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parse/train/HkCsm6lRb/HkCsm6lRb.md

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+# GENERATIVE MODELS OF VISUALLY GROUNDED IMAGINATION
+
+Ramakrishna Vedantam∗ Georgia Tech vrama@gatech.edu
+
+Ian Fischer   
+Google Inc.   
+iansf@google.com Jonathan Huang   
+Google Inc.   
+jonathanhuang@google.com
+
+Kevin Murphy Google Inc. kpmurphy@google.com
+
+# ABSTRACT
+
+It is easy for people to imagine what a man with pink hair looks like, even if they have never seen such a person before. We call the ability to create images of novel semantic concepts visually grounded imagination. In this paper, we show how we can modify variational auto-encoders to perform this task. Our method uses a novel training objective, and a novel product-of-experts inference network, which can handle partially specified (abstract) concepts in a principled and efficient way. We also propose a set of easy-to-compute evaluation metrics that capture our intuitive notions of what it means to have good visual imagination, namely correctness, coverage, and compositionality (the $3 \ : C ' s$ ). Finally, we perform a detailed comparison of our method with two existing joint image-attribute VAE methods (the JMVAE method of Suzuki et al. (2017) and the BiVCCA method of Wang et al. (2016b)) by applying them to two datasets: the MNIST-with-attributes dataset (which we introduce here), and the CelebA dataset (Liu et al., 2015).
+
+# 1 INTRODUCTION
+
+Consider the following two-party communication game: a speaker thinks of a visual concept $C$ , such as “men with black hair”, and then generates a description $\mathbf { y }$ of this concept, which she sends to a listener; the listener interprets the description y, by creating an internal representation $\mathbf { z }$ , which captures its “meaning”. We can think of $\mathbf { z }$ as representing a set of “mental images” which depict the concept $C$ . To test whether the listener has correctly “understood” the concept, we ask him to draw a set of real images $S = \{ \mathbf { x } _ { s } : s = 1 : S \}$ , which depict the concept $C$ . He then sends these back to the speaker, who checks to see if the images correctly match the concept $C$ . We call this process visually grounded imagination.
+
+In this paper, we represent concept descriptions in terms of a fixed length vector of discrete attributes $\mathcal { A }$ . This allows us to specify an exponentially large set of concepts using a compact, combinatorial representation. In particular, by specifying different subsets of attributes, we can generate concepts at different levels of granularity or abstraction. We can arrange these concepts into a compositional abstraction hierarchy, as shown in Figure 1. This is a directed acyclic graph (DAG) in which nodes represent concepts, and an edge from a node to its parent is added whenever we drop one of the attributes from the child’s concept definition. Note that we dont make any assumptions about the order in which the attributes are dropped (that is, dropping the attribute “smiling” is just as valid as dropping “female” in Figure 1). Thus, the tree shown in the figure is just a subset extracted from the full DAG of concepts, shown for illustration purposes.
+
+We can describe a concept by creating the attribute vector $\mathbf { y } _ { \mathcal { O } }$ , in which we only specify the value of the attributes in the subset $\mathcal { O } \subseteq A$ ; the remaining attributes are unspecified, and are assumed to take all possible legal values. For example, consider the following concepts, in order of increasing abstraction: $C _ { m s b } =$ (male, smiling, blackhair), $C _ { * s b } = ( *$ , smiling, blackhair), and $C _ { * * b } = ( * , * , \mathrm { b l a c k h a i r } )$ , where the attributes are gender, smiling or not, and hair color, and $^ *$ represents “don’t care”. A good model should be able to generate images from different levels of the abstraction hierarchy, as shown in Figure 1. (This is in contrast to most prior work on conditional generative models of images, which assume that all attributes are fully specified, which corresponds to sampling only from leaf nodes in the hierarchy.)
+
+![](images/df5ae9ce9cf733cf7373f02ab07cee5213fc9e1882dff1aa502a1f29fdf95c60.jpg)  
+Figure 1: A compositional abstraction hierarchy for faces, derived from 3 attributes: hair color, smiling or not, and gender. We show a set of sample images generated by our model, when trained on CelebA, for different nodes in this hierarchy.
+
+In Section 2, we show how we can extend the variational autoencoder (VAE) framework of Kingma & Welling (2014) to create models which can perform this task. The first extension is to modify the model to the “multi-modal” setting where we have both an image, x, and an attribute vector, y. More precisely, we assume a joint generative model of the form $\begin{array} { r } { p ( \mathbf { x } , \mathbf { y } , \mathbf { z } ) = p ( \mathbf { z } ) p ( \mathbf { x } | \mathbf { z } ) p ( \mathbf { y } | \mathbf { z } ) } \end{array}$ , where $p ( \mathbf { z } )$ is the prior over the latent variable $\mathbf { z }$ , $p ( \mathbf { x } | \mathbf { z } )$ is our image decoder, and $p ( \mathbf { y } \vert \mathbf { z } )$ is our description decoder. We additionally assume that the description decoder factorizes over the specified attributes in the description, so $\begin{array} { r } { p ( \mathbf { \dot { y } } _ { \mathcal { O } } | \mathbf { z } ) = \prod _ { k \in \mathcal { O } } p ( y _ { k } | \mathbf { z } ) } \end{array}$ .
+
+We further extend the VAE by devising a novel objective function, which we call the TELBO, for training the model from paired data, $\bar { \cal D } \bar { \bf \Delta } = \{ ( { \bf x } _ { n } , { \bf y } _ { n } ) \}$ . However, at test time, we will allow unpaired data (either just a description or just an image). Hence we fit three inference networks: $q ( \mathbf { z } | \mathbf { x } , \mathbf { y } )$ , $q ( \mathbf { z } | \mathbf { x } )$ and $q ( \mathbf { z } | \mathbf { y } )$ . This way we can embed an image or a description into the same shared latent space (using $q ( \mathbf { z } | \mathbf { x } )$ and $q ( \mathbf { z } | \mathbf { y } )$ , respectively); this lets us “translate” images into descriptions or vice versa, by computing $\begin{array} { r } { p ( \mathbf { y } | \mathbf { x } ) = \int d \mathbf { z } \ p ( \mathbf { y } | \mathbf { z } ) q ( \mathbf { z } | \mathbf { x } ) } \end{array}$ and $\begin{array} { r } { p ( \mathbf { x } | \mathbf { y } ) = \int d \mathbf { z } \ p ( \mathbf { x } | \mathbf { z } ) q ( \mathbf { z } | \mathbf { y } ) } \end{array}$ .
+
+To handle abstract concepts (i.e., partially observed attribute vectors), we use a method based on the product of experts (POE) (Hinton, 2002). In particular, our inference network for attributes has the form $\begin{array} { r } { q ( \mathbf { z } | \mathbf { y } _ { \mathcal { O } } ) \propto p ( \mathbf { z } ) \prod _ { k \in \mathcal { O } } q ( \mathbf { z } | \mathbf { y } _ { k } ) } \end{array}$ . If no attributes are specified, the posterior is equal to the prior. As we condition on more attributes, the posterior becomes narrower, which corresponds to specifying a more precise concept. This enables us to generate a more diverse set of images to represent abstract concepts, and a less diverse set of images to represent concrete concepts, as we show below.
+
+Section 3 discusses how to evaluate the performance of our method in an objective way. Specifically, we first “ground” the description by generating a set of images, $\begin{array} { r } { \mathcal { S } ( \mathbf { y } _ { \mathcal { O } } ) = \{ \mathbf { x } ^ { s } \sim p ( \mathbf { x } | \mathbf { y } _ { \mathcal { O } } ) : s = 1 : } \end{array}$ $S \}$ . We then check that all the sampled images in $\scriptstyle { S ( \mathbf { y } _ { \mathcal { O } } ) }$ are consistent with the specified attributes $\mathbf { y } _ { \mathcal { O } }$ (we call this correctness). We also check that the set of images “spans” the extension of the concept, by exhibiting suitable diversity (c.f. (Young et al., 2014)). Concretely, we check that the attributes that were not specified (e.g., gender in $C _ { * s b }$ above) vary across the different images; we call this coverage. Finally, we want the set of images to have high correctness and coverage even if the concept $\mathbf { y } _ { \mathcal { O } }$ has a combination of attribute values that have not been seen in training. For example, if we train on $C _ { m s b } =$ (male, smiling, blackhair), and ${ C _ { f n b } = }$ (female, notsmiling, blackhair), we should be able to test on $C _ { m n b } =$ (male, notsmiling, blackhair), and $C _ { f s b } =$ (female, smiling, blackhair). We will call this property compositionality. Being able to generate plausible images in response to truly compositionally novel queries is the essence of imagination. Together, we call these criteria the 3 C’s of visual imagination.
+
+Section 5 reports experimental results on two different datasets. The first dataset is a modified version of MNIST, which we call MNIST-with-attributes (or MNIST-A), in which we “render” modified versions of a single MNIST digit on a $6 4 \mathrm { x } 6 4$ canvas, varying its location, orientation and size. The second dataset is CelebA (Liu et al., 2015), which consists of over $2 0 0 \mathrm { k }$ face images, annotated with 40 binary attributes. We show that our method outperforms previous methods on these datasets.
+
+The contributions of this paper are threefold. First, we present a novel extension to VAEs in the multimodal setting, introducing a principled new training objective (the TELBO), and deriving an interpretation of a previously proposed objective (JMVAE) (Wang et al., 2016a) as a valid alternative in Appendix A.1. Second, we present a novel way to handle missing data in inference networks based on a product of experts. Third, we present novel criteria (the 3 C’s) for evaluating conditional generative models of images, that extends prior work by considering the notion of visual abstraction and imagination.
+
+# 2 METHODS
+
+We start by describing standard VAEs, to introduce notation. We then discuss our extensions to handle the multimodal and the missing input settings.
+
+Standard VAEs. A variational autoencoder (Kingma & Welling, 2014) is a latent variable model of the form $p _ { \pmb { \theta } } ( \mathbf { x } , \mathbf { z } ) = p _ { \pmb { \theta } } ( \mathbf { z } ) p _ { \pmb { \theta } } ( \mathbf { x } | \mathbf { z } )$ , where $p _ { \pmb { \theta } } ( \mathbf { z } )$ is the prior (we assume it is Gaussian, $p _ { \pmb { \theta } } ( \mathbf { z } ) =$ $\mathcal { N } ( \mathbf { z } | \mathbf { 0 } , \mathbf { I } )$ , although this assumption can be relaxed), and $p _ { \pmb { \theta } } ( \mathbf { x } | \mathbf { z } )$ is the likelihood (sometimes called the decoder), usually represented by a neural network. To perform approximate posterior inference, we fit an inference network (sometimes called the encoder) of the form $q _ { \phi } ( \mathbf { z } | \mathbf { x } )$ , so as to maximize $\mathcal { L } ( \pmb \theta , \phi ) = \mathbb { E } _ { \hat { p } ( \mathbf { x } ) } [ \mathrm { e l b o } ( \mathbf { x } , \pmb \theta , \phi ) ]$ , where $\begin{array} { r } { \hat { p } ( \mathbf x ) = \frac { 1 } { N } \sum _ { n = 1 } ^ { N } \delta _ { \mathbf x _ { n } } ( \mathbf x ) } \end{array}$ is the empirical distribution, and ELBO is the evidence lower bound:
+
+$$
+\mathrm { e l b o } _ { \lambda , \beta } ( \mathbf { x } , \pmb { \theta } , \phi ) = \mathbb { E } _ { q _ { \phi } ( \mathbf { z } | \mathbf { x } , \phi ) } \left[ \lambda \log p _ { \theta } ( \mathbf { x } | \mathbf { z } ) \right] - \beta \mathrm { K L } ( q _ { \phi } ( \mathbf { z } | \mathbf { x } ) , p _ { \theta } ( \mathbf { z } ) )
+$$
+
+Here $\mathrm { K L } ( p , q )$ is the Kullback Leibler divergence between distributions $p$ and $q$ . By default, $\beta =$ $\lambda = 1$ , in which case we will just write $\mathrm { e l b o } ( { \bf x } , \pmb { \theta } , \phi )$ . However, by using $\beta > 1$ we can encourage the posterior to be closer to the factorial prior $p ( \mathbf { z } ) = \mathcal { N } ( \mathbf { z } | \mathbf { 0 } , \mathbf { I } )$ , which encouarges the latent factors to be “disentangled”, as proved in Achille $\&$ Soatto (2017); this is known as the $\beta$ -VAE trick (Higgins et al., 2017a). And allowing $\lambda > 1$ will be useful later, when we have multiple modalities.
+
+Joint VAEs and the TELBO. We extend the VAE to model images and attributes by defining the joint distribution $\begin{array} { r } { p _ { \pmb { \theta } } ( \mathbf { x } , \mathbf { y } , \mathbf { z } ) = p _ { \pmb { \theta } } ( \mathbf { z } ) p _ { \pmb { \theta } } ( \mathbf { x } | \mathbf { z } ) p _ { \pmb { \theta } } ( \mathbf { y } | \mathbf { z } ) } \end{array}$ , where $p _ { \pmb { \theta } } ( \mathbf { x } | \mathbf { z } )$ is the image decoder (we use the DCGAN architecture from Radford et al. (2016)), and $p _ { \pmb { \theta } } ( \mathbf { y } | \mathbf { z } )$ is an MLP for the attribute vector. The corresponding training objective which we want to maximize becomes ${ \mathcal { L } } ( \theta , \phi ) =$ $\mathbb { E } _ { \hat { p } ( \mathbf { x } , \mathbf { y } ) } \left[ \mathrm { e l b o } ( \mathbf { x } , \mathbf { y } , \pmb { \theta } , \phi ) \right]$ , where $\begin{array} { r } { \hat { p } ( \mathbf x , \mathbf y ) = \frac { 1 } { N } \sum _ { n = 1 } ^ { N } \delta _ { \mathbf x _ { n } } ( \mathbf x ) \delta _ { \mathbf y _ { n } } ( \mathbf y _ { n } ) } \end{array}$ is the empirical distribution derived from paired data, and the joint ELBO is given by
+
+$$
+\begin{array} { r l r } & { } & { \mathrm { e l b o } _ { \lambda _ { x } , \lambda _ { y } , \beta } ( { \bf x } , { \bf y } , \pmb { \theta } _ { x } , \pmb { \theta } _ { y } , \phi ) = \mathbb { E } _ { q _ { \phi } ( { \bf z } | { \bf x } , { \bf y } ) } \left[ \lambda _ { x } \log p _ { \pmb { \theta } _ { x } } ( { \bf x } | { \bf z } ) + \lambda _ { y } \log p _ { \pmb { \theta } _ { y } } ( { \bf y } | { \bf z } ) \right] \quad } \\ & { } & { - \beta \mathrm { K L } ( q _ { \phi } ( { \bf z } | { \bf x } , { \bf y } ) , p _ { \theta } ( { \bf z } ) ) } \end{array}
+$$
+
+We call this the JVAE (joint VAE) model. We usually set $\beta = 1$ , but set $\lambda _ { y } / \lambda _ { x } > 1$ to to scale up the likelihood from the low dimensional attribute vector, $p _ { \pmb { \theta } } ( \mathbf { y } | \mathbf { z } )$ , to match the likelihood from the high dimensional image, $p _ { \pmb { \theta } } ( \mathbf { x } | \mathbf { z } )$ .
+
+Having fit the joint model above, we can proceed to train unpaired inference networks $q _ { \phi _ { x } } ( \mathbf { z } | \mathbf { x } )$ and $q _ { \phi _ { y } } ( \mathbf { z } | \mathbf { y } )$ , so we can embed images and attributes into the same shared latent space. Keeping the $p$ family fixed from the joint model, a natural objective to fit, say, $q _ { \phi _ { x } } ( \mathbf { z } | \mathbf { x } )$ is to maximize the following:1
+
+$$
+\begin{array} { r l } & { \mathcal { L } ( \phi _ { x } | \theta ) = - \mathbb { E } _ { \hat { p } ( \mathbf { x } ) } \left[ \mathrm { K L } ( q _ { \phi _ { x } } ( \mathbf { z } | \mathbf { x } ) , p _ { \theta _ { x } } ( \mathbf { z } | \mathbf { x } ) ) \right] } \\ & { \quad \quad \quad = \displaystyle \int \int d \mathbf { x } d \mathbf { z } \hat { p } ( \mathbf { x } ) q _ { \phi _ { x } } ( \mathbf { z } | \mathbf { x } ) \left[ - \log q _ { \phi _ { x } } ( \mathbf { z } | \mathbf { x } ) - \log p _ { \theta _ { x } } ( \mathbf { x } ) + \log p _ { \theta _ { x } } ( \mathbf { x } | \mathbf { z } ) + \log p _ { \theta } ( \mathbf { z } ) \right] } \\ & { \quad \quad \quad = \mathbb { E } _ { \hat { p } ( \mathbf { x } ) } \left[ \mathrm { e l b o } ( \mathbf { x } , \theta _ { x } , \phi _ { x } ) \right] - \mathbb { E } _ { \hat { p } ( \mathbf { x } ) } \left[ \log p _ { \theta _ { x } } ( \mathbf { x } ) \right] } \end{array}
+$$
+
+![](images/3626395a8ff14eb60506a0ca4184450c7060dd974d57a894b45710f69c9fe7d6.jpg)  
+Figure 2: Illustration of the product of experts inference network. Each expert votes for a part of latent space implied by its observed attribute. The final posterior is the intersection of these regions. When all attributes are observed, the posterior will be a narrowly defined Gaussian, but when some attributes are missing, the posterior will be broader. Right: we illustrate how inclusion of the “universal expert” $p ( \mathbf { z } )$ in the product ensures that the posterior is always well-conditioned (close to spherical), even when we are missing some attributes.
+
+where the last term is constant wrt $\phi _ { x }$ and the model family $p$ , and hence can be dropped. We can use a similar method to fit $q _ { \phi _ { y } } ( \mathbf { z } | \mathbf { y } )$ . Combining these gives the following triple ELBO (TELBO) objective:
+
+$$
+\begin{array} { r l } & { { \mathcal { L } } ( \theta _ { x } , \theta _ { y } , \phi , \phi _ { x } , \phi _ { y } ) = \mathbf { E } _ { \hat { p } ( \mathbf { x } , \mathbf { y } ) } \left[ \mathrm { e l b o } _ { 1 , \lambda , 1 } ( \mathbf { x } , \mathbf { y } , \theta _ { x } , \theta _ { y } , \phi ) \right. } \\ & { ~ \left. ~ + \mathrm { e l b o } _ { 1 , 1 } ( \mathbf { x } , \theta _ { x } , \phi _ { x } ) + \mathrm { e l b o } _ { \gamma , 1 } ( \mathbf { y } , \theta _ { y } , \phi _ { y } ) \right] } \end{array}
+$$
+
+where $\lambda$ and $\gamma$ scale the log likelihood terms $\log p ( \mathbf { y } | \mathbf { z } )$ ; we set these parameters using a validation set. Since we are training the generative model only on aligned data, and simply retrofitting inference networks, we freeze the $p _ { \pmb { \theta } _ { x } } ( \mathbf { x } | \mathbf { z } )$ and $p _ { \pmb { \theta } _ { y } } ( \mathbf { y } | \mathbf { z } )$ terms when training the last two ELBO terms above, and just optimize $q _ { \phi _ { x } } ( \mathbf { z } | \mathbf { x } )$ and $q _ { \phi _ { y } } ( \mathbf { z } | \mathbf { y } )$ terms. This enables us to optimize all terms in Equation (2) jointly. Alternatively, we can first fit the joint model, and then fit the unimodal inference networks.2 In Section 4, we compare this to other methods for training joint VAEs that have been proposed in the literature.
+
+Handling missing attributes. In order to handle missing attributes at test time, we use a product of experts model, where each attribute instantiates an expert. We are motivated by prior work (Williams & Nash, 2018) which shows that for a linear factor analysis model, the posterior distribution $p ( \mathbf { z } | \mathbf { y } )$ is a product of $K$ -dimensional Gaussians, one for each visible dimension. Since our model is just a nonlinear extension of factor analysis, we choose the form of the approximate posterior of our inference network, $q ( \mathbf { z } | \mathbf { y } )$ , to be a product of Gaussians, one for each visible feature: $\begin{array} { r } { q ( \mathbf { z } | \mathbf { y } _ { \mathcal { O } } ) \propto p ( \mathbf { z } ) \prod _ { k \in \mathcal { O } } q ( \mathbf { z } | y _ { k } ) } \end{array}$ , where $q ( \mathbf { z } | y _ { k } ) { \overset { \cdot } { = } } { \mathcal { N } } ( \mathbf { z } | \mu _ { k } ( y _ { k } ) , \mathbf { C } _ { k } ( y _ { k } ) )$ is the kth Gaussian “expert”, and $p ( \mathbf { z } ) = \mathcal { N } ( \mathbf { z } | \boldsymbol { \mu } _ { 0 } = \mathbf { 0 } , \mathbf { C } _ { 0 } = \mathbf { I } )$ is the prior. A similar model was concurrently proposed in Bouchacourt et al. (2018) to perform inference for a set of images. Unlike the product of experts model in (Hinton, 2002), our model multiplies Gaussians, not Bernoullis, so the product has a closed form solution namely $q ( \mathbf { z } | \mathbf { y } _ { \mathcal { O } } ) = \mathcal { N } ( \mathbf { z } | \mu , \mathbf { C } )$ , where $\begin{array} { r } { \mathbf { C } ^ { - 1 } = \sum _ { k } \mathbf { C } _ { k } ^ { - 1 } } \end{array}$ and $\begin{array} { r } { \pmb { \mu } = \mathbf { C } ( \sum _ { k } \mathbf { C } _ { k } ^ { - 1 } \pmb { \mu } _ { k } ) } \end{array}$ , and the sum is over all the observed attributes. Intuitively, y imposes an increasing number of constraints on $\mathbf { z }$ as more of it is observed, as explained in Williams & Agakov (2002). In our setting, if we do not observe any attributes, the posterior reduces to the prior. As we observe more attributes, the posterior becomes narrower, since the (positive definite) precision matrices, $\mathbf { C } ^ { - 1 }$ add up, reflecting the increased specificity of the concept being specified, as illustrated in Figure 2 (middle) (see also Williams & Agakov (2002)). We always include the prior term, $p ( \mathbf { z } )$ , in the product, since without it, the posterior $q _ { \phi _ { y } } ( \mathbf { z } | \mathbf { y } _ { \mathcal { O } } )$ may not be well-conditioned when we are missing attributes, as illustrated in Figure 2 (right). For more implementation-level details on the model architectures, see Appendix A.4.
+
+# 3 EVALUATION METRICS: THE 3C’S OF VISUAL IMAGINATION
+
+To evaluate the quality of a set of generated images, $S ( \mathbf { y } _ { \mathcal { O } } ) = \{ \mathbf { x } _ { s } \sim p ( \mathbf { x } | \mathbf { y } _ { \mathcal { O } } ) : s = 1 : S \}$ , we apply a multi-label classifier to each image, to convert it to a predicted attribute vector, $\hat { \mathbf { y } } ( \mathbf x )$ . This attribute classifier is trained on a large dataset of images and attributes, and is held constant across all methods that are being evaluated. It plays the role of a human observer. This is similar in spirit to generative adversarial networks (Goodfellow et al., 2014), that declare a generated image to be good enough if a binary classifier cannot distinguish it from a real image. (Both approaches avoid the problems mentioned in Theis et al. (2016) related to evaluating generative image models in terms of their likelihood.) However, the attribute classifier checks not only that the images look realistic, but also that they have the desired attributes.
+
+To quantify this, we define the correctness as the fraction of attributes for each generated image that match those specified in the concept’s description: correctness $( \bar { s } , \mathbf { y } \bar { \omega } ) \ =$ $\begin{array} { r } { \frac { 1 } { | \mathcal { S } | } \sum _ { \mathbf { x } \in \mathcal { S } } \overset { 1 } { \underset { | \mathcal { O } | } { \mid } } \sum _ { k \in \mathcal { O } } \mathbb { I } ( \hat { y } ( \mathbf { x } ) _ { k } = \overset { \cdot } { y } _ { k } ) } \end{array}$ . However, we also want to measure the diversity of values for the unspecified or missing attributes, $\mathcal { M } = \mathcal { A } \backslash \mathcal { O }$ . We do this by comparing $q _ { k }$ , the empirical distribution over values for attribute $k$ induced by the generated set $s$ , to $p _ { k }$ , the true distribution for this attribute induced by the training set. We measure the difference between these distributions using the Jensen-Shannon divergence, since it is symmetric and satisfies $0 \leq \mathrm { J S } ( p , q ) \leq 1$ . We then define the coverage as follows: $\begin{array} { r } { \mathbf { \bar { c o v e r a g e } } ( S , \mathbf { y } _ { \mathcal { O } } ) \mathbf { \bar { = } } \frac { 1 } { | \mathcal { M } | } \sum _ { k \in \mathcal { M } } ( 1 - \mathbf { J } \mathbf { S } ( p _ { k } ^ { - } , q _ { k } ) ) } \end{array}$ . If desired, we can combine correctness and coverage into a single number, by computing the JS divergence between $p _ { k }$ and $q _ { k }$ for all attributes, where, for observed attributes, $p _ { k }$ is a delta function and $q _ { k }$ is the empirical distribution (we call this JS-overall). This gives us a convenient way to pick hyperparameters. However, for analysis, we find it helpful to report correctness and coverage separately.
+
+Note that our metric is different from the inception score proposed in Salimans et al. (2016). That is defined as follows: inception $\underline { { \mathbf { \Pi } } } = \exp \left( \mathbb { E } _ { \hat { p } ( \mathbf { x } ) } \left[ \dot { \mathrm { K L } } ( p ( y | \mathbf { x } ) , \dot { p } ( y \dot { ) } ) \right] \right)$ , where $y$ is a class label. Expanding the term inside the exponential, we get
+
+$$
+\sum _ { \mathbf { x } } p ( \mathbf { x } ) \left[ \sum _ { y } p ( y | \mathbf { x } ) \log p ( y | \mathbf { x } ) \right] - \sum _ { \mathbf { x } } \sum _ { y } p ( \mathbf { x } , y ) \log p ( y ) = \mathbb { E } _ { \hat { p } ( \mathbf { x } ) } \left[ - H ( y | \mathbf { x } ) \right] + H ( y )
+$$
+
+A high inception score means that the distribution $p ( y | \mathbf { x } )$ has low entropy, so the generated images match some class, but that the marginal $p ( y )$ has high entropy, so the images are diverse. However, the inception score was created to evaluate unconditional generative models of images, so it does not check if the generated images are consistent with the concept $\mathbf { y } _ { \mathcal { O } }$ , and the degree of diversity does not vary in response to the level of abstraction of the concept.
+
+Finally, we can assess how well the model understands compositionality, by checking correctness of its generated images in response to test concepts $\mathbf { y } _ { \mathcal { O } }$ that differ in at least one attribute from the training concepts. We call this a compositional split of the data. This is much harder than a standard iid split, since we are asking the model to predict the effects of novel combinations of attributes, which it has not seen before (and which might actually be impossible). Note that abstraction is different from compositionality – in abstraction we are asking the model to predict the effects of dropping certain attributes instead of predicting novel combinations of attributes.
+
+# 4 RELATED WORK
+
+In this section, we briefly mention some of the most closely related prior work.
+
+Conditional models. Many conditional generative image models of the form $p ( \mathbf { y } \vert \mathbf { x } )$ have been proposed recently, where y can be a class label (e.g., (Radford et al., 2016)), a vector of attributes (e.g., (Yan et al., 2016)), a sentence (e.g., (Reed et al., 2016)), another image (e.g., (Isola et al., 2017)), etc. Such models are usually based on VAEs or GANs. However, we are more interested in learning a shared latent space from either descriptions y or images x, which means we need to use a joint, symmetric, model.
+
+Joint models. Several papers use the same joint VAE model as us, but they differ in how it is trained. In particular, the BiVCCA objective of Wang et al. (2016b) has the form ${ \mathcal { L } } ( \theta , \phi ) =$
+
+$\mathbb { E } _ { \hat { p } ( \mathbf { x } , \mathbf { y } ) } \left[ J ( \mathbf { x } , \mathbf { y } , \pmb { \theta } , \phi ) \right]$ , where
+
+$$
+\begin{array} { r } { J ( \mathbf { x } , \mathbf { y } , \pmb { \theta } , \phi ) = \mu \left( E _ { q _ { \phi _ { x } } ( \mathbf { z } | \mathbf { x } ) } [ \log p \pmb { \theta } _ { x } ( \mathbf { x } | \mathbf { z } ) + \lambda \log p \pmb { \theta } _ { y } ( \mathbf { y } | \mathbf { z } ) ] - \mathrm { K L } ( q _ { \phi _ { x } } ( \mathbf { z } | \mathbf { x } ) , p \pmb { \theta } ( \mathbf { z } ) ) \right) } \\ { + ( 1 - \mu ) \left( E _ { q _ { \phi _ { y } } ( \mathbf { z } | \mathbf { y } ) } [ \log p \pmb { \theta } _ { x } ( \mathbf { x } | \mathbf { z } ) + \lambda \log p \pmb { \theta } _ { y } ( \mathbf { y } | \mathbf { z } ) ] - \mathrm { K L } ( q _ { \phi _ { y } } ( \mathbf { z } | \mathbf { y } ) , p \pmb { \theta } ( \mathbf { z } ) ) \right) } \end{array}
+$$
+
+This method results in the model generating the mean image corresponding to each concept, due to the $E _ { q _ { \phi _ { y } } ( \mathbf { z } | \mathbf { y } ) } \log p _ { \theta } ( \mathbf { x } , \mathbf { y } | \mathbf { z } )$ term, which requires that $\mathbf { z }$ ’s sampled from $q _ { \phi _ { y } } ( \mathbf { z } | \mathbf { y } _ { n } )$ be good at generating all the different ${ \bf x } _ { n }$ ’s which co-occur with ${ \bf y } _ { n }$ . We show this empirically in Section 5. This problem can be partially compensated for by increasing $\mu$ , but that reduces the $\operatorname { K L } ( q _ { \phi } ( \mathbf { z } | \mathbf { y } ) , p _ { \theta } ( \mathbf { z } ) )$ penalty, which is required to ensure $q _ { \phi _ { y } } ( \mathbf { z } | \mathbf { y } )$ is a broad distribution with good coverage of the concept.
+
+The JMVAE objective of Suzuki et al. (2017) has the form $\mathcal { L } ( \pmb { \theta } , \phi ) = \mathbb { E } _ { \hat { p } ( \mathbf { x } , \mathbf { y } ) } \left[ J ( \mathbf { x } , \mathbf { y } , \pmb { \theta } , \phi ) \right]$ , where
+
+$$
+I ( \mathbf { x } , \mathbf { y } , \boldsymbol { \theta } , \boldsymbol { \phi } ) = \mathrm { e l b o } _ { 1 , \boldsymbol { \lambda } , 1 } ( \mathbf { x } , \mathbf { y } , \boldsymbol { \theta } , \boldsymbol { \phi } ) - \alpha \left[ \mathrm { K L } ( q _ { \boldsymbol { \phi } } ( \mathbf { z } | \mathbf { x } , \mathbf { y } ) , q _ { \boldsymbol { \phi } _ { y } } ( \mathbf { z } | \mathbf { y } ) ) + \mathrm { K L } ( q _ { \boldsymbol { \phi } } ( \mathbf { z } | \mathbf { x } , \mathbf { y } ) , q _ { \boldsymbol { \phi } _ { x } } ( \mathbf { z } | \mathbf { x } ) ) \right]
+$$
+
+At first glance, forcing $q _ { \phi } ( \mathbf { z } | \mathbf { y } )$ to be close to $q _ { \phi } ( \mathbf { z } | \mathbf { x } , \mathbf { y } )$ seems undesirable, since the latter will typically be close to a delta function, since there is little posterior uncertainty in $\mathbf { z }$ once we see the image x. However, in Appendix A.1, we use results from Hoffman $\&$ Johnson (2016) to show that $\mathbb { E } _ { \hat { p } ( \mathbf { x } , \mathbf { y } ) } \left[ \mathrm { K L } ( q _ { \phi } ( \mathbf { z } | \mathbf { x } , \mathbf { y } ) , q _ { \phi _ { y } } ( \mathbf { z } | \mathbf { y } ) ) \right]$ can be written in terms of $\operatorname { K L } ( q _ { \phi } ^ { \mathrm { a v g } } ( \mathbf { z } | \mathbf { y } ) , q _ { \phi _ { y } } ( \mathbf { z } | \mathbf { y } ) )$ , where $q _ { \phi } ^ { \mathrm { a v g } } ( \mathbf { z } | \mathbf { y } ) = \mathbf { \bar { \mathbb { E } } } _ { \hat { p } ( \mathbf { x } | \mathbf { y } ) } \left[ q _ { \phi } ( \mathbf { z } | \mathbf { x } , \mathbf { y } ) \right]$ is the aggregated posterior over $\mathbf { z }$ induced by all images $\mathbf { x }$ which are associated with description $\mathbf { y }$ . This ensures that $q _ { \phi _ { y } } ( \mathbf { z } | \mathbf { y } )$ will cover the embeddings of all the images associated with concept $\mathbf { y }$ . However, since there is no $\mathrm { K L } ( q _ { \phi _ { y } } ( \mathbf { z } | \mathbf { y } ) , p _ { \pmb { \theta } } ( \mathbf { z } ) )$ term, the diversity of the samples is slightly reduced for novel concepts compared to TELBO, as we show empirically in Section 5. On the flip side, the benefit of using the aggregated posterior to fit the $q ( \mathbf { z } | \mathbf { y } )$ inference network is that one can expect sharper images, as this ensures we will sample $\mathbf { z } \sim q ( \mathbf { z } | \mathbf { y } )$ which have been seen by the image decoder $p _ { \pmb { \theta } } ( \mathbf { x } | \mathbf { z } )$ during joint training. If the aggregated posterior does not exactly match the prior (which is known to happen in VAE-type models, see Hoffman & Johnson (2016)) then regularizing with respect to the prior (as TELBO does) can generate samples in parts of space not seen by the image decoder, which can potentially lead to less “correct” samples. Again, our empirical findings in Section 5 confirm this tradeoff between correctness and coverage implicit in choices of TELBO vs. JMVAE.
+
+The SCAN method of Higgins et al. (2017b) first fits a standard $\beta$ -VAE model (Higgins et al., 2017a) on unlabeled images (or rather, features derived from images using a pre-trained denoising autoencoder) by maximizing $\mathcal { L } ( \pmb { \theta } _ { x } , \pmb { \phi } _ { x } ) = \mathbb { E } _ { \hat { p } ( \mathbf { x } ) }$ $[ \mathrm { e l b o } _ { 1 , \beta _ { x } } ( \mathbf { x } , \pmb { \theta } _ { x } , \phi _ { x } ) ]$ . They then fit a second VAE by maximizing $\mathcal { L } ( \pmb { \theta } _ { y } , \pmb { \phi } _ { y } ) = \mathbb { E } _ { \hat { p } ( \mathbf { x } , \mathbf { y } ) } \left[ J ( \mathbf { x } , \mathbf { y } , \pmb { \theta } _ { y } , \phi _ { y } , \phi _ { x } ) \right]$ , where
+
+$$
+J ( \mathbf { x } , \mathbf { y } , \pmb { \theta } _ { y } , \phi _ { y } , \phi _ { x } ) = \mathrm { { e l b o } } _ { 1 , \beta _ { y } } ( \mathbf { y } , \pmb { \theta } _ { y } , \phi _ { y } ) - \alpha \mathrm { { K L } } ( q _ { \phi _ { x } } ( \mathbf { z } | \mathbf { x } ) , q _ { \phi _ { y } } ( \mathbf { z } | \mathbf { y } ) )
+$$
+
+This is very similar to JMVAE, since $q _ { \phi _ { x } } ( \mathbf { z } | \mathbf { x } ) \approx q _ { \phi } ( \mathbf { z } | \mathbf { x } , \mathbf { y } )$ , when $\displaystyle ( \mathbf { x } , \mathbf { y } )$ is a matching pair of images and labels. An important difference, however, is that SCAN treats the attribute vectors $\mathbf { y }$ as atomic symbols; this has the advantage that there is no need to handle missing inputs, but the disadvantage that they cannot infer the meaning of unseen attribute combinations at test time, unless they are “taught” them by having them paired with images. Also, they rely on $\beta _ { x } > 1$ as a way to get compositionality, assuming that a disentangled latent space will suffice. However, in Appendix A.3, we show that unsupervised learning of the latent space given images alone can result in poor results when some of the attributes in the compositional concept hierarchy are non-visual, such as parity of an MNIST digit. Our approach always takes the labels into consideration when learning the latent space, permitting well-organized latent spaces even in the presence of non-visual concepts (c.f. the difference between PCA and LDA).
+
+Handling missing inputs. Conditional generative models of images, of the form $p ( \mathbf { x } | \mathbf { y } )$ , have problems with missing input attributes, as do inference networks $q ( \mathbf { z } | \mathbf { y } )$ for VAEs. Hoffman (2017) uses MCMC to fit a latent Gaussian model, which can in principle handle missing data; however, he initializes the Markov chain with the posterior mode computed by an inference network, which cannot easily handle missing inputs. One approach we can use, if we have a joint model, is to estimate or impute the missing values, as follows: $\hat { \mathbf { y } } = \arg \operatorname* { m a x } _ { \mathbf { y } _ { \mathcal { M } } } p ( \mathbf { y } _ { \mathcal { M } } | \mathbf { y } _ { \mathcal { O } } )$ , where $p ( \mathbf { y } _ { \mathcal { M } } , \mathbf { y } _ { \mathcal { O } } )$ models dependencies between attributes. We can then sample images using $p ( \mathbf { x } | \hat { \mathbf { y } } )$ . This approach was used in Yan et al. (2016) to handle the case where some of the pixels being passed into an inference network were not observed. However, conditioning on an imputed value will give different results from not conditioning on the missing inputs; only the latter will increase the posterior uncertainty in order to correctly represent less precise concepts with broader support.
+
+Gaussian embeddings. There are many papers that embed images and text into points in a vector space. However, we want to represent concepts of different levels of abstraction, and therefore want to map images and text to regions of latent space. There are some prior works that use Gaussian embeddings for words (Vilnis & McCallum, 2015; Athiwaratkun & Wilson, 2017), sometimes in conjunction with images (Mukherjee & Hospedales, 2016; Ren et al., 2016). Our method differs from these approaches in several ways. First, we maximize the likelihood of $\displaystyle ( \mathbf { x } , \mathbf { y } )$ pairs, whereas the above methods learn a Gaussian embedding using a contrastive loss. Second, our PoE formulation ensures that the covariance of the posterior $q ( \mathbf { z } | \mathbf { y } _ { \mathcal { O } } )$ is adaptive to the data that we condition on. In particular, it becomes narrower as we observe more attributes (because the precision matrices sum up), which is a property not shared by other embedding methods.
+
+Abstraction and compositionality. Young et al. (2014) represent the extension of a concept (described by a noun phrase) in terms of a set of images whose captions match the phrase. By contrast, we use a parametric probability distribution in a latent space that can generate new images. Vendrov et al. (2016) use order embeddings, where they explicitly learn subsumption-like relationships by learning a space that respects a partial order. In contrast, we reason about generality of concepts via the uncertainty induced by their latent representation. There has been some work on compositionality in the language/vision literature (see e.g., Atzmon et al. (2016); Johnson et al. (2017); Agrawal et al. (2017)), but none of these papers use generative models, which is arguably a much more stringent test of whether a model has truly “understood” the meaning of the components which are being composed.
+
+# 5 EXPERIMENTAL RESULTS
+
+In this section, we fit the JVAE model to two different datasets (MNIST-A and CelebA), using the TELBO objective, as well as BiVCCA and JMVAE. We measure the quality of the resulting model using the 3 C’s, and show that our method of handling missing data behaves in a qualitatively reasonable way.
+
+# 5.1 MNIST-A
+
+Dataset. In this section, we report results on the MNIST-A dataset. This is created by modifying the original MNIST dataset as follows. We first create a compositional concept hierarchy using 4 discrete attributes, corresponding to class label (10 values), location (4 values), orientation (3 values), and size (2 values). Thus there are $1 0 { \times } 2 { \times } 3 { \times } 4 = 2 4 0$ unique concepts in total. We then sample $\sim 2 9 0$ example images of each concept, and create both an iid and compositional split of the data. See Appendix A.2 for details.
+
+Models and algorithms. We train the JVAE model on this dataset using TELBO, BiVCCA and JMVAE objectives. We use Adam (Kingma & Ba, 2015) for optimization, with a learning rate of 0.0001, and a minibatch size of 64. We train all models for 250,000 steps (we generally found that the models do not tend to overfit in our experiments). Our models typically take around a day to train on NVIDIA Titan X GPUs. For the image models, $p ( \mathbf { x } | \mathbf { z } )$ and $q ( \mathbf { z } | \mathbf { x } )$ , we use the DCGAN architecture from Radford et al. (2016). Our generated images are of size $6 4 \times 6 4$ , as in Radford et al. (2016). For the attribute models, $p ( y _ { k } | \mathbf { z } )$ and $q ( \mathbf { z } | y _ { k } )$ , we use MLPs. For the joint inference network, $q ( \mathbf { z } | \mathbf { x } , \mathbf { y } )$ , we use a CNN combined with an MLP. We use $d = 1 0$ latent dimensions for all models. We choose the hyperparameters for each method so as to maximize JS-overall, which is an overall measure of correctness and coverage (see Section 3) on a validation set of attribute queries. See Appendix A.4 for further details on the model architectures.
+
+Evaluation. To measure correctness and coverage, we first train the observation classifier on the full iid dataset, where it gets to an accuracy of $9 1 . 1 8 \%$ for class label, $9 0 . 5 6 \%$ for scale, $9 2 . 2 3 \%$ for orientation, and $100 \%$ for location. Consequently, it is a reliable way to assess the quality of samples from various generative models (see Appendix A.5 for details). We then compute correctness and coverage on the iid dataset, and coverage on the comp dataset.
+
+![](images/216614ff4d58de7325957c620c11d08b9ac4d0b78fb017fed0bf9166d2831af1.jpg)  
+Query: 0, small, clockwise, top-right   
+Figure 3: Samples from attribute vectors seen at training time, generated by the 3 different models. We plot the posterior mean of each pixel, $\mathbb { E } \left[ \mathbf { x } | \mathbf { z } _ { s } \right]$ , where $\mathbf { z } _ { s } \sim \hat { q } _ { \phi _ { y } } ( \mathbf { z } | \mathbf { y } )$ . The caption at the top of each little image is the predicted attribute values. The border of the generated image is red if any of the attributes are predicted incorrectly. (The observation classifier is fed sampled images, not the mean image that we are showing here.)
+
+Familiar concrete concepts. We start by assessing the quality of the models in the simplest setting, which is where the test concepts are fully specified (i.e., all attributes are known), and the concepts have been seen before in the training set (i.e., we are using the iid split). Figure 4a shows the correctness scores for the three methods. (Since the test concepts are fully grounded, coverage is not well defined, since there are no missing attributes.) We see that TELBO has a correctness of $8 2 . 0 8 \%$ , which is close to that of JMVAE $( 8 5 . 1 5 \% )$ ; both methods significantly outperform BiVCCA $( 6 7 . 3 8 \% )$ . To gain more insight, Figure 3 shows some samples from each of these methods for a leaf concept chosen at random. We see that the images generated by BiVCCA are very blurry, for reasons we discussed in Section 4. Note that these blurry images are correctly detected by the attribute classifier.3 We also see that the JMVAE samples all look good (in this example). Most of the samples from TELBO are also good, although there is one error (correctly detected by the attribute classifier).
+
+<table><tr><td>Method</td><td>#Attributes</td><td>Coverage (%)</td><td>Correctness (%)</td></tr><tr><td colspan="4">iid split</td></tr><tr><td>TELBO</td><td rowspan="3">4</td><td></td><td>82.08±0.56</td></tr><tr><td>JMVAE</td><td></td><td>85.15 ±0.26</td></tr><tr><td>BiVCCA</td><td></td><td>67.38 ±0.69</td></tr><tr><td>TELBO</td><td rowspan="3">3</td><td>91.14 ± 0.53</td><td>81.63 ± 0.38</td></tr><tr><td>JMVAE</td><td>88.52 ± 0.37</td><td>82.00 ±0.37</td></tr><tr><td>BiVCCA</td><td>85.28 ±0.68</td><td>70.68 ± 0.87</td></tr><tr><td>TELBO</td><td rowspan="3">2</td><td>90.32 ± 0.57</td><td>82.03 ± 1.37</td></tr><tr><td>JMVAE</td><td>87.89 ±0.69</td><td>81.02 ± 1.05</td></tr><tr><td>BiVCCA</td><td>85.09 ±0.76</td><td>72.33 ± 2.31</td></tr><tr><td>TELBO</td><td rowspan="3">1</td><td>90.94 ±0.19</td><td>83.67 ± 1.70</td></tr><tr><td>JMVAE</td><td>88.70 ±0.35</td><td>81.58 ± 1.78</td></tr><tr><td>BiVCCA</td><td>85.53 ± 0.27</td><td>68.36 ± 2.21</td></tr><tr><td colspan="5">Compositional split</td></tr><tr><td>TELBO</td><td rowspan="3">4</td><td></td><td>75.61 ± 1.43</td></tr><tr><td>JMVAE</td><td></td><td>76.86 ± 1.30</td></tr><tr><td>BiVCCA</td><td></td><td>68.58 ± 1.02</td></tr></table>
+
+![](images/67115530c872ea1da2927de9b0adb176a254c84c012a930b88a6d53d578942b5.jpg)  
+(b) Mean images generated by TELBO and JMVAE in response to queries at different levels of abstraction, starting from abstract (top) to refined (bottom).
+
+(a) Evaluation of different approaches on the test set. Higher numbers are better. We report standard deviation across 5 splits of the test set.
+
+Figure 4: (a) We show quantitaive results on the 3C’s on MNIST-A. (b) Qualitative results on MNIST-A for various queries. For refined/fully specified queries, we can see that both TELBO and JMVAE produce good correctness, i.e., the images produced follow constraints placed by the specified attributes. When the attribute ‘orientation’ is unspecified, we see that TELBO produces upright and counter clockwise digits, while JMVAE produces clockwise and upright digits. Finally, when we leave the digit unspecified (top), we see that TELBO appears to generate a more diverse set of digits (9, 3, 8, 6) while JMVAE produces 0 and 3.
+
+Novel abstract concepts. Next we assess the quality of the models when the test concepts are abstract, i.e., one or more attributes are not specified. (Note that the model was never trained on such abstract concepts.) Figure 4a shows that the correctness scores for JMVAE seems to drop somewhat (from about $85 \%$ to about $8 1 . 5 \%$ ), although it remains steady for TELBO and BiVCCA. We also see that the coverage of TELBO is higher than the other methods, due to the use of the $\mathrm { K L } ( q _ { \phi _ { y } } ( \mathbf { z } | \mathbf { y } ) , p _ { \pmb { \theta } } ( \mathbf { z } ) )$ regularizer, as we discussed in Section 4. Figure 4b illustrates how the methods respond to concepts of different levels of abstraction. The samples from the TELBO seem to be more diverse, which is consistent with the numbers in Figure 4a.
+
+Compositionally novel concrete concepts. Finally we assess the quality of the models when the test concepts are fully specified, but have not been seen before (i.e., we are using the comp split). Figure 4a shows some quantitative results. We see that the correctness for TELBO and JMVAE has dropped from about $82 \%$ to about $7 5 \%$ , since this task is much harder, and requires “strong generalization”. However, as before, we see that both TELBO and JMVAE outperform BiVCCA, which has a correctness of about $69 \%$ . See Appendix A.7 qualitative results and more details.
+
+# 5.2 CELEBA
+
+In this section, we report results on the CelebA dataset (Liu et al., 2015). In particular, we use the version that was used in Perarnau et al. (2016), which selects 18 visually distinctive attributes, and generate images of size $6 4 \times 6 4$ ; see Appendix A.8 for more details on the CelebA dataset and Appendix A.4 for details of the model architectures. Figure 5 shows some sample qualitative results. On the top left, we show some images which were generated by the three methods given the concept shown in the left column. TELBO and JMVAE generate realistic and diverse images. That is, the generated images are generally of males, with mouth slightly open and smiling attributes present in the images. On the other hand, BiVCCA just generates the mean image. On the bottom left, we show what happens when we drop some attributes, thus specifying more abstract concepts. We see that when we drop the gender, we get a mixture of both male and female images for both TELBO and JMVAE. Going further, when we drop the “smiling” attribute, we see that the samples now comprise of people who are smiling as well as not smiling, and we see a mixture of genders in the samples. Further, while we see a greater diversity in the samples, we also notice a slight drop in image quality (presumably because none of the approaches has seen supervision with just ‘abstract’ concepts). See Appendix A.9 for more qualitative examples on CelebA. On the top right, we show some examples of visual imagination, where we ask the models to generate images from the concept “bald female”, which does not occur in the training set.4 (We omit the results from BiVCCA, which are uniformly poor.) We see that both TELBO and JMVAE can sometimes do a fairly reasonable job (although these are admittedly cherry picked results). Finally, the bottom right illustrates an interesting bias in the dataset: if we ask the model to generate images where we do not specify the value of the eyeglasses attribute, nearly all of the samples fail to included glasses, since the prior probability of this attribute is rare (about $6 \%$ ).
+
+# 6 CONCEPT NAMING WITH IMAGINATION MODELS
+
+In this section, we demonstrate initial results which show that our imagination models can be used for concept naming, where the task is to assign a label to a set of images illustrating the concept depicted by the images. A similar problem has been studied in previous work such as Tenenbaum (1999) and Jia et al. (2013). Tenenbaum (1999) studies a set naming problem with integers (instead of images), and show that construct a likelihood function given a hypothesis set that can capture notions of the minimal/smallest hypothesis that explains the observed samples in the set. Jia et al. (2013) extend this approach to concept-naming on images, incorporating perceptual uncertainty (in recognizing the contents of an image) using a confusion matrix weighted likelihood term. While this approach first extracts labels for each image and then performs concept naming, here we test how well our generative model itself is able to generalize to concept naming without ever performing explicit classification on the images.
+
+![](images/b6b6e00f19c0762479d47d8ed6c0293c054705b7c21da5444678f934722760b6.jpg)  
+Figure 5: Sample CelebA results. Left: we show the attributes specified to be present or absent when generating images. Middle: we show 10 samples each generated from TELBO, JMVAE and BiVCCA. We see that TELBO and JMVAE genreate better samples than BiVCCA which collapses to the mean. Middle, bottom: We show five samples from TELBO and JMVAE in response to queries with unspecified attributes, and see that both approaches generate a mix in the samples, generalizing meaningfully across unspecified attributes.
+
+In more detail, the problem setup in concept naming is as follows: we are given as input a set $\mathcal { X }$ of images, each of which corresponds to a concept in the compositional abstraction hierarchy Figure 1. The task is to assign a label $\mathbf { y } \in \mathcal { V }$ to the set of images. One of the key challenges in concept learning is to understand “how far” to generalize in the concept hierarchy given a limited number of positive examples (Tenenbaum, 1999). That is, given a small set of images with 7 in the top-left corner and bottom-right corner, one must infer that the concept is “7” as opposed to “7, top-left”. In other words, we wish to find the least common ancestor (in the concept hierarchy) corresponding to all the images in the set, given any number of images in the set, so that we can be consistent with the set. We consider two heuristic solutions to this problem:
+
+1. Concept-NB: In this approach we compute arg maxy $p ( \mathbf { y } | \boldsymbol { \mathcal { X } } )$ , where $p ( \mathbf { y } | \boldsymbol { \mathcal { X } } )$ is computed using the naive bayes assumption:
+
+$$
+p ( \mathbf { y } | \mathcal { X } ) \propto p ( \mathbf { y } ) \Pi _ { \mathbf { x } _ { n } \in \mathcal { X } } p ( \mathbf { x } _ { n } | \mathbf { y } ) = p ( \mathbf { y } ) \Pi _ { \mathbf { x } _ { n } \in \mathcal { X } } \int d \mathbf { z } _ { n } p ( \mathbf { x } _ { n } | \mathbf { z } _ { n } ) q ( \mathbf { z } _ { n } | \mathbf { y } )
+$$
+
+where $p ( y )$ is chosen to be uniform across all concepts, and the integrals are approximated using Monte Carlo.
+
+2. Concept-Latent: In this approach, instead of working in the observed space, we work in the latent space. That is, we pick a $\mathrm { r g m i n } _ { \mathbf { y } } \mathrm { K L } ( q ( \mathbf { z } | \mathcal { X } ) | \bar { q ( \mathbf { z } | \mathbf { y } ) } )$ , where $q ( \mathbf { z } | \mathcal { X } )$ is approximated using $\bar { \sum _ { \mathbf { x } \in \mathcal { X } } q ( \mathbf { z } | \mathbf { x } ) }$ , which is a mixture of gaussians. The KL divergence can be computed analytically by considering the first two moments of the gaussian mixture5.
+
+# 6.1 EXPERIMENTAL SETUP
+
+We use the MNIST-A dataset for the concept naming studies. We consider the fully specified attribute labels in the MNIST-A hierarchy, and consider differrent patterns of missingness (corresponding to different nodes in the abstraction hirearchy) by dropping attributes. Specifically, we ignore the case where no attribute is specified, and consider a uniform distribution over the rest of the $\mathsf { \bar { ( 2 ^ { 4 } - 1 = 1 5 ) } }$ ) patterns of missingness. Now, for each fully specified attribute pattern in the iid split of MNIST-A, we sample four missingness patterns and repeat across all fully specified attributes to form a bank of 960 candidate names that a model must choose. We randomly select three subsets of 100 candidate names (and the corresponding images) to form the query set for concept naming, namely tuples of $( \mathbf { y } , { \mathcal { X } } )$ . Specifically, given all the images in the eval set for a concept $\mathbf { y }$ , we form $\mathcal { X }$ using a randomly sampled subset of 5 images. We report the accuracy metric, measuring how often the selected concept for a set $\mathcal { X }$ matches the ground truth concept, across three different splits of 100 datapoints.
+
+Table 1: Accuracy of Imagination models on Concept Naming. Higher is better.   
+
+<table><tr><td>Approach</td><td>Concept-Latent (%)</td><td>Concept-NB (%)</td></tr><tr><td>TELBO</td><td>35.66 ± 2.05</td><td>17.66 ± 1.70</td></tr><tr><td>JMVAE</td><td>54.66 ± 4.92</td><td>13.33 ± 2.05</td></tr><tr><td>BiVCCA</td><td>28.00 ± 4.54</td><td>18.00 ± 1.40</td></tr><tr><td>Random</td><td>0.28±0.00</td><td>0.28±0.00</td></tr><tr><td>Most Frequent</td><td>6.33 ± 1.88</td><td>6.33 ± 1.88</td></tr></table>
+
+![](images/d8dbbd7899942a27afafc0400945fd04332ca9dcd37f0361b8f6a7b42039a005.jpg)  
+Figure 6: A qualitative illustration of some of the examples from concept naming models. Top-left: an example of a sample that is correctly named by a Concept-NB model. However, the Concept-NB model is not that strong and often gets simple concepts such as digits incorrect, making mistakes between 6 and 0, for example (bottom-left). This is likely because the only way in which the Concept-NB approach reasons about the set is not via a "meaningful" low dimensional latent variable but via a sampling distribution on a high dimensional space of images. The Concept-Latent model is able to do better on the same set of images, and classify the set as the concept “6”. Finally, we show a failure case of the model where it incorrectly classifies the digits as being large (there is a small digit in the set), and ignores the fact that all of the digits are in the top-left.
+
+# 6.2 RESULTS
+
+We evaluate the best versions of TELBO, JMVAE, and BiVCCA on the iid split of MNIST-A for concept naming (Table 1). In general, we find that Concept-NB approaches perform significantly worse than Concept-Latent approaches. For example, the best Concept-NB approach (using TELBO/BiVCCA objective) gets to an accuracy of around $1 8 \%$ , while Concept-Latent using JMVAE gets to $5 4 . 6 6 \pm 4 . 9 2 \%$ . In general, these numbers are better than a random chance baseline which would get to $0 . 2 8 \%$ (picking one of 348 effective options, after collating the 960 candidate names based on missingness patterns), while picking the most frequent (ground truth) fully-specified y depicted across an image set gets to $6 . 3 \bar { 3 } \pm 1 . \bar { 8 } 8 \%$ . Figure 6 shows some qualitative examples from Concept-NB as well as Concept-Latent models for concept / set classification. We observe that the Concept-Latent models are much more powerful than using Concept-NB in terms of naming the concept based on few positive examples from the support set.
+
+# 7 CONCLUSIONS AND FUTURE WORK
+
+We have shown how to create generative models which can “imagine” compositionally novel concrete and abstract visual concepts. In the future we would like to explore richer forms of description, beyond attribute vectors, such as natural language text, as well as compositional descriptions of scenes, which will require dealing with a variable number of objects.
+
+# ACKNOWLEDGMENTS
+
+We would like to thank Hernan Moraldo for his help in writing the JVAE library, Alex Alemi for valuable insights on TELBO and JMVAE, and Sergio Guadarrama and Harsh Satija for numerous discussions around the project. Finally we would like to thank Devi Parikh for advice on the CelebA experiments, and Stefan Lee and Yash Goyal for feedback on an initial version of this draft.
+
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+
+# A APPENDIX
+
+A.1 ANALYSIS OF JMVAE OBJECTIVE
+
+The JMVAE objective of (Suzuki et al., 2017) has the form
+
+$$
+J ( \mathbf { x } , \mathbf { y } , \theta , \phi ) = \operatorname { e l b o } ( \mathbf { x } , \mathbf { y } , \theta , \phi ) - \alpha \left[ \mathrm { K L } ( q _ { \phi } ( \mathbf { z } | \mathbf { x } , \mathbf { y } ) , q _ { \phi _ { y } } ( \mathbf { z } | \mathbf { y } ) ) + \mathrm { K L } ( q _ { \phi } ( \mathbf { z } | \mathbf { x } , \mathbf { y } ) , q _ { \phi _ { x } } ( \mathbf { z } | \mathbf { x } ) ) \right]
+$$
+
+Let us focus on the $\mathrm { K L } ( q _ { \phi } ( \mathbf { z } | \mathbf { x } , \mathbf { y } ) | q _ { \phi _ { y } } ( \mathbf { z } | \mathbf { y } ) )$ term. Let $\mathcal { V }$ be the set of unique labels (attribute vectors) in the training set, $\mathcal { X } _ { i }$ be the indices of the images associated with label $\mathbf { y } _ { i }$ , and let $N _ { i } = | \mathcal { X } _ { i } |$ be the size of that set. Then we can write
+
+$$
+\mathbb { E } _ { \hat { p } ( \mathbf { x } , \mathbf { y } ) } \left[ \mathrm { K L } ( q _ { \phi } ( \mathbf { z } | \mathbf { x } , \mathbf { y } ) | q _ { \phi _ { y } } ( \mathbf { z } | \mathbf { y } ) ) \right] = \frac { 1 } { | \mathcal { y } | } \sum _ { i \in \mathcal { Y } } \frac { 1 } { N _ { i } } \sum _ { n \in \mathbf { X } _ { i } } \mathrm { K L } ( q _ { \phi } ( \mathbf { z } | \mathbf { x } _ { n } , \mathbf { y } _ { i } ) , q _ { \phi _ { y } } ( \mathbf { z } | \mathbf { y } _ { i } ) )
+$$
+
+As explained in (Hoffman & Johnson, 2016), we can rewrite this by treating the index $n \in$ $\{ 1 , \cdots , N _ { i } \}$ as a random variable, with prior $q ( n | \mathbf { y } _ { i } ) = 1 / N _ { i }$ . Also, let us define the likelihood $q ( \mathbf { z } | n , \mathbf { y } _ { i } ) \stackrel { - } { = } q _ { \phi } ( \mathbf { z } | \mathbf { x } _ { n } , \mathbf { y } _ { i } )$ . Using this notation, we can show that the above average KL becomes
+
+$$
+\frac { 1 } { | \mathcal { D } | } \sum _ { i \in \mathcal { Y } } \Big \{ \mathrm { K L } ( q _ { \phi } ^ { \mathrm { a v g } } ( \mathbf { z } | \mathbf { y } _ { i } ) | q _ { \phi _ { y } } ( \mathbf { z } | \mathbf { y } _ { i } ) ) + \log N _ { i } - \mathbb { E } _ { q _ { \phi _ { y } } ( \mathbf { z } | \mathbf { y } _ { i } ) } \left[ \mathbb { H } ( q ( n | \mathbf { z } , \mathbf { y } _ { i } ) ) \right] \Big \}
+$$
+
+where
+
+$$
+q _ { \phi } ^ { \mathrm { a v g } } ( \mathbf { z } | \mathbf { y } _ { i } ) = \frac { 1 } { N _ { i } } \sum _ { n \in \mathcal { X } _ { i } } q _ { \phi } ( \mathbf { z } | \mathbf { x } _ { n } , \mathbf { y } _ { i } )
+$$
+
+is the average of the posteriors for that concept, and $q ( n | \mathbf { z } , \mathbf { y } _ { i } )$ is the posterior over the indices for all the possible examples from the set $\mathcal { X } _ { i }$ , given that the latent code is $\mathbf { z }$ and the description is $\mathbf { y } _ { i }$ .
+
+The $\operatorname { K L } ( q _ { \phi } ^ { \mathrm { a v g } } ( \mathbf { z } | \mathbf { y } _ { i } ) | q _ { \phi _ { y } } ( \mathbf { z } | \mathbf { y } _ { i } ) )$ term in Equation (4) tells us that JMVAE encourages the inference network for descriptions, $q _ { \phi _ { y } } ( \mathbf { z } | \mathbf { y } _ { i } )$ , to be close to the average of the posteriors induced by each of the images ${ \bf x } _ { n }$ associated with $\mathbf { y } _ { i }$ . Since each $q _ { \phi } ( \mathbf { z } | \mathbf { x } _ { n } , \mathbf { y } _ { i } )$ is close to a delta function (since there is little posterior uncertainty when conditioning on an image), we are essentially requiring that $q _ { \phi } ( \mathbf { z } | \mathbf { y } _ { i } )$ cover the embeddings of each of these images.
+
+# A.2 DETAILS ON THE MNIST-A DATASET
+
+We created the MNIST-A dataset as follows. Given an image in the original MNIST dataset, we first sample a discrete scale label (big or small), an orientation label (clockwise, upright, and anticlockwise), and a location label (top-left, top-right, bottom-left, bottom-right).
+
+Next, we converted this vector of discrete attributes into a vector of continuous transformation parameters, using the procedure described below:
+
+• Scale: For big, we sample scale values from a Gaussian centered at 0.9 with a standard deviation of 0.1, while for small we sample from a Gaussian centered at 0.6 with a standard deviation of 0.1. In all cases, we reject and draw a sample again if we get values outside the range [0.4, 1.0], to avoid artifacts from upsampling or problems with illegible (small) digits. Orientation: For the clockwise label, we sample the amount of rotation to apply for a digit from a Gaussian centered at $+ 4 5$ degrees, with a standard deviation of 10 degrees. For anti-clockwise, we use a Gaussian at -45 degrees, with a standard deviation of 10 degrees. For upright, we set the rotation to be 0 degrees always. Location: For location, we place Gaussians at the centers of the four quadrants in the image, and then apply an offset of image_size/16 to shift the centers a bit towards the corresponding corners. We then use a standard deviation of image_size/16 and sample locations for centers of the digits. We reject and draw the sample again if we find that the location for the center would place the extremities of the digit outside of the canvas.
+
+Finally, we generate the image as follows. We first take an empty black canvas of size $6 4 \times 6 4$ , rotate the original $2 8 \times 2 8$ MNIST image, and then scale and translate the image and paste it on the canvas. (We use bicubic interpolation for scaling and resizing the images.) Finally, we use the method of (Salakhutdinov & Murray, 2008) to binarize the images. See Figure 7 for example images generated in this way.
+
+We repeat the above process of sampling labels, and applying corresponding transformations, to generate images 10 times for each image in the original MNIST dataset. Each trial samples labels from a uniform categorical distribution over the sample space for the corresponding attribute. Thus, we get a new MNIST-A dataset with 700,000 images from the original MNIST dataset of 70,000 images. We split the images into a train, val and test set of $85 \%$ , $5 \%$ , and $10 \%$ of the data respectively to create the IID split. To create the compositional split, we split the $1 0 { \times } 2 { \times } 3 { \times } 4 = 2 4 0$ possible label combinations by the sample train/val/test split, giving us splits of the dataset with non-overlapping label combinations.
+
+![](images/33f8b05ab8fce936e7f3835a7b292e98cde56931f9c9e1a8753bb9a2c64cf1a8.jpg)  
+Figure 7: Example binary images from our MNIST-A dataset.
+
+# A.3 $\beta$ -VAE vs.JOINT VAE
+
+![](images/0af5e0e3d9ed36265cfe96b0d79354e35c4e4ec83bd117cfc12e8cf76c61b051.jpg)  
+Figure 8: Visualization of the benefit of semantic annotations for learning a good latent space. Each small digit is a single sample generated from $p ( x | z )$ from the corresponding point $z$ in latent space. (a) $\beta$ -VAE fit to images without annotations. The color of a point $z$ is inferred from looking at the attributes of the training image that maps to this point of space using $q ( z | x )$ . Note that the red region (corresponding to the concept of large and even digits) is almost non existent. (b) Joint-VAE fit to images with annotations. The color of a point $z$ is inferred from $p ( y | z )$ .
+
+$\beta$ -VAE Higgins et al. (2017a) is an approach that aims to learn disentangled latent spaces. It does this by modifying the ELBO objective, so that it scales the $\mathrm { K L } ( q ( \mathbf { z } | \mathbf { x } ) , p ( \mathbf { z } ) )$ term by a factor $\beta > 1$ . This gives rise to disentangled spaces since the prior $p ( \mathbf { z } ) = \mathcal { N } ( \mathbf { z } | \mathbf { 0 } , \mathbf { I } )$ is factorized (see (Achille & Soatto, 2017) for details). However, to learn latent spaces that correspond to high level concepts, this is not sufficient: we need to use labeled data as well.
+
+To illustrate this, we set up an experiment where we learn a 2d latent space for standard MNIST digit images, but where we replace the label with two binary attributes: parity (odd vs.even) and magnitude (value $< 5$ or $> = 5$ ). We call this dataset MNIST-2bit.
+
+In Figure 8(a), we show the results of fitting a 2d $\beta$ -VAE model (Higgins et al., 2017a) to the images in MNIST-2bit, ignoring the attributes. We perform a hyperparameter sweep over $\beta$ , and pick the one that gives the best looking latent space (this corresponds to a value of $\beta = 1 0$ ). At each point $z$ in the latent 2d space, we show a single image sampled from $p ( x | z )$ . To derive the colors for each point in latent space, we proceed as follows: we embed each training image $x$ (with label $y ( x ) )$ into latent space, by computing $\hat { z } ( x ) = E _ { q ( z | x ) } [ z ]$ . We then associate label $y ( x )$ with this point in space. To derive the label for an arbitrary point $z$ , we lookup the closest embedded training image (using $\ell _ { 2 }$ distance in $z$ space), and use its corresponding label. We see that the latent space is useful for autoencoding (since the generated images look good), but it does not capture the relevant semantic properties of parity and magnitude. In fact, we argue that there is no way of forcing the model to learn a latent space that captures such high level conceptual properties from images alone.
+
+In Figure 8(b), we show the results of fitting a joint VAE model to MNIST-2bit, by optimizing $\mathrm { e l b o } ( x , y )$ on images and attributes (i.e., we do not include the uni-modality $\operatorname { e l b o } ( x )$ and $\operatorname { e l b o } ( y )$ terms in this experiment.) Now the color codes are derived from $p ( y | z )$ rather than using nearest neighbor retrieval. We see that the latent space autoencodes well, and also captures the 4 relevant types of concepts. In particular, the regions are all convex and linearly seperable, which facilitates the learning of a good imagination function $q ( z | y )$ , interpolation, retrieval, and other latent-space tasks.
+
+A skeptic might complain that we have created an arbitrary partitioning of the data, that is unrelated to the appearance of the objects, and that learning such concepts is therefore “unnatural”. But consider an agent interacting with an environment by touching digits on a screen. Suppose the amount of reward they get depends on whether the digit that they touch is small or big, or odd or even. In such an environment, it would be very useful for the agent to structure its internal representation to capture the concepts of magnitude and parity, rather than in terms of low level visual similarity. (In fact, (Scarf et al., 2011) showed that pigeons can learn simple numerical concepts, such as magnitude, by rewarding them for doing exactly this!) Language can be considered as the realization of such concepts, which enables agents to share useful information about their common environments more easily.
+
+# A.4 DETAILS OF THE NEURAL NETWORK ARCHITECTURES
+
+As explained in the main paper, we fit the joint graphical model $p ( x , y , z ) = p ( z ) p ( x | z ) p ( y | z )$ with inference networks $q ( z | x , y )$ , $q ( z | x )$ , and $q ( z | y )$ . Thus, our overall model is made up of three encoders (denoted with $q$ ) and two decoders (denoted with $p$ ). Across all models we use the exponential linear unit (ELU) which is a leaky non-linearity often used to train VAEs. We explain the architectures in more detail below.
+
+# MNIST-A model architecture
+
+• Image decoder, $p ( x | z )$ : Our architecture for the image decoder exactly follows the standard DCGAN architecture from (Radford et al., 2016), where the input to the model is the latent state of the VAE.
+
+• Label decoder, $p ( y | z )$ : Our label decoder assumes a factorized output space $p ( y | z ) =$ $\textstyle \prod _ { k \in { \mathcal { A } } } p ( y _ { k } | z )$ , where $y _ { k }$ is each individual attribute. We parameterize each $p ( y _ { k } | z )$ with a two-layer MLP with 128 hidden units each. We apply a small amount of $\ell _ { 2 }$ regularization to the weight matrices.
+
+• Image and Label encoder, $q ( z | x , y )$ : Our architecture (Figure 9) for the image-label encoder first separately processes the images and the labels, and then concatenates them downstream in the network and then passes the concatenated features through a multi-layered perceptron. More specifically, we have convolutional layers which process image into 32, 64, 128, 16 feature maps with strides $1 , 2 , 2 , 2$ in the corresponding layers. We use batch normalization in the convolutional layers before applying the ELU non-linearity. On the label encoder side, we first encode the each attribute label into a 32d continuous vector and then pass each individual attribute vector through a 2-layered MLP with 512 hidden dimensions each. For example, for MNIST-A we have 4 attributes, which gives us 4 vectors of 512d. We then concatenate these vectors and pass it through a two layer MLP. Finally we concatenate this label feature with the image feature after the convolutional layers (after flattening the conv-features) and then pass the result through a 2 layer MLP to predict the mean $( \mu )$ and standard deviation $( \sigma )$ for the latent space gaussian. Following standard practice, we predict $\log \sigma$ for the standard deviation in order to get values which are positive.
+
+# µ 
+
+![](images/3c8666b4b928e365d467fb0e5f9ee430477aaa538630478bba8cd6275380d78a.jpg)  
+Figure 9: Architecture for the $q ( z | x , y )$ network in our JVAE models for MNIST-A. Images are ( $6 4 \times 6 4 \times 1 )$ , class has 10 possible values, scale has 2 possible values, orientation has 3 possible values, and location has 4 possible values.
+
+• Image encoder, $q ( z | x )$ : The image encoder (Figure 10a) uses the same architecture to process the image as the image feature extractor in $q ( z | x , y )$ network described above. After the conv-features, we pass the result through a 3-layer MLP to get the latent state mean and standard deviation vectors following the procedure described above.
+
+• Label encoder, $q ( z | y )$ : The label encoder (Figure 10b) part of the architecture uses the same design choices to process the labels as the label encoder part in the $q ( z | x , y )$ network. After obtaining the concatenated label feature vectors, we pass the result through a 4-layered MLP with 512 hidden dimensions each and then finally obtain the mean $( \mu )$ and $\log \sigma$ values for each dimension in the latent state of the VAE.
+
+MNIST-A Observation Classifier Model We next describe the architecuture of the observation classifier we use for evaluating the 3C’s on the MNIST-A dataset. The observation classifier is a convolutional neural network, with the first convolutional layer with filters of size $5 \times 5$ , and 32 channels, followed by a $2 \times 2$ pooling layer applied with a stride of 2. This is followed by another convolutional layer with $5 \times 5$ filter size and 64 output channels. This is followed by another $2 \times 2$ pooling layer of stride 2. After this, the network has four heads (corresponding to each attribute), each of which is an MLP with a single hidden layer (of size 1024), with dropout applied to the activations. The final layer of the MLP outputs the logits for classifying each attribute into the corresponding categorical labels associated with it. We train this model from scratch on the MNIST-A dataset using stochastic gradient descent, batch size of 64 and a learning rate of $1 0 ^ { - 4 }$ .
+
+CelebA model architecture Our design choices for CelebA closely mirror the models we built for MNIST-A. One primary difference is that we use a latent dimensionality of 18 in our CelebA experiments which matches the number of attributes we model. Meanwhile, the architectures of the image encoder, image decoder (i.e.DCGAN), are exactly identical to what is described above for MNIST-A execept that encoders take as input a 3-channel RGB image, while decoders produce a
+
+![](images/9e12d7fa6e49bc69072418dfb93ff708face4629172583c460b31e4c20242522.jpg)  
+Figure 10: Archtectures for the single input inference networks for MNIST-A.
+
+3-channel output. We replace the Bernoulli likelihood with Quantized Normal likelihood (which is basically gaussian likelihood with uniform noise).
+
+In terms of the label encoder $q ( z | y )$ , we follow Figure 10b quite closely, except that we get as input 18 categorical (embedded) class labels as input, and we process the labels through a single hidden layer before concatenation and two hidden layers post concatenation (as opposed to two and four used in Figure 10b).
+
+Finally, the joint encoder $q ( z | x , y )$ , is again based heavily on Figure 9 where we feed as input 18 labels as opposed to 4, process them through a single layer mlp of 512d, concatenate them, and then pass the result through a two hidden layer mlp of $5 1 2 \mathrm { d }$ . At this point we concatenate the result with the image feature through the image feature head in Figure 9. Finally, we process the feature through another 512d single hidden layer mlp to produce the $\mu , \sigma$ values.
+
+# A.5 OUTPUTS OF OBSERVATION CLASSIFIER ON GENERATED IMAGES
+
+Figure 11 shows some images sampled from our TELBO model trained on MNIST-A. It also shows the attributes that are predicted by the attribute classifier. We see that the classifier often produces reasonable results that we as humans would also agree with. Thus, it acts as a reasonable proxy for humans classifying the labels for the generated images.
+
+# A.6 HYPERPARAMTER CHOICES FOR TELBO, JMVAE, BIVCCA ON MNIST-A
+
+We discuss more hyperparameter choices for the different objectives and how they impact performance on the MNIST-A dataset. Across all the objectives we set $\lambda _ { x } { = } 1$ , and vary $\lambda _ { y }$ . In addition, we also discuss how the private hyperparamter choices for each loss, $\gamma$ for TELBO, $\alpha$ for JMVAE, as in Wang et al. (2016a)) and $\mu$ for BiVCCA affect performance. We use the JS-overall metric for picking hyperparameters, as explained in the main paper.
+
+![](images/cde83191ec4daa6010cc90c7a347bfde0d62bb6512dfca0f4e7d11dc7eb652b3.jpg)  
+Figure 11: Randomly sampled images from the TELBO model when fed randomly sampled concepts from the iid training set. We also show the outputs of the observation classifier for the images. Note that we visualize mean images above (since they tend to be more human interpretable) but the classifier is fed samples from the model. Figure best viewed by zooming in.
+
+Query: 6, small, clockwise, bottom-right
+
+![](images/fefa3f8d74de211cd7f7d83aeb79bb43beef5def80d74fb90dfdbd1002ae226b.jpg)  
+Figure 12: Compositional generalization on MNIST-A. Models are given the unseen compositional query shown at the top and each of the three columns shows the mean of the image distribution generated by the models. Images marked with a red box are those that the observation classifier detected as being incorrect. We also show the classification result from the observation classifier on top of each image. We see that TELBO and JMVAE both do really well, while BiVCCA is substantially poorer.
+
+1. Effect of $\lambda _ { y }$ : We search for $\lambda _ { y }$ values in the set $\{ 1 , 5 0 , 1 0 0 \}$ for all objectives. In general, we find the setting of $\lambda _ { y }$ in the elbo terms to be critical for good performance (especially on correctness). For example, at $\lambda _ { y } { = } 1$ , we find that correctness numbers for the best performing TELBO model drop to 60.47 $( \pm 0 . 3 4 )$ (from 82.08 $( \pm 0 . 5 6 )$ at $\lambda _ { y } = 5 0$ ) on the validation set for iid queries. Similar trends can be observed for the JMVAE and BiVCCA objectives as well (with $\lambda _ { y } = 1 0$ being the best setting for BiVCCA, $\lambda _ { y } = 5 0$ for JMVAE). We have seen qualitative evidence which shows that the likelihood scaling for $\lambda _ { y }$ affects how disentangled the latent space is along the specified attributes. When the latent space is not grouped or organized as per high-level attributes (see Figure 8 for example), the posterior distribution for a given concept is multimodal, which is hard for a gaussian inference network $q ( \mathbf { z } | \mathbf { y } )$ to capture. This leads to poor correctness values.
+
+2. Effect of $\gamma$ : In addition to the $\lambda _ { y }$ scaling term which is common across all objectives, TELBO has a $\gamma$ scaling factor which controls how we scale the $\log p ( y | z )$ term in the $\mathrm { e l b o } _ { \gamma , 1 } ( \mathbf { y } , \pmb { \theta } _ { y } , \phi _ { y } )$ term. We sweep values of $\{ 1 , 5 0 , 1 0 0 \}$ for this parameter. In general, we find that the effect of this term is smaller on the performance than the $\lambda _ { y }$ term. Based on the setting of this parameter, we find that, for example, the correctness values for fully specified queries change from 82.08 $( \pm 0 . 5 6 )$ at $\gamma { = } 5 0$ to 80.27 $( \pm 0 . 3 8 )$ at $\gamma { = } 1$ on validation set for iid queries.
+
+3. Effect of $\alpha$ : We generally find that $\alpha { = } 1 . 0$ works best for JMVAE across the different choices explored in Wang et al. (2016a), namely, $\{ 0 . 0 1 , 0 . 1 , 1 . 0 \}$ . For example, decreasing the value of $\alpha$ to 0.1 or 0.01 reduces correctness for fully sepcified queries from 85.63 $( \pm 0 . 2 9 )$ t o 77.58 $( \pm 0 . 2 3 )$ at 0.1 and 74.57 $( \pm 0 . 4 4 )$ at 0.01 respectively on the validation set for iid queries.
+
+4. Effect of $\mu$ : For BiVCCA, we ran a search for $\mu$ over $\{ 0 . 3 , 0 . 5 , 0 . 7 \}$ , running each training experiment four times, and picked the best hyperparameter choice across the runs. We found that $\mu { = } 0 . 7$ was the best value, however the performance difference across different choices was not very large. Intuitively, higher values of $\mu$ should lead to improved performance compared to lower values of $\mu$ . This is because lower values of $\mu$ mean that we put more weight on the elbo term with a $q ( \mathbf { z } | \mathbf { x } )$ inference network than the one with a $q ( \mathbf { z } | \mathbf { y } )$ inference network, which results in sharper samples.
+
+# A.7 COMPOSITIONAL GENRALIZATION ON MNIST-A: QUALITATIVE RESULTS AND DETAILS
+
+We next show some examples of compositional generalization on MNIST-A on a validation set of queries. For the compositinal experiments we reused the parameters of the best models on the iid splits for all the models, and trained the models for $\sim 1 6 0 K$ iterations. All other design choices were the same. Figure 12 shows some qualitative results.
+
+![](images/8cf44736649a82d0dee93d3dd6407bf1e6b14acd4091e65573ebad1c27c4069d.jpg)
+
+Figure 13: Set of all 9 images labelled as bald ${ } = 1$ and $\mathtt { m a l e = 0 }$ in the CelebA dataset. We can see that in all the cases the labels are inaccurate for the image, probably due to annotator error.
+
+![](images/88f1a8db5dd48aabca30b221c9b09a1011be7beaee7d5f327fd0b1494f8e613a.jpg)  
+Figure 14: TELBO creates more diverse images than JMVAE. At the top we show the set of attributes which are present and absent in the input query. Below, we show the results of generation with all the attributes specified, drawing 10 samples each. We see that both TELBO and JMVAE create accurate images satisfying the constraints. Note that the concept “male” is set to “absent” in the query, which in CelebA means that “female” is present. Next, we unspecify whether the image should contain a male or a female. We see that in this setting, TELBO has a better mixing of male and female images (fourth, sixth, eighth and ninth images in the third row are male), than JMVAE which just produces a single male image (the ninth image in the fourth row).
+
+# A.8 DETAILS ON CELEBA
+
+CelebA consists of 202,599 face colored images and 40 attribute binary vectors. We use the version of this dataset that was used in (Perarnau et al., 2016); this uses a subset of 18 visually distinctive attributes, and preprocesses each image so they are aligned, cropped, and scaled down to $6 4 \times 6 4$ . We use the official train and test partitions, 182K for training and 20K for testing. Note that this is an iid split, so the attribute vectors in the test set all occur in the training set, even though the images and people are unique. In total, the original dataset with 40 attributes specified a set of 96486 unique visual concepts, while our dataset of 18 attributes spans 3690 different visual concepts.
+
+In Section 5.2, we claim that our generations of “Bald” and “Female” images are from a compositionally novel concept. Our claim comes with a minor caveat/clarification: the concept $\mathtt { b a l d } { = } 1$ and $\mathtt { m a l e = 0 }$ does occur in 9 training examples, but they are all incorrect labelings, as shown in Figure 13! Further, we see that the images generated from our model (shown in Figure 5) are qualitatively very different from any of the images here, showing that the model has not memorized these examples.
+
+# A.9 MORE RESULTS ON CELEBA
+
+Finally, we show further qualitative examples of performance on the CelebA dataset. We focus on the TELBO and JMVAE objectives here, since BiVCCA generally produces poor samples (see Figure 5). Figure 14 (middle) shows some example generations for the concept specified by the attributes (top). We see that both TELBO and JMVAE produce correct images when provided the full attribute queries (first two rows). However, when we stop specifying attribute “male” or “not male” (female), we see that TELBO provides more diverse samples, spanning both male and female (compared to JMVAE). This ties into the explanation in Appendix A.1, where we show how one can interpret JMVAE as optimizing for the $\bar { \mathrm { K L } } ( q _ { \phi } ^ { \mathrm { a v g } } ( \mathbf { z } | \mathbf { y } _ { i } ) | \bar { q } _ { \phi _ { y } } ( \mathbf { z } | \mathbf { y } _ { i } ) )$ to fit the unimodal inference network $q _ { \phi _ { y } } ( \mathbf { z } | \mathbf { y } _ { i } )$ . Since JMVAE only reasons about the “aggregate” posterior as opposed to the prior (which TELBO reasons about), it has the tendency to generate less diverse samples when shown unseen concepts.

parse/train/HyxPx3R9tm/HyxPx3R9tm.md

@@ -0,0 +1,529 @@
+# VARIATIONAL DISCRIMINATOR BOTTLENECK: IMPROVING IMITATION LEARNING, INVERSE RL, AND GANS BY CONSTRAINING INFORMATION FLOW
+
+Xue Bin Peng & Angjoo Kanazawa & Sam Toyer & Pieter Abbeel & Sergey Levine
+
+University of California, Berkeley {xbpeng,kanazawa,sdt,pabbeel,svlevine}@berkeley.edu
+
+# ABSTRACT
+
+Adversarial learning methods have been proposed for a wide range of applications, but the training of adversarial models can be notoriously unstable. Effectively balancing the performance of the generator and discriminator is critical, since a discriminator that achieves very high accuracy will produce relatively uninformative gradients. In this work, we propose a simple and general technique to constrain information flow in the discriminator by means of an information bottleneck. By enforcing a constraint on the mutual information between the observations and the discriminator’s internal representation, we can effectively modulate the discriminator’s accuracy and maintain useful and informative gradients. We demonstrate that our proposed variational discriminator bottleneck (VDB) leads to significant improvements across three distinct application areas for adversarial learning algorithms. Our primary evaluation studies the applicability of the VDB to imitation learning of dynamic continuous control skills, such as running. We show that our method can learn such skills directly from raw video demonstrations, substantially outperforming prior adversarial imitation learning methods. The VDB can also be combined with adversarial inverse reinforcement learning to learn parsimonious reward functions that can be transferred and re-optimized in new settings. Finally, we demonstrate that VDB can train GANs more effectively for image generation, improving upon a number of prior stabilization methods. (Video1)
+
+# 1 INTRODUCTION
+
+Adversarial learning methods provide a promising approach to modeling distributions over highdimensional data with complex internal correlation structures. These methods generally use a discriminator to supervise the training of a generator in order to produce samples that are indistinguishable from the data. A particular instantiation is generative adversarial networks, which can be used for high-fidelity generation of images (Goodfellow et al., 2014; Karras et al., 2017) and other highdimensional data (Vondrick et al., 2016; Xie et al., 2018; Donahue et al., 2018). Adversarial methods can also be used to learn reward functions in the framework of inverse reinforcement learning (Finn et al., 2016a; Fu et al., 2017), or to directly imitate demonstrations (Ho & Ermon, 2016). However, they suffer from major optimization challenges, one of which is balancing the performance of the generator and discriminator. A discriminator that achieves very high accuracy can produce relatively uninformative gradients, but a weak discriminator can also hamper the generator’s ability to learn. These challenges have led to widespread interest in a variety of stabilization methods for adversarial learning algorithms (Arjovsky et al., 2017; Kodali et al., 2017; Berthelot et al., 2017).
+
+In this work, we propose a simple regularization technique for adversarial learning, which constrains the information flow from the inputs to the discriminator using a variational approximation to the information bottleneck. By enforcing a constraint on the mutual information between the input observations and the discriminator’s internal representation, we can encourage the discriminator to learn a representation that has heavy overlap between the data and the generator’s distribution, thereby effectively modulating the discriminator’s accuracy and maintaining useful and informative gradients for the generator. Our approach to stabilizing adversarial learning can be viewed as an adaptive variant of instance noise (Salimans et al., 2016; Sønderby et al., 2016; Arjovsky & Bottou, 2017). However, we show that the adaptive nature of this method is critical. Constraining the mutual information between the discriminator’s internal representation and the input allows the regularizer to directly limit the discriminator’s accuracy, which automates the choice of noise magnitude and applies this noise to a compressed representation of the input that is specifically optimized to model the most discerning differences between the generator and data distributions.
+
+![](images/79dae658a21f3a99bd5d512bd39e4ed96a241605dc27ac431de015f01936586c.jpg)  
+Figure 1: Our method is general and can be applied to a broad range of adversarial learning tasks. Left: Motion imitation with adversarial imitation learning. Middle: Image generation. Right: Learning transferable reward functions through adversarial inverse reinforcement learning.
+
+The main contribution of this work is the variational discriminator bottleneck (VDB), an adaptive stochastic regularization method for adversarial learning that substantially improves performance across a range of different application domains, examples of which are available in Figure 1. Our method can be easily applied to a variety of tasks and architectures. First, we evaluate our method on a suite of challenging imitation tasks, including learning highly acrobatic skills from mocap data with a simulated humanoid character. Our method also enables characters to learn dynamic continuous control skills directly from raw video demonstrations, and drastically improves upon previous work that uses adversarial imitation learning. We further evaluate the effectiveness of the technique for inverse reinforcement learning, which recovers a reward function from demonstrations in order to train future policies. Finally, we apply our framework to image generation using generative adversarial networks, where employing VDB improves the performance in many cases.
+
+# 2 RELATED WORK
+
+Recent years have seen an explosion of adversarial learning techniques, spurred by the success of generative adversarial networks (GANs) (Goodfellow et al., 2014). A GAN framework is commonly composed of a discriminator and a generator, where the discriminator’s objective is to classify samples as real or fake, while the generator’s objective is to produce samples that fool the discriminator. Similar frameworks have also been proposed for inverse reinforcement learning (IRL) (Finn et al., 2016b) and imitation learning (Ho & Ermon, 2016). The training of adversarial models can be extremely unstable, with one of the most prevalent challenges being balancing the interplay between the discriminator and the generator (Berthelot et al., 2017). The discriminator can often overpower the generator, easily differentiating between real and fake samples, thus providing the generator with uninformative gradients for improvement (Che et al., 2016). Alternative loss functions have been proposed to mitigate this problem (Mao et al., 2016; Zhao et al., 2016; Arjovsky et al., 2017). Regularizers have been incorporated to improve stability and convergence, such as gradient penalties (Kodali et al., 2017; Gulrajani et al., 2017a; Mescheder et al., 2018), reconstruction loss (Che et al., 2016), and a myriad of other heuristics (Sønderby et al., 2016; Salimans et al., 2016; Arjovsky & Bottou, 2017; Berthelot et al., 2017). Task-specific architectural designs can also substantially improve performance (Radford et al., 2015; Karras et al., 2017). Similarly, our method also aims to regularize the discriminator in order to improve the feedback provided to the generator. But instead of explicit regularization of gradients or architecture-specific constraints, we apply a general information bottleneck to the discriminator, which previous works have shown to encourage networks to ignore irrelevant cues (Achille & Soatto, 2017). We hypothesize that this then allows the generator to focus on improving the most discerning differences between real and fake samples.
+
+Adversarial techniques have also been applied to inverse reinforcement learning (Fu et al., 2017), where a reward function is recovered from demonstrations, which can then be used to train policies to reproduce a desired skill. Finn et al. (2016a) showed an equivalence between maximum entropy IRL and GANs. Similar techniques have been developed for adversarial imitation learning (Ho & Ermon, 2016; Merel et al., 2017), where agents learn to imitate demonstrations without explicitly recovering a reward function. One advantage of adversarial methods is that by leveraging a discriminator in place of a reward function, they can be applied to imitate skills where reward functions can be difficult to engineer. However, the performance of policies trained through adversarial methods still falls short of those produced by manually designed reward functions, when such reward functions are available (Rajeswaran et al., 2017; Peng et al., 2018). We show that our method can significantly improve upon previous works that use adversarial techniques, and produces results of comparable quality to those from state-of-the-art approaches that utilize manually engineered reward functions.
+
+Our variational discriminator bottleneck is based on the information bottleneck (Tishby & Zaslavsky, 2015), a technique for regularizing internal representations to minimize the mutual information with the input. Intuitively, a compressed representation can improve generalization by ignoring irrelevant distractors present in the original input. The information bottleneck can be instantiated in practical deep models by leveraging a variational bound and the reparameterization trick, inspired by a similar approach in variational autoencoders (VAE) (Kingma & Welling, 2013). The resulting variational information bottleneck approximates this compression effect in deep networks (Alemi et al., 2016; Achille & Soatto, 2017). A similar bottleneck has also been applied to learn disentangled representations (Higgins et al., 2017). Building on the success of VAEs and GANs, a number of efforts have been made to combine the two. Makhzani et al. (2016) used adversarial discriminators during the training of VAEs to encourage the marginal distribution of the latent encoding to be similar to the prior distribution, similar techniques include Mescheder et al. (2017) and Chen et al. (2018). Conversely, Larsen et al. (2016) modeled the generator of a GAN using a VAE. Zhao et al. (2016) used an autoencoder instead of a VAE to model the discriminator, but does not enforce an information bottleneck on the encoding. While instance noise is widely used in modern architectures (Salimans et al., 2016; Sønderby et al., 2016; Arjovsky & Bottou, 2017), we show that explicitly enforcing an information bottleneck leads to improved performance over simply adding noise for a variety of applications.
+
+# 3 PRELIMINARIES
+
+In this section, we provide a review of the variational information bottleneck proposed by Alemi et al. (2016) in the context of supervised learning. Our variational discriminator bottleneck is based on the same principle, and can be instantiated in the context of GANs, inverse RL, and imitation learning. Given a dataset $\left\{ \mathbf { x } _ { i } , \mathbf { y } _ { i } \right\}$ , with features $\mathbf { x } _ { i }$ and labels $\mathbf { y } _ { i }$ , the standard maximum likelihood estimate $q ( \mathbf { y } _ { i } | \mathbf { x } _ { i } )$ can be determined according to
+
+$$
+\operatorname* { m i n } _ { \boldsymbol { q } } \quad \mathbb { E } _ { \mathbf { x } , \mathbf { y } \sim p ( \mathbf { x } , \mathbf { y } ) } \left[ - \log q ( \mathbf { y } | \mathbf { x } ) \right] .
+$$
+
+Unfortunately, this estimate is prone to overfitting, and the resulting model can often exploit idiosyncrasies in the data (Krizhevsky et al., 2012; Srivastava et al., 2014). Alemi et al. (2016) proposed regularizing the model using an information bottleneck to encourage the model to focus only on the most discriminative features. The bottleneck can be incorporated by first introducing an encoder $E ( { \bf z } | { \bf x } )$ that maps the features $\mathbf { x }$ to a latent distribution over $Z$ , and then enforcing an upper bound $I _ { c }$ on the mutual information between the encoding and the original features $I ( X , Z )$ . This results in the following regularized objective $J ( \boldsymbol { q } , E )$
+
+$$
+\begin{array} { r l } { J ( q , E ) = \underset { q , E } { \mathrm { m i n } } } & { \mathbb { E } _ { \mathbf { x } , \mathbf { y } \sim p ( \mathbf { x } , \mathbf { y } ) } \left[ \mathbb { E } _ { \mathbf { z } \sim E ( \mathbf { z } \mid \mathbf { x } ) } \left[ - \log q ( \mathbf { y } \vert \mathbf { z } ) \right] \right] } \\ { \mathrm { s . t . } } & { I ( X , Z ) \le I _ { c } . } \end{array}
+$$
+
+Note that the model $q ( \mathbf { y } \vert \mathbf { z } )$ now maps samples from the latent distribution $\mathbf { z }$ to the label $\mathbf { y }$ . The mutual information is defined according to
+
+$$
+I ( X , Z ) = \int p ( \mathbf { x } , \mathbf { z } ) \log { \frac { p ( \mathbf { x } , \mathbf { z } ) } { p ( \mathbf { x } ) p ( \mathbf { z } ) } } \ d \mathbf { x } \ d \mathbf { z } \ = \int p ( \mathbf { x } ) E ( \mathbf { z } | \mathbf { x } ) \log { \frac { E ( \mathbf { z } | \mathbf { x } ) } { p ( \mathbf { z } ) } } \ d \mathbf { x } \ d \mathbf { z } \ ,
+$$
+
+where $p ( \mathbf { x } )$ is the distribution given by the dataset. Computing the marginal distribution $\begin{array} { r } { p ( \mathbf { z } ) = \int E ( \mathbf { z } | \mathbf { x } ) p ( \mathbf { x } ) d \mathbf { x } } \end{array}$ can be challenging. Instead, a variational lower bound can be obtained by using an approximation $r ( \mathbf { z } )$ of the marginal. Since KL $[ p ( \mathbf { z } ) | | r ( \mathbf { z } ) ] \geq 0$ , $\begin{array} { r } { \int p ( \mathbf { z } ) \log p ( \mathbf { z } ) d \mathbf { z } \geq } \end{array}$ $\begin{array} { r } { \int p ( \mathbf { z } ) \log r ( \mathbf { z } ) d \mathbf { z } } \end{array}$ , an upper bound on $I ( X , Z )$ can be obtained via the KL divergence,
+
+$$
+I ( X , Z ) \leq \int p ( \mathbf { x } ) E ( \mathbf { z } | \mathbf { x } ) \log { \frac { E ( \mathbf { z } | \mathbf { x } ) } { r ( \mathbf { z } ) } } d \mathbf { x } d \mathbf { z } = \mathbb { E } _ { \mathbf { x } \sim p ( \mathbf { x } ) } \left[ \mathrm { K L } \left[ E ( \mathbf { z } | \mathbf { x } ) | | r ( \mathbf { z } ) \right] \right] .
+$$
+
+![](images/b3e8e858d6dcdf4c97bb70738e3b1c011f0ac9d11f19000401897e1000d8ecdd.jpg)  
+Figure 2: Left: Overview of the variational discriminator bottleneck. The encoder first maps samples $\mathbf { x }$ to a latent distribution $E ( { \bf z } | { \bf x } )$ . The discriminator is then trained to classify samples $\mathbf { z }$ from the latent distribution. An information bottleneck $I ( X , Z ) \leq I _ { c }$ is applied to $Z$ . Right: Visualization of discriminators trained to differentiate two Gaussians with different KL bounds $I _ { c }$ .
+
+This provides an upper bound on the regularized objective $\tilde { J } ( q , E ) \ge J ( q , E )$ ,
+
+$$
+\begin{array} { r l } { \tilde { J } ( q , E ) = \underset { q , E } { \mathrm { m i n } } } & { \mathbb { E } _ { \mathbf { x } , \mathbf { y } \sim p ( \mathbf { x } , \mathbf { y } ) } \left[ \mathbb { E } _ { \mathbf { z } \sim E ( \mathbf { z } \mid \mathbf { x } ) } \left[ - \log q ( \mathbf { y } \vert \mathbf { z } ) \right] \right] } \\ { \mathrm { s . t . } } & { \mathbb { E } _ { \mathbf { x } \sim p ( \mathbf { x } ) } \left[ \mathrm { K L } \left[ E ( \mathbf { z } \vert \mathbf { x } ) \vert \vert r ( \mathbf { z } ) \right] \right] \leq I _ { c } . } \end{array}
+$$
+
+To solve this problem, the constraint can be subsumed into the objective with a coefficient $\beta$
+
+$$
+\begin{array} { r l } { \underset { \boldsymbol { q } , E } { \mathop { \operatorname* { m i n } } } } & { \mathbb { E } _ { \mathbf { x } , \mathbf { y } \sim p ( \mathbf { x } , \mathbf { y } ) } \left[ \mathbb { E } _ { \mathbf { z } \sim E ( \mathbf { z } | \mathbf { x } ) } \left[ - \log q ( \mathbf { y } | \mathbf { z } ) \right] \right] + \beta \left( \mathbb { E } _ { \mathbf { x } \sim p ( \mathbf { x } ) } \left[ \mathrm { K L } \left[ E ( \mathbf { z } | \mathbf { x } ) | | r ( \mathbf { z } ) \right] \right] - I _ { c } \right) . } \end{array}
+$$
+
+Alemi et al. (2016) evaluated the method on supervised learning tasks, and showed that models trained with a VIB can be less prone to overfitting and more robust to adversarial examples.
+
+# 4 VARIATIONAL DISCRIMINATOR BOTTLENECK
+
+To outline our method, we first consider a standard GAN framework consisting of a discriminator $D$ and a generator $G$ , where the goal of the discriminator is to distinguish between samples from the target distribution $p ^ { * } ( \mathbf { x } )$ and samples from the generator $G ( \mathbf { x } )$ ,
+
+$$
+\begin{array} { r l } { \underset { G } { \mathop { \operatorname* { m a x } } } \underset { D } { \mathop { \operatorname* { m i n } } } } & { \mathbb { E } _ { \mathbf { x } \sim p ^ { * } ( \mathbf { x } ) } \left[ - \log \left( D ( \mathbf { x } ) \right) \right] + \mathbb { E } _ { \mathbf { x } \sim G ( \mathbf { x } ) } \left[ - \log \left( 1 - D ( \mathbf { x } ) \right) \right] . } \end{array}
+$$
+
+We incorporate a variational information bottleneck by introducing an encoder $E$ into the discriminator that maps a sample $\mathbf { x }$ to a stochastic encoding $\mathbf { z } \sim E ( \mathbf { z } | \mathbf { x } )$ , and then apply a constraint $I _ { c }$ on the mutual information $I ( X , Z )$ between the original features and the encoding. $D$ is then trained to classify samples drawn from the encoder distribution. A schematic illustration of the framework is available in Figure 2. The regularized objective $J ( D , E )$ for the discriminator is given by
+
+$$
+\begin{array} { r l } { J ( D , E ) = \underset { D , E } { \operatorname* { m i n } } } & { \mathbb { E } _ { x \sim p ^ { * } ( \mathbf { x } ) } \left[ \mathbb { E } _ { \mathbf { z } \sim E ( \mathbf { z } \mid \mathbf { x } ) } \left[ - \log \left( D ( \mathbf { z } ) \right) \right] \right] + \mathbb { E } _ { \mathbf { x } \sim G ( \mathbf { x } ) } \left[ \mathbb { E } _ { \mathbf { z } \sim E ( \mathbf { z } \mid \mathbf { x } ) } \left[ - \log \left( 1 - D ( \mathbf { z } ) \right) \right] \right] } \\ { \mathrm { s . t . } } & { \mathbb { E } _ { \mathbf { x } \sim \tilde { p } ( \mathbf { x } ) } \left[ \mathrm { K L } \left[ E ( \mathbf { z } | \mathbf { x } ) | | r ( \mathbf { z } ) \right] \right] \leq I _ { c } , } \end{array}
+$$
+
+with $\tilde { p } = { \textstyle \frac { 1 } { 2 } } p ^ { * } + { \textstyle \frac { 1 } { 2 } } G$ being a mixture of the target distribution and the generator. We refer to this regularizer as the variational discriminator bottleneck (VDB). To optimize this objective, we can introduce a Lagrange multiplier $\beta$ ,
+
+$$
+\begin{array} { r l } { I ( D , E ) = \underset { D , E } { \mathrm { m i n } } \underset { \beta \geq 0 } { \mathrm { m a x } } } & { \mathbb { E } _ { \mathbf { x } \sim p ^ { * } ( \mathbf { x } ) } \left[ \mathbb { E } _ { \mathbf { z } \sim E ( \mathbf { z } \mid \mathbf { x } ) } \left[ - \log \left( D ( \mathbf { z } ) \right) \right] \right] + \mathbb { E } _ { \mathbf { x } \sim G ( \mathbf { x } ) } \left[ \mathbb { E } _ { \mathbf { z } \sim E ( \mathbf { z } \mid \mathbf { x } ) } \left[ - \log \left( 1 - D ( \mathbf { z } ) \right) \right] \right. } \\ & { \left. + \beta \left( \mathbb { E } _ { \mathbf { x } \sim \tilde { p } ( \mathbf { x } ) } \left[ \mathrm { K L } \left[ E ( \mathbf { z } | \mathbf { x } ) \right| | r ( \mathbf { z } ) \right] \right] - I _ { c } \right) . } \end{array}
+$$
+
+As we will discuss in Section 4.1 and demonstrate in our experiments, enforcing a specific mutual information budget between $\mathbf { x }$ and $\mathbf { z }$ is critical for good performance. We therefore adaptively update $\beta$ via dual gradient descent to enforce a specific constraint $I _ { c }$ on the mutual information,
+
+$$
+\begin{array} { r l } & { D , E \gets \arg \operatorname* { m i n } _ { D , E } \mathcal { L } ( D , E , \beta ) } \\ & { \beta \gets \operatorname* { m a x } \big ( 0 , \beta + \alpha _ { \beta } \big ( \mathbb { E } _ { \mathbf { x } \sim \tilde { p } ( \mathbf { x } ) } [ \mathrm { K L } [ E ( \mathbf { z } | \mathbf { x } ) | | r ( \mathbf { z } ) ] ] - I _ { c } \big ) \big ) , } \end{array}
+$$
+
+where $\mathcal { L } ( D , E , \beta )$ is the Lagrangian
+
+$$
+\begin{array} { r l } & { \mathcal { L } ( D , E , \beta ) = \mathbb { E } _ { \mathbf { x } \sim p ^ { * } ( \mathbf { x } ) } [ \mathbb { E } _ { \mathbf { z } \sim E ( \mathbf { z } | \mathbf { x } ) } [ - \log ( D ( \mathbf { z } ) ) ] ] + \mathbb { E } _ { \mathbf { x } \sim G ( \mathbf { x } ) } [ \mathbb { E } _ { \mathbf { z } \sim E ( \mathbf { z } | \mathbf { x } ) } [ - \log ( 1 - D ( \mathbf { z } ) ) ] ] } \\ & { \quad \quad \quad \quad + \beta ( \mathbb { E } _ { \mathbf { x } \sim \hat { p } ( \mathbf { x } ) } [ \mathrm { K L } [ E ( \mathbf { z } | \mathbf { x } ) | | r ( \mathbf { z } ) ] ] - I _ { c } ) , } \end{array}
+$$
+
+and $\alpha _ { \beta }$ is the stepsize for the dual variable in dual gradient descent (Boyd & Vandenberghe, 2004). In practice, we perform only one gradient step on $D$ and $E$ , followed by an update to $\beta$ . We refer to a GAN that incorporates a VDB as a variational generative adversarial network (VGAN).
+
+In our experiments, the prior $r ( \mathbf { z } ) = \mathcal { N } ( 0 , I )$ is modeled with a standard Gaussian. The encoder $E ( \mathbf { z } | \mathbf { x } ) \overset { - } { = } \mathcal { N } ( \mu _ { E } ( \mathbf { x } ) , \boldsymbol { \Sigma } _ { E } ^ { - } ( \mathbf { x } ) )$ models a Gaussian distribution in the latent variables $Z$ , with mean $\mu _ { E } ( { \bf x } )$ and diagonal covariance matrix $\Sigma _ { E } ( { \bf x } )$ . When computing the KL loss, each batch of data contains an equal number of samples from $p ^ { * } ( x )$ and $G ( x )$ . We use a simplified objective for the generator,
+
+$$
+\operatorname* { m a x } _ { G } ~ \mathbb { E } _ { \mathbf { x } \sim G ( \mathbf { x } ) } \left[ - \log \left( 1 - D ( \mu _ { E } ( \mathbf { x } ) ) \right) \right] .
+$$
+
+where the $\mathrm { K L }$ penalty is excluded from the generator’s objective. Instead of computing the expectation over $Z$ , we found that approximating the expectation by evaluating $D$ at the mean $\mu _ { E } ( { \bf x } )$ of the encoder’s distribution was sufficient for our tasks. The discriminator is modeled with a single linear unit followed by a sigmoid $D ( \mathbf { z } ) = \sigma ( \mathbf { w } _ { D } ^ { T } \mathbf { z } + \mathbf { b } _ { D } )$ , with weights $\mathbf { w } _ { D }$ and bias $\mathbf { b } _ { D }$ .
+
+# 4.1 DISCUSSION AND ANALYSIS
+
+To interpret the effects of the VDB, we consider the results presented by Arjovsky & Bottou (2017), which show that for two distributions with disjoint support, the optimal discriminator can perfectly classify all samples and its gradients will be zero almost everywhere. Thus, as the discriminator converges to the optimum, the gradients for the generator vanishes accordingly. To address this issue, Arjovsky & Bottou (2017) proposed applying continuous noise to the discriminator inputs, thereby ensuring that the distributions have continuous support everywhere. In practice, if the original distributions are sufficiently distant from each other, the added noise will have negligible effects. As shown by Mescheder et al. (2017), the optimal choice for the variance of the noise to ensure convergence can be quite delicate. In our method, by first using a learned encoder to map the inputs to an embedding and then applying an information bottleneck on the embedding, we can dynamically adjust the variance of the noise such that the distributions not only share support in the embedding space, but also have significant overlap. Since the minimum amount of information required for binary classification is 1 bit, by selecting an information constraint $I _ { c } < 1 $ , the discriminator is prevented from from perfectly differentiating between the distributions. To illustrate the effects of the VDB, we consider a simple task of training a discriminator to differentiate between two Gaussian distributions. Figure 2 visualizes the decision boundaries learned with different bounds $I _ { c }$ on the mutual information. Without a VDB, the discriminator learns a sharp decision boundary, resulting in vanishing gradients for much of the space. But as $I _ { c }$ decreases and the bound tightens, the decision boundary is smoothed, providing more informative gradients that can be leveraged by the generator.
+
+Taking this analysis further, we can extend Theorem 3.2 from Arjovsky & Bottou (2017) to analyze the VDB, and show that the gradient of the generator will be non-degenerate for a small enough constraint $I _ { c }$ , under some additional simplifying assumptions. The result in Arjovsky & Bottou (2017) states that the gradient consists of vectors that point toward samples on the data manifold, multiplied by coefficients that depend on the noise. However, these coefficients may be arbitrarily small if the generated samples are far from real samples, and the noise is not large enough. This can still cause the generator gradient to vanish. In the case of the VDB, the constraint ensures that these coefficients are always bounded below. Due to space constraints, this result is presented in Appendix A.
+
+# 4.2 VAIL: VARIATIONAL ADVERSARIAL IMITATION LEARNING
+
+To extend the VDB to imitation learning, we start with the generative adversarial imitation learning (GAIL) framework (Ho & Ermon, 2016), where the discriminator’s objective is to differentiate between the state distribution induced by a target policy $\pi ^ { * } ( \mathbf { s } )$ and the state distribution of the agent’s policy $\pi ( \mathbf { s } )$ ,
+
+$$
+\operatorname* { m a x } _ { \pi } \operatorname* { m i n } _ { D } \quad \mathbb { E } _ { \mathbf { s } \sim \pi ^ { * } ( \mathbf { s } ) } \left[ - \log \left( D ( \mathbf { s } ) \right) \right] + \mathbb { E } _ { \mathbf { s } \sim \pi ( \mathbf { s } ) } \left[ - \log \left( 1 - D ( \mathbf { s } ) \right) \right] .
+$$
+
+![](images/30d57dddd51de39696f83d45f9cb8ba9fe5f6258906497db5bb38f50fa1a90ca.jpg)  
+Figure 3: Simulated humanoid performing various skills. VAIL is able to closely imitate a broad range of skills from mocap data.
+
+The discriminator is trained to maximize the likelihood assigned to states from the target policy, while minimizing the likelihood assigned to states from the agent’s policy. The discriminator also serves as the reward function for the agent, which encourages the policy to visit states that, to the discriminator, appear indistinguishable from the demonstrations. Similar to the GAN framework, we can incorporate a VDB into the discriminator,
+
+$$
+\begin{array} { r l } & { I ( D , E ) = \underset { D , E } { \mathrm { m i n } } \underset { \beta \geq 0 } { \mathrm { m a x } } \mathbb { E } _ { \mathbf { s } \sim \pi ^ { * } ( \mathbf { s } ) } [ \mathbb { E } _ { \mathbf { z } \sim E ( \mathbf { z } \mid \mathbf { s } ) } [ - \log ( D ( \mathbf { z } ) ) ] ] + \mathbb { E } _ { \mathbf { s } \sim \pi ( \mathbf { s } ) } [ \mathbb { E } _ { \mathbf { z } \sim E ( \mathbf { z } \mid \mathbf { s } ) } [ - \log ( 1 - D ( \mathbf { z } ) ) ] ] } \\ & { \qquad + \beta ( \mathbb { E } _ { \mathbf { s } \sim \pi ( \mathbf { s } ) } [ \mathrm { K L } [ E ( \mathbf { z } \mid \mathbf { s } ) ] \mid \mid r ( \mathbf { z } ) ] ] - I _ { c } ) . } \end{array}
+$$
+
+where $\tilde { \pi } = { \textstyle \frac { 1 } { 2 } } \pi ^ { * } + { \textstyle \frac { 1 } { 2 } } \pi$ represents a mixture of the target policy and the agent’s policy. The reward for $\pi$ is then specified by the discriminator $r _ { t } = - \mathrm { l o g } \bar { ( 1 - D ( \mu _ { E } ( \mathbf { s } ) ) ) }$ . We refer to this method as variational adversarial imitation learning (VAIL).
+
+# 4.3 VAIRL: VARIATIONAL ADVERSARIAL INVERSE REINFORCEMENT LEARNING
+
+The VDB can also be applied to adversarial inverse reinforcement learning (Fu et al., 2017) to yield a new algorithm which we call variational adversarial inverse reinforcement learning (VAIRL). AIRL operates in a similar manner to GAIL, but with a discriminator of the form
+
+$$
+D ( { \bf s } , { \bf a } , { \bf s } ^ { \prime } ) = \frac { \exp \left( f ( { \bf s } , { \bf a } , { \bf s } ^ { \prime } ) \right) } { \exp \left( f ( { \bf s } , { \bf a } , { \bf s } ^ { \prime } ) \right) + \pi ( { \bf a } | { \bf s } ) } ,
+$$
+
+where $f ( \mathbf { s } , \mathbf { a } , \mathbf { s } ^ { \prime } ) = g ( \mathbf { s } , \mathbf { a } ) + \gamma h ( \mathbf { s } ^ { \prime } ) - h ( \mathbf { s } )$ , with $g$ and $h$ being learned functions. Under certain restrictions on the environment, Fu et al. show that if $g ( \mathbf { s } , \mathbf { a } )$ is defined to depend only on the current state s, the optimal $g ( \mathbf { s } )$ recovers the expert’s true reward function $r ^ { * } ( \mathbf { s } )$ up to a constant $g ^ { * } ( \mathbf { s } ) =$ $r ^ { * } ( \mathbf { s } ) + \mathrm { c o n s i }$ . In this case, the learned reward can be re-used to train policies in environments with different dynamics, and will yield the same policy as if the policy was trained under the expert’s true reward. In contrast, GAIL’s discriminator typically cannot be re-optimized in this way $\mathrm { F u }$ et al., 2017). In VAIRL, we introduce stochastic encoders $E _ { g } ( \mathbf { z } _ { g } | \mathbf { s } ) , E _ { h } ( \mathbf { z } _ { h } | \mathbf { s } )$ , and $g ( \mathbf { z } _ { g } ) , h ( \mathbf { z } _ { h } )$ are modified to be functions of the encoding. We can reformulate Equation 13 as
+
+$$
+D ( { \bf s } , { \bf a } , { \bf z } ) = \frac { \exp \left( f ( { \bf z } _ { g } , { \bf z } _ { h } , { \bf z } _ { h } ^ { \prime } ) \right) } { \exp \left( f ( { \bf z } _ { g } , { \bf z } _ { h } , { \bf z } _ { h } ^ { \prime } ) \right) + \pi ( { \bf a } | { \bf s } ) } ,
+$$
+
+for ${ \bf z } = ( { \bf z } _ { g } , { \bf z } _ { h } , { \bf z } _ { h } ^ { \prime } )$ and $f ( \mathbf { z } _ { g } , \mathbf { z } _ { h } , \mathbf { z } _ { h } ^ { \prime } ) = D _ { g } ( \mathbf { z } _ { g } ) + \gamma D _ { h } ( \mathbf { z } _ { h } ^ { \prime } ) - D _ { h } ( \mathbf { z } _ { h } )$ . We then obtain a modified objective of the form
+
+$$
+\begin{array} { r l } { J ( D , E ) = \underset { D , E } { \mathrm { m i n } } \underset { \beta \geq 0 } { \mathrm { m a x } } } & { \mathbb { E } _ { \mathbf { s } , \mathbf { s } ^ { \prime } \sim \pi ^ { * } ( \mathbf { s } , \mathbf { s } ^ { \prime } ) } \left[ \mathbb { E } _ { \mathbf { z } \sim E ( \mathbf { z } \mid \mathbf { s } , \mathbf { s } ^ { \prime } ) } \left[ - \log \left( D ( \mathbf { s } , \mathbf { a } , \mathbf { z } ) \right) \right] \right] } \\ & { + \mathbb { E } _ { \mathbf { s } , \mathbf { s } ^ { \prime } \sim \pi ( \mathbf { s } , \mathbf { s } ^ { \prime } ) } \left[ \mathbb { E } _ { \mathbf { z } \sim E ( \mathbf { z } \mid \mathbf { s } , \mathbf { s } ^ { \prime } ) } \left[ - \log \left( 1 - D ( \mathbf { s } , \mathbf { a } , \mathbf { z } ) \right) \right] \right] } \\ & { + \beta \left( \mathbb { E } _ { \mathbf { s } , \mathbf { s } ^ { \prime } \sim \pi ( \mathbf { s } , \mathbf { s } ^ { \prime } ) } \left[ \mathrm { K L } \left[ E ( \mathbf { z } \mid \mathbf { s } , \mathbf { s } ^ { \prime } ) | | r ( \mathbf { z } ) \right] \right] - I _ { c } \right) , } \end{array}
+$$
+
+where $\pi ( s , s ^ { \prime } )$ denotes the joint distribution of successive states from a policy, and $E ( \mathbf { z } | \mathbf { s } , \mathbf { s } ^ { \prime } ) =$ $E _ { g } ( \mathbf { z } _ { g } | \mathbf { s } ) { \cdot } E _ { h } ( \mathbf { z } _ { h } | \mathbf { s } ) { \cdot } E _ { h } ( \mathbf { z } _ { h } ^ { \prime } | \bar { \mathbf { s } } ^ { \prime } )$ .
+
+![](images/f8345f9a0288395a6d57696ed487a1c948855dd61361b99c00b0067aa21d534b.jpg)  
+Figure 4: Learning curves comparing VAIL to other methods for motion imitation. Performance is measured using the average joint rotation error between the simulated character and the reference motion. Each method is evaluated with 3 random seeds.
+
+<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>Backflip</td><td rowspan=1 colspan=1>Cartwheel</td><td rowspan=1 colspan=1>Dance</td><td rowspan=1 colspan=1>Run</td><td rowspan=1 colspan=1>Spinkick</td></tr><tr><td rowspan=1 colspan=1>BC</td><td rowspan=1 colspan=1>3.01</td><td rowspan=1 colspan=1>2.88</td><td rowspan=1 colspan=1>2.93</td><td rowspan=1 colspan=1>2.63</td><td rowspan=1 colspan=1>2.88</td></tr><tr><td rowspan=1 colspan=1>Merel et al., 2017</td><td rowspan=1 colspan=1>1.33 ± 0.03</td><td rowspan=1 colspan=1>1.47 ± 0.12</td><td rowspan=1 colspan=1>2.61 ± 0.30</td><td rowspan=1 colspan=1>0.52 ± 0.04</td><td rowspan=1 colspan=1>1.82 ± 0.35</td></tr><tr><td rowspan=1 colspan=1>GAIL</td><td rowspan=1 colspan=1>0.74±0.15</td><td rowspan=1 colspan=1>0.84± 0.05</td><td rowspan=1 colspan=1>1.31 ± 0.16</td><td rowspan=1 colspan=1>0.17±0.03</td><td rowspan=1 colspan=1>1.07 ± 0.03</td></tr><tr><td rowspan=1 colspan=1>GAIL -noise</td><td rowspan=1 colspan=1>0.42±0.02</td><td rowspan=1 colspan=1>0.92±0.07</td><td rowspan=1 colspan=1>0.96±0.08</td><td rowspan=1 colspan=1>0.21±0.05</td><td rowspan=1 colspan=1>0.95±0.14</td></tr><tr><td rowspan=1 colspan=1>GAIL - noise z</td><td rowspan=1 colspan=1>0.67±0.12</td><td rowspan=1 colspan=1>0.72± 0.04</td><td rowspan=1 colspan=1>1.14 ± 0.08</td><td rowspan=1 colspan=1>0.14±0.03</td><td rowspan=1 colspan=1>0.64±0.09</td></tr><tr><td rowspan=1 colspan=1>GAIL - GP</td><td rowspan=1 colspan=1>0.62±0.09</td><td rowspan=1 colspan=1>0.69 ±0.05</td><td rowspan=1 colspan=1>0.80± 0.32</td><td rowspan=1 colspan=1>0.12 ± 0.02</td><td rowspan=1 colspan=1>0.64± 0.04</td></tr><tr><td rowspan=1 colspan=1>VAIL (ours)</td><td rowspan=1 colspan=1>0.36±0.13</td><td rowspan=1 colspan=1>0.40±0.08</td><td rowspan=1 colspan=1>0.40±0.21</td><td rowspan=1 colspan=1>0.13±0.01</td><td rowspan=1 colspan=1>0.34± 0.05</td></tr><tr><td rowspan=1 colspan=1>VAIL - GP (ours)</td><td rowspan=1 colspan=1>0.46±0.17</td><td rowspan=1 colspan=1>0.31 ± 0.02</td><td rowspan=1 colspan=1>0.15±0.01</td><td rowspan=1 colspan=1>0.10±0.01</td><td rowspan=1 colspan=1>0.31 ± 0.02</td></tr><tr><td rowspan=1 colspan=1>Peng et al., 2018</td><td rowspan=1 colspan=1>0.26</td><td rowspan=1 colspan=1>0.21</td><td rowspan=1 colspan=1>0.20</td><td rowspan=1 colspan=1>0.14</td><td rowspan=1 colspan=1>0.19</td></tr></table>
+
+Table 1: Average joint rotation error (radians) on humanoid motion imitation tasks. VAIL outperforms the other methods for all skills evaluated, except for policies trained using the manuallydesigned reward function from (Peng et al., 2018).
+
+# 5 EXPERIMENTS
+
+We evaluate our method on adversarial learning problems in imitation learning, inverse reinforcement learning, and image generation. In the case of imitation learning, we show that the VDB enables agents to learn complex motion skills from a single demonstration, including visual demonstrations provided in the form of video clips. We also show that the VDB improves the performance of inverse RL methods. Inverse RL aims to reconstruct a reward function from a set demonstrations, which can then used to perform the task in new environments, in contrast to imitation learning, which aims to recover a policy directly. Our method is also not limited to control tasks, and we demonstrate its effectiveness for unconditional image generation.
+
+# 5.1 VAIL: VARIATIONAL ADVERSARIAL IMITATION LEARNING
+
+The goal of the motion imitation tasks is to train a simulated character to mimic demonstrations provided by mocap clips recorded from human actors. Each mocap clip provides a sequence of target states $\big \{ \mathbf { s } _ { 0 } ^ { * } , \mathbf { s } _ { 1 } ^ { * } , . . . , \mathbf { s } _ { T } ^ { * } \big \}$ that the character should track at each timestep. We use a similar experimental setup as Peng et al. (2018), with a 34 degrees-of-freedom humanoid character. We found that the discriminator architecture can greatly affect the performance on complex skills. The particular architecture we employ differs substantially from those used in prior work (Merel et al., 2017), details of which are available in Appendix C. The encoding $Z$ is 128D and an information constraint of $I _ { c } = 0 . 5$ is applied for all skills, with a dual stepsize of $\alpha _ { \beta } = 1 0 ^ { - 5 }$ . All policies are trained using PPO (Schulman et al., 2017).
+
+The motions learned by the policies are best seen in the supplementary video. Snapshots of the character’s motions are shown in Figure 3. Each skill is learned from a single demonstration. VAIL is able to closely reproduce a variety of skills, including those that involve highly dynamics flips and complex contacts. We compare VAIL to a number of other techniques, including state-only GAIL (Ho & Ermon, 2016), GAIL with instance noise applied to the discriminator inputs (GAIL - noise), GAIL with instance noise applied to the last hidden layer (GAIL - noise z), and GAIL with a gradient penalty applied to the discriminator (GAIL - GP) (Mescheder et al., 2018). Since the VDB helps to prevent vanishing gradients, while GP mitigates exploding gradients, the two techniques can be seen as being complementary. Therefore, we also train a model that combines both VAIL and GP (VAIL -
+
+![](images/7497cb73e50bbfbd813c2ffce4161f7254aec761aabb16358d90c1638d8e73f8.jpg)  
+Figure 5: Left: Snapshots of the video demonstration and the simulated character trained with VAIL. The policy learns to run by directly imitating the video. Right: Saliency maps that visualize the magnitude of the discriminator’s gradient with respect to all channels of the RGB input images from both the demonstration and the simulation. Pixel values are normalized between [0, 1].
+
+![](images/5f2f5cf0687f611bc03d408752373a1aaf35da53c395f876a101449a9d723a44.jpg)  
+Figure 6: Left: Learning curves comparing policies for the video imitation task trained using a pixel-wise loss as the reward, GAIL, and VAIL. Only VAIL successfully learns to run from a video demonstration. Middle: Effect of training with fixed values of $\beta$ and adaptive $\beta$ $I _ { c } = 0 . 5$ ). Right:. KL loss over the course of training with adaptive $\beta$ . The dual gradient descent update for $\beta$ effectively enforces the VDB constraint $I _ { c }$ .
+
+GP). Implementation details for combining the VDB and GP are available in Appendix B. Learning curves for the various methods are shown in Figure 10 and Table 1 summarizes the performance of the final policies. Performance is measured in terms of the average joint rotation error between the simulated character and the reference motion. We also include a reimplementation of the method described by Merel et al. (2017). For the purpose of our experiments, GAIL denotes policies trained using our particular architecture but without a VDB, and Merel et al. (2017) denotes policies trained using an architecture that closely mirror those from previous work. Furthermore, we include comparisons to policies trained using the handcrafted reward from Peng et al. (2018), as well as policies trained via behavioral cloning (BC). Since mocap data does not provide expert actions, we use the policies from Peng et al. (2018) as oracles to provide state-action demonstrations, which are then used to train the BC policies via supervised learning. Each BC policy is trained with $1 0 \mathrm { k }$ samples from the oracle policies, while all other policies are trained from just a single demonstration, the equivalent of approximately 100 samples.
+
+VAIL consistently outperforms previous adversarial methods, and VAIL - GP achieves the best performance overall. Simply adding instance noise to the inputs (Salimans et al., 2016) or hidden layer without the KL constraint (Sønderby et al., 2016) leads to worse performance, since the network can learn a latent representation that renders the effects of the noise negligible. Though training with the handcrafted reward still outperforms the adversarial methods, VAIL demonstrates comparable performance to the handcrafted reward without manual reward or feature engineering, and produces motions that closely resemble the original demonstrations. The method from Merel et al. (2017) was able to imitate simple skills such as running, but was unable to reproduce more acrobatic skills such as the backflip and spinkick. In the case of running, our implementation produces more natural gaits than the results reported in Merel et al. (2017). Behavioral cloning is unable to reproduce any of the skills, despite being provided with substantially more demonstration data than the other methods.
+
+Video Imitation: While our method achieves substantially better results on motion imitation when compared to prior work, previous methods can still produce reasonable behaviors. However, if the demonstrations are provided in terms of the raw pixels from video clips, instead of mocap data, the imitation task becomes substantially harder. The goal of the agent is therefore to directly imitate the skill depicted in the video. This is also a setting where manually engineering rewards is impractical, since simple losses like pixel distance do not provide a semantically meaningful measure of similarity. Figure 6 compares learning curves of policies trained with VAIL, GAIL, and policies trained using a reward function defined by the average pixel-wise difference between the frame $M _ { t } ^ { * }$ from the video demonstration and a rendered image $M _ { t }$ of the agent at each timestep $t$ , $\begin{array} { r } { r _ { t } = 1 - \frac { 1 } { 3 \times 6 4 ^ { 2 } } | | M _ { t } ^ { * } - M _ { t } | | ^ { 2 } } \end{array}$ . Each frame is represented by a $6 4 \times 6 4$ RGB image.
+
+![](images/9a3768a8c49373d423021d089034c2c03922c0410f3b165f6f1fe6e355dafe56.jpg)  
+Figure 7: Left: C-Maze and S-Maze. When trained on the training maze on the left, AIRL learns a reward that overfits to the training task, and which cannot be transferred to the mirrored maze on the right. In contrast, VAIRL learns a smoother reward function that enables more-reliable transfer. Right: Performance on flipped test versions of our two training mazes. We report mean return ( $\pm$ std. dev.) over five runs, and the mean return for the expert used to generate demonstrations.
+
+<table><tr><td rowspan=2 colspan=1>Method</td><td rowspan=1 colspan=2>Transferenvironments</td></tr><tr><td rowspan=1 colspan=1>C-maze</td><td rowspan=1 colspan=1>S-maze</td></tr><tr><td rowspan=1 colspan=1>GAIL</td><td rowspan=1 colspan=1>-24.6±7.2</td><td rowspan=1 colspan=1>1.0±1.3</td></tr><tr><td rowspan=1 colspan=1>VAIL</td><td rowspan=1 colspan=1>-65.6±18.9</td><td rowspan=1 colspan=1>20.8±39.7</td></tr><tr><td rowspan=1 colspan=1>AIRL</td><td rowspan=1 colspan=1>-15.3±7.8</td><td rowspan=1 colspan=1>-0.2±0.1</td></tr><tr><td rowspan=1 colspan=1>AIRL - GP</td><td rowspan=1 colspan=1>-9.14±0.4</td><td rowspan=1 colspan=1>-0.14±0.3</td></tr><tr><td rowspan=1 colspan=1>VAIRL (β = 0)</td><td rowspan=1 colspan=1>-25.5±7.2</td><td rowspan=1 colspan=1>62.3±33.2</td></tr><tr><td rowspan=1 colspan=1>VAIRL (ours)</td><td rowspan=1 colspan=1>-10.0±2.2</td><td rowspan=1 colspan=1>74.0±38.7</td></tr><tr><td rowspan=1 colspan=1>VAIRL - GP (ours)</td><td rowspan=1 colspan=1>-9.18±0.4</td><td rowspan=1 colspan=1>156.5±5.6</td></tr><tr><td rowspan=1 colspan=1>TRPO expert</td><td rowspan=1 colspan=1>-5.1</td><td rowspan=1 colspan=1>153.2</td></tr></table>
+
+Both GAIL and the pixel-loss are unable to learn the running gait. VAIL is the only method that successfully learns to imitate the skill from the video demonstration. Snapshots of the video demonstration and the simulated motion is available in Figure 5. To further investigate the effects of the VDB, we visualize the gradient of the discriminator with respect to images from the video demonstration and simulation. Saliency maps for discriminators trained with VAIL and GAIL are available in Figure 5. The VAIL discriminator learns to attend to spatially coherent image patches around the character, while the GAIL discriminator exhibits less structure. The magnitude of the gradients from VAIL also tend to be significantly larger than those from GAIL, which may suggests that VAIL is able to mitigate the problem of vanishing gradients present in GAIL.
+
+Adaptive Constraint: To evaluate the effects of the adaptive $\beta$ updates, we compare policies trained with different fixed values of $\beta$ and policies where $\beta$ is updated adaptively to enforce a desired information constraint $I _ { c } = 0 . 5$ . Figure 6 illustrates the learning curves and the KL loss over the course of training. When $\beta$ is too small, performance reverts to that achieved by GAIL. Large values of $\beta$ help to smooth the discriminator landscape and improve learning speed during the early stages of training, but converges to a worse performance. Policies trained using dual gradient descent to adaptively update $\beta$ consistently achieves the best performance overall.
+
+# 5.2 VAIRL: VARIATIONAL ADVERSARIAL INVERSE REINFORCEMENT LEARNING
+
+Next, we use VAIRL to recover reward functions from demonstrations. Unlike the discriminator learned by VAIL, the reward function recovered by VAIRL can be re-optimized to train new policies from scratch in the same environment. In some cases, it can also be used to transfer similar behaviour to different environments. In Figure 7, we show the results of applying VAIRL to the C-maze from Fu et al. (2017), and a more complex S-maze; the simple 2D observation spaces of these tasks make it easy to interpret the recovered reward functions. In both mazes, the expert is trained to navigate from a start position at the bottom of the maze to a fixed target position at the top. We use each method to obtain an imitation policy and to approximate the expert’s reward on the original maze. The recovered reward is then used to train a new policy to solve a left–right flipped version of the training maze. On the C-maze, we found that plain AIRL—without a gradient penalty— would sometimes overfit and fail to transfer to the new environment, as evidenced by the reward visualization in Figure 7 (left) and the higher return variance in Figure 7 (right). In contrast, by incorporating a VDB into AIRL, VAIRL learns a substantially smoother reward function that is more suitable for transfer. Furthermore, we found that in the S-maze with two internal walls, AIRL was too unstable to acquire a meaningful reward function. This was true even with the use of a gradient penalty. In contrast, VAIRL was able to learn a reasonable reward in most cases without a gradient penalty, and its performance improved even further with the addition of a gradient penalty. To evaluate the effects of the VDB, we observe that the performance of VAIRL drops on both tasks when the KL constraint is disabled $\mathcal { B } = 0$ ), suggesting that the improvements from the VDB cannot be attributed entirely to the noise introduced by the sampling process for z. Further details of these experiments and illustrations of the recovered reward functions are available in Appendix D.
+
+![](images/c86c34eae7b7dab70e64c321c0f81908e190aa387a041d3cc2f92e4b2ee908cc.jpg)
+
+![](images/78b0c0503897ba4746233494cfd31890b086ca770f3439471530fef4c520776d.jpg)  
+Figure 8: Comparison of VGAN and other methods on CIFAR-10, with performance evaluated using the Frechet Inception Distance (FID). ´   
+Figure 9: VGAN samples on CIFAR-10, CelebA $1 2 8 \times 1 2 8$ , and CelebAHQ $1 0 2 4 \times 1 0 2 4$ .
+
+# 5.3 VGAN: VARIATIONAL GENERATIVE ADVERSARIAL NETWORKS
+
+Finally, we apply the VDB to image generation with generative adversarial networks, which we refer to as VGAN. Experiment are conducted on CIFAR-10 (Krizhevsky et al.), CelebA (Liu et al. (2015)), and CelebAHQ (Karras et al., 2018) datasets. We compare our approach to recent stabilization techniques: WGAN-GP (Gulrajani et al., 2017b), instance noise (Sønderby et al., 2016; Arjovsky & Bottou, 2017), spectral normalization (SN) (Miyato et al., 2018), and gradient penalty (GP) (Mescheder et al., 2018), as well as the original GAN (Goodfellow et al., 2014) on CIFAR10. To measure performance, we report the Frechet Inception Distance (FID) (Heusel et al., 2017), ´ which has been shown to be more consistent with human evaluation. All methods are implemented using the same base model, built on the resnet architecture of Mescheder et al. (2018). Aside from tuning the KL constraint $I _ { c }$ for VGAN, no additional hyperparameter optimization was performed to modify the settings provided by Mescheder et al. (2018). The performance of the various methods on CIFAR-10 are shown in Figure 8. While vanilla GAN and instance noise are prone to diverging as training progresses, VGAN remains stable. Note that instance noise can be seen as a non-adaptive version of VGAN without constraints on $I _ { c }$ . This experiment again highlights that there is a significant improvement from imposing the information bottleneck over simply adding instance noise. Combining both VDB and gradient penalty (VGAN - GP) achieves the best performance overall with an FID of 18.1. We also experimented with combining the VDB with SN, but this combination is prone to diverging. See Figure 9 for samples of images generated with our approach. Please refer to Appendix E for experimental details and more results.
+
+# 6 CONCLUSION
+
+We present the variational discriminator bottleneck, a general regularization technique for adversarial learning. Our experiments show that the VDB is broadly applicable to a variety of domains, and yields significant improvements over previous techniques on a number of challenging tasks. While our experiments have produced promising results for video imitation, the results have been primarily with videos of synthetic scenes. We believe that extending the technique to imitating realworld videos is an exciting direction. Another exciting direction for future work is a more in-depth theoretical analysis of the method, to derive convergence and stability results or conditions.
+
+# ACKNOWLEDGEMENTS
+
+We would like to thank the anonymous reviewers for their helpful feedback, and AWS and NVIDIA for providing computational resources. This research was funded by an NSERC Postgraduate Scholarship, a Berkeley Fellowship for Graduate Study, BAIR, Huawei, and ONR PECASE N000141612723.
+
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+SUPPLEMENTARY MATERIAL
+
+# A ANALYSIS AND PROOFS
+
+In this appendix, we show that the gradient of the generator when the discriminator is augmented with the VDB is non-degenerate, under some mild additional assumptions. First, we assume a pointwise constraint of the form $\mathrm { K L } [ E ( \mathbf { z } | \mathbf { x } ) \| r ( \mathbf { z } ) ] \leq I _ { c }$ for all $\mathbf { x }$ . In reality, we use an average KL constraint, since we found it to be more convenient to optimize, though a pointwise constraint is also possible to enforce by using the largest constraint violation to increment $\beta$ . We could likely also extend the analysis to the average constraint, though we leave this to future work. The main theorem can then be stated as follows:
+
+Theorem A.1. Let $g ( \mathbf { u } )$ denote the generator’s mapping from a noise vector $\mathbf { u } \sim p ( \mathbf { u } )$ to a point in $X$ . Given the generator distribution $G ( \mathbf { x } )$ and data distribution $p ^ { * } ( \mathbf { x } )$ , a VDB with an encoder $E ( \mathbf { z } | \mathbf { x } ) = \mathcal { N } ( \mu _ { E } ( \mathbf { x } ) , \Sigma )$ , and $\mathrm { K L } [ E ( \mathbf { z } | \mathbf { x } ) \| r ( \mathbf { z } ) ] \leq I _ { c } ,$ the gradient passed to the generator has the form
+
+$$
+\begin{array} { l } { \nabla _ { g } \mathbb { E } _ { { \mathbf { u } } \sim p ( { \mathbf { u } } ) } \left[ \log \left( 1 - D ^ { * } ( \mu _ { E } ( g ( { \mathbf { u } } ) ) ) \right) \right] } \\ { = \mathbb { E } _ { { \mathbf { u } } \sim p ( { \mathbf { u } } ) } \left[ a ( { \mathbf { u } } ) \int E ( \mu _ { E } ( g ( { \mathbf { u } } ) ) | { \mathbf { x } } ) \nabla _ { g } | | \mu _ { E } ( g ( { \mathbf { u } } ) ) - \mu _ { E } ( { \mathbf { x } } ) | | ^ { 2 } d p ^ { * } ( { \mathbf { x } } ) \right. } \\ { \left. - b ( { \mathbf { u } } ) \int E ( \mu _ { E } ( g ( { \mathbf { u } } ) ) | { \mathbf { x } } ) \nabla _ { g } | | \mu _ { E } ( g ( { \mathbf { u } } ) ) - \mu _ { E } ( { \mathbf { x } } ) | | ^ { 2 } d G ( { \mathbf { x } } ) \right] } \end{array}
+$$
+
+where $D ^ { * } ( \mathbf { z } )$ is the optimal discriminator, $a ( \mathbf { x } )$ and $b ( \mathbf { x } )$ are positive functions, and we always have $E ( \mu _ { E } ( g ( \mathbf { u } ) ) | \mathbf { x } ) > C ( I _ { c } )$ , where $C ( I _ { c } )$ is a continuous monotonic function, and $C ( I _ { c } )  \delta > 0$ as $I _ { c } \to 0$ .
+
+Analysis for an encoder with an input-dependent variance $\Sigma ( \mathbf { x } )$ is also possible, but more involved. We’ll further assume below for notational simplicity that $\Sigma$ is diagonal with diagonal values $\sigma ^ { 2 }$ . This assumption is not required, but substantially simplifies the linear algebra. Analogously to Theorem 3.2 from Arjovsky & Bottou (2017), this theorem states that the gradient of the generator points in the direction of points in the data distribution, and away from points in the generator distribution. However, going beyond the theorem in Arjovsky & Bottou (2017), this result states that the coefficients on these vectors, given by $E ( \mu _ { E } ( g ( \mathbf { u } ) ) | \mathbf { x } )$ , are always bounded below by a value that approaches a positive constant $\delta$ as we decrease $I _ { c }$ , meaning that the gradient does not vanish. The proof of the first part of this theorem is essentially identical to the proof presented by Arjovsky & Bottou (2017), but accounting for the fact that the noise is now injected into the latent space of the VDB, rather than being added directly to $\mathbf { x }$ . This result assumes that $E ( { \bf z } | { \bf x } )$ has a learned but input-independent variance $\Sigma = \sigma ^ { 2 } I$ , though the proof can be repeated for an input-dependent or non-diagonal $\Sigma$ :
+
+Proof. Overloading $p ^ { * } ( \mathbf { x } )$ and $G ( \mathbf { x } )$ , let $p ^ { * } ( \mathbf { z } )$ and $G ( \mathbf { z } )$ be the distribution of embeddings $\mathbf { z }$ under the real data and generator respectively. $p ^ { * } ( \mathbf { z } )$ is then given by
+
+$$
+p ^ { * } ( \mathbf { z } ) = \mathbb { E } _ { \mathbf { x } \sim p ^ { * } ( \mathbf { x } ) } \left[ E ( \mathbf { z } | \mathbf { x } ) \right] = \int E ( \mathbf { z } | \mathbf { x } ) d p ^ { * } ( \mathbf { x } ) ,
+$$
+
+and similarly for $G ( z )$
+
+$$
+G ( \mathbf { z } ) = \mathbb { E } _ { \mathbf { x } \sim G ( \mathbf { x } ) } \left[ E ( \mathbf { z } | \mathbf { x } ) \right] = \int E ( \mathbf { z } | \mathbf { x } ) d G ( \mathbf { x } ) ,
+$$
+
+From Arjovsky $\&$ Bottou (2017), the optimal discriminator between $p ^ { * } ( \mathbf { z } )$ and $G ( \mathbf { z } )$ is
+
+$$
+D ^ { * } ( \mathbf { z } ) = \frac { p ^ { * } ( \mathbf { z } ) } { p ^ { * } ( \mathbf { z } ) + G ( \mathbf { z } ) }
+$$
+
+The gradient passed to the generator then has the form
+
+$$
+\begin{array} { r l } & { \nabla _ { g } \mathbb { E } _ { \mathbf { u } \sim p ( \mathbf { u } ) } \left[ \log \left( 1 - D ^ { * } ( \mu _ { E } ( g ( \mathbf { u } ) ) ) \right) \right] } \\ & { \qquad = \mathbb { E } _ { \mathbf { u } \sim p ( \mathbf { u } ) } \left[ \nabla _ { g } \log \left( G ( \mu _ { E } ( g ( \mathbf { u } ) ) ) \right) - \nabla _ { g } \log \left( p ^ { * } ( \mu _ { E } ( g ( \mathbf { u } ) ) ) + G ( \mu _ { E } ( g ( \mathbf { u } ) ) ) \right) \right] } \\ & { \qquad = \mathbb { E } _ { \mathbf { u } \sim p ( \mathbf { u } ) } \left[ \frac { \nabla _ { g } G ( \mu _ { E } ( g ( \mathbf { u } ) ) ) } { G ( \mu _ { E } ( g ( \mathbf { u } ) ) ) } - \frac { \nabla _ { g } p ^ { * } ( \mu _ { E } ( g ( \mathbf { u } ) ) ) + \nabla _ { g } G ( \mu _ { E } ( g ( \mathbf { u } ) ) ) } { p ^ { * } ( \mu _ { E } ( g ( \mathbf { u } ) ) ) + G ( \mu _ { E } ( g ( \mathbf { u } ) ) ) } \right] } \\ & { \qquad = \mathbb { E } _ { \mathbf { u } \sim p ( \mathbf { u } ) } \left[ \frac { 1 } { p ^ { * } ( \mu _ { E } ( g ( \mathbf { u } ) ) ) + G ( \mu _ { E } ( g ( \mathbf { u } ) ) ) } \nabla _ { g } \left[ - p ^ { * } ( \mu _ { E } ( g ( \mathbf { u } ) ) ) \right] \right. } \\ & { \qquad \left. - \frac { 1 } { p ^ { * } ( \mu _ { E } ( g ( \mathbf { u } ) ) ) + G ( \mu _ { E } ( g ( \mathbf { u } ) ) ) } \frac { p ^ { * } ( \mu _ { E } ( g ( \mathbf { u } ) ) ) } { G ( \mu _ { E } ( g ( \mathbf { u } ) ) ) } \nabla _ { g } \left[ - G ( \mu _ { E } ( g ( \mathbf { u } ) ) ) \right] \right] . } \end{array}
+$$
+
+Let
+
+$$
+\begin{array} { l } { { \displaystyle a ( { \bf u } ) = \frac { 1 } { 2 \sigma ^ { 2 } } \frac { 1 } { p ^ { * } \left( \mu _ { E } ( g ( { \bf u } ) ) \right) + G \left( \mu _ { E } ( g ( { \bf u } ) ) \right) } \ ~ } } \\ { { \displaystyle b ( { \bf u } ) = \frac { 1 } { 2 \sigma ^ { 2 } } \frac { 1 } { p ^ { * } \left( \mu _ { E } ( g ( { \bf u } ) ) \right) + G \left( \mu _ { E } ( g ( { \bf u } ) ) \right) } \frac { p ^ { * } \left( \mu _ { E } ( g ( { \bf u } ) ) \right) } { G \left( \mu _ { E } ( g ( { \bf u } ) ) \right) } . } } \end{array}
+$$
+
+We then have
+
+$$
+\begin{array} { r l } & {  \varepsilon _ { \mathrm { S P } } ^ { ( \varepsilon ) } ( \varepsilon ) \{ 2 \sigma ^ { 2 } ( \varepsilon ) ( \varepsilon ) ( \eta ) \}  } \\ & { \quad - \varepsilon _ { \mathrm { S P } } ( \varepsilon )  \varepsilon _ { \mathrm { P } } ^ { ( \varepsilon ) } ( \varepsilon ) \{ 2 \sigma ^ { 2 } ( \varepsilon ) ( \eta ) \}  = \int _ { - \varepsilon } ^ { \varepsilon } \gamma _ { \varepsilon }  \varepsilon _ { \mathrm { P } } ( \varepsilon ) ( \varepsilon ) ( \eta )  - 2 \eta ^ { \varepsilon }  \varepsilon _ { \mathrm { P } } ( \varepsilon ) ( \eta )  \int _ { - \varepsilon } ^ { \varepsilon }  \varepsilon _ { \mathrm { P } } ( \varepsilon ) ( \varepsilon ) ( \eta )  \int _ { - \varepsilon } ^ { \varepsilon }  \varepsilon _ { \mathrm { P } } ( \varepsilon ) ( \varepsilon ) ( \eta )  } \\ & { \quad - \varepsilon _ { \mathrm { P } } ( \varepsilon )  \varepsilon _ { \mathrm { P } } ^ { ( \varepsilon ) } ( \varepsilon ) \{ 2 \sigma ( \varepsilon ) ( \eta ) \}  - \varepsilon _ { \mathrm { P } } ^ { ( \varepsilon ) }  \varepsilon _ { \mathrm { P } } ( \varepsilon ) ( \eta )   \varepsilon _ { \mathrm { P } } ( \varepsilon ) ( \eta )  } \\ & { \quad - \varepsilon _ { \mathrm { P } } ( \varepsilon )  \varepsilon _ { \mathrm { P } } ^ { ( \varepsilon ) } ( \varepsilon ) \{ 2 \sigma ( \varepsilon ) ( \eta ) \}  } \\ & { \quad - \varepsilon _ { \mathrm { S P } } ( \varepsilon )  \varepsilon _ { \mathrm { P } } ^ { ( \varepsilon ) } ( \varepsilon ) \{ 2 \sigma ^ { 2 } ( \varepsilon ) ( \eta ) \} \int _ { - \varepsilon } ^ { \varepsilon } \varepsilon _ { \mathrm { P } } ^ { \varepsilon } \varepsilon _ { \mathrm { P } } ^ { ( \varepsilon ) } ( \varepsilon ) \{ 2 \sigma ( \varepsilon ) ( \eta ) \} } \\ &  \quad - \varepsilon _ { \mathrm { P } } ( \varepsilon )  \varepsilon _ { \mathrm { P } } ^ { ( \varepsilon ) } ( \varepsilon ) \{ 2 \sigma ^ { 2 } ( \eta ) \}  + \varepsilon _ { \mathrm { P } } ^ { ( \varepsilon ) }  \varepsilon _ { \mathrm { P } } ^ { ( \varepsilon ) } ( \varepsilon _ { \mathrm { P } } ^ { ( \varepsilon ) } ( \varepsilon _ { \mathrm { P } } ^ { ( \varepsilon ) } ) - \mu _  \ \end{array}
+$$
+
+Similar to the result from Arjovsky & Bottou (2017), the gradient of the generator drives the generator’s samples in the embedding space $\mu _ { E } ( g ( { \bf u } ) )$ towards embeddings of the points from the dataset $\mu _ { E } ( { \bf x } )$ weighted by their likelihood $E ( \mu _ { E } ( g ( \mathbf { u } ) ) | \mathbf { x } )$ under the real data. For an arbitrary encoder $E$ , real and fake samples in the embedding may be far apart. As such, the coefficients $E ( \mu _ { E } ( g ( \mathbf { u } ) ) | \mathbf { x } )$ can be arbitrarily small, thereby resulting in vanishing gradients for the generator.
+
+The second part of the theorem states that $C ( I _ { c } )$ is a continuous monotonic function, and $C ( I _ { c } )  \delta > 0$ as $I _ { c } \to 0$ . This is the main result, and relies on the fact that $\mathrm { K L } [ E ( \mathbf { z } | \mathbf { x } ) | | r ( \mathbf { z } ) ] \leq I _ { c }$ . The intuition behind this result is that, for any two inputs $\mathbf { x }$ and $\mathbf { y }$ , their encoded distributions $E ( { \bf z } | { \bf x } )$ and $E ( \mathbf { z } | \mathbf { y } )$ have means that cannot be more than some distance apart, and that distance shrinks with $I _ { c }$ . This allows us to bound $E ( \mu _ { E } ( \mathbf { y } ) ) | \mathbf { x } )$ below by $C ( I _ { c } )$ , which ensures that the coefficients on the vectors in the theorem above are always at least as large as $C ( I _ { c } )$ .
+
+Proof. Let $r ( \mathbf { z } ) = \mathcal { N } ( 0 , I )$ be the prior distribution and suppose the $\mathrm { K L }$ divergence for all $\mathbf { x }$ in the dataset and all $g ( \mathbf { u } )$ generated by the generator are bounded by $I _ { c }$
+
+$$
+\begin{array} { r l } & { \mathrm { K L } \left[ E ( \mathbf { z } | \mathbf { x } ) | | r ( \mathbf { z } ) \right] \leq I _ { c } , \quad \forall \mathbf { x } , \ \mathbf { x } \sim p ^ { * } ( \mathbf { x } ) } \\ & { \mathrm { K L } \left[ E ( \mathbf { z } | g ( \mathbf { u } ) ) | | r ( \mathbf { z } ) \right] \leq I _ { c } , \quad \forall \mathbf { u } , \ \mathbf { u } \sim p ( \mathbf { u } ) . } \end{array}
+$$
+
+From the definition of the KL-divergence we can bound the length of all embedding vectors,
+
+$$
+\begin{array} { r } { \mathrm { K L } \left[ E ( \mathbf { z } | \mathbf { x } ) | | r ( \mathbf { z } ) \right] = \displaystyle \frac { 1 } { 2 } \left( K \sigma ^ { 2 } + \mu _ { E } ( \mathbf { x } ) ^ { T } \mu _ { E } ( \mathbf { x } ) - K - K \log \sigma ^ { 2 } \right) \leq I _ { c } } \\ { | | \mu _ { E } ( \mathbf { x } ) | | ^ { 2 } \leq 2 I _ { c } - K \sigma ^ { 2 } + K + K \log \sigma ^ { 2 } , } \end{array}
+$$
+
+and similarly for $| | \mu _ { E } ( g ( \mathbf { u } ) ) | | ^ { 2 }$ , with $K$ denoting the dimension of $Z$ . A lower bound on $E ( \mu _ { E } ( g ( \mathbf { u } ) ) | \mathbf { x } )$ , where $\mathbf { u } \sim p ( \mathbf { u } )$ and $\mathbf { x } \sim p ^ { * } ( \mathbf { x } )$ , can then be determined by
+
+$$
+\begin{array} { l } { { \displaystyle { E ( \mu _ { E } ( g ( { \bf u } ) ) \vert { \bf x } ) } ) = - \frac { 1 } { 2 \sigma ^ { 2 } } \left( \mu _ { E } ( g ( { \bf u } ) ) - \mu _ { E } ( { \bf x } ) \right) ^ { T } \left( \mu _ { E } ( g ( { \bf u } ) ) - \mu _ { E } ( { \bf x } ) \right) - \frac { K } { 2 } \log \sigma ^ { 2 } - \frac { K } { 2 } \log 2 \pi } } \\ { { \displaystyle \approx \vert \vert \mu _ { E } ( { \bf x } ) \vert \vert ^ { 2 } , \vert \vert \mu _ { E } ( g ( { \bf u } ) ) \vert \vert ^ { 2 } \le 2 I _ { c } - K \sigma ^ { 2 } + K + K \log \sigma ^ { 2 } , \ ~ } } \\ { { \displaystyle \qquad \vert \vert \mu _ { E } ( g ( { \bf u } ) ) - \mu _ { E } ( { \bf x } ) \vert \vert ^ { 2 } \le 8 I _ { c } - 4 K \sigma ^ { 2 } + 4 K + 4 K \log \sigma ^ { 2 } , } } \end{array}
+$$
+
+and it follows that
+
+$$
+- \frac { 1 } { 2 \sigma ^ { 2 } } \left( \mu _ { E } ( g ( { \bf u } ) ) - \mu _ { E } ( { \bf x } ) \right) ^ { T } \left( \mu _ { E } ( g ( { \bf u } ) ) - \mu _ { E } ( { \bf x } ) \right) \ge - 4 \sigma ^ { - 2 } I _ { c } + 2 K - 2 K \sigma ^ { - 2 } - 2 K \sigma ^ { - 2 } \log \sigma ^ { - 2 } .
+$$
+
+The likelihood is therefore bounded below by
+
+$$
+\begin{array} { c } { { \displaystyle \log \left( E ( \mu _ { E } ( g ( \mathbf { u } ) ) | \mathbf { x } ) \right) \geq - 4 \sigma ^ { - 2 } I _ { c } + 2 K - 2 K \sigma ^ { - 2 } - 2 K \sigma ^ { - 2 } \log { \sigma ^ { - 2 } } - \frac { K } { 2 } \log { \sigma ^ { 2 } } - \frac { K } { 2 } \log { 2 \pi } } } \\ { { \displaystyle \mathrm { S i n c e - } \sigma ^ { - 2 } - \sigma ^ { - 2 } \log { \sigma ^ { - 2 } } \geq - 1 , \hfill } } \\ { { \displaystyle \log \left( E ( \mu _ { E } ( g ( \mathbf { u } ) ) | \mathbf { x } ) \right) \geq - 4 \sigma ^ { - 2 } I _ { c } - \frac { K } { 2 } \log { \sigma ^ { 2 } } - \frac { K } { 2 } \log { 2 \pi } } } \end{array}
+$$
+
+From the KL constraint, we can derive a lower bound $\ell ( I _ { c } )$ and an upper bound ${ \cal U } ( I _ { c } )$ on $\sigma ^ { 2 }$ .
+
+$$
+\begin{array} { l } { { { \displaystyle { \frac { 1 } { 2 } } \left( K \sigma ^ { 2 } + \mu _ { E } ( { \bf x } ) ^ { T } \mu _ { E } ( { \bf x } ) - K - K \log \sigma ^ { 2 } \right) \le I _ { c } } } } \\ { { \displaystyle { \sigma ^ { 2 } - 1 - \log \sigma ^ { 2 } \le \frac { 2 I _ { c } } { K } } } } \\ { { \displaystyle { \log \sigma ^ { 2 } \ge - \frac { 2 I _ { c } } { K } - 1 } } } \\ { { \displaystyle { \sigma ^ { 2 } \ge \exp \left( - \frac { 2 I _ { c } } { K } - 1 \right) = \ell ( I _ { c } ) } } } \end{array}
+$$
+
+For the upper bound, since $\sigma ^ { 2 } - \log \sigma ^ { 2 } > { \textstyle { \frac { 1 } { 2 } } } \sigma ^ { 2 }$ ,
+
+$$
+\begin{array} { l } { { \displaystyle { \sigma ^ { 2 } - 1 - \log \sigma ^ { 2 } \leq \frac { 2 I _ { c } } { K } } } } \\ { { \displaystyle { \frac { 1 } { 2 } \sigma ^ { 2 } - 1 < \frac { 2 I _ { c } } { K } } } } \\ { { \displaystyle { \sigma ^ { 2 } < \frac { 4 I _ { c } } { K } + 2 = \mathcal { U } ( I _ { c } ) } } } \end{array}
+$$
+
+Substituting $\ell ( I _ { c } )$ and ${ \cal U } ( I _ { c } )$ into Equation 14, we arrive at the following lower bound
+
+$$
+E ( \mu _ { E } ( g ( \mathbf { u } ) ) | \mathbf { x } ) > \exp \left( - 4 I _ { c } \exp \left( \frac { 2 I _ { c } } { K } + 1 \right) - \frac { K } { 2 } \log \left( \frac { 4 I _ { c } } { K } + 2 \right) - \frac { K } { 2 } \log 2 \pi \right) = C ( I _ { c } ) .
+$$
+
+# B GRADIENT PENALTY
+
+To combine VDB with gradient penalty, we use the reparameterization trick to backprop through the encoder when computing the gradient of the discriminator with respect to the inputs.
+
+$$
+\begin{array} { r l } { J ( D , E ) = \underset { D , E } { \operatorname* { m i n } } } & { \mathbb { E } _ { x \sim p ^ { * } ( \mathbf { x } ) } \left[ \mathbb { E } _ { \mathbf { z } \sim E ( \mathbf { z } \mid \mathbf { x } ) } \left[ - \log \left( D ( \mathbf { z } ) \right) \right] \right] + \mathbb { E } _ { \mathbf { x } \sim G ( \mathbf { x } ) } \left[ \mathbb { E } _ { \mathbf { z } \sim E ( \mathbf { z } \mid \mathbf { x } ) } \left[ - \log \left( 1 - D ( \mathbf { z } ) \right) \right] \right] } \\ & { \quad + w _ { G P } \mathbb { E } _ { x \sim p ^ { * } ( \mathbf { x } ) } \left[ \mathbb { E } _ { \epsilon \sim \mathcal { N } ( 0 , I ) } \left[ \frac { 1 } { 2 } | | \nabla _ { x } D ( \mu _ { E } ( x ) + \Sigma _ { E } ( x ) \epsilon ) | | ^ { 2 } \right] \right] } \\ { \mathrm { s . t . } \quad } & { \mathbb { E } _ { \mathbf { x } \sim \tilde { p } ( \mathbf { x } ) } \left[ \mathrm { K L } \left[ E ( \mathbf { z } | \mathbf { x } ) | | r ( \mathbf { z } ) \right] \right] \leq I _ { c } , } \end{array}
+$$
+
+The coefficient $w _ { G P }$ weights the gradient penalty in the objective, $w _ { G P } = 1 0$ for the image generation, $w _ { G P } = 1$ for motion imitation, and $w _ { G P } = 0 . 1$ (C-maze) or $w _ { G P } = 0 . 0 1$ (S-maze) for the IRL tasks. The gradient penalty is applied only to real samples $p ^ { * } ( x )$ . We have experimented with apply the penalty to both real and fake samples, but found that performance was worse than penalizing only gradients from real samples. This is consistent with the GP implementation from Mescheder et al. (2018).
+
+# C IMITATION LEARNING
+
+Experimental Setup: The goal of the motion imitation tasks is to train a simulated agent to mimic a demonstration provided in the form of a mocap clip recorded from a human actor. We use a similar experimental setup as Peng et al. (2018), with a 34 degrees-of-freedom humanoid character. The state s consists of features that represent the configuration of the character’s body (link positions and velocities). We also include a phase variable $\phi \in [ 0 , 1 ]$ among the state features, which records the character’s progress along the motion and helps to synchronize the character with the reference motion. With 0 and 1 denoting the start and end of the motion respectively. The action a sampled from the policy $\pi ( \mathbf { a } | \mathbf { s } )$ specifies target poses for PD controller positioned at each joint. Given a state, the policy specifies a Gaussian distribution over the action space $\pi ( \mathbf { a } | \mathbf { s } ) = \mathcal { N } ( \mu ( \mathbf { s } ) , \Sigma )$ , with a state-dependent mean $\mu ( \mathbf { s } )$ and fixed diagonal covariance matrix $\Sigma$ . $\mu ( \mathbf { s } )$ is modeled using a 3- layered fully-connected network with 1024 and 512 hidden units, followed by a linear output layer that specifies the mean of the Gaussian. ReLU activations are used for all hidden layers. The value function is modeled with a similar architecture but with a single linear output unit. The policy is queried at $3 0 H z$ . Physics simulation is performed at $1 . 2 \mathrm { k H z }$ using the Bullet physics engine Bullet (2015).
+
+Given the rewards from the discriminator, PPO (Schulman et al., 2017) is used to train the policy, with a stepsize of $2 . 5 \times 1 0 ^ { - 6 }$ for the policy, a stepsize of 0.01 for the value function, and a stepsize of $1 0 ^ { - 5 }$ for the discirminator. Gradient descent with momentum 0.9 is used for all models. The PPO clipping threshold is set to 0.2. When evaluating the performance of the policies, each episode is simulated for a maximum horizon of $2 0 s$ . Early termination is triggered whenever the character’s torso contacts the ground, leaving the policy is a maximum error of $\pi$ radians for all remaining timesteps.
+
+Phase-Functioned Discriminator: Unlike the policy and value function, which are modeled with standard fully-connected networks, the discriminator is modeled by a phase-functioned neural network (PFNN) to explicitly model the time-dependency of the reference motion (Holden et al., 2017). While the parameters of a network are generally fixed, the parameters of a PFNN are functions of the phase variable $\phi$ . The parameters $\theta$ of the network for a given $\phi$ is determined by a weighted combination of a set of fixed parameters $\{ \theta _ { 0 } , \theta _ { 1 } , . . . , \theta _ { k } \}$ ,
+
+$$
+\theta = \sum _ { i = 0 } ^ { k } w _ { i } ( \phi ) \theta _ { i } ,
+$$
+
+where $w _ { i } ( \phi )$ is a phase-dependent weight for $\theta _ { i }$ . In our implementation, we use $k = 5$ sets of parameters and $w _ { i } ( \phi )$ is designed to linearly interpolate between two adjacent sets of parameters for each phase $\phi$ , where each set of parameters $\theta _ { i }$ corresponds to a discrete phase value $\phi _ { i }$ spaced
+
+![](images/d75d7a0543068769a72e9d9e205a9e3d3f0f65701de88a9b90fd8d4a628e0114.jpg)  
+Figure 10: Learning curves comparing VAIL to other methods for motion imitation. Performance is measured using the average joint rotation error between the simulated character and the reference motion. Each method is evaluated with 3 random seeds.
+
+![](images/5caec7daf65168a42ba692b0bb339931c433a0f09dde3fe83412da4879b5ec7a.jpg)  
+Figure 11: Learning curves comparing VAIL with a discriminator modeled by a phase-functioned neural network (PFNN), to modeling the discriminator with a fully-conneted network that receives the phase-variable $\phi$ as part of the input (no PFNN), and a discriminator modeled with a fullyconnected network but does not receive $\phi$ as an input (no phase).
+
+uniformly between $[ 0 , 1 ]$ . For a given value of $\phi$ , the parameters of the discriminator are determined according to
+
+$$
+\theta = w _ { i } ( \phi ) \theta _ { i } + w _ { i + 1 } ( \phi ) \theta _ { i + 1 }
+$$
+
+where $\theta _ { i }$ and $\theta _ { i + 1 }$ correspond to the phase values $\phi _ { i } ~ \le ~ \phi ~ < ~ \phi _ { i + 1 }$ that form the endpoints of the phase interval that contains $\phi$ . A PFNN is used for all motion imitation experiments, both VAIL and GAIL, except for those that use the approach proposed by Merel et al. (2017), which use standard fully-connected networks for the discriminator. Figure 11 compares the performance of VAIL when the discriminator is modeled with a phase-functioned neural network (with PFNN) to discriminators modeled with standard fully-connected networks. We increased the size of the layers of the fully-connected nets to have a similar number of parameters as a PFNN. We evaluate the performance of fully-connected nets that receive the phase variable $\phi$ as part of the input (no PFNN), and fully-connected nets that do not receive $\phi$ as an input. The phase-functioned discriminator leads to significant performance improvements across all tasks evaluated. Policies trained without a phase variable performs worst overall, suggesting that phase information is critical for performance. All methods perform well on simpler skills, such as running, but the additional phase structure introduced by the PFNN proved to be vital for successful imitation of more complex skills, such as the dance and backflip.
+
+Next we compare the accuracy of discriminators trained using different methods. Figure 12 illustrates accuracy of the discriminators over the course of training. Discriminators trained via GAIL quickly overpowers the policy, and learns to accurately differentiate between samples, even when instance noise is applied to the inputs. VAIL without the KL constraint slows the discriminator’s progress, but nonetheless reaches near perfect accuracy with a larger number of samples. Once the KL constraint is enforced, the information bottleneck constrains the performance of the discriminator, converging to approximately $8 0 \%$ accuracy. Figure 12 also visualizes the value of $\beta$ over the course of training for motion imitation tasks, along with the loss of the KL term in the objective. The dual gradient descent update effectively enforces the VDB constraint $I _ { c }$ .
+
+![](images/b6c41654bddeb136cfb4678e2198801a461d01e727e3636d2a6f1dc1cd9fd3ca.jpg)  
+Figure 12: Left: Accuracy of the discriminator trained using different methods for imitating the dance skill. Middle:. Value of the dual variable $\beta$ over the course of training. Right: KL loss over the course of training. The dual gradient descent update for $\beta$ effectively enforces the VDB constraint $I _ { c }$ .
+
+Video Imitation: In the video imitation tasks, we use a simplified 2D biped character in order to avoid issues that may arise due to depth ambiguity from monocular videos. The biped character has a total of 12 degrees-of-freedom, with similar state and action parameters as the humanoid. The video demonstrations are generated by rendering a reference motion into a sequence of video frames, which are then provided to the agent as a demonstration. The goal of the agent is to imitate the motion depicted in the video, without access to the original reference motion, and the reference motion is used only to evaluate performance.
+
+# D INVERSE REINFORCEMENT LEARNING
+
+# D.1 EXPERIMENTAL SETUP
+
+Environments We evaluate on two maze tasks, as illustrated in Figure 13. The C-maze is taken from Fu et al. (2017): in this maze, the agent starts at a random point within a small fixed distance of the mean start position. The agent has a continuous, 2D action space which allows it to accelerate in the $x$ or $y$ directions, and is able to observe its $x$ and $y$ position, but not its velocity. The ground truth reward is $r _ { t } = - d _ { t } - 1 0 ^ { - 3 } \| a _ { t } \| ^ { 2 }$ , where $d _ { t }$ is the agent’s distance to the goal, and $a _ { t }$ is its action (this action penalty is assumed to be zero in Figure 13). Episodes terminate after 100 steps; for evaluation, we report the undiscounted mean sum of rewards over each episode The S-maze is larger variant of the same environment with an extra wall between the agent and its goal. To make the S-maze easier to solve for the expert, we added further reward shaping to encourage the agent to pass between the gaps between walls. We also increased the maximum control forces relative to the C-maze to enable more rapid exploration. Environments will be released along with the rest of our VAIRL implementations.
+
+Hyperparameters Policy networks for all methods were two-layer ReLU MLPs with 32 hidden units per layer. Reward and discriminator networks were similar, but with 32-unit mean and standard deviation layers inserted before the final layer for VDB methods. To generate expert demonstrations, we trained a TRPO (Schulman et al., 2015) agent on the ground truth reward for the training environment for 200 iterations, and saved 107 trajectories from each of the policies corresponding to the five final iterations. TRPO used a batch size of 10,000, a step size of 0.01, and entropy bonus with a coefficient of 0.1 to increase diversity. After generating demonstrations, we trained the IRL and imitation methods on a training maze for 200 iterations; again, our policy optimizer was TRPO with the same hyperparameters used to generate demonstrations. Between each policy update, we did 100 discriminator updates using Adam with a learning rate of $5 \times 1 0 ^ { - 5 }$ and batch size of 32. For the C-maze our VAIRL runs used a target KL of $I _ { C } = 0 . 5$ , while for the more complex S-maze we use a tighter target of $I _ { C } = 0 . 0 5$ . For the test C-maze, we trained new policies against the recovered reward using TRPO with the hyperparameters described above; for the test S-maze, we modified these parameters to use a batch size of 50,000 and learning rate of 0.001 for 400 iterations.
+
+![](images/12bc128902971697948715d483d597b425c86c8befea48689d912e331c19ad9a.jpg)  
+Figure 13: Left: The C-maze used for training and its mirror version used for testing. Colour contours show the ground truth reward function that we use to train the expert and evaluate transfer quality, while the red and green dots show the initial and goal positions, respectively. Right: The analogous diagram for the S-maze.
+
+# D.2 RECOVERED REWARD FUNCTIONS
+
+Figure 14 and 15 show the reward functions recovered by each IRL baseline on the C-maze and S-maze, respectively, along with sample trajectories for policies trained to optimize those rewards. Notice that VAIRL tends to recover smoother reward functions that match the ground truth reward more closely than the baselines. Addition of a gradient penalty enhances this effect for both AIRL and VAIRL. This is especially true in S-maze, where combining a gradient penalty with a variational discriminator bottleneck leads to a smooth reward that gradually increases as the agent nears its goal position at the top of the maze.
+
+# E IMAGE GENERATION
+
+We provide further experiment on image generation and details of the experimental setup.
+
+# E.1 EXPERIMENTAL SETUP:
+
+We use the non-saturating objective of Goodfellow et al. (2014) for all models except WGANGP. Following (Lucic et al., 2017), we compute FID on samples of size $1 0 0 0 0 ^ { 2 }$ . We base our implementation on (Mescheder et al., 2018), where we do not use any batch normalization for both the generator and the discriminator. We use RMSprop (Hinton et al.) and a fixed learning rate for all experiments.
+
+For convolutional GAN, variational discriminative bottleneck is implemented as a 1x1 convolution on the final embedding space that outputs a Gaussian distribution over $Z$ parametrized with a mean and a diagonal covariance matrix. For all image experiments, we preserve the dimensionality of the latent space. All experiments use adaptive $\beta$ update with a dual stepsize of $\alpha _ { \beta } = 1 0 ^ { - 5 }$ . We will make our code public. Similarly to VGAN, instance noise Sønderby et al. (2016); Arjovsky & Bottou (2017) is added to the final embedding space of the discriminator right before applying the classifier. Instance noise can be interpreted as a non-adaptive VGAN without a information constraint.
+
+Architecture: For CIFAR-10, we use a resnet-based architecture adapted from (Mescheder et al., 2018) detailed in Tables 2, 3, and 4. For CelebA and CelebAHQ, we use the same architecture used in (Mescheder et al., 2018).
+
+![](images/7d3bc0d020f3fd296b0a917213f8ec69575561699ec2437043741ab56e1b2aef.jpg)  
+Figure 14: Visualizations of recovered reward functions transferred to the mirrored C-maze. Also shown are trajectories executed by policies trained to maximize the corresponding reward in the new environment.
+
+![](images/5caf8907963ae0367361848e6ba59de05031977dd75a83acfbebdb3e3a78005c.jpg)  
+Figure 15: Visualizations of recovered reward functions transferred to the mirrored S-maze, like Figure 14.
+
+Table 2: CIFAR-10 Generator   
+
+<table><tr><td rowspan=1 colspan=1>Layer</td><td rowspan=1 colspan=1>Output size</td><td rowspan=1 colspan=1>Filter</td></tr><tr><td rowspan=1 colspan=1>FCReshape</td><td rowspan=1 colspan=1>256·4·4256×4×4</td><td rowspan=1 colspan=1>256→256·4·4</td></tr><tr><td rowspan=1 colspan=1>Resnet-blockUpsample</td><td rowspan=1 colspan=1>128×4×4128×8×8</td><td rowspan=1 colspan=1>256→128</td></tr><tr><td rowspan=1 colspan=1>Resnet-blockUpsample</td><td rowspan=1 colspan=1>64×8×864 × 16 × 16</td><td rowspan=1 colspan=1>128→64</td></tr><tr><td rowspan=1 colspan=1>Resnet-blockUpsample</td><td rowspan=1 colspan=1>32 ×16×1632× 32×32</td><td rowspan=1 colspan=1>64→32</td></tr><tr><td rowspan=1 colspan=1>Resnet-blockConv2D</td><td rowspan=1 colspan=1>32×32×323× 32× 32</td><td rowspan=1 colspan=1>32→32</td></tr></table>
+
+Table 3: CIFAR-10 Discriminator   
+
+<table><tr><td rowspan=1 colspan=1>Layer</td><td rowspan=1 colspan=1>Output size</td><td rowspan=1 colspan=1>Filter</td></tr><tr><td rowspan=1 colspan=1>Conv2D</td><td rowspan=1 colspan=1>32 × 32 × 32</td><td rowspan=1 colspan=1>3→32</td></tr><tr><td rowspan=1 colspan=1>Resnet-blockAvgPool2D</td><td rowspan=1 colspan=1>64×32×3264 ×16 ×16</td><td rowspan=1 colspan=1>32→64</td></tr><tr><td rowspan=1 colspan=1>Resnet-blockAvgPool2D</td><td rowspan=1 colspan=1>128×16×16128×8×8</td><td rowspan=1 colspan=1>64→128</td></tr><tr><td rowspan=1 colspan=1>Resnet-blockAvgPool2D</td><td rowspan=1 colspan=1>256×8×8256×4×4</td><td rowspan=1 colspan=1>128→256</td></tr><tr><td rowspan=1 colspan=1>FC</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>256·4·4→1</td></tr></table>
+
+# E.2 RESULTS
+
+CIFAR-10: We compare our approach with recent stabilization techniques: WGAN-GP (Gulrajani et al., 2017b), instance noise (Sønderby et al., 2016; Arjovsky & Bottou, 2017), spectral normalization (Miyato et al., 2018), and gradient penalty (Mescheder et al., 2018). We train report the networks at $7 5 0 \mathrm { k }$ iterations. We use $I _ { c } = 0 . 1$ , and a coefficient of $w _ { G P } = 1 0$ for the gradient penalty, which is the same as the value used by the implementation from Mescheder et al. (2018). See Figure 16 for visual comparisons of randomly generated samples.
+
+CelebA: On the CelebA (Liu et al., 2015) dataset, we generate images of size $1 2 8 \times 1 2 8$ with $I _ { c } = 0 . 2$ . On this dataset we do not see a big improvement upon the other baselines. This is likely because the architecture has been effectively tuned for this task, reflected by the fact that even the vanilla GAN trains fine on this dataset. All GAN, GP, and VGAN-GP obtain a similar FID scores of 7.64, 7.76, 7.25 respectively. See Figure 17 for more qualitative results with our approach.
+
+CelebAHQ: VGAN can also be trained on on CelebAHQ Karras et al. (2018) at 1024 by 1024 resolution directly, without progressive growing (Karras et al., 2018). We use $I _ { c } = 0 . 1$ and train with VGAN-GP. We train on a single Tesla V100, which fits a batch size of 8 in our experiments. Previous approaches (Karras et al., 2018; Mescheder et al., 2018) use a larger batch size and train over multiple GPUs. The model was trained for $3 0 0 \mathrm { k }$ iterations.
+
+Table 4: CIFAR-10 Discriminator with VDB   
+
+<table><tr><td rowspan=1 colspan=1>Layer</td><td rowspan=1 colspan=1>Output size</td><td rowspan=1 colspan=1>Filter</td></tr><tr><td rowspan=1 colspan=1>Conv2D</td><td rowspan=1 colspan=1>32 ×32×32</td><td rowspan=1 colspan=1>3→32</td></tr><tr><td rowspan=1 colspan=1>Resnet-blockAvgPool2D</td><td rowspan=1 colspan=1>64×32×3264×16×16</td><td rowspan=1 colspan=1>32→64</td></tr><tr><td rowspan=1 colspan=1>Resnet-blockAvgPool2D</td><td rowspan=1 colspan=1>128×16×16128×8×8</td><td rowspan=1 colspan=1>64→128</td></tr><tr><td rowspan=1 colspan=1>Resnet-blockAvgPool2D</td><td rowspan=1 colspan=1>256×8×8256×4×4</td><td rowspan=1 colspan=1>128→256</td></tr><tr><td rowspan=1 colspan=1>1×1Conv2D</td><td rowspan=1 colspan=1>2:256×4×4</td><td rowspan=1 colspan=1>256→2:256</td></tr><tr><td rowspan=1 colspan=1>Sampling</td><td rowspan=1 colspan=1>256×4×4</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>FC</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>256·4·4→1</td></tr></table>
+
+![](images/093328ea8485a78bfbb984c256407e8d540314cb0606b17307debb5f2f4316cc.jpg)  
+Figure 16: Random results on CIFAR-10 (Krizhevsky et al.): GAN (Goodfellow et al., 2014) FID: 63.6, instance noise (Sønderby et al., 2016; Arjovsky & Bottou, 2017) FID: 30.7, spectral normalization (SN) (Miyato et al., 2018) FID: 23.9, gradient penalty (GP) (Mescheder et al., 2018) FID: 22.6, WGAN-GP Gulrajani et al. (2017b) FID: 19.9, and the proposed VGAN-GP FID: 18.1. The samples produced by VGAN-GP (right) look the most realistic where objects like vehicles may be discerned.
+
+![](images/0f4c1610d1c621991c4b65e2d319bf2da62a7f55619f72224c958e5aeb05677f.jpg)  
+Figure 17: Random VGAN samples on CelebA $1 2 8 \times 1 2 8$ at $3 0 0 \mathrm { k }$ iterations.
+
+![](images/82b55f54f72b88903dca2d47813f7ee3a93f22bbf756b31c3c5ed8a68f59a2c8.jpg)  
+Figure 18: VGAN samples on CelebA HQ (Karras et al., 2018) $1 0 2 4 \times 1 0 2 4$ resolution at $3 0 0 \mathrm { k }$ iterations. Models are trained from scratch at full resolution, without the progressive scheme proposed by Karras et al. (2017).

parse/train/P42rXLGZQ07/P42rXLGZQ07.md

@@ -0,0 +1,422 @@
+# DIRECT EVOLUTIONARY OPTIMIZATION OF VARIATIONAL AUTOENCODERS WITH BINARY LATENTS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Discrete latent variables are considered important to model the generation process of real world data, which has motivated research on Variational Autoencoders (VAEs) with discrete latents. However, standard VAE training is not possible in this case, which has motivated different strategies to manipulate discrete distributions in order to train discrete VAEs similarly to conventional ones. Here we ask if it is also possible to keep the discrete nature of the latents fully intact by applying a direct discrete optimization for the encoding model. The studied approach is consequently strongly diverting from standard VAE training by altogether sidestepping absolute standard VAE mechanisms such as sampling approximation, reparameterization trick and amortization. Discrete optimization is realized in a variational setting using truncated posteriors in conjunction with evolutionary algorithms (using a recently suggested approach). For VAEs with binary latents, we first show how such a discrete variational method (A) ties into gradient ascent for network weights and (B) uses the decoder network to select latent states for training. More conventional amortized training is, as may be expected, more efficient than direct discrete optimization, and applicable to large neural networks. However, we here find direct optimization to be efficiently scalable to hundreds of latent variables using smaller networks. More importantly, we find the effectiveness of direct optimization to be highly competitive in ‘zero-shot’ learning (where high effectiveness for small networks is required). In contrast to large supervised neural networks, the here investigated VAEs can, e.g., denoise a single image without previous training on clean data and/or training on large image datasets. More generally, the studied approach shows that training of VAEs is indeed possible without sampling-based approximation and reparameterization, which may be interesting for the analysis of VAE-training in general. In the regime of few data, direct optimization, furthermore, makes VAEs competitive for denoising where they have previously been outperformed by non-generative approaches.
+
+# 1 INTRODUCTION AND RELATED WORK
+
+Variational autoencoders (Kingma & Welling, 2014; Rezende et al., 2014) are prominent and very actively researched models for unsupervised learning. VAEs, in their many different variations, have successfully been applied to a large number of tasks including semi-supervised learning (e.g. Maaløe et al., 2016), anomaly detection (e.g. An & Cho, 2015; Kiran et al., 2018), sentence interpolation (Bowman et al., 2016), music interpolation (Roberts et al., 2018) and drug response prediction (Rampasek et al., 2017). The success of VAEs rests on a series of methods that enable the derivation of scalable training algorithms to optimize their model parameters (discussed further below). A desired feature when applying VAEs to a given problem is that their latent variables (i.e., the encoder output variables) correspond to meaningful properties of the data, ideally to those latent causes that have originally generated the data. However, many real-world datasets suggest the use of discrete latents as they often describe the data generation process more naturally. For instance, the presence or absence of objects in images is best described by binary latents (e.g. Jojic & Frey, 2001). Discrete latents are also a popular choice in modeling sounds; for instance, describing piano sounds may naturally involve binary latents: keys are pressed or not (e.g. Titsias & Lazaro-Gredilla, ´ 2011; Goodfellow et al., 2013; Sheikh et al., 2014). The success of standard forms of VAEs has
+
+consequently spurred research on novel formulations that feature discrete latents (e.g. Rolfe, 2016;   
+Khoshaman & Amin, 2018; Roy et al., 2018; Sadeghi et al., 2019; Vahdat et al., 2019).
+
+The objective of VAE training is the optimization of a generative data model which parameterizes a given data distribution. Typically we seek model parameters $\Theta$ of a VAE that maximize the data log-likelihood, $\begin{array} { r } { \mathrm { L } ( \Theta ) = \sum _ { n } \log \left( p _ { \Theta } ( \vec { x } ^ { ( n ) } ) \right) } \end{array}$ , where we denote by $\vec { x } ^ { ( 1 : N ) }$ a set of $N$ observed data points, and where $p _ { \Theta } ( \vec { x } )$ denotes the modeled data distribution. Like conventional autoencoders (e.g., Bengio et al., 2007), VAEs use a deep neural network (DNN) to generate (or decode) observables $\vec { x }$ from a latent code $\vec { z }$ . Unlike conventional autoencoders, however, the generation of data $\vec { x }$ is not deterministic but it takes the form of a probabilistic generative model.
+
+For VAEs with binary latent variables, as they will be of interest here, we consider the following VAE generative model:
+
+$$
+\begin{array} { r } { p _ { \Theta } ( \vec { z } ) = \mathrm { B e r n } ( \vec { z } ; \vec { \pi } ) = \prod _ { h } \big ( \pi _ { h } ^ { z _ { h } } ( 1 - \pi _ { h } ) ^ { ( 1 - z _ { h } ) } \big ) , \qquad p _ { \Theta } ( \vec { x } \mid \vec { z } ) = \mathcal { N } \big ( \vec { x } ; \vec { \mu } ( \vec { z } ; W ) , \sigma ^ { 2 } \mathbb { I } \big ) , } \end{array}
+$$
+
+where $\vec { z } \in \{ 0 , 1 \} ^ { H }$ is a binary code and the non-linear function $\vec { \mu } ( \vec { z } ; W )$ is a DNN that outputs the mean of the Gaussian distribution. $p _ { \Theta } ( \vec { x } \vert \vec { z } )$ is commonly referred to as decoder. The set of model parameters is $\Theta = \{ \vec { \pi } , W , \sigma ^ { 2 } \}$ , where $W$ incorporates DNN weights and biases. We assume homoscedasticity of the Gaussian distribution, but note that there is no obstacle to generalizing the model by inserting a DNN non-linearity that outputs a correlation matrix. Similarly, the algorithm could easily be generalized to different noise distributions should the task at hand call for it. For the purpose of this work, however, we will focus on as elementary as possible VAEs, with the form shown in Eqn. (1).
+
+Given standard or binary-latent VAEs, essentially all learning algorithms seek to approximately maximize the log-likelihood using the following series of methods (we elaborate in the appendix):
+
+(A) Instead of the log-likelihood, a variational lower-bound (a.k.a. ELBO) is optimized.   
+(B) VAE posteriors are approximated by an encoding model, that is a specific distribution (often Gaussian) parameterized by one or more DNNs.   
+(C) The variational parameters of the encoder are optimized using gradient ascent on the lower bound, where the gradient is evaluated based on sampling and reparameterization trick to obtain sufficiently low-variance and yet efficiently computable estimates.   
+(D) Using samples from the encoder, the parameters of the decoder are optimized using gradient ascent on the variational lower bound.
+
+Optimization procedures for VAEs with discrete latents follow the same steps (Points A to D). However, discrete or binary latents pose substantial further obstacles in learning, mainly due to the fact that backpropagation through discrete variables is generally not possible (Rolfe, 2016; Bengio et al., 2013). In order to maintain the general VAE framework for encoder optimization, different groups have therefore suggested different possible solutions: work by Rolfe (2016), for instance, extends VAEs with discrete latents by auxiliary continuous latents such that gradients can still be computed. Work on the concrete distribution (Maddison et al., 2016) or Gumbel-softmax distribution (Jang et al., 2016) proposes newly defined continuous distributions that contain discrete distributions as limit cases. Work by Lorberbom et al. (2019) merges the Gumbel-Max reparameterization with the use of direct loss minimization for gradient estimation, enabling efficient training on structured latent spaces. Finally, work by van den Oord et al. (2017), and Roy et al. (2018) combines VAEs with a vector quantization (VQ) stage in the latent layer. Latents become discrete through quantization but gradients for learning are adapted from latent values before they are processed by the VQ stage. All methods have the goal of treating discrete distributions such that standard VAE training as developed for continuous latents can still be applied. These techniques interact during training with the standard methods (Points A-D) already in place for VAE optimization. Furthermore, they add further types of design decisions and hyper-parameters, for example parameters for annealing from softened discrete distributions to the (hard) original distributions for discrete latents.
+
+For discrete VAEs, it may consequently be a desirable goal to investigate alternative, more direct optimization procedures that do not require a softening of discrete distributions or the use of other indirect solutions. Such a direct approach is challenging, however, because once DNNs are used to define the encoding model (Point B) standard tricks to estimate gradients (Point C) seem unavoidable. A direct optimization procedure, as is investigated here, consequently has to substantially change VAE training. For the data model (1) we will maintain the variational setting and a decoding model with DNNs as non-linearity (Points A and D). However, we will not use an encoder model parameterized by DNNs (Point B). Instead, the variational bound will be increased w.r.t. the encoder model by using a discrete optimization approach. The procedure does not require gradients to be computed for the encoder such that discrete latents are addressed without the use of reparameterization trick and sampling approximations.
+
+# 2 TRUNCATED VARIATIONAL OPTIMIZATION
+
+Let us consider the variational lower bound of the likelihood. If we denote by $q _ { \Phi } ^ { ( n ) } ( { \vec { z } } )$ the variational distributions with parameters $\boldsymbol { \Phi } ^ { ( n ) }$ , and by $\begin{array} { r } { \left. h ( \vec { z } ) \right. _ { q _ { \Phi } ^ { ( n ) } } = \sum _ { \vec { z } } q _ { \Phi } ^ { ( n ) } ( \vec { z } ) h ( \vec { z } ) } \end{array}$ expectation values w.r.t. to $q _ { \Phi } ^ { ( n ) } ( { \vec { z } } )$ , then the lower bound can be written as:
+
+$$
+\begin{array} { r } { \mathcal { F } ( \Phi , \Theta ) = \sum _ { n } \big \langle \log \big ( p _ { \Theta } ( \vec { x } ^ { ( n ) } \vert \vec { z } ) p _ { \Theta } ( \vec { z } ) \big ) \big \rangle _ { q _ { \Phi } ^ { ( n ) } } - \sum _ { n } \big \langle \log \big ( q _ { \Phi } ^ { ( n ) } ( \vec { z } ) \big ) \big \rangle _ { q _ { \Phi } ^ { ( n ) } } , } \end{array}
+$$
+
+The general challenge for the maximization of $\mathcal { F } ( \Phi , \Theta )$ is the optimization of the encoding model $q _ { \Phi } ^ { ( n ) }$ . VAEs with discrete latents add to this challenge the problem of taking gradients w.r.t. discrete latents. If we seek to avoid derivatives w.r.t. discrete variables, we have to define an alternative encoding model $q _ { \Phi } ^ { ( n ) }$ but such an encoding has to remain sufficiently efficient. Considering prior work on generative models with discrete latents, variational distributions based on truncated posteriors offer themselves as such an alternative (Lucke & Sahani, 2008). Truncated posterior approxima- ¨ tions have been shown to be functionally competitive (e.g. Sheikh et al., 2014; Hughes & Sudderth, 2016; Shelton et al., 2017), and they are able to efficiently train also very large scale models with hundreds or thousands of latents (e.g. Shelton et al., 2011; Sheikh & Lucke, 2016; Forster & L ¨ ucke, ¨ 2018). However, the important question for training discrete VAEs is if or how truncated variational distributions can be used in gradient-based optimization of neural network parameters. We here, for the first time, address this question noting that all previous approaches relied on closed-form (or pseudo-closed form) parameter update equations in an expectation-maximization learning paradigm.
+
+Optimization of the Decoding Model. In order to optimize the parameters $W$ of the decoder DNN $\vec { \mu } ( \vec { z } , W )$ , the gradient of the variational bound (2) w.r.t. $W$ has to be computed. We consequently need, for any VAE, a sufficiently precise and efficient approximation of the expectation value w.r.t. the encoder $q _ { \Phi } ^ { ( n ) } ( \vec { z } )$ . Gradient estimation is of central importance for deep unsupervised learning, and approaches, e.g., for variance reduction of estimators have played an important role and are dedicated solely to this purpose (e.g., Williams, 1992). Reparameterization finally emerged as a key method because it allowed for sufficiently low-variance estimation of gradients based, e.g., on Gaussian middle-layer units (Kingma & Welling, 2014; Rezende et al., 2014).
+
+For discrete VAEs, however, reparameterization requires the introduction of additional manipulations of discrete distributions that we here seek to fully avoid.
+
+Instead of using reparameterization or variance reduction, we will compute gradients based on truncated posterior as variational distributions. A truncated posterior has the following form:
+
+$$
+\begin{array} { r l } { q _ { \Phi } ^ { ( n ) } ( \vec { z } ) : = } & { \frac { p _ { \Theta } ( \vec { z } | \vec { x } ^ { ( n ) } ) } { \sum _ { \vec { z } ^ { \prime } \in \Phi ^ { ( n ) } } p _ { \Theta } \left( \vec { z } ^ { \prime } | \vec { x } ^ { ( n ) } \right) } = \frac { p _ { \Theta } ( \vec { x } ^ { ( n ) } | \vec { z } ) p _ { \Theta } ( \vec { z } ) } { \sum _ { \vec { z } ^ { \prime } \in \Phi ^ { ( n ) } } p _ { \Theta } ( \vec { x } ^ { ( n ) } | \vec { z } ^ { \prime } ) p _ { \Theta } \left( \vec { z } ^ { \prime } \right) } \mathrm { i f } \vec { z } \in \Phi ^ { ( n ) } , } \end{array}
+$$
+
+where for all $\vec { z } \not \in \boldsymbol { \Phi } ^ { ( n ) }$ the probability $q _ { \Phi } ^ { ( n ) } ( \vec { z } )$ equals zero. That is, a variational distribution q(n)Φ (\~z) is proportional to the true posteriors in a subset $\boldsymbol { \Phi } ^ { ( n ) }$ , which acts as its variational parameter.
+
+We can now compute the gradient of (2) w.r.t. the decoder weights $W$ which results in:
+
+$$
+\vec { \nabla } _ { W } \mathcal { F } ( \Phi , \Theta ) = - \frac { 1 } { 2 \sigma ^ { 2 } } \sum _ { n } \sum _ { \vec { z } \in \Phi ^ { ( n ) } } q _ { \Phi } ^ { ( n ) } ( \vec { z } ) ~ \vec { \nabla } _ { W } \| \vec { x } ^ { ( n ) } - \vec { \mu } ( \vec { z } , W ) \| ^ { 2 } .
+$$
+
+The right-hand-side has salient similarities to the standard gradient ascent for VAE decoders. Especially the familiar gradient of the mean squared error (MSE) shows that, e.g., standard automatic differentiation tools can be applied. However, the decisive difference are the weighting factors $q _ { \Phi } ^ { ( n ) } ( \vec { z } )$ . Considering (3), in order to compute the weighting factors we require all $\vec { z } \in \boldsymbol { \Phi } ^ { ( n ) }$ to be passed through the decoder DNN. As all states of Φ(n) anyway have to be passed through the decoder for the MSE term of (4), the overall computational complexity is not higher than an estimation of the gradient with samples instead of states in $\boldsymbol { \Phi } ^ { ( n ) }$ (we elaborate in Appendix A).
+
+To complete the decoder optimization, update equations for variance $\sigma ^ { 2 }$ and prior parameters $\vec { \pi }$ can be computed in closed-form (compare, e.g., Shelton et al., 2011) and are given by:
+
+$$
+\begin{array} { r } { \sigma ^ { 2 , \mathrm { n e w } } = \frac { 1 } { D N } \displaystyle \sum _ { n } \displaystyle \sum _ { \vec { z } \in \Phi ^ { ( n ) } } q _ { \Phi } ^ { ( n ) } ( \vec { z } ) \| \vec { x } ^ { ( n ) } - \vec { \mu } ( \vec { z } , W ) \| ^ { 2 } , \qquad \vec { \pi } ^ { \mathrm { n e w } } = \frac { 1 } { N } \displaystyle \sum _ { n } \displaystyle \sum _ { \vec { z } \in \Phi ^ { ( n ) } } q _ { \Phi } ^ { ( n ) } ( \vec { z } ) \vec { z } , } \end{array}
+$$
+
+where $N$ is the number of samples in the training dataset and $D$ is the number of observables.
+
+Optimization of the Encoding Model. After having established that the decoder can be optimized efficiently and by using standard DNN methods, the important question is if the encoder can be trained efficiently. Encoder optimization is usually based on a reformulation of the variational bound (2) given by:
+
+$$
+\begin{array} { r } { \mathcal { F } ( \Phi , \Theta ) = \sum _ { n } \big \langle \log \big ( p _ { \Theta } ( \vec { x } ^ { ( n ) } \mid \vec { z } ) \big ) \big \rangle _ { q _ { \Phi } ^ { ( n ) } } - \sum _ { n } D _ { \mathrm { K L } } \big ( q _ { \Phi } ^ { ( n ) } ( \vec { z } ) , p _ { \Theta } ( \vec { z } ) \big ) . } \end{array}
+$$
+
+Centrally for this work, truncated posteriors allow a specific alternative reformulation of the bound that enables efficient optimization. The reformulation recombines the entropy term of the original form (2) with the first expectation value into a single term, and is given by (see Lucke, 2019, for ¨ details):
+
+$$
+\mathcal { F } ( \Phi , \Theta ) = \sum _ { n } \log \big ( \sum _ { \vec { z } \in \Phi ^ { ( n ) } } p _ { \Theta } ( \vec { x } ^ { ( n ) } | \vec { z } ) p _ { \Theta } ( \vec { z } ) \big ) .
+$$
+
+Thanks to the simplified form of the bound, the variational parameters $\boldsymbol { \Phi } ^ { ( n ) }$ of the encoding model can now be sought using direct discrete optimization procedures. More concretely, because of the specific form (7), pairwise comparisons of joint probabilities are sufficient to maximize the lower bound: if we update the set $\Phi ^ { ( { \bar { n } } ) }$ for a given $\vec { x } ^ { ( n ) }$ by replacing a state $\vec { z } ^ { \mathrm { o l d } } \in \boldsymbol { \Phi } ^ { ( n ) }$ with a state $\vec { z } ^ { \mathrm { n e w } } \not \in \boldsymbol { \Phi } ^ { ( n ) }$ , then $\mathcal { F } ( \Phi , \Theta )$ increases if and only if:
+
+$$
+\log \left( p _ { \Theta } ( \vec { x } ^ { ( n ) } , \vec { z } ^ { \mathrm { n e w } } ) \right) > \log \left( p _ { \Theta } ( \vec { x } ^ { ( n ) } , \vec { z } ^ { \mathrm { o l d } } ) \right) .
+$$
+
+To obtain intuition for the pairwise comparison, consider its form when inserting the binary VAE (1) into the left- and right-hand sides. Eliding terms that do not depend on $\vec { z }$ we obtain:
+
+$$
+\begin{array} { r } { \widetilde { \log p } _ { \Theta } ( \vec { x } , \vec { z } ) = - \| \vec { x } - \vec { \mu } ( \vec { z } , W ) \| ^ { 2 } - 2 \sigma ^ { 2 } \sum _ { h } \widetilde { \pi } _ { h } z _ { h } } \end{array}
+$$
+
+where $\tilde { \pi } _ { h } = \log \left( ( 1 - \pi _ { h } ) / \pi _ { h } \right)$ . The expression assumes an even more familiar form if we restrict ourselves for a moment to sparse priors $\pi < { \frac { 1 } { 2 } }$ , i.e., $\tilde { \pi } > 0$ . Criterion (8) then becomes:
+
+$$
+\begin{array} { r } { \| \vec { x } ^ { ( n ) } - \vec { \mu } ( \vec { z } ^ { \mathrm { n e w } } , W ) \| ^ { 2 } + 2 \sigma ^ { 2 } \tilde { \pi } | \vec { z } ^ { \mathrm { n e w } } | ~ < ~ \| \vec { x } ^ { ( n ) } - \vec { \mu } ( \vec { z } ^ { \mathrm { o l d } } , W ) \| ^ { 2 } + 2 \sigma ^ { 2 } \tilde { \pi } | \vec { z } ^ { \mathrm { o l d } } | } \end{array}
+$$
+
+where $\begin{array} { r } { | \vec { z } | = \sum _ { h = 1 } ^ { H } z _ { h } } \end{array}$ . Such functions are routinely encountered in sparse coding or compressive sensing (Eldar & Kutyniok, 2012): for each set $\boldsymbol { \Phi } ^ { ( n ) }$ we seek those states $\vec { z }$ that are reconstructing $\vec { x } ^ { ( n ) }$ well while being sparse $\vec { z }$ with few non-zero bits). For VAEs, $\vec { \mu } ( \vec { z } ^ { \mathrm { n e w } } , W )$ is a DNN and as such much more flexible in matching the distribution of observables $\vec { x }$ than can be expected from linear mappings. Furthermore, criteria like (10) usually emerge for maximum a-posteriori (MAP) training in sparse coding (Olshausen & Field, 1996). In contrast, we here seek a population of states $\vec { z }$ in $\Phi ^ { ( n ) }$ for each data point. It is a consequence of the reformulated lower bound (7) that it remains optimal to evaluate joint probabilities (as for MAP) although the constructed population of states $\boldsymbol { \Phi } ^ { ( n ) }$ can capture (unlike MAP training) a rich posterior structure.
+
+But how can new states ${ \vec { z } } ^ { \mathrm { n e w } }$ that optimize $\boldsymbol { \Phi } ^ { ( n ) }$ be found efficiently in high-dimensional latent spaces? Random search and search by sampling has recently been explored for elementary generative models (Lucke et al., 2018). Here we will follow another recent suggestion (Guiraud et al., ¨ 2018) and make use of a search based on evolutionary algorithms (EAs). In this setting we interpret sets $\boldsymbol { \Phi } ^ { ( n ) }$ as populations of binary genomes $\vec { z }$ and base the fitness function on Eqn. (9).
+
+Concretely, using $\boldsymbol { \Phi } ^ { ( n ) }$ as initial parent pool, we apply the following genetic operators in sequence: firstly, parent selection picks $N _ { p }$ states from the parent pool. In our numerical experiments we used
+
+A.Parent selectionB.MutationC.New population
+
+![](images/9ac5796e15c2abf03775e735e88196f8ed4ac0a48e8267479695a7378d10fa67.jpg)  
+Figure 1: The optimization process of the variational parameters $\Phi ^ { ( n ) }$ using evolutionary search. A. Some states are selected as parents. B. Each child undergoes mutation. C. Children are merged with the original population and the least fit are discarded.
+
+fitness-proportional parent selection, for which we add an offset (constant w.r.t. $\vec { z }$ ) to the fitness values in order to make them strictly non-negative. Each of the children undergoes mutation: one or more bits are flipped to further increase offspring diversity. In our experiments we perform random uniform selection of the bits to flip. Crossover could also be employed to increase offspring diversity. We repeat the procedure using the children generated this way as the parent pool, giving birth to multiple generations of candidate states. Finally, we update $\boldsymbol { \Phi } ^ { ( n ) }$ by substituting individuals with low fitness with candidates with higher fitness. The whole procedure can be seen as an evolutionary algorithm with perfect memory or very strong elitism (individuals with higher fitness never drop out of the gene pool). Note that the improvement of the variational lower bound depends on generating as many as possible different children with high fitness over the course of training.
+
+We point out that the EAs optimize each $\boldsymbol { \Phi } ^ { ( n ) }$ independently, so this technique can be applied to large datasets in conjunction with stochastic or batch gradient descent on the model parameters $\Theta$ : it does not require to keep the full dataset (or all sets $\boldsymbol { \Phi } ^ { ( n ) }$ ) in memory at a given time. Fig 1 shows how EAs produce new states that are used to update each set $\boldsymbol { \Phi } ^ { ( n ) }$ . The full training procedure for binary VAEs is summarized in Algorithm 1.
+
+# Algorithm 1 Training Truncated Variational Autoencoders
+
+Initialize model parameters $\Theta = \{ W , \vec { \pi } , \sigma ^ { 2 } \}$   
+Initialize each Φ(n) with $S$ distinct latent states   
+repeat for all batches in dataset do for sample $_ n$ in batch do $\Phi ^ { n e w } = \Phi ^ { ^ { ( n ) } }$ for all generations do Φnew mutation (crossover (selection $( \Phi ^ { n e w } ) )$ ) $\boldsymbol { \Phi } ^ { ( n ) } = \boldsymbol { \Phi } ^ { ( n ) } \cup \boldsymbol { \Phi } ^ { n e w }$ end for Truncate $\boldsymbol { \Phi } ^ { ( n ) }$ to $S$ fittest elements based on (9) end for Use Adam to update $W$ using objective (4) end for Use (5) to update $\vec { \pi }$ , $\sigma ^ { 2 }$   
+until parameters $\Theta$ have sufficiently converged
+
+![](images/2bc6269339c0e4affc63a21c98617219d8cf4f53bbc0d2971dd59f7f4f440ed2.jpg)  
+Figure 2: Graphical representation of the model architecture used in numerical experiments.
+
+# 3 NUMERICAL EXPERIMENTS
+
+Having defined the training procedure, we numerically investigated its properties. After first verifying that the procedure can recover generating parameters using ground-truth data (see Appendix B), we conducted experiments to address the following two standard questions:
+
+(1) How efficient, i.e. how scalable, is the direct discrete optimization of binary VAEs? (2) How effective is the procedure, i.e., how well does it perform for a given VAE model?
+
+In all numerical experiments, the training of the DNN parameters based on (4) is performed with mini-batches, the Adam optimizer (Kingma & Ba, 2014) and decaying or cyclical learning rate scheduling (Smith, 2017). Xavier/Glorot initialization (Glorot $\&$ Bengio, 2010) is used for the DNN weights, while biases are always zero-initialized. Parameters $\vec { \pi }$ and $\sigma ^ { 2 }$ are updated via Eqn. (5) and initialized to $\textstyle { \frac { 1 } { H } }$ $H$ is the size of $\vec { \pi }$ ) and 0.01 respectively. Hyper-parameter optimization was conducted manually and, for the more complex datasets, it also made use of black box Bayesian optimization based on Gaussian Processes (Nogueira, 2019). We will refer to the binary VAE trained with the method described above as Truncated Variational Autoencoder (TVAE) as the use of truncated posteriors is the main distinguishing feature.
+
+<table><tr><td></td><td>g=15</td><td>g=25</td><td>g=50</td></tr><tr><td>MTMKL</td><td>34.29</td><td>31.88</td><td>28.08</td></tr><tr><td>GSC</td><td>32.68</td><td>31.10</td><td>28.02</td></tr><tr><td>VAR-BSC</td><td>32.25</td><td>31.15</td><td>28.62</td></tr><tr><td>TVAE</td><td>34.27 ± .02</td><td>32.65 ± .06</td><td>29.61± .02</td></tr></table>
+
+Scalability and improvement on linear models. Let us first numerically investigate scalability properties of TVAE especially in comparison with linear models. After verifying parameter recovery for ground-truth data (see Appendix B), we used natural data in the form of image patches as an intermediately large scale and natural dataset. Concretely, we used 100,000 whitened image patches of $1 6 \times 1 6$ pixels extracted from a standard image database (van Hateren & van der Schaaf, 1998) and pre-processed as in Guiraud et al. (2018).
+
+![](images/a89cb8777ccb03c1af036831a57ce347819b931b4fcc37a3ff12df3f99e95270.jpg)  
+Table 1: Denoising performance in PSNR (dB) for the ‘house’ image under controlled conditions $\scriptstyle { \mathcal { D } } = 8 \times 8$ , $H { = } 6 4$ for all algorithms).
+
+The simplest possible VAEs would use linear mappings for the decoder $\vec { \mu } ( \vec { z } , W )$ . For standard Gaussian latents, a linear VAE can recover probabilistic PCA solutions (e.g. Lucas et al., 2019). For Bernoulli latents, we would recover binary sparse coding (Haft et al., 2004; Shelton et al., 2011) solutions. We therefore start training (using $H = 3 0 0$ latents) with a linear VAE. After 100 epochs the weights of the linear mapping were used to initialize the bottom layer of a deeper decoder network with three layers of 300, 300 and $1 6 \times 1 6 = 2 5 6$ units. The weights of the deeper layers were simply initialized to the identity matrix. Furthermore, prior and variance were optimized. The described setup guarantees a common starting point for linear and non-linear VAEs such that the difference provided by deeper decoder DNNs can be highlighted. Fig. 3 shows the variational bounds during learning of the linear VAE compared to the non-linear VAE for a typical experiment. The non-linear VAE can be observed to quickly and significantly optimize the lower bound beyond a linear VAE. We will later (when we are not interested in comparisons to linear VAEs) simply optimize the weights of non-linear TVAE directly as we did not observed an advantage by first optimizing a linear VAE.
+
+![](images/be5972d41617605cdac7f1e82da43139d1803097042d0f2204b119824795367a.jpg)  
+Figure 3: ELBO gain of TVAE compared to linear VAE with binary latents (on $1 6 \times 1 6$ image patches).   
+Figure 4: TVAE denoising of house image with noise level $\sigma ~ = ~ 5 0$ . The denoised image has $\mathrm { P S N R } { = } 3 0 . 0 3$ , the best of the runs of Tab. 2.
+
+Compared to shallow linear models, we observed a similar efficiency and scalability of TVAE. The main additional computational costs are given by passing the latent states through the a full decoder DNN instead of just through a linear mapping. The sets of states used could be kept small, at size $S = | \boldsymbol \Phi ^ { ^ { ( n ) } } | = 6 \dot { 4 }$ , such that $N \times ( | \Phi ^ { ( n ) } | + | \bar { \Phi _ { \mathrm { n e w } } ^ { ( n ) } } | )$ states had to be evaluated for each epoch. This compares to $N \times M$ states that would be used for standard VAE training (given $M$ samples are drawn per data point). Differently to standard VAE training the $\boldsymbol { \Phi } ^ { ( n ) }$ have to be remembered across iterations. For very large datasets, the additional $\mathcal { O } ( N \times | \Phi ^ { ( n ) } | \times H )$ memory demand can be distributed over compute nodes, however. To further investigate scalability, we went to up to $H { = } 1 0 0 0$ latent variables (while using 100 units in the DNN middle layer). TVAE training time remained in line with the theoretical linear scaling with $H$ while the variational bound further increased.
+
+Effectiveness: Image Denoising. As we have observed, scaling to large latent spaces does not pose a problem for the presented approach. It is clear, however, that memory and computational cost
+
+Table 2: Denoising performance in PSNR (dB) for the ‘house’ image for different algorithms with different optimized hyper-parameters. The top category only requires the noisy image. The middle requires additional information such as noise level (KSVD, WNNM, BM3D) or additional noisy images with matched noise level $( \mathrm { n } 2 \mathrm { v } ^ { \dag } )$ . The bottom requires large clean datasets.
+
+<table><tr><td></td><td>g=15</td><td>g=25</td><td>g=50</td></tr><tr><td>N2v*</td><td>32.05</td><td>29.20</td><td>25.42</td></tr><tr><td>MTMKL</td><td>34.29</td><td>31.88</td><td>28.08</td></tr><tr><td>GSC</td><td>33.78</td><td>32.01</td><td>28.35</td></tr><tr><td>S5C</td><td>33.50</td><td>32.08</td><td>28.35</td></tr><tr><td>VAR-BSC</td><td>33.50</td><td>32.32</td><td>28.91</td></tr><tr><td>TVAE</td><td>34.27± .02</td><td>32.65±.06</td><td>29.98 ± .05</td></tr><tr><td>N2vt</td><td>33.91</td><td>32.10</td><td>28.94</td></tr><tr><td>KSVD</td><td>34.32</td><td>32.15</td><td>27.95</td></tr><tr><td>WNNM</td><td>35.13</td><td>33.22</td><td>30.33</td></tr><tr><td>BM3D</td><td>34.94</td><td>32.86</td><td>29.37</td></tr><tr><td>EPLL</td><td>34.17</td><td>32.17</td><td>29.12</td></tr><tr><td>BDGAN</td><td>34.57</td><td>33.28</td><td>30.61</td></tr><tr><td>DPDNN</td><td>35.40</td><td>33.54</td><td>31.04</td></tr></table>
+
+increase with the number of data points. Above, we processed 100, 000 data points which is still feasible for the small DNNs used. However, larger DNNs increase computational load significantly because $N \times ( | \Phi ^ { ( n ) } | + | \Phi _ { \mathrm { n e w } } ^ { ( n ) } | )$ latent states have to be passed through the decoder. Furthermore, larger DNNs require more data points to not overfit which further increases computational load of our $N$ -dependent method. In many applications, there is, however, anyway relatively few data available which makes the application of large DNNs prohibitive. One example is the task of ‘zeroshot’ denoising, i.e., denoising of an image when only the image itself is available. Learning without clean data recently became very popular. The task is currently addressed using approaches based on standard feed-forward DNNs whose training objectives have been altered to allow for training on noisy images (e.g. Lehtinen et al., 2018; Krull et al., 2019). Deep generative models are, on the other hand, more naturally suited for training on noisy data as their learning objective can be used directly. Shocher et al. (2018) also argue that smaller DNNs are sufficient for the ‘zero-shot’ setting. Because of its recent popularity and suitability for approaches with smaller DNNs, we consequently focus on ‘zero-shot’ denoising. As a very significant additional benefit, the task allows for directly comparing the VAE approach to a large range of other approaches that have recently been suggested. Most notably we can compare to non-deep generative models, large feed-forward DNNs (Zhu et al., 2019; Dong et al., 2019) and DNNs dedicated to learning from noisy data (Lehtinen et al., 2018; Krull et al., 2019).
+
+The one denoising benchmark that offers the broadest possible comparison to other methods is probably the ‘house’ image (Fig. 4 left). The standard benchmark settings for ‘house’ make use of additive Gaussian white noise with standard deviations $\sigma \in \{ 1 5 , 2 5 , 5 0 \}$ . First, consider the comparison in Tab. 1 where all models used the same patch size of $D = 8 \times 8$ pixels and $H = 6 4$ latent variables (Appendix B for details). Tab. 1 lists the different approaches in terms of the standard measure of peak signal-to-noise ratio (PSNR). Values for MTMKL (Titsias & Lazaro-Gredilla, ´ 2011), GSC (Sheikh et al., 2014) and S5C (Sheikh & Lucke, 2016) were taken from the respective ¨ original publications (which all established new state-of-the-art results when first published). As can be observed, TVAE significantly improves performance for high noise levels. TVAE is able to learn the best data representation for denoising and represents the state-of-the-art in this controlled setting (i.e., fixed $D$ and $H$ ). The decoder DNN of TVAE provides the decisive performance advantage: TVAE significantly improves performance compared to the linear Binary Sparse Coding (var-BSC, Henniges et al., 2010; Shelton et al., 2011), confirming that the high lower bounds of TVAE on natural images translate into improved performance on a concrete benchmark. For $\sigma = 2 5$ and $\sigma = 5 0$ , TVAE also significantly improves on MTMKL, GSC, and S5C. These three approaches are based on a spike-and-slab sparse coding model (also compare Goodfellow et al., 2012). Despite the less flexible Bernoulli prior, the decoder DNN of TVAE provides the highest PSNR values for high noise levels.
+
+In order to further extend our comparison, in the last experiment we considered the denoising task without controlling for equal conditions. Concretely, we allowed for any approach that performs denoising on the benchmark including approaches that are trained on large image datasets and/or use different patch sizes (including multi-scale and whole image processing). Note that different approaches may employ very different sets of hyper-parameters that can be optimized for denoising performance: for sparse coding approaches, hyper-parameters include patch and dictionary sizes; for DNN approaches they include all network and training scheme hyper-parameters. By allowing for comparison in this less controlled setting, we can include a number of recent approaches including large DNNs trained on clean data and training schemes specifically targeted to noisy training data. Tab. 2 shows the denoising performance for the three noise levels we investigated, with results for other algorithms taken from their corresponding original publications unless specified otherwise. For WNNM and EPLL we cite values from Zhang et al. (2017). The results reported for noise2void (n2v, Krull et al., 2019) were produced specifically for this work (see Appendix B).
+
+Note that the best performing approaches in Tab. 2 cannot be trained on noisy data: EPLL (Zoran & Weiss, 2011), BDGAN (Zhu et al., 2019) and DPDNN (Dong et al., 2019) all make use of clean training data (typically hundreds of thousands of data points or more). For denoising, EPLL also requires the ground-truth noise level of the test image. Ground-truth noise level information is also required by KSVD (Elad & Aharon, 2006) and WNNM (Gu et al., 2014). As noisy data is very frequently occurring, removing the requirement of clean data has been of considerable interest with, e.g., approaches like noise2noise (n2n, Lehtinen et al., 2018) and noise2void being very actively discussed currently. The n2n approach can achieve denoising performance on noisy training data which is almost as high as the performance of a given DNN when trained on clean data. It would thus outperform all approaches in Tab. 2 except for the bottom three. However, n2n requires different noise realizations of the very same underlying image. noise2void aims to remove this artificial assumption. Considering Tab. 2, PSNR values of TVAE were consistently higher than those of $\mathbf { n } 2 \mathbf { v }$ even if $\mathrm { n } 2 \mathrm { v }$ was trained on external data with matched-noise level $( { \bf n } 2 { \bf v } ^ { \dagger }$ in Tab. 2). Performance of TVAE is $0 . 2 \mathrm { d B }$ lower than BM3D for $\sigma = 2 5$ and $0 . 6 \mathrm { d B }$ higher for $\sigma = 5 0$ , which makes it, for large noise levels, the state-of-the-art on this benchmark in the ‘zero-shot’ setting (i.e., the setting n2n and $\mathrm { n } 2 \mathrm { v }$ aim to address).
+
+# 4 DISCUSSION
+
+We investigated a novel way to train VAEs with binary latents. In order to avoid derivatives w.r.t. stochastic discrete latents, we here changed the standard training setup substantially. Updates of the decoder DNN now involve a weighted sum over states (4) and the encoder DNN is replaced by a discrete evolutionary optimization. The direct optimization of the encoder replaces methods that are usually considered indispensable for the training of VAEs: sampling approximation and reparameterization trick. Furthermore, the here investigated encoding model does not use a joint mapping for all datapoints to latent space, i.e., the approach is not amortized. While amortization can be advantageous as information can be shared across datapoints, disadvantages in terms of less tight lower bounds have also been pointed out (e.g. Kim et al., 2018; Cremer et al., 2018). Related to this point, standard VAE training usually involves factored Gaussian approximations of VAE posteriors which can introduce biases (e.g., discussion by Vertes & Sahani, 2018). The investigation ´ of alternatives may therefore, more generally, shed further light on the consequences of specific approximation choices used to define VAE encoders.
+
+The price we pay for not using amortization is efficiency: we optimize variational parameters for each data point which is, of course, more costly. However, direct optimization can scale VAEs to large latent spaces if smaller DNNs are used. When the use of large DNNs is anyway prohibitive because of limited data, the here studied approach can play out its effectiveness. For the recently popular task of ‘zero-shot’ denoising, we observed state-of-the-art results in a domain where VAEs have not been reported to be competitive before. The competitive performance is presumably due to the approach not being subject to an amortization gap as well as not being based on factored variational distributions.
+
+Our conclusion is consequently that direct discrete optimization can serve as an alternative for training discrete VAEs. In a sense, the approach can be considered as a brute-force optimization which is slower than conventional amortized training but more effective for scales at which it can be applied. To our knowledge, the approach is also the first training method for VAEs that is not using samplingbased gradient estimates, and the first which makes VAEs competitive for ‘zero-shot’ denoising.
+
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+Leslie N Smith. Cyclical learning rates for training neural networks. In 2017 IEEE Winter Conference on Applications of Computer Vision (WACV), pp. 464–472. IEEE, 2017.
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+Arash Vahdat, Evgeny Andriyash, and William G Macready. Learning undirected posteriors by backpropagation through mcmc updates. arXiv preprint arXiv:1901.03440, 2019.
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+Kai Zhang, Wangmeng Zuo, Yunjin Chen, Deyu Meng, and Lei Zhang. Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising. IEEE Transactions on Image Processing, 26(7):3142–3155, 2017.
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+Kai Zhang, Wangmeng Zuo, and Lei Zhang. FFDNet: Toward a fast and flexible solution for CNN based image denoising. IEEE Transactions on Image Processing, 2018.
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+
+Daniel Zoran and Yair Weiss. From Learning Models of Natural Image Patches to Whole Image Restoration. In IEEE International Conference on Computer Vision, pp. 479–486, 2011.
+
+# A DETAILS OF ENCODER AND DECODER OPTIMIZATION
+
+![](images/2b7855043a370020906d659795f7c6fcd5c7b6e3f849b7fdfc411ca3e69fc1bf.jpg)  
+Figure 5: From left to right: generic VAE decoding model, continuous-latent VAE model with Gaussian noise and the binary-latent VAE model of Eqn. (1), in plate notation.
+
+See Fig. 5 for a graphical comparison between the decoding models of a vanilla VAE and the binary VAE considered here (1). Fig. 6 graphically illustrates different steps to optimize standard VAEs, and additional steps suggested by different contributions in order to optimize discrete VAEs.
+
+For the optimization of the binary VAE (1), consider the original form of the lower bound, Eqn. (2). When taking derivatives of $\mathcal { F } ( \Phi , \Theta )$ w.r.t. $\Theta$ we can ignore the entropy term1. For the binary VAE model of Eqn. (1) the gradient of the lower bound w.r.t. $W$ is then given by:
+
+$$
+\begin{array} { r c l } { { \displaystyle \vec { \nabla } _ { W } \mathcal { F } ( \Phi , \Theta ) } } & { { = } } & { { \displaystyle \sum _ { n } \vec { \nabla } _ { W } \big \langle \log \big ( p _ { \Theta } ( \vec { x } ^ { ( n ) } \mid \vec { z } ) p _ { \Theta } ( \vec { z } ) \big ) \big \rangle _ { q _ { \Phi } ^ { ( n ) } } } } \\ { { } } & { { = } } & { { \displaystyle \sum _ { n } \vec { \nabla } _ { W } \big \langle \log \big ( p _ { \Theta } ( \vec { x } ^ { ( n ) } \mid \vec { z } ) \big ) \big \rangle _ { q _ { \Phi } ^ { ( n ) } } = \displaystyle \sum _ { n } \vec { \nabla } _ { W } \big \langle \log \big ( \mathcal { N } ( \vec { x } ^ { ( n ) } ; \vec { \mu } ( \vec { z } , W ) , \sigma ^ { 2 } \mathbb { I } \big ) \big \rangle _ { q _ { \Phi } ^ { ( n ) } } } } \\ { { } } & { { = } } & { { \displaystyle - \frac { 1 } { 2 \sigma ^ { 2 } } \sum _ { n } \vec { \nabla } _ { W } \sum _ { \vec { z } \in \Phi ^ { ( n ) } } q _ { \Phi } ^ { ( n ) } ( \vec { z } ) \| \vec { x } ^ { ( n ) } - \vec { \mu } ( \vec { z } , W ) \| ^ { 2 } } } \\ { { } } & { { = } } & { { \displaystyle - \frac { 1 } { 2 \sigma ^ { 2 } } \sum _ { n } \sum _ { \vec { z } \in \Phi ^ { ( n ) } } q _ { \Phi } ^ { ( n ) } ( \vec { z } ) \vec { \nabla } _ { W } \| \vec { x } ^ { ( n ) } - \vec { \mu } ( \vec { z } , W ) \| ^ { 2 } , } } \end{array}
+$$
+
+where the weighting factors $q _ { \Phi } ^ { ( n ) } ( { \vec { z } } )$ are by using (3) and (1) given by:
+
+$$
+\begin{array} { r c l } { { q _ { \Phi } ^ { ( n ) } ( \vec { z } ) } } & { { = } } & { { \displaystyle \frac { p _ { \Theta } ( \vec { x } ^ { ( n ) } \mid \vec { z } ) p _ { \Theta } ( \vec { z } ) } { \sum _ { \vec { z } ^ { \prime } \in \Phi ^ { ( n ) } } p _ { \Theta } ( \vec { x } ^ { ( n ) } \mid \vec { z } ^ { \prime } ) p _ { \Theta } ( \vec { z } ^ { \prime } ) } } } \\ { { } } & { { = } } & { { \displaystyle \frac { \exp \big ( - \frac { 1 } { 2 \sigma ^ { 2 } } \| \vec { x } ^ { ( n ) } - \vec { \mu } ( \vec { z } , W ) \| ^ { 2 } - \tilde { \vec { \pi } } ^ { T } \vec { z } \big ) } { \sum _ { \vec { z } ^ { \prime } \in \Phi ^ { ( n ) } } \exp \big ( - \frac { 1 } { 2 \sigma ^ { 2 } } \| \vec { x } ^ { ( n ) } - \vec { \mu } ( \vec { z } ^ { \prime } , W ) \| ^ { 2 } - \tilde { \vec { \pi } } ^ { T } \vec { z } ^ { \prime } \big ) } } } \end{array}
+$$
+
+for all of the $\vec { z } \in \boldsymbol { \Phi } ^ { ( n ) }$ , rs ere , th $\begin{array} { r } { \tilde { \pi } _ { h } = \log \left( \frac { 1 - \pi _ { h } } { \pi _ { h } } \right) } \end{array}$ . Note that the ated as constan $q _ { \Phi } ^ { ( n ) } ( \vec { z } )$ are evaluated at theor the gradient w.r.t. rent values. $\Theta$ $W$
+
+It may be interesting to compare the gradient estimate (11) to the gradient estimate of conventional VAE training. For this consider a standard encoder given by an amortized variational distribution which we shall denote by $\tilde { q } _ { \Phi } ^ { ( n ) } ( \vec { z } )$ . The distribution $\tilde { q } _ { \Phi } ^ { ( n ) } ( \vec { z } )$ could be a Gaussian whose mean and variance are set by passing data point $\vec { x } ^ { ( n ) }$ through encoder DNNs. For discrete VAEs, $\tilde { q } _ { \Phi } ^ { ( n ) } ( \vec { z } )$ can be thought of as an analog discrete distribution. If we now take gradients of (6) w.r.t. $W$ and estimate using samples from $\tilde { q } _ { \Phi } ^ { ( n ) } ( \vec { z } )$ , we obtain the familiar form:
+
+$$
+\begin{array} { r c l } { \displaystyle \vec { \nabla } _ { W } \mathcal { F } ( \Phi , \Theta ) } & { = } & { \displaystyle \sum _ { n } \vec { \nabla } _ { W } \Big \langle \log \big ( p _ { \Theta } ( \vec { x } ^ { ( n ) } \mid \vec { z } ) p _ { \Theta } ( \vec { z } ) \big ) \Big \rangle _ { \vec { q } _ { \Phi } ^ { ( n ) } } } \\ & { = } & { \displaystyle \sum _ { n } \vec { \nabla } _ { W } \Big \langle \log \big ( \mathcal { N } ( \vec { x } ^ { ( n ) } ; \vec { \mu } ( \vec { z } , W ) , \sigma ^ { 2 } \mathbb { I } ) \big \rangle _ { \vec { q } _ { \Phi } ^ { ( n ) } } } \\ & { \approx } & { \displaystyle - \frac { 1 } { 2 \sigma ^ { 2 } } \sum _ { n } \frac { 1 } { M } \sum _ { m = 1 } ^ { M } \vec { \nabla } _ { W } \| \vec { x } ^ { ( n ) } - \vec { \mu } ( \vec { z } ^ { ( m ) } , W ) \| ^ { 2 } , \mathrm { ~ w h e r e ~ } \vec { z } ^ { ( m ) } \sim \tilde { q } _ { \Phi } ^ { ( n ) } ( \vec { z } ) } \end{array}
+$$
+
+We can slightly rewrite this expression to obtain:
+
+$$
+\vec { \nabla } _ { W } \mathcal { F } ( \Phi , \Theta ) ~ \approx ~ - \frac { 1 } { 2 \sigma ^ { 2 } } \sum _ { n } \sum _ { \vec { z } \sim \tilde { q } _ { \Phi } ^ { ( n ) } } \left( \frac { 1 } { M } \right) \vec { \nabla } _ { W } \| \vec { x } ^ { ( n ) } - \vec { \mu } ( \vec { z } , W ) \| ^ { 2 } ,
+$$
+
+If we now compare with the gradient using the truncated approximation $q _ { \Phi } ^ { ( n ) } ( { \vec { z } } )$
+
+$$
+\vec { \nabla } _ { W } \mathcal { F } ( \Phi , \Theta ) = - \frac { 1 } { 2 \sigma ^ { 2 } } \sum _ { n } \sum _ { \vec { z } \in \Phi ^ { ( n ) } } q _ { \Phi } ^ { ( n ) } ( \vec { z } ) \vec { \nabla } _ { W } \| \vec { x } ^ { ( n ) } - \vec { \mu } ( \vec { z } , W ) \| ^ { 2 } ,
+$$
+
+one can discuss analogous roles played by the subsets $\boldsymbol { \Phi } ^ { ( n ) }$ (the variational parameters of q(n)Φ (\~z)) and by a standard encoder $\tilde { q } _ { \Phi } ^ { ( n ) }$ . The states in a subset Φ(n) are used to estimate the gradient similar to the samples from a standard encoder $\tilde { q } _ { \Phi } ^ { ( n ) } ( \vec { z } )$ . The size of $\boldsymbol { \Phi } ^ { ( n ) }$ can consequently be thought of as analog to the number of samples used in a conventional estimation of the gradient. Standard VAE training estimates the gradient by weighting all samples equally (with $( 1 / M )$ ) and the gradient direction is approximated using sufficiently many samples drawn from the current $\tilde { q } _ { \Phi } ^ { ( n ) } ( \vec { z } )$ . In contrast, truncated gradient estimation uses the states in $\boldsymbol { \Phi } ^ { ( n ) }$ , and the gradient is computed using a weighted summation with weights $q _ { \Phi } ^ { ( n ) } ( \vec { z } )$ . These weights are computed by passing the states $\vec { z }$ through the decoder network. The gradient is then, notably, not a stochastic estimation but exact: gradient ascent is guaranteed (for small steps) to always monotonically increase the variational lower bound.
+
+Computational Complexity. To add to the discussion of computational complexity of TVAE compared to standard VAE training, consider again Eqns. 13 and 14. If as many samples $M$ are used, per data point, as there are states in each $\boldsymbol { \Phi } ^ { ( n ) }$ , then both sums have the same number of summands. The evaluation of the gradients of the mean square error (MSE) is consequently precisely the same for both approaches. The additional weighting factors $q _ { \Phi } ^ { ( n ) } ( \vec { z } )$ have to be computed for TVAE. However, the weighting factors just represent a small overhead because the evaluation of the decoder DNN for the states in $\boldsymbol { \Phi } ^ { ( n ) }$ is a computation that can be reused from the updates of $\boldsymbol { \Phi } ^ { ( n ) }$
+
+The main computational differences are in the updates of Φ(n) compared to the update of encoder DNNs for conventional VAEs. Once the parameters $\Theta = ( W , \sigma ^ { 2 } , { \bar { \vec { \pi } } } )$ are updated using (14), new states for $\boldsymbol { \Phi } ^ { ( n ) }$ have to be sought based on criterion (9). In practice and for each $n$ , we generate $M ^ { \prime }$ new states according to the applied evolutionary procedure. To select the best states we have to pass all these $M ^ { \prime }$ new states through the decoder DNN to evaluate (9). Furthermore, we have to pass all M states already in Φ(n) through the DNN to re-evaluate (9) because the parameters $\Theta$ have changed. In summary, we do require $\mathcal { O } ( N \times \left( M + M ^ { \prime } \right) )$ passes through the decoder DNN. Selecting the $M$ best states from the $( M + M ^ { \prime } )$ states does not add complexity as this can be done in $\mathcal { O } ( M + M ^ { \prime } )$ for each $n$ (Blum et al., 1973). The EA does add to the computational load but parent selection and mutation only add a constant offset for each of the considered states.
+
+For comparison with standard VAEs, if we use $M$ samples of an encoder $\tilde { q } _ { \Phi } ^ { ( n ) } ( \vec { z } )$ , we require $\mathcal { O } ( M \times$ ) passes through the decoder DNN to update the parameters according to (13). For the encoder update, one requires $N \times { \tilde { M } }$ passes through encoder and decoder DNN to estimate the gradient w.r.t. the encoder weights (if we draw $\tilde { M }$ samples for each data point from a conventional encoder distribution $\tilde { q } _ { \Phi } ^ { ( n ) } ( \vec { z } )$ . The additional overhead to actually draw the samples is usually negligible.
+
+Hence, the computational complexity of TVAE training is comparable if $M \approx M ^ { \prime } \approx \tilde { M }$ . However, conventional VAE training is amortized, i.e., the update of encoder weights uses information from all data points $n$ . In contrast, TVAE training is not amortized, i.e., the $\boldsymbol { \Phi } ^ { ( n ) }$ are updated per data point. The advantage of amortization is that in practice, weights of a conventional encoder can converge faster or (alternatively) less samples $\tilde { M }$ are required. Considering the observed runtimes, more efficient conventional VAE training can presumably in large parts attributed to faster convergence using amortization. Furthermore, the used number of samples $M$ for conventional VAE training is usually smaller than best working sizes of Φ(n) (we used, e.g., |Φ(n) | $| { \Phi } ^ { ^ { ( n ) } } | = 6 4$ and $| { \boldsymbol { \Phi } } ^ { ( n ) } | = 2 0 0$ for denoising, see Tab. 3); and the required storage of $\boldsymbol { \Phi } ^ { ( n ) }$ results in overhead computations. On the other hand, amortization also has disadvantages (e.g. Kim et al., 2018; Cremer et al., 2018). The competitive performance for denoising may consequently be attributed at least in part to TVAE not being subject to an amortization gap.
+
+![](images/28fcf2926d4d74e4346e81bf3b7039d35652023639a4ede0fe18b3fa6a2251ca.jpg)  
+Figure 6: Standard series of methods applied to optimize the encoding model of VAEs. Left: methods applied for encoding models of standard VAEs. Middle: additional methods applied to maintain the standard procedure of encoding model optimization also for discrete latent variables. Right: alternative approach to optimize the VAE encoding model using direct discrete optimization.
+
+# B DETAILS ON THE NUMERICAL EXPERIMENTS
+
+# B.1 VERIFICATION ON GROUND-TRUTH DATA
+
+We first evaluated TVAE training on artificial datasets with known ground-truth parameters and loglikelihood, in order to verify the correct functioning of the algorithm and to investigate possible local optima effects. The dataset consisted of $5 0 0 4 \mathrm { x } 4$ images generated by linear superposition of vertical and horizontal bars, with a small amount of Gaussian noise. The DNN’s input and middle layers had 8 units each. The $\Phi ^ { ( n ) }$ variational sets consisted of 64 hidden states each. Fig. 7 shows the evolution of the run that achieved the highest ELBO value out of ten. All parameters were correctly recovered, and the ELBO value was consistent with actual ground-truth log-likelihood.
+
+Such a simple test, however, can also be solved by linear models. In order to demonstrate that TVAEs can solve non-linear problems, taking advantage of the neural network non-linearity embedded in the generative model, we introduced correlations between pairs of bars: the bars combinations shown in the first two datapoints from the left in Fig. 8 were discouraged from appearing together. We employed the same evolutionary scheme and again we selected the run with highest peak ELBO value out of ten. The model correctly learns that certain combinations of bars are much more unlikely than others, and correctly estimates their likelihood.
+
+![](images/2a6b04221e9ae44435f5a6d9105878e4322c3702f6543e98f05d8e32a9dce63d.jpg)  
+Figure 7: TVAE training on simple bars data: noiseless output of the TVAE’s DNN for the 8 possible one-hot input vectors over several training epochs. Generating parameters are in the last row.
+
+![](images/18ac5c6b9ab26009afa724c2a4f8213b294d061feb31af9d3055cb0d8f1edbce.jpg)  
+Figure 8: Correlated bars test. The plot shows the ratio between inferred and ground-truth loglikelihoods $\log p _ { \Theta } ( \vec { x } )$ of datapoints with interesting bar combinations. The inferred values are reported below the datapoints themselves.
+
+Fig. 9 offers some more insight into the correlated bars test experiment described. The left section of the figure shows the generative parameters for the dataset used: $W _ { 0 }$ is the $8 \mathrm { x } 8$ weight matrix of the top-to-middle layer: this makes it so that the activation of the first latent variable inhibits activation of the second, and activation of the last latent variable inhibits activation of the last. Concretely, this results in a dataset where these specific bars combinations are discouraged from appearing. The weights $W _ { 1 }$ , visualized as $8 4 \mathrm { x } 4$ matrices, generate the actual bars. $\sigma ^ { 2 }$ was set to 0.01 and the dataset contained an average of two superimposing bars per datapoint $\pi _ { h } = 2 / 8$ for each $h$ ).
+
+The middle section of the figure shows the ELBO values (averages over all batches for each epoch) as training progresses. The cyclic learning rate schedule is responsible for the oscillatory behavior.
+
+The right section shows some example datapoints together with samples from the trained TVAE model that reached the highest ELBO value out of the ten runs.
+
+![](images/dc0cbe1af3b8b15e73702eaecae44249051888aa76aa16c38dc8063e30376cb5.jpg)  
+Figure 9: From left to right: generative parameters for the correlated bars test; ELBO values over epochs for 10 runs; example datapoints and samples from the generative model.
+
+# B.2 DENOISING
+
+Given a trained TVAE with parameters $\Theta$ , we estimated the value of a pixel in a single patch as $x _ { d } ^ { \mathrm { e s t } } = \langle x _ { d } \rangle _ { p \Theta ( x _ { d } | \vec { x } ) }$ . When using $\begin{array} { r } { p _ { \Theta } ( x _ { d } \mid \vec { x } ) = \sum _ { \{ \vec { z } \} } p _ { \Theta } ( x _ { d } \mid \vec { z } ) p _ { \Theta } ( \vec { z } \mid \vec { x } ) } \end{array}$ we obtain:
+
+$$
+x _ { d } ^ { \mathrm { e s t } } = \Big \langle \langle x _ { d } \rangle _ { p _ { \Theta } ( x _ { d } | \vec { z } ) } \Big \rangle _ { p _ { \Theta } ( \vec { z } | \vec { x } ) } = \langle \mu _ { d } ( \vec { z } ) \rangle _ { p _ { \Theta } ( \vec { z } | \vec { x } ) } .
+$$
+
+The expectation value on the right-hand-side of Eqn. (15) is then approximated based on the encoding parameters $\boldsymbol { \Phi } ^ { ( n ) }$ using truncated posteriors. Finally, we took a weighted average of the estimates of a pixel value in different patches (see, e.g., Burger et al., 2012) in order to generate the pixel values of the full denoised image.
+
+In Tab. 3 we list the exact hyper-parameters used to obtain the PSNR values reported. In parentheses, the parameters for the run on data with noise level $\sigma = 5 0$ and unconstrained hyper-parameters are given, when they differ from the other experiments.
+
+Table 3: Hyper-parameters for the denoising experiments on the house image.   
+
+<table><tr><td>Neural network units Input (H) 64 (512) Middle 64 (512)</td></tr><tr><td>Output (D) 64 (144) Cyclic Learning Rates</td></tr><tr><td>Min 1.r. 0.0001</td></tr><tr><td>Max l.r. 0.01 (0.05)</td></tr><tr><td>Epochs/cycle 20 Batch size 32</td></tr><tr><td>Evolutionaryparameters</td></tr><tr><td>Parents 10 (5) Children 9 (4) Generations 4(1)</td></tr></table>
+
+Table 4: Denoising performance of $\mathbf { n } 2 \mathbf { v }$ in PSNR (dB) for the ‘house’ image with AWG noise. For comparison, we additionally list the performance of TVAE (numbers copied from Tab. 2). PSNR values for $\mathrm { n } 2 \mathrm { v } ^ { \star }$ are obtained by training only on the noisy image (i.e., in the same setting as used for MTMKL, GSC, var-BSC and TVAE in Tab. 2. More training data improves performance for $\mathbf { n } 2 \mathbf { v } .$ . PSNR values for $\mathrm { n } 2 \mathrm { \bar { v } } ^ { \dagger }$ show performance if additional training data in the form of noisy images with AWG noise $\sigma = 2 5$ is used. Further improvements (especially for high noise) are obtained if the $\mathrm { n } 2 \mathrm { v }$ network is trained on training data with a noise level that matches the noise of the test set $\mathrm { ( s e e \ n 2 v ^ { \ddagger } ) }$ ). For instance, we used for $\mathrm { n } 2 \mathrm { v } ^ { \ddag }$ training data with $\sigma = 5 0$ to denoise the ‘house’ with $\sigma = 5 0$ . See text for further details.
+
+<table><tr><td></td><td>0=15</td><td>g=25</td><td>q=50</td></tr><tr><td>n2v*</td><td>32.05</td><td>29.20</td><td>25.42</td></tr><tr><td>n2vt</td><td>32.93</td><td>32.10</td><td>20.96</td></tr><tr><td>n2vt</td><td>33.91</td><td>32.10</td><td>28.94</td></tr><tr><td>TVAE</td><td>34.27 ± .02</td><td>32.65 ± .06</td><td>29.98 ± .05</td></tr></table>
+
+To evaluate the performance on standard denoising benchmarks, we first compared TVAE to related probabilistic sparse coding approaches such as MTMKL, GSC and var-BSC (Tab. 1). MTMKL and GSC use the data model of spike-and-slab sparse coding and for training mean-field and truncated posterior approximations with pre-selection are used, respectively. Compared to MTMKL and GSC, var-BSC uses a less complex data model and a training scheme also based on evolutionary optimization (Guiraud et al., 2018). The denoising performance observed in the scenario with controlled conditions (Tab. 1) shows that for high noise level $( \sigma = 5 0 $ ), var-BSC achieves higher PSNR values than MTMKL and GSC although the method uses a simpler data model. This observation demonstrates the effectiveness of the evolutionary training method used by var-BSC. However, PSNR values for TVAE are significantly higher due to the higher flexibility in modeling the data distribution provided by the used DNN.
+
+In a second step, Tab. 2 compared the performance of TVAE with respect to different denoising approaches including deterministic sparse coding (KSVD), a mixture model approach (EPLL), a non-local image processing method (WNNM) and state-of-the-art denoising methods based on deep neural networks (BDGAN and DPDNN). These approaches can be distinguished, e.g., by the amount of employed training data and by the requirement for clean data.
+
+TVAE as well as MTMKL, GSC and var-BSC do not require clean images for training. Furthermore, all these approaches can be trained if only the single noisy image is available (‘zero-shot’ learning; compare, e.g., Shocher et al., 2018; Imamura et al., 2019). Instead, EPLL, BDGAN and DPDNN use clean training data (typically tens or hundreds of thousands of data points are collected for training).
+
+Approaches such as noise2noise $\mathtt { n 2 n }$ Lehtinen et al., 2018) and noise2void (n2v Krull et al., 2019) occupy a middle ground: they can be trained on noisy data but they typically require much larger amounts of data than, e.g., TVAE or MTMKL. In the original n2v publication, for instance, 400 (noisy) $1 8 0 \times 1 8 0$ BSD (Martin et al., 2001) images were used to create a training dataset (this procedure also involved data augmentation; compare Krull et al. 2019). For our comparison with results of Tab. 2, we used the standard, publicly available code for $\mathbf { n } 2 \mathbf { v }$ together with the default training set $\sigma = 2 5$ ) employed in the original n2v publication. We then applied the trained $\mathbf { n } 2 \mathbf { v }$ network to denoise the ‘house’ image with $\sigma = 2 5$ . The resulting PSNR value was $3 2 . 1 0 d B$ which is $0 . 7 6 d B$ lower than the PSNR value for BM3D $( 3 2 . 8 6 d B )$ . The difference is consistent with an on average $0 . 8 8 d B$ lower performance of $\mathbf { n } 2 \mathbf { v }$ compared to BM3D on the BSD68 test set (see Krull et al., 2019). The same network can also be used to denoise an image with lower or higher noise level. The $\mathbf { n } 2 \mathbf { v }$ network trained on $\sigma = 2 5$ does, for instance, result in PSNR values of $3 2 . 9 3 d B$ for the ‘house’ image with $\sigma = 1 5$ and in $2 0 . 9 6 d B$ for the ‘house’ image with $\sigma = 5 0$ (see $n 2 v ^ { \dagger }$ in Tab. 4). Especially for high noise levels performance can be much improved, however, if the n2v network is trained using images with the same noise level as the test image. In order to do so, we followed the procedure described in the $\mathbf { n } 2 \mathbf { v }$ publication while adapting the noise level of $\sigma = 1 5$ in one case and $\sigma = 5 0$ for the other case. Trained on a dataset with matched noise, we then denoised the ‘house’ image with $\sigma = 1 5$ in the one, and $\sigma = 5 0$ in the other case (results listed as $\mathrm { n } 2 \mathrm { v } ^ { \ddag }$ in Tab. 4). The PSNR values obtained for ‘house’ in this matched-noise-level scenario are much higher compared to the scenario with unmatched noise level (e.g., for $\sigma = 5 0$ the PSNR improvement is approximately 8 dB). The much lower performance for mismatched noise for $\mathbf { n } 2 \mathbf { v }$ is in this respect consistent with observations for standard DNN denoising for which training with the ground-truth noise level has been pointed out as important for performance (Chaudhury & Roy, 2017; Zhang et al., 2018).
+
+The $\mathrm { n } 2 \mathrm { v }$ approach can avoid having to know the exact noise level, e.g., if it is trained on just the single noisy image. In a last experiment, we hence investigated this ‘zero-shot’ denoising feature of n2v and applied the algorithm to denoise the ‘house’ image while using the same noisy image for training that we seek to denoise (we took the publicly available code of $\mathrm { n } 2 \mathrm { v }$ as an example and manually adjusted hyperparameters as follows: we set the ”Percentage of pixel to manipulate per patch” to a value of 0.4, as ”Number of training epochs” we used 400 and we set the ”Number of parameter update steps per epoch” to 33). The obtained PSNR values are listed as $\mathrm { n } 2 \mathrm { v } ^ { \ast }$ in Tab. 4.
+
+From Tab. 4 it can be observed, that for all considered training settings of $\mathrm { n } 2 \mathrm { v }$ and all noise levels, PSNR values of TVAE are consistently higher than those of $\mathbf { n } 2 \mathbf { v }$ even if $\mathrm { n } 2 \mathrm { v }$ is trained on external data with matched-noise level. Additional parameter tuning may improve performance of $\mathrm { n } 2 \mathrm { v } ^ { \ast }$ to a certain extent but PSNRs are in general much lower than $\mathrm { \hat { n } } 2 \mathrm { v } ^ { \ddag }$ . While we followed for $\mathrm { n } 2 \mathrm { v } ^ { \ddag }$ the standard hyperparameter setting of the original paper/code publication of $\mathbf { n } 2 \mathbf { v }$ (Krull et al., 2019), we cannot exclude further improvements with parameter fine tuning for the ‘house’ benchmark. However, we remark that the difference of $\mathrm { n } 2 \mathrm { v } ^ { \ddag }$ and BM3D for the ‘house’ benchmark is on the very same range as the differences between n2v and BM3D as reported on the BSD data set in the original n2v publication. The stronger performing BM3D is according to denoising performance the preferable comparison and as such included in Tab. 2. In terms of efficiency, the $\mathrm { n } 2 \mathrm { v }$ approach is in general (once trained) faster than BM3D as well as TVAE, however.
+
+PSNR values of noise2noise $( \mathtt { n 2 n } )$ are usually very closely aligned with PSNR values achievable by feed-forward DNNs. More concretely, $\mathfrak { n } 2 \mathfrak { n }$ uses, for instance, a RED30 network (Mao et al., 2016) which achieves 31.07 dB PSNR on the BSD300 data set if trained on clean data. If directly trained on noisy data, RED30 achieves 31.06 dB (Lehtinen et al., 2018). n2n is thus strongly performing in terms of PSNR. The caveat of n2n compared to n2v is, however, that the noisy data n2n uses is rather artificial. The pairs of images n2n is trained on consist of two different noise realization of the same underlying clean image. For real data, such a setting is only approximately occurring at most, which has motivated the n2v approach.
+
+Like n2v, BDGAN and DPDNN are optimized for specific noise levels (specific standard deviations are used to generate the noisy training examples). EPLL is trained exclusively on clean image patches; for denoising, the algorithm requires the ground-truth noise level of the test image as input parameter. Ground-truth noise level information is also required by KSVD and WNNM.
+
+Like all approaches in the top category of Tab. 2, TVAE does not require ground-truth noise level information, nor clean images, nor large amounts of training data. For the ‘zero-shot’ setting, TVAE is consequently the best performing system on the ‘house’ benchmark. Such a high performance is notably achieved using a basic DNN and relatively small patch sizes of $D = 8 \times 8$ (for $\sigma = 1 5$ and $\sigma = 2 5$ ) or $D = 1 2 \times 1 2$ (for $\sigma = 5 0$ ). All feed-forward DNNs for denoising use much larger patches (e.g., n2v use $6 4 \times 6 4$ ). That a competitive denoising performance can be achieved for small patches, in general, argues in favor for VAE approaches to denoising. Indeed, TVAE even comes close to state-of-the-art approaches (BDGAN and DPDNN) that use very intricate DNN architectures and large amounts of clean training data. We believe that such results underline the potential of the here investigated approach although the novelty of the approach is the focus rather than extensive benchmarking.
+
+On the other hand, an important limitation of TVAE is its computational demand. For our experiments on the ‘house’ image with noise level $\sigma = 5 0$ in Tab. 2 we used $N = 6 0 0 2 5$ patches of $D = 1 2 \times 1 2$ pixels, which amounts to all possible non-overlapping square patches of that size that can be extracted from the image. For training and denoising we used a TVAE with $H = 5 1 2$ latent variables, sizes of $| { \Phi } ^ { ( n ) } | = 6 4 $ , and 512 units in the DNN middle layer of the decoder. TVAE training required 49 seconds per training epoch when executing on a single NVIDIA Titan $\mathrm { X p }$ GPU and $2 . 5 \mathrm { G B }$ of GPU memory. We ran for 500 epochs which required between seven and eight hours on the single GPU. We did not observe significant changes in variational bound values or in denoising performance after 500 epochs in any of the experiments we conducted for Tabs 1 and 2. Runtime complexity increased linear with the number of data points $N$ , with the dimensionality of the data $D$ , with the number of the latents $H$ , and with the size of the DNN used. Runtimes also increased approximately proportional w.r.t. the size of $\boldsymbol { \Phi } ^ { ( n ) }$ . Empirically we observed a sublinear scaling with |Φ(n) | presumably because of significant overhead computations: for example, increasing from $| { \boldsymbol { \Phi } } ^ { ( n ) } | = 6 4$ to $| { \Phi } ^ { ( n ) } | = 1 2 8$ (while keeping all other parameters as above) computational time increases from 49 seconds per training epoch to 75 seconds.
+
+For noise levels $\sigma = 1 5$ and $\sigma = 2 5$ in Tab. 2 we used smaller patch sizes $( D = 8 \times 8 )$ ) and fewer stochastic latents $H = 6 4$ ) but larger $\boldsymbol { \Phi } ^ { ( n ) }$ (i.e., $| { \Phi } ^ { ( n ) } | = 2 0 \bar { 0 } )$ . In general, if the patch size $D$ is increased, more structure has to be captured. This can be done either by increasing the size of the stochastic latents $H$ or by using larger DNNs. Both, in turn, requires more training data in order to estimate the increased number of parameters. In the current setup, the sizes of $D$ which are currently feasible are comparably small. The denoising performance based on small patches is, however, notably very high.
+
+For comparison, n2v uses up to $D = 6 4 \times 6 4$ and also all other feed-forward DNN approaches use significantly larger patch sizes than TVAE (and the other approaches in category 1). Still, n2v can be trained efficiently on large patches requiring approximately 19 hours on a NVIDIA Tesla K80 GPU for training on approximately $3 \mathrm { k }$ noisy images of shape $1 8 0 \mathrm { x } 1 8 0$ and seconds for the denoising of one 256x256 image. The higher computational demand of TVAE is also the reason why averaging across databases with many images (such as BSD68) or applications to large single images quickly becomes infeasible. As a novel approach, TVAE is, however, far from being fully optimized algorithmically compared to large feed-forward approaches, and there is certainly further potential to improve training efficiency.
+
+While denoising is, in general, well suited for deep generative models, performance for standard image denoising is by far not as common as such results for standard DNNs (which may also be related to efficiency aspects). An exception is a recent GAN approach (BDGAN; Zhu et al., 2019). VAEs are often evaluated using binarized MNIST with approximate log-likelihoods for comparison; that benchmark, however, is not consistent with the Gaussian noise model used here and does not allow a direct comparison with feed-forward DNNs which are the state-of-the-art.

parse/train/SUyxNGzUsH/SUyxNGzUsH.md

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+# VILNMN: A NEURAL MODULE NETWORK APPROACHTO VIDEO-GROUNDED LANGUAGE TASKS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Neural module networks (NMN) have achieved success in image-grounded tasks such as Visual Question Answering (VQA) on synthetic images. However, very limited work on NMN has been studied in the video-grounded language tasks. These tasks extend the complexity of traditional visual tasks with the additional visual temporal variance. Motivated by recent NMN approaches on image-grounded tasks, we introduce Visio-Linguistic Neural Module Network (VilNMN) to model the information retrieval process in video-grounded language tasks as a pipeline of neural modules. VilNMN first decomposes all language components to explicitly resolve any entity references and detect corresponding action-based inputs from the question. The detected entities and actions are used as parameters to instantiate neural module networks and extract visual cues from the video. Our experiments show that VilNMN can achieve promising performance on two video-grounded language tasks: video QA and video-grounded dialogues.
+
+# 1 INTRODUCTION
+
+Vision-language tasks have been studied to build intelligent systems that can perceive information from multiple modalities, such as images, videos, and text. Extended from imaged-grounded tasks, e.g. (Antol et al., 2015), recently Jang et al. (2017); Lei et al. (2018) propose to use video as the grounding features. This modification poses a significant challenge to previous image-based models with the additional temporal variance through video frames. Recently Alamri et al. (2019) further develop video-grounded language research into the dialogue domain. In the proposed task, videogrounded dialogues, the dialogue agent is required to answer questions about a video over multiple dialogue turns. Using Figure 1 as an example, to answer questions correctly, a dialogue agent has to resolve references in dialogue context, e.g. “he” and “it”, and identify the original entity, e.g. “a boy" and “a backpack". In addition, the dialogue agent also needs to identify the actions of these entities, e.g. “carrying a backpack” to retrieve information along the temporal dimension of the video.
+
+Current state-of-the-art approaches to video-grounded language tasks, e.g. (Le et al., 2019b; Fan et al., 2019) have achieved remarkable performance through the use of deep neural networks to retrieve grounding video signals based on language inputs. However, these approaches often assume the reasoning structure, including resolving references of entities and detecting the corresponding actions to retrieve visual cues, is implicitly learned. An explicit reasoning structure becomes more beneficial as the tasks complicates in two scenarios: video with complex spatial and temporal dynamics, and language inputs with sophisticated semantic dependencies, e.g. questions positioned in a dialogue context. In these cases, it becomes challenging to interpret model outputs, assess model reasoning capability, and identify errors in neural network models.
+
+Similar challenges have been observed in image-grounded tasks in which deep neural networks often exhibit shallow understanding capability as they simply exploit superficial visual cues (Agrawal et al., 2016; Goyal et al., 2017; Feng et al., 2018; Serrano & Smith, 2019). Andreas et al. (2016b) propose neural model networks (NMNs) by decomposing a question into sub-sequences called program and assembling a network of neural operations. Motivated by this line of research, we propose an NMN approach to video-grounded language tasks. Our approach benefits from integrating neural networks with a compositional reasoning structure to exploit low-level information signals in video. An example of the reasoning structure can be seen on the right side of Figure 1.
+
+![](images/08dfa3054943d0f06eb58bf4e0e2e48e82ee934f1926997c0219cf2c8c6783ea.jpg)  
+Figure 1: A sample video-grounded dialogue: Inputs are question, dialogue history, video with caption, visual and audio input, and the output is the answer to the question. On the right side, we demonstrate an example symbolic reasoning process a dialogue agent can perform to extract textual and visual clues for the answer.
+
+We propose Visio-Linguistic Neural Module Network (VilNMN) for video-grounded language tasks. VilNMN leverages entity-based dialogue representations as inputs to neural operations on spatial and temporal-level visual features. Previous approaches exploit question-level and token-level representations to extract question-dependent information from video (Jang et al., 2017; Fan et al., 2019; Le et al., 2019b). In complex videos with many entities or actions, these approaches might not be optimal to locate the right features. To exploit object-level features, VilNMN is trained to identify relevant entities first, and then to extract the temporal steps using detected actions of these entities.
+
+VilNMN is also trained to resolve any co-references in language inputs, e.g. questions in a dialogue context, to identify the original entities. Previous approaches to video-grounded dialogues often obtain question global representations in relation to dialogue context. These approaches might be suitable to represent general semantics in open-domain or chit-chat dialogues (Serban et al., 2016; Li et al., 2016) but they are not ideal to detect fine-grained entity-based information as the dialogue context evolves over time.
+
+In summary, we introduce a neural module network approach to video-grounded language tasks through a reasoning pipeline with entity and action representations applied on the spatio-temporal dynamics of video. To cater to complex semantic inputs in language inputs, e.g. dialogues, our approach also allows models to resolve entity references to incorporate question representations with fine-grained entity information. In our evaluation, we achieve competitive performance on the large-scale benchmark Audio-visual Scene-aware Dialogues (AVSD) (Alamri et al., 2019). We also adapt VilNMN for video QA and obtain the state-of-the-art on the TGIF-QA benchmark (Jang et al., 2017) across all tasks. Our experiments and ablation analysis indicate a potential direction to develop compositional and interpretable neural models for video-grounded language tasks.
+
+# 2 RELATED WORK
+
+Video QA has been a proxy for evaluating a model’s understanding capability of language and video and the task is treated as a visual information retrieval task. Jang et al. (2017); Gao et al. (2018); Jiang et al. (2020) propose to learn attention guided by question global representation to retrieve spatial-level and temporal-level visual features. Li et al. (2019); Fan et al. (2019); Jiang & Han (2020) model interaction between all pairs of question token-level representations and temporal-level features of input video. Extended from video QA, video-grounded dialogue is an emerging task that combines dialogue response generation and video-language understanding research. Nguyen et al. (2018); Hori et al. (2019); Hori et al. (2019); Sanabria et al. (2019); Le et al. (2019a;b) extend traditional QA models by adding dialogue history neural encoders. Kumar et al. (2019) enhances dialogue features with topic-level representations to express the general topic in each dialogue. Sanabria et al. (2019) considers the task as a video summary task and concatenates question and dialogue history into a single sequence and proposes to transfer parameter weights from a large-scale video summary model. Different from prior work, we dissect the question sequence and explicitly detect and decode any entities and their references. Our models also benefit from the additional insights on how models learn to use component linguistic inputs for extraction of visual information.
+
+![](images/c1042d57aa0064953fc1ccfac4878bbe8349871223ee844ed91ddeaf1edb3ed5.jpg)  
+Figure 2: VilNMN includes 4 major components: (1) encoders that encode dialogue and video components into continuous vector representations; (2) question parsers that parse question of the current dialogue turn into compositional programs for dialogue and video understanding; (3) an inventory of neural modules that operate on dialogue and video input components; and (4) a response decoder that generates natural language sequence using dialogue-based and video-based execution outputs.
+
+Extending from the line of research on neural semantic parsing (Jia & Liang, 2016; Liang et al., 2017), Andreas et al. (2016b;a) introduce NMNs to address visual QA by decomposing questions into linguistic sub-structures, known as programs, to instantiate a network of neural modules. NMN models have achieved significant success in synthetic image domains where complex reasoning process is required (Johnson et al., 2017b; Hu et al., 2018; Han et al., 2019). Our work is related to the recent work that extends NMN models to real data domains. For instance, Kottur et al. (2018); Jiang & Bansal (2019); Gupta et al. (2020) extend NMNs to visual dialogues and reading comprehension tasks. In this paper, we introduce a new approach that exploits NMN to learn dependencies between the lexical composition in language inputs and the spatio-temporal dynamics in videos. This is not present in prior NMN models which are designed to apply on a two-dimensional image input without temporal variance. In video represented as sequence of images, each represented by object-level features, applying prior NMN models require aggregating frame-level features, e.g. through average pooling, resulting in potential loss of information. An alternative solution is a late-fusion method in which an NMN model performs reasoning structure programs on sampled video frames only. An object tracking mechanism or attention mechanism is then used to fuse the output representations. Instead, we propose to construct a reasoning structure with multi-step interaction between the space-time information in video with entity-action detected in text.
+
+# 3 METHOD
+
+In this section, we present the design of our model, called Visio-Linguistic Neural Module Networks (VilNMN). An overview of the model can be seen in Figure 2. The input to the model consists of a dialogue $\mathcal { D }$ which is grounded on a video $\nu$ . The input components include the question of current dialogue turn $\mathcal { Q }$ , dialogue history $\mathcal { H }$ , and the features of input video, including visual and audio input. The output is a dialogue response, denoted as $\mathcal { R }$ . Each text input component is a sequence of words $w _ { 1 } , . . . , \bar { w } _ { m } \in \mathbb { V } ^ { i n }$ , the input vocabulary. Similarly, the output response $\mathcal { R }$ is a sequence of tokens $w _ { 1 } , . . . , w _ { n } \in \mathbb { V } ^ { o u t }$ , the output vocabulary.
+
+To learn compositional programs, we follow Johnson et al. (2017a); Hu et al. (2017) and consider program generation as a sequence-to-sequence task. Different from prior approaches, our models are trained to fully generate the parameters of component modules in text. This approach is appropriate as reasoning programs in real data domains such as current video-grounded dialogues are usually shorter than those for synthetic data (Johnson et al., 2017a) and thus, program generation takes less computational cost. However, module parameters, i.e. entities and actions, contain much higher semantic variance than synthetic data, and our approach facilitates better transparency and interpretability. We adopt a simple template $\mathrm { \langle \langle p a r a m _ { 1 } \rangle \langle m o d u l e _ { 1 } \rangle \langle p a r a m _ { 2 } \rangle \langle m o d u l e _ { 2 } \rangle . }$ .” as the target sequence. The resulting target sequences for dialogue and video understanding programs are sequences $\mathcal { P } _ { \mathrm { d i a l } }$ and $\mathcal { P } _ { \mathrm { v i d } }$ respectively.
+
+Table 1: Description of the modules and their functionalities. We denote $P$ as the parameter to instantiate each module, $H$ as the dialogue history, $Q$ as the question of the current dialogue turn, and $V$ as video input.   
+
+<table><tr><td>Module</td><td>Input</td><td>Output</td><td>Description</td></tr><tr><td>find</td><td>P,H</td><td>Hent</td><td>Forrelated entities in question,select the relevant tokens from dialogue history</td></tr><tr><td>summarize</td><td>Hent,Q</td><td>Qctx</td><td>Based on contextual entity representations,summarise the question semantics</td></tr><tr><td>where</td><td>P,V</td><td>Vent</td><td>Select the relevant spatial position corresponding to original (resolved) entities</td></tr><tr><td>when</td><td>P,Vent</td><td>Vent+act</td><td>Select the relevant entity-aware temporal steps corresponding to the action parameter</td></tr><tr><td>describe</td><td>P,Vent+act</td><td>Vctx Vctx</td><td>Select visual entity-action features based on non-binary question types</td></tr><tr><td>exist</td><td>Q,Vent+act</td><td></td><td>Select visual entity-action features based on binary (yes/no) question types</td></tr></table>
+
+# 3.1 ENCODERS
+
+Text Encoder. A text encoder is shared to encode text inputs, including dialogue history, questions, and captions. The text encoder converts each text sequence $\mathcal { X } = w _ { 1 } , . . . , w _ { m }$ into a sequence of embeddings $\boldsymbol { X } \in \mathbb { R } ^ { m \times d }$ . We use a trainable embedding matrix to map token indices to vector representations of $d$ dimensions through a mapping function $\phi$ . These vectors are then integrated with ordering information of tokens through a positional encoding function with layer normalization (Ba et al., 2016; Vaswani et al., 2017). The embedding and positional representations are combined through element-wise summation. The encoded dialogue history and question of the current turn are defined as $H = \operatorname { N o r m } ( \phi ( \mathcal { H } ) + \operatorname { P E } ( \mathcal { H } ) ) \in \mathbb { R } ^ { L _ { \mathrm { H } } \times d }$ and $Q = \dot { \mathrm { N o r m } } ( \dot { \phi ( \mathcal { Q } ) } + \mathrm { P E } ( \mathcal { Q } ) ) \in \mathbb { R } ^ { L _ { \mathrm { Q } } \times d }$ .
+
+To decode program and response sequences auto-repressively, a special token “_sos” is concatenated as the first token $w _ { 0 }$ . The decoded token $w _ { 1 }$ is then appended to $w _ { 0 }$ as input to decode $w _ { 2 }$ and so on. Similarly to input source sequences, at decoding time step $j$ , the input target sequence is encoded to obtain representations for dialogue understanding program $P _ { \mathrm { d i a l } } | _ { 0 } ^ { j - 1 }$ , video understanding program $P _ { \mathrm { v i d } } | _ { 0 } ^ { j - 1 }$ , and system response $R | _ { 0 } ^ { j - 1 }$ . We combine vocabulary of input and output sequences and share the embedding matrix $E \in \mathbb { R } ^ { | \mathbb { V } | \times d }$ where $\mathbb { V } = \mathbb { V } ^ { i n } \cap \mathbb { V } ^ { o u t }$ .
+
+Video Encoder. To encode video input, we use pre-trained models to extract visual features and audio features. We denote $F$ as the sampled video frames or video clips. For object-level visual features, we denote $O$ as the maximum number of objects considered in each frame. The resulting output from a pretrained object detection model is $Z _ { \mathrm { o b j } } \in \mathbb { R } ^ { F \times O \times d _ { \mathrm { v i s } } }$ . We concatenate each object representation with the corresponding coordinates projected to $d _ { \mathrm { v i s } }$ dimensions. We also make use of a CNN-based pre-trained model to obtain features of temporal dimension $Z _ { \mathrm { c n n } } \in \mathbb { R } ^ { F \times d _ { \mathrm { v i s } } }$ . The audio feature is obtained through a pretrained audio model, $\boldsymbol { Z _ { \mathrm { a u d } } ^ { \star } } \in \mathbb { R } ^ { F \times d _ { \mathrm { a u d } } }$ . We passed all video features through a linear transformation layer with ReLU activation to the same embedding dimension $d$ .
+
+# 3.2 NEURAL MODULES
+
+We introduce neural modules that are used to assemble an executable program constructed by the generated sequence from question parsers. We provide an overview of neural modules in Table 1 and demonstrate dialogue understanding and video understanding modules in Figure 3 and 4 respectively. Each module parameter, e.g. “a backpack”, is extracted from the parsed program. For each parameter, we denote $P \in \mathbb { R } ^ { d }$ as the average pooling of component token embeddings.
+
+find $( \pmb { \mathbb { P } } , \pmb { \mathbb { H } } ) \to \pmb { \mathbb { H } } _ { \mathrm { e n t } }$ . This module handles entity tracing by obtaining a distribution over tokens in dialogue history. We use an entity-to-dialogue-history attention mechanism applied from an entity $P _ { i }$ to all tokens in dialogue history. Any neural network that learn to generate attention between two tensors is applicable .e.g. (Bahdanau et al., 2015; Vaswani et al., 2017). The attention matrix normalized by softmax, $A _ { \mathrm { f i n d , i } } \in \mathbb { R } ^ { L _ { \mathrm { H } } }$ , is used to compute the weighted sum of dialogue history token representations. The output is combined with entity embedding $P _ { i }$ to obtain contextual entity representation $H _ { \mathrm { e n t , i } } \in \mathbb { R } ^ { d }$ .
+
+summarize $( \mathbf { \delta H } _ { \mathrm { e n t } } , \mathbf { \delta Q } ) \to \mathbf { \delta Q } _ { \mathrm { c t x } }$ . For each contextual entity representation $H _ { \mathrm { e n t , i } }$ , $i = 1 , . . . , N _ { \mathrm { e n t } }$ , it is projected to $L _ { \mathrm { Q } }$ dimensions and is combined with question token embeddings through elementwise summation to obtain entity-aware question representation $Q _ { \mathrm { e n t , i } } \in \mathbb { R } ^ { L _ { \mathrm { Q } } \times d }$ . It is fed to a one-dimensional CNN with max pooling layer (Kim, 2014) to obtain a contextual entity-aware question representation. We denote the final output as $Q _ { \mathrm { c t x } } \in \mathbb { R } ^ { N _ { \mathrm { e n t } } \times d }$ .
+
+While previous models usually focus on global or token-level dependencies (Hori et al., 2019; Le et al., 2019b) to encode question features, our modules compress fine-grained question representations at entity level. Specifically, find and summarize modules can generate entity-dependent local and global representations of question semantics. We show that our modularized approach can achieve better performance and transparency than traditional approaches to encode dialogue context (Serban et al., 2016; Vaswani et al., 2017) (Section 4).
+
+![](images/f85b6a3de3efb5d106598bd63ca55e04399de8753853806ee05d04fae2725123.jpg)  
+Figure 3: find and summarize neural modules for dialogue understanding
+
+where $( \pmb { \mathrm { p } } , \pmb { \mathrm { v } } )  \pmb { \mathrm { v } } _ { \mathrm { e n t } }$ . Similar to the find module, this module handle entity-based attention to the video input. However, the entity representation $P$ in this case is parameterized by the original entity in dialogue rather than in question (See Section 3.3 for more description). Each entity $P _ { i }$ is stacked to match the number of sampled video frames/clips $F$ . An attention network is used to obtain entity-to-object attention matrix $A _ { \mathrm { w h e r e , i } } ^ { \bullet } \in \mathbb { R } ^ { F \times O }$ . The attended feature are compressed through weighted sum pooling along the spatial dimension, resulting in $V _ { \mathrm { e n t , i } } \in \mathbb { R } ^ { F \times d }$ , $i = 1 , . . . , N _ { \mathrm { e n t } }$ .
+
+when $( \mathbb { P } , \pmb { \nabla } _ { \mathrm { e n t } } )  \pmb { \nabla } _ { \mathrm { e n t + a c t } }$ . This module follows a similar architecture as the where module. However, the action parameter $P _ { i }$ is stacked to match $N _ { \mathrm { e n t } }$ dimensions. The attention matrix $A _ { \mathrm { w h e n , i } } \in \mathbb { R } ^ { F }$ is then used to compute the visual entity-action representations through weighted sum along the temporal dimension. We denote the output for all actions $P _ { i }$ as $V _ { \mathrm { e n t + a c t } } \in \mathbb { R } ^ { N _ { \mathrm { e n t } } \times N _ { \mathrm { a c t } } \times d }$ describe $( \mathbb { P } , \nabla _ { \mathrm { e n t + a c t } } ) \to \mathbf { \nabla } \mathbf { V } _ { \mathrm { c t x } }$ . This module is a linear transformation to compute $V _ { \mathrm { c t x } } ~ =$ ${ W _ { \mathrm { d e s c } } } ^ { T } [ V _ { \mathrm { e n t + a c t } } ; P _ { \mathrm { s t a c k } } ] \in \mathbb { R } ^ { N _ { \mathrm { e n t } } \times N _ { \mathrm { a c t } } \times d }$ where $W _ { \mathrm { d e s c } } \in \mathrm { R } ^ { 2 d \times d }$ , $P _ { \mathrm { s t a c k } }$ is the stacked representations of parameter embedding $P$ to $N _ { \mathrm { e n t } } \times N _ { \mathrm { a c t } }$ dimensions, and $[ ; ]$ is the concatenation operation.
+
+The exist module is a special case of describe module where the parameter $P$ is the average pooled question embeddings. The above where module is applied to object-level features. For temporal-based features such as CNN-based and audio features, the same neural operation is applied along the temporal dimension. Each resulting entity-aware output is then incorporated to frame-level features through element-wise summation (Please refer to Appendix A.1).
+
+![](images/fccd4959bc59d2dd1c8e856c9749101d47a87e9632762e7aba9a6f87cfbf6d66.jpg)  
+Figure 4: where and when neural modules for video understanding
+
+An advantage of our architecture is that it separates dialogue and video understanding. We adopt a transparent approach to solve linguistic entity references during the dialogue understanding phase. The resolved entities are fed to the video understanding phase to learn entity-action dynamics in video. We show that our approach is robust when dialogue evolves to many turns and video extends over time (Please see Section 4 and Appendix C).
+
+# 3.3 DECODERS
+
+Question parsers. The parsers decompose questions into sub-sequences to construct compositional reasoning programs for dialogue and video understanding. Each parser is an attention-based Transformer decoder. Given the encoded question $Q$ , to decode program for dialogue understanding, the contextual signals are integrated through 2 attention layers: one attention on previously generated tokens, and the other on question tokens.
+
+To generate programs for video understanding, the contextual signals are learned and incorporated in a similar manner. However, to exploit dialogue contextual cues, the execution output of dialogue understanding neural modules $Q _ { \mathrm { c t x } }$ (See Section 3.2) is incorporated to each vector in $P _ { \mathrm { v i d } }$ through an additional attention layer. This layer integrates the entity-dependent contextual representations from $Q _ { \mathrm { c t x } }$ to explicitly decode the original entities for video understanding programs.
+
+Response Decoder. System response is decoded by incorporating the dialogue context and video context outputs from the corresponding reasoning programs to target token representations. We follows a vanilla Transformer decoder architecture (Le et al., 2019b), which consists of 3 attention layers: self-attention to attend on existing tokens, attention to $Q _ { \mathrm { c t x } }$ from dialogue understanding program execution, and attention to $V _ { \mathrm { c t x } }$ from video understanding program execution.
+
+Optimization. We use the standard cross-entropy losses for prediction of dialogue and video understanding programs and output responses. We optimize models by joint training to minimize: $\begin{array} { r } { \mathcal { L } = \alpha \mathcal { L } _ { \mathrm { d i a l } } + \beta \mathcal { L } _ { \mathrm { v i d } } + \mathcal { L } _ { \mathrm { r e s } } = \alpha \sum _ { j } - \log ( \mathbf { P } _ { \mathrm { d i a l } } ( \mathcal { P } _ { \mathrm { d i a l } , \mathbf { j } } ) ) + \beta \sum _ { l } - \log ( \mathbf { P } _ { \mathrm { v i d e o } } ( \mathcal { P } _ { \mathrm { v i d e o } , 1 } ) ) + \sum _ { n } - \log ( \mathbf { P } _ { \mathrm { r e s } } ( \mathcal { R } _ { \mathrm { n } } ) ) } \end{array}$ where $\mathbf { P }$ is the probability distribution of an output token. The probability is computed by passing output representations from the parsers and decoder to a linear layer $\dot { W } \in \mathbb { R } ^ { d \times V }$ with softmax activation. We share the parameters between $W$ and embedding matrix $E$ . The hyper-parameters $\alpha \geq 0$ and $\beta \geq$ are fine-tuned during training.
+
+# 4 EXPERIMENTS
+
+Datasets. We use the AVSD benchmark from the $7 ^ { t h }$ Dialogue System Technology Challenge (DSTC7) (Hori et al., 2019). In the experiments with AVSD, we consider two settings: one with video summary and one without video summary as input. In the setting with video summary, the summary is concatenated to the dialogue history before the first dialogue turn. We also adapt VilNMN to the video QA benchmark TGIF-QA (Jang et al., 2017). Different from AVSD, TGIF-QA contains a diverse set of tasks, which address different visual aspects in video.
+
+Training Procedure. We follow prior approaches (Hu et al., 2017; 2018; Kottur et al., 2018) by obtaining the annotations of the programs through a language parser (Hu et al., 2016) and a reference resolution model (Clark & Manning, 2016). During training, we directly use these soft labels of programs and the given ground-truth responses to train the models. The labels are augmented with label smoothing technique (Szegedy et al., 2016). During inference time, we generate all programs and responses from given dialogues and videos. We run beam search to enumerate programs for dialogue and video understanding and dialogue responses. (Please see Appendix B for more details).
+
+AVSD Results. We evaluate model performance by the objective metrics based on word overlapping, including BLEU (Papineni et al., 2002), METEOR (Banerjee & Lavie, 2005), ROUGE-L (Lin, 2004), and CIDEr (Vedantam et al., 2015), between each generated response and 6 reference gold responses. As seen in Table 2, our models outperform most of existing approaches. In particular, the performance of our model in the setting without video summary input is comparable to the GPT-based RLM (Li et al., 2020) with much smaller model size. The Student-Teacher baseline (Hori et al., 2019) specifically focuses on the performance gap between models with and without textual signals from video summary through a dual network of expert and student models. Instead, VilNMN reduces this performance gap through efficiently extracting relevant visual/audio information based on fine-grained entity and action signals. We also found that VilNMN applied on object-level features is competitive to the model applied on CNN-based features. The flexibility of VilNMN neural programs can also be seen in the experiment when the video understanding program is applied on the caption input as a visual feature.
+
+Ablation Analysis. We experiment with several variants of VilMNM (either NMN or non-NMNbased) in the setting with CNN based features and video summary input As can be seen in Table 3, our approach to video and dialogue understanding through compositional reasoning programs exhibits better performance than non-compositional approaches. Compared to the approaches that directly process frame-level features in videos (Row B) or token-level features in dialogues (Row C, D), our full VilNMN (Row A) considers entity-level and action-level information extraction and thus, avoids unnecessary and possibly noisy extraction. Compared to the approaches that obtain dialogue contextual cues through a hierarchical encoding architecture (Row E, F) such as (Serban et al., 2016; Hori et al., 2019), VilNMN directly addresses the challenge of entity references in dialogues. As mentioned, we hypothesize that the hierarchical encoding architecture is more appropriate for less entity-sensitive dialogues such as chit-chat and open-domain dialogues. Please see Appendix C for additional analysis of performance breakdown by dialogue turns and video lengths.
+
+Table 2: AVSD test results: The visual features are: I (I3D), ResNeXt-101 (RX), Faster-RCNN (FR), C (caption as a video input). The audio features are: VGGish (V), AclNet (A). Xon PT denotes models using pretrained weights and/or additional finetuning. Best and second best results are bold and underlined respectively.   
+
+<table><tr><td>Model</td><td>PT</td><td>Visual</td><td>Audio</td><td>BLEU4</td><td>METEOR</td><td>ROUGE-L</td><td>CIDEr</td></tr><tr><td colspan="6">Audio/Visual only (without Video Summary/Caption)</td><td></td><td></td></tr><tr><td>Baseline (Hori et al.,2019)</td><td></td><td>I</td><td>■</td><td>0.305</td><td>0.217</td><td>0.481</td><td>0.733</td></tr><tr><td>Baseline (Hori et al.,2019)</td><td></td><td>I</td><td>V</td><td>0.309</td><td>0.215</td><td>0.487</td><td>0.746</td></tr><tr><td>Baseline+GRU+Attn.(Le et al.,2019a)</td><td></td><td>I</td><td>V</td><td>0.315</td><td>0.239</td><td>0.509</td><td>0.848</td></tr><tr><td>FGA (Schwartz et al.,2019)</td><td></td><td>I</td><td>V</td><td></td><td></td><td></td><td>0.806</td></tr><tr><td>JMAN (Chu et al., 2020)</td><td></td><td>I</td><td>-</td><td>0.309</td><td>0.240</td><td>0.520</td><td>0.890</td></tr><tr><td>Student-Teacher (Hori et al.,2019)</td><td></td><td>I</td><td>V</td><td>0.371</td><td>0.248</td><td>0.527</td><td>0.966</td></tr><tr><td>MTN (Le et al.,2019b)</td><td></td><td>I</td><td>-</td><td>0.343</td><td>0.247</td><td>0.520</td><td>0.936</td></tr><tr><td>MTN (Le et al., 2019b)</td><td></td><td>I</td><td>V</td><td>0.368</td><td>0.259</td><td>0.537</td><td>0.964</td></tr><tr><td>MSTN (Lee et al.,2020)</td><td></td><td>I</td><td>V</td><td>0.379</td><td>0.261</td><td>0.548</td><td>1.028</td></tr><tr><td>RLM-GPT2 (Li et al., 2020)</td><td>√</td><td>I</td><td>V</td><td>0.402</td><td>0.254</td><td>0.544</td><td>1.052</td></tr><tr><td>VilNMN</td><td>=</td><td>I</td><td>-</td><td>0.397</td><td>0.262</td><td>0.550</td><td>1.059</td></tr><tr><td>VilNMN</td><td></td><td>FR</td><td>■</td><td>0.388</td><td>0.259</td><td>0.549</td><td>1.040</td></tr><tr><td>VilNMN</td><td></td><td>-</td><td>V</td><td>0.381</td><td>0.252</td><td>0.534</td><td>1.004</td></tr><tr><td>VilNMN</td><td></td><td>I</td><td>V</td><td>0.396</td><td>0.263</td><td>0.549</td><td>1.059</td></tr><tr><td colspan="6">Audio/Visual only (with Video Summary/Caption)</td><td></td><td></td></tr><tr><td>TopicEmb (Kumar et al.,2019)</td><td></td><td>I</td><td>A</td><td>0.329</td><td>0.223</td><td>0.488</td><td>0.762</td></tr><tr><td>Baseline+GRU+Attn.(Le et al.,2019a)</td><td></td><td>I</td><td>V</td><td>0.310</td><td>0.242</td><td>0.515</td><td>0.856</td></tr><tr><td>JMAN (Chu et al., 2020)</td><td></td><td>I</td><td>-</td><td>0.334</td><td>0.239</td><td>0.533</td><td>0.941</td></tr><tr><td>FA+HRED (Nguyen et al., 2018)</td><td>=</td><td>I</td><td>V</td><td>0.360</td><td>0.249</td><td>0.544</td><td>0.997</td></tr><tr><td>VideoSum (Sanabria etal.,2019)</td><td>=</td><td>RX</td><td>-</td><td>0.394</td><td>0.267</td><td>0.563</td><td>1.094</td></tr><tr><td>VideoSum+How2 (Sanabria et al.,2019)</td><td>√</td><td>RX</td><td>-</td><td>0.387</td><td>0.266</td><td>0.564</td><td>1.087</td></tr><tr><td>MSTN (Lee et al.,2020)</td><td></td><td>I</td><td>V</td><td>0.377</td><td>0.275</td><td>0.566</td><td>1.115</td></tr><tr><td>Student-Teacher (Hori et al.,2019)</td><td></td><td>I</td><td>V</td><td>0.405</td><td>0.273</td><td>0.566</td><td>1.118</td></tr><tr><td>MTN (Le et al.,2019b)</td><td></td><td>I</td><td>-</td><td>0.392</td><td>0.269</td><td>0.559</td><td>1.066</td></tr><tr><td>MTN (Le et al.,2019b)</td><td></td><td>I</td><td>V</td><td>0.410</td><td>0.274</td><td>0.569</td><td>1.129</td></tr><tr><td>VGD-GPT2 (Le &amp; Hoi,2020)</td><td>√</td><td>I</td><td>V</td><td>0.436</td><td>0.282</td><td>0.579</td><td>1.194</td></tr><tr><td>RLM-GPT2 (Li et al., 2020)</td><td>√</td><td>I</td><td>V</td><td>0.459</td><td>0.294</td><td>0.606</td><td>1.308</td></tr><tr><td>VilNMN</td><td>■</td><td>I</td><td>1</td><td>0.421</td><td>0.277</td><td>0.574</td><td>1.171</td></tr><tr><td>VilNMN</td><td></td><td>FR</td><td>-</td><td>0.421</td><td>0.275</td><td>0.571</td><td>1.148</td></tr><tr><td>VilNMN</td><td></td><td>I</td><td>V</td><td>0.421</td><td>0.277</td><td>0.573</td><td>1.167</td></tr><tr><td>VilNMN</td><td></td><td>I+C</td><td>V</td><td>0.429</td><td>0.278</td><td>0.578</td><td>1.188</td></tr></table>
+
+Table 3: Ablation analysis of VilNMN with different model variants on the test split of the AVSD benchmark   
+
+<table><tr><td>#</td><td>ModelVariant</td><td>BLEU4</td><td>CIDEr</td></tr><tr><td>A</td><td>FullVilNMN</td><td>0.421</td><td>1.171</td></tr><tr><td>B</td><td>No video NMNs; + vanilla text-→video attention</td><td>0.415</td><td>1.159</td></tr><tr><td>C</td><td>→No dial.NMNs;+ response-→history attention</td><td>0.412</td><td>1.151</td></tr><tr><td>D</td><td>→No dial.NMNs;+ response-→concat(history+question) attention</td><td>0.411</td><td>1.133</td></tr><tr><td>E</td><td>No dial.NMNs;+ HREDLsTM(history) + question attn.</td><td>0.414</td><td>1.153</td></tr><tr><td>F</td><td>→No dial.NMNs; + HREDGRU(history) + question attn.</td><td>0.415</td><td>1.138</td></tr></table>
+
+Intepretability. A difference of VilNMN from previous approaches to video-grounded dialogues is the model interpretability based on the predicted dialogue and video programs. From Figure 5, we observe that in cases where predicted dialogue programs and video program match or are close to the gold labels, the model can generate generally correct responses. In cases of wrong predicted responses, we can further look at how the model understands the questions based on predicted programs. In the $3 ^ { r d }$ turn of example 1, the output response is missing a minor detail as compared to the label response because the video program fails to capture the parameter “rooftop”. These subtle yet important details can determine whether output responses can fully address user queries. Similarly, in example 2, the model answers the question “what room” instead of question about “an object”. For additional qualitative analysis, please see Appendix D.
+
+TGIF-QA Results. In TGIF-QA experiments, we report the result using the L2 loss in Count task and accuracy in other tasks. From Table 4, VilNMN outperforms all baseline models in all tasks by a large margin. Compared to AVSD experiments, the TGIF-QA experiments emphasize video understanding ability of the models, removing the requirement for dialogue understanding and natural language generation. This is demonstrated through higher performance gaps between VilNMN with generated programs and soft label programs as compared to ones in AVSD experiments. We also observe that an attention layer attending to question is important during the response decoding phase in TGIF-QA as there is no dialogue context $Q _ { \mathrm { c t x } }$ in Video QA tasks.
+
+![](images/b76cdb81c535fa60c73e2ed0fbce53a5355fe5d0af1cb511403543cd18aa93a9.jpg)
+
+<table><tr><td rowspan=1 colspan=1>Turn</td><td rowspan=1 colspan=1>Question</td><td rowspan=1 colspan=1>Dialogue NMN</td><td rowspan=1 colspan=1>Video NMN</td><td rowspan=1 colspan=1>Response</td></tr><tr><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>...what are theydoing in the scene ？are they on arooftop？</td><td rowspan=1 colspan=1>Predicted:find(they)，find(the scene)→summarize()×Gold: find(they)→summarize()</td><td rowspan=1 colspan=1>Predicted: where(two men in thevideo),where(the scene)→when(doing inthe scene)→describe(what)Gold:where(two men)，where (rooftop)→when(doing in the scene)-describe(what)</td><td rowspan=1 colspan=1>Predicted: one is washinga chair and the other istaking picturesXGold: yes,on a secondfloor roof deck ,one man iswashing a chair,anotherman is either filming ortaking pictures</td></tr><tr><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>does he sit in thechair after washingit？</td><td rowspan=1 colspan=1>Predicted:find (he),find(the chair)→summarize()Gold: find(he),find(the chair)→summarize()</td><td rowspan=1 colspan=1>Predicted: where(one man in the video)，where(a chair)→ when(sit in the chairafter washing it)-exist()Gold:where(one man)，where(a chair)→when(sit in the chair after washingit)-exist()</td><td rowspan=1 colspan=1>Predicted: no, he does notsit in the chair√Gold: no he does not,thereis a pipe with water runningall over</td></tr></table>
+
+![](images/b2ce15a7da399a54da904703872d8a7d046bcc27067064f947a84577c92b87c0.jpg)
+
+![](images/181282280ef506314f5ed0b0226febd77d1a2bd127b0b5c7121f865b44526cc7.jpg)
+
+Figure 5: Intepretability of model outputs from a dialogue in the test split of the AVSD benchmark.   
+Table 4: Experiment results on the TGIF-QA benchmark. The visual features are: ResNet-152 (R), C3D (C), Flow CNN from two-stream model (F), VGG (V), ResNeXt-101 (RX).   
+
+<table><tr><td>Model</td><td>Visual</td><td>Count (Loss)</td><td>Action (Acc)</td><td>Transition (Acc)</td><td>FrameQA (Acc)</td></tr><tr><td>VIS(avg) (Ren et al.,2015a)</td><td>R</td><td>4.80</td><td>0.488</td><td>0.348</td><td>0.350</td></tr><tr><td>MCB (aggr) (Fukui et al.,2016)</td><td>R</td><td>5.17</td><td>0.589</td><td>0.243</td><td>0.257</td></tr><tr><td>Yu et al. (Yu et al., 2017)</td><td>R</td><td>5.13</td><td>0.561</td><td>0.640</td><td>0.396</td></tr><tr><td>ST-VQA (t) (Gao et al., 2018)</td><td>R+F</td><td>4.32</td><td>0.629</td><td>0.694</td><td>0.495</td></tr><tr><td>Co-Mem (Gao et al.,2018)</td><td>R+F</td><td>4.10</td><td>0.682</td><td>0.743</td><td>0.515</td></tr><tr><td>PSAC (Li et al.,2019)</td><td>R</td><td>4.27</td><td>0.704</td><td>0.769</td><td>0.557</td></tr><tr><td>HME (Fan et al.,2019)</td><td>R+C</td><td>4.02</td><td>0.739</td><td>0.778</td><td>0.538</td></tr><tr><td>STA (Gao et al.,2019)</td><td>R</td><td>4.25</td><td>0.723</td><td>0.790</td><td>0.566</td></tr><tr><td>CRN+MAC (Le et al.,2019c)</td><td>R</td><td>4.23</td><td>0.713</td><td>0.787</td><td>0.592</td></tr><tr><td>MQL (Lei et al., 2020)</td><td>V</td><td>-</td><td>-</td><td>-</td><td>0.598</td></tr><tr><td>QueST (Jiang et al.,2020)</td><td>R</td><td>4.19</td><td>0.759</td><td>0.810</td><td>0.597</td></tr><tr><td>HGA (Jiang &amp; Han,2020)</td><td>R+C</td><td>4.09</td><td>0.754</td><td>0.810</td><td>0.551</td></tr><tr><td>GCN (Huang et al., 2020)</td><td>R+C</td><td>3.95</td><td>0.743</td><td>0.811</td><td>0.563</td></tr><tr><td>HCRN (Le et al.,2020)</td><td>R+RX</td><td>3.82</td><td>0.750</td><td>0.814</td><td>0.559</td></tr><tr><td>VilNMN</td><td>R</td><td>2.65</td><td>0.845</td><td>0.887</td><td>0.747</td></tr><tr><td>→soft label programs</td><td>R</td><td>1.90</td><td>0.857</td><td>0.898</td><td>0.780</td></tr><tr><td>→- res-to-question attn.</td><td>R</td><td>3.28</td><td>0.801</td><td>0.776</td><td>0.679</td></tr></table>
+
+# 5 CONCLUSION
+
+While conventional neural network approaches have achieved notable successes in video-grounded dialogues and video QA, they often rely on superficial pattern learning principles between contextual cues from questions/dialogues and videos. In this work, we introduce Visio-Linguistic Neural Module Network (VilNMN). VilNMN consists of dialogue and video understanding neural modules, each of which performs entity and action-level operations on language and video components. Our comprehensive experiments on AVSD and TGIF-QA benchmarks show that our models can achieve competitive performance while promoting a compositional and interpretable learning approach.
+
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+
+# A ADDITIONAL MODEL DETAILS
+
+# A.1 NEURAL MODULES ON TEMPORAL FEATURES
+
+To adapt our neural modules to temporal features, we apply the same neural architectures in all modules except for the where module. In object-level features, this module operates on object-based or spatial-based level. We can apply this module to temporal-based features similarly simply by not stacking the parameter and pooling the attended features along the temporal dimension. For an entity parameter $P _ { i }$ , the attention matrix in this case is an entity-to-temporal-step matrix $A _ { \mathrm { w h e r e , i } } \in \mathbb { R } ^ { \tilde { F } }$ and the resulting pooled feature is $V _ { \mathrm { e n t , i } } \in \mathbb { R } ^ { d }$ . Before feeding this representation to a when module, we incorporate each $V _ { \mathrm { e n t , i } }$ into feature of each temporal step through an MLP layer and element-wise summation, resulting in overview of the where $V _ { \mathrm { e n t , i } } ^ { s t a c k } \in \mathbb { R } ^ { \mathrm { F } \times d }$ where poral $F$ is the number of sampled video frames/clips. Antures can be seen in Figure 6.
+
+![](images/00f0a2cc042a838be019c96ec5a63325ec1007abd58e16de83f1037dc0c8f8c8.jpg)  
+Figure 6: Adaptation of the where module to temporal-based features
+
+We adapt this module in a similar manner to other temporal-level features such as audio and textual features such as video caption. We keep the same architecture in the when module. We denote the resulting output from the when module for all actions $P _ { i }$ is $V _ { \mathrm { a c t } } \in \mathbb { R } ^ { N _ { \mathrm { a c t } } \times d }$ . We concatenate this to the output from the previous where module $V _ { \mathrm { e n t } }$ to obtain $V _ { \mathrm { e n t + a c t } } \in \mathbb { R } ^ { ( N _ { \mathrm { e n t } } + N _ { \mathrm { a c t } } ) \times d }$ . This is used as input to the describe or exist module.
+
+# A.2 QUESTION PARSER
+
+The parsers decompose questions into sub-sequences to construct compositional reasoning programs for dialogue and video understanding. Each parser is an attention-based Transformer decoder. The Transformer attention is a multi-head attention on query, key, and value tensors, denoted as Attention(Query, Key, Value). For each token in the Query sequence , the distribution over tokens in the Key sequence is used to obtain the weighted sum of the corresponding representations in the Value sequence.
+
+$$
+{ \mathrm { A t t e n t i o n } } ( Q u e r y , K e y , V a l u e ) = { \mathrm { s o f t m a x } } ( { \frac { Q u e r y K e y ^ { T } } { \sqrt { d _ { k e y } } } } ) V a l u e \in \mathbb { R } ^ { L _ { \mathrm { q u e r y } } \times d _ { \mathrm { q u e r y } } }
+$$
+
+Each attention is followed by a feed-forward network applied on each position identically. We exploit the multi-head and feed-forward architecture, which show good performance in NLP tasks such as NMT and QA (Vaswani et al., 2017; Dehghani et al., 2019), to efficiently incorporate contextual cues from dialogue components to parse question into reasoning programs. Given the encoded question $Q$ , to decode program for dialogue understanding, the contextual signals are integrated through 2 attention layers: one attention on previously generated tokens, and the other on question tokens. At time step $j$ , we denote the output from an attention layer as $A _ { \mathrm { d i a l , j } }$ .
+
+$$
+\begin{array} { r l } & { A _ { \mathrm { d i a l } } ^ { ( 1 ) } = \mathrm { A t t e n t i o n } ( P _ { \mathrm { d i a l } } | _ { 0 } ^ { j - 1 } , P _ { \mathrm { d i a l } } | _ { 0 } ^ { j - 1 } , P _ { \mathrm { d i a l } } | _ { 0 } ^ { j - 1 } ) \in \mathbb { R } ^ { j \times d } } \\ & { A _ { \mathrm { d i a l } } ^ { ( 2 ) } = \mathrm { A t t e n t i o n } ( A _ { \mathrm { d i a l } } ^ { ( 1 ) } , Q , Q ) \in \mathbb { R } ^ { j \times d } } \end{array}
+$$
+
+Similarly, to generate programs for video understanding, the contextual signals are learned and incorporated in a similar manner. However, to exploit dialogue contextual cues, the execution output of dialogue understanding neural modules $Q _ { \mathrm { c t x } }$ is incorporated to each vector in $P _ { \mathrm { d i a l } }$ through an additional attention layer. This layer integrates the resolved entity information to decode the original entities for video understanding. It is equivalent to a reasoning process that converts the question from its original multi-turn semantics to single-turn semantics.
+
+$$
+{ \cal A } _ { \mathrm { v i d } } ^ { ( 3 ) } = \mathrm { A t t e n t i o n } ( A _ { \mathrm { v i d } } ^ { ( 2 ) } , Q _ { \mathrm { c t x } } , Q _ { \mathrm { c t x } } ) \in \mathbb { R } ^ { j \times d }
+$$
+
+# A.3 NON-NMN MODELS
+
+For ablation analysis, we evaluate several variants of VilNMN, based on the following categories:
+
+To test the contribution of our NMN approach for video understanding, we remove the parser for video understanding program and related neural modules and replace them with pure neural network architecture (Model $B$ ). Specifically, we remove neural modules where, when, describe, and exist. We then directly use video feature embeddings $V$ as $V _ { \mathrm { c t x } }$ as input to the original attention layer in response decoder similarly to (Hori et al., 2019; Sanabria et al., 2019).
+
+$$
+A _ { \mathrm { r e s } } ^ { ( 3 ) } = \mathrm { A t t e n t i o n } ( A _ { \mathrm { r e s } } ^ { ( 2 ) } , V , V ) \in \mathbb { R } ^ { j \times d }
+$$
+
+To further test the contribution of NMN architecture for dialogue understanding, we similarly remove the question parser for dialogue understanding program and neural modules find and describe. We then directly use the dialogue history embeddings $H$ and question embeddings $Q$ as inputs to the response decoder in two different ways. First, we replace the original attention on dialogue context $Q _ { \mathrm { c t x } }$ with two attention layers to attend on dialogue history and question sequentially (Model $C$ ). As noted by Le et al. (2019b), question input contains much more relevant signals than dialogue history and attention operation should be separated from the one on dialogue history.
+
+$$
+\begin{array} { r } { A _ { \mathrm { r e s } } ^ { ( 2 a ) } = \mathrm { A t t e n t i o n } ( A _ { \mathrm { r e s } } ^ { ( 1 ) } , H , H ) \in \mathbb { R } ^ { j \times d } } \\ { A _ { \mathrm { r e s } } ^ { ( 2 b ) } = \mathrm { A t t e n t i o n } ( A _ { \mathrm { r e s } } ^ { ( 2 a ) } , Q , Q ) \in \mathbb { R } ^ { j \times d } } \end{array}
+$$
+
+Alternatively, we simply concatenate dialogue and question embeddings similarly to (Hori et al., 2019; Sanabria et al., 2019) and use it as input to the original attention layer (Model $D$ ).
+
+$$
+A _ { \mathrm { r e s } } ^ { ( 2 ) } = \mathrm { A t t e n t i o n } ( A _ { \mathrm { r e s } } ^ { ( 1 ) } , [ H ; Q ] , [ H ; Q ] ) \in \mathbb { R } ^ { j \times d }
+$$
+
+To use more sophisticated neural models for dialogue understanding, we further adopt the hierarchical encoding architecture with question attention (Li et al., 2016; Serban et al., 2016; Hori et al., 2019). Each dialogue turn $\mathcal { H } _ { t }$ , including a pair of human utterance and system response, is processed separately by a word-level RNN such as LSTM (Model $E$ ) or GRU (Model $F$ ). A sentence-level RNN is used to sequentially process the last hidden states obtained previously turn by turn. The output in each recurrent step is fed to an attention layer such as (Bahdanau et al., 2015; Vaswani et al., 2017) to obtain question-aware representations of dialogue history.
+
+$$
+\begin{array} { r l } & { H _ { t } ^ { \mathrm { w o r d } } = \mathrm { R N N } ( H _ { t } ) \in \mathbb { R } ^ { d } } \\ & { H _ { t } ^ { \mathrm { s e n t } } = \mathrm { R N N } ( H _ { t } ^ { \mathrm { w o r d } } ) \in \mathbb { R } ^ { d } } \\ & { \qquad H = [ H _ { t } ^ { \mathrm { s e n t } } ] | _ { t = 1 } ^ { T - 1 } \in \mathbb { R } ^ { d \times ( T - 1 ) } } \\ & { Q _ { \mathrm { c t x } } = \mathrm { A t t e n t i o n } ( Q , H , H ) \in \mathbb { R } ^ { L _ { Q } \times d } } \end{array}
+$$
+
+where $T$ is the current dialogue turn. The output is treated as $Q _ { \mathrm { c t x } }$ and is fed to the corresponding attention layer in the response decoder.
+
+# B ADDITIONAL EXPERIMENT DETAILS
+
+# B.1 DATASETS
+
+We use the AVSD benchmark from DSTC7 (Hori et al., 2019) which consists of dialogues grounded on the Charades videos (Sigurdsson et al., 2016). Each dialogue contains up to 10 dialogue turns, each turn consists of a question and expected response about a given video. For visual features, we use the 3D CNN based features from a pretrained I3D model (Carreira & Zisserman, 2017) and object-level features from a pretrained FasterRNN model (Ren et al., 2015b). The audio features are obtained from a pretrained VGGish model (Hershey et al., 2017). In the experiments with AVSD, we consider two settings: one with video summary and one without video summary as input. In the setting with video summary, the summary is concatenated to the dialogue history before the first dialogue turn. We also adapt VilNMN to the video QA benchmark TGIF-QA (Jang et al., 2017). Different from AVSD, TGIF-QA contains a diverse set of QA tasks:
+
+• Count: open-ended task which counts the number of repetitions of an action • Action: multiple-choice (MC) task which asks about a certain action occurring for a fixed number of times • Transition: MC task which emphasizes temporal transition in video • Frame: open-ended task which can be answered from visual contents of one of video frames
+
+For the TGIF-QA benchmark, we use the extracted features from a pretrained ResNet model (He et al., 2016).
+
+Table 5: Summary of DSTC7 AVSD and TGIF-QA benchmark   
+
+<table><tr><td colspan="2">#</td><td>Train</td><td>Val.</td><td>Test</td></tr><tr><td rowspan="3">AVSD</td><td>Dialogs</td><td>7,659</td><td>1,787</td><td>1,710</td></tr><tr><td>Turns</td><td>153,180</td><td>35,740</td><td>13,490</td></tr><tr><td>Words</td><td>1,450,754</td><td>339,006</td><td>110,252</td></tr><tr><td rowspan="4">TGIFQA</td><td>Count QA</td><td>24,159</td><td>2,684</td><td>3,554</td></tr><tr><td>Action QA</td><td>18,428</td><td>2,047</td><td>2,274</td></tr><tr><td>Trans. QA</td><td>47,434</td><td>5,270</td><td>6,232</td></tr><tr><td>Frame QA</td><td>35,453</td><td>3,939</td><td>13,691</td></tr></table>
+
+# B.2 TRAINING DETAILS
+
+We use a training batch size of 32 and embedding dimension $d = 1 2 8$ in all experiments. Where Transformer attention is used, we fix the number of attention heads to 8 in all attention layers. In neural modules with MLP layers, the MLP network is fixed to 2 linear layers with a ReLU activation in between. In neural modules with CNN, we adopt a vanilla CNN architecture for text classification (without the last MLP layer) where the number of input channels is 1, the kernel sizes are $\{ 3 , 4 , 5 \}$ , and the number of output channels is $d$ . We initialize models with uniform distribution (Glorot & Bengio, 2010). During training, we adopt the Adam optimizer (Kingma & Ba, 2015) and a decaying learning rate Vaswani et al. (2017) where we fix the warm-up steps to 15K training steps. We employ dropout (Srivastava et al., 2014) of 0.2 at all networks except the last linear layers of question parsers and response decoder. We train models up to 50 epochs and select the best models based on the average loss per epoch in the validation set.
+
+# C ADDITIONAL RESULTS
+
+To evaluate model robustness, we report the relative performance by calculating the difference of CIDEr in experimental settings against the most basic setting. Specifically, we compare against performance of output responses in the first dialogue turn position (i.e. $2 ^ { n \dot { d } }  – 1 0 ^ { t h }$ turn vs. the $1 ^ { s t }$ turn), or responses grounded on the shortest video length range (video ranges are intervals of $0 – 1 0 ^ { t h }$ , $1 0 \ – 2 0 ^ { t h }$ percentile and so on). We report the results of the model variants A, B, and E (See the Ablation Analysis section in the main paper and Appendix A.3 for model description). First, as can be seen in Figure 7, for various dialogue turn positions, we observe that the original VilNMN (model A) suffers less than model E when dialogues extend over time up the $8 ^ { t h }$ turn. This explains the contribution of dialogue understanding modules in solving entities even when the dialogues grow longer. Secondly, as compared to model B, we observe that the Full VilNMN (model A) is less affected as the videos grounding the dialogues grow longer. The difference is clear when the video length increases up to 33 seconds.
+
+We also report the absolute scores and compare model variants. In Table 6a, we compare model variants B and E. We observe that model B generally performs better than model E in overall, especially in higher turn positions, i.e. from the $4 ^ { t h }$ turn to $8 ^ { t h }$ turn. Interestingly, we note some mixed results in very low turn position, i.e. the $2 ^ { n d }$ and $3 ^ { r d }$ turn, and very high turn position, i.e. the $1 0 ^ { t h }$ turn. Potentially, in very high turn position, the neural based approach such as hierarchical RNN can better capture the global dependencies within dialogue context than the entity-based compositional NMN method.
+
+In Table 6b, we compare model variants A and B. We note that the performance gap between model A and B is quite distinct, with 7/10 cases of video ranges in which model A outperforms. However, similarly to our prior observations in experiments by dialogue turn, in lower ranges (i.e. 1-23 seconds) and higher ranges (37-75 seconds), model A performs not as well as model B. There are additional factors that we will need to examine further to explain the results, such as the complexity of the questions for these short and long-range videos. Potentially, our question parser for video understanding program needs more sophisticated composition method to retrieve information from these video ranges.
+
+![](images/398de27efc81f7e3642d1abc7c7b4de47175198fb1e45e3f27e2fc1e430a106c.jpg)  
+Figure 7: Performance of model variants A, B, and E, by dialogue turn position and video length. The performance is calculated relatively to performance of the most basic setting, i.e. responses of the first dialogue turn $\Delta \mathrm { C I D E r _ { t u r n \_ i } } = \mathrm { C I D E r _ { t u r n \_ i } } - \mathrm { C I D E r _ { t u r n \_ 1 } }$ , or responses grounding on the lowest video range (0 to 23 seconds) $\Delta \mathrm { C I D E r _ { r a n g e \mathrm { _ i } } } = \mathrm { C I D E r _ { r a n g e \mathrm { _ i } } } - \mathrm { C I D E r _ { 0 - 2 3 } }$ .
+
+Table 6: Performance breakdown in BLEU4 and CIDEr   
+(a) by dialogue turn between model variants B and E.   
+
+<table><tr><td colspan="3">BLEU4</td><td colspan="2">CIDEr</td></tr><tr><td>turn position</td><td>Model B</td><td>Model E</td><td>Model B</td><td>Model E</td></tr><tr><td>1</td><td>0.579</td><td>0.587</td><td>1.623</td><td>1.650</td></tr><tr><td>2</td><td>0.429</td><td>0.430</td><td>1.155</td><td>1.142</td></tr><tr><td>3</td><td>0.275</td><td>0.289</td><td>0.867</td><td>0.846</td></tr><tr><td>4</td><td>0.309</td><td>0.305</td><td>0.859</td><td>0.855</td></tr><tr><td>5</td><td>0.355</td><td>0.335</td><td>1.088</td><td>1.023</td></tr><tr><td>6</td><td>0.357</td><td>0.329</td><td>1.044</td><td>0.950</td></tr><tr><td>7</td><td>0.342</td><td>0.325</td><td>0.896</td><td>0.847</td></tr><tr><td>8</td><td>0.361</td><td>0.332</td><td>1.025</td><td>0.973</td></tr><tr><td>9</td><td>0.383</td><td>0.431</td><td>1.043</td><td>1.182</td></tr><tr><td>10</td><td>0.395</td><td>0.371</td><td>0.931</td><td>0.977</td></tr></table>
+
+(b) by video length range (in seconds) between model variants A and B.   
+
+<table><tr><td colspan="3">BLEU4</td><td colspan="2">CIDEr</td></tr><tr><td>video range (seconds)</td><td>Model A</td><td>Model B</td><td>Model A</td><td>Model B</td></tr><tr><td>1-23</td><td>0.432</td><td>0.447</td><td>1.298</td><td>1.355</td></tr><tr><td>23-28</td><td>0.436</td><td>0.433</td><td>1.264</td><td>1.165</td></tr><tr><td>28-30</td><td>0.398</td><td>0.376</td><td>1.203</td><td>1.164</td></tr><tr><td>30-30.6</td><td>0.441</td><td>0.418</td><td>1.220</td><td>1.202</td></tr><tr><td>30.6-31</td><td>0.413</td><td>0.411</td><td>1.250</td><td>1.166</td></tr><tr><td>31-31.6</td><td>0.439</td><td>0.451</td><td>1.249</td><td>1.295</td></tr><tr><td>31.6-32</td><td>0.430</td><td>0.419</td><td>1.217</td><td>1.192</td></tr><tr><td>32-33</td><td>0.468</td><td>0.445</td><td>1.343</td><td>1.237</td></tr><tr><td>33-37</td><td>0.388</td><td>0.381</td><td>1.149</td><td>1.124</td></tr><tr><td>37-75</td><td>0.356</td><td>0.365</td><td>0.910</td><td>0.962</td></tr></table>
+
+# D QUALITATIVE ANALYSIS
+
+We extract the predicted programs and responses for some example dialogues in Figure 8, 9, 10, and 11 and report our observations:
+
+• We observe that when the predicted programs are correct, the output responses generally match the ground-truth (See the $1 ^ { s t }$ and $2 ^ { n d }$ turn in Figure 8, and the $1 ^ { s t }$ and $4 ^ { { \bar { t } } h }$ turn in Figure 10) or close to the ground-truth responses ( $1 ^ { s t }$ turn in Figure 9).   
+• When the output responses do not match the ground truth, we can understand the model mistakes by interpreting the predicted programs. For example, in the $3 ^ { r d }$ turn in Figure 8, the output response describes a room because the predicted video program focuses on the entity “what room” instead of the entity “an object” in the question. Another example is the $3 ^ { \dot { r } d }$ turn in Figure 10 where the entity “rooftop” is missing in the video program. These mismatches can deviate the information retrieved from the video during video program execution, leading to wrong output responses with wrong visual contents.   
+We also note that in some cases, one or both of the predicted programs are incorrect, but the predicted responses still match the ground-truth responses. This might be explained as the predicted module parameters are not exactly the same as the ground truth but they are close enough (e.g. $4 ^ { t h }$ turn in Figure 8). Sometimes, our model predicted programs that are more appropriate than the ground truth. For example, in the $2 ^ { { \bar { n } } d }$ turn in Figure 9, the program is added with a where module parameterized by the entity “the shopping bag” which was solved from the reference “them” mentioned in the question.   
+• We observe that for complex questions that involve more than one queries (e.g. the $3 ^ { r d }$ turn in Figure 10), it becomes more challenging to decode an appropriate video understanding program and generate responses that can address all queries.   
+• In Figure 11, we demonstrate some output examples of VilNMN and compare with two baselines: Baseline (Hori et al., 2019) and MTN (Le et al., 2019b). We noted that VilNMN can include important entities relevant to the current dialogue turn to construct output responses while other models might miss some entity details, e.g. “them/dishes” in example A and “the magazine” in example B. These small yet important details can determine the correctness of dialogue responses.
+
+![](images/5e39bc95a0ceb8e3fb3d5469b7dd1efdd1618424facc2e40e32d72bd986dd909.jpg)
+
+![](images/4418977275529630c424ae9a011e6eb6302163738aca0446dbfdcd78de7e81e8.jpg)
+
+![](images/5ae7eb0ce394e9a325452d0eab7ff74705e152d97186a68063a0e73b0f9fe07c.jpg)
+
+![](images/d13d1be16d3744b012c2f4f68213120b43ec9510bcc6defd08e26fb98b5bceec.jpg)
+
+![](images/bcaefdb7909d644203e7c3e16b22b646729f276040913b46cc890612139a86e9.jpg)
+
+Figure 10: Example output programs and responses from a dialogue in the test split of the AVSD benchmark.   
+
+<table><tr><td rowspan=1 colspan=1>Turn</td><td rowspan=1 colspan=1>Question</td><td rowspan=1 colspan=1>Dialogue UnderstandingProgram</td><td rowspan=1 colspan=1>Video UnderstandingProgram</td><td rowspan=1 colspan=1>Response</td></tr><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>hi how many peopleare in the video fromstart to finish ?</td><td rowspan=1 colspan=1>Predicted: summari ze()Gold: summarize ()</td><td rowspan=1 colspan=1>Predicted: describe (how manypeople)Gold:describe (how manypeople)</td><td rowspan=1 colspan=1>Predicted: there are twopeople in the video.Gold: hello,there aretwo men</td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>are they carrying ona conversationbetween themselves？</td><td rowspan=1 colspan=1>Predicted:find(they)→summarize()Gold:find(they)，find(themselves)→summarize()</td><td rowspan=1 colspan=1>Predicted: where(two men in the←video)→when(carrying on aconversation betweenthemselves)-exist()Gold:where(two men)→ when(carrying a conversationbetween themselves)-exist()</td><td rowspan=1 colspan=1>Predicted: yes,they arecarrying a conversation×Gold: no,backgroundnoise is birdsong andshuffling noises</td></tr><tr><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>what are they doingin the scene ?arethey on a rooftop ?</td><td rowspan=1 colspan=1>Predicted: find(they)，find(the scene)→summarize() ×Gold: find(they)→summarize()</td><td rowspan=1 colspan=1>Predicted:where(two men in thevideo),where(the scene)→when(doing in the scene)→describe(what) ×Gold: where(two men),where(rooftop)→when(doingin the scene)-describe(what)</td><td rowspan=1 colspan=1>Predicted: one iswashing a chair and theother is taking pictures×Gold: yes,on a secondfloor roof deck ,oneman is washing a chair,another man is eitherfilming or taking pictures</td></tr><tr><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>does he sit in thechair after washingit？</td><td rowspan=1 colspan=1>Predicted: find(he),find(the chair)→summarize()Gold: find (he),find(the chair)→summarize()</td><td rowspan=1 colspan=1>Predicted: where(one man in thevideo)，where(a chair)→when(sit in the chair afterwashing it)→exist()Gold:where(one man)，where(achair)→ when(sit in thechair after washing it)→exist()</td><td rowspan=1 colspan=1>Predicted: no,he doesnot sit in the chairGold: no he does not,there is a pipe withwater running all over</td></tr></table>
+
+![](images/b020b5ccf752b45a137b3877cf43d0b9573cdb8ca1bd4a82fc366971a6c29af8.jpg)  
+Figure 11: Intepretability of example outputs from VilNMN and baselines models (Hori et al., 2019; Le et al., 2019b)

parse/train/_b8l7rVPe8z/_b8l7rVPe8z.md

@@ -0,0 +1,392 @@
+# RELEVANCE ATTACK ON DETECTORS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+This paper focuses on high-transferable adversarial attacks on detectors, which are hard to attack in a black-box manner, because of their multiple-output characteristics and the diversity across architectures. To pursue a high attack transferability, one plausible way is to find a common property across detectors, which facilitates the discovery of common weaknesses. We are the first to suggest that the relevance map for detectors is such a property. Based on it, we design a Relevance Attack on Detectors (RAD), which achieves a state-of-the-art transferability, exceeding existing results by above $20 \%$ . On MS COCO, the detection mAPs for all 8 black-box architectures are more than halved and the segmentation mAPs are also significantly influenced. Given the great transferability of RAD, we generate the first adversarial dataset for object detection, i.e., Adversarial Objects in COntext (AOCO), which helps to quickly evaluate and improve the robustness of detectors.
+
+# 1 INTRODUCTION
+
+Adversarial attacks (Szegedy et al. (2014); Goodfellow et al. (2015); Carlini & Wagner (2017); M ˛adry et al. (2017); Baluja & Fischer (2017); Su et al. (2019)) have revealed the fragility of Deep Neural Networks (DNNs) by fooling them with elaborately-crafted imperceptible perturbations. Among them, the black-box attack, i.e., attacking without knowledge of their inner structure and weights, is much harder, more aggressive and closer to real-world scenarios. For classifiers, there exist some promising black-box attacks (Papernot et al. (2016); Brendel et al. (2018); Dong et al. (2018); Xie et al. (2019); Lin et al. (2020); Chen et al. (2020)). It is also severe to attack object detection (Zhang & Wang (2019)) in a black-box manner, e.g., hiding certain objects from unknown detectors (Thys et al. (2019)). By that, life-concerning systems based on detection such as autonomous driving and security surveillance would be greatly influence.
+
+To the best of our knowledge, no existing attack is specifically designed for black-box transferability in detectors, because they have multiple-outputs and a high diversity across architectures. In such situations, adversarial samples do not transfer well (Su et al. (2018)), and most attacks only decrease mAP of black-box detectors by 5 to $10 \%$ (Xie et al. (2017); Li et al. (2018c;b)). To overcome this, we propose one plausible way to find common properties across detectors, which facilitates the discovery of common weaknesses. Based on them, the designed attack can threaten variable victims.
+
+In this paper, we adopt the relevance map as a common property, on which different detectors have similar interpretable results, as shown in Fig. 1. Based on relevance maps, we design a Relevance Attack on Detectors (RAD). RAD focuses on suppressing the relevance map rather than directly attacking the prediction as in existing works (Xie et al. (2017); Li et al. (2018c;a)). Because the relevance maps are quite similar across models, those of black-box models are influenced and misled as well in attack, leading to the great transferability. Although some works have adopted the relevance map as an indicator or reference of success attacks (Dong et al. (2019); Zhang & Zhu (2019); Chen et al. (2020); Wu et al. (2020a)), there is no work to directly attack the relevance maps of detectors to the best of our knowledge.
+
+In our comprehensive evaluation, RAD achieves the state-of-the-art transferability on 8 black-box models for COCO) dataset (Lin et al. (2014)), nearly halving the detection mAP. Interestingly, the adversarial samples of RAD also greatly influence the performance of instance segmentation, even only detectors are attacked. Given the high transferability of RAD, we create Adversarial Objects in COntext (AOCO), the first adversarial dataset for object detection. AOCO contains 10K samples that significantly decrease the performance of black-box models for detection and segmentation. AOCO may serve as a benchmark to test the robustness of a DNN or improve it by adversarial training.
+
+![](images/532cf52146221673a00f7d8064e690df7110aa3752fa5b8559f71cea00249ecd.jpg)  
+Figure 1: Relevance maps for models with different architectures. Three models not only predict the “stop sign” right, but also share similar relevance maps.
+
+# CONTRIBUTIONS
+
+• We propose a novel attack framework on relevance maps for detectors. We extend network visualization methods to detectors, find out the most suitable nodes to attack by relevance maps, and explore on the best update techniques to increase the transferability. • We evaluate RAD comprehensively and find its state-of-the-art transferability, which exceeds existing results by above $20 \%$ . Detection mAPs are more than halved, invalidating the stateof-the-art detectors to a large extent. By RAD, we create the first adversarial dataset for object detection, i.e., AOCO. As a potential benchmark, AOCO is generated from COCO and contains 10K high-transferable samples. AOCO helps to quickly evaluate and improve the robustness of detectors.
+
+# 2 RELATED WORK
+
+Since (Szegedy et al. (2014)), there have been lots of promising adversarial attacks (Goodfellow et al. (2015); Carlini & Wagner (2017); M ˛adry et al. (2017)). Generally, they fix the network weights and change the input slightly to optimize the attack loss. The network then predicts incorrectly on adversarial samples with a high confidence. (Papernot et al. (2016; 2017)) find that adversarial samples crafted by attacking a white-box surrogate model may transfer to other black-box models as well. Input modification (Xie et al. (2019); Dong et al. (2019); Lin et al. (2020)) or other optimization ways (Dong et al. (2018); Lin et al. (2020)) are validated to be effective in enhancing the transferability.
+
+(Xie et al. (2017)) extends adversarial attacks to detectors. It proposes to attack on densely generated bounding boxes. After that, losses about localization and classification are designed (Li et al. (2018c)) for attacking detectors. (Lu et al. (2017)) and (Li et al. (2018a)) propose to attack detectors in a restricted area. Existing works achieve good results in white-box scenarios, but are not specifically designed for transferability. The adversarial impact on black-box models is quite limited, i.e., a 5 to $10 \%$ decrease from the original mAP, even when two models only differ in backbone (Xie et al. (2017); Li et al. (2018c;b)). (Wang et al. (2020)) discusses black-box attacks towards detectors based on queries rather than the transferability as we do. The performance is satisfactory, but it requires over 30K queries, which is easy to be discovered by the model owner. Besides, physical attacks on white-box detectors are also feasible (Huang et al. (2020); Wu et al. (2020b); Xu et al. (2020)).
+
+For the great transferability, we propose to attack on relevance maps, which are calculated by network visualization methods (Zeiler & Fergus (2014); Selvaraju et al. (2017); Shrikumar et al. (2017)). They are originally developed to interpret how DNNs predict and help users gain trust on them. Specifically, they display how the input contributes to a certain node output in a pixel-wise manner. Typical works include Layer-wise Relevance Propagation (LRP) (Bach et al. (2015)), Contrastive LRP (Gu et al. (2018)) and Softmax Gradient LRP (SGLRP) (Iwana et al. (2019)). These methods encourage the reference of relevance maps in attack (Dong et al. (2019); Zhang & Zhu (2019); Chen et al. (2020); Wu et al. (2020a)), and also inspire us. However, none of them attack on relevance maps for detectors.
+
+RAD differs from (Ghorbani et al. (2019); Zhang et al. (2020)) in the goal. RAD misleads detectors by suppressing relevance maps. In contrast, (Ghorbani et al. (2019)) misleads the relevance maps while keeping the prediction unchanged. (Zhang et al. (2020)) also misleads DNNs, but it keeps the relevance maps unchanged.
+
+# 3 RELEVANCE ATTACK ON DETECTORS
+
+We propose an attack specifically designed for black-box transferability, named Relevance Attack on Detectors (RAD). RAD suppresses multi-node relevance maps for several bounding boxes. Since the relevance map is commonly shared by different detectors as shown in Fig. 1, attacking on it in the white-box surrogate model achieves a high transferability towards black-box models. In this section, we first provide a high-level overview of RAD, and analyze the potential reasons of its transferability. Then we thoroughly discuss three crucial concrete issues in RAD.
+
+• In Section 3.3, we specify the calculation of relevance maps for detectors, where current visualization methods are not applicable.   
+• In Section 3.4, we introduce the proper nodes to attack by RAD.   
+• In Section 3.5, we explore on the suitable techniques to update samples in RAD.
+
+# 3.1 WHAT IS RAD?
+
+We present the framework of RAD in Fig. 2. Initialized by the original sample $x _ { 0 }$ , the adversarial sample $x _ { k }$ in the $k ^ { \mathrm { { t h } } }$ iteration is forward propagated in the surrogate model, getting the prediction $f ( x _ { k } )$ . Current attacks generally suppress the prediction values of all attacked output nodes in $T$ . In contrast, RAD suppresses the corresponding relevance map $h ( \boldsymbol { x } _ { k } , T )$ . To restrain that, gradients of $h ( \boldsymbol { x } _ { k } , T )$ back propagate to $x _ { k }$ , which is then modified to $x _ { k + 1 }$ .
+
+![](images/d176c9609254b18a9a20a0467bc55eedcff17b807a627c6824363aab7ff06f3a.jpg)  
+Figure 2: Framework of RAD. $x _ { k }$ is the sample in iteration $k$ and $f ( x _ { k } )$ is the network prediction for it. $h ( \boldsymbol { x } _ { k } , T )$ stands for the relevance map for all attacked nodes in $T$ . RAD works by repeating processes denoted by “black”, “red” and “blue” arrows in turn.
+
+It is notable that RAD is a complete framework to attack detectors, and its each component requires a special design. Besides the calculation of relevance maps of detectors, other components in RAD, e.g., the attacked nodes or the update techniques, also need a customized analysis. The reason is that no existing work directly attacks the relevance of detectors, and the experience in attacking predictions is not totally applicable here. For example, (Zhang & Wang (2019)) emphasizes classification loss and localization loss equally, but the former is validated to be significantly better in attacking the relevance in Section 3.4.
+
+# 3.2 WHY RAD TRANSFERS?
+
+RAD’s transferability comes from the attack goal: changing the common properties, i.e., the relevance maps. As shown in Fig. 3, the relevance maps are clear and structured for the original sample in both detectors. After RAD, the relevance maps are induced to be meaningless without a correct focus, leading to wrong predictions, i.e., no or false detection. Because relevance maps transfer well across models, those for black-box detectors are also significantly influenced, causing a great performance drop, which is illustrated visually in Section 4.2.
+
+![](images/249f41e82d757b53efe52a9a203cba6c688813a5d70a898f39c29cb9afebf958.jpg)  
+Figure 3: RAD’s transferability origins from the change of relevance maps. The image contains a person and a skateboard. By attacking on relevance maps, both surrogate models make extremely confusing predictions.
+
+RAD also attacks quite “precisely”, i.e., the perturbation pattern is significantly focused on distinct areas and has a clear structure as shown in Fig. 4. That is to say, RAD accurately locates the most discriminating parts of a sample and concentrates the perturbation on them, leading to a great transferability when the perturbations is equally bounded.
+
+![](images/63ab5ec8395fab7e9a3955a560451309ecf40525acdc32dadc207a0abe714629.jpg)  
+Figure 4: The original image and the adversarial perturbations $\times 5$ in magnitude for demonstration) generated by Dfool (Lu et al. (2017)), DAG (Xie et al. (2017)), and RAD (from left to right)
+
+3.3 WHAT IS THE RELEVANCE MAPS FOR DETECTORS?
+
+We analyze the potential of RAD above, below we make it feasible by addressing three crucial issues.   
+To conduct the relevance attack, we first need to know the relevance maps for detectors.
+
+Currently, there have been lots of methods to calculate the relevance maps for classifiers as described in Section 2, but none of them are suitable for detectors. We take SGLRP (Iwana et al. (2019)) as an example to explain this and then to modify, because it excels in discriminating ability against irrelevant regions of a certain target node.
+
+SGLRP visualizes how the input contributes to one output node in a pixel-wise way by backpropagating the relevance from the output to the input based on Deep Taylor Decomposition (DTD) as illustrated in Appendix A. $R ^ { ( L ) }$ is the initial relevance in the output layer $L$ and its $n ^ { \mathrm { t h } }$ component is calculated as
+
+$$
+R _ { n } ^ { ( L ) } = { \left\{ \begin{array} { l l } { y _ { n } \left( 1 - y _ { n } \right) } & { n = t , } \\ { - y _ { n } y _ { t } } & { n \neq t , } \end{array} \right. }
+$$
+
+where $y _ { n }$ is the predicted probability of class $n$ , and $y _ { t }$ is that for the single-node target $t$ . The pixel-wise relevance map $h ( x , t )$ for the single-node target $t$ is calculated by back propagating the relevance $R$ from the final layer to the input following rules specified in (Iwana et al. (2019)).
+
+In detectors, we need the pixel-wise contributions from the input to $m$ bounding boxes. This multi-node relevance map could not be directly calculated by (1), so we naturally modify SGLRP as
+
+$$
+R _ { n } ^ { ( L ) } = \left\{ \begin{array} { l l } { y _ { n } \left( 1 - y _ { n } \right) } & { n \in T , } \\ { - \frac { 1 } { m } y _ { n } \sum _ { i = 1 } ^ { m } y _ { t _ { i } } } & { n \notin T , } \end{array} \right.
+$$
+
+where $y _ { t _ { i } }$ is the predicted probabilities for one target node $t _ { i }$ . $T$ is the set containing all target nodes $\{ t _ { 1 } , t _ { 2 } , . . . , t _ { m } \}$ . With iNNvestigate Library (Alber et al. (2019)) to implement Multi-Node SGLRP and Deep Learning platforms supporting auto-gradient, the gradients from RAD loss $L _ { \mathrm { R A D } } ( x ) = h ( x , T )$ to sample $x$ could be obtained according to the calculation rules of relevance maps in Appendix A.
+
+We illustrate the difference between SGLRP and our Multi-Node SGLRP in Fig. 5. SGLRP only displays the relevance map for one bounding box, e.g., “TV”, “chair” and “bottle”. Multi- Node SGLRP, in contrast, visualizes the overall relevance.
+
+![](images/a45412ecf81682fd77f346e329b0e1b5ce65e40a7e142423efbe4588b2d04216.jpg)  
+Figure 5: Difference between relevance maps from SGLRP and Multi-Node SGLRP. The relevance maps are for YOLOv3 (Redmon & Farhadi (2018)).
+
+# 3.4 WHERE TO ATTACK?
+
+Besides the calculation of relevance maps, it is also important to choose a proper node set $T$ to attack.   
+Specifically, we need to select certain bounding boxes and the corresponding output nodes for RAD.
+
+Heuristically, the most “obvious” bounding boxes are desired to be eliminated, so we select the bounding boxes with the highest confidence, following (Xie et al. (2017)). Concretely, it is feasible to statically choose $m$ bounding boxes to attack in each iteration, or dynamically attack all bounding boxes whose confidence exceeds a threshold. In our evaluation, the two strategies differ a little in performance and are not sensitive to hyper-parameter as demonstrated in Appendix C. This shows that RAD does not require a sophisticated tuning of parameters, which is user-friendly. In our following experiments, we statically attack $m = 2 0$ nodes.
+
+After selecting bounding boxes, we could attack their size, leading them to shrink; or their localization, leading them to shift; or their confidence, leading them to be misclassified. To adopt the best strategy, we conduct a toy experiment by attacking YOLOv3 (Redmon & Farhadi (2018)), denoted as M2 and other models are specified in Appendix B. Given the results in Table 1, the classification loss induces a better black-box transferability. This may because detectors generally include a pre-trained classification as the feature extractor, and relevance maps are believed to be an indicator of success attacks (Dong et al. (2019); Zhang & Zhu (2019)).
+
+Table 1: Detection mAP in RAD with different attacked nodes   
+
+<table><tr><td>Strategy</td><td>M1</td><td>M2</td><td>M3</td><td>M4</td><td>M5</td><td>M6</td><td>M7</td><td>M8</td><td>M9</td></tr><tr><td>No Attack</td><td>29.3</td><td>33.4</td><td>38.1</td><td>40.7</td><td>42.1</td><td>42.5</td><td>45.7</td><td>46.9</td><td>53.9</td></tr><tr><td>Size</td><td>26.0</td><td>14.7</td><td>31.9</td><td>32.5</td><td>35.6</td><td>35.4</td><td>38.6</td><td>40.0</td><td>47.8</td></tr><tr><td>Localization</td><td>22.8</td><td>6.4</td><td>27.4</td><td>28.1</td><td>31.7</td><td>30.8</td><td>34.4</td><td>35.9</td><td>45.1</td></tr><tr><td>Classification</td><td>18.1</td><td>1.2</td><td>19.9</td><td>20.5</td><td>24.3</td><td>22.6</td><td>26.4</td><td>28.2</td><td>39.9</td></tr></table>
+
+# 3.5 HOW TO UPDATE?
+
+By the relevance map $h ( x , T )$ for certain attacked nodes $T$ , we are able to attack, i.e., update the original sample to become adversarial with the guidance of the attack gradients $g ( x )$ as
+
+$$
+g ( x ) = \frac { \partial L _ { \mathrm { R A D } } ( x ) } { \partial x } = \frac { \partial h ( x , T ) } { \partial x } .
+$$
+
+Some update techniques are validated to be effective for enhancing the transferability in classification. For example, Scale-Invariant (SI) (Lin et al. (2020)) proposes to average the attack gradients by scale copies of the samples as
+
+$$
+g _ { \mathrm { s i } } ( x ) = \frac { 1 } { k } \sum _ { i = 0 } ^ { k } g ( x / 2 ^ { i } ) .
+$$
+
+Besides SI, Diverse Input (DI) (Xie et al. (2019)), Translation-Invariant (TI) (Dong et al. (2019)) are also promising in classification. We are curious about whether they also work well in object detection. To explore on this, we adopt these techniques in RAD as the setting suggested by their designers (see Appendix E). From the results in Table 2, we discover that SI is quite effective, further decreasing the mAP from the baseline to a large extent. Accordingly, we adopt (4) to update the sample in RAD.
+
+Table 2: Detection mAP in RAD with different update techniques   
+
+<table><tr><td>Technique</td><td>M1</td><td>M2</td><td>M3</td><td>M4</td><td>M5</td><td>M6</td><td>M7</td><td>M8</td><td>M9</td></tr><tr><td>None</td><td>18.1</td><td>1.2</td><td>19.9</td><td>20.5</td><td>24.3</td><td>22.6</td><td>26.4</td><td>28.2</td><td>39.9</td></tr><tr><td>DI</td><td>18.1</td><td>1.0</td><td>19.9</td><td>20.5</td><td>23.9</td><td>22.4</td><td>26.3</td><td>27.9</td><td>39.6</td></tr><tr><td>TI</td><td>17.0</td><td>2.4</td><td>20.8</td><td>20.8</td><td>25.2</td><td>23.0</td><td>27.9</td><td>29.7</td><td>41.5</td></tr><tr><td>SI</td><td>14.6</td><td>0.7</td><td>16.3</td><td>17.0</td><td>20.4</td><td>19.1</td><td>22.3</td><td>23.8</td><td>35.0</td></tr></table>
+
+With the calculated gradient, we update the sample as
+
+$$
+x _ { k + 1 } = \mathrm { c l i p } _ { \varepsilon } \left( x _ { k } - \alpha \frac { g _ { \mathrm { s i } } ( x _ { k } ) } { | | g _ { \mathrm { s i } } ( x _ { k } ) | | _ { 1 } / N } \right) ,
+$$
+
+where $\alpha$ stands for the step length. $x$ is $\ell _ { \infty }$ -norm bounded by $\varepsilon$ from the original sample in each iteration as in (Xie et al. (2019); Dong et al. (2019); Lin et al. (2020)). Gradient $g ( x )$ is normalized by its average $\ell _ { 1 }$ -norm,i.e., $| | g ( x ) | | _ { 1 } / N$ to prevent numerical errors and control the degree of perturbations. $N$ is the dimension of the image, i.e., $N = h e i g h t \times w i d t h \times c h a n n e l .$ Division by $N$ is necessary because $\ell _ { 1 }$ -norm sums all components of the tensor $x$ , which is too large as a normalization factor. We do not adopt the mainstream sign method because it is not suitable to generate small perturbations as shown in other attacks in detectors (Xie et al. (2017)).
+
+# 4 EXPERIMENTS
+
+In this section, we evaluate the performance of RAD, especially its transferability. The results are presented numerically and visually. In comprehensive evaluation, RAD achieves a great transferability in across models and even across tasks.
+
+# 4.1 SETUP
+
+Our experiments are based on Keras (Chollet et al. (2015)), Tensorflow (Abadi et al. (2015)) and PyTorch (Paszke et al. (2019)) in 4 NVIDIA GeForce RTX 2080Ti GPUs. Library iNNvestigate (Alber et al. (2019)) is used to implement Multi-Node SGLRP.
+
+We conduct experiments on MS COCO 2017 dataset (Lin et al. (2014)), which is a large-scale benchmark for object detection, instance segmentation and image captioning. For a fair evaluation, we generate adversarial samples from all 5K samples in its validation set and test several black-box models on their mAP, a standard criteria in many works (He et al. (2017); Chen et al. (2019a)).
+
+All attacks are conducted with the step length $\alpha = 2$ for 10 iterations and the perturbation is $\ell _ { \infty }$ - bounded in $\varepsilon = 1 6$ to guarantee the imperceptibility as in (Dong et al. (2019)). To validate that the mAP drop comes from the attack instead of resizing or perturbation, we add large Gaussian noises $( \sigma = 9$ ) to the resized images, and report it as “Ablation”.
+
+We choose 8 typical detectors ranging from the first end-to-end detector to recent ones for attack and test. The variety of model guarantees the validity of results. We specify their information in Appendix B and the corresponding pre-processing in Appendix E.
+
+# 4.2 VISUAL RESULTS OF RAD
+
+We visualize several predictions on the same adversarial sample by black-box models in Fig. 6 to intuitively illustrate the transferability of RAD. The objects in the image, e.g., the laptop and keyboard, are quite large and obvious to detect. However, with a small perturbation from RAD, 5 black-box models all fail to detect the laptop, keyboard and mouse. Surprisingly, 4 of them even detect a non-existent “bed”, which is neither relevant nor similar in the image. The attack process of RAD is analyzed in Appendix F.
+
+![](images/f3ddcdba9a9215b81c2eaf82d628a1c96039b1c11d3762d6ba4245add204ea1f.jpg)  
+Figure 6: RAD has a great transferability. The same adversarial sample generated by attacking Mask R-CNN fools all 5 black-box detectors.
+
+# 4.3 RAD’S TRANSFERABILITY IN OBJECT DETECTION
+
+To evaluate the in-domain transferability of detection attacks and cross-domain transferability of classification attacks, we test the detection mAP of 8 models in COCO adversarial samples generated in the same setting.
+
+For detection attacks, adversarial samples are crafted by attacking surrogate model M2 (YOLOv3 Redmon & Farhadi (2018)). Results of attacking other surrogates are reported in Appendix D. For classification attacks, we use the model output on the clean sample as the label. By several state-of-the-art attacks on surrogate classifiers (InceptionV3 Szegedy et al. (2016) here as in Xie et al. (2019); Dong et al. (2019)), the adversarial samples are generated and tested the mAP as the transferability towards detectors. Details of implementation are described in Appendix E.
+
+We present the results in Table 3. Among the classification attacks and detection ones, cross-domain attack (Naseer et al. (2019)) is effective, but RAD is more aggressive. RAD enjoys a state-of-theart transferability towards most black-box models, outperforming other methods for above $20 \%$ . The detection mAPs are more than halved, making state-of-the-art detectors worse than the early single-shot detector (SSD Liu et al. (2016), M1).
+
+Table 3: Detection mAP in different attacks   
+
+<table><tr><td></td><td>Method</td><td>M1</td><td>M2</td><td>M3</td><td>M4</td><td>M5</td><td>M6</td><td>M7</td><td>M8</td><td>M9</td></tr><tr><td>Reference</td><td>No Attack Ablation</td><td>29.3 24.9</td><td>33.4 31.4</td><td>38.1 31.2</td><td>40.7 31.6</td><td>42.1 35.0</td><td>42.5 34.3</td><td>45.7 37.5</td><td>46.9 38.8</td><td>53.9 48.6</td></tr><tr><td rowspan="4">Classification Attack</td><td>PGD SI-PGD</td><td>26.4 27.5</td><td>30.4 31.6</td><td>34.4 36.1</td><td>35.4 37.1</td><td>38.4</td><td>38.3</td><td>41.7</td><td>43.1</td><td>51.1</td></tr><tr><td>MI-DI-PGD</td><td>22.9</td><td>26.2</td><td></td><td></td><td>40.0</td><td>40.1</td><td>43.5</td><td>44.8</td><td>52.4</td></tr><tr><td>MI-TI-PGD</td><td></td><td></td><td>29.3</td><td>30.0</td><td>33.2</td><td>32.1</td><td>36.0</td><td>37.5</td><td>48.0</td></tr><tr><td>CD-painting</td><td>20.1 16.4</td><td>23.7 20.8</td><td>24.9 21.3</td><td>25.4 22.8</td><td>30.1 26.6</td><td>27.4 24.5</td><td>32.8 28.9</td><td>34.5 29.5</td><td>47.1 42.3</td></tr><tr><td rowspan="3">Detection Attack</td><td>CD-comics</td><td>16.6</td><td>21.6</td><td>21.7</td><td>22.7</td><td>26.8</td><td>24.3</td><td>29.1</td><td>42.3</td><td>43.7</td></tr><tr><td>Dfool</td><td>23.3</td><td>2.5</td><td>29.2</td><td>29.8</td><td>33.3</td><td>32.9</td><td>36.5</td><td>38.0</td><td>47.5</td></tr><tr><td>Loc DAG</td><td>21.9 20.8</td><td>0.2</td><td>25.8</td><td>26.6</td><td>29.8</td><td>29.4</td><td>33.2</td><td>33.2</td><td>45.2</td></tr><tr><td>Ours</td><td>RAD</td><td>14.6</td><td>0.6 0.7</td><td>22.8 16.3</td><td>23.4 17.0</td><td>26.8 20.4</td><td>25.6 19.1</td><td>28.9 22.3</td><td>31.0 23.8</td><td>40.6 35.0</td></tr></table>
+
+The influence of $\varepsilon$ on detection mAP in RAD are displayed in Fig. 7. With the $\ell _ { \infty }$ bound increases, the resulting mAP greatly decreases for all black-box models especially for $\varepsilon$ from 8 to 12.
+
+![](images/a2da1b2cd3f09b13287065c5673de5c813feaeb1b97bca2660057b3e05804b22.jpg)  
+Figure 7: The influence of $\varepsilon$ on detection mAP in RAD
+
+# 4.4 RAD’S TRANSFERABILITY TO INSTANCE SEGMENTATION
+
+Detection and segmentation are similar in some aspects, so they could be implemented in one network (He et al. (2017); Cai & Vasconcelos (2018); Chen et al. (2019a)). Also, adversarial samples for object detection tend to transfer to instance segmentation (Xie et al. (2017)). Accordingly, we evaluate this cross-task transferability by RAD on surrogate detectors YOLOv3 (Redmon & Farhadi (2018), M2), RetinaNet (Lin et al. (2017), M3) and Mask R-CNN (He et al. (2017), M5). From the results in Table 4, we find that RAD also greatly hurts the performance of instance segmentation, leading to a drop on mAP of over $70 \%$ . This inspire the segmentation attackers to indirectly attack detectors.
+
+Table 4: Segmentation mAP of RAD   
+
+<table><tr><td></td><td colspan="2">mAP</td><td colspan="2">mAP50</td><td colspan="2"></td><td colspan="2">mAP75</td><td></td></tr><tr><td> Surrogate</td><td>M5</td><td>M7</td><td>M8</td><td>M5</td><td>M7</td><td>M8</td><td>M5</td><td>M7</td><td>M8</td></tr><tr><td>None</td><td>38.0</td><td>39.4</td><td>40.8</td><td>60.6</td><td>61.3</td><td>63.3</td><td>40.9</td><td>42.9</td><td>44.1</td></tr><tr><td>Ablation</td><td>31.0</td><td>31.9</td><td>33.5</td><td>51.2</td><td>51.0</td><td>53.7</td><td>32.4</td><td>34.3</td><td>35.4</td></tr><tr><td>M2</td><td>17.9</td><td>18.6</td><td>20.3</td><td>31.6</td><td>31.7</td><td>34.5</td><td>18.0</td><td>18.9</td><td>20.7</td></tr><tr><td>M3</td><td>11.6</td><td>11.9</td><td>12.9</td><td>19.2</td><td>19.1</td><td>20.7</td><td>12.1</td><td>12.6</td><td>13.7</td></tr><tr><td>M5</td><td>1.2</td><td>11.1</td><td>11.8</td><td>2.4</td><td>17.9</td><td>18.9</td><td>1.0</td><td>11.9</td><td>12.6</td></tr></table>
+
+# 5 ADVERSARIAL OBJECTS IN CONTEXT
+
+Given the great transferability of RAD, we create Adversarial Objects in COntext (AOCO), the first adversarial dataset for object detection. AOCO dataset serves as a potential benchmark to evaluate the robustness of detectors, which will be beneficial to network designers. It will also be useful for adversarial training, as the most effective practice to improve the robustness of DNNs Zhang et al. (2019); Tramèr et al. (2018). Notice that there is no other adversarial dataset for detection at all. This is not because the dataset is useless, but due to the low transferability of attack methods such that the examples are detector-dependent. Now we have achieved high transferability and can then make such an adversarial dataset publicly available.
+
+AOCO is generated from the full COCO 2017 validation set (Lin et al. (2014)) with $5 \mathrm { k }$ samples. It contains 5K adversarial samples for evaluating object detection (AOCO detection) and 5K for instance segmentation (AOCO segmentation). All 10K samples in AOCO are crafted by RAD. The surrogate model we attack is YOLOv3 for AOCO detection and Mask R-CNN for AOCO segmentation given the results in Table 3 and Table 4.
+
+We measure the perturbation $\Delta x$ in AOCO by Root Mean Squared Error (RMSE) as in (Xie et al. (2017); Liu et al. (2017)). It is calculated as $\sqrt { \textstyle \sum _ { i } ( \Delta x _ { i } ) ^ { 2 } / N }$ in a pixel-wise way, and $N$ is the size of the image. Performance of AOCO is reported in Table 5. The RMSE in AOCO is lower than that in (Wu et al. (2019)), and the perturbation is quite imperceptible. Details and samples of AOCO are presented in Appendix G.
+
+Table 5: Detection mAP and segmentation mAP on COCO and AOCO   
+
+<table><tr><td></td><td>RMSE</td><td>M1</td><td>M2</td><td>M3</td><td>M4</td><td>M5</td><td>M6</td><td>M7</td><td>M8</td><td>M9</td></tr><tr><td>COCO detection AOCO detection</td><td>0.000 6.469</td><td>29.3 14.6</td><td>33.4 0.7</td><td>38.1 16.3</td><td>40.7 17</td><td>42.1 20.4</td><td>42.5 19.1</td><td>45.7 22.3</td><td>46.9 23.8</td><td>53.9 35.0</td></tr><tr><td>COCO segmentation</td><td>0.000</td><td>N</td><td></td><td></td><td></td><td>38.0</td><td></td><td>39.4</td><td>40.8</td><td>\\</td></tr><tr><td> AOCO segmentation</td><td>6.606</td><td></td><td>广</td><td>广</td><td>\\</td><td>1.2</td><td>一</td><td>11.1</td><td>11.8</td><td></td></tr></table>
+
+# 6 CONCLUSION
+
+To pursue a high transferability, this paper proposes Relevance Attack on Detectors (RAD), which works by suppressing the multi-node relevance, a common property across detectors calculated by our Multi-Node SGLRP. We also thoroughly discuss where to attack and the how to update in attacking relevance maps. RAD achieves a state-of-the-art transferability towards 8 diverse black-box models, exceeding existing results by above $20 \%$ , and also significantly hurts the instance segmentation. Given the great transferability of RAD, we generate the first adversarial dataset for object detection, i.e., Adversarial Objects in COntext (AOCO), which helps to quickly evaluate and improve the robustness of detectors. Also, attacking other common properties is promising for a good transferability.
+
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+# A RELEVANCE BACK-PROPAGATION RULES (DTD)
+
+DTD-based network visualization methods, such as LRP, CLRP and SGLRP, back-propagate the relevance from the output layer to the input layer according to the rules specified in this section. Their only difference is the relevance in the initial output layer $\bar { R } _ { n } ^ { ( L ) }$ .
+
+For each layer $l$ in a DNN with $L$ layers in total, suppose layer $l$ has $N$ nodes and layer $l + 1$ has $M$ nodes, the relevance $R _ { n } ^ { ( L ) }$ at node $n$ in layer $l$ is defined recursively by
+
+$$
+R _ { n } ^ { ( l ) } = \sum _ { m } \frac { a _ { n } ^ { ( l ) } w _ { n , m } ^ { + ( l ) } } { \sum _ { n ^ { \prime } } a _ { n ^ { \prime } } ^ { ( l ) } w _ { n ^ { \prime } , m } ^ { + ( l ) } } R _ { m } ^ { ( l + 1 ) } ,
+$$
+
+for nodes with definite positive values (such as after ReLU), and
+
+$$
+R _ { n } ^ { ( l ) } = \sum _ { m } \frac { z _ { n } ^ { ( l ) } w _ { n , m } ^ { ( l ) } - b _ { n } ^ { ( l ) } w _ { n , m } ^ { + ( l ) } - h _ { n } ^ { ( l ) } w _ { n , m } ^ { - ( l ) } } { \sum _ { n ^ { \prime } } z _ { n ^ { \prime } } ^ { ( l ) } w _ { n ^ { \prime } , m } ^ { ( l ) } - b _ { n ^ { \prime } } ^ { ( l ) } w _ { n ^ { \prime } , m } ^ { + ( l ) } - h _ { n ^ { \prime } } ^ { ( l ) } w _ { n ^ { \prime } , m } ^ { - ( l ) } } R _ { m } ^ { ( l + 1 ) } ,
+$$
+
+for nodes that may have negative values. In the formulas above, $a _ { n } ^ { ( l ) }$ is the post-activation output of node in layer $l$ and $z _ { n } ^ { ( l ) }$ is the pre-activation one. The range $[ b _ { n } ^ { ( l ) } , h _ { n } ^ { ( l ) } ]$ stands for the minimum and maximum of $z _ { n } ^ { ( l ) }$ . Finally, $w _ { n , m } ^ { + ( l ) } = \operatorname* { m a x } \left( w _ { n , m , 0 } ^ { ( l ) } \right)$ and $w _ { n , m } ^ { - ( l ) } = \operatorname* { m i n } \left( w _ { n , m , 0 } ^ { ( l ) } \right)$ .
+
+According to the propagation rules above as mentioned in (Iwana et al. (2019)), we could naturally obtain the attack gradients as
+
+$$
+\frac { \partial R _ { m } ^ { ( l + 1 ) } } { \partial R _ { n } ^ { ( l ) } } = \left\{ \begin{array} { l l } { \big ( \sum _ { m } \frac { a _ { n } ^ { ( l ) } w _ { n , m } ^ { + ( l ) } } { \sum _ { n ^ { \prime } } a _ { n ^ { \prime } } ^ { ( l ) } w _ { n ^ { \prime } , m } ^ { + ( l ) } } \big ) ^ { - 1 } , \mathrm { f o r ~ n o d e s ~ w i t h ~ d e f i n i t e ~ p o s i t i v e ~ v a l u e s } } \\ { \big ( \sum _ { m } \frac { z _ { n } ^ { ( l ) } w _ { n , m } ^ { ( l ) } - b _ { n } ^ { ( l ) } w _ { n , m } ^ { + ( l ) } - h _ { n } ^ { ( l ) } w _ { n , m } ^ { - ( l ) } } { \sum _ { n ^ { \prime } } z _ { n ^ { \prime } } ^ { ( l ) } w _ { n ^ { \prime } , m } ^ { ( l ) } - b _ { n ^ { \prime } } ^ { ( l ) } w _ { n ^ { \prime } , m } ^ { + ( l ) } - h _ { n ^ { \prime } } ^ { ( l ) } w _ { n ^ { \prime } , m } ^ { - ( l ) } } \big ) ^ { - 1 } , \mathrm { o t h e r w i s e } } \end{array} \right. .
+$$
+
+# B MODEL INFORMATION
+
+Table 6 presents the models’ information from our evaluation and MMdetection (Chen et al. (2019b)).
+
+Table 6: Model backbone and mAPs   
+
+<table><tr><td>ID</td><td>Model</td><td>Backbone</td><td>mAP</td></tr><tr><td>M1</td><td>SSD512 (Liu et al. (2016))</td><td>VGG16</td><td>29.3</td></tr><tr><td>M2</td><td>YOLOv3 (Redmon &amp; Farhadi (2018))</td><td>Darknet</td><td>33.4</td></tr><tr><td>M3</td><td>RetinaNet (Lin et al. (2017))</td><td>ResNet-101</td><td>38.1</td></tr><tr><td>M4</td><td>Faster R-CNN (Ren et al. (2015))</td><td>ResNeXt-101-64*4d</td><td>40.7</td></tr><tr><td>M5</td><td>Mask R-CNN (He et al. (2017))</td><td>ResNeXt-101-64*4d</td><td>42.1</td></tr><tr><td>M6</td><td>Cascade RCNN (Cai &amp; Vasconcelos (2018))</td><td>ResNet-101</td><td>42.5</td></tr><tr><td>M7</td><td>Cascade Mask R-CNN (Cai &amp; Vasconcelos (2018))</td><td>ResNeXt-101-64*4d</td><td>45.7</td></tr><tr><td>M8</td><td>Hyrbrid Task Cascade (Chen et al. (2019a))</td><td>ResNeXt-101-64*4d</td><td>46.9</td></tr><tr><td>M9</td><td>EfficientDet (Tan et al. (2020))</td><td>EfficientNet+BiFPN</td><td>53.9</td></tr></table>
+
+# C INFLUENCE OF HYPER-PARAMETERS IN NODE SELECTION
+
+Performance of RAD is not sensitive to hyper-parameter, no matter the strategy to select bounding boxes is dynamic or static as Table 7. Attackers are not bothered to tune them carefully. The parameter for dynamic strategy refers to the pre-softmax confidence threshold to select a bounding box. The parameter for static strategy refers to the fixed number of selected bounding boxes in each iteration.
+
+# D RAD ON MORE SURROGATES
+
+The results on attacking more surrogates by RAD are reported in Table 8.
+
+Table 7: Detection mAP in different hyper-parameters in RAD   
+
+<table><tr><td>Strategy</td><td>Parameter</td><td>M1</td><td>M2</td><td>M3</td><td>M4</td><td>M5</td><td>M6</td><td>M7</td><td>M8</td><td>M9</td></tr><tr><td rowspan="3">Dynamic</td><td>-1</td><td>18.3</td><td>1.2</td><td>20.0</td><td>20.6</td><td>24.3</td><td>22.8</td><td>26.7</td><td>28.3</td><td>40.0</td></tr><tr><td>-2</td><td>18.2</td><td>1.2</td><td>20.1</td><td>20.7</td><td>24.2</td><td>22.8</td><td>26.6</td><td>28.0</td><td>39.8</td></tr><tr><td>-3</td><td>18.4</td><td>1.3</td><td>20.3</td><td>20.8</td><td>24.3</td><td>22.9</td><td>26.7</td><td>28.5</td><td>40.2</td></tr><tr><td rowspan="3">Static</td><td>10</td><td>18.2</td><td>1.1</td><td>19.9</td><td>20.5</td><td>24.2</td><td>22.8</td><td>26.2</td><td>28.0</td><td>39.8</td></tr><tr><td>20</td><td>18.1</td><td>1.2</td><td>19.9</td><td>20.5</td><td>24.3</td><td>22.6</td><td>26.4</td><td>28.2</td><td>39.9</td></tr><tr><td>30</td><td>18.2</td><td>1.3</td><td>20.1</td><td>20.7</td><td>24.3</td><td>22.8</td><td>26.0</td><td>27.9</td><td>40.1</td></tr></table>
+
+Table 8: Detection mAP of RAD on different surrogates   
+
+<table><tr><td> Surrogate</td><td>M1</td><td>M2</td><td>M3</td><td>M4</td><td>M5</td><td>M6</td><td>M7</td><td>M8</td><td>M9</td></tr><tr><td>YOLOv3</td><td>14.6</td><td>0.7</td><td>16.3</td><td>17.0</td><td>20.4</td><td>19.1</td><td>22.3</td><td>23.8</td><td>35.0</td></tr><tr><td>Mask R-CNN</td><td>20.4</td><td>24.2</td><td>25.7</td><td>26.5</td><td>1.1</td><td>28.9</td><td>33.1</td><td>34.9</td><td>45.4</td></tr><tr><td>RetinaNet</td><td>20.7</td><td>6.1</td><td>2.3</td><td>25.7</td><td>29.3</td><td>28.2</td><td>31.7</td><td>33.8</td><td>44.1</td></tr></table>
+
+# E IMPLEMENTATION DETAILS
+
+# E.1 PRE-PROCESSING
+
+To pre-process, we resize the image with its long side as 416 for YOLOv3 or RetinaNet and 448 for Mask R-CNN, and then zero-pad it to a square. The resolution is kept relatively the same for a fair evaluation. Images are normalized to [0,1] in YOLOv3 or subtracted by the mean of COCO training set in RetinaNet and Mask R-CNN. Accordingly, samples in AOCO detection have the long side 416 and that for AOCO segmentation is 448.
+
+# E.2 TRANSFER-ENHANCING UPDATE TECHNIQUES
+
+DI (Xie et al. (2019)) transforms the image for 4 times with probability $p$ $\mathbf { \boldsymbol { p } } = 1$ for better transferability as suggested) and averaging the gradients. The transformation is to resize the image to $0 . 9 \times$ its size and randomly padding the outer areas with white pixels. SI (Lin et al. (2020)) divides the sample numerically by the power 2 for 4 times and averages the 4 obtained gradients. TI (Dong et al. (2019)) translates the image to calculate the augmented gradients. To implement it efficiently, it adopts a kernel to simulate the averaging of gradients. We choose the kernel size 15 as suggested. MI (Dong et al. (2018)) uses momentum optimization (parameter $\mu = 1$ as suggested) for a better transferability and a faster attack. Cross-domain attack (Naseer et al. (2019)) uses extra datasets (paintings, denoted as CD-paintings, and comics, denoted as CD-comics) to train a perturbation generator with the relative loss. The adopted surrogate model is also InceptionV3 for consistency. All perturbations are resized to fit the sample size.
+
+# E.3 DETECTION ATTACKS
+
+For DAG (Xie et al. (2017)), we follow the setting of generating dense proposals. The classification probabilities of 3000 bounding boxes with highest confidence are attacked. But we alter its optimization to (5) because its original update produces quite small perturbation, leading to a poor transferability, which is unfair for comparison. Dfool (Lu et al. (2017)) suppresses the classification confidence for the original bounding boxes, which is the same in our experiment. Localization loss is shown to be useful in (Zhang & Wang (2019)), and here we suppress the width and height of the original bounding boxes.
+
+# F VISUAL RESULTS OF RAD PROCESS
+
+By RAD, the relevance map is attacked to be meaningless and loss its focus. In Fig. 8, the initial prediction is correct and the relevance map is clear. RAD constantly misleads the relevance map to be unstructured without outline of objects. Finally, all bounding boxes vanish.
+
+![](images/95ea87cc03ebb9e5d0232975306185915289b6e281665271ac23ef4ee01c0c56.jpg)  
+Figure 8: Transition of prediction and relevance map in RAD (from top to bottom and left to right).
+
+# G MORE ABOUT AOCO
+
+We report the mAP50 and mAP75 of AOCO in Table 9 and Table 10. We show in Fig. 9 the visual comparative results.
+
+Table 9: Detection mAP50 and segmentation mAP50 on COCO and AOCO   
+
+<table><tr><td></td><td>M1</td><td>M2</td><td>M3</td><td>M4</td><td>M5</td><td>M6</td><td>M7</td><td>M8</td><td>M9</td></tr><tr><td>COCO detection AOCO detection</td><td>49.2 26.7</td><td>56.4 1.6</td><td>58.1 27.6</td><td>62.0 29.1</td><td>63.8 34.5</td><td>60.7 29.9</td><td>64.1 34.3</td><td>66.0 37.4</td><td>74.3 51.7</td></tr><tr><td>COCO segmentation</td><td></td><td></td><td>人</td><td>一</td><td>60.6</td><td>一</td><td>61.3</td><td>63.3</td><td>一</td></tr><tr><td>AOCO segmentation</td><td></td><td></td><td></td><td></td><td>2.4</td><td>一</td><td>17.9</td><td>18.9</td><td>一</td></tr></table>
+
+Table 10: Detection mAP75 and segmentation mAP75 on COCO and AOCO   
+
+<table><tr><td></td><td>M1</td><td>M2</td><td>M3</td><td>M4</td><td>M5</td><td>M6</td><td>M7</td><td>M8</td><td>M9</td></tr><tr><td>COCO detection AOCO detection</td><td>30.8 14.2</td><td>35.8 0.6</td><td>40.6 16.5</td><td>44.6 17.1</td><td>46.3 20.8</td><td>46.3 19.9</td><td>50.0 23.3</td><td>51.2 24.6</td><td>59.9 37.4</td></tr><tr><td>COCO segmentation</td><td></td><td></td><td>/</td><td>/</td><td>40.9</td><td>一</td><td>42.9</td><td>44.1</td><td>一</td></tr><tr><td>AOCO segmentation</td><td></td><td></td><td></td><td></td><td>1.0</td><td>/</td><td>11.9</td><td>12.6</td><td></td></tr></table>
+
+![](images/93f75b8037c78908db07fd96e74d56484522f7c49490b9643c1b77d5e76d509b.jpg)  
+Figure 9: Detection and segmentation results in COCO and AOCO by YOLOv3 and Mask R-CNN. For COCO, both networks predict correctly. For AOCO segmentation results, the top image contains two big masks for “chair” and “potted plan”; the second image contains one false mask for “sports ball”; the bottom image contains “dog” in green, “car” in purple and “elephant” in red.

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