, D. Held, A. Tamar, and P. Abbeel. Constrained Policy Optimization. In International Conference on Machine Learning, 2017.   \n[2] R. Agarwal, M. Schwarzer, P. S. Castro, A. C. Courville, and M. G. Bellemare. Deep Reinforcement Learning at the Edge of the Statistical Precipice. In Advances in Neural Information Processing Systems, 2021.   \n[3] A. Ajay, A. Kumar, P. Agrawal, S. Levine, and O. Nachum. OPAL: Offline Primitive Discovery for Accelerating Offline Reinforcement Learning. In International Conference on Learning Representations, 2021.   \n[4] C. Bai, L. Wang, Z. Yang, Z.-H. Deng, A. Garg, P. Liu, and Z. Wang. Pessimistic Bootstrapping for Uncertainty-Driven Offline Reinforcement Learning. In International Conference on Learning Representations, 2022.   \n[5] D. Bang and H. Shim. MGGAN: Solving Mode Collapse Using Manifold-Guided Training. IEEE/CVF International Conference on Computer Vision Workshops, pages 2347–2356, 2021.   \n[6] D. Bau, J.-Y. Zhu, J. Wulff, W. S. Peebles, H. Strobelt, B. Zhou, and A. Torralba. Seeing What a GAN Cannot Generate. In IEEE/CVF International Conference on Computer Vision, pages 4501–4510, 2019.   \n[7] L. Chen, K. Lu, A. Rajeswaran, K. Lee, A. Grover, M. Laskin, P. Abbeel, A. Srinivas, and I. Mordatch. Decision Transformer: Reinforcement Learning via Sequence Modeling. ArXiv, abs/2106.01345, 2021.   \n[8] X. Chen, Z. Zhou, Z. Wang, C. Wang, Y. Wu, Q. Deng, and K. W. Ross. BAIL: Best-Action Imitation Learning for Batch Deep Reinforcement Learning. In Advances in Neural Information Processing Systems, 2020.   \n[9] I. Csiszár and J. Körner. Information Theory - Coding Theorems for Discrete Memoryless Systems, Second Edition. Cambridge University Press, 2011.   \n[10] C. P. Diehl, T. Sievernich, M. Krüger, F. Hoffmann, and T. Bertram. UMBRELLA: Uncertainty-Aware Model-Based Offline Reinforcement Learning Leveraging Planning. ArXiv, abs/2111.11097, 2021.   \n[11] R. Fakoor, J. Mueller, P. Chaudhari, and A. Smola. Continuous Doubly Constrained Batch Reinforcement Learning. In Advances in Neural Information Processing Systems, 2021.   \n[12] J. Fu, A. Kumar, O. Nachum, G. Tucker, and S. Levine. D4RL: Datasets for Deep Data-Driven Reinforcement Learning. ArXiv, abs/2004.07219, 2020.   \n[13] S. Fujimoto and S. S. Gu. A Minimalist Approach to Offline Reinforcement Learning. In Advances in Neural Information Processing Systems, 2021.   \n[14] S. Fujimoto, D. Meger, and D. Precup. Off-Policy Deep Reinforcement Learning without Exploration. In International Conference on Machine Learning, 2019.   \n[15] S. Fujimoto, H. van Hoof, and D. Meger. Addressing Function Approximation Error in Actor-Critic Methods. In International Conference on Machine Learning, 2018.   \n[16] C. Gelada and M. G. Bellemare. Off-Policy Deep Reinforcement Learning by Bootstrapping the Covariate Shift. In AAAI Conference on Artificial Intelligence, volume 33, 2019.   \n[17] S. K. S. Ghasemipour, D. Schuurmans, and S. S. Gu. EMaQ: Expected-Max Q-Learning Operator for Simple Yet Effective Offline and Online RL. In International Conference on Machine Learning, 2021.   \n[18] D. Ghosh, A. Ajay, P. Agrawal, and S. Levine. Offline RL Policies Should be Trained to be Adaptive. ArXiv, abs/2207.02200, 2022.   \n[19] I. J. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. C. Courville, and Y. Bengio. Generative Adversarial Nets. In Advances in Neural Information Processing Systems, 2014.   \n[20] T. Haarnoja, A. Zhou, P. Abbeel, and S. Levine. Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor. In International Conference on Machine Learning, 2018.   \n[21] T. Hester, M. Vecerík, O. Pietquin, M. Lanctot, T. Schaul, B. Piot, D. Horgan, J. Quan, A. Sendonaris, I. Osband, G. Dulac-Arnold, J. P. Agapiou, J. Z. Leibo, and A. Gruslys. Deep Q-learning From Demonstrations. In AAAI Conference on Artificial Intelligence, 2018.   \n[22] M. Janner, J. Fu, M. Zhang, and S. Levine. When to Trust Your Model: Model-Based Policy Optimization. In Advances in Neural Information Processing Systems, 2019.   \n[23] M. Janner, Q. Li, and S. Levine. Offline Reinforcement Learning as One Big Sequence Modeling Problem. In Advances in Neural Information Processing Systems, 2021.   \n[24] Y. Jin, Z. Yang, and Z. Wang. Is Pessimism Provably Efficient for Offline RL? In International Conference on Machine Learning, pages 5084–5096. PMLR, 2021.   \n[25] S. M. Kakade and J. Langford. Approximately Optimal Approximate Reinforcement Learning. In International Conference on Machine Learning, 2002.   \n[26] B. Kang, Z. Jie, and J. Feng. Policy Optimization with Demonstrations. In International Conference on Machine Learning, 2018.   \n[27] R. Kidambi, A. Rajeswaran, P. Netrapalli, and T. Joachims. MOReL: Model-Based Offline Reinforcement Learning. In Advances in Neural Information Processing Systems, 2020.   \n[28] D. P. Kingma and J. Ba. Adam: A Method for Stochastic Optimization. In International Conference on Learning Representation, 2015.   \n[29] D. P. Kingma and M. Welling. Auto-Encoding Variational Bayes. ArXiv, abs/1312.6114, 2013.   \n[30] V. R. Konda and V. S. Borkar. Actor-Critic - Type Learning Algorithms for Markov Decision Processes. SIAM J. Control. Optim., 38:94–123, 1999.   \n[31] V. R. Konda and J. N. Tsitsiklis. Actor-Critic Algorithms. In Advances in Neural Information Processing Systems, 1999.   \n[32] I. Kostrikov, A. Nair, and S. Levine. Offline Reinforcement Learning with Implicit Q-Learning. In International Conference on Learning Representations, 2022.   \n[33] I. Kostrikov, J. Tompson, R. Fergus, and O. Nachum. Offline Reinforcement Learning with Fisher Divergence Critic Regularization. In International Conference on Machine Learning, 2021.   \n[34] A. Kumar, J. Fu, G. Tucker, and S. Levine. Stabilizing Off-Policy Q-Learning via Bootstrapping Error Reduction. In Advances in Neural Information Processing Systems, 2019.   \n[35] A. Kumar, A. Zhou, G. Tucker, and S. Levine. Conservative Q-Learning for Offline Reinforcement Learning. In Advances in Neural Information Processing Systems, 2020.   \n[36] S. Lange, T. Gabel, and M. A. Riedmiller. Batch Reinforcement Learning. In Reinforcement Learning, 2012.   \n[37] B.-J. Lee, J. Lee, and K.-E. Kim. Representation Balancing Offline Model-based Reinforcement Learning. In International Conference on Learning Representations, 2021.   \n[38] S. Lee, Y. Seo, K. Lee, P. Abbeel, and J. Shin. Offline-to-Online Reinforcement Learning via Balanced Replay and Pessimistic Q-Ensemble. In Conference on Robot Learning, 2021.   \n[39] Y. Liu, A. Swaminathan, A. Agarwal, and E. Brunskill. Off-Policy Policy Gradient with State Distribution Correction. ArXiv, abs/1904.08473, 2019.   \n[40] X. Ma, Y. Yang, H. Hu, J. Yang, C. Zhang, Q. Zhao, B. Liang, and Q. Liu. Offline Reinforcement Learning with Value-based Episodic Memory. In International Conference on Learning Representations, 2022.   \n[41] Y. J. Ma, D. Jayaraman, and O. Bastani. Conservative Offline Distributional Reinforcement Learning. In Advances in Neural Information Processing Systems, volume 34, 2021.   \n[42] T. Matsushima, H. Furuta, Y. Matsuo, O. Nachum, and S. Gu. Deployment-Efficient Reinforcement Learning via Model-Based Offline Optimization. In International Conference on Learning Representations, 2021.   \n[43] L. Meng, M. Wen, Y. Yang, C. Le, X. Li, W. Zhang, Y. Wen, H. Zhang, J. Wang, and B. Xu. Offline Pre-trained Multi-Agent Decision Transformer: One Big Sequence Model Tackles All SMAC Tasks. ArXiv, abs/2112.02845, 2021.   \n[44] O. Nachum, Y. Chow, B. Dai, and L. Li. DualDICE: Behavior-Agnostic Estimation of Discounted Stationary Distribution Corrections. In Advances in Neural Information Processing Systems, 2019.   \n[45] A. Nair, M. Dalal, A. Gupta, and S. Levine. Accelerating Online Reinforcement Learning with Offline Datasets. ArXiv, abs/2006.09359, 2020.   \n[46] Y. Ovadia, E. Fertig, J. Ren, Z. Nado, D. Sculley, S. Nowozin, J. V. Dillon, B. Lakshminarayanan, and J. Snoek. Can You Trust Your Model’s Uncertainty? Evaluating Predictive Uncertainty Under Dataset Shift. In Advances in Neural Information Processing Systems, 2019.   \n[47] D. Precup, R. S. Sutton, and S. Dasgupta. Off-Policy Temporal Difference Learning with Function Approximation. In International Conference on Machine Learning, 2001.   \n[48] R. Rafailov, T. Yu, A. Rajeswaran, and C. Finn. Offline Reinforcement Learning from Images with Latent Space Models. In Learning for Dynamics and Control (L4DC), 2021.   \n[49] A. Rajeswaran, V. Kumar, A. Gupta, J. Schulman, E. Todorov, and S. Levine. Learning Complex Dexterous Manipulation with Deep Reinforcement Learning and Demonstrations. ArXiv, abs/1709.10087, 2018.   \n[50] P. Rashidinejad, B. Zhu, C. Ma, J. Jiao, and S. Russell. Bridging Offline reinforcement Learning and Imitation Learning: A Tale of Pessimism. Advances in Neural Information Processing Systems, 34, 2021.   \n[51] K. Sohn, H. Lee, and X. Yan. Learning Structured Output Representation using Deep Conditional Generative Models. In Advances in Neural Information Processing Systems, 2015.   \n[52] A. Srivastava, L. Valkov, C. Russell, M. U. Gutmann, and C. Sutton. VEEGAN: Reducing Mode Collapse in GANs using Implicit Variational Learning. In Advances in Neural Information Processing Systems, 2017.   \n[53] R. S. Sutton and A. G. Barto. Reinforcement learning: An introduction. MIT press, 2018.   \n[54] R. S. Sutton, A. R. Mahmood, and M. White. An Emphatic Approach to the Problem of Off-policy Temporal-Difference Learning. Journal of Machine Learning Research, 17:2603–2631, 2016.   \n[55] J. Wang, W. Li, H. Jiang, G. Zhu, S. Li, and C. Zhang. Offline Reinforcement Learning with Reverse Model-based Imagination. In Advances in Neural Information Processing Systems, 2021.   \n[56] Q. Wang, J. Xiong, L. Han, P. Sun, H. Liu, and T. Zhang. Exponentially Weighted Imitation Learning for Batched Historical Data. In Advances in Neural Information Processing Systems, 2018.   \n[57] Z. Wang, A. Novikov, K. Zolna, J. T. Springenberg, S. E. Reed, B. Shahriari, N. Siegel, J. Merel, C. Gulcehre, N. M. O. Heess, and N. de Freitas. Critic Regularized Regression. In Advances in Neural Information Processing Systems, 2020.   \n[58] Y. Wu, G. Tucker, and O. Nachum. Behavior Regularized Offline Reinforcement Learning. ArXiv, abs/1911.11361, 2019.   \n[59] Y. Wu, S. Zhai, N. Srivastava, J. M. Susskind, J. Zhang, R. Salakhutdinov, and H. Goh. Uncertainty Weighted Actor-Critic for Offline Reinforcement Learning. In International Conference on Machine Learning, 2021.   \n[60] T. Xie, C.-A. Cheng, N. Jiang, P. Mineiro, and A. Agarwal. Bellman-Consistent Pessimism for Offline Reinforcement Learning. Advances in Neural Information Processing Systems, 34, 2021.   \n[61] S. Yang, Z. Wang, H. Zheng, Y. Feng, and M. Zhou. A Regularized Implicit Policy for Offline Reinforcement Learning. ArXiv, abs/2202.09673, 2022.   \n[62] Y. Yang, X. Ma, L. Chenghao, Z. Zheng, Q. Zhang, G. Huang, J. Yang, and Q. Zhao. Believe what you see: Implicit constraint approach for offline multi-agent reinforcement learning. Advances in Neural Information Processing Systems, 34, 2021.   \n[63] T. Yu, A. Kumar, R. Rafailov, A. Rajeswaran, S. Levine, and C. Finn. COMBO: Conservative Offline Model-Based Policy Optimization. In Advances in Neural Information Processing Systems, 2021.   \n[64] T. Yu, G. Thomas, L. Yu, S. Ermon, J. Y. Zou, S. Levine, C. Finn, and T. Ma. MOPO: Model-based Offline Policy Optimization. In Advances in Neural Information Processing Systems, 2020.   \n[65] A. Zanette, M. J. Wainwright, and E. Brunskill. Provable Benefits of Actor-Critic Methods for Offline Reinforcement Learning. In Advances in Neural Information Processing Systems, 2021.   \n[66] Y. Zhao, R. Boney, A. Ilin, J. Kannala, and J. Pajarinen. Adaptive Behavior Cloning Regularization for Stable Offline-to-Online Reinforcement Learning, 2022.   \n[67] W. Zhou, S. Bajracharya, and D. Held. PLAS: Latent Action Space for Offline Reinforcement Learning. In Conference on Robot Learning, 2020. ",
+        "bbox": [
+            178,
+            452,
+            826,
+            915
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            83,
+            828,
+            920
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            64,
+            828,
+            911
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            92,
+            826,
+            458
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Checklist ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            481,
+            254,
+            497
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "1. For all authors... ",
+        "bbox": [
+            214,
+            507,
+            330,
+            520
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   \n(b) Did you describe the limitations of your work? [Yes]   \n(c) Did you discuss any potential negative societal impacts of your work? [N/A]   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
+        "bbox": [
+            240,
+            523,
+            825,
+            594
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "2. If you are including theoretical results... ",
+        "bbox": [
+            214,
+            599,
+            467,
+            612
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes] ",
+        "bbox": [
+            240,
+            616,
+            686,
+            645
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "3. If you ran experiments... ",
+        "bbox": [
+            214,
+            648,
+            377,
+            661
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes]   \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]   \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes]   \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] ",
+        "bbox": [
+            240,
+            666,
+            825,
+            775
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ",
+        "bbox": [
+            215,
+            779,
+            767,
+            792
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "(a) If your work uses existing assets, did you cite the creators? [Yes]   \n(b) Did you mention the license of the assets? [Yes]   \n(c) Did you include any new assets either in the supplemental material or as a URL? [No]   \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [No]   \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] ",
+        "bbox": [
+            240,
+            795,
+            826,
+            895
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "5. If you used crowdsourcing or conducted research with human subjects... ",
+        "bbox": [
+            212,
+            898,
+            658,
+            911
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   \n(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] ",
+        "bbox": [
+            240,
+            92,
+            826,
+            174
+        ],
+        "page_idx": 13
+    }
+]

parse/dev/XzTtHjgPDsT/XzTtHjgPDsT.md

@@ -0,0 +1,479 @@
+# COORDINATION AMONG NEURAL MODULESTHROUGH A SHARED GLOBAL WORKSPACE
+
+Anirudh Goyal 1, Aniket Didolkar1, Alex Lamb 5, Kartikeya Badola 6, Nan Rosemary Ke 2, Nasim Rahaman 1, 3, Jonathan Binas 1, Charles Blundell 2, Michael Mozer 4, Yoshua Bengio 1
+
+# ABSTRACT
+
+Deep learning has seen a movement away from representing examples with a monolithic hidden state towards a richly structured state. For example, Transformers segment by position, and object-centric architectures decompose images into entities. In all these architectures, interactions between different elements are modeled via pairwise interactions: Transformers make use of self-attention to incorporate information from other positions and object-centric architectures make use of graph neural networks to model interactions among entities. We consider how to improve on pairwise interactions in terms of global coordination and a coherent, integrated representation that can be used for downstream tasks. In cognitive science, a global workspace architecture has been proposed in which functionally specialized components share information through a common, bandwidth-limited communication channel. We explore the use of such a communication channel in the context of deep learning for modeling the structure of complex environments. The proposed method includes a shared workspace through which communication among different specialist modules takes place but due to limits on the communication bandwidth, specialist modules must compete for access. We show that capacity limitations have a rational basis in that (1) they encourage specialization and compositionality and (2) they facilitate the synchronization of otherwise independent specialists.
+
+# 1 INTRODUCTION
+
+Deep Learning has seen a movement towards more structured models with cleaner separation between different pieces of information often handled by different components. The induced structure, and separation of knowledge has improved generalization, model-size scaling, and long-range dependencies (Berner et al., 2019; Vinyals et al., 2019; Brown et al., 2020). This opens up questions about how to achieve coherence and coordination between different components in such architectures. Looking back to the 1980s, the focus in AI was much less on learning and more on constructing articulated, multi-component architectures and examining how intelligence might emerge from interactions among this collection of simple, functionally specialized components (Fodor, 1983; Braitenberg, 1986; Minsky, 1988; Brooks, 1991). Each
+
+![](images/8777a35dd3e32207b005a7fb22b142a13eb9a0fe8291741211b601e34b432f41.jpg)  
+Figure 1: Step 1: an ensemble of specialist modules doing their own default processing; at a particular computational stage, depending upon the input, a subset of the specialists becomes active. Step 2: the active specialists get to write information in a shared global workspace. Step 3: the contents of the workspace are broadcast to all specialists.
+
+of these specialist modules is on the scale of a typical component of a computer program, like a subroutine that implements a narrow, prespecified function from certain input contents to certain output contents. Through appropriate communication and coordination, a set of specialists can achieve complex, dynamic, and flexible behavior patterns.
+
+As a concrete illustration, consider the task of driving a car in terms of specialists. One specialist might monitor the position of the car with respect to lines on the road, and another specialist might adjust the steering direction based on the perceptual data. In addition, there might be specialists which provide alerts when certain events occur, such as loud sounds, reaching a critical intersection on a route, or coming into close proximity to the car in front. To execute the task of driving the car properly, all these specialists need to interact coherently and broadcast their individual information to each other.
+
+Arguably, modern ML and AI has yet to develop broad architectural frameworks for learning both the specialist modules and how they should interact, while the classical view lacks an articulate story about how learning could take place successfully in such frameworks. In this article, we revisit this classical view with modern machine learning tools based on end-to-end learning and differentiable memory and attention mechanisms. Inspired by the Global Workspace Theory (Baars, 1993; Dehaene et al., 1998; Shanahan and Baars, 2005; Shanahan, 2006; 2010; 2012; Dehaene et al., 2017) from cognitive neuroscience, we argue that more flexibility and generalization emerge through an architecture of specialists if their training encourages them to communicate effectively with one another via the bottleneck of a shared workspace (Figure. 1).
+
+Distributed specialist modules. From a computational perspective, articulated multi-component architectures composed of sparsely interacting specialist modules show desirable scaling properties (e.g., more specialists can seamlessly be added), increased robustness (the system can tolerate the removal of or changes in individual specialists), and efficiency (information is processed predominantly locally, reducing the cost of communication between specialists). However, modularization also requires mechanisms to establish sharing of compatible representations across specialists, a form of shared internal language. While portions of a task might be solved by independent specialists, synchronization is critical particularly when there are statistical, functional, or causal dependencies among the specialists.
+
+Coherence through a shared workspace. In cognitive neuroscience, the Global Workspace Theory (GWT) (Baars, 1993; Dehaene et al., 2017) suggests an architecture allowing specialist modules to interact. The key claim of GWT is the existence of a shared representation—sometimes called a blackboard, sometimes a workspace—that can be modified by any specialist and that is broadcast to all specialists, along with the notion that write access is limited to maintain coherence. Our interpretation of this restriction on write access is that it stems from an assumption on the form of the joint distribution between high-level concepts. In this paper, we explore a communication and coordination scheme similar to the one proposed by GWT for modern neural network architectures like Transformers (Vaswani et al., 2017; Dehghani et al., 2018; Parmar et al., 2018; Radford et al., 2019; Brown et al., 2020) and attention-based modular architectures (Goyal et al., 2019; Rahaman et al., 2020; Mittal et al., 2020a; Goyal et al., 2020; Madan et al., 2021).
+
+In terms of our driving example, the workspace could be used to override default behaviors by giving high priority to specialist modules which provide alerts of various sorts (loud sounds, presence of a child on the street), allowing specialists which respond to such alerts to take control of behavior over default driving routines. This scenario implies that prioritization of signals in a shared workspace is critical.
+
+A shared communication channel necessitates common representations. For a multitude of specialist modules to cooperate, a common language is necessary (Baars, 1997). For example, in the driving scenario, alerts may come from auditory or visual processing specialists, but regardless of the source, a signal for danger must be placed in the workspace to override default behavior, whether that behavior is controlled by a radio-tuning specialist or a steering specialist. Although specialist modules can be pre-wired to have compatible communication interfaces, we will model an architecture in which an ensemble of specialist modules is trained in coordination, which should lead to a shared language (Colagrosso and Mozer, 2005). Internally, individual specialists can use whatever form of representations that serves them, but their inputs and outputs require alignment with other specialists in order to synchronize. For example, an unusual event such as a rough thud under the wheels might not have been previously experienced, but the mere signalling of novelty could override default specialists. Without a global communication channel, specialists would have to learn to communicate through pairwise interactions, which might limit coordination of behavior in novel situations: global communication ensures exchangeability of knowledge to achieve systematic generalization.
+
+![](images/b200dda54bf250d4c34d4b3b1ed61035292dec53a713c40bf25491b1c4b8d5d7.jpg)  
+Figure 2: Using a Shared Workspace for creating global coherence in RIMs, Transformers, TIMs and Universal Transformers (UT). (Top Half) All four of these architectures use pairwise communication (using key-value attention) to establish coherence between individual specialist modules. In the case of RIMs (Goyal et al., 2019) and TIMs (Lamb et al., 2021), these specialists are independent modules that compete with each other in order to take control over the state update based on a given input. In the case of Transformers (Vaswani et al., 2017) and Universal Transformers (Dehghani et al., 2018), each specialist is associated with a different position. Activated specialists are denoted by a blue shade and the intensity depends on the degree of activation. In the case of Universal Transformers, the state update dynamics for each position is shared across all layers and all positions (denoted by a yellow triangle). (Bottom Half) We replace pairwise communication with a shared workspace to create global coherence between different specialists. Communication using the shared workspace is a two-step process (as denoted by 1 and 2 in the figures). In the first step (1), specialists compete for write access to the shared workspace, resulting in a subset of them being activated (in blue), and only the activated specialists perform the write operation on the workspace. In the second step (2), the contents of the shared workspace are broadcast to all the specialists.
+
+# 2 SYNCHRONIZING NEURAL MODULES THROUGH A SHARED WORKSPACE
+
+We investigate a neural architecture reminiscent of the GW model, where a number of sparsely communicating specialist modules interact via a shared working memory. In particular, we extend the Transformer (Vaswani et al., 2017), attention and slot-based modular architectures (Goyal et al., 2019) by adding a shared workspace and allowing modules (each representing an entity) to compete for write access in each computational stage.
+
+Key-value attention. Key-value attention defines the backbone of updates to the hidden states in the proposed model. This form of attention is widely used in self-attention models and performs well on a wide array of tasks (Bahdanau et al., 2014; Vaswani et al., 2017; Santoro et al., 2018). Key-value attention selects an input value based on the match of a query vector to a key vector associated with each value. To allow differentiability and thus easier learnability, selection is soft and computes a convex combination of all the values. Such a mechanism makes it possible to change on-the-fly both the source of input and how the shared workspace is updated. It also makes the outputs of the specialists and the elements of the memory permutation invariant: they should be considered as an unordered set of elements to be selected by an attention mechanism from the contents of specialists. More precisely, soft attention uses the product of a query (represented as a matrix $Q$ of dimensionality $N _ { r } \times d$ , with $N _ { r }$ queries, and $d$ the dimension of each query) with a set of $N _ { o }$ objects each associated with a key as a row in matrix $K ^ { T }$ $( N _ { o } \times d )$ . After normalization with a softmax the resulting convex weights are used to combine the values $V _ { i }$ (row $i$ of matrix $V$ ): where the softmax is applied to each row of its argument matrix, yielding a set of convex weights. For our experiments, we use multihead dot product attention.
+
+Neural modules with pairwise interactions. Our approach to synchronizing neural modules is highly general and mostly agnostic to the task, domain, or specific choice of architecture, with the only requirement being that the model consists of multiple specialist modules which either operate independently or have sparse interactions requiring to only match pairs of modules at a time. Our goal is to explore how introducing a shared workspace can help these modules to become better synchronized and coordinated. We show the utility of the shared workspace for synchronization in (a) Transformers (Vaswani et al., 2017), in which all interactions between positions are performed via attention, and (b) slot-based architectures like Recurrent Independent Mechanisms or RIMs (Goyal et al., 2019) in which all pairwise interactions between modules are performed via attention. In the context of slot-based architectures, each slot’s content is associated with a specialist module, whereas in Transformers different entities each associated with a different position acts as a specialist module (Figure 2).
+
+Both Transformers and RIMs utilize a self-attention mechanism for sharing information between modules, typically implemented in a pairwise manner, i.e., each specialist attends to every other specialist. Instead, we facilitate information sharing among specialist modules through a limited capacity shared workspace. In this framework at each computational stage, different specialists compete for write access to the common workspace. The contents of the workspace, in turn, are broadcast to all specialist modules simultaneously.
+
+Notation. The input is processed through a sequence of computational stages indexed by $t$ , and at each stage, $n _ { s }$ entities are operated on (i.e., $n _ { s }$ different modules in slot-based architectures like RIMs or $n _ { s }$ different positions in the case of Transformers). Each of these $n _ { s }$ specialist modules has a distinct internal $n _ { h }$ -dimensional state $\mathbf { \Delta } _ { h _ { t } ^ { k } }$ , for $k \in \{ 1 , . . . , n _ { s } \}$ . The specialist modules communicate with each other via a shared workspace divided into $n _ { m }$ memory slots, each consisting of a vector of $n _ { l }$ elements, denoted $M = [ \pmb { m } _ { 1 } ; \dots \pmb { m } _ { j } ; \dots \pmb { m } _ { n _ { m } } ]$ . The shared workspace is updated across different computational stages i.e., different time-steps in recurrent architecture and different layers in the case of Transformers. At each computational stage $t$ , different specialists compete for writing in the shared workspace, but all specialists can read from the current state of the workspace. In the case of an autoregressive task, we can restrict the information sharing to previous positions and keep a separate version of the workspace for each position.
+
+# 2.1 SPECIFICS OF THE SHARED WORKSPACE.
+
+Step 1: Process Input to obtain an entity representation for each specialist. The first step is external to the proposed method, and involves processing the input to form the initial representation vector for each of the different specialists. Different common deep learning architectures can be used to form the representation of different specialists. For example, Transformers start with a matrix $n _ { s } \times n _ { h }$ whose rows are initialized as the $n _ { h }$ -dimensional embeddings of the input at each position of the sequence. Slot-Based Recurrent architectures like RIMs consist of a single-layer recurrent structure where the hidden state $\mathbf { h } _ { t }$ at computational stage $t$ is decomposed into the substates of the $n _ { s }$ specialists, $\mathbf { h } _ { t } ^ { k }$ for $k = 1 , . . . n _ { s }$ .
+
+In the proposed scheme, within each computational stage, the updates of the hidden state of different specialists follow a two-step process. First, specialists compete and write to a shared workspace. Second, information from the workspace gets broadcast to all the specialists, as detailed next.
+
+Step 2: Writing Information in the shared workspace. The specialists compete to write into the shared workspace, whose contents need to be updated in the context of new information received from different specialists. This step ensures that only the critically important signals make it to the shared workspace, therefore preventing the workspace from being cluttered. Let matrix $\pmb { R }$ represent the combined state of all the specialists (i.e. $h _ { t } ^ { k } \mathbf { \Sigma } ^ { \ast } \forall k \in \{ 1 , \dots , \mathbf { \bar { n } } _ { s } \}$ as the rows of $\pmb { R }$ ). In order to implement the competition between specialists to write into the workspace, we use a key-query-value attention mechanism. In this case, the query is a function of the state of the current workspace memory content, represented by matrix $M$ (with one row per slot of the memory), i.e $\widetilde { Q } = M \widetilde { W } ^ { q }$ . Keys and values are a function of the information from the specialists i.e., a function of $\pmb { R }$ . We apply dot product attention to get the updated memory matrix: $\begin{array} { r } { M \gets \operatorname { s o f t m a x } \left( \frac { \widetilde { Q } ( R \widetilde { W } ^ { e } ) ^ { \mathrm { T } } } { \sqrt { d _ { e } } } \right) R \widetilde { W } ^ { v } } \end{array}$ . The use of a regular softmax to write into $M$ leads to a standard soft competition among different specialists to write in the shared workspace. One can also use a top- $k$ softmax (Ke et al., 2018) to select a fixed number of specialists allowed to write in the shared workspace: based on the pre-softmax values, a fixed number of $k$ specialists which have the highest values are selected, and get access to write in the shared workspace. Selection with a top- $k$ softmax is a hybrid between hard and soft selection. We denote the set of thus selected specialists as $\mathcal { F } _ { t }$ . We note that we can apply the attention mechanism multiple times to distill information from different specialists into the shared workspace. Here, the contents of the shared workspace are updated in the gated way as proposed in RMC (Santoro et al., 2018). We ask the reader to refer to appendix section $\textrm { C }$ for more details.
+
+Step 3: Broadcast of information from the shared workspace. Each specialist then updates its state using the information broadcast from the shared workspace. We again utilize an attention mechanism to perform this consolidation. All the specialists create queries ${ \widehat { q } } _ { k } = h _ { t } ^ { k } { \widehat { W } } ^ { q }$ , which are matched with the keys $\widehat { \pmb { \kappa } } _ { j } = ( \pmb { m } _ { j } \widehat { W } ^ { e } ) ^ { \mathrm { T } } \quad \forall k \in \{ 1 , \dots , n _ { s } \}$ , $j \in \{ 1 , \dots , n _ { m } \}$ from the updated memory slots, forming attention weights $\begin{array} { r } { s _ { k , j } = \mathrm { s o f t m a x } \left( \frac { \widehat { q } _ { k } \widehat { \kappa } _ { j } } { \sqrt { d _ { e } } } \right) } \end{array}$ The memory slot values generated by each slot of the shared workspace and the attention weights are then used to update the state of all the specialists: $\begin{array} { r } { \pmb { h } _ { t } ^ { k }  \pmb { h } _ { t } ^ { k } + \sum _ { j } s _ { k , j } \widehat { \pmb { v } } _ { j } } \end{array}$ where $\widehat { \pmb { v } } _ { j } = \pmb { m } _ { j } \widehat { \pmb { W } } ^ { v } \quad \forall k \in \{ 1 , \dots , \overset { \cdot } { n } _ { s } \}$ . After receiving the broadcast information from the workspace, each specialist update their state by applying some dynamics function i.e., one step update of LSTM or GRU units in the case of recurrent architectures, and a feedforward layer in the case of Transformers. This yields the new value $\boldsymbol { h } _ { t + 1 } ^ { k }$ for the $k \mathrm { . }$ -th specialist, from which we start the next stage $( t + 1 )$ .
+
+Replacing pairwise interactions among neural modules with interaction facilitated by the shared workspace allows for the following:
+
+1. Higher-order $( H O )$ interaction among neural modules. The two-step write-read process first allows each memory slot to store a ‘filtered summary’ of the current input where the ‘filter’ is determined by the previous state of that slot (‘Query’ for the write step). Neural modules then summarize the information contained in these slots and update their state. Hence unlike pairwise interaction, messages passed among neural modules in the shared workspace setting also include HO interaction terms; those consisting of more than 2 modules at a time. Naturally, HO interaction require that messages passed among neural modules lie in the same representation space, which is precisely what we aim to achieve by allowing message passing only via a singular global channel.
+
+2. Dynamic filtering due to persistence of memory. With a shared workspace (SW), contents of the memory slot play a key role in filtering and summarizing the information contained in the input at a given time step. Persistence of memory throughout an episode 1) would allow the memory layer to summarize and filter information based on what it has seen thus far 2) should ideally lead to better generalization as the model is able to dynamically modify its filtering machinery for a particular input. In contrast, “inducing points” in Set Transformers (Lee et al., 2019) are fixed after training and hence the bottleneck cannot adjust itself on the fly for any new input. We present comparisons on several tasks in section 4. They show the importance of these two properties by comparing performance of SW with a) $2 \times \mathrm { S e l f }$ -Attention (to simulate HO interaction without global communication) b) a version without memory persistence, in Appendix D.
+
+Computational Complexity of using shared workspace for synchronizing different specialists. To encourage a coherent global coordination, Transformers and slot-based recurrent architectures rely on pairwise interactions captured via an attention mechanism. Unfortunately, such attention mechanisms scale quadratically with the number of specialists. Here, we propose a method which uses a shared workspace to create global coherence between different specialists and in the process, replaces the pairwise interactions of conventional dot-product attention. The computational complexity of the proposed method is thus linear in the number of specialists. In our experimentation, the number of memory slots is practically constant, which suggests a very favourable scaling behavior, and certainly much less than quadratic. As a point of reference, what would correspond to the number of slots in human working memory (Baars, 1993) is indeed very small (less than 10 slots).
+
+# 3 RELATED WORK
+
+This work taps into a line of reasoning put forward by historical works, such as Minsky (1988); Braitenberg (1986); Fodor (1983), wherein it is argued that in order to be able to deal with a wide spectrum of conditions and tasks, an intelligent system should be comprised of many interacting specialized modules or programs, rather than a single “one-size-fits-all” entity. While modular architectures have been the subject of a number of research directions, (Jacobs et al., 1991; Bottou and Gallinari, 1991; Ronco et al., 1997; Reed and De Freitas, 2015; Andreas et al., 2016; Rosenbaum et al., 2017; Fernando et al., 2017; Shazeer et al., 2017; Rosenbaum et al., 2019; Goyal and Bengio,
+
+2020), we focus here on a mechanism for achieving coherence and synchronization between specialist modules via a global workspace shared between all specialists.
+
+Prior works have explored incorporating slot-based memory in the context of recurrent neural networks (Graves et al., 2014; 2016; Santoro et al., 2018). In the context of transformers, Burtsev and Sapunov (2020) introduce memory tokens that are processed in addition to sequence tokens, whereas Dai et al. (2019) (Transformer-XL) propose to partition a long sequence to smaller segments and use the activations of the previous segment in memory while processing the current segment. Building on the latter, Rae et al. (2019) propose to store activations from prior segments in a compressed memory. However, these methods do not restrict memory writes to be sparse and competitive. Recent advances in this direction include the global neuronal workspace (GNW) model (Dehaene and Changeux, 2011), which identifies the global workspace with a large network of excitatory pyramidal neurons with long-range axonal processes connecting prefrontal and parietal cortices. Further, deploying a shared workspace to establish coherence between different specialists as opposed to using all-pair communication has an added benefit, in that it allows us to tackle the $O ( n ^ { 2 } )$ complexity of selfattention. This makes our work related to previous work on reducing the computational complexity of dot product attention in Transformers. Lee et al. (2019) introduce the $I S A B$ module, which maps between sets and comprises two dot-product attention layers. In the first layer, a set of trainable parameters are used as queries and the elements of the input set as keys; in the second layer, the output of the first layer is used as keys and the input set as queries. However, unlike in this work, the intermediate states (corresponding to the output of the first layer) are not maintained across layers. Concurrent to our work, (Jaegle et al., 2021) also introduced the idea of using a latent bottleneck for addressing quadratic complexity by learning a bottleneck but there are important differences. For example. in Perceiver the latent bottleneck iteratively queries the information about different positions, and does not maintain the representation of the different specialists. More precisely, in our proposed method different specialists write information in the workspace and then information gets read from the shared workspace. In Perceiver, the latent bottleneck iteratively reads information from the set of positions. We also show the applicability of the proposed idea both for slot based models and Transformers.
+
+The proposed model can also be seen as integrating out different ideas popular in modular architectures (Andreas et al., 2016; Goyal et al., 2019), memory networks (Graves et al., 2014; Santoro et al., 2018) and mixture of experts (Jacobs et al., 1991), and hence combining some of their benefits in a unified architecture. The proposed model is factored as a set of specialists (incorporating modularity). The proposed model achieves coordination among different specialists via the use of a shared workspace (in the Neural Turing machines, there is only a single specialist i.e., without any modularity). Multiple experts can be active at the same time (generally not the case with a mixture of experts).
+
+# 4 EXPERIMENTS
+
+Here we briefly outline the tasks on which we applied the idea of the shared workspace and direct the reader to the appendix for some more experiments (Appendix G), full details on each task and details on hyperparameter settings for the model. The experiments have the following goals: (a) Demonstrate that the use of the shared workspace can improve results on a wide array of challenging benchmark tasks, with the goal of demonstrating the practical utility and breadth of the technique. (b) Show that the shared workspace addresses coherence between different specialists by achieving improved performance without requiring all pairwise interactions. Finally, to show wide applicability of our model, we integrate SW in TIMs (Lamb et al., 2021), SCOFF (Goyal et al., 2020) and BRIMs (Mittal et al., 2020b) and show improvements over the default communication method used in each.
+
+Making sense of the visual input. Using a shared workspace introduces a bottleneck in sharing of information between specialists. Since the size of the workspace is limited and generally much lower than the number of specialists, there is a limit to the amount of information that can be exchanged among specialists. We hypothesize that mediating communication through a limited capacity workspace should encourage the model to look at relevant information that is important for the downstream objective. We test this hypothesis on a set of visually challenging benchmarks. For our experiments, we use either Transformers or RIMs as a backbone. We consider variants of Transformers based on different subsets of important properties. Transformers [TR]: Self-attention based multi-layer architecture (Vaswani et al., 2017) with shared parameters across layers. Set transformer [ISAB]: Transformers where self attention is replaced by ISAB module (Lee et al., 2019). Sparse
+
+![](images/6a3bacda9a45376fac80a0586fbea6a1d8172cfb3e0d81c02b64d9e33eb59ce0.jpg)  
+Figure 3: Detecting Equilateral Triangles. Here, we compare the performance of the Transformers with shared workspace to other Transformer baselines. Here, we plot the test accuracy for each model.
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>Top-1 %</td><td rowspan=1 colspan=1>Top-5 %</td></tr><tr><td rowspan=1 colspan=1>ISABSTRTR</td><td rowspan=1 colspan=1>65.3±0.02570.6±0.0870.83±0.44</td><td rowspan=1 colspan=1>83.6±0.01187.33±0.0687.8±0.08</td></tr><tr><td rowspan=1 colspan=1>TR+HC</td><td rowspan=1 colspan=1>70.17±0.31</td><td rowspan=1 colspan=1>88.33±0.2</td></tr><tr><td rowspan=1 colspan=1>TR+HSW (OURS)</td><td rowspan=1 colspan=1>71.07±0.04</td><td rowspan=1 colspan=1>88.6±0.49</td></tr><tr><td rowspan=1 colspan=1>TR+ SSW (OURS)</td><td rowspan=1 colspan=1>71.33±0.34</td><td rowspan=1 colspan=1>88.3±0.05</td></tr></table>
+
+Table 1: Comparison on CATER Object Tracking. Here, we compare the Top-1 and Top-5 accuracy of Transformers with shared workspace and Transformers with self-attention. We can see that Transformers with a shared workspace outperform those with pairwise selfattention.
+
+Transformers [STR]: Transformers with sparse factorizations of the attention matrix (Child et al., 2019). High Capacity Transformers $[ \mathrm { T R + H C } ]$ : Same as TR but with different parameters across layers. Transformers with Shared Workspace with soft-competition $\scriptstyle \left[ \mathrm { T R + S S W } \right]$ : Transformers with different positions competing with each other to write in shared workspace using soft-competition. Transformers with Shared Workspace with top- $k$ competition $\mathrm { [ T R + H S W ] }$ : Transformers with different positions competing with each other to write in shared workspace using top- $k$ competition. For a more detailed description of all the tasks described below, we ask the reader to appendix section E.
+
+Detecting Equilateral Triangles. We first use a simple toy task to test our hypothesis where the model should detect equilateral triangles in images (Ahmad and Omohundro, 2009). Each image is of size $6 4 \times 6 4$ and contains 3 randomly placed clusters of points. For equilateral triangles, the midpoints of these clusters are equidistant from each other. This is a binary classification task where the model has to predict whether the three given clusters form an equilateral triangle or not. To feed an image into a Transformer, we follow the same methodology as used in vision Transformers (Dosovitskiy et al., 2020). We first divide an image into equal sized $4 \times 4$ patches and treat each patch as a different input position of the Transformer.
+
+To solve this task correctly, the model only needs to attend to relevant information i.e., to patches that contain the cluster of points. Therefore, using a limited capacity shared workspace should be useful here. Our results (presented in Figure
+
+![](images/22cb27bec2d479651d4cb37b2e18cd5125c3ca4f9066ec0b36284d49723faf73.jpg)  
+Figure 4: Comparison on Sort-of-CLEVR relational reasoning. Speed of convergence for relational and non-relational questions in the sort-ofclevr dataset. We can see that the proposed model converges much faster than the baselines in both cases.
+
+3) confirm this hypothesis. We can see that Transformers with shared workspace attention converge much faster and reach higher accuracy as compared to the baseline Transformer. Our method also outperforms Set Transformer by a significant margin.
+
+Multi MNIST Generation. In this task, we train an Image Transformer (Parmar et al., 2018) (pixelby-pixel, raster-order generative model) for next-pixel prediction on the “MultiMNIST dataset” where each image consists of 4 independently sampled MNIST digits stacked horizontally to form one image (see Figure 10 for demonstration). The main aim of this task is to observe the inductive biases that allow for specialization of mechanisms in TIMs (Lamb et al., 2021). Each image in the MultiMNIST dataset can be broken down into different sets of independent spatial components. Since the digits which make up the image are independently selected, the joint distribution of pixel intensities in any one of the four sections of the image is statistically independent of the pixel intensities in any other section of the image. Moreover each section of the image can be further broken down into independent spatial components: one that pertains to the background and one that pertains to the foreground. One can expect that architectures that are made up of sparsely interacting different mechanisms to naturally capture this statistical independence by dividing labour among different mechanisms. While, for monolithic architectures, a major portion of their training time will be spent in learning these statistical independencies from scratch. We find that replacing the pairwise communication in TIMs with a shared workspace $( \mathrm { T I M s } + \mathrm { S W } )$ ) leads to better and more interpretable division of labor among specialists as shown in Figure 5. From the figure, It is clear that the TIMs model is unable to divide labour among specialists with mechanism 2 being activation for all the pixels in the image. On the other hand, we can see that TIMs $+ \ S W$ is able to divide labor among specialists with each mechanism focusing on a different aspect of the image. We can see that mechanism 2 gets activated for the digits which are present towards the centre of each of the 4 columns while mechanisms 3 and 4 cover the background of the digits, with mechanism 3 covering the area between adjacent digits and mechanism 4 covering the area above and below the digits. Thus, we can see that using a shared workspace aids the division of labor among different specialists. We also find that TIMs $+ \thinspace S \mathbf { W }$ results in the least cross-entropy loss in the test set when compared to TIMs and Image Transformers (Parmar et al., 2018). Results shown in appendix Table 5.
+
+CATER: Object Tracking. Cater is a spatiotemporal reasoning video dataset introduced in Girdhar and Ramanan (2019). Each video contains 3D objects organized in a $6 \times 6$ grid. Each object affords certain actions that can be performed on them. These actions result in movement of the concerned objects and change in their positions. Some of these actions include: rotate, pick-place, slide, contain. Throughout the duration of the video, a number of these actions are performed to get the final state of the grid. Note that only a single object undergoes an action, at any instant. The task that we focus on here is called localization. In this task, the goal is to predict the location of the target object in the final frame. In this case the target object is called a snitch. The snitch as well as the other objects move across the $6 \times 6$ grid. In some scenarios, the snitch may be covered by other objects hence hiding it from the view. In such cases, tracking the movement of the snitch across frames becomes essential. Therefore, capturing long-range temporal dependencies is essential to solve this task.
+
+The information exchange limit enforced by the limited capacity of the shared workspace should
+
+![](images/c064a124eb1294be068a2021c283a70485ee5d0c268b63d04399caaf43dcf5b9.jpg)  
+Figure 5: This figure shows the mechanism activation map for all 4 mechanims used in the multimnist generation task for both TIMs and TIMs $^ +$ SW. Both the images in the figure correspond to the activation maps from 4 different examples. Each activation map contains 4 mechanisms shown from left to right in a single row. Each mechanism is shown using a $3 2 \times 3 2$ image, a particular pixel in a mechanism activation map is shown in white if that mechanism was used during the generation of that pixel while generating the image.
+
+be useful here as well. For CATER, in some frames the snitch is not visible as it is covered by other objects. Therefore, ideally the model only needs to attend to frames in which the snitch is visible. Additionally, if the snitch is visible throughout the video in all frames, then to accurately predict the final position of the snitch, the model only needs to attend to the final frame of the video and can completely ignore the initial frames. The results for this task are presented in Table 1. We also experimented with both soft competition $\mathrm { T R } { + } \mathrm { S } \mathrm { S } \mathrm { W }$ and hard competition $\mathrm { T R } { + } \mathrm { H S } \mathrm { W }$ , with only $k = 5$ specialists writing into the shared workspace. We can see that models with a shared workspace outperform those with pairwise multihead attention thus confirming our hypothesis about the benefits of a shared workspace for this task. As shown in Table 1 proposed method convincingly outperforms the Set Transformer.
+
+Relational Reasoning $:$ Sort-of-CLEVR. In relational reasoning, the model is tasked with answering questions about certain properties of various objects and their relations with other objects. The model is presented with an image and a question for that image. This task has a clear sparse structure as in order to answer the questions correctly, it needs to only reason about a specific subset of objects that the question mentions. For this task, we use the Sort-of-CLEVR dataset (Santoro et al., 2017).
+
+Each image in Sort-of-CLEVR is of size $7 5 \times 7 5$ and contains 6 randomly placed geometrical shapes of 6 possible colors and 2 possible shapes. Each image comes with 10 relational questions and 10 nonrelational questions. Non-relational questions only consider properties of individual objects. On the other hand, relational questions consider relations among multiple objects. For more details about the question see appendix Figure 8. The input to the model consists of the image and the corresponding question. We first obtain a sequence of equal-sized patches for the image as in vision Transformers (Dosovitskiy et al., 2020). We concatenate the resulting patch sequence with the representation of the question and pass the combined sequence through the Transformer. Sort-of-CLEVR has a finite number of possible answers, hence this task is setup as a classification task.
+
+We present the results for this task in Figure 4. We observe that the Transformers with the shared workspace converge faster and outperform the baselines for relational as well as non-relational questions. The superior performance with shared memory can be attributed to the inherent sparsity of this task. For instance, in non-relational questions, the model only needs to attend to a single object referenced in the question to answer it correctly, while relational questions only consider a small subset of objects in the image, thus sparsity is helpful for both these types of questions. Therefore, the limited capacity of the shared workspace forces the model to attend to only relevant information.
+
+Shared Workspace for Physical Reasoning. In this task, we consider a set of bouncing balls and the model is tasked with predicting the trajectory of the balls at each step. In order to solve this task, a coherent picture of where and which objects will collide needs to be established by the learner. We use the bouncing-ball dataset from Van Steenkiste et al. (2018). We train the model for
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>Num. Slots</td><td rowspan=1 colspan=1>ARI个</td><td rowspan=1 colspan=1>MSE↓</td></tr><tr><td rowspan=1 colspan=1>SCOFF</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>0.276±0.001</td><td rowspan=1 colspan=1>0.083±0.0</td></tr><tr><td rowspan=1 colspan=1>SCOFF +SW</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>0.154±0.007</td><td rowspan=1 colspan=1>0.135±0.002</td></tr><tr><td rowspan=1 colspan=1>SCOFF + SW</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>0.487±0.085</td><td rowspan=1 colspan=1>0.059±0.0</td></tr><tr><td rowspan=1 colspan=1>SCOFF + SW</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>0.915±0.0</td><td rowspan=1 colspan=1>0.035±0.0</td></tr><tr><td rowspan=1 colspan=1>SCOFF +SW</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>0.891±0.001</td><td rowspan=1 colspan=1>0.039±0.0</td></tr><tr><td rowspan=1 colspan=1>SCOFF+ SW</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>0.351±0.001</td><td rowspan=1 colspan=1>0.08±0.0</td></tr></table>
+
+Table 2: Here we show the performance of SCOFF augmented with shared workspace attention on the bouncing balls task. We also analyse the effect of varying number of slots in the shared workspace. This also shows that by increasing the number of slots performance decreases hence validating claims regarding bandwidth limited communication channel via shared workspace.
+
+next-step prediction. We compare the proposed approach against SCOFF (Goyal et al., 2020). The results of our comparison are shown in Table 2. We use the ARI and MSE metric for comparison. ARI measures how well the different balls are segregated into different slots, higher ARI means better segregation. We can see that using a shared workspace results in higher ARI as compared to pairwise communication in SCOFF. Thus, using a shared workspace results in better division of labor among specialists. We also compare the proposed method against other baselines in appendix section F.1.
+
+Shared Workspace for Atari Video Games. We start by training RIMs, RIMs $^ +$ shared workspace (SW) on three "source" games (Pong, River Raid, and Seaquest) and test if the learned features transfer to a different subset of randomly selected "target" games (Alien, Asterix, Boxing, Centipede, Gopher, Hero, James Bond, Krull, Robotank, Road Runner, Star Gunner, and Wizard of Wor). We take a sufficient number of specialists in RIMs (10). We train on source games for 10M steps, and then fine-tune on transfer games for 10M more steps. We choose these games as they were also used in the original RIMs paper (Goyal et al., 2019). Using a suite of 36 game pairs, we find that RIMs $+ \thinspace \mathrm { S W }$ outperforms RIMs on both game A (a median performance ratio of 1.13; mean of 1.16) and game B (a median performance ratio of 1.11; mean of 1.15). The improved performance with RIMs $^ +$ SW is due to better forward transfer (knowledge acquired for game A facilitates the learning of game B) and reduced backward interference (knowledge acquired for game B does not disrupt knowledge acquired for game A), presumably thanks to a more appropriate modularization of knowledge.
+
+# 5 CONCLUSION
+
+Inspired by cognitive neuroscience global workspace theories, we have proposed a shared workspace model for establishing coherence among modular neural specialists while exchanging information in a systematic way. We show that using a limited capacity shared workspace as a bottleneck for mediating communication among specialists results in better performance across a wide range of visual reasoning benchmarks as compared to the pairwise interactions typically used in self-attention schemes. The proposed approach combines several key properties: knowledge and expertise is divided among specialists, they compete to post new contents to the workspace, and after being updated, the shared workspace is accessible to all specialists for their own updates.
+
+# ETHICS STATEMENT
+
+The authors do not foresee any negative social impacts of this work, but of course the accumulation of improvements in ML could be misused as it may give more power to nefarious agents.
+
+# REPRODUCIBILITY STATEMENT
+
+We use Algorithms 1 and 2 for our experiments, we will be releasing the code after the review process.   
+We also provide our code in the supplementary material.
+
+# REFERENCES
+
+S. Ahmad and S. Omohundro. Equilateral triangles: A challenge for connectionist vision. 2009.
+
+Jacob Andreas, Marcus Rohrbach, Trevor Darrell, and Dan Klein. Neural module networks. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 39–48, 2016.
+
+Bernard J Baars. A cognitive theory of consciousness. Cambridge University Press, 1993.
+
+Bernard J Baars. In the theatre of consciousness. global workspace theory, a rigorous scientific theory of consciousness. Journal of Consciousness Studies, 4(4):292–309, 1997.
+
+Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. arXiv preprint arXiv:1409.0473, 2014.
+
+Christopher Berner, Greg Brockman, Brooke Chan, Vicki Cheung, Przemysław D˛ebiak, Christy Dennison, David Farhi, Quirin Fischer, Shariq Hashme, Chris Hesse, et al. Dota 2 with large scale deep reinforcement learning. arXiv preprint arXiv:1912.06680, 2019.
+
+Léon Bottou and Patrick Gallinari. A framework for the cooperation of learning algorithms. In Advances in neural information processing systems, pages 781–788, 1991.
+
+Valentino Braitenberg. Vehicles: Experiments in synthetic psychology. MIT press, 1986.
+
+Rodney A Brooks. Intelligence without representation. Artificial intelligence, 47(1-3):139–159, 1991.
+
+Tom B Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. Language models are few-shot learners. arXiv preprint arXiv:2005.14165, 2020.
+
+Mikhail S Burtsev and Grigory V Sapunov. Memory transformer. arXiv preprint arXiv:2006.11527, 2020.
+
+Rewon Child, Scott Gray, Alec Radford, and Ilya Sutskever. Generating long sequences with sparse transformers. arXiv preprint arXiv:1904.10509, 2019.
+
+Michael D. Colagrosso and Michael C Mozer. Theories of access consciousness. In L. K. Saul, Y. Weiss, and L. Bottou, editors, Advances in Neural Information Processing Systems 17, pages 289–296. MIT Press, 2005. URL http://papers.nips.cc/paper/ 2715-theories-of-access-consciousness.pdf.
+
+Zihang Dai, Zhilin Yang, Yiming Yang, Jaime Carbonell, Quoc V Le, and Ruslan Salakhutdinov. Transformer-xl: Attentive language models beyond a fixed-length context. arXiv preprint arXiv:1901.02860, 2019.
+
+S. Dehaene, H. Lau, and S. Kouider. What is consciousness, and could machines have it? Science, 358(6362):486–492, 2017.
+
+Stanislas Dehaene and Jean-Pierre Changeux. Experimental and theoretical approaches to conscious processing. Neuron, 70(2):200–227, 2011.
+
+Stanislas Dehaene, Michel Kerszberg, and Jean-Pierre Changeux. A neuronal model of a global workspace in effortful cognitive tasks. Proceedings of the national Academy of Sciences, 95(24): 14529–14534, 1998.
+
+Mostafa Dehghani, Stephan Gouws, Oriol Vinyals, Jakob Uszkoreit, and Łukasz Kaiser. Universal transformers. arXiv preprint arXiv:1807.03819, 2018.
+
+Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, et al. An image is worth 16x16 words: Transformers for image recognition at scale. arXiv preprint arXiv:2010.11929, 2020.
+
+Chrisantha Fernando, Dylan Banarse, Charles Blundell, Yori Zwols, David Ha, Andrei A Rusu, Alexander Pritzel, and Daan Wierstra. Pathnet: Evolution channels gradient descent in super neural networks. arXiv preprint arXiv:1701.08734, 2017.
+
+Jerry A Fodor. The modularity of mind. MIT press, 1983.
+
+Rohit Girdhar and Deva Ramanan. CATER: A diagnostic dataset for compositional actions and temporal reasoning. CoRR, abs/1910.04744, 2019. URL http://arxiv.org/abs/1910. 04744.
+
+Anirudh Goyal and Yoshua Bengio. Inductive biases for deep learning of higher-level cognition. arXiv preprint arXiv:2011.15091, 2020.
+
+Anirudh Goyal, Alex Lamb, Jordan Hoffmann, Shagun Sodhani, Sergey Levine, Yoshua Bengio, and Bernhard Schölkopf. Recurrent independent mechanisms. arXiv preprint arXiv:1909.10893, 2019.
+
+Anirudh Goyal, Alex Lamb, Phanideep Gampa, Philippe Beaudoin, Sergey Levine, Charles Blundell, Yoshua Bengio, and Michael Mozer. Object files and schemata: Factorizing declarative and procedural knowledge in dynamical systems. arXiv preprint arXiv:2006.16225, 2020.
+
+Alex Graves, Greg Wayne, and Ivo Danihelka. Neural turing machines. CoRR, abs/1410.5401, 2014. URL http://arxiv.org/abs/1410.5401.
+
+Alex Graves, Greg Wayne, Malcolm Reynolds, Tim Harley, Ivo Danihelka, Agnieszka GrabskaBarwinska, Sergio Gómez Colmenarejo, Edward Grefenstette, Tiago Ramalho, John Agapiou, et al. ´ Hybrid computing using a neural network with dynamic external memory. Nature, 538(7626): 471–476, 2016.
+
+Robert A Jacobs, Michael I Jordan, Steven J Nowlan, and Geoffrey E Hinton. Adaptive mixtures of local experts. Neural computation, 3(1):79–87, 1991.
+
+Andrew Jaegle, Felix Gimeno, Andrew Brock, Andrew Zisserman, Oriol Vinyals, and Joao Carreira. Perceiver: General perception with iterative attention. arXiv preprint arXiv:2103.03206, 2021.
+
+Andrej Karpathy. karpathy/mingpt, Aug 2020. URL https://github.com/karpathy/ minGPT.
+
+Nan Rosemary Ke, Anirudh Goyal ALIAS PARTH GOYAL, Olexa Bilaniuk, Jonathan Binas, Michael C Mozer, Chris Pal, and Yoshua Bengio. Sparse attentive backtracking: Temporal credit assignment through reminding. In Advances in neural information processing systems, pages 7640–7651, 2018.
+
+Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
+
+Alex Lamb, Di He, Anirudh Goyal, Guolin Ke, Chien-Feng Liao, Mirco Ravanelli, and Yoshua Bengio. Transformers with competitive ensembles of independent mechanisms, 2021. URL https://openreview.net/forum?id $\underline { { \underline { { \mathbf { \Pi } } } } } =$ 1TIrbngpW0x.
+
+Juho Lee, Yoonho Lee, Jungtaek Kim, Adam Kosiorek, Seungjin Choi, and Yee Whye Teh. Set transformer: A framework for attention-based permutation-invariant neural networks. In International Conference on Machine Learning, pages 3744–3753, 2019.
+
+Kanika Madan, Nan Rosemary Ke, Anirudh Goyal, Bernhard Schölkopf, and Yoshua Bengio. Meta attention networks: Meta-learning attention to modulate information between recurrent independent mechanisms. In International Conference on Learning Representations, 2021. URL https: //openreview.net/forum?id $\underline { { \underline { { \mathbf { \Pi } } } } } =$ Lc28QAB4ypz.
+
+Marvin Minsky. Society of mind. Simon and Schuster, 1988.
+
+Sarthak Mittal, Alex Lamb, Anirudh Goyal, Vikram Voleti, Murray Shanahan, Guillaume Lajoie, Michael Mozer, and Yoshua Bengio. Learning to combine top-down and bottom-up signals in recurrent neural networks with attention over modules. In International Conference on Machine Learning, pages 6972–6986. PMLR, 2020a.
+
+Sarthak Mittal, Alex Lamb, Anirudh Goyal, Vikram Voleti, Murray Shanahan, Guillaume Lajoie, Michael Mozer, and Yoshua Bengio. Learning to combine top-down and bottom-up signals in recurrent neural networks with attention over modules. In International Conference on Machine Learning, pages 6972–6986. PMLR, 2020b.
+
+Myle Ott, Sergey Edunov, Alexei Baevski, Angela Fan, Sam Gross, Nathan $\mathrm { N g }$ , David Grangier, and Michael Auli. fairseq: A fast, extensible toolkit for sequence modeling. In Proceedings of NAACL-HLT 2019: Demonstrations, 2019.
+
+Niki Parmar, Ashish Vaswani, Jakob Uszkoreit, Lukasz Kaiser, Noam Shazeer, Alexander Ku, and Dustin Tran. Image transformer. In International Conference on Machine Learning, pages 4055–4064. PMLR, 2018.
+
+Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, and Ilya Sutskever. Language models are unsupervised multitask learners. OpenAI blog, 1(8):9, 2019.
+
+Jack W Rae, Anna Potapenko, Siddhant M Jayakumar, and Timothy P Lillicrap. Compressive transformers for long-range sequence modelling. arXiv preprint arXiv:1911.05507, 2019.
+
+Nasim Rahaman, Anirudh Goyal, Muhammad Waleed Gondal, Manuel Wuthrich, Stefan Bauer, Yash Sharma, Yoshua Bengio, and Bernhard Schölkopf. S2rms: Spatially structured recurrent modules. arXiv preprint arXiv:2007.06533, 2020.
+
+Scott Reed and Nando De Freitas. Neural programmer-interpreters. arXiv preprint arXiv:1511.06279, 2015.
+
+Eric Ronco, Henrik Gollee, and Peter J Gawthrop. Modular neural networks and self-decomposition. Technical Report CSC-96012, 1997.
+
+Clemens Rosenbaum, Tim Klinger, and Matthew Riemer. Routing networks: Adaptive selection of non-linear functions for multi-task learning. arXiv preprint arXiv:1711.01239, 2017.
+
+Clemens Rosenbaum, Ignacio Cases, Matthew Riemer, and Tim Klinger. Routing networks and the challenges of modular and compositional computation. arXiv preprint arXiv:1904.12774, 2019.
+
+Adam Santoro, David Raposo, David G Barrett, Mateusz Malinowski, Razvan Pascanu, Peter Battaglia, and Timothy Lillicrap. A simple neural network module for relational reasoning. In Advances in neural information processing systems, pages 4967–4976, 2017.
+
+Adam Santoro, Ryan Faulkner, David Raposo, Jack Rae, Mike Chrzanowski, Theophane Weber, Daan Wierstra, Oriol Vinyals, Razvan Pascanu, and Timothy Lillicrap. Relational recurrent neural networks. In Advances in Neural Information Processing Systems, pages 7299–7310, 2018.
+
+Murray Shanahan. A cognitive architecture that combines internal simulation with a global workspace. Consciousness and cognition, 15(2):433–449, 2006.
+
+Murray Shanahan. Embodiment and the inner life: Cognition and Consciousness in the Space of Possible Minds. Oxford University Press, USA, 2010.
+
+Murray Shanahan. The brain’s connective core and its role in animal cognition. Philosophical Transactions of the Royal Society B: Biological Sciences, 367(1603):2704–2714, 2012.
+
+Murray Shanahan and Bernard Baars. Applying global workspace theory to the frame problem. Cognition, 98(2):157–176, 2005.
+
+Noam Shazeer, Azalia Mirhoseini, Krzysztof Maziarz, Andy Davis, Quoc Le, Geoffrey Hinton, and Jeff Dean. Outrageously large neural networks: The sparsely-gated mixture-of-experts layer. arXiv preprint arXiv:1701.06538, 2017.
+
+Sjoerd Van Steenkiste, Michael Chang, Klaus Greff, and Jürgen Schmidhuber. Relational neural expectation maximization: Unsupervised discovery of objects and their interactions. arXiv preprint arXiv:1802.10353, 2018.
+
+Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in neural information processing systems, pages 5998–6008, 2017.
+
+Oriol Vinyals, Igor Babuschkin, Wojciech M Czarnecki, Michaël Mathieu, Andrew Dudzik, Junyoung Chung, David H Choi, Richard Powell, Timo Ewalds, Petko Georgiev, et al. Grandmaster level in starcraft ii using multi-agent reinforcement learning. Nature, 575(7782):350–354, 2019.
+
+# Part I
+
+# Appendix
+
+A PSEUDO CODES
+
+Alg. 1 shows the integration of shared workspace with RIMs (Goyal et al., 2019). We replace the direct module to module interaction via attention in RIMs, with shared workspace. Specialists compete to write in the shared workspace, and the contents of the workspace are broadcasted to all the specialists.
+
+Alg. 2 shows the integration of the shared workspace with TIMs (Lamb et al., 2021). Again we replace the direct module to module communication in TIMs, with a shared workspace.
+
+# Algorithm 1: Shared Workspace integration with RIMs
+
+Input: Current sequence element, $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ and previous state of the specialist, $\{ h _ { t - 1 , k } \}$ , for $k \in \{ 1 , \ldots , n _ { s } \}$ and structure of memory as a matrix $M$ with row wise compartmentalized memories, where $m _ { i }$ refers to the state of slot $_ { i }$ (total number of slots is $n _ { m }$ ).
+
+Step 1: Process image by position $p$ with fully convolutional net • $\pmb { c } _ { p } = [ \mathrm { C N N } ( \pmb { x } _ { t } ) ] _ { p }$ • $\boldsymbol { z } _ { t } = [ c _ { p } \boldsymbol { e } _ { p } ]$ (concatenate encoding of position to CNN output)
+
+Step 2: Specialists compete to be selected to update the workspace based on current input   
+• qk = ht−1,kW q   
+• $\begin{array} { r } { s _ { k } = \operatorname { s o f t m a x } \left( \frac { q _ { k } \kappa } { \sqrt { d _ { e } } } \right) } \end{array}$ , where $\pmb { \kappa } = ( z _ { t } \pmb { W } ^ { e } ) ^ { \mathrm { T } }$   
+• Construct a set $\dot { \mathcal { F } } _ { t }$ which contains the indices of the $n _ { \mathrm { s e l } }$ specialists that have the largest $s _ { k }$ $\bar { \boldsymbol { h } } _ { t , k } = \left\{ \begin{array} { l l } { g _ { k } \left( \boldsymbol { s } _ { k } \boldsymbol { z } _ { t } \boldsymbol { W } ^ { v } , \boldsymbol { h } _ { t - 1 , k } \right) \quad } & { k \in \mathcal { F } _ { t } , } \\ { \boldsymbol { h } _ { t - 1 , k } \quad } & { k \notin \mathcal { F } _ { t } , } \end{array} \right.$   
+• $\pmb { a } _ { k } = s _ { k } z _ { t } W ^ { v } \forall k \in \mathcal { F } _ { t }$ (Scaled Dot Product Attention)   
+Step 3: Activated specialists write in a shared workspace   
+• $\widetilde { Q } = M \widetilde { W } ^ { q }$   
+• $\pmb { R } = [ M ; \pmb { A } ]$ where $\pmb { A }$ is the matrix whose rows are the ${ \pmb a } _ { k } \forall k \in \mathcal { F } _ { t }$   
+• $\begin{array} { r } { M \gets \operatorname { s o f t m a x } \left( \frac { \widetilde { Q } ( R \widetilde { W } ^ { e } ) ^ { \mathrm { T } } } { \sqrt { d _ { e } } } \right) R \widetilde { W } ^ { v } } \end{array}$
+
+# Step 4: Broadcast of information from the shared workspac
+
+$$
+\begin{array} { r l } & { \widehat { \widehat { q } } _ { k } = \widehat { h } _ { t , k } \widehat { W } ^ { q } \quad \forall k \in \{ 1 , \ldots , \widehat { n } _ { s } \} } \\ & { s _ { k , j } = \operatorname { s o f t m a x } \left( \frac { \widehat { q } _ { k } \widehat { \kappa } _ { j } } { \sqrt { d _ { e } } } \right) \mathrm { ~ w h e r e ~ } \widehat { \kappa } _ { j } = ( m _ { j } \widehat { W } ^ { e } ) ^ { \mathrm { T } } \quad \forall k \in \{ 1 , \ldots , n _ { s } \} , j \in \{ 1 , \ldots , n _ { m } \} } \\ & { h _ { t , k } = \widehat { h } _ { t , k } + \sum _ { j } s _ { k , j } \widehat { v } _ { j } \mathrm { ~ w h e r e ~ } \widehat { v } _ { j } = m _ { j } \widehat { W } ^ { v } \quad \forall k \in \{ 1 , \ldots , n _ { s } \} } \end{array}
+$$
+
+# B HYPERPARAMETERS
+
+Table 3 lists the different hyper-parameters.
+
+# Parameters in RIMs+SW:
+
+RIMs with shared workspace has three set of parameters:
+
+• Parameters corresponding to Input attention Parameters for the attention for the $k$ -th specialist $\theta _ { k } = ( W _ { k } ^ { \bar { q } } , W ^ { e } , \bar { W } ^ { v } )$ corresponding to query, keys, and values respectively. Each specialist has different query parameters but share the same keys and values (which are function of the input). In the table it corresponds to the inp keys, inp values, inp heads respectively.
+
+• Writing in a shared workspace: Parameters corresponding to the writing in the memory. Here, we follow the similar mechanisms as in RMC(Santoro et al., 2018), where shared
+
+# Algorithm 2: Shared Workspace integration with TIMs
+
+Notation: Consider $\pmb { h } _ { l }$ as the output of the ${ l ^ { t h } }$ transformer layer. Let sequence length of original input be $T$ and embedding dimension of transformer be $D$ . Let the transformer be composed of $n _ { b }$ mechanisms and memory be denoted as a matrix $M$ with row wise compartmentalized memories, where $m _ { i }$ refers to the state of slot $_ { i }$ (total number of slots is $n _ { m }$ ). Consider $\begin{array} { r } { { h } _ { l } ^ { k } = { \bf \dot { h } } _ { l } [ : } \end{array}$ , $( k - 1 ) D / n _ { b } : k D / n _ { b } ]$ to be the hidden state of mechanism indexed $k$ at layer $l$ .
+
+Initialization: Convert the raw input $X \in \mathbb { R } ^ { T \times v o c a b \_ s i z e }$ to   
+$h _ { 0 } = .$ positional_encodin $\jmath + \bar { E } m b e d d i n g ( X )$ where $\pmb { h } _ { 0 } \in \mathbb { R } ^ { T \times D }$ . Initialize memory matrix $M$ which remains common for all layers in the transformer.
+
+Input to the layer l: $\pmb { h } _ { l - 1 }$ having shape $\mathbb { R } ^ { T \times D }$
+
+# Step 1: Mechanisms compete to be selected to update the workspace based on the input they receive from the previous layer
+
+• W c ∈ RD/nb×1• $c _ { k } = { \pmb h } _ { l - 1 } ^ { k } W _ { k } ^ { c } \quad \forall k \in \{ 1 , \dots , n _ { b } \}$ • $c = s o f t m a x ( c o n c a t ( c _ { 1 } , . . , c _ { n _ { b } } ) )$ , $\boldsymbol { c } \in \mathbb { R } ^ { T \times n _ { b } }$
+
+• For each time step $t$ in the original sequence of length $T$ , we use the soft score $c$ to select the top $n _ { s e l }$ mechanisms which would self-attend and write to the memory. Hence generating set $\mathcal { F } _ { t }$ which stores the indices of $n _ { s e l }$ mechanisms for position $t \in \{ 1 , 2 , . . . , T \}$ . Also construct $\boldsymbol { c } _ { k } ^ { * } \in \mathbb { R } ^ { T \times D / n _ { b } }$ where
+
+$$
+c _ { k } ^ { * } [ t , : ] = \left\{ { c [ t ] [ k ] } \atop { 0 \quad k \notin \mathcal { F } _ { t } , } \right.
+$$
+
+Step 2: Selected mechanisms self-attend and update their hidden state $\bullet \ : \dot { r e s i d u a l _ { k } } = { \pmb h } _ { l - 1 } ^ { k }$ • $\bar { h } _ { l } ^ { k } = c _ { k } ^ { * } \odot S e l f A t t e n t i o n ( h _ { l - 1 } ^ { k } ) + r e s i d u a l _ { k } \quad \forall k \in \{ 1 , \ldots , n _ { b } \}$
+
+# Step 3: Selected mechanisms write on the shared workspace
+
+• Memory matrix $M$ was last modified by mechanisms of layer $l - 1$   
+• Let $\mathbf { \Delta } _ { a _ { k } } \dot { = } c _ { k } ^ { * } \odot \bar { h } _ { l } ^ { k }$ and $\mathbf { a } = c o n c a t ( \mathbf { a } _ { 1 } , . . , \mathbf { a } _ { n _ { b } } )$ . Absorb the first dimension (corresponding to position in the sequence) in the batch dimension by reshaping $\textbf { \em a }$ . Perform the same steps as in algorithm 1.   
+• $\widetilde { Q } = M \widetilde { W } ^ { q }$   
+• $\pmb { R } = [ M ; \pmb { A } ]$ where $\pmb { A } = \pmb { a } \pmb { W } ^ { v }$   
+• $\begin{array} { r } { M \gets \operatorname { s o f t m a x } \left( \frac { { \widetilde Q } ( R \widetilde W ^ { e } ) ^ { \mathrm { T } } } { \sqrt { d _ { e } } } \right) R \widetilde W ^ { v } } \end{array}$
+
+# Step 4: Broadcast of information from the shared workspace
+
+• Reshape the new memory to bring back the sequence dimension. Perform the same steps as in algorithm 1. • $\widehat { \pmb q } _ { k } = \dot { \bar { \pmb h } } _ { l } ^ { k } \widehat { \pmb W } ^ { q } \quad \forall k \in \{ 1 , \dots , n _ { b } \}$   
+• $\begin{array} { r } { s _ { k , j } = \mathrm { s o f t m a x } \left( \frac { \widehat { q } _ { k } \widehat { \kappa } _ { j } } { \sqrt { d _ { e } } } \right) } \end{array}$ where $\widehat { \kappa } _ { j } = ( m _ { j } \widehat { W } ^ { e } ) ^ { \mathrm { T } } \quad \forall k \in \{ 1 , \ldots , n _ { b } \} , \ j \in \{ 1 , \ldots , n _ { m } \}$   
+• $\begin{array} { r } { \pmb { h } _ { l } ^ { k } = \pmb { \bar { h } } _ { l } ^ { k } + \sum _ { j } s _ { k , j } \pmb { \widehat { v } } _ { j } } \end{array}$ where $\widehat { \pmb { v } } _ { j } = { \pmb { m } } _ { j } \widehat { \pmb { W } } ^ { v } \quad \forall k \in \{ 1 , \dots , n _ { b } \}$
+
+![](images/7a50dbe883119edf1ef4b31846842e37e39410c4f5272374f6928ccc0e9e54e4.jpg)  
+Figure 6: A demonstration of the detecting equilateral triangles task.
+
+Table 3: Generic Hyperparameters for the proposed model (for RIMs)   
+
+<table><tr><td>Parameter</td><td>Value 6</td></tr><tr><td>Number of specialists (ns) Size of each specialist Number of memory slots (nm)</td><td>85</td></tr><tr><td>Optimizer learning rate</td><td>Adam(Kingma and Ba, 2014) 1:10-4</td></tr><tr><td>batch size</td><td>64</td></tr><tr><td>Inp keys</td><td>64</td></tr><tr><td>Inp Values</td><td>85</td></tr><tr><td>Inp Heads</td><td>4</td></tr><tr><td>Inp Dropout</td><td>0.1</td></tr><tr><td>Number of memory slots</td><td>4</td></tr><tr><td>Number of memory heads</td><td></td></tr><tr><td></td><td>1</td></tr><tr><td>Size of attention head</td><td>32</td></tr><tr><td>Key size</td><td>32</td></tr><tr><td>Number of MLP layers in Attention</td><td>3</td></tr><tr><td>Gate Style</td><td>&#x27;unit’</td></tr><tr><td>Memory Attention Heads</td><td>4</td></tr><tr><td>Memory Attention keys</td><td>32</td></tr><tr><td></td><td>32</td></tr><tr><td>Memory Attention Values</td><td></td></tr></table>
+
+workspace is seen as a Matrix with row wise compartmentalized memories (i.e slots) i.e $\widetilde { W } ^ { q }$ , $\hat { \overline { { W } } } ^ { e }$ , $\widetilde { W } ^ { v }$ . In the table it corresponds to number of memory slots, number of memory heads, size of attention head, key size and number of mlp layers in attention. These are the same hyper-paramter as in RMC (Santoro et al., 2018). We tried two different set of hyper-parameters (a) where we only have a single slot and (b) where we have 4 slots.
+
+• Broadcast of Information from the shared workspace: In this process, the information in the workspace gets broadcasted to all the specialists such that each specialist produces a query, and the keys and values are a function of the memory state. Each specialist gets information from the memory according to its query, and this information is used to update the state of each specialist in a residual fashion. This corresponds to the parameters of $\widehat { W } ^ { v }$ , $\widehat { W } ^ { q }$ , $\widehat { W } ^ { e }$ in the table i.e memory attention heads, memory attention keys, and memory attention values. We did not do any hyper-parameter search for these hyper-parameters.
+
+# Resources Used:
+
+• For vision tasks like Sort-of-clever, Equilateral triangle, CIFAR classification, it takes about 6 hours to run 200 epochs on V100 (32G) GPU.
+
+• It takes about 2 days to train the proposed model on bouncing ball task for 100 epochs on V100 (32G) GPU. We did not do any hyper-parameter search specific to a particular dataset (i.e 4Balls or 678Balls or Curtain Task). We ran the proposed model for different number of memory slots (i.e 2/4/8) for all the different datasets.
+
+• For Starcraft task, it takes about 5 days to train on V100 (16G) GPU with batch size of 4.
+
+# C IMPLEMENTATION DETAILS
+
+Writing Information in the shared workspace. While writing information to the shared workspace, we update the workspace using a gating mechanism as proposed in Santoro et al. (2018). The gating mechanism consists of input and forget gates. Let $\bar { M } ^ { t - 1 }$ and $M ^ { t }$ be the previous and updated memory matrix respectively. Let $M$ be the result of the attention mechanism as described in step 2 of section 2.1. Let $X _ { 1 \dots n _ { s } }$ be the input to $n _ { s }$ specialists. The gating mechanism can be formulated as follows.
+
+$$
+\begin{array} { l } { { \displaystyle \bar { X } = \frac { 1 } { n _ { s } } \sum _ { i = 1 } ^ { n _ { s } } \mathrm { r e l u } ( X _ { i } \times W ^ { 1 } ) } \ ~ } \\ { { \displaystyle K = \bar { X } + \mathrm { t a n h } ( M ^ { t - 1 } ) } \ ~ } \\ { { \displaystyle I = \mathrm { s i g m o i d } ( K W ^ { I } ) } \ ~ } \\ { { \displaystyle F = \mathrm { s i g m o i d } ( K W ^ { F } ) } \ ~ } \\ { { \displaystyle M ^ { t } = I \times \mathrm { t a n h } ( M ) + F \times M ^ { t - 1 } } } \end{array}
+$$
+
+Here, $\pmb { I }$ and $\pmb { F }$ indicate the input and forget gates respectively. Note that $W ^ { 1 }$ is shared across all $n _ { s }$ specialists.
+
+# D PROPERTIES OF SHARED WORKSPACE
+
+In section 2, we claim that higher-order interaction terms and effects due to persistence of memory are key contributors to Shared Workspace performance. We support those claims here:
+
+Shared Workspace vs repeated self attention Higher-order interaction can be simulated by repeating the self-attention step multiple times at the same layer/time-step. However, due to the absence of a global communication channel, there is no constraint that the messages passed among the neural modules should lie in the same representation space. We modify a standard transformer where we repeat the self-attention step two times in every layer. We expect that $2 \times \mathrm { S e l f }$ Attention will perform worse than SW. We also run a model where both self-attention as well as shared workspace is used by the transformer to update its state.
+
+Persistence of Memory To check whether persistence is crucial for our model to perform well, we run a model where we re-initialize the shared workspace at every layer. Again we expect that removing memory persistence should result in a drop in performance and speed of convergence.
+
+We run these models on sort-of-clevr dataset and present the results in figure 7
+
+We note that removing persistence of memory results in significantly slower convergence. Replacing SW with $2 \times \mathbf { S } \mathbf { A }$ results in a significant drop in performance.
+
+![](images/7d04817b9b8a4d2d9c59791011021b1b07be31c9268175327f0b50a1bb66bd69.jpg)  
+Figure 7: Comparison on Sort-of-CLEVR relational reasoning. Speed of convergence for relational and non-relational questions in the sort-of-clevr dataset. We can see that the Shared Workspace model converges faster and generalizes better as compared to all the other models. Here SW refers to shared workspace, $2 \times \mathbf { S } \mathbf { A }$ refers to applying self-attention twice in the same layer, $\mathrm { S W } { + } \mathrm { S A }$ refers using both Shared Workspace and Self Attention in each transformer layer.
+
+# Relationalquestions:
+
+1.What is the shape ofthe object closest to the red object $2 \Rightarrow$ square   
+2.What isthe shape of theobject furthest totheorange object $\wr \Rightarrow$ circle   
+3.How many objects have same shape with the blue object?=3
+
+![](images/8d9954974cd8f8d486c5e79fb7e44932d44ca39c88091c1dbf1101f64c3c1f4e.jpg)
+
+# Non-relational questions:
+
+1.What is the shape of the red object $\Rightarrow$ Circle   
+2.Is green object placed on the left side of the image?⇒yes   
+3.Is orange object placed on the upside of the image?= no
+
+Figure 8: A sample from the sort-of-clevr dataset.
+
+# E TRANSFORMER TASKS
+
+# E.1 DETECTING EQUILATERAL TRIANGLES
+
+A demonstration of this task can be found in figure 6. We use images of size $6 4 \times 6 4$ for this task. Our training dataset consists of 50000 examples and we evaluate on 10000 examples. We follow the same setup as vision transformers Dosovitskiy et al. (2020) for this task. We divide the image into patches of size $4 \times 4$ , this sequence of patches is fed as input to a 4-layered transformer along with the CLS token which is used for classification. We set hidden dim to 256 and ffn dim to 512. For the proposed model $\mathrm { T R } { + } { \cal S } { \cal S } { \cal W } _ { ; }$ , $\mathrm { T R } { + } \mathrm { H S W }$ ), We use a query and key size of 32, and value size of 64. We use 4 heads during reading from and writing into the shared workspace which consist of 8 memory slots. For the baseline models (TR, $\mathrm { T R } + \mathrm { H C }$ , STR), we use query, key and value size of 64 and 4 heads. For training, we use a batch size of 64. We train the model for 200 epochs using Adam optimizer with a learning rate of 0.0001. We anneal the learning rate using cosine annealing.
+
+# E.2 SORT-OF-CLEVR
+
+Figure 8 shows a sample from this dataset. The images in this dataset are of size $7 5 \times 7 5$ . Each question is encoded into 11 bits. The first 6 bits indicate color, the next 2 bits indicate question type (relational or non-relational), and the remaining 3 bits indicate question subtype (according to figure 8). We use a 4-layered transformer for this task with hidden dim set to 256 and ffn dim set to 512. For the proposed model $\mathrm { T R } { + } \mathrm { S S W } _ { \mathrm { \Omega } }$ , $\mathrm { T R } { + } \mathrm { H S W }$ ), We use a query and key size of 32, and value size of 64. We use 4 heads during reading from and writing into the shared workspace which consists of 8 memory slots. For the baseline models (TR, $\mathrm { T R } + \mathrm { H C }$ , STR), we use query, key and value size of 64 and 4 heads. We encode the 11 bit question into a 256 dimensional vector representation and concatenate it with the sequence of $1 5 \times 1 5$ sized patched obtained from the image.
+
+We use the representation corresponding to the CLS token for classification. We train the model using cross-entropy loss. We use a batch size of 64 and train the model for 100 epochs. We use Adam optimizer with a learning rate of 0.0001 for training.
+
+# E.3 CATER: OBJECT TRACKING
+
+Each CATER video consists of about 300 frames of size $2 2 4 \times 2 2 4$ . We first sample frames at a sampling rate of 6 which results in 50 frames. From these 50 frames, we stack 5 consecutive frames together and pass each stack through a 18 layered resnet. The corresponding sequence of 10 frames is passed as input to the transformer. This task is setup as a classification task where we have to predict which cell in the $6 \times 6$ grid contains the snitch in the final frame. We use a 6-layered transformer with hidden dim set to 512 and ffn dim set to 2048. For the proposed model $\mathrm { T R } { + } \mathrm { S S W }$ , $\mathrm { T R + H S W }$ ), We use a query and key size of 32, and value size of 64. We use 8 heads during reading from and writing into the shared workspace which consists of 8 memory slots. For the baseline models (TR, $\mathrm { T R } + \mathrm { H C }$ , STR), we use query, key and value size of 64 and 8 heads.
+
+# F RIMS TASKS
+
+# F.1 BOUNCING BALL
+
+The dataset consists of 50,000 training examples and 10,000 test examples showing ${ \sim } 5 0$ frames of either 4 solid balls bouncing in a confined square geometry (4Balls), 6-8 balls bouncing in a confined geometry (678Balls), 3 balls bouncing in a confined geometry with an occluded region (Curtain), or balls of different colors (Colored 4Balls) and (Colored 678Balls). We train baselines as well as the proposed shared workspace extension (e.g., RIMs $+ \ S \mathbf { W } _ { \mathbf { \alpha } }$ ). As shown in Fig. 9, we study the performance of the proposed model compared with LSTM, RIMs and RMC. The first 10 frames of ground truth are fed in and then the system is rolled out for the next 35 time steps. During the rollout phase, the proposed method performs better than the baselines in accurately predicting the dynamics of the balls as reflected by cross entropy (CE).
+
+We trained baselines as well as proposed model for about 100 epochs. We use the same architecture for encoder as well as decoder as in (Van Steenkiste et al., 2018). Hyper-parameters specific to the proposed architecture are listed in Tab. 3.
+
+![](images/cdad1d828ccd48f59731ebd6fa5dcc3a9c8292bdf3ef7a320b7804214b58cfe5.jpg)  
+Figure 9: Bouncing ball motion: Prediction error comparison of the proposed method, LSTM, RIMs and RMC baseline. Given 10 frames of ground truth, the model predicts the rollout over the next 35 steps. Here, we present the BCE for the 30th frame and $4 5 \mathrm { t h }$ frame. The proposed SW extension performs better than other baselines in accurately predicting the dynamics, with an increasing advantage as the number of unrolled steps (30 vs 45) and balls ((a) vs (b)) increases. Results are an average over 5 random seeds.
+
+# G INTEGRATING SW WITH MORE ARCHITECTURES
+
+# G.1 TIMS
+
+TIMs was proposed by Lamb et al. (2021). A transformer network is divided into ‘independent mechanisms’ which update their state via sharing information between positions and sharing information between mechanisms. The information sharing step between mechanisms can be replaced by SW to create TIMs $+ \mathbf { S } \mathbf { W } .$ .
+
+# G.1.1 MULTIMNIST GENERATION
+
+In this task, we train an Image Transformer Parmar et al. (2018) (pixel-by-pixel, raster-order generative model) for next pixel prediction task on the “MultiMNIST dataset”
+
+Table 4: Hyperparameters for MultiMNIST Task   
+
+<table><tr><td>Parameter</td><td>Value</td></tr><tr><td>Common Parameters</td><td></td></tr><tr><td>Optimizer</td><td>Adam(Kingma and Ba, 2014)</td></tr><tr><td>Learning rate</td><td>1:10-3 12</td></tr><tr><td>Batch size</td><td>8</td></tr><tr><td>Number of attention heads</td><td></td></tr><tr><td>TR</td><td></td></tr><tr><td>Size of transformer layer</td><td>256</td></tr><tr><td>TIMs</td><td></td></tr><tr><td>Number of mechanisms Size of mechanism</td><td>4 48</td></tr><tr><td></td><td></td></tr><tr><td>TIMs+SW</td><td></td></tr><tr><td>Number of mechanisms</td><td>4 40</td></tr><tr><td>Size of mechanism</td><td>2</td></tr><tr><td>Number of memory slots</td><td>160</td></tr><tr><td>Size of memory slots Memory Attention Heads</td><td>8</td></tr><tr><td>Gate Style</td><td></td></tr><tr><td></td><td>&#x27;unit&#x27;</td></tr><tr><td>Number of MLP layers in Attention</td><td>2</td></tr></table>
+
+Each $3 2 \times 3 2$ image in this dataset is made up of four randomly selected (and augmented) MNIST digits (resized to $3 2 \times 8 \time 1 0 \mathrm { \Omega }$ ) placed side-by-side as shown in figure 10. The digits themselves are selected independently of one-another.
+
+The main aim of creating such a task is to observe the working of independent mechanisms in architectures such as TIMs (Lamb et al., 2021). Each image in the MultiMNIST dataset can be broken down into different sets of independent spatial components. Since the digits which make up the image are independently selected, the joint distribution of pixel intensities in any one of the four sections of the image is statistically independent of the pixel intensities in any other section of the image. Moreover each section of the image can be further broken down into independent spatial components: one that pertains to the background and one that pertains to the foreground.
+
+It is expected that a monolithic architecture (having a single computational unit) would have to devote a significant portion of its training to learn the statistical independence between the different constituents of the image. On the other hand, architectures made up of sparsely interacting independent mechanisms have a natural way of capturing such statistical independence. A division of labour where each mechanism is focused on the generation of a distinct independent constituent of the image should allow for better generalization on the test set. Once the generation of a constituent is completed, the task can be handed over to some other mechanism based on current position in the image.
+
+For this experiment we train a standard transformer with shared parameters across all layers (denoted by TR), TIMs (Lamb et al., 2021) with 4 mechanisms, and a modified version of TIMs with 4 mechanisms where the pair-wise communication between the mechanisms is replaced by communication via a shared workspace (denoted by $\mathrm { T I M s } { + } \mathrm { S W } )$ .
+
+Training. We follow the minGPT Image Transformer setup Karpathy (2020) for our experiments. All three of the configurations have 8 layers, 8 heads for multi-headed attention and use the exact same parameter initialization and base architecture. We train all three of the models for 20 epochs.
+
+In the TR model, all of the 8 monolithic layers share the same set of parameters. In TIMs and $\mathrm { T I M s } { + } \mathrm { S W } ,$ , the first two layers are the standard monolithic layers having shared parameters. The middle four layers in both of these architectures are modular layers with four mechanisms. These four
+
+<table><tr><td>Model</td><td>Loss</td></tr><tr><td>TR</td><td>0.000058</td></tr><tr><td>TIMs (4 mechanisms)</td><td>0.000050</td></tr><tr><td>TIMs+SW (4 mechanisms)</td><td>0.000042</td></tr></table>
+
+Table 5: MultiMNIST Generation Task: We report cross-entropy loss between the generated pixel values and the true pixel values on the test set of MultiMNIST Generation Task (smaller numbers are better)
+
+![](images/e90997e1b64a95e8d0a3e05255fe0979319d39aca32c250ae67f23d07690db60.jpg)  
+Figure 10: A randomly selected batch of 16 images from the MultiMNIST generation dataset (4 rows and 4 columns)
+
+layers share the same set of parameters. In the case of $\mathrm { T I M s } { + } { \cal { S } } \mathrm { W } ,$ the four mechanisms in these layers communicate via a shared workspace (having 2 memory slots). This shared workspace is common for all four middles layers and is absent in TIMs where the mechanisms communicate via pair-wise competition as proposed in the original paper. TIMs and $\mathrm { T I M s } { + } \mathrm { S W }$ architectures are concluded by two more monolithic layers which again share the same parameters.
+
+For all three models to have comparable number of parameters, we chose the transformer embedding dimension to be 256 for TR model, 192 for TIMs model and 160 for $\mathrm { T I M s } { + } \mathrm { S W }$ model. In TIMs and $\mathrm { T I M s } { + } \mathrm { S W } ,$ the embedding dimension is divided equally among the four specialists. Each memory slot in the shared workspace of the $\mathrm { T I M s } { + } \mathrm { S W }$ model has a 160 dimensional embedding and the model uses four heads to perform read and write operations on the shared workspace. Total number of parameters for all three architectures lie between 1M and 1.8M.
+
+Results. We observe the best cross-entropy loss in 20 epochs on the test set of the MultiMNIST dataset for the next pixel prediction task in the table 5. We further plot the sixth layer “mechanism activation score” of TIMs and $\mathrm { T I M s } { + } S \mathrm { W }$ while generating the first four images of the test set in the best epoch (shown in figure 5).
+
+# G.1.2 USING WORKSPACE FOR LANGUAGE MODELLING
+
+We train our models on the WikiText-103 dataset by posing a language modeling problem. The dataset is divided into train, test and validation sets which are composed out of 28,475, 60 and 60 articles respectively. The total number of tokens in the train set is more than 103 million, hence the name of the dataset. This dataset retains numbers, punctuation and case.
+
+Training. We train our models for 15 epochs for the next word prediction task on the WikiText-103 dataset and report the perplexity on the validation set. We show the results using TIMs (Lamb et al., 2021) with 4 mechanisms and TIMs $+ \mathbf { S } \mathbf { W }$ with 4 mechanisms (where we replace the pairwise communication in TIMs with communication via a shared workspace like in the MultiMNIST experiment). We modify the FAIRSEQ Ott et al. (2019) transformer language model class for all of our experiments.
+
+For $\mathrm { T I M s } { + } \mathrm { S W }$ , we train and test two different variants: TIMs+SSW uses soft attention to generate the activation scores of competing independent mechanisms whereas TIMs+HSW uses top-k attention with ${ \bf k } = 2$ .
+
+Since in this test, our aim is to compare the performance of the two models for the language modeling task, the architectures are only made up of a transformer decoder. In both of the models, there are 8 transformer decoder layers divided into 3 sets. The first 2 layers are standard monolithic decoder layers which share the same parameters. The next 4 layers are modular layers (TIMs layers or $\mathrm { T I M s } { + } \mathrm { S W }$ layers depending on the model choice). These layers also share the same parameters among themselves. The last 2 layers are again standard monolothic decoder layers, both sharing the same parameters.
+
+The inputs to the network are 1024 dimensional word embeddings, input to a transformer layer of dimension 1024 and feed forward dimension of 2048.
+
+Both of the networks have 8 attention heads with head dimension of 128. The total transformer layer size of $8 \times 1 2 8 = 1 0 2 4$ is equally divided among the four mechanisms. In the case of TIMs, these mechanisms (in layers 3,4,5) interact via pair-wise communication, whereas in TIMs+SSW and TIMs $+ \mathrm { H S W }$ , these mechanisms interact via a shared workspace. The shared workspace has 2 memory slots, each 1024 dimensional, having 4 attention heads for reading and writing.
+
+Table 6: Hyperparameters for WikiText-103 Language Modeling Task   
+
+<table><tr><td>Parameter Value</td></tr><tr><td>Common Parameters</td></tr><tr><td>Optimizer Adam(Kingma and Ba, 2014)</td></tr><tr><td>Learning rate 5:10-4</td></tr><tr><td>Adam betas 0.99, 0.98</td></tr><tr><td>Weight decay</td></tr><tr><td>0.01 lr scheduler ‘inverse square root&#x27;</td></tr><tr><td>Max tokens per gpu 3078</td></tr><tr><td>Batch size multiple 8 8</td></tr><tr><td>Number of attention heads</td></tr><tr><td>Transformer layer size 1024</td></tr><tr><td>Number of Mechanisms</td></tr><tr><td>Update frequency</td></tr><tr><td>Number of warmup updates 4000</td></tr><tr><td>Starting Warmup lr 1·10-7</td></tr><tr><td>TIMs+SSW</td></tr><tr><td>Number of memory slots 2 1024 4</td></tr><tr><td>Size of memory slots</td></tr><tr><td>Memory Attention Heads &#x27;unit&#x27;</td></tr><tr><td>Gate Style</td></tr><tr><td>Number of MLP layers in Attention 3 False</td></tr><tr><td>top-k competition</td></tr><tr><td>TIMs+HSW</td></tr><tr><td>Number of memory slots 2</td></tr><tr><td>Size of memory slots 1024</td></tr><tr><td>Memory Attention Heads</td></tr><tr><td>4 Gate Style &#x27;unit&#x27;</td></tr><tr><td>Number of MLP layers in Attention 3</td></tr><tr><td>top-k competition True, k=2</td></tr></table>
+
+Results. We plot the perplexity (per epoch) on the validation set. All models have comparable number of parameters (within a $10 \%$ difference). We note that TIMs performs poorly on this dataset but adding shared workspace improves the performance consistently. We also note that sparsity indeed helps as TIMs+HSW performed the best.
+
+![](images/32c62abdcace816019c28982ab5507952cc6a9e65279b97c4b160ab9c0c8e49d.jpg)  
+Figure 11: Per epoch validation perplexity for TIMs, $\mathrm { T I M s } { + } \mathrm { S S W }$ , $\mathrm { T I M s } { + } \mathrm { H S W }$ for wikitext-103 language modeling task

parse/dev/toR64fsPir/toR64fsPir_content_list.json

@@ -0,0 +1,1311 @@
+[
+    {
+        "type": "text",
+        "text": "Structure-Preserving Embedding of Multi-layer Networks ",
+        "text_level": 1,
+        "bbox": [
+            210,
+            122,
+            789,
+            171
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
+        "bbox": [
+            423,
+            222,
+            578,
+            276
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            313,
+            535,
+            329
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 This paper investigates structure-preserving embedding for multi-layer networks   \n2 with community structure. We propose a novel generative tensor-based latent space   \n3 model (TLSM) that allows heterogeneity among vertices. It embeds vertices into   \n4 a low-dimensional latent space so that vertices within the same community are   \n5 close to each other in the ambient space, and captures layer heterogeneity through   \n6 a layer-effect factor matrix. With a general and flexible tensor decomposition   \n7 on the expected network adjacency tensor, TLSM is dedicated to preserving the   \n8 original vertex relations and layer-specific effects in the network embedding. An   \n9 efficient alternative updating scheme is developed to estimate the model parameters   \n10 and conduct community detection simultaneously. Theoretically, we establish the   \n11 asymptotic consistencies of TLSM in terms of both multi-layer network estimation   \n12 and community detection. The theoretical results are supported by extensive   \n13 numerical experiments on both synthetic and real-life multi-layer networks. ",
+        "bbox": [
+            148,
+            345,
+            766,
+            525
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "14 1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            551,
+            312,
+            569
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "15 Network has arisen as one of the most common structures to represent the relations among entities.   \n16 In many complex systems, entities can be multi-relational in that they may interact with each other   \n17 under various circumstances. A multi-layer network, which consists of a common vertex set across all   \n18 network layers representing the entities and an edge set at each layer to characterize a particular type   \n19 of relation among entities, is faithful to represent these relations. Examples of multi-layer networks   \n20 include social networks of multiple interaction channels [42, 15], biological networks of different   \n21 collaboration schemes [49, 31, 29] and world trading networks [1, 37] of various goods.   \n22 In this paper, we propose a structure-preserving embedding framework for multi-layer networks   \n23 via a tensor-based latent space model. Specifically, TLSM utilizes the factorization of network   \n24 adjacency tensor as a building block, embeds the vertices into a low dimensional latent space, and   \n25 captures the heterogeneity among different layers through a layer-effect factor matrix. Consequently,   \n26 the community structure of the multi-layer network can be detected from a network embedding   \n27 perspective, such that vertices within the same community are closer to one another in the ambient   \n28 space than those in different communities. In addition, one key feature of TLSM is that it introduces   \n29 a sparsity factor into the vanilla logit transformation of the network adjacency tensor, which allows   \n30 TLSM to model sparse multi-layer networks in a more explicit fashion and accommodate relatively   \n31 sparser multi-layer networks as the ones considered in literature [22]. More importantly, this sparsity   \n32 factor can be estimated from the network adjacency tensor directly.   \n33 The main contribution of this paper is three-fold. First, the proposed TLSM is flexible and general   \n34 in that it includes many popular network models as special cases. It also relaxes the layer-wise   \n35 positive semi-definite condition that has been frequently employed in literature [6, 35]. Second, a   \n36 joint modeling framework is constructed for TLSM, consisting of the multi-layer network likelihood   \n37 and a clustering type penalty, to estimate the multi-layer network and conduct community detection   \n38 simultaneously. Its advantages are supported by extensive numerical experiments on both synthetic   \n39 and real-life multi-layer networks. Third, the asymptotic consistencies of TLSM are established in   \n40 terms of both multi-layer network estimation and community detection. Notably, the established   \n41 theoretical results imply that the proposed methods can accommodate the sparsest multi-layer   \n42 networks considered in literature.   \n43 The rest of the paper is organized as follows. The remaining of Section 1 discusses related works and   \n44 introduces necessary notations. Section 2 presents the proposed TLSM and its estimation scheme with   \n45 an efficient algorithm. In Section 3, we establish the asymptotic consistencies of TLSM. Extensive   \n46 numerical performance of TLSM on synthetic and real-life multi-layer networks as well as ablation   \n47 studies on two novel components of the proposed method are carried out in Section 4. Section 5   \n48 concludes the paper. The supplementary materials contains technique proofs and necessary lemmas,   \n49 additional simulation studies, detailed parameter tuning process, among others. ",
+        "bbox": [
+            147,
+            583,
+            825,
+            680
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            686,
+            825,
+            838
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            844,
+            823,
+            900
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            145,
+            92,
+            825,
+            174
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            180,
+            825,
+            279
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "50 1.1 Related work ",
+        "text_level": 1,
+        "bbox": [
+            147,
+            294,
+            305,
+            309
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "51 While there is a growing number of literature focusing on community detection in single-layer   \n52 network [48, 28, 13], community detection in multi-layer network is still in its infancy. One classical   \n53 approach is to detect community structure in each layer separately [4, 5], which fails to leverage   \n54 the homogeneity across different layers. Another approach is to aggregate multi-layer networks   \n55 into a single-layer one [41, 12, 35], which heavily relies on the assumption of homogeneous linking   \n56 pattern across multiple layers. Recently, [26] proposed to aggregate the biased-adjusted version of   \n57 the squared adjacency matrix in each layer to alleviate the information loss in aggregation. yet it   \n58 requires the average node degree to grow at a sub-optimal order.   \n59 In terms of multi-layer network generative models, [34] extended the seminal stochastic block   \n60 model (SBM; 19) to the multi-layer stochastic block model (MLSBM; 34), where the probability for   \n61 any two vertices to form an edge in a given layer depends only on their community memberships.   \n62 Clearly, MLSBM heavily relies on the assumption of homogeneous vertices within communities.   \n63 The framework of MLSBM has also been incorporated in degree-corrected network estimation [36],   \n64 spectral clustering [6, 35, 26], least square estimation [27] and likelihood-based approaches [45]. In   \n65 addition, network response regression model [46] and tensor factorization methods [8, 22] have also   \n66 been proposed to detect community structures in multi-layer networks.   \n67 To allow heterogeneous vertices, the latent space model [18] and random dot product graph model   \n68 [3] have been extended to multi-layer networks[47, 32, 2]. In addition, graph neural network and   \n69 graph convolutional networks has been extended to multi-layer network for learning the multi-layer   \n70 network embedding [14, 23, 17, 39]. ",
+        "bbox": [
+            145,
+            319,
+            825,
+            430
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            145,
+            435,
+            825,
+            547
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            554,
+            825,
+            609
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "71 1.2 Notations ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            626,
+            279,
+            640
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "72 Throughout the paper, we use boldface calligraphic Euler scripts $( A )$ to denote tensors, boldface   \n73 capital letters $( A )$ or Greece letters $( \\alpha , \\beta )$ to denote matrices, boldface lowercase letters $( a )$ to   \n74 denote vectors, and regular letters $( a )$ to denote scalars. For an order three tensor $\\pmb { \\mathcal { A } } \\in \\mathbb { R } ^ { I _ { 1 } \\times I _ { 2 } \\times I _ { 3 } }$ ,   \n75 $\\mathcal { A } _ { i , . , . } \\in \\mathbb { R } ^ { I _ { 2 } \\times I _ { 3 } } , \\mathcal { A } _ { . , j , \\cdot } \\in \\mathbb { R } ^ { I _ { 1 } \\times I _ { 3 } }$ , and $\\pmb { \\mathscr { A } } _ { . , . , m } \\in \\mathbb { R } ^ { I _ { 1 } \\times I _ { 2 } }$ are the $i$ -th horizontal slide, $j$ -th lateral slide   \n76 and $m$ -th frontal slide of $\\mathcal { A }$ , respectively. Similarly, for a matrix $\\pmb { A }$ , $A _ { i , }$ . denotes its $i$ -th row and $A _ { . , j }$   \n77 denotes its $j$ -th column. For a vector $\\textbf { \\em a }$ , $\\mathrm { d i a g } ( a )$ stands for the diagonal matrix whose diagonal is $\\textbf { \\em a }$ .   \n78 We use $| | \\cdot | | , | | \\cdot | | _ { \\infty }$ , and $| | \\cdot | | _ { F }$ to denote the $l _ { 2 }$ -norm, $l _ { \\infty }$ -norm of a vector, and the Frobenius norm   \n79 of matrix or tensor, respectively. For any integer $n$ , denote $[ n ] = \\{ 1 , 2 , . . . , n \\}$ . ",
+        "bbox": [
+            147,
+            650,
+            826,
+            762
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "80 81 1 product betsuch that its or -th $\\pmb { \\mathcal { A } } \\in \\mathbb { R } ^ { I _ { 1 } \\times I _ { 2 } \\times I _ { 3 } }$ an as $U \\in \\mathbb { R } ^ { J _ { 1 } \\times I _ { 1 } }$ $\\pmb { A } \\times _ { 1 } \\pmb { U } \\in$ $\\mathbb { R } ^ { J _ { 1 } \\times I _ { 2 } \\times I _ { 3 } }$ $( j _ { 1 } , i _ { 2 } , i _ { 3 } )$ $\\begin{array} { r } { ( \\pmb { \\mathscr { A } } \\times _ { 1 } \\pmb { U } ) _ { j _ { 1 } , i _ { 2 } , i _ { 3 } } = \\sum _ { i _ { 1 } = 1 } ^ { I _ { 1 } } \\pmb { \\mathscr { A } } _ { i _ { 1 } , i _ { 2 } , i _ { 3 } } U _ { j _ { 1 } , i _ { 1 } } } \\end{array}$ The mode-2 or mode-3 product between $\\pmb { A }$ and any matrix of appropriate dimension are defined 83 similarly. The CANDECOMP/PARAFAC (CP) decomposition of $\\pmb { A }$ has the form ",
+        "bbox": [
+            147,
+            766,
+            825,
+            827
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fd73287583f9da777c1ee7f8f5d820903c9a757ad2e07c6113e97d028ed1aba2.jpg",
+        "text": "$$\n\\pmb { \\mathcal { A } } = \\sum _ { r = 1 } ^ { R } \\pmb { a } ^ { ( r ) } \\circ \\pmb { b } ^ { ( r ) } \\circ \\pmb { c } ^ { ( r ) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            408,
+            830,
+            588,
+            875
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where 84 $\\pmb { a } ^ { ( r ) } \\in \\mathbb { R } ^ { I _ { 1 } }$ , $\\boldsymbol { b } ^ { ( r ) } \\in \\mathbb { R } ^ { I _ { 2 } }$ , and $\\boldsymbol { c } ^ { ( r ) } \\in \\mathbb { R } ^ { I _ { 3 } }$ for $r \\in [ R ]$ , and $\\circ$ stands for the vector outer product. The CP-rank [24] of the tensor 85 $\\pmb { a } ^ { ( r ) } \\circ \\pmb { b } ^ { ( r ) } \\circ \\pmb { c } ^ { ( r ) }$ is defined to be 1, for $r \\in [ R ]$ . The minimal number ",
+        "bbox": [
+            142,
+            878,
+            825,
+            912
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "86 of rank-1 tensors in the CP decomposition of $\\pmb { A }$ is called the CP-rank of $\\pmb { A }$ . Let $\\pmb { \\mathcal { T } } \\in \\{ 0 , 1 \\} ^ { R \\times R \\times R }$   \n87 be the identity tensor such that $\\pmb { \\mathcal { T } } _ { i _ { 1 } , i _ { 2 } , i _ { 3 } } = 1$ if $i _ { 1 } = i _ { 2 } = i _ { 3 }$ and 0 otherwise, and let $\\pmb { A } \\in \\mathbb { R } ^ { I _ { 1 } \\times R }$ ,   \n88 $\\boldsymbol { B } \\in \\mathbb { R } ^ { I _ { 2 } \\times R }$ , and $C \\in \\mathbb { R } ^ { I _ { 3 } \\times R }$ such that $\\mathbf { \\boldsymbol { A } } _ { \\cdot , r } = \\mathbf { \\boldsymbol { a } } ^ { ( r ) }$ , $\\mathbf { \\delta } _ { B _ { \\cdot , r } } = \\mathbf { \\delta } _ { \\mathbf { \\delta } } \\mathbf { \\delta } _ { B _ { \\cdot , r } } ^ { ( r ) }$ , and $\\boldsymbol { C } _ { \\cdot , r } = \\boldsymbol { c } ^ { ( r ) }$ . Equation (1)   \n89 then can be equivalently written as $\\pmb { \\mathcal { A } } = \\pmb { \\mathcal { T } } \\times _ { 1 } \\pmb { A } \\times _ { 2 } \\pmb { B } \\times _ { 3 } \\pmb { C }$ . ",
+        "bbox": [
+            147,
+            90,
+            826,
+            150
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "90 2 Structure-preserving embedding ",
+        "text_level": 1,
+        "bbox": [
+            147,
+            169,
+            478,
+            186
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "91 In this paper, we consider multi-layer networks that can be represented as an undirected and un  \n92 weighted $M$ -layer graph $\\mathcal { G } = ( V , \\mathcal { E } )$ , where $V = [ n ]$ consists of the common $n$ vertices across   \n93 different layers, and $\\mathcal { E } = \\{ E ^ { ( m ) } \\} _ { m = 1 } ^ { M }$ with $E ^ { ( m ) } \\subset V \\times V$ representing the $m$ -th relation network   \n94 among vertices. A order three adjacency tensor $\\pmb { \\mathcal { A } } = ( a _ { i , j , m } ) \\in \\{ 0 , 1 \\} ^ { n \\times n \\times M }$ is then defined to   \n95 represent $\\mathcal { G }$ with entries $a _ { i , j , m } = 1$ if $( i , j ) \\in E ^ { ( m ) }$ and 0 otherwise. ",
+        "bbox": [
+            147,
+            200,
+            826,
+            275
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.1 Tensor-based latent space model ",
+        "text_level": 1,
+        "bbox": [
+            166,
+            290,
+            437,
+            306
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "97 To fully characterize the multi-layer network structure, we propose the following generative tensor  \n8 based latent space model (TLSM). For any $i \\leq j \\in [ n ]$ , and $m \\in [ M ]$ , ",
+        "bbox": [
+            155,
+            316,
+            823,
+            347
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b436436b1887ffdea938f8902e031dfd0448815099254d2b5919edccd4d1aa1b.jpg",
+        "text": "$$\n\\begin{array} { r l } & { a _ { i , j , m } = a _ { j , i , m } \\overset { i n d . } { \\sim } \\mathrm { B e r n o u l l i } ( p _ { i , j , m } ) , \\mathrm { ~ w i t h ~ } } \\\\ & { \\theta _ { i , j , m } = \\log \\Big ( \\frac { p _ { i , j , m } } { s _ { n } - p _ { i , j , m } } \\Big ) , \\mathrm { ~ a n d ~ } } \\\\ & { \\Theta = \\mathbb { Z } \\times _ { 1 } \\alpha \\times _ { 2 } \\alpha \\times _ { 3 } \\beta , \\alpha \\in \\Omega _ { \\alpha } , \\beta \\in \\Omega _ { \\beta } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            348,
+            352,
+            647,
+            426
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "99 where $\\boldsymbol { \\mathscr { x } }$ is the order three $R$ -dimensional identity tensor. Basically, (2) follows the standard routine   \n100 in the multi-layer network literature [34, 35, 27, 22] to model that $a _ { i , j , m } = a _ { j , i , m }$ are independently   \n101 generated from a Bernoulli distribution, for $i \\leq j \\in [ n ]$ and $m \\in [ M ]$ . Denote $\\pmb { \\mathcal { P } } = ( p _ { i , j , m } ) \\in$   \n102 $\\mathbb { R } ^ { n \\times n \\times M }$ as the network underlying probability tensor, and then $\\Theta = ( \\theta _ { i , j , m } ) \\in \\mathbb { R } ^ { n \\times n \\times M }$ is   \n103 the entry-wise transformation of $\\mathcal { P }$ by (3). We call the transformation (3) as the modified logit   \n104 transformation in that the constant 1 in the standard logit transformation is replaced by a sparsity   \n105 factor $s _ { n }$ , which may vanish with $n$ and $M$ . We further assume all entries of $\\mathcal { P }$ are of the order $s _ { n }$ ; that   \n106 is, there exists a constant $\\textstyle { \\frac { 1 } { 2 } } \\leq \\xi < 1$ such that $( 1 - \\xi ) s _ { n } \\leq p _ { i , j , m } \\leq \\xi s _ { n }$ , for $i , j \\in [ n ]$ and $m \\in [ M ]$   \n107 Thus, the in $s _ { n }$ essval $\\begin{array} { r } { [ - \\log \\frac { \\xi } { 1 - \\xi } , \\log \\frac { \\xi } { 1 - \\xi } ] } \\end{array}$ overall network sparsity and the entries of . More importantly, (4) models the CP d $\\Theta$ are ensured toomposition of $\\Theta$ cate inby the   \n109 factor matrices $\\pmb { \\alpha } \\in \\mathbb { R } ^ { n \\times R }$ and $\\mathbf { \\boldsymbol { \\beta } } \\in \\mathbb { R } ^ { M \\times R }$ with CP-rank $R$ , which can greatly reduce the number of   \n110 free parameters from $n ( n + 1 ) M / 2$ to $( n + M ) R$ . Throughout the paper, the CP-rank $R$ is allowed   \n111 to diverge with $n$ . In the CP decomposition of $\\Theta$ , $_ \\alpha$ is the vertex latent position matrix with each row   \n112 $\\alpha _  i , $ . serving as the embedding of vertex $i$ , and $\\beta$ captures heterogeneity across different layers. Herein,   \n113 we define the constraint sets for $_ { \\pmb { \\alpha } }$ and $\\beta$ as $\\begin{array} { r } { \\Omega _ { \\alpha } = \\{ \\alpha \\in \\mathbb { R } ^ { n \\times R } : | | \\alpha _ { i , \\cdot } | | \\leq \\sqrt { \\log \\frac { \\xi } { 1 - \\xi } } } \\end{array}$ , for $i \\in [ n ] \\}$   \n114 and $\\Omega _ { \\beta } = \\{ \\beta \\in \\mathbb { R } ^ { M \\times R } : | | \\beta _ { \\cdot , r } | | = 1 , r \\in [ R ] \\}$ . Note that the constraint on $\\beta$ is necessary for   \n115 model identification, and detailed discussion will be presented shortly. The constraint set $\\Omega _ { \\alpha } \\times \\Omega _ { \\beta }$   \n116 is sufficient to maintain the bounded condition of $\\Theta$ since a general Hölder inequality yields that   \n117 $\\begin{array} { r } { | \\theta _ { i , j , m } | = | \\pmb { \\mathcal { Z } } \\times _ { 1 } \\pmb { \\alpha } _ { i , . } ^ { T } \\times _ { 2 } \\pmb { \\alpha } _ { j , . } ^ { T } \\times _ { 3 } \\beta _ { m , . } ^ { T } | \\le | | \\pmb { \\alpha } _ { i , . } | | | | \\pmb { \\alpha } _ { j , . } | | | | \\beta _ { m , . } | | _ { \\infty } \\le \\log \\frac { \\xi } { 1 - \\xi } } \\end{array}$ . To conclude this   \n118 paragraph, we remake that the parameter $\\xi$ is introduced for theoretical purpose and it is not treated as   \n119 a tuning parameter. One can choose $\\xi$ sufficiently close to 1 in empirical studies so that the restriction   \n120 on $_ { \\pmb { \\alpha } }$ will be alleviated.   \n121 We make several essential observations of the proposed TLSM. First and foremost, TLSM is flexible   \n122 and general. It includes the celebrated MLSBM [34, 43, 35, 27, 26, 36, 22] as special case. Specif  \n123 ically, suppose the vertices comes form $K$ disjoint communities, the standard MLSBM assumes   \n124 that the underlying network probability tensor ${ \\pmb { \\mathcal { P } } } = { \\pmb { \\mathcal { B } } } \\times _ { 1 } { \\pmb { Z } } \\times _ { 2 } { \\pmb { Z } }$ , where $\\pmb { \\mathscr { B } } \\in \\mathbb { R } ^ { K \\times K \\times M }$ is a   \n125 semi-symmetric core probability tensor with $\\pmb { \\mathscr { B } } _ { k _ { 1 } , k _ { 2 } , m } = \\pmb { \\mathscr { B } } _ { k _ { 2 } , k _ { 1 } , m }$ for $k _ { 1 } , k _ { 2 } \\in [ K ]$ and $m \\in [ M ]$ ,   \n126 and $Z \\in \\{ 0 , 1 \\} ^ { n \\times K }$ is the community membership matrix with $Z _ { i , k } = 1$ if vertex $i$ comes from the   \n127 $k$ -th community and 0 otherwise. That is, the probability of any vertex pair to form an edge in a   \n128 particular layer depends only on their community memberships. Equivalently, under the modified   \n129 logit transformation (3), we have $\\Theta = \\widetilde { \\pmb { \\mathscr { B } } } \\times _ { 1 } { Z } \\times _ { 2 } { Z }$ , where $\\widetilde { B }$ is the entry-wise transformation   \n130 of $_ { \\pmb { B } }$ under (3). Taking $R$ to be the CP-rank of $\\widetilde { B }$ , the CP-decomposition of $\\widetilde { B }$ then has the form   \n131 $\\widetilde { \\pmb { \\mathscr { B } } } = \\pmb { \\mathscr { T } } \\times _ { 1 } \\pmb { C } \\times _ { 2 } \\pmb { C } \\times _ { 3 } \\ \\pmb { \\beta }$ for some matrix $C \\in \\mathbb { R } ^ { K \\times R }$ and $\\beta \\in \\mathbb { R } ^ { M \\times R }$ due to semi-symmetry.   \n132 This leads to the CP decomposition of $\\Theta$ has the form (4) with $\\mathbf { \\alpha } _ { \\alpha } = Z C$ . It is clear that MLSBM   \n133 requires vertices within the same community are homogeneous and exchangeable, while TLSM   \n134 allows vertices to have different embeddings even when they are in the same community.   \n135 Second, TLSM is identifiable when both $_ { \\pmb { \\alpha } }$ and $\\beta$ have full column ranks. When both $_ { \\pmb { \\alpha } }$ and $\\beta$   \n136 have full column ranks, the Kruskal’s $\\mathbf { k }$ -ranks [25] of $_ { \\pmb { \\alpha } }$ and $\\beta$ satisfy $k _ { \\alpha } = k _ { \\beta } = R$ , then $\\Theta$ has   \n137 CP-rank $R$ . Hence, $k _ { \\alpha } + k _ { \\alpha } + k _ { \\beta } \\geq 2 R + 2$ as long as $R \\geq 2$ . By Theorem 1 of [40], the fixed   \n138 column $l _ { 2 }$ -norm constraint of $\\beta$ implies that the tensor factorization in (4) is unique up to column   \n139 permutations of $_ { \\pmb { \\alpha } }$ and $\\beta$ and column sign flip of $_ \\alpha$ . It is important to remark that the community   \n140 structure encoded in $_ { \\pmb { \\alpha } }$ remains unchanged under any column permutation or sign flip.   \n141 Third, introducing a sparsity factor $s _ { n }$ via a modified logit transformation into the TLSM is non  \n142 trivial. We take a single-layer network as an example to illustrate the limitation of the standard   \n143 logit transformation in handling sparse network. Suppose a vanilla logit link is used to connect   \n144 the network underlying probability matrix $_ { r }$ and its transformation $\\Theta$ , and the latent space model   \n145 usually assumes that $\\breve { \\Theta } = \\alpha \\alpha ^ { T }$ . A sparse network requires the entries of $\\Theta$ diverge to negative   \n146 infinite due to the small magnitude of edge probability, which leads to unstable estimation of $_ { \\pmb { \\alpha } }$ in   \n147 numerical experiments. Moreover, this may conflict with the assumption that vertices within the same   \n148 community tend to be close in the embedding space and their inner product is likely to be positive.   \n149 These difficulties can be naturally circumvented when an appropriate $s _ { n }$ is chosen in (3). ",
+        "bbox": [
+            140,
+            431,
+            825,
+            761
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            766,
+            826,
+            912
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            88,
+            825,
+            147
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            152,
+            825,
+            237
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            242,
+            825,
+            368
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "150 2.2 Regularized likelihood ",
+        "text_level": 1,
+        "bbox": [
+            142,
+            383,
+            370,
+            398
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Given a network adjacency tensor $\\mathcal { A }$ and number of communities $K$ , our goal is to estimate the multi-layer network embedding $( \\alpha , \\beta )$ and conduct community detection on the vertices. Throughout this paper, we assume the number of potential communities $K$ is given and may diverge with $n$ . Under the TLSM framework, with slight abuse of notation, we denote the average negative log-likelihood function of the multi-layer network $\\mathcal { G }$ is $\\mathcal { L } ( \\alpha , \\beta ; \\mathcal { A } ) = \\mathcal { L } ( \\Theta ; \\mathcal { A } )$ with ",
+        "bbox": [
+            173,
+            409,
+            825,
+            479
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fd40f56fe21cadb24977808b0416a1c85722a2a2ba2497354c5059b80edb0542.jpg",
+        "text": "$$\n\\mathcal { L } ( \\Theta ; \\pmb { A } ) = \\frac { 1 } { \\varphi ( n , M ) } \\sum _ { m = 1 } ^ { M } \\sum _ { i \\leq j } L ( \\theta _ { i , j , m } ; a _ { i , j , m } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            338,
+            486,
+            656,
+            531
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where 151 $\\varphi ( n , M ) = { \\textstyle { \\frac { 1 } { 2 } } } n ( n { + } 1 ) M$ is the number of potential edges, and $\\begin{array} { r } { L ( \\theta ; a ) = \\log \\left( 1 + \\frac { s _ { n } } { 1 - s _ { n } + e ^ { - \\theta } } \\right) - } \\end{array}$ 152 $\\begin{array} { r } { a \\log \\left( \\frac { s _ { n } } { 1 - s _ { n } + e ^ { - \\theta } } \\right) } \\end{array}$ is a negative log-density of a Bernoulli random variable $a$ . We now introduce a 153 novel regularization term to detect the potential communities in $\\mathcal { G }$ , ",
+        "bbox": [
+            142,
+            537,
+            825,
+            587
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ae36dfa0403882847bfe6e74c0fd4ef5581db00cb37cdc9afbf3edd7e340e10e.jpg",
+        "text": "$$\nJ ( \\alpha ) = \\operatorname* { m i n } _ { Z \\in \\Gamma , C \\in \\mathbb { R } ^ { K \\times R } } \\frac { 1 } { n } \\| \\alpha - Z C \\| _ { F } ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            369,
+            593,
+            627,
+            625
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "154 where $C$ encodes the vertex embedding centers and ${ \\Gamma } ~ \\subset ~ \\{ 0 , 1 \\} ^ { n \\times K }$ is the set of all possible   \n155 community membership matrices; that is, for any $Z \\in \\Gamma$ , each row of $z$ consists of only one 1   \n156 indicating the community membership and all others entries being 0. This leads to the proposed   \n157 regularized cost function, ",
+        "bbox": [
+            140,
+            632,
+            825,
+            689
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8c0ea5f253f1aa6b340d087082e3a70350d2ca486563223707a3cb08ec9ace23.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal L _ { \\lambda } ( \\boldsymbol { \\alpha } , \\beta ; \\boldsymbol { \\mathcal { A } } ) = \\mathcal L ( \\boldsymbol { \\alpha } , \\beta ; \\boldsymbol { \\mathcal { A } } ) + \\lambda _ { n } J ( \\boldsymbol { \\alpha } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            366,
+            695,
+            629,
+            713
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "158 where $\\lambda _ { n }$ is a positive tuning parameter that strikes the balance between network estimation and   \n159 community detection in the cost function. It is clear that the embeddings of vertices with similar   \n160 linking pattern will be pushed towards the same center, and thus close to each other in the ambient   \n161 space, leading to the desired community structure in $\\mathcal { G }$ . ",
+        "bbox": [
+            142,
+            719,
+            825,
+            776
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.3 Projected gradient descent algorithm ",
+        "text_level": 1,
+        "bbox": [
+            161,
+            791,
+            472,
+            808
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "163 We develop a scalable projected gradient descent (PGD) algorithm to optimize the penalized cost   \n164 function (6), which is highly non-convex and can be solved only locally. PGD, which alternatively   \n165 conducts gradient step and projection step, is one of the most popular and computationally fast   \n166 algorithm in tackling non-convex optimization problem [7, 33, 47, 9]. ",
+        "bbox": [
+            143,
+            816,
+            825,
+            875
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To compute the gradients of 167 $_ \\alpha$ and $\\beta$ , we introduce the following notations. Define $\\pmb { \\mathcal { T } } \\in \\mathbb { R } ^ { n \\times n \\times M }$ with entries 168 $\\begin{array} { r } { \\pmb { \\mathcal { T } } _ { i , j , m } = \\frac { \\exp ( - \\theta _ { i , j , m } ) } { 1 - s _ { n } + \\exp ( - \\theta _ { i , j , m } ) } ( p _ { i , j , m } - a _ { i , j , m } ) } \\end{array}$ , and $\\boldsymbol { X } _ { \\mathcal { T } ( 2 , 3 ) } ^ { \\alpha , \\beta } \\in \\mathbb { R } ^ { n \\times R }$ whose $i$ -th row ",
+        "bbox": [
+            145,
+            878,
+            825,
+            916
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "169 170 al elements of the slic. Similarly, we define $( \\mathcal { T } \\times _ { 2 } \\alpha ^ { T } \\times _ { 3 } \\beta ^ { T } ) _ { i , . , . }$ $X _ { \\mathcal { T } ( 2 , 3 ) } ^ { \\alpha , \\beta } ( i , r ) ~ =$ $( \\pmb { \\mathcal { T } } \\times _ { 2 } \\pmb { \\alpha } ^ { T } \\times _ { 3 } \\beta ^ { T } ) _ { i , r , r }$ $X _ { \\mathcal { T } ( 1 , 2 ) } ^ { \\alpha , \\alpha } \\in \\mathbb { R } ^ { R \\times M }$ $\\boldsymbol { X } _ { \\mathcal { T } ( 3 ) } ^ { \\beta } \\in \\mathbb { R } ^ { n \\times R }$ $X _ { T ( 1 , 2 ) } \\in$ 171 $\\mathbb { R } ^ { n \\times M }$ , such that $X _ { \\mathcal { T } ( 1 , 2 ) } ^ { \\alpha , \\alpha } ( r , m ) = ( \\mathcal { T } \\times _ { 1 } \\alpha ^ { T } \\times _ { 2 } \\alpha ^ { T } ) _ { r , r , m }$ , $X _ { \\mathcal { T } ( 3 ) } ^ { \\beta } ( i , r ) = ( \\mathcal { T } \\times _ { 3 } \\beta ^ { T } ) _ { i , i , r }$ , and 172 $X _ { \\mathcal { T } ( 1 , 2 ) } ( i , m ) = \\mathcal { T } _ { i , i , m }$ . Consequently, when the vertex membership matrix $z$ and the community 173 center matrix $C$ are fixed, we can derive the gradients of $\\mathcal { L } _ { \\lambda } ( \\alpha , \\beta ; \\mathcal { A } )$ with respect to $_ { \\pmb { \\alpha } }$ and $\\beta$ , as $\\frac { 1 } { \\varphi ( n , M ) } \\big ( X _ { \\mathcal { T } ( 2 , 3 ) } ^ { \\alpha , \\beta } + X _ { \\mathcal { T } ( 3 ) } ^ { \\beta } \\ast \\alpha \\big ) + 2 \\lambda _ { n } ( \\alpha - Z C )$ and $\\frac { 1 } { 2 \\varphi ( n , M ) } \\big ( ( X _ { \\mathcal { T } ( 1 , 2 ) } ^ { \\alpha , \\alpha } ) ^ { T } + X _ { \\mathcal { T } ( 1 , 2 ) } ^ { T } ( \\alpha * \\alpha ) \\big ) ,$ 174 respectively. Herein, \\* denotes the Hadamard product (entry-wise product) between two matrices. ",
+        "bbox": [
+            138,
+            88,
+            826,
+            228
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Let 175 $( { \\tilde { \\alpha } } , { \\tilde { \\beta } } )$ denote the solution given by one-step gradient descent, we then project $( { \\tilde { \\alpha } } , { \\tilde { \\beta } } )$ onto 176 $\\Omega _ { \\alpha } \\times \\Omega _ { \\beta }$ in the following steps. ",
+        "bbox": [
+            147,
+            234,
+            826,
+            265
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Step 1. Multiply the $r$ -th column of $\\tilde { \\alpha } _ { . , r }$ by $| | \\tilde { \\beta } _ { . , r } | | ^ { 1 / 2 }$ for $r \\in [ R ]$ . Denote the resultant matrix as $\\tilde { \\alpha } ^ { \\prime }$ ",
+        "bbox": [
+            160,
+            270,
+            825,
+            289
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Step 2. Regularize each row of $_ { \\pmb { \\alpha } }$ as $\\begin{array} { r } { \\pmb { \\alpha } _ { i , . } = \\tilde { \\pmb { \\alpha } } _ { i , . } ^ { \\prime } \\operatorname* { m i n } \\{ \\sqrt { \\log \\frac { \\xi } { 1 - \\xi } } , | | \\tilde { \\pmb { \\alpha } } _ { i , . } ^ { \\prime } | | \\} / | | \\tilde { \\pmb { \\alpha } } _ { i , . } ^ { \\prime } | | , \\mathbf { f } } \\end{array}$ or $i \\in [ n ]$ . ",
+        "bbox": [
+            163,
+            295,
+            782,
+            319
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Step 3. Normalize the columns of 179 $\\beta$ as $\\beta _ { . , r } = \\tilde { \\beta } _ { . , r } / | | \\tilde { \\beta } _ { . , r } | |$ , for $r \\in [ R ]$ . ",
+        "bbox": [
+            147,
+            325,
+            643,
+            344
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Next, when $( \\alpha , \\beta )$ are given, we apply a $( 1 + \\delta )$ -approximation K-means algorithm on $\\tilde { \\alpha }$ to update the vertex community membership matrix $z$ and community center matrix $C$ . ",
+        "bbox": [
+            165,
+            348,
+            823,
+            377
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "182 The above steps will be alternatively conducted until convergence or reaching the maximum number   \n183 of iterations. We further summarized the developed alternative updated scheme in Algorithm 1 in   \n184 Appendix A of the supplementary materials   \n185 Several remarks on the algorithm are in order. First, Algorithm 1 can only be guaranteed to converge   \n186 to a stationary point but not any local minimizer. We hence employ a transformed higher order   \n187 orthogonal iteration (HOOI) algorithm for warm initialization in all the numerical experiments in   \n188 Section 4 and 5. Specifically, given a user-specific value $\\tau$ , we define $\\widetilde { \\Theta }$ to mimic the magnitude   \n189 of $\\Theta$ such that $\\widetilde { \\Theta } _ { i , j , m } = - \\tau$ if $a _ { i , j , m } = 0$ and $\\widetilde { \\Theta } _ { i , j , m } = \\tau$ otherwise. A standard HOOI algorithm   \n190 [11] is applied to $\\Theta$ to obtain $\\pmb { \\alpha } ^ { ( 0 ) }$ and $\\beta ^ { ( 0 ) }$ . We set $\\tau = 1 0 0$ in all the numerical experiments.   \n191 Second, the sparsity factor $s _ { n }$ is an intrinsic quantity of the multi-layer network data, and it should be   \n192 estimated from the network directly. Note that the minimal and maximal probabilities for any vertex   \n193 pair to form an edge in any layer are $p _ { \\operatorname* { m i n } } = ( 1 - \\xi ) s _ { n }$ and $p _ { \\operatorname* { m a x } } = \\xi s _ { n }$ , respectively. Interestingly,   \n194 $p _ { \\operatorname* { m i n } } + p _ { \\operatorname* { m a x } } = s _ { n }$ , which does not depend on $\\xi$ any more. Therefore, we propose to estimate $s _ { n }$ as ",
+        "bbox": [
+            142,
+            382,
+            825,
+            425
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            431,
+            826,
+            579
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/56ded90da5c22f1c9bb409437045b127787b470df32cfa6c1f7b8ab13dde0b99.jpg",
+        "text": "$$\n\\hat { s } _ { n } = \\operatorname* { m i n } _ { i \\in [ n ] } \\frac { 1 } { n M } \\sum _ { m = 1 } ^ { M } \\sum _ { j = 1 } ^ { n } a _ { i , j , m } + \\operatorname* { m a x } _ { i \\in [ n ] } \\frac { 1 } { n M } \\sum _ { m = 1 } ^ { M } \\sum _ { j = 1 } ^ { n } a _ { i , j , m } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            303,
+            579,
+            694,
+            625
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "195 which is the sum of the minimal and maximal frequencies of a vertex to form edges with all other   \n196 vertices in all layers. Third, to optimally choose $\\lambda _ { n }$ , we extend the network cross-validation by   \n197 edge sampling scheme in [30] to multi-layer networks. The detailed tuning procedure is relegated to   \n198 Appendix B in the supplementary materials. ",
+        "bbox": [
+            140,
+            625,
+            825,
+            681
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 Asymptotic theory ",
+        "text_level": 1,
+        "bbox": [
+            168,
+            699,
+            359,
+            717
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.1 Consistency in estimating $\\Theta ^ { * }$ ",
+        "text_level": 1,
+        "bbox": [
+            168,
+            728,
+            413,
+            744
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "201 Let $\\begin{array} { r } { \\lambda = \\left\\{ \\Theta = \\mathbb { Z } \\times _ { 1 } { \\pmb \\alpha } \\times _ { 2 } { \\pmb \\alpha } \\times _ { 3 } \\beta : { \\pmb \\alpha } \\in \\Omega _ { \\pmb { \\alpha } } , \\beta \\in \\Omega _ { \\beta } \\right\\} } \\end{array}$ } be the parameter space of the problem and   \n202 203 $\\Theta ^ { * } = \\mathcal { T } \\times _ { 1 } \\pmb { \\alpha } ^ { * } \\times _ { 2 } \\pmb { \\alpha } ^ { * } \\times _ { 3 } \\beta ^ { * }$ $\\begin{array} { r } { K L ( \\boldsymbol { \\Theta } ^ { * } | | \\boldsymbol { \\Theta } ) = \\varphi ^ { - 1 } ( n , M ) \\sum _ { m = 1 } ^ { M } \\sum _ { i \\leq j } E \\bigl ( L ( \\theta _ { i , j , m } ; a _ { i , j , m } ) - L ( \\theta _ { i , j , m } ^ { * } ; a _ { i , j , m } ) \\bigr ) } \\end{array}$ ty tensor. Denote be the averaged   \n204 Kullback–Leibler divergence of the network generation distributions parametrized by and , for   \n205 any $\\mathbf { \\Theta } \\Theta \\in \\Omega$ . The following large deviation inequality is derived to quantify the behavior of $\\mathcal { L } _ { \\lambda } ( \\Theta ; \\mathbf { \\mathcal { A } } )$   \n206 for any $\\Theta$ in the neighborhood of $\\Theta ^ { * }$ defined by $\\dot { K } L ( \\Theta ^ { * } | | \\Theta )$ . ",
+        "bbox": [
+            142,
+            753,
+            825,
+            843
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proposition 1. Suppose 207 $\\lambda _ { n } J ( \\alpha ^ { * } ) \\leq \\epsilon _ { n }$ , and $( n + M ) R \\varphi ^ { - 1 } ( n , M ) \\epsilon _ { n } ^ { - 1 } \\log ( \\epsilon _ { n } ^ { - 1 / 2 } ) \\leq c _ { 1 }$ for some constant 208 $c _ { 1 }$ . Then with probability at lease $\\begin{array} { r } { 1 - 2 \\exp \\Big ( - \\frac { \\varphi ( n , M ) \\epsilon _ { n } } { 1 5 6 \\frac { \\xi } { 1 - \\xi } + 2 8 \\log 2 } \\Big ) } \\end{array}$ , we have ",
+        "bbox": [
+            142,
+            847,
+            825,
+            887
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e208ff90300d7ef95c1302b1a18a29e82a5d7cde13a20851e86b33a3f30e9283.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\lambda } ( \\Theta ^ { * } ; \\mathcal { A } ) \\leq \\operatorname* { i n f } _ { \\substack { \\{ \\Theta \\in \\Omega \\vert K L ( \\Theta ^ { * } \\vert \\vert \\Theta ) \\geq 4 \\epsilon _ { n } \\} } } \\mathcal { L } _ { \\lambda } ( \\Theta ; \\mathcal { A } ) - \\epsilon _ { n } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            320,
+            890,
+            676,
+            915
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "209 Proposition 1 basically states that any estimators with sufficiently small objective value should   \n210 be close enough to $\\Theta ^ { * }$ in terms of $K \\dot { L } ( \\Theta ^ { * } | | \\Theta )$ . We next study the asymptotic behavior of these   \n211 estimators more precisely. Let $( \\hat { \\alpha } , \\hat { \\beta } ) \\in \\Omega _ { \\alpha } \\times \\Omega _ { \\beta }$ be any estimator of $( \\alpha ^ { * } , \\beta ^ { * } )$ such that ",
+        "bbox": [
+            140,
+            90,
+            825,
+            137
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ef7a779c586e3ed7fe85fe6e1e62dee90cc9586904e8779c1d72e385e96b200f.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal L _ { \\lambda } ( \\hat { \\alpha } , \\hat { \\beta } ; \\mathcal A ) \\le \\mathcal L _ { \\lambda } ( \\alpha ^ { * } , \\beta ^ { * } ; \\mathcal A ) + \\epsilon _ { n } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            374,
+            140,
+            620,
+            159
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "and denote 212 $\\widehat { \\Theta } = \\mathcal { T } \\times _ { 1 } \\hat { \\alpha } \\times _ { 2 } \\hat { \\alpha } \\times _ { 3 } \\hat { \\beta }$ . we have the following theorem. ",
+        "bbox": [
+            140,
+            161,
+            630,
+            178
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1. Under the condition of Proposition $^ { l }$ , $i f \\left( { \\hat { \\alpha } } , { \\hat { \\beta } } \\right)$ satisfies (8), then with probability at least $\\begin{array} { r } { 1 - 2 \\exp \\Big ( - \\frac { \\varphi ( n , M ) \\epsilon _ { n } } { 1 5 6 \\frac { \\xi } { 1 - \\xi } + 2 8 \\log 2 } \\Big ) } \\end{array}$ , we have ",
+        "bbox": [
+            173,
+            181,
+            826,
+            223
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a24396f6ccb9a84cdadb07e30de34f05df15036dded7dd2defadee29f5fc51c2.jpg",
+        "text": "$$\n\\frac { 1 } { n \\sqrt { M } } \\| \\widehat { \\Theta } - \\Theta ^ { * } \\| _ { F } \\leq \\frac { 4 \\sqrt { 2 } \\sqrt { \\epsilon _ { n } } } { ( 1 - \\xi ) \\sqrt { \\xi s _ { n } } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            374,
+            224,
+            622,
+            262
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "213 The condition that $\\lambda _ { n } J ( \\Theta ^ { * } ) ~ \\le ~ \\epsilon _ { n }$ in Proposition 1 is mild. It implies that the true em  \n214 beddings of vertices within the same community are close to one another. We remark that   \n215 $\\lambda _ { n } J ( \\Theta ^ { * } )$ exactly equals to zero under the MLSBM discussed in Section 2.2. The condition that   \n216 $( n + M ) R \\varphi ^ { - 1 } ( n , M ) \\epsilon _ { n } ^ { - 1 } \\log ( \\epsilon _ { n } ^ { - 1 / 2 } )$ vanishes with $n$ is also mild. When $R = O ( 1 )$ , we can take any   \n217 ϵn such that ϵn ≫ log nn min{n,M} . Consequently, to ensure $\\widehat { \\Theta }$ converges to $\\Theta ^ { * }$ , Theorem 1 implies the   \n218 smallest sparsity factor one can take is $\\begin{array} { r } { s _ { n } \\gg \\epsilon _ { n } \\gg \\frac { \\log n } { n \\operatorname* { m i n } \\{ n , M \\} } } \\end{array}$ log nn min{n,M} , which means that the average degree   \n219 of a vertex in any particular layer can be as small as $n s _ { n }$ . We remark that a common assumption   \n220 $M = O ( n )$ that appears in literature, such as [27] and [22], is not necessary in our theory. If we   \n221 further assume $\\bar { M } \\stackrel {  } { = } O ( n )$ , we find that the average degree of a vertex in any layer under the   \n222 proposed TLSM set up can be smaller than that in [27] by a factor $( M \\log n ) ^ { - 1 / 2 }$ and in [22] by a   \n223 factor $( \\log n ) ^ { - 3 }$ , showing that our theoretical result accommodates sparser multi-layer networks. ",
+        "bbox": [
+            138,
+            270,
+            826,
+            440
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.2 Consistency in community detection ",
+        "text_level": 1,
+        "bbox": [
+            166,
+            455,
+            464,
+            470
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We now turn to establish the consistency of community detection in multi-layer network $\\mathcal { G }$ . Let $\\psi ^ { * } : [ n ] \\ \\longrightarrow \\ [ K ]$ be the true community assignment function such that $\\begin{array} { r l } { \\psi ^ { * } } & { { } = } \\end{array}$ $\\begin{array} { r l } & { \\arg \\operatorname* { m i n } _ { \\psi } \\operatorname* { m i n } _ { C _ { 1 } , \\ldots , C _ { K } } \\sum _ { i = 1 } ^ { n } \\| \\pmb { \\alpha } _ { i } ^ { * } - C _ { \\psi _ { i } } \\| ^ { 2 } } \\end{array}$ , and then the community detection error of any estimated community assignment function $\\hat { \\psi }$ can be evaluated by the minimum scaled Hamming distance between $\\hat { \\psi }$ and $\\psi ^ { * }$ under permutations, which is defined as ",
+        "bbox": [
+            171,
+            479,
+            825,
+            556
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fa71d719108632ebc5ce5652e3cbd73ccf6d7c36709063f102400bdbb4ab7d27.jpg",
+        "text": "$$\n\\operatorname { e r r } ( \\psi ^ { * } , \\hat { \\psi } ) = \\operatorname* { m i n } _ { \\pi \\in S _ { K } } \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\mathbf { 1 } \\{ \\psi _ { i } ^ { * } \\neq \\pi ( \\hat { \\psi } _ { i } ) \\} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            356,
+            558,
+            640,
+            599
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "230 where $\\mathbf { 1 } \\{ \\cdot \\}$ is the indicator function and $S _ { K }$ is the symmetric group of degree $K$ . Such a scaled   \n231 or unscaled Hamming distance has become a popular metric in quantifying the performance of   \n232 community detection [21, 22].   \n233 Denote $N _ { k } ^ { * } = \\{ i : \\psi _ { i } ^ { * } = k \\}$ be the $k$ -th true underlying community whose cardinality is $n _ { k }$ . Let   \n234 $C ^ { * } \\in \\mathbb { R } ^ { K \\times R }$ be the true underlying community centers of the network embedding with $C _ { k . } ^ { * } =$   \n235 $\\begin{array} { r } { \\frac { 1 } { n _ { k } } \\sum _ { \\psi _ { i } ^ { * } = k } \\alpha _ { i . } ^ { * } } \\end{array}$ , and let $\\pmb { \\mathcal { B } } ^ { \\ast } = \\pmb { \\mathcal { T } } \\times _ { 1 } \\pmb { C } ^ { \\ast } \\times _ { 2 } \\pmb { C } ^ { \\ast } \\times _ { 3 } \\pmb { \\beta } ^ { \\ast }$ . The following assumptions are made to ensure   \n236 that communities within the multi-layer networks are asymptotically identifiable. ",
+        "bbox": [
+            142,
+            601,
+            825,
+            643
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            142,
+            648,
+            825,
+            710
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Assumption A. Assume the difference between any two distinct horizontal slides of 37 ${ \\pmb { \\beta } } ^ { * }$ satisfies that ",
+        "bbox": [
+            155,
+            713,
+            823,
+            728
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/56d4274d8147e4f4c696984727e3467e58bd89fd92f28f410a3a842656ea46a6.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { k , k ^ { \\prime } \\in [ K ] , k \\neq k ^ { \\prime } } \\frac { 1 } { \\sqrt { K M } } \\| \\pmb { \\mathscr { B } } _ { k , . , . } ^ { * } - \\pmb { \\mathscr { B } } _ { k ^ { \\prime } , . , . } ^ { * } \\| _ { F } \\geq \\gamma _ { n } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            344,
+            729,
+            650,
+            762
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "238 where $\\gamma _ { n } > 0$ may vanish with $n$ ",
+        "bbox": [
+            142,
+            763,
+            393,
+            779
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Assumption B. Assume the tuning parameter $\\lambda _ { n }$ satisfies that ",
+        "bbox": [
+            173,
+            781,
+            586,
+            796
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/13f2f021007ecfc352f6d2f624968f2ae49b49d0f9262b92d2a7cfa38c6439ca.jpg",
+        "text": "$$\n\\begin{array} { r } { \\lambda _ { n } \\epsilon _ { n } s _ { n } ^ { - 2 } ( \\log s _ { n } ^ { - 1 } ) ^ { - 1 } \\geq c _ { 2 } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            406,
+            796,
+            588,
+            815
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "for an absolute constant $c _ { 2 }$ that does not depend on any model parameter. ",
+        "bbox": [
+            171,
+            816,
+            704,
+            832
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Assumption C. Denote $n _ { \\mathrm { m i n } } = \\mathrm { m i n } _ { k \\in [ K ] } n _ { k }$ as the minimal community size. Assume ",
+        "bbox": [
+            163,
+            833,
+            736,
+            849
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/17baccceeb972a9ed41df09e86d48755a25bbf834e7736f136e84c155c082e10.jpg",
+        "text": "$$\n\\frac { \\gamma _ { n } n _ { \\mathrm { m i n } } \\sqrt { \\cal K } } { n } \\geq c _ { \\xi } \\sqrt { \\frac { \\epsilon _ { n } } { s _ { n } } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            419,
+            852,
+            576,
+            887
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where cξ =240 $\\begin{array} { r } { c _ { \\xi } = \\frac { 4 \\sqrt 2 } { ( 1 - \\xi ) \\sqrt \\xi } + c _ { 3 } \\sqrt { \\frac { ( 1 + \\delta ) \\operatorname* { m i n } \\{ M , R \\} } { M } } } \\end{array}$ and $c _ { 3 }$ is a constant that depends on $\\xi$ only. ",
+        "bbox": [
+            137,
+            890,
+            746,
+            915
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "241 Assumption A is the minimal community separation requirement, and similar assumption has been   \n242 employed in [27] with a constant $\\gamma _ { n }$ . Together with the condition $\\lambda _ { n } J ( \\alpha ^ { * } ) \\leq \\epsilon _ { n }$ in Proposition 1,   \n243 Assumption B gives a feasible interval for $\\lambda _ { n }$ . Assumption $\\textrm { C }$ allows for unbalanced communities   \n244 with vanishing $n _ { \\mathrm { m i n } } / n$ if the network is not too sparse. Note that $c _ { \\xi }$ can be further bounded by   \n245 $\\begin{array} { r } { \\frac { 4 \\sqrt { 2 } } { ( 1 - \\xi ) \\sqrt { \\xi } } + c _ { 3 } \\sqrt { 1 + \\delta } } \\end{array}$ , and the first term of $c _ { \\xi }$ will dominate the second term if $R = o ( M )$ . ",
+        "bbox": [
+            140,
+            90,
+            826,
+            169
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2. Suppose all the assumptions in Theorem $^ { l }$ as well as Assumptions $A , B$ and $C$ are satisfied, it holds true that ",
+        "bbox": [
+            171,
+            172,
+            823,
+            200
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/eb959de8c6d61f2cd3e7fa8e6ffe37bd51152ccb1a3502113ec6287df10396f8.jpg",
+        "text": "$$\ne r r ( \\psi ^ { * } , \\hat { \\psi } ) \\leq \\frac { c _ { \\xi } ^ { 2 } n \\epsilon _ { n } } { n _ { \\mathrm { m i n } } K \\gamma _ { n } ^ { 2 } s _ { n } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            406,
+            198,
+            589,
+            234
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "with probability at least 246 $\\begin{array} { r } { 1 - \\frac { 1 } { n ^ { 2 } } - 2 \\exp \\Big ( - \\frac { \\varphi ( n , M ) \\epsilon _ { n } } { 1 5 6 \\frac { \\xi } { 1 - \\xi } + 2 8 \\log 2 } \\Big ) . } \\end{array}$ ",
+        "bbox": [
+            140,
+            238,
+            581,
+            265
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2 assures that the community structure in a multi-layer network can be consistently recovered by the proposed TLSM. As a theoretical example, we consider a sparse case with $\\begin{array} { r } { s _ { n } = \\dot { \\frac { ( \\log n ) ^ { 1 + \\tau _ { 1 } } } { n \\operatorname* { m i n } \\{ n , M \\} } } } \\end{array}$ , where $0 < \\tau _ { 1 } < 1$ , $n _ { \\mathrm { m a x } } = O ( n _ { \\mathrm { m i n } } )$ , $\\begin{array} { r } { \\frac { 1 } { \\sqrt { n } } | | \\alpha ^ { * } - Z ^ { * } C ^ { * } | | _ { F } \\leq ( \\log n ) ^ { - 3 / 2 } } \\end{array}$ , and both $\\gamma _ { n }$ , $R$ and $K$ are of constant orders. With $\\begin{array} { r } { \\lambda _ { n } = \\frac { ( \\log n ) ^ { 2 + 2 \\tau _ { 1 } } } { n \\operatorname* { m i n } \\{ n , M \\} } } \\end{array}$ , Theorems 1 and 2 imply that $\\begin{array} { r } { \\epsilon _ { n } = \\frac { ( \\log n ) ^ { 1 + \\tau _ { 2 } } } { n \\operatorname* { m i n } \\{ n , M \\} } } \\end{array}$ with $0 < \\tau _ { 2 } < \\tau _ { 1 }$ and $e r r ( \\psi ^ { * } , \\hat { \\psi } ) = o _ { p } ( 1 )$ . ",
+        "bbox": [
+            165,
+            273,
+            826,
+            369
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "52 4 Numerical experiments ",
+        "text_level": 1,
+        "bbox": [
+            156,
+            387,
+            400,
+            405
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section, we evaluate the numerical performance of the proposed TLSM in a variety of synthetic as well as real-life multi-layer networks, compare it against four competitors in literature, including the mean adjacency spectral embeddings (MASE; 16), least square estimation (LSE; 27), Tucker decomposition with HOSVD initialization (HOSVD-Tucker; 22), and spectral kernel (SPECK; 35), and conduct some ablation studies. The implementations of LSE and SPECK are available at the authors’ personal websites, HOSVD-Tucker is implemented in the routine “tucker\" of the Python package “tensorly\", and TLSM and MASE are implemented in Python by ourselves. ",
+        "bbox": [
+            173,
+            417,
+            825,
+            516
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.1 Synthetic networks ",
+        "text_level": 1,
+        "bbox": [
+            165,
+            531,
+            346,
+            546
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The multi-layer network $\\mathcal { A } = ( a _ { i , j , m } ) \\in \\{ 0 , 1 \\} ^ { n \\times n \\times M }$ is generated as follows. First, we randomly select $K = 4$ elements uniformly from $\\{ 2 . 5 * ( b _ { 1 } , b _ { 2 } , \\ldots , b _ { R } ) : b _ { r } \\in \\{ - 1 , 1 \\} , r \\in [ R ] \\}$ as community centers, which are denoted as $c _ { k }$ , $k \\in [ K ]$ . Second, the latent space embedding of vertex $i$ is generated as $\\pmb { \\alpha } _ { i } = \\pmb { c } _ { \\psi _ { i } } + \\pmb { e } _ { i }$ with $\\pmb { e } _ { i } \\sim N ( \\mathbf { 0 } _ { R } , 1 . 5 * I _ { R } )$ , and $\\psi _ { i } \\in [ K ]$ are independently drawn from the multinomial distribution $\\mathbf { M u l t i } ( 1 ; \\frac { 1 } { K } \\mathbf { 1 } _ { K } )$ . Third, we generate $\\beta = [ \\beta _ { 1 } , \\ldots , \\beta _ { M } ] ^ { T }$ with $\\beta _ { m , r }$ being independent standard normal random varibeles, for $m \\in [ M ]$ and $r \\in [ R ]$ . We then rescale the column norms of $\\beta$ to be 1 for model identifiability. Finally, we generate $\\mathcal { A }$ according to the proposed TLSM with $s _ { n } = 0 . 1$ . For the sake of fair comparisons, the embedding dimension $R$ is set as $K$ in all scenarios. We aim to illustrate the community detection performance of all methods as the number of vertices and number of layers increase. To this end, we consider $( n , M ) \\in \\{ 2 0 0 , 4 0 0 , 6 0 0 , 8 0 0 \\} \\times \\{ 5 , 1 0 , 1 5 , 2 0 \\}$ . The averaged hamming errors and their standard errors over 50 independent experiments of all methods are reported in Table 1. ",
+        "bbox": [
+            173,
+            555,
+            825,
+            724
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "273 It is evident that TLSM consistently outperforms its competitors, and the performances of LSE   \n274 and HOSVD-Tucker are better than those of MASE and SPECK. This is expected since TLSM,   \n275 LSE and HOSVD-Tucker work on the multi-layer network adjacency tensor directly, while MASE   \n276 and SPECK are matrix aggregation methods that suffer form information loss. Furthermore, as the   \n277 number of vertices and number of layers increase, the community detection errors of all methods   \n278 decrease rapidly. Notably, TLSM and LSE converge faster than the other methods, and attain stable   \n279 performance even for relatively small $n$ and $M$ . Additional simulation studies for various network   \n280 sparsity and unbalanced community sizes are relegated to Appendix C in the supplementary materials. ",
+        "bbox": [
+            142,
+            729,
+            825,
+            842
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2 Real-life networks ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            858,
+            339,
+            872
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "282 We also apply the proposed TLSM method to analyze three real-life multi-layer networks, including   \n283 a social network in the department of Computer Science at Aarhus University (AUCS) [38], a yeast   \n284 Saccharomyces cerevisiae gene co-expression (YSCGC) network [44], and a worldwide agriculture   \n285 trading network (WAT) [10]. Specifically, we conduct community detection on the first two networks   \n286 whose vertex community memberships are available, and carry out a link prediction task on the third   \n287 network whose vertex community memberships are unavailable. ",
+        "bbox": [
+            147,
+            882,
+            825,
+            911
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/fa1fd74e4853d91ae7929c771b5dc9410680f0b0df89b32231ce08426314335b.jpg",
+        "table_caption": [
+            "Table 1: The averaged hamming errors of various methods with their standard errors in Scenario I. The best performer in each case is bold-faced. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>n</td><td>M</td><td>TLSM</td><td>LSE</td><td>MASE</td><td>HOSVD-Tucker</td><td>SPECK</td></tr><tr><td rowspan=\"4\">200</td><td>5</td><td>0.1180(0.0147)</td><td>0.1405(0.0118)</td><td>0.5086(0.0136)</td><td>0.1623(0.0126)</td><td>0.4254(0.0138)</td></tr><tr><td>10</td><td>0.0585(0.0046)</td><td>0.0751(0.0050)</td><td>0.4949(0.0131)</td><td>0.1148(0.0106)</td><td>0.2996(0.0141)</td></tr><tr><td>15</td><td>0.0551(0.0067)</td><td>0.0593(0.0045)</td><td>0.4910(0.0176)</td><td>0.1040(0.0115)</td><td>0.2505(0.0142)</td></tr><tr><td>20</td><td>0.0510(0.0037)</td><td>0.0588(0.0043)</td><td>0.4977(0.0161)</td><td>0.1023(0.0110)</td><td>0.1942(0.0156)</td></tr><tr><td rowspan=\"4\">400</td><td>5</td><td>0.0653(0.0066)</td><td>0.1019(0.0087)</td><td>0.3845(0.0193)</td><td>0.1220(0.0106)</td><td>0.3766(0.0195)</td></tr><tr><td>10</td><td>0.0608(0.0063)</td><td>0.0636(0.0037)</td><td>0.3859(0.0160)</td><td>0.1012(0.0092)</td><td>0.2244(0.0191)</td></tr><tr><td>15</td><td>0.0511(0.0031)</td><td>0.0595(0.0036)</td><td>0.3844(0.0221)</td><td>0.0787(0.0051)</td><td>0.1490(0.0123)</td></tr><tr><td>20</td><td>0.0536(0.0047)</td><td>0.0551(0.0036)</td><td>0.3985(0.0185)</td><td>0.0795(0.0063)</td><td>0.1409(0.0131)</td></tr><tr><td rowspan=\"4\">600</td><td>5</td><td>0.0607(0.0029)</td><td>0.0909(0.0040)</td><td>0.3665(0.0186)</td><td>0.1221(0.0108)</td><td>0.3038(0.0193)</td></tr><tr><td>10</td><td>0.0567(0.0029)</td><td>0.0688(0.0031)</td><td>0.3726(0.0179)</td><td>0.1003(0.0081)</td><td>0.1651(0.0127)</td></tr><tr><td>15</td><td>0.0558(0.0027)</td><td>0.0630(0.0030)</td><td>0.3803(0.0167)</td><td>0.0918(0.0076)</td><td>0.1231(0.0076)</td></tr><tr><td>20</td><td>0.0548(0.0028)</td><td>0.0586(0.0029)</td><td>0.3814(0.0185)</td><td>0.0883(0.0078)</td><td>0.1150(0.0088)</td></tr><tr><td rowspan=\"4\">800</td><td>5</td><td>0.0556(0.0056)</td><td>0.0768(0.0055)</td><td>0.3012(0.0194)</td><td>0.1003(0.0103)</td><td>0.2733(0.0171)</td></tr><tr><td>10</td><td>0.0560(0.0063)</td><td>0.0583(0.0034)</td><td>0.3004(0.0177)</td><td>0.0788(0.0065)</td><td>0.1424(0.0127)</td></tr><tr><td>15</td><td>0.0498(0.0030)</td><td>0.0539(0.0033)</td><td>0.3179(0.0195)</td><td>0.0812(0.0068)</td><td>0.1146(0.0098)</td></tr><tr><td>20</td><td>0.0485(0.0031)</td><td>0.0516(0.0032)</td><td>0.3184(0.0218)</td><td>0.0803(0.0075)</td><td>0.0979(0.0078)</td></tr></table>",
+        "bbox": [
+            187,
+            137,
+            805,
+            361
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            372,
+            825,
+            429
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The AUCS dataset is publicly available at http://multilayer.it.uu.se/datasets.html, and it is a $6 1 \\times 6 1 \\times 5$ multi-layer network that records pairwise relationships of 5 types among 61 persons in AUCS, including current working relationships, repeated leisure activities, regularly eating lunch together, co-authorship of a publication, and friendship on Facebook. Since 54 persons in the dataset come from 7 research groups and the other 7 persons do not belong to any group, the dataset consists of 8 communities corresponding to 7 research groups and an outlier community. Applying TLSM and its competitors to the dataset, the number of misclassified vertices by TLSM, LSE, MASE, HOSVD-Tucker and SPECK, are 8, 21, 19, 23, 18, respectively. Clearly, TLSM significantly outperforms its competitors by at least reducing $1 6 . 3 9 \\%$ of community detection error. ",
+        "bbox": [
+            171,
+            435,
+            825,
+            560
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The YSCGC dataset is publicly available at https://www.ncbi.nlm.nih.gov/pmc/articles/ $\\mathtt { P M C 1 5 6 5 9 0 } /$ , and contains 205 genes of 4 functional categories, including protein metabolism and modification, carbohydrate metabolism and catabolism, nucleobase, nucleoside, nucleotide and nucleic acide metabolism, as well as transportation. We regard these four functional category labels as the community memberships of the genes. Further, the gene expression responses are measured by 20 systematic perturbations with varying genetic and environmental conditions in 4 replicated hybridizations. We thus constructed a gene co-expression network $\\mathcal { A } = ( a _ { i , j , m } ) \\in$ $\\mathbb { R } ^ { 2 0 \\bar { 5 } \\times 2 0 5 \\times 4 }$ based on the similarities of their expressions, where each layer represents one replicated hybridization. Specifically, the similarity between genes $i$ and $j$ in the $m$ -th replication is measured by $w _ { i , j , m } = \\mathrm { e x p } \\big ( - \\| \\pmb { x } _ { i } ^ { ( m ) } - \\pmb { x } _ { j } ^ { ( m ) } \\| \\big )$ , where $\\pmb { x } _ { i } ^ { ( m ) } \\in \\mathbb { R } ^ { 2 0 }$ contains the expression levels of 20 perturbations in the $m$ -th replicated hybridization for $i \\in [ 2 0 5 ]$ and $m \\in [ 4 ]$ . The binary value $a _ { i , j , m }$ is obtained by thresholding $w _ { i , j , m }$ with the thresholding value being the $60 \\%$ quantile of all elements in $\\{ w _ { i , j , m } : i \\le j \\in [ 2 0 5 ] , m \\in [ 4 ] \\}$ . Applying TLSM and its competitors to this dataset, the number of misclassified vertices by TLSM, LSE, MASE, HOSVD-Tucker and SPECK, are 6, 9, 12, 48, 13, respectively. TLSM again outperforms its competitors in this YSCGC dataset. ",
+        "bbox": [
+            161,
+            566,
+            825,
+            780
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "312 The WAT dataset is publicly available at http://www.fao.org, and includes 364 agriculture   \n313 product trading relationships among 214 countries in 2010. To process the data, we extract 130 major   \n314 countries whose average degrees are greater than 9 from the 32 densest connected agriculture product   \n315 trading relations, leading to a $1 3 0 \\times 1 3 0 \\times 3 2$ multi-layer network. Investigating the eigen-structure   \n316 of the mode-1 matricization of the network adjacency tensor, we identify an elbow point [20] at the   \n317 7th largest eigen-value, suggesting there are 6 potential communities among the countries, and thus   \n318 we set $K = 6$ . The corresponding eigen-value plot is attached in Appendex D of the supplementary   \n319 materials. We then randomly selected $8 0 \\%$ of the entries of the adjacency tensor as the training set,   \n320 and conduct link prediction on the remaining $2 0 \\%$ of the entries. Specifically, we employ TLSM   \n321 and the adaptations of its competitors to estimate the network expected tensor $\\mathcal { P }$ and generate   \n322 estimations for the missing entries by independent Bernoulli random variables accordingly. The   \n323 averaged link prediction accuracy of TLSM, LSE, MASE, HOSVD-Tucker and SPECK over 50   \n324 independent replications are $7 9 . 6 0 \\%$ , $7 6 . 6 6 \\%$ , $7 5 . 9 6 \\%$ , $7 7 . 7 8 \\%$ and $7 9 . 0 8 \\%$ , respectively, where the   \n325 link prediction accuracy is defined as the percentile of the correctly predicted entries. Clearly, all 5   \n326 methods are comparative in terms of link prediction, while TLSM still deliver highest averaged link   \n327 prediction accuracy. ",
+        "bbox": [
+            140,
+            786,
+            825,
+            911
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            92,
+            825,
+            189
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "328 4.3 Ablation studies ",
+        "text_level": 1,
+        "bbox": [
+            143,
+            208,
+            325,
+            222
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this subsection, we carry out some ablation studies on two novel components of the proposed method, namely the sparsity factor $s _ { n }$ and the community-inducing regularizer $J ( \\alpha )$ . To study the effectiveness of $s _ { n }$ , we generate a $3 0 0 \\times 3 0 0 \\times 5$ multi-layer network with 3 communities and the true network sparsity $s _ { n } = 0 . 3$ . The blue curve in the left panel of Figure 1 shows the average Hamming error of 50 independent replications given by the proposed method when employing $\\hat { s } _ { n } \\in \\{ 0 . 0 5 i : i \\in [ 2 0 ] \\}$ in the optimization algorithm, and the red line indicates the averaged Hamming error of the proposed method with $\\hat { s } _ { n }$ estimated via the proposed data-adapted estimation scheme. It is clear that the Hamming error at $s _ { n } = 1$ is much larger than that when $s _ { n }$ is close to 0.3, showing the advantages of the modified logit transformation by $s _ { n }$ over the standard logit transformation when the network indeed reveals sparse pattern. Moreover, we observe that the red line is even lower than the minimum Hamming error in the blue curve. This further confirms the effectiveness of the proposed data-adapted estimation scheme for estimating $s _ { n }$ . ",
+        "bbox": [
+            169,
+            234,
+            825,
+            400
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/c331ba0dbd825a27deb1e7af3e215fd861180ab5cae8dc4dc8e45ed78969d719.jpg",
+        "image_caption": [
+            "Figure 1: Ablation studies on $s _ { n }$ (left) and community-inducing regularizer (right). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            189,
+            416,
+            784,
+            577
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "341 To study the effectiveness of the community-inducing regularizer in the proposed objective function,   \n342 we generate an $n \\times n \\times 5$ multi-layer network with 2 communities, for $\\overline { { n } } \\in \\{ 5 0 , 1 0 \\mathrm { { 0 } } , 2 0 0 , 4 0 0 \\}$ . In   \n343 the right panel of Figure 1, the black pillars indicate the network estimation error $\\frac { 1 } { n \\sqrt { 5 } } \\| \\widehat { \\Theta } - \\Theta ^ { * } \\| _ { F }$   \n344 given by the proposed method with $\\lambda _ { n } = 0$ which corresponds to the absence of $J ( \\alpha )$ , while the   \n345 red ones indicate the counterparts given by the proposed method with $\\lambda _ { n }$ is selected by network   \n346 cross-validation. There is a clear improvement when the community-inducing regularizer is enforced   \n347 in all scenarios, particularly for small $n$ . This showcases the helpfulness of the community-inducing   \n348 regularizer in detecting network community structure. ",
+        "bbox": [
+            140,
+            613,
+            825,
+            732
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "349 5 Conclusions ",
+        "text_level": 1,
+        "bbox": [
+            143,
+            753,
+            305,
+            770
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "50 In this paper, we propose a novel tensor-based latent space model for community detection in   \n51 multi-layer networks. The model embeds vertices into a low-dimensional latent space and views   \n52 the community structure from an network embedding perspective, so that heterogeneous structures   \n53 in different network layers can be properly integrated. The proposed model is formulated as a   \n54 regularization framework, which conducts multi-layer network estimation and community detection   \n55 simultaneously. The advantages of the proposed method are supported by extensive numerical   \n56 experiments and theoretical results. Particularly, the asymptotic consistencies of the proposed method   \n57 are established in terms of both multi-layer network estimation and community detection, even for   \n58 relatively sparse networks. ",
+        "bbox": [
+            151,
+            786,
+            825,
+            911
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "359 References [1] Luiz GA Alves, Giuseppe Mangioni, Isabella Cingolani, Francisco Aparecido Rodrigues, Pietro Panzarasa, and Yamir Moreno. The nested structural organization of the worldwide trade multi-layer network. Scientific reports, 9(1):1–14, 2019. [2] Jesús Arroyo, Avanti Athreya, Joshua Cape, Guodong Chen, Carey E Priebe, and Joshua T Vogelstein. Inference for multiple heterogeneous networks with a common invariant subspace. Journal of Machine Learning Research, 22(142):1–49, 2021. [3] Avanti Athreya, Donniell E Fishkind, Minh Tang, Carey E Priebe, Youngser Park, Joshua T Vogelstein, Keith Levin, Vince Lyzinski, and Yichen Qin. Statistical inference on random dot product graphs: a survey. The Journal of Machine Learning Research, 18(1):8393–8484, 2017. [4] Matteo Barigozzi, Giorgio Fagiolo, and Giuseppe Mangioni. Identifying the community structure of the international-trade multi-network. Physica A: statistical mechanics and its applications, 390(11):2051–2066, 2011. [5] Michele Berlingerio, Fabio Pinelli, and Francesco Calabrese. Abacus: frequent pattern miningbased community discovery in multidimensional networks. Data Mining and Knowledge Discovery, 27(3):294–320, 2013. [6] Sharmodeep Bhattacharyya and Shirshendu Chatterjee. Spectral clustering for multiple sparse networks: I. arXiv preprint arXiv:1805.10594, 2018. [7] Han Chen, Garvesh Raskutti, and Ming Yuan. Non-convex projected gradient descent for generalized low-rank tensor regression. Journal of Machine Learning Research, 20:1–37, 2019. [8] Zitai Chen, Chuan Chen, Zibin Zheng, and Yi Zhu. Tensor decomposition for multilayer networks clustering. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pages 3371–3378, 2019. [9] Eric C Chi, Brian R Gaines, Will Wei Sun, Hua Zhou, and Jian Yang. Provable convex co-clustering of tensors. Journal of Machine Learning Research, 21(214):1–58, 2020. [10] Manlio De Domenico, Vincenzo Nicosia, Alexandre Arenas, and Vito Latora. Structural reducibility of multilayer networks. Nature communications, 6(1):1–9, 2015.   \n386 [11] Lieven De Lathauwer, Bart De Moor, and Joos Vandewalle. On the best rank-1 and rank-(r 1, r 2,..., rn) approximation of higher-order tensors. SIAM journal on Matrix Analysis and Applications, 21(4):1324–1342, 2000. [12] Xiaowen Dong, Pascal Frossard, Pierre Vandergheynst, and Nikolai Nefedov. Clustering with multi-layer graphs: A spectral perspective. IEEE Transactions on Signal Processing, 60(11):5820–5831, 2012. [13] Junxian Geng, Anirban Bhattacharya, and Debdeep Pati. Probabilistic community detection with unknown number of communities. Journal of the American Statistical Association, 114(526):893–905, 2019.   \n395 [14] Mahsa Ghorbani, Mahdieh Soleymani Baghshah, and Hamid R Rabiee. Mgcn: semi-supervised classification in multi-layer graphs with graph convolutional networks. In Proceedings of the 2019 IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining, pages 208–211, 2019. [15] Derek Greene and Pádraig Cunningham. Producing a unified graph representation from multiple social network views. In Proceedings of the 5th annual ACM web science conference, pages 118–121, 2013. [16] Qiuyi Han, Kevin Xu, and Edoardo Airoldi. Consistent estimation of dynamic and multi-layer block models. In International Conference on Machine Learning, pages 1511–1520. PMLR, 2015. [17] Xin He, Qiong Liu, and You Yang. Mv-gnn: Multi-view graph neural network for compression artifacts reduction. IEEE Transactions on Image Processing, 29:6829–6840, 2020.   \n[18] Peter D Hoff, Adrian E Raftery, and Mark S Handcock. Latent space approaches to social network analysis. Journal of the American Statistical Association, 97(460):1090–1098, 2002.   \n[19] Paul W Holland, Kathryn Blackmond Laskey, and Samuel Leinhardt. Stochastic blockmodels: First steps. Social networks, 5(2):109–137, 1983.   \n[20] Pengsheng Ji and Jiashun Jin. Coauthorship and citation networks for statisticians. The Annals of Applied Statistics, 10(4):1779–1812, 2016.   \n[21] Jiashun Jin. Fast community detection by score. Ann. Statist., 43(1):57–89, 02 2015.   \n[22] Bing-Yi Jing, Ting Li, Zhongyuan Lyu, and Dong Xia. Community detection on mixture multilayer networks via regularized tensor decomposition. The Annals of Statistics, 49(6):3181– 3205, 2021.   \n[23] Muhammad Raza Khan and Joshua E Blumenstock. Multi-gcn: Graph convolutional networks for multi-view networks, with applications to global poverty. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pages 606–613, 2019.   \n[24] Tamara G. Kolda and Brett W. Bader. Tensor decompositions and applications. SIAM Review, 51:455–500, 2009.   \n[25] Joseph B Kruskal. Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics. Linear algebra and its applications, 18(2):95– 138, 1977.   \n[26] Jing Lei. Tail bounds for matrix quadratic forms and bias adjusted spectral clustering in multi-layer stochastic block models. arXiv preprint arXiv:2003.08222, 2020.   \n[27] Jing Lei, Kehui Chen, and Brian Lynch. Consistent community detection in multi-layer network data. Biometrika, 107(1):61–73, 2020.   \n[28] Jing Lei and Alessandro Rinaldo. Consistency of spectral clustering in stochastic block models. The Annals of Statistics, 43(1):215–237, 2015.   \n[29] Dong Li, Zhisong Pan, Guyu Hu, Graham Anderson, and Shan He. Active module identification from multilayer weighted gene co-expression networks: a continuous optimization approach. IEEE/ACM transactions on computational biology and bioinformatics, 2020.   \n[30] Tianxi Li, Elizaveta Levina, and Ji Zhu. Network cross-validation by edge sampling. Biometrika, 107(2):257–276, 2020.   \n[31] Xueming Liu, Enrico Maiorino, Arda Halu, Kimberly Glass, Rashmi B Prasad, Joseph Loscalzo, Jianxi Gao, and Amitabh Sharma. Robustness and lethality in multilayer biological molecular networks. Nature communications, 11(1):1–12, 2020.   \n[32] Zhongyuan Lyu, Dong Xia, and Yuan Zhang. Latent space model for higher-order networks and generalized tensor decomposition. arXiv preprint arXiv:2106.16042, 2021.   \n[33] Zhuang Ma, Zongming Ma, and Hongsong Yuan. Universal latent space model fitting for large networks with edge covariates. Journal of Machine Learning Research, 21(4):1–67, 2020.   \n[34] Subhadeep Paul and Yuguo Chen. Consistent community detection in multi-relational data through restricted multi-layer stochastic blockmodel. Electronic Journal of Statistics, 10(2):3807–3870, 2016.   \n[35] Subhadeep Paul and Yuguo Chen. Spectral and matrix factorization methods for consistent community detection in multi-layer networks. Ann. Statist., 48(1):230–250, 02 2020.   \n[36] Subhadeep Paul and Yuguo Chen. Null models and community detection in multi-layer networks. Sankhya A, pages 1–55, 2021.   \n[37] Zhuo-Ming Ren, An Zeng, and Yi-Cheng Zhang. Bridging nestedness and economic complexity in multilayer world trade networks. Humanities and Social Sciences Communications, 7(1):1–8, 2020.   \n[38] Luca Rossi and Matteo Magnani. Towards effective visual analytics on multiplex and multilayer networks. Chaos, Solitons & Fractals, 72:68–76, 2015.   \n[39] Uday Shankar Shanthamallu, Jayaraman J Thiagarajan, Huan Song, and Andreas Spanias. Gramme: Semisupervised learning using multilayered graph attention models. IEEE transactions on neural networks and learning systems, 31(10):3977–3988, 2019.   \n[40] Nicholas D Sidiropoulos and Rasmus Bro. On the uniqueness of multilinear decomposition of n-way arrays. Journal of Chemometrics: A Journal of the Chemometrics Society, 14(3):229–239, 2000.   \n[41] Wei Tang, Zhengdong Lu, and Inderjit S Dhillon. Clustering with multiple graphs. In 2009 Ninth IEEE International Conference on Data Mining, pages 1016–1021. IEEE, 2009.   \n[42] Edwin JCG Van Den Oord and Ronan Van Rossem. Differences in first graders’ school adjustment: The role of classroom characteristics and social structure of the group. Journal of School Psychology, 40(5):371–394, 2002.   \n[43] James D Wilson, John Palowitch, Shankar Bhamidi, and Andrew B Nobel. Community extraction in multilayer networks with heterogeneous community structure. The Journal of Machine Learning Research, 18(1):5458–5506, 2017.   \n[44] Ka Yee Yeung, Mario Medvedovic, and Roger E Bumgarner. Clustering gene-expression data with repeated measurements. Genome biology, 4(5):1–17, 2003.   \n[45] Yubai Yuan and Annie Qu. Community detection with dependent connectivity. The Annals of Statistics, 49(4):2378–2428, 2021.   \n[46] Jingfei Zhang, Will Wei Sun, and Lexin Li. Network response regression for modeling population of networks with covariates. arXiv preprint arXiv:1810.03192, 2018.   \n[47] Xuefei Zhang, Songkai Xue, and Ji Zhu. A flexible latent space model for multilayer networks. In International Conference on Machine Learning, pages 11288–11297. PMLR, 2020.   \n[48] Yunpeng Zhao, Elizaveta Levina, and Ji Zhu. Consistency of community detection in networks under degree-corrected stochastic block models. The Annals of Statistics, 40(4):2266–2292, 2012.   \n[49] Wei Zheng, Dingjie Wang, and Xiufen Zou. Control of multilayer biological networks and applied to target identification of complex diseases. BMC bioinformatics, 20(1):1–12, 2019. ",
+        "bbox": [
+            147,
+            54,
+            828,
+            917
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            155,
+            71,
+            830,
+            919
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            160,
+            92,
+            828,
+            622
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Checklist ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            642,
+            254,
+            659
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "1. For all authors... ",
+        "bbox": [
+            214,
+            670,
+            339,
+            684
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] See the abstract and the third paragrath of the introduction.   \n(b) Did you describe the limitations of your work? [Yes] The optimization algorithm can only be guaranteed to converge to a stationary point.   \n(c) Did you discuss any potential negative societal impacts of your work? [No] There should be no negative societal impacts.   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
+        "bbox": [
+            238,
+            689,
+            825,
+            823
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "2. If you are including theoretical results... ",
+        "bbox": [
+            214,
+            828,
+            493,
+            843
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Section 3. (b) Did you include complete proofs of all theoretical results? [Yes] All technical proofs are provided in Appendix E of the supplementary materials. ",
+        "bbox": [
+            238,
+            847,
+            825,
+            892
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "3. If you ran experiments... ",
+        "bbox": [
+            214,
+            897,
+            393,
+            911
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] The URLs for data are included in Section 4.2, and codes with instructions are included in the supplementary materials.   \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section 2.3 and Appendix B in the supplementary materials.   \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] We show the standard erros in Table 1 and $9 5 \\%$ confident intervals of additional simulation studies in Appendix C in the supplementary materials.   \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [No] ",
+        "bbox": [
+            238,
+            90,
+            825,
+            251
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ",
+        "bbox": [
+            215,
+            256,
+            823,
+            271
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "(a) If your work uses existing assets, did you cite the creators? [Yes] We used publicly available datasets and cite the creators.   \n(b) Did you mention the license of the assets? [Yes] All datasets we used are publicly available.   \n(c) Did you include any new assets either in the supplemental material or as a URL? [No]   \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [No]   \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No] All data we used do not contains personally identifiable information or offensive content. ",
+        "bbox": [
+            238,
+            273,
+            825,
+            422
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "5. If you used crowdsourcing or conducted research with human subjects... ",
+        "bbox": [
+            214,
+            426,
+            705,
+            441
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   \n(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] ",
+        "bbox": [
+            238,
+            445,
+            825,
+            535
+        ],
+        "page_idx": 12
+    }
+]