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+# INVASE: INSTANCE-WISE VARIABLE SELECTION USING NEURAL NETWORKS
+
+Jinsung Yoon   
+Department of Electrical and Computer Engineering   
+UCLA, California, USA   
+jsyoon0823@g.ucla.edu   
+James Jordon   
+Engineering Science Department   
+University of Oxford, UK   
+james.jordon@wolfson.ox.ac.uk   
+Mihaela van der Schaar   
+University of Cambridge, UK   
+Department of Electrical and Computer Engineering, UCLA, California, USA   
+Alan Turing Institute, London, UK   
+mihaela@ee.ucla.edu
+
+# ABSTRACT
+
+The advent of big data brings with it data with more and more dimensions and thus a growing need to be able to efficiently select which features to use for a variety of problems. While global feature selection has been a well-studied problem for quite some time, only recently has the paradigm of instance-wise feature selection been developed. In this paper, we propose a new instance-wise feature selection method, which we term INVASE. INVASE consists of 3 neural networks, a selector network, a predictor network and a baseline network which are used to train the selector network using the actor-critic methodology. Using this methodology, INVASE is capable of flexibly discovering feature subsets of a different size for each instance, which is a key limitation of existing state-of-the-art methods. We demonstrate through a mixture of synthetic and real data experiments that INVASE significantly outperforms state-of-the-art benchmarks.
+
+# 1 INTRODUCTION
+
+High-dimensional data is becoming more readily available, and it brings with it a growing need to be able to efficiently select which features to use for a variety of problems. When doing predictions, it is well known that using too many variables with too few samples can lead to overfitting, which can significantly hinder the performance of predictive models. In the realm of interpretability, the large dimensionality of the data is often too much information to present to a human who may be using the machine learning model as a support system. Understanding which features are most relevant to an outcome or to a model output is an important first step in improving predictions and interpretability and many works exist that tackle feature selection on a global level. However, in the heterogeneous data we typically encounter, the prediction made by a model (and indeed the true label) may rely on a different subset of the features for different subgroups within the data [14]. In this paper we propose a novel instance-wise feature selection method, INVASE (INstance-wise VAriable SElection), which attempts to learn which subset of the features is relevant for each sample, allowing us to display the minimal information required to explain each prediction and also to reduce overfitting of predictive models.
+
+Discovering a global subset of relevant features for a particular task is a well-studied problem and there are several existing methods for solving it such as Sequential Correlation Feature Selection [11], Mutual Information Feature Selection [21], Knockoff models [3], and more [10; 16]. However, global feature selection suffers from a key limitation - the features discovered by global feature selection are the same for all samples. In many cases, in particular when populations are highly heterogeneous, the relevant features may differ across samples [33; 32]. For instance, different patient subgroups have different relevant features for predicting heart failure [14]. Instance-wise feature selection methods such as [4; 27] instead try to discover the features that are relevant for each sample. When the goal is to provide an interpretable explanation of the predictions made, a key challenge is in ensuring that we do not over-explain by providing too much information (i.e. choosing too many features). Naturally, by performing feature selection on an individualized level we are able to select features that are more relevant to each sample, rather than having to choose the top $k$ features globally, which may not explain the predictions for some samples very well, but simply perform well on average across all samples.
+
+In this paper, we propose a novel instance-wise feature selection method which we term INVASE. We draw influence from actor-critic models [22] to solve the problem of backpropagating through subset sampling. Our model consists of 3 neural networks: a selector network, a predictor network and a baseline network. During training, each of these are trained iteratively, with the selector network being trained to minimize a Kullback-Leibler (KL) divergence between the full conditional distribution and the selected-features-only conditional distribution of the outcome. Our model is capable of discovering a different number of relevant variables for each sample which is a key limitation in existing instance-wise approaches (such as [4]). We show significant improvements over the state-of-the-art in both synthetic data and real-word data in terms of true positive rates, false discovery rates, and show better predictive performance with respect to several prediction metrics. Our model can also be easily extended to handle both continuous and discrete outputs and time-series inputs (see the Appendix for details).
+
+# 1.1 RELATED WORKS
+
+There are many existing works on global variable selection (see [10] for a good summary paper). [21] and [11] use max-dependency min-redundancy criteria [17] with mutual information and Pearson correlation, respectively. [3] uses multiple hypothesis testing for global variable selection. As noted above, these global selection methods are not capable of learning sample-specific relevance.
+
+Instance-wise variable selection is also closely related to model interpretation methods. Some previous works are based on backpropagation from the output of the predictive model to the input variables [29]. DeepLIFT [27] decomposes the output of the neural network on a reference input to compute the contribution of each input variable. However, both methods need white-box access to the pre-trained predictive models to compute the gradient and decomposition. [2] approximates the predictive models using a Parzen window approximator when there is only black-box access to the predictive models. Some other works are based on input perturbation such as [1], [15], [30] and [5]. [18] uses Shapley values to compute the variable importance, and [24] uses locally linear models to explain the linear dependency for each sample. [19] tries to interpret tree ensemble models using Shapley values but cannot generalize to other predictive models such as neural networks.
+
+Our work is most closely related to L2X (Learning to Explain) [4]. However, there are 3 key differences between our work and theirs. In L2X, they try to maximize a lower bound of the mutual information between the target $Y$ and the selected input variables $X _ { S }$ . In contrast, we try to minimize the KL divergence between the conditional distributions $Y | X$ and $Y | X _ { S }$ . In order to be able to backpropagate through subset sampling, L2X use the Gumbel-softmax trick [13] to approximately discretize the continuous outputs of the neural network. In our work, we use methods from actorcritic models [22] to bypass backpropagation through the sampling and instead use the predictor network to provide a reward to the selector network. Finally, due to the Gumbel-softmax used in L2X, the number of variables to be detected must be fixed in advance and is necessarily the same for every sample. The actor-critic methodology used in our model has no such limitations and so we are able to flexibly select a different number of relevant variables for each sample and instead induce sparsity using an $l _ { 0 }$ penalty term. In fact, using the actor-critic methodology allows us to directly use the $l _ { 0 }$ penalty term (which is not differentiable and therefore not practical to use in general). A summary table highlighting the key features of all of the related works can be found in the Appendix.
+
+# 2 PROBLEM FORMULATION
+
+Let ${ \mathcal { X } } = { \mathcal { X } } _ { 1 } \times \ldots \times { \mathcal { X } } _ { d }$ be a $d$ -dimensional feature space and $\boldsymbol { \mathcal { V } } = \{ 1 , . . . , c \}$ be a discrete label space1. Let $\mathbf { X } = ( X _ { 1 } , . . . , X _ { d } ) \in \mathcal { X }$ and $Y \in \mathcal { D }$ be random variables with joint density (or mass) $p$ and marginal densities (or masses) $p _ { X }$ and $p _ { Y }$ respectively. We will refer to $\mathbf { s } \in \{ 0 , 1 \} ^ { d }$ as the selection vector, where $s _ { i } = 1$ will indicate that variable $i$ is selected, and $s _ { i } ~ = ~ 0$ will indicate that variable $i$ is not selected. Let $^ *$ be any point not in any of the spaces $\mathcal { X } _ { 1 } , . . . , \mathcal { X } _ { d }$ and define ${ { \mathcal { X } } _ { i } ^ { * } } = { { \mathcal { X } } _ { i } } \cup \left\{ * \right\}$ and $\mathcal { X } ^ { * } = \mathcal { X } _ { 1 } ^ { * } \times \ldots \times \mathcal { X } _ { d } ^ { * }$ . Given $\mathbf { x } \in \mathcal { X }$ we will write $\mathbf { x } ^ { ( \mathbf { s } ) }$ to denote the suppressed feature vector defined by
+
+$$
+x _ { i } ^ { ( \mathbf { s } ) } = \left\{ { \atop * \operatorname { i f } s _ { i } = 0 } \right.
+$$
+
+so that $^ *$ represents that a feature is not selected.
+
+In the global feature selection literature, the goal is to find the smallest s (i.e. the one with fewest 1s) such that $\mathbb { E } ( Y | \mathbf { X } ^ { ( \mathbf { s } ) } ) = \mathbb { E } ( Y | \mathbf { X } )$ , or equivalently such that the conditional distribution of $Y$ given $\mathbf { X } ^ { ( \mathbf { s } ) }$ is the same as $Y$ given all of $\mathbf { X }$ . Note that this definition is given fully in terms of random variables, rather than realizations of those random variables.
+
+In contrast, our problem necessarily needs to be defined in terms of realizations since we are aiming to select features for a given realization. We will write $\mathbf { x }$ to denote realizations of the random variable $\mathbf { X }$ . Then we formalize our problem as one of finding a selector function, $S : \mathcal { X }  \{ 0 , 1 \} ^ { d }$ such that for almost every $\mathbf { x } \in \mathcal { X }$ (w.r.t. $p _ { X }$ ) we have
+
+$$
+( Y | \mathbf { X } ^ { ( S ( \mathbf { x } ) ) } = \mathbf { x } ^ { ( S ( \mathbf { x } ) ) } ) \overset { d . } { = } ( Y | \mathbf { X } = \mathbf { x } )
+$$
+
+where $\underline { { \underline { { d . } } } }$ denotes equality in distribution and $S ( \mathbf { x } )$ is minimal (i.e. fewest 1s) such that (1) holds.
+
+We suppose that we have a dataset $\mathbfcal { D } = \{ ( \mathbf { x } _ { j } , y _ { j } ) \} _ { j = 1 } ^ { n }$ consisting of $n$ i.i.d. realizations of the pair $( \mathbf { X } , Y )$ .2 Note that $Y$ can be viewed as having either come from a dataset, in which case the problem is of selecting predictive features, or as having come from a predictive model, in which case the problem is of explaining the model’s predictions.
+
+# 2.1 OPTIMIZATION PROBLEM
+
+In order to learn a suitable selector function, we transform the constraint (1) into a soft constraint using the Kullback-Leibler (KL) divergence which, for random variables $W$ and $V$ with densities $p _ { W }$ and $p _ { V }$ is defined as
+
+$$
+K L ( W | | V ) = \mathbb { E } \left[ \log \left( \frac { p _ { W } ( W ) } { p _ { V } ( W ) } \right) \right] .
+$$
+
+We define the following loss for our selector function $S$
+
+$$
+\mathcal { L } ( S ) = \mathbb { E } _ { \mathbf { x } \sim p _ { X } } \left[ K L ( Y | \mathbf { X } = \mathbf { x } | | Y | \mathbf { X } ^ { ( S ( \mathbf { x } ) ) } = \mathbf { x } ^ { ( S ( \mathbf { x } ) ) } ) + \lambda | | S ( \mathbf { x } ) | | \right]
+$$
+
+where $| | \cdot | |$ simply denotes the number of non-zero entries of a vector (or equivalently in this case, the number of 1s) and $\lambda$ is a hyper-parameter that trades off between the constraint in (1) and the number of selected features. The KL divergence in (2) can be rewritten as
+
+$$
+\begin{array} { l } { { \displaystyle K L ( Y | { \bf X } = { \bf x } | | Y | { \bf X } ^ { ( S ( { \bf x } ) ) } = { \bf x } ^ { ( S ( { \bf x } ) ) } ) = \mathbb { E } _ { y \sim Y | { \bf X } = { \bf x } } \left[ \log \left( \frac { p _ { Y } ( y | { \bf x } ) } { p _ { Y } ( y | { \bf x } ^ { ( S ( { \bf x } ) ) } ) } \right) \right] } \ ~ } \\ { { \displaystyle = \mathbb { E } _ { y \sim Y | { \bf X } = { \bf x } } \left[ \log ( p _ { Y } ( y | { \bf x } ) ) - \log ( p _ { Y } ( y | { \bf x } ^ { ( S ( { \bf x } ) ) } ) ) \right] } \ ~ } \\ { { \displaystyle = \int _ { y } p _ { Y } ( y | { \bf x } ) \left[ \log ( p _ { Y } ( y | { \bf x } ) ) - \log ( p _ { Y } ( y | { \bf x } ^ { ( S ( { \bf x } ) ) } ) ) \right] d y } , } \end{array}
+$$
+
+where $p _ { Y } ( \cdot | \cdot )$ denotes the appropriate conditional densities of $Y$ . We will write
+
+$$
+l ( \mathbf { x } , \mathbf { s } ) = \int _ { \mathcal { V } } p _ { Y } ( y | \mathbf { x } ) \left[ \log ( p _ { Y } ( y | \mathbf { x } ) ) - \log ( p _ { Y } ( y | \mathbf { x } ^ { ( s ) } ) ) \right] d y
+$$
+
+so that our final loss can be written as
+
+$$
+\mathcal { L } ( S ) = \mathbb { E } _ { \mathbf { x } \sim p _ { X } } \left[ l ( \mathbf { x } , S ( \mathbf { x } ) ) + \lambda | | S ( \mathbf { x } ) | | \right]
+$$
+
+where $| | \cdot | |$ denotes the $l _ { 0 }$ (pseudo-)norm.
+
+# 3 PROPOSED MODEL
+
+There are two main challenges in minimizing the loss in (4). First, the output space of the selector function $( \{ 0 , 1 \} ^ { d } )$ is large - its size increases exponentially with the dimension of the feature space; thus a complete search is impractical in high dimensional settings (and it should be noted that it is in high dimensional settings where feature selection is most necessary). Second, we do not have access to the densities $p _ { Y } \big ( \cdot | \mathbf { x } ^ { ( S ( \mathbf { x } ) ) } \big )$ and $p _ { Y } ( y \vert \mathbf { x } )$ required to compute (4).
+
+# 3.1 LOSS ESTIMATION
+
+To approximate the densities in (3), we introduce a pair of functions $f ^ { \phi } : \mathcal { X } ^ { * } \times \{ 0 , 1 \} ^ { d } \to [ 0 , 1 ] ^ { c }$ parametrized by $\phi$ and $f ^ { \gamma } : \mathcal { X } \to [ 0 , 1 ] ^ { c }$ parametrized by $\gamma$ that will estimate $p _ { Y } \big ( \cdot | \mathbf { x } ^ { ( S ( \mathbf { x } ) ) } \big )$ and $p _ { Y } ( \cdot | \mathbf { x } )$ respectively.
+
+# 3.1.1 PREDICTOR NETWORK
+
+We refer to $f ^ { \phi }$ as the predictor network. This will take as input a suppressed3 feature vector $\mathbf { x } ^ { ( \mathbf { s } ) }$ and its corresponding selection vector s and will output a probability distribution (using a softmax layer) over the $c$ -dimensional output space.
+
+$f ^ { \phi }$ is trained to minimize the cross entropy loss given by
+
+$$
+l _ { 1 } ( \phi ) = - \mathbb { E } _ { ( \mathbf { x } , y ) \sim p , \mathbf { s } \sim \pi _ { \theta } ( \mathbf { x } , \cdot ) } \Big [ \sum _ { i = 1 } ^ { c } y _ { i } \log ( f _ { i } ^ { \phi } ( \mathbf { x } ^ { ( \mathbf { s } ) } , \mathbf { s } ) ) \Big ]
+$$
+
+where $y _ { i }$ is the $i$ th component of the one-hot encoding of $y$ and $\pi _ { \theta }$ is the distribution induced by our selector network which will be defined in the following section. $f ^ { \phi }$ is implemented as a fully connected neural network4.
+
+# 3.1.2 BASELINE NETWORK
+
+We refer to $f ^ { \gamma }$ as the baseline network, which is standard in the actor-critic literature for variance reduction. $f ^ { \gamma }$ is implemented as a fully connected neural network and is trained to minimize
+
+$$
+l _ { 3 } ( \gamma ) = - \mathbb { E } _ { ( \mathbf { x } , y ) \sim p } \Big [ \sum _ { i = 1 } ^ { c } y _ { i } \log ( f _ { i } ^ { \gamma } ( \mathbf { x } ) ) \Big ] .
+$$
+
+For fixed $\phi , \gamma$ we define our loss estimator, $\hat { l }$ , by
+
+$$
+\hat { l } ( \mathbf { x } , \mathbf { s } ) = - \left[ \sum _ { i = 1 } ^ { c } y _ { i } \log ( f _ { i } ^ { \phi } ( \mathbf { x ^ { ( s ) } } , \mathbf { s } ) ) - \sum _ { i = 1 } ^ { c } y _ { i } \log ( f _ { i } ^ { \gamma } ( \mathbf { x } ) ) \right] .
+$$
+
+# 3.2 SELECTOR FUNCTION OPTIMIZATION
+
+We approximate the selector function $S : \mathcal { X }  \{ 0 , 1 \} ^ { d }$ by using a single neural network, $\hat { S } ^ { \theta } : \mathcal { X } $ $[ 0 , 1 ] ^ { \bar { d } }$ parameterized by weights $\theta$ , that outputs a probability for selecting each feature (i.e. the ith component of $\hat { S } ^ { \theta } ( \mathbf { x } )$ will denote the probability with which we select the ith feature). The selector network induces a probability distribution over the selection space $( \{ 0 , 1 \} ^ { d } )$ , with the probability of a given joint selection vector $\mathbf { s } \in \{ 0 , 1 \} ^ { d }$ being given by5
+
+$$
+\pi _ { \boldsymbol \theta } ( \mathbf { x } , \mathbf { s } ) = \Pi _ { i = 1 } ^ { d } \hat { S } _ { i } ^ { \theta } ( \mathbf { x } ) ^ { s _ { i } } ( 1 - \hat { S } _ { i } ^ { \theta } ( \mathbf { x } ) ) ^ { 1 - s _ { i } } .
+$$
+
+![](images/dab836e1e2e287b6c85e1510eb3ffcad658b59e83e87918824fa99ded51229df.jpg)  
+Figure 1: Block diagram of INVASE. Instances are fed into the selector network which outputs a vector of selection probabilities. The selection vector is then sampled according to these probabilities. The predictor network then receives the selected features and makes a prediction and the baseline network is given the entire feature vector and makes a prediction. Each of these networks are trained using backpropagation using the real label. The loss of the baseline network is then subtracted from the prediction network’s loss and this is used to update the selector network.
+
+Using this, we define the following loss for our selector network
+
+$$
+\begin{array} { l } { l _ { 2 } ( \theta ) = \mathbb { E } _ { ( \mathbf { x } , y ) \sim p } \left[ \mathbb { E } _ { \mathbf { s } \sim \pi _ { \theta } ( \mathbf { x } , \cdot ) } \Big [ \hat { l } ( \mathbf { x } , \mathbf { s } ) + \lambda | | \mathbf { s } | | _ { 0 } \Big ] \right] } \\ { \displaystyle = \int _ { \mathcal { X } \times \mathcal { Y } } p ( \mathbf { x } , y ) \left( \sum _ { \mathbf { s } \in \{ 0 , 1 \} ^ { d } } \pi _ { \theta } ( \mathbf { x } , \mathbf { s } ) \left( \hat { l } ( \mathbf { x } , \mathbf { s } ) + \lambda | | \mathbf { s } | | _ { 0 } \right) \right) d x d y . } \end{array}
+$$
+
+Taking the gradient of this loss with respect to $\theta$ gives us
+
+$$
+\begin{array} { l } { { \displaystyle \nabla _ { \theta } l _ { 2 } ( \theta ) = \int _ { \mathcal { X } \times \mathcal { Y } } p ( { \bf x } , y ) \left( \sum _ { { \bf s \in \{ 0 , 1 \} } ^ { d } } \nabla _ { \theta } \pi _ { \theta } ( { \bf x } , { \bf s } ) \left( \hat { l } ( { \bf x } , { \bf s } ) + \lambda | | { \bf s } | | _ { 0 } \right) \right) d x d y } \ ~ } \\ { { \displaystyle ~ = \int _ { \mathcal { X } \times \mathcal { Y } } p ( { \bf x } , y ) \left( \sum _ { { \bf s \in \{ 0 , 1 \} } ^ { d } } \frac { \nabla _ { \theta } \pi _ { \theta } ( { \bf x } , { \bf s } ) } { \pi _ { \theta } ( { \bf x } , { \bf s } ) } \pi _ { \theta } ( { \bf x } , { \bf s } ) \left( \hat { l } ( { \bf x } , { \bf s } ) + \lambda | | { \bf s } | | _ { 0 } \right) \right) d x d y } \ ~ } \\ { { \displaystyle ~ = \int _ { \mathcal { X } \times \mathcal { Y } } p ( { \bf x } , y ) \left( \sum _ { { \bf s \in \{ 0 , 1 \} } ^ { d } } \nabla _ { \theta } \log \pi _ { \theta } ( { \bf x } , { \bf s } ) \pi _ { \theta } ( { \bf x } , { \bf s } ) \left( \hat { l } ( { \bf x } , { \bf s } ) + \lambda | | { \bf s } | | _ { 0 } \right) \right) d x d y } \ ~ } \\  { \displaystyle ~ = \mathbb { E } _ { ( { \bf x } , y ) \sim p } \left[ \mathbb { E } _ { { \bf s } \sim \pi _ { \theta } ( { \bf x } , { \bf s } ) } \left[ \left( \hat { l } ( { \bf x } , { \bf s } ) + \lambda | | { \bf s } | | _ { 0 } \right) \nabla _ { \theta } \log \pi _ { \theta } ( { \bf x } , { \bf s } ) \right] \right] . } \end{array}
+$$
+
+We update each of ${ \hat { S } } ^ { \theta }$ , $f ^ { \phi }$ and $f ^ { \gamma }$ iteratively using stochastic gradient descent. Pseudo-code of INVASE is given in Algorithm 1 and a block representation of INVASE can be found in Fig. 1.
+
+# 4 EXPERIMENTS
+
+In this section, we quantitatively evaluate INVASE against various state-of-the-art benchmarks on both synthetic and real-world datasets. We evaluate our performance both at identifying ground truth relevance and at enhancing predictions. We compare our model with 4 global variable selection models: Knockoffs [3], Tree Ensembles (Tree) [7], Sequential Correlation Feature Selection (SCFS) [11], and LASSO regularized linear model; and 3 instance-wise feature selection methods: L2X
+
+# Algorithm 1 Pseudo-code of INVASE
+
+1: Inputs: learning rates $\alpha , \beta > 0$ , mini-batch size $n _ { m b } > 0$ , dataset $\mathcal { D }$   
+2: Initialize parameters $\theta , \phi , \gamma$   
+3: while Converge do   
+4: Sample a mini-batch from the dataset $( \mathbf { x } _ { j } , y _ { j } ) _ { j = 1 } ^ { n _ { m b } } \sim \mathcal { D }$   
+5: for $j = 1 , . . . , n _ { m b }$ do   
+6: Calculate selection probabilities $( p _ { 1 } ^ { j } , . . . , p _ { d } ^ { j } )  \hat { S } ^ { \theta } ( \mathbf { x } _ { j } )$   
+7: Sample selection vector   
+8: for $i = 1 , . . . , d$ do $s _ { i } ^ { j } \sim B e r ( p _ { i } ^ { j } )$   
+9: Calculate loss $\hat { l } _ { j } ( \mathbf { x } _ { j } , \mathbf { s } _ { j } ) \gets - \left[ \sum _ { i = 1 } ^ { c } y _ { i } ^ { j } \log ( f _ { i } ^ { \phi } ( \mathbf { x } _ { j } ^ { ( \mathbf { s } _ { j } ) } , \mathbf { s } _ { j } ) ) - \sum _ { i = 1 } ^ { c } y _ { i } ^ { j } \log ( f _ { i } ^ { \gamma } ( \mathbf { x } _ { j } ) ) \right]$
+
+10: Update the selector network parameters $\theta$
+
+$$
+\theta  \theta - \alpha \frac { 1 } { n _ { m b } } \sum _ { j = 1 } ^ { n _ { m b } } ( \hat { l } _ { j } ( \mathbf { x } _ { j } , \mathbf { s } _ { j } ) + \lambda | | \mathbf { s } _ { j } | | ) \nabla _ { \theta } \log \pi _ { \theta } ( \mathbf { x } _ { j } , \mathbf { s } _ { j } )
+$$
+
+11: Update the predictor network parameters $\phi$
+
+$$
+\phi  \phi - \beta \frac { 1 } { n _ { m b } } \sum _ { j = 1 } ^ { n _ { m b } } \sum _ { i = 1 } ^ { c } y _ { i } ^ { j } \times \nabla _ { \phi } \log ( f _ { i } ^ { \phi } ( \mathbf { x } _ { j } ^ { ( \mathbf { s } _ { j } ) } , \mathbf { s } _ { j } ) )
+$$
+
+12: Update the baseline network parameters $\gamma$
+
+$$
+\gamma  \gamma - \beta \frac { 1 } { n _ { m b } } \sum _ { j = 1 } ^ { n _ { m b } } \sum _ { i = 1 } ^ { c } y _ { i } ^ { j } \times \nabla _ { \gamma } \log ( f _ { i } ^ { \gamma } ( \mathbf { x } _ { j } ) )
+$$
+
+[4], LIME [24], and Shapley [18]. The details of benchmark implementation can be found in the appendix. Implementation of INVASE can be found at https://github.com/jsyoon0823/ INVASE.
+
+# 4.1 SYNTHETIC DATA EXPERIMENTS
+
+# 4.1.1 EXPERIMENTAL SETTINGS
+
+For our first set of experiments, we use the same synthetic data generation models as in L2X [4]. The input features are generated from an 11-dimensional67 Gaussian distribution with no correlations across the features $( \mathbf { X } \sim \mathcal { N } ( \mathbf { 0 } , \mathbf { I } ) )$ . The label $Y$ is sampled as a Bernoulli random variable with $\begin{array} { r } { \mathbb { P } ( Y = 1 | \mathbf { X } ) = \frac { 1 } { 1 + \mathrm { l o g i t } ( \mathbf { X } ) } } \end{array}$ , where logit $( \mathbf { X } )$ is varied to create 3 different synthetic datasets:
+
+• Syn1: $\exp ( X _ { 1 } X _ { 2 } )$   
+• Syn2: $\textstyle \exp ( \sum _ { i = 3 } ^ { 6 } X _ { i } ^ { 2 } - 4 )$   
+• Syn3: $- 1 0 \times \sin 2 X _ { 7 } + 2 | X _ { 8 } | + X _ { 9 } + \exp ( - X _ { 1 0 } )$
+
+In each of these datasets, the label depends on the same subset of features for every sample. To highlight the capability of INVASE to detect instance-wise dependence, we generate 3 further synthetic datasets as follows:
+
+• Syn4: If $X _ { 1 1 } < 0$ , logit follows Syn1, otherwise, logit follows Syn2.   
+• Syn5: If $X _ { 1 1 } < 0$ , logit follows Syn1, otherwise, logit follows Syn3.   
+• Syn6: If $X _ { 1 1 } < 0$ , logit follows Syn2, otherwise, logit follows Syn3.
+
+Note that in Syn4 and Syn5, the number of relevant features is different for different samples.
+
+For each of Syn1 to Syn6 we draw 20,000 samples from the data generation model and separate each into trainingwe try to fin $( \mathcal { D } _ { t r a i n } { \bf \bar { \alpha } } = ( { \bf x } _ { i } , y _ { i } ) _ { i = 1 } ^ { 1 0 0 0 0 } )$ and testingures for each $( \mathcal { D } _ { t e s t } = ( \mathbf { \bar { x } } _ { j } ^ { - } , y _ { j } ) _ { j = 1 } ^ { 1 0 0 0 0 } )$ sets. For each methodfor Syn1, Syn2, Syn3, $k$ $k = 4$ Syn4, Syn5 and $k = 5$ for Syn6), note, however, that $k$ is not given as an input to INVASE (but is necessary for other methods). The performance metrics we use are the true positive rate (TPR) (higher is better) and false discovery rate $\mathrm { ( F D R ) } ^ { 8 }$ (lower is better) to measure the performance of the methods when the focus is on discovery (i.e. discovering which features are relevant) and we use Area Under the Receiver Operating Characteristic Curve (AUROC), Area Under the Precision Recall Curve (AUPRC) and accuracy when the focus is on predictions.
+
+# 4.1.2 DISCOVERY
+
+Table 1: Relevant feature discovery results for Synthetic datasets with 11 features   
+
+<table><tr><td>Dataset</td><td colspan="2">Syn1</td><td colspan="2">Syn2</td><td colspan="2">Syn3</td><td colspan="2">Syn4</td><td colspan="2">Syn5</td><td colspan="2">Syn6</td></tr><tr><td>Metrics (%)</td><td>TPR</td><td>FDR</td><td>TPR</td><td>FDR</td><td>TPR</td><td>FDR</td><td>TPR</td><td>FDR</td><td>TPR</td><td>FDR</td><td>TPR</td><td>FDR</td></tr><tr><td>INVASE</td><td>100.0</td><td>0.0</td><td>100.0</td><td>0.0</td><td>92.0</td><td>0.0</td><td>99.8</td><td>10.3</td><td>84.8</td><td>1.1</td><td>90.1</td><td>7.4</td></tr><tr><td>L2X LIME</td><td>100.0 13.8</td><td>0.0 86.2</td><td>100.0 100.0</td><td>0.0 0.0</td><td>69.4 98.1</td><td>30.6 1.9</td><td>79.5 40.7</td><td>21.8 49.4</td><td>74.8 41.1</td><td>26.3 50.6</td><td>83.3 50.5</td><td>16.7 49.5</td></tr><tr><td>Shapley Knockoff</td><td>60.4 10.0</td><td>39.6 70.0</td><td>93.3 8.7</td><td>6.7 36.2</td><td>90.9 81.2</td><td>9.1 17.5</td><td>65.2 38.8</td><td>31.9 35.1</td><td>62.9 41.0</td><td>33.7 51.1</td><td>71.2 56.6</td><td>28.8 42.1</td></tr><tr><td>Tree</td><td>100.0</td><td>0.0</td><td>100.0</td><td>0.0</td><td>100.0</td><td>0.0</td><td>54.7</td><td>39.0</td><td>56.8</td><td></td><td></td><td>40.0</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>37.5</td><td>60.0</td><td></td></tr><tr><td>SCFS</td><td>23.5</td><td></td><td>39.5</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td>76.5</td><td></td><td>60.5</td><td>78.3</td><td>22.0</td><td>48.9</td><td>52.4</td><td>42.4</td><td>51.2</td><td>56.1</td><td>43.9</td></tr><tr><td>LASSO</td><td>19.0</td><td>81.0</td><td>39.8</td><td>60.2</td><td>78.3</td><td>21.7</td><td>49.9</td><td>50.9</td><td>45.5</td><td>48.2</td><td>56.4</td><td>43.6</td></tr></table>
+
+As demonstrated by Table 1, our method is capable of detecting relevant features on a global level (Syn1, Syn2 and Syn3) as well as on an instance-wise level (Syn4, Syn5 and Syn6) outperforming all other methods in both cases (both global and instance-wise methods). The particularly poor performance of some global feature selection methods in Syn1, Syn2 and Syn3 (where there is no instance-wise relevance) is due to the non-linearity of the relationship between features and labels, further details can be found in the Appendix.
+
+The results for Syn4, Syn5 and Syn6 demonstrate that INVASE is capable of detecting a different number of relevant features for each sample when necessary - the performance improvement over L2X is greater in Syn4 and Syn5 than Syn6. In particular, in Syn4, L2X is forced to overselect features when $X _ { 1 1 } ~ < ~ 0$ and underselect when $X _ { 1 1 } \geq 0$ thus resulting in higher FDR and lower TPR, respectively. To highlight this, in Table 2 we report the group specific FDR and TPR on Syn4 and Syn5 when setting $k = 3 , 4 , 5$ , where Group 1 refers to samples with $X _ { 1 1 } < 0$ and Group 2 to samples with $X _ { 1 1 } \geq 0$ .
+
+For $k = 3$ in Syn4, we see that INVASE and L2X have comparable FDR in Group 1, since the total number of relevant features for each sample is 3 $( X _ { 1 } , X _ { 2 } , X _ { 1 1 } )$ . However, when we increase $k$ , we see that the FDR increases for L2X as it is forced to select more than 3 features, which necessarily means that the FDR must be at least $40 \%$ even if L2X was finding the relevant features perfectly. On the other hand, for Group 2 we see that the TPR is low for $k = 3$ since necessarily, L2X cannot possibly select all of the 5 relevant features. INVASE, however, is able to select the correct number in both and hence enjoys low FDR and high TPR.
+
+Syn5 reinforces the conclusions we drew for L2X in Syn4. Interestingly, though, for INVASE, we found that $X _ { 1 1 }$ was almost never selected for Group 1 in Syn5. We believe this is because the lack of overlap between the relevant features for each group means that the predictor network can essentially learn two separate networks - one for each group. This is because it is possible to create two subnetworks with non-overlapping weights that each take as input the features of a given group. $X _ { 1 1 }$ is therefore unnecessary for prediction. Note, however, that $X _ { 1 1 }$ is highly relevant for the selector network in deciding which features to pass on and so it is not true that $X _ { 1 1 }$ isn’t relevant, but simply that the selector network does not need to “pass on” its relevance to the predictor network. To investigate this further, results for settings where the features overlap between groups (and so it is not possible to disentangle the networks) can be found in the Appendix.
+
+Table 2: Detailed comparison of INVASE with L2X in Syn4 and Syn5, highlighting the capability of INVASE to select a flexible number of features for each sample. Group 1: $X _ { 1 1 } < 0$ , Group 2: $X _ { 1 1 } \geq 0$   
+
+<table><tr><td>Datasets</td><td colspan="4">Syn4</td><td colspan="4">Syn5</td></tr><tr><td>Group</td><td>1</td><td></td><td colspan="2">2</td><td colspan="2">1</td><td colspan="2">2</td></tr><tr><td>Metrics (%)</td><td>TPR</td><td>FDR</td><td>TPR</td><td>FDR</td><td>TPR</td><td>FDR</td><td>TPR</td><td>FDR</td></tr><tr><td>INVASE</td><td>99.5</td><td>24.6</td><td>100.0</td><td>0.4</td><td>1 69.2</td><td>1.6</td><td>99.8</td><td>0.6</td></tr><tr><td>L2X (k = 3)</td><td>71.1</td><td>28.9</td><td>57.2</td><td>4.6</td><td>65.5</td><td>34.5</td><td>55.4</td><td>7.7</td></tr><tr><td>L2X (k = 4)</td><td>81.0</td><td>39.2</td><td>74.9</td><td>6.3</td><td>76.2</td><td>42.9</td><td>72.4</td><td>9.4</td></tr><tr><td>L2X(k = 5)</td><td>89.9</td><td>46.0</td><td>84.6</td><td>15.4</td><td>87.5</td><td>47.5</td><td>82.1</td><td>17.9</td></tr></table>
+
+# 4.1.3 PREDICTION
+
+In this experiment we analyze the effect of using feature selection as a pre-processing step for prediction. We first perform feature selection (either instance-wise or global) and then train a 3-layer fully connected network with Batch Normalization [12] in every layer (to avoid overfitting) to perform predictions on top of the (feature-selected) data. In this setting we compare the two global feature selection methods (LASSO and Tree) and one instance-wise feature selection method (L2X). Furthermore, we also compare with the predictive model without any feature selections (w/o FS) and the predictive model with ground truth globally relevant features9 (with Global). In particular, this allows us to demonstrate that the improvements in prediction performance are not just because the global feature selection performed implicitly by INVASE is better than the other global feature selection methods but are also due to the fact that we select features on an instance-wise level. Experiments here are conducted on synthetic data with 100 features but the same labelling procedures as above.
+
+As can be seen in Table 3, there is a significant performance improvement when discarding all of the irrelevant features (with Global). However, neither of the global feature selection methods (Tree and Lasso) are capable of achieving this improvement. On the other hand, INVASE is capable of achieving (and beating - in Syn4 and Syn6) this improvement, demonstrating its capability both at selecting features globally better than existing methods but also at improving on global selection with instance-wise selection (where relevant), to provide further improvements. On the other hand, $L 2 X$ performs worse than the global methods in Syn1-3, demonstrating an inability to perform even global feature selection in this higher dimensional setting (this is supported by the high dimensional discovery results in the Appendix), and in Syn4-6 is performing worse than with Global (which now is not even optimal).
+
+Furthermore, even though we include Batch Normalization to avoid overfitting, with a small number of samples and high number of dimensions, the 3-layer fully connected network still suffers from overfitting as demonstrated by the significant difference in performance between w/o FS and with Global. This demonstrates the necessity of feature selection as a pre-processing step. Lastly, in comparison to with Global, with INVASE achieves performance gains in Syn4 and Syn6. It quantitatively shows that instance-wise feature selection can further improves the predictive model from ground truth global feature selection.
+
+Table 3: Prediction performance comparison with and without feature selection methods (L2X, LASSO, Tree, INVASE, and Global). Global is using ground-truth globally relevant features for each dataset   
+
+<table><tr><td rowspan="2">Dataset</td><td colspan="6">AUROC</td></tr><tr><td>w/o FS</td><td>with Global</td><td>with INVASE</td><td>with Tree</td><td>with L2X</td><td>with LASSO</td></tr><tr><td>Syn1</td><td>.578±.004</td><td>.686±.005</td><td>.690±.006</td><td>.574±.101</td><td>.498±.005</td><td>.498±.006</td></tr><tr><td>Syn2</td><td>.789±.003</td><td>.873±.003</td><td>.877±.003</td><td>.872±.003</td><td>.823±.029</td><td>.555±.061</td></tr><tr><td>Syn3</td><td>.854±.004</td><td>.900±.003</td><td>.902±.003</td><td>.899±.001</td><td>.862±.009</td><td>.886±.003</td></tr><tr><td>Syn4</td><td>.558±.021</td><td>.774±.006</td><td>.787±.004</td><td>.684±.017</td><td>.678±.024</td><td>.514±.031</td></tr><tr><td>Syn5</td><td>.662±.013</td><td>.784±.005 .858±.004</td><td>.784±.005</td><td>.741±.004</td><td>.709±.008</td><td>.691±.024</td></tr><tr><td>Syn6</td><td>.692±.015</td><td></td><td>.877±.003</td><td>.771±.031</td><td>.827±.017</td><td>.727±.025</td></tr><tr><td rowspan="2">Dataset</td><td colspan="6">AUPRC</td></tr><tr><td>w/o FS</td><td>with Global</td><td>with INVASE</td><td>with Tree</td><td>with L2X</td><td>with LASSO</td></tr><tr><td>Syn1</td><td>.567±.007</td><td>.690±.006</td><td>.694±.006</td><td>.577±.102</td><td>.498±.007</td><td>.499±.008</td></tr><tr><td>Syn2</td><td>.799±.005</td><td>.878±.005</td><td>.886±.004</td><td>.878±.004</td><td>.817±.031</td><td>.591±.037</td></tr><tr><td>Syn3</td><td>.861±.003</td><td>.905±.002</td><td>.907±.003</td><td>.904±.002</td><td>.860±.012</td><td>.890±.002</td></tr><tr><td>Syn4</td><td>.572±.019</td><td>.794±.006</td><td>.804±.004</td><td>.681±.031</td><td>.672±.025</td><td>.536±.025</td></tr><tr><td>Syn5</td><td>.665±.019</td><td>.796±.005</td><td>.797±.006</td><td>.765±.003</td><td>.719±.011</td><td>.680±.040</td></tr><tr><td>Syn6</td><td>.709±.018</td><td>.870±.005</td><td>.886±.004</td><td>.779±.027</td><td>.835±.017</td><td>.757±.036</td></tr></table>
+
+# 4.2 REAL-WORLD DATA EXPERIMENTS
+
+# 4.2.1 DATA DESCRIPTION
+
+In this section we use two real-world datasets to perform a series of further experiments. The first, the Meta-Analysis Global Group in Chronic Heart Failure (MAGGIC) dataset [23], has 40,409 patients each with 31 measured features. The label is all-cause mortality. The second, the Prostate, Lung, Colorectal and Ovarian (PLCO) Cancer Screening Trial in the US and the European Randomized Study of Screening for Prostate Cancer (ERSPC) dataset [8; 26] contains 38,001 each with 106 measured features. The label in this dataset is mortality due to prostate cancer. We refer to this as the PLCO dataset.
+
+The first experiment we carried out was to create semi-synthetic datasets by using the labelling procedures Syn1-6 from above but with the features now coming from real data (instead of being i.i.d. Gaussian). The results of this experiment can be found in the Appendix.
+
+# 4.2.2 THE DISCOVERED FEATURE IMPORTANCE IN MAGGIC DATASET
+
+In this next experiment, we visualize the ability of INVASE to select features on an individualized level. Fig. 2(left) shows the selection probability (given by INVASE) of each feature for 20 randomly selected patients in the MAGGIC dataset. Fig. 2(right) shows the selection probability of each feature averaged over different binary splits of the data (i.e. when split into Male and Female). In Table 4, we also report the mean and variance of the number of selected features in each subgroup.
+
+As can be seen, INVASE discovers significantly different features for both individuals and for different subgroups of the dataset.
+
+# 4.2.3 RESULTS: PREDICTION USING REAL DATA VARIABLES WITH REAL LABEL
+
+Evaluating the performance of feature selection methods on real data is difficult, since ground truth relevance is often not known. We therefore cannot use TPR and FDR to evaluate the performance on real data. In our final experiment, therefore, we instead focus on prediction performance exactly as in 4.1.3 (except now both the features and label come from real data).
+
+![](images/d1951e735951992fdbed0a447435470530b7108fcf6a6da51afe008e69ba6a2e.jpg)  
+Figure 2: Left: The feature importance for each of 20 randomly selected patients in the MAGGIC dataset. Right: The average feature importance for different binary splits in the MAGGIC dataset.
+
+Table 4: Selection probability of overall and patient subgroups by INVASE in MAGGIC dataset. (Mean $\pm$ Std)   
+
+<table><tr><td rowspan=1 colspan=1>Overall</td><td rowspan=1 colspan=1>Male</td><td rowspan=1 colspan=1>Diabetes</td><td rowspan=1 colspan=1>Hypertension</td><td rowspan=1 colspan=1>Smoker</td><td rowspan=1 colspan=1>Heart Failure</td></tr><tr><td rowspan=3 colspan=1>42.5±18.4</td><td rowspan=1 colspan=2>43.5±10.7   53.2±10.8</td><td rowspan=1 colspan=2>46.6±9.3       41.0±12.1</td><td rowspan=1 colspan=1>51.8±11.1</td></tr><tr><td rowspan=1 colspan=1>Female</td><td rowspan=1 colspan=1>Non-diabetes</td><td rowspan=1 colspan=2>Non-hypertension Non-smoker</td><td rowspan=1 colspan=1>No Heart Failure</td></tr><tr><td rowspan=1 colspan=1>40.8±15.6</td><td rowspan=1 colspan=1>39.3±8.0</td><td rowspan=1 colspan=1>40.0±9.3</td><td rowspan=1 colspan=1>43.2±7.0</td><td rowspan=1 colspan=1>39.6±6.9</td></tr></table>
+
+Table 5: Prediction performance for MAGGIC and PLCO dataset.   
+
+<table><tr><td rowspan=1 colspan=1>Datasets</td><td rowspan=1 colspan=1>Metrics</td><td rowspan=1 colspan=2>AUROC    AUPRC</td><td rowspan=1 colspan=2>AUROC    AUPRC</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Labels</td><td rowspan=1 colspan=2>3year</td><td rowspan=1 colspan=2>5year</td></tr><tr><td rowspan=1 colspan=1>MAGGIC</td><td rowspan=1 colspan=1>INVASEWithout INVASE</td><td rowspan=1 colspan=1>.722±.005.720±.006</td><td rowspan=1 colspan=1>.655±.010.639±.009</td><td rowspan=1 colspan=2>.740±.005  .867±.006.730±.006  .855±.004</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Labels</td><td rowspan=1 colspan=2>5year</td><td rowspan=1 colspan=2>10 year</td></tr><tr><td rowspan=1 colspan=1>PLCO</td><td rowspan=1 colspan=1>INVASEWithout INVASE</td><td rowspan=1 colspan=1>.637±.007.629±.008</td><td rowspan=1 colspan=1>.329±.013.324±.011</td><td rowspan=1 colspan=1>.673±.007.657±.006</td><td rowspan=1 colspan=1>.506±.006.485±.008</td></tr></table>
+
+As can be seen in Table 5, INVASE consistently improves prediction performance in each of the two settings (different time horizons) in each dataset.
+
+# 5 FUTURE WORK
+
+While this paper has focused on discovering relevant features in the static setting, this could also be extended to apply in the temporal setting. One such avenue of exploration for this would be to replace each of the networks with an RNN. Particular care will need to be taken in defining the problem, though; do we treat each stream as a feature or each time point of each stream? We leave this investigation to future work.
+
+# ACKNOWLEDGEMENT
+
+The authors would like to thank the reviewers for their helpful comments. The research presented in this paper was supported by the Office of Naval Research (ONR) and the NSF (Grant number: ECCS1462245, ECCS1533983, and ECCS1407712).
+
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+
+# APPENDIX
+
+# SUMMARY OF RELATED WORKS
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Key ideas</td><td rowspan=1 colspan=1>Experimentsshown</td><td rowspan=1 colspan=1>Global/Instance-wise</td><td rowspan=1 colspan=1>Modelagnostic</td><td rowspan=1 colspan=1>#of relevantfeatures</td></tr><tr><td rowspan=1 colspan=1>SCFS[11]</td><td rowspan=1 colspan=1>Max-dependency min-redundancycriteria with Pearson correlations</td><td rowspan=1 colspan=1>Feature selection</td><td rowspan=1 colspan=1>Global</td><td rowspan=1 colspan=1>Yes</td><td rowspan=1 colspan=1>Not needed</td></tr><tr><td rowspan=1 colspan=1>MIFS[21]</td><td rowspan=1 colspan=1>Max-dependency min-redundancycriteria with Mutual Information</td><td rowspan=1 colspan=1>Feature selection</td><td rowspan=1 colspan=1>Global</td><td rowspan=1 colspan=1>Yes</td><td rowspan=1 colspan=1>Not needed</td></tr><tr><td rowspan=1 colspan=1>LASSO[31]</td><td rowspan=1 colspan=1>Linear regressionwith l1-norm penalty</td><td rowspan=1 colspan=1>Feature selectionPrediction</td><td rowspan=1 colspan=1>Global</td><td rowspan=1 colspan=1>Yes</td><td rowspan=1 colspan=1>Not needed</td></tr><tr><td rowspan=1 colspan=1>Knock-off[3]</td><td rowspan=1 colspan=1>Comparison between knock-offvariablesand real variables</td><td rowspan=1 colspan=1>Feature selectionHypothesis test</td><td rowspan=1 colspan=1>Global</td><td rowspan=1 colspan=1>Yes</td><td rowspan=1 colspan=1>Not needed</td></tr><tr><td rowspan=1 colspan=1>L2X[4]</td><td rowspan=1 colspan=1>Mutual Information maximizationwith Gumbel-softmax</td><td rowspan=1 colspan=1>Interpretation</td><td rowspan=1 colspan=1>Instance-wise</td><td rowspan=1 colspan=1>Yes</td><td rowspan=1 colspan=1>Should begiven</td></tr><tr><td rowspan=1 colspan=1>LIME[24]</td><td rowspan=1 colspan=1>Locally linearapproximation</td><td rowspan=1 colspan=1>Interpretation</td><td rowspan=1 colspan=1>Instance-wise</td><td rowspan=1 colspan=1>Yes</td><td rowspan=1 colspan=1>Should begiven</td></tr><tr><td rowspan=1 colspan=1>Shapley[18]</td><td rowspan=1 colspan=1>Shapley value estimationto quantify feature importance</td><td rowspan=1 colspan=1>Feature selection</td><td rowspan=1 colspan=1>Instance-wise</td><td rowspan=1 colspan=1>Yes</td><td rowspan=1 colspan=1>Should begiven</td></tr><tr><td rowspan=1 colspan=1>DeepLIFT[27]</td><td rowspan=1 colspan=1>Decompose the output ofNN on a reference input</td><td rowspan=1 colspan=1>Interpretation</td><td rowspan=1 colspan=1>Instance-wise</td><td rowspan=1 colspan=1>No</td><td rowspan=1 colspan=1>Should begiven</td></tr><tr><td rowspan=1 colspan=1>Saliency[29]</td><td rowspan=1 colspan=1>Backpropagation from theoutput of the NN to the input</td><td rowspan=1 colspan=1>Interpretation</td><td rowspan=1 colspan=1>Instance-wise</td><td rowspan=1 colspan=1>No</td><td rowspan=1 colspan=1>Should begiven</td></tr><tr><td rowspan=1 colspan=1>Tree SHAP[19]</td><td rowspan=1 colspan=1>Shapley value estimationonly for tree-ensemble models</td><td rowspan=1 colspan=1>Interpretation</td><td rowspan=1 colspan=1>Instance-wise</td><td rowspan=1 colspan=1>No</td><td rowspan=1 colspan=1>Should begiven</td></tr><tr><td rowspan=1 colspan=1>Pixel-wise[1]</td><td rowspan=1 colspan=1>Measuring the effects onthe output using input perturbation</td><td rowspan=1 colspan=1>Interpretation</td><td rowspan=1 colspan=1>Instance-wise</td><td rowspan=1 colspan=1>No</td><td rowspan=1 colspan=1>Should begiven</td></tr><tr><td rowspan=1 colspan=1>INVASE(Ours)</td><td rowspan=1 colspan=1>Minimize KL divergence usingdeep NN influenced byactor-critic models</td><td rowspan=1 colspan=1>Feature selectionInterpretationPrediction</td><td rowspan=1 colspan=1>Instance-wise</td><td rowspan=1 colspan=1>Yes</td><td rowspan=1 colspan=1>Not needed</td></tr></table>
+
+Table 6: Summary of the related works. (NN: Neural networks, KL: Kullback-Leibler)
+
+# EXTENDING INVASE TO REGRESSION
+
+To extend our model to the setting where $Y$ is continuous (regression problem), we replace the estimated loss with the reconstruction error as follows.
+
+$$
+\hat { l } ( { \bf x } , { \bf s } ) = - | | y - f ^ { \phi } ( { \bf x } , { \bf s } ) | | _ { 2 }
+$$
+
+where $f ^ { \phi } : \mathcal { X }  \mathbb { R }$ is now the (continuous) predictor function trained to minimize the $\ell _ { 2 }$ -norm between its outputs and the real labels. As noted in [9], when the distribution of $Y$ given $\mathbf { X }$ is Gaussian, minimizing the $l _ { 2 }$ -norm is equivalent to minimizing the KL divergence.
+
+# DETAILS OF INVASE
+
+In the experiments, the depth of the selector, predictor, and baseline networks is set to 3. The number of hidden nodes in each layer is $d$ and $2 d$ , respectively. We use either ReLu or SeLu as the activation functions of each layer except for the output layer where we use the sigmoid activation function for the selector network and softmax activation function for the predictor and baseline networks. The number of samples in each mini-batch is 1000 for the selector, predictor, and baseline networks. We use cross-validation to select $\lambda$ among $\{ 0 . 1 , 0 . 3 , 0 . 5 , 1 , 2 , 5 , 1 0 \}$ . We use tensorflow to implement INVASE. The source-code can be found at https://github.com/iclr2018invase/ INVASE/.
+
+# DETAILS OF BENCHMARKS
+
+We use the following links for the implementations of 7 benchmarks.
+
+• L2X: https://github.com/Jianbo-Lab/L2X   
+• LIME: https://github.com/marcotcr/lime   
+• Shapley: https://github.com/slundberg/shap   
+• Knock-off: http://web.stanford.edu/group/candes/knockoffs/ software/knockoff/   
+• Tree: http://scikit-learn.org/stable/modules/generated/ sklearn.ensemble.ExtraTreesClassifier.html   
+• LASSO: http://scikit-learn.org/stable/modules/linear_model. html#lasso
+
+For L2X, we use the same network settings used in INVASE for fair comparisons. For SCFS, we explicitly implement from the reference ([11]).
+
+# HIGH DIMENSIONAL DISCOVERY
+
+To demonstrate the scalability of our method, we run an experiment in which we increase the total number of features to 100. The features are generated as a 100-dimensional Gaussian with no correlations $( \mathcal { N } ( \mathbf { 0 } , \mathbf { I } ) )$ and the relationships between features and label remains as in Table 1 in the main manucript (i.e. we are adding 89 additional noisy signals that have no effect on the label).
+
+Table 7: Relevant feature discovery for synthetic datasets with 100 features   
+
+<table><tr><td rowspan="2">Dataset</td><td colspan="2">Syn1</td><td colspan="2">Syn2</td><td colspan="2">Syn3</td><td colspan="2">Syn4</td><td colspan="2">Syn5</td><td colspan="2">Syn6</td></tr><tr><td>Metrics (%) TPR</td><td>FDR</td><td>TPR</td><td>FDR</td><td>TPR</td><td>FDR</td><td>TPR</td><td>FDR</td><td>TPR</td><td>FDR</td><td>TPR</td><td>FDR</td></tr><tr><td>INVASE</td><td>100.0</td><td>0.0</td><td>100.0</td><td>0.0</td><td>100.0</td><td>0.0</td><td>66.3</td><td>40.5</td><td>73.2</td><td>23.7</td><td>90.5</td><td>15.4</td></tr><tr><td>L2X</td><td>6.1</td><td>93.9</td><td>81.4</td><td>18.6</td><td>57.7</td><td>42.3</td><td>48.5</td><td>46.4</td><td>35.4</td><td>60.8</td><td>66.3</td><td>33.7</td></tr><tr><td>LIME</td><td>0.0</td><td>100.0 95.6</td><td>100.0 95.1</td><td>0.0</td><td>92.7</td><td>7.3</td><td>43.8</td><td>47.4</td><td>42.3</td><td>50.1</td><td>50.1</td><td>49.9</td></tr><tr><td>Shapley</td><td>4.4</td><td></td><td></td><td>4.9</td><td>88.8</td><td>11.2</td><td>50.2</td><td>43.4</td><td>49.9</td><td>44.2</td><td>62.5</td><td>37.5</td></tr><tr><td>Knock off</td><td>0.0</td><td>64.9</td><td>3.7</td><td>71.2</td><td>74.9</td><td>24.9</td><td>28.2</td><td>59.8</td><td>33.1</td><td>59.4</td><td>46.9</td><td>53.0</td></tr><tr><td>Tree</td><td>49.9</td><td>50.1</td><td>100.0</td><td>0.0</td><td>100.0</td><td>0.0</td><td>40.7</td><td>49.5</td><td>56.7</td><td>37.5</td><td>58.4</td><td>41.6</td></tr><tr><td>SCFS</td><td>2.5</td><td>97.5</td><td>5.3</td><td>94.7</td><td>74.9</td><td>25.1</td><td>27.0</td><td>74.6</td><td>30.6</td><td>62.1</td><td>38.3</td><td>61.7</td></tr><tr><td>LASSO</td><td>2.5</td><td>97.5</td><td>4.0</td><td>96.0</td><td>75.3</td><td>24.7</td><td>28.3</td><td>73.2</td><td>36.0</td><td>56.9</td><td>45.9</td><td>54.1</td></tr></table>
+
+As can be seen in Table 7, INVASE also works consistently better than all other benchmarks in all 6 synthetic datasets in this setting. In fact, we see a significant reduction in performance (compared to the 11 feature setting) for L2X in Syn1, with the TPR dropping more than $90 \%$ leading to an almost complete failure of the method to detect any relevant features. In particular, we see that L2X does not scale as well as INVASE with the dimensionality of the data, which is particularly limiting for a feature selection method.
+
+We also compare the CPU times of the algorithm for training and testing with other instance-wise feature selection benchmarks to show the scalability in terms of computational complexity. As can be seen in Table 8, INVASE is much faster (10 times) than LIME and Shapley methods and comparable with L2X; we see that INVASE takes approximately $50 \%$ longer to run than L2X, which can be accounted for by the addition of a 3rd network (the baseline network) in INVASE that is not present in L2X. Note, however, that this baseline network can be trained in parallel with the predictor network and we believe that doing so would lead to both INVASE and L2X having the same run-time.
+
+Table 8: Comparison of CPU clock time across different instance-wise feature selection methods on average across Syn1 to Syn6 with 100 features and 10,000 samples on training/testing, respectively   
+
+<table><tr><td rowspan=1 colspan=1>Methods</td><td rowspan=1 colspan=1>INVASE</td><td rowspan=1 colspan=1>L2X</td><td rowspan=1 colspan=1>Shapley</td><td rowspan=1 colspan=1>LIME</td></tr><tr><td rowspan=1 colspan=1>TrainTest</td><td rowspan=1 colspan=1>1327.69s0.38s</td><td rowspan=1 colspan=1>939.82s0.78s</td><td rowspan=1 colspan=1>12801.21s0.06s</td><td rowspan=1 colspan=1>-18931.98s</td></tr></table>
+
+# HYPER-PARAMETER ANALYSIS
+
+In the following experiment, we provide results for various values of the hyper-parameter, $\lambda$ , in the Syn4, Syn5, and Syn6 100-dimensional setting. Table 9 gives the results in terms of TPR and FDR. Note that in the other experiments, we select the hyper-parameter $\lambda$ which maximizes the predictor accuracy in terms of AUROC.
+
+<table><tr><td>Datasets</td><td colspan="2">Syn4</td><td colspan="2">Syn5</td><td colspan="2">Syn6</td></tr><tr><td>X/Metris (%)</td><td>TPR</td><td>FDR</td><td>TPR</td><td>FDR</td><td>TPR</td><td>FDR</td></tr><tr><td>0.1</td><td>98.0</td><td>94.3</td><td>90.0</td><td>93.4</td><td>99.2</td><td>92.3</td></tr><tr><td>0.3</td><td>93.7</td><td>87.9</td><td>84.2</td><td>88.9</td><td>96.9</td><td>86.7</td></tr><tr><td>0.5</td><td>99.0</td><td>43.1</td><td>88.3</td><td>50.6</td><td>99.6</td><td>31.7</td></tr><tr><td>1</td><td>66.3</td><td>40.5</td><td>73.2</td><td>23.7</td><td>90.5</td><td>15.4</td></tr><tr><td>2</td><td>0.0</td><td>0.0</td><td>25.4</td><td>4.1</td><td>67.1</td><td>3.6</td></tr><tr><td>5</td><td>0.0</td><td>0.0</td><td>7.5</td><td>2.7</td><td>7.6</td><td>2.5</td></tr><tr><td>10</td><td>0.0</td><td>0.0</td><td>0.0</td><td>0.0</td><td>0.0</td><td>0.0</td></tr></table>
+
+Table 9: Relevant feature discovery results for various values of the hyper-parameter $\lambda$ in the Syn4, Syn5, and Syn6 100-dimensional setting.
+
+# ADDITIONAL RESULTS ON COMPLEX SYNTHETIC DATASETS
+
+In the main paper, the relevant subset for a sample in each of our variable synthetic datasets (Syn4- 6) depended on $X _ { 1 1 }$ only, which was unused in the rest of the model (i.e. $X _ { 1 1 }$ determined only the relevant subset, and was otherwise unused as a predictive variable). In this set of experiments, we investigate the effect of having the subset relevance depend on a variable that is also used in the model itself (Syn4A, Syn5A, Syn6A). We then investigate the effect of having more than one variable being used to determine subset relevance (Syn4B, Syn5B, Syn6B, Syn7). The results for these are reported in Tables 10 and 11, respectively.
+
+The input features are generated from a 100-dimensional Gaussian distribution with no correlations across the features $( \mathbf { X } \sim \mathcal { N } ( \mathbf { 0 } , \mathbf { I } ) )$ . $Y$ is generated according to $\begin{array} { r } { \mathbb { P } ( Y = 1 | \mathbf { X } ) = \frac { 1 } { 1 + \mathrm { l o g i t } ( \mathbf { X } ) } } \end{array}$ with the logit value for each synthetic dataset now defined as follows:
+
+• Syn4A: If $X _ { 1 } < 0$ , ${ \mathrm { l o g i t } } = \exp ( X _ { 1 } X _ { 2 } )$ , otherwise, logit $= \textstyle \exp ( \sum _ { i = 3 } ^ { 6 } X _ { i } ^ { 2 } - 4 )$ .   
+• Syn5A: If $X _ { 1 } < 0$ , ${ \mathrm { l o g i t } } = \exp ( X _ { 1 } X _ { 2 } )$ , otherwise, logi $\mathrm { t = - 1 0 } \times \sin 2 X _ { 7 } + 2 | X _ { 8 } | + X _ { 9 } +$ $\exp ( - X _ { 1 0 } )$ .   
+• $X _ { 7 } < 0$ , $\scriptstyle { \mathrm { l o g i t } } = \exp ( \sum _ { i = 3 } ^ { 6 } X _ { i } ^ { 2 } - 4 )$ , otherwise, logi $\mathrel { \mathop { : = } } - 1 0 \times \sin 2 X _ { 7 } + 2 | X _ { 8 } | +$ $X _ { 9 } + \exp ( - X _ { 1 0 } )$
+
+Table 10: Relevant feature discovery results for complex synthetic datasets (Syn4A, 5A, 6A) with 100 features   
+
+<table><tr><td>Dataset 1</td><td>Syn4A</td><td>Syn5A</td><td></td><td>Syn6A</td><td></td></tr><tr><td>Metrics (%)</td><td>TPR</td><td>FDR</td><td>TPR</td><td>FDR</td><td>TPR</td><td>FDR</td></tr><tr><td>INVASE+</td><td>77.5</td><td>14.5</td><td>85.9</td><td>8.8</td><td>89.9</td><td>7.3</td></tr><tr><td>L2X</td><td>65.0</td><td>39.3</td><td>48.0</td><td>57.4</td><td>74.4</td><td>35.5</td></tr><tr><td>LIME Shapley</td><td>56.3 71.8</td><td>49.2 39.8</td><td>58.2 71.0</td><td>48.8 41.3</td><td>58.9 68.9</td><td>47.8 38.2</td></tr><tr><td>Knock off</td><td>59.8</td><td>62.6</td><td>55.0</td><td>49.9</td><td>65.0</td><td>40.0</td></tr><tr><td>Tree</td><td>61.3</td><td>46.9</td><td>75.6</td><td>39.4</td><td>66.9</td><td>40.0</td></tr><tr><td>SCFS</td><td>52.8</td><td>66.9</td><td>55.3</td><td>50.6</td><td>50.4</td><td>51.8</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>LASSO</td><td>61.0</td><td>61.2</td><td>55.0</td><td>50.0</td><td>53.9</td><td>48.8</td></tr></table>
+
+• Syn4B: If $X _ { 1 } X _ { 3 } < 0$ , ${ \mathrm { l o g i t } } = \exp ( X _ { 1 } X _ { 2 } )$ , otherwise, logit $= \textstyle \exp ( \sum _ { i = 3 } ^ { 6 } X _ { i } ^ { 2 } - 4 )$
+
+• Syn5B: If $X _ { 1 } X _ { 7 } < 0$ , ${ \mathrm { l o g i t } } = \exp ( X _ { 1 } X _ { 2 } )$ , otherwise, logit $= - 1 0 \times \sin 2 X _ { 7 } + 2 | X _ { 8 } | +$ $X _ { 9 } + \exp ( - X _ { 1 0 } )$ .
+
+• Syn6B: If $X _ { 3 } X _ { 7 } \ < \ 0$ , logit $= \textstyle \exp ( \sum _ { i = 3 } ^ { 6 } X _ { i } ^ { 2 } - 4 )$ , otherwise, logit $= - 1 0 \times \sin 2 X _ { 7 } +$ $2 | X _ { 8 } | + X _ { 9 } + \exp ( - X _ { 1 0 } )$ .
+
+• Syn7:
+
+– If $X _ { 1 } < 0 , X _ { 2 } < 0$ , ${ \mathrm { l o g i t } } = \exp ( X _ { 1 } X _ { 2 } )$   
+– If $X _ { 1 } < 0 , X _ { 2 } \geq 0$ , $\scriptstyle { \mathrm { l o g i t } } = \exp ( \sum _ { i = 3 } ^ { 6 } X _ { i } ^ { 2 } - 4 )$ .   
+– If $X _ { 1 } \geq 0 , X _ { 2 } < 0$ , $\mathrm { l o g i t = - 1 0 \times \sin 2 X _ { 7 } + 2 } | X _ { 8 } | + X _ { 9 } + \exp ( - X _ { 1 0 } ) .$ – If $X _ { 1 } \geq 0 , X _ { 2 } \geq 0$ $\begin{array} { r } { ) , \mathrm { l o g i t } = 0 . 5 \times \exp ( X _ { 1 } X _ { 2 } ) + 0 . 5 \times \exp ( \sum _ { i = 3 } ^ { 4 } X _ { i } ^ { 2 } - 2 ) . } \end{array}$
+
+# RESULTS ON SEMI-SYNTHETIC DATASETS
+
+In this experiment, we use real features (which have correlation across features) but generate the labels as in the synthetic experiments from the main paper, using Syn1-Syn6. This allows us to know the ground truth relevance of the features, and calculate TPR and FDR, while using unknown and correlated feature distributions (instead of the unrealistic setting of i.i.d. Gaussian used in the fully synthetic experiment). The results for the MAGGIC and PLCO datasets are given below.
+
+Table 11: Relevant feature discovery results for complex synthetic datasets (Syn4B, 5B, 6B, 7) with 100 features   
+
+<table><tr><td>Dataset</td><td colspan="2">Syn4B</td><td colspan="2">Syn5B</td><td colspan="2">Syn6B</td><td colspan="2">Syn7</td></tr><tr><td>Metrics (%)</td><td>TPR</td><td>FDR</td><td>TPR</td><td>FDR</td><td>TPR</td><td>FDR</td><td>TPR</td><td>FDR</td></tr><tr><td>INVASE+</td><td>65.5</td><td>30.1</td><td>85.0</td><td>15.0</td><td>86.5</td><td>27.8</td><td>86.8</td><td>32.4</td></tr><tr><td>L2X</td><td>43.2</td><td>53.4</td><td>50.3</td><td>50.3</td><td>44.6</td><td>55.4</td><td>35.3</td><td>70.9</td></tr><tr><td>LIME</td><td>56.8</td><td>37.2</td><td>71.9</td><td>27.2</td><td>69.8</td><td>30.2</td><td>56.4</td><td>51.0</td></tr><tr><td>Shapley</td><td>51.4</td><td>43.5</td><td>77.2</td><td>24.1</td><td>69.3</td><td>30.7</td><td>61.8</td><td>45.6</td></tr><tr><td>Knock off</td><td>5.3</td><td>87.4</td><td>73.3</td><td>25.2</td><td>59.9</td><td>40.1</td><td>54.1</td><td>60.0</td></tr><tr><td>Tree</td><td>56.7</td><td>37.4</td><td>73.9</td><td>25.0</td><td>70.1</td><td>29.9</td><td>71.6</td><td>40.3</td></tr><tr><td>SCFS</td><td>3.7</td><td>96.2</td><td>72.3</td><td>26.3</td><td>61.1</td><td>38.9</td><td>22.9</td><td>77.5</td></tr><tr><td>LASSO</td><td>4.2</td><td>95.6</td><td>73.3</td><td>25.0</td><td>60.1</td><td>39.9</td><td>24.9</td><td>75.8</td></tr></table>
+
+Table 12: Relevant feature discovery for real datasets with synthetic labels using MAGGIC dataset   
+
+<table><tr><td rowspan="2">Dataset</td><td colspan="2">Syn1</td><td colspan="2">Syn2</td><td colspan="2">Syn3</td><td colspan="2">Syn4</td><td colspan="2">Syn5</td><td colspan="2">Syn6</td></tr><tr><td>Metrics (%) TPR</td><td>FDR</td><td>TPR</td><td>FDR</td><td>TPR</td><td>FDR</td><td>TPR</td><td>FDR</td><td>TPR</td><td>FDR</td><td>TPR</td><td>FDR</td></tr><tr><td>INVASE</td><td>100.0</td><td>0.0</td><td>100.0</td><td>0.0</td><td>100.0</td><td>0.0</td><td>85.9</td><td>0.0</td><td>72.9</td><td>0.1</td><td>81.0</td><td>13.2</td></tr><tr><td>L2X LIME</td><td>68.8 46.9</td><td>31.2 53.1</td><td>99.9 99.9</td><td>0.1 0.1</td><td>83.0 87.2</td><td>17.0 12.8</td><td>60.0 63.6</td><td>31.3 24.4</td><td>68.3 50.2</td><td>22.3 37.6</td><td>73.5 68.7</td><td>26.5 31.3</td></tr><tr><td>Shapley Knock off</td><td>73.9 27.5</td><td>26.1 65.0</td><td>94.5 77.5</td><td>5.5 22.5</td><td>81.0 100.0</td><td>19.0 0.0</td><td>65.3</td><td>23.9 34.4</td><td>61.2</td><td>29.0</td><td>69.9</td><td>30.1</td></tr><tr><td>Tree</td><td>100.0</td><td>0.0</td><td>100.0</td><td>0.0</td><td>100.0</td><td>0.0</td><td>57.0 56.3</td><td>29.7</td><td>56.1 51.6</td><td>29.8 40.2</td><td>58.0 46.7</td><td>42.0 53.3</td></tr><tr><td>SCFS</td><td>30.0</td><td>70.0</td><td>53.0</td><td>47.0</td><td>100.0</td><td>0.0</td><td>52.0</td><td>39.9</td><td>54.0</td><td>32.4</td><td>64.5</td><td>35.5</td></tr><tr><td>LASSO</td><td></td><td></td><td>75.0</td><td>25.0</td><td>100.0</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>25.0</td><td>75.0</td><td></td><td></td><td></td><td>0.0</td><td>60.7</td><td>33.1</td><td>56.1</td><td>29.8</td><td>58.2</td><td>41.8</td></tr></table>
+
+Table 13: Relevant feature discovery for real datasets with synthetic labels using PLCO dataset   
+
+<table><tr><td>Dataset</td><td colspan="2">Syn1</td><td colspan="2">Syn2</td><td colspan="2">Syn3</td><td colspan="2">Syn4</td><td colspan="2">Syn5</td><td colspan="2">Syn6</td></tr><tr><td>Metrics (%)</td><td>TPR</td><td>FDR</td><td>TPR</td><td>FDR</td><td>TPR</td><td>FDR</td><td>TPR</td><td>FDR</td><td>TPR</td><td>FDR</td><td>TPR</td><td>FDR</td></tr><tr><td>INVASE</td><td>35.9</td><td>0.0</td><td>100.0</td><td>0.0</td><td>84.0</td><td>7.0</td><td>59.2</td><td>38.6</td><td>64.6</td><td>31.7</td><td>70.0</td><td>29.9</td></tr><tr><td>L2X LIME</td><td>0.0 1.0</td><td>100.0 99.0</td><td>62.2 70.3</td><td>37.8 29.7</td><td>43.6 74.9</td><td>56.4 25.1</td><td>41.9 43.5</td><td>55.4 55.9</td><td>21.5 26.8</td><td>76.7 68.9</td><td>66.9 56.8</td><td>33.1 43.2</td></tr><tr><td>Shapley Knock off</td><td>5.4</td><td>94.6</td><td>68.5</td><td>31.5</td><td>67.9</td><td>32.1</td><td>32.7</td><td>69.4</td><td>39.6</td><td>58.6</td><td>48.5</td><td>51.5</td></tr><tr><td>Tree</td><td>15.0 0.0</td><td>50.0 100.0</td><td>85.0 71.0</td><td>15.0 29.0</td><td>100.0 75.0</td><td>0.0 25.0</td><td>46.1 34.5</td><td>52.1 66.3</td><td>34.5</td><td>58.3 54.7</td><td>60.0</td><td>40.0</td></tr><tr><td>SCFS</td><td>10.0</td><td>90.0</td><td>61.0</td><td>39.0</td><td>93.8</td><td>6.2</td><td>43.2</td><td>55.7</td><td>43.8</td><td></td><td>36.9</td><td>63.1</td></tr><tr><td>LASSO</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>31.0</td><td>63.6</td><td>55.5</td><td>44.5</td></tr><tr><td></td><td>0.0</td><td>100.0</td><td>72.5</td><td>27.5</td><td>100.0</td><td>0.0</td><td>39.2</td><td>60.8</td><td>33.2</td><td>68.2</td><td>45.0</td><td>55.0</td></tr></table>
+
+As demonstrated in Tables 12 and 13, INVASE outperforms all other methods across all 6 of the synthetic-label settings using real features. This also demonstrates the capability of INVASE in settings where there are unknown correlation structures in the features.
+
+# 5.1 PREDICTIVE PERFORMANCE COMPARISON ON REAL-WORLD DATASETS
+
+In this experiment, we evaluate the predictive performance gains of using each feature selection method as a pre-processing step on the two real datasets, MAGGIC and PLCO (as was done for synthetic data in Section 4.1.3). For each method, we first perform feature selection and then train a predictive model on top of the feature-selected data, where the model has the same architecture as the INVASE predictor network (to create a fair comparison of methods). As can be seen in Table 14, INVASE significantly outperform the other approaches.
+
+<table><tr><td>Datasets</td><td colspan="4">MAGGIC</td><td colspan="4">PLCO</td></tr><tr><td>Labels</td><td colspan="2">3-year</td><td colspan="2">5-year</td><td colspan="2">5-year</td><td colspan="2">10-year</td></tr><tr><td>Metrics</td><td>AUROC</td><td>AUPRC</td><td>AUROC</td><td>AUPRC</td><td>AUROC</td><td>AUPRC</td><td>AUROC</td><td>AUPRC</td></tr><tr><td>INVASE</td><td>0.722</td><td>0.655</td><td>0.740</td><td>0.867</td><td>0.637</td><td>0.329</td><td>0.673</td><td>0.506</td></tr><tr><td>L2X</td><td>0.609</td><td>0.529</td><td>0.607</td><td>0.794</td><td>0.558</td><td>0.170</td><td>0.583</td><td>0.365</td></tr><tr><td>LIME</td><td>0.637</td><td>0.5596</td><td>0.634</td><td>0.808</td><td>0.597</td><td>0.183</td><td>0.601</td><td>0.374</td></tr><tr><td>Shapley</td><td>0.641</td><td>0.557</td><td>0.617</td><td>0.797</td><td>0.614</td><td>0.194</td><td>0.615</td><td>0.381</td></tr><tr><td>Knockoff</td><td>0.686</td><td>0.614</td><td>0.711</td><td>0.853</td><td>0.619</td><td>0.230</td><td>0.658</td><td>0.475</td></tr><tr><td>Tree</td><td>0.678</td><td>0.604</td><td>0.708</td><td>0.850</td><td>0.632</td><td>0.269</td><td>0.655</td><td>0.469</td></tr><tr><td>SCFS</td><td>0.683</td><td>0.623</td><td>0.723</td><td>0.857</td><td>0.632</td><td>0.231</td><td>0.632</td><td>0.444</td></tr><tr><td>LASSO</td><td>0.692</td><td>0.615</td><td>0.709</td><td>0.847</td><td>0.623</td><td>0.218</td><td>0.656</td><td>0.467</td></tr></table>
+
+Table 14: Predictive Performance Comparison on two real-world datasets (MAGGIC and PLCO) in terms of AUROC and AUPRC
+
+CORRELATIONS BETWEEN FEATURES AND LABELS IN THE SYNTHETIC AND SEMI-SYNTHETIC EXPERIMENTS   
+Table 15: Correlation between features and labels in Synthetic datasets with 100 features. Ground truth (in the global sense) relevant features are given in bold. Features with correlation $> 0 . 0 5$ are highlighted in red.   
+
+<table><tr><td>Variables</td><td>Syn1</td><td>Syn2</td><td>Syn3</td><td>Syn4</td><td>Syn5</td><td>Syn6</td></tr><tr><td>X1 X</td><td>0.003 0.001</td><td>0.008 0.005</td><td>0.006 0.006</td><td>0.009 0.005</td><td>0.007 0.015</td><td>0.006 0.005</td></tr><tr><td>X3</td><td>0.006 0.006</td><td>0.011</td><td>0.001</td><td>0.017</td><td>0.016</td><td>0.010</td></tr><tr><td>X4</td><td></td><td>0.003</td><td>0.003</td><td>0.002</td><td>0.000</td><td>0.002</td></tr><tr><td>X5</td><td>0.003</td><td>0.015</td><td>0.022</td><td>0.004</td><td>0.017</td><td>0.028</td></tr><tr><td>X6</td><td>0.003</td><td>0.004</td><td>0.005</td><td>0.002</td><td>0.004</td><td>0.005</td></tr><tr><td>X7</td><td>0.013</td><td>0.009</td><td>0.481</td><td>0.002</td><td>0.242</td><td>0.235</td></tr><tr><td>X8</td><td>0.010</td><td>0.008</td><td>0.012</td><td>0.003</td><td>0.010</td><td>0.022</td></tr><tr><td>X9</td><td>0.001</td><td>0.003</td><td>0.239</td><td>0.002</td><td>0.115</td><td>0.121</td></tr><tr><td>X10</td><td>0.002</td><td>0.003</td><td>0.308</td><td>0.003</td><td>0.149</td><td>0.144</td></tr><tr><td>X11</td><td>0.014</td><td>0.012</td><td>0.004</td><td>0.028</td><td>0.018</td><td>0.002</td></tr></table>
+
+As can be seen in Table 15, among 33 relevant features, only 9 features have more than 0.05 (linear) correlation with the label. In particular, using a linear model, it is very hard to discover the relevant features. However, Knock-off (based on LASSO and linear correlations), LASSO, and SCFS are linear models, resulting in a poor performance in our experiments. The above table results are directly reflected in the results given in the main manuscript.
+
+Table 16: Correlation between features and labels in MAGGIC datasets. Ground truth relevant features are described in bold. Features with correlation $> 0 . 0 5$ are described in red   
+
+<table><tr><td>Variables</td><td>Syn1</td><td>Syn2</td><td>Syn3</td><td>Syn4</td><td>Syn5</td><td>Syn6</td></tr><tr><td>X1</td><td>0.028 0.002</td><td>0.030 0.011</td><td>0.070 0.009</td><td>0.011 0.001</td><td>0.044 0.001</td><td>0.026 0.012</td></tr><tr><td>X</td><td>0.018</td><td>0.079</td><td>0.008</td><td>0.038</td><td>0.006</td><td>0.046</td></tr><tr><td>X3 X4</td><td>0.005</td><td>0.113</td><td>0.006</td><td>0.056</td><td>0.001</td><td>0.055</td></tr><tr><td></td><td>0.006</td><td>0.034</td><td></td><td>0.013</td><td></td><td></td></tr><tr><td>X5 X6</td><td>0.019</td><td>0.114</td><td>0.032 0.027</td><td>0.099</td><td>0.016</td><td>0.036</td></tr><tr><td></td><td>0.005</td><td>0.010</td><td></td><td>0.000</td><td>0.018</td><td>0.004</td></tr><tr><td>X7</td><td></td><td></td><td>0.367</td><td></td><td>0.262</td><td>0.272</td></tr><tr><td>X8</td><td>0.020</td><td>0.030</td><td>0.112</td><td>0.023</td><td>0.075</td><td>0.082</td></tr><tr><td>X9</td><td>0.025</td><td>0.022</td><td>0.299</td><td>0.006</td><td>0.216</td><td>0.200</td></tr><tr><td>X10</td><td>0.07</td><td>0.043</td><td>0.328</td><td>0.027</td><td>0.222</td><td>0.206</td></tr><tr><td>X11</td><td>0.009</td><td>0.006</td><td>0.034</td><td>0.046</td><td>0.018</td><td>0.058</td></tr></table>
+
+We do the same analysis for the MAGGIC dataset; results are given in Table 16. We see that here the linear correlation with the label is stronger and this is reflected in Tables 12 and 13, where all of the linear models performed better than in the fully-synthetic settings. However, we note that although they had a better performance, in most cases it was still not comparable with INVASE.
+
+DEFINITION OF TPR AND FDR
+
+Figure 3: The definitions of True Positive Rate (TPR) and False Discovery Rate (FDR)   
+
+<table><tr><td rowspan=2 colspan=2></td><td rowspan=1 colspan=2>True Condition</td></tr><tr><td rowspan=1 colspan=1>Positive</td><td rowspan=1 colspan=1>Negative</td></tr><tr><td rowspan=2 colspan=1>PredictedCondition</td><td rowspan=1 colspan=1>Positive</td><td rowspan=1 colspan=1>TruePositive</td><td rowspan=1 colspan=1>FalsePositive</td></tr><tr><td rowspan=1 colspan=1>Negative</td><td rowspan=1 colspan=1>FalseNegative</td><td rowspan=1 colspan=1>TrueNegative</td></tr></table>
+
+୘୰୳ୣ ୔୭ୱ୧୲୧୴ୣ   
+True Positive Rate (TPR) $=$ ୘୰୳ୣ ୔୭ୱ୧୲୧୴ୣା୊ୟ୪ୱ��� ୒ୣ୥ୟ୲୧୴ୣ ୊ୟ୪ୱୣ ୔୭ୱ୧୲୧୴ୣ   
+False Discovery Rate (FDR) $=$ ୘୰୳ୣ ୔୭ୱ୧୲୧୴ୣ ୊ୟ୪ୱୣ ୔୭ୱ୧୲୧୴ୣ $^ +$
+
+# COMPUTER VISION
+
+Another natural application of INVASE is in computer vision. To briefly demonstrate the applicability and capability of INVASE to computer vision, we conduct two experiments using the Kaggle Dogs vs. Cats dataset (https://www.kaggle.com/c/dogs-vs-cats) [6] and the Oxford Pet dataset (http://www.robots.ox.ac.uk/˜vgg/data/pets/) [20]. The goal is to select a set $1 6 \times 1 6$ patches of each image that maximize the predictive capability of a model. In order to apply INVASE to this problem, we simply treat each $1 6 \times 1 6$ patch as a feature.
+
+We use the U-Net [25] architecture for the selector network and the VGG network [28] architecture for the predictor and baseline networks. Below we give qualitative results of INVASE applied to these datasets, where we see that INVASE successfully identifies patches of each image in which the animal’s face is visible.
+
+![](images/4f36cf05a53bf3d7e37f5224237ed6e770fab264b6463ced2d9ee180b9a626b6.jpg)  
+Figure 4: Selected $1 6 \mathrm { ~ x ~ } 1 6$ patches by INVASE on Kaggle Dogs vs. Cats dataset - Cats
+
+![](images/aa7861ab9378342c4713f4385cc0c535523fb2e28b91aa84c9951a7e10e9b6d0.jpg)  
+Figure 5: Selected $1 6 \times 1 6$ patches by INVASE on Kaggle Dogs vs. Cats dataset - Dogs
+
+![](images/b5bc222942ad195921fdf291082c28ac5a6e4f8c2d9edef9031184a9a0ca0f60.jpg)  
+Figure 6: Selected $1 6 \times 1 6$ patches by INVASE on Oxford Pet dataset - Cats
+
+![](images/c0d45684db148f7fe42f9625cf27b015fa3f4dba35149e59b0d12a308dbc8f23.jpg)  
+Figure 7: Selected $1 6 \mathrm { ~ x ~ } 1 6$ patches by INVASE on Oxford Pet dataset - Dogs

parse/train/ByxT7TNFvH/ByxT7TNFvH.md

@@ -0,0 +1,279 @@
+# SEMANTICALLY-GUIDED REPRESENTATION LEARNING FOR SELF-SUPERVISED MONOCULAR DEPTH
+
+Vitor Guizilini1 Rui Hou1,2 Jie Li1 Rares, Ambrus, 1 Adrien Gaidon1
+
+1Toyota Research Institute (TRI) {first.last}@tri.global
+
+2University of Michigan rayhou@umich.edu
+
+# ABSTRACT
+
+Self-supervised learning is showing great promise for monocular depth estimation, using geometry as the only source of supervision. Depth networks are indeed capable of learning representations that relate visual appearance to 3D properties by implicitly leveraging category-level patterns. In this work we investigate how to leverage more directly this semantic structure to guide geometric representation learning, while remaining in the self-supervised regime. Instead of using semantic labels and proxy losses in a multi-task approach, we propose a new architecture leveraging fixed pretrained semantic segmentation networks to guide self-supervised representation learning via pixel-adaptive convolutions. Furthermore, we propose a two-stage training process to overcome a common semantic bias on dynamic objects via resampling. Our method improves upon the state of the art for self-supervised monocular depth prediction over all pixels, fine-grained details, and per semantic categories.†
+
+# 1 INTRODUCTION
+
+Accurate depth estimation is a key problem in computer vision and robotics, as it is instrumental for perception, navigation, and planning. Although perceiving depth typically requires dedicated sensors (e.g., stereo rigs, LiDAR), learning to predict depth from monocular imagery can provide useful cues for a wide array of tasks (Michels et al., 2005; Kendall et al., 2018; Manhardt et al., 2019; Lee et al., 2019). Going beyond supervised learning from direct measurements (Eigen et al., 2014), self-supervised methods exploit geometry as supervision (Guo et al., 2018; Pillai et al., 2019; Zou et al., 2018; Yang et al., 2017), therefore having the potential to leverage large scale datasets of raw videos to outperform supervised methods (Guizilini et al., 2019).
+
+Although depth from a single image is an ill-posed inverse problem, monocular depth networks are able to make accurate predictions by learning representations connecting the appearance of scenes and objects with their geometry in Euclidean 3D space. Due to perspective, there is indeed an equivariance relationship between the visual appearance of an object in 2D and its depth, when conditioned on the object’s category. For instance, a car 25 meters away appears smaller (on the image plane) than a car only 5 meters away but bigger than a truck 50 meters away. Current depth estimation methods either do not leverage this structure explicitly or rely on strong semantic supervision to jointly optimize geometric consistency and a semantic proxy task in a multi-task objective (Ochs et al., 2019; Chen et al., 2019), thus departing from the self-supervised paradigm.
+
+In this paper, we explore how we can leverage semantic information to improve monocular depth prediction in a self-supervised way. Our main contribution is a novel architecture that uses a fixed pre-trained semantic segmentation network to guide geometric representation learning in a selfsupervised monocular depth network. In contrast to standard convolutional layers, our architecture uses pixel-adaptive convolutions (Su et al., 2019) to learn semantic-dependent representations that can better capture the aforementioned equivariance property. Leveraging semantics may nonetheless introduce category-specific biases. Our second contribution is a two-stage training process where we automatically detect the presence of a common bias on dynamic objects (projections at infinity) and resample the training set to de-bias it. Our method improves upon the state of the art in selfsupervised monocular depth estimation on the standard KITTI benchmark (Geiger et al., 2013), both on average over pixels, over classes, and for dynamic categories in particular.
+
+![](images/b09c90637e486163bce4d7333340755790cf27ff41a391d503abe5d3014d643d.jpg)  
+Figure 1: Example of a pointcloud generated using our proposed semantically-guided architecture, colored by RGB values from the input image and corresponding predicted semantic labels.
+
+# 2 RELATED WORK
+
+Since the seminal work of Eigen et al. (2014), substantial progress has been done to improve the accuracy of supervised depth estimation from monocular images, including the use of Conditional Random Fields (CRFs) (Li et al., 2015), joint optimization of surface normals (Qi et al., 2018), fusion of multiple depth maps (Lee et al., 2018), and ordinal classification (Fu et al., 2018). Consequently, as supervised techniques for depth estimation advanced rapidly, the availability of largescale depth labels became a bottleneck, especially for outdoor applications. Garg et al. (2016) and Godard et al. (2017) provided an alternative self-supervised strategy involving stereo cameras, where Spatial Transformer Networks (Jaderberg et al., 2015) can be used to geometrically warp, in a differentiable way, the right image into a synthesized left image, using the predicted depth from the left image. The photometric consistency loss between the resulting synthesized and original left images can then be minimized in an end-to-end manner using a Structural Similarity term (Wang et al., 2004) and additional depth regularization terms. Following Godard et al. (2017) and Ummenhofer et al. (2017), Zhou et al. (2017) generalized this to the purely monocular setting, where a depth and a pose networks are simultaneously learned from unlabeled monocular videos. Rapid progress in terms of architectures and objective functions (Yin & Shi, 2018; Mahjourian et al., 2018; Casser et al., 2019; Zou et al., 2018; Klodt & Vedaldi, 2018; Wang et al., 2018; Yang et al., 2018) have since then turned monocular depth estimation into one of the most successful applications of self-supervised learning, even outperforming supervised methods (Guizilini et al., 2019).
+
+The introduction of semantic information to improve depth estimates has been explored in prior works, and can be broadly divided into two categories. The first one uses semantic (or instance) information to mask out or properly model dynamic portions of the image, which are not accounted for in the photometric loss calculation. Guney & Geiger ¨ (2015) leveraged object knowledge in a Markov Random Field (MRF) to resolve stereo ambiguities, while Bai et al. (2016) used a conjunction of instance-level segmentation and epipolar constraints to reduce uncertainty in optical flow estimation. Casser et al. (2019) used instance-level masks to estimate motion models for different objects in the environment, and account for their external motion in the resulting warped image. The second category attempts to learn both tasks in a single framework, and uses consistency losses to ensure that both are optimized simultaneously and regularize each other, so the information contained in one task can be transferred to improve the other. For instance, Ochs et al. (2019) estimated depth with an ordinal classification loss similar to the standard semantic classification loss, and used empirical weighting to combine them into a single loss for optimization. Similarly, Chen et al. (2019) used a unified conditional decoder that can generate either semantic or depth estimates, and both outputs are used to generate a series of losses also combined using empirical weighting to generate the final loss to be optimized.
+
+Our approach focuses instead on representation learning, exploiting semantic features into the selfsupervised depth network by using a pretrained semantic segmentation network to guide the generation of depth features. This is done using pixel-adaptive convolutions, recently proposed in Su et al. (2019) and applied to tasks such as depth upsampling using RGB images for feature guidance. We show that different depth networks can be readily modified to leverage this semantic feature guidance, ranging from widely used ResNets (He et al., 2016) to the current state-of-the-art PackNet (Guizilini et al., 2019), with a consistent gain in performance across these architectures.
+
+# 3 SELF-SUPERVISED STRUCTURE-FROM-MOTION
+
+Our semantically-guided architecture is developed within a self-supervised monocular depth estimation setting, commonly known as structure-from-motion (SfM). Learning in a self-supervised structure-from-motion setting requires two networks: a monocular depth model $f _ { D } : I  D$ , that outputs a depth prediction $\hat { D } = f _ { D } ( I ( p ) )$ for every pixel $p$ in the target image $I$ ; and a monocular ego-motion estimator $f _ { \mathbf { x } } : ( I _ { t } , I _ { S } ) \to \mathbf { x } _ { t \to S }$ , that predicts the $6 \mathrm { D o F }$ transformations for all $s \in S$ given by $\mathbf { x } _ { t  s } = ( \mathbf { \Sigma } _ { \mathbf { 0 } } ^ { \mathbf { R } \textbf { t } } ) \in \mathbf { S } \mathbf { E } ( 3 )$ between the target image $I _ { t }$ and a set of temporal context source images $I _ { s } \in I _ { S }$ . In all reported experiments we use $I _ { t - 1 }$ and $I _ { t + 1 }$ as source images.
+
+# 3.1 THE SELF-SUPERVISED OBJECTIVE LOSS
+
+We train the depth and pose networks simultaneously, using the same protocols and losses as described in Guizilini et al. (2019). Our self-supervised objective loss consists of an appearance matching term ${ \mathcal { L } } _ { p }$ that is imposed between the synthesized $\bar { \hat { I } } _ { t }$ and original $I _ { t }$ target images, and a depth regularization term $\mathcal { L } _ { s }$ that ensures edge-aware smoothing in the depth estimates $\hat { D } _ { t }$ . The final objective loss is averaged per pixel, pyramid scale and image batch, and is defined as:
+
+$$
+\mathcal { L } ( I _ { t } , \hat { I } _ { t } ) = \mathcal { L } _ { p } ( I _ { t } , \hat { I } _ { t } ) + \lambda _ { 1 } \mathcal { L } _ { s } ( \hat { D } _ { t } )
+$$
+
+where $\lambda _ { 1 }$ is a weighting coefficient between the photometric ${ \mathcal { L } } _ { p }$ and depth smoothness $\mathcal { L } _ { s }$ loss terms. Following Godard et al. (2017) and Zhou et al. (2017), the similarity between synthesized $\hat { I } _ { t }$ and original $I _ { t }$ target images is estimated using a Structural Similarity (SSIM) term (Wang et al., 2004) combined with an L1 loss term, inducing the following overall photometric loss:
+
+$$
+\mathcal { L } _ { p } ( I _ { t } , \hat { I } _ { t } ) = \alpha \frac { 1 - \mathrm { S S I M } ( I _ { t } , \hat { I } _ { t } ) } { 2 } + ( 1 - \alpha ) \parallel I _ { t } - \hat { I } _ { t } \parallel
+$$
+
+In order to regularize the depth in low gradient regions, we incorporate an edge-aware term similar to Godard et al. (2017). This loss is weighted for each of the pyramid levels, decaying by a factor of 2 on each downsampling, starting with a weight of 1 for the $0 ^ { \mathrm { { i } \mathrm { { \bar { h } } } } }$ pyramid level.
+
+$$
+\mathcal { L } _ { s } ( \hat { D } _ { t } ) = | \delta _ { x } \hat { D } _ { t } | e ^ { - | \delta _ { x } I _ { t } | } + | \delta _ { y } \hat { D } _ { t } | e ^ { - | \delta _ { y } I _ { t } | }
+$$
+
+We also incorporate some of the insights introduced in Godard et al. (2018), namely auto-masking, minimum reprojection error, and inverse depth map upsampling to further improve depth estimation performance in our self-supervised monocular setting.
+
+# 3.2 DEPTH AND POSE NETWORKS
+
+Our baseline depth and pose networks are based on the PackNet architecture introduced by Guizilini et al. (2019), which proposes novel packing and unpacking blocks to respectively downsample and upsample feature maps during the encoding and decoding stages. This network was selected due to its state-of-the-art performance in the task of self-supervised monocular depth estimation, so we can analyze if our proposed architecture is capable of further improving the current state-of-the-art. However, there are no restrictions as to which models our proposed semantically-guided architecture can be applied to, and in Section 5.4 we study its application to different depth networks.
+
+# 4 SEMANTICALLY-GUIDED GEOMETRIC REPRESENTATION LEARNING
+
+In this section, we describe our method to inject semantic information into a self-supervised depth network via its augmentation with semantic-aware convolutions. Our proposed architecture is depicted in Figure 2 and is composed of two networks: a primary one, responsible for the generation of depth predictions $\hat { D } = f _ { D } ( I ( p ) )$ ; and a secondary one, capable of producing semantic predictions. Only the first network is optimized during self-supervised learning; the semantic network is initialized from pretrained weights and is not further optimized. This is in contrast to the common practice of supervised (ImageNet) pretraining of depth encoders (Godard et al., 2018; Casser et al., 2019; Zou et al., 2018): here instead of fine-tuning from pre-trained weights, we preserve these secondary weights to guide the feature learning process of the primary depth network. Our approach also differs from learning without forgetting (Li & Hoiem, 2017) by leveraging fixed intermediate feature representations as a way to maintain consistent semantic guidance throughout training.
+
+![](images/768e7a4db114cab117418b58a69ddf5a06d6900d9c4ab615ba74d01856d69ab0.jpg)  
+Figure 2: Diagram of our proposed architecture for self-supervised monocular depth estimation with semantically-guided feature learning. The semantic network is fixed and initialized from pretrained weights, while the depth network is trained end-to-end in a self-supervised way, including pixel-adaptive convolutions (Guidance) on its decoder to learn semantic-dependent geometric features.
+
+# 4.1 SEMANTICALLY-GUIDED DEPTH FEATURES
+
+We leverage the information from the pretrained semantic network in the depth network through the use of pixel-adaptive convolutions (Su et al., 2019). They were recently proposed to address some limitations inherent to the standard convolution operation, namely its translation invariance making it content-agnostic. While this significantly reduces the number of parameters of the resulting network, this might also lead to sub-optimal solutions under certain conditions important for geometric representation learning. For example, spatially-shared filters globally average the loss gradients over the entire image, forcing the network to learn weights that cannot leverage location-specific information beyond their limited receptive fields. Content-agnostic filters are unable to distinguish between different pixels that are visually similar (i.e. dark areas due to shadows or black objects) or generalize to similar objects that are visually different (i.e. cars with varying colors). In this work, we use pixel-adaptive convolutions to produce semantic-aware depth features, where the fixed information encoded in the semantic network is used to disambiguate geometric representations for the generation of multi-level depth features.
+
+As shown in Figure 2, we extract multi-level feature maps from the semantic network. For each feature map, we apply a $3 \times 3$ and a $1 \times 1$ convolutional layer followed by Group Normalization (Wu & He, 2018) and ELU non-linearities (Clevert et al., 2016). These processed semantic feature maps are then used as guidance on their respective pixel-adaptive convolutional layers, following the formulation proposed in $\mathrm { S u }$ et al. (2019):
+
+$$
+\mathbf { v } _ { i } ^ { \prime } = \sum _ { j \in \Omega ( i ) } K ( \mathbf { f } _ { i } , \mathbf { f } _ { j } ) \mathbf { W } [ \mathbf { p } _ { i } - \mathbf { p } _ { j } ] \mathbf { v } _ { j } + \mathbf { b }
+$$
+
+In the above equation, $\mathbf { f } \in \mathcal { R } ^ { D }$ are processed features from the semantic network that will serve to guide the pixel-adaptive convolutions from the depth network, $\mathbf { p } = ( x , y ) ^ { T }$ are pixel coordinates, with $[ { \bf p } _ { i } - { \bf p } _ { j } ]$ denoting 2D spatial offsets between pixels, $\mathbf { W } _ { k \times k }$ are convolutional weights with kernel size $k , \check { \Omega } _ { i }$ defines a $k \times k$ convolutional window around $i$ , $\mathbf { v }$ is the input signal to be convolved, and $\mathbf { b } \in \mathcal { R } ^ { 1 }$ is a bias term. $K$ is the kernel used to calculate the correlation between guiding features, here chosen to be the standard Gaussian kernel:
+
+$$
+K ( \mathbf { f } _ { i } , \mathbf { f } _ { j } ) = \exp \left( - \frac { 1 } { 2 } ( \mathbf { f } _ { i } - \mathbf { f } _ { j } ) ^ { T } \Sigma _ { i j } ^ { - 1 } ( \mathbf { f } _ { i } - \mathbf { f } _ { j } ) \right)
+$$
+
+where $\Sigma _ { i j }$ is the covariance matrix between features $\mathbf { f } _ { i }$ and $\mathbf { f } _ { j }$ , here chosen to be a diagonal matrix $\sigma ^ { 2 } \cdot I _ { D }$ , with $\sigma$ as an extra learnable parameter for each convolutional filter. These kernel evaluations can be seen as a secondary set of weights applied to the standard convolutional weights, changing their impact on the resulting depth features depending on the content stored in the guiding semantic features. For example, the information contained in depth features pertaining to the sky should not be used to generate depth features describing a pedestrian, and this behavior is now captured as a larger distance between their corresponding semantic features, which in turn produces smaller weights for that particular convolutional filter. Note that the standard convolution can be considered a special case of the pixel-adaptive convolution, where $\forall ~ i j , K ( \mathbf { f } _ { i } , \mathbf { f } _ { j } ) ~ = ~ 1$ .
+
+![](images/c1133bd34f68068e6cc07f5e3946ece1ab608146cce40ed0d3a8c96d349ad49c.jpg)  
+Figure 3: Qualitative results of our proposed two-stage training to address the infinite depth problem. Top images were obtained evaluating the first-stage depth network, and bottom images were obtained using the second-stage depth network, trained with a filtered dataset.
+
+# 4.2 SEMANTIC GUIDANCE NETWORK
+
+As the secondary network used to provide semantic guidance for the generation of depth features, we use a Feature Pyramid Network (FPN) with ResNet backbone (Lin et al., 2017). This architecture has been shown to be efficient for both semantic and instance-level predictions towards panoptic segmentation (Kirillov et al., 2019; Li et al., 2018; Xiong et al., 2019; Porzi et al., 2019). While our proposed semantically-guided architecture is not restricted to any particular network, we chose this particular implementation to facilitate the future exploration of different sources for guidance information. Architectural details follow the protocols described in Li et al. (2018), and unless mentioned otherwise the same pretrained model was used in all reported experiments. The semantic network is assumed fixed, pretrained on a held out dataset different than the raw data used for selfsupervised learning, i.e. we do not require any semantic ground truth on the target dataset.
+
+# 4.3 TWO-STAGE TRAINING
+
+One well-known limitation of the self-supervised photometric loss is its inability to model dynamic objects, due to a static world assumption that only accounts for camera ego-motion (Godard et al., 2018; Casser et al., 2019). A resulting common failure mode is the infinite depth problem, which is caused by the presence of objects moving at the same speed as the camera. This typically causes distinct holes in the predicted depth maps, with arbitrarily large values where these objects should be. This severely hinders the applicability of such models in real-world applications, particularly for automated driving, where the ability to detect and properly model dynamic objects is crucial. Moreover, this limitation may be further accentuated in our proposed semantically-guided architecture, as the infinite depth problem occurs mostly on dynamic categories (i.e. cars and motorcycles) and the semantic-aware features may reinforce this bias.
+
+We propose a simple and efficient two-stage training method to detect and remove this bias from the training set. In the first stage, we learn a standard depth network on all available training data. This network, exhibiting the infinite depth problem, is then used to resample the dataset by automatically filtering out sequences with infinite depth predictions that violate a basic geometric prior. We indeed find that depth predictions for pixels corresponding to the nearby ground plane are generally robust. This enables getting a coarse estimate of the ground plane using RANSAC and detecting the number of pixels whose predicted depth projects them significantly below the ground. If that number is above a threshold, then the corresponding image is subsequently ignored (we found a conservative threshold of 10 to work well in all our experiments, filtering out roughly $5 \%$ of the KITTI training dataset). During the second stage, we retrain the network on the subsampled dataset (from scratch to avoid the previous local optimum). As this subsampled dataset is de-biased, the network learns better depth estimates on dynamic objects. This process can be repeated, but we find that two stages are enough to remove any traces of infinite depth in our experiments, as shown in Figure 3.
+
+# 5 EXPERIMENTAL RESULTS
+
+# 5.1 DATASETS
+
+We use the standard KITTI benchmark (Geiger et al., 2013) for self-supervised training and evaluation. More specifically, we adopt the training, validation and test splits used in Eigen et al. (2014) with the pre-processing from Zhou et al. (2017) to remove static frames, which is more suitable for
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Superv.</td><td colspan="4">Lower is Better</td><td colspan="3">Higher is Better</td></tr><tr><td>Abs Rel</td><td>Sq Rel</td><td>RMSE</td><td>RMSEtog</td><td>δ&lt;1.25δ&lt;1.25²</td><td></td><td>δ&lt;1.253</td></tr><tr><td>Garg et al. (2016)</td><td>M</td><td>0.152</td><td>1.226</td><td>5.849</td><td>0.246</td><td>0.784</td><td>0.921</td><td>0.967</td></tr><tr><td>Zou et al. (2018)</td><td>M</td><td>0.150</td><td>1.124</td><td>5.507</td><td>0.223</td><td>0.806</td><td>0.933</td><td>0.973</td></tr><tr><td>Godard et al. (2017)</td><td>M</td><td>0.141</td><td>1.186</td><td>5.677</td><td>0.238</td><td>0.809</td><td>0.928</td><td>0.969</td></tr><tr><td>Zhan et al. (2018)</td><td>M</td><td>0.135</td><td>1.132</td><td>5.585</td><td>0.229</td><td>0.820</td><td>0.933</td><td>0.971</td></tr><tr><td>Godard et al. (2018) (R18)</td><td>M</td><td>0.115</td><td>0.903</td><td>4.863</td><td>0.193</td><td>0.877</td><td>0.959</td><td>0.981</td></tr><tr><td>Godard et al. (2018) (R50)</td><td>M</td><td>0.112</td><td>0.851</td><td>4.754</td><td>0.190</td><td>0.881</td><td>0.960</td><td>0.981</td></tr><tr><td>Guizilini et al. (2019) (MR)</td><td>M</td><td>0.108</td><td>0.727</td><td>4.426</td><td>0.184</td><td>0.885</td><td>0.963</td><td>0.983</td></tr><tr><td>Guizilini et al. (2019) (HR)</td><td>M</td><td>0.104</td><td>0.758</td><td>4.386</td><td>0.182</td><td>0.895</td><td>0.964</td><td>0.982</td></tr><tr><td>Casser et al. (2019)</td><td>S+Inst</td><td>0.141</td><td>1.025</td><td>5.290</td><td>0.215</td><td>0.816</td><td>0.945</td><td>0.979</td></tr><tr><td>Chen et al. (2019)</td><td>S+Sem</td><td>0.118</td><td>0.905</td><td>5.096</td><td>0.211</td><td>0.839</td><td>0.945</td><td>0.977</td></tr><tr><td>Ochs et al. (2019)</td><td>D+Sem</td><td>0.116</td><td>0.945</td><td>4.916</td><td>0.208</td><td>0.861</td><td>0.952</td><td>0.968</td></tr><tr><td>Ours (MR)</td><td>M+Sem</td><td>0.102</td><td>0.698</td><td>4.381</td><td>0.178</td><td>0.896</td><td>0.964</td><td>0.984</td></tr><tr><td>Ours (HR)</td><td>M+Sem</td><td>0.100</td><td>0.761</td><td>4.270</td><td>0.175</td><td>0.902</td><td>0.965</td><td>0.982</td></tr></table>
+
+Table 1: Quantitative performance comparison of our proposed architecture on KITTI for depths up to $8 0 \mathrm { m }$ . $M$ refers to methods that train using monocular images, S refers to methods that train using stereo pairs, $D$ refers to methods that use ground-truth depth supervision, Sem refers to methods that include semantic information, and Inst refers to methods that include semantic and instance information. $M R$ indicates $6 4 0 \times 1 9 2$ input images, and $H R$ indicates $1 2 8 0 \mathrm { ~ x ~ } 3 8 4$ input images. Our proposed architecture is able to further improve the current state of the art in self-supervised monocular depth estimation, and outperforms other methods that exploit semantic information (including ground truth labels) by a substantial margin.
+
+monocular self-supervised learning. This results in 39810 images for training, 4424 for validation, and 697 for evaluation. Following common practice, we pretrain our depth and pose networks on the CityScapes dataset (Cordts et al., 2016), consisting of 88250 unlabeled images. Unless noted otherwise, input images are downsampled to $6 4 0 \times 1 9 2$ resolution and output inverse depth maps are upsampled to full resolution using bilinear interpolation. Our fixed semantic segmentation network is pretrained on Cityscapes, achieving a mIoU of $7 5 \%$ on the validation set.
+
+# 5.2 IMPLEMENTATION DETAILS
+
+We implement our models with PyTorch (Paszke et al., 2017) and follow the same training protocols of Guizilini et al. (2019) when optimizing our depth and pose networks. The initial training stage is conducted on the CityScapes dataset for 50 epochs, with a batch size of 4 per GPU and initial depth and pose learning rates of $2 \cdot 1 0 ^ { - 4 }$ and $5 \cdot { \bar { 1 } } 0 ^ { - 4 }$ respectively, that are halved every 20 epochs. Afterwards, the depth and pose networks are fine-tuned on KITTI for 30 epochs, with the same parameters and halving the learning rates after every 12 epochs. This fine-tuning stage includes the proposed architecture, where information from the fixed semantic network, pretrained separately, is used to directly guide the generation of depth features. There is no direct supervision at any stage during depth training, all semantic information is derived from the fixed secondary network.
+
+When pretraining the semantic segmentation network, we use a ResNet-50 backbone with Imagenet (Deng et al., 2009) pretrained weights and optimize the network for $4 8 k$ iterations on the CityScapes dataset with a learning rate of 0.01, momentum of 0.9, weight decay of $1 0 ^ { - 4 }$ , and a batch size of 1 per GPU. Random scaling between (0.7, 1.3), random horizontal flipping, and a crop size of $1 0 0 0 \times 2 0 0 0$ are used for data augmentation. We decay the learning rate by a factor of 10 at iterations $3 6 k$ and $4 4 k$ . Once training is complete, the semantic segmentation network is fixed and becomes the only source of semantic information when fine-tuning the depth and pose networks on KITTI.
+
+# 5.3 DEPTH ESTIMATION PERFORMANCE
+
+Our depth estimation results are summarized in Table 1, where we compare our proposed architecture with other published works. From these results we can see that the introduction of semanticallyguided geometric representation learning further improves upon the current state of the art in selfsupervised monocular depth estimation from Guizilini et al. (2019), which served as our baseline. Our approach also outperforms other methods that leverage semantic information by a substantial margin, even those using ground-truth KITTI semantic segmentation and depth labels during train
+
+<table><tr><td rowspan="2">Network</td><td rowspan="2">SEM</td><td rowspan="2">TST</td><td colspan="4">Lower is Better</td><td colspan="2">Higher is Better δ &lt;1.25 δ&lt;1.252 δ&lt; 1.253</td><td rowspan="2">Class-Avg. Abs Rel</td></tr><tr><td>Abs Rel</td><td>Sq Rel RMSE RMSElog</td><td></td><td></td><td></td><td></td></tr><tr><td>ResNet-18</td><td>√</td><td></td><td>0.120 0.117</td><td>0.896 0.854</td><td>4.869 4.714</td><td>0.198 0.191</td><td>0.868 0.957 0.873</td><td>0.981 0.963 0.981</td><td>0.149 0.139</td></tr><tr><td>ResNet-50</td><td>√</td><td></td><td>0.117 0.113</td><td>0.900 0.831</td><td>4.826 4.663</td><td>0.196 0.189</td><td>0.873 0.967 0.878 0.971</td><td>0.980 0.983</td><td>0.144 0.136</td></tr><tr><td>PackNet</td><td>√</td><td>√</td><td>0.108 0.103 0.102</td><td>0.727 0.710 0.698</td><td>4.426 4.301 4.381</td><td>0.184 0.179 0.178</td><td>0.885 0.963 0.895 0.964 0.896 0.963</td><td>0.983 0.984 0.984</td><td>0.132 0.121 0.117</td></tr></table>
+
+Table 2: Ablative analysis of our semantic guidance (SEM) and two-stage-training (TST) contributions. The last column indicates class-average Abs. Rel. obtained by averaging all class-specific depth errors in Figure 4, while other columns indicate pixel-average metrics.
+
+ing (Ochs et al., 2019). Furthermore, in Figure 5 we also present qualitative results showing the improvements in depth estimation generated by our proposed framework, compared to our baseline. Note how our semantically-guided architecture produces sharper boundaries and better object delineation, especially in structures further away or not clearly distinguishable in the input image.
+
+# 5.4 ABLATIVE ANALYSIS
+
+# 5.4.1 DIFFERENT DEPTH NETWORKS
+
+To better evaluate our main contribution, we provide an ablative analysis showing how it generalizes to different depth networks. To this end, we consider two variations of the widely used ResNet architecture as the encoder for our depth network: ResNet-18 and ResNet-50 (the same pretrained semantic network was used in all experiments). Depth estimation results considering these variations are shown in Table 2, where we can see that our proposed semantically-guided architecture is able to consistently improve the performance of different depth networks, for all considered metrics.
+
+# 5.4.2 CLASS-SPECIFIC DEPTH PERFORMANCE
+
+To further showcase the benefits of our semantically-guided architecture, we also provide classspecific evaluation metrics, as shown in Figure 4. As we do not have ground-truth semantic segmentation for these images, we use the prediction of the semantic network to bin pixels per predicted category, and evaluate only on those pixels. From these results we can see that our proposed architecture consistently improves depth performance for pixels across all predicted classes, especially those containing fine-grained structures and sharp boundaries, e.g. poles and traffic signs.
+
+![](images/029bba1944c05b59fd31cd1b470c4d6e1b61b263ba0128d9dbf7c9034beaab1f.jpg)  
+Figure 4: Class-specific depth evaluation for our proposed architecture (blue), relative to our baseline (red). The rightmost column indicates class-average depth metrics, obtained by averaging all individual classes. The introduction of semantically-guided features, in conjunction with our proposed two-stage training methodology to address the infinite depth problem, consistently improved depth results for all considered classes (lower is better).
+
+![](images/8b4a389f1dbad0e331cbee9e393af3ad825e78c305d86b9334b90a768bca6149.jpg)  
+Figure 5: Qualitative results of our proposed architecture. The left, middle, and right columns show respectively input images, baseline predicted depth maps (Guizilini et al., 2019), and the depths maps obtained using our proposed architecture. Our semantic-aware depth network predicts sharper boundaries and fine-grained details on distant objects. The dotted lines indicate class-average errors, obtained by averaging all the class-specific depth errors.
+
+We also measure the impact of our two-stage training process, which is expected to address the infinite depth problem in dynamic objects. Although we find the pixel-average difference in performance to not be significant (see Table 2), there is a significant improvement in class-average depth estimation, from 0.121 to 0.117 Abs-Rel. This is because the number of pixels affected by the infinite depth problem is vastly smaller than the total number of pixels. However, when considering class-average depth evaluation, the improvement over classes such as cars (0.200 to 0.177 Abs-Rel) and motorcycles (0.091 to 0.069) becomes statistically significant. This further exemplifies the importance of fine-grained metrics in depth evaluation, so these underlying behaviors can be properly observed and accounted for in the development of new techniques.
+
+# 6 CONCLUSION
+
+This paper introduces a novel architecture for self-supervised monocular depth estimation that leverages semantic information from a fixed pretrained network to guide the generation of multi-level depth features via pixel-adaptive convolutions. Our monodepth network learns semantic-aware geometric representations that can disambiguate photometric ambiguities in a self-supervised learning structure-from-motion context. Furthermore, we introduce a two-stage training process that resamples training data to overcome a common bias on dynamic objects resulting in predicting them at infinite depths. Our experiments on challenging real-world data shows that our proposed architecture consistently improves the performance of different monodepth architectures, thus establishing a new state of the art in self-supervised monocular depth estimation. Future directions of research include leveraging other sources of guidance (i.e. instance masks, optical flow, surface normals), as well as avenues for self-supervised fine-tuning of the semantic network.
+
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+Nan Yang, Rui Wang, Jorg St ¨ uckler, and Daniel Cremers. Deep virtual stereo odometry: Leveraging ¨ deep depth prediction for monocular direct sparse odometry. arXiv:1807.02570, 2018. 2
+
+Zhenheng Yang, Peng Wang, Wei Xu, Liang Zhao, and Ramakant Nevatia. Unsupervised learning of geometry with edge-aware depth-normal consistency. arXiv preprint arXiv:1711.03665, 2017. 1
+
+Zhichao Yin and Jianping Shi. Geonet: Unsupervised learning of dense depth, optical flow and camera pose. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), volume 2, 2018. 2
+
+Huangying Zhan, Ravi Garg, Chamara Saroj Weerasekera, Kejie Li, Harsh Agarwal, and Ian Reid. Unsupervised learning of monocular depth estimation and visual odometry with deep feature reconstruction. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 340–349, 2018. 6
+
+Tinghui Zhou, Matthew Brown, Noah Snavely, and David G Lowe. Unsupervised learning of depth and ego-motion from video. In CVPR, volume 2, pp. 7, 2017. 2, 3, 5
+
+Yuliang Zou, Zelun Luo, and Jia-Bin Huang. Df-net: Unsupervised joint learning of depth and flow using cross-task consistency. In European Conference on Computer Vision, 2018. 1, 2, 3, 6
+
+# A PRE-TRAINING THE SEMANTIC SEGMENTATION NETWORK
+
+The introduction of a semantic segmentation network to the depth estimation task increases the depth estimation performance, however it also increases model complexity (e.g. number of trainable parameters). To investigate that the increased performance for the depth estimation task is indeed due to the semantic features encoded in the secondary network, we perform an in-depth analysis (summarized in Table 3) where we explore the impact of pre-training the semantic segmentation network before it is used to guide the generation of depth features. From these results we can see that the presence of semantic information encoded in the secondary network indeed leads to an increase in performance, and that fine-tuning this secondary network for the speficic task of depth estimation actually decreases performance.
+
+In the first two rows an untrained semantic network is utilized, with only its encoder initialized from ImageNet (Deng et al., 2009) weights. Two different scenarios are explored: in the first one (D) only the depth network is fine-tuned in a self-supervised fashion, while in $\mathbf { D + S }$ both networks are fine-tuned together in the same way. As expected, using untrained features as guidance leads to significantly worse results, since there is no structure encoded in the secondary network and the primary network needs to learn to filter out all this spurious information. When both networks are fine-tuned simultaneously, results improve because now the added complexity from the secondary network can be leveraged for the task of depth estimation, however there is still no improvement over the baseline.
+
+Next, the semantic network was pre-trained on only half of the CityScapes (Cordts et al., 2016) dataset (samples chosen randomly), leading to a worse semantic segmentation performance (validation mIoU of around $7 0 \%$ vs. $\dot { 7 } 5 \%$ for the fully trained one). This partial pre-training stage was enough to enable the transfer of useful information between networks, leading to improvements over the baseline. Interestingly, fine-tuning both networks for the task of depth estimation actually hurt performance this time, which we attribute to forgetting the information contained in the secondary network, as both networks are optimized for the depth task. When the semantic network is pretrained with all of CityScapes (last two rows), these effects are magnified, with fine-tuning only the depth network leading to our best reported performance (Table 1) and fine-tuning both networks again leading to results similar to the baseline.
+
+Table 3: Analysis of the impact of pre-training the semantic segmentation network. On the PreTrain column, $I$ indicates ImageNet (Deng et al., 2009) pretraining and CS indicates CityScapes (Cordts et al., 2016) pretraining, with $1 / 2$ indicating the use of only half the dataset (samples chosen randomly). In the Fine-Tune column, $D$ indicates fine-tuning the depth network and $s$ indicates finetuning the semantic network (note that this is a self-supervised fine-tuning for the depth task, using the objective described in Section 3.1).   
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Pre-Train</td><td rowspan="2">Fine-Tune</td><td colspan="4">Lower is Better</td><td colspan="3">Higher is Better</td></tr><tr><td>Abs Rel</td><td>Sq Rel</td><td>RMSE</td><td>RMSElog</td><td>δ &lt;1.25 δ&lt;1.252</td><td></td><td>δ&lt;1.253</td></tr><tr><td>Baseline</td><td></td><td>D</td><td>0.108</td><td>0.727</td><td>4.426</td><td>0.184</td><td>0.885</td><td>0.963</td><td>0.983</td></tr><tr><td rowspan="6">Proposed</td><td>I</td><td>D+S</td><td>0.116</td><td>0.847</td><td>4.751</td><td>0.192</td><td>0.879</td><td>0.960</td><td>0.981</td></tr><tr><td>I</td><td>D</td><td>0.197</td><td>1.323</td><td>6.114</td><td>0.265</td><td>0.776</td><td>0.918</td><td>0.966</td></tr><tr><td>CS (1/2)</td><td>D+S</td><td>0.109</td><td>0.737</td><td>4.389</td><td>0.185</td><td>0.884</td><td>0.962</td><td>0.982</td></tr><tr><td>CS (1/2)</td><td>D</td><td>0.104</td><td>0.716</td><td>4.322</td><td>0.180</td><td>0.893</td><td>0.964</td><td>0.984</td></tr><tr><td>Cs</td><td>D+S</td><td>0.107</td><td>0.741</td><td>4.407</td><td>0.183</td><td>0.883</td><td>0.963</td><td>0.983</td></tr><tr><td>Cs</td><td>D</td><td>0.102</td><td>0.698</td><td>4.381</td><td>0.178</td><td>0.896</td><td>0.964</td><td>0.984</td></tr></table>
+
+# B UNCERTAINTY AND GENERALIZATION TO DIFFERENT OBJECTS
+
+In a self-supervised setting, increasing the number of unlabeled videos used for depth training is expected to lead to an increasing specialization away from the domain in which the semantic network was pre-trained. This might result in harmful guidance if our method is not robust to this gap. However, our approach does not use semantic predictions directly, but rather the decoded features of the semantic network themselves, which represent general appearance information that should be more robust to this domain gap. To validate our hypothesis, we further explore the impact of erroneous semantic information in the performance of our proposed semantically-guided depth framework. In Figure 6 we present qualitative results highlighting situations in which our pretrained semantic network failed to generate correct semantic predictions for certain objects in the scene, and yet our proposed framework was still able to properly recover depth values for that portion of the environment. These exemplify possible scenarios for erroneous semantic prediction.
+
+• Imprecise boundaries: in the first row, we can see that the semantic segmentation network does not correctly detect the traffic sign, yet the semantically-guided depth network predicts its shape and depth accurately.   
+• Wrong classification: in the second row, the truck was mistakenly classified as partially “road” and “building”, however our semantically-guided depth network was still able to properly recover its overall shape with sharp delineation that was not available from its semantic contour. A similar scenario happens in the same image, with “fence” being partially labeled as “bicycle”.   
+• Missing ontology: there is no “trash can” class on the CityScapes ontology, however in the third row our semantically-guided depth network was able to correctly reconstruct such object even though it was classified as “fence”, similarly to its surroundings.   
+• Object Hallucination: in the fourth row, the contour of a “person” was erroneously introduced in the image and correctly removed by our semantically-guided framework.
+
+These examples are evidence that our proposed framework is able to reason over the uncertainty inherent to semantic classification, leveraging this information when accurate to achieve the results reported in this paper, but also discarding it if necessary to generate a better reconstruction according to the self-supervised photometric loss.
+
+![](images/1571107d15766292cb50733ff1a095273dd604cd102c3c0235ac0a84b6b5dd4e.jpg)  
+Figure 6: Examples of erroneous semantic predictions that still led to accurate depth predictions using our proposed semantically-guided depth framework.
+
+# C GENERALIZATION TO DIFFERENT DATASETS
+
+In the previous sections, we show that our proposed framework is robust to a degraded semantic network, both by pretraining the semantic network with fewer annotated labels (Appendix A) and also by providing evidence that the depth network is able to reason over erroneous predictions to still generate accurate reconstructions (Appendix B). We now go one step further and analyze how our proposed semantically-guided framework generalizes to a dataset that was used neither during pre-training nor for fine-tuning. To this end, we evaluate our KITTI depth model on the recently released NuScenes dataset (Caesar et al., 2019). The official NuScenes validation split is used, containing 6019 images from the front camera with ground-truth depth maps generated by LiDAR reprojection. Results presented in Table 4 provide additional evidence that our method indeed results in generalization improvements, even on significantly different data from different platforms
+
+and environments (Karlsruhe, Germany for KITTI vs Boston, USA and Singapore for NuScenes), outperforming the state of the art methods and our baseline (Guizilini et al., 2019).   
+Table 4: Generalization capability of different networks, trained on both KITTI and CityScapes datasets and evaluated on the NuScenes (Caesar et al., 2019) dataset. Our proposed semanticallyguided architecture is able to further improve upon the baseline from Guizilini et al. (2019), which only used unlabeled image sequences for self-supervised depth training.   
+
+<table><tr><td>Method</td><td>Abs Rel</td><td>Sq Rel</td><td>RMSE</td><td>RMSElog</td><td></td><td>δ&lt;1.25δ&lt;1.25²</td><td>δ&lt;1.253</td></tr><tr><td>Godard et al. (2018) (R18)</td><td>0.212</td><td>1.918</td><td>7.958</td><td>0.323</td><td>0.674</td><td>0.898</td><td>0.954</td></tr><tr><td>Godard et al. (2018) (R50)</td><td>0.210</td><td>2.017</td><td>8.111</td><td>0.328</td><td>0.697</td><td>0.903</td><td>0.960</td></tr><tr><td>Guizilini et al. (2019) (MR)</td><td>0.187</td><td>1.852</td><td>7.636</td><td>0.289</td><td>0.742</td><td>0.917</td><td>0.961</td></tr><tr><td>Ours (MR)</td><td>0.181</td><td>1.505</td><td>7.237</td><td>0.271</td><td>0.765</td><td>0.931</td><td>0.969</td></tr></table>

parse/train/ByxT7TNFvH/ByxT7TNFvH_content_list.json

@@ -0,0 +1,1495 @@
+[
+    {
+        "type": "text",
+        "text": "SEMANTICALLY-GUIDED REPRESENTATION LEARNING FOR SELF-SUPERVISED MONOCULAR DEPTH ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Vitor Guizilini1 Rui Hou1,2 Jie Li1 Rares, Ambrus, 1 Adrien Gaidon1 ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1Toyota Research Institute (TRI) {first.last}@tri.global ",
+        "bbox": [
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+            218
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "2University of Michigan rayhou@umich.edu ",
+        "bbox": [
+            552,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            452,
+            243,
+            544,
+            258
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Self-supervised learning is showing great promise for monocular depth estimation, using geometry as the only source of supervision. Depth networks are indeed capable of learning representations that relate visual appearance to 3D properties by implicitly leveraging category-level patterns. In this work we investigate how to leverage more directly this semantic structure to guide geometric representation learning, while remaining in the self-supervised regime. Instead of using semantic labels and proxy losses in a multi-task approach, we propose a new architecture leveraging fixed pretrained semantic segmentation networks to guide self-supervised representation learning via pixel-adaptive convolutions. Furthermore, we propose a two-stage training process to overcome a common semantic bias on dynamic objects via resampling. Our method improves upon the state of the art for self-supervised monocular depth prediction over all pixels, fine-grained details, and per semantic categories.† ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            467,
+            336,
+            482
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Accurate depth estimation is a key problem in computer vision and robotics, as it is instrumental for perception, navigation, and planning. Although perceiving depth typically requires dedicated sensors (e.g., stereo rigs, LiDAR), learning to predict depth from monocular imagery can provide useful cues for a wide array of tasks (Michels et al., 2005; Kendall et al., 2018; Manhardt et al., 2019; Lee et al., 2019). Going beyond supervised learning from direct measurements (Eigen et al., 2014), self-supervised methods exploit geometry as supervision (Guo et al., 2018; Pillai et al., 2019; Zou et al., 2018; Yang et al., 2017), therefore having the potential to leverage large scale datasets of raw videos to outperform supervised methods (Guizilini et al., 2019). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Although depth from a single image is an ill-posed inverse problem, monocular depth networks are able to make accurate predictions by learning representations connecting the appearance of scenes and objects with their geometry in Euclidean 3D space. Due to perspective, there is indeed an equivariance relationship between the visual appearance of an object in 2D and its depth, when conditioned on the object’s category. For instance, a car 25 meters away appears smaller (on the image plane) than a car only 5 meters away but bigger than a truck 50 meters away. Current depth estimation methods either do not leverage this structure explicitly or rely on strong semantic supervision to jointly optimize geometric consistency and a semantic proxy task in a multi-task objective (Ochs et al., 2019; Chen et al., 2019), thus departing from the self-supervised paradigm. ",
+        "bbox": [
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+            616,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we explore how we can leverage semantic information to improve monocular depth prediction in a self-supervised way. Our main contribution is a novel architecture that uses a fixed pre-trained semantic segmentation network to guide geometric representation learning in a selfsupervised monocular depth network. In contrast to standard convolutional layers, our architecture uses pixel-adaptive convolutions (Su et al., 2019) to learn semantic-dependent representations that can better capture the aforementioned equivariance property. Leveraging semantics may nonetheless introduce category-specific biases. Our second contribution is a two-stage training process where we automatically detect the presence of a common bias on dynamic objects (projections at infinity) and resample the training set to de-bias it. Our method improves upon the state of the art in selfsupervised monocular depth estimation on the standard KITTI benchmark (Geiger et al., 2013), both on average over pixels, over classes, and for dynamic categories in particular. ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/b09c90637e486163bce4d7333340755790cf27ff41a391d503abe5d3014d643d.jpg",
+        "image_caption": [
+            "Figure 1: Example of a pointcloud generated using our proposed semantically-guided architecture, colored by RGB values from the input image and corresponding predicted semantic labels. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            306,
+            344,
+            323
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Since the seminal work of Eigen et al. (2014), substantial progress has been done to improve the accuracy of supervised depth estimation from monocular images, including the use of Conditional Random Fields (CRFs) (Li et al., 2015), joint optimization of surface normals (Qi et al., 2018), fusion of multiple depth maps (Lee et al., 2018), and ordinal classification (Fu et al., 2018). Consequently, as supervised techniques for depth estimation advanced rapidly, the availability of largescale depth labels became a bottleneck, especially for outdoor applications. Garg et al. (2016) and Godard et al. (2017) provided an alternative self-supervised strategy involving stereo cameras, where Spatial Transformer Networks (Jaderberg et al., 2015) can be used to geometrically warp, in a differentiable way, the right image into a synthesized left image, using the predicted depth from the left image. The photometric consistency loss between the resulting synthesized and original left images can then be minimized in an end-to-end manner using a Structural Similarity term (Wang et al., 2004) and additional depth regularization terms. Following Godard et al. (2017) and Ummenhofer et al. (2017), Zhou et al. (2017) generalized this to the purely monocular setting, where a depth and a pose networks are simultaneously learned from unlabeled monocular videos. Rapid progress in terms of architectures and objective functions (Yin & Shi, 2018; Mahjourian et al., 2018; Casser et al., 2019; Zou et al., 2018; Klodt & Vedaldi, 2018; Wang et al., 2018; Yang et al., 2018) have since then turned monocular depth estimation into one of the most successful applications of self-supervised learning, even outperforming supervised methods (Guizilini et al., 2019). ",
+        "bbox": [
+            174,
+            339,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The introduction of semantic information to improve depth estimates has been explored in prior works, and can be broadly divided into two categories. The first one uses semantic (or instance) information to mask out or properly model dynamic portions of the image, which are not accounted for in the photometric loss calculation. Guney & Geiger ¨ (2015) leveraged object knowledge in a Markov Random Field (MRF) to resolve stereo ambiguities, while Bai et al. (2016) used a conjunction of instance-level segmentation and epipolar constraints to reduce uncertainty in optical flow estimation. Casser et al. (2019) used instance-level masks to estimate motion models for different objects in the environment, and account for their external motion in the resulting warped image. The second category attempts to learn both tasks in a single framework, and uses consistency losses to ensure that both are optimized simultaneously and regularize each other, so the information contained in one task can be transferred to improve the other. For instance, Ochs et al. (2019) estimated depth with an ordinal classification loss similar to the standard semantic classification loss, and used empirical weighting to combine them into a single loss for optimization. Similarly, Chen et al. (2019) used a unified conditional decoder that can generate either semantic or depth estimates, and both outputs are used to generate a series of losses also combined using empirical weighting to generate the final loss to be optimized. ",
+        "bbox": [
+            174,
+            597,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our approach focuses instead on representation learning, exploiting semantic features into the selfsupervised depth network by using a pretrained semantic segmentation network to guide the generation of depth features. This is done using pixel-adaptive convolutions, recently proposed in Su et al. (2019) and applied to tasks such as depth upsampling using RGB images for feature guidance. We show that different depth networks can be readily modified to leverage this semantic feature guidance, ranging from widely used ResNets (He et al., 2016) to the current state-of-the-art PackNet (Guizilini et al., 2019), with a consistent gain in performance across these architectures. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3 SELF-SUPERVISED STRUCTURE-FROM-MOTION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            604,
+            118
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our semantically-guided architecture is developed within a self-supervised monocular depth estimation setting, commonly known as structure-from-motion (SfM). Learning in a self-supervised structure-from-motion setting requires two networks: a monocular depth model $f _ { D } : I  D$ , that outputs a depth prediction $\\hat { D } = f _ { D } ( I ( p ) )$ for every pixel $p$ in the target image $I$ ; and a monocular ego-motion estimator $f _ { \\mathbf { x } } : ( I _ { t } , I _ { S } ) \\to \\mathbf { x } _ { t \\to S }$ , that predicts the $6 \\mathrm { D o F }$ transformations for all $s \\in S$ given by $\\mathbf { x } _ { t  s } = ( \\mathbf { \\Sigma } _ { \\mathbf { 0 } } ^ { \\mathbf { R } \\textbf { t } } ) \\in \\mathbf { S } \\mathbf { E } ( 3 )$ between the target image $I _ { t }$ and a set of temporal context source images $I _ { s } \\in I _ { S }$ . In all reported experiments we use $I _ { t - 1 }$ and $I _ { t + 1 }$ as source images. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 THE SELF-SUPERVISED OBJECTIVE LOSS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We train the depth and pose networks simultaneously, using the same protocols and losses as described in Guizilini et al. (2019). Our self-supervised objective loss consists of an appearance matching term ${ \\mathcal { L } } _ { p }$ that is imposed between the synthesized $\\bar { \\hat { I } } _ { t }$ and original $I _ { t }$ target images, and a depth regularization term $\\mathcal { L } _ { s }$ that ensures edge-aware smoothing in the depth estimates $\\hat { D } _ { t }$ . The final objective loss is averaged per pixel, pyramid scale and image batch, and is defined as: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/761f2abe68dfa81b034447e742baf1132ea0810702d321c69fdb07bf013cb13a.jpg",
+        "text": "$$\n\\mathcal { L } ( I _ { t } , \\hat { I } _ { t } ) = \\mathcal { L } _ { p } ( I _ { t } , \\hat { I } _ { t } ) + \\lambda _ { 1 } \\mathcal { L } _ { s } ( \\hat { D } _ { t } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            352,
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+            371
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\lambda _ { 1 }$ is a weighting coefficient between the photometric ${ \\mathcal { L } } _ { p }$ and depth smoothness $\\mathcal { L } _ { s }$ loss terms. Following Godard et al. (2017) and Zhou et al. (2017), the similarity between synthesized $\\hat { I } _ { t }$ and original $I _ { t }$ target images is estimated using a Structural Similarity (SSIM) term (Wang et al., 2004) combined with an L1 loss term, inducing the following overall photometric loss: ",
+        "bbox": [
+            173,
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f2a66c686ee12d86d751182aa0618de9112625ef825ac1c804f306732f83ea01.jpg",
+        "text": "$$\n\\mathcal { L } _ { p } ( I _ { t } , \\hat { I } _ { t } ) = \\alpha \\frac { 1 - \\mathrm { S S I M } ( I _ { t } , \\hat { I } _ { t } ) } { 2 } + ( 1 - \\alpha ) \\parallel I _ { t } - \\hat { I } _ { t } \\parallel\n$$",
+        "text_format": "latex",
+        "bbox": [
+            320,
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+            468
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In order to regularize the depth in low gradient regions, we incorporate an edge-aware term similar to Godard et al. (2017). This loss is weighted for each of the pyramid levels, decaying by a factor of 2 on each downsampling, starting with a weight of 1 for the $0 ^ { \\mathrm { { i } \\mathrm { { \\bar { h } } } } }$ pyramid level. ",
+        "bbox": [
+            174,
+            469,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bdd6a57a38eb218b5f7baba4a91ebe102970a1a99333a6272f6df20bd96d94e2.jpg",
+        "text": "$$\n\\mathcal { L } _ { s } ( \\hat { D } _ { t } ) = | \\delta _ { x } \\hat { D } _ { t } | e ^ { - | \\delta _ { x } I _ { t } | } + | \\delta _ { y } \\hat { D } _ { t } | e ^ { - | \\delta _ { y } I _ { t } | }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            356,
+            515,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We also incorporate some of the insights introduced in Godard et al. (2018), namely auto-masking, minimum reprojection error, and inverse depth map upsampling to further improve depth estimation performance in our self-supervised monocular setting. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 DEPTH AND POSE NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            594,
+            423,
+            609
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our baseline depth and pose networks are based on the PackNet architecture introduced by Guizilini et al. (2019), which proposes novel packing and unpacking blocks to respectively downsample and upsample feature maps during the encoding and decoding stages. This network was selected due to its state-of-the-art performance in the task of self-supervised monocular depth estimation, so we can analyze if our proposed architecture is capable of further improving the current state-of-the-art. However, there are no restrictions as to which models our proposed semantically-guided architecture can be applied to, and in Section 5.4 we study its application to different depth networks. ",
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+    {
+        "type": "text",
+        "text": "4 SEMANTICALLY-GUIDED GEOMETRIC REPRESENTATION LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section, we describe our method to inject semantic information into a self-supervised depth network via its augmentation with semantic-aware convolutions. Our proposed architecture is depicted in Figure 2 and is composed of two networks: a primary one, responsible for the generation of depth predictions $\\hat { D } = f _ { D } ( I ( p ) )$ ; and a secondary one, capable of producing semantic predictions. Only the first network is optimized during self-supervised learning; the semantic network is initialized from pretrained weights and is not further optimized. This is in contrast to the common practice of supervised (ImageNet) pretraining of depth encoders (Godard et al., 2018; Casser et al., 2019; Zou et al., 2018): here instead of fine-tuning from pre-trained weights, we preserve these secondary weights to guide the feature learning process of the primary depth network. Our approach also differs from learning without forgetting (Li & Hoiem, 2017) by leveraging fixed intermediate feature representations as a way to maintain consistent semantic guidance throughout training. ",
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+        "img_path": "images/768e7a4db114cab117418b58a69ddf5a06d6900d9c4ab615ba74d01856d69ab0.jpg",
+        "image_caption": [
+            "Figure 2: Diagram of our proposed architecture for self-supervised monocular depth estimation with semantically-guided feature learning. The semantic network is fixed and initialized from pretrained weights, while the depth network is trained end-to-end in a self-supervised way, including pixel-adaptive convolutions (Guidance) on its decoder to learn semantic-dependent geometric features. "
+        ],
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+    {
+        "type": "text",
+        "text": "4.1 SEMANTICALLY-GUIDED DEPTH FEATURES ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "We leverage the information from the pretrained semantic network in the depth network through the use of pixel-adaptive convolutions (Su et al., 2019). They were recently proposed to address some limitations inherent to the standard convolution operation, namely its translation invariance making it content-agnostic. While this significantly reduces the number of parameters of the resulting network, this might also lead to sub-optimal solutions under certain conditions important for geometric representation learning. For example, spatially-shared filters globally average the loss gradients over the entire image, forcing the network to learn weights that cannot leverage location-specific information beyond their limited receptive fields. Content-agnostic filters are unable to distinguish between different pixels that are visually similar (i.e. dark areas due to shadows or black objects) or generalize to similar objects that are visually different (i.e. cars with varying colors). In this work, we use pixel-adaptive convolutions to produce semantic-aware depth features, where the fixed information encoded in the semantic network is used to disambiguate geometric representations for the generation of multi-level depth features. ",
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+        "type": "text",
+        "text": "As shown in Figure 2, we extract multi-level feature maps from the semantic network. For each feature map, we apply a $3 \\times 3$ and a $1 \\times 1$ convolutional layer followed by Group Normalization (Wu & He, 2018) and ELU non-linearities (Clevert et al., 2016). These processed semantic feature maps are then used as guidance on their respective pixel-adaptive convolutional layers, following the formulation proposed in $\\mathrm { S u }$ et al. (2019): ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/f3fef7cb20207b1c3ef03f527c9aea2361576831a29f017ee6708df9c24376ac.jpg",
+        "text": "$$\n\\mathbf { v } _ { i } ^ { \\prime } = \\sum _ { j \\in \\Omega ( i ) } K ( \\mathbf { f } _ { i } , \\mathbf { f } _ { j } ) \\mathbf { W } [ \\mathbf { p } _ { i } - \\mathbf { p } _ { j } ] \\mathbf { v } _ { j } + \\mathbf { b }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the above equation, $\\mathbf { f } \\in \\mathcal { R } ^ { D }$ are processed features from the semantic network that will serve to guide the pixel-adaptive convolutions from the depth network, $\\mathbf { p } = ( x , y ) ^ { T }$ are pixel coordinates, with $[ { \\bf p } _ { i } - { \\bf p } _ { j } ]$ denoting 2D spatial offsets between pixels, $\\mathbf { W } _ { k \\times k }$ are convolutional weights with kernel size $k , \\check { \\Omega } _ { i }$ defines a $k \\times k$ convolutional window around $i$ , $\\mathbf { v }$ is the input signal to be convolved, and $\\mathbf { b } \\in \\mathcal { R } ^ { 1 }$ is a bias term. $K$ is the kernel used to calculate the correlation between guiding features, here chosen to be the standard Gaussian kernel: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/342bdf167552c4191233cf0175ff696f26e046329d7bb2de0a3adfc72f586e04.jpg",
+        "text": "$$\nK ( \\mathbf { f } _ { i } , \\mathbf { f } _ { j } ) = \\exp \\left( - \\frac { 1 } { 2 } ( \\mathbf { f } _ { i } - \\mathbf { f } _ { j } ) ^ { T } \\Sigma _ { i j } ^ { - 1 } ( \\mathbf { f } _ { i } - \\mathbf { f } _ { j } ) \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where $\\Sigma _ { i j }$ is the covariance matrix between features $\\mathbf { f } _ { i }$ and $\\mathbf { f } _ { j }$ , here chosen to be a diagonal matrix $\\sigma ^ { 2 } \\cdot I _ { D }$ , with $\\sigma$ as an extra learnable parameter for each convolutional filter. These kernel evaluations can be seen as a secondary set of weights applied to the standard convolutional weights, changing their impact on the resulting depth features depending on the content stored in the guiding semantic features. For example, the information contained in depth features pertaining to the sky should not be used to generate depth features describing a pedestrian, and this behavior is now captured as a larger distance between their corresponding semantic features, which in turn produces smaller weights for that particular convolutional filter. Note that the standard convolution can be considered a special case of the pixel-adaptive convolution, where $\\forall ~ i j , K ( \\mathbf { f } _ { i } , \\mathbf { f } _ { j } ) ~ = ~ 1$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/c1133bd34f68068e6cc07f5e3946ece1ab608146cce40ed0d3a8c96d349ad49c.jpg",
+        "image_caption": [
+            "Figure 3: Qualitative results of our proposed two-stage training to address the infinite depth problem. Top images were obtained evaluating the first-stage depth network, and bottom images were obtained using the second-stage depth network, trained with a filtered dataset. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.2 SEMANTIC GUIDANCE NETWORK ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As the secondary network used to provide semantic guidance for the generation of depth features, we use a Feature Pyramid Network (FPN) with ResNet backbone (Lin et al., 2017). This architecture has been shown to be efficient for both semantic and instance-level predictions towards panoptic segmentation (Kirillov et al., 2019; Li et al., 2018; Xiong et al., 2019; Porzi et al., 2019). While our proposed semantically-guided architecture is not restricted to any particular network, we chose this particular implementation to facilitate the future exploration of different sources for guidance information. Architectural details follow the protocols described in Li et al. (2018), and unless mentioned otherwise the same pretrained model was used in all reported experiments. The semantic network is assumed fixed, pretrained on a held out dataset different than the raw data used for selfsupervised learning, i.e. we do not require any semantic ground truth on the target dataset. ",
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+    {
+        "type": "text",
+        "text": "4.3 TWO-STAGE TRAINING ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "One well-known limitation of the self-supervised photometric loss is its inability to model dynamic objects, due to a static world assumption that only accounts for camera ego-motion (Godard et al., 2018; Casser et al., 2019). A resulting common failure mode is the infinite depth problem, which is caused by the presence of objects moving at the same speed as the camera. This typically causes distinct holes in the predicted depth maps, with arbitrarily large values where these objects should be. This severely hinders the applicability of such models in real-world applications, particularly for automated driving, where the ability to detect and properly model dynamic objects is crucial. Moreover, this limitation may be further accentuated in our proposed semantically-guided architecture, as the infinite depth problem occurs mostly on dynamic categories (i.e. cars and motorcycles) and the semantic-aware features may reinforce this bias. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We propose a simple and efficient two-stage training method to detect and remove this bias from the training set. In the first stage, we learn a standard depth network on all available training data. This network, exhibiting the infinite depth problem, is then used to resample the dataset by automatically filtering out sequences with infinite depth predictions that violate a basic geometric prior. We indeed find that depth predictions for pixels corresponding to the nearby ground plane are generally robust. This enables getting a coarse estimate of the ground plane using RANSAC and detecting the number of pixels whose predicted depth projects them significantly below the ground. If that number is above a threshold, then the corresponding image is subsequently ignored (we found a conservative threshold of 10 to work well in all our experiments, filtering out roughly $5 \\%$ of the KITTI training dataset). During the second stage, we retrain the network on the subsampled dataset (from scratch to avoid the previous local optimum). As this subsampled dataset is de-biased, the network learns better depth estimates on dynamic objects. This process can be repeated, but we find that two stages are enough to remove any traces of infinite depth in our experiments, as shown in Figure 3. ",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5.1 DATASETS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We use the standard KITTI benchmark (Geiger et al., 2013) for self-supervised training and evaluation. More specifically, we adopt the training, validation and test splits used in Eigen et al. (2014) with the pre-processing from Zhou et al. (2017) to remove static frames, which is more suitable for ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/fadcee4f355440bacfe89f4e2c7a0a5a4bb1ad8ded3e322dc5351cda42632e1c.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">Superv.</td><td colspan=\"4\">Lower is Better</td><td colspan=\"3\">Higher is Better</td></tr><tr><td>Abs Rel</td><td>Sq Rel</td><td>RMSE</td><td>RMSEtog</td><td>δ&lt;1.25δ&lt;1.25²</td><td></td><td>δ&lt;1.253</td></tr><tr><td>Garg et al. (2016)</td><td>M</td><td>0.152</td><td>1.226</td><td>5.849</td><td>0.246</td><td>0.784</td><td>0.921</td><td>0.967</td></tr><tr><td>Zou et al. (2018)</td><td>M</td><td>0.150</td><td>1.124</td><td>5.507</td><td>0.223</td><td>0.806</td><td>0.933</td><td>0.973</td></tr><tr><td>Godard et al. (2017)</td><td>M</td><td>0.141</td><td>1.186</td><td>5.677</td><td>0.238</td><td>0.809</td><td>0.928</td><td>0.969</td></tr><tr><td>Zhan et al. (2018)</td><td>M</td><td>0.135</td><td>1.132</td><td>5.585</td><td>0.229</td><td>0.820</td><td>0.933</td><td>0.971</td></tr><tr><td>Godard et al. (2018) (R18)</td><td>M</td><td>0.115</td><td>0.903</td><td>4.863</td><td>0.193</td><td>0.877</td><td>0.959</td><td>0.981</td></tr><tr><td>Godard et al. (2018) (R50)</td><td>M</td><td>0.112</td><td>0.851</td><td>4.754</td><td>0.190</td><td>0.881</td><td>0.960</td><td>0.981</td></tr><tr><td>Guizilini et al. (2019) (MR)</td><td>M</td><td>0.108</td><td>0.727</td><td>4.426</td><td>0.184</td><td>0.885</td><td>0.963</td><td>0.983</td></tr><tr><td>Guizilini et al. (2019) (HR)</td><td>M</td><td>0.104</td><td>0.758</td><td>4.386</td><td>0.182</td><td>0.895</td><td>0.964</td><td>0.982</td></tr><tr><td>Casser et al. (2019)</td><td>S+Inst</td><td>0.141</td><td>1.025</td><td>5.290</td><td>0.215</td><td>0.816</td><td>0.945</td><td>0.979</td></tr><tr><td>Chen et al. (2019)</td><td>S+Sem</td><td>0.118</td><td>0.905</td><td>5.096</td><td>0.211</td><td>0.839</td><td>0.945</td><td>0.977</td></tr><tr><td>Ochs et al. (2019)</td><td>D+Sem</td><td>0.116</td><td>0.945</td><td>4.916</td><td>0.208</td><td>0.861</td><td>0.952</td><td>0.968</td></tr><tr><td>Ours (MR)</td><td>M+Sem</td><td>0.102</td><td>0.698</td><td>4.381</td><td>0.178</td><td>0.896</td><td>0.964</td><td>0.984</td></tr><tr><td>Ours (HR)</td><td>M+Sem</td><td>0.100</td><td>0.761</td><td>4.270</td><td>0.175</td><td>0.902</td><td>0.965</td><td>0.982</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "Table 1: Quantitative performance comparison of our proposed architecture on KITTI for depths up to $8 0 \\mathrm { m }$ . $M$ refers to methods that train using monocular images, S refers to methods that train using stereo pairs, $D$ refers to methods that use ground-truth depth supervision, Sem refers to methods that include semantic information, and Inst refers to methods that include semantic and instance information. $M R$ indicates $6 4 0 \\times 1 9 2$ input images, and $H R$ indicates $1 2 8 0 \\mathrm { ~ x ~ } 3 8 4$ input images. Our proposed architecture is able to further improve the current state of the art in self-supervised monocular depth estimation, and outperforms other methods that exploit semantic information (including ground truth labels) by a substantial margin. ",
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+    {
+        "type": "text",
+        "text": "monocular self-supervised learning. This results in 39810 images for training, 4424 for validation, and 697 for evaluation. Following common practice, we pretrain our depth and pose networks on the CityScapes dataset (Cordts et al., 2016), consisting of 88250 unlabeled images. Unless noted otherwise, input images are downsampled to $6 4 0 \\times 1 9 2$ resolution and output inverse depth maps are upsampled to full resolution using bilinear interpolation. Our fixed semantic segmentation network is pretrained on Cityscapes, achieving a mIoU of $7 5 \\%$ on the validation set. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.2 IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We implement our models with PyTorch (Paszke et al., 2017) and follow the same training protocols of Guizilini et al. (2019) when optimizing our depth and pose networks. The initial training stage is conducted on the CityScapes dataset for 50 epochs, with a batch size of 4 per GPU and initial depth and pose learning rates of $2 \\cdot 1 0 ^ { - 4 }$ and $5 \\cdot { \\bar { 1 } } 0 ^ { - 4 }$ respectively, that are halved every 20 epochs. Afterwards, the depth and pose networks are fine-tuned on KITTI for 30 epochs, with the same parameters and halving the learning rates after every 12 epochs. This fine-tuning stage includes the proposed architecture, where information from the fixed semantic network, pretrained separately, is used to directly guide the generation of depth features. There is no direct supervision at any stage during depth training, all semantic information is derived from the fixed secondary network. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "When pretraining the semantic segmentation network, we use a ResNet-50 backbone with Imagenet (Deng et al., 2009) pretrained weights and optimize the network for $4 8 k$ iterations on the CityScapes dataset with a learning rate of 0.01, momentum of 0.9, weight decay of $1 0 ^ { - 4 }$ , and a batch size of 1 per GPU. Random scaling between (0.7, 1.3), random horizontal flipping, and a crop size of $1 0 0 0 \\times 2 0 0 0$ are used for data augmentation. We decay the learning rate by a factor of 10 at iterations $3 6 k$ and $4 4 k$ . Once training is complete, the semantic segmentation network is fixed and becomes the only source of semantic information when fine-tuning the depth and pose networks on KITTI. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.3 DEPTH ESTIMATION PERFORMANCE ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Our depth estimation results are summarized in Table 1, where we compare our proposed architecture with other published works. From these results we can see that the introduction of semanticallyguided geometric representation learning further improves upon the current state of the art in selfsupervised monocular depth estimation from Guizilini et al. (2019), which served as our baseline. Our approach also outperforms other methods that leverage semantic information by a substantial margin, even those using ground-truth KITTI semantic segmentation and depth labels during train",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/b3112d78d10a26174e02c3bc30a24a88918102e8b993042d38659c646c8ad47a.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Network</td><td rowspan=\"2\">SEM</td><td rowspan=\"2\">TST</td><td colspan=\"4\">Lower is Better</td><td colspan=\"2\">Higher is Better δ &lt;1.25 δ&lt;1.252 δ&lt; 1.253</td><td rowspan=\"2\">Class-Avg. Abs Rel</td></tr><tr><td>Abs Rel</td><td>Sq Rel RMSE RMSElog</td><td></td><td></td><td></td><td></td></tr><tr><td>ResNet-18</td><td>√</td><td></td><td>0.120 0.117</td><td>0.896 0.854</td><td>4.869 4.714</td><td>0.198 0.191</td><td>0.868 0.957 0.873</td><td>0.981 0.963 0.981</td><td>0.149 0.139</td></tr><tr><td>ResNet-50</td><td>√</td><td></td><td>0.117 0.113</td><td>0.900 0.831</td><td>4.826 4.663</td><td>0.196 0.189</td><td>0.873 0.967 0.878 0.971</td><td>0.980 0.983</td><td>0.144 0.136</td></tr><tr><td>PackNet</td><td>√</td><td>√</td><td>0.108 0.103 0.102</td><td>0.727 0.710 0.698</td><td>4.426 4.301 4.381</td><td>0.184 0.179 0.178</td><td>0.885 0.963 0.895 0.964 0.896 0.963</td><td>0.983 0.984 0.984</td><td>0.132 0.121 0.117</td></tr></table>",
+        "bbox": [
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+            101,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 2: Ablative analysis of our semantic guidance (SEM) and two-stage-training (TST) contributions. The last column indicates class-average Abs. Rel. obtained by averaging all class-specific depth errors in Figure 4, while other columns indicate pixel-average metrics. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "ing (Ochs et al., 2019). Furthermore, in Figure 5 we also present qualitative results showing the improvements in depth estimation generated by our proposed framework, compared to our baseline. Note how our semantically-guided architecture produces sharper boundaries and better object delineation, especially in structures further away or not clearly distinguishable in the input image. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.4 ABLATIVE ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.4.1 DIFFERENT DEPTH NETWORKS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To better evaluate our main contribution, we provide an ablative analysis showing how it generalizes to different depth networks. To this end, we consider two variations of the widely used ResNet architecture as the encoder for our depth network: ResNet-18 and ResNet-50 (the same pretrained semantic network was used in all experiments). Depth estimation results considering these variations are shown in Table 2, where we can see that our proposed semantically-guided architecture is able to consistently improve the performance of different depth networks, for all considered metrics. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.4.2 CLASS-SPECIFIC DEPTH PERFORMANCE",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To further showcase the benefits of our semantically-guided architecture, we also provide classspecific evaluation metrics, as shown in Figure 4. As we do not have ground-truth semantic segmentation for these images, we use the prediction of the semantic network to bin pixels per predicted category, and evaluate only on those pixels. From these results we can see that our proposed architecture consistently improves depth performance for pixels across all predicted classes, especially those containing fine-grained structures and sharp boundaries, e.g. poles and traffic signs. ",
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+    {
+        "type": "image",
+        "img_path": "images/029bba1944c05b59fd31cd1b470c4d6e1b61b263ba0128d9dbf7c9034beaab1f.jpg",
+        "image_caption": [
+            "Figure 4: Class-specific depth evaluation for our proposed architecture (blue), relative to our baseline (red). The rightmost column indicates class-average depth metrics, obtained by averaging all individual classes. The introduction of semantically-guided features, in conjunction with our proposed two-stage training methodology to address the infinite depth problem, consistently improved depth results for all considered classes (lower is better). "
+        ],
+        "image_footnote": [],
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+        "image_caption": [
+            "Figure 5: Qualitative results of our proposed architecture. The left, middle, and right columns show respectively input images, baseline predicted depth maps (Guizilini et al., 2019), and the depths maps obtained using our proposed architecture. Our semantic-aware depth network predicts sharper boundaries and fine-grained details on distant objects. The dotted lines indicate class-average errors, obtained by averaging all the class-specific depth errors. "
+        ],
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "We also measure the impact of our two-stage training process, which is expected to address the infinite depth problem in dynamic objects. Although we find the pixel-average difference in performance to not be significant (see Table 2), there is a significant improvement in class-average depth estimation, from 0.121 to 0.117 Abs-Rel. This is because the number of pixels affected by the infinite depth problem is vastly smaller than the total number of pixels. However, when considering class-average depth evaluation, the improvement over classes such as cars (0.200 to 0.177 Abs-Rel) and motorcycles (0.091 to 0.069) becomes statistically significant. This further exemplifies the importance of fine-grained metrics in depth evaluation, so these underlying behaviors can be properly observed and accounted for in the development of new techniques. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "This paper introduces a novel architecture for self-supervised monocular depth estimation that leverages semantic information from a fixed pretrained network to guide the generation of multi-level depth features via pixel-adaptive convolutions. Our monodepth network learns semantic-aware geometric representations that can disambiguate photometric ambiguities in a self-supervised learning structure-from-motion context. Furthermore, we introduce a two-stage training process that resamples training data to overcome a common bias on dynamic objects resulting in predicting them at infinite depths. Our experiments on challenging real-world data shows that our proposed architecture consistently improves the performance of different monodepth architectures, thus establishing a new state of the art in self-supervised monocular depth estimation. Future directions of research include leveraging other sources of guidance (i.e. instance masks, optical flow, surface normals), as well as avenues for self-supervised fine-tuning of the semantic network. ",
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+        "text": "A PRE-TRAINING THE SEMANTIC SEGMENTATION NETWORK ",
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+        "text": "The introduction of a semantic segmentation network to the depth estimation task increases the depth estimation performance, however it also increases model complexity (e.g. number of trainable parameters). To investigate that the increased performance for the depth estimation task is indeed due to the semantic features encoded in the secondary network, we perform an in-depth analysis (summarized in Table 3) where we explore the impact of pre-training the semantic segmentation network before it is used to guide the generation of depth features. From these results we can see that the presence of semantic information encoded in the secondary network indeed leads to an increase in performance, and that fine-tuning this secondary network for the speficic task of depth estimation actually decreases performance. ",
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+        "type": "text",
+        "text": "In the first two rows an untrained semantic network is utilized, with only its encoder initialized from ImageNet (Deng et al., 2009) weights. Two different scenarios are explored: in the first one (D) only the depth network is fine-tuned in a self-supervised fashion, while in $\\mathbf { D + S }$ both networks are fine-tuned together in the same way. As expected, using untrained features as guidance leads to significantly worse results, since there is no structure encoded in the secondary network and the primary network needs to learn to filter out all this spurious information. When both networks are fine-tuned simultaneously, results improve because now the added complexity from the secondary network can be leveraged for the task of depth estimation, however there is still no improvement over the baseline. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Next, the semantic network was pre-trained on only half of the CityScapes (Cordts et al., 2016) dataset (samples chosen randomly), leading to a worse semantic segmentation performance (validation mIoU of around $7 0 \\%$ vs. $\\dot { 7 } 5 \\%$ for the fully trained one). This partial pre-training stage was enough to enable the transfer of useful information between networks, leading to improvements over the baseline. Interestingly, fine-tuning both networks for the task of depth estimation actually hurt performance this time, which we attribute to forgetting the information contained in the secondary network, as both networks are optimized for the depth task. When the semantic network is pretrained with all of CityScapes (last two rows), these effects are magnified, with fine-tuning only the depth network leading to our best reported performance (Table 1) and fine-tuning both networks again leading to results similar to the baseline. ",
+        "bbox": [
+            174,
+            397,
+            825,
+            536
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+    },
+    {
+        "type": "table",
+        "img_path": "images/39674a6eb79324a71abaab952b2fa2a5c0e792c6121c26ae828662acaccea15b.jpg",
+        "table_caption": [
+            "Table 3: Analysis of the impact of pre-training the semantic segmentation network. On the PreTrain column, $I$ indicates ImageNet (Deng et al., 2009) pretraining and CS indicates CityScapes (Cordts et al., 2016) pretraining, with $1 / 2$ indicating the use of only half the dataset (samples chosen randomly). In the Fine-Tune column, $D$ indicates fine-tuning the depth network and $s$ indicates finetuning the semantic network (note that this is a self-supervised fine-tuning for the depth task, using the objective described in Section 3.1). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">Pre-Train</td><td rowspan=\"2\">Fine-Tune</td><td colspan=\"4\">Lower is Better</td><td colspan=\"3\">Higher is Better</td></tr><tr><td>Abs Rel</td><td>Sq Rel</td><td>RMSE</td><td>RMSElog</td><td>δ &lt;1.25 δ&lt;1.252</td><td></td><td>δ&lt;1.253</td></tr><tr><td>Baseline</td><td></td><td>D</td><td>0.108</td><td>0.727</td><td>4.426</td><td>0.184</td><td>0.885</td><td>0.963</td><td>0.983</td></tr><tr><td rowspan=\"6\">Proposed</td><td>I</td><td>D+S</td><td>0.116</td><td>0.847</td><td>4.751</td><td>0.192</td><td>0.879</td><td>0.960</td><td>0.981</td></tr><tr><td>I</td><td>D</td><td>0.197</td><td>1.323</td><td>6.114</td><td>0.265</td><td>0.776</td><td>0.918</td><td>0.966</td></tr><tr><td>CS (1/2)</td><td>D+S</td><td>0.109</td><td>0.737</td><td>4.389</td><td>0.185</td><td>0.884</td><td>0.962</td><td>0.982</td></tr><tr><td>CS (1/2)</td><td>D</td><td>0.104</td><td>0.716</td><td>4.322</td><td>0.180</td><td>0.893</td><td>0.964</td><td>0.984</td></tr><tr><td>Cs</td><td>D+S</td><td>0.107</td><td>0.741</td><td>4.407</td><td>0.183</td><td>0.883</td><td>0.963</td><td>0.983</td></tr><tr><td>Cs</td><td>D</td><td>0.102</td><td>0.698</td><td>4.381</td><td>0.178</td><td>0.896</td><td>0.964</td><td>0.984</td></tr></table>",
+        "bbox": [
+            186,
+            549,
+            808,
+            685
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B UNCERTAINTY AND GENERALIZATION TO DIFFERENT OBJECTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            809,
+            740,
+            825
+        ],
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+    {
+        "type": "text",
+        "text": "In a self-supervised setting, increasing the number of unlabeled videos used for depth training is expected to lead to an increasing specialization away from the domain in which the semantic network was pre-trained. This might result in harmful guidance if our method is not robust to this gap. However, our approach does not use semantic predictions directly, but rather the decoded features of the semantic network themselves, which represent general appearance information that should be more robust to this domain gap. To validate our hypothesis, we further explore the impact of erroneous semantic information in the performance of our proposed semantically-guided depth framework. In Figure 6 we present qualitative results highlighting situations in which our pretrained semantic network failed to generate correct semantic predictions for certain objects in the scene, and yet our proposed framework was still able to properly recover depth values for that portion of the environment. These exemplify possible scenarios for erroneous semantic prediction. ",
+        "bbox": [
+            174,
+            840,
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+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            176,
+            103,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• Imprecise boundaries: in the first row, we can see that the semantic segmentation network does not correctly detect the traffic sign, yet the semantically-guided depth network predicts its shape and depth accurately.   \n• Wrong classification: in the second row, the truck was mistakenly classified as partially “road” and “building”, however our semantically-guided depth network was still able to properly recover its overall shape with sharp delineation that was not available from its semantic contour. A similar scenario happens in the same image, with “fence” being partially labeled as “bicycle”.   \n• Missing ontology: there is no “trash can” class on the CityScapes ontology, however in the third row our semantically-guided depth network was able to correctly reconstruct such object even though it was classified as “fence”, similarly to its surroundings.   \n• Object Hallucination: in the fourth row, the contour of a “person” was erroneously introduced in the image and correctly removed by our semantically-guided framework. ",
+        "bbox": [
+            217,
+            184,
+            825,
+            377
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "These examples are evidence that our proposed framework is able to reason over the uncertainty inherent to semantic classification, leveraging this information when accurate to achieve the results reported in this paper, but also discarding it if necessary to generate a better reconstruction according to the self-supervised photometric loss. ",
+        "bbox": [
+            176,
+            388,
+            825,
+            444
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/1571107d15766292cb50733ff1a095273dd604cd102c3c0235ac0a84b6b5dd4e.jpg",
+        "image_caption": [
+            "Figure 6: Examples of erroneous semantic predictions that still led to accurate depth predictions using our proposed semantically-guided depth framework. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            469,
+            820,
+            699
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C GENERALIZATION TO DIFFERENT DATASETS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            753,
+            578,
+            770
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In the previous sections, we show that our proposed framework is robust to a degraded semantic network, both by pretraining the semantic network with fewer annotated labels (Appendix A) and also by providing evidence that the depth network is able to reason over erroneous predictions to still generate accurate reconstructions (Appendix B). We now go one step further and analyze how our proposed semantically-guided framework generalizes to a dataset that was used neither during pre-training nor for fine-tuning. To this end, we evaluate our KITTI depth model on the recently released NuScenes dataset (Caesar et al., 2019). The official NuScenes validation split is used, containing 6019 images from the front camera with ground-truth depth maps generated by LiDAR reprojection. Results presented in Table 4 provide additional evidence that our method indeed results in generalization improvements, even on significantly different data from different platforms ",
+        "bbox": [
+            173,
+            785,
+            825,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/bec1078c5287f03bdd3c650a6e65b61fa474c55eb675fac2632764ab582173d1.jpg",
+        "table_caption": [
+            "and environments (Karlsruhe, Germany for KITTI vs Boston, USA and Singapore for NuScenes), outperforming the state of the art methods and our baseline (Guizilini et al., 2019). ",
+            "Table 4: Generalization capability of different networks, trained on both KITTI and CityScapes datasets and evaluated on the NuScenes (Caesar et al., 2019) dataset. Our proposed semanticallyguided architecture is able to further improve upon the baseline from Guizilini et al. (2019), which only used unlabeled image sequences for self-supervised depth training. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Abs Rel</td><td>Sq Rel</td><td>RMSE</td><td>RMSElog</td><td></td><td>δ&lt;1.25δ&lt;1.25²</td><td>δ&lt;1.253</td></tr><tr><td>Godard et al. (2018) (R18)</td><td>0.212</td><td>1.918</td><td>7.958</td><td>0.323</td><td>0.674</td><td>0.898</td><td>0.954</td></tr><tr><td>Godard et al. (2018) (R50)</td><td>0.210</td><td>2.017</td><td>8.111</td><td>0.328</td><td>0.697</td><td>0.903</td><td>0.960</td></tr><tr><td>Guizilini et al. (2019) (MR)</td><td>0.187</td><td>1.852</td><td>7.636</td><td>0.289</td><td>0.742</td><td>0.917</td><td>0.961</td></tr><tr><td>Ours (MR)</td><td>0.181</td><td>1.505</td><td>7.237</td><td>0.271</td><td>0.765</td><td>0.931</td><td>0.969</td></tr></table>",
+        "bbox": [
+            196,
+            145,
+            795,
+            226
+        ],
+        "page_idx": 13
+    }
+]

parse/train/HkL7n1-0b/HkL7n1-0b.md

@@ -0,0 +1,402 @@
+# WASSERSTEIN AUTO-ENCODERS
+
+Ilya Tolstikhin MPI for Intelligent Systems Tubingen, Germany ¨ ilya@tue.mpg.de
+
+Olivier Bousquet   
+Google Brain   
+Zurich, Switzerland ¨   
+obousquet@google.com   
+Sylvain Gelly   
+Google Brain   
+Zurich, Switzerland¨   
+sylvaingelly@google.com
+
+Bernhard Scholkopf ¨ MPI for Intelligent Systems Tubingen, Germany ¨ bs@tue.mpg.de
+
+# ABSTRACT
+
+We propose the Wasserstein Auto-Encoder (WAE)—a new algorithm for building a generative model of the data distribution. WAE minimizes a penalized form of the Wasserstein distance between the model distribution and the target distribution, which leads to a different regularizer than the one used by the Variational Auto-Encoder (VAE) (Kingma & Welling, 2014). This regularizer encourages the encoded training distribution to match the prior. We compare our algorithm with several other techniques and show that it is a generalization of adversarial auto-encoders (AAE) (Makhzani et al., 2016). Our experiments show that WAE shares many of the properties of VAEs (stable training, encoder-decoder architecture, nice latent manifold structure) while generating samples of better quality, as measured by the FID score.
+
+# 1 INTRODUCTION
+
+The field of representation learning was initially driven by supervised approaches, with impressive results using large labelled datasets. Unsupervised generative modeling, in contrast, used to be a domain governed by probabilistic approaches focusing on low-dimensional data. Recent years have seen a convergence of those two approaches. In the new field that formed at the intersection, variational auto-encoders (VAEs) (Kingma & Welling, 2014) constitute one well-established approach, theoretically elegant yet with the drawback that they tend to generate blurry samples when applied to natural images. In contrast, generative adversarial networks (GANs) (Goodfellow et al., 2014) turned out to be more impressive in terms of the visual quality of images sampled from the model, but come without an encoder, have been reported harder to train, and suffer from the “mode collapse” problem where the resulting model is unable to capture all the variability in the true data distribution. There has been a flurry of activity in assaying numerous configurations of GANs as well as combinations of VAEs and GANs. A unifying framework combining the best of GANs and VAEs in a principled way is yet to be discovered.
+
+This work builds up on the theoretical analysis presented in Bousquet et al. (2017). Following Arjovsky et al. (2017); Bousquet et al. (2017), we approach generative modeling from the optimal transport (OT) point of view. The OT cost (Villani, 2003) is a way to measure a distance between probability distributions and provides a much weaker topology than many others, including $f$ -divergences associated with the original GAN algorithms (Nowozin et al., 2016). This is particularly important in applications, where data is usually supported on low dimensional manifolds in the input space $\mathcal { X }$ . As a result, stronger notions of distances (such as $f$ -divergences, which capture the density ratio between distributions) often max out, providing no useful gradients for training. In contrast, OT was claimed to have a nicer behaviour (Arjovsky et al., 2017; Gulrajani et al., 2017) although it requires, in its GAN-like implementation, the addition of a constraint or a regularization term into the objective.
+
+![](images/dfbfcc5b3c4f8ecaf5bcf4d077a22aa437f7aa7791df942b76efddc7ab3c527b.jpg)  
+Figure 1: Both VAE and WAE minimize two terms: the reconstruction cost and the regularizer penalizing discrepancy between $P _ { Z }$ and distribution induced by the encoder $Q$ . VAE forces $Q ( Z | X = x )$ to match $P _ { Z }$ for all the different input examples $x$ drawn from $P _ { X }$ . This is illustrated on picture (a), where every single red ball is forced to match $P _ { Z }$ depicted as the white shape. Red balls start intersecting, which leads to problems with reconstruction. In contrast, WAE forces the continuous mixture $\begin{array} { r } { \dot { Q _ { Z } } : = \int Q ( Z | X ) \dot { d } P _ { X } } \end{array}$ to match $P _ { Z }$ , as depicted with the green ball in picture (b). As a result latent codes of different examples get a chance to stay far away from each other, promoting a better reconstruction.
+
+In this work we aim at minimizing OT $W _ { c } ( P _ { X } , P _ { G } )$ between the true (but unknown) data distribution $P _ { X }$ and a latent variable model $P _ { G }$ specified by the prior distribution $P _ { Z }$ of latent codes $Z \in { \mathcal { Z } }$ and the generative model $P _ { G } ( X | Z )$ of the data points $X \in { \mathcal { X } }$ given $Z$ . Our main contributions are listed below (cf. also Figure 1):
+
+• A new family of regularized auto-encoders (Algorithms 1, 2 and Eq. 4), which we call Wasserstein Auto-Encoders (WAE), that minimize the optimal transport $W _ { c } ( P _ { X } , P _ { G } )$ for any cost function $c$ . Similarly to VAE, the objective of WAE is composed of two terms: the $c$ -reconstruction cost and a regularizer $\mathcal { D } _ { Z } ( P _ { Z } , Q _ { Z } )$ penalizing a discrepancy between two distributions in $\mathcal { Z }$ : $P _ { Z }$ and a distribution of encoded data points, i.e. $Q _ { Z } \backslash = \dot { \mathbb { E } } _ { P _ { X } } [ Q ( Z | X ) ]$ . When $c$ is the squared cost and $\mathcal { D } _ { Z }$ is the GAN objective, WAE coincides with adversarial auto-encoders of Makhzani et al. (2016). Empirical evaluation of WAE on MNIST and CelebA datasets with squared cost $c ( x , y ) =$ $\| x - y \| _ { 2 } ^ { 2 }$ . Our experiments show that WAE keeps the good properties of VAEs (stable training, encoder-decoder architecture, and a nice latent manifold structure) while generating samples of better quality, approaching those of GANs.   
+• We propose and examine two different regularizers $\mathcal { D } _ { Z } ( P _ { Z } , Q _ { Z } )$ . One is based on GANs and adversarial training in the latent space $\mathcal { Z }$ . The other uses the maximum mean discrepancy, which is known to perform well when matching high-dimensional standard normal distributions $P _ { Z }$ (Gretton et al., 2012). Importantly, the second option leads to a fully adversary-free min-min optimization problem.   
+• Finally, the theoretical considerations presented in Bousquet et al. (2017) and used here to derive the WAE objective might be interesting in their own right. In particular, Theorem 1 shows that in the case of generative models, the primal form of $W _ { c } ( P _ { X } , P _ { G } )$ is equivalent to a problem involving the optimization of a probabilistic encoder $Q ( Z | X )$ .
+
+The paper is structured as follows. In Section 2 we review a novel auto-encoder formulation for OT between $P _ { X }$ and the latent variable model $P _ { G }$ derived in Bousquet et al. (2017). Relaxing the resulting constrained optimization problem we arrive at an objective of Wasserstein auto-encoders. We propose two different regularizers, leading to WAE-GAN and WAE-MMD algorithms. Section 3 discusses the related work. We present the experimental results in Section 4 and conclude by pointing out some promising directions for future work.
+
+# 2 PROPOSED METHOD
+
+Our new method minimizes the optimal transport cost $W _ { c } ( P _ { X } , P _ { G } )$ based on the novel auto-encoder formulation (see Theorem 1 below). In the resulting optimization problem the decoder tries to accurately reconstruct the encoded training examples as measured by the cost function $c$ . The encoder tries to simultaneously achieve two conflicting goals: it tries to match the encoded distribution of training examples ${ \dot { Q } } _ { Z } : = \mathbb { E } _ { P _ { X } } [ Q ( Z | X ) ]$ to the prior $P _ { Z }$ as measured by any specified divergence $\mathcal { D } _ { Z } ( Q _ { Z } , P _ { Z } )$ , while making sure that the latent codes provided to the decoder are informative enough to reconstruct the encoded training examples. This is schematically depicted on Fig. 1.
+
+# 2.1 PRELIMINARIES AND NOTATIONS
+
+We use calligraphic letters (i.e. $\mathcal { X } )$ for sets, capital letters (i.e. $X )$ for random variables, and lower case letters (i.e. $x _ { \mathrm { . } }$ ) for their values. We denote probability distributions with capital letters (i.e. $P ( X ) )$ and corresponding densities with lower case letters (i.e. $p ( x ) )$ ). In this work we will consider several measures of discrepancy between probability distributions $P _ { X }$ and $P _ { G }$ . The class of $f$ -d e & Miescke, 2008) is defined b $\begin{array} { r } { D _ { f } ( P _ { X } \| P _ { G } ) : = \int f \big ( \frac { p _ { X } ( x ) } { p _ { G } ( x ) } \big ) p _ { G } ( x ) d x } \end{array}$ $f \colon ( 0 , \infty ) \to { \mathcal { R } }$ $f ( 1 ) = 0$   
+Kullback-Leibler $D _ { \mathrm { K L } }$ and Jensen-Shannon $D _ { \mathrm { J S } }$ divergences.
+
+# 2.2 OPTIMAL TRANSPORT AND ITS DUAL FORMULATIONS
+
+A rich class of divergences between probability distributions is induced by the optimal transport (OT) problem (Villani, 2003). Kantorovich’s formulation of the problem is given by
+
+$$
+W _ { c } ( P _ { X } , P _ { G } ) : = \operatorname* { i n f } _ { \Gamma \in { \mathcal { P } } ( X \sim P _ { X } , Y \sim P _ { G } ) } \mathbb { E } _ { ( X , Y ) \sim \Gamma } [ c ( X , Y ) ] ,
+$$
+
+where $c ( x , y ) \colon \mathcal { X } \times \mathcal { X } \to \mathcal { R } _ { + }$ is any measurable cost function and ${ \mathcal { P } } ( X \sim P _ { X } , Y \sim P _ { G } )$ is a set of all joint distributions of $( X , Y )$ with marginals $P _ { X }$ and $P _ { G }$ respectively. A particularly interesting case is when $( \mathcal { X } , d )$ is a metric space and $c ( x , y ) = d ^ { p } ( x , y )$ for $p \geq 1$ . In this case $W _ { p }$ , the $p$ -th root of $W _ { c }$ , is called the $p$ -Wasserstein distance.
+
+When $c ( x , y ) = d ( x , y )$ the following Kantorovich-Rubinstein duality holds1:
+
+$$
+W _ { 1 } ( P _ { X } , P _ { G } ) = \operatorname* { s u p } _ { f \in { \mathcal { F } } _ { L } } \mathbb { E } _ { X \sim P _ { X } } [ f ( X ) ] - \mathbb { E } _ { Y \sim P _ { G } } [ f ( Y ) ] ,
+$$
+
+where $\mathcal { F } _ { L }$ is the class of all bounded 1-Lipschitz functions on $( \mathcal { X } , d )$
+
+# 2.3 APPLICATION TO GENERATIVE MODELS: WASSERSTEIN AUTO-ENCODERS
+
+One way to look at modern generative models like VAEs and GANs is to postulate that they are trying to minimize certain discrepancy measures between the data distribution $P _ { X }$ and the model $P _ { G }$ . Unfortunately, most of the standard divergences known in the literature, including those listed above, are hard or even impossible to compute, especially when $P _ { X }$ is unknown and $P _ { G }$ is parametrized by deep neural networks. Previous research provides several tricks to address this issue.
+
+In case of minimizing the KL-divergence $D _ { \mathrm { K L } } ( P _ { X } , P _ { G } )$ , or equivalently maximizing the marginal log-likelihood $E _ { P _ { X } } [ \bar { \log { p _ { G } ( X ) } } ]$ , the famous variational lower bound provides a theoretically grounded framework successfully employed by VAEs (Kingma & Welling, 2014; Mescheder et al., 2017). More generally, if the goal is to minimize the $f$ -divergence $D _ { f } ( P _ { X } , P _ { G } )$ (with one example being $D _ { \mathrm { K L } } ,$ ), one can resort to its dual formulation and make use of $f$ -GANs and the adversarial training (Nowozin et al., 2016). Finally, OT cost $W _ { c } ( P _ { X } , P _ { G } )$ is yet another option, which can be, thanks to the celebrated Kantorovich-Rubinstein duality (2), expressed as an adversarial objective as implemented by the Wasserstein-GAN (Arjovsky et al., 2017). We include an extended review of all these methods in Supplementary A.
+
+In this work we will focus on latent variable models $P _ { G }$ defined by a two-step procedure, where first a code $Z$ is sampled from a fixed distribution $P _ { Z }$ on a latent space $\mathcal { Z }$ and then $Z$ is mapped to the image $X \in \mathcal { X } \overset { = } { = } \mathcal { R } ^ { d }$ with a (possibly random) transformation. This results in a density of the form
+
+$$
+p _ { G } ( x ) : = \int _ { \mathcal { Z } } p _ { G } ( x | \boldsymbol { z } ) p _ { z } ( \boldsymbol { z } ) d \boldsymbol { z } , \quad \forall \boldsymbol { x } \in \mathcal { X } ,
+$$
+
+assuming all involved densities are properly defined. For simplicity we will focus on non-random decoders, i.e. generative models $P _ { G } ( X | Z )$ deterministically mapping $Z$ to $X = G ( Z )$ for a given map $G \colon { \mathcal { Z } } \to { \mathcal { X } }$ . Similar results for random decoders can be found in Supplementary B.1.
+
+It turns out that under this model, the OT cost takes a simpler form as the transportation plan factors through the map $G$ : instead of finding a coupling $\Gamma$ in (1) between two random variables living in the $\mathcal { X }$ space, one distributed according to $P _ { X }$ and the other one according to $P _ { G }$ , it is sufficient to find a conditional distribution $Q ( Z | X )$ such that its $Z$ marginal $Q _ { Z } ( Z ) \setminus : = \mathbb { E } _ { X \sim P _ { X } } \left[ Q ( Z | X ) \right]$ is identical to the prior distribution $P _ { Z }$ . This is the content of the theorem below proved in Bousquet et al. (2017). To make this paper self contained we repeat the proof in Supplementary B.
+
+Theorem 1 For $P _ { G }$ as defined above with deterministic $P _ { G } ( X | Z )$ and any function $G \colon { \mathcal { Z } } \to { \mathcal { X } }$
+
+$$
+\operatorname* { i n f } _ { \substack { \Gamma \in \mathcal { P } ( X \sim P _ { X } , Y \sim P _ { G } ) } } \mathbb { E } _ { ( X , Y ) \sim \Gamma } \left[ c \big ( X , Y \big ) \right] = \operatorname* { i n f } _ { Q \colon Q z = P _ { Z } } \mathbb { E } _ { P _ { X } } \mathbb { E } _ { Q ( Z | X ) } \left[ c \big ( X , G ( Z ) \big ) \right] ,
+$$
+
+where $Q _ { Z }$ is the marginal distribution of $Z$ when $X \sim P _ { X }$ and $Z \sim Q ( Z | X )$
+
+This result allows us to optimize over random encoders $Q ( Z | X )$ instead of optimizing over all couplings between $X$ and $Y$ . Of course, both problems are still constrained. In order to implement a numerical solution we relax the constraints on $Q _ { Z }$ by adding a penalty to the objective. This finally leads us to the WAE objective:
+
+$$
+D _ { \mathrm { W A E } } ( P _ { X } , P _ { G } ) : = \operatorname* { i n f } _ { Q ( Z | X ) \in \mathcal { Q } } \mathbb { E } _ { P _ { X } } \mathbb { E } _ { Q ( Z | X ) } \left[ c \bigl ( X , G ( Z ) \bigr ) \right] + \lambda \cdot \mathcal { D } _ { Z } ( Q _ { Z } , P _ { Z } ) ,
+$$
+
+where $\mathcal { Q }$ is any nonparametric set of probabilistic encoders, $\mathcal { D } _ { Z }$ is an arbitrary divergence between $Q _ { Z }$ and $P _ { Z }$ , and $\lambda > 0$ is a hyperparameter. Similarly to VAE, we propose to use deep neural networks to parametrize both encoders $Q$ and decoders $G$ . Note that as opposed to VAEs, the WAE formulation allows for non-random encoders deterministically mapping inputs to their latent codes.
+
+We propose two different penalties $\mathcal { D } _ { Z } ( Q _ { Z } , P _ { Z } )$
+
+GAN-based $\mathcal { D } _ { Z }$ . The first option is to choose $\mathcal { D } _ { Z } ( Q _ { Z } , P _ { Z } ) = D _ { \mathrm { J S } } ( Q _ { Z } , P _ { Z } )$ and use the adversarial training to estimate it. Specifically, we introduce an adversary (discriminator) in the latent space $\mathcal { Z }$ trying to separate2 “true” points sampled from $P _ { Z }$ and “fake” ones sampled from $Q _ { Z }$ (Goodfellow et al., 2014). This results in the WAE-GAN described in Algorithm 1. Even though WAE-GAN falls back to the min-max problem, we move the adversary from the input (pixel) space $\mathcal { X }$ to the latent space $\mathcal { Z }$ . On top of that, $P _ { Z }$ may have a nice shape with a single mode (for a Gaussian prior), in which case the task should be easier than matching an unknown, complex, and possibly multi-modal distributions as usually done in GANs. This is also a reason for our second penalty:
+
+MMD-based $\mathcal { D } _ { Z }$ . For a positive-definite reproducing kernel $k \colon \mathcal { Z } \times \mathcal { Z }  \mathcal { R }$ the following expression is called the maximum mean discrepancy (MMD):
+
+$$
+\mathrm { M M D } _ { k } ( P _ { Z } , Q _ { Z } ) = \big \lVert \int _ { \mathcal { Z } } k ( z , \cdot ) d P _ { Z } ( z ) - \int _ { \mathcal { Z } } k ( z , \cdot ) d Q _ { Z } ( z ) \big \rVert _ { \mathcal { H } _ { k } } ,
+$$
+
+where $\mathcal { H } _ { k }$ is the RKHS of real-valued functions mapping $\mathcal { Z }$ to $\mathcal { R }$ . If $k$ is characteristic then ${ \mathrm { M M D } } _ { k }$ defines a metric and can be used as a divergence measure. We propose to use $\mathcal { D } _ { Z } ( P _ { Z } , Q _ { Z } ) =$ $\mathrm { M M D } _ { k } ( P _ { Z } , Q _ { Z } )$ . Fortunately, MMD has an unbiased U-statistic estimator, which can be used in conjunction with stochastic gradient descent (SGD) methods. This results in the WAE-MMD described in Algorithm 2. It is well known that the maximum mean discrepancy performs well when matching high-dimensional standard normal distributions (Gretton et al., 2012) so we expect this penalty to work especially well working with the Gaussian prior $P _ { Z }$ .
+
+<table><tr><td>ALGORITHM1Wasserstein Auto-Encoder with GAN-based penalty (WAE-GAN).</td><td>ALGORITHM 2Wasserstein Auto-Encoder withMMD-based penalty (WAE-MMD).</td></tr><tr><td>Require:Regularization coefficient 入&gt;0. Initialize the parameters of the encoder Q, decoder Gθ,and latent discriminator Dy. while(,θ) not converged do Sample {x1,...,xn} from the training set Sample {z1,...,zn} from the prior Pz</td><td>Require:Regularization coefficient入&gt; O,char- acteristic positive-definite kernel k. Initialize the parameters of the encoder Q𝜙, decoder Gθ,and latent discriminator D. while (,0) not converged do</td></tr><tr><td>Sample zi from Q(Z|xi) fori=1,...,n Update D, by ascending:</td><td>Sample {x1,...,xn} from the training set Sample {z1,..., zn} from the prior Pz Sample z from Q(Z|xi) fori=1,...,n</td></tr><tr><td>入 n ? logDγ(zi)+log(1-Dγ(i)) n i=1</td><td>Update Q and Ge by descending: 1 n 入</td></tr><tr><td>Update Q and Ge by descending: 1 n c(xi,Ge(zi))-λ·logDγ(zi) n</td><td>c(xi,Gθ(zi)） &gt; k(ze,zj） n(n-1) n i=1 +j</td></tr><tr><td>i=1 end while</td><td>入 2入 &gt; k（2e，） + &gt; k(ze,2j） n</td></tr><tr><td></td><td>n(n-1) l≠j e,j</td></tr></table>
+
+We point out once again that the encoders $Q _ { \phi } ( Z | x )$ in Algorithms 1 and 2 can be non-random, i.e. deterministically mapping input points to the latent codes. In this case $Q _ { \phi } ( Z | x ) = \delta _ { \mu _ { \phi } ( x ) }$ fo r a function $\mu _ { \phi } \colon \mathcal { X } \to \mathcal { Z }$ and in order to sample $\tilde { z } _ { i }$ from $Q _ { \phi } ( Z | x _ { i } )$ we just need to return $\mu _ { \phi } ( x _ { i } )$ .
+
+# 3 RELATED WORK
+
+Literature on auto-encoders Classical unregularized auto-encoders minimize only the reconstruction cost. This results in different training points being encoded into non-overlapping zones chaotically scattered all across the $\mathcal { Z }$ space with “holes” in between where the decoder mapping $P _ { G } ( X | Z )$ has never been trained. Overall, the encoder $Q ( Z | X )$ trained in this way does not provide a useful representation and sampling from the latent space $\mathcal { Z }$ becomes hard (Bengio et al., 2013).
+
+Variational auto-encoders (Kingma & Welling, 2014) minimize a variational bound on the KLdivergence $D _ { \mathrm { K L } } ( P _ { X } , P _ { G } )$ which is composed of the reconstruction cost plus the regularizer $\mathbb { E } _ { P _ { X } } \left[ D _ { \mathrm { K L } } ( Q ( Z | X ) , P _ { Z } ) \right]$ . The regularizer captures how distinct the image by the encoder of each training example is from the prior $P _ { Z }$ , which is not guaranteeing that the overall encoded distribution $\mathbb { E } _ { P _ { X } }$ $\mathinner { \langle { Q \mathopen { \left( Z \vert X \right) } } \rvert }$ matches $P _ { Z }$ like WAE does. Also, VAEs require non-degenerate (i.e. nondeterministic) Gaussian encoders and random decoders for which the term $\log p _ { G } ( x | z )$ can be computed and differentiated with respect to the parameters. Later Mescheder et al. (2017) proposed a way to use VAE with non-Gaussian encoders. WAE minimizes the optimal transport $W _ { c } ( P _ { X } , P _ { G } )$ and allows both probabilistic and deterministic encoder-decoder pairs of any kind.
+
+The VAE regularizer can be also equivalently written (Hoffman & Johnson, 2016) as a sum of $D _ { \mathrm { K L } } ( Q _ { Z } , P _ { Z } )$ and a mutual information $\mathbb { I } _ { Q } ( X , Z )$ between the images $X$ and latent codes $Z$ jointly distributed according to $P _ { X } \times Q ( Z | X )$ . This observation provides another intuitive way to explain a difference between our algorithm and VAEs: WAEs simply drop the mutual information term $\mathbb { I } _ { Q } ( X , Z )$ in the VAE regularizer.
+
+When used with $c ( x , y ) = \| x - y \| _ { 2 } ^ { 2 }$ WAE-GAN is equivalent to adversarial auto-encoders (AAE) proposed by Makhzani et al. (2016). Theory of Bousquet et al. (2017) (and in particular Theorem 1) thus suggests that AAEs minimize the 2-Wasserstein distance between $P _ { X }$ and $P _ { G }$ . This provides the first theoretical justification for AAEs known to the authors. WAE generalizes AAE in two ways: first, it can use any cost function $c$ in the input space $\mathcal { X }$ ; second, it can use any discrepancy measure $\mathcal { D } _ { Z }$ in the latent space $\mathcal { Z }$ (for instance MMD), not necessarily the adversarial one of WAE-GAN.
+
+Finally, Zhao et al. (2017b) independently proposed a regularized auto-encoder objective similar to Bousquet et al. (2017) and our (4) based on very different motivations and arguments. Following
+
+VAEs their objective (called InfoVAE) defines the reconstruction cost in the image space implicitly through the negative log likelihood term $- \log p _ { G } ( x | z )$ , which should be properly normalized for all $z \in { \mathcal { Z } }$ . In theory VAE and InfoVAE can both induce arbitrary cost functions, however in practice this may require an estimation of the normalizing constant (partition function) which can3 be different for different values of $z$ . WAEs specify the cost $c ( x , y )$ explicitly and don’t constrain it in any way.
+
+Literature on OT Genevay et al. (2016) address computing the OT cost in large scale using SGD and sampling. They approach this task either through the dual formulation, or via a regularized version of the primal. They do not discuss any implications for generative modeling. Our approach is based on the primal form of OT, we arrive at regularizers which are very different, and our main focus is on generative modeling.
+
+The WGAN (Arjovsky et al., 2017) minimizes the 1-Wasserstein distance $W _ { 1 } ( P _ { X } , P _ { G } )$ for generative modeling. The authors approach this task from the dual form. Their algorithm comes without an encoder and can not be readily applied to any other cost $W _ { c }$ , because the neat form of the Kantorovich-Rubinstein duality (2) holds only for $W _ { 1 }$ . WAE approaches the same problem from the primal form, can be applied for any cost function $c$ , and comes naturally with an encoder.
+
+In order to compute the values (1) or (2) of OT we need to handle non-trivial constraints, either on the coupling distribution $\Gamma$ or on the function $f$ being considered. Various approaches have been proposed in the literature to circumvent this difficulty. For $W _ { 1 }$ Arjovsky et al. (2017) tried to implement the constraint in the dual formulation (2) by clipping the weights of the neural network $f$ . Later Gulrajani et al. (2017) proposed to relax the same constraint by penalizing the objective of (2) with a term $\lambda \cdot \mathbb { E } \left( \left\| \nabla f ( X ) \right\| - 1 \right) ^ { 2 }$ which should not be greater than 1 if $f \in \mathcal { F } _ { L }$ . In a more general OT setting of $W _ { c }$ Cuturi (2013) proposed to penalize the objective of (1) with the KLdivergence $\lambda \cdot D _ { \mathrm { K L } } ( \Gamma , P \otimes Q )$ between the coupling distribution and the product of marginals. Genevay et al. (2016) showed that this entropic regularization drops the constraints on functions in the dual formulation as opposed to (2). Finally, in the context of unbalanced optimal transport it has been proposed to relax the constraint in (1) by regularizing the objective with $\lambda \cdot \left( D _ { f } ( \Gamma _ { X } , \mathbf { \bar { \Gamma } } ^ { } P ) + \right.$ $D _ { f } ( \Gamma _ { Y } , Q ) )$ (Chizat et al., 2015; Liero et al., 2015), where $\Gamma _ { X }$ and $\Gamma _ { Y }$ are marginals of $\Gamma$ . In this paper we propose to relax OT in a way similar to the unbalanced optimal transport, i.e. by adding additional divergences to the objective. However, we show that in the particular context of generative modeling, only one extra divergence is necessary.
+
+Literature on GANs Many of the GAN variations (including $f$ -GAN and WGAN) come without an encoder. Often it may be desirable to reconstruct the latent codes and use the learned manifold, in which cases these models are not applicable.
+
+There have been many other approaches trying to blend the adversarial training of GANs with autoencoder architectures (Zhao et al., $2 0 1 7 \mathrm { a }$ ; Dumoulin et al., 2017; Ulyanov et al., 2017; Berthelot et al., 2017). The approach proposed by Ulyanov et al. (2017) is perhaps the most relevant to our work. The authors use the discrepancy between $Q _ { Z }$ and the distribution $\mathbb { E } _ { Z ^ { \prime } \sim P _ { Z } } [ Q \left( Z | G ( Z ^ { \prime } ) \right) ]$ of auto-encoded noise vectors as the objective for the max-min game between the encoder and decoder respectively. While the authors showed that the saddle points correspond to $P _ { X } = P _ { G }$ , they admit that encoders and decoders trained in this way have no incentive to be reciprocal. As a workaround they propose to include an additional reconstruction term to the objective. WAE does not necessarily lead to a min-max game, uses a different penalty, and has a clear theoretical foundation.
+
+Several works used reproducing kernels in context of GANs. Li et al. (2015); Dziugaite et al. (2015) use MMD with a fixed kernel $k$ to match $P _ { X }$ and $P _ { G }$ directly in the input space $\mathcal { X }$ . These methods have been criticised to require larger mini-batches during training: estimating $\mathrm { M M D } _ { k } ( P _ { X } , P _ { G } )$ requires number of samples roughly proportional to the dimensionality of the input space $\mathcal { X }$ (Reddi et al., 2015) which is typically larger than $1 0 ^ { 3 }$ . Li et al. (2017) take a similar approach but further train $k$ adversarially so as to arrive at a meaningful loss function. WAE-MMD uses MMD to match $Q _ { Z }$ to the prior $P _ { Z }$ in the latent space $\mathcal { Z }$ . Typically $\mathcal { Z }$ has no more than 100 dimensions and $P _ { Z }$ is Gaussian, which allows us to use regular mini-batch sizes to accurately estimate MMD.
+
+![](images/ba6eb123cbe974ac8bc98667fc71d8053d7e708a6c4e49eac087dd03ed3734c6.jpg)  
+Figure 2: VAE (left column), WAE-MMD (middle column), and WAE-GAN (right column) trained on MNIST dataset. In “test reconstructions” odd rows correspond to the real test points.
+
+# 4 EXPERIMENTS
+
+In this section we empirically evaluate4 the proposed WAE model. We would like to test if WAE can simultaneously achieve (i) accurate reconstructions of data points, (ii) reasonable geometry of the latent manifold, and (iii) random samples of good (visual) quality. Importantly, the model should generalize well: requirements (i) and (ii) should be met on both training and test data. We trained WAE-GAN and WAE-MMD (Algorithms 1 and 2) on two real-world datasets: MNIST (LeCun et al., 1998) consisting of $7 0 \mathrm { k }$ images and CelebA (Liu et al., 2015) containing roughly 203k images.
+
+Experimental setup In all reported experiments we used Euclidian latent spaces $\mathcal { Z } \ = \ \mathcal { R } ^ { d _ { z } }$ for various $d _ { z }$ depending on the complexity of the dataset, isotropic Gaussian prior distributions $P _ { Z } ( Z ) = \mathcal { N } ( Z ; \mathbf { 0 } , \sigma _ { z } ^ { 2 } \cdot I _ { d } )$ over $\mathcal { Z }$ , and a squared cost function $c ( \bar { x } , y ) = \| x - \bar { y } \| _ { 2 } ^ { 2 }$ for data points $x , y \in \mathcal { X } = \mathcal { R } ^ { d _ { x } }$ . We used deterministic encoder-decoder pairs, Adam (Kingma & Lei, 2014) with $\beta _ { 1 } = 0 . 5 , \beta _ { 2 } = 0 . 9 9 9$ , and convolutional deep neural network architectures for encoder mapping $\mu _ { \phi } \colon \mathcal { X } \to \mathcal { Z }$ and decoder mapping $G _ { \theta } \colon { \mathcal { Z } } \to { \mathcal { X } }$ similar to the DCGAN ones reported by Radford et al. (2016) with batch normalization (Ioffe & Szegedy, 2015). We tried various values of $\lambda$ and noticed that $\lambda = 1 0$ seems to work good across all datasets we considered.
+
+Since we are using deterministic encoders, choosing $d _ { z }$ larger than intrinsic dimensionality of the dataset would force the encoded distribution $Q _ { Z }$ to live on a manifold in $\mathcal { Z }$ . This would make matching $Q _ { Z }$ to $P _ { Z }$ impossible if $P _ { Z }$ is Gaussian and may lead to numerical instabilities. We use $d _ { z } = 8$ for MNIST and $d _ { z } = 6 4$ for CelebA which seems to work reasonably well.
+
+![](images/5f9154f809ae2eb7b1b03217d0ca3922df4a01f90d50874364f5a3fe74e9b6da.jpg)  
+Figure 3: VAE (left column), WAE-MMD (middle column), and WAE-GAN (right column) trained on CelebA dataset. In “test reconstructions” odd rows correspond to the real test points.
+
+We also report results of VAEs. VAEs used the same latent spaces as discussed above and standard Gaussian priors $P _ { Z } = \mathcal { N } ( \mathbf { 0 } , I _ { d } )$ . We used Gaussian encoders $Q ( Z | X ) = \mathcal { N } \big ( Z ; \mu _ { \phi } ( X ) , \Sigma ( X ) \big )$ with mean $\mu _ { \phi }$ and diagonal covariance $\Sigma$ . For both MNIST and CelebA we used Bernoulli decoders parametrized by $G _ { \theta }$ . Functions $\mu _ { \phi } , \Sigma$ , and $G _ { \theta }$ were parametrized by deep nets of the same architectures as used in WAE.
+
+WAE-GAN and WAE-MMD specifics In WAE-GAN we used discriminator $D$ composed of several fully connected layers with ReLu. We tried WAE-MMD with the RBF kernel but observed that it fails to penalize the outliers of $Q _ { Z }$ because of the quick tail decay. If the codes $\tilde { z } = \mu _ { \phi } ( x )$ for some of the training points $x \in \mathcal { X }$ end up far away from the support of $P _ { Z }$ (which may happen in the early stages of training) the corresponding terms in the U-statistic $k ( z , \tilde { z } ) = e ^ { - \| \tilde { z } - z \| _ { 2 } ^ { 2 } / \sigma _ { k } ^ { 2 } }$ will quickly approach zero and provide no gradient for those outliers. This could be avoided by choosing the kernel bandwidth $\sigma _ { k } ^ { 2 }$ in a data-dependent manner, however in this case per-minibatch U-statistic would not provide an unbiased estimate for the gradient. Instead, we used the inverse multiquadratics kernel $\bar { k } ( x , y ) = C / ( C + \| x - y \| _ { 2 } ^ { 2 } )$ which is also characteristic and has much heavier tails. In all experiments we used $C = 2 d _ { z } \sigma _ { z } ^ { 2 }$ , which is the expected squared distance between two multivariate Gaussian vectors drawn from $P _ { Z }$ . This significantly improved the performance compared to the RBF kernel (even the one with $\sigma _ { k } ^ { 2 } = 2 d _ { z } \sigma _ { z } ^ { 2 }$ ). Trained models are presented in Figures 2 and 3. Further details are presented in Supplementary C.
+
+Random samples are generated by sampling $P _ { Z }$ and decoding the resulting noise vectors $z$ into $G _ { \theta } ( z )$ . As expected, in our experiments we observed that for both WAE-GAN and WAE-MMD the quality of samples strongly depends on how accurately $Q _ { Z }$ matches $P _ { Z }$ . To see this, notice that during training the decoder function $G _ { \theta }$ is presented only with encoded versions $\mu _ { \phi } ( X )$ of the data points $X \sim P _ { X }$ . Indeed, the decoder is trained on samples from $Q _ { Z }$ and thus there is no reason to expect good results when feeding it with samples from $P _ { Z }$ . In our experiments we noticed that even slight differences between $Q _ { Z }$ and $P _ { Z }$ may affect the quality of samples. In some cases WAE-GAN seems to lead to a better matching and generates better samples than WAE-MMD. However, due to adversarial training WAE-GAN is highly unstable, while WAE-MMD has a very stable training much like VAE.
+
+In order to quantitatively assess the quality of the generated images, we use the Frechet Inception Distance´ introduced by Heusel et al. (2017) and report the results on CelebA in Table 1. These results confirm that the sampled images from WAE are of better quality than from VAE, and WAE-GAN gets a slightly better score than WAEMMD, which correlates with visual inspection of the images.
+
+Table 1: FID scores for samples on CelebA (smaller is better).   
+
+<table><tr><td>Algorithm</td><td>FID</td></tr><tr><td>VAE</td><td>82</td></tr><tr><td>WAE-MMD</td><td>55</td></tr><tr><td>WAE-GAN</td><td>42</td></tr></table>
+
+Test reconstructions and interpolations. We take random points $x$ from the held out test set and report their auto-encoded versions $G _ { \theta } ( \mu _ { \phi } ( x ) )$ . Next, pairs $( x , y )$ of different data points are sampled randomly from the held out test set and encoded: $z _ { x } = \mu _ { \phi } ( x )$ , $z _ { y } = \mu _ { \phi } ( y )$ . We linearly interpolate between $z _ { x }$ and $z _ { y }$ with equally-sized steps in the latent space and show decoded images.
+
+# 5 CONCLUSION
+
+Using the optimal transport cost, we have derived Wasserstein auto-encoders—a new family of algorithms for building generative models. We discussed their relations to other probabilistic modeling techniques. We conducted experiments using two particular implementations of the proposed method, showing that in comparison to VAEs, the images sampled from the trained WAE models are of better quality, without compromising the stability of training and the quality of reconstruction. Future work will include further exploration of the criteria for matching the encoded distribution $Q _ { Z }$ to the prior distribution $P _ { Z }$ , assaying the possibility of adversarially training the cost function $c$ in the input space $\mathcal { X }$ , and a theoretical analysis of the dual formulations for WAE-GAN and WAE-MMD.
+
+# ACKNOWLEDGMENTS
+
+The authors are thankful to Carl Johann Simon-Gabriel, Mateo Rojas-Carulla, Arthur Gretton, Paul Rubenstein, and Fei Sha for stimulating discussions.
+
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+
+# A IMPLICIT GENERATIVE MODELS: A SHORT TOUR OF GANS AND VAES
+
+Even though GANs and VAEs are quite different—both in terms of the conceptual frameworks and empirical performance—they share important features: (a) both can be trained by sampling from the model $P _ { G }$ without knowing an analytical form of its density and (b) both can be scaled up with SGD. As a result, it becomes possible to use highly flexible implicit models $P _ { G }$ defined by a twostep procedure, where first a code $Z$ is sampled from a fixed distribution $P _ { Z }$ on a latent space $\mathcal { Z }$ and then $Z$ is mapped to the image $G ( Z ) \in \mathcal { X } = \mathcal { R } ^ { d }$ with a (possibly random) transformation $G \colon { \mathcal { Z } } \to { \mathcal { X } }$ . This results in latent variable models $P _ { G }$ of the form (3).
+
+These models are indeed easy to sample and, provided $G$ can be differentiated analytically with respect to its parameters, $P _ { G }$ can be trained with SGD. The field is growing rapidly and numerous variations of VAEs and GANs are available in the literature. Next we introduce and compare several of them.
+
+The original generative adversarial network (GAN) Goodfellow et al. (2014) approach minimizes
+
+$$
+D _ { \mathrm { G A N } } ( P _ { X } , P _ { G } ) = \operatorname* { s u p } _ { T \in { \mathcal T } } \mathbb { E } _ { X \sim P _ { X } } [ \log T ( X ) ] + \mathbb { E } _ { Z \sim P _ { Z } } [ \log \left( 1 - T ( G ( Z ) ) \right) ]
+$$
+
+with respect to a deterministic decoder $G \colon { \mathcal { Z } } \to { \mathcal { X } }$ , where $\tau$ is any non-parametric class of choice. It is known that $D _ { \mathrm { G A N } } ( P _ { X } , P _ { G } ) \leq 2 \cdot D _ { \mathrm { J S } } ( P _ { X } , P _ { G } ) - \log ( 4 )$ and the inequality turns into identity in the nonparametric limit, that is when the class $\tau$ becomes rich enough to represent all functions mapping $\mathcal { X }$ to $( 0 , 1 )$ . Hence, GANs are minimizing a lower bound on the JS-divergence. However, GANs are not only linked to the JS-divergence: the $f$ -GAN approach Nowozin et al. (2016) showed that a slight modification $D _ { \mathrm { f , G A N } }$ of the objective (5) allows to lower bound any desired $f$ -divergence in a similar way. In practice, both decoder $G$ and discriminator $T$ are trained in alternating SGD steps. Stopping criteria as well as adequate evaluation of the trained GAN models remain open questions.
+
+Recently, the authors of Arjovsky et al. (2017) argued that the 1-Wasserstein distance $W _ { 1 }$ , which is known to induce a much weaker topology than $D _ { \mathrm { J S } }$ , may be better suited for generative modeling. When $P _ { X }$ and $P _ { G }$ are supported on largely disjoint low-dimensional manifolds (which may be the case in applications), $D _ { \mathrm { K L } } , D _ { \mathrm { J S } }$ , and other strong distances between $P _ { X }$ and $P _ { G }$ max out and no longer provide useful gradients for $P _ { G }$ . This “vanishing gradient” problem necessitates complicated scheduling between the $G / T$ updates. In contrast, $W _ { 1 }$ is still sensible in these cases and provides stable gradients. The Wasserstein GAN (WGAN) minimizes
+
+$$
+D _ { \mathrm { W G A N } } ( P _ { X } , P _ { G } ) = \operatorname* { s u p } _ { T \in \mathcal { W } } \mathbb { E } _ { X \sim P _ { X } } [ T ( X ) ] - \mathbb { E } _ { Z \sim P _ { Z } } \left[ T ( G ( Z ) ) \right] ,
+$$
+
+where $\mathcal { W }$ is any subset of 1-Lipschitz functions on $\mathcal { X }$ . It follows from (2) that $D _ { \mathrm { W G A N } } ( P _ { X } , P _ { G } ) \leq$ $W _ { 1 } ( P _ { X } , P _ { G } )$ and thus WGAN is minimizing a lower bound on the 1-Wasserstein distance.
+
+Variational auto-encoders (VAE) Kingma & Welling (2014) utilize models $P _ { G }$ of the form (3) and minimize
+
+$$
+D _ { \mathrm { V A E } } \big ( P _ { X } , P _ { G } \big ) = \operatorname* { i n f } _ { Q ( Z | X ) \in \mathcal { Q } } \mathbb { E } _ { P _ { X } } \left[ D _ { \mathrm { K L } } \big ( Q ( Z | X ) , P _ { Z } \big ) - \mathbb { E } _ { Q ( Z | X ) } [ \log p _ { G } ( X | Z ) ] \right]
+$$
+
+with respect to a random decoder mapping $P _ { G } ( X | Z )$ . The conditional distribution $P _ { G } ( X | Z )$ is often parametrized by a deep net $G$ and can have any form as long as its density $p _ { G } ( x | z )$ can be computed and differentiated with respect to the parameters of $G$ . A typical choice is to use Gaussians $P _ { G } ( X | Z ) = \mathcal { N } ( X ; G ( Z ) , \sigma ^ { \hat { 2 } } \cdot I )$ . If $\mathcal { Q }$ is the set of all conditional probability distributions $Q ( Z | X )$ , the objective of VAE coincides with the negative marginal log-likelihood $D _ { \mathrm { V A E } } ( P _ { X } , P _ { G } ) = - \mathbb { E } _ { P _ { X } } [ \log P _ { G } ( X ) ]$ . However, in order to make the $D _ { \mathrm { K L } }$ term of (6) tractable in closed form, the original implementation of VAE uses a standard normal $P _ { Z }$ and restricts $\mathcal { Q }$ to a class of Gaussian distributions $Q ( Z | X ) = { \mathcal { N } } { \big ( } Z ; \mu ( X ) , \Sigma ( X ) { \big ) }$ with mean $\mu$ and diagonal covariance $\Sigma$ parametrized by deep nets. As a consequence, VAE is minimizing an upper bound on the negative log-likelihood or, equivalently, on the KL-divergence $D _ { \mathrm { K L } } ( P _ { X } , P _ { G } )$ .
+
+One possible way to reduce the gap between the true negative log-likelihood and the upper bound provided by $D _ { \mathrm { V A E } }$ is to enlarge the class $\mathcal { Q }$ . Adversarial variational Bayes (AVB) Mescheder et al. (2017) follows this argument by employing the idea of GANs. Given any point $x \in \mathcal { X }$ , a noise $\epsilon \sim \mathcal { N } ( 0 , 1 )$ , and any fixed transformation $e \colon \mathcal { X } \times \mathcal { R }  \mathcal { Z }$ , a random variable $e ( x , \epsilon )$
+
+implicitly defines one particular conditional distribution $Q _ { e } ( Z | X = x )$ . AVB allows $\mathcal { Q }$ to contain all such distributions for different choices of $e$ , replaces the intractable term $D _ { \mathrm { K L } } \big ( Q _ { e } ( Z | X ) , P _ { Z } \big )$ in (6) by the adversarial approximation $D _ { \mathrm { f , G A N } }$ corresponding to the KL-divergence, and proposes to minimize5
+
+$$
+D _ { \mathrm { A V B } } ( P _ { X } , P _ { G } ) = \operatorname* { i n f } _ { Q _ { \epsilon } ( Z | X ) \in \mathcal { Q } } \mathbb { E } _ { P _ { X } } \left[ D _ { \mathrm { f } , \operatorname { G A N } } \big ( Q _ { e } ( Z | X ) , P _ { Z } \big ) - \mathbb { E } _ { Q _ { \epsilon } ( Z | X ) } [ \log p _ { G } ( X | Z ) ] \right] .
+$$
+
+The $D _ { \mathrm { K L } }$ term in (6) may be viewed as a regularizer. Indeed, VAE reduces to the classical unregularized auto-encoder if this term is dropped, minimizing the reconstruction cost of the encoder-decoder pair $Q ( Z | X ) , P _ { G } ( X | Z )$ . This often results in different training points being encoded into nonoverlapping zones chaotically scattered all across the $\mathcal { Z }$ space with “holes” in between where the decoder mapping $P _ { G } ( X | Z )$ has never been trained. Overall, the encoder $Q ( Z | X )$ trained in this way does not provide a useful representation and sampling from the latent space $\mathcal { Z }$ becomes hard Bengio et al. (2013).
+
+Adversarial auto-encoders (AAE) Makhzani et al. (2016) replace the $D _ { \mathrm { K L } }$ term in (6) with another regularizer:
+
+$$
+D _ { \mathrm { A A E } } ( P _ { X } , P _ { G } ) = \operatorname* { i n f } _ { Q ( Z | X ) \in Q } D _ { \mathrm { G A N } } ( Q _ { Z } , P _ { Z } ) - \mathbb { E } _ { P _ { X } } \mathbb { E } _ { Q ( Z | X ) } [ \log p _ { G } ( X | Z ) ] ,
+$$
+
+where $Q _ { Z }$ is the marginal distribution of $Z$ when first $X$ is sampled from $P _ { X }$ and then $Z$ is sampled from $Q ( Z | X )$ , also known as the aggregated posterior Makhzani et al. (2016). Similarly to AVB, there is no clear link to log-likelihood, as $D _ { \mathrm { A A E } } \leq D _ { \mathrm { A V B } }$ . The authors of Makhzani et al. (2016) argue that matching $Q _ { Z }$ to $P _ { Z }$ in this way ensures that there are no “holes” left in the latent space $\mathcal { Z }$ and $P _ { G } ( X | Z )$ generates reasonable samples whenever $Z \sim P _ { Z }$ . They also report an equally good performance of different types of conditional distributions $Q ( Z | X )$ , including Gaussians as used in VAEs, implicit models $Q _ { e }$ as used in AVB, and deterministic encoder mappings, i.e. $Q ( Z | X ) =$ $\delta _ { \mu ( X ) }$ with $\mu \colon \mathcal X \to \mathcal Z$ .
+
+# B PROOF OF THEOREM 1 AND FURTHER DETAILS
+
+We will consider certain sets of joint probability distributions of three random variables $( X , Y , Z ) \in$ $\mathcal { X } \times \mathcal { X } \times \mathcal { Z }$ . The reader may wish to think of $X$ as true images, $Y$ as images sampled from the model, and $Z$ as latent codes. We denote by $P _ { G , Z } ( Y , Z )$ a joint distribution of a variable pair $( Y , Z )$ , where $Z$ is first sampled from $P _ { Z }$ and next $Y$ from $P _ { G } ( Y | Z )$ . Note that $P _ { G }$ defined in (3) and used throughout this work is the marginal distribution of $Y$ when $( Y , Z ) \sim P _ { G , Z }$ .
+
+In the optimal transport problem (1), we consider joint distributions $\Gamma ( X , Y )$ which are called couplings between values of $X$ and $Y$ . Because of the marginal constraint, we can write $\Gamma ( X , Y ) =$ $\bar { \Gamma } ( \bar { Y ( X ) } P _ { X } ( X )$ and we can consider $\Gamma ( Y | X )$ as a non-deterministic mapping from $X$ to $Y$ . Theorem 1. shows how to factor this mapping through $\mathcal { Z }$ , i.e., decompose it into an encoding distribution $Q ( Z | X )$ and the generating distribution $P _ { G } ( Y | Z )$ .
+
+As in Section 2.2, $\mathcal { P } ( X \sim P _ { X } , Y \sim P _ { G } )$ denotes the set of all joint distributions of $( X , Y )$ with marginals $P _ { X } , P _ { G }$ , and likewise for $\mathcal { P } ( X \sim P _ { X } , Z \sim P _ { Z } )$ . The set of all joint distributions of $( X , Y , Z )$ such that $X \sim P _ { X }$ , $( Y , Z ) \sim P _ { G , Z }$ , and $( Y \perp \perp X ) | Z$ will be denoted by ${ \mathcal { P } } _ { X , Y , Z }$ . Finally, we denote by $\mathcal { P } _ { X , Y }$ and $\mathcal { P } _ { X , Z }$ the sets of marginals on $( X , Y )$ and $( X , Z )$ (respectively) induced by distributions in ${ \mathcal { P } } _ { X , Y , Z }$ . Note that $\mathcal { P } ( P _ { X } , P _ { G } )$ , ${ \mathcal { P } } _ { X , Y , Z }$ , and $\mathcal { P } _ { X , Y }$ depend on the choice of conditional distributions $P _ { G } ( Y | Z )$ , while $\mathcal { P } _ { X , Z }$ does not. In fact, it is easy to check that $\mathcal { P } _ { X , Z } =$ $\mathcal { P } ( X \sim P _ { X } , Z \sim P _ { Z } )$ . From the definitions it is clear that $\mathcal { P } _ { X , Y } \subseteq \mathcal { P } ( P _ { X } , P _ { G } )$ and we immediately get the following upper bound:
+
+$$
+W _ { c } ( P _ { X } , P _ { G } ) \leq W _ { c } ^ { \dagger } ( P _ { X } , P _ { G } ) : = \operatorname* { i n f } _ { P \in \mathcal { P } _ { X , Y } } \mathbb { E } _ { ( X , Y ) \sim P } \left[ c ( X , Y ) \right] .
+$$
+
+If $P _ { G } ( Y | Z )$ are Dirac measures (i.e., $Y = G ( Z ) )$ , it turns out that $\mathcal { P } _ { X , Y } = \mathcal { P } ( P _ { X } , P _ { G } )$ :
+
+Lemma 1 $\mathcal { P } _ { X , Y } \subseteq \mathcal { P } ( P _ { X } , P _ { G } )$ with identity $i f ^ { 6 } ~ P _ { G } ( Y | Z = z )$ are Dirac for all $z \in { \mathcal { Z } }$ .
+
+Proof The first assertion is obvious. To prove the identity, note that when $Y$ is a deterministic function of $Z$ , for any $A$ in the sigma-algebra induced by $Y$ we have $\mathbb { E } \left[ \mathbb { 1 } _ { [ Y \in A ] } | X , Z \right] = \mathbb { E } \left[ \mathbb { 1 } _ { [ Y \in A ] } | Z \right]$ . This implies $( Y \perp \perp X ) | Z$ and concludes the proof.
+
+We are now in place to prove Theorem 1. Lemma 1 obviously leads to
+
+$$
+W _ { c } ( P _ { X } , P _ { G } ) = W _ { c } ^ { \dagger } ( P _ { X } , P _ { G } ) .
+$$
+
+The tower rule of expectation, and the conditional independence property of ${ \mathcal { P } } _ { X , Y , Z }$ implies
+
+$$
+\begin{array} { r l } & { W _ { c } ^ { \dagger } ( P _ { X } , P _ { G } ) = \underset { P \in \mathcal { P } _ { X , Y , Z } } { \operatorname* { i n f } } \mathbb { E } _ { ( X , Y , Z ) \sim P } \left[ c ( X , Y ) \right] } \\ & { \quad \quad \quad = \underset { P \in \mathcal { P } _ { X , Y , Z } } { \operatorname* { i n f } } \mathbb { E } _ { P _ { Z } } \mathbb { E } _ { X \sim P ( X \mid Z ) } \mathbb { E } _ { Y \sim P ( Y \mid Z ) } [ c ( X , Y ) ] } \\ & { \quad \quad \quad = \underset { P \in \mathcal { P } _ { X , Y , Z } } { \operatorname* { i n f } } \mathbb { E } _ { P _ { Z } } \mathbb { E } _ { X \sim P ( X \mid Z ) } \left[ c \left( X , G ( Z ) \right) \right] } \\ & { \quad \quad \quad = \underset { P \in \mathcal { P } _ { X , Z } } { \operatorname* { i n f } } \mathbb { E } _ { ( X , Z ) \sim P } \left[ c \left( X , G ( Z ) \right) \right] . } \end{array}
+$$
+
+It remains to notice that $\mathcal { P } _ { X , Z } = \mathcal { P } ( X \sim P _ { X } , Z \sim P _ { Z } )$ as stated earlier.
+
+# B.1 RANDOM DECODERS $P _ { G } ( Y | Z )$
+
+If the decoders are non-deterministic, Lemma 1 provides only the inclusion of sets $\mathcal { P } _ { X , Y } \subseteq \mathcal { P } ( P _ { X } , P _ { G } )$ and we get the following upper bound on the OT:
+
+Corollary 1 Let $\mathcal { X } ~ = ~ \mathcal { R } ^ { d }$ and assume the conditional distributions $P _ { G } ( Y | Z = z )$ have mean values $G ( z ) \in \mathcal { R } ^ { d }$ and marginal variances $\sigma _ { 1 } ^ { 2 } , \ldots , \sigma _ { d } ^ { 2 } \geq 0$ for all $z \in { \mathcal { Z } }$ , where $G \colon { \mathcal { Z } } \to { \mathcal { X } }$ . Take $c ( x , y ) = \| x - y \| _ { 2 } ^ { 2 }$ . Then
+
+$$
+W _ { c } ( P _ { X } , P _ { G } ) \leq W _ { c } ^ { \dagger } ( P _ { X } , P _ { G } ) = \sum _ { i = 1 } ^ { d } \sigma _ { i } ^ { 2 } + \operatorname* { i n f } _ { P \in \mathcal { P } ( X \setminus P _ { X } , Z \sim P _ { Z } ) } \mathbb { E } _ { ( X , Z ) \sim P } \big [ \| X - G ( Z ) \| ^ { 2 } \big ] .
+$$
+
+Proof First inequality follows from (9). For the identity we proceed similarly to the proof of Theorem 1 and write
+
+$$
+W _ { c } ^ { \dagger } ( P _ { X } , P _ { G } ) = \operatorname* { i n f } _ { P \in \mathcal { P } _ { X , Y , Z } } \mathbb { E } _ { P _ { Z } } \mathbb { E } _ { X \sim P ( X \mid Z ) } \mathbb { E } _ { Y \sim P ( Y \mid Z ) } \left[ \left\| X - Y \right\| ^ { 2 } \right] .
+$$
+
+Note that
+
+$$
+\begin{array} { l } { \mathbb { E } _ { Y \sim P ( Y \vert Z ) } \big [ \| X - Y \| ^ { 2 } \big ] = \mathbb { E } _ { Y \sim P ( Y \vert Z ) } \big [ \| X - G ( Z ) + G ( Z ) - Y \| ^ { 2 } \big ] } \\ { = \| X - G ( Z ) \| ^ { 2 } + \mathbb { E } _ { Y \sim P ( Y \vert Z ) } \big [ \langle X - G ( Z ) , G ( Z ) - Y \rangle \big ] + \mathbb { E } _ { Y \sim P ( Y \vert Z ) } \| G ( Z ) - Y \| ^ { 2 } } \\ { = \| X - G ( Z ) \| ^ { 2 } + \displaystyle \sum _ { i = 1 } ^ { d } \sigma _ { i } ^ { 2 } . } \end{array}
+$$
+
+Together with (11) and the fact that $\mathcal { P } _ { X , Z } = \mathcal { P } ( X \sim P _ { X } , Z \sim P _ { Z } )$ this concludes the proof.
+
+# C FURTHER DETAILS ON EXPERIMENTS
+
+# C.1 MNIST
+
+We use mini-batches of size 100 and trained the models for 100 epochs. We used $\lambda = 1 0$ and $\sigma _ { z } ^ { 2 } = 1$ . For the encoder-decoder pair we set $\alpha = 1 0 ^ { - 3 }$ for Adam in the beginning and for the
+
+adversary in WAE-GAN to $\alpha = 5 \times 1 0 ^ { - 4 }$ . After 30 epochs we decreased both by factor of 2, and after first 50 epochs further by factor of 5.
+
+Both encoder and decoder used fully convolutional architectures with 4x4 convolutional filters.
+
+Encoder architecture:
+
+$$
+\begin{array} { r l } { x \in \mathcal { R } ^ { 2 8 \times 2 8 }  \mathrm { C o n v _ { 1 2 8 } }  \mathrm { B N }  \mathrm { R e L U } } & { } \\ {  \mathrm { C o n v _ { 2 5 6 } }  \mathrm { B N }  \mathrm { R e L U } } & { } \\ {  \mathrm { C o n v _ { 5 1 2 } }  \mathrm { B N }  \mathrm { R e L U } } & { } \\ {  \mathrm { C o n v _ { 1 0 2 4 } }  \mathrm { B N }  \mathrm { R e L U }  \mathrm { F C } _ { 8 } } \end{array}
+$$
+
+Decoder architecture:
+
+$$
+\begin{array} { r l } & { z \in \mathcal { R } ^ { 8 }  \mathrm { F C } _ { 7 \times 7 \times 1 0 2 4 } } \\ & { \qquad \mathrm { F S C o n v } _ { 5 1 2 }  \mathrm { B N }  \mathrm { R e L U } } \\ & { \qquad \mathrm { F S C o n v } _ { 2 5 6 }  \mathrm { B N }  \mathrm { R e L U }  \mathrm { F S C o n v } _ { 1 } } \end{array}
+$$
+
+Adversary architecture for WAE-GAN:
+
+$$
+\begin{array} { r l } & { z \in \mathcal { R } ^ { 8 }  \mathrm { F C } _ { 5 1 2 }  \mathrm { R e L U } } \\ & { ~  \mathrm { F C } _ { 5 1 2 }  \mathrm { R e L U } } \\ & { ~  \mathrm { F C } _ { 5 1 2 }  \mathrm { R e L U } } \\ & { ~  \mathrm { F C } _ { 5 1 2 }  \mathrm { R e L U }  \mathrm { F C } _ { 1 } } \end{array}
+$$
+
+Here $\operatorname { C o n v } _ { k }$ stands for a convolution with $k$ filters, $\mathrm { F S C o n v } _ { k }$ for the fractional strided convolution with $k$ filters (first two of them were doubling the resolution, the third one kept it constant), BN for the batch normalization, ReLU for the rectified linear units, and $\mathrm { F C } _ { \mathrm { k } }$ for the fully connected layer mapping to $\mathcal { R } ^ { k }$ . All the convolutions in the encoder used vertical and horizontal strides 2 and SAME padding.
+
+Finally, we used two heuristics. First, we always pretrained separately the encoder for several minibatch steps before the main training stage so that the sample mean and covariance of $Q _ { Z }$ would try to match those of $P _ { Z }$ . Second, while training we were adding a pixel-wise Gaussian noise truncated at 0.01 to all the images before feeding them to the encoder, which was meant to make the encoders random. We played with all possible ways of combining these two heuristics and noticed that together they result in slightly (almost negligibly) better results compared to using only one or none of them.
+
+Our VAE model used cross-entropy loss (Bernoulli decoder) and otherwise same architectures and hyperparameters as listed above.
+
+# C.2 CELEBA
+
+We pre-processed CelebA images by first taking a $1 4 0 \mathrm { x } 1 4 0$ center crops and then resizing to the $6 4 \mathrm { x } 6 4$ resolution. We used mini-batches of size 100 and trained the models for various number of epochs (up to 250). All reported WAE models were trained for 55 epochs and VAE for 68 epochs. For WAE-MMD we used $\lambda = 1 0 0$ and for WAE-GAN $\lambda = 1$ . Both used $\sigma _ { z } ^ { 2 } = 2$ .
+
+For WAE-MMD the learning rate of Adam was initially set to $\alpha = 1 0 ^ { - 3 }$ . For WAE-GAN the learning rate of Adam for the encoder-decoder pair was initially set to $\alpha = 3 \times 1 0 ^ { - 4 }$ and for the adversary to $1 0 ^ { - 3 }$ . All learning rates were decreased by factor of 2 after 30 epochs, further by factor of 5 after 50 first epochs, and finally additional factor of 10 after 100 first epochs.
+
+Both encoder and decoder used fully convolutional architectures with $5 \mathrm { x } 5$ convolutional filters.
+
+Encoder architecture:
+
+$$
+\begin{array} { r l } & { x \in \mathcal { R } ^ { 6 4 \times 6 4 \times 3 } \to \mathrm { C o n v _ { 1 2 8 } } \to \mathrm { B N } \to \mathrm { R e L U } } \\ & { \qquad \to \mathrm { C o n v _ { 2 5 6 } } \to \mathrm { B N } \to \mathrm { R e L U } } \\ & { \qquad \to \mathrm { C o n v _ { 5 1 2 } } \to \mathrm { B N } \to \mathrm { R e L U } } \\ & { \qquad \to \mathrm { C o n v _ { 1 0 2 4 } } \to \mathrm { B N } \to \mathrm { R e L U } \to \mathrm { F C } _ { 6 4 } } \end{array}
+$$
+
+Decoder architecture:
+
+$$
+\begin{array} { r l } { z \in \mathcal { R } ^ { 6 4 }  \mathrm { F C } _ { 8 \times 8 \times 1 0 2 4 } } & { } \\ {  \mathrm { F S C o n v } _ { 5 1 2 }  \mathrm { B N }  \mathrm { R e L U } } \\ {  \mathrm { F S C o n v } _ { 2 5 6 }  \mathrm { B N }  \mathrm { R e L U } } & { } \\ {  \mathrm { F S C o n v } _ { 1 2 8 }  \mathrm { B N }  \mathrm { R e L U }  \mathrm { F S C o n v } _ { 1 } } \end{array}
+$$
+
+Adversary architecture for WAE-GAN:
+
+$$
+\begin{array} { r l } { z \in \mathcal { R } ^ { 6 4 } \to \mathrm { F C } _ { 5 1 2 } \to \mathrm { R e L U } } \\ { \to \mathrm { F C } _ { 5 1 2 } \to \mathrm { R e L U } } \\ { \to \mathrm { F C } _ { 5 1 2 } \to \mathrm { R e L U } } \\ { \to \mathrm { F C } _ { 5 1 2 } \to \mathrm { R e L U } \to \mathrm { F C } _ { 1 } } \end{array}
+$$
+
+For WAE-GAN we used a heuristic proposed in Supplementary $\mathrm { I V }$ of Mescheder et al. (2017). Notice that the theoretically optimal discriminator would result in $D ^ { * } ( z ) = \log p _ { Z } ( z ) - \log q _ { Z } ( z )$ , where $p _ { Z }$ and $q _ { Z }$ are densities of $P _ { Z }$ and $Q _ { Z }$ respectively. In our experiments we added the log prior $\log p _ { Z } ( z )$ explicitly to the adversary output as we know it analytically. This should hopefully make it easier for the adversary to learn the remaining $Q _ { Z }$ density term.
+
+Our VAE model used a cross-entropy reconstruction loss (Bernoulli decoder) and $\alpha = 1 0 ^ { - 4 }$ as the initial Adam learning rate and the same decay schedule as explained above. Otherwise all the architectures and hyperparameters were as explained above.

parse/train/HkL7n1-0b/HkL7n1-0b_content_list.json

@@ -0,0 +1,1956 @@
+[
+    {
+        "type": "text",
+        "text": "WASSERSTEIN AUTO-ENCODERS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            99,
+            570,
+            121
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ilya Tolstikhin MPI for Intelligent Systems Tubingen, Germany ¨ ilya@tue.mpg.de ",
+        "bbox": [
+            184,
+            145,
+            367,
+            202
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Olivier Bousquet   \nGoogle Brain   \nZurich, Switzerland ¨   \nobousquet@google.com   \nSylvain Gelly   \nGoogle Brain   \nZurich, Switzerland¨   \nsylvaingelly@google.com ",
+        "bbox": [
+            614,
+            145,
+            813,
+            202
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            184,
+            222,
+            411,
+            279
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Bernhard Scholkopf ¨ MPI for Intelligent Systems Tubingen, Germany ¨ bs@tue.mpg.de ",
+        "bbox": [
+            529,
+            222,
+            712,
+            277
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            315,
+            544,
+            330
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We propose the Wasserstein Auto-Encoder (WAE)—a new algorithm for building a generative model of the data distribution. WAE minimizes a penalized form of the Wasserstein distance between the model distribution and the target distribution, which leads to a different regularizer than the one used by the Variational Auto-Encoder (VAE) (Kingma & Welling, 2014). This regularizer encourages the encoded training distribution to match the prior. We compare our algorithm with several other techniques and show that it is a generalization of adversarial auto-encoders (AAE) (Makhzani et al., 2016). Our experiments show that WAE shares many of the properties of VAEs (stable training, encoder-decoder architecture, nice latent manifold structure) while generating samples of better quality, as measured by the FID score. ",
+        "bbox": [
+            233,
+            352,
+            764,
+            505
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            546,
+            336,
+            563
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The field of representation learning was initially driven by supervised approaches, with impressive results using large labelled datasets. Unsupervised generative modeling, in contrast, used to be a domain governed by probabilistic approaches focusing on low-dimensional data. Recent years have seen a convergence of those two approaches. In the new field that formed at the intersection, variational auto-encoders (VAEs) (Kingma & Welling, 2014) constitute one well-established approach, theoretically elegant yet with the drawback that they tend to generate blurry samples when applied to natural images. In contrast, generative adversarial networks (GANs) (Goodfellow et al., 2014) turned out to be more impressive in terms of the visual quality of images sampled from the model, but come without an encoder, have been reported harder to train, and suffer from the “mode collapse” problem where the resulting model is unable to capture all the variability in the true data distribution. There has been a flurry of activity in assaying numerous configurations of GANs as well as combinations of VAEs and GANs. A unifying framework combining the best of GANs and VAEs in a principled way is yet to be discovered. ",
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+        "text": "This work builds up on the theoretical analysis presented in Bousquet et al. (2017). Following Arjovsky et al. (2017); Bousquet et al. (2017), we approach generative modeling from the optimal transport (OT) point of view. The OT cost (Villani, 2003) is a way to measure a distance between probability distributions and provides a much weaker topology than many others, including $f$ -divergences associated with the original GAN algorithms (Nowozin et al., 2016). This is particularly important in applications, where data is usually supported on low dimensional manifolds in the input space $\\mathcal { X }$ . As a result, stronger notions of distances (such as $f$ -divergences, which capture the density ratio between distributions) often max out, providing no useful gradients for training. In contrast, OT was claimed to have a nicer behaviour (Arjovsky et al., 2017; Gulrajani et al., 2017) although it requires, in its GAN-like implementation, the addition of a constraint or a regularization term into the objective. ",
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+            "Figure 1: Both VAE and WAE minimize two terms: the reconstruction cost and the regularizer penalizing discrepancy between $P _ { Z }$ and distribution induced by the encoder $Q$ . VAE forces $Q ( Z | X = x )$ to match $P _ { Z }$ for all the different input examples $x$ drawn from $P _ { X }$ . This is illustrated on picture (a), where every single red ball is forced to match $P _ { Z }$ depicted as the white shape. Red balls start intersecting, which leads to problems with reconstruction. In contrast, WAE forces the continuous mixture $\\begin{array} { r } { \\dot { Q _ { Z } } : = \\int Q ( Z | X ) \\dot { d } P _ { X } } \\end{array}$ to match $P _ { Z }$ , as depicted with the green ball in picture (b). As a result latent codes of different examples get a chance to stay far away from each other, promoting a better reconstruction. "
+        ],
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+        "type": "text",
+        "text": "In this work we aim at minimizing OT $W _ { c } ( P _ { X } , P _ { G } )$ between the true (but unknown) data distribution $P _ { X }$ and a latent variable model $P _ { G }$ specified by the prior distribution $P _ { Z }$ of latent codes $Z \\in { \\mathcal { Z } }$ and the generative model $P _ { G } ( X | Z )$ of the data points $X \\in { \\mathcal { X } }$ given $Z$ . Our main contributions are listed below (cf. also Figure 1): ",
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+        "type": "text",
+        "text": "• A new family of regularized auto-encoders (Algorithms 1, 2 and Eq. 4), which we call Wasserstein Auto-Encoders (WAE), that minimize the optimal transport $W _ { c } ( P _ { X } , P _ { G } )$ for any cost function $c$ . Similarly to VAE, the objective of WAE is composed of two terms: the $c$ -reconstruction cost and a regularizer $\\mathcal { D } _ { Z } ( P _ { Z } , Q _ { Z } )$ penalizing a discrepancy between two distributions in $\\mathcal { Z }$ : $P _ { Z }$ and a distribution of encoded data points, i.e. $Q _ { Z } \\backslash = \\dot { \\mathbb { E } } _ { P _ { X } } [ Q ( Z | X ) ]$ . When $c$ is the squared cost and $\\mathcal { D } _ { Z }$ is the GAN objective, WAE coincides with adversarial auto-encoders of Makhzani et al. (2016). Empirical evaluation of WAE on MNIST and CelebA datasets with squared cost $c ( x , y ) =$ $\\| x - y \\| _ { 2 } ^ { 2 }$ . Our experiments show that WAE keeps the good properties of VAEs (stable training, encoder-decoder architecture, and a nice latent manifold structure) while generating samples of better quality, approaching those of GANs.   \n• We propose and examine two different regularizers $\\mathcal { D } _ { Z } ( P _ { Z } , Q _ { Z } )$ . One is based on GANs and adversarial training in the latent space $\\mathcal { Z }$ . The other uses the maximum mean discrepancy, which is known to perform well when matching high-dimensional standard normal distributions $P _ { Z }$ (Gretton et al., 2012). Importantly, the second option leads to a fully adversary-free min-min optimization problem.   \n• Finally, the theoretical considerations presented in Bousquet et al. (2017) and used here to derive the WAE objective might be interesting in their own right. In particular, Theorem 1 shows that in the case of generative models, the primal form of $W _ { c } ( P _ { X } , P _ { G } )$ is equivalent to a problem involving the optimization of a probabilistic encoder $Q ( Z | X )$ . ",
+        "bbox": [
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+        "type": "text",
+        "text": "The paper is structured as follows. In Section 2 we review a novel auto-encoder formulation for OT between $P _ { X }$ and the latent variable model $P _ { G }$ derived in Bousquet et al. (2017). Relaxing the resulting constrained optimization problem we arrive at an objective of Wasserstein auto-encoders. We propose two different regularizers, leading to WAE-GAN and WAE-MMD algorithms. Section 3 discusses the related work. We present the experimental results in Section 4 and conclude by pointing out some promising directions for future work. ",
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+    {
+        "type": "text",
+        "text": "2 PROPOSED METHOD ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our new method minimizes the optimal transport cost $W _ { c } ( P _ { X } , P _ { G } )$ based on the novel auto-encoder formulation (see Theorem 1 below). In the resulting optimization problem the decoder tries to accurately reconstruct the encoded training examples as measured by the cost function $c$ . The encoder tries to simultaneously achieve two conflicting goals: it tries to match the encoded distribution of training examples ${ \\dot { Q } } _ { Z } : = \\mathbb { E } _ { P _ { X } } [ Q ( Z | X ) ]$ to the prior $P _ { Z }$ as measured by any specified divergence $\\mathcal { D } _ { Z } ( Q _ { Z } , P _ { Z } )$ , while making sure that the latent codes provided to the decoder are informative enough to reconstruct the encoded training examples. This is schematically depicted on Fig. 1. ",
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+    {
+        "type": "text",
+        "text": "2.1 PRELIMINARIES AND NOTATIONS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We use calligraphic letters (i.e. $\\mathcal { X } )$ for sets, capital letters (i.e. $X )$ for random variables, and lower case letters (i.e. $x _ { \\mathrm { . } }$ ) for their values. We denote probability distributions with capital letters (i.e. $P ( X ) )$ and corresponding densities with lower case letters (i.e. $p ( x ) )$ ). In this work we will consider several measures of discrepancy between probability distributions $P _ { X }$ and $P _ { G }$ . The class of $f$ -d e & Miescke, 2008) is defined b $\\begin{array} { r } { D _ { f } ( P _ { X } \\| P _ { G } ) : = \\int f \\big ( \\frac { p _ { X } ( x ) } { p _ { G } ( x ) } \\big ) p _ { G } ( x ) d x } \\end{array}$ $f \\colon ( 0 , \\infty ) \\to { \\mathcal { R } }$ $f ( 1 ) = 0$   \nKullback-Leibler $D _ { \\mathrm { K L } }$ and Jensen-Shannon $D _ { \\mathrm { J S } }$ divergences. ",
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+    {
+        "type": "text",
+        "text": "2.2 OPTIMAL TRANSPORT AND ITS DUAL FORMULATIONS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "A rich class of divergences between probability distributions is induced by the optimal transport (OT) problem (Villani, 2003). Kantorovich’s formulation of the problem is given by ",
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+        "type": "equation",
+        "img_path": "images/d4e9d952b993474bb949c7d0c4312c3cb190b0ca448977e1d3bb429ba4007def.jpg",
+        "text": "$$\nW _ { c } ( P _ { X } , P _ { G } ) : = \\operatorname* { i n f } _ { \\Gamma \\in { \\mathcal { P } } ( X \\sim P _ { X } , Y \\sim P _ { G } ) } \\mathbb { E } _ { ( X , Y ) \\sim \\Gamma } [ c ( X , Y ) ] ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where $c ( x , y ) \\colon \\mathcal { X } \\times \\mathcal { X } \\to \\mathcal { R } _ { + }$ is any measurable cost function and ${ \\mathcal { P } } ( X \\sim P _ { X } , Y \\sim P _ { G } )$ is a set of all joint distributions of $( X , Y )$ with marginals $P _ { X }$ and $P _ { G }$ respectively. A particularly interesting case is when $( \\mathcal { X } , d )$ is a metric space and $c ( x , y ) = d ^ { p } ( x , y )$ for $p \\geq 1$ . In this case $W _ { p }$ , the $p$ -th root of $W _ { c }$ , is called the $p$ -Wasserstein distance. ",
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+    {
+        "type": "text",
+        "text": "When $c ( x , y ) = d ( x , y )$ the following Kantorovich-Rubinstein duality holds1: ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/18b31c033b1150941ed4f04e8e9e413b9559579c3c06661edc65fd560d526be3.jpg",
+        "text": "$$\nW _ { 1 } ( P _ { X } , P _ { G } ) = \\operatorname* { s u p } _ { f \\in { \\mathcal { F } } _ { L } } \\mathbb { E } _ { X \\sim P _ { X } } [ f ( X ) ] - \\mathbb { E } _ { Y \\sim P _ { G } } [ f ( Y ) ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { F } _ { L }$ is the class of all bounded 1-Lipschitz functions on $( \\mathcal { X } , d )$ ",
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+    {
+        "type": "text",
+        "text": "2.3 APPLICATION TO GENERATIVE MODELS: WASSERSTEIN AUTO-ENCODERS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "One way to look at modern generative models like VAEs and GANs is to postulate that they are trying to minimize certain discrepancy measures between the data distribution $P _ { X }$ and the model $P _ { G }$ . Unfortunately, most of the standard divergences known in the literature, including those listed above, are hard or even impossible to compute, especially when $P _ { X }$ is unknown and $P _ { G }$ is parametrized by deep neural networks. Previous research provides several tricks to address this issue. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In case of minimizing the KL-divergence $D _ { \\mathrm { K L } } ( P _ { X } , P _ { G } )$ , or equivalently maximizing the marginal log-likelihood $E _ { P _ { X } } [ \\bar { \\log { p _ { G } ( X ) } } ]$ , the famous variational lower bound provides a theoretically grounded framework successfully employed by VAEs (Kingma & Welling, 2014; Mescheder et al., 2017). More generally, if the goal is to minimize the $f$ -divergence $D _ { f } ( P _ { X } , P _ { G } )$ (with one example being $D _ { \\mathrm { K L } } ,$ ), one can resort to its dual formulation and make use of $f$ -GANs and the adversarial training (Nowozin et al., 2016). Finally, OT cost $W _ { c } ( P _ { X } , P _ { G } )$ is yet another option, which can be, thanks to the celebrated Kantorovich-Rubinstein duality (2), expressed as an adversarial objective as implemented by the Wasserstein-GAN (Arjovsky et al., 2017). We include an extended review of all these methods in Supplementary A. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this work we will focus on latent variable models $P _ { G }$ defined by a two-step procedure, where first a code $Z$ is sampled from a fixed distribution $P _ { Z }$ on a latent space $\\mathcal { Z }$ and then $Z$ is mapped to the image $X \\in \\mathcal { X } \\overset { = } { = } \\mathcal { R } ^ { d }$ with a (possibly random) transformation. This results in a density of the form ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3019688d70fec01fd4fd85529d1fd5d624b67899e63195cca6422579a2a212b2.jpg",
+        "text": "$$\np _ { G } ( x ) : = \\int _ { \\mathcal { Z } } p _ { G } ( x | \\boldsymbol { z } ) p _ { z } ( \\boldsymbol { z } ) d \\boldsymbol { z } , \\quad \\forall \\boldsymbol { x } \\in \\mathcal { X } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "assuming all involved densities are properly defined. For simplicity we will focus on non-random decoders, i.e. generative models $P _ { G } ( X | Z )$ deterministically mapping $Z$ to $X = G ( Z )$ for a given map $G \\colon { \\mathcal { Z } } \\to { \\mathcal { X } }$ . Similar results for random decoders can be found in Supplementary B.1. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "It turns out that under this model, the OT cost takes a simpler form as the transportation plan factors through the map $G$ : instead of finding a coupling $\\Gamma$ in (1) between two random variables living in the $\\mathcal { X }$ space, one distributed according to $P _ { X }$ and the other one according to $P _ { G }$ , it is sufficient to find a conditional distribution $Q ( Z | X )$ such that its $Z$ marginal $Q _ { Z } ( Z ) \\setminus : = \\mathbb { E } _ { X \\sim P _ { X } } \\left[ Q ( Z | X ) \\right]$ is identical to the prior distribution $P _ { Z }$ . This is the content of the theorem below proved in Bousquet et al. (2017). To make this paper self contained we repeat the proof in Supplementary B. ",
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+    {
+        "type": "text",
+        "text": "Theorem 1 For $P _ { G }$ as defined above with deterministic $P _ { G } ( X | Z )$ and any function $G \\colon { \\mathcal { Z } } \\to { \\mathcal { X } }$ ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/8760809b4a0490a2ea8ce6048266b52ba82d2494672ba947592689aae44a6fa0.jpg",
+        "text": "$$\n\\operatorname* { i n f } _ { \\substack { \\Gamma \\in \\mathcal { P } ( X \\sim P _ { X } , Y \\sim P _ { G } ) } } \\mathbb { E } _ { ( X , Y ) \\sim \\Gamma } \\left[ c \\big ( X , Y \\big ) \\right] = \\operatorname* { i n f } _ { Q \\colon Q z = P _ { Z } } \\mathbb { E } _ { P _ { X } } \\mathbb { E } _ { Q ( Z | X ) } \\left[ c \\big ( X , G ( Z ) \\big ) \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $Q _ { Z }$ is the marginal distribution of $Z$ when $X \\sim P _ { X }$ and $Z \\sim Q ( Z | X )$ ",
+        "bbox": [
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+        "type": "text",
+        "text": "This result allows us to optimize over random encoders $Q ( Z | X )$ instead of optimizing over all couplings between $X$ and $Y$ . Of course, both problems are still constrained. In order to implement a numerical solution we relax the constraints on $Q _ { Z }$ by adding a penalty to the objective. This finally leads us to the WAE objective: ",
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+        "type": "equation",
+        "img_path": "images/5360a52f9d5d2fbad8fcb31a5590250f91d309607420486ac7e194c2be63a660.jpg",
+        "text": "$$\nD _ { \\mathrm { W A E } } ( P _ { X } , P _ { G } ) : = \\operatorname* { i n f } _ { Q ( Z | X ) \\in \\mathcal { Q } } \\mathbb { E } _ { P _ { X } } \\mathbb { E } _ { Q ( Z | X ) } \\left[ c \\bigl ( X , G ( Z ) \\bigr ) \\right] + \\lambda \\cdot \\mathcal { D } _ { Z } ( Q _ { Z } , P _ { Z } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\mathcal { Q }$ is any nonparametric set of probabilistic encoders, $\\mathcal { D } _ { Z }$ is an arbitrary divergence between $Q _ { Z }$ and $P _ { Z }$ , and $\\lambda > 0$ is a hyperparameter. Similarly to VAE, we propose to use deep neural networks to parametrize both encoders $Q$ and decoders $G$ . Note that as opposed to VAEs, the WAE formulation allows for non-random encoders deterministically mapping inputs to their latent codes. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We propose two different penalties $\\mathcal { D } _ { Z } ( Q _ { Z } , P _ { Z } )$ ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "GAN-based $\\mathcal { D } _ { Z }$ . The first option is to choose $\\mathcal { D } _ { Z } ( Q _ { Z } , P _ { Z } ) = D _ { \\mathrm { J S } } ( Q _ { Z } , P _ { Z } )$ and use the adversarial training to estimate it. Specifically, we introduce an adversary (discriminator) in the latent space $\\mathcal { Z }$ trying to separate2 “true” points sampled from $P _ { Z }$ and “fake” ones sampled from $Q _ { Z }$ (Goodfellow et al., 2014). This results in the WAE-GAN described in Algorithm 1. Even though WAE-GAN falls back to the min-max problem, we move the adversary from the input (pixel) space $\\mathcal { X }$ to the latent space $\\mathcal { Z }$ . On top of that, $P _ { Z }$ may have a nice shape with a single mode (for a Gaussian prior), in which case the task should be easier than matching an unknown, complex, and possibly multi-modal distributions as usually done in GANs. This is also a reason for our second penalty: ",
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+    {
+        "type": "text",
+        "text": "MMD-based $\\mathcal { D } _ { Z }$ . For a positive-definite reproducing kernel $k \\colon \\mathcal { Z } \\times \\mathcal { Z }  \\mathcal { R }$ the following expression is called the maximum mean discrepancy (MMD): ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/1d5147f8e2176e1d2251c84ae0803c2fb6f0af4b7beee6cc89f3b9b737729f82.jpg",
+        "text": "$$\n\\mathrm { M M D } _ { k } ( P _ { Z } , Q _ { Z } ) = \\big \\lVert \\int _ { \\mathcal { Z } } k ( z , \\cdot ) d P _ { Z } ( z ) - \\int _ { \\mathcal { Z } } k ( z , \\cdot ) d Q _ { Z } ( z ) \\big \\rVert _ { \\mathcal { H } _ { k } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { H } _ { k }$ is the RKHS of real-valued functions mapping $\\mathcal { Z }$ to $\\mathcal { R }$ . If $k$ is characteristic then ${ \\mathrm { M M D } } _ { k }$ defines a metric and can be used as a divergence measure. We propose to use $\\mathcal { D } _ { Z } ( P _ { Z } , Q _ { Z } ) =$ $\\mathrm { M M D } _ { k } ( P _ { Z } , Q _ { Z } )$ . Fortunately, MMD has an unbiased U-statistic estimator, which can be used in conjunction with stochastic gradient descent (SGD) methods. This results in the WAE-MMD described in Algorithm 2. It is well known that the maximum mean discrepancy performs well when matching high-dimensional standard normal distributions (Gretton et al., 2012) so we expect this penalty to work especially well working with the Gaussian prior $P _ { Z }$ . ",
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+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>ALGORITHM1Wasserstein Auto-Encoder with GAN-based penalty (WAE-GAN).</td><td>ALGORITHM 2Wasserstein Auto-Encoder withMMD-based penalty (WAE-MMD).</td></tr><tr><td>Require:Regularization coefficient 入&gt;0. Initialize the parameters of the encoder Q, decoder Gθ,and latent discriminator Dy. while(,θ) not converged do Sample {x1,...,xn} from the training set Sample {z1,...,zn} from the prior Pz</td><td>Require:Regularization coefficient入&gt; O,char- acteristic positive-definite kernel k. Initialize the parameters of the encoder Q𝜙, decoder Gθ,and latent discriminator D. while (,0) not converged do</td></tr><tr><td>Sample zi from Q(Z|xi) fori=1,...,n Update D, by ascending:</td><td>Sample {x1,...,xn} from the training set Sample {z1,..., zn} from the prior Pz Sample z from Q(Z|xi) fori=1,...,n</td></tr><tr><td>入 n ? logDγ(zi)+log(1-Dγ(i)) n i=1</td><td>Update Q and Ge by descending: 1 n 入</td></tr><tr><td>Update Q and Ge by descending: 1 n c(xi,Ge(zi))-λ·logDγ(zi) n</td><td>c(xi,Gθ(zi)） &gt; k(ze,zj） n(n-1) n i=1 +j</td></tr><tr><td>i=1 end while</td><td>入 2入 &gt; k（2e，） + &gt; k(ze,2j） n</td></tr><tr><td></td><td>n(n-1) l≠j e,j</td></tr></table>",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We point out once again that the encoders $Q _ { \\phi } ( Z | x )$ in Algorithms 1 and 2 can be non-random, i.e. deterministically mapping input points to the latent codes. In this case $Q _ { \\phi } ( Z | x ) = \\delta _ { \\mu _ { \\phi } ( x ) }$ fo r a function $\\mu _ { \\phi } \\colon \\mathcal { X } \\to \\mathcal { Z }$ and in order to sample $\\tilde { z } _ { i }$ from $Q _ { \\phi } ( Z | x _ { i } )$ we just need to return $\\mu _ { \\phi } ( x _ { i } )$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Literature on auto-encoders Classical unregularized auto-encoders minimize only the reconstruction cost. This results in different training points being encoded into non-overlapping zones chaotically scattered all across the $\\mathcal { Z }$ space with “holes” in between where the decoder mapping $P _ { G } ( X | Z )$ has never been trained. Overall, the encoder $Q ( Z | X )$ trained in this way does not provide a useful representation and sampling from the latent space $\\mathcal { Z }$ becomes hard (Bengio et al., 2013). ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Variational auto-encoders (Kingma & Welling, 2014) minimize a variational bound on the KLdivergence $D _ { \\mathrm { K L } } ( P _ { X } , P _ { G } )$ which is composed of the reconstruction cost plus the regularizer $\\mathbb { E } _ { P _ { X } } \\left[ D _ { \\mathrm { K L } } ( Q ( Z | X ) , P _ { Z } ) \\right]$ . The regularizer captures how distinct the image by the encoder of each training example is from the prior $P _ { Z }$ , which is not guaranteeing that the overall encoded distribution $\\mathbb { E } _ { P _ { X } }$ $\\mathinner { \\langle { Q \\mathopen { \\left( Z \\vert X \\right) } } \\rvert }$ matches $P _ { Z }$ like WAE does. Also, VAEs require non-degenerate (i.e. nondeterministic) Gaussian encoders and random decoders for which the term $\\log p _ { G } ( x | z )$ can be computed and differentiated with respect to the parameters. Later Mescheder et al. (2017) proposed a way to use VAE with non-Gaussian encoders. WAE minimizes the optimal transport $W _ { c } ( P _ { X } , P _ { G } )$ and allows both probabilistic and deterministic encoder-decoder pairs of any kind. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The VAE regularizer can be also equivalently written (Hoffman & Johnson, 2016) as a sum of $D _ { \\mathrm { K L } } ( Q _ { Z } , P _ { Z } )$ and a mutual information $\\mathbb { I } _ { Q } ( X , Z )$ between the images $X$ and latent codes $Z$ jointly distributed according to $P _ { X } \\times Q ( Z | X )$ . This observation provides another intuitive way to explain a difference between our algorithm and VAEs: WAEs simply drop the mutual information term $\\mathbb { I } _ { Q } ( X , Z )$ in the VAE regularizer. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "When used with $c ( x , y ) = \\| x - y \\| _ { 2 } ^ { 2 }$ WAE-GAN is equivalent to adversarial auto-encoders (AAE) proposed by Makhzani et al. (2016). Theory of Bousquet et al. (2017) (and in particular Theorem 1) thus suggests that AAEs minimize the 2-Wasserstein distance between $P _ { X }$ and $P _ { G }$ . This provides the first theoretical justification for AAEs known to the authors. WAE generalizes AAE in two ways: first, it can use any cost function $c$ in the input space $\\mathcal { X }$ ; second, it can use any discrepancy measure $\\mathcal { D } _ { Z }$ in the latent space $\\mathcal { Z }$ (for instance MMD), not necessarily the adversarial one of WAE-GAN. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Finally, Zhao et al. (2017b) independently proposed a regularized auto-encoder objective similar to Bousquet et al. (2017) and our (4) based on very different motivations and arguments. Following ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "VAEs their objective (called InfoVAE) defines the reconstruction cost in the image space implicitly through the negative log likelihood term $- \\log p _ { G } ( x | z )$ , which should be properly normalized for all $z \\in { \\mathcal { Z } }$ . In theory VAE and InfoVAE can both induce arbitrary cost functions, however in practice this may require an estimation of the normalizing constant (partition function) which can3 be different for different values of $z$ . WAEs specify the cost $c ( x , y )$ explicitly and don’t constrain it in any way. ",
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+    },
+    {
+        "type": "text",
+        "text": "Literature on OT Genevay et al. (2016) address computing the OT cost in large scale using SGD and sampling. They approach this task either through the dual formulation, or via a regularized version of the primal. They do not discuss any implications for generative modeling. Our approach is based on the primal form of OT, we arrive at regularizers which are very different, and our main focus is on generative modeling. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The WGAN (Arjovsky et al., 2017) minimizes the 1-Wasserstein distance $W _ { 1 } ( P _ { X } , P _ { G } )$ for generative modeling. The authors approach this task from the dual form. Their algorithm comes without an encoder and can not be readily applied to any other cost $W _ { c }$ , because the neat form of the Kantorovich-Rubinstein duality (2) holds only for $W _ { 1 }$ . WAE approaches the same problem from the primal form, can be applied for any cost function $c$ , and comes naturally with an encoder. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In order to compute the values (1) or (2) of OT we need to handle non-trivial constraints, either on the coupling distribution $\\Gamma$ or on the function $f$ being considered. Various approaches have been proposed in the literature to circumvent this difficulty. For $W _ { 1 }$ Arjovsky et al. (2017) tried to implement the constraint in the dual formulation (2) by clipping the weights of the neural network $f$ . Later Gulrajani et al. (2017) proposed to relax the same constraint by penalizing the objective of (2) with a term $\\lambda \\cdot \\mathbb { E } \\left( \\left\\| \\nabla f ( X ) \\right\\| - 1 \\right) ^ { 2 }$ which should not be greater than 1 if $f \\in \\mathcal { F } _ { L }$ . In a more general OT setting of $W _ { c }$ Cuturi (2013) proposed to penalize the objective of (1) with the KLdivergence $\\lambda \\cdot D _ { \\mathrm { K L } } ( \\Gamma , P \\otimes Q )$ between the coupling distribution and the product of marginals. Genevay et al. (2016) showed that this entropic regularization drops the constraints on functions in the dual formulation as opposed to (2). Finally, in the context of unbalanced optimal transport it has been proposed to relax the constraint in (1) by regularizing the objective with $\\lambda \\cdot \\left( D _ { f } ( \\Gamma _ { X } , \\mathbf { \\bar { \\Gamma } } ^ { } P ) + \\right.$ $D _ { f } ( \\Gamma _ { Y } , Q ) )$ (Chizat et al., 2015; Liero et al., 2015), where $\\Gamma _ { X }$ and $\\Gamma _ { Y }$ are marginals of $\\Gamma$ . In this paper we propose to relax OT in a way similar to the unbalanced optimal transport, i.e. by adding additional divergences to the objective. However, we show that in the particular context of generative modeling, only one extra divergence is necessary. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Literature on GANs Many of the GAN variations (including $f$ -GAN and WGAN) come without an encoder. Often it may be desirable to reconstruct the latent codes and use the learned manifold, in which cases these models are not applicable. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "There have been many other approaches trying to blend the adversarial training of GANs with autoencoder architectures (Zhao et al., $2 0 1 7 \\mathrm { a }$ ; Dumoulin et al., 2017; Ulyanov et al., 2017; Berthelot et al., 2017). The approach proposed by Ulyanov et al. (2017) is perhaps the most relevant to our work. The authors use the discrepancy between $Q _ { Z }$ and the distribution $\\mathbb { E } _ { Z ^ { \\prime } \\sim P _ { Z } } [ Q \\left( Z | G ( Z ^ { \\prime } ) \\right) ]$ of auto-encoded noise vectors as the objective for the max-min game between the encoder and decoder respectively. While the authors showed that the saddle points correspond to $P _ { X } = P _ { G }$ , they admit that encoders and decoders trained in this way have no incentive to be reciprocal. As a workaround they propose to include an additional reconstruction term to the objective. WAE does not necessarily lead to a min-max game, uses a different penalty, and has a clear theoretical foundation. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Several works used reproducing kernels in context of GANs. Li et al. (2015); Dziugaite et al. (2015) use MMD with a fixed kernel $k$ to match $P _ { X }$ and $P _ { G }$ directly in the input space $\\mathcal { X }$ . These methods have been criticised to require larger mini-batches during training: estimating $\\mathrm { M M D } _ { k } ( P _ { X } , P _ { G } )$ requires number of samples roughly proportional to the dimensionality of the input space $\\mathcal { X }$ (Reddi et al., 2015) which is typically larger than $1 0 ^ { 3 }$ . Li et al. (2017) take a similar approach but further train $k$ adversarially so as to arrive at a meaningful loss function. WAE-MMD uses MMD to match $Q _ { Z }$ to the prior $P _ { Z }$ in the latent space $\\mathcal { Z }$ . Typically $\\mathcal { Z }$ has no more than 100 dimensions and $P _ { Z }$ is Gaussian, which allows us to use regular mini-batch sizes to accurately estimate MMD. ",
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/ba6eb123cbe974ac8bc98667fc71d8053d7e708a6c4e49eac087dd03ed3734c6.jpg",
+        "image_caption": [
+            "Figure 2: VAE (left column), WAE-MMD (middle column), and WAE-GAN (right column) trained on MNIST dataset. In “test reconstructions” odd rows correspond to the real test points. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section we empirically evaluate4 the proposed WAE model. We would like to test if WAE can simultaneously achieve (i) accurate reconstructions of data points, (ii) reasonable geometry of the latent manifold, and (iii) random samples of good (visual) quality. Importantly, the model should generalize well: requirements (i) and (ii) should be met on both training and test data. We trained WAE-GAN and WAE-MMD (Algorithms 1 and 2) on two real-world datasets: MNIST (LeCun et al., 1998) consisting of $7 0 \\mathrm { k }$ images and CelebA (Liu et al., 2015) containing roughly 203k images. ",
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+    {
+        "type": "text",
+        "text": "Experimental setup In all reported experiments we used Euclidian latent spaces $\\mathcal { Z } \\ = \\ \\mathcal { R } ^ { d _ { z } }$ for various $d _ { z }$ depending on the complexity of the dataset, isotropic Gaussian prior distributions $P _ { Z } ( Z ) = \\mathcal { N } ( Z ; \\mathbf { 0 } , \\sigma _ { z } ^ { 2 } \\cdot I _ { d } )$ over $\\mathcal { Z }$ , and a squared cost function $c ( \\bar { x } , y ) = \\| x - \\bar { y } \\| _ { 2 } ^ { 2 }$ for data points $x , y \\in \\mathcal { X } = \\mathcal { R } ^ { d _ { x } }$ . We used deterministic encoder-decoder pairs, Adam (Kingma & Lei, 2014) with $\\beta _ { 1 } = 0 . 5 , \\beta _ { 2 } = 0 . 9 9 9$ , and convolutional deep neural network architectures for encoder mapping $\\mu _ { \\phi } \\colon \\mathcal { X } \\to \\mathcal { Z }$ and decoder mapping $G _ { \\theta } \\colon { \\mathcal { Z } } \\to { \\mathcal { X } }$ similar to the DCGAN ones reported by Radford et al. (2016) with batch normalization (Ioffe & Szegedy, 2015). We tried various values of $\\lambda$ and noticed that $\\lambda = 1 0$ seems to work good across all datasets we considered. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Since we are using deterministic encoders, choosing $d _ { z }$ larger than intrinsic dimensionality of the dataset would force the encoded distribution $Q _ { Z }$ to live on a manifold in $\\mathcal { Z }$ . This would make matching $Q _ { Z }$ to $P _ { Z }$ impossible if $P _ { Z }$ is Gaussian and may lead to numerical instabilities. We use $d _ { z } = 8$ for MNIST and $d _ { z } = 6 4$ for CelebA which seems to work reasonably well. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/5f9154f809ae2eb7b1b03217d0ca3922df4a01f90d50874364f5a3fe74e9b6da.jpg",
+        "image_caption": [
+            "Figure 3: VAE (left column), WAE-MMD (middle column), and WAE-GAN (right column) trained on CelebA dataset. In “test reconstructions” odd rows correspond to the real test points. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We also report results of VAEs. VAEs used the same latent spaces as discussed above and standard Gaussian priors $P _ { Z } = \\mathcal { N } ( \\mathbf { 0 } , I _ { d } )$ . We used Gaussian encoders $Q ( Z | X ) = \\mathcal { N } \\big ( Z ; \\mu _ { \\phi } ( X ) , \\Sigma ( X ) \\big )$ with mean $\\mu _ { \\phi }$ and diagonal covariance $\\Sigma$ . For both MNIST and CelebA we used Bernoulli decoders parametrized by $G _ { \\theta }$ . Functions $\\mu _ { \\phi } , \\Sigma$ , and $G _ { \\theta }$ were parametrized by deep nets of the same architectures as used in WAE. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "WAE-GAN and WAE-MMD specifics In WAE-GAN we used discriminator $D$ composed of several fully connected layers with ReLu. We tried WAE-MMD with the RBF kernel but observed that it fails to penalize the outliers of $Q _ { Z }$ because of the quick tail decay. If the codes $\\tilde { z } = \\mu _ { \\phi } ( x )$ for some of the training points $x \\in \\mathcal { X }$ end up far away from the support of $P _ { Z }$ (which may happen in the early stages of training) the corresponding terms in the U-statistic $k ( z , \\tilde { z } ) = e ^ { - \\| \\tilde { z } - z \\| _ { 2 } ^ { 2 } / \\sigma _ { k } ^ { 2 } }$ will quickly approach zero and provide no gradient for those outliers. This could be avoided by choosing the kernel bandwidth $\\sigma _ { k } ^ { 2 }$ in a data-dependent manner, however in this case per-minibatch U-statistic would not provide an unbiased estimate for the gradient. Instead, we used the inverse multiquadratics kernel $\\bar { k } ( x , y ) = C / ( C + \\| x - y \\| _ { 2 } ^ { 2 } )$ which is also characteristic and has much heavier tails. In all experiments we used $C = 2 d _ { z } \\sigma _ { z } ^ { 2 }$ , which is the expected squared distance between two multivariate Gaussian vectors drawn from $P _ { Z }$ . This significantly improved the performance compared to the RBF kernel (even the one with $\\sigma _ { k } ^ { 2 } = 2 d _ { z } \\sigma _ { z } ^ { 2 }$ ). Trained models are presented in Figures 2 and 3. Further details are presented in Supplementary C. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Random samples are generated by sampling $P _ { Z }$ and decoding the resulting noise vectors $z$ into $G _ { \\theta } ( z )$ . As expected, in our experiments we observed that for both WAE-GAN and WAE-MMD the quality of samples strongly depends on how accurately $Q _ { Z }$ matches $P _ { Z }$ . To see this, notice that during training the decoder function $G _ { \\theta }$ is presented only with encoded versions $\\mu _ { \\phi } ( X )$ of the data points $X \\sim P _ { X }$ . Indeed, the decoder is trained on samples from $Q _ { Z }$ and thus there is no reason to expect good results when feeding it with samples from $P _ { Z }$ . In our experiments we noticed that even slight differences between $Q _ { Z }$ and $P _ { Z }$ may affect the quality of samples. In some cases WAE-GAN seems to lead to a better matching and generates better samples than WAE-MMD. However, due to adversarial training WAE-GAN is highly unstable, while WAE-MMD has a very stable training much like VAE. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In order to quantitatively assess the quality of the generated images, we use the Frechet Inception Distance´ introduced by Heusel et al. (2017) and report the results on CelebA in Table 1. These results confirm that the sampled images from WAE are of better quality than from VAE, and WAE-GAN gets a slightly better score than WAEMMD, which correlates with visual inspection of the images. ",
+        "bbox": [
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+            215,
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+            325
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/bb53b755aa9224ba9656ef109a4e1d971d6a6d75d2817d17ef00984c94322e3a.jpg",
+        "table_caption": [
+            "Table 1: FID scores for samples on CelebA (smaller is better). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Algorithm</td><td>FID</td></tr><tr><td>VAE</td><td>82</td></tr><tr><td>WAE-MMD</td><td>55</td></tr><tr><td>WAE-GAN</td><td>42</td></tr></table>",
+        "bbox": [
+            617,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Test reconstructions and interpolations. We take random points $x$ from the held out test set and report their auto-encoded versions $G _ { \\theta } ( \\mu _ { \\phi } ( x ) )$ . Next, pairs $( x , y )$ of different data points are sampled randomly from the held out test set and encoded: $z _ { x } = \\mu _ { \\phi } ( x )$ , $z _ { y } = \\mu _ { \\phi } ( y )$ . We linearly interpolate between $z _ { x }$ and $z _ { y }$ with equally-sized steps in the latent space and show decoded images. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            410,
+            318,
+            426
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Using the optimal transport cost, we have derived Wasserstein auto-encoders—a new family of algorithms for building generative models. We discussed their relations to other probabilistic modeling techniques. We conducted experiments using two particular implementations of the proposed method, showing that in comparison to VAEs, the images sampled from the trained WAE models are of better quality, without compromising the stability of training and the quality of reconstruction. Future work will include further exploration of the criteria for matching the encoded distribution $Q _ { Z }$ to the prior distribution $P _ { Z }$ , assaying the possibility of adversarially training the cost function $c$ in the input space $\\mathcal { X }$ , and a theoretical analysis of the dual formulations for WAE-GAN and WAE-MMD. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            571,
+            326,
+            584
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The authors are thankful to Carl Johann Simon-Gabriel, Mateo Rojas-Carulla, Arthur Gretton, Paul Rubenstein, and Fei Sha for stimulating discussions. ",
+        "bbox": [
+            176,
+            594,
+            823,
+            623
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            646,
+            285,
+            661
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "M. Arjovsky, S. Chintala, and L. Bottou. Wasserstein GAN, 2017.   \nY. Bengio, A. Courville, and P. Vincent. Representation learning: A review and new perspectives. Pattern Analysis and Machine Intelligence, 35, 2013.   \nD. Berthelot, T. Schumm, and L. Metz. Began: Boundary equilibrium generative adversarial networks, 2017.   \nO. Bousquet, S. Gelly, I. Tolstikhin, C. J. Simon-Gabriel, and B. Scholkopf. From optimal transport ¨ to generative modeling: the VEGAN cookbook, 2017.   \nLenaic Chizat, Gabriel Peyre, Bernhard Schmitzer, and Franc¸ois-Xavier Vialard. Unbalanced opti- ´ mal transport: geometry and kantorovich formulation. arXiv preprint arXiv:1508.05216, 2015.   \nM. Cuturi. Sinkhorn distances: Lightspeed computation of optimal transport. In Advances in Neural Information Processing Systems, pp. 2292–2300, 2013.   \nV. Dumoulin, I. Belghazi, B. Poole, A. Lamb, M. Arjovsky, O. Mastropietro, and A. Courville. Adversarially learned inference. In ICLR, 2017. ",
+        "bbox": [
+            171,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "G. K. Dziugaite, D. M. Roy, and Z. Ghahramani. Training generative neural networks via maximum mean discrepancy optimization. In UAI, 2015. ",
+        "bbox": [
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+        "text": "A IMPLICIT GENERATIVE MODELS: A SHORT TOUR OF GANS AND VAES ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Even though GANs and VAEs are quite different—both in terms of the conceptual frameworks and empirical performance—they share important features: (a) both can be trained by sampling from the model $P _ { G }$ without knowing an analytical form of its density and (b) both can be scaled up with SGD. As a result, it becomes possible to use highly flexible implicit models $P _ { G }$ defined by a twostep procedure, where first a code $Z$ is sampled from a fixed distribution $P _ { Z }$ on a latent space $\\mathcal { Z }$ and then $Z$ is mapped to the image $G ( Z ) \\in \\mathcal { X } = \\mathcal { R } ^ { d }$ with a (possibly random) transformation $G \\colon { \\mathcal { Z } } \\to { \\mathcal { X } }$ . This results in latent variable models $P _ { G }$ of the form (3). ",
+        "bbox": [
+            173,
+            132,
+            825,
+            232
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "These models are indeed easy to sample and, provided $G$ can be differentiated analytically with respect to its parameters, $P _ { G }$ can be trained with SGD. The field is growing rapidly and numerous variations of VAEs and GANs are available in the literature. Next we introduce and compare several of them. ",
+        "bbox": [
+            174,
+            237,
+            825,
+            292
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The original generative adversarial network (GAN) Goodfellow et al. (2014) approach minimizes ",
+        "bbox": [
+            173,
+            299,
+            823,
+            315
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a59ac1d737ca8f03b353ef156dcb0aa8414c6da98612047cea30f74c36c0bb94.jpg",
+        "text": "$$\nD _ { \\mathrm { G A N } } ( P _ { X } , P _ { G } ) = \\operatorname* { s u p } _ { T \\in { \\mathcal T } } \\mathbb { E } _ { X \\sim P _ { X } } [ \\log T ( X ) ] + \\mathbb { E } _ { Z \\sim P _ { Z } } [ \\log \\left( 1 - T ( G ( Z ) ) \\right) ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            250,
+            320,
+            746,
+            347
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "with respect to a deterministic decoder $G \\colon { \\mathcal { Z } } \\to { \\mathcal { X } }$ , where $\\tau$ is any non-parametric class of choice. It is known that $D _ { \\mathrm { G A N } } ( P _ { X } , P _ { G } ) \\leq 2 \\cdot D _ { \\mathrm { J S } } ( P _ { X } , P _ { G } ) - \\log ( 4 )$ and the inequality turns into identity in the nonparametric limit, that is when the class $\\tau$ becomes rich enough to represent all functions mapping $\\mathcal { X }$ to $( 0 , 1 )$ . Hence, GANs are minimizing a lower bound on the JS-divergence. However, GANs are not only linked to the JS-divergence: the $f$ -GAN approach Nowozin et al. (2016) showed that a slight modification $D _ { \\mathrm { f , G A N } }$ of the objective (5) allows to lower bound any desired $f$ -divergence in a similar way. In practice, both decoder $G$ and discriminator $T$ are trained in alternating SGD steps. Stopping criteria as well as adequate evaluation of the trained GAN models remain open questions. ",
+        "bbox": [
+            173,
+            352,
+            825,
+            478
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Recently, the authors of Arjovsky et al. (2017) argued that the 1-Wasserstein distance $W _ { 1 }$ , which is known to induce a much weaker topology than $D _ { \\mathrm { J S } }$ , may be better suited for generative modeling. When $P _ { X }$ and $P _ { G }$ are supported on largely disjoint low-dimensional manifolds (which may be the case in applications), $D _ { \\mathrm { K L } } , D _ { \\mathrm { J S } }$ , and other strong distances between $P _ { X }$ and $P _ { G }$ max out and no longer provide useful gradients for $P _ { G }$ . This “vanishing gradient” problem necessitates complicated scheduling between the $G / T$ updates. In contrast, $W _ { 1 }$ is still sensible in these cases and provides stable gradients. The Wasserstein GAN (WGAN) minimizes ",
+        "bbox": [
+            173,
+            483,
+            825,
+            582
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/11516a3278e745fdf04fed42048535c16a3a3250cb51ccdedea0ec2928ff1e7a.jpg",
+        "text": "$$\nD _ { \\mathrm { W G A N } } ( P _ { X } , P _ { G } ) = \\operatorname* { s u p } _ { T \\in \\mathcal { W } } \\mathbb { E } _ { X \\sim P _ { X } } [ T ( X ) ] - \\mathbb { E } _ { Z \\sim P _ { Z } } \\left[ T ( G ( Z ) ) \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            282,
+            587,
+            712,
+            614
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { W }$ is any subset of 1-Lipschitz functions on $\\mathcal { X }$ . It follows from (2) that $D _ { \\mathrm { W G A N } } ( P _ { X } , P _ { G } ) \\leq$ $W _ { 1 } ( P _ { X } , P _ { G } )$ and thus WGAN is minimizing a lower bound on the 1-Wasserstein distance. ",
+        "bbox": [
+            174,
+            619,
+            820,
+            648
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Variational auto-encoders (VAE) Kingma & Welling (2014) utilize models $P _ { G }$ of the form (3) and minimize ",
+        "bbox": [
+            169,
+            655,
+            823,
+            684
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/73b2d7fec2d7342b3db2f58e91ee4dc807956f98a6dfe86d99e0a7e7e00b668d.jpg",
+        "text": "$$\nD _ { \\mathrm { V A E } } \\big ( P _ { X } , P _ { G } \\big ) = \\operatorname* { i n f } _ { Q ( Z | X ) \\in \\mathcal { Q } } \\mathbb { E } _ { P _ { X } } \\left[ D _ { \\mathrm { K L } } \\big ( Q ( Z | X ) , P _ { Z } \\big ) - \\mathbb { E } _ { Q ( Z | X ) } [ \\log p _ { G } ( X | Z ) ] \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            220,
+            688,
+            776,
+            715
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "with respect to a random decoder mapping $P _ { G } ( X | Z )$ . The conditional distribution $P _ { G } ( X | Z )$ is often parametrized by a deep net $G$ and can have any form as long as its density $p _ { G } ( x | z )$ can be computed and differentiated with respect to the parameters of $G$ . A typical choice is to use Gaussians $P _ { G } ( X | Z ) = \\mathcal { N } ( X ; G ( Z ) , \\sigma ^ { \\hat { 2 } } \\cdot I )$ . If $\\mathcal { Q }$ is the set of all conditional probability distributions $Q ( Z | X )$ , the objective of VAE coincides with the negative marginal log-likelihood $D _ { \\mathrm { V A E } } ( P _ { X } , P _ { G } ) = - \\mathbb { E } _ { P _ { X } } [ \\log P _ { G } ( X ) ]$ . However, in order to make the $D _ { \\mathrm { K L } }$ term of (6) tractable in closed form, the original implementation of VAE uses a standard normal $P _ { Z }$ and restricts $\\mathcal { Q }$ to a class of Gaussian distributions $Q ( Z | X ) = { \\mathcal { N } } { \\big ( } Z ; \\mu ( X ) , \\Sigma ( X ) { \\big ) }$ with mean $\\mu$ and diagonal covariance $\\Sigma$ parametrized by deep nets. As a consequence, VAE is minimizing an upper bound on the negative log-likelihood or, equivalently, on the KL-divergence $D _ { \\mathrm { K L } } ( P _ { X } , P _ { G } )$ . ",
+        "bbox": [
+            173,
+            720,
+            825,
+            862
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "One possible way to reduce the gap between the true negative log-likelihood and the upper bound provided by $D _ { \\mathrm { V A E } }$ is to enlarge the class $\\mathcal { Q }$ . Adversarial variational Bayes (AVB) Mescheder et al. (2017) follows this argument by employing the idea of GANs. Given any point $x \\in \\mathcal { X }$ , a noise $\\epsilon \\sim \\mathcal { N } ( 0 , 1 )$ , and any fixed transformation $e \\colon \\mathcal { X } \\times \\mathcal { R }  \\mathcal { Z }$ , a random variable $e ( x , \\epsilon )$ ",
+        "bbox": [
+            174,
+            867,
+            823,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "implicitly defines one particular conditional distribution $Q _ { e } ( Z | X = x )$ . AVB allows $\\mathcal { Q }$ to contain all such distributions for different choices of $e$ , replaces the intractable term $D _ { \\mathrm { K L } } \\big ( Q _ { e } ( Z | X ) , P _ { Z } \\big )$ in (6) by the adversarial approximation $D _ { \\mathrm { f , G A N } }$ corresponding to the KL-divergence, and proposes to minimize5 ",
+        "bbox": [
+            174,
+            102,
+            825,
+            162
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/234379459d36ea6a3949d78f6e2e2e101d53bbec5ea6e1e381a3f7e12fbfbb4f.jpg",
+        "text": "$$\nD _ { \\mathrm { A V B } } ( P _ { X } , P _ { G } ) = \\operatorname* { i n f } _ { Q _ { \\epsilon } ( Z | X ) \\in \\mathcal { Q } } \\mathbb { E } _ { P _ { X } } \\left[ D _ { \\mathrm { f } , \\operatorname { G A N } } \\big ( Q _ { e } ( Z | X ) , P _ { Z } \\big ) - \\mathbb { E } _ { Q _ { \\epsilon } ( Z | X ) } [ \\log p _ { G } ( X | Z ) ] \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            187,
+            167,
+            789,
+            194
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The $D _ { \\mathrm { K L } }$ term in (6) may be viewed as a regularizer. Indeed, VAE reduces to the classical unregularized auto-encoder if this term is dropped, minimizing the reconstruction cost of the encoder-decoder pair $Q ( Z | X ) , P _ { G } ( X | Z )$ . This often results in different training points being encoded into nonoverlapping zones chaotically scattered all across the $\\mathcal { Z }$ space with “holes” in between where the decoder mapping $P _ { G } ( X | Z )$ has never been trained. Overall, the encoder $Q ( Z | X )$ trained in this way does not provide a useful representation and sampling from the latent space $\\mathcal { Z }$ becomes hard Bengio et al. (2013). ",
+        "bbox": [
+            173,
+            207,
+            825,
+            306
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Adversarial auto-encoders (AAE) Makhzani et al. (2016) replace the $D _ { \\mathrm { K L } }$ term in (6) with another regularizer: ",
+        "bbox": [
+            171,
+            311,
+            823,
+            340
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/11bed2c41516f97c0b74ab2c2d62f438bc2924db0fe43e1adcfa303325913ed1.jpg",
+        "text": "$$\nD _ { \\mathrm { A A E } } ( P _ { X } , P _ { G } ) = \\operatorname* { i n f } _ { Q ( Z | X ) \\in Q } D _ { \\mathrm { G A N } } ( Q _ { Z } , P _ { Z } ) - \\mathbb { E } _ { P _ { X } } \\mathbb { E } _ { Q ( Z | X ) } [ \\log p _ { G } ( X | Z ) ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            240,
+            347,
+            756,
+            372
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "where $Q _ { Z }$ is the marginal distribution of $Z$ when first $X$ is sampled from $P _ { X }$ and then $Z$ is sampled from $Q ( Z | X )$ , also known as the aggregated posterior Makhzani et al. (2016). Similarly to AVB, there is no clear link to log-likelihood, as $D _ { \\mathrm { A A E } } \\leq D _ { \\mathrm { A V B } }$ . The authors of Makhzani et al. (2016) argue that matching $Q _ { Z }$ to $P _ { Z }$ in this way ensures that there are no “holes” left in the latent space $\\mathcal { Z }$ and $P _ { G } ( X | Z )$ generates reasonable samples whenever $Z \\sim P _ { Z }$ . They also report an equally good performance of different types of conditional distributions $Q ( Z | X )$ , including Gaussians as used in VAEs, implicit models $Q _ { e }$ as used in AVB, and deterministic encoder mappings, i.e. $Q ( Z | X ) =$ $\\delta _ { \\mu ( X ) }$ with $\\mu \\colon \\mathcal X \\to \\mathcal Z$ . ",
+        "bbox": [
+            173,
+            378,
+            825,
+            491
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B PROOF OF THEOREM 1 AND FURTHER DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            511,
+            598,
+            526
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We will consider certain sets of joint probability distributions of three random variables $( X , Y , Z ) \\in$ $\\mathcal { X } \\times \\mathcal { X } \\times \\mathcal { Z }$ . The reader may wish to think of $X$ as true images, $Y$ as images sampled from the model, and $Z$ as latent codes. We denote by $P _ { G , Z } ( Y , Z )$ a joint distribution of a variable pair $( Y , Z )$ , where $Z$ is first sampled from $P _ { Z }$ and next $Y$ from $P _ { G } ( Y | Z )$ . Note that $P _ { G }$ defined in (3) and used throughout this work is the marginal distribution of $Y$ when $( Y , Z ) \\sim P _ { G , Z }$ . ",
+        "bbox": [
+            173,
+            541,
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+            613
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In the optimal transport problem (1), we consider joint distributions $\\Gamma ( X , Y )$ which are called couplings between values of $X$ and $Y$ . Because of the marginal constraint, we can write $\\Gamma ( X , Y ) =$ $\\bar { \\Gamma } ( \\bar { Y ( X ) } P _ { X } ( X )$ and we can consider $\\Gamma ( Y | X )$ as a non-deterministic mapping from $X$ to $Y$ . Theorem 1. shows how to factor this mapping through $\\mathcal { Z }$ , i.e., decompose it into an encoding distribution $Q ( Z | X )$ and the generating distribution $P _ { G } ( Y | Z )$ . ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "As in Section 2.2, $\\mathcal { P } ( X \\sim P _ { X } , Y \\sim P _ { G } )$ denotes the set of all joint distributions of $( X , Y )$ with marginals $P _ { X } , P _ { G }$ , and likewise for $\\mathcal { P } ( X \\sim P _ { X } , Z \\sim P _ { Z } )$ . The set of all joint distributions of $( X , Y , Z )$ such that $X \\sim P _ { X }$ , $( Y , Z ) \\sim P _ { G , Z }$ , and $( Y \\perp \\perp X ) | Z$ will be denoted by ${ \\mathcal { P } } _ { X , Y , Z }$ . Finally, we denote by $\\mathcal { P } _ { X , Y }$ and $\\mathcal { P } _ { X , Z }$ the sets of marginals on $( X , Y )$ and $( X , Z )$ (respectively) induced by distributions in ${ \\mathcal { P } } _ { X , Y , Z }$ . Note that $\\mathcal { P } ( P _ { X } , P _ { G } )$ , ${ \\mathcal { P } } _ { X , Y , Z }$ , and $\\mathcal { P } _ { X , Y }$ depend on the choice of conditional distributions $P _ { G } ( Y | Z )$ , while $\\mathcal { P } _ { X , Z }$ does not. In fact, it is easy to check that $\\mathcal { P } _ { X , Z } =$ $\\mathcal { P } ( X \\sim P _ { X } , Z \\sim P _ { Z } )$ . From the definitions it is clear that $\\mathcal { P } _ { X , Y } \\subseteq \\mathcal { P } ( P _ { X } , P _ { G } )$ and we immediately get the following upper bound: ",
+        "bbox": [
+            173,
+            694,
+            825,
+            808
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/766228ca76f59ab0764a694e387d78f86a68c940c93d4443b7b327b70efb8a9e.jpg",
+        "text": "$$\nW _ { c } ( P _ { X } , P _ { G } ) \\leq W _ { c } ^ { \\dagger } ( P _ { X } , P _ { G } ) : = \\operatorname* { i n f } _ { P \\in \\mathcal { P } _ { X , Y } } \\mathbb { E } _ { ( X , Y ) \\sim P } \\left[ c ( X , Y ) \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            287,
+            813,
+            710,
+            840
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "If $P _ { G } ( Y | Z )$ are Dirac measures (i.e., $Y = G ( Z ) )$ , it turns out that $\\mathcal { P } _ { X , Y } = \\mathcal { P } ( P _ { X } , P _ { G } )$ : ",
+        "bbox": [
+            171,
+            845,
+            751,
+            863
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Lemma 1 $\\mathcal { P } _ { X , Y } \\subseteq \\mathcal { P } ( P _ { X } , P _ { G } )$ with identity $i f ^ { 6 } ~ P _ { G } ( Y | Z = z )$ are Dirac for all $z \\in { \\mathcal { Z } }$ . ",
+        "bbox": [
+            171,
+            102,
+            751,
+            119
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Proof The first assertion is obvious. To prove the identity, note that when $Y$ is a deterministic function of $Z$ , for any $A$ in the sigma-algebra induced by $Y$ we have $\\mathbb { E } \\left[ \\mathbb { 1 } _ { [ Y \\in A ] } | X , Z \\right] = \\mathbb { E } \\left[ \\mathbb { 1 } _ { [ Y \\in A ] } | Z \\right]$ . This implies $( Y \\perp \\perp X ) | Z$ and concludes the proof. ",
+        "bbox": [
+            173,
+            128,
+            825,
+            174
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We are now in place to prove Theorem 1. Lemma 1 obviously leads to ",
+        "bbox": [
+            173,
+            200,
+            635,
+            217
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5fad4c089ed46e041e888e1354250fd12c2da03691113069d45e7c34d2eb98f7.jpg",
+        "text": "$$\nW _ { c } ( P _ { X } , P _ { G } ) = W _ { c } ^ { \\dagger } ( P _ { X } , P _ { G } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            397,
+            222,
+            599,
+            241
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The tower rule of expectation, and the conditional independence property of ${ \\mathcal { P } } _ { X , Y , Z }$ implies ",
+        "bbox": [
+            173,
+            246,
+            776,
+            261
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/653bed1e8cad7ed1de42342b9a7c0aa95c3d1fe17f75f4d5d3812d5573a3ca76.jpg",
+        "text": "$$\n\\begin{array} { r l } & { W _ { c } ^ { \\dagger } ( P _ { X } , P _ { G } ) = \\underset { P \\in \\mathcal { P } _ { X , Y , Z } } { \\operatorname* { i n f } } \\mathbb { E } _ { ( X , Y , Z ) \\sim P } \\left[ c ( X , Y ) \\right] } \\\\ & { \\quad \\quad \\quad = \\underset { P \\in \\mathcal { P } _ { X , Y , Z } } { \\operatorname* { i n f } } \\mathbb { E } _ { P _ { Z } } \\mathbb { E } _ { X \\sim P ( X \\mid Z ) } \\mathbb { E } _ { Y \\sim P ( Y \\mid Z ) } [ c ( X , Y ) ] } \\\\ & { \\quad \\quad \\quad = \\underset { P \\in \\mathcal { P } _ { X , Y , Z } } { \\operatorname* { i n f } } \\mathbb { E } _ { P _ { Z } } \\mathbb { E } _ { X \\sim P ( X \\mid Z ) } \\left[ c \\left( X , G ( Z ) \\right) \\right] } \\\\ & { \\quad \\quad \\quad = \\underset { P \\in \\mathcal { P } _ { X , Z } } { \\operatorname* { i n f } } \\mathbb { E } _ { ( X , Z ) \\sim P } \\left[ c \\left( X , G ( Z ) \\right) \\right] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            284,
+            267,
+            712,
+            376
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "It remains to notice that $\\mathcal { P } _ { X , Z } = \\mathcal { P } ( X \\sim P _ { X } , Z \\sim P _ { Z } )$ as stated earlier. ",
+        "bbox": [
+            173,
+            381,
+            648,
+            397
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "B.1 RANDOM DECODERS $P _ { G } ( Y | Z )$ ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            412,
+            429,
+            428
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "If the decoders are non-deterministic, Lemma 1 provides only the inclusion of sets $\\mathcal { P } _ { X , Y } \\subseteq \\mathcal { P } ( P _ { X } , P _ { G } )$ and we get the following upper bound on the OT: ",
+        "bbox": [
+            171,
+            439,
+            823,
+            468
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Corollary 1 Let $\\mathcal { X } ~ = ~ \\mathcal { R } ^ { d }$ and assume the conditional distributions $P _ { G } ( Y | Z = z )$ have mean values $G ( z ) \\in \\mathcal { R } ^ { d }$ and marginal variances $\\sigma _ { 1 } ^ { 2 } , \\ldots , \\sigma _ { d } ^ { 2 } \\geq 0$ for all $z \\in { \\mathcal { Z } }$ , where $G \\colon { \\mathcal { Z } } \\to { \\mathcal { X } }$ . Take $c ( x , y ) = \\| x - y \\| _ { 2 } ^ { 2 }$ . Then ",
+        "bbox": [
+            174,
+            479,
+            825,
+            525
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3191801fdce877583d57ab64b281fe1a4a2f12b841d0bc7daead3630c2b31298.jpg",
+        "text": "$$\nW _ { c } ( P _ { X } , P _ { G } ) \\leq W _ { c } ^ { \\dagger } ( P _ { X } , P _ { G } ) = \\sum _ { i = 1 } ^ { d } \\sigma _ { i } ^ { 2 } + \\operatorname* { i n f } _ { P \\in \\mathcal { P } ( X \\setminus P _ { X } , Z \\sim P _ { Z } ) } \\mathbb { E } _ { ( X , Z ) \\sim P } \\big [ \\| X - G ( Z ) \\| ^ { 2 } \\big ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            191,
+            529,
+            781,
+            571
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Proof First inequality follows from (9). For the identity we proceed similarly to the proof of Theorem 1 and write ",
+        "bbox": [
+            173,
+            583,
+            826,
+            612
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/435b3f67c7089dcb32fdf511c52506db9c9f938851293f40e5095673d00dd171.jpg",
+        "text": "$$\nW _ { c } ^ { \\dagger } ( P _ { X } , P _ { G } ) = \\operatorname* { i n f } _ { P \\in \\mathcal { P } _ { X , Y , Z } } \\mathbb { E } _ { P _ { Z } } \\mathbb { E } _ { X \\sim P ( X \\mid Z ) } \\mathbb { E } _ { Y \\sim P ( Y \\mid Z ) } \\left[ \\left\\| X - Y \\right\\| ^ { 2 } \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            272,
+            616,
+            725,
+            643
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Note that ",
+        "bbox": [
+            173,
+            648,
+            238,
+            662
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6635a165dd4c7884d61bad12d4d8eafea9a3505f1a3845db267635dbf26e8001.jpg",
+        "text": "$$\n\\begin{array} { l } { \\mathbb { E } _ { Y \\sim P ( Y \\vert Z ) } \\big [ \\| X - Y \\| ^ { 2 } \\big ] = \\mathbb { E } _ { Y \\sim P ( Y \\vert Z ) } \\big [ \\| X - G ( Z ) + G ( Z ) - Y \\| ^ { 2 } \\big ] } \\\\ { = \\| X - G ( Z ) \\| ^ { 2 } + \\mathbb { E } _ { Y \\sim P ( Y \\vert Z ) } \\big [ \\langle X - G ( Z ) , G ( Z ) - Y \\rangle \\big ] + \\mathbb { E } _ { Y \\sim P ( Y \\vert Z ) } \\| G ( Z ) - Y \\| ^ { 2 } } \\\\ { = \\| X - G ( Z ) \\| ^ { 2 } + \\displaystyle \\sum _ { i = 1 } ^ { d } \\sigma _ { i } ^ { 2 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            205,
+            666,
+            792,
+            753
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Together with (11) and the fact that $\\mathcal { P } _ { X , Z } = \\mathcal { P } ( X \\sim P _ { X } , Z \\sim P _ { Z } )$ this concludes the proof. ",
+        "bbox": [
+            168,
+            757,
+            779,
+            775
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C FURTHER DETAILS ON EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            814,
+            517,
+            830
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C.1 MNIST ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            844,
+            272,
+            859
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We use mini-batches of size 100 and trained the models for 100 epochs. We used $\\lambda = 1 0$ and $\\sigma _ { z } ^ { 2 } = 1$ . For the encoder-decoder pair we set $\\alpha = 1 0 ^ { - 3 }$ for Adam in the beginning and for the ",
+        "bbox": [
+            174,
+            871,
+            826,
+            900
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "adversary in WAE-GAN to $\\alpha = 5 \\times 1 0 ^ { - 4 }$ . After 30 epochs we decreased both by factor of 2, and after first 50 epochs further by factor of 5. ",
+        "bbox": [
+            173,
+            103,
+            825,
+            132
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Both encoder and decoder used fully convolutional architectures with 4x4 convolutional filters. ",
+        "bbox": [
+            173,
+            138,
+            795,
+            154
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Encoder architecture: ",
+        "bbox": [
+            174,
+            160,
+            315,
+            174
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4ec08ef2fb77f6182905c1b2463089a01ddd69b21d39057868d34af6a063a0a8.jpg",
+        "text": "$$\n\\begin{array} { r l } { x \\in \\mathcal { R } ^ { 2 8 \\times 2 8 }  \\mathrm { C o n v _ { 1 2 8 } }  \\mathrm { B N }  \\mathrm { R e L U } } & { } \\\\ {  \\mathrm { C o n v _ { 2 5 6 } }  \\mathrm { B N }  \\mathrm { R e L U } } & { } \\\\ {  \\mathrm { C o n v _ { 5 1 2 } }  \\mathrm { B N }  \\mathrm { R e L U } } & { } \\\\ {  \\mathrm { C o n v _ { 1 0 2 4 } }  \\mathrm { B N }  \\mathrm { R e L U }  \\mathrm { F C } _ { 8 } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
+            175,
+            673,
+            248
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Decoder architecture: ",
+        "bbox": [
+            174,
+            257,
+            316,
+            272
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5f2642888c10b61733327ce52932487c854bc6e0a19397bc95bd54e41612f99b.jpg",
+        "text": "$$\n\\begin{array} { r l } & { z \\in \\mathcal { R } ^ { 8 }  \\mathrm { F C } _ { 7 \\times 7 \\times 1 0 2 4 } } \\\\ & { \\qquad \\mathrm { F S C o n v } _ { 5 1 2 }  \\mathrm { B N }  \\mathrm { R e L U } } \\\\ & { \\qquad \\mathrm { F S C o n v } _ { 2 5 6 }  \\mathrm { B N }  \\mathrm { R e L U }  \\mathrm { F S C o n v } _ { 1 } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            273,
+            678,
+            329
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Adversary architecture for WAE-GAN: ",
+        "bbox": [
+            176,
+            337,
+            433,
+            352
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e1f139d7840af0aceff1b6d56897c99b40596ad65bd6ff3df7eeac4824de9bbf.jpg",
+        "text": "$$\n\\begin{array} { r l } & { z \\in \\mathcal { R } ^ { 8 }  \\mathrm { F C } _ { 5 1 2 }  \\mathrm { R e L U } } \\\\ & { ~  \\mathrm { F C } _ { 5 1 2 }  \\mathrm { R e L U } } \\\\ & { ~  \\mathrm { F C } _ { 5 1 2 }  \\mathrm { R e L U } } \\\\ & { ~  \\mathrm { F C } _ { 5 1 2 }  \\mathrm { R e L U }  \\mathrm { F C } _ { 1 } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            375,
+            353,
+            620,
+            426
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Here $\\operatorname { C o n v } _ { k }$ stands for a convolution with $k$ filters, $\\mathrm { F S C o n v } _ { k }$ for the fractional strided convolution with $k$ filters (first two of them were doubling the resolution, the third one kept it constant), BN for the batch normalization, ReLU for the rectified linear units, and $\\mathrm { F C } _ { \\mathrm { k } }$ for the fully connected layer mapping to $\\mathcal { R } ^ { k }$ . All the convolutions in the encoder used vertical and horizontal strides 2 and SAME padding. ",
+        "bbox": [
+            173,
+            434,
+            825,
+            506
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Finally, we used two heuristics. First, we always pretrained separately the encoder for several minibatch steps before the main training stage so that the sample mean and covariance of $Q _ { Z }$ would try to match those of $P _ { Z }$ . Second, while training we were adding a pixel-wise Gaussian noise truncated at 0.01 to all the images before feeding them to the encoder, which was meant to make the encoders random. We played with all possible ways of combining these two heuristics and noticed that together they result in slightly (almost negligibly) better results compared to using only one or none of them. ",
+        "bbox": [
+            173,
+            511,
+            825,
+            609
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Our VAE model used cross-entropy loss (Bernoulli decoder) and otherwise same architectures and hyperparameters as listed above. ",
+        "bbox": [
+            173,
+            616,
+            823,
+            645
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "C.2 CELEBA ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            660,
+            277,
+            675
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We pre-processed CelebA images by first taking a $1 4 0 \\mathrm { x } 1 4 0$ center crops and then resizing to the $6 4 \\mathrm { x } 6 4$ resolution. We used mini-batches of size 100 and trained the models for various number of epochs (up to 250). All reported WAE models were trained for 55 epochs and VAE for 68 epochs. For WAE-MMD we used $\\lambda = 1 0 0$ and for WAE-GAN $\\lambda = 1$ . Both used $\\sigma _ { z } ^ { 2 } = 2$ . ",
+        "bbox": [
+            174,
+            686,
+            825,
+            742
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "For WAE-MMD the learning rate of Adam was initially set to $\\alpha = 1 0 ^ { - 3 }$ . For WAE-GAN the learning rate of Adam for the encoder-decoder pair was initially set to $\\alpha = 3 \\times 1 0 ^ { - 4 }$ and for the adversary to $1 0 ^ { - 3 }$ . All learning rates were decreased by factor of 2 after 30 epochs, further by factor of 5 after 50 first epochs, and finally additional factor of 10 after 100 first epochs. ",
+        "bbox": [
+            173,
+            750,
+            825,
+            806
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Both encoder and decoder used fully convolutional architectures with $5 \\mathrm { x } 5$ convolutional filters. ",
+        "bbox": [
+            169,
+            811,
+            794,
+            827
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Encoder architecture: ",
+        "bbox": [
+            174,
+            834,
+            315,
+            848
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0b4a05953840371a154045fa91646c9656f16a1d3250d31dbf4bb2d438b78793.jpg",
+        "text": "$$\n\\begin{array} { r l } & { x \\in \\mathcal { R } ^ { 6 4 \\times 6 4 \\times 3 } \\to \\mathrm { C o n v _ { 1 2 8 } } \\to \\mathrm { B N } \\to \\mathrm { R e L U } } \\\\ & { \\qquad \\to \\mathrm { C o n v _ { 2 5 6 } } \\to \\mathrm { B N } \\to \\mathrm { R e L U } } \\\\ & { \\qquad \\to \\mathrm { C o n v _ { 5 1 2 } } \\to \\mathrm { B N } \\to \\mathrm { R e L U } } \\\\ & { \\qquad \\to \\mathrm { C o n v _ { 1 0 2 4 } } \\to \\mathrm { B N } \\to \\mathrm { R e L U } \\to \\mathrm { F C } _ { 6 4 } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            313,
+            848,
+            686,
+            922
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Decoder architecture: ",
+        "bbox": [
+            174,
+            103,
+            316,
+            117
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3e05e6baaffed94a4836e3264097296730055df532749b1f7b051889c00086f8.jpg",
+        "text": "$$\n\\begin{array} { r l } { z \\in \\mathcal { R } ^ { 6 4 }  \\mathrm { F C } _ { 8 \\times 8 \\times 1 0 2 4 } } & { } \\\\ {  \\mathrm { F S C o n v } _ { 5 1 2 }  \\mathrm { B N }  \\mathrm { R e L U } } \\\\ {  \\mathrm { F S C o n v } _ { 2 5 6 }  \\mathrm { B N }  \\mathrm { R e L U } } & { } \\\\ {  \\mathrm { F S C o n v } _ { 1 2 8 }  \\mathrm { B N }  \\mathrm { R e L U }  \\mathrm { F S C o n v } _ { 1 } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            312,
+            123,
+            681,
+            196
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Adversary architecture for WAE-GAN: ",
+        "bbox": [
+            174,
+            209,
+            433,
+            224
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/efe2c9405faf9db2c3548f178daba26ef0ed251f5b288574f892bd82ed35f8f0.jpg",
+        "text": "$$\n\\begin{array} { r l } { z \\in \\mathcal { R } ^ { 6 4 } \\to \\mathrm { F C } _ { 5 1 2 } \\to \\mathrm { R e L U } } \\\\ { \\to \\mathrm { F C } _ { 5 1 2 } \\to \\mathrm { R e L U } } \\\\ { \\to \\mathrm { F C } _ { 5 1 2 } \\to \\mathrm { R e L U } } \\\\ { \\to \\mathrm { F C } _ { 5 1 2 } \\to \\mathrm { R e L U } \\to \\mathrm { F C } _ { 1 } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            372,
+            228,
+            624,
+            303
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "For WAE-GAN we used a heuristic proposed in Supplementary $\\mathrm { I V }$ of Mescheder et al. (2017). Notice that the theoretically optimal discriminator would result in $D ^ { * } ( z ) = \\log p _ { Z } ( z ) - \\log q _ { Z } ( z )$ , where $p _ { Z }$ and $q _ { Z }$ are densities of $P _ { Z }$ and $Q _ { Z }$ respectively. In our experiments we added the log prior $\\log p _ { Z } ( z )$ explicitly to the adversary output as we know it analytically. This should hopefully make it easier for the adversary to learn the remaining $Q _ { Z }$ density term. ",
+        "bbox": [
+            173,
+            314,
+            825,
+            386
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Our VAE model used a cross-entropy reconstruction loss (Bernoulli decoder) and $\\alpha = 1 0 ^ { - 4 }$ as the initial Adam learning rate and the same decay schedule as explained above. Otherwise all the architectures and hyperparameters were as explained above. ",
+        "bbox": [
+            174,
+            391,
+            823,
+            434
+        ],
+        "page_idx": 15
+    }
+]

parse/train/r1IRctqxg/r1IRctqxg.md

@@ -0,0 +1,256 @@
+# SAMPLE IMPORTANCE IN TRAINING DEEP NEURAL NETWORKS
+
+Tianxiang Gao, Vladimir Jojic Department of Computer Science University of North Carolina at Chapel Hill Chapel Hill, NC, 27599 , USA {tgao,vjojic}@cs.unc.edu
+
+# ABSTRACT
+
+The contribution of each sample during model training varies across training iterations and the model’s parameters. We define the concept of sample importance as the change in parameters induced by a sample. In this paper, we explored the sample importance in training deep neural networks using stochastic gradient descent. We found that “easy” samples – samples that are correctly and confidently classified at the end of the training – shape parameters closer to the output, while the “hard” samples impact parameters closer to the input to the network. Further, “easy” samples are relevant in the early training stages, and “hard” in the late training stage. Further, we show that constructing batches which contain samples of comparable difficulties tends to be a poor strategy compared to maintaining a mix of both hard and easy samples in all of the batches. Interestingly, this contradicts some of the results on curriculum learning which suggest that ordering training examples in terms of difficulty can lead to better performance.
+
+# 1 INTRODUCTION
+
+Sample importance is the sample’s contribution to the parameter change during training. In statistics, the concept “leverage” of a point is used (St Laurent & Cook (1992)) to measure the impact of a sample on the training of a model. In the context of SVM, the most important samples are the support vectors as they define the separating hyperplane. Understanding the importance of the samples can help us interpret trained models and structure training to speed up convergence and improve prediction accuracy. For example, Curriculum learning (CL) from Bengio et al. (2009) shows that training with easy samples first, then gradually transitioning to difficult samples can improve the learning. In CL, the “easiness” of a sample is predefined either manually or using an evaluation model. Self-paced learning (SPL) (Kumar et al. (2010)) shows that it is possible to learn from samples in order of easiness. In this framework, easiness is related to the prediction error and can be estimated from the model. However, easiness of a sample may not be sufficient to decide when it should be introduced to a learner. Maintaining diversity among the training samples can have a substantial effect on the training (Jiang et al. (2014)).
+
+In this work, we explore the sample importance in deep neural networks. Deep learning methods have been successfully applied in many tasks and routinely achieve better generalization error than classical shallow methods (LeCun et al. (2015)). One of the key characteristics of a deep network is its capacity to construct progressively more complex features throughout its layers (Lee et al. (2011)). An intuitive question arises: which samples contribute the most to the training of the different layer’s parameters? From literature Saxe et al. (2011), we know that even randomly generated filters can compute features that lead to good performance – presumably on easy samples. However, to learn hard samples correctly, the model may need to construct complex features, which require both more training time and refined filters from bottom layers. Hence, we hypothesized that the hard samples shape the bottom layers – closer to the input – and easy samples shape the top layers – closer to the output.
+
+Motivated by the above hypothesis, we analyzed the sample importance in a 3 layer ReLU network on two standard datasets. The results reveal several interesting facts about the sample importance in easy and hard samples:
+
+1. Easy and hard samples impact the parameters in different training stages. The biggest impact of easy samples on parameters are mostly during the early training stage, while the impact of hard samples become large in the late training stage.
+
+2. Easy and hard samples impact the parameters in different layers. Easy samples impact have larger impact on top layer parameters, while hard samples shape the bottom layer parameters.
+
+3. Mixing hard samples with easy samples in each batch helps training. We conducted batches with homogeneous or mixed “easiness”. We found that use of homogeneous batches hinders the training. Hence, it is preferable for network to see both easy and hard samples during all stages of training.
+
+Next, we are going to give the definition of sample importance in Section 2. The empirical analysis for sample importance in the deep neural network in two real datasets is discussed in Section 3. Extension about sample importance is showed in Section 4.
+
+# 2 SAMPLE IMPORTANCE
+
+In this section, we are going to introduce the terminology and provide a quantitative measurement of sample importance for a training procedure.
+
+# 2.1 SAMPLE WEIGHT
+
+In supervised learning, a model is trained by optimizing an objective over a set of observed training samples $\left( \mathbf { x } _ { i } , y _ { i } \right)$ . Let $f ( \mathbf { x } _ { i } , \pmb { \theta } )$ be the output of a model for parameter $\pmb { \theta }$ . The training objective can be written as:
+
+$$
+\sum _ { i = 1 } ^ { n } L ( y _ { i } , f ( \mathbf { x } _ { i } , \pmb { \theta } ) ) + R ( \pmb { \theta } ) ,
+$$
+
+where $L ( y _ { i } , f ( \mathbf { x } _ { i } , \pmb { \theta } ) )$ is the loss on sample $i$ , and $R ( \pmb \theta )$ is the regularization on the parameters. In order to highlight contribution of each sample, we can introduce sample specific weights $v _ { i } \in [ 0 , 1 ]$ which scale sample’s contribution to the loss. Hence, the objective in (1) can be rewritten as:
+
+$$
+\sum _ { i = 1 } ^ { n } v _ { i } L ( y _ { i } , f ( \mathbf { x } _ { i } , \pmb { \theta } ) ) + R ( \pmb { \theta } ) ,
+$$
+
+We define the weight $v _ { i }$ as the sample weight. Similar definitions on $v _ { i }$ has been proposed in Self-paced learning (SPL) Kumar et al. (2010).
+
+In Stochastic Gradient descend (SGD) methods, parameters $\pmb \theta$ are updated with a certain step size $\eta$ in each iteration with regard to a set of training samples. If we allow different sample weights in different iterations, a single update can be written as:
+
+$$
+\pmb { \theta } ^ { t + 1 } = \pmb { \theta } ^ { t } - \eta \sum _ { i = 1 } ^ { n } v _ { i } ^ { t } \mathbf { g } _ { i } ^ { t } - \eta \mathbf { r } ^ { t } ,
+$$
+
+where $\theta ^ { t }$ is the parameter vector at epoch $t$ , $\begin{array} { r } { \mathbf { g } _ { i } ^ { t } = \frac { \partial } { \partial \pmb { \theta } ^ { t } } L _ { i } ( y _ { i } , f ( \mathbf { x } _ { i } , \pmb { \theta } ^ { t } ) ) , \mathbf { r } ^ { t } = \frac { \partial } { \partial \pmb { \theta } ^ { t } } R ( \pmb { \theta } ^ { t } ) } \end{array}$ , and $\boldsymbol { v } _ { i } ^ { t }$ is the weight of ith sample at iteration $t$ .
+
+# 2.2 SAMPLE IMPORTANCE
+
+If we change the weight of a sample $i$ at iteration $t$ , how would such change impact the parameter training in that iteration? We can answer this question by calculating the first order derivative of parameter change $\Delta \pmb { \theta } ^ { t } = \pmb { \theta } ^ { t + 1 } - \pmb { \theta } ^ { t }$ with regard to sample weight $\boldsymbol { v } _ { i } ^ { t }$ :
+
+$$
+{ \boldsymbol { \phi } } _ { i } ^ { t } = { \frac { \partial } { \partial v _ { i } ^ { t } } } \Delta { \pmb { \theta } } ^ { t } = - \eta \mathbf { g } _ { i } ^ { t } .
+$$
+
+We call $\boldsymbol { \phi } _ { i } ^ { t }$ the parameter affectibility by ith sample at iteration $t$ . $\boldsymbol { \phi } _ { i } ^ { t }$ is a vector consists of parameter affectibility from all parameters in the network. Specifically, $\boldsymbol { \phi } _ { i , j } ^ { t }$ is the parameter affectibility for $j$ th parameter in the network. $\boldsymbol { \phi } _ { i } ^ { t }$ reflects the relationship between parameter change and different samples.
+
+Typical deep networks contains millions of parameters. Hence, we are going to focus on groups of parameters of interests. We define ith sample’s importance for parameters of $d \mathrm { t h }$ layer of as:
+
+$$
+\beta _ { i , d } ^ { t } = \sum _ { j \in \mathcal { Q } _ { d } } ( \phi _ { i , j } ^ { t } ) ^ { 2 } ,
+$$
+
+where $\mathcal { Q } _ { d }$ is a set consists of the indexes of all parameters in layer $d$ . Hence, sample’s importance for all the parameters in the model is:
+
+$$
+\alpha _ { i } ^ { t } = \sum _ { j } ( \phi _ { i , j } ^ { t } ) ^ { 2 } ,
+$$
+
+Obviously, we have $\begin{array} { r } { \alpha _ { i } ^ { t } = \sum _ { d = 1 } ^ { D } \beta _ { i , d } ^ { t } } \end{array}$
+
+The sum of sample’s importance across all iterations is defined as overall importance of a sample:
+
+$$
+\tau _ { i } = \sum _ { t } \alpha _ { i } ^ { t }
+$$
+
+In general, for each sample $i$ , computing $\beta _ { i , d } ^ { t }$ allows us to decompose its influence in the model’s training across training stages and different layers.
+
+We note that the sample importance is a high-level measurement of the samples influence on parameters at each iteration of the update. This quantity is not an accurate measurement of the relationship between a sample and final trained model. Refinements of this concept are discussed in Section 4.
+
+# 3 EMPIRICAL ANALYSIS OF SAMPLE IMPORTANCE
+
+We are going to explore the samples’ importance for different layers at different epoch through a series of empirical experiments on two standard datasets.
+
+# 3.1 EXPERIMENT SETUP
+
+Dataset All the analysis are performed on two standard datasets: MNIST 1 (LeCun et al. (1998)), a benchmark dataset that contains handwritten digit images. Each sample is a $2 8 \times 2 8$ image from 10 classes. We used 50000 samples for training and 10000 samples for testing. CIFAR-10 2 (Krizhevsky & Hinton (2009)), a dataset contains $3 2 \times 3 2$ tiny color images from 10 classes. Each sample has 3072 features. We used 50000 samples for training and 10000 samples for testing.
+
+Architecture We used a multilayer feed forward neural network with 3 hidden layers of 512 hidden nodes with rectified linear units (ReLU) activation function, a linear output layer, and a softmax layer on top for classification. The weights in each hidden layer are initialized according to Glorot & Bengio (2010). For hyper-parameters, we used learning rate of 0.1, batch size of 100, 50 total epochs, and weight-decay of $1 e - 5$ . No momentum or learning decay was used. All the code are based on a common deep learning package Theano from Bergstra et al. (2010); Bastien et al. (2012).
+
+# 3.2 SAMPLE IMPORTANCE IS STABLE WITH RESPECT TO DIFFERENT INITIALIZATIONS
+
+Firstly, we want to explore whether the sample importance is stable under different initializations. We used three different random seeds to initialize the network parameters and calculated the sample importance every five epochs. We computed the Spearman’s rank correlation between sample importance to the model, $\alpha _ { i } ^ { t }$ , in each pair of initializations. This correlation remains high in all epochs, above 0.9, as shown in Figure 1. This indicates that the sample importance is relatively stable to initialization of the network. Hence, all the following analysis are based on the results from initialization seed 1. (Details of training and test error for the chosen model can be viewed in Appendix Figure 9).
+
+![](images/0a3fc09ab1365aa5c2ccb287411ae343403bce7f9654cd12222a19101f92d457.jpg)  
+Figure 1: Does initialization affect sample’s importance? Sample importance is preserved between initializations of the network. For each epoch, and a pair of initializations, we computed Spearman Correlation of samples’ importance. Across all epochs, the correlation is of greater than 0.9 in both MNIST and CIFAR-10. Early epochs show higher consistency between ranks of sample importance across different initializations.
+
+# 3.3 DECOMPOSITION OF SAMPLE IMPORTANCE
+
+To better understand and visualize the sample importance, we firstly calculate the overall sample importance at each epoch as $\begin{array} { r } { A ^ { t } = \sum _ { i = 1 } ^ { n } \dot { \alpha _ { i } ^ { t } } } \end{array}$ . Similarly, the overall sample importance to layer $d$ is $\begin{array} { r } { B _ { d } ^ { t } = \sum _ { i = 1 } ^ { n } \beta _ { i , d } ^ { t } } \end{array}$ . We show the overall sample importance and its decomposition in layers for two datasets in Figure 2. Firstly, we found that even with a fixed learning rate, the overall sample importance is different under different epochs. Output layer always has the largest average sample importance per parameter, and its contribution reaches the maximum in the early training stage and then drops. Each layer contributes differently to the total sample importance. In both MNIST and CIFAR-10, the 2nd layer contributes more than the 3rd layer. In CIFAR-10, the 1st layer’s provides largest contribution the total sample importance, as it contains much more parameters than other layers. Interestingly, all classes do not provide the same amount of sample importance.
+
+We found that most samples have small sample importance (Appendix Figure 10). To visualize the contribution of different samples, we split the samples based on their total importance into three groups: $10 \%$ , top $10 \%$ – top $20 \%$ most important samples, and other $80 \%$ samples. We show the decomposition of importance contribution in each layer in Figure 3. In MNIST, the top $10 \%$ samples contribute almost all the sample importance. In CIFAR-10, most important samples contribute more in lower layers rather than output layer. This result indicates that top $20 \%$ most important samples contribute to the majority of the sample importance.
+
+# 3.4 SAMPLE IMPORTANCE AND NEGATIVE LOG-LIKELIHOOD
+
+Negative log likelihood (NLL) is the loss metric we used for training objective. It has been used to measure the “easiness” of a sample in Curriculum learning Bengio et al. (2009) and Self-paced learning Kumar et al. (2010). Intuitively, the samples with large NLL should also have large sample importance (SI). However, in our experiment, we found that this is not always the case. In Figure 4, we found that 1) NLL and SI become more correlated as training goes on. However, 2) NLL is not predictive of the SI. There are many points with high NLL but small SI, and otherwise.
+
+![](images/7b01f6ee56ad410d554315b0e829197fb194fba35538582c41bb8eb261a1288f.jpg)  
+Figure 2: Which classes and at which stage shape the network’s layer’s parameters? Parameters of different layers are learned at different times. Parameters in Output layers are learned mostly during the early training stage. In the lower layers, parameters are learned predominantly during the middle and late training stage. All classes do not contribute equally to training of the model.
+
+![](images/c65928096336387e31f6986d8a5c1157a406686b28f9f2fb2cc62ef0c29e1cd8.jpg)  
+Figure 3: Are all data samples equally important for all layers? The top $20 \%$ most important samples contributes to the majority of parameter learning, especially in lower layers. “L1” to “L3” stands for Layer 1 to Layer 3. “Out” stands for output layer.
+
+# 3.5 CLUSTERING SAMPLES BASED ON SAMPLE IMPORTANCE
+
+To better visualize the importance of different samples, we provide three representative clusters of samples for each dataset. In MNIST, we clustered all digit “5” samples into 20 clusters based on their epoch-specific, layer-specific sample importance. In CIFAR-10, we clustered all “horse” samples into 30 clusters using the same features as MNIST. Kmeans algorithm is used for clustering.
+
+![](images/366cabd6209561647cb6d00ed6cdc1387b95f239c1a5c25a3d36e0ae64232719.jpg)  
+Figure 4: Is Sample Importance correlated with Negative log-likelihood of a sample? Sample importance is positively correlated with negative log-likelihood. As training goes on, their correlation becomes higher. However, there remain many samples with high NLL and low SI, and vice versa. Left column: correlation between sample importance and negative log likelihood for all samples across epochs. Right column: scatter plot for NLL in the last epoch and all epoch sample importance for each sample.
+
+MNIST In Figure 5, we showed 3 example clusters on digit $\cdot 5 ^ { , }$ . In the cluster of easy samples, where NLL converges very fast, most of the sample importance is concentrated in the first few epochs in output layer parameters. The cluster of medium samples has a slow NLL convergence compared to the easy cluster. The biggest impact is in middle training stage and more towards bottom layer parameters. Hard samples hardly converge even during the late stage of training. As training goes on, the sample importance for the bottom layer parameters become larger.
+
+CIFAR-10 In Figure 6, we showed 3 examples clusters on class “horse”. We observed very similar sample importance changing pattern as for the MNIST examples for easy, medium and hard clusters. Comparing to MNIST, all three clusters in CIFAR-10 have a very large impact on the parameters in the bottom layer. We note that the CIFAR-10 has almost 4 times larger number of parameters $( 3 0 7 5 \times 5 1 2 \approx 1 5 7 4 k )$ in the first layer than MNIST $7 8 5 \times 5 1 2 \approx 4 0 1 k$ ).
+
+# 3.6 BATCH ORDER AND SAMPLE IMPORTANCE
+
+With the observations from empirical analysis on sample importance, we know that time – iteration – and place – layer – of sample’s impact varies according to its “easiness” . We wanted to know whether constructing batches based on the sample importance or negative log likelihood would make a difference in training. Hence, we designed an experiment to explore how the different construction of batches influence the training. We note that the information used to structure the batches (negative log-likelihood and sample importance) – was obtained from a full training run.
+
+We split all 50, 000 samples into $b = 5 0 0$ batch subsets $\{ B _ { 1 } , B _ { 2 } , \ldots , B _ { b } \}$ . Each batch has batch size $| B _ { i } | = 1 0 0$ . In our experiment, each training sample must be in exactly one batch. There is no intersection between batches.
+
+During training, in each epoch, we update the parameters with each batch in order $1 , 2 , \ldots , b$ iteratively.
+
+We used seven different batch construction methods in this experiment:
+
+![](images/beca3d0da8b2003b41df3f1588b013088625e0966c9a1ccd126e7c77463862f6.jpg)  
+Figure 5: When and where does an MNIST sample make the biggest impact? For “easy” samples, their biggest impact is on output layers and during the early training stage. As sample’s difficulty increases (medium and hard), the biggest impact moves to lower layers and in the late training stage. Each row is a sample cluster. In each row, from left to right: example images in the cluster; average sample importance and layer-wise decomposition across epochs; A boxplot of average training negative log likelihood across epochs.
+
+1. Rand: Randomly constructed batch. All $5 0 \mathrm { k }$ samples are randomly split into $b$ batches before training. The batches and orders stay fixed during training.   
+2. NLO: Negative Log-likelihood Order. We sort all the samples based on their final NLL from low to high. The batches are constructed based on the sorted samples. First 100 samples with least NLL are in $\boldsymbol { B } _ { 1 }$ , 101 to 200 samples are in $B _ { 2 }$ , and so on. Hence, during training, the batches with small NLL will be trained first.   
+3. RNLO Reverse-Negative Log-likelihood Order. We construct the batches same as NLO. During training, we update the batches in reverse order $B _ { b } , B _ { b - 1 } , \ldots , B _ { 1 }$ . Hence, the batches with large NLL will be trained first.   
+4. NLM Negative Log-likelihood Mixed. We sort all the samples based on their final NLL from low to high. Next, for each sample $i$ in the sorted sequence, we put that sample into batch $j = ( i { \bmod { b } } ) + 1$ . This ordering constructs batches out of samples with diverse levels of NLL.   
+5. SIO: Sample Importance Order. Similar to NLO, except that we sort all the samples based on their sum sample importance over all epochs from low to high. Hence, batches with small sample importance will be trained first.   
+6. RSIO Reverse-Sample Importance Order. We construct the batches same as SIO. During training, we update the batches in reverse order $B _ { b } , B _ { b - 1 } , \ldots , B _ { 1 }$ . Hence, during training, the batches with large sample importance will be trained first.
+
+![](images/06ecd2bfd33d18b39d48776d6ab4984ee42b58766a17f2e0d0d3d92cd9424928.jpg)  
+Figure 6: When and where does a CIFAR-10 sample make the biggest impact? For “easy” samples, their biggest impact is on the first layer during the early training stage. As samples’s difficulty increases (medium and hard), the biggest impact moves to lower layers and to late training stage. Each row is a sample cluster. In each row, from left to right: example images in the cluster; average sample importance and layer-wise decomposition across epochs; A boxplot of average training negative log likelihood across epochs.
+
+7. SIM Sample Important Mixed. Similar to NLM, but we sort the samples based on overall sample importance. Thus, batches contain samples with divers sample importance.
+
+We performed five different runs (with different random initializations) on MNIST and CIFAR-10. The result is shown in Figure 7. From the result, we found that: 1) In both MNIST and CIFAR-10, Rand, SIS, and NLS have the lowest test error compared to all other methods. This indicates that diverse batches are helpful for training. 2) NLO and SIO got the worst performance in CIFAR10. Their training error even goes up after the early stage. RNLO and RSIO have same batch constructions as NLO and SIO, but their performances are drastically different. This indicates that the order of batches during training is important. Further, training on easy samples first and hard later seems to be counter-productive.
+
+To better understand the impact of different batch construction, we performed the principle component analysis on the learned parameters in each epoch (Figure 8). In MNIST, the impact of batch construction is not very significant. In CIFAR-10, batch construction and even the order of batch training do have a large impact on the training.
+
+Our experiment result shows a different conclusion to Curriculum Learning and Self-paced learning, where easy samples are trained on before introducing hard samples. We found that constructing and ordering the batches – hard to easy and easy to hard – seems to hinder the performance of learning. Having hard samples mixed in with the easy ones in each batch helps the training.
+
+Also, the results show that we want to learn from the hard samples in early epochs and “see” hard samples more frequently, even if their major impact on parameters is during the late stage. As hard examples are few compared to easy samples and hard examples need a longer time to train, we do want to mix the hard samples into each batch to start learning from those samples early and learn longer.
+
+![](images/42716aa645f0c758ca13ad616d456e30f3c7f969dca4da6b441665f267b2d69c.jpg)  
+Figure 7: Does organizing batches by “easiness” affect training? When batches are constructed with homogeneous easiness, the training performance become worse. Batches with mixed easiness have lower test error. The solid color line represents the mean over 5 runs. The error bar indicates the standard error over 5 runs.
+
+![](images/2356039eb64f765e3572f2159baff9807ba4501ba29d7c5578292085327cf2f9.jpg)  
+Figure 8: Do parameters converge differently under different batch construction? In MNIST, the converging path for all batch constructions are very similar. In CIFAR-10, batch construction with mixed easiness (Rand, NLM, SIM) has a very different converging path with all other methods. Notably, we found that even with same batch constructions but just reversed order (NLO vs. RNLO, SIO vs. RSIO), the parameters converge to different points. Each circle dotted line shows the path of the first two principle components of all parameters in different epochs. Note that in CIFAR-10, the paths of Rand, NLS and SIS are very similar and they are overlapped in the plot.
+
+# 4 EXTENSIONS OF SAMPLE IMPORTANCE
+
+We calculated the sample importance in each iteration in Stochastic Gradient Descent. However, such quantity only reflects the impact on the change in parameters within each iteration. The influence of a sample at a particular iteration can be accumulated through updates and impact the final model. Here, we are going to derive the exact calculation of the sample’s influence to the model. We rewrite the Objective (2) in Section 2 here:
+
+$$
+\operatorname* { m i n } _ { \pmb { \theta } } \sum _ { i = 1 } ^ { n } v _ { i } L ( y _ { i } , f ( \mathbf { x } _ { i } , \pmb { \theta } ) ) + R ( \pmb { \theta } ) ,
+$$
+
+Here, we deem the sample weight $v _ { i }$ is fixed across all iterations. The update rule for stochastic gradient descent in each iteration is:
+
+$$
+\pmb { \theta } ^ { t + 1 } = \pmb { \theta } ^ { t } - \eta \sum _ { i } v _ { i } \mathbf { g } _ { i } ^ { t } - \eta \mathbf { r } _ { i } ^ { t }
+$$
+
+The derivative of $\pmb { \theta } ^ { t + 1 }$ with respect to sample weight $v _ { i }$ is:
+
+$$
+\begin{array} { r c l } { { \displaystyle \frac { \partial } { \partial v _ { i } } \pmb { \theta } ^ { t + 1 } } } & { { = } } & { { \displaystyle \frac { \partial } { \partial v _ { i } } \pmb { \theta } ^ { t } - \eta \mathbf { g } _ { i } ^ { t } - \eta H ( \pmb { \theta } ^ { t } ) \frac { \partial } { \partial v _ { i } } \pmb { \theta } ^ { t } } } \\ { { \displaystyle \frac { \partial } { \partial v _ { i } } \pmb { \theta } ^ { 1 } } } & { { = } } & { { \displaystyle - \eta \mathbf { g } _ { i } ^ { 0 } , } } \end{array}
+$$
+
+where $H ( \pmb \theta ^ { t } )$ is the Hessian matrix of the objective in (2) with regard to all parameters in iteration $t$ . If we iterate the updates until convergence, then we can assume that $\theta ^ { T }$ is a fix-point, $\pmb { \theta } ^ { \ast } = \pmb { \theta } ^ { T + 1 } =$ $\theta ^ { T }$ , and we obtain:
+
+$$
+\frac { \partial } { \partial v _ { i } } \pmb { \theta } ^ { T + 1 } - \frac { \partial } { \partial v _ { i } } \pmb { \theta } ^ { T } = - \eta \mathbf { g } _ { i } ^ { t } - \eta H ( \pmb { \theta } ^ { * } ) \frac { \partial } { \partial v _ { i } } \pmb { \theta } ^ { * }
+$$
+
+Hence, the derivative of parameters in the final model with regard to a sample weight is:
+
+$$
+\frac { \partial } { \partial v _ { i } } \pmb { \theta } ^ { * } = - H ( \pmb { \theta } ^ { * } ) ^ { - 1 } \mathbf { g } _ { i } ^ { T }
+$$
+
+Equation (4) indicates that we can calculate the sample specific impact on final trained model by using the parameters learned at the convergence point. In deep learning methods, due to early stopping, fix point might not be achieved, and Equation (4) might not be an accurate.
+
+For any target quantity $\boldsymbol { \mathcal { T } } ( \boldsymbol { \theta } ^ { * } )$ that depends on the final trained parameter $\pmb { \theta } ^ { * }$ , we can calculate the impact of a particular sample on that target as:
+
+$$
+\frac { \partial } { \partial v _ { i } } T ( \pmb { \theta } ^ { * } ) = \frac { \partial } { \partial \pmb { \theta } ^ { * } } T ( \pmb { \theta } ^ { * } ) \frac { \partial } { \partial v _ { i } } \pmb { \theta } ^ { * }
+$$
+
+For example, if we are interested in the sum of predictions on a set of samples ${ \mathcal { T } } ( \theta ^ { * } ) ~ =$ $\textstyle \sum _ { i \in { \mathcal { S } } _ { c } } f ( \mathbf { x } _ { i } , { \pmb \theta } ^ { t } )$ , we can use Equation (5) to calculate the derivative:
+
+$$
+\begin{array} { r c l } { \displaystyle \frac { \partial } { \partial v _ { i } } \mathcal { T } ( \pmb { \theta } ^ { * } ) } & { = } & { \displaystyle \sum _ { i \in \mathcal { S } _ { c } } \frac { \partial } { \partial \pmb { \theta } ^ { * } } f ( \mathbf { x } _ { i } , \pmb { \theta } ^ { t } ) \frac { \partial } { \partial v _ { i } } \pmb { \theta } ^ { * } } \end{array}
+$$
+
+We note evaluating the exact impact of a sample, as shown above, is computationally cumbersome for all but the simplest models.
+
+# 5 DISCUSSION
+
+Samples’ impact on the deep network’s parameters vary across stages of training and network’s layers. In our work, we found that easy samples predominantly shape parameters the top layers at the early training stages, while hard samples predominantly shape the parameters of the bottom layers at the late training stage. Our experiments show that it is important to mix hard samples into different batches rather than keep them together in the same batch and away from other examples.
+
+There are many future extensions to the current work. Firstly, we want to expand our sample importance analysis to different deep learning structures, like Convolution Neural Network and Recurrent Neural Networks. Secondly, we want to use the sample importance as a guidance to extract a minimal subset of samples that are sufficient to achieve performance comparable to a network trained on the full dataset.
+
+# REFERENCES
+
+Fred´ eric Bastien, Pascal Lamblin, Razvan Pascanu, James Bergstra, Ian J. Goodfellow, Arnaud ´ Bergeron, Nicolas Bouchard, and Yoshua Bengio. Theano: new features and speed improvements. Deep Learning and Unsupervised Feature Learning NIPS 2012 Workshop, 2012.
+
+Yoshua Bengio, Jer´ ome Louradour, Ronan Collobert, and Jason Weston. Curriculum learning. In ˆ Proceedings of the 26th annual international conference on machine learning, pp. 41–48. ACM, 2009.
+
+James Bergstra, Olivier Breuleux, Fred´ eric Bastien, Pascal Lamblin, Razvan Pascanu, Guillaume ´ Desjardins, Joseph Turian, David Warde-Farley, and Yoshua Bengio. Theano: a CPU and GPU math expression compiler. In Proceedings of the Python for Scientific Computing Conference (SciPy), June 2010. Oral Presentation.
+
+Xavier Glorot and Yoshua Bengio. Understanding the difficulty of training deep feedforward neural networks. In International Conference on Artificial Intelligence and Statistics, pp. 249–256, 2010.
+
+Lu Jiang, Deyu Meng, Shoou-I Yu, Zhenzhong Lan, Shiguang Shan, and Alexander Hauptmann. Self-paced learning with diversity. In Advances in Neural Information Processing Systems, pp. 2078–2086, 2014.
+
+Alex Krizhevsky and Geoffrey Hinton. Learning multiple layers of features from tiny images, 2009.
+
+M Pawan Kumar, Benjamin Packer, and Daphne Koller. Self-paced learning for latent variable models. In Advances in Neural Information Processing Systems, pp. 1189–1197, 2010.
+
+Yann LeCun, Corinna Cortes, and Christopher JC Burges. The MNIST database of handwritten digits, 1998.
+
+Yann LeCun, Yoshua Bengio, and Geoffrey Hinton. Deep learning. Nature, 521(7553):436–444, 2015.
+
+Honglak Lee, Roger Grosse, Rajesh Ranganath, and A. Y. Ng. Unsupervised learning of hierarchical representations with convolutional deep belief networks. Communications of the ACM, 54(10): 95–103, 2011.
+
+Andrew Saxe, Pang W Koh, Zhenghao Chen, Maneesh Bhand, Bipin Suresh, and Andrew Y Ng. On random weights and unsupervised feature learning. In Proceedings of the 28th international conference on machine learning (ICML-11), pp. 1089–1096, 2011.
+
+Roy T St Laurent and R Dennis Cook. Leverage and superleverage in nonlinear regression. Journal of the American Statistical Association, 87(420):985–990, 1992.
+
+# APPENDIX
+
+![](images/b50b1bc5189d4cb3fa5ad2950a2e80b6a0c2b3a70f01fab8b5c6bf2838fbfbfa.jpg)  
+Figure 9: The training and test error on MNIST (first row) and CIFAR-10 (second row). The left column showed the average class-specific negative log likelihood.
+
+![](images/f8db2eb88900f5393079c3638df2a7dc58b9a63367a73c5ed63a70e9013e6e1f.jpg)  
+Figure 10: Histogram of total sample importance.

parse/train/r1IRctqxg/r1IRctqxg_content_list.json

@@ -0,0 +1,1286 @@
+[
+    {
+        "type": "text",
+        "text": "SAMPLE IMPORTANCE IN TRAINING DEEP NEURAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
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+            98,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Tianxiang Gao, Vladimir Jojic Department of Computer Science University of North Carolina at Chapel Hill Chapel Hill, NC, 27599 , USA {tgao,vjojic}@cs.unc.edu ",
+        "bbox": [
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+            241
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
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+            291
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The contribution of each sample during model training varies across training iterations and the model’s parameters. We define the concept of sample importance as the change in parameters induced by a sample. In this paper, we explored the sample importance in training deep neural networks using stochastic gradient descent. We found that “easy” samples – samples that are correctly and confidently classified at the end of the training – shape parameters closer to the output, while the “hard” samples impact parameters closer to the input to the network. Further, “easy” samples are relevant in the early training stages, and “hard” in the late training stage. Further, we show that constructing batches which contain samples of comparable difficulties tends to be a poor strategy compared to maintaining a mix of both hard and easy samples in all of the batches. Interestingly, this contradicts some of the results on curriculum learning which suggest that ordering training examples in terms of difficulty can lead to better performance. ",
+        "bbox": [
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+            316,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            544,
+            336,
+            560
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Sample importance is the sample’s contribution to the parameter change during training. In statistics, the concept “leverage” of a point is used (St Laurent & Cook (1992)) to measure the impact of a sample on the training of a model. In the context of SVM, the most important samples are the support vectors as they define the separating hyperplane. Understanding the importance of the samples can help us interpret trained models and structure training to speed up convergence and improve prediction accuracy. For example, Curriculum learning (CL) from Bengio et al. (2009) shows that training with easy samples first, then gradually transitioning to difficult samples can improve the learning. In CL, the “easiness” of a sample is predefined either manually or using an evaluation model. Self-paced learning (SPL) (Kumar et al. (2010)) shows that it is possible to learn from samples in order of easiness. In this framework, easiness is related to the prediction error and can be estimated from the model. However, easiness of a sample may not be sufficient to decide when it should be introduced to a learner. Maintaining diversity among the training samples can have a substantial effect on the training (Jiang et al. (2014)). ",
+        "bbox": [
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+            583,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this work, we explore the sample importance in deep neural networks. Deep learning methods have been successfully applied in many tasks and routinely achieve better generalization error than classical shallow methods (LeCun et al. (2015)). One of the key characteristics of a deep network is its capacity to construct progressively more complex features throughout its layers (Lee et al. (2011)). An intuitive question arises: which samples contribute the most to the training of the different layer’s parameters? From literature Saxe et al. (2011), we know that even randomly generated filters can compute features that lead to good performance – presumably on easy samples. However, to learn hard samples correctly, the model may need to construct complex features, which require both more training time and refined filters from bottom layers. Hence, we hypothesized that the hard samples shape the bottom layers – closer to the input – and easy samples shape the top layers – closer to the output. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Motivated by the above hypothesis, we analyzed the sample importance in a 3 layer ReLU network on two standard datasets. The results reveal several interesting facts about the sample importance in easy and hard samples: ",
+        "bbox": [
+            174,
+            103,
+            823,
+            146
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1. Easy and hard samples impact the parameters in different training stages. The biggest impact of easy samples on parameters are mostly during the early training stage, while the impact of hard samples become large in the late training stage. ",
+        "bbox": [
+            174,
+            152,
+            821,
+            195
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2. Easy and hard samples impact the parameters in different layers. Easy samples impact have larger impact on top layer parameters, while hard samples shape the bottom layer parameters. ",
+        "bbox": [
+            171,
+            202,
+            823,
+            231
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3. Mixing hard samples with easy samples in each batch helps training. We conducted batches with homogeneous or mixed “easiness”. We found that use of homogeneous batches hinders the training. Hence, it is preferable for network to see both easy and hard samples during all stages of training. ",
+        "bbox": [
+            174,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Next, we are going to give the definition of sample importance in Section 2. The empirical analysis for sample importance in the deep neural network in two real datasets is discussed in Section 3. Extension about sample importance is showed in Section 4. ",
+        "bbox": [
+            174,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 SAMPLE IMPORTANCE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            362,
+            390,
+            378
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this section, we are going to introduce the terminology and provide a quantitative measurement of sample importance for a training procedure. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 SAMPLE WEIGHT ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            439,
+            336,
+            454
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In supervised learning, a model is trained by optimizing an objective over a set of observed training samples $\\left( \\mathbf { x } _ { i } , y _ { i } \\right)$ . Let $f ( \\mathbf { x } _ { i } , \\pmb { \\theta } )$ be the output of a model for parameter $\\pmb { \\theta }$ . The training objective can be written as: ",
+        "bbox": [
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+            464,
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6c9700eb2b2cae7d3282a001c5cf3d7cfe57b076643c802dab9b5964942bf55a.jpg",
+        "text": "$$\n\\sum _ { i = 1 } ^ { n } L ( y _ { i } , f ( \\mathbf { x } _ { i } , \\pmb { \\theta } ) ) + R ( \\pmb { \\theta } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            401,
+            502,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $L ( y _ { i } , f ( \\mathbf { x } _ { i } , \\pmb { \\theta } ) )$ is the loss on sample $i$ , and $R ( \\pmb \\theta )$ is the regularization on the parameters. In order to highlight contribution of each sample, we can introduce sample specific weights $v _ { i } \\in [ 0 , 1 ]$ which scale sample’s contribution to the loss. Hence, the objective in (1) can be rewritten as: ",
+        "bbox": [
+            174,
+            546,
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/908eabf6f381664fd44fceefb840a8d6b32a49e06bd6bb8d0623fbfe011e5f72.jpg",
+        "text": "$$\n\\sum _ { i = 1 } ^ { n } v _ { i } L ( y _ { i } , f ( \\mathbf { x } _ { i } , \\pmb { \\theta } ) ) + R ( \\pmb { \\theta } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            395,
+            594,
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+            636
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We define the weight $v _ { i }$ as the sample weight. Similar definitions on $v _ { i }$ has been proposed in Self-paced learning (SPL) Kumar et al. (2010). ",
+        "bbox": [
+            173,
+            640,
+            823,
+            670
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In Stochastic Gradient descend (SGD) methods, parameters $\\pmb \\theta$ are updated with a certain step size $\\eta$ in each iteration with regard to a set of training samples. If we allow different sample weights in different iterations, a single update can be written as: ",
+        "bbox": [
+            173,
+            676,
+            825,
+            718
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f911e47349428d3009a00bd32a2c526871c2ec2abfcf626fe3b2210dd57be4e3.jpg",
+        "text": "$$\n\\pmb { \\theta } ^ { t + 1 } = \\pmb { \\theta } ^ { t } - \\eta \\sum _ { i = 1 } ^ { n } v _ { i } ^ { t } \\mathbf { g } _ { i } ^ { t } - \\eta \\mathbf { r } ^ { t } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            395,
+            723,
+            601,
+            765
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $\\theta ^ { t }$ is the parameter vector at epoch $t$ , $\\begin{array} { r } { \\mathbf { g } _ { i } ^ { t } = \\frac { \\partial } { \\partial \\pmb { \\theta } ^ { t } } L _ { i } ( y _ { i } , f ( \\mathbf { x } _ { i } , \\pmb { \\theta } ^ { t } ) ) , \\mathbf { r } ^ { t } = \\frac { \\partial } { \\partial \\pmb { \\theta } ^ { t } } R ( \\pmb { \\theta } ^ { t } ) } \\end{array}$ , and $\\boldsymbol { v } _ { i } ^ { t }$ is the weight of ith sample at iteration $t$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "2.2 SAMPLE IMPORTANCE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            818,
+            369,
+            833
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "If we change the weight of a sample $i$ at iteration $t$ , how would such change impact the parameter training in that iteration? We can answer this question by calculating the first order derivative of parameter change $\\Delta \\pmb { \\theta } ^ { t } = \\pmb { \\theta } ^ { t + 1 } - \\pmb { \\theta } ^ { t }$ with regard to sample weight $\\boldsymbol { v } _ { i } ^ { t }$ : ",
+        "bbox": [
+            174,
+            844,
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+            888
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/78153153295e27c983546c99d6faeb9380990239d25380c67002d86e081ab9d0.jpg",
+        "text": "$$\n{ \\boldsymbol { \\phi } } _ { i } ^ { t } = { \\frac { \\partial } { \\partial v _ { i } ^ { t } } } \\Delta { \\pmb { \\theta } } ^ { t } = - \\eta \\mathbf { g } _ { i } ^ { t } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            418,
+            895,
+            580,
+            928
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We call $\\boldsymbol { \\phi } _ { i } ^ { t }$ the parameter affectibility by ith sample at iteration $t$ . $\\boldsymbol { \\phi } _ { i } ^ { t }$ is a vector consists of parameter affectibility from all parameters in the network. Specifically, $\\boldsymbol { \\phi } _ { i , j } ^ { t }$ is the parameter affectibility for $j$ th parameter in the network. $\\boldsymbol { \\phi } _ { i } ^ { t }$ reflects the relationship between parameter change and different samples. ",
+        "bbox": [
+            174,
+            102,
+            825,
+            165
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Typical deep networks contains millions of parameters. Hence, we are going to focus on groups of parameters of interests. We define ith sample’s importance for parameters of $d \\mathrm { t h }$ layer of as: ",
+        "bbox": [
+            173,
+            171,
+            825,
+            200
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8626b5054cbdde3860883ba9c05d0db064409126f521bc710cb5c5be7c6c90de.jpg",
+        "text": "$$\n\\beta _ { i , d } ^ { t } = \\sum _ { j \\in \\mathcal { Q } _ { d } } ( \\phi _ { i , j } ^ { t } ) ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            431,
+            205,
+            563,
+            241
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { Q } _ { d }$ is a set consists of the indexes of all parameters in layer $d$ . Hence, sample’s importance for all the parameters in the model is: ",
+        "bbox": [
+            174,
+            246,
+            825,
+            275
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a5f3836f8436def08a638eb774f585b8a6c938e4ea65ef386dd35259b0013a1d.jpg",
+        "text": "$$\n\\alpha _ { i } ^ { t } = \\sum _ { j } ( \\phi _ { i , j } ^ { t } ) ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            441,
+            281,
+            555,
+            316
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Obviously, we have $\\begin{array} { r } { \\alpha _ { i } ^ { t } = \\sum _ { d = 1 } ^ { D } \\beta _ { i , d } ^ { t } } \\end{array}$ ",
+        "bbox": [
+            174,
+            323,
+            419,
+            342
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The sum of sample’s importance across all iterations is defined as overall importance of a sample: ",
+        "bbox": [
+            173,
+            348,
+            812,
+            363
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/24968536f95e16a13035e3f1082522378fa5bfb112efbdcb7d333c36ee2a3446.jpg",
+        "text": "$$\n\\tau _ { i } = \\sum _ { t } \\alpha _ { i } ^ { t }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            459,
+            368,
+            539,
+            402
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In general, for each sample $i$ , computing $\\beta _ { i , d } ^ { t }$ allows us to decompose its influence in the model’s training across training stages and different layers. ",
+        "bbox": [
+            171,
+            415,
+            823,
+            446
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We note that the sample importance is a high-level measurement of the samples influence on parameters at each iteration of the update. This quantity is not an accurate measurement of the relationship between a sample and final trained model. Refinements of this concept are discussed in Section 4. ",
+        "bbox": [
+            174,
+            453,
+            823,
+            496
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 EMPIRICAL ANALYSIS OF SAMPLE IMPORTANCE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            516,
+            606,
+            531
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We are going to explore the samples’ importance for different layers at different epoch through a series of empirical experiments on two standard datasets. ",
+        "bbox": [
+            174,
+            546,
+            825,
+            574
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 EXPERIMENT SETUP ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            592,
+            357,
+            607
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Dataset All the analysis are performed on two standard datasets: MNIST 1 (LeCun et al. (1998)), a benchmark dataset that contains handwritten digit images. Each sample is a $2 8 \\times 2 8$ image from 10 classes. We used 50000 samples for training and 10000 samples for testing. CIFAR-10 2 (Krizhevsky & Hinton (2009)), a dataset contains $3 2 \\times 3 2$ tiny color images from 10 classes. Each sample has 3072 features. We used 50000 samples for training and 10000 samples for testing. ",
+        "bbox": [
+            173,
+            617,
+            825,
+            688
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Architecture We used a multilayer feed forward neural network with 3 hidden layers of 512 hidden nodes with rectified linear units (ReLU) activation function, a linear output layer, and a softmax layer on top for classification. The weights in each hidden layer are initialized according to Glorot & Bengio (2010). For hyper-parameters, we used learning rate of 0.1, batch size of 100, 50 total epochs, and weight-decay of $1 e - 5$ . No momentum or learning decay was used. All the code are based on a common deep learning package Theano from Bergstra et al. (2010); Bastien et al. (2012). ",
+        "bbox": [
+            174,
+            703,
+            825,
+            787
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 SAMPLE IMPORTANCE IS STABLE WITH RESPECT TO DIFFERENT INITIALIZATIONS ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            804,
+            776,
+            818
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Firstly, we want to explore whether the sample importance is stable under different initializations. We used three different random seeds to initialize the network parameters and calculated the sample importance every five epochs. We computed the Spearman’s rank correlation between sample importance to the model, $\\alpha _ { i } ^ { t }$ , in each pair of initializations. This correlation remains high in all epochs, above 0.9, as shown in Figure 1. This indicates that the sample importance is relatively stable to initialization of the network. Hence, all the following analysis are based on the results from initialization seed 1. (Details of training and test error for the chosen model can be viewed in Appendix Figure 9). ",
+        "bbox": [
+            174,
+            829,
+            825,
+            886
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            825,
+            160
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/0a3fc09ab1365aa5c2ccb287411ae343403bce7f9654cd12222a19101f92d457.jpg",
+        "image_caption": [
+            "Figure 1: Does initialization affect sample’s importance? Sample importance is preserved between initializations of the network. For each epoch, and a pair of initializations, we computed Spearman Correlation of samples’ importance. Across all epochs, the correlation is of greater than 0.9 in both MNIST and CIFAR-10. Early epochs show higher consistency between ranks of sample importance across different initializations. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            274,
+            188,
+            723,
+            358
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3 DECOMPOSITION OF SAMPLE IMPORTANCE ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            503,
+            509,
+            517
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To better understand and visualize the sample importance, we firstly calculate the overall sample importance at each epoch as $\\begin{array} { r } { A ^ { t } = \\sum _ { i = 1 } ^ { n } \\dot { \\alpha _ { i } ^ { t } } } \\end{array}$ . Similarly, the overall sample importance to layer $d$ is $\\begin{array} { r } { B _ { d } ^ { t } = \\sum _ { i = 1 } ^ { n } \\beta _ { i , d } ^ { t } } \\end{array}$ . We show the overall sample importance and its decomposition in layers for two datasets in Figure 2. Firstly, we found that even with a fixed learning rate, the overall sample importance is different under different epochs. Output layer always has the largest average sample importance per parameter, and its contribution reaches the maximum in the early training stage and then drops. Each layer contributes differently to the total sample importance. In both MNIST and CIFAR-10, the 2nd layer contributes more than the 3rd layer. In CIFAR-10, the 1st layer’s provides largest contribution the total sample importance, as it contains much more parameters than other layers. Interestingly, all classes do not provide the same amount of sample importance. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We found that most samples have small sample importance (Appendix Figure 10). To visualize the contribution of different samples, we split the samples based on their total importance into three groups: $10 \\%$ , top $10 \\%$ – top $20 \\%$ most important samples, and other $80 \\%$ samples. We show the decomposition of importance contribution in each layer in Figure 3. In MNIST, the top $10 \\%$ samples contribute almost all the sample importance. In CIFAR-10, most important samples contribute more in lower layers rather than output layer. This result indicates that top $20 \\%$ most important samples contribute to the majority of the sample importance. ",
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+    {
+        "type": "text",
+        "text": "3.4 SAMPLE IMPORTANCE AND NEGATIVE LOG-LIKELIHOOD ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Negative log likelihood (NLL) is the loss metric we used for training objective. It has been used to measure the “easiness” of a sample in Curriculum learning Bengio et al. (2009) and Self-paced learning Kumar et al. (2010). Intuitively, the samples with large NLL should also have large sample importance (SI). However, in our experiment, we found that this is not always the case. In Figure 4, we found that 1) NLL and SI become more correlated as training goes on. However, 2) NLL is not predictive of the SI. There are many points with high NLL but small SI, and otherwise. ",
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+        "img_path": "images/7b01f6ee56ad410d554315b0e829197fb194fba35538582c41bb8eb261a1288f.jpg",
+        "image_caption": [
+            "Figure 2: Which classes and at which stage shape the network’s layer’s parameters? Parameters of different layers are learned at different times. Parameters in Output layers are learned mostly during the early training stage. In the lower layers, parameters are learned predominantly during the middle and late training stage. All classes do not contribute equally to training of the model. "
+        ],
+        "image_footnote": [],
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+    {
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+        "img_path": "images/c65928096336387e31f6986d8a5c1157a406686b28f9f2fb2cc62ef0c29e1cd8.jpg",
+        "image_caption": [
+            "Figure 3: Are all data samples equally important for all layers? The top $20 \\%$ most important samples contributes to the majority of parameter learning, especially in lower layers. “L1” to “L3” stands for Layer 1 to Layer 3. “Out” stands for output layer. "
+        ],
+        "image_footnote": [],
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+    },
+    {
+        "type": "text",
+        "text": "3.5 CLUSTERING SAMPLES BASED ON SAMPLE IMPORTANCE ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To better visualize the importance of different samples, we provide three representative clusters of samples for each dataset. In MNIST, we clustered all digit “5” samples into 20 clusters based on their epoch-specific, layer-specific sample importance. In CIFAR-10, we clustered all “horse” samples into 30 clusters using the same features as MNIST. Kmeans algorithm is used for clustering. ",
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+    {
+        "type": "image",
+        "img_path": "images/366cabd6209561647cb6d00ed6cdc1387b95f239c1a5c25a3d36e0ae64232719.jpg",
+        "image_caption": [
+            "Figure 4: Is Sample Importance correlated with Negative log-likelihood of a sample? Sample importance is positively correlated with negative log-likelihood. As training goes on, their correlation becomes higher. However, there remain many samples with high NLL and low SI, and vice versa. Left column: correlation between sample importance and negative log likelihood for all samples across epochs. Right column: scatter plot for NLL in the last epoch and all epoch sample importance for each sample. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "MNIST In Figure 5, we showed 3 example clusters on digit $\\cdot 5 ^ { , }$ . In the cluster of easy samples, where NLL converges very fast, most of the sample importance is concentrated in the first few epochs in output layer parameters. The cluster of medium samples has a slow NLL convergence compared to the easy cluster. The biggest impact is in middle training stage and more towards bottom layer parameters. Hard samples hardly converge even during the late stage of training. As training goes on, the sample importance for the bottom layer parameters become larger. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "CIFAR-10 In Figure 6, we showed 3 examples clusters on class “horse”. We observed very similar sample importance changing pattern as for the MNIST examples for easy, medium and hard clusters. Comparing to MNIST, all three clusters in CIFAR-10 have a very large impact on the parameters in the bottom layer. We note that the CIFAR-10 has almost 4 times larger number of parameters $( 3 0 7 5 \\times 5 1 2 \\approx 1 5 7 4 k )$ in the first layer than MNIST $7 8 5 \\times 5 1 2 \\approx 4 0 1 k$ ). ",
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+    {
+        "type": "text",
+        "text": "3.6 BATCH ORDER AND SAMPLE IMPORTANCE ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "With the observations from empirical analysis on sample importance, we know that time – iteration – and place – layer – of sample’s impact varies according to its “easiness” . We wanted to know whether constructing batches based on the sample importance or negative log likelihood would make a difference in training. Hence, we designed an experiment to explore how the different construction of batches influence the training. We note that the information used to structure the batches (negative log-likelihood and sample importance) – was obtained from a full training run. ",
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+    {
+        "type": "text",
+        "text": "We split all 50, 000 samples into $b = 5 0 0$ batch subsets $\\{ B _ { 1 } , B _ { 2 } , \\ldots , B _ { b } \\}$ . Each batch has batch size $| B _ { i } | = 1 0 0$ . In our experiment, each training sample must be in exactly one batch. There is no intersection between batches. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "During training, in each epoch, we update the parameters with each batch in order $1 , 2 , \\ldots , b$ iteratively. ",
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+    {
+        "type": "text",
+        "text": "We used seven different batch construction methods in this experiment: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/beca3d0da8b2003b41df3f1588b013088625e0966c9a1ccd126e7c77463862f6.jpg",
+        "image_caption": [
+            "Figure 5: When and where does an MNIST sample make the biggest impact? For “easy” samples, their biggest impact is on output layers and during the early training stage. As sample’s difficulty increases (medium and hard), the biggest impact moves to lower layers and in the late training stage. Each row is a sample cluster. In each row, from left to right: example images in the cluster; average sample importance and layer-wise decomposition across epochs; A boxplot of average training negative log likelihood across epochs. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "1. Rand: Randomly constructed batch. All $5 0 \\mathrm { k }$ samples are randomly split into $b$ batches before training. The batches and orders stay fixed during training.   \n2. NLO: Negative Log-likelihood Order. We sort all the samples based on their final NLL from low to high. The batches are constructed based on the sorted samples. First 100 samples with least NLL are in $\\boldsymbol { B } _ { 1 }$ , 101 to 200 samples are in $B _ { 2 }$ , and so on. Hence, during training, the batches with small NLL will be trained first.   \n3. RNLO Reverse-Negative Log-likelihood Order. We construct the batches same as NLO. During training, we update the batches in reverse order $B _ { b } , B _ { b - 1 } , \\ldots , B _ { 1 }$ . Hence, the batches with large NLL will be trained first.   \n4. NLM Negative Log-likelihood Mixed. We sort all the samples based on their final NLL from low to high. Next, for each sample $i$ in the sorted sequence, we put that sample into batch $j = ( i { \\bmod { b } } ) + 1$ . This ordering constructs batches out of samples with diverse levels of NLL.   \n5. SIO: Sample Importance Order. Similar to NLO, except that we sort all the samples based on their sum sample importance over all epochs from low to high. Hence, batches with small sample importance will be trained first.   \n6. RSIO Reverse-Sample Importance Order. We construct the batches same as SIO. During training, we update the batches in reverse order $B _ { b } , B _ { b - 1 } , \\ldots , B _ { 1 }$ . Hence, during training, the batches with large sample importance will be trained first. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/06ecd2bfd33d18b39d48776d6ab4984ee42b58766a17f2e0d0d3d92cd9424928.jpg",
+        "image_caption": [
+            "Figure 6: When and where does a CIFAR-10 sample make the biggest impact? For “easy” samples, their biggest impact is on the first layer during the early training stage. As samples’s difficulty increases (medium and hard), the biggest impact moves to lower layers and to late training stage. Each row is a sample cluster. In each row, from left to right: example images in the cluster; average sample importance and layer-wise decomposition across epochs; A boxplot of average training negative log likelihood across epochs. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "7. SIM Sample Important Mixed. Similar to NLM, but we sort the samples based on overall sample importance. Thus, batches contain samples with divers sample importance. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We performed five different runs (with different random initializations) on MNIST and CIFAR-10. The result is shown in Figure 7. From the result, we found that: 1) In both MNIST and CIFAR-10, Rand, SIS, and NLS have the lowest test error compared to all other methods. This indicates that diverse batches are helpful for training. 2) NLO and SIO got the worst performance in CIFAR10. Their training error even goes up after the early stage. RNLO and RSIO have same batch constructions as NLO and SIO, but their performances are drastically different. This indicates that the order of batches during training is important. Further, training on easy samples first and hard later seems to be counter-productive. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To better understand the impact of different batch construction, we performed the principle component analysis on the learned parameters in each epoch (Figure 8). In MNIST, the impact of batch construction is not very significant. In CIFAR-10, batch construction and even the order of batch training do have a large impact on the training. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Our experiment result shows a different conclusion to Curriculum Learning and Self-paced learning, where easy samples are trained on before introducing hard samples. We found that constructing and ordering the batches – hard to easy and easy to hard – seems to hinder the performance of learning. Having hard samples mixed in with the easy ones in each batch helps the training. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Also, the results show that we want to learn from the hard samples in early epochs and “see” hard samples more frequently, even if their major impact on parameters is during the late stage. As hard examples are few compared to easy samples and hard examples need a longer time to train, we do want to mix the hard samples into each batch to start learning from those samples early and learn longer. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/42716aa645f0c758ca13ad616d456e30f3c7f969dca4da6b441665f267b2d69c.jpg",
+        "image_caption": [
+            "Figure 7: Does organizing batches by “easiness” affect training? When batches are constructed with homogeneous easiness, the training performance become worse. Batches with mixed easiness have lower test error. The solid color line represents the mean over 5 runs. The error bar indicates the standard error over 5 runs. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/2356039eb64f765e3572f2159baff9807ba4501ba29d7c5578292085327cf2f9.jpg",
+        "image_caption": [
+            "Figure 8: Do parameters converge differently under different batch construction? In MNIST, the converging path for all batch constructions are very similar. In CIFAR-10, batch construction with mixed easiness (Rand, NLM, SIM) has a very different converging path with all other methods. Notably, we found that even with same batch constructions but just reversed order (NLO vs. RNLO, SIO vs. RSIO), the parameters converge to different points. Each circle dotted line shows the path of the first two principle components of all parameters in different epochs. Note that in CIFAR-10, the paths of Rand, NLS and SIS are very similar and they are overlapped in the plot. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4 EXTENSIONS OF SAMPLE IMPORTANCE ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We calculated the sample importance in each iteration in Stochastic Gradient Descent. However, such quantity only reflects the impact on the change in parameters within each iteration. The influence of a sample at a particular iteration can be accumulated through updates and impact the final model. Here, we are going to derive the exact calculation of the sample’s influence to the model. We rewrite the Objective (2) in Section 2 here: ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8ca9c38e6c5150d63505cc23867c20c4d293b98739219cf5bd6679bd8c8ccc7f.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\pmb { \\theta } } \\sum _ { i = 1 } ^ { n } v _ { i } L ( y _ { i } , f ( \\mathbf { x } _ { i } , \\pmb { \\theta } ) ) + R ( \\pmb { \\theta } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Here, we deem the sample weight $v _ { i }$ is fixed across all iterations. The update rule for stochastic gradient descent in each iteration is: ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6cc13f76fb0dc2b8abc82ce34dff334fff073e9fe51d4eef1884b1df7628cc5a.jpg",
+        "text": "$$\n\\pmb { \\theta } ^ { t + 1 } = \\pmb { \\theta } ^ { t } - \\eta \\sum _ { i } v _ { i } \\mathbf { g } _ { i } ^ { t } - \\eta \\mathbf { r } _ { i } ^ { t }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The derivative of $\\pmb { \\theta } ^ { t + 1 }$ with respect to sample weight $v _ { i }$ is: ",
+        "bbox": [
+            173,
+            181,
+            560,
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+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/55051f6288701aec5bc042041645cddaf7911962d699ba048cc340f009ad8f71.jpg",
+        "text": "$$\n\\begin{array} { r c l } { { \\displaystyle \\frac { \\partial } { \\partial v _ { i } } \\pmb { \\theta } ^ { t + 1 } } } & { { = } } & { { \\displaystyle \\frac { \\partial } { \\partial v _ { i } } \\pmb { \\theta } ^ { t } - \\eta \\mathbf { g } _ { i } ^ { t } - \\eta H ( \\pmb { \\theta } ^ { t } ) \\frac { \\partial } { \\partial v _ { i } } \\pmb { \\theta } ^ { t } } } \\\\ { { \\displaystyle \\frac { \\partial } { \\partial v _ { i } } \\pmb { \\theta } ^ { 1 } } } & { { = } } & { { \\displaystyle - \\eta \\mathbf { g } _ { i } ^ { 0 } , } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            346,
+            204,
+            653,
+            271
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "where $H ( \\pmb \\theta ^ { t } )$ is the Hessian matrix of the objective in (2) with regard to all parameters in iteration $t$ . If we iterate the updates until convergence, then we can assume that $\\theta ^ { T }$ is a fix-point, $\\pmb { \\theta } ^ { \\ast } = \\pmb { \\theta } ^ { T + 1 } =$ $\\theta ^ { T }$ , and we obtain: ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8ccd2bd9ca63663dc86b7ef0562ad15de38ed0ccf52a47be5041531fd09326a6.jpg",
+        "text": "$$\n\\frac { \\partial } { \\partial v _ { i } } \\pmb { \\theta } ^ { T + 1 } - \\frac { \\partial } { \\partial v _ { i } } \\pmb { \\theta } ^ { T } = - \\eta \\mathbf { g } _ { i } ^ { t } - \\eta H ( \\pmb { \\theta } ^ { * } ) \\frac { \\partial } { \\partial v _ { i } } \\pmb { \\theta } ^ { * }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            331,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Hence, the derivative of parameters in the final model with regard to a sample weight is: ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d30cb4417f117e660ccdc2a6eee39501c9d3a39ed7f1bff4e22e84cfa00b7d0f.jpg",
+        "text": "$$\n\\frac { \\partial } { \\partial v _ { i } } \\pmb { \\theta } ^ { * } = - H ( \\pmb { \\theta } ^ { * } ) ^ { - 1 } \\mathbf { g } _ { i } ^ { T }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            416,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Equation (4) indicates that we can calculate the sample specific impact on final trained model by using the parameters learned at the convergence point. In deep learning methods, due to early stopping, fix point might not be achieved, and Equation (4) might not be an accurate. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "For any target quantity $\\boldsymbol { \\mathcal { T } } ( \\boldsymbol { \\theta } ^ { * } )$ that depends on the final trained parameter $\\pmb { \\theta } ^ { * }$ , we can calculate the impact of a particular sample on that target as: ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/99669177b9d03a381895ef1d3e8c83623a18afe29a5d5ceb5234ccb87dc95e73.jpg",
+        "text": "$$\n\\frac { \\partial } { \\partial v _ { i } } T ( \\pmb { \\theta } ^ { * } ) = \\frac { \\partial } { \\partial \\pmb { \\theta } ^ { * } } T ( \\pmb { \\theta } ^ { * } ) \\frac { \\partial } { \\partial v _ { i } } \\pmb { \\theta } ^ { * }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            544,
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+            577
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "For example, if we are interested in the sum of predictions on a set of samples ${ \\mathcal { T } } ( \\theta ^ { * } ) ~ =$ $\\textstyle \\sum _ { i \\in { \\mathcal { S } } _ { c } } f ( \\mathbf { x } _ { i } , { \\pmb \\theta } ^ { t } )$ , we can use Equation (5) to calculate the derivative: ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8e4ed9df320fa4f39c893488bb13c88063209723ed58f1f12dd7b772e9e01530.jpg",
+        "text": "$$\n\\begin{array} { r c l } { \\displaystyle \\frac { \\partial } { \\partial v _ { i } } \\mathcal { T } ( \\pmb { \\theta } ^ { * } ) } & { = } & { \\displaystyle \\sum _ { i \\in \\mathcal { S } _ { c } } \\frac { \\partial } { \\partial \\pmb { \\theta } ^ { * } } f ( \\mathbf { x } _ { i } , \\pmb { \\theta } ^ { t } ) \\frac { \\partial } { \\partial v _ { i } } \\pmb { \\theta } ^ { * } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            631,
+            640,
+            671
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We note evaluating the exact impact of a sample, as shown above, is computationally cumbersome for all but the simplest models. ",
+        "bbox": [
+            173,
+            694,
+            823,
+            724
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "5 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            744,
+            312,
+            761
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Samples’ impact on the deep network’s parameters vary across stages of training and network’s layers. In our work, we found that easy samples predominantly shape parameters the top layers at the early training stages, while hard samples predominantly shape the parameters of the bottom layers at the late training stage. Our experiments show that it is important to mix hard samples into different batches rather than keep them together in the same batch and away from other examples. ",
+        "bbox": [
+            173,
+            776,
+            825,
+            847
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "There are many future extensions to the current work. Firstly, we want to expand our sample importance analysis to different deep learning structures, like Convolution Neural Network and Recurrent Neural Networks. Secondly, we want to use the sample importance as a guidance to extract a minimal subset of samples that are sufficient to achieve performance comparable to a network trained on the full dataset. ",
+        "bbox": [
+            174,
+            853,
+            825,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
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+    {
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+    {
+        "type": "text",
+        "text": "APPENDIX ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/b50b1bc5189d4cb3fa5ad2950a2e80b6a0c2b3a70f01fab8b5c6bf2838fbfbfa.jpg",
+        "image_caption": [
+            "Figure 9: The training and test error on MNIST (first row) and CIFAR-10 (second row). The left column showed the average class-specific negative log likelihood. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            222,
+            140,
+            774,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/f8db2eb88900f5393079c3638df2a7dc58b9a63367a73c5ed63a70e9013e6e1f.jpg",
+        "image_caption": [
+            "Figure 10: Histogram of total sample importance. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            277,
+            561,
+            717,
+            728
+        ],
+        "page_idx": 11
+    }
+]

parse/train/rknt2Be0-/rknt2Be0-.md

@@ -0,0 +1,340 @@
+# COMPOSITIONAL OBVERTER COMMUNICATION LEARNING FROM RAW VISUAL INPUT
+
+Edward Choi ∗   
+Georgia Institute of Technology   
+Atlanta, GA, USA   
+mp2893@gatech.edu   
+Angeliki Lazaridou & Nando de Freitas   
+DeepMind   
+London, UK   
+{angeliki, nandodefreitas}@google.com
+
+# ABSTRACT
+
+One of the distinguishing aspects of human language is its compositionality, which allows us to describe complex environments with limited vocabulary. Previously, it has been shown that neural network agents can learn to communicate in a highly structured, possibly compositional language based on disentangled input (e.g. handengineered features). Humans, however, do not learn to communicate based on well-summarized features. In this work, we train neural agents to simultaneously develop visual perception from raw image pixels, and learn to communicate with a sequence of discrete symbols. The agents play an image description game where the image contains factors such as colors and shapes. We train the agents using the obverter technique where an agent introspects to generate messages that maximize its own understanding. Through qualitative analysis, visualization and a zero-shot test, we show that the agents can develop, out of raw image pixels, a language with compositional properties, given a proper pressure from the environment.
+
+# 1 INTRODUCTION
+
+One of the key requirements for artificial general intelligence (AGI) to thrive in the real world is its ability to communicate with humans in natural language. Natural language processing (NLP) has been an active field of research for a long time, and the introduction of deep learning (LeCun et al., 2015) enabled great progress in NLP tasks such as translation, image captioning, text generation and visual question answering (Cho et al., 2014; Bahdanau et al., 2014; Vinyals et al., 2015; Karpathy & Fei-Fei, 2015; Hu et al., 2017; Serban et al., 2016; Lewis et al., 2017; Antol et al., 2015). However, training machines in a supervised manner with a large dataset has its limits when it comes to communication. Supervised methods are effective for capturing statistical associations between discrete symbols (i.e. words, letters). The essence of communication is more than just predicting the most likely word to come next; it is a means to coordinate with others and potentially achieve a common goal (Austin, 1975; Clark, 1996; Wittgenstein, 1953).
+
+An alternative path to teaching machines the art of communication is to give them a specific task and encourage them to learn how to communicate on their own. This approach will encourage the agents to use languages grounded to task-related entities as well as communicate with other agents, which is one of the ways humans learn to communicate (Bruner, 1981). Recently, there have been several notable works that demonstrated the emergence of communication between neural network agents. Even though each work produced very interesting results of its own, in all cases, communication was either achieved with a single discrete symbol (as opposed to a sequence of discrete symbols) (Foerster et al., 2016; Lazaridou et al., 2017) or via a continuous value (Sukhbaatar et al., 2016; Jorge et al., 2016). Not only is human communication un-differentiable, but also using a single discrete symbol is quite far from natural language communication. One of the key features of human language is its compositional nature; the meaning of a complex expression is determined by its structure and the meanings of its constituents (Frege, 1892). More recently, Mordatch & Abbeel (2017) and Kottur et al. (2017) trained the agents to communicate in grounded, compositional language. In both studies, however, inputs given to the agents were hand-engineered features (disentangled input) rather than raw perceptual signals that we receive as humans.
+
+In this work, we train neural agents to simultaneously develop visual perception from raw image pixels, and learn to communicate with a sequence of discrete symbols. Unlike previous works, our setup poses greater challenges to the agents since visual understanding and discrete communication have to be induced from scratch in parallel. We place the agents in a two-person image description game, where images contain objects of various color and shape. Inspired by the pioneering work of Batali (1998), we employ a communication philosophy named obverter to train the agents. Having its root in the theory of mind (Premack & Woodruff, 1978) and human language development (Milligan et al., 2007), the obverter technique motivates an agent to search over messages and generate the ones that maximize their own understanding. The contribution of our work can be summarized as follows:
+
+• We train artificial agents to learn to disentangle raw image pixels and communicate in compositional language at the same time.   
+• We describe how the obverter technique, a differentiable learning algorithm for discrete communication, could be employed in a communication game with raw visual input.   
+• We visualize how the agents are perceiving the images and show that they learn to disentangle color and shape without any explicit supervision other than the communication one.   
+• Experiment results suggest that the agents could develop, out of raw image input, a language with compositional properties, given a proper pressure from the environment (i.e. the image description game).
+
+Finally, while our exposition follows a multi-agent perspective, it is also possible to interpret our results in the single-agent setting. We are effectively learning a neural network that is able to learn disentangled compositional representations of visual scenes, without any supervision. Subject to the constraints imposed by their environment, our agents learn disentangled concepts, and how to compose these to form new concepts. This is an important milestone in the path to AGI.
+
+# 2 METHOD
+
+# 2.1 THE TWO-PERSON IMAGE DESCRIPTION GAME
+
+![](images/fe848c0db6f5de8bab346ead880e705a22d31e75c43a4509f5ea6f47130e08f7.jpg)  
+Figure 1: The two-person image description game. Speaker observes an image and generates a message (i.e. a sequence of discrete symbols). The listener, after observing a separate image and the message, must correctly decide whether it is seeing the same object as the speaker (left side; output 1) or not (right side; output 0).
+
+![](images/b0f08c116783796164e0f37b02eb2fea29ddc93f0d918fda9b7fa3e6ebcdda8e.jpg)  
+Figure 2: Example images of the dataset. There are eight colors (blue, red, white, gray, yellow, green, cyan, magenta), and five shapes (box, sphere, cylinder, capsule, ellipsoid), giving us total 40 combinations.
+
+We choose a straightforward image description game with two factors (color and shape) so that we can perform extensive analysis on the outcome confidently, based on full control of the experiment. In a single round of the two-person image description game, one agent becomes the speaker and the other the listener. The speaker is given a random image, and generates a message to describe it. The listener is also given a random image, possibly the same image as the speaker’s. After hearing the message from the speaker, the listener must decide if it is seeing the same object as the speaker (Figure 1). Note that an image is the raw pixels given to the agents, and an object is the thing described by the image. Therefore two different images can depict the same object. In each round the agents change roles of being the speaker and the listener.
+
+We generated synthetic images using Mujoco physics simulator1. The example images are shown in Figure 2. Each image depicts a single object with a specific color and shape in $1 2 8 \times 1 2 8$ resolution. There are eight colors (blue, red, white, gray, yellow, green, cyan, magenta) and five shapes (box, sphere, cylinder, capsule, ellipsoid), giving us 40 combinations. We generated 100 variations for each of the 40 object type. Note that the position of the object varies in each image, changing the object size and the orientation. Therefore even if the speaker and the listener are given the same object type, the actual images are very likely to be different, preventing the agents from using pixel-specific information, rather than object-related information to win the game.
+
+# 2.2 MODEL ARCHITECTURE
+
+![](images/7902c027c72b5a000bf12271e28b3cfff9c03cda8af5899e656377dae25107a7.jpg)  
+Figure 3: Agent model architecture. The visual module processes the image, and the language module generates or consumes messages. The decision module accepts embeddings from both modules and produces the output. The solid arrows indicate modifying the output from the previous layer. The dotted arrows indicate copying the output from the previous layer.
+
+Aside from using disentangled input, another strong assumption made in previous works(Batali, 1998; Mordatch & Abbeel, 2017) was that the agents had access to the true intention of the speaker. In Batali (1998), the listener was trained to modify its RNN hidden vector as closely to the speaker’s intention (meaning vector; please see Table 5 in Appendix A) as possible. In Mordatch & Abbeel (2017), each agent had an auxiliary task to predict the goals of all other agents. In both cases, the true meaning/goal vector was used to update the model parameters, exposing the disentangled information to the agents. In order to relax this assumption and encourage the agents to develop communication with minimal guidance, our model uses no other signal than whether the listener made a correct decision.
+
+Figure 3 depicts the agent model architecture. We use a convolutional neural network followed by a fully-connected layer to process the image. A single RNN, specifically the gated recurrent units (GRU) (Cho et al., 2014), is used for both generating and consuming messages (the message generation using the obverter strategy is described in the next section). When consuming a message, the image embedding from the visual module and the message embedding from the language module are concatenated and processed by another fully-connected layers (i.e. decision module) with the sigmoid output $\hat { y }$ , 0 being “My (listener) image is different from the speaker’s” and 1 being “My image is the same as the speaker’s”. Further details of the model architecture (e.g. number of layers) are described in Appendix C.
+
+# 2.3 OBVERTER TECHNIQUE
+
+Although our work is inspired by Batali (1998) (see Appendix A for the description of Batali (1998)), obverter technique is a general message generation philosophy used/discussed in a number of communication and language evolution studies (Hurford, 1989; Oliphant & Batali, 1997; Smith, 2001; Kirby & Hurford, 2002), which has its root in the theory of mind. Theory of mind (Premack & Woodruff, 1978) observes that a human has direct access only to one’s own mind and not to the others’. Therefore we typically assume that the mind of others is analogous to ours, and such assumption is reflected in the functional use of language (Bruner, 1981). For example, if we want to convey a piece of information to the listener2, it is best to speak in a way that maximizes the listener’s understanding. However, since we cannot directly observe the listener’s state of mind, we cannot exactly solve this optimization problem. Therefore we posit that the listener’s mind operates in a similar manner as ours, and speak in a way that maximizes our understanding, thus approximately solving the optimization problem. This is exactly what the obverter technique tries to achieve.
+
+When an agent becomes the teacher (i.e. speaker), the model parameters are fixed. The image is converted to an embedding via the visual module. After initializing its RNN hidden layer to zeros, the teacher at each timestep evaluates $\hat { y }$ for all possible symbols and selects the one that maximizes $\hat { y }$ The RNN hidden vector induced by the chosen symbol is used in the next timestep. This is repeated until $\hat { y }$ becomes bigger than the predefined threshold, or the maximum message length is reached (see Appendix D for algorithm). Therefore the teacher, through introspection, greedily selects characters at each timestep to generate a message such that the consistency between the image and the message is as clear to itself as possible. When an agent becomes the learner (i.e. listener), its parameters are updated via back-propagating the cross entropy loss between its output $\hat { y }$ and the true label $y$ . Therefore the agents must learn to communicate only from the true label indicating whether the teacher and the learner are seeing the same object.
+
+We remind the reader that only the learner’s RNN parameters are updated, and the teacher uses its fixed RNN. Therefore an agent uses only one RNN for both speaking and listening, guaranteeing self-consistency (see Appendix B for a detailed comparison between the obverter technique and the RL-based approach). Furthermore, because the teacher’s parameters are fixed, message generation can easily be extended to be more exploratory. Although in this work we deterministically selected a character in each timestep, one can, for example, sample characters proportionally to $\hat { y }$ and still use gradient descent for training the agents. Using a more exploratory message generation strategy could help us discover a more optimal communication language when dealing with complex tasks.
+
+Another feature of the obverter technique is that it observes the principle of least effort (Zipf, 1949). Because the teacher stops generating symbols as soon as $\hat { y }$ reaches the threshold, it does not waste any more effort trying to perfect the message. The same principle was implemented in one way or another in previous works, such as choosing the shortest among the generated strings (Kirby & Hurford, 2002) or imposing a small cost for generating a message (Mordatch & Abbeel, 2017).
+
+# 2.4 ENVIRONMENTAL PRESSURE FOR COMPOSITIONAL COMMUNICATION
+
+During the early stages of research, we noticed that randomly sampling object pairs (one for the teacher, one for the learner) lead to agents focusing only on colors and ignoring shapes. When the teacher’s object is fixed, there are 40 (8 colors $\times 5$ shapes) possibilities on the learner’s side. If the teacher only talks about the color of the object, the learner can correctly decide for 36 out of 40 possible object types. The learner makes incorrect decisions only when the teacher and the learner are given objects of the same color but different shapes, resulting in $9 0 \%$ accuracy on average. This is actually what we observed; the accuracy plateaued between 0.9 and 0.92 during the training, and the messages were more or less the same for objects with the same color. Therefore when constructing a mini-batch of images, we set $2 5 \%$ to be the object pairs of the same color and shape, $30 \%$ the same shape but different colors, $20 \%$ the same color but different shapes. The remaining $2 5 \%$ object pairs were picked randomly3.
+
+Vocabulary size (i.e. number of unique symbols) and the maximum message length were also influential to the final outcome. We noticed that a larger vocabulary and a longer message length helped the agents achieve a high communication accuracy more easily. But the resulting messages were more challenging to analyze for compositional patterns. In all our experiments we used 5 and 20 respectively for the vocabulary size and the maximum message length, similar to what Batali (1998) used. This suggests that the environment plays as important, if not more, role as the model architecture in the emergence of complex communication as discussed by previous studies (Kirby et al., 2014; Bratman et al., 2010; Kottur et al., 2017) and should be a main consideration for future efforts. Further details regarding hyperparameters are described in Appendix E.
+
+# 3 EXPERIMENTS
+
+In this section, we first study the convergence behavior during the training phase. Then we analyze the language developed by the agents in terms of compositionality. As stated in the introduction, in compositional language, the meaning of a complex expression is determined by its structure and the meanings of its constituents. With this definition in mind, we focus on two aspects of the inter-agent communication to evaluate its compositional properties: the structure (i.e. grammar) of the communication, and zero-shot performance (i.e. generalizing to novel stimuli). These two aspects, which are both necessary conditions for any language to be considered compositional, have been used by previous works to study the compositional nature of artificial communication (Batali, 1998; Mordatch & Abbeel, 2017; Kottur et al., 2017).
+
+To evaluate the structure of the messages, we study the evolution of the communication as training proceeds, and try to derive a grammar for expressing colors and shapes. To evaluate the zero-shot capabilities, we test if the agents can compose consistent messages for objects they have not seen during the training. Moreover, we visualize the image embeddings from the visual modules of both agents to understand how they are recognizing colors and shapes, the results of which, for a better view of the figures, are provided in Appendix H.
+
+# 3.1 CONVERGENCE BEHAVIOR
+
+![](images/e4918ef563341b6e90dc729d2d56d92b62fe35943fcf79abaaa0650a641662d6.jpg)  
+Figure 4: Progress during the training (best seen in color). (Top) We plot the training accuracy, training loss, average message length and average message distinctness in each round. (Bottom) We plot the perplexities and the Jaccard similarity of the messages spoken by both agents in each round. Note that the average message length and the perplexities are divided by 20 to match the y-axis range with other metrics.
+
+Figure 4 shows the convergence behavior during the training. Training accuracy was calculated by rounding the learner’s sigmoid output by 0.5. Message distinctness was calculated by dividing the number of unique messages in the mini-batch by the size of the mini-batch. Ideally there should be, on average, 40 distinct messages in the mini-batch of 50 images, therefore giving us 0.8 distinctness. Every 10 round, both agents were given the same 1, 000 randomly sampled images to generate 1, 000 message pairs. Then perplexity was calculated for each object type and averaged, thus indicating the average number of distinct messages used by the agents to describe a single object type (note that perplexities in the plot was divided by 20). Jaccard similarity between both agents’ messages was also calculated for each object type and averaged.
+
+At the beginning, the listener (i.e. learner) always decides it is not seeing the same object as the speaker, giving us 0.75 accuracy4. But after 7, 000 rounds, accuracy starts to go beyond 0.9. Loss is negatively correlated with accuracy until round 15, 000, where it starts to fluctuate. Accuracy, however, remains high due to how accuracy is measured; by rounding the learner’s output by 0.5. Although we could occasionally observe some patterns in the messages when both accuracy and loss were high, a lower loss generally resulted in a clearer communication structure (i.e. grammar) and better zero-shot performance. The loss fluctuation also indicates some instability in the training process, which is a potential direction for future work. Message distinctness starts at near 0, indicating the agents are generating the same message for all object types. After round 7, 000, where both message distinctness and message length reach their maximum, both start to decrease. But message distinctness never goes as high as the ideal 0.8, meaning that the agents are occasionally using the same message for different object types, as will be shown in the following section.
+
+Both perplexities and Jaccard similarity show seemingly meaningless fluctuation at early rounds. After round 7, 000, perplexities and Jaccard similarity show negatively correlated behavior, meaning that not only is each agent using consistent messages to describe each object type, but also both agents are using very similar messages to describe each object type. We found perplexity and Jaccard similarity to be an important indicator of the degree of the communication structure. During rounds $7 , 0 0 0 \sim 8 , 0 0 0$ , performance was excellent in terms of loss and accuracy, but perplexity was high and Jaccard similarity low, indicating the agents were assigning incoherent strings to each object type just to win the game. Similar behavior was observed in the early stages of language evolution simulation in Kirby & Hurford (2002) where words represented some meanings but had no structure (i.e. protolanguage). It seems that artificial communication acquires compositional properties after the emergence of protolanguage regardless of whether the input is entangled or disentangled.
+
+# 3.2 GRAMMAR ANALYSIS
+
+We choose agents from different training rounds to highlight how the language becomes more structured over time. Table 1 shows agents’ messages in the beginning (round 40), when the training accuracy starts pushing beyond $9 0 \%$ (round 6, 940), when agents settle on a common language (round 16, 760).
+
+In round 40, both agents are respectively producing the same message for all object types as mentioned in section 3.1. We might say the messages are structured, but considering that the listener always answers 0 in early rounds, we cannot say the agents are communicating. In round 6, 940, which is roughly when the agents begin to communicate more efficiently, training accuracy is significantly higher than round 40. However, perplexities show that both agents are assigning many names to a single object type (40-80 names depending on the object type), indicating that the agents are focusing on pixel-level differences between images of the same object type. Table 1 shows, as an example, the messages used by both agents to describe the red sphere. Due to high perplexity, it is difficult to capture the underlying grammar of the messages even with regular expression. Furthermore, as Jaccard similarity indicates, both agents are generating completely different messages for the same object type. In round 16, 760, as the perplexities and Jaccard similarity tell us, the agents came to share a very narrow set of names for each object type (1-4 names depending on the object type). Moreover, the names of the same-colored objects and same-shaped objects clearly seem to follow a pattern. Overall, each of the three phases (round 40, round 6,940, round 16,760) seem to represent the development of visual perception, learning to communicate, and emergence of structure.
+
+We found the messages in round 16, 760 could be decomposed in a similar manner as Table 6 in Appendix A. The top of Table 2 shows a possible decomposition of the messages from round 16, 760 and the bottom shows the rules for each color and shape derived from the decomposition. According to our analysis, the agents use the first part of the message (i.e. prefix) to specify a shape, and the second part (i.e. suffix) to specify a color. However, they use two different strings to specify a shape. For example, the agents use either aaaa or bbbbb to describe a box. The strings used for specifying colors show slightly weaker regularity. For example, red is always described by either the suffix c or suffix e, but magenta is described by the suffix bb, bd, and sometimes b or bc. a used for ¯ gray objects
+
+Round 40 (Training accuracy: $6 6 . 1 \%$ , Agent0 perplexity:1.0, Agent1 perplexity:1.0, Jaccard similarity:0.0)   
+
+<table><tr><td rowspan=1 colspan=1>■</td><td rowspan=1 colspan=1>All objectsdescribed by Agent 0</td><td rowspan=1 colspan=1>All objectsdescribed byAgent1</td></tr><tr><td rowspan=1 colspan=1>Message</td><td rowspan=1 colspan=1>dddddddddddddddddddd</td><td rowspan=1 colspan=1>bbbbbbbbbbbbbbbbbbbb</td></tr></table>
+
+# Round 6,940
+
+(Training accuracy: $9 3 . 1 \%$ , Agent0 perplexity:9.90, Agent1 perplexity:17.73, Jaccard similarity:0.0)   
+
+<table><tr><td>■</td><td>Redsphere described byagent 0</td><td>Redspheredescribedbyagent1</td></tr><tr><td rowspan="9">Messages</td><td>aeceaeaeaaaeeeeeeeee</td><td>aaedacacaaaaaaaaaaaa</td></tr><tr><td>aacceeaaaaeeeeeeeeee</td><td>aaccdaacaaaaaaaaaaaa</td></tr><tr><td>aaccceaaaaaaaaeeeeee</td><td>aaabcdadaaaaaaaaaaaa</td></tr><tr><td>aeeeeaaaaeeeeeeeeeee</td><td>aaeeacaeaaaaaaaaaaaa</td></tr><tr><td>aeaceeaeaeeeeeeeeeee</td><td>aaccdadaaaaaaacaaaaa</td></tr><tr><td>aceacacaaaaaaeeeeeee</td><td>aaedaceaaaaaaaaaaaaa</td></tr><tr><td>abeeeeaeeeeeeeeeeeee</td><td>aaeaccaeaaaaaaaaaaaa</td></tr><tr><td>aacceeaeeeeeeeeeeeee</td><td>aaceacaacaaaaaaaaaaa</td></tr><tr><td>aacceeaeaaeeeeeeeeee</td><td>aacdacdaaaaaaaaaaaaa</td></tr><tr><td>aeeacacaaaeeccceeeee</td><td></td><td>aaccdadaaaaacaaaaaaa</td></tr></table>
+
+Round 16,760 (Training accuracy: $9 8 . 0 \%$ , Agent0 perplexity:1.69, Agent1 perplexity:1.62, Jaccard similarity:0.82)   
+Table 1: (Top) Messages used by both agents when speaking about any object in round 40. (Middle) Ten most frequent messages used by each agent to describe a red sphere in round 6, 940. (Bottom) Messages most often used by both agents for each object type in round 16, 760. Brackets indicate the variation often seen at the last character. Underscores indicate blanks.   
+
+<table><tr><td>:</td><td>Box</td><td>Sphere</td><td>Cylinder</td><td>Capsule</td><td>Ellipsoid</td></tr><tr><td>Blue</td><td>bbbbbbb{b.d}</td><td>bb{b,d}</td><td>bbbbbb{b,d}</td><td>bbbbb{c,d}</td><td>bbbb{-,c,d}</td></tr><tr><td>Red White</td><td>aaaa{c,e}</td><td>aa{c,e}</td><td>aaa{c,e}</td><td>a{c,e}</td><td>c,e</td></tr><tr><td></td><td>bbbbbb</td><td>b,d</td><td>bbbb{b,d}</td><td>bbb{b.d}</td><td>bb{-,c,d}</td></tr><tr><td>Gray</td><td>aaa</td><td>a</td><td>aa</td><td>C</td><td>b,bd</td></tr><tr><td>Yellow</td><td>aaaaaa</td><td>aaaa</td><td>aaaaa</td><td>aaa</td><td>a{a,e}</td></tr><tr><td>Green</td><td>aaaa{a,ad}</td><td>aa{a,ad}</td><td>aaa{a.ad}</td><td>a{a,ad}</td><td>a</td></tr><tr><td>Cyan</td><td>ax20</td><td>bbb{b,d}</td><td>bbbbbbb{b,d}</td><td>bbbbbbd</td><td>bbbbb{b.d}</td></tr><tr><td>Magenta</td><td>bbbbbbb</td><td>b{b.d</td><td>bbbbb{b,d}</td><td>bbbbd</td><td>bbb{-.c,d}</td></tr></table>
+
+represents deletion of the prefix a. Note that removing prefixes beyond their length causes the pattern to break. For example, gray box, gray sphere and gray cylinder use the same a¯aa to express the color, ¯ but gray capsule and gray ellipsoid use irregular suffixes.
+
+Despite some irregularities and one exceptional case (cyan box), the messages provide strong evidence that the agents learned to properly recognize color and shape from raw pixel input (see Appendix H for studying what the visual module learned), mapped each color and shape to prefixes and suffixes, and are able to compose meaningful messages to describe a given image to one another. Communication accuracy for each object type is described in Appendix F. Communication examples and their analysis are given in Appendix G.
+
+# 3.3 ZERO-SHOT EVALUATION
+
+If the agents have truly learned to compose a message that can be divided into a color part and a shape part, then they should be able to accurately describe an object they have not seen before, which is another necessary condition for a compositional language. Therefore, we hold out five objects (the shaded cells in Table 3) from the dataset during the training and observe how agents describe five novel objects during the test phase. The agents were chosen from round 19, 980, which showed a high accuracy $( 9 7 . 8 \% )$ , low perplexities (1.48, 1.65) and a high Jaccard similarity (0.75). Table 3 shows a potential decomposition of the messages used by the agents (original messages are described by Table 8 in Appendix I). We can observe that there is clearly a structure in the communication, although some messages show somewhat weaker patterns compared to when the agents were trained with all object types (Table 2). Suffixes for specifying yellow and magenta are especially irregular, even when we consider the effects of b and ¯ e. However, the messages describing the held-out object ¯ types show clear structure with the exception of yellow ellipsoid.
+
+<table><tr><td>■</td><td>Box</td><td>Sphere</td><td>Cylinder</td><td>Capsule</td><td>Ellipsoid</td></tr><tr><td>Blue Red</td><td>bbbbb bb{b,d} aaaa {c,e}</td><td>bb{b,d} aa {c,e}</td><td>bbbb bb{b.d} aaa {c,e}</td><td>bbb bb{c,d} a{c,e}</td><td>bb bb{-.c,d} {c,e}</td></tr><tr><td>White</td><td>bbbbbb</td><td>{b.d}</td><td>bbbb{b.d}</td><td>bbb {b.d}</td><td>bb{-c,d}</td></tr><tr><td>Gray</td><td>aaaa aaa</td><td>aa aaa</td><td>aaa aaa</td><td>a aac</td><td>aa{b,bd}</td></tr><tr><td>Yellow</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>aaaa aa</td><td>aaaa</td><td>aaaaa</td><td>aaa</td><td>a{ae}</td></tr><tr><td>Green</td><td>aaaa {a,ad}</td><td>aa {a,ad}</td><td>aaa {a,ad}</td><td>a{a,ad}</td><td>a</td></tr><tr><td>Cyan</td><td>a×20</td><td>bbb{b,d}</td><td>bbbb bbb{b,d}</td><td>bbb bbbd</td><td>bb bbb{b.d}</td></tr><tr><td>Magenta</td><td>bbbbbbb</td><td>b{b,d}</td><td>bbbbb{b.d}</td><td>bbb bd</td><td>bbb{-c,d}</td></tr></table>
+
+Define Blue, White, Cyan, Magenta as color group 0, rest as color group 1.   
+
+<table><tr><td rowspan=1 colspan=1>■</td><td rowspan=1 colspan=1>Rule</td></tr><tr><td rowspan=1 colspan=1>Blue</td><td rowspan=3 colspan=1>End with bbb or bbdEnd with c or eEnd with b or dEnd with aaaEnd with aaEnd with a or adEnd with bbbb or bbbdEnd with bb or bd</td></tr><tr><td rowspan=1 colspan=1>RedWhite</td></tr><tr><td rowspan=1 colspan=1>Gray</td><td rowspan=1 colspan=1>GrayYellowGreenCyanMagenta</td></tr><tr><td rowspan=1 colspan=1>BoxSphereCylinderCapsuleEllipsoid</td><td rowspan=1 colspan=1>Start with bbbbb for color group O, start with aaaa for color group 1Start with aa for color group 1Start with bbbb for color group O, start with aaa for color group 1Start with bbb for color group O,start with a for color group 1Start with bb for color group 0</td></tr></table>
+
+Table 2: (Top) Potential composition analysis of the messages from round 16,760 (bottom of Table 1). Italic symbols are used to specify shapes and roman symbols are used to specify colors. a indicates ¯ deleting a single prefix a. (Bottom) Rules for each color and shape derived from the top table.   
+Table 3: Potential analysis of the messages observed in the zero-shot test. Gray cells indicate object types unseen during the training phase. Italic symbols are used to specify shapes and roman symbols to specify colors. b indicates deleting a single prefix¯ $b$ . e indicates deleting a single prefix ¯ $e$ . Underscores indicate blanks.   
+
+<table><tr><td rowspan=1 colspan=1>■</td><td rowspan=1 colspan=3>Box</td><td rowspan=1 colspan=2>Sphere</td><td rowspan=1 colspan=3>Cylinder</td><td rowspan=1 colspan=1>Capsule</td><td rowspan=1 colspan=1>Ellipsoid</td></tr><tr><td rowspan=7 colspan=1>BlueRedWhiteGrayYellowGreen</td><td rowspan=1 colspan=3>eeeeee e{e,ee}</td><td rowspan=1 colspan=2>eeeeee</td><td rowspan=2 colspan=3>eeeee e{e,ed}eeeee eeee{e,a}</td><td rowspan=3 colspan=1>eee e{e,ed}eee eeee{b,ba}bb {b.d}</td><td rowspan=5 colspan=1>eee{e,a}ee eeee{a,c}bbb {d.c}ee eee{b.d}</td></tr><tr><td rowspan=6 colspan=3>eeeeee eeee{e.a}bbbbbeeeeeeeeeebbbb bb{c,d}eeeeee {e,a}</td><td rowspan=1 colspan=2>eeee eeee{e,ea}</td><td rowspan=1 colspan=2>eeee</td></tr><tr><td rowspan=1 colspan=2>bb</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=2></td><td rowspan=2 colspan=1></td><td rowspan=2 colspan=1>eeee eee{-,a}</td><td rowspan=1 colspan=1></td><td rowspan=2 colspan=3>eeeee eee{e,a}</td><td rowspan=2 colspan=1>eee eee{e,a}</td></tr><tr><td rowspan=4 colspan=2>bbbe{e,ec}eeee {e,a}eeee ee{e,a}bb{c.d</td></tr><tr><td rowspan=3 colspan=3>bbb bb{c,d}eeeee {e,a}eeeee ee{e,ea}bbbb{b.d}</td><td rowspan=2 colspan=1>bb bb{a.c}eee{e,a}</td><td rowspan=1 colspan=1>bbb bb{a,e}</td></tr><tr><td rowspan=2 colspan=1>eee{e,a}eee ee{e,a}bb bb</td><td rowspan=2 colspan=1>ee {e,a}ee eeabbb bb{c.a}</td></tr><tr><td rowspan=1 colspan=1>CyanMagenta</td><td rowspan=1 colspan=3>eeeeee eeeabbbb b{b.d}</td><td rowspan=1 colspan=1></td></tr></table>
+
+In order to assess the communication accuracy when held-out objects are involved, we conducted another test with the agents from round 19, 980. Each held-out object was given to the speaker, the listener, or both. In the first two cases, the held-out object was paired with all 40 object types and each pair was tested 10 times. In the last case, the held-out object was tested against itself 10 times. In all cases, the agents switched roles after 5 times. Table 4 shows communication accuracies for each case. We can see the agents can successfully communicate most of the time even when given novel objects. The last column shows that the listener is not simply producing 0 to maximize its chance to win the game. It is also notable that the objects described without $\bar { \mathsf b }$ or e show better performance in general. ¯ We noticed the communication accuracy for held-out objects seems relatively weak considering the messages used to describe them strongly showed structure. (Table 3). This, however, results from the grammar (i.e. structure) being not as straightforward as Table 2, especially with short messages (i.e. frequent use of $\bar { \mathsf b }$ and e). The same tendency can be observed for non-held-out objects as described ¯ by the per-object communication accuracy Table 9 in Appendix J.
+
+Table 4: Communication accuracy when agents were given objects not seen during the training.   
+
+<table><tr><td></td><td>Given to speaker</td><td>Given to listener</td><td>Given to both</td></tr><tr><td>BlueBox</td><td>0.97</td><td>0.97</td><td>1.00</td></tr><tr><td>Red Sphere</td><td>0.92</td><td>0.91</td><td>0.80</td></tr><tr><td>White Cylinder</td><td>0.95</td><td>0.95</td><td>0.90</td></tr><tr><td>Gray Capsule</td><td>0.91</td><td>0.91</td><td>1.00</td></tr><tr><td>Yellow Ellipsoid</td><td>0.91</td><td>0.90</td><td>1.00</td></tr></table>
+
+From the grammar analysis in the previous section, we have shown that the emerged language strongly follows a well-defined grammar. In the zero-shot test, the agents demonstrated that they can successfully describe novel object, although not perfectly, by also following a similar grammar. Both are, as stated in the beginning of section 3, necessary conditions for any communication to be considered compositional. Therefore we can safely conclude that the emerged language in this work possesses some qualifications to be considered compositional.
+
+# 4 DISCUSSION
+
+In this work, we used the obverter technique to train neural network agents to communicate in a two-person image description game. Through qualitative analysis, visualization and the zero-shot test, we have shown that even though the agents receive raw perception in the form of image pixels, under the right environment pressures, the emerged language had properties consistent with the ones found in compositional languages.
+
+As an evaluation strategy, we followed previous works and focused on assessing the necessary conditions of compositional languages. However, the exact definition of compositional language is still somewhat debatable, and, to the best of our knowledge, there is no reliable way to mathematically quantify the degree of compositionality of an arbitrary language. Therefore, in order to encourage active research and discussion among researchers in this domain, we propose for future work, a quantitatively measurable definition of compositionality. We believe compositionality of a language is not binary (e.g. language A is compositional/not compositional), but a spectrum. For example, human language has some aspects that are compositional (e.g., syntactic constructions, most morphological combinations) and some that are not (e.g., irregular verb tenses in English, character-level word composition). It is also important to clearly define grounded language and compositional language. If one agent says abc (eat red apple) and another says cba (apple red eat), and they both understand each other, are they speaking compositional language? We believe such questions should be asked and addressed to shape the definition of compositionality.
+
+In addition to the definition/evaluation of compositional languages, there are numerous directions of future work. Observing the emergence of a compositional language among more than two agents is an apparent next step. Designing an environment to motivate the agents to disentangle more than two factors is also an interesting direction. Training agents to consider the context (i.e. pragmatics), such as giving each agent several images instead of one, is another exciting future work.
+
+# ACKNOWLEDGMENTS
+
+We would like to thank Scott Reed for discussions on pragmatics, Tom Le Paine for advising the visual module architecture, Jakob Foerster for discussions on grammar induction, Sookyung Kim and Joonseok Lee for discussions on human language and compositionality, Phil Blunsom and Jimeng Sun for helpful comments on the manuscript.
+
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+![](images/81623d6fa1690cf173bcc5d807c2fd91fc367f71b2d863a4604fa77be6c9f3a2.jpg)  
+Table 5: Meaning vectors are composed of (a) subject vectors and (b) predicate vectors. (c) shows 10 out of 100 possible meaning vectors.
+
+# A EMERGENCE OF GRAMMAR, BATALI (1998)
+
+In Batali (1998), the author successfully trained neural agents to develop a structured (i.e. grammatical) language using disentangled meaning vectors as the input. Using 10 subject vectors and 10 predicate vectors, all represented as explicit binary vectors, total 100 meaning vectors could be composed(Table 5). Each digit in the subject vector 5a serves a clear role, respectively representing speaker $( s p )$ , hearer $( h r )$ , other $( o t )$ , and plural $( p l )$ . The predicate vector values, on the other hand, are randomly chosen so that each predicate vector will have three 1’s and three 0’s. The combination of ten subject vectors and ten predicate vectors allows 100 meaning vectors.
+
+The author used twenty neural agents for the experiment. Each agent was implemented with the vanilla recurrent neural networks (RNN), where the hidden vector h’s size was 10, same as the size of the meaning vector m in order to treat h as the agent’s understanding of m. In each training round a single learner (i.e. listener) and ten teachers (i.e. speaker) were randomly chosen. Each teacher, given all $1 0 0 \mathbf { m }$ ’s in random order, generates a message $\mathbf { s } ^ { 5 }$ for each $\mathbf { m }$ and sends it to the learner. The messages are generated using the obverter techinque, which is described in Algorithm 1. The learner is trained to minimize the mean squared error (MSE) between $\mathbf { h }$ (after consuming the s) and m. After the learner has learned from all ten teachers, the next round begins, repeating the process until the error goes below some threshold.
+
+1 $\mathbf { h } ^ { ( 0 ) } = \mathbf { 0 }$ //Initialize RNN hidden layer with zeros;   
+2 $\mathbf { s } = \left[ \mathbf { \epsilon } \right]$ //Initialize the message vector;   
+3 $t = 0$ //Timestep index;   
+4 $\mathbf { V } = \mathbf { I } \in \mathbb { R } ^ { 4 \times 4 }$ //Each row $\mathbf { v } _ { 0 } , \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } , \mathbf { v } _ { 3 }$ corresponds to $a , b , c , d$ ;   
+5 while $| \mathbf { s } | < m a x$ message length do   
+6 $\mathbf { h } _ { \mathbf { v } _ { i } } ^ { ( t ) } = \sigma ( \mathbf { v } _ { i } \mathbf { W } _ { i } + \mathbf { h } ^ { ( t - 1 ) } \mathbf { W } _ { h } + \mathbf { b } )$ ;   
+7 $i = \mathrm { a r g m i n } _ { i } \vert \vert \mathbf { m } - \mathbf { h } _ { \mathbf { v } _ { i } } ^ { ( t ) } \vert \vert ^ { 2 }$ ;   
+8 $\mathbf { h } ^ { ( t ) } = \mathbf { h } _ { \mathbf { v } _ { i } } ^ { ( t ) }$ ;   
+9 Append $i$ to s;   
+10 if $| | \mathbf { m } - \mathbf { h } ^ { ( t ) } | | ^ { 2 } < t h r e s h o l d \mathbf { t } 1$ hen   
+11 Terminate;
+
+When the training was complete, the author was able to find strong patterns in the messages used by the agents (Table 6). Note that the messages using predicates tired, scared, sick and happy especially follow a very clear pattern. Batali also conducted a zero-shot test where the agents were trained without the diagonal elements in Table 6 and tested with all 100 meaning vectors. The agents were able to successfully communicate even when held-out meaning vectors were used, but the
+
+<table><tr><td>=</td><td>one</td><td>they</td><td>you</td><td>yall</td><td>yup</td><td>me</td><td>we</td><td>mip</td><td>yumi</td><td>all</td></tr><tr><td>tired</td><td>cda</td><td>cdab</td><td>cdc</td><td>cdcb</td><td>cdba</td><td>cd</td><td>cdd</td><td>cddb</td><td>cdcd</td><td>cdb</td></tr><tr><td>scared</td><td>caa</td><td>caab</td><td>cac</td><td>cacb</td><td>caba</td><td>ca</td><td>cad</td><td>cadb</td><td>cacd</td><td>cab</td></tr><tr><td>sick</td><td>daa</td><td>daab</td><td>dac</td><td>dacb</td><td>daba</td><td>da</td><td>dad</td><td>dadb</td><td>dacd</td><td>dab</td></tr><tr><td>happy sad</td><td>baa</td><td>baab abab</td><td>bca</td><td>bcab</td><td>baac</td><td>ba</td><td>badc</td><td>bab</td><td>bac</td><td>babc</td></tr><tr><td>excited</td><td>aba cba</td><td>cbab</td><td>ac</td><td>acb cacb</td><td>abac cbca</td><td>a</td><td>abdc ccdc</td><td>abb</td><td>abc</td><td>abbc</td></tr><tr><td>angry</td><td>bb</td><td>bbb</td><td>cca bc</td><td></td><td></td><td>c</td><td></td><td>cb</td><td>ccb</td><td>cbc</td></tr><tr><td></td><td></td><td></td><td></td><td>bcb</td><td>bbc</td><td>b</td><td>bddc</td><td>bdb</td><td>bdc</td><td>bdbc</td></tr><tr><td>silly</td><td>aa</td><td>aaab</td><td>aca</td><td>acab</td><td>adba</td><td>add</td><td>addc</td><td>adad</td><td>adc</td><td>adbc</td></tr><tr><td>thirsty</td><td>dbaa</td><td>dbab</td><td>dca</td><td>dcba</td><td>dbca</td><td>dda</td><td>ddac</td><td>dbad</td><td>dcad</td><td>dbacd</td></tr><tr><td>hungry</td><td>dbb</td><td>dbbd</td><td>dc</td><td>dcb</td><td>dbc</td><td>dd</td><td>ddc</td><td>dbd</td><td>dcd</td><td>dbcd</td></tr></table>
+
+<table><tr><td>-</td><td>one</td><td>they</td><td>you</td><td>yall</td><td>yup</td><td>me</td><td>we</td><td>mip</td><td>yumi</td><td>all</td></tr><tr><td>tired</td><td>cda</td><td>cdab</td><td>cdc</td><td>cdcb</td><td>cdba</td><td>cd</td><td>cdd</td><td>cddb</td><td>cdcd</td><td>cdb</td></tr><tr><td>scared</td><td>caa</td><td>caab</td><td>cac</td><td>cacb</td><td>caba</td><td>ca</td><td>cad</td><td>cadb</td><td>cacd</td><td>cab</td></tr><tr><td>sick</td><td>daa</td><td>daab</td><td>dac</td><td>dacb</td><td>daba</td><td>da</td><td>dad</td><td>dadb</td><td>dacd</td><td>dab</td></tr><tr><td>happy</td><td>baa</td><td>baab</td><td>bca</td><td>bcab</td><td>baac</td><td>ba</td><td>badc</td><td>bab</td><td>bac</td><td>babc</td></tr><tr><td>sad excited</td><td>aba</td><td>abab</td><td>ac</td><td>acb</td><td>abac</td><td>a</td><td>abdc</td><td>abb</td><td>abc</td><td>abbc</td></tr><tr><td></td><td>cba</td><td>cbab</td><td>cca</td><td>cacb</td><td>cbca</td><td>C</td><td>ccdc</td><td>cb</td><td>ccb</td><td>cbc</td></tr><tr><td>angry</td><td>cb</td><td>cbb</td><td>CC</td><td>ccb</td><td>cbc</td><td>C</td><td>cddc</td><td>cdb</td><td>cdc</td><td>cdbc</td></tr><tr><td>silly</td><td>(aa)</td><td>(aaab)</td><td>(aca)</td><td>(acab)</td><td>adba</td><td>add</td><td>addc</td><td>adad</td><td>adc</td><td>adbc</td></tr><tr><td>thirsty</td><td>dbaa</td><td>dbab</td><td>dca</td><td>dcba</td><td>dbca</td><td>dda</td><td>ddac</td><td>dbad</td><td>dcad</td><td>dbacd</td></tr><tr><td>hungry</td><td>dbb</td><td>dbbd</td><td>dc</td><td>dcb</td><td>dbc</td><td>dd</td><td>ddc</td><td>dbd</td><td>dcd</td><td>dbcd</td></tr></table>
+
+Table 6: (Top) Messages used by a majority of the population for each of the given meanings. (Bottom) A potential analysis of the system in terms of a root plus modifications. Italic symbols are used to specify predicates and roman symbols are used to specify subjects. Messages in parentheses cannot be made to fit into this analysis.
+
+messages used for the held-out meaning vectors did not show as strong compositional patterns as the non-zero-shot case.
+
+# B COMPARISON BETWEEN THE OBVERTER TECHINQUE AND THEREINFORCEMENT LEARNING-BASED APPROACH
+
+The obverter technique allows us to generate messages that encourage the agents to use a shared language, even a highly structured one, via using a single RNN for both speaking and listening.
+
+This is quite different from other RL-based related works (Lazaridou et al., 2017; Mordatch & Abbeel, 2017; Foerster et al., 2016; Jorge et al., 2016; Kottur et al., 2017) where each agent has separate components (e.g. two RNNs) for generating messages and consuming messages. This is necessary typically because the message generation module and the message consumption module have different input/output requirements. The message generation module accepts some input related to the task (e.g. goal description vector, question embedding, or image embedding) and generates discrete symbols. The message consumption module, on the other hand, accepts discrete symbols (i.e. the message) and generates some output related to the task (e.g. some prediction or some action to take). Therefore, when a neural agent speaks in the RL-based approach, its message generation process is completely separated from its own listening process, but tied to the listening process of another agent (i.e. listener)6. This means an agent may not have internal consistency; what an agent speaks may not make sense to itself. However, agents in the RL-based setting do converge on a common language because, during the training, the error signal flows from the listener to the speaker directly.
+
+Obverter approach, on the other hand, requires that each agent has a single component for both message generation and message consumption. This single component accepts discrete symbols and generates some output related to the task. This guarantees internal consistency because an agent’s message generation process is tied to its own message consumption process; it will only generate messages that make sense to itself. In the obverter setting, the error signal does not flow between agents directly, but agents converge on a common language by taking turns to be the listener; the listener tries to understand what the speaker says, so that when the listener becomes the speaker, it can generate messages that make sense to itself and, at the same time, will be understood by the former speaker (now listener).
+
+The advantage of obverter approach over RL-based approach is that it is motivated by the theory of mind and more resembles the acquisition/development process of human language. Having a single mechanism for both speaking and listening, and training oneself to be a good listener leads to the emergence of self-consistent, shared language. However, obverter technique requires that all agents perform the same task, which means all agents must have identical model architectures. This is because, during the message generation process, the speaker internally simulates what the listener will go though when it hears the message. Therefore we cannot play an asymmetrical game such as where the speaker sees only one image and generates a message but the listener is given multiple images and must choose one after hearing the message. RL-based approaches do not have this problem since there are separate modules for speaking and listening.
+
+We believe obverter technique could be the better choice for certain tasks regarding human mind emulation. But it certainly is not the tool for every occasion. The RL-based approach is a robust tool for any general task that may or may not involve human-like communication. We conclude this section with a possible future research direction that combines the strengths of both approaches to enable communication in more interesting and complicated tasks.
+
+# C MODEL ARCHITECTURE DETAILS
+
+We used TensorFlow and the Sonnet library for all implementation.
+
+# C.1 VISUAL MODULE
+
+We used an eight-layer convolutional neural network. We used 32 filters with the kernel size 3 for every layer. The strides were $[ 2 , 1 , 1 , 2 , 1 , 2 , 1 , 2 ]$ for each layer. We used rectified linear unit (ReLU) as the activation function for every layer. Batch normalization was used for every layer. We did not use the bias parameters since we used Batch normalization. For padding, we used the TensorFlow VALID padding option for every layer. The fully connected layer that follows the convolutional neural network was of 256 dimensions, with ReLU as the activation function.
+
+# C.2 LANGUAGE MODULE
+
+We used a single layer Gated Recurrent Units (GRU) to implement the language module. The size of the hidden layer was 64.
+
+# C.3 DECISION MODULE
+
+We used a two-layer feedforward neural network. The first layer reduces the dimensionality to 128 with ReLU as the activation function, then the second layer generates a scalar value with sigmoid as the activation function.
+
+# D MESSAGE GENERATION ALGORITHM USED IN OUR WORK
+
+# E TRAINING DETAILS
+
+Both agents’ model parameters are randomly initialized. The training process consists of rounds where teacher/learner roles are changed, and each round consists of multiple games where learner’s model parameters are updated. In each game, the teacher, given a mini-batch of images, generates corresponding messages. The learner, given a separate mini-batch of images and the messages from the teacher, decides whether it is seeing the same object type as the teacher. Learner’s model parameters are updated to minimize the cross entropy loss. After playing a predefined number of games, we move on to the next round where two agents change their roles.
+
+1 $\mathbf { h } ^ { ( 0 ) } = \mathbf { 0 }$ //Initialize GRU hidden layer with zeros;   
+2 $\mathbf { s } = \left[ \mathbf { \epsilon } \right]$ //Initialize the message vector;   
+3 $t = 0$ //Timestep index;   
+4 $\mathbf { V } = \mathbf { I } \in \mathbb { R } ^ { 5 \times 5 }$ //Each row $\mathbf { v } _ { 0 } , \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } , \mathbf { v } _ { 3 } , \mathbf { v } _ { 4 }$ corresponds to $a , b , c , d , e$ ;   
+5 $\mathbf { x } =$ image;   
+6 $\mathbf { z } = V i s u a l M o d u l e ( \mathbf { x } )$ ;   
+7 while $| \mathbf { s } | < m a x$ message length do   
+8 $\mathbf { h } _ { \mathbf { v } _ { i } } ^ { ( t ) } = G R U ( \mathbf { v } _ { i } , \mathbf { h } ^ { ( t - 1 ) } )$ ;   
+9 $\mathbf { h } ^ { ( t ) } = \mathbf { h } _ { \mathbf { v } _ { i } } ^ { ( t ) }$ $i = \mathrm { a r g m a x } _ { i }$ DecisionModule $( [ \mathbf { h } _ { \mathbf { v } _ { i } } ^ { ( t ) } , \mathbf { z } ] )$ ;   
+11 Append $i$ to s;   
+12 if DecisionModule $( [ \mathbf { h } _ { \mathbf { v } _ { i } } ^ { ( t ) } , \mathbf { z } ] ) >$ threshold then   
+13 Terminate;
+
+We found twenty games per round, with fifty images per mini-batch to work well. We repeat the rounds for 20, 000 times. Further rounds did not improve the results, or even degraded the performance. For vocabulary size (i.e. number of unique symbols) and the maximum message length, we used 5 and 20 respectively, similar to what Batali (1998) used. Note that when generating a message using the obverter technique, the generation process stops as soon as the speaker’s (i.e. teacher) output $\hat { y }$ becomes bigger than some threshold. In our work, we experimented with various values from 0.5 to 0.95, and found higher values to work better than lower values. We used 0.95 for all our final experiments.
+
+# F COMMUNICATION ACCURACY FOR EACH OBJECT TYPE
+
+Algorithm 2: Message generation process used in our work.   
+Table 7: Accuracy when each object type is given to the speaker.   
+
+<table><tr><td>■</td><td>Box</td><td>Sphere</td><td>Cylinder</td><td>Capsule</td><td>Ellipsoid</td></tr><tr><td>Blue</td><td>97.75</td><td>97.00</td><td>95.00</td><td>93.75</td><td>93.50</td></tr><tr><td>Red</td><td>95.50</td><td>93.00</td><td>95.75</td><td>95.25</td><td>97.00</td></tr><tr><td>White</td><td>95.25</td><td>96.25</td><td>93.50</td><td>94.50</td><td>97.00</td></tr><tr><td>Gray</td><td>93.00</td><td>95.00</td><td>94.25</td><td>96.75</td><td>97.25</td></tr><tr><td>Yellow</td><td>98.00</td><td>95.00</td><td>95.50</td><td>94.25</td><td>94.25</td></tr><tr><td>Green</td><td>96.00</td><td>93.50</td><td>95.00</td><td>94.50</td><td>95.25</td></tr><tr><td>Cyan</td><td>97.50</td><td>94.50</td><td>97.00</td><td>94.00</td><td>94.75</td></tr><tr><td>Magenta</td><td>95.25</td><td>96.75</td><td>94.75</td><td>94.50</td><td>95.25</td></tr></table>
+
+We conducted a separate test with the agents from round 16, 760 to assess the communication accuracy for each object type. The agents were given 1, 600 total object pairs $( 4 0 \times 4 0 )$ . Each object pair was tested 10 times, where after 5 times the agents switched speaker/listener roles. The average accuracy was $9 5 . 4 \%$ , and only 88 out of 1, 600 object pairs were communicated with accuracy lower than 0.8.
+
+Table 7 describes the accuracy when each object type was given to the speaker. We can observe that the accuracy is higher for objects that are described with less overlapping messages. For example, yellow box is communicated with the accuracy of $9 8 \%$ , and it is described with aaaaaa, which is not used for any other object types. Gray box, on the other hand, is communicated with accuracy $9 3 \%$ . It is described with aaa, which is also used for yellow capsule and green sphere, both of which are communicated with low accuracies as well.
+
+![](images/a3223aef4b5cd4670d5f3f12325fbeb3c69cf08d9078f75c33cd75f24d8f0912.jpg)  
+Figure 5: Ten communication examples when the speaker is given a blue box. Five examples on the left show when the listener is given blue objects. Five examples on the right show when the listener is given objects of different colors.
+
+# G COMMUNICATION EXAMPLE AND ANALYSIS
+
+Figure 5 provides ten examples of communication when the speaker is given a blue box and the listener is given various object types. The listener’s belief (i.e. score) that it is seeing the same image as the speaker changes each time it consumes a symbol. It is notable that most of the time the score jumps between 0 and 1, rather than gradually changing in between. This is natural given that messages that differ by only a single character can mean different objects (e.g. blue box and blue cylinder). This phenomenon can also be seen in human language. For example, blue can and blue cat differ by a single alphabet, but the semantics are completely different.
+
+Object types that are described by similar messages as blue box, such as blue cylinder and magenta box cause marginal confusion to the listener such that prediction scores for both objects are not complete zeros. There are also cases where two completely different objects are described by the same message as mentioned in Section 3.1. From Table 2 we can see that blue box and cyan cylinder are described by the same message $b b b b b b \{ b , d \}$ , although the messages were composed using different rules. Therefore the listener generates high scores for both objects, occasionally losing the game when the agents are given this specific object pair (1 out of 40 chance). This can be seen as a side effect coming from the principle of least effort which motivates the agents to win the game most of the time while minimizing the effort to generate messages.
+
+# H VISUALIZATION OF IMAGE EMBEDDINGS
+
+Section 3.2 provides strong evidence that the agents are properly recognizing the color and shape of an object. In this section, we study the visual module of both agents to study how they are processing the pixel input. We give each agent 1600 images, 40 per object type, and take their image embeddings (output of the fully-connected layer in the visual module). We use t-SNE (Maaten & Hinton, 2008) to reduce the dimensionality to 2D, and generate scatter plots as shown by Figure 6. The top row and the bottom row are the same scatter plots, but the dots are colored with different labels; the top row shows the color of each object, and the bottom row shows the shape of each object.
+
+![](images/2bfef7d4a774b0f9d6527840974612b010564b9945e307d3641516c2479550bc.jpg)  
+Figure 6: Scatter plots of the image embedding from the agents’ visual module. T-SNE was used to reduce the dimension to 2D.
+
+It is notable that the image embeddings form clear clusters, and all clusters are quite disentangled from one another. This means that the agents have learned to differentiate objects by their color and shape, and each color and shape have a specific place in the agents’ mind. It is also impressive that not only both agents learned similar relationships between colors and shapes(Figure 6 (a) and (b) show similar clusters, as do (d) and (e)), but also they learned similar absolute values for colors and shapes. Even when we plot image embeddings from both agents together (Figure 6 (c) and (f)) the cluster qualities are kept almost identical (with slightly higher number of clusters) to when we plot them separately. Therefore when one agent thinks of color red, and utters a message to describe it, the other agent hears the message and think of something red as well. This is, of course, what we wanted to achieve by using the obverter technique.
+
+The fact that a couple of color or shapes are occasionally in the same cluster suggests the agents have not perfectly disentangled colors and shapes. For perfect disentanglement, we believe some modifications to the image description game is required, which can be an important topic for future work. However, studying the cluster sheds some light on why the agents generate specific messages for each object type. For example, in Figure 6 (a) and (b), blue, white, cyan and magenta are often intertwined or their respective clusters are located nearby, as is the case for red, gray, yellow and green. This suggests the reason agents use prefix $^ b$ ’s to specify shapes for former color group and prefix a’s to specify shapes for the latter color group. Additionally, in Figure 6 (d) and (e), Box and cylinder are often located nearby, and sphere and ellipsoid show similar behavior7. We conjecture that this is the reason the messages describing box and cylinder are similarly long, and the messages describing sphere and ellipsoid tend to be similarly short.
+
+I ORIGINAL MESSAGES FROM THE ZERO-SHOT TEST   
+Table 8: Messages most often used by the two agents when speaking about a given object. Shaded cells indicate the objects not seen during the training. Brackets indicate the variation often seen at the last character.   
+
+<table><tr><td>■</td><td>Box</td><td>Sphere</td><td>Cylinder</td><td>Capsule</td><td>Ellipsoid</td></tr><tr><td>Blue Red</td><td>eeeeeee{e,ee}</td><td>eeeeee</td><td>eeeeee{e,ed}</td><td>eeee{e,ed}</td><td>eee{e,a}</td></tr><tr><td>White</td><td>ee{e,a} bbbbb</td><td>e,ea</td><td>e{e,a}</td><td>b,ba</td><td>a,c</td></tr><tr><td></td><td></td><td>bb e,ea</td><td>bbb{b,d}</td><td>bb{b,d}</td><td>bbb{d.c}</td></tr><tr><td>Gray</td><td>eeee</td><td>e{e,ec}</td><td>ee{e,a} b{c,d}</td><td>e,a</td><td>b,d</td></tr><tr><td>Yellow</td><td>bb{c.d}</td><td>eeee{e,a}</td><td>eeeee{e,a}</td><td>a,c</td><td>a,e</td></tr><tr><td>Green</td><td>eeeeee{e,a}</td><td></td><td></td><td>eee{e,a}</td><td>ee{e,a}</td></tr><tr><td>Cyan</td><td>eeeeeeeeea,a×20</td><td>eeeeee{e,a}</td><td>eeeeeee{e,ea}</td><td>eeeee{e,a}</td><td>eeeea</td></tr><tr><td>Magenta</td><td>bbb{b,d}</td><td>c,d</td><td>bb{b,d}</td><td>bb</td><td>b{c.a}</td></tr></table>
+
+J COMMUNICATION ACCURACY FOR EACH OBJECT TYPE IN ZERO-SHOT TEST   
+Table 9: Accuracy when each object type is given to the speaker. Shaded cells indicate the objects not seen during the training.   
+
+<table><tr><td></td><td rowspan=1 colspan=1>■</td><td rowspan=1 colspan=1>Box</td><td rowspan=1 colspan=1>Sphere</td><td rowspan=1 colspan=1>Cylinder</td><td rowspan=1 colspan=1>Capsule</td><td rowspan=1 colspan=1>Ellipsoid</td></tr><tr><td></td><td rowspan=1 colspan=1>Blue</td><td rowspan=1 colspan=1>96.75</td><td rowspan=1 colspan=1>95.00</td><td rowspan=1 colspan=1>94.75</td><td rowspan=1 colspan=1>95.75</td><td rowspan=1 colspan=1>96.50</td></tr><tr><td></td><td rowspan=2 colspan=1>RedWhite</td><td rowspan=1 colspan=1>94.25</td><td rowspan=1 colspan=1>90.00</td><td rowspan=1 colspan=1>94.25</td><td rowspan=1 colspan=1>96.75</td><td rowspan=3 colspan=1>91.2595.7596.00</td></tr><tr><td></td><td rowspan=1 colspan=1>99.00</td><td rowspan=1 colspan=1>96.00</td><td rowspan=1 colspan=1>95.75</td><td rowspan=1 colspan=1>97.25</td></tr><tr><td></td><td rowspan=2 colspan=1>GrayYellow</td><td rowspan=2 colspan=1>93.5098.75</td><td rowspan=1 colspan=1>91.00</td><td rowspan=1 colspan=1>94.75</td><td rowspan=1 colspan=1>91.50</td></tr><tr><td></td><td rowspan=1 colspan=1>94.75</td><td rowspan=1 colspan=1>95.75</td><td rowspan=1 colspan=1>92.00</td><td rowspan=1 colspan=1>91.50</td></tr><tr><td></td><td rowspan=1 colspan=1>Green</td><td rowspan=1 colspan=1>95.00</td><td rowspan=1 colspan=1>95.00</td><td rowspan=1 colspan=1>95.00</td><td rowspan=1 colspan=1>95.00</td><td rowspan=1 colspan=1>94.50</td></tr><tr><td rowspan=2 colspan=2>CyanMagenta</td><td rowspan=1 colspan=1>Cyan</td><td rowspan=1 colspan=1>92.00</td><td rowspan=1 colspan=1>95.00</td><td rowspan=1 colspan=1>97.50</td><td rowspan=1 colspan=1>95.00</td></tr><tr><td rowspan=1 colspan=1>95.25</td><td rowspan=1 colspan=1>96.00</td><td rowspan=1 colspan=1>97.00</td><td rowspan=1 colspan=1>96.00</td><td rowspan=1 colspan=1>95.75</td></tr></table>
+
+In the same manner as Appendix F, we conducted a separate test with the agents from round 19, 980 to assess the communication accuracy for each object type. The agents were given 1, 600 total object pairs $( 4 0 \times 4 0 )$ . Each object pair was tested 10 times, where after 5 times the agents switched speaker/listener roles. The average accuracy was $9 4 . 7 3 \%$ , and 103 out of 1, 600 object pairs were communicated with accuracy lower than 0.8.
+
+Table 9 describes the accuracy when each object type was given the speaker. Shaded cells indicate objects not seen during the training. Here we can observe the same tendency as the one seen in Appendix F; the accuracy is higher for objects that are described with less overlapping messages.
+
+# K AN EXAMPLE OF USING NEGATION TO PASS A ZERO-SHOT TEST
+
+Lets assume agent0 is aware of red circle, blue square and green triangle. If agent0 came upon a blue circle for the first time and had to describe it to agent1, the efficient way would be to say blue circle. But it could also say blue not square not triangle. If agent1 had a similar knowledge as agent0 did, then both agents would have a successful communication. However, it is debatable whether saying blue not square not triangle is as compositional as blue circle.

parse/train/rknt2Be0-/rknt2Be0-_content_list.json

@@ -0,0 +1,1734 @@
+[
+    {
+        "type": "text",
+        "text": "COMPOSITIONAL OBVERTER COMMUNICATION LEARNING FROM RAW VISUAL INPUT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            99,
+            743,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Edward Choi ∗   \nGeorgia Institute of Technology   \nAtlanta, GA, USA   \nmp2893@gatech.edu   \nAngeliki Lazaridou & Nando de Freitas   \nDeepMind   \nLondon, UK   \n{angeliki, nandodefreitas}@google.com ",
+        "bbox": [
+            183,
+            170,
+            392,
+            227
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            431,
+            170,
+            794,
+            227
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            262,
+            544,
+            277
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "One of the distinguishing aspects of human language is its compositionality, which allows us to describe complex environments with limited vocabulary. Previously, it has been shown that neural network agents can learn to communicate in a highly structured, possibly compositional language based on disentangled input (e.g. handengineered features). Humans, however, do not learn to communicate based on well-summarized features. In this work, we train neural agents to simultaneously develop visual perception from raw image pixels, and learn to communicate with a sequence of discrete symbols. The agents play an image description game where the image contains factors such as colors and shapes. We train the agents using the obverter technique where an agent introspects to generate messages that maximize its own understanding. Through qualitative analysis, visualization and a zero-shot test, we show that the agents can develop, out of raw image pixels, a language with compositional properties, given a proper pressure from the environment. ",
+        "bbox": [
+            233,
+            295,
+            766,
+            474
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            502,
+            336,
+            517
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "One of the key requirements for artificial general intelligence (AGI) to thrive in the real world is its ability to communicate with humans in natural language. Natural language processing (NLP) has been an active field of research for a long time, and the introduction of deep learning (LeCun et al., 2015) enabled great progress in NLP tasks such as translation, image captioning, text generation and visual question answering (Cho et al., 2014; Bahdanau et al., 2014; Vinyals et al., 2015; Karpathy & Fei-Fei, 2015; Hu et al., 2017; Serban et al., 2016; Lewis et al., 2017; Antol et al., 2015). However, training machines in a supervised manner with a large dataset has its limits when it comes to communication. Supervised methods are effective for capturing statistical associations between discrete symbols (i.e. words, letters). The essence of communication is more than just predicting the most likely word to come next; it is a means to coordinate with others and potentially achieve a common goal (Austin, 1975; Clark, 1996; Wittgenstein, 1953). ",
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+        "type": "text",
+        "text": "An alternative path to teaching machines the art of communication is to give them a specific task and encourage them to learn how to communicate on their own. This approach will encourage the agents to use languages grounded to task-related entities as well as communicate with other agents, which is one of the ways humans learn to communicate (Bruner, 1981). Recently, there have been several notable works that demonstrated the emergence of communication between neural network agents. Even though each work produced very interesting results of its own, in all cases, communication was either achieved with a single discrete symbol (as opposed to a sequence of discrete symbols) (Foerster et al., 2016; Lazaridou et al., 2017) or via a continuous value (Sukhbaatar et al., 2016; Jorge et al., 2016). Not only is human communication un-differentiable, but also using a single discrete symbol is quite far from natural language communication. One of the key features of human language is its compositional nature; the meaning of a complex expression is determined by its structure and the meanings of its constituents (Frege, 1892). More recently, Mordatch & Abbeel (2017) and Kottur et al. (2017) trained the agents to communicate in grounded, compositional language. In both studies, however, inputs given to the agents were hand-engineered features (disentangled input) rather than raw perceptual signals that we receive as humans. ",
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+    {
+        "type": "text",
+        "text": "In this work, we train neural agents to simultaneously develop visual perception from raw image pixels, and learn to communicate with a sequence of discrete symbols. Unlike previous works, our setup poses greater challenges to the agents since visual understanding and discrete communication have to be induced from scratch in parallel. We place the agents in a two-person image description game, where images contain objects of various color and shape. Inspired by the pioneering work of Batali (1998), we employ a communication philosophy named obverter to train the agents. Having its root in the theory of mind (Premack & Woodruff, 1978) and human language development (Milligan et al., 2007), the obverter technique motivates an agent to search over messages and generate the ones that maximize their own understanding. The contribution of our work can be summarized as follows: ",
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+        "text": "• We train artificial agents to learn to disentangle raw image pixels and communicate in compositional language at the same time.   \n• We describe how the obverter technique, a differentiable learning algorithm for discrete communication, could be employed in a communication game with raw visual input.   \n• We visualize how the agents are perceiving the images and show that they learn to disentangle color and shape without any explicit supervision other than the communication one.   \n• Experiment results suggest that the agents could develop, out of raw image input, a language with compositional properties, given a proper pressure from the environment (i.e. the image description game). ",
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+        "text": "Finally, while our exposition follows a multi-agent perspective, it is also possible to interpret our results in the single-agent setting. We are effectively learning a neural network that is able to learn disentangled compositional representations of visual scenes, without any supervision. Subject to the constraints imposed by their environment, our agents learn disentangled concepts, and how to compose these to form new concepts. This is an important milestone in the path to AGI. ",
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+    {
+        "type": "text",
+        "text": "2 METHOD ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "2.1 THE TWO-PERSON IMAGE DESCRIPTION GAME ",
+        "text_level": 1,
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+    {
+        "type": "image",
+        "img_path": "images/fe848c0db6f5de8bab346ead880e705a22d31e75c43a4509f5ea6f47130e08f7.jpg",
+        "image_caption": [
+            "Figure 1: The two-person image description game. Speaker observes an image and generates a message (i.e. a sequence of discrete symbols). The listener, after observing a separate image and the message, must correctly decide whether it is seeing the same object as the speaker (left side; output 1) or not (right side; output 0). "
+        ],
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+    {
+        "type": "image",
+        "img_path": "images/b0f08c116783796164e0f37b02eb2fea29ddc93f0d918fda9b7fa3e6ebcdda8e.jpg",
+        "image_caption": [
+            "Figure 2: Example images of the dataset. There are eight colors (blue, red, white, gray, yellow, green, cyan, magenta), and five shapes (box, sphere, cylinder, capsule, ellipsoid), giving us total 40 combinations. "
+        ],
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+    {
+        "type": "text",
+        "text": "We choose a straightforward image description game with two factors (color and shape) so that we can perform extensive analysis on the outcome confidently, based on full control of the experiment. In a single round of the two-person image description game, one agent becomes the speaker and the other the listener. The speaker is given a random image, and generates a message to describe it. The listener is also given a random image, possibly the same image as the speaker’s. After hearing the message from the speaker, the listener must decide if it is seeing the same object as the speaker (Figure 1). Note that an image is the raw pixels given to the agents, and an object is the thing described by the image. Therefore two different images can depict the same object. In each round the agents change roles of being the speaker and the listener. ",
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+    {
+        "type": "text",
+        "text": "We generated synthetic images using Mujoco physics simulator1. The example images are shown in Figure 2. Each image depicts a single object with a specific color and shape in $1 2 8 \\times 1 2 8$ resolution. There are eight colors (blue, red, white, gray, yellow, green, cyan, magenta) and five shapes (box, sphere, cylinder, capsule, ellipsoid), giving us 40 combinations. We generated 100 variations for each of the 40 object type. Note that the position of the object varies in each image, changing the object size and the orientation. Therefore even if the speaker and the listener are given the same object type, the actual images are very likely to be different, preventing the agents from using pixel-specific information, rather than object-related information to win the game. ",
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+    {
+        "type": "text",
+        "text": "2.2 MODEL ARCHITECTURE ",
+        "text_level": 1,
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+    {
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+        "img_path": "images/7902c027c72b5a000bf12271e28b3cfff9c03cda8af5899e656377dae25107a7.jpg",
+        "image_caption": [
+            "Figure 3: Agent model architecture. The visual module processes the image, and the language module generates or consumes messages. The decision module accepts embeddings from both modules and produces the output. The solid arrows indicate modifying the output from the previous layer. The dotted arrows indicate copying the output from the previous layer. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Aside from using disentangled input, another strong assumption made in previous works(Batali, 1998; Mordatch & Abbeel, 2017) was that the agents had access to the true intention of the speaker. In Batali (1998), the listener was trained to modify its RNN hidden vector as closely to the speaker’s intention (meaning vector; please see Table 5 in Appendix A) as possible. In Mordatch & Abbeel (2017), each agent had an auxiliary task to predict the goals of all other agents. In both cases, the true meaning/goal vector was used to update the model parameters, exposing the disentangled information to the agents. In order to relax this assumption and encourage the agents to develop communication with minimal guidance, our model uses no other signal than whether the listener made a correct decision. ",
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+    {
+        "type": "text",
+        "text": "Figure 3 depicts the agent model architecture. We use a convolutional neural network followed by a fully-connected layer to process the image. A single RNN, specifically the gated recurrent units (GRU) (Cho et al., 2014), is used for both generating and consuming messages (the message generation using the obverter strategy is described in the next section). When consuming a message, the image embedding from the visual module and the message embedding from the language module are concatenated and processed by another fully-connected layers (i.e. decision module) with the sigmoid output $\\hat { y }$ , 0 being “My (listener) image is different from the speaker’s” and 1 being “My image is the same as the speaker’s”. Further details of the model architecture (e.g. number of layers) are described in Appendix C. ",
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+    {
+        "type": "text",
+        "text": "2.3 OBVERTER TECHNIQUE ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Although our work is inspired by Batali (1998) (see Appendix A for the description of Batali (1998)), obverter technique is a general message generation philosophy used/discussed in a number of communication and language evolution studies (Hurford, 1989; Oliphant & Batali, 1997; Smith, 2001; Kirby & Hurford, 2002), which has its root in the theory of mind. Theory of mind (Premack & Woodruff, 1978) observes that a human has direct access only to one’s own mind and not to the others’. Therefore we typically assume that the mind of others is analogous to ours, and such assumption is reflected in the functional use of language (Bruner, 1981). For example, if we want to convey a piece of information to the listener2, it is best to speak in a way that maximizes the listener’s understanding. However, since we cannot directly observe the listener’s state of mind, we cannot exactly solve this optimization problem. Therefore we posit that the listener’s mind operates in a similar manner as ours, and speak in a way that maximizes our understanding, thus approximately solving the optimization problem. This is exactly what the obverter technique tries to achieve. ",
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+    {
+        "type": "text",
+        "text": "When an agent becomes the teacher (i.e. speaker), the model parameters are fixed. The image is converted to an embedding via the visual module. After initializing its RNN hidden layer to zeros, the teacher at each timestep evaluates $\\hat { y }$ for all possible symbols and selects the one that maximizes $\\hat { y }$ The RNN hidden vector induced by the chosen symbol is used in the next timestep. This is repeated until $\\hat { y }$ becomes bigger than the predefined threshold, or the maximum message length is reached (see Appendix D for algorithm). Therefore the teacher, through introspection, greedily selects characters at each timestep to generate a message such that the consistency between the image and the message is as clear to itself as possible. When an agent becomes the learner (i.e. listener), its parameters are updated via back-propagating the cross entropy loss between its output $\\hat { y }$ and the true label $y$ . Therefore the agents must learn to communicate only from the true label indicating whether the teacher and the learner are seeing the same object. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "We remind the reader that only the learner’s RNN parameters are updated, and the teacher uses its fixed RNN. Therefore an agent uses only one RNN for both speaking and listening, guaranteeing self-consistency (see Appendix B for a detailed comparison between the obverter technique and the RL-based approach). Furthermore, because the teacher’s parameters are fixed, message generation can easily be extended to be more exploratory. Although in this work we deterministically selected a character in each timestep, one can, for example, sample characters proportionally to $\\hat { y }$ and still use gradient descent for training the agents. Using a more exploratory message generation strategy could help us discover a more optimal communication language when dealing with complex tasks. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Another feature of the obverter technique is that it observes the principle of least effort (Zipf, 1949). Because the teacher stops generating symbols as soon as $\\hat { y }$ reaches the threshold, it does not waste any more effort trying to perfect the message. The same principle was implemented in one way or another in previous works, such as choosing the shortest among the generated strings (Kirby & Hurford, 2002) or imposing a small cost for generating a message (Mordatch & Abbeel, 2017). ",
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+    {
+        "type": "text",
+        "text": "2.4 ENVIRONMENTAL PRESSURE FOR COMPOSITIONAL COMMUNICATION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "During the early stages of research, we noticed that randomly sampling object pairs (one for the teacher, one for the learner) lead to agents focusing only on colors and ignoring shapes. When the teacher’s object is fixed, there are 40 (8 colors $\\times 5$ shapes) possibilities on the learner’s side. If the teacher only talks about the color of the object, the learner can correctly decide for 36 out of 40 possible object types. The learner makes incorrect decisions only when the teacher and the learner are given objects of the same color but different shapes, resulting in $9 0 \\%$ accuracy on average. This is actually what we observed; the accuracy plateaued between 0.9 and 0.92 during the training, and the messages were more or less the same for objects with the same color. Therefore when constructing a mini-batch of images, we set $2 5 \\%$ to be the object pairs of the same color and shape, $30 \\%$ the same shape but different colors, $20 \\%$ the same color but different shapes. The remaining $2 5 \\%$ object pairs were picked randomly3. ",
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+    {
+        "type": "text",
+        "text": "Vocabulary size (i.e. number of unique symbols) and the maximum message length were also influential to the final outcome. We noticed that a larger vocabulary and a longer message length helped the agents achieve a high communication accuracy more easily. But the resulting messages were more challenging to analyze for compositional patterns. In all our experiments we used 5 and 20 respectively for the vocabulary size and the maximum message length, similar to what Batali (1998) used. This suggests that the environment plays as important, if not more, role as the model architecture in the emergence of complex communication as discussed by previous studies (Kirby et al., 2014; Bratman et al., 2010; Kottur et al., 2017) and should be a main consideration for future efforts. Further details regarding hyperparameters are described in Appendix E. ",
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+    {
+        "type": "text",
+        "text": "3 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we first study the convergence behavior during the training phase. Then we analyze the language developed by the agents in terms of compositionality. As stated in the introduction, in compositional language, the meaning of a complex expression is determined by its structure and the meanings of its constituents. With this definition in mind, we focus on two aspects of the inter-agent communication to evaluate its compositional properties: the structure (i.e. grammar) of the communication, and zero-shot performance (i.e. generalizing to novel stimuli). These two aspects, which are both necessary conditions for any language to be considered compositional, have been used by previous works to study the compositional nature of artificial communication (Batali, 1998; Mordatch & Abbeel, 2017; Kottur et al., 2017). ",
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+        "type": "text",
+        "text": "To evaluate the structure of the messages, we study the evolution of the communication as training proceeds, and try to derive a grammar for expressing colors and shapes. To evaluate the zero-shot capabilities, we test if the agents can compose consistent messages for objects they have not seen during the training. Moreover, we visualize the image embeddings from the visual modules of both agents to understand how they are recognizing colors and shapes, the results of which, for a better view of the figures, are provided in Appendix H. ",
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+    {
+        "type": "text",
+        "text": "3.1 CONVERGENCE BEHAVIOR ",
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+    {
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+        "img_path": "images/e4918ef563341b6e90dc729d2d56d92b62fe35943fcf79abaaa0650a641662d6.jpg",
+        "image_caption": [
+            "Figure 4: Progress during the training (best seen in color). (Top) We plot the training accuracy, training loss, average message length and average message distinctness in each round. (Bottom) We plot the perplexities and the Jaccard similarity of the messages spoken by both agents in each round. Note that the average message length and the perplexities are divided by 20 to match the y-axis range with other metrics. "
+        ],
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+    {
+        "type": "text",
+        "text": "Figure 4 shows the convergence behavior during the training. Training accuracy was calculated by rounding the learner’s sigmoid output by 0.5. Message distinctness was calculated by dividing the number of unique messages in the mini-batch by the size of the mini-batch. Ideally there should be, on average, 40 distinct messages in the mini-batch of 50 images, therefore giving us 0.8 distinctness. Every 10 round, both agents were given the same 1, 000 randomly sampled images to generate 1, 000 message pairs. Then perplexity was calculated for each object type and averaged, thus indicating the average number of distinct messages used by the agents to describe a single object type (note that perplexities in the plot was divided by 20). Jaccard similarity between both agents’ messages was also calculated for each object type and averaged. ",
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+    {
+        "type": "text",
+        "text": "At the beginning, the listener (i.e. learner) always decides it is not seeing the same object as the speaker, giving us 0.75 accuracy4. But after 7, 000 rounds, accuracy starts to go beyond 0.9. Loss is negatively correlated with accuracy until round 15, 000, where it starts to fluctuate. Accuracy, however, remains high due to how accuracy is measured; by rounding the learner’s output by 0.5. Although we could occasionally observe some patterns in the messages when both accuracy and loss were high, a lower loss generally resulted in a clearer communication structure (i.e. grammar) and better zero-shot performance. The loss fluctuation also indicates some instability in the training process, which is a potential direction for future work. Message distinctness starts at near 0, indicating the agents are generating the same message for all object types. After round 7, 000, where both message distinctness and message length reach their maximum, both start to decrease. But message distinctness never goes as high as the ideal 0.8, meaning that the agents are occasionally using the same message for different object types, as will be shown in the following section. ",
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+    {
+        "type": "text",
+        "text": "Both perplexities and Jaccard similarity show seemingly meaningless fluctuation at early rounds. After round 7, 000, perplexities and Jaccard similarity show negatively correlated behavior, meaning that not only is each agent using consistent messages to describe each object type, but also both agents are using very similar messages to describe each object type. We found perplexity and Jaccard similarity to be an important indicator of the degree of the communication structure. During rounds $7 , 0 0 0 \\sim 8 , 0 0 0$ , performance was excellent in terms of loss and accuracy, but perplexity was high and Jaccard similarity low, indicating the agents were assigning incoherent strings to each object type just to win the game. Similar behavior was observed in the early stages of language evolution simulation in Kirby & Hurford (2002) where words represented some meanings but had no structure (i.e. protolanguage). It seems that artificial communication acquires compositional properties after the emergence of protolanguage regardless of whether the input is entangled or disentangled. ",
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+    {
+        "type": "text",
+        "text": "3.2 GRAMMAR ANALYSIS ",
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+    {
+        "type": "text",
+        "text": "We choose agents from different training rounds to highlight how the language becomes more structured over time. Table 1 shows agents’ messages in the beginning (round 40), when the training accuracy starts pushing beyond $9 0 \\%$ (round 6, 940), when agents settle on a common language (round 16, 760). ",
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+    {
+        "type": "text",
+        "text": "In round 40, both agents are respectively producing the same message for all object types as mentioned in section 3.1. We might say the messages are structured, but considering that the listener always answers 0 in early rounds, we cannot say the agents are communicating. In round 6, 940, which is roughly when the agents begin to communicate more efficiently, training accuracy is significantly higher than round 40. However, perplexities show that both agents are assigning many names to a single object type (40-80 names depending on the object type), indicating that the agents are focusing on pixel-level differences between images of the same object type. Table 1 shows, as an example, the messages used by both agents to describe the red sphere. Due to high perplexity, it is difficult to capture the underlying grammar of the messages even with regular expression. Furthermore, as Jaccard similarity indicates, both agents are generating completely different messages for the same object type. In round 16, 760, as the perplexities and Jaccard similarity tell us, the agents came to share a very narrow set of names for each object type (1-4 names depending on the object type). Moreover, the names of the same-colored objects and same-shaped objects clearly seem to follow a pattern. Overall, each of the three phases (round 40, round 6,940, round 16,760) seem to represent the development of visual perception, learning to communicate, and emergence of structure. ",
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+    {
+        "type": "text",
+        "text": "We found the messages in round 16, 760 could be decomposed in a similar manner as Table 6 in Appendix A. The top of Table 2 shows a possible decomposition of the messages from round 16, 760 and the bottom shows the rules for each color and shape derived from the decomposition. According to our analysis, the agents use the first part of the message (i.e. prefix) to specify a shape, and the second part (i.e. suffix) to specify a color. However, they use two different strings to specify a shape. For example, the agents use either aaaa or bbbbb to describe a box. The strings used for specifying colors show slightly weaker regularity. For example, red is always described by either the suffix c or suffix e, but magenta is described by the suffix bb, bd, and sometimes b or bc. a used for ¯ gray objects ",
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+    {
+        "type": "table",
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+        "table_caption": [
+            "Round 40 (Training accuracy: $6 6 . 1 \\%$ , Agent0 perplexity:1.0, Agent1 perplexity:1.0, Jaccard similarity:0.0) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>■</td><td rowspan=1 colspan=1>All objectsdescribed by Agent 0</td><td rowspan=1 colspan=1>All objectsdescribed byAgent1</td></tr><tr><td rowspan=1 colspan=1>Message</td><td rowspan=1 colspan=1>dddddddddddddddddddd</td><td rowspan=1 colspan=1>bbbbbbbbbbbbbbbbbbbb</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Round 6,940 ",
+        "text_level": 1,
+        "bbox": [
+            455,
+            169,
+            539,
+            180
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/2e2f6d38361b624e84533ab1404050752c80d29d1359536ccff6bde3e0194449.jpg",
+        "table_caption": [
+            "(Training accuracy: $9 3 . 1 \\%$ , Agent0 perplexity:9.90, Agent1 perplexity:17.73, Jaccard similarity:0.0) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>■</td><td>Redsphere described byagent 0</td><td>Redspheredescribedbyagent1</td></tr><tr><td rowspan=\"9\">Messages</td><td>aeceaeaeaaaeeeeeeeee</td><td>aaedacacaaaaaaaaaaaa</td></tr><tr><td>aacceeaaaaeeeeeeeeee</td><td>aaccdaacaaaaaaaaaaaa</td></tr><tr><td>aaccceaaaaaaaaeeeeee</td><td>aaabcdadaaaaaaaaaaaa</td></tr><tr><td>aeeeeaaaaeeeeeeeeeee</td><td>aaeeacaeaaaaaaaaaaaa</td></tr><tr><td>aeaceeaeaeeeeeeeeeee</td><td>aaccdadaaaaaaacaaaaa</td></tr><tr><td>aceacacaaaaaaeeeeeee</td><td>aaedaceaaaaaaaaaaaaa</td></tr><tr><td>abeeeeaeeeeeeeeeeeee</td><td>aaeaccaeaaaaaaaaaaaa</td></tr><tr><td>aacceeaeeeeeeeeeeeee</td><td>aaceacaacaaaaaaaaaaa</td></tr><tr><td>aacceeaeaaeeeeeeeeee</td><td>aacdacdaaaaaaaaaaaaa</td></tr><tr><td>aeeacacaaaeeccceeeee</td><td></td><td>aaccdadaaaaacaaaaaaa</td></tr></table>",
+        "bbox": [
+            192,
+            194,
+            800,
+            335
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/517503722029e95167c4f50fb278299fcd9199e1a5f19f39b6c93f3be6ba0bfd.jpg",
+        "table_caption": [
+            "Round 16,760 (Training accuracy: $9 8 . 0 \\%$ , Agent0 perplexity:1.69, Agent1 perplexity:1.62, Jaccard similarity:0.82) ",
+            "Table 1: (Top) Messages used by both agents when speaking about any object in round 40. (Middle) Ten most frequent messages used by each agent to describe a red sphere in round 6, 940. (Bottom) Messages most often used by both agents for each object type in round 16, 760. Brackets indicate the variation often seen at the last character. Underscores indicate blanks. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>:</td><td>Box</td><td>Sphere</td><td>Cylinder</td><td>Capsule</td><td>Ellipsoid</td></tr><tr><td>Blue</td><td>bbbbbbb{b.d}</td><td>bb{b,d}</td><td>bbbbbb{b,d}</td><td>bbbbb{c,d}</td><td>bbbb{-,c,d}</td></tr><tr><td>Red White</td><td>aaaa{c,e}</td><td>aa{c,e}</td><td>aaa{c,e}</td><td>a{c,e}</td><td>c,e</td></tr><tr><td></td><td>bbbbbb</td><td>b,d</td><td>bbbb{b,d}</td><td>bbb{b.d}</td><td>bb{-,c,d}</td></tr><tr><td>Gray</td><td>aaa</td><td>a</td><td>aa</td><td>C</td><td>b,bd</td></tr><tr><td>Yellow</td><td>aaaaaa</td><td>aaaa</td><td>aaaaa</td><td>aaa</td><td>a{a,e}</td></tr><tr><td>Green</td><td>aaaa{a,ad}</td><td>aa{a,ad}</td><td>aaa{a.ad}</td><td>a{a,ad}</td><td>a</td></tr><tr><td>Cyan</td><td>ax20</td><td>bbb{b,d}</td><td>bbbbbbb{b,d}</td><td>bbbbbbd</td><td>bbbbb{b.d}</td></tr><tr><td>Magenta</td><td>bbbbbbb</td><td>b{b.d</td><td>bbbbb{b,d}</td><td>bbbbd</td><td>bbb{-.c,d}</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "represents deletion of the prefix a. Note that removing prefixes beyond their length causes the pattern to break. For example, gray box, gray sphere and gray cylinder use the same a¯aa to express the color, ¯ but gray capsule and gray ellipsoid use irregular suffixes. ",
+        "bbox": [
+            174,
+            594,
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+            636
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Despite some irregularities and one exceptional case (cyan box), the messages provide strong evidence that the agents learned to properly recognize color and shape from raw pixel input (see Appendix H for studying what the visual module learned), mapped each color and shape to prefixes and suffixes, and are able to compose meaningful messages to describe a given image to one another. Communication accuracy for each object type is described in Appendix F. Communication examples and their analysis are given in Appendix G. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "3.3 ZERO-SHOT EVALUATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            755,
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+            768
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "If the agents have truly learned to compose a message that can be divided into a color part and a shape part, then they should be able to accurately describe an object they have not seen before, which is another necessary condition for a compositional language. Therefore, we hold out five objects (the shaded cells in Table 3) from the dataset during the training and observe how agents describe five novel objects during the test phase. The agents were chosen from round 19, 980, which showed a high accuracy $( 9 7 . 8 \\% )$ , low perplexities (1.48, 1.65) and a high Jaccard similarity (0.75). Table 3 shows a potential decomposition of the messages used by the agents (original messages are described by Table 8 in Appendix I). We can observe that there is clearly a structure in the communication, although some messages show somewhat weaker patterns compared to when the agents were trained with all object types (Table 2). Suffixes for specifying yellow and magenta are especially irregular, even when we consider the effects of b and ¯ e. However, the messages describing the held-out object ¯ types show clear structure with the exception of yellow ellipsoid. ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "table",
+        "img_path": "images/dacd9fe559f2ed7b036c0b7cab2d949618cf2c40cf80ad45e099c0e858753566.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>■</td><td>Box</td><td>Sphere</td><td>Cylinder</td><td>Capsule</td><td>Ellipsoid</td></tr><tr><td>Blue Red</td><td>bbbbb bb{b,d} aaaa {c,e}</td><td>bb{b,d} aa {c,e}</td><td>bbbb bb{b.d} aaa {c,e}</td><td>bbb bb{c,d} a{c,e}</td><td>bb bb{-.c,d} {c,e}</td></tr><tr><td>White</td><td>bbbbbb</td><td>{b.d}</td><td>bbbb{b.d}</td><td>bbb {b.d}</td><td>bb{-c,d}</td></tr><tr><td>Gray</td><td>aaaa aaa</td><td>aa aaa</td><td>aaa aaa</td><td>a aac</td><td>aa{b,bd}</td></tr><tr><td>Yellow</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>aaaa aa</td><td>aaaa</td><td>aaaaa</td><td>aaa</td><td>a{ae}</td></tr><tr><td>Green</td><td>aaaa {a,ad}</td><td>aa {a,ad}</td><td>aaa {a,ad}</td><td>a{a,ad}</td><td>a</td></tr><tr><td>Cyan</td><td>a×20</td><td>bbb{b,d}</td><td>bbbb bbb{b,d}</td><td>bbb bbbd</td><td>bb bbb{b.d}</td></tr><tr><td>Magenta</td><td>bbbbbbb</td><td>b{b,d}</td><td>bbbbb{b.d}</td><td>bbb bd</td><td>bbb{-c,d}</td></tr></table>",
+        "bbox": [
+            217,
+            101,
+            777,
+            218
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/6f670dc99afe992b1e6c0df384ee179a8d2c33499e913f45b4b7f17f66ccdb8a.jpg",
+        "table_caption": [
+            "Define Blue, White, Cyan, Magenta as color group 0, rest as color group 1. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>■</td><td rowspan=1 colspan=1>Rule</td></tr><tr><td rowspan=1 colspan=1>Blue</td><td rowspan=3 colspan=1>End with bbb or bbdEnd with c or eEnd with b or dEnd with aaaEnd with aaEnd with a or adEnd with bbbb or bbbdEnd with bb or bd</td></tr><tr><td rowspan=1 colspan=1>RedWhite</td></tr><tr><td rowspan=1 colspan=1>Gray</td><td rowspan=1 colspan=1>GrayYellowGreenCyanMagenta</td></tr><tr><td rowspan=1 colspan=1>BoxSphereCylinderCapsuleEllipsoid</td><td rowspan=1 colspan=1>Start with bbbbb for color group O, start with aaaa for color group 1Start with aa for color group 1Start with bbbb for color group O, start with aaa for color group 1Start with bbb for color group O,start with a for color group 1Start with bb for color group 0</td></tr></table>",
+        "bbox": [
+            246,
+            242,
+            746,
+            420
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/33a781e3e5d4062ed9e9d59d3c69194abbcc7c1739d074727eba66f914b690dc.jpg",
+        "table_caption": [
+            "Table 2: (Top) Potential composition analysis of the messages from round 16,760 (bottom of Table 1). Italic symbols are used to specify shapes and roman symbols are used to specify colors. a indicates ¯ deleting a single prefix a. (Bottom) Rules for each color and shape derived from the top table. ",
+            "Table 3: Potential analysis of the messages observed in the zero-shot test. Gray cells indicate object types unseen during the training phase. Italic symbols are used to specify shapes and roman symbols to specify colors. b indicates deleting a single prefix¯ $b$ . e indicates deleting a single prefix ¯ $e$ . Underscores indicate blanks. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>■</td><td rowspan=1 colspan=3>Box</td><td rowspan=1 colspan=2>Sphere</td><td rowspan=1 colspan=3>Cylinder</td><td rowspan=1 colspan=1>Capsule</td><td rowspan=1 colspan=1>Ellipsoid</td></tr><tr><td rowspan=7 colspan=1>BlueRedWhiteGrayYellowGreen</td><td rowspan=1 colspan=3>eeeeee e{e,ee}</td><td rowspan=1 colspan=2>eeeeee</td><td rowspan=2 colspan=3>eeeee e{e,ed}eeeee eeee{e,a}</td><td rowspan=3 colspan=1>eee e{e,ed}eee eeee{b,ba}bb {b.d}</td><td rowspan=5 colspan=1>eee{e,a}ee eeee{a,c}bbb {d.c}ee eee{b.d}</td></tr><tr><td rowspan=6 colspan=3>eeeeee eeee{e.a}bbbbbeeeeeeeeeebbbb bb{c,d}eeeeee {e,a}</td><td rowspan=1 colspan=2>eeee eeee{e,ea}</td><td rowspan=1 colspan=2>eeee</td></tr><tr><td rowspan=1 colspan=2>bb</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=2></td><td rowspan=2 colspan=1></td><td rowspan=2 colspan=1>eeee eee{-,a}</td><td rowspan=1 colspan=1></td><td rowspan=2 colspan=3>eeeee eee{e,a}</td><td rowspan=2 colspan=1>eee eee{e,a}</td></tr><tr><td rowspan=4 colspan=2>bbbe{e,ec}eeee {e,a}eeee ee{e,a}bb{c.d</td></tr><tr><td rowspan=3 colspan=3>bbb bb{c,d}eeeee {e,a}eeeee ee{e,ea}bbbb{b.d}</td><td rowspan=2 colspan=1>bb bb{a.c}eee{e,a}</td><td rowspan=1 colspan=1>bbb bb{a,e}</td></tr><tr><td rowspan=2 colspan=1>eee{e,a}eee ee{e,a}bb bb</td><td rowspan=2 colspan=1>ee {e,a}ee eeabbb bb{c.a}</td></tr><tr><td rowspan=1 colspan=1>CyanMagenta</td><td rowspan=1 colspan=3>eeeeee eeeabbbb b{b.d}</td><td rowspan=1 colspan=1></td></tr></table>",
+        "bbox": [
+            179,
+            492,
+            813,
+            611
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            176,
+            707,
+            821,
+            736
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In order to assess the communication accuracy when held-out objects are involved, we conducted another test with the agents from round 19, 980. Each held-out object was given to the speaker, the listener, or both. In the first two cases, the held-out object was paired with all 40 object types and each pair was tested 10 times. In the last case, the held-out object was tested against itself 10 times. In all cases, the agents switched roles after 5 times. Table 4 shows communication accuracies for each case. We can see the agents can successfully communicate most of the time even when given novel objects. The last column shows that the listener is not simply producing 0 to maximize its chance to win the game. It is also notable that the objects described without $\\bar { \\mathsf b }$ or e show better performance in general. ¯ We noticed the communication accuracy for held-out objects seems relatively weak considering the messages used to describe them strongly showed structure. (Table 3). This, however, results from the grammar (i.e. structure) being not as straightforward as Table 2, especially with short messages (i.e. frequent use of $\\bar { \\mathsf b }$ and e). The same tendency can be observed for non-held-out objects as described ¯ by the per-object communication accuracy Table 9 in Appendix J. ",
+        "bbox": [
+            173,
+            743,
+            826,
+            924
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/b88809ab7238f10aacf21f76006badba3743f4dadd2459290f80dd87b5acee65.jpg",
+        "table_caption": [
+            "Table 4: Communication accuracy when agents were given objects not seen during the training. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Given to speaker</td><td>Given to listener</td><td>Given to both</td></tr><tr><td>BlueBox</td><td>0.97</td><td>0.97</td><td>1.00</td></tr><tr><td>Red Sphere</td><td>0.92</td><td>0.91</td><td>0.80</td></tr><tr><td>White Cylinder</td><td>0.95</td><td>0.95</td><td>0.90</td></tr><tr><td>Gray Capsule</td><td>0.91</td><td>0.91</td><td>1.00</td></tr><tr><td>Yellow Ellipsoid</td><td>0.91</td><td>0.90</td><td>1.00</td></tr></table>",
+        "bbox": [
+            267,
+            101,
+            725,
+            181
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "From the grammar analysis in the previous section, we have shown that the emerged language strongly follows a well-defined grammar. In the zero-shot test, the agents demonstrated that they can successfully describe novel object, although not perfectly, by also following a similar grammar. Both are, as stated in the beginning of section 3, necessary conditions for any communication to be considered compositional. Therefore we can safely conclude that the emerged language in this work possesses some qualifications to be considered compositional. ",
+        "bbox": [
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+            234,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            340,
+            310,
+            357
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this work, we used the obverter technique to train neural network agents to communicate in a two-person image description game. Through qualitative analysis, visualization and the zero-shot test, we have shown that even though the agents receive raw perception in the form of image pixels, under the right environment pressures, the emerged language had properties consistent with the ones found in compositional languages. ",
+        "bbox": [
+            174,
+            373,
+            825,
+            443
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "As an evaluation strategy, we followed previous works and focused on assessing the necessary conditions of compositional languages. However, the exact definition of compositional language is still somewhat debatable, and, to the best of our knowledge, there is no reliable way to mathematically quantify the degree of compositionality of an arbitrary language. Therefore, in order to encourage active research and discussion among researchers in this domain, we propose for future work, a quantitatively measurable definition of compositionality. We believe compositionality of a language is not binary (e.g. language A is compositional/not compositional), but a spectrum. For example, human language has some aspects that are compositional (e.g., syntactic constructions, most morphological combinations) and some that are not (e.g., irregular verb tenses in English, character-level word composition). It is also important to clearly define grounded language and compositional language. If one agent says abc (eat red apple) and another says cba (apple red eat), and they both understand each other, are they speaking compositional language? We believe such questions should be asked and addressed to shape the definition of compositionality. ",
+        "bbox": [
+            174,
+            450,
+            825,
+            630
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In addition to the definition/evaluation of compositional languages, there are numerous directions of future work. Observing the emergence of a compositional language among more than two agents is an apparent next step. Designing an environment to motivate the agents to disentangle more than two factors is also an interesting direction. Training agents to consider the context (i.e. pragmatics), such as giving each agent several images instead of one, is another exciting future work. ",
+        "bbox": [
+            174,
+            637,
+            825,
+            707
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            727,
+            326,
+            738
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We would like to thank Scott Reed for discussions on pragmatics, Tom Le Paine for advising the visual module architecture, Jakob Foerster for discussions on grammar induction, Sookyung Kim and Joonseok Lee for discussions on human language and compositionality, Phil Blunsom and Jimeng Sun for helpful comments on the manuscript. ",
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+        "page_idx": 8
+    },
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+        "image_caption": [
+            "Table 5: Meaning vectors are composed of (a) subject vectors and (b) predicate vectors. (c) shows 10 out of 100 possible meaning vectors. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            209,
+            99,
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+        ],
+        "page_idx": 11
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+    {
+        "type": "text",
+        "text": "A EMERGENCE OF GRAMMAR, BATALI (1998) ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            330,
+            575,
+            348
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In Batali (1998), the author successfully trained neural agents to develop a structured (i.e. grammatical) language using disentangled meaning vectors as the input. Using 10 subject vectors and 10 predicate vectors, all represented as explicit binary vectors, total 100 meaning vectors could be composed(Table 5). Each digit in the subject vector 5a serves a clear role, respectively representing speaker $( s p )$ , hearer $( h r )$ , other $( o t )$ , and plural $( p l )$ . The predicate vector values, on the other hand, are randomly chosen so that each predicate vector will have three 1’s and three 0’s. The combination of ten subject vectors and ten predicate vectors allows 100 meaning vectors. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The author used twenty neural agents for the experiment. Each agent was implemented with the vanilla recurrent neural networks (RNN), where the hidden vector h’s size was 10, same as the size of the meaning vector m in order to treat h as the agent’s understanding of m. In each training round a single learner (i.e. listener) and ten teachers (i.e. speaker) were randomly chosen. Each teacher, given all $1 0 0 \\mathbf { m }$ ’s in random order, generates a message $\\mathbf { s } ^ { 5 }$ for each $\\mathbf { m }$ and sends it to the learner. The messages are generated using the obverter techinque, which is described in Algorithm 1. The learner is trained to minimize the mean squared error (MSE) between $\\mathbf { h }$ (after consuming the s) and m. After the learner has learned from all ten teachers, the next round begins, repeating the process until the error goes below some threshold. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "1 $\\mathbf { h } ^ { ( 0 ) } = \\mathbf { 0 }$ //Initialize RNN hidden layer with zeros;   \n2 $\\mathbf { s } = \\left[ \\mathbf { \\epsilon } \\right]$ //Initialize the message vector;   \n3 $t = 0$ //Timestep index;   \n4 $\\mathbf { V } = \\mathbf { I } \\in \\mathbb { R } ^ { 4 \\times 4 }$ //Each row $\\mathbf { v } _ { 0 } , \\mathbf { v } _ { 1 } , \\mathbf { v } _ { 2 } , \\mathbf { v } _ { 3 }$ corresponds to $a , b , c , d$ ;   \n5 while $| \\mathbf { s } | < m a x$ message length do   \n6 $\\mathbf { h } _ { \\mathbf { v } _ { i } } ^ { ( t ) } = \\sigma ( \\mathbf { v } _ { i } \\mathbf { W } _ { i } + \\mathbf { h } ^ { ( t - 1 ) } \\mathbf { W } _ { h } + \\mathbf { b } )$ ;   \n7 $i = \\mathrm { a r g m i n } _ { i } \\vert \\vert \\mathbf { m } - \\mathbf { h } _ { \\mathbf { v } _ { i } } ^ { ( t ) } \\vert \\vert ^ { 2 }$ ;   \n8 $\\mathbf { h } ^ { ( t ) } = \\mathbf { h } _ { \\mathbf { v } _ { i } } ^ { ( t ) }$ ;   \n9 Append $i$ to s;   \n10 if $| | \\mathbf { m } - \\mathbf { h } ^ { ( t ) } | | ^ { 2 } < t h r e s h o l d \\mathbf { t } 1$ hen   \n11 Terminate; ",
+        "bbox": [
+            158,
+            627,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "When the training was complete, the author was able to find strong patterns in the messages used by the agents (Table 6). Note that the messages using predicates tired, scared, sick and happy especially follow a very clear pattern. Batali also conducted a zero-shot test where the agents were trained without the diagonal elements in Table 6 and tested with all 100 meaning vectors. The agents were able to successfully communicate even when held-out meaning vectors were used, but the ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/004fdabe8b0e0e6a8ff74df89bf7550173c82bf401750b5a2790f081cd9bf041.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>=</td><td>one</td><td>they</td><td>you</td><td>yall</td><td>yup</td><td>me</td><td>we</td><td>mip</td><td>yumi</td><td>all</td></tr><tr><td>tired</td><td>cda</td><td>cdab</td><td>cdc</td><td>cdcb</td><td>cdba</td><td>cd</td><td>cdd</td><td>cddb</td><td>cdcd</td><td>cdb</td></tr><tr><td>scared</td><td>caa</td><td>caab</td><td>cac</td><td>cacb</td><td>caba</td><td>ca</td><td>cad</td><td>cadb</td><td>cacd</td><td>cab</td></tr><tr><td>sick</td><td>daa</td><td>daab</td><td>dac</td><td>dacb</td><td>daba</td><td>da</td><td>dad</td><td>dadb</td><td>dacd</td><td>dab</td></tr><tr><td>happy sad</td><td>baa</td><td>baab abab</td><td>bca</td><td>bcab</td><td>baac</td><td>ba</td><td>badc</td><td>bab</td><td>bac</td><td>babc</td></tr><tr><td>excited</td><td>aba cba</td><td>cbab</td><td>ac</td><td>acb cacb</td><td>abac cbca</td><td>a</td><td>abdc ccdc</td><td>abb</td><td>abc</td><td>abbc</td></tr><tr><td>angry</td><td>bb</td><td>bbb</td><td>cca bc</td><td></td><td></td><td>c</td><td></td><td>cb</td><td>ccb</td><td>cbc</td></tr><tr><td></td><td></td><td></td><td></td><td>bcb</td><td>bbc</td><td>b</td><td>bddc</td><td>bdb</td><td>bdc</td><td>bdbc</td></tr><tr><td>silly</td><td>aa</td><td>aaab</td><td>aca</td><td>acab</td><td>adba</td><td>add</td><td>addc</td><td>adad</td><td>adc</td><td>adbc</td></tr><tr><td>thirsty</td><td>dbaa</td><td>dbab</td><td>dca</td><td>dcba</td><td>dbca</td><td>dda</td><td>ddac</td><td>dbad</td><td>dcad</td><td>dbacd</td></tr><tr><td>hungry</td><td>dbb</td><td>dbbd</td><td>dc</td><td>dcb</td><td>dbc</td><td>dd</td><td>ddc</td><td>dbd</td><td>dcd</td><td>dbcd</td></tr></table>",
+        "bbox": [
+            222,
+            101,
+            772,
+            242
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/41a1513b8f0051663e2b6f09172f21f3ea517adf28bdd283c3cc3c0bb5f347b6.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>-</td><td>one</td><td>they</td><td>you</td><td>yall</td><td>yup</td><td>me</td><td>we</td><td>mip</td><td>yumi</td><td>all</td></tr><tr><td>tired</td><td>cda</td><td>cdab</td><td>cdc</td><td>cdcb</td><td>cdba</td><td>cd</td><td>cdd</td><td>cddb</td><td>cdcd</td><td>cdb</td></tr><tr><td>scared</td><td>caa</td><td>caab</td><td>cac</td><td>cacb</td><td>caba</td><td>ca</td><td>cad</td><td>cadb</td><td>cacd</td><td>cab</td></tr><tr><td>sick</td><td>daa</td><td>daab</td><td>dac</td><td>dacb</td><td>daba</td><td>da</td><td>dad</td><td>dadb</td><td>dacd</td><td>dab</td></tr><tr><td>happy</td><td>baa</td><td>baab</td><td>bca</td><td>bcab</td><td>baac</td><td>ba</td><td>badc</td><td>bab</td><td>bac</td><td>babc</td></tr><tr><td>sad excited</td><td>aba</td><td>abab</td><td>ac</td><td>acb</td><td>abac</td><td>a</td><td>abdc</td><td>abb</td><td>abc</td><td>abbc</td></tr><tr><td></td><td>cba</td><td>cbab</td><td>cca</td><td>cacb</td><td>cbca</td><td>C</td><td>ccdc</td><td>cb</td><td>ccb</td><td>cbc</td></tr><tr><td>angry</td><td>cb</td><td>cbb</td><td>CC</td><td>ccb</td><td>cbc</td><td>C</td><td>cddc</td><td>cdb</td><td>cdc</td><td>cdbc</td></tr><tr><td>silly</td><td>(aa)</td><td>(aaab)</td><td>(aca)</td><td>(acab)</td><td>adba</td><td>add</td><td>addc</td><td>adad</td><td>adc</td><td>adbc</td></tr><tr><td>thirsty</td><td>dbaa</td><td>dbab</td><td>dca</td><td>dcba</td><td>dbca</td><td>dda</td><td>ddac</td><td>dbad</td><td>dcad</td><td>dbacd</td></tr><tr><td>hungry</td><td>dbb</td><td>dbbd</td><td>dc</td><td>dcb</td><td>dbc</td><td>dd</td><td>ddc</td><td>dbd</td><td>dcd</td><td>dbcd</td></tr></table>",
+        "bbox": [
+            209,
+            253,
+            782,
+            395
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Table 6: (Top) Messages used by a majority of the population for each of the given meanings. (Bottom) A potential analysis of the system in terms of a root plus modifications. Italic symbols are used to specify predicates and roman symbols are used to specify subjects. Messages in parentheses cannot be made to fit into this analysis. ",
+        "bbox": [
+            173,
+            405,
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+            462
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "messages used for the held-out meaning vectors did not show as strong compositional patterns as the non-zero-shot case. ",
+        "bbox": [
+            176,
+            491,
+            823,
+            518
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B COMPARISON BETWEEN THE OBVERTER TECHINQUE AND THEREINFORCEMENT LEARNING-BASED APPROACH",
+        "text_level": 1,
+        "bbox": [
+            174,
+            541,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The obverter technique allows us to generate messages that encourage the agents to use a shared language, even a highly structured one, via using a single RNN for both speaking and listening. ",
+        "bbox": [
+            174,
+            592,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "This is quite different from other RL-based related works (Lazaridou et al., 2017; Mordatch & Abbeel, 2017; Foerster et al., 2016; Jorge et al., 2016; Kottur et al., 2017) where each agent has separate components (e.g. two RNNs) for generating messages and consuming messages. This is necessary typically because the message generation module and the message consumption module have different input/output requirements. The message generation module accepts some input related to the task (e.g. goal description vector, question embedding, or image embedding) and generates discrete symbols. The message consumption module, on the other hand, accepts discrete symbols (i.e. the message) and generates some output related to the task (e.g. some prediction or some action to take). Therefore, when a neural agent speaks in the RL-based approach, its message generation process is completely separated from its own listening process, but tied to the listening process of another agent (i.e. listener)6. This means an agent may not have internal consistency; what an agent speaks may not make sense to itself. However, agents in the RL-based setting do converge on a common language because, during the training, the error signal flows from the listener to the speaker directly. ",
+        "bbox": [
+            174,
+            626,
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+            820
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Obverter approach, on the other hand, requires that each agent has a single component for both message generation and message consumption. This single component accepts discrete symbols and generates some output related to the task. This guarantees internal consistency because an agent’s message generation process is tied to its own message consumption process; it will only generate messages that make sense to itself. In the obverter setting, the error signal does not flow between agents directly, but agents converge on a common language by taking turns to be the listener; the listener tries to understand what the speaker says, so that when the listener becomes the speaker, it can generate messages that make sense to itself and, at the same time, will be understood by the former speaker (now listener). ",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The advantage of obverter approach over RL-based approach is that it is motivated by the theory of mind and more resembles the acquisition/development process of human language. Having a single mechanism for both speaking and listening, and training oneself to be a good listener leads to the emergence of self-consistent, shared language. However, obverter technique requires that all agents perform the same task, which means all agents must have identical model architectures. This is because, during the message generation process, the speaker internally simulates what the listener will go though when it hears the message. Therefore we cannot play an asymmetrical game such as where the speaker sees only one image and generates a message but the listener is given multiple images and must choose one after hearing the message. RL-based approaches do not have this problem since there are separate modules for speaking and listening. ",
+        "bbox": [
+            174,
+            180,
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+            319
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We believe obverter technique could be the better choice for certain tasks regarding human mind emulation. But it certainly is not the tool for every occasion. The RL-based approach is a robust tool for any general task that may or may not involve human-like communication. We conclude this section with a possible future research direction that combines the strengths of both approaches to enable communication in more interesting and complicated tasks. ",
+        "bbox": [
+            174,
+            327,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C MODEL ARCHITECTURE DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            416,
+            483,
+            431
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We used TensorFlow and the Sonnet library for all implementation. ",
+        "bbox": [
+            174,
+            446,
+            612,
+            460
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C.1 VISUAL MODULE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            477,
+            336,
+            491
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We used an eight-layer convolutional neural network. We used 32 filters with the kernel size 3 for every layer. The strides were $[ 2 , 1 , 1 , 2 , 1 , 2 , 1 , 2 ]$ for each layer. We used rectified linear unit (ReLU) as the activation function for every layer. Batch normalization was used for every layer. We did not use the bias parameters since we used Batch normalization. For padding, we used the TensorFlow VALID padding option for every layer. The fully connected layer that follows the convolutional neural network was of 256 dimensions, with ReLU as the activation function. ",
+        "bbox": [
+            174,
+            503,
+            825,
+            587
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C.2 LANGUAGE MODULE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            603,
+            362,
+            617
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We used a single layer Gated Recurrent Units (GRU) to implement the language module. The size of the hidden layer was 64. ",
+        "bbox": [
+            173,
+            628,
+            825,
+            657
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C.3 DECISION MODULE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            674,
+            351,
+            689
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We used a two-layer feedforward neural network. The first layer reduces the dimensionality to 128 with ReLU as the activation function, then the second layer generates a scalar value with sigmoid as the activation function. ",
+        "bbox": [
+            174,
+            700,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D MESSAGE GENERATION ALGORITHM USED IN OUR WORK ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            763,
+            683,
+            779
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "E TRAINING DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            795,
+            369,
+            810
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Both agents’ model parameters are randomly initialized. The training process consists of rounds where teacher/learner roles are changed, and each round consists of multiple games where learner’s model parameters are updated. In each game, the teacher, given a mini-batch of images, generates corresponding messages. The learner, given a separate mini-batch of images and the messages from the teacher, decides whether it is seeing the same object type as the teacher. Learner’s model parameters are updated to minimize the cross entropy loss. After playing a predefined number of games, we move on to the next round where two agents change their roles. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "1 $\\mathbf { h } ^ { ( 0 ) } = \\mathbf { 0 }$ //Initialize GRU hidden layer with zeros;   \n2 $\\mathbf { s } = \\left[ \\mathbf { \\epsilon } \\right]$ //Initialize the message vector;   \n3 $t = 0$ //Timestep index;   \n4 $\\mathbf { V } = \\mathbf { I } \\in \\mathbb { R } ^ { 5 \\times 5 }$ //Each row $\\mathbf { v } _ { 0 } , \\mathbf { v } _ { 1 } , \\mathbf { v } _ { 2 } , \\mathbf { v } _ { 3 } , \\mathbf { v } _ { 4 }$ corresponds to $a , b , c , d , e$ ;   \n5 $\\mathbf { x } =$ image;   \n6 $\\mathbf { z } = V i s u a l M o d u l e ( \\mathbf { x } )$ ;   \n7 while $| \\mathbf { s } | < m a x$ message length do   \n8 $\\mathbf { h } _ { \\mathbf { v } _ { i } } ^ { ( t ) } = G R U ( \\mathbf { v } _ { i } , \\mathbf { h } ^ { ( t - 1 ) } )$ ;   \n9 $\\mathbf { h } ^ { ( t ) } = \\mathbf { h } _ { \\mathbf { v } _ { i } } ^ { ( t ) }$ $i = \\mathrm { a r g m a x } _ { i }$ DecisionModule $( [ \\mathbf { h } _ { \\mathbf { v } _ { i } } ^ { ( t ) } , \\mathbf { z } ] )$ ;   \n11 Append $i$ to s;   \n12 if DecisionModule $( [ \\mathbf { h } _ { \\mathbf { v } _ { i } } ^ { ( t ) } , \\mathbf { z } ] ) >$ threshold then   \n13 Terminate; ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We found twenty games per round, with fifty images per mini-batch to work well. We repeat the rounds for 20, 000 times. Further rounds did not improve the results, or even degraded the performance. For vocabulary size (i.e. number of unique symbols) and the maximum message length, we used 5 and 20 respectively, similar to what Batali (1998) used. Note that when generating a message using the obverter technique, the generation process stops as soon as the speaker’s (i.e. teacher) output $\\hat { y }$ becomes bigger than some threshold. In our work, we experimented with various values from 0.5 to 0.95, and found higher values to work better than lower values. We used 0.95 for all our final experiments. ",
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+    },
+    {
+        "type": "text",
+        "text": "F COMMUNICATION ACCURACY FOR EACH OBJECT TYPE ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/2b0aae3d0c28893d77741dbe8bd14fdbc46cd11b53335f3d7760608d62e3e0c7.jpg",
+        "table_caption": [
+            "Algorithm 2: Message generation process used in our work. ",
+            "Table 7: Accuracy when each object type is given to the speaker. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>■</td><td>Box</td><td>Sphere</td><td>Cylinder</td><td>Capsule</td><td>Ellipsoid</td></tr><tr><td>Blue</td><td>97.75</td><td>97.00</td><td>95.00</td><td>93.75</td><td>93.50</td></tr><tr><td>Red</td><td>95.50</td><td>93.00</td><td>95.75</td><td>95.25</td><td>97.00</td></tr><tr><td>White</td><td>95.25</td><td>96.25</td><td>93.50</td><td>94.50</td><td>97.00</td></tr><tr><td>Gray</td><td>93.00</td><td>95.00</td><td>94.25</td><td>96.75</td><td>97.25</td></tr><tr><td>Yellow</td><td>98.00</td><td>95.00</td><td>95.50</td><td>94.25</td><td>94.25</td></tr><tr><td>Green</td><td>96.00</td><td>93.50</td><td>95.00</td><td>94.50</td><td>95.25</td></tr><tr><td>Cyan</td><td>97.50</td><td>94.50</td><td>97.00</td><td>94.00</td><td>94.75</td></tr><tr><td>Magenta</td><td>95.25</td><td>96.75</td><td>94.75</td><td>94.50</td><td>95.25</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We conducted a separate test with the agents from round 16, 760 to assess the communication accuracy for each object type. The agents were given 1, 600 total object pairs $( 4 0 \\times 4 0 )$ . Each object pair was tested 10 times, where after 5 times the agents switched speaker/listener roles. The average accuracy was $9 5 . 4 \\%$ , and only 88 out of 1, 600 object pairs were communicated with accuracy lower than 0.8. ",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Table 7 describes the accuracy when each object type was given to the speaker. We can observe that the accuracy is higher for objects that are described with less overlapping messages. For example, yellow box is communicated with the accuracy of $9 8 \\%$ , and it is described with aaaaaa, which is not used for any other object types. Gray box, on the other hand, is communicated with accuracy $9 3 \\%$ . It is described with aaa, which is also used for yellow capsule and green sphere, both of which are communicated with low accuracies as well. ",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/a3223aef4b5cd4670d5f3f12325fbeb3c69cf08d9078f75c33cd75f24d8f0912.jpg",
+        "image_caption": [
+            "Figure 5: Ten communication examples when the speaker is given a blue box. Five examples on the left show when the listener is given blue objects. Five examples on the right show when the listener is given objects of different colors. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "G COMMUNICATION EXAMPLE AND ANALYSIS",
+        "text_level": 1,
+        "bbox": [
+            174,
+            578,
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+            593
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Figure 5 provides ten examples of communication when the speaker is given a blue box and the listener is given various object types. The listener’s belief (i.e. score) that it is seeing the same image as the speaker changes each time it consumes a symbol. It is notable that most of the time the score jumps between 0 and 1, rather than gradually changing in between. This is natural given that messages that differ by only a single character can mean different objects (e.g. blue box and blue cylinder). This phenomenon can also be seen in human language. For example, blue can and blue cat differ by a single alphabet, but the semantics are completely different. ",
+        "bbox": [
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+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Object types that are described by similar messages as blue box, such as blue cylinder and magenta box cause marginal confusion to the listener such that prediction scores for both objects are not complete zeros. There are also cases where two completely different objects are described by the same message as mentioned in Section 3.1. From Table 2 we can see that blue box and cyan cylinder are described by the same message $b b b b b b \\{ b , d \\}$ , although the messages were composed using different rules. Therefore the listener generates high scores for both objects, occasionally losing the game when the agents are given this specific object pair (1 out of 40 chance). This can be seen as a side effect coming from the principle of least effort which motivates the agents to win the game most of the time while minimizing the effort to generate messages. ",
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "H VISUALIZATION OF IMAGE EMBEDDINGS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Section 3.2 provides strong evidence that the agents are properly recognizing the color and shape of an object. In this section, we study the visual module of both agents to study how they are processing the pixel input. We give each agent 1600 images, 40 per object type, and take their image embeddings (output of the fully-connected layer in the visual module). We use t-SNE (Maaten & Hinton, 2008) to reduce the dimensionality to 2D, and generate scatter plots as shown by Figure 6. The top row and the bottom row are the same scatter plots, but the dots are colored with different labels; the top row shows the color of each object, and the bottom row shows the shape of each object. ",
+        "bbox": [
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+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/2bfef7d4a774b0f9d6527840974612b010564b9945e307d3641516c2479550bc.jpg",
+        "image_caption": [
+            "Figure 6: Scatter plots of the image embedding from the agents’ visual module. T-SNE was used to reduce the dimension to 2D. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "It is notable that the image embeddings form clear clusters, and all clusters are quite disentangled from one another. This means that the agents have learned to differentiate objects by their color and shape, and each color and shape have a specific place in the agents’ mind. It is also impressive that not only both agents learned similar relationships between colors and shapes(Figure 6 (a) and (b) show similar clusters, as do (d) and (e)), but also they learned similar absolute values for colors and shapes. Even when we plot image embeddings from both agents together (Figure 6 (c) and (f)) the cluster qualities are kept almost identical (with slightly higher number of clusters) to when we plot them separately. Therefore when one agent thinks of color red, and utters a message to describe it, the other agent hears the message and think of something red as well. This is, of course, what we wanted to achieve by using the obverter technique. ",
+        "bbox": [
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+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "The fact that a couple of color or shapes are occasionally in the same cluster suggests the agents have not perfectly disentangled colors and shapes. For perfect disentanglement, we believe some modifications to the image description game is required, which can be an important topic for future work. However, studying the cluster sheds some light on why the agents generate specific messages for each object type. For example, in Figure 6 (a) and (b), blue, white, cyan and magenta are often intertwined or their respective clusters are located nearby, as is the case for red, gray, yellow and green. This suggests the reason agents use prefix $^ b$ ’s to specify shapes for former color group and prefix a’s to specify shapes for the latter color group. Additionally, in Figure 6 (d) and (e), Box and cylinder are often located nearby, and sphere and ellipsoid show similar behavior7. We conjecture that this is the reason the messages describing box and cylinder are similarly long, and the messages describing sphere and ellipsoid tend to be similarly short. ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "table",
+        "img_path": "images/4381fb9534e586f1c124e2c94e60600a3fa1ced6341dcf861bac5e549c080214.jpg",
+        "table_caption": [
+            "I ORIGINAL MESSAGES FROM THE ZERO-SHOT TEST ",
+            "Table 8: Messages most often used by the two agents when speaking about a given object. Shaded cells indicate the objects not seen during the training. Brackets indicate the variation often seen at the last character. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>■</td><td>Box</td><td>Sphere</td><td>Cylinder</td><td>Capsule</td><td>Ellipsoid</td></tr><tr><td>Blue Red</td><td>eeeeeee{e,ee}</td><td>eeeeee</td><td>eeeeee{e,ed}</td><td>eeee{e,ed}</td><td>eee{e,a}</td></tr><tr><td>White</td><td>ee{e,a} bbbbb</td><td>e,ea</td><td>e{e,a}</td><td>b,ba</td><td>a,c</td></tr><tr><td></td><td></td><td>bb e,ea</td><td>bbb{b,d}</td><td>bb{b,d}</td><td>bbb{d.c}</td></tr><tr><td>Gray</td><td>eeee</td><td>e{e,ec}</td><td>ee{e,a} b{c,d}</td><td>e,a</td><td>b,d</td></tr><tr><td>Yellow</td><td>bb{c.d}</td><td>eeee{e,a}</td><td>eeeee{e,a}</td><td>a,c</td><td>a,e</td></tr><tr><td>Green</td><td>eeeeee{e,a}</td><td></td><td></td><td>eee{e,a}</td><td>ee{e,a}</td></tr><tr><td>Cyan</td><td>eeeeeeeeea,a×20</td><td>eeeeee{e,a}</td><td>eeeeeee{e,ea}</td><td>eeeee{e,a}</td><td>eeeea</td></tr><tr><td>Magenta</td><td>bbb{b,d}</td><td>c,d</td><td>bb{b,d}</td><td>bb</td><td>b{c.a}</td></tr></table>",
+        "bbox": [
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+            137,
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+            255
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "table",
+        "img_path": "images/a20c0aaf7f9221bafb1a13cbcfdbb633fec04b47610d7e533553f9733bbff3f3.jpg",
+        "table_caption": [
+            "J COMMUNICATION ACCURACY FOR EACH OBJECT TYPE IN ZERO-SHOT TEST ",
+            "Table 9: Accuracy when each object type is given to the speaker. Shaded cells indicate the objects not seen during the training. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td rowspan=1 colspan=1>■</td><td rowspan=1 colspan=1>Box</td><td rowspan=1 colspan=1>Sphere</td><td rowspan=1 colspan=1>Cylinder</td><td rowspan=1 colspan=1>Capsule</td><td rowspan=1 colspan=1>Ellipsoid</td></tr><tr><td></td><td rowspan=1 colspan=1>Blue</td><td rowspan=1 colspan=1>96.75</td><td rowspan=1 colspan=1>95.00</td><td rowspan=1 colspan=1>94.75</td><td rowspan=1 colspan=1>95.75</td><td rowspan=1 colspan=1>96.50</td></tr><tr><td></td><td rowspan=2 colspan=1>RedWhite</td><td rowspan=1 colspan=1>94.25</td><td rowspan=1 colspan=1>90.00</td><td rowspan=1 colspan=1>94.25</td><td rowspan=1 colspan=1>96.75</td><td rowspan=3 colspan=1>91.2595.7596.00</td></tr><tr><td></td><td rowspan=1 colspan=1>99.00</td><td rowspan=1 colspan=1>96.00</td><td rowspan=1 colspan=1>95.75</td><td rowspan=1 colspan=1>97.25</td></tr><tr><td></td><td rowspan=2 colspan=1>GrayYellow</td><td rowspan=2 colspan=1>93.5098.75</td><td rowspan=1 colspan=1>91.00</td><td rowspan=1 colspan=1>94.75</td><td rowspan=1 colspan=1>91.50</td></tr><tr><td></td><td rowspan=1 colspan=1>94.75</td><td rowspan=1 colspan=1>95.75</td><td rowspan=1 colspan=1>92.00</td><td rowspan=1 colspan=1>91.50</td></tr><tr><td></td><td rowspan=1 colspan=1>Green</td><td rowspan=1 colspan=1>95.00</td><td rowspan=1 colspan=1>95.00</td><td rowspan=1 colspan=1>95.00</td><td rowspan=1 colspan=1>95.00</td><td rowspan=1 colspan=1>94.50</td></tr><tr><td rowspan=2 colspan=2>CyanMagenta</td><td rowspan=1 colspan=1>Cyan</td><td rowspan=1 colspan=1>92.00</td><td rowspan=1 colspan=1>95.00</td><td rowspan=1 colspan=1>97.50</td><td rowspan=1 colspan=1>95.00</td></tr><tr><td rowspan=1 colspan=1>95.25</td><td rowspan=1 colspan=1>96.00</td><td rowspan=1 colspan=1>97.00</td><td rowspan=1 colspan=1>96.00</td><td rowspan=1 colspan=1>95.75</td></tr></table>",
+        "bbox": [
+            287,
+            373,
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+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "In the same manner as Appendix F, we conducted a separate test with the agents from round 19, 980 to assess the communication accuracy for each object type. The agents were given 1, 600 total object pairs $( 4 0 \\times 4 0 )$ . Each object pair was tested 10 times, where after 5 times the agents switched speaker/listener roles. The average accuracy was $9 4 . 7 3 \\%$ , and 103 out of 1, 600 object pairs were communicated with accuracy lower than 0.8. ",
+        "bbox": [
+            174,
+            547,
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+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Table 9 describes the accuracy when each object type was given the speaker. Shaded cells indicate objects not seen during the training. Here we can observe the same tendency as the one seen in Appendix F; the accuracy is higher for objects that are described with less overlapping messages. ",
+        "bbox": [
+            174,
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+            666
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "K AN EXAMPLE OF USING NEGATION TO PASS A ZERO-SHOT TEST",
+        "text_level": 1,
+        "bbox": [
+            176,
+            688,
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+            703
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Lets assume agent0 is aware of red circle, blue square and green triangle. If agent0 came upon a blue circle for the first time and had to describe it to agent1, the efficient way would be to say blue circle. But it could also say blue not square not triangle. If agent1 had a similar knowledge as agent0 did, then both agents would have a successful communication. However, it is debatable whether saying blue not square not triangle is as compositional as blue circle. ",
+        "bbox": [
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+        ],
+        "page_idx": 17
+    }
+]