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+    {
+        "type": "text",
+        "text": "A CRITICAL ANALYSIS OF SELF-SUPERVISION, ORWHAT WE CAN LEARN FROM A SINGLE IMAGE",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Yuki M. Asano Christian Rupprecht ",
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+    {
+        "type": "text",
+        "text": "Andrea Vedaldi ",
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+    {
+        "type": "text",
+        "text": "Visual Geometry Group   \nUniversity of Oxford   \n{yuki,chrisr,vedaldi}@robots.ox.ac.uk ",
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+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+        "text": "We look critically at popular self-supervision techniques for learning deep convolutional neural networks without manual labels. We show that three different and representative methods, BiGAN, RotNet and DeepCluster, can learn the first few layers of a convolutional network from a single image as well as using millions of images and manual labels, provided that strong data augmentation is used. However, for deeper layers the gap with manual supervision cannot be closed even if millions of unlabelled images are used for training. We conclude that: (1) the weights of the early layers of deep networks contain limited information about the statistics of natural images, that (2) such low-level statistics can be learned through self-supervision just as well as through strong supervision, and that (3) the low-level statistics can be captured via synthetic transformations instead of using a large image dataset. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Despite tremendous progress in supervised learning, learning without external supervision remains difficult. Self-supervision has recently emerged as one of the most promising approaches to address this limitation. Self-supervision builds on the fact that convolutional neural networks (CNNs) transfer well between tasks (Shin et al., 2016; Oquab et al., 2014; Girshick, 2015; Huh et al., 2016). The idea then is to pre-train networks via pretext tasks that do not require expensive manual annotations and can be automatically generated from the data itself. Once pre-trained, networks can be applied to a target task by using only a modest amount of labelled data. ",
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+        "type": "text",
+        "text": "Early successes in self-supervision have encouraged authors to develop a large variety of pretext tasks, from colorization to rotation estimation and image autoencoding. Recent papers have shown performance competitive with supervised learning by learning complex neural networks on very large image datasets. Nevertheless, for a given model complexity, pre-training by using an off-theshelf annotated image datasets such as ImageNet remains much more efficient. ",
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+        "text": "In this paper, we aim to investigate the effectiveness of current self-supervised approaches by characterizing how much information they can extract from a given dataset of images. Since deep networks learn a hierarchy of representations, we further break down this investigation on a per-layer basis. We are motivated by the fact that the first few layers of most networks extract low-level information (Yosinski et al., 2014), and thus learning them may not require the high-level semantic information captured by manual labels. ",
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+        "text": "Concretely, in this paper we answer the following simple question: “is self-supervision able to exploit the information contained in a large number of images in order to learn different parts of a neural network?” ",
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+        "text": "We contribute two key findings. First, we show that as little as a single image is sufficient, when combined with self-supervision and data augmentation, to learn the first few layers of standard deep networks as well as using millions of images and full supervision (Figure 1). Hence, while selfsupervised learning works well for these layers, this may be due more to the limited complexity of such features than the strength of the supervisory technique. This also confirms the intuition that early layers in a convolutional network amounts to low-level feature extractors, analogous to early learned and hand-crafted features for visual recognition (Olshausen & Field, 1997; Lowe, 2004; Dalal & Triggs, 2005). Finally, it demonstrates the importance of image transformations in learning such low-level features as opposed to image diversity.1 ",
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+        "img_path": "images/170a22e5e719a53efa60993ee606484b35b4465c1a6a9fe2204b521c8d69c3b5.jpg",
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+        "image_caption": [
+            "Figure 1: Single-image self-supervision. We show that several self-supervision methods can be used to train the first few layers of a deep neural networks using a single training image, such as this Image A, B or even C (above), provided that sufficient data augmentation is used. "
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+        "type": "text",
+        "text": "Our second finding is about the deeper layers of the network. For these, self-supervision remains inferior to strong supervision even if millions of images are used for training. Our finding is that this is unlikely to change with the addition of more data. In particular, we show that training these layers with self-supervision and a single image already achieves as much as two thirds of the performance that can be achieved by using a million different images. ",
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+        "text": "We show that these conclusions hold true for three different self-supervised methods, BiGAN (Donahue et al., 2017), RotNet (Gidaris et al., 2018) and DeepCluster (Caron et al., 2018), which are representative of the spectrum of techniques that are currently popular. We find that performance as a function of the amount of data is dependent on the method, but all three methods can indeed leverage a single image to learn the first few layers of a deep network almost “perfectly”. ",
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+        "text": "Overall, while our results do not improve self-supervision per-se, they help to characterize the limitations of current methods and to better focus on the important open challenges. ",
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+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+        "text": "Our paper relates to three broad areas of research: (a) self-supervised/unsupervised learning, (b) learning from a single sample, and (c) designing/learning low-level feature extractors. We discuss closely related work for each. ",
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+        "text": "Self-supervised learning: A wide variety of proxy tasks, requiring no manual annotations, have been proposed for the self-training of deep convolutional neural networks. These methods use various cues and tasks namely, in-painting (Pathak et al., 2016), patch context and jigsaw puzzles (Doersch et al., 2015; Noroozi & Favaro, 2016; Noroozi et al., 2018; Mundhenk et al., 2017), clustering (Caron et al., 2018), noise-as-targets (Bojanowski & Joulin, 2017), colorization (Zhang et al., 2016; Larsson et al., 2017), generation (Jenni & Favaro, 2018; Ren & Lee, 2018; Donahue et al., 2017), geometry (Dosovitskiy et al., 2016; Gidaris et al., 2018) and counting (Noroozi et al., 2017). The idea is that the pretext task can be constructed automatically and easily on images alone. Thus, methods often modify information in the images and require the network to recover them. Inpainting or colorization techniques fall in this category. However these methods have the downside that the features are learned on modified images which potentially harms the generalization to unmodified ones. For example, colorization uses a gray scale image as input, thus the network cannot learn to extract color information, which can be important for other tasks. ",
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+    {
+        "type": "text",
+        "text": "Slightly less related are methods that use additional information to learn features. Here, often temporal information is used in the form of videos. Typical pretext tasks are based on temporalcontext (Misra et al., 2016; Wei et al., 2018; Lee et al., 2017; Sermanet et al., 2018), spatio-temporal cues (Isola et al., 2015; Gao et al., 2016; Wang et al., 2017), foreground-background segmentation via video segmentation (Pathak et al., 2017), optical-flow (Gan et al., 2018; Mahendran et al., 2018), future-frame synthesis (Srivastava et al., 2015), audio prediction from video (de Sa, 1994; Owens et al., 2016), audio-video alignment (Arandjelovic & Zisserman ´ , 2017), ego-motion estimation (Jayaraman & Grauman, 2015), slow feature analysis with higher order temporal coherence (Jayaraman & Grauman, 2016), transformation between frames (Agrawal et al., 2015) and patch tracking in videos (Wang & Gupta, 2015). Since we are interested in learning features from as little data as one image, we cannot make use of methods that rely on video input. ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Our contribution inspects three unsupervised feature learning methods that use very different means of extracting information from the data: BiGAN (Donahue et al., 2017) utilizes a generative adversarial task, RotNet (Gidaris et al., 2018) exploits the photographic bias in the dataset and DeepCluster (Caron et al., 2018) learns stable feature representations under a number of image transformations by proxy labels obtained from clustering. These are described in more detail in the Methods section. ",
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+    {
+        "type": "text",
+        "text": "Learning from a single sample: In some applications of computer vision, the bold idea of learning from a single sample comes out of necessity. For general object tracking, methods such as max margin correlation filters (Rodriguez et al., 2013) learn robust tracking templates from a single sample of the patch. A single image can also be used to learn and interpolate multi-scale textures with a GAN framework (Rott Shaham et al., 2019). Single sample learning was pursued by the semi-parametric exemplar SVM model (Malisiewicz et al., 2011). They learn one SVM per positive sample separating it from all negative patches mined from the background. While only one sample is used for the positive set, the negative set consists of thousands of images and is a necessary component of their method. The negative space was approximated by a multi-dimensional Gaussian by the Exemplar LDA (Hariharan et al., 2012). These SVMs, one per positive sample, are pooled together using a max aggregation. We differ from both of these approaches in that we do not use a large collection of negative images to train our model. Instead we restrict ourselves to a single or a few images with a systematic augmentation strategy. ",
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+    {
+        "type": "text",
+        "text": "Classical learned and hand-crafted low-level feature extractors: Learning and hand-crafting features pre-dates modern deep learning approaches and self-supervision techniques. For example the classical work of (Olshausen & Field, 1997) shows that edge-like filters can be learned via sparse coding of just 10 natural scene images. SIFT (Lowe, 2004) and HOG (Dalal & Triggs, 2005) have been used extensively before the advent of convolutional neural networks and, in many ways, they resemble the first layers of these networks. The scatter transform of Bruna & Mallat (2013); Oyallon et al. (2017) is an handcrafted design that aims at replacing at least the first few layers of a deep network. While these results show that effective low-level features can be handcrafted, this is insufficient to clarify the power and limitation of self-supervision in deep networks. For instance, it is not obvious whether deep networks can learn better low level features than these, how many images may be required to learn them, and how effective self-supervision may be in doing so. For instance, as we also show in the experiments, replacing low-level layers in a convolutional networks with handcrafted features such as Oyallon et al. (2017) may still decrease the overall performance of the model. Furthermore, this says little about deeper layers, which we also investigate. ",
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+    {
+        "type": "text",
+        "text": "In this work we show that current deep learning methods learn slightly better low-level representations than hand crafted features such as the scattering transform. Additionally, these representations can be learned from one single image with augmentations and without supervision. The results show how current self-supervised learning approaches that use one million images yield only relatively small gains when compared to what can be achieved from one image and augmentations, and motivates a renewed focus on augmentations and incorporating prior knowledge into feature extractors. ",
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+    {
+        "type": "text",
+        "text": "3 METHODS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We discuss first our data and data augmentation strategy (section 3.1) and then we summarize the three different methods for unsupervised feature learning used in the experiments (section 3.2). ",
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+        "text": "3.1 DATA ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our goal is to understand the performance of representation learning methods as a function of the image data used to train them. To make comparisons as fair as possible, we develop a protocol where only the nature of the training data is changed, but all other parameters remain fixed. ",
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+        "text": "In order to do so, given a baseline method trained on $d$ source images, we replace those with another set of $d$ images. Of these, now only $N \\ll d$ are source images (i.e. i.i.d. samples), while the remaining $d - N$ are augmentations of the source ones. Thus, the amount of information in the training data is controlled by $N$ and we can generate a continuum of datasets that vary from one extreme, utilizing a single source image $N = 1$ , to the other extreme, using all $N { = } d$ original training set images. For example, if the baseline method is trained on ImageNet, then $d = 1 , 2 8 1 , 1 6 7$ . When $N = 1$ , it means that we train the method using a single source image and generate the remaining 1,281,166 images via augmentation. Other baselines use CIFAR-10/100 images, so in those cases $d = 5 0 , 0 0 0$ instead. ",
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+        "type": "text",
+        "text": "The data augmentation protocol, is an extreme version of augmentations already employed by most deep learning protocols. Each method we test, in fact, already performs some data augmentation internally. Thus, when the method is applied on our augmented data, this can be equivalently thought of as incrementing these “native” augmentations by concatenating them with our own. ",
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+        "type": "text",
+        "text": "Choice of augmentations. Next, we describe how the $N$ source images are expanded to additional $d - N$ images so that the models can be trained on exactly $d$ images, independent from the choice of $N$ . The idea is to use an aggressive form of data augmentation involving cropping, scaling, rotation, contrast changes, and adding noise. These transformations are representative of invariances that one may wish to incorporate in the features. Augmentation can be seen as imposing a prior on how we expect the manifold of natural images to look like. When training with very few images, these priors become more important since the model cannot extract them directly from data. ",
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+        "type": "text",
+        "text": "Given a source image of size size $H \\times W$ , we first extract a certain number of random patches of size $( w , h )$ , where $w \\leq W$ and $h \\leq H$ satisfy the additional constraints $\\beta \\le \\frac { w h } { W H }$ and $\\begin{array} { r } { \\gamma \\leq \\frac { h } { w } \\leq \\gamma ^ { - 1 } } \\end{array}$ . Thus, the smallest size of the crops is limited to be at least $\\beta W H$ and at most the whole image. Additionally, changes to the aspect ratio are limited by $\\gamma$ . In practice we use $\\beta = 1 0 ^ { - 3 }$ and $\\textstyle \\gamma = { \\frac { 3 } { 4 } }$ . ",
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+        "type": "text",
+        "text": "Second, good features should not change much by small image rotations, so images are rotated (before cropping to avoid border artifacts) by $\\alpha \\in \\mathsf { \\Gamma } ( - 3 5 , 3 5 )$ degrees. Due to symmetry in image statistics, images are also flipped left-to-right with $50 \\%$ probability. ",
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+    {
+        "type": "text",
+        "text": "Illumination changes are common in natural images, we thus expect image features to be robust to color and contrast changes. Thus, we employ a set of linear transformations in RGB space to model this variability in real data. Additionally, the color/intensity of single pixels should not affect the feature representation, as this does not change the contents of the image. To this end, color jitter with additive brightness, contrast and saturation are sampled from three uniform distributions in (0.6, 1.4) and hue noise from $( - 0 . 1 , 0 . 1 )$ is applied to the image patches. Finally, the cropped and transformed patches are scaled to the color range $( - 1 , 1 )$ and then rescaled to full $S \\times S$ resolution to be supplied to each representation learning method, using bilinear interpolation. This formulation ensures that the patches are created in the target resolution $S$ , independent from the size and aspect ratio $W , H$ of the source image. ",
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+        "type": "text",
+        "text": "Real samples. The images used for the $N { = } 1$ and $N { = } 1 0$ experiments are shown in Figure 1 and the appendix respectively (this is all the training data used in such experiments). For the special case of using a single training image, i.e. $N { = } 1$ , we have chosen one photographic $( 2 5 6 0 \\times 1 9 2 0 )$ ) and one drawn image $( 6 0 0 \\times 2 2 5 )$ , which we call Image A and Image $B$ , respectively. The two images were manually selected as they contain rich texture and are diverse, but their choice was not optimized for performance. We test only two images due to the cost of running a full set of experiments (each image is expanded up to $1 . 2 \\mathbf { M }$ times for training some of the models, as explained above). However, this is sufficient to prove our main points. We also test another $( 1 1 6 5 \\times 5 8 5 )$ ) Image $C$ to ablate the “crowdedness” of an image, as this latter contains large areas covering no objects. While resolution matters to some extent as a bigger image contains more pixels, the information within is still far more correlated, and thus more redundant than sampling several smaller images. In particular, the resolution difference in Image $\\mathtt { A }$ and $_ \\mathrm { B }$ appears to be negligible in our experiments. For CIFAR-10, where $S = 3 2$ we only use Image B due to the resolution difference. In direct comparison, Image B is the size of about 132 CIFAR images which is still much less than $d = 5 0 { , } 0 0 0$ . For $N > 1$ , we select the source images randomly from each method’s training set. ",
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+    {
+        "type": "text",
+        "text": "3.2 REPRESENTATION LEARNING METHODS ",
+        "text_level": 1,
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+        "text": "Generative models. Generative Adversarial Networks (GANs) (Goodfellow et al., 2014) learn to generate images using an adversarial objective: a generator network maps noise samples to image samples, approximating a target image distribution and a discriminator network is tasked with distinguishing generated and real samples. Generator and discriminator are pitched one against the other and learned together; when an equilibrium is reached, the generator produces images indistinguishable (at least from the viewpoint of the discriminator) from real ones. ",
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+    {
+        "type": "text",
+        "text": "Bidirectional Generative Adversarial Networks (BiGAN) (Donahue et al., 2017; Dumoulin et al., 2016) are an extension of GANs designed to learn a useful image representation as an approximate inverse of the generator through joint inference on an encoding and the image. This method’s native augmentation uses random crops and random horizontal flips to learn features from $S = 1 2 8$ sized images. As opposed to the other two methods discussed below it employs leaky ReLU nonlinearities as is typical in GAN discriminators. ",
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+        "text": "Rotation. Most image datasets contain pictures that are ‘upright’ as this is how humans prefer to take and look at them. This photographer bias can be understood as a form of implicit data labelling. RotNet (Gidaris et al., 2018) exploits this by tasking a network with predicting the upright direction of a picture after applying to it a random rotation multiple of 90 degrees (in practice this is formulated as a 4-way classification problem). The authors reason that the concept of ‘upright’ requires learning high level concepts in the image and hence this method is not vulnerable to exploiting low-level visual information, encouraging the network to learn more abstract features. In our experiments, we test this hypothesis by learning from impoverished datasets that may lack the photographer bias. The native augmentations that RotNet uses on the $S { = } 2 5 6$ inputs only comprise horizontal flips and non-scaled random crops to $2 2 4 \\times 2 2 4$ . ",
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+    {
+        "type": "text",
+        "text": "Clustering. DeepCluster (Caron et al., 2018) is a recent state-of-the-art unsupervised representation learning method. This approach alternates $k$ -means clustering to produce pseudo-labels for the data and feature learning to fit the representation to these labels. The authors attribute the success of the method to the prior knowledge ingrained in the structure of the convolutional neural network (Ulyanov et al., 2018). ",
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+        "type": "text",
+        "text": "The method alternatives between a clustering step, in which $k$ -means is applied on the PCA-reduced features with ID for each i $k = 1 0 ^ { 4 }$ , and a learning step, in which the network is trainedder a set of augmentations (random resized crops with usterand $\\beta { \\dot { = } } 0 . 0 8 , \\gamma = { \\textstyle \\frac { 3 } { 4 } }$ horizontal flips) that constitute its native augmentations used on top of the $S { = } 2 5 6$ input images. ",
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+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+        "text": "We evaluate the representation learning methods on ImageNet and CIFAR-10/100 using linear probes (Section 4.1). After ablating various choices of transformations in our augmentation protocol (Section 4.2), we move to the core question of the paper: whether a large dataset is beneficial to unsupervised learning, especially for learning early convolutional features (Section 4.3). ",
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+    {
+        "type": "text",
+        "text": "4.1 LINEAR PROBES AND BASELINE ARCHITECTURE ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In order to quantify if a neural network has learned useful feature representations, we follow the standard approach of using linear probes (Zhang et al., 2017). This amounts to solving a difficult task such as ImageNet classification by training a linear classifier on top of pre-trained feature representations, which are kept fixed. Linear classifiers heavily rely on the quality of the representation since their discriminative power is low. ",
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+        "type": "text",
+        "text": "We apply linear probes to all intermediate convolutional layers of networks and train on the ImageNet LSVRC-12 (Deng et al., 2009) and CIFAR-10/100 (Krizhevsky, 2009) datasets, which are the standard benchmarks for evaluation in self-supervised learning. Our base encoder architecture is AlexNet (Krizhevsky et al., 2012) with BatchNorm, since this is a good representative model and is most often used in other unsupervised learning work for the purpose of benchmarking. This model has five convolutional blocks (each comprising a linear convolution later followed by ReLU and optionally max pooling). We insert the probes right after the ReLU layer in each block, and denote these entry points conv1 to conv5. Applying the linear probes at each convolutional layer allows studying the quality of the representation learned at different depths of the network. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/8f49cf50080a267d745206dbd0f3fd7fb56e91ae459841585e50dd72d5746eb5.jpg",
+        "table_caption": [
+            "Table 1: Ablating data augmentation using MonoGAN (left). Training a linear classifier on the features extracted at different depths of the network for CIFAR-10. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"4\">CIFAR-10</td></tr><tr><td></td><td>conv1</td><td>conv2</td><td>conv3</td><td>conv4</td></tr><tr><td>(a） Fully sup.</td><td>66.5</td><td>70.1</td><td>72.4</td><td>75.9</td></tr><tr><td>(b) Random feat.</td><td>57.8</td><td>55.5</td><td>54.2</td><td>47.3</td></tr><tr><td>（c） No aug.</td><td>57.9</td><td>56.2</td><td>54.2</td><td>47.8</td></tr><tr><td>d） Jitter</td><td>58.9</td><td>58.0</td><td>57.0</td><td>49.8</td></tr><tr><td>通 Rotation</td><td>61.4</td><td>58.8</td><td>56.1</td><td>47.5</td></tr><tr><td>(f) Scale</td><td>67.9</td><td>69.3</td><td>67.9</td><td>59.1</td></tr><tr><td>（g） Rot.&amp;jitter</td><td>64.9</td><td>63.6</td><td>61.0</td><td>53.4</td></tr><tr><td>? Rot.&amp; scale</td><td>67.6</td><td>69.9</td><td>68.0</td><td>60.7</td></tr><tr><td>i Jitter &amp; scale</td><td>68.1</td><td>71.3</td><td>69.5</td><td>62.4</td></tr><tr><td>i All</td><td>68.1</td><td>72.3</td><td>70.8</td><td>63.5</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/652e1c5ace61ceee759f48ca8164a750521ce098c222c7bbe403837161179ea0.jpg",
+        "table_caption": [
+            "Table 2: ImageNet LSVRC-12 linear probing evaluation (below). A linear classifier is trained on the (downsampled) activations of each layer in the pretrained model. We report classification accuracy averaged over 10 crops. The ‡ indicated that numbers are taken from (Zhang et al., 2017). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method, Reference</td><td rowspan=\"2\"></td><td colspan=\"6\">ILSVRC-12</td></tr><tr><td>#images</td><td>conv1</td><td>conv2</td><td>conv3</td><td>conv4</td><td>conv5</td></tr><tr><td>(a)</td><td>Full-supervision‡</td><td>1,281,167</td><td>19.3</td><td>36.3</td><td>44.2</td><td>48.3</td><td>50.5</td></tr><tr><td>(b)</td><td>(Oyallon et al., 2017): Scattering</td><td>0</td><td>-</td><td>18.9</td><td>-</td><td>1</td><td>1</td></tr><tr><td>(c)</td><td>Random*</td><td>0</td><td>11.6</td><td>17.1</td><td>16.9</td><td>16.3</td><td>14.1</td></tr><tr><td>(d)</td><td>(Krahenbuhl et al., 2016):k-means‡</td><td>~160</td><td>17.5</td><td>23.0</td><td>24.5</td><td>23.2</td><td>20.6</td></tr><tr><td>(e)</td><td>(Donahue et al., 2017): BiGAN‡</td><td>1,281,167</td><td>17.7</td><td>24.5</td><td>31.0</td><td>29.9</td><td>28.0</td></tr><tr><td>(f)</td><td>mono,Image A</td><td>1</td><td>20.4</td><td>30.9</td><td>33.4</td><td>28.4</td><td>16.0</td></tr><tr><td>（g)</td><td>mono, Image B</td><td>1</td><td>20.5</td><td>30.4</td><td>31.6</td><td>27.0</td><td>16.8</td></tr><tr><td>(h)</td><td>deka</td><td>10</td><td>16.2</td><td>16.5</td><td>16.5</td><td>13.1</td><td>7.5</td></tr><tr><td>(i</td><td>kilo</td><td>1,000</td><td>16.1</td><td>17.7</td><td>18.3</td><td>17.6</td><td>13.5</td></tr><tr><td>i</td><td>(Gidaris et al., 2018): RotNet</td><td>1,281,167</td><td>18.8</td><td>31.7</td><td>38.7</td><td>38.2</td><td>36.5</td></tr><tr><td>()</td><td>mono,Image A</td><td>1</td><td>19.9</td><td>30.2</td><td>30.6</td><td>27.6</td><td>21.9</td></tr><tr><td>1</td><td>mono, Image B</td><td>1</td><td>17.8</td><td>27.6</td><td>27.9</td><td>25.4</td><td>20.2</td></tr><tr><td>(m)</td><td>deka</td><td>10</td><td>19.6</td><td>30.7</td><td>32.6</td><td>28.9</td><td>22.6</td></tr><tr><td>(n)</td><td>kilo</td><td>1,000</td><td>21.0</td><td>33.5</td><td>36.5</td><td>34.0</td><td>29.4</td></tr><tr><td>(0)</td><td>(Caron et al., 2018): DeepCluster</td><td>1,281,167</td><td>18.0</td><td>32.5</td><td>39.2</td><td>37.2</td><td>30.6</td></tr><tr><td>(p)</td><td>mono, Image A</td><td>1</td><td>20.7</td><td>31.5</td><td>32.5</td><td>28.5</td><td>21.0</td></tr><tr><td>(q)</td><td>mono,Image B</td><td>1</td><td>19.7</td><td>30.1</td><td>31.6</td><td>28.5</td><td>20.4</td></tr><tr><td>(r)</td><td>mono, Image C</td><td>1</td><td>18.9</td><td>29.2</td><td>31.5</td><td>28.9</td><td>23.5</td></tr><tr><td>(s）</td><td>deka</td><td>10</td><td>18.5</td><td>29.0</td><td>31.1</td><td>28.2</td><td>21.9</td></tr><tr><td>(t</td><td>kilo</td><td>1,000</td><td>19.5</td><td>29.8</td><td>33.0</td><td>31.7</td><td>26.8</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/c6ff862e7d503119a0134343e3d173d41752a58b13a62c81d6d1a18dd3608986.jpg",
+        "table_caption": [
+            "Table 3: CIFAR-10/100. Accuracy of linear classifiers on different network layers. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Dataset</td><td colspan=\"4\"></td><td colspan=\"4\">CIFAR-100</td></tr><tr><td>conv1</td><td>conv2</td><td>CIFAR-10 conv3</td><td>conv4</td><td>conv1</td><td>conv2</td><td>conv3</td><td>conv4</td></tr><tr><td>Fully supervised</td><td>66.5</td><td>70.1</td><td>72.4</td><td>75.9</td><td>38.7</td><td>43.6</td><td>44.4</td><td>46.5</td></tr><tr><td>Random</td><td>57.8</td><td>55.5</td><td>54.2</td><td>47.3</td><td>30.9</td><td>29.8</td><td>28.6</td><td>24.1</td></tr><tr><td>RotNet</td><td>64.4</td><td>65.6</td><td>65.6</td><td>59.1</td><td>36.0</td><td>35.9</td><td>34.2</td><td>25.8</td></tr><tr><td>GAN (CIFAR-10)</td><td>67.7</td><td>73.0</td><td>72.5</td><td>69.2</td><td>39.6</td><td>46.0</td><td>45.1</td><td>39.9</td></tr><tr><td>GAN (CIFAR-100)</td><td>1</td><td>1</td><td>1</td><td>1</td><td>38.1</td><td>42.2</td><td>44.0</td><td>46.6</td></tr><tr><td>MonoGAN</td><td>68.1</td><td>72.3</td><td>70.8</td><td>63.5</td><td>39.9</td><td>46.9</td><td>44.5</td><td>38.8</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Details. While linear probes are conceptually straightforward, there are several technical details that affect the final accuracy by a few percentage points. Unfortunately, prior work has used several slightly different setups, so that comparing results of different publications must be done with caution. To make matters more difficult, not all papers released evaluation source code. We prove this standardized testing code here2. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In our implementation, we follow the original proposal (Zhang et al., 2017) in pooling each representation to a vector with 9600, 9216, 9600, 9600, 9216 dimensions for $\\mathtt { c o n v l - 5 }$ using adaptive max-pooling, and absorb the batch normalization weights into the preceding convolutions. For evaluation on ImageNet we follow RotNet to train linear probes: images are resized such that the shorter edge has a length of 256 pixels, random crops of $2 2 4 \\times 2 2 4$ are computed and flipped horizontally with $5 0 \\%$ probability. Learning lasts for 36 epochs and the learning rate schedule starts from 0.01 and is divided by five at epochs 5, 15 and 25. The top-1 accuracy of the linear classifier is then measured on the ImageNet validation subset. This uses DeepCluster’s protocol, extracting 10 crops for each validation image (four at the corners and one at the center along with their horizontal flips) and averaging the prediction scores before the accuracy is computed. For CIFAR-10/100 data, we follow the same learning rate schedule and for both training and evaluation we do not reduce the dimensionality of the representations and keep the images’ original size of $3 2 \\times 3 2$ . ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2 EFFECT OF AUGMENTATIONS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In order to better understand which image transformations are important to learn a good feature representations, we analyze the impact of augmentation settings. For speed, these experiments are conducted using the CIFAR-10 images $Q = 5 0$ , 000 in the training set) and with the smaller source Image $_ \\mathrm { B }$ and a GAN using the Wasserstein GAN formulation with gradient penalty (Gulrajani et al., 2017). The encoder is a smaller AlexNet-like CNN consisting of four convolutional layers (kernel sizes: $7 , 5 , 3 , 3$ ; strides: 3, 2, 2, 1) followed by a single fully connected layer as the discriminator. Given that the GAN is trained on a single image (w/ augmentations), we call this setting MonoGAN. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 1 reports all $2 ^ { 3 }$ combinations of the three main augmentations (scale, rotation, and jitter) and a randomly initialized network baseline (see Table 1 (b)) using the linear probes protocol discussed above. Without data augmentation the model only achieves marginally better performance than the random network (which also achieves a non-negligible level of performance (Ulyanov et al., 2017; Caron et al., 2018)). This is understandable since the dataset literally consists of a single training image cloned $d$ times. Color jitter and rotation slightly improve the performance of all probes by 1- $2 \\%$ points, but random rescaling adds at least ten points at every depth (see Table 1 (f,h,i)) and is the most important single augmentation. A similar conclusion can be drawn when two augmentations are combined, although there are diminishing returns as more augmentations are combined. Overall, we find all three types of augmentations are of importance when training in the ultra-low data setting. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.3 BENCHMARK EVALUATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            560,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We analyze how performance varies as a function $N$ , the number of actual samples that are used to generated the augmented datasets, and compare it to the gold-standard setup (in terms of choice of training data) defined in the papers that introduced each method. The evaluation is again based on linear probes (Section 4.1). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Mono is enough. From Table 2 we make the following observations. Training with just a single source image (f,g,k,l,p,q) is much better than random initialization (c) for all layers. Notably, these models also outperform Gabor-like filters from Scattering networks (Bruna & Mallat, 2013), which are hand crafted image features, replacing the first two convolutional layers as in (Oyallon et al., 2017). Using the same protocol as in the paper, this only achieves an accuracy of $1 8 . 9 \\%$ compared to (p)’s conv $2 > 3 0 \\%$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "More importantly, when comparing within pretext task, even with one image we are able to improve the quality of conv1–conv3 features compared to full (unsupervised) ImageNet training for GAN based self-supervision (e-i). For the other methods $( \\mathrm { j \\cdot }$ -n, o-s) we reach and also surpass the performance for the first layer and are within $1 . 5 \\%$ points for the second. Given that the best unsupervised performance for conv2 is 32.5, our method using a single source Image A (Table 2, p) is remarkably close with 31.5. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Image contents. While we surpass the GAN based approach of (Donahue et al., 2017) for both single source images, we find more nuanced results for the other two methods: For RotNet, as expected, the photographic bias cannot be extracted from a single image. Thus its performance is low with little training data and increases together with the number of images (Table 2, j-n). When comparing Image A and B trained networks for RotNet, we find that the photograph yields better performance than the hand drawn animal image. This indicates that the method can extract rotation information from low level image features such as patches which is at first counter intuitive. Considering that the hand-drawn image does not work well, we can assume that lighting and shadows even in small patches can indeed give important cues on the up direction which can be learned even from a single (real) image. DeepCluster shows poor performance in conv1 which we can improve upon in the single image setting (Table 2, o-r). Naturally, the image content matters: a trivial image without any image gradient (e.g. picture of a white wall) would not provide enough signal for any method. To better understand this issue, we also train DeepCluster on the much less cluttered Image C to analyze how much the image influences our claims. We find that even though this image contains large parts of sky and sea, the performance is only slightly lower than that of Image A. This finding indicates that the augmentations can even compensate for large untextured areas and the exact choice of image is not critical. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/47da3d26c9e9068651078be6be738408c464663bb72932691e02010a521e8e9a.jpg",
+        "image_caption": [
+            "Figure 2: conv1 filters trained using a single image. The 96 learned $( 3 \\times 1 1 \\times 1 1 )$ filters for the first layer of AlexNet are shown for each single training image and method along with their linear classifier performance. For visualization, each filter is normalized to be in the range of $( - 1 , 1 )$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "More than one image. While BiGAN fails to converge for $N \\in \\{ 1 0 , 1 0 0 0 \\}$ , most likely due to issues in learning from a distribution which is neither whole images nor only patches, we find that both RotNet and DeepCluster improve their performance in deeper layers when increasing the number of training images. However, for conv1 and conv2, a single image is enough. In deeper layers, DeepCluster seems to require large amounts of source images to yield the reported results as the deka- and kilo- variants start improving over the single image case (Table 2, o-t). This need for data also explains the gap between the two input images which have different resolutions. Summarizing Table 2, we can conclude that learning conv1, conv2 and for the most part conv3 (33.4 vs. 39.4) on over 1M images does not yield a significant performance increase over using one single training image — a highly unexpected result. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Generalization. In Table 3, we show the results of training linear classifiers for the CIFAR-10 dataset and compare against various baselines. We find that the GAN trained on the smaller Image B outperforms all other methods including the fully-supervised trained one for the first convolutional layer. We also outperform the same architecture trained on the full CIFAR-10 training set using RotNet, which might be due to the fact that either CIFAR images do not contain much information about the orientation of the picture or because they do not contain as many objects as in ImageNet. While the GAN trained on the whole dataset outperforms the MonoGAN on the deeper layers, the gap stays very small until the last layer. These findings are also reflected in the experiments on the CIFAR-100 dataset shown in Table 3. We find that our method obtains the best performance for the first two layers, even against the fully supervised version. The gap between our mono variant and the other methods increases again with deeper layers, hinting to the fact that we cannot learn very high level concepts in deeper layers from just one single image. These results corroborate the finding that our method allows learning very generalizable early features that are not domain dependent. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.4 QUALITATIVE ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Visual comparison of weights. In Figure 2, we compare the learned filters of all first-layer convolutions of an AlexNet trained with the different methods and a single image. First, we find that the filters closely resemble those obtained via supervised training: Gabor-like edge detectors and various color blobs. Second, we find that the look is not easily predictive of its performance, e.g. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "while generatively learned filters (BiGAN) show many edge detectors, its linear probes performance is about the same as that of DeepCluster which seems to learn many somewhat redundant point features. However, we also find that some edge detectors are required, as we can confirm from RotNet and DeepCluster trained on Image B, which yield less crisp filters and worse performances. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/2c370a153095606da7a5c1e8357c729b3eb85f8f83933eccd9bd9e4d360e376a.jpg",
+        "table_caption": [
+            "Table 4: Finetuning experiments The pretrained model’s first two convolutions are left frozen (or replaced by the Scattering transform) and the nework is retrained using ImageNet LSVRC-12 training set. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Top-1</td></tr><tr><td>Full sup.</td><td>59.4</td></tr><tr><td>Random</td><td>42.6</td></tr><tr><td>Scattering</td><td>49.2</td></tr><tr><td>BiGAN, A</td><td>51.4</td></tr><tr><td>RotNet, A</td><td>49.5</td></tr><tr><td>DeepCluster A</td><td>52.5</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/28641601f8b033094919f1f4d9ec2805f166320615575ce7240f102b217c3607.jpg",
+        "image_caption": [
+            "Figure 3: Style transfer with single-image pretraining. We show two style transfer results using the Image A trained BiGAN and the ImageNet pretrained AlexNet. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            501,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Fine-tuning instead of freezing. In Tab. 4, we show the results of retraining a network with the first two convolutional filters, or the scattering transform from (Oyallon et al., 2017), left frozen. We observe that our single image trained DeepCluster and BiGAN models achieve performances closes to the supervised benchmark. Notably, the scattering transform as a replacement for conv1-2 performs slightly worse than the analyzed single image methods. We also show in the appendix the results of retraining a network initialized with the first two convolutional layers obtained from a single image and subsequently linearly probing the model. The results are shown in Appendix Tab. 5 and we find that we can recover the performance of fully-supervised networks, i.e. the first two convolutional filters trained from just a single image generalize well and do not get stuck in an image specific minimum. ",
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+        "text": "Neural style transfer. Lastly, we show how our features trained on only a single image can be used for other applications. In Figure 3 we show two basic style transfers using the method of (Gatys et al., 2016) from an official PyTorch tutorial3. Image content and style are separated and the style is transferred from the source to target image using all CNN features, not just the shallow layers. We visually compare the results of using our features and from full ImageNet supervision. We find almost no visual differences in the stylized images and can conclude that our early features are equally powerful as fully supervised ones for this task. ",
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+        "text": "5 CONCLUSIONS ",
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+        "text": "We have made the surprising observation that we can learn good and generalizable features through self-supervision from one single source image, provided that sufficient data augmentation is used. Our results complement recent works (Mahajan et al., 2018; Goyal et al., 2019) that have investigated self-supervision in the very large data regime. Our main conclusion is that these methods succeed perfectly in capturing the simplest image statistics, but that for deeper layers a gap exist with strong supervision which is compensated only in limited manner by using large datasets. This novel finding motivates a renewed focus on the role of augmentations in self-supervised learning and critical rethinking of how to better leverage the available data. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENTS. ",
+        "text_level": 1,
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+        "text": "We thank Aravindh Mahendran for fruitful discussions. Yuki Asano gratefully acknowledges support from the EPSRC Centre for Doctoral Training in Autonomous Intelligent Machines & Systems (EP/L015897/1). The work is supported by ERC IDIU-638009. ",
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+        "type": "text",
+        "text": "REFERENCES   \nPulkit Agrawal, Joao Carreira, and Jitendra Malik. Learning to see by moving. In Proc. ICCV, pp. 37–45. IEEE, 2015. 3   \nR. Arandjelovic and A. Zisserman. Look, listen and learn. In ´ Proc. ICCV, 2017. 3   \nPiotr Bojanowski and Armand Joulin. Unsupervised learning by predicting noise. In Proc. ICML, pp. 517–526. PMLR, 2017. 2   \nJoan Bruna and Stephane Mallat. Invariant scattering convolution networks. ´ IEEE transactions on pattern analysis and machine intelligence, 35(8):1872–1886, 2013. 3, 7   \nM. Caron, P. Bojanowski, A. Joulin, and M. Douze. Deep clustering for unsupervised learning of visual features. In Proc. ECCV, 2018. 2, 3, 5, 6, 7, 13   \nN. Dalal and B Triggs. Histogram of Oriented Gradients for Human Detection. In Proc. CVPR, volume 2, pp. 886–893, 2005. 2, 3   \nVirginia R de Sa. Learning classification with unlabeled data. In NIPS, pp. 112–119, 1994. 3   \nJ. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei. Imagenet: A large-scale hierarchical image database. In Proc. CVPR, 2009. 5   \nCarl Doersch, Abhinav Gupta, and Alexei A Efros. Unsupervised visual representation learning by context prediction. In Proc. ICCV, pp. 1422–1430, 2015. 2   \nJeff Donahue, Philipp Krhenbhl, and Trevor Darrell. Adversarial feature learning. Proc. ICLR, 2017. 2, 3, 5, 6, 7   \nA. Dosovitskiy, P. Fischer, J. T. Springenberg, M. Riedmiller, and T. Brox. Discriminative unsupervised feature learning with exemplar convolutional neural networks. IEEE PAMI, 38(9):1734–1747, Sept 2016. ISSN 0162-8828. doi: 10.1109/TPAMI.2015.2496141. 2   \nV. Dumoulin, I. Belghazi, B. Poole, O. Mastropietro, A. Lamb, M. Arjovsky, and A. Courville. Adversarially learned inference. arXiv preprint arXiv:1606.00704, 2016. 5   \nD. Erhan, Y. Bengio, A. Courville, and P. Vincent. Visualizing higher-layer features of a deep network. Technical Report 1341, University of Montreal, Jun 2009. 13   \nChuang Gan, Boqing Gong, Kun Liu, Hao Su, and Leonidas J Guibas. Geometry guided convolutional neural networks for self-supervised video representation learning. In Proc. CVPR, 2018. 3   \nRouhan Gao, Dinesh Jayaraman, and Kristen Grauman. Object-centric representation learning from unlabeled videos. In Proc. ACCV, 2016. 3   \nLeon A Gatys, Alexander S Ecker, and Matthias Bethge. Image style transfer using convolutional neural networks. In Proc. CVPR, pp. 2414–2423, 2016. 2, 9   \nR. Geirhos, P. Rubisch, C. Michaelis, M. Bethge, F. A. Wichmann, and W. Brendel. Imagenet-trained CNNs are biased towards texture; increasing shape bias improves accuracy and robustness. In International Conference on Learning Representations, 2019. 14   \nSpyros Gidaris, Praveen Singh, and Nikos Komodakis. Unsupervised representation learning by predicting image rotations. In Proc. ICLR, 2018. 2, 3, 5, 6   \nR. B. Girshick. Fast R-CNN. In Proc. ICCV, 2015. 1   \nIan Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In NIPS, pp. 2672–2680, 2014. 5   \nPriya Goyal, Dhruv Mahajan, Abhinav Gupta, and Ishan Misra. Scaling and benchmarking self-supervised visual representation learning. arXiv preprint arXiv:1905.01235, 2019. 9   \nIshaan Gulrajani, Faruk Ahmed, Martin Arjovsky, Vincent Dumoulin, and Aaron C Courville. Improved training of wasserstein gans. In NIPS, pp. 5767–5777, 2017. 7   \nB. Hariharan, J. Malik, and D. Ramanan. Discriminative decorrelation for clustering and classification. In Proc. ECCV, 2012. 3   \nM. Huh, P. Agrawal, and A. A. Efros. What makes imagenet good for transfer learning? arXiv preprint arXiv:1608.08614, 2016. 1   \nPhillip Isola, Daniel Zoran, Dilip Krishnan, and Edward H Adelson. Learning visual groups from cooccurrences in space and time. In Proc. ICLR, 2015. 3   \nDinesh Jayaraman and Kristen Grauman. Learning image representations tied to ego-motion. In Proc. ICCV, 2015. 3   \nDinesh Jayaraman and Kristen Grauman. Slow and steady feature analysis: higher order temporal coherence in video. In Proc. CVPR, 2016. 3   \nSimon Jenni and Paolo Favaro. Self-supervised feature learning by learning to spot artifacts. In Proc. CVPR, 2018. 2   \nJ. Johnson, A. Alahi, and L. Fei-Fei. Perceptual losses for real-time style transfer and super-resolution. In Proc. ECCV, pp. 694–711, 2016. 2   \nR. Kat, R. Jevnisek, and S. Avidan. Matching pixels using co-occurrence statistics. In Proc. ICCV, pp. 1751– 1759, 2018. 2   \nPhilipp Krahenb ¨ uhl, Carl Doersch, Jeff Donahue, and Trevor Darrell. Data-dependent initializations of convo-¨ lutional neural networks. Proc. ICLR, 2016. 6   \nA. Krizhevsky, I. Sutskever, and G. E. Hinton. ImageNet classification with deep convolutional neural networks. In NIPS, pp. 1106–1114, 2012. 5   \nAlex Krizhevsky. Learning multiple layers of features from tiny images. Technical report, Citeseer, 2009. 5   \nGustav Larsson, Michael Maire, and Gregory Shakhnarovich. Colorization as a proxy task for visual understanding. In Proc. CVPR, 2017. 2   \nHsin-Ying Lee, Jia-Bin Huang, Maneesh Kumar Singh, and Ming-Hsuan Yang. Unsupervised representation learning by sorting sequence. In Proc. ICCV, 2017. 2   \nD. Lowe. Distinctive image features from scale-invariant keypoints. IJCV, 60(2):91–110, 2004. 2, 3   \nDhruv Mahajan, Ross Girshick, Vignesh Ramanathan, Kaiming He, Manohar Paluri, Yixuan Li, Ashwin Bharambe, and Laurens van der Maaten. Exploring the limits of weakly supervised pretraining. In Proceedings of the European Conference on Computer Vision (ECCV), pp. 181–196, 2018. 9   \nA. Mahendran, J. Thewlis, and A. Vedaldi. Cross pixel optical-flow similarity for self-supervised learning. In Proc. ACCV, 2018. 3   \nT. Malisiewicz, A. Gupta, and A. A. Efros. Ensemble of exemplar-SVMs for object detection and beyond. In Proc. ICCV, 2011. 3   \nIshan Misra, C. Lawrence Zitnick, and Martial Hebert. Shuffle and learn: Unsupervised learning using temporal order verification. In Proc. ECCV, 2016. 2   \nT Mundhenk, Daniel Ho, and Barry Y. Chen. Improvements to context based self-supervised learning. In Proc. CVPR, 2017. 2   \nMehdi Noroozi and Paolo Favaro. Unsupervised learning of visual representations by solving jigsaw puzzles. In Proc. ECCV, pp. 69–84. Springer, 2016. 2   \nMehdi Noroozi, Hamed Pirsiavash, and Paolo Favaro. Representation learning by learning to count. In Proc. ICCV, 2017. 2   \nMehdi Noroozi, Ananth Vinjimoor, Paolo Favaro, and Hamed Pirsiavash. Boosting self-supervised learning via knowledge transfer. In Proc. CVPR, 2018. 2   \nChris Olah, Arvind Satyanarayan, Ian Johnson, Shan Carter, Ludwig Schubert, Katherine Ye, and Alexander Mordvintsev. The building blocks of interpretability. Distill, 3(3):e10, 2018. 13   \nB. A. Olshausen and D. J. Field. Sparse coding with an overcomplete basis set: A strategy employed by V1? Vision Research, 37(23):3311–3325, 1997. 2, 3   \nM. Oquab, L. Bottou, I. Laptev, and J. Sivic. Learning and Transferring Mid-Level Image Representations using Convolutional Neural Networks. In Proc. CVPR, 2014. 1   \nAndrew Owens, Phillip Isola, Josh H. McDermott, Antonio Torralba, Edward H. Adelson, and William T. Freeman. Visually indicated sounds. In Proc. CVPR, pp. 2405–2413, 2016. 3   \nEdouard Oyallon, Eugene Belilovsky, and Sergey Zagoruyko. Scaling the scattering transform: Deep hybrid networks. pp. 5618–5627, 2017. 3, 6, 7, 9   \nDeepak Pathak, Philipp Krahenbuhl, Jeff Donahue, Trevor Darrell, and Alexei A Efros. Context encoders: Feature learning by inpainting. In Proc. CVPR, pp. 2536–2544, 2016. 2   \nDeepak Pathak, Ross Girshick, Piotr Dollar, Trevor Darrell, and Bharath Hariharan. Learning features by ´ watching objects move. In Proc. CVPR, 2017. 3   \nZhongzheng Ren and Yong Jae Lee. Cross-domain self-supervised multi-task feature learning using synthetic imagery. In Proc. CVPR, 2018. 2   \nA. Rodriguez, V. Naresh Boddeti, BVK V. Kumar, and A. Mahalanobis. Maximum margin correlation filter: A new approach for localization and classification. IEEE Transactions on Image Processing, 22(2):631–643, 2013. 3   \nTamar Rott Shaham, Tali Dekel, and Tomer Michaeli. Singan: Learning a generative model from a single natural image. In Computer Vision (ICCV), IEEE International Conference on, 2019. 3   \nPierre Sermanet et al. Time-contrastive networks: Self-supervised learning from video. In Proc. Intl. Conf. on Robotics and Automation, 2018. 2   \nH. Shin, H. R. Roth, M. Gao, L. Lu, Z. Xu, I. Nogues, J. Yao, D. Mollura, and R. M. Summers. Deep convolutional neural networks for computer-aided detection: Cnn architectures, dataset characteristics and transfer learning. IEEE Trans. on Medical Imaging, 35(5):1285–1298, May 2016. ISSN 0278-0062. 1   \nN. Srivastava, E. Mansimov, and R. Salakhudinov. Unsupervised learning of video representations using lstms. In Proc. ICML, 2015. 3   \nI. Talmi, R. Mechrez, and L. Zelnik-Manor. Template matching with deformable diversity similarity. In Proc. CVPR, pp. 175–183, 2017. 2   \nD. Ulyanov, A. Vedaldi, and V. Lempitsky. Improved texture networks: Maximizing quality and diversity in feed-forward stylization and texture synthesis. In Proc. CVPR, 2017. 7   \nD. Ulyanov, A. Vedaldi, and V. Lempitsky. Deep image prior. In Proc. CVPR, 2018. 5   \nX. Wang and A. Gupta. Unsupervised learning of visual representations using videos. In Proc. ICCV, pp. 2794–2802, 2015. 3   \nXiaolong Wang, Kaiming He, and Abhinav Gupta. Transitive invariance for self-supervised visual representation learning. In Proc. ICCV, 2017. 3   \nD. Wei, J. Lim, A. Zisserman, and W. T. Freeman. Learning and using the arrow of time. In Proc. CVPR, 2018. 2   \nJ. Yosinski, J. Clune, Y. Bengio, and H. Lipson. How transferable are features in deep neural networks? In NIPS, pp. 3320–3328, 2014. 1   \nM. D. Zeiler and Rob Fergus. Visualizing and understanding convolutional networks. In Proc. ECCV, 2014. 13   \nRichard Zhang, Phillip Isola, and Alexei A Efros. Colorful image colorization. In Proc. ECCV, pp. 649–666. Springer, 2016. 2   \nRichard Zhang, Phillip Isola, and Alexei A. Efros. Split-brain autoencoders: Unsupervised learning by crosschannel prediction. In Proc. CVPR, 2017. 5, 6, 7, 13 ",
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+        "type": "text",
+        "text": "A APPENDIX ",
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+    {
+        "type": "text",
+        "text": "A.1 IMAGENET TRAINING IMAGES ",
+        "text_level": 1,
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+        "img_path": "images/046e21f8bf046efcf2184e9d1086c428f27cb732dddd0a050c1fa62dee848678.jpg",
+        "image_caption": [
+            "Figure 4: ImageNet images for the $N { = } 1 0$ experiments. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "The images used for the $N { = } 1 0$ experiments are shown in fig. 4. ",
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+    {
+        "type": "text",
+        "text": "A.2 VISUAL COMPARISON OF FILTERS ",
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+        "img_path": "images/2f5f075fffec40a9fe974918f1470779b27b8abdc86af77cd7e6cd3a644973b5.jpg",
+        "image_caption": [
+            "Figure 5: Filter visualization. We show activation maximization (left) and retrieval of top 9 activated images from the training set of ImageNet (right) for four random non-cherrypicked target filters. From top to bottom: conv1-5 of the BiGAN trained on a single image A. The filter visualization is obtained by learning a (regularized) input image that maximizes the response to the target filter using the library Lucid (Olah et al., 2018). "
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+        "type": "text",
+        "text": "In order to understand what deeper neurons are responding to in our model, we visualize random neurons via activation maximization (Erhan et al., 2009; Zeiler & Fergus, 2014) in each layer. Additionally, we retrieve the top-9 images in the ImageNet training set that activate each neuron most in Figure 5. Since the mono networks are not trained on the ImageNet dataset, it can be used here for visualization. From the first convolutional layer we find typical neurons strongly reacting to oriented edges. In layers 2-4 we find patterns such as grids (conv2:3), and textures such as leopard skin (conv2:2) and round grid cover (conv4:4). Confirming our hypothesis that the neural network is only extracting patterns and not semantic information, we do not find any neurons particularly specialized to certain objects even in higher levels as for example dog faces or similar which can be fund in supervised networks. This finding aligns with the observations of other unsupervised methods (Caron et al., 2018; Zhang et al., 2017). As most neurons extract simple patterns and textures, the surprising effectiveness of training a network using a single image can be explained by the recent finding that even CNNs trained on ImageNet rely on texture (as opposed to shape) information to classify (Geirhos et al., 2019). ",
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+        "table_caption": [
+            "Table 5: Finetuning experiments Models are initialized using conv1 and conv2 from various single image trained models and the whole network is fine-tuned using ImageNet LSVRC-12 training set. Accuracy is averaged over 10 crops. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>c1</td><td>c2 c3</td><td>c4</td><td>c5</td></tr><tr><td>Full sup.</td><td>19.3 36.3 44.2</td><td>48.3 50.5</td><td></td></tr><tr><td>BiGAN, A</td><td>22.5 37.6 44.2</td><td>47.6 48.3</td><td></td></tr><tr><td>RotNet, A</td><td>22.0 38.2</td><td>44.8 49.2 51.8</td><td></td></tr><tr><td>DeepCluster, A 21.8 35.9</td><td>43.6</td><td>48.8</td><td>50.4</td></tr></table>",
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+        "text": "A.3 RETRAINING FROM SINGLE IMAGE INITIALIZATION",
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+        "type": "text",
+        "text": "In Table 5, we initialize AlexNet models using the first two convolutional filters learned from a single image and retrain them using ImageNet. We find that the networks recover their performance fully and the first filters do not make the network stuck in a bad local minimum despite having been trained on a single image from a different distribution. The difference from the BiGAN to the full supervision model is likely due to it using a smaller input resolution (112 instead of 224), as the BiGAN’s output resolution is limited. ",
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+        "text": "A.4 LINEAR PROBES ON IMAGENET ",
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+        "text": "We show two plots of the ImageNet linear probes results (Table 2 of the paper) in fig. 6. On the left we plot performance per layer in absolute scale. Naturally the performance of the supervised model improves with depth, while all unsupervised models degrade after conv3. From the relative plot on the right, it becomes clear that with our training scheme, we can even slightly surpass supervised performance on conv1 presumably since our model is trained with sometimes very small patches, thus receiving an emphasis on learning good low level filters. The gap between all self-supervised methods and the supervised baseline increases with depth, due to the fact that the supervised model is trained for this specific task, whereas the self-supervised models learn from a surrogate task without labels. ",
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+            "Figure 6: Linear Classifiers on ImageNet. Classification accuracies of linear classifiers trained on the representations from Table 2 are shown in absolute scale. "
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+        "text": "In figs. 7 to 10 we show example patches generated by our augmentation strategy for the datasets with different N. Even though the images and patches are very different in color and shape distribu",
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parse/train/BJxhLAuxg/BJxhLAuxg.md

@@ -0,0 +1,232 @@
+# A DEEP LEARNING APPROACH FOR JOINT VIDEOFRAME AND REWARD PREDICTION IN ATARI GAMES
+
+Felix Leibfried ∗   
+Max Planck Institute for Intelligent Systems   
+Max Planck Institute for Biological Cybernetics   
+Graduate Training Center of Neuroscience   
+Tuebingen, Germany   
+felix.leibfried@gmail.com
+
+Nate Kushman & Katja Hofmann
+
+Microsoft Research   
+Cambridge, UK   
+nkushman@microsoft.com   
+katja.hofmann@microsoft.com
+
+# ABSTRACT
+
+Reinforcement learning is concerned with learning to interact with environments that are initially unknown. State-of-the-art reinforcement learning approaches, such as DQN, are model-free and learn to act effectively across a wide range of environments such as Atari games, but require huge amounts of data. Modelbased techniques are more data-efficient, but need to acquire explicit knowledge about the environment dynamics or the reward structure.
+
+In this paper we take a step towards using model-based techniques in environments with high-dimensional visual state space when system dynamics and the reward structure are both unknown and need to be learned, by demonstrating that it is possible to learn both jointly. Empirical evaluation on five Atari games demonstrate accurate cumulative reward prediction of up to 200 frames. We consider these positive results as opening up important directions for model-based RL in complex, initially unknown environments.
+
+# 1 INTRODUCTION
+
+When humans or animals receive reward for taking a particular action in a given situation, the probability is increased that they will act similarly in similar situations in the future. This is described by principles such as the law of effect (Thorndike, 1898), operant conditioning (Skinner, 1938) and trial-and-error learning (Thorpe, 1979) in behaviorist psychology, and has inspired a discipline of artificial intelligence called reinforcement learning (RL, Sutton & Barto (1998)). RL is concerned with finding optimal behavior policies in order to maximize agents’ cumulative future reward.
+
+Approaches to RL can be divided into model-free and model-based approaches. In model-free approaches, agents learn by trial and error but do not aim to explicitly capture the dynamics of the environment or the structure of the reward function underlying the environment. State-of-the-art modelfree approaches, such as DQN (Mnih et al., 2015), effectively approximate so-called Q-values, i.e., the value of taking specific actions in a given state, using deep neural networks. The impressive effectiveness of these approaches comes from their ability to learn complex policies directly from high-dimensional input (e.g., video frames). Despite their effectiveness, model-free approaches require large amounts of training data that have to be collected through direct interactions with the environment, which makes them expensive to apply in settings where interactions are costly (such as most real-world applications). Additionally, model-free RL requires access to reward observations during training, which is problematic in environments with sparse reward structure—unless coupled with an explicit exploration mechanism.
+
+RL approaches that explicitly learn statistics about the environment or the reward are generally referred to as model-based—in a more narrow definition these statistics comprise environment dynamics and the reward function. In recent work, model-based techniques were successfully used to learn statistics about cumulative future reward (Veness et al., 2015) and to improve exploration by favoring actions that are likely to lead to novel states (Bellemare et al., 2016; Oh et al., 2015), resulting in substantially more data efficient learning compared to model-free approaches. When an accurate model of the true environment dynamics and the true reward function is available, modelbased approaches, such as planning via Monte-Carlo tree search (Browne et al., 2012) outperform model-free state-of-the-art approaches (Guo et al., 2014).
+
+A key open question is whether effective model-based RL is possible in complex settings where the environment dynamics and the reward function are initially unknown, and the agent has to acquire such knowledge through experience. In this paper, we take a step towards addressing this question by extending recent work on video frame prediction (Oh et al., 2015), which has been demonstrated to effectively learn system dynamics, to enable joint prediction of future states and rewards using a single latent representation. We propose a network architecture and training procedure for joint state and reward prediction, and evaluate our approach in the Arcade Learning Environment (ALE, Bellemare et al. (2013)).
+
+Our empirical results on five Atari games demonstrate that our approach can successfully predict cumulative reward up to roughly 200 frames. We complement our quantitative results with a detailed error analysis by visualizing example predictions. Our results are the first to demonstrate the feasibility of using a learned dynamics and reward model for accurate planning. We see this as a significant step towards data efficient RL in high-dimensional environments without prior knowledge.
+
+# 2 RELATED WORK AND MOTIVATION
+
+Two lines of research are related to the work presented in this paper: model-based RL and optimal control theory. Model-based RL utilizes a given or learned model of some aspect of a task to, e.g., reduce data or exploration requirements (Bellemare et al., 2016; Oh et al., 2015; Veness et al., 2015). Optimal control theory describes mathematical principles for deriving control policies in continuous action spaces that maximize cumulative future reward in scenarios with known system dynamics and known reward structure (Bertsekas, 2007; 2005).
+
+There has been recent interest in combining principles from optimal control theory and model-based learning in settings where no information on system dynamics is available a priori and instead has to be acquired from visual data (Finn et al., 2016; Wahlstrom et al., 2015; Watter et al., 2015). The ¨ general idea behind these approaches is to learn a compressed latent representation of the visual state space from raw images through autoencoder networks (Bengio, 2009) and to utilize the acquired latent representation to infer system dynamics. System dynamics are then used to specify a planning problem which can be solved by optimization techniques to derive optimal policies. Watter et al. (2015) introduce an approach for learning system dynamics from raw visual data by jointly training a variational autoencoder (Kingma & Welling, 2014; Rezende et al., 2014) and a state prediction model that operates in the autoencoder’s compressed latent state representation. A similar approach for jointly learning a compressed state representation and a predictive model is pursued by Wahlstrom et al. (2015).Finn et al. (2016) devise a sequential approach that first learns a latent state ¨ representation from visual data and that subsequently exploits this latent representation to augment a robot’s initial state space describing joint angles and end-effector positions. The augmented state space is then used to improve estimates of local system dynamics for planning.
+
+The approaches presented above assume knowledge of the functional form of the true reward signal and are hence not directly applicable in settings like ALE (and many real-world settings) where the reward function is initially unknown. Planning in such settings therefore necessitates learning both system dynamics and reward function in order to infer optimal behavioral policies. Recent work by Oh et al. (2015) introduced an approach for learning environment dynamics from pixel images and demonstrated that this enabled successful video frame prediction over up to 400 frames. In our current paper, we extend this recent work to enable reward prediction as well by modifying the network’s architecture and training objective accordingly. The modification of the training objective bears a positive side effect: since our network must optimize a compound loss consisting of the video frame reconstruction loss and the reward loss, reward-relevant aspects in the video frames to which the reconstruction loss alone might be insensitive are explicitly captured by the optimization objective. In the subsequent section, we elucidate the approach from Oh et al. (2015) as well as our extensions for reward prediction in more detail.
+
+![](images/debc0e4e13acb719598c45070f146487cd9562cd8f9454fa101138d9cea47c15.jpg)  
+Figure 1: Network architecture for joint video frame and reward prediction. The architecture comprises three stages: an encoding stage mapping current input frames to some compressed latent representation, a transformation stage integrating the current action into the latent representation through element-wise vector multiplication denoted by $" \times "$ , and a final predictive stage for reconstructing the frame of the next time step and the current reward. The network uses three different types of neuron layers (’Conv’ for convolutional, ’Deconv’ for deconvolutional and ’Fc’ for forward connection) in combination with three different types of activation functions (’ReLU’, ’Softmax’ and ’Lin’ for linear activations). The dimensional extend of individual layers is either depicted beneath or within layers. The network part coloured in red highlights the extension for reward prediction.
+
+# 3 NETWORK ARCHITECTURE AND TRAINING
+
+The deep network proposed by Oh et al. (2015) for video frame prediction in Atari games aims at learning a function that predicts the video frame $\mathbf { s } _ { t + 1 }$ at the next time step $t + 1$ , given the current history of frames $\mathbf { S } _ { t - h + 1 : t }$ with time horizon $h$ and the current action $\mathbf { a } _ { t }$ taken by the agent—see Section 3.1. Here, we extend this work to enable joint video frame and reward prediction such that the network anticipates the current reward $\mathbf { r } _ { t }$ as well—see Sections 3.2 and 3.3.
+
+# 3.1 VIDEO FRAME PREDICTION
+
+The video-frame-predictive architecture from Oh et al. (2015) comprises three informationprocessing stages: an encoding stage that maps input frames to some compressed latent representation, a transformation stage that integrates the current action into the compressed latent representation, and a decoding stage that maps the compressed latent representation to the predicted next frame—see Figure 1. The initial encoding stage is a sequence of convolutional and forward operations that map the current frame history $\mathbf { S } _ { t - h + 1 : t }$ —a three-dimensional tensor—to a compressed feature vector $\mathbf { h } _ { t } ^ { \mathrm { e n c } }$ . The transformation stage converts this compressed feature vector $\mathbf { h } _ { t } ^ { \mathrm { e n c } }$ into an action-conditional representation $\mathbf { h } _ { t } ^ { \mathrm { d e c } }$ in vectorized form by integrating the current action $\mathbf { a } _ { t }$ . The current action $\mathbf { a } _ { t }$ is represented as a one-hot vector with length varying from game to game since there are at least 3 and at most 18 actions in ALE. The integration of the current action into the compressed feature vector includes an element-wise vector multiplication—depicted as $\because \mathbf { \nabla } _ { \times } ,$ in Figure 1—with the particularity that the two neuron layers involved in this element-wise multiplication are the only layers in the entire network without bias parameters, see Section 3.2 in Oh et al. (2015). Finally, the decoding stage performs a series of forward and deconvolutional operations (Dosovitskiy et al., 2015; Zeiler et al., 2010) by mapping the action-conditional representation $\mathbf { h } _ { t } ^ { \mathrm { d e c } }$ of the current frame history $\mathbf { S } _ { t - h + 1 : t }$ and the current action $\mathbf { a } _ { t }$ to the predicted video frame $\mathbf { s } _ { t + 1 }$ of the next time step $t + 1$ . Note that this necessitates a reshape operation at the beginning of the decoding cascade in order to transform the vectorized hidden representation into a three-dimensional tensor. The whole network uses linear and rectified linear units (Glorot et al., 2011) only. In all our experiments, following DQN (Mnih et al., 2015), the video frames processed by the network are $8 4 \times 8 4$ grey-scale images down-sampled from the full-resolution $2 1 0 \times 1 6 0$ Atari RGB images from ALE. Following Mnih et al. (2015) and Oh et al. (2015), the history frame time horizon $h$ is set to 4.
+
+# 3.2 REWARD PREDICTION
+
+In this section we detail our proposed network architecture for joint state and reward prediction. Our model assumes ternary rewards which result from reward clipping in line with Mnih et al. (2015). Original game scores in ALE are integers that can vary significantly between different Atari games and the corresponding original rewards are clipped to assume one of three values: $- 1$ for negative rewards, 0 for no reward and 1 for positive rewards. Because of reward clipping, rewards can be represented as vectors $\mathbf { r } _ { t }$ in one-hot encoding of size 3.
+
+In Figure 1, our extension of the video-frame-predictive architecture from Oh et al. (2015) to enable reward prediction is highlighted in red. We add an additional softmax layer to predict the current reward $\mathbf { r } _ { t }$ with information contained in the action-conditional encoding $\dot { \mathbf { h } } _ { t } ^ { \mathrm { d e c } }$ . The motivation behind this extension is twofold. First, our extension makes it possible to jointly train the network with a compound objective that emphasizes both video frame reconstruction and reward prediction, and thus encourages the network to not abstract away reward-relevant features to which the reconstruction loss alone might be insensitive. Second, this formulation facilitates the future use of the model for reward prediction through virtual roll-outs in the compressed latent space, without the computational expensive necessity of reconstructing video frames explicitly—note that this requires another ”shortcut” predictive model to map from $\mathbf { h } _ { t } ^ { \mathrm { d e c } }$ to $\mathbf { h } _ { t + 1 } ^ { \mathrm { e n c } }$ .
+
+Following previous work (Oh et al., 2015; Mnih et al., 2015), actions are chosen by the agent on every fourth frame and are repeated on frames that were skipped. Skipped frames and repeated actions are hence not part of the data sets used to train and test the predictive network on, and original reward values are accumulated over four frames before clipping.
+
+# 3.3 TRAINING
+
+Training the model for joint video frame and reward prediction requires trajectory samples $\left\{ \left( \mathbf { s } _ { n } ^ { ( i ) } , \mathbf { a } _ { n } ^ { ( i ) } , \mathbf { r } _ { n } ^ { ( i ) } \right) _ { n = 1 } ^ { N } \right\} _ { i = 1 } ^ { I }$ collected by some agent playing the Atari game, where $i$ is an index over trajectories and $n$ is a time index over samples within one trajectory $i$ . The parameter $I$ denotes the number of trajectories in the training set or the minibatch respectively and the parameter $N$ denotes the length of an individual trajectory. In our case, we use agents trained according to Mnih et al. (2015) in order to collect trajectory samples.
+
+The original training objective in Oh et al. (2015) consists of a video frame reconstruction loss in terms of a squared loss function aimed at minimizing the quadratic $l ^ { 2 }$ -norm of the difference vector between the ground truth image and its action-conditional reconstruction. We extend this training objective to enable joint reward prediction. This results in a compound training loss consisting of the original video frame reconstruction loss and a reward prediction loss given by the cross entropy (Simard et al., 2003) between the ground truth reward and the corresponding prediction:
+
+$$
+L _ { K } ( \boldsymbol \theta ) = \frac { 1 } { 2 \cdot I \cdot T \cdot K } \sum _ { i = 1 } ^ { I } \sum _ { t = 0 } ^ { T - 1 } \sum _ { k = 1 } ^ { K } ( \underbrace { | \mathbf { s } _ { t + k } ^ { ( i ) } - \hat { \mathbf { s } } _ { t + k } ^ { ( i ) } | | _ { 2 } ^ { 2 } } _ { \mathrm { v i d e o f r a m e r e c o n s t u c t i o n 1 0 s s } } + \underbrace { \lambda \cdot ( - 1 ) \sum _ { l = 1 } ^ { 3 } \mathbf { r } _ { t + k } ^ { ( i ) } [ l ] \cdot \ln \mathbf { p } _ { t + k } ^ { ( i ) } [ l ] } _ { \mathrm { r e w a r d p r e d i c i t i o n 1 l o s s } } ) ,
+$$
+
+where $\hat { \mathbf { s } } _ { t + k } ^ { ( i ) }$ denotes the $k$ -step look ahead frame prediction with target video frame $\mathbf { s } _ { t + k } ^ { ( i ) }$ and p(i)t+k denotes the in red in Fi -step look ahead probability vare1—with target reward vector $\mathbf { r } _ { t + k } ^ { ( i ) }$ f the reward-p. The paramete $\lambda > 0$ softmax layer—depicted controls the trade-off between video frame reconstruction and reward loss. The parameter $T$ determines how often a single trajectory sample $i$ is unrolled into the future, and $K$ determines the look ahead prediction horizon dictating how far the network predicts into the future by using its own video frame predicted output as input for the next time step. Following Oh et al. (2015) and Michalski et al. (2014), we apply a curriculum learning (Bengio et al., 2009) scheme by successively increasing $K$ in the course of training such that the network initially learns to predict over a short time horizon and becomes fine-tuned on longer-term predictions as training advances (see Section A.1 for details). The network parameters $\theta$ are updated by stochastic gradient descent, derivatives of the training objective w.r.t. $\theta$ are computed with backpropagation through time (Werbos, 1988).
+
+# 4 RESULTS
+
+In our evaluations, we investigate cumulative reward predictions quantitatively and qualitatively on five different Atari games (Q\*bert, Seaquest, Freeway, Ms Pacman and Space Invaders). The quantitative analysis comprises evaluating the cumulative reward prediction error—see Section 4.1. The qualitative analysis comprises visualizations of example predictions in Seaquest—see Section 4.2.
+
+4.1 QUANTITATIVE REWARD PREDICTION ANALYSIS: CUMULATIVE REWARD ERROR
+
+Our quantitative evaluation examines whether our joint model of system dynamics and reward function results in a shared latent representation that enables accurate cumulative reward prediction. We assess cumulative reward prediction on test sets consisting of approximately 50,000 video frames per game, including actions and rewards. Each network is evaluated on 1,000 trajectories—suitable to analyze up to 100-step ahead prediction—drawn randomly from the test set. Look ahead prediction is measured in terms of the cumulative reward error which is the difference between ground truth cumulative reward and predicted cumulative reward. For each game, this results in 100 empirical distributions over the cumulative reward error—one distribution for each look ahead step—consisting of 1,000 samples each (one for each trajectory). We compare our model predictions to a baseline model that samples rewards from the marginal reward distribution observed on the test set for each game. Note that negative reward values are absent in the games investigated for this study.
+
+Figure 2 illustrates 20 of the 100 empirical cumulative reward error distributions in all games for our network model in blue and for the baseline model in red (histograms, bottom), together with the median and the 5 to 95 percentiles of the cumulative reward error over look ahead steps (top). Across all games, we observe that our joint state and reward prediction model accurately predicts future cumulative rewards at least 20 look ahead steps, and that it predicts future rewards substantially more accurately than the baseline model. This is evidenced by cumulative reward error distributions that maintain a unimodal form with mode zero and do not flatten out as quickly as the distributions for the random-prediction baseline model. Best results are achieved in Freeway and $\boldsymbol { \mathrm { Q } } ^ { * } \boldsymbol { \mathrm { b e r t } }$ where the probability of zero cumulative reward error at 51 look ahead steps is still around $8 0 \%$ and $6 0 \%$ respectively—see Figure 2. Note that 51 look ahead steps correspond to 204 frames because the underlying DQN agent, collecting trajectory samples for training and testing our model, skipped every fourth frame when choosing an action—see Section 3.2. Lowest performance is obtained in Seaquest where the probability of zero cumulative reward error at 26 steps (104 frames) is around $4 0 \%$ and begins to flatten out soon thereafter—see Figure 2. Running the ALE emulator at a frequency of 60fps, 26 steps correspond to more than 1 second real-time game play because of frame skipping. Since our model is capable of predicting 26 steps ahead in less than 1 second, our model enables real-time planning and could be therefore utilized in an online fashion.
+
+We now turn our attention to error analysis. While the look ahead step at which errors become prominent differs substantially from game to game, we find that overall our model underestimates cumulative reward. This can be seen in the asymmetry towards positive cumulative reward error values when inspecting the 5 to 95 percentile intervals in the first plot per each game in Figure 2. We identify a likely cause in (pseudo-)stochastic transitions inherent in these games. Considering Seaquest as our running example, objects such as divers and submarines can enter the scene randomly from the right and from the left and at the same time have an essential impact on which rewards the agent can potentially collect. In the ground truth trajectories, the agent’s actions are reactions to these objects. If the predicted future trajectory deviates from the ground truth, targeted actions such as shooting will miss their target, leading to underestimating true reward. We analyze this effect in more detail in Section 4.2.
+
+All our experiments were conducted in triplicate with different initial random seeds. Different initial random seeds did not have a significant impact on cumulative reward prediction in all games except Freeway—see Section A.5 for a detailed analysis. So far, we discussed results concerning reward prediction only. In the appendix, we also evaluate the joint performance of reward and video frame prediction on the test set in terms of the optimization objective as in Oh et al. (2015), where the authors report successful video frame reconstruction up to approximately 100 steps (400 frames), and observe similar results—see Section A.6.
+
+In the previous section, we identified stochasticity in state transitions as a likely cause for relatively low performance in long-term cumulative reward prediction in games such as Seaquest. In Seaquest objects may randomly enter a scene in a non-deterministic fashion. Errors in predicting these events result in predicted possible futures that do not match actually observed future states, resulting in inaccurate reward predictions. Here, we support this hypothesis by visualizations in Seaquest illustrating joint video frame and reward prediction for a single network over 20 steps (80 frames)—see Figure 3 where ground truth video frames are compared to predicted video frames in terms of error maps. Error maps emphasize the difference between ground truth and predicted frames through squared error values between pixels in black or white depending on whether objects are absent or present by mistake in the network’s prediction. Actions, ground truth rewards and model-predicted rewards are shown between state transitions. Peculiarities in the prediction process are shown in red.
+
+In step 2, the model predicts reward by mistake because the agent barely misses its target. Steps 4 to 6 report how the model predicts reward correctly but is off by one time step. Steps 7 to 14 depict problems caused by objects randomly entering the scene from the right which the model cannot predict. Steps 26 to 30 show how the model has problems to predict rewards at steps 26 and 28 as these rewards are attached to objects the model failed to notice entering the scene earlier.
+
+# 5 CONCLUSION AND FUTURE WORK
+
+In this paper, we extended recent work on video frame prediction (Oh et al., 2015) in Atari games to enable reward prediction. Our approach can be used to jointly predict video frames and cumulative rewards up to a horizon of approximately 200 frames in five different games ( $\mathbf { Q } ^ { * }$ bert, Seaquest, Freeway, Ms Pacman and Space Invaders). We achieved best results in Freeway and $\mathbf { Q } ^ { * }$ bert where the probability of zero cumulative reward error after 200 frames is still around $8 0 \%$ and $6 0 \%$ respectively, and worst results in Seaquest where the probability of zero cumulative reward error after 100 frames is around $4 0 \%$ . Our study fits into the general line of research using autoencoder networks to learn a latent representation from visual data (Finn et al., 2016; Goroshin et al., 2015; Gregor et al., 2015; Kulkarni et al., 2015; Srivastava et al., 2015; Wahlstrom et al., 2015; Watter et al., 2015; ¨ Kingma & Welling, 2014; Rezende et al., 2014; Lange et al., 2012; Hinton et al., 2011; Ranzato et al., 2007), and extends this line of research by showing that autoencoder networks are capable of learning a combined representation for system dynamics and the reward function in reinforcement learning settings with high-dimensional visual state spaces—a first step towards applying modelbased techniques for planning in environments where the reward function is not initially known.
+
+Our positive results open up intriguing directions for future work. Our long-term goal is the integration of model-based and model-free approaches for effective interactive learning and planning in complex environments. Directions for achieving this long-standing challenge include the Dyna method (Sutton, 1990), which uses a predictive model to artificially augment expensive training data, and has been shown to lead to substantial reductions in data requirements in tabular RL approaches. Alternatively, the model could be could be utilized for planning via Monte-Carlo tree search (Guo et al., 2014; Browne et al., 2012). We hypothesize that such an approach would be particularly beneficial in multi-task or life-long learning scenarios where the reward function changes but the environment dynamics are stationary. Testing this hypothesis requires a flexible learning framework where the reward function and the artificial environment can be changed by the experimenter in an arbitrary fashion, which is not possible in ALE where the environment and the reward function are fixed per game. A learning environment providing such a flexibility is the recently released Malmo platform for Minecraft (Johnson et al., 2016) where researchers can create user-defined en-¨ vironments and tasks in order to evaluate the performance of artificial agents. In the shorter-term, we envision improving the prediction performance of our network by regularization methods such as dropout and max norm regularization (Srivastava et al., 2014)—a state-of-the-art regularizer in supervised learning—and by modifying the optimization objective to enforce similarity between hidden encodings in multi-step ahead prediction and one-step ahead prediction—see Watter et al. (2015). Finally, extensions of our model to non-deterministic state transitions through dropout and variational autoencoder schemes (Kingma & Welling, 2014; Rezende et al., 2014) is a promising direction to alleviate the limitations highlighted in Section 4.2—paving the way for models that adequately predict and reason over alternative possible future trajectories.
+
+![](images/02740b7abbfcbfdaf1dce245f49cc57b86ddcfd6ce7d81361d8571db31ad4b30.jpg)  
+two plots for each game. The top plot per game shows how the median and the 5 to 95 percentiles of the cumulative reward error evolve over look ahead steps for both our model (in blue) and a baseline model that samples rewards from the marginal reward distribution of the test set (in red). Each vertical slice of this concise representation corresponds to a single empirical distribution over the cumulative reward error. We depict these for every fifth look ahead step in the compound plots below for both models. These empirical error distributions demonstrate successful cumulative reward prediction over at least 20 steps (80 frames) in all five games as evidenced by their zero-centered and unimodal shape in the first column of each compound plot per game.
+
+![](images/5ca73ff0f11ebf0f135df66feefe058cd29d3977e94fcfe61f6728d039d6d497.jpg)  
+Figure 3: Example predictions in Seaquest. Ground truth video frames, model predictions and error maps emphasizing differences between ground truth and predicted frames—in form of the squared error between pixel values—are compared column-wise. Error maps highlight objects in black or white respectively depending on whether these objects are absent by mistake or present by mistake in the model’s prediction. Actions taken by the agent as well as ground truth rewards (’rew’) and reward predictions (’pred’) are shown below video and error frames. Peculiarities in the prediction process are marked in red. The figure demonstrates how our predictive model fails to anticipate objects that randomly enter the scene from the right and rewards associated to these objects.
+
+# REFERENCES
+
+M G Bellemare, Y Naddaf, J Veness, and M Bowling. The Arcade Learning Environment: an evaluation platform for general agents. Journal of Artificial Intelligence Research, 47:253–279, 2013.
+
+M G Bellemare, S Srinivasan, G Ostrovski, T Schaul, D Saxton, and R Munos. Unifying count-based exploration and intrinsic motivation. arXiv preprint arXiv:1606.01868, 2016.
+
+Y Bengio. Learning deep architectures for AI. Foundations and Trends in Machine Learning, 2(1): 1–127, 2009.
+
+Y Bengio, J Louradour, R Collobert, and J Weston. Curriculum learning. In Proceedings of the International Conference on Machine Learning, 2009.
+
+D P Bertsekas. Dynamic programming & optimal control, volume 1. Athena Scientific, 2005.
+
+D P Bertsekas. Dynamic programming & optimal control, volume 2. Athena Scientific, 2007.
+
+C Browne, E Powley, D Whitehouse, S Lucas, P I Cowling, P Rohlfshagen, S Tavener, D Perez, S Samothrakis, and S Colton. A survey of monte carlo tree search methods. IEEE Transactions on Computational Intelligence and AI in Games, 4(1):1–49, 2012.
+
+A Dosovitskiy, J T Springenberg, and T Brox. Learning to generate chairs with convolutional neural networks. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2015.
+
+C Finn, X Y Tan, Y Duan, T Darrell, S Levine, and P Abbeel. Deep spatial autoencoders for visuomotor learning. In Proceedings of the IEEE International Conference on Robotics and Automation, 2016.
+
+X Glorot and Y Bengio. Understanding the difficulty of training deep feedforward neural networks. In Proceedings of the International Conference on Artificial Intelligence and Statistics, 2010.
+
+X Glorot, A Bordes, and Y Bengio. Deep sparse rectifier neural networks. In Proceedings of the International Conference on Artificial Intelligence and Statistics, 2011.
+
+R Goroshin, M Mathieu, and Y LeCun. Learning to linearize under uncertainty. Advances in Neural Information Processing Systems, 2015.
+
+K Gregor, I Danihelka, A Graves, D J Rezende, and D Wierstra. DRAW: a recurrent neural network for image generation. In Proceedings of the International Conference on Machine Learning, 2015.
+
+X Guo, S Singh, H Lee, R Lewis, and X Wang. Deep learning for real-time Atari game play using offline Monte-Carlo tree search planning. In Advances in Neural Information Processing Systems, 2014.
+
+G E Hinton, A Krizhevsky, and S D Wang. Transforming auto-encoders. In Proceedings of the International Conference on Artificial Neural Networks, 2011.
+
+M Johnson, K Hofmann, T Hutton, and D Bignell. The Malmo platform for artificial intelligence experimentation. In Proceedings of the International Joint Conference on Artificial Intelligence, 2016.
+
+D P Kingma and J Ba. Adam: a method for stochastic optimization. In Proceedings of the International Conference on Learning Representations, 2015.
+
+D P Kingma and M Welling. Auto-encoding variational Bayes. In Proceedings of the International Conference on Learning Representations, 2014.
+
+T D Kulkarni, W F Whitney, P Kohli, and J B Tenenbaum. Deep convolutional inverse graphics network. In Advances in Neural Information Processing Systems, 2015.
+
+S Lange, M Riedmiller, and A Voigtlander. Autonomous reinforcement learning on raw visual input ¨ data in a real world application. In Proceedings of the International Joint Conference on Neural Networks, 2012.
+
+V Michalski, R Memisevic, and K Konda. Modeling deep temporal dependencies with recurrent grammar cells. In Advances in Neural Information Processing Systems, 2014.
+
+V Mnih, K Kavukcuoglu, D Silver, A A Rusu, J Veness, M G Bellemare, A Graves, M Riedmiller, A K Fidjeland, G Ostrovski, S Petersen, C Beattie, A Sadik, I Antonoglou, H King, D Kumaran, D Wierstra, S Legg, and D Hassabis. Human-level control through deep reinforcement learning. Nature, 518(7540):529–533, 2015.
+
+J Oh, X Guo, H Lee, R Lewis, and S Singh. Action-conditional video prediction using deep networks in Atari games. In Advances in Neural Information Processing Systems, 2015.
+
+R Pascanu, T Mikolov, and Y Bengio. On the difficulty of training recurrent neural networks. In Proceedings of the International Conference on Machine Learning, 2013.
+
+M Ranzato, F J Huang, Y-L Boureau, and Y LeCun. Unsupervised learning of invariant feature hierarchies with applications to object recognition. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2007.
+
+D J Rezende, S Mohamed, and D Wierstra. Stochastic backpropagation and approximate inference in deep generative models. In Proceedings of the International Conference on Machine Learning, 2014.
+
+P Y Simard, D Steinkraus, and J C Platt. Best practices for convolutional neural networks applied to visual document analysis. In Proceedings of the International Conference on Document Analysis and Recognition, 2003.
+
+B F Skinner. The behavior of organisms: an experimental analysis. Appleton-Century-Crofts, 1938.
+
+N Srivastava, G E Hinton, A Krizhevsky, I Sutskever, and R Salakhutdinov. Dropout : a simple way to prevent neural networks from overfitting. Journal of Machine Learning Research, 15: 1929–1958, 2014.
+
+N Srivastava, E Mansimov, and R Salakhutdinov. Unsupervised learning of video representations using LSTMs. In Proceedings of the International Conference on Machine Learning, 2015.
+
+R S Sutton. Integrated architectures for learning, planning, and reacting based on approximating dynamic programming. In Proceedings of the International Conference on Machine Learning, 1990.
+
+R S Sutton and A G Barto. Reinforcement learning: an introduction. MIT Press, 1998.
+
+E L Thorndike. Animal intelligence: an experimental study of the associative processes in animals. The Psychological Review: Monograph Supplements, 2(4):1–107, 1898.
+
+W H Thorpe. The origins and rise of ethology. Heinemann Educational Books, 1979.
+
+J Veness, M G Bellemare, M Hutter, A Chua, and G Desjardins. Compress and control. In Proceedings of the AAAI Conference on Artificial Intelligence, 2015.
+
+N Wahlstrom, T B Sch ¨ on, and M P Deisenroth. From pixels to torques: policy learning with deep ¨ dynamical models. arXiv preprint arXiv:1502.02251, 2015.
+
+M Watter, J T Springenberg, J Boedecker, and M Riedmiller. Embed to control: a locally linear latent dynamics model for control from raw images. In Advances in Neural Information Processing Systems, 2015.
+
+P J Werbos. Generalization of backpropagation with application to a recurrent gas market model. Neural Networks, 1(4):339–356, 1988.
+
+M D Zeiler, D Krishnan, G W Taylor, and R Fergus. Deconvolutional networks. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2010.
+
+# A APPENDIX
+
+# A.1 TRAINING DETAILS
+
+We performed all our experiments in Python with Chainer and adhered to the instructions in Oh et al. (2015) as close as possible. Trajectory samples for learning the network parameters were obtained from a previously trained DQN agent according to Mnih et al. (2015). The dataset for training comprised around 500, 000 video frames per game in addition to actions chosen by the DQN agent and rewards collected during game play. Video frames used as network input were $8 4 \times 8 4$ grey-scale images with pixel values between 0 and 255 down-sampled from the full-resolution $2 1 0 \times 1 6 0$ ALE RGB images. We applied a further preprocessing step by dividing each pixel by 255 and subtracting mean pixel values from each image leading to final pixel values $\in \ [ - 1 ; 1 ]$ . A detailed network architecture is shown in Figure 1 in the main paper. All weights in the network were initialized according to Glorot & Bengio (2010) except for those two layers that participate in the element-wise multiplication in Figure 1: the weights of the action-processing layer were initialized uniformly in the range $[ - 0 . 1 ; 0 . { \bar { 1 } } ]$ and the weights of the layer receiving the latent encoding of the input video frames were initialized uniformly in the range $[ - 1 ; 1 ]$ . Training was performed for $1 , 5 0 0 , 0 0 0$ minibatch iterations with a curriculum learning scheme increasing the look ahead parameter $K$ every 500, 000 iterations from 1 to 3 to 5. When increasing the look ahead parameter $K$ for the first time after 500, 000 iterations, the minibatch size $I$ was also altered from 32 to 8 as was the learning rate for parameter updates from $1 0 ^ { - 4 }$ to $1 0 ^ { - 5 }$ . Throughout the entire curriculum scheme, the time horizon parameter determining the number of times a single trajectory is unrolled into the future was $T = 4$ . The optimizer for updating weights was Adam (Kingma & Ba, 2015) with gradient momentum 0.9, squared gradient momentum 0.95 and epsilon parameter $1 0 ^ { - 8 }$ . In evaluation mode, network outputs were clipped to $[ - 1 ; 1 ]$ so that strong activations could not accumulate over roll-out time in the network.
+
+In our experiments, we modified the reward prediction loss slightly in order to prevent exploding gradient values by replacing the term $- \ln p$ with a first-order Taylor approximation for $p$ -values smaller than $e ^ { - 1 0 } { \mathrm { - a } } $ similar technique is used in DQN (Mnih et al., 2015) to improve the stability of the optimization algorithm. To identify optimal values for the reward weight $\lambda$ , we performed initial experiments on Ms Pacman without applying the aforementioned curriculum learning scheme instead using a fixed look ahead parameter $K = 1$ . We evaluated the effect of different $\lambda$ -values $\in \{ 0 . 1 , 1 , \bar { 1 0 } , 1 0 0 \}$ on the training objective and identified $\lambda = 1$ for conducting further experiments—see Section A.2. After identifying an optimal reward weight, we conducted additional initial experiments without curriculum learning with fixed look ahead parameter $K = 1$ on all of the five different Atari games used in this paper. We observed periodic oscillations in the reward prediction loss of the training objective in Seaquest, which was fixed by adding gradient clipping (Pascanu et al., 2013) with threshold parameter 1 to our optimization procedure—experiments investigating the effect of gradient clipping in Seaquest are reported in Section A.3. The fine-tuning effect of curriculum learning on the training objective in our final experiments is shown in Section A.4 for all of the five analysed Atari games.
+
+# A.2 EFFECT OF REWARD WEIGHT IN MS PACMAN
+
+To identify optimal values for the reward weight $\lambda$ , we conducted initial experiments in Ms Pacman without curriculum learning and a fixed look ahead horizon $K = 1$ . We tested four different $\lambda$ - values $\in \{ 0 . 1 , 1 , 1 0 , 1 0 0 \}$ and investigated how the frame reconstruction loss and the reward loss of the training objective evolve over minibatch iterations—see Figure 4. Best results were obtained for $\lambda = 1$ and for $\lambda = 1 0$ , whereas values of $\lambda = 0 . 1$ and $\lambda = 1 0 0$ lead to significantly slower convergence and worse overall training performance respectively.
+
+# A.3 EFFECT OF GRADIENT CLIPPING IN SEAQUEST
+
+After identifying an optimal value for the reward weight, see Section A.2, we observed oscillations in the reward loss of the training objective in Seaquest—see first column in Figure 5—which was solved by adding gradient clipping to our optimization procedure—see second and third column in Figure 5. We tested two different values for the gradient clipping threshold (5 and 1) both of which worked, but for a value of 1 the oscillation vanished completely.
+
+![](images/720d06db36adf5fcea69e6c832cc20f28160a2bac7d37f237596cd85afe2ab0c.jpg)  
+Figure 4: Effect of reward weight on training loss in Ms Pacman. Each of the four panels depicts one experiment with a different reward weight $\lambda$ . Each panel shows how the training loss evolves over minibatch iterations in terms of two subplots reporting video frame reconstruction and reward loss respectively. Each experiment was conducted three times with different initial random seeds depicted in blue, green and red. Graphs were smoothed with an exponential window of size 1000.
+
+![](images/36937623cd79e22cff1c94043eb54f0eaf9f923351d0908c7abcabc68ad664cb.jpg)  
+Figure 5: Effect of gradient clipping on training loss in Seaquest. The three panels compare experiments with no reward clipping to those with reward clipping using the threshold values 5 and 1 respectively. Subplots within each panel are similar to those in Figure 4 but display in the first row the evolution of the compound training loss in addition to the frame reconstruction and reward loss.
+
+# A.4 EFFECT OF CURRICULUM LEARNING
+
+In our final experiments with curriculum learning, the networks were trained for 1, 500, 000 minibatch iterations in total but the look ahead parameter $K$ was gradually increased every 500, 000 iterations from 1 to 3 to 5. The networks were hence initially trained on one-step ahead prediction only and later on fine-tuned on further-step ahead prediction. Figure 6 shows how the training objective evolves over iterations. The characteristic ”bumps” in the training objective every 500, 000 iterations as training evolves demonstrate improvements in long-term predictions in all games except Freeway where the training objective assumed already very low values within the first 500, 000 iterations and might have been therefore insensitive to further fine-tuning by curriculum learning.
+
+![](images/42260c8ebdefcaf5957906d4d18479b03baf8e2d6e219b90b9840e87cb4e0056.jpg)  
+Figure 6: Effect of curriculum learning on five different Atari games. Each panel corresponds to a different game, individual panels are structured in the same way as are those in Figure 5
+
+# A.5 EFFECT OF RANDOM SEEDS
+
+We conducted three different experiments per game with different initial random seeds. The effect of different initial random seeds on the cumulative reward error is summarized in Figure 7 which reports how the median and the 5 to 95 percentiles of the cumulative reward error evolve over look ahead steps in the different experiments per game. Note that the results of the first column in Figure 7 are shown in Figure 2 from the main paper together with a more detailed analysis depicting empirical cumulative reward error distributions for some look ahead steps. The random initial seed does not seem to have a significant impact on the cumulative reward prediction except for Freeway where the network in the third experiment starts to considerably overestimate cumulative rewards at around 30 to 40 look ahead steps.
+
+In order to investigate this reward overestimation in Freeway further, we analyse visualizations of joint video frame and reward prediction for this particular seed (similar in style to Figure 3 from Section 4.2 in the main paper). The results are shown in Figure 8 where a peculiar situation occurs after 31 predicted look ahead steps. In Freeway, the agent’s job is to cross a busy road from the bottom to the top without bumping into a car in order to receive reward. If the agent bumps into a car, the agent is propelled downwards further away from the reward-yielding top. This propelled downwards movement happens even when the agent tries to move upwards. Exactly that kind of situation is depicted at the beginning of Figure 8 and occurs for this particular prediction after 31 steps. Our predictive model is however not able to correctly predict the aforementioned downwards movement caused by the agent hitting the car, which is highlighted in red throughout steps 31 to 35 documenting an increasing gap between ground truth and predicted agent position as the propelled downwards movement of the ground truth agent continues. In the course of further prediction, the network model assumes the agent to reach the reward-yielding top side of the road way too early which results in a sequence of erroneous positive reward predictions throughout steps 41 to 50, and as a side effect seemingly that the predictive model loses track of other objects in the scene. Concluding, this finding may serve as a possible explanation for cumulative reward overestimation for that particular experiment in Freeway.
+
+![](images/2c7c9ade8930ab6ee1fa7c50fd28ae1624b117ea38fbd937c24eace3779d16f4.jpg)  
+Figure 7: Effect of different initial random seeds on cumulative reward error. The plots show how the cumulative reward error evolves over look ahead steps in terms of the median and the 5 to 95 percentiles for our network model (blue) as well as the baseline model (red) in each experiment. Each row refers to a different game, each column refers to a different experiment per game initialized with a different random seed. The first column of this figure is presented in Figure 2 of the main paper explaining the results in more detail by additionally illustrating empirical distributions over the cumulative reward error for some look ahead steps.
+
+![](images/42cc60475d758ae8825e1726997fafe061eb4ad7e22120c1289032a7d756b091.jpg)  
+Figure 8: Example predictions in Freeway over 20 steps. The figure is similar in nature to Figure 3 from the main paper with the only difference that predictions are depicted from time step 31 onwards.
+
+# A.6 LOSS ON TEST SET
+
+In the main paper, our analysis focuses on evaluating how well our model serves the purpose of cumulative reward prediction. Here, we evaluate network performance in terms of both the video frame reconstruction loss as well as the reward prediction loss on the test set following the analysis conducted in Oh et al. (2015). For each game, we sample 300 minibatches of size $I = 5 0$ from the underlying test set and compute the test loss over $K = 1 0 0$ look ahead steps with the formula presented in the main paper in Section 3.3 used for learning network parameters, but without averaging over look ahead steps because we aim to illustrate the test loss as a function of look ahead steps—statistics of this analysis are plotted in Figure 9.
+
+Best overall test loss is achieved in Freeway and for initial look ahead steps (up to roughly between 40 and 60 steps) in Q\*bert, which is in accordance with results for cumulative reward prediction from the main paper. Also in line with results from the main paper is the finding that the reward loss on the test set is worse in Seaquest, Ms Pacman and Space Invaders when compared to $\mathrm { Q ^ { * } }$ bert (up to approximately 40 steps) and Freeway. Worst video frame reconstruction loss is observed for Space Invaders in compliance with Oh et al. (2015) where the authors report that there are objects in the scene moving at a period of 9 time steps which is hard to predict by a network only taking the last 4 frames from the last 4 steps as input for future predictions. At first sight, it might seem a bit surprising that the reward prediction loss in Space Invaders is significantly lower than in Seaquest and Ms Pacman for long-term ahead prediction despite the higher frame reconstruction loss in Space Invaders. A possible explanation for this paradox might be the frequency at which rewards are collected—this frequency is significantly higher in Seaquest and Ms Pacman than in Space Invaders. A reward prediction model with bias towards zero rewards—as indicated by the main results in the paper—might therefore err less often in absolute terms when rewards are collected at a lower frequency and may hence achieve lower overall reward reconstruction loss.
+
+![](images/d68110c680a4ecb1d049be4cc34c6c6ee9b3614e4c0d331724fcd64d8d0f138b.jpg)  
+Figure 9: Loss on test set over look ahead steps. Each row reports the loss on the test set over 100 look ahead steps for a different game. The first column illustrates the compound loss consisting of the video frame reconstruction loss (second column) and the reward prediction loss (third column). The loss on the test set is computed according to Oh et al. (2015) similar to the training loss for learning network parameters, however with a different look ahead parameter $K = 1 0 0$ and a different minibatch size $I = 5 0$ , and without averaging over look ahead steps since we aim to plot the test loss as a function of look ahead steps. For each game, the test loss is computed for 300 minibatches resulting in an empirical distribution with 300 loss values per look ahead step. The figure shows the mean (in green), the median (in red), the 5 to 95 percentiles (in shaded blue) as well as minimum and maximum elements (in black dashed lines) of these empirical distributions.

parse/train/BJxhLAuxg/BJxhLAuxg_content_list.json

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+    {
+        "type": "text",
+        "text": "A DEEP LEARNING APPROACH FOR JOINT VIDEOFRAME AND REWARD PREDICTION IN ATARI GAMES",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Felix Leibfried ∗   \nMax Planck Institute for Intelligent Systems   \nMax Planck Institute for Biological Cybernetics   \nGraduate Training Center of Neuroscience   \nTuebingen, Germany   \nfelix.leibfried@gmail.com ",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "Nate Kushman & Katja Hofmann ",
+        "bbox": [
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+            184
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Microsoft Research   \nCambridge, UK   \nnkushman@microsoft.com   \nkatja.hofmann@microsoft.com ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
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+            290,
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+            305
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Reinforcement learning is concerned with learning to interact with environments that are initially unknown. State-of-the-art reinforcement learning approaches, such as DQN, are model-free and learn to act effectively across a wide range of environments such as Atari games, but require huge amounts of data. Modelbased techniques are more data-efficient, but need to acquire explicit knowledge about the environment dynamics or the reward structure. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper we take a step towards using model-based techniques in environments with high-dimensional visual state space when system dynamics and the reward structure are both unknown and need to be learned, by demonstrating that it is possible to learn both jointly. Empirical evaluation on five Atari games demonstrate accurate cumulative reward prediction of up to 200 frames. We consider these positive results as opening up important directions for model-based RL in complex, initially unknown environments. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            534,
+            334,
+            549
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "When humans or animals receive reward for taking a particular action in a given situation, the probability is increased that they will act similarly in similar situations in the future. This is described by principles such as the law of effect (Thorndike, 1898), operant conditioning (Skinner, 1938) and trial-and-error learning (Thorpe, 1979) in behaviorist psychology, and has inspired a discipline of artificial intelligence called reinforcement learning (RL, Sutton & Barto (1998)). RL is concerned with finding optimal behavior policies in order to maximize agents’ cumulative future reward. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Approaches to RL can be divided into model-free and model-based approaches. In model-free approaches, agents learn by trial and error but do not aim to explicitly capture the dynamics of the environment or the structure of the reward function underlying the environment. State-of-the-art modelfree approaches, such as DQN (Mnih et al., 2015), effectively approximate so-called Q-values, i.e., the value of taking specific actions in a given state, using deep neural networks. The impressive effectiveness of these approaches comes from their ability to learn complex policies directly from high-dimensional input (e.g., video frames). Despite their effectiveness, model-free approaches require large amounts of training data that have to be collected through direct interactions with the environment, which makes them expensive to apply in settings where interactions are costly (such as most real-world applications). Additionally, model-free RL requires access to reward observations during training, which is problematic in environments with sparse reward structure—unless coupled with an explicit exploration mechanism. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "RL approaches that explicitly learn statistics about the environment or the reward are generally referred to as model-based—in a more narrow definition these statistics comprise environment dynamics and the reward function. In recent work, model-based techniques were successfully used to learn statistics about cumulative future reward (Veness et al., 2015) and to improve exploration by favoring actions that are likely to lead to novel states (Bellemare et al., 2016; Oh et al., 2015), resulting in substantially more data efficient learning compared to model-free approaches. When an accurate model of the true environment dynamics and the true reward function is available, modelbased approaches, such as planning via Monte-Carlo tree search (Browne et al., 2012) outperform model-free state-of-the-art approaches (Guo et al., 2014). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "A key open question is whether effective model-based RL is possible in complex settings where the environment dynamics and the reward function are initially unknown, and the agent has to acquire such knowledge through experience. In this paper, we take a step towards addressing this question by extending recent work on video frame prediction (Oh et al., 2015), which has been demonstrated to effectively learn system dynamics, to enable joint prediction of future states and rewards using a single latent representation. We propose a network architecture and training procedure for joint state and reward prediction, and evaluate our approach in the Arcade Learning Environment (ALE, Bellemare et al. (2013)). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our empirical results on five Atari games demonstrate that our approach can successfully predict cumulative reward up to roughly 200 frames. We complement our quantitative results with a detailed error analysis by visualizing example predictions. Our results are the first to demonstrate the feasibility of using a learned dynamics and reward model for accurate planning. We see this as a significant step towards data efficient RL in high-dimensional environments without prior knowledge. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK AND MOTIVATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            393,
+            500,
+            410
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Two lines of research are related to the work presented in this paper: model-based RL and optimal control theory. Model-based RL utilizes a given or learned model of some aspect of a task to, e.g., reduce data or exploration requirements (Bellemare et al., 2016; Oh et al., 2015; Veness et al., 2015). Optimal control theory describes mathematical principles for deriving control policies in continuous action spaces that maximize cumulative future reward in scenarios with known system dynamics and known reward structure (Bertsekas, 2007; 2005). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "There has been recent interest in combining principles from optimal control theory and model-based learning in settings where no information on system dynamics is available a priori and instead has to be acquired from visual data (Finn et al., 2016; Wahlstrom et al., 2015; Watter et al., 2015). The ¨ general idea behind these approaches is to learn a compressed latent representation of the visual state space from raw images through autoencoder networks (Bengio, 2009) and to utilize the acquired latent representation to infer system dynamics. System dynamics are then used to specify a planning problem which can be solved by optimization techniques to derive optimal policies. Watter et al. (2015) introduce an approach for learning system dynamics from raw visual data by jointly training a variational autoencoder (Kingma & Welling, 2014; Rezende et al., 2014) and a state prediction model that operates in the autoencoder’s compressed latent state representation. A similar approach for jointly learning a compressed state representation and a predictive model is pursued by Wahlstrom et al. (2015).Finn et al. (2016) devise a sequential approach that first learns a latent state ¨ representation from visual data and that subsequently exploits this latent representation to augment a robot’s initial state space describing joint angles and end-effector positions. The augmented state space is then used to improve estimates of local system dynamics for planning. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The approaches presented above assume knowledge of the functional form of the true reward signal and are hence not directly applicable in settings like ALE (and many real-world settings) where the reward function is initially unknown. Planning in such settings therefore necessitates learning both system dynamics and reward function in order to infer optimal behavioral policies. Recent work by Oh et al. (2015) introduced an approach for learning environment dynamics from pixel images and demonstrated that this enabled successful video frame prediction over up to 400 frames. In our current paper, we extend this recent work to enable reward prediction as well by modifying the network’s architecture and training objective accordingly. The modification of the training objective bears a positive side effect: since our network must optimize a compound loss consisting of the video frame reconstruction loss and the reward loss, reward-relevant aspects in the video frames to which the reconstruction loss alone might be insensitive are explicitly captured by the optimization objective. In the subsequent section, we elucidate the approach from Oh et al. (2015) as well as our extensions for reward prediction in more detail. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/debc0e4e13acb719598c45070f146487cd9562cd8f9454fa101138d9cea47c15.jpg",
+        "image_caption": [
+            "Figure 1: Network architecture for joint video frame and reward prediction. The architecture comprises three stages: an encoding stage mapping current input frames to some compressed latent representation, a transformation stage integrating the current action into the latent representation through element-wise vector multiplication denoted by $\" \\times \"$ , and a final predictive stage for reconstructing the frame of the next time step and the current reward. The network uses three different types of neuron layers (’Conv’ for convolutional, ’Deconv’ for deconvolutional and ’Fc’ for forward connection) in combination with three different types of activation functions (’ReLU’, ’Softmax’ and ’Lin’ for linear activations). The dimensional extend of individual layers is either depicted beneath or within layers. The network part coloured in red highlights the extension for reward prediction. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 NETWORK ARCHITECTURE AND TRAINING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The deep network proposed by Oh et al. (2015) for video frame prediction in Atari games aims at learning a function that predicts the video frame $\\mathbf { s } _ { t + 1 }$ at the next time step $t + 1$ , given the current history of frames $\\mathbf { S } _ { t - h + 1 : t }$ with time horizon $h$ and the current action $\\mathbf { a } _ { t }$ taken by the agent—see Section 3.1. Here, we extend this work to enable joint video frame and reward prediction such that the network anticipates the current reward $\\mathbf { r } _ { t }$ as well—see Sections 3.2 and 3.3. ",
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+    {
+        "type": "text",
+        "text": "3.1 VIDEO FRAME PREDICTION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The video-frame-predictive architecture from Oh et al. (2015) comprises three informationprocessing stages: an encoding stage that maps input frames to some compressed latent representation, a transformation stage that integrates the current action into the compressed latent representation, and a decoding stage that maps the compressed latent representation to the predicted next frame—see Figure 1. The initial encoding stage is a sequence of convolutional and forward operations that map the current frame history $\\mathbf { S } _ { t - h + 1 : t }$ —a three-dimensional tensor—to a compressed feature vector $\\mathbf { h } _ { t } ^ { \\mathrm { e n c } }$ . The transformation stage converts this compressed feature vector $\\mathbf { h } _ { t } ^ { \\mathrm { e n c } }$ into an action-conditional representation $\\mathbf { h } _ { t } ^ { \\mathrm { d e c } }$ in vectorized form by integrating the current action $\\mathbf { a } _ { t }$ . The current action $\\mathbf { a } _ { t }$ is represented as a one-hot vector with length varying from game to game since there are at least 3 and at most 18 actions in ALE. The integration of the current action into the compressed feature vector includes an element-wise vector multiplication—depicted as $\\because \\mathbf { \\nabla } _ { \\times } ,$ in Figure 1—with the particularity that the two neuron layers involved in this element-wise multiplication are the only layers in the entire network without bias parameters, see Section 3.2 in Oh et al. (2015). Finally, the decoding stage performs a series of forward and deconvolutional operations (Dosovitskiy et al., 2015; Zeiler et al., 2010) by mapping the action-conditional representation $\\mathbf { h } _ { t } ^ { \\mathrm { d e c } }$ of the current frame history $\\mathbf { S } _ { t - h + 1 : t }$ and the current action $\\mathbf { a } _ { t }$ to the predicted video frame $\\mathbf { s } _ { t + 1 }$ of the next time step $t + 1$ . Note that this necessitates a reshape operation at the beginning of the decoding cascade in order to transform the vectorized hidden representation into a three-dimensional tensor. The whole network uses linear and rectified linear units (Glorot et al., 2011) only. In all our experiments, following DQN (Mnih et al., 2015), the video frames processed by the network are $8 4 \\times 8 4$ grey-scale images down-sampled from the full-resolution $2 1 0 \\times 1 6 0$ Atari RGB images from ALE. Following Mnih et al. (2015) and Oh et al. (2015), the history frame time horizon $h$ is set to 4. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 REWARD PREDICTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
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+            117
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this section we detail our proposed network architecture for joint state and reward prediction. Our model assumes ternary rewards which result from reward clipping in line with Mnih et al. (2015). Original game scores in ALE are integers that can vary significantly between different Atari games and the corresponding original rewards are clipped to assume one of three values: $- 1$ for negative rewards, 0 for no reward and 1 for positive rewards. Because of reward clipping, rewards can be represented as vectors $\\mathbf { r } _ { t }$ in one-hot encoding of size 3. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In Figure 1, our extension of the video-frame-predictive architecture from Oh et al. (2015) to enable reward prediction is highlighted in red. We add an additional softmax layer to predict the current reward $\\mathbf { r } _ { t }$ with information contained in the action-conditional encoding $\\dot { \\mathbf { h } } _ { t } ^ { \\mathrm { d e c } }$ . The motivation behind this extension is twofold. First, our extension makes it possible to jointly train the network with a compound objective that emphasizes both video frame reconstruction and reward prediction, and thus encourages the network to not abstract away reward-relevant features to which the reconstruction loss alone might be insensitive. Second, this formulation facilitates the future use of the model for reward prediction through virtual roll-outs in the compressed latent space, without the computational expensive necessity of reconstructing video frames explicitly—note that this requires another ”shortcut” predictive model to map from $\\mathbf { h } _ { t } ^ { \\mathrm { d e c } }$ to $\\mathbf { h } _ { t + 1 } ^ { \\mathrm { e n c } }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Following previous work (Oh et al., 2015; Mnih et al., 2015), actions are chosen by the agent on every fourth frame and are repeated on frames that were skipped. Skipped frames and repeated actions are hence not part of the data sets used to train and test the predictive network on, and original reward values are accumulated over four frames before clipping. ",
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+    {
+        "type": "text",
+        "text": "3.3 TRAINING ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Training the model for joint video frame and reward prediction requires trajectory samples $\\left\\{ \\left( \\mathbf { s } _ { n } ^ { ( i ) } , \\mathbf { a } _ { n } ^ { ( i ) } , \\mathbf { r } _ { n } ^ { ( i ) } \\right) _ { n = 1 } ^ { N } \\right\\} _ { i = 1 } ^ { I }$ collected by some agent playing the Atari game, where $i$ is an index over trajectories and $n$ is a time index over samples within one trajectory $i$ . The parameter $I$ denotes the number of trajectories in the training set or the minibatch respectively and the parameter $N$ denotes the length of an individual trajectory. In our case, we use agents trained according to Mnih et al. (2015) in order to collect trajectory samples. ",
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+    {
+        "type": "text",
+        "text": "The original training objective in Oh et al. (2015) consists of a video frame reconstruction loss in terms of a squared loss function aimed at minimizing the quadratic $l ^ { 2 }$ -norm of the difference vector between the ground truth image and its action-conditional reconstruction. We extend this training objective to enable joint reward prediction. This results in a compound training loss consisting of the original video frame reconstruction loss and a reward prediction loss given by the cross entropy (Simard et al., 2003) between the ground truth reward and the corresponding prediction: ",
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+    {
+        "type": "equation",
+        "img_path": "images/df4d7927af964fb2f09632a0ba487752aa23c82611929006ae599961eac5a2ef.jpg",
+        "text": "$$\nL _ { K } ( \\boldsymbol \\theta ) = \\frac { 1 } { 2 \\cdot I \\cdot T \\cdot K } \\sum _ { i = 1 } ^ { I } \\sum _ { t = 0 } ^ { T - 1 } \\sum _ { k = 1 } ^ { K } ( \\underbrace { | \\mathbf { s } _ { t + k } ^ { ( i ) } - \\hat { \\mathbf { s } } _ { t + k } ^ { ( i ) } | | _ { 2 } ^ { 2 } } _ { \\mathrm { v i d e o f r a m e r e c o n s t u c t i o n 1 0 s s } } + \\underbrace { \\lambda \\cdot ( - 1 ) \\sum _ { l = 1 } ^ { 3 } \\mathbf { r } _ { t + k } ^ { ( i ) } [ l ] \\cdot \\ln \\mathbf { p } _ { t + k } ^ { ( i ) } [ l ] } _ { \\mathrm { r e w a r d p r e d i c i t i o n 1 l o s s } } ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\hat { \\mathbf { s } } _ { t + k } ^ { ( i ) }$ denotes the $k$ -step look ahead frame prediction with target video frame $\\mathbf { s } _ { t + k } ^ { ( i ) }$ and p(i)t+k denotes the in red in Fi -step look ahead probability vare1—with target reward vector $\\mathbf { r } _ { t + k } ^ { ( i ) }$ f the reward-p. The paramete $\\lambda > 0$ softmax layer—depicted controls the trade-off between video frame reconstruction and reward loss. The parameter $T$ determines how often a single trajectory sample $i$ is unrolled into the future, and $K$ determines the look ahead prediction horizon dictating how far the network predicts into the future by using its own video frame predicted output as input for the next time step. Following Oh et al. (2015) and Michalski et al. (2014), we apply a curriculum learning (Bengio et al., 2009) scheme by successively increasing $K$ in the course of training such that the network initially learns to predict over a short time horizon and becomes fine-tuned on longer-term predictions as training advances (see Section A.1 for details). The network parameters $\\theta$ are updated by stochastic gradient descent, derivatives of the training objective w.r.t. $\\theta$ are computed with backpropagation through time (Werbos, 1988). ",
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+    {
+        "type": "text",
+        "text": "4 RESULTS ",
+        "text_level": 1,
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+        "text": "In our evaluations, we investigate cumulative reward predictions quantitatively and qualitatively on five different Atari games (Q\\*bert, Seaquest, Freeway, Ms Pacman and Space Invaders). The quantitative analysis comprises evaluating the cumulative reward prediction error—see Section 4.1. The qualitative analysis comprises visualizations of example predictions in Seaquest—see Section 4.2. ",
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+    {
+        "type": "text",
+        "text": "4.1 QUANTITATIVE REWARD PREDICTION ANALYSIS: CUMULATIVE REWARD ERROR ",
+        "bbox": [
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+        "type": "text",
+        "text": "Our quantitative evaluation examines whether our joint model of system dynamics and reward function results in a shared latent representation that enables accurate cumulative reward prediction. We assess cumulative reward prediction on test sets consisting of approximately 50,000 video frames per game, including actions and rewards. Each network is evaluated on 1,000 trajectories—suitable to analyze up to 100-step ahead prediction—drawn randomly from the test set. Look ahead prediction is measured in terms of the cumulative reward error which is the difference between ground truth cumulative reward and predicted cumulative reward. For each game, this results in 100 empirical distributions over the cumulative reward error—one distribution for each look ahead step—consisting of 1,000 samples each (one for each trajectory). We compare our model predictions to a baseline model that samples rewards from the marginal reward distribution observed on the test set for each game. Note that negative reward values are absent in the games investigated for this study. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Figure 2 illustrates 20 of the 100 empirical cumulative reward error distributions in all games for our network model in blue and for the baseline model in red (histograms, bottom), together with the median and the 5 to 95 percentiles of the cumulative reward error over look ahead steps (top). Across all games, we observe that our joint state and reward prediction model accurately predicts future cumulative rewards at least 20 look ahead steps, and that it predicts future rewards substantially more accurately than the baseline model. This is evidenced by cumulative reward error distributions that maintain a unimodal form with mode zero and do not flatten out as quickly as the distributions for the random-prediction baseline model. Best results are achieved in Freeway and $\\boldsymbol { \\mathrm { Q } } ^ { * } \\boldsymbol { \\mathrm { b e r t } }$ where the probability of zero cumulative reward error at 51 look ahead steps is still around $8 0 \\%$ and $6 0 \\%$ respectively—see Figure 2. Note that 51 look ahead steps correspond to 204 frames because the underlying DQN agent, collecting trajectory samples for training and testing our model, skipped every fourth frame when choosing an action—see Section 3.2. Lowest performance is obtained in Seaquest where the probability of zero cumulative reward error at 26 steps (104 frames) is around $4 0 \\%$ and begins to flatten out soon thereafter—see Figure 2. Running the ALE emulator at a frequency of 60fps, 26 steps correspond to more than 1 second real-time game play because of frame skipping. Since our model is capable of predicting 26 steps ahead in less than 1 second, our model enables real-time planning and could be therefore utilized in an online fashion. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We now turn our attention to error analysis. While the look ahead step at which errors become prominent differs substantially from game to game, we find that overall our model underestimates cumulative reward. This can be seen in the asymmetry towards positive cumulative reward error values when inspecting the 5 to 95 percentile intervals in the first plot per each game in Figure 2. We identify a likely cause in (pseudo-)stochastic transitions inherent in these games. Considering Seaquest as our running example, objects such as divers and submarines can enter the scene randomly from the right and from the left and at the same time have an essential impact on which rewards the agent can potentially collect. In the ground truth trajectories, the agent’s actions are reactions to these objects. If the predicted future trajectory deviates from the ground truth, targeted actions such as shooting will miss their target, leading to underestimating true reward. We analyze this effect in more detail in Section 4.2. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "All our experiments were conducted in triplicate with different initial random seeds. Different initial random seeds did not have a significant impact on cumulative reward prediction in all games except Freeway—see Section A.5 for a detailed analysis. So far, we discussed results concerning reward prediction only. In the appendix, we also evaluate the joint performance of reward and video frame prediction on the test set in terms of the optimization objective as in Oh et al. (2015), where the authors report successful video frame reconstruction up to approximately 100 steps (400 frames), and observe similar results—see Section A.6. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the previous section, we identified stochasticity in state transitions as a likely cause for relatively low performance in long-term cumulative reward prediction in games such as Seaquest. In Seaquest objects may randomly enter a scene in a non-deterministic fashion. Errors in predicting these events result in predicted possible futures that do not match actually observed future states, resulting in inaccurate reward predictions. Here, we support this hypothesis by visualizations in Seaquest illustrating joint video frame and reward prediction for a single network over 20 steps (80 frames)—see Figure 3 where ground truth video frames are compared to predicted video frames in terms of error maps. Error maps emphasize the difference between ground truth and predicted frames through squared error values between pixels in black or white depending on whether objects are absent or present by mistake in the network’s prediction. Actions, ground truth rewards and model-predicted rewards are shown between state transitions. Peculiarities in the prediction process are shown in red. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In step 2, the model predicts reward by mistake because the agent barely misses its target. Steps 4 to 6 report how the model predicts reward correctly but is off by one time step. Steps 7 to 14 depict problems caused by objects randomly entering the scene from the right which the model cannot predict. Steps 26 to 30 show how the model has problems to predict rewards at steps 26 and 28 as these rewards are attached to objects the model failed to notice entering the scene earlier. ",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION AND FUTURE WORK ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this paper, we extended recent work on video frame prediction (Oh et al., 2015) in Atari games to enable reward prediction. Our approach can be used to jointly predict video frames and cumulative rewards up to a horizon of approximately 200 frames in five different games ( $\\mathbf { Q } ^ { * }$ bert, Seaquest, Freeway, Ms Pacman and Space Invaders). We achieved best results in Freeway and $\\mathbf { Q } ^ { * }$ bert where the probability of zero cumulative reward error after 200 frames is still around $8 0 \\%$ and $6 0 \\%$ respectively, and worst results in Seaquest where the probability of zero cumulative reward error after 100 frames is around $4 0 \\%$ . Our study fits into the general line of research using autoencoder networks to learn a latent representation from visual data (Finn et al., 2016; Goroshin et al., 2015; Gregor et al., 2015; Kulkarni et al., 2015; Srivastava et al., 2015; Wahlstrom et al., 2015; Watter et al., 2015; ¨ Kingma & Welling, 2014; Rezende et al., 2014; Lange et al., 2012; Hinton et al., 2011; Ranzato et al., 2007), and extends this line of research by showing that autoencoder networks are capable of learning a combined representation for system dynamics and the reward function in reinforcement learning settings with high-dimensional visual state spaces—a first step towards applying modelbased techniques for planning in environments where the reward function is not initially known. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Our positive results open up intriguing directions for future work. Our long-term goal is the integration of model-based and model-free approaches for effective interactive learning and planning in complex environments. Directions for achieving this long-standing challenge include the Dyna method (Sutton, 1990), which uses a predictive model to artificially augment expensive training data, and has been shown to lead to substantial reductions in data requirements in tabular RL approaches. Alternatively, the model could be could be utilized for planning via Monte-Carlo tree search (Guo et al., 2014; Browne et al., 2012). We hypothesize that such an approach would be particularly beneficial in multi-task or life-long learning scenarios where the reward function changes but the environment dynamics are stationary. Testing this hypothesis requires a flexible learning framework where the reward function and the artificial environment can be changed by the experimenter in an arbitrary fashion, which is not possible in ALE where the environment and the reward function are fixed per game. A learning environment providing such a flexibility is the recently released Malmo platform for Minecraft (Johnson et al., 2016) where researchers can create user-defined en-¨ vironments and tasks in order to evaluate the performance of artificial agents. In the shorter-term, we envision improving the prediction performance of our network by regularization methods such as dropout and max norm regularization (Srivastava et al., 2014)—a state-of-the-art regularizer in supervised learning—and by modifying the optimization objective to enforce similarity between hidden encodings in multi-step ahead prediction and one-step ahead prediction—see Watter et al. (2015). Finally, extensions of our model to non-deterministic state transitions through dropout and variational autoencoder schemes (Kingma & Welling, 2014; Rezende et al., 2014) is a promising direction to alleviate the limitations highlighted in Section 4.2—paving the way for models that adequately predict and reason over alternative possible future trajectories. ",
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+    {
+        "type": "image",
+        "img_path": "images/02740b7abbfcbfdaf1dce245f49cc57b86ddcfd6ce7d81361d8571db31ad4b30.jpg",
+        "image_caption": [
+            "two plots for each game. The top plot per game shows how the median and the 5 to 95 percentiles of the cumulative reward error evolve over look ahead steps for both our model (in blue) and a baseline model that samples rewards from the marginal reward distribution of the test set (in red). Each vertical slice of this concise representation corresponds to a single empirical distribution over the cumulative reward error. We depict these for every fifth look ahead step in the compound plots below for both models. These empirical error distributions demonstrate successful cumulative reward prediction over at least 20 steps (80 frames) in all five games as evidenced by their zero-centered and unimodal shape in the first column of each compound plot per game. "
+        ],
+        "image_footnote": [],
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+        "page_idx": 6
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+        "img_path": "images/5ca73ff0f11ebf0f135df66feefe058cd29d3977e94fcfe61f6728d039d6d497.jpg",
+        "image_caption": [
+            "Figure 3: Example predictions in Seaquest. Ground truth video frames, model predictions and error maps emphasizing differences between ground truth and predicted frames—in form of the squared error between pixel values—are compared column-wise. Error maps highlight objects in black or white respectively depending on whether these objects are absent by mistake or present by mistake in the model’s prediction. Actions taken by the agent as well as ground truth rewards (’rew’) and reward predictions (’pred’) are shown below video and error frames. Peculiarities in the prediction process are marked in red. The figure demonstrates how our predictive model fails to anticipate objects that randomly enter the scene from the right and rewards associated to these objects. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            287,
+            117
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "M G Bellemare, Y Naddaf, J Veness, and M Bowling. The Arcade Learning Environment: an evaluation platform for general agents. Journal of Artificial Intelligence Research, 47:253–279, 2013. ",
+        "bbox": [
+            174,
+            126,
+            823,
+            167
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "M G Bellemare, S Srinivasan, G Ostrovski, T Schaul, D Saxton, and R Munos. Unifying count-based exploration and intrinsic motivation. arXiv preprint arXiv:1606.01868, 2016. ",
+        "bbox": [
+            171,
+            178,
+            823,
+            208
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Y Bengio. Learning deep architectures for AI. Foundations and Trends in Machine Learning, 2(1): 1–127, 2009. ",
+        "bbox": [
+            174,
+            217,
+            823,
+            246
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Y Bengio, J Louradour, R Collobert, and J Weston. Curriculum learning. In Proceedings of the International Conference on Machine Learning, 2009. ",
+        "bbox": [
+            176,
+            256,
+            825,
+            286
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "D P Bertsekas. Dynamic programming & optimal control, volume 1. Athena Scientific, 2005. ",
+        "bbox": [
+            178,
+            295,
+            787,
+            311
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "D P Bertsekas. Dynamic programming & optimal control, volume 2. Athena Scientific, 2007. ",
+        "bbox": [
+            178,
+            320,
+            787,
+            337
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "C Browne, E Powley, D Whitehouse, S Lucas, P I Cowling, P Rohlfshagen, S Tavener, D Perez, S Samothrakis, and S Colton. A survey of monte carlo tree search methods. IEEE Transactions on Computational Intelligence and AI in Games, 4(1):1–49, 2012. ",
+        "bbox": [
+            174,
+            344,
+            825,
+            388
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "A Dosovitskiy, J T Springenberg, and T Brox. Learning to generate chairs with convolutional neural networks. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2015. ",
+        "bbox": [
+            173,
+            397,
+            825,
+            440
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "C Finn, X Y Tan, Y Duan, T Darrell, S Levine, and P Abbeel. Deep spatial autoencoders for visuomotor learning. In Proceedings of the IEEE International Conference on Robotics and Automation, 2016. ",
+        "bbox": [
+            176,
+            450,
+            823,
+            493
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "X Glorot and Y Bengio. Understanding the difficulty of training deep feedforward neural networks. In Proceedings of the International Conference on Artificial Intelligence and Statistics, 2010. ",
+        "bbox": [
+            173,
+            503,
+            823,
+            534
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "X Glorot, A Bordes, and Y Bengio. Deep sparse rectifier neural networks. In Proceedings of the International Conference on Artificial Intelligence and Statistics, 2011. ",
+        "bbox": [
+            173,
+            541,
+            823,
+            571
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "R Goroshin, M Mathieu, and Y LeCun. Learning to linearize under uncertainty. Advances in Neural Information Processing Systems, 2015. ",
+        "bbox": [
+            176,
+            580,
+            825,
+            611
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "K Gregor, I Danihelka, A Graves, D J Rezende, and D Wierstra. DRAW: a recurrent neural network for image generation. In Proceedings of the International Conference on Machine Learning, 2015. ",
+        "bbox": [
+            174,
+            619,
+            825,
+            662
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "X Guo, S Singh, H Lee, R Lewis, and X Wang. Deep learning for real-time Atari game play using offline Monte-Carlo tree search planning. In Advances in Neural Information Processing Systems, 2014. ",
+        "bbox": [
+            173,
+            672,
+            826,
+            715
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "G E Hinton, A Krizhevsky, and S D Wang. Transforming auto-encoders. In Proceedings of the International Conference on Artificial Neural Networks, 2011. ",
+        "bbox": [
+            174,
+            724,
+            825,
+            755
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "M Johnson, K Hofmann, T Hutton, and D Bignell. The Malmo platform for artificial intelligence experimentation. In Proceedings of the International Joint Conference on Artificial Intelligence, 2016. ",
+        "bbox": [
+            176,
+            763,
+            823,
+            808
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "D P Kingma and J Ba. Adam: a method for stochastic optimization. In Proceedings of the International Conference on Learning Representations, 2015. ",
+        "bbox": [
+            171,
+            818,
+            823,
+            847
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "D P Kingma and M Welling. Auto-encoding variational Bayes. In Proceedings of the International Conference on Learning Representations, 2014. ",
+        "bbox": [
+            176,
+            856,
+            823,
+            886
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "T D Kulkarni, W F Whitney, P Kohli, and J B Tenenbaum. Deep convolutional inverse graphics network. In Advances in Neural Information Processing Systems, 2015. ",
+        "bbox": [
+            176,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "S Lange, M Riedmiller, and A Voigtlander. Autonomous reinforcement learning on raw visual input ¨ data in a real world application. In Proceedings of the International Joint Conference on Neural Networks, 2012. ",
+        "bbox": [
+            174,
+            103,
+            821,
+            146
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "V Michalski, R Memisevic, and K Konda. Modeling deep temporal dependencies with recurrent grammar cells. In Advances in Neural Information Processing Systems, 2014. ",
+        "bbox": [
+            171,
+            154,
+            823,
+            184
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "V Mnih, K Kavukcuoglu, D Silver, A A Rusu, J Veness, M G Bellemare, A Graves, M Riedmiller, A K Fidjeland, G Ostrovski, S Petersen, C Beattie, A Sadik, I Antonoglou, H King, D Kumaran, D Wierstra, S Legg, and D Hassabis. Human-level control through deep reinforcement learning. Nature, 518(7540):529–533, 2015. ",
+        "bbox": [
+            179,
+            190,
+            821,
+            247
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "J Oh, X Guo, H Lee, R Lewis, and S Singh. Action-conditional video prediction using deep networks in Atari games. In Advances in Neural Information Processing Systems, 2015. ",
+        "bbox": [
+            171,
+            256,
+            820,
+            286
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "R Pascanu, T Mikolov, and Y Bengio. On the difficulty of training recurrent neural networks. In Proceedings of the International Conference on Machine Learning, 2013. ",
+        "bbox": [
+            173,
+            292,
+            825,
+            323
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "M Ranzato, F J Huang, Y-L Boureau, and Y LeCun. Unsupervised learning of invariant feature hierarchies with applications to object recognition. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2007. ",
+        "bbox": [
+            176,
+            330,
+            821,
+            373
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "D J Rezende, S Mohamed, and D Wierstra. Stochastic backpropagation and approximate inference in deep generative models. In Proceedings of the International Conference on Machine Learning, 2014. ",
+        "bbox": [
+            176,
+            381,
+            825,
+            424
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "P Y Simard, D Steinkraus, and J C Platt. Best practices for convolutional neural networks applied to visual document analysis. In Proceedings of the International Conference on Document Analysis and Recognition, 2003. ",
+        "bbox": [
+            174,
+            433,
+            825,
+            476
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "B F Skinner. The behavior of organisms: an experimental analysis. Appleton-Century-Crofts, 1938. ",
+        "bbox": [
+            171,
+            483,
+            823,
+            500
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "N Srivastava, G E Hinton, A Krizhevsky, I Sutskever, and R Salakhutdinov. Dropout : a simple way to prevent neural networks from overfitting. Journal of Machine Learning Research, 15: 1929–1958, 2014. ",
+        "bbox": [
+            174,
+            506,
+            825,
+            550
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "N Srivastava, E Mansimov, and R Salakhutdinov. Unsupervised learning of video representations using LSTMs. In Proceedings of the International Conference on Machine Learning, 2015. ",
+        "bbox": [
+            174,
+            558,
+            823,
+            588
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "R S Sutton. Integrated architectures for learning, planning, and reacting based on approximating dynamic programming. In Proceedings of the International Conference on Machine Learning, 1990. ",
+        "bbox": [
+            176,
+            594,
+            823,
+            638
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "R S Sutton and A G Barto. Reinforcement learning: an introduction. MIT Press, 1998. ",
+        "bbox": [
+            173,
+            646,
+            745,
+            661
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "E L Thorndike. Animal intelligence: an experimental study of the associative processes in animals. The Psychological Review: Monograph Supplements, 2(4):1–107, 1898. ",
+        "bbox": [
+            176,
+            669,
+            823,
+            699
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "W H Thorpe. The origins and rise of ethology. Heinemann Educational Books, 1979. ",
+        "bbox": [
+            176,
+            705,
+            736,
+            722
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "J Veness, M G Bellemare, M Hutter, A Chua, and G Desjardins. Compress and control. In Proceedings of the AAAI Conference on Artificial Intelligence, 2015. ",
+        "bbox": [
+            173,
+            729,
+            821,
+            758
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "N Wahlstrom, T B Sch ¨ on, and M P Deisenroth. From pixels to torques: policy learning with deep ¨ dynamical models. arXiv preprint arXiv:1502.02251, 2015. ",
+        "bbox": [
+            171,
+            767,
+            823,
+            796
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "M Watter, J T Springenberg, J Boedecker, and M Riedmiller. Embed to control: a locally linear latent dynamics model for control from raw images. In Advances in Neural Information Processing Systems, 2015. ",
+        "bbox": [
+            173,
+            804,
+            823,
+            847
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "P J Werbos. Generalization of backpropagation with application to a recurrent gas market model. Neural Networks, 1(4):339–356, 1988. ",
+        "bbox": [
+            171,
+            854,
+            821,
+            885
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "M D Zeiler, D Krishnan, G W Taylor, and R Fergus. Deconvolutional networks. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2010. ",
+        "bbox": [
+            169,
+            892,
+            825,
+            921
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            297,
+            117
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.1 TRAINING DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            133,
+            354,
+            148
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We performed all our experiments in Python with Chainer and adhered to the instructions in Oh et al. (2015) as close as possible. Trajectory samples for learning the network parameters were obtained from a previously trained DQN agent according to Mnih et al. (2015). The dataset for training comprised around 500, 000 video frames per game in addition to actions chosen by the DQN agent and rewards collected during game play. Video frames used as network input were $8 4 \\times 8 4$ grey-scale images with pixel values between 0 and 255 down-sampled from the full-resolution $2 1 0 \\times 1 6 0$ ALE RGB images. We applied a further preprocessing step by dividing each pixel by 255 and subtracting mean pixel values from each image leading to final pixel values $\\in \\ [ - 1 ; 1 ]$ . A detailed network architecture is shown in Figure 1 in the main paper. All weights in the network were initialized according to Glorot & Bengio (2010) except for those two layers that participate in the element-wise multiplication in Figure 1: the weights of the action-processing layer were initialized uniformly in the range $[ - 0 . 1 ; 0 . { \\bar { 1 } } ]$ and the weights of the layer receiving the latent encoding of the input video frames were initialized uniformly in the range $[ - 1 ; 1 ]$ . Training was performed for $1 , 5 0 0 , 0 0 0$ minibatch iterations with a curriculum learning scheme increasing the look ahead parameter $K$ every 500, 000 iterations from 1 to 3 to 5. When increasing the look ahead parameter $K$ for the first time after 500, 000 iterations, the minibatch size $I$ was also altered from 32 to 8 as was the learning rate for parameter updates from $1 0 ^ { - 4 }$ to $1 0 ^ { - 5 }$ . Throughout the entire curriculum scheme, the time horizon parameter determining the number of times a single trajectory is unrolled into the future was $T = 4$ . The optimizer for updating weights was Adam (Kingma & Ba, 2015) with gradient momentum 0.9, squared gradient momentum 0.95 and epsilon parameter $1 0 ^ { - 8 }$ . In evaluation mode, network outputs were clipped to $[ - 1 ; 1 ]$ so that strong activations could not accumulate over roll-out time in the network. ",
+        "bbox": [
+            174,
+            160,
+            825,
+            465
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In our experiments, we modified the reward prediction loss slightly in order to prevent exploding gradient values by replacing the term $- \\ln p$ with a first-order Taylor approximation for $p$ -values smaller than $e ^ { - 1 0 } { \\mathrm { - a } } $ similar technique is used in DQN (Mnih et al., 2015) to improve the stability of the optimization algorithm. To identify optimal values for the reward weight $\\lambda$ , we performed initial experiments on Ms Pacman without applying the aforementioned curriculum learning scheme instead using a fixed look ahead parameter $K = 1$ . We evaluated the effect of different $\\lambda$ -values $\\in \\{ 0 . 1 , 1 , \\bar { 1 0 } , 1 0 0 \\}$ on the training objective and identified $\\lambda = 1$ for conducting further experiments—see Section A.2. After identifying an optimal reward weight, we conducted additional initial experiments without curriculum learning with fixed look ahead parameter $K = 1$ on all of the five different Atari games used in this paper. We observed periodic oscillations in the reward prediction loss of the training objective in Seaquest, which was fixed by adding gradient clipping (Pascanu et al., 2013) with threshold parameter 1 to our optimization procedure—experiments investigating the effect of gradient clipping in Seaquest are reported in Section A.3. The fine-tuning effect of curriculum learning on the training objective in our final experiments is shown in Section A.4 for all of the five analysed Atari games. ",
+        "bbox": [
+            174,
+            473,
+            825,
+            680
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.2 EFFECT OF REWARD WEIGHT IN MS PACMAN",
+        "text_level": 1,
+        "bbox": [
+            176,
+            699,
+            534,
+            713
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "To identify optimal values for the reward weight $\\lambda$ , we conducted initial experiments in Ms Pacman without curriculum learning and a fixed look ahead horizon $K = 1$ . We tested four different $\\lambda$ - values $\\in \\{ 0 . 1 , 1 , 1 0 , 1 0 0 \\}$ and investigated how the frame reconstruction loss and the reward loss of the training objective evolve over minibatch iterations—see Figure 4. Best results were obtained for $\\lambda = 1$ and for $\\lambda = 1 0$ , whereas values of $\\lambda = 0 . 1$ and $\\lambda = 1 0 0$ lead to significantly slower convergence and worse overall training performance respectively. ",
+        "bbox": [
+            174,
+            726,
+            825,
+            809
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.3 EFFECT OF GRADIENT CLIPPING IN SEAQUEST ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            827,
+            542,
+            842
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "After identifying an optimal value for the reward weight, see Section A.2, we observed oscillations in the reward loss of the training objective in Seaquest—see first column in Figure 5—which was solved by adding gradient clipping to our optimization procedure—see second and third column in Figure 5. We tested two different values for the gradient clipping threshold (5 and 1) both of which worked, but for a value of 1 the oscillation vanished completely. ",
+        "bbox": [
+            174,
+            854,
+            823,
+            924
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/720d06db36adf5fcea69e6c832cc20f28160a2bac7d37f237596cd85afe2ab0c.jpg",
+        "image_caption": [
+            "Figure 4: Effect of reward weight on training loss in Ms Pacman. Each of the four panels depicts one experiment with a different reward weight $\\lambda$ . Each panel shows how the training loss evolves over minibatch iterations in terms of two subplots reporting video frame reconstruction and reward loss respectively. Each experiment was conducted three times with different initial random seeds depicted in blue, green and red. Graphs were smoothed with an exponential window of size 1000. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            235,
+            99,
+            758,
+            386
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/36937623cd79e22cff1c94043eb54f0eaf9f923351d0908c7abcabc68ad664cb.jpg",
+        "image_caption": [
+            "Figure 5: Effect of gradient clipping on training loss in Seaquest. The three panels compare experiments with no reward clipping to those with reward clipping using the threshold values 5 and 1 respectively. Subplots within each panel are similar to those in Figure 4 but display in the first row the evolution of the compound training loss in addition to the frame reconstruction and reward loss. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            169,
+            501,
+            805,
+            657
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.4 EFFECT OF CURRICULUM LEARNING",
+        "text_level": 1,
+        "bbox": [
+            176,
+            777,
+            473,
+            791
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In our final experiments with curriculum learning, the networks were trained for 1, 500, 000 minibatch iterations in total but the look ahead parameter $K$ was gradually increased every 500, 000 iterations from 1 to 3 to 5. The networks were hence initially trained on one-step ahead prediction only and later on fine-tuned on further-step ahead prediction. Figure 6 shows how the training objective evolves over iterations. The characteristic ”bumps” in the training objective every 500, 000 iterations as training evolves demonstrate improvements in long-term predictions in all games except Freeway where the training objective assumed already very low values within the first 500, 000 iterations and might have been therefore insensitive to further fine-tuning by curriculum learning. ",
+        "bbox": [
+            174,
+            811,
+            825,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/42260c8ebdefcaf5957906d4d18479b03baf8e2d6e219b90b9840e87cb4e0056.jpg",
+        "image_caption": [
+            "Figure 6: Effect of curriculum learning on five different Atari games. Each panel corresponds to a different game, individual panels are structured in the same way as are those in Figure 5 "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            171,
+            98,
+            808,
+            419
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.5 EFFECT OF RANDOM SEEDS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            492,
+            411,
+            506
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We conducted three different experiments per game with different initial random seeds. The effect of different initial random seeds on the cumulative reward error is summarized in Figure 7 which reports how the median and the 5 to 95 percentiles of the cumulative reward error evolve over look ahead steps in the different experiments per game. Note that the results of the first column in Figure 7 are shown in Figure 2 from the main paper together with a more detailed analysis depicting empirical cumulative reward error distributions for some look ahead steps. The random initial seed does not seem to have a significant impact on the cumulative reward prediction except for Freeway where the network in the third experiment starts to considerably overestimate cumulative rewards at around 30 to 40 look ahead steps. ",
+        "bbox": [
+            173,
+            517,
+            825,
+            642
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In order to investigate this reward overestimation in Freeway further, we analyse visualizations of joint video frame and reward prediction for this particular seed (similar in style to Figure 3 from Section 4.2 in the main paper). The results are shown in Figure 8 where a peculiar situation occurs after 31 predicted look ahead steps. In Freeway, the agent’s job is to cross a busy road from the bottom to the top without bumping into a car in order to receive reward. If the agent bumps into a car, the agent is propelled downwards further away from the reward-yielding top. This propelled downwards movement happens even when the agent tries to move upwards. Exactly that kind of situation is depicted at the beginning of Figure 8 and occurs for this particular prediction after 31 steps. Our predictive model is however not able to correctly predict the aforementioned downwards movement caused by the agent hitting the car, which is highlighted in red throughout steps 31 to 35 documenting an increasing gap between ground truth and predicted agent position as the propelled downwards movement of the ground truth agent continues. In the course of further prediction, the network model assumes the agent to reach the reward-yielding top side of the road way too early which results in a sequence of erroneous positive reward predictions throughout steps 41 to 50, and as a side effect seemingly that the predictive model loses track of other objects in the scene. Concluding, this finding may serve as a possible explanation for cumulative reward overestimation for that particular experiment in Freeway. ",
+        "bbox": [
+            173,
+            650,
+            825,
+            886
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/2c7c9ade8930ab6ee1fa7c50fd28ae1624b117ea38fbd937c24eace3779d16f4.jpg",
+        "image_caption": [
+            "Figure 7: Effect of different initial random seeds on cumulative reward error. The plots show how the cumulative reward error evolves over look ahead steps in terms of the median and the 5 to 95 percentiles for our network model (blue) as well as the baseline model (red) in each experiment. Each row refers to a different game, each column refers to a different experiment per game initialized with a different random seed. The first column of this figure is presented in Figure 2 of the main paper explaining the results in more detail by additionally illustrating empirical distributions over the cumulative reward error for some look ahead steps. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            102,
+            808,
+            789
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/42cc60475d758ae8825e1726997fafe061eb4ad7e22120c1289032a7d756b091.jpg",
+        "image_caption": [
+            "Figure 8: Example predictions in Freeway over 20 steps. The figure is similar in nature to Figure 3 from the main paper with the only difference that predictions are depicted from time step 31 onwards. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            141,
+            820,
+            835
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.6 LOSS ON TEST SET ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            351,
+            117
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "In the main paper, our analysis focuses on evaluating how well our model serves the purpose of cumulative reward prediction. Here, we evaluate network performance in terms of both the video frame reconstruction loss as well as the reward prediction loss on the test set following the analysis conducted in Oh et al. (2015). For each game, we sample 300 minibatches of size $I = 5 0$ from the underlying test set and compute the test loss over $K = 1 0 0$ look ahead steps with the formula presented in the main paper in Section 3.3 used for learning network parameters, but without averaging over look ahead steps because we aim to illustrate the test loss as a function of look ahead steps—statistics of this analysis are plotted in Figure 9. ",
+        "bbox": [
+            173,
+            128,
+            825,
+            241
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Best overall test loss is achieved in Freeway and for initial look ahead steps (up to roughly between 40 and 60 steps) in Q\\*bert, which is in accordance with results for cumulative reward prediction from the main paper. Also in line with results from the main paper is the finding that the reward loss on the test set is worse in Seaquest, Ms Pacman and Space Invaders when compared to $\\mathrm { Q ^ { * } }$ bert (up to approximately 40 steps) and Freeway. Worst video frame reconstruction loss is observed for Space Invaders in compliance with Oh et al. (2015) where the authors report that there are objects in the scene moving at a period of 9 time steps which is hard to predict by a network only taking the last 4 frames from the last 4 steps as input for future predictions. At first sight, it might seem a bit surprising that the reward prediction loss in Space Invaders is significantly lower than in Seaquest and Ms Pacman for long-term ahead prediction despite the higher frame reconstruction loss in Space Invaders. A possible explanation for this paradox might be the frequency at which rewards are collected—this frequency is significantly higher in Seaquest and Ms Pacman than in Space Invaders. A reward prediction model with bias towards zero rewards—as indicated by the main results in the paper—might therefore err less often in absolute terms when rewards are collected at a lower frequency and may hence achieve lower overall reward reconstruction loss. ",
+        "bbox": [
+            174,
+            247,
+            825,
+            455
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/d68110c680a4ecb1d049be4cc34c6c6ee9b3614e4c0d331724fcd64d8d0f138b.jpg",
+        "image_caption": [
+            "Figure 9: Loss on test set over look ahead steps. Each row reports the loss on the test set over 100 look ahead steps for a different game. The first column illustrates the compound loss consisting of the video frame reconstruction loss (second column) and the reward prediction loss (third column). The loss on the test set is computed according to Oh et al. (2015) similar to the training loss for learning network parameters, however with a different look ahead parameter $K = 1 0 0$ and a different minibatch size $I = 5 0$ , and without averaging over look ahead steps since we aim to plot the test loss as a function of look ahead steps. For each game, the test loss is computed for 300 minibatches resulting in an empirical distribution with 300 loss values per look ahead step. The figure shows the mean (in green), the median (in red), the 5 to 95 percentiles (in shaded blue) as well as minimum and maximum elements (in black dashed lines) of these empirical distributions. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            44,
+            805,
+            801
+        ],
+        "page_idx": 16
+    }
+]

parse/train/HyTqHL5xg/HyTqHL5xg.md

@@ -0,0 +1,384 @@
+# DEEP VARIATIONAL BAYES FILTERS: UNSUPERVISED LEARNING OF STATE SPACE MODELS FROM RAW DATA
+
+Maximilian Karl, Maximilian Soelch, Justin Bayer, Patrick van der Smagt Data Lab, Volkswagen Group, 80805, München, Germany zip([maximilian.karl, maximilian.soelch], [@volkswagen.de])
+
+# ABSTRACT
+
+We introduce Deep Variational Bayes Filters (DVBF), a new method for unsupervised learning and identification of latent Markovian state space models. Leveraging recent advances in Stochastic Gradient Variational Bayes, DVBF can overcome intractable inference distributions via variational inference. Thus, it can handle highly nonlinear input data with temporal and spatial dependencies such as image sequences without domain knowledge. Our experiments show that enabling backpropagation through transitions enforces state space assumptions and significantly improves information content of the latent embedding. This also enables realistic long-term prediction.
+
+# 1 INTRODUCTION
+
+Estimating probabilistic models for sequential data is central to many domains, such as audio, natural language or physical plants, Graves (2013); Watter et al. (2015); Chung et al. (2015); Deisenroth & Rasmussen (2011); Ko & Fox (2011). The goal is to obtain a model $p ( \mathbf { x } _ { 1 : T } )$ that best reflects a data set of observed sequences $\mathbf { x } _ { \mathrm { 1 : } T }$ . Recent advances in deep learning have paved the way to powerful models capable of representing high-dimensional sequences with temporal dependencies, e.g., Graves (2013); Watter et al. (2015); Chung et al. (2015); Bayer & Osendorfer (2014).
+
+Time series for dynamic systems have been studied extensively in systems theory, cf. McGoff et al. (2015) and sources therein. In particular, state space models have shown to be a powerful tool to analyze and control the dynamics. Two tasks remain a significant challenge to this day: Can we identify the governing system from data only? And can we perform inference from observables to the latent system variables? These two tasks are competing: A more powerful representation of system requires more computationally demanding inference, and efficient inference, such as the well-known Kalman filters, Kalman & Bucy (1961), can prohibit sufficiently complex system classes.
+
+Leveraging a recently proposed estimator based on variational inference, stochastic gradient variational Bayes (SGVB, Kingma & Welling (2013); Rezende et al. (2014)), approximate inference of latent variables becomes tractable. Extensions to time series have been shown in Bayer & Osendorfer (2014); Chung et al. (2015). Empirically, they showed considerable improvements in marginal data likelihood, i.e., compression, but lack full-information latent states, which prohibits, e.g., long-term sampling. Yet, in a wide range of applications, full-information latent states should be valued over compression. This is crucial if the latent spaces are used in downstream applications.
+
+Our contribution is, to our knowledge, the first model that (i) enforces the latent state-space model assumptions, allowing for reliable system identification, and plausible long-term prediction of the observable system, (ii) provides the corresponding inference mechanism with rich dependencies, (iii) inherits the merit of neural architectures to be trainable on raw data such as images or other sensory inputs, and (iv) scales to large data due to optimization of parameters based on stochastic gradient descent, Bottou (2010). Hence, our model has the potential to exploit systems theory methodology for downstream tasks, e.g., control or model-based reinforcement learning, Sutton (1996).
+
+# 2 BACKGROUND AND RELATED WORK
+
+2.1 PROBABILISTIC MODELING AND FILTERING OF DYNAMICAL SYSTEMS
+
+We consider non-linear dynamical systems with observations $\mathbf { x } _ { t } \in \mathcal { X } \subset \mathbb { R } ^ { n _ { x } }$ , depending on control inputs (or actions) $\mathbf { u } _ { t } \in \mathcal { U } \subset \mathbb { R } ^ { n _ { u } }$ . Elements of $\mathcal { X }$ can be high-dimensional sensory data, e.g., raw images. In particular they may exhibit complex non-Markovian transitions. Corresponding timediscrete sequences of length $\mathrm { T }$ are denoted as $\mathbf { x } _ { 1 : T } = ( \mathbf { x } _ { 1 } , \mathbf { x } _ { 2 } , \ldots , \mathbf { x } _ { T } )$ and $\mathbf { u } _ { 1 : T } = ( \mathbf { u } _ { 1 } , \mathbf { u } _ { 2 } , \ldots , \mathbf { u } _ { T } )$ .
+
+We are interested in a probabilistic model1 $p ( \mathbf { x } _ { 1 : T } \mid \mathbf { u } _ { 1 : T } )$ . Formally, we assume the graphical model
+
+$$
+p ( \mathbf { x } _ { 1 : T } \mid \mathbf { u } _ { 1 : T } ) = \int p ( \mathbf { x } _ { 1 : T } \mid \mathbf { z } _ { 1 : T } , \mathbf { u } _ { 1 : T } ) p ( \mathbf { z } _ { 1 : T } \mid \mathbf { u } _ { 1 : T } ) \mathrm { d } \mathbf { z } _ { 1 : T } ,
+$$
+
+where $\mathbf { z } _ { 1 : T }$ , $\mathbf { z } _ { t } \in \mathcal { Z } \subset \mathbb { R } ^ { n _ { z } }$ , denotes the corresponding latent sequence. That is, we assume a generative model with an underlying latent dynamical system with emission model $p ( \mathbf { x } _ { 1 : T } \mid \mathbf { z } _ { 1 : T } , \mathbf { u } _ { 1 : T } )$ and transition model $p ( \mathbf { z } _ { 1 : T } \mid \mathbf { u } _ { 1 : T } )$ . We want to learn both components, i.e., we want to perform latent system identification. In order to be able to apply the identified system in downstream tasks, we need to find efficient posterior inference distributions $p ( \mathbf { z } _ { 1 : T } \mid \mathbf { x } _ { 1 : T } )$ . Three common examples are prediction, filtering, and smoothing: inference of $\mathbf { z } _ { t }$ from $\mathbf { x } _ { 1 : t - 1 }$ , $\mathbf { x } _ { 1 : t }$ , or $\mathbf { x } _ { 1 : T }$ , respectively. Accurate identification and efficient inference are generally competing tasks, as a wider generative model class typically leads to more difficult or even intractable inference.
+
+The transition model is imperative for achieving good long-term results: a bad transition model can lead to divergence of the latent state. Accordingly, we put special emphasis on it through a Bayesian treatment. Assuming that the transitions may differ for each time step, we impose a regularizing prior distribution on a set of transition parameters $\beta _ { 1 : T }$ :
+
+$$
+( 1 ) = \iint p ( { \bf x } _ { 1 : T } \mid { \bf z } _ { 1 : T } , { \bf u } _ { 1 : T } ) p ( { \bf z } _ { 1 : T } \mid \beta _ { 1 : T } , { \bf u } _ { 1 : T } ) p ( \beta _ { 1 : T } ) \mathrm { d } \beta _ { 1 : T } \mathrm { d } { \bf z } _ { 1 : T }
+$$
+
+To obtain state-space models, we impose assumptions on emission and state transition model,
+
+$$
+\begin{array} { l } { \displaystyle p ( \mathbf { x } _ { 1 : T } \mid \mathbf { z } _ { 1 : T } , \mathbf { u } _ { 1 : T } ) = \prod _ { t = 1 \atop t = 1 } ^ { T } p ( \mathbf { x } _ { t } \mid \mathbf { z } _ { t } ) , } \\ { \displaystyle p ( \mathbf { z } _ { 1 : T } \mid \beta _ { 1 : T } , \mathbf { u } _ { 1 : T } ) = \prod _ { t = 0 } ^ { T - 1 } p ( \mathbf { z } _ { t + 1 } \mid \mathbf { z } _ { t } , \mathbf { u } _ { t } , \beta _ { t } ) . } \end{array}
+$$
+
+Equations (3) and (4) assume that the current state $\mathbf { z } _ { t }$ contains all necessary information about the current observation $\mathbf { x } _ { t }$ , as well as the next state $\mathbf { z } _ { t + 1 }$ (given the current control input $\mathbf { u } _ { t }$ and transition parameters $\beta _ { t }$ ). That is, in contrast to observations, $\mathbf { z } _ { t }$ exhibits Markovian behavior.
+
+A typical example of these assumptions are Linear Gaussian Models (LGMs), i.e., both state transition and emission model are affine transformations with Gaussian offset noise,
+
+$$
+\begin{array} { r l r } { \mathbf { z } _ { t + 1 } = \mathbf { F } _ { t } \mathbf { z } _ { t } + \mathbf { B } _ { t } \mathbf { u } _ { t } + \mathbf { w } _ { t } \quad } & { } & { \mathbf { w } _ { t } \sim \mathcal { N } ( \mathbf { 0 } , \mathbf { Q } _ { t } ) , } \\ { \mathbf { x } _ { t } = \mathbf { H } _ { t } \mathbf { z } _ { t } + \mathbf { y } _ { t } \quad } & { } & { \mathbf { y } _ { t } \sim \mathcal { N } ( \mathbf { 0 } , \mathbf { R } _ { t } ) . } \end{array}
+$$
+
+Typically, state transition matrix $\mathbf { F } _ { t }$ and control-input matrix $\mathbf { B } _ { t }$ are assumed to be given, so that $\boldsymbol { \beta } _ { t } = \mathbf { w } _ { t }$ . Section 3.3 will show that our approach allows other variants such as $\beta _ { t } = \left( \mathbf { F } _ { t } , \mathbf { B } _ { t } , \mathbf { w } _ { t } \right)$ . Under the strong assumptions (5) and (6) of LGMs, inference is provably solved optimally by the well-known Kalman filters. While extensions of Kalman filters to nonlinear dynamical systems exist, Julier & Uhlmann (1997), and are successfully applied in many areas, they suffer from two major drawbacks: firstly, its assumptions are restrictive and are violated in practical applications, leading to suboptimal results. Secondly, parameters such as $\mathbf { F } _ { t }$ and $\mathbf { B } _ { t }$ have to be known in order to perform posterior inference. There have been efforts to learn such system dynamics, cf. Ghahramani $\&$ Hinton (1996); Honkela et al. (2010) based on the expectation maximization (EM) algorithm or Valpola & Karhunen (2002), which uses neural networks. However, these algorithms are not applicable in cases where the true posterior distribution is intractable. This is the case if, e.g., image sequences are used, since the posterior is then highly nonlinear—typical mean-field assumptions on the approximate posterior are too simplified. Our new approach will tackle both issues, and moreover learn both identification and inference jointly by exploiting Stochastic Gradient Variational Bayes.
+
+# 2.2 STOCHASTIC GRADIENT VARIATIONAL BAYES (SGVB) FOR TIME SERIES DISTRIBUTIONS
+
+Replacing the bottleneck layer of a deterministic auto-encoder with stochastic units $\mathbf { z }$ , the variational auto-encoder (VAE, Kingma & Welling (2013); Rezende et al. (2014)) learns complex marginal data distributions on $\mathbf { x }$ in an unsupervised fashion from simpler distributions via the graphical model
+
+$$
+p ( \mathbf { x } ) = \int p ( \mathbf { x } , \mathbf { z } ) \mathrm { d } \mathbf { z } = \int p ( \mathbf { x } \mid \mathbf { z } ) p ( \mathbf { z } ) \mathrm { d } \mathbf { z } .
+$$
+
+In VAEs, $p ( \mathbf { x } \mid \mathbf { z } ) \equiv p _ { \theta } ( \mathbf { x } \mid \mathbf { z } )$ is typically parametrized by a neural network with parameters $\theta$ . Within this framework, models are trained by maximizing a lower bound to the marginal data log-likelihood via stochastic gradients:
+
+$$
+\ln p ( \mathbf { x } ) \geq \mathbb { E } _ { q _ { \phi } ( \mathbf { z } | \mathbf { x } ) } [ \ln p _ { \theta } ( \mathbf { x } \mid \mathbf { z } ) ] - \mathrm { K L } ( q _ { \phi } ( \mathbf { z } \mid \mathbf { x } ) \mid \mid p ( \mathbf { z } ) ) = : \mathcal { L } _ { \mathrm { S G V B } } ( \mathbf { x } , \phi , \theta )
+$$
+
+This is provably equivalent to minimizing the KL-divergence between the approximate posterior or recognition model $q _ { \phi } ( \mathbf { z } \mid \mathbf { x } )$ and the true, but usually intractable posterior distribution $p ( \mathbf { z } \mid \mathbf { x } )$ . $q _ { \phi }$ is parametrized by a neural network with parameters $\phi$ .
+
+The principle of VAEs has been transferred to time series, Bayer & Osendorfer (2014); Chung et al. (2015). Both employ nonlinear state transitions in latent space, but violate eq. (4): Observations are directly included in the transition process. Empirically, reconstruction and compression work well. The state space $\mathcal { Z }$ , however, does not reflect all information available, which prohibits plausible generative long-term prediction. Such phenomena with generative models have been explained in Theis et al. (2015).
+
+In Krishnan et al. (2015), the state-space assumptions (3) and (4) are softly encoded in the Deep Kalman Filter (DKF) model. Despite that, experiments, cf. section 4, show that their model fails to extract information such as velocity (and in general time derivatives), which leads to similar problems with prediction.
+
+Johnson et al. (2016) give an algorithm for general graphical model variational inference, not tailored to dynamical systems. In contrast to previously discussed methods, it does not violate eq. (4). The approaches differ in that the recognition model outputs node potentials in combination with message passing to infer the latent state. Our approach focuses on learning dynamical systems for controlrelated tasks and therefore uses a neural network for inferring the latent state directly instead of an inference subroutine.
+
+Others have been specifically interested in applying variational inference for controlled dynamical systems. In Watter et al. (2015) (Embed to Control—E2C), a VAE is used to learn the mappings to and from latent space. The regularization is clearly motivated by eq. (7). Still, it fails to be a mathematically correct lower bound to the marginal data likelihood. More significantly, their recognition model requires all observations that contain information w.r.t. the current state. This is nothing short of an additional temporal i.i.d. assumption on data: Multiple raw samples need to be stacked into one training sample such that all latent factors (in particular all time derivatives) are present within one sample. The task is thus greatly simplified, because instead of time-series, we learn a static auto-encoder on the processed data.
+
+A pattern emerges: good prediction should boost compression. Still, previous methods empirically excel at compression, while prediction will not work. We conjecture that this is caused by previous methods trying to fit the latent dynamics to a latent state that is beneficial for reconstruction. This encourages learning of a stationary auto-encoder with focus of extracting as much from a single observation as possible. Importantly, it is not necessary to know the entire sequence for excellent reconstruction of single time steps. Once the latent states are set, it is hard to adjust the transition to them. This would require changing the latent states slightly, and that comes at a cost of decreasing the reconstruction (temporarily). The learning algorithm is stuck in a local optimum with good reconstruction and hence good compression only. Intriguingly, E2C bypasses this problem with its data augmentation.
+
+![](images/3575706f643090530f958bb9ba2559b160e8016b28d587187d9d19d1e13fff7e.jpg)  
+Figure 1: Left: Graphical model for one transition under state-space model assumptions. The updated latent state $\mathbf { z } _ { t + 1 }$ depends on the previous state $\mathbf { z } _ { t }$ , control input $\mathbf { u } _ { t }$ , and transition parameters $\beta _ { t }$ . $\mathbf { z } _ { t + 1 }$ contains all information for generating observation $\mathbf { x } _ { t + 1 }$ . Diamond nodes indicate a deterministic dependency on parent nodes. Right: Inference performed during training (or while filtering). Past observations are indirectly used for inference as $\mathbf { z } _ { t }$ contains all information about them.
+
+This leads to a key contribution of this paper: We force the latent space to fit the transition—reversing the direction, and thus achieving the state-space model assumptions and full information in the latent states.
+
+# 3 DEEP VARIATIONAL BAYES FILTERS
+
+# 3.1 REPARAMETRIZING THE TRANSITION
+
+The central problem for learning latent states system dynamics is efficient inference of a latent space that obeys state-space model assumptions. If the latter are fulfilled, the latent space must contain all information. Previous approaches emphasized good reconstruction, so that the space only contains information necessary for reconstruction of one time step. To overcome this, we establish gradient paths through transitions over time so that the transition becomes the driving factor for shaping the latent space, rather than adjusting the transition to the recognition model’s latent space. The key is to prevent the recognition model $\bar { q _ { \phi } } ( \mathbf { z } _ { 1 : T } \mid \mathbf { x } _ { 1 : T } )$ from directly drawing the latent state $\mathbf { z } _ { t }$ .
+
+Similar to the reparametrization trick from Kingma & Welling (2013); Rezende et al. (2014) for making the Monte Carlo estimate differentiable w.r.t. the parameters, we make the transition differentiable w.r.t. the last state and its parameters:
+
+$$
+\mathbf { z } _ { t + 1 } = f ( \mathbf { z } _ { t } , \mathbf { u } _ { t } , \boldsymbol { \beta } _ { t } )
+$$
+
+Given the stochastic parameters $\beta _ { t }$ , the state transition is deterministic (which in turn means that by marginalizing $\beta _ { t }$ , we still have a stochastic transition). The immediate and crucial consequence is that errors in reconstruction of $\mathbf { x } _ { t }$ from $\mathbf { z } _ { t }$ are backpropagated directly through time.
+
+This reparametrization has a couple of other important implications: the recognition model no longer infers latent states $\mathbf { z } _ { t }$ , but transition parameters $\beta _ { t }$ . In particular, the gradient $\partial { \mathbf z } _ { t + 1 } / \partial { \mathbf z } _ { t }$ is well-defined from (8)—gradient information can be backpropagated through the transition.
+
+This is different from the method used in Krishnan et al. (2015), where the transition only occurs in the KL-divergence term of their loss function (a variant of eq. (7)). No gradient from the generative model is backpropagated through the transitions.
+
+Much like in eq. (5), the stochastic parameters includes a corrective offset term $\mathbf { w } _ { t }$ , which emphasizes the notion of the recognition model as a filter. In theory, the learning algorithm could still learn the transition as $\mathbf { z } _ { t + 1 } = \mathbf { w } _ { t }$ . However, the introduction of $\beta _ { t }$ also enables us to regularize the transition with meaningful priors, which not only prevents overfitting the recognition model, but also enforces meaningful manifolds in the latent space via transition priors. Ignoring the potential of the transition over time yields large penalties from these priors. Thus, the problems outlined in Section 2 are overcome by construction.
+
+To install such transition priors, we split $\boldsymbol { \beta } _ { t } = \left( \mathbf { w } _ { t } , \mathbf { v } _ { t } \right)$ . The interpretation of $\mathbf { w } _ { t }$ is a sample-specific process noise which can be inferred from incoming data, like in eq. (5). On the other hand, $\mathbf { v } _ { t }$ are universal transition parameters, which are sample-independent (and are only inferred from data during training). This corresponds to the idea of weight uncertainty in Hinton $\&$ Van Camp (1993). This interpretation leads to a natural factorization assumption on the recognition model:
+
+![](images/2f06dc66adf13f953b2b283f158982e7fca52642d7eaad76f111e49d154273f4.jpg)  
+(b) One particular example of a latent transition: local linearity.
+
+![](images/9d5530af0592aa3f830396220bcbb3bd0b1e56b2fc31b89d38a842b3e0406bf6.jpg)  
+Figure 2: Left: General architecture for DVBF. Stochastic transition parameters $\beta _ { t }$ are inferred via the recognition model, e.g., a neural network. Based on a sampled $\beta _ { t }$ , the state transition is computed deterministically. The updated latent state $\mathbf { z } _ { t + 1 }$ is used for predicting $\mathbf { x } _ { t + 1 }$ . For details, see section 3.1. Right: Zoom into latent space transition (red box in left figure). One exemplary transition is shown, the locally linear transition from section 3.3.
+
+$$
+q _ { \phi } ( \beta _ { 1 : T } \mid \mathbf { x } _ { 1 : T } ) = q _ { \phi } ( \mathbf { w } _ { 1 : T } \mid \mathbf { x } _ { 1 : T } ) q _ { \phi } ( \mathbf { v } _ { 1 : T } )
+$$
+
+When using the fully trained model for generative sampling, i.e., sampling without input, the universal state transition parameters can still be drawn from $q _ { \phi } ( \mathbf { v } _ { 1 : T } )$ , whereas $\mathbf { w } _ { 1 : T }$ is drawn from the prior in the absence of input data.
+
+Figure 1 shows the underlying graphical model and the inference procedure. Figure 2a shows a generic view on our new computational architecture. An example of a locally linear transition parametrization will be given in section 3.3.
+
+# 3.2 THE LOWER BOUND OBJECTIVE FUNCTION
+
+In analogy to eq. (7), we now derive a lower bound to the marginal likelihood $p ( \mathbf { x } _ { 1 : T } \mid \mathbf { u } _ { 1 : T } )$ . After reflecting the Markov assumptions (3) and (4) in the factorized likelihood (2), we have:
+
+$$
+p ( \mathbf { x } _ { 1 : T } \mid \mathbf { u } _ { 1 : T } ) = \int \int p ( \beta _ { 1 : T } ) \prod _ { t = 1 } ^ { T } p _ { \theta } ( \mathbf { x } _ { t } \mid \mathbf { z } _ { t } ) \prod _ { t = 0 } ^ { T - 1 } p ( \mathbf { z } _ { t + 1 } \mid \mathbf { z } _ { t } , \mathbf { u } _ { t } , \beta _ { t } ) \mathrm { d } \beta _ { 1 : T } \mathrm { d } \mathbf { z } _ { 1 : T }
+$$
+
+Due to the deterministic transition given $\beta _ { t + 1 }$ , the last term is a product of Dirac distributions and the overall distribution simplifies greatly:
+
+$$
+\begin{array} { l } { p ( \mathbf { x } _ { 1 : T } \mid \mathbf { u } _ { 1 : T } ) = \displaystyle \int p ( \boldsymbol { \beta } _ { 1 : T } ) \prod _ { t = 1 } ^ { T } p _ { \boldsymbol { \theta } } ( \mathbf { x } _ { t } \mid \mathbf { z } _ { t } ) \Big \vert _ { \mathbf { z } _ { t } = f \left( \mathbf { z } _ { t - 1 } , \mathbf { u } _ { t - 1 } , \boldsymbol { \beta } _ { t - 1 } \right) } \mathrm { d } \boldsymbol { \beta } _ { 1 : T } } \\ { \displaystyle \left( = \int p ( \boldsymbol { \beta } _ { 1 : T } ) p _ { \boldsymbol { \theta } } ( \mathbf { x } _ { 1 : T } \mid \mathbf { z } _ { 1 : T } ) \mathrm { d } \boldsymbol { \beta } _ { 1 : T } \right) } \end{array}
+$$
+
+The last formulation is for notational brevity: the term $p _ { \theta } ( \mathbf { x } _ { 1 : T } \mid \mathbf { z } _ { 1 : T } )$ is not independent of $\beta _ { 1 : T }$ and $\mathbf { u } _ { 1 : T }$ . We now derive the objective function, a lower bound to the data likelihood:
+
+$$
+\begin{array}{c} \ln p ( \mathbf { x } _ { 1 : T } \mid \mathbf { u } _ { 1 : T } ) = \ln \int p ( \beta _ { 1 : T } ) p _ { \theta } ( \mathbf { x } _ { 1 : T } \mid \mathbf { z } _ { 1 : T } ) { \frac { q _ { \phi } ( \beta _ { 1 : T } \mid \mathbf { x } _ { 1 : T } , \mathbf { u } _ { 1 : T } ) } { q _ { \phi } ( \beta _ { 1 : T } \mid \mathbf { x } _ { 1 : T } , \mathbf { u } _ { 1 : T } ) } } \mathrm { d } \beta _ { 1 : T }  \\ { \geq \int q _ { \phi } ( \beta _ { 1 : T } \mid \mathbf { x } _ { 1 : T } , \mathbf { u } _ { 1 : T } ) \ln \left( p _ { \theta } ( \mathbf { x } _ { 1 : T } \mid \mathbf { z } _ { 1 : T } ) { \frac { p ( \beta _ { 1 : T } ) } { q _ { \phi } ( \beta _ { 1 : T } \mid \mathbf { x } _ { 1 : T } , \mathbf { u } _ { 1 : T } ) } } \right) \mathrm { d } \beta _ { 1 : T } } \\ { = \mathbb { E } _ { q _ { \phi } } [ \ln p _ { \theta } ( \mathbf { x } _ { 1 : T } \mid \mathbf { z } _ { 1 : T } ) - \ln q _ { \phi } ( \beta _ { 1 : T } \mid \mathbf { x } _ { 1 : T } , \mathbf { u } _ { 1 : T } ) + \ln p ( \beta _ { 1 : T } ) ] } \\ { = \mathbb { E } _ { q _ { \phi } } [ \ln p _ { \theta } ( \mathbf { x } _ { 1 : T } \mid \mathbf { z } _ { 1 : T } ) ] - \mathrm { K L } ( q _ { \phi } ( \beta _ { 1 : T } \mid \mathbf { x } _ { 1 : T } , \mathbf { u } _ { 1 : T } ) \mid \mid p ( \beta _ { 1 : T } ) ) } & { ( \ln p ( \beta _ { 1 : T } ) ) } \\ { = : { \mathcal { L } } _ { \mathrm { D V B F } } ( \mathbf { x } _ { 1 : T } , \theta , \phi \mid \mathbf { u } _ { 1 : T } ) } \end{array}
+$$
+
+Our experiments show that an annealed version of (10) is beneficial to the overall performance:
+
+$$
+( \mathbf { 1 0 ^ { \prime } } ) = \mathbb { E } _ { q _ { \phi } } [ c _ { i } \ln p _ { \theta } ( \mathbf { x } _ { 1 : T } ~ \vert ~ \mathbf { z } _ { 1 : T } ) - \ln q _ { \phi } ( \beta _ { 1 : T } ~ \vert ~ \mathbf { x } _ { 1 : T } , \mathbf { u } _ { 1 : T } ) + c _ { i } \ln p ( \mathbf { w } _ { 1 : T } ) + \ln p ( \mathbf { v } _ { 1 : T } ) ]
+$$
+
+Here, $c _ { i } = \operatorname* { m a x } ( 1 , 0 . 0 1 + i / T _ { A } )$ is an inverse temperature that increases linearly in the number of gradient updates $i$ until reaching 1 after $T _ { A }$ annealing iterations. Similar annealing schedules have been applied in, e.g., Ghahramani $\&$ Hinton (2000); Mandt et al. (2016); Rezende & Mohamed (2015), where it is shown that they smooth the typically highly non-convex error landscape. Additionally, the transition prior $p ( \mathbf { v } _ { 1 : T } )$ was estimated during optimization, i.e., through an empirical Bayes approach. In all experiments, we used isotropic Gaussian priors.
+
+# 3.3 EXAMPLE: LOCALLY LINEAR TRANSITIONS
+
+We have derived a learning algorithm for time series with particular focus on general transitions in latent space. Inspired by Watter et al. (2015), this section will show how to learn a particular instance: locally linear state transitions. That is, we set eq. (8) to
+
+$$
+\begin{array} { r } { \mathbf { z } _ { t + 1 } = \mathbf { A } _ { t } \mathbf { z } _ { t } + \mathbf { B } _ { t } \mathbf { u } _ { t } + \mathbf { C } _ { t } \mathbf { w } _ { t } , \qquad t = 1 , \ldots , T , } \end{array}
+$$
+
+where $\mathbf { w } _ { t }$ is a stochastic sample from the recognition model and $\mathbf { A } _ { t } , \mathbf { B } _ { t }$ , and $\mathbf { C } _ { t }$ are matrices of matching dimensions. They are stochastic functions of $\mathbf { z } _ { t }$ and $\mathbf { u } _ { t }$ (thus local linearity). We draw
+
+$$
+\mathbf { v } _ { t } = \Big \{ \mathbf { A } _ { t } ^ { ( i ) } , \mathbf { B } _ { t } ^ { ( i ) } , \mathbf { C } _ { t } ^ { ( i ) } \mid i = 1 , \ldots , M \Big \} ,
+$$
+
+from $q _ { \phi } ( \mathbf { v } _ { t } )$ , i.e., $M$ triplets of matrices, each corresponding to data-independent, but learned globally linear system. These can be learned as point estimates. We employed a Bayesian treatment as in Blundell et al. (2015). We yield $\mathbf { A } _ { t } , \mathbf { B } _ { t }$ , and $\mathbf { C } _ { t }$ as state- and control-dependent linear combinations:
+
+$$
+\begin{array}{c} \begin{array} { r l r l r l } & { \quad } & & { \quad } & & { \boldsymbol { \alpha } _ { t } = f _ { \psi } ( \mathbf { z } _ { t } , \mathbf { u } _ { t } ) \in \mathbb { R } ^ { M } } \\ & { \quad } & & { \quad } & { \quad } & { \quad } \\ & { \quad } & { \quad } & { \quad } & { \quad } & { \quad } \\ & { \quad } & { \quad } & { \quad } & { \quad } & { \quad } & { \quad } \\ & { \quad } & { \quad } & { \quad } & { \quad } & { \quad } & { \quad } \end{array} \qquad \begin{array} { r l } & { \quad } & { \quad } & { \boldsymbol { \alpha } _ { t } = f _ { \psi } ( \mathbf { z } _ { t } , \mathbf { u } _ { t } ) \in \mathbb { R } ^ { M } } \\ & { \quad } & { \quad } \\ & { \quad } & { \quad } \\ & { \quad } & { \quad } & { \quad } \end{array} \qquad \mathbf { C } _ { t } = \sum _ { i = 1 } ^ { M } { \boldsymbol { \alpha } _ { t } ^ { ( i ) } } \mathbf { C } _ { t } ^ { ( i ) }  \end{array}
+$$
+
+The computation is depicted in fig. 2b. The function $f _ { \psi }$ can be, e.g., a (deterministic) neural network with weights $\psi$ . As a subset of the generative parameters $\theta$ , $\psi$ is part of the trainable parameters of our model. The weight vector $\pmb { \alpha } _ { t }$ is shared between the three matrices. There is a correspondence to eq. (5): ${ \bf A } _ { t }$ and $\mathbf { F } _ { t }$ , $\mathbf { B } _ { t }$ and $\mathbf { B } _ { t }$ , as well as $\mathbf { C } _ { t } \mathbf { C } _ { t } ^ { \top }$ and $\mathbf { Q } _ { t }$ are related.
+
+We used this parametrization of the state transition model for our experiments. It is important that the parametrization is up to the user and the respective application.
+
+# 4 EXPERIMENTS AND RESULTS
+
+In this section we validate that DVBF with locally linear transitions (DVBF-LL) (section 3.3) outperforms Deep Kalman Filters (DKF, Krishnan et al. (2015)) in recovering latent spaces with full information. 2 We focus on environments that can be simulated with full knowledge of the
+
+2We do not include E2C, Watter et al. (2015), due to the need for data modification and its inability to provide a correct lower bound as mentioned in section 2.2.
+
+![](images/89343f9239b5f95d0b3f05f420ceb14168472d729834fda8850d44f8baa77701.jpg)  
+Figure 3: (a) Our DVBF-LL model trained on pendulum image sequences. The upper plots show the latent space with coloring according to the ground truth with angles on the left and angular velocities on the right. The lower plots show regression results for predicting ground truth from the latent representation. The latent space plots show clearly that all information for representing the full state of a pendulum is encoded in each latent state. (b) DKF from Krishnan et al. (2015) trained on the same pendulum dataset. The latent space plot shows that DKF fails to learn velocities of the pendulum. It is therefore not able to capture all information for representing the full pendulum state.
+
+ground truth latent dynamical system. The experimental setup is described in the Supplementary Material. We published the code for DVBF and a link will be made available at https://brml. org/projects/dvbf.
+
+# 4.1 DYNAMIC PENDULUM
+
+In order to test our algorithm on truly non-Markovian observations of a dynamical system, we simulated a dynamic torque-controlled pendulum governed by the differential equation
+
+$$
+m l ^ { 2 } { \ddot { \varphi } } ( t ) = - \mu { \dot { \varphi } } ( t ) + m g l \sin \varphi ( t ) + u ( t ) ,
+$$
+
+$m = l = 1 , \mu = 0 . 5 , g = 9 . 8 1$ , via numerical integration, and then converted the ground-truth angle $\varphi$ into an image observation in $\mathcal { X }$ . The one-dimensional control corresponds to angle acceleration (which is proportional to joint torque). Angle and angular velocity fully describe the system.
+
+Figure 3 shows the latent spaces for identical input data learned by DVBF-LL and DKF, respectively, colored with the ground truth in the top row. It should be noted that latent samples are shown, not means of posterior distributions. The state-space model was allowed to use three latent dimensions. As we can see in fig. 3a, DVBF-LL learned a two-dimensional manifold embedding, i.e., it encoded the angle in polar coordinates (thus circumventing the discontinuity of angles modulo $2 \pi$ ). The bottom row shows ordinary least-squares regressions (OLS) underlining the performance: there exists a high correlation between latent states and ground-truth angle and angular velocity for DVBF-LL. On the contrary, fig. 3b verifies our prediction that DKF is equally capable of learning the angle, but extracts little to no information on angular velocity.
+
+The OLS regression results shown in table 1 validate this observation.3 Predicting $\sin ( \varphi )$ and $\cos ( \varphi )$ , i.e., polar coordinates of the ground-truth angle $\varphi$ , works almost equally well for DVBF-LL and DKF, with DVBF-LL slightly outperforming DKF. For predicting the ground truth velocity $\dot { \varphi }$ , DVBF-LL shows remarkable performance. DKF, instead, contains hardly any information, resulting in a very low goodness-of-fit score of $R ^ { 2 } = 0 . 0 3 5$ .
+
+Table 1: Results for pendulum OLS regressions of all latent states on respective dependent variable.   
+
+<table><tr><td colspan="3">DVBF-LL</td><td colspan="2">DKF</td></tr><tr><td></td><td></td><td>Log-Likelihood</td><td>R² Log-Likelihood</td><td>R²</td></tr><tr><td>Dependent</td><td>sin()</td><td>3990.8</td><td>0.961 0.982</td><td>1737.6 0.929</td></tr><tr><td>ground truth</td><td>cos()</td><td>7231.1</td><td>6614.2</td><td>0.979</td></tr><tr><td>variable</td><td>6</td><td>-11139 0.916</td><td>-20289</td><td>0.035</td></tr></table>
+
+![](images/886df3e2f7dd268db086c2c03abc30a59759b198fe32da3389af3e840dbf5cc5.jpg)  
+(b) Reconstructive latent walk.
+
+![](images/d58aeaeeefb189a121cf98cce5b00a9f7e03e587fd265f3883b8988c3b5125e1.jpg)  
+(a) Generative latent walk.   
+(c) Ground truth (top), reconstructions (middle), generative samples (bottom) from identical initial latent state.   
+Figure 4: (a) Latent space walk in generative mode. (b) Latent space walk in filtering mode. (c) Ground truth and samples from recognition and generative model. The reconstruction sampling has access to observation sequence and performs filtering. The generative samples only get access to the observations once for creating the initial state while all subsequent samples are predicted from this single initial state. The red bar indicates the length of training sequences. Samples beyond show the generalization capabilities for sequences longer than during training. The complete sequence can be found in the Appendix in fig. 7.
+
+<table><tr><td rowspan=1 colspan=31>1            5               10               15              20                      40              45</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>`</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>.</td><td rowspan=1 colspan=1>,</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>,</td><td rowspan=1 colspan=1>,</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1>.</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>.</td><td rowspan=1 colspan=1>.</td><td rowspan=1 colspan=1>.</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>、</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>·</td><td rowspan=1 colspan=1>·</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>·</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>、</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>,</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1>、</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1>.</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>.</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>.</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>,</td><td rowspan=1 colspan=1>,</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>,</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr></table>
+
+Figure 4 shows that the strong relation between ground truth and latent state is beneficial for generative sampling. All plots show 100 time steps of a pendulum starting from the exact same latent state and not being actuated. The top row plots show a purely generative walk in the latent space on the left, and a walk in latent space that is corrected by filtering observations on the right. We can see that both follow a similar trajectory to an attractor. The generative model is more prone to noise when approaching the attractor.
+
+The bottom plot shows the first 45 steps of the corresponding observations (top row), reconstructions (middle row), and generative samples (without correcting from observations). Interestingly, DVBF works very well even though the sequence is much longer than all training sequences (indicated by the red line).
+
+Table (2) shows values of the lower bound to the marginal data likelihood (for DVBF-LL, this corresponds to eq. (11)). We see that DVBF-LL outperforms DKF in terms of compression, but only with a slight margin, which does not reflect the better generative sampling as Theis et al. (2015) argue.
+
+![](images/c9482434209e7dde8771c88440c90feb407a9a4f7b03d8c70dac223a41f3459b.jpg)  
+Figure 5: (a) Two dimensions of 4D bouncing ball latent space. Ground truth x and y coordinates are combined into a regular $3 \times 3$ checkerboard coloring. This checkerboard is correctly extracted by the embedding. (b) Remaining two latent dimensions. Same latent samples, colored with ball velocities in x and y direction (left and right image, respectively). The smooth, perpendicular coloring indicates that the ground truth value is stored in the latent dimension.
+
+# 4.2 BOUNCING BALL
+
+The bouncing ball experiment features a ball rolling within a bounding box in a plane. The system has a two-dimensional control input, added to the directed velocity of the ball. If the ball hits the wall, it bounces off, so that the true dynamics are highly dependent on the current position and velocity of the ball. The system’s state is four-dimensional, two dimensions each for position and velocity.
+
+Consequently, we use a DVBF-LL with four latent dimensions. Figure 5 shows that DVBF again captures the entire system dynamics in the latent space. The checkerboard is quite a remarkable result: the ground truth position of the ball lies within the 2D unit square, the bounding box. In order to visualize how ground truth reappears in the learned latent states, we show the warping of the ground truth bounding box into the latent space. To this end, we partitioned (discretized) the ground truth unit square into a regular 3x3 checkerboard with respective coloring. We observed that DVBF learned to extract the 2D position from the 256 pixels, and aligned them in two dimensions of the latent space in strong correspondence to the physical system. The algorithm does the exact same pixel-to-2D inference that a human observer automatically does when looking at the image.
+
+![](images/0fee90b828d2c622647068fb4a7868c7ca217d5d2ed1f6ad3072d29a40f8c53b.jpg)  
+Figure 6: Ground truth (top), reconstructions (middle), generative samples (bottom) from identical initial latent state for the two bouncing balls experiment. Red bar indicates length of training sequences.
+
+# 4.3 TWO BOUNCING BALLS
+
+Another more complex environment4 features two balls in a bounding box. We used a 10-dimensional latent space to fully capture the position and velocity information of the balls. Reconstruction and generative samples are shown in fig. 6. Same as in the pendulum example we get a generative model with stable predictions beyond training data sequence length.
+
+# 5 CONCLUSION
+
+We have proposed Deep Variational Bayes Filters (DVBF), a new method to learn state space models from raw non-Markovian sequence data. DVBFs perform latent dynamic system identification, and subsequently overcome intractable inference. As DVBFs make use of stochastic gradient variational Bayes they naturally scale to large data sets. In a series of vision-based experiments we demonstrated that latent states can be recovered which identify the underlying physical quantities. The generative model showed stable long-term predictions far beyond the sequence length used during training.
+
+# ACKNOWLEDGEMENTS
+
+Part of this work was conducted at Chair of Robotics and Embedded Systems, Department of Informatics, Technische Universität München, Germany, and supported by the TACMAN project, EC Grant agreement no. 610967, within the FP7 framework programme.
+
+We would like to thank Jost Tobias Springenberg, Adam Kosiorek, Moritz Münst, and anonymous reviewers for valuable input.
+
+# REFERENCES
+
+Justin Bayer and Christian Osendorfer. Learning stochastic recurrent networks. arXiv preprint arXiv:1411.7610, 2014.
+
+Charles Blundell, Julien Cornebise, Koray Kavukcuoglu, and Daan Wierstra. Weight uncertainty in neural networks. arXiv preprint arXiv:1505.05424, 2015.
+
+Léon Bottou. Large-scale machine learning with stochastic gradient descent. In Proceedings of COMPSTAT’2010, pp. 177–186. Springer, 2010.
+
+Junyoung Chung, Kyle Kastner, Laurent Dinh, Kratarth Goel, Aaron C. Courville, and Yoshua Bengio. A recurrent latent variable model for sequential data. CoRR, abs/1506.02216, 2015. URL http://arxiv.org/abs/1506.02216.
+
+Marc Deisenroth and Carl E Rasmussen. Pilco: A model-based and data-efficient approach to policy search. In Proceedings of the 28th International Conference on machine learning (ICML-11), pp. 465–472, 2011.
+
+Zoubin Ghahramani and Geoffrey E Hinton. Parameter estimation for linear dynamical systems. Technical report, Technical Report CRG-TR-96-2, University of Toronto, Dept. of Computer Science, 1996.
+
+Zoubin Ghahramani and Geoffrey E Hinton. Variational learning for switching state-space models. Neural computation, 12(4):831–864, 2000.
+
+Alex Graves. Generating sequences with recurrent neural networks. arXiv preprint arXiv:1308.0850, 2013.
+
+Geoffrey E Hinton and Drew Van Camp. Keeping the neural networks simple by minimizing the description length of the weights. In Proceedings of the sixth annual conference on Computational learning theory, pp. 5–13. ACM, 1993.
+
+4We used the script attached to Sutskever & Hinton (2007) for generating our datasets.
+
+Antti Honkela, Tapani Raiko, Mikael Kuusela, Matti Tornio, and Juha Karhunen. Approximate riemannian conjugate gradient learning for fixed-form variational bayes. Journal of Machine Learning Research, 11(Nov):3235–3268, 2010.
+
+Matthew J Johnson, David Duvenaud, Alexander B Wiltschko, Sandeep R Datta, and Ryan P Adams. Structured VAEs: Composing probabilistic graphical models and variational autoencoders. arXiv preprint arXiv:1603.06277, 2016.
+
+Simon J Julier and Jeffrey K Uhlmann. New extension of the kalman filter to nonlinear systems. In AeroSense’97, pp. 182–193. International Society for Optics and Photonics, 1997.
+
+Rudolph E Kalman and Richard S Bucy. New results in linear filtering and prediction theory. Journal of basic engineering, 83(1):95–108, 1961.
+
+Diederik P Kingma and Max Welling. Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114, 2013.
+
+Jonathan Ko and Dieter Fox. Learning gp-bayesfilters via gaussian process latent variable models. Autonomous Robots, 30(1):3–23, 2011.
+
+Rahul G Krishnan, Uri Shalit, and David Sontag. Deep Kalman filters. arXiv preprint arXiv:1511.05121, 2015.
+
+Stephan Mandt, James McInerney, Farhan Abrol, Rajesh Ranganath, and David Blei. Variational tempering. In Proceedings of the 19th International Conference on Artificial Intelligence and Statistics, pp. 704–712, 2016.
+
+Kevin McGoff, Sayan Mukherjee, Natesh Pillai, et al. Statistical inference for dynamical systems: A review. Statistics Surveys, 9:209–252, 2015.
+
+Danilo J. Rezende, Shakir Mohamed, and Daan Wierstra. Stochastic backpropagation and approximate inference in deep generative models. In Tony Jebara and Eric P. Xing (eds.), Proceedings of the 31st International Conference on Machine Learning (ICML-14), pp. 1278–1286. JMLR Workshop and Conference Proceedings, 2014. URL http://jmlr.org/proceedings/ papers/v32/rezende14.pdf.
+
+Danilo Jimenez Rezende and Shakir Mohamed. Variational inference with normalizing flows. arXiv preprint arXiv:1505.05770, 2015.
+
+Ilya Sutskever and Geoffrey E. Hinton. Learning multilevel distributed representations for high-dimensional sequences. In Marina Meila and Xiaotong Shen (eds.), Proceedings of the Eleventh International Conference on Artificial Intelligence and Statistics (AISTATS-07), volume 2, pp. 548–555. Journal of Machine Learning Research - Proceedings Track, 2007. URL http://jmlr.csail.mit.edu/proceedings/papers/v2/sutskever07a/ sutskever07a.pdf.
+
+Leonid Kuvayev Rich Sutton. Model-based reinforcement learning with an approximate, learned model. In Proceedings of the ninth Yale workshop on adaptive and learning systems, pp. 101–105, 1996.
+
+Lucas Theis, Aäron van den Oord, and Matthias Bethge. A note on the evaluation of generative models. arXiv preprint arXiv:1511.01844, 2015.
+
+Harri Valpola and Juha Karhunen. An unsupervised ensemble learning method for nonlinear dynamic state-space models. Neural computation, 14(11):2647–2692, 2002.
+
+Manuel Watter, Jost Springenberg, Joschka Boedecker, and Martin Riedmiller. Embed to control: A locally linear latent dynamics model for control from raw images. In Advances in Neural Information Processing Systems, pp. 2728–2736, 2015.
+
+# A SUPPLEMENTARY TO LOWER BOUND
+
+# A.1 ANNEALED KL-DIVERGENCE
+
+We used the analytical solution of the annealed KL-divergence in eq. (10) for optimization:
+
+$$
+\begin{array} { r l r } & { } & { \mathbb { E } _ { q _ { \phi } } [ - \ln q _ { \phi } ( { \bf w } _ { 1 : T } \mid { \bf x } _ { 1 : T } , { \bf u } _ { 1 : T } ) + c _ { i } \ln p ( { \bf w } _ { 1 : T } ) ] = } \\ & { } & { c _ { i } \frac { 1 } { 2 } \ln ( 2 \pi \sigma _ { p } ^ { 2 } ) - \frac { 1 } { 2 } \ln ( 2 \pi \sigma _ { q } ^ { 2 } ) + c _ { i } \frac { \sigma _ { q } ^ { 2 } + ( \mu _ { q } - \mu _ { p } ) ^ { 2 } } { 2 \sigma _ { p } ^ { 2 } } - \frac { 1 } { 2 } } \end{array}
+$$
+
+# B SUPPLEMENTARY TO IMPLEMENTATION
+
+# B.1 EXPERIMENTAL SETUP
+
+In all our experiments, we use sequences of 15 raw images of the respective system with $1 6 \times 1 6$ pixels each, i.e., observation space $\mathcal { X } \subset \mathbb { R } ^ { 2 5 6 }$ , as well as control inputs of varying dimension and interpretation depending on the experiment. We used training, validation and test sets with 500 sequences each. Control input sequences were drawn randomly (“motor babbling”). Additional details about the implementation can be found in the published code at https://brml.org/ projects/dvbf.
+
+# B.2 ADDITIONAL EXPERIMENT PLOTS
+
+![](images/e2917c69eb71d5423688cb5a5187f15f229e6017c11ecfbeb490972b4bf65861.jpg)
+
+Figure 7: Ground truth and samples from recognition and generative model. Complete version of fig. 4 with all missing samples present.
+
+B.3 IMPLEMENTATION DETAILS FOR DVBF IN PENDULUM EXPERIMENT
+
+• Input: 15 timesteps of $1 6 ^ { 2 }$ observation dimensions and 1 action dimension   
+• Latent Space: 3 dimensions   
+• Observation Network $p ( \mathbf { x } _ { t } | \mathbf { z } _ { t } ) = \mathcal { N } ( \mathbf { x } _ { t } ; \mu ( \mathbf { z } _ { t } ) , \boldsymbol { \sigma } )$ : $1 2 8 \mathrm { R e L U + 1 6 ^ { 2 } }$ identity output   
+• Recognition Model: $1 2 8 { \mathrm { R e L U } } + 6$ identity output
+
+$$
+\begin{array} { r } { q ( \mathbf { w } _ { t } | \mathbf { z } _ { t } , \mathbf { x } _ { t + 1 } , \mathbf { u } _ { t } ) = \mathcal { N } ( \mathbf { w } _ { t } ; \boldsymbol { \mu } , \boldsymbol { \sigma } ) , } \\ { ( \boldsymbol { \mu } , \boldsymbol { \sigma } ) = f ( \mathbf { z } _ { t } , \mathbf { x } _ { t + 1 } , \mathbf { u } _ { t } ) } \end{array}
+$$
+
+• Transition Network ${ \pmb { \alpha } } _ { t } ( { \bf z } _ { t } )$ : 16 softmax output   
+• Initial Network $\mathbf { w } _ { 1 } \sim p ( \mathbf { x } _ { 1 : T } )$ : Fast Dropout BiRNN with: $1 2 8 { \mathrm { R e L U } } + 3$ identity output   
+• Initial Transition ${ \bf z } _ { 1 } ( { \bf w } _ { 1 } )$ : $1 2 8 { \mathrm { R e L U } } + 3$ identity output   
+• Optimizer: adadelta, 0.1 step rate   
+• Inverse temperature: $c _ { 0 } = 0 . 0 1$ , updated every 250th gradient update, $T _ { A } = 1 0 ^ { 5 }$ iterations   
+• Batch-size: 500
+
+B.4 IMPLEMENTATION DETAILS FOR DVBF IN BOUNCING BALL EXPERIMENT
+
+• Input: 15 timesteps of $1 6 ^ { 2 }$ observation dimensions and 2 action dimension   
+• Latent Space: 4 dimensions   
+• Observation Network $p ( \mathbf { x } _ { t } | \mathbf { z } _ { t } ) = \mathcal { N } ( \mathbf { x } _ { t } ; \mu ( \mathbf { z } _ { t } ) , \boldsymbol { \sigma } )$ : $1 2 8 \mathrm { R e L U + 1 6 ^ { 2 } }$ identity output   
+• Recognition Model: $1 2 8 { \mathrm { R e L U } } + 8$ identity output
+
+$$
+\begin{array} { r } { q ( \mathbf { w } _ { t } | \mathbf { z } _ { t } , \mathbf { x } _ { t + 1 } , \mathbf { u } _ { t } ) = \mathcal { N } ( \mathbf { w } _ { t } ; \boldsymbol { \mu } , \boldsymbol { \sigma } ) , } \\ { ( \boldsymbol { \mu } , \boldsymbol { \sigma } ) = f ( \mathbf { z } _ { t } , \mathbf { x } _ { t + 1 } , \mathbf { u } _ { t } ) } \end{array}
+$$
+
+• Transition Network ${ \pmb { \alpha } } _ { t } ( { \bf z } _ { t } )$ : 16 softmax output   
+• Initial Network $\mathbf { w } _ { 1 } \sim p ( \mathbf { x } _ { 1 : T } )$ : Fast Dropout BiRNN with: $1 2 8 { \mathrm { R e L U } } + 4$ identity output   
+• Initial Transition ${ \bf z } _ { 1 } ( { \bf w } _ { 1 } )$ : $1 2 8 { \mathrm { R e L U } } + 4$ identity output   
+Optimizer: adadelta, 0.1 step rate   
+• Inverse temperature: $c _ { 0 } = 0 . 0 1$ , updated every 250th gradient update, $T _ { A } = 1 0 ^ { 5 }$ iterations   
+• Batch-size: 500
+
+B.5 IMPLEMENTATION DETAILS FOR DVBF IN TWO BOUNCING BALLS EXPERIMENT
+
+• Input: 15 timesteps of $2 0 ^ { 2 }$ observation dimensions and 2000 samples   
+• Latent Space: 10 dimensions   
+• Observation Network $p ( \mathbf { x } _ { t } | \mathbf { z } _ { t } ) = \mathcal { N } ( \mathbf { x } _ { t } ; \mu ( \mathbf { z } _ { t } ) , \boldsymbol { \sigma } )$ : $1 2 8 \mathrm { R e L U + 2 0 ^ { 2 } }$ sigmoid output   
+• Recognition Model: $1 2 8 \mathrm { R e L U } + 2 0 $ identity output
+
+$$
+\begin{array} { r } { q ( \mathbf { w } _ { t } | \mathbf { z } _ { t } , \mathbf { x } _ { t + 1 } , \mathbf { u } _ { t } ) = \mathcal { N } ( \mathbf { w } _ { t } ; \boldsymbol { \mu } , \boldsymbol { \sigma } ) , } \\ { ( \boldsymbol { \mu } , \boldsymbol { \sigma } ) = f ( \mathbf { z } _ { t } , \mathbf { x } _ { t + 1 } , \mathbf { u } _ { t } ) } \end{array}
+$$
+
+• Transition Network ${ \pmb { \alpha } } _ { t } ( { \bf z } _ { t } )$ : 64 softmax output   
+• Initial Network $\mathbf { w } _ { 1 } \sim p ( \mathbf { x } _ { 1 : T } )$ : MLP with: $1 2 8 \mathrm { R e L U } + 1 0$ identity output   
+• Initial Transition ${ \bf z } _ { 1 } ( { \bf w } _ { 1 } )$ : $1 2 8 \mathrm { R e L U } + 1 0$ identity output   
+• Optimizer: adam, 0.001 step rate   
+• Inverse temperature: $c _ { 0 } = 0 . 0 1$ , updated every gradient update, $T _ { A } = 2 ~ 1 0 ^ { 5 }$ iterations   
+• Batch-size: 80
+
+B.6 IMPLEMENTATION DETAILS FOR DKF IN PENDULUM EXPERIMENT
+
+• Input: 15 timesteps of $1 6 ^ { 2 }$ observation dimensions and 1 action dimension   
+• Latent Space: 3 dimensions   
+• Observation Network $p ( \mathbf { x } _ { t } | \mathbf { z } _ { t } ) = { \mathcal { N } } ( \mathbf { x } _ { t } ; \mu ( \mathbf { z } _ { t } ) , \sigma ( \mathbf { z } _ { t } ) ) \colon 1 2 8 { \mathrm { ~ S i g m o i d } } + 1 2 8 { \mathrm { ~ S i g m o i d } } + 2 1 6 ^ { 2 }$ identity output   
+• Recognition Model: Fast Dropout BiRNN 128 Sigmoid $+ ~ 1 2 8$ Sigmoid $^ { + 3 }$ identity output   
+• Transition Network $p ( \mathbf { z } _ { t } | \mathbf { z } _ { t - 1 } , \mathbf { u } _ { t - 1 } )$ : 128 Sigmoid $+ ~ 1 2 8$ Sigmoid $+ ~ 6$ output   
+• Optimizer: adam, 0.001 step rate   
+• Inverse temperature: $c _ { 0 } = 0 . 0 1$ , updated every $2 5 \mathrm { t h }$ gradient update, $T _ { A } = 2 0 0 0$ iterations   
+• Batch-size: 500

parse/train/HyTqHL5xg/HyTqHL5xg_content_list.json

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+    {
+        "type": "text",
+        "text": "DEEP VARIATIONAL BAYES FILTERS: UNSUPERVISED LEARNING OF STATE SPACE MODELS FROM RAW DATA ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            98,
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+    {
+        "type": "text",
+        "text": "Maximilian Karl, Maximilian Soelch, Justin Bayer, Patrick van der Smagt Data Lab, Volkswagen Group, 80805, München, Germany zip([maximilian.karl, maximilian.soelch], [@volkswagen.de]) ",
+        "bbox": [
+            183,
+            194,
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+            237
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+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+            454,
+            273,
+            544,
+            289
+        ],
+        "page_idx": 0
+    },
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+        "type": "text",
+        "text": "We introduce Deep Variational Bayes Filters (DVBF), a new method for unsupervised learning and identification of latent Markovian state space models. Leveraging recent advances in Stochastic Gradient Variational Bayes, DVBF can overcome intractable inference distributions via variational inference. Thus, it can handle highly nonlinear input data with temporal and spatial dependencies such as image sequences without domain knowledge. Our experiments show that enabling backpropagation through transitions enforces state space assumptions and significantly improves information content of the latent embedding. This also enables realistic long-term prediction. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "page_idx": 0
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+        "text": "Estimating probabilistic models for sequential data is central to many domains, such as audio, natural language or physical plants, Graves (2013); Watter et al. (2015); Chung et al. (2015); Deisenroth & Rasmussen (2011); Ko & Fox (2011). The goal is to obtain a model $p ( \\mathbf { x } _ { 1 : T } )$ that best reflects a data set of observed sequences $\\mathbf { x } _ { \\mathrm { 1 : } T }$ . Recent advances in deep learning have paved the way to powerful models capable of representing high-dimensional sequences with temporal dependencies, e.g., Graves (2013); Watter et al. (2015); Chung et al. (2015); Bayer & Osendorfer (2014). ",
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+        "text": "Time series for dynamic systems have been studied extensively in systems theory, cf. McGoff et al. (2015) and sources therein. In particular, state space models have shown to be a powerful tool to analyze and control the dynamics. Two tasks remain a significant challenge to this day: Can we identify the governing system from data only? And can we perform inference from observables to the latent system variables? These two tasks are competing: A more powerful representation of system requires more computationally demanding inference, and efficient inference, such as the well-known Kalman filters, Kalman & Bucy (1961), can prohibit sufficiently complex system classes. ",
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+        "type": "text",
+        "text": "Leveraging a recently proposed estimator based on variational inference, stochastic gradient variational Bayes (SGVB, Kingma & Welling (2013); Rezende et al. (2014)), approximate inference of latent variables becomes tractable. Extensions to time series have been shown in Bayer & Osendorfer (2014); Chung et al. (2015). Empirically, they showed considerable improvements in marginal data likelihood, i.e., compression, but lack full-information latent states, which prohibits, e.g., long-term sampling. Yet, in a wide range of applications, full-information latent states should be valued over compression. This is crucial if the latent spaces are used in downstream applications. ",
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+        "text": "Our contribution is, to our knowledge, the first model that (i) enforces the latent state-space model assumptions, allowing for reliable system identification, and plausible long-term prediction of the observable system, (ii) provides the corresponding inference mechanism with rich dependencies, (iii) inherits the merit of neural architectures to be trainable on raw data such as images or other sensory inputs, and (iv) scales to large data due to optimization of parameters based on stochastic gradient descent, Bottou (2010). Hence, our model has the potential to exploit systems theory methodology for downstream tasks, e.g., control or model-based reinforcement learning, Sutton (1996). ",
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+        "type": "text",
+        "text": "2 BACKGROUND AND RELATED WORK ",
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+        "text": "2.1 PROBABILISTIC MODELING AND FILTERING OF DYNAMICAL SYSTEMS ",
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+        "text": "We consider non-linear dynamical systems with observations $\\mathbf { x } _ { t } \\in \\mathcal { X } \\subset \\mathbb { R } ^ { n _ { x } }$ , depending on control inputs (or actions) $\\mathbf { u } _ { t } \\in \\mathcal { U } \\subset \\mathbb { R } ^ { n _ { u } }$ . Elements of $\\mathcal { X }$ can be high-dimensional sensory data, e.g., raw images. In particular they may exhibit complex non-Markovian transitions. Corresponding timediscrete sequences of length $\\mathrm { T }$ are denoted as $\\mathbf { x } _ { 1 : T } = ( \\mathbf { x } _ { 1 } , \\mathbf { x } _ { 2 } , \\ldots , \\mathbf { x } _ { T } )$ and $\\mathbf { u } _ { 1 : T } = ( \\mathbf { u } _ { 1 } , \\mathbf { u } _ { 2 } , \\ldots , \\mathbf { u } _ { T } )$ . ",
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+        "text": "We are interested in a probabilistic model1 $p ( \\mathbf { x } _ { 1 : T } \\mid \\mathbf { u } _ { 1 : T } )$ . Formally, we assume the graphical model ",
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+        "img_path": "images/d4968207dfaac3d3caba4c27897752320952d44474c31f7f81e4a25095440e31.jpg",
+        "text": "$$\np ( \\mathbf { x } _ { 1 : T } \\mid \\mathbf { u } _ { 1 : T } ) = \\int p ( \\mathbf { x } _ { 1 : T } \\mid \\mathbf { z } _ { 1 : T } , \\mathbf { u } _ { 1 : T } ) p ( \\mathbf { z } _ { 1 : T } \\mid \\mathbf { u } _ { 1 : T } ) \\mathrm { d } \\mathbf { z } _ { 1 : T } ,\n$$",
+        "text_format": "latex",
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+        "text": "where $\\mathbf { z } _ { 1 : T }$ , $\\mathbf { z } _ { t } \\in \\mathcal { Z } \\subset \\mathbb { R } ^ { n _ { z } }$ , denotes the corresponding latent sequence. That is, we assume a generative model with an underlying latent dynamical system with emission model $p ( \\mathbf { x } _ { 1 : T } \\mid \\mathbf { z } _ { 1 : T } , \\mathbf { u } _ { 1 : T } )$ and transition model $p ( \\mathbf { z } _ { 1 : T } \\mid \\mathbf { u } _ { 1 : T } )$ . We want to learn both components, i.e., we want to perform latent system identification. In order to be able to apply the identified system in downstream tasks, we need to find efficient posterior inference distributions $p ( \\mathbf { z } _ { 1 : T } \\mid \\mathbf { x } _ { 1 : T } )$ . Three common examples are prediction, filtering, and smoothing: inference of $\\mathbf { z } _ { t }$ from $\\mathbf { x } _ { 1 : t - 1 }$ , $\\mathbf { x } _ { 1 : t }$ , or $\\mathbf { x } _ { 1 : T }$ , respectively. Accurate identification and efficient inference are generally competing tasks, as a wider generative model class typically leads to more difficult or even intractable inference. ",
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+        "text": "The transition model is imperative for achieving good long-term results: a bad transition model can lead to divergence of the latent state. Accordingly, we put special emphasis on it through a Bayesian treatment. Assuming that the transitions may differ for each time step, we impose a regularizing prior distribution on a set of transition parameters $\\beta _ { 1 : T }$ : ",
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+        "img_path": "images/f97ad239be9a355bc266668b9da6c72491506ce1f273f55467d8db9cab826c51.jpg",
+        "text": "$$\n( 1 ) = \\iint p ( { \\bf x } _ { 1 : T } \\mid { \\bf z } _ { 1 : T } , { \\bf u } _ { 1 : T } ) p ( { \\bf z } _ { 1 : T } \\mid \\beta _ { 1 : T } , { \\bf u } _ { 1 : T } ) p ( \\beta _ { 1 : T } ) \\mathrm { d } \\beta _ { 1 : T } \\mathrm { d } { \\bf z } _ { 1 : T }\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "To obtain state-space models, we impose assumptions on emission and state transition model, ",
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+        "img_path": "images/46f16639f1049ca8e5aea2202adbb02f179280b9ac0062fa808591fd1f7fae07.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle p ( \\mathbf { x } _ { 1 : T } \\mid \\mathbf { z } _ { 1 : T } , \\mathbf { u } _ { 1 : T } ) = \\prod _ { t = 1 \\atop t = 1 } ^ { T } p ( \\mathbf { x } _ { t } \\mid \\mathbf { z } _ { t } ) , } \\\\ { \\displaystyle p ( \\mathbf { z } _ { 1 : T } \\mid \\beta _ { 1 : T } , \\mathbf { u } _ { 1 : T } ) = \\prod _ { t = 0 } ^ { T - 1 } p ( \\mathbf { z } _ { t + 1 } \\mid \\mathbf { z } _ { t } , \\mathbf { u } _ { t } , \\beta _ { t } ) . } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Equations (3) and (4) assume that the current state $\\mathbf { z } _ { t }$ contains all necessary information about the current observation $\\mathbf { x } _ { t }$ , as well as the next state $\\mathbf { z } _ { t + 1 }$ (given the current control input $\\mathbf { u } _ { t }$ and transition parameters $\\beta _ { t }$ ). That is, in contrast to observations, $\\mathbf { z } _ { t }$ exhibits Markovian behavior. ",
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+        "type": "text",
+        "text": "A typical example of these assumptions are Linear Gaussian Models (LGMs), i.e., both state transition and emission model are affine transformations with Gaussian offset noise, ",
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+        "img_path": "images/4346d7e55504a061320e2fcfe4e78736293df217192d3fd74698e0c26851b04c.jpg",
+        "text": "$$\n\\begin{array} { r l r } { \\mathbf { z } _ { t + 1 } = \\mathbf { F } _ { t } \\mathbf { z } _ { t } + \\mathbf { B } _ { t } \\mathbf { u } _ { t } + \\mathbf { w } _ { t } \\quad } & { } & { \\mathbf { w } _ { t } \\sim \\mathcal { N } ( \\mathbf { 0 } , \\mathbf { Q } _ { t } ) , } \\\\ { \\mathbf { x } _ { t } = \\mathbf { H } _ { t } \\mathbf { z } _ { t } + \\mathbf { y } _ { t } \\quad } & { } & { \\mathbf { y } _ { t } \\sim \\mathcal { N } ( \\mathbf { 0 } , \\mathbf { R } _ { t } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "type": "text",
+        "text": "Typically, state transition matrix $\\mathbf { F } _ { t }$ and control-input matrix $\\mathbf { B } _ { t }$ are assumed to be given, so that $\\boldsymbol { \\beta } _ { t } = \\mathbf { w } _ { t }$ . Section 3.3 will show that our approach allows other variants such as $\\beta _ { t } = \\left( \\mathbf { F } _ { t } , \\mathbf { B } _ { t } , \\mathbf { w } _ { t } \\right)$ . Under the strong assumptions (5) and (6) of LGMs, inference is provably solved optimally by the well-known Kalman filters. While extensions of Kalman filters to nonlinear dynamical systems exist, Julier & Uhlmann (1997), and are successfully applied in many areas, they suffer from two major drawbacks: firstly, its assumptions are restrictive and are violated in practical applications, leading to suboptimal results. Secondly, parameters such as $\\mathbf { F } _ { t }$ and $\\mathbf { B } _ { t }$ have to be known in order to perform posterior inference. There have been efforts to learn such system dynamics, cf. Ghahramani $\\&$ Hinton (1996); Honkela et al. (2010) based on the expectation maximization (EM) algorithm or Valpola & Karhunen (2002), which uses neural networks. However, these algorithms are not applicable in cases where the true posterior distribution is intractable. This is the case if, e.g., image sequences are used, since the posterior is then highly nonlinear—typical mean-field assumptions on the approximate posterior are too simplified. Our new approach will tackle both issues, and moreover learn both identification and inference jointly by exploiting Stochastic Gradient Variational Bayes. ",
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "2.2 STOCHASTIC GRADIENT VARIATIONAL BAYES (SGVB) FOR TIME SERIES DISTRIBUTIONS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Replacing the bottleneck layer of a deterministic auto-encoder with stochastic units $\\mathbf { z }$ , the variational auto-encoder (VAE, Kingma & Welling (2013); Rezende et al. (2014)) learns complex marginal data distributions on $\\mathbf { x }$ in an unsupervised fashion from simpler distributions via the graphical model ",
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+        "img_path": "images/c570d54879ca96547bc8a517bef95c6b2c5d0083a9c631f8575b0845c774509b.jpg",
+        "text": "$$\np ( \\mathbf { x } ) = \\int p ( \\mathbf { x } , \\mathbf { z } ) \\mathrm { d } \\mathbf { z } = \\int p ( \\mathbf { x } \\mid \\mathbf { z } ) p ( \\mathbf { z } ) \\mathrm { d } \\mathbf { z } .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "In VAEs, $p ( \\mathbf { x } \\mid \\mathbf { z } ) \\equiv p _ { \\theta } ( \\mathbf { x } \\mid \\mathbf { z } )$ is typically parametrized by a neural network with parameters $\\theta$ . Within this framework, models are trained by maximizing a lower bound to the marginal data log-likelihood via stochastic gradients: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/891adf64cc5daa021c5286634b5acd9d3638370cf4b22b0114c64ad308d75a19.jpg",
+        "text": "$$\n\\ln p ( \\mathbf { x } ) \\geq \\mathbb { E } _ { q _ { \\phi } ( \\mathbf { z } | \\mathbf { x } ) } [ \\ln p _ { \\theta } ( \\mathbf { x } \\mid \\mathbf { z } ) ] - \\mathrm { K L } ( q _ { \\phi } ( \\mathbf { z } \\mid \\mathbf { x } ) \\mid \\mid p ( \\mathbf { z } ) ) = : \\mathcal { L } _ { \\mathrm { S G V B } } ( \\mathbf { x } , \\phi , \\theta )\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "This is provably equivalent to minimizing the KL-divergence between the approximate posterior or recognition model $q _ { \\phi } ( \\mathbf { z } \\mid \\mathbf { x } )$ and the true, but usually intractable posterior distribution $p ( \\mathbf { z } \\mid \\mathbf { x } )$ . $q _ { \\phi }$ is parametrized by a neural network with parameters $\\phi$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The principle of VAEs has been transferred to time series, Bayer & Osendorfer (2014); Chung et al. (2015). Both employ nonlinear state transitions in latent space, but violate eq. (4): Observations are directly included in the transition process. Empirically, reconstruction and compression work well. The state space $\\mathcal { Z }$ , however, does not reflect all information available, which prohibits plausible generative long-term prediction. Such phenomena with generative models have been explained in Theis et al. (2015). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In Krishnan et al. (2015), the state-space assumptions (3) and (4) are softly encoded in the Deep Kalman Filter (DKF) model. Despite that, experiments, cf. section 4, show that their model fails to extract information such as velocity (and in general time derivatives), which leads to similar problems with prediction. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Johnson et al. (2016) give an algorithm for general graphical model variational inference, not tailored to dynamical systems. In contrast to previously discussed methods, it does not violate eq. (4). The approaches differ in that the recognition model outputs node potentials in combination with message passing to infer the latent state. Our approach focuses on learning dynamical systems for controlrelated tasks and therefore uses a neural network for inferring the latent state directly instead of an inference subroutine. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Others have been specifically interested in applying variational inference for controlled dynamical systems. In Watter et al. (2015) (Embed to Control—E2C), a VAE is used to learn the mappings to and from latent space. The regularization is clearly motivated by eq. (7). Still, it fails to be a mathematically correct lower bound to the marginal data likelihood. More significantly, their recognition model requires all observations that contain information w.r.t. the current state. This is nothing short of an additional temporal i.i.d. assumption on data: Multiple raw samples need to be stacked into one training sample such that all latent factors (in particular all time derivatives) are present within one sample. The task is thus greatly simplified, because instead of time-series, we learn a static auto-encoder on the processed data. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "A pattern emerges: good prediction should boost compression. Still, previous methods empirically excel at compression, while prediction will not work. We conjecture that this is caused by previous methods trying to fit the latent dynamics to a latent state that is beneficial for reconstruction. This encourages learning of a stationary auto-encoder with focus of extracting as much from a single observation as possible. Importantly, it is not necessary to know the entire sequence for excellent reconstruction of single time steps. Once the latent states are set, it is hard to adjust the transition to them. This would require changing the latent states slightly, and that comes at a cost of decreasing the reconstruction (temporarily). The learning algorithm is stuck in a local optimum with good reconstruction and hence good compression only. Intriguingly, E2C bypasses this problem with its data augmentation. ",
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+        "img_path": "images/3575706f643090530f958bb9ba2559b160e8016b28d587187d9d19d1e13fff7e.jpg",
+        "image_caption": [
+            "Figure 1: Left: Graphical model for one transition under state-space model assumptions. The updated latent state $\\mathbf { z } _ { t + 1 }$ depends on the previous state $\\mathbf { z } _ { t }$ , control input $\\mathbf { u } _ { t }$ , and transition parameters $\\beta _ { t }$ . $\\mathbf { z } _ { t + 1 }$ contains all information for generating observation $\\mathbf { x } _ { t + 1 }$ . Diamond nodes indicate a deterministic dependency on parent nodes. Right: Inference performed during training (or while filtering). Past observations are indirectly used for inference as $\\mathbf { z } _ { t }$ contains all information about them. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This leads to a key contribution of this paper: We force the latent space to fit the transition—reversing the direction, and thus achieving the state-space model assumptions and full information in the latent states. ",
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+    {
+        "type": "text",
+        "text": "3 DEEP VARIATIONAL BAYES FILTERS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "3.1 REPARAMETRIZING THE TRANSITION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The central problem for learning latent states system dynamics is efficient inference of a latent space that obeys state-space model assumptions. If the latter are fulfilled, the latent space must contain all information. Previous approaches emphasized good reconstruction, so that the space only contains information necessary for reconstruction of one time step. To overcome this, we establish gradient paths through transitions over time so that the transition becomes the driving factor for shaping the latent space, rather than adjusting the transition to the recognition model’s latent space. The key is to prevent the recognition model $\\bar { q _ { \\phi } } ( \\mathbf { z } _ { 1 : T } \\mid \\mathbf { x } _ { 1 : T } )$ from directly drawing the latent state $\\mathbf { z } _ { t }$ . ",
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+        "type": "text",
+        "text": "Similar to the reparametrization trick from Kingma & Welling (2013); Rezende et al. (2014) for making the Monte Carlo estimate differentiable w.r.t. the parameters, we make the transition differentiable w.r.t. the last state and its parameters: ",
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+    {
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+        "img_path": "images/e8760aa7ae63034c40da90e1b44b24dbae86079030ab536fc189dd2c0e60de10.jpg",
+        "text": "$$\n\\mathbf { z } _ { t + 1 } = f ( \\mathbf { z } _ { t } , \\mathbf { u } _ { t } , \\boldsymbol { \\beta } _ { t } )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Given the stochastic parameters $\\beta _ { t }$ , the state transition is deterministic (which in turn means that by marginalizing $\\beta _ { t }$ , we still have a stochastic transition). The immediate and crucial consequence is that errors in reconstruction of $\\mathbf { x } _ { t }$ from $\\mathbf { z } _ { t }$ are backpropagated directly through time. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This reparametrization has a couple of other important implications: the recognition model no longer infers latent states $\\mathbf { z } _ { t }$ , but transition parameters $\\beta _ { t }$ . In particular, the gradient $\\partial { \\mathbf z } _ { t + 1 } / \\partial { \\mathbf z } _ { t }$ is well-defined from (8)—gradient information can be backpropagated through the transition. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This is different from the method used in Krishnan et al. (2015), where the transition only occurs in the KL-divergence term of their loss function (a variant of eq. (7)). No gradient from the generative model is backpropagated through the transitions. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Much like in eq. (5), the stochastic parameters includes a corrective offset term $\\mathbf { w } _ { t }$ , which emphasizes the notion of the recognition model as a filter. In theory, the learning algorithm could still learn the transition as $\\mathbf { z } _ { t + 1 } = \\mathbf { w } _ { t }$ . However, the introduction of $\\beta _ { t }$ also enables us to regularize the transition with meaningful priors, which not only prevents overfitting the recognition model, but also enforces meaningful manifolds in the latent space via transition priors. Ignoring the potential of the transition over time yields large penalties from these priors. Thus, the problems outlined in Section 2 are overcome by construction. ",
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+    {
+        "type": "text",
+        "text": "To install such transition priors, we split $\\boldsymbol { \\beta } _ { t } = \\left( \\mathbf { w } _ { t } , \\mathbf { v } _ { t } \\right)$ . The interpretation of $\\mathbf { w } _ { t }$ is a sample-specific process noise which can be inferred from incoming data, like in eq. (5). On the other hand, $\\mathbf { v } _ { t }$ are universal transition parameters, which are sample-independent (and are only inferred from data during training). This corresponds to the idea of weight uncertainty in Hinton $\\&$ Van Camp (1993). This interpretation leads to a natural factorization assumption on the recognition model: ",
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+        "image_caption": [
+            "(b) One particular example of a latent transition: local linearity. "
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+        "image_caption": [
+            "Figure 2: Left: General architecture for DVBF. Stochastic transition parameters $\\beta _ { t }$ are inferred via the recognition model, e.g., a neural network. Based on a sampled $\\beta _ { t }$ , the state transition is computed deterministically. The updated latent state $\\mathbf { z } _ { t + 1 }$ is used for predicting $\\mathbf { x } _ { t + 1 }$ . For details, see section 3.1. Right: Zoom into latent space transition (red box in left figure). One exemplary transition is shown, the locally linear transition from section 3.3. "
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+    {
+        "type": "equation",
+        "img_path": "images/0e95ebe86b573067d8096a1a603aedb86a75a0eb210c74a12541fc913ad5070d.jpg",
+        "text": "$$\nq _ { \\phi } ( \\beta _ { 1 : T } \\mid \\mathbf { x } _ { 1 : T } ) = q _ { \\phi } ( \\mathbf { w } _ { 1 : T } \\mid \\mathbf { x } _ { 1 : T } ) q _ { \\phi } ( \\mathbf { v } _ { 1 : T } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "When using the fully trained model for generative sampling, i.e., sampling without input, the universal state transition parameters can still be drawn from $q _ { \\phi } ( \\mathbf { v } _ { 1 : T } )$ , whereas $\\mathbf { w } _ { 1 : T }$ is drawn from the prior in the absence of input data. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Figure 1 shows the underlying graphical model and the inference procedure. Figure 2a shows a generic view on our new computational architecture. An example of a locally linear transition parametrization will be given in section 3.3. ",
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+    {
+        "type": "text",
+        "text": "3.2 THE LOWER BOUND OBJECTIVE FUNCTION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In analogy to eq. (7), we now derive a lower bound to the marginal likelihood $p ( \\mathbf { x } _ { 1 : T } \\mid \\mathbf { u } _ { 1 : T } )$ . After reflecting the Markov assumptions (3) and (4) in the factorized likelihood (2), we have: ",
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+        "type": "equation",
+        "img_path": "images/af9ae5913e6d4a3023ea7e055ba04f05e374fca6889cf54e846207aa7396eeb0.jpg",
+        "text": "$$\np ( \\mathbf { x } _ { 1 : T } \\mid \\mathbf { u } _ { 1 : T } ) = \\int \\int p ( \\beta _ { 1 : T } ) \\prod _ { t = 1 } ^ { T } p _ { \\theta } ( \\mathbf { x } _ { t } \\mid \\mathbf { z } _ { t } ) \\prod _ { t = 0 } ^ { T - 1 } p ( \\mathbf { z } _ { t + 1 } \\mid \\mathbf { z } _ { t } , \\mathbf { u } _ { t } , \\beta _ { t } ) \\mathrm { d } \\beta _ { 1 : T } \\mathrm { d } \\mathbf { z } _ { 1 : T }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Due to the deterministic transition given $\\beta _ { t + 1 }$ , the last term is a product of Dirac distributions and the overall distribution simplifies greatly: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/88e7ae4797b5633519c77572e0e967194d68b3b2e519dfdaa4d60a2adfc57735.jpg",
+        "text": "$$\n\\begin{array} { l } { p ( \\mathbf { x } _ { 1 : T } \\mid \\mathbf { u } _ { 1 : T } ) = \\displaystyle \\int p ( \\boldsymbol { \\beta } _ { 1 : T } ) \\prod _ { t = 1 } ^ { T } p _ { \\boldsymbol { \\theta } } ( \\mathbf { x } _ { t } \\mid \\mathbf { z } _ { t } ) \\Big \\vert _ { \\mathbf { z } _ { t } = f \\left( \\mathbf { z } _ { t - 1 } , \\mathbf { u } _ { t - 1 } , \\boldsymbol { \\beta } _ { t - 1 } \\right) } \\mathrm { d } \\boldsymbol { \\beta } _ { 1 : T } } \\\\ { \\displaystyle \\left( = \\int p ( \\boldsymbol { \\beta } _ { 1 : T } ) p _ { \\boldsymbol { \\theta } } ( \\mathbf { x } _ { 1 : T } \\mid \\mathbf { z } _ { 1 : T } ) \\mathrm { d } \\boldsymbol { \\beta } _ { 1 : T } \\right) } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "The last formulation is for notational brevity: the term $p _ { \\theta } ( \\mathbf { x } _ { 1 : T } \\mid \\mathbf { z } _ { 1 : T } )$ is not independent of $\\beta _ { 1 : T }$ and $\\mathbf { u } _ { 1 : T }$ . We now derive the objective function, a lower bound to the data likelihood: ",
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+        "type": "equation",
+        "img_path": "images/1d26543005830c88b8ffd0233159e77be2c5238b1f739952af428cdee5ee0ecc.jpg",
+        "text": "$$\n\\begin{array}{c} \\ln p ( \\mathbf { x } _ { 1 : T } \\mid \\mathbf { u } _ { 1 : T } ) = \\ln \\int p ( \\beta _ { 1 : T } ) p _ { \\theta } ( \\mathbf { x } _ { 1 : T } \\mid \\mathbf { z } _ { 1 : T } ) { \\frac { q _ { \\phi } ( \\beta _ { 1 : T } \\mid \\mathbf { x } _ { 1 : T } , \\mathbf { u } _ { 1 : T } ) } { q _ { \\phi } ( \\beta _ { 1 : T } \\mid \\mathbf { x } _ { 1 : T } , \\mathbf { u } _ { 1 : T } ) } } \\mathrm { d } \\beta _ { 1 : T }  \\\\ { \\geq \\int q _ { \\phi } ( \\beta _ { 1 : T } \\mid \\mathbf { x } _ { 1 : T } , \\mathbf { u } _ { 1 : T } ) \\ln \\left( p _ { \\theta } ( \\mathbf { x } _ { 1 : T } \\mid \\mathbf { z } _ { 1 : T } ) { \\frac { p ( \\beta _ { 1 : T } ) } { q _ { \\phi } ( \\beta _ { 1 : T } \\mid \\mathbf { x } _ { 1 : T } , \\mathbf { u } _ { 1 : T } ) } } \\right) \\mathrm { d } \\beta _ { 1 : T } } \\\\ { = \\mathbb { E } _ { q _ { \\phi } } [ \\ln p _ { \\theta } ( \\mathbf { x } _ { 1 : T } \\mid \\mathbf { z } _ { 1 : T } ) - \\ln q _ { \\phi } ( \\beta _ { 1 : T } \\mid \\mathbf { x } _ { 1 : T } , \\mathbf { u } _ { 1 : T } ) + \\ln p ( \\beta _ { 1 : T } ) ] } \\\\ { = \\mathbb { E } _ { q _ { \\phi } } [ \\ln p _ { \\theta } ( \\mathbf { x } _ { 1 : T } \\mid \\mathbf { z } _ { 1 : T } ) ] - \\mathrm { K L } ( q _ { \\phi } ( \\beta _ { 1 : T } \\mid \\mathbf { x } _ { 1 : T } , \\mathbf { u } _ { 1 : T } ) \\mid \\mid p ( \\beta _ { 1 : T } ) ) } & { ( \\ln p ( \\beta _ { 1 : T } ) ) } \\\\ { = : { \\mathcal { L } } _ { \\mathrm { D V B F } } ( \\mathbf { x } _ { 1 : T } , \\theta , \\phi \\mid \\mathbf { u } _ { 1 : T } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our experiments show that an annealed version of (10) is beneficial to the overall performance: ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "equation",
+        "img_path": "images/68d6606f230929ebb0dceeccb13bbe97839363d47d8d39c2ed1cc74367de6a42.jpg",
+        "text": "$$\n( \\mathbf { 1 0 ^ { \\prime } } ) = \\mathbb { E } _ { q _ { \\phi } } [ c _ { i } \\ln p _ { \\theta } ( \\mathbf { x } _ { 1 : T } ~ \\vert ~ \\mathbf { z } _ { 1 : T } ) - \\ln q _ { \\phi } ( \\beta _ { 1 : T } ~ \\vert ~ \\mathbf { x } _ { 1 : T } , \\mathbf { u } _ { 1 : T } ) + c _ { i } \\ln p ( \\mathbf { w } _ { 1 : T } ) + \\ln p ( \\mathbf { v } _ { 1 : T } ) ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Here, $c _ { i } = \\operatorname* { m a x } ( 1 , 0 . 0 1 + i / T _ { A } )$ is an inverse temperature that increases linearly in the number of gradient updates $i$ until reaching 1 after $T _ { A }$ annealing iterations. Similar annealing schedules have been applied in, e.g., Ghahramani $\\&$ Hinton (2000); Mandt et al. (2016); Rezende & Mohamed (2015), where it is shown that they smooth the typically highly non-convex error landscape. Additionally, the transition prior $p ( \\mathbf { v } _ { 1 : T } )$ was estimated during optimization, i.e., through an empirical Bayes approach. In all experiments, we used isotropic Gaussian priors. ",
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+    {
+        "type": "text",
+        "text": "3.3 EXAMPLE: LOCALLY LINEAR TRANSITIONS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We have derived a learning algorithm for time series with particular focus on general transitions in latent space. Inspired by Watter et al. (2015), this section will show how to learn a particular instance: locally linear state transitions. That is, we set eq. (8) to ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/8d10cbee324d1ff819209078c5f9909359d74fcb8d38aae19f1f99f908db5e4e.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbf { z } _ { t + 1 } = \\mathbf { A } _ { t } \\mathbf { z } _ { t } + \\mathbf { B } _ { t } \\mathbf { u } _ { t } + \\mathbf { C } _ { t } \\mathbf { w } _ { t } , \\qquad t = 1 , \\ldots , T , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { w } _ { t }$ is a stochastic sample from the recognition model and $\\mathbf { A } _ { t } , \\mathbf { B } _ { t }$ , and $\\mathbf { C } _ { t }$ are matrices of matching dimensions. They are stochastic functions of $\\mathbf { z } _ { t }$ and $\\mathbf { u } _ { t }$ (thus local linearity). We draw ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/8f4b7522861531ac14c7de501ee91d39a4a1a81baa1658ae073dd80f4f3470ae.jpg",
+        "text": "$$\n\\mathbf { v } _ { t } = \\Big \\{ \\mathbf { A } _ { t } ^ { ( i ) } , \\mathbf { B } _ { t } ^ { ( i ) } , \\mathbf { C } _ { t } ^ { ( i ) } \\mid i = 1 , \\ldots , M \\Big \\} ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "from $q _ { \\phi } ( \\mathbf { v } _ { t } )$ , i.e., $M$ triplets of matrices, each corresponding to data-independent, but learned globally linear system. These can be learned as point estimates. We employed a Bayesian treatment as in Blundell et al. (2015). We yield $\\mathbf { A } _ { t } , \\mathbf { B } _ { t }$ , and $\\mathbf { C } _ { t }$ as state- and control-dependent linear combinations: ",
+        "bbox": [
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+    {
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+        "img_path": "images/64057e6290652167d9bcdc372f79e505b7df2268ca402eef6e065cc7a8dadb6a.jpg",
+        "text": "$$\n\\begin{array}{c} \\begin{array} { r l r l r l } & { \\quad } & & { \\quad } & & { \\boldsymbol { \\alpha } _ { t } = f _ { \\psi } ( \\mathbf { z } _ { t } , \\mathbf { u } _ { t } ) \\in \\mathbb { R } ^ { M } } \\\\ & { \\quad } & & { \\quad } & { \\quad } & { \\quad } \\\\ & { \\quad } & { \\quad } & { \\quad } & { \\quad } & { \\quad } \\\\ & { \\quad } & { \\quad } & { \\quad } & { \\quad } & { \\quad } & { \\quad } \\\\ & { \\quad } & { \\quad } & { \\quad } & { \\quad } & { \\quad } & { \\quad } \\end{array} \\qquad \\begin{array} { r l } & { \\quad } & { \\quad } & { \\boldsymbol { \\alpha } _ { t } = f _ { \\psi } ( \\mathbf { z } _ { t } , \\mathbf { u } _ { t } ) \\in \\mathbb { R } ^ { M } } \\\\ & { \\quad } & { \\quad } \\\\ & { \\quad } & { \\quad } \\\\ & { \\quad } & { \\quad } & { \\quad } \\end{array} \\qquad \\mathbf { C } _ { t } = \\sum _ { i = 1 } ^ { M } { \\boldsymbol { \\alpha } _ { t } ^ { ( i ) } } \\mathbf { C } _ { t } ^ { ( i ) }  \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "The computation is depicted in fig. 2b. The function $f _ { \\psi }$ can be, e.g., a (deterministic) neural network with weights $\\psi$ . As a subset of the generative parameters $\\theta$ , $\\psi$ is part of the trainable parameters of our model. The weight vector $\\pmb { \\alpha } _ { t }$ is shared between the three matrices. There is a correspondence to eq. (5): ${ \\bf A } _ { t }$ and $\\mathbf { F } _ { t }$ , $\\mathbf { B } _ { t }$ and $\\mathbf { B } _ { t }$ , as well as $\\mathbf { C } _ { t } \\mathbf { C } _ { t } ^ { \\top }$ and $\\mathbf { Q } _ { t }$ are related. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We used this parametrization of the state transition model for our experiments. It is important that the parametrization is up to the user and the respective application. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS AND RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "In this section we validate that DVBF with locally linear transitions (DVBF-LL) (section 3.3) outperforms Deep Kalman Filters (DKF, Krishnan et al. (2015)) in recovering latent spaces with full information. 2 We focus on environments that can be simulated with full knowledge of the ",
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+    {
+        "type": "text",
+        "text": "2We do not include E2C, Watter et al. (2015), due to the need for data modification and its inability to provide a correct lower bound as mentioned in section 2.2. ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/89343f9239b5f95d0b3f05f420ceb14168472d729834fda8850d44f8baa77701.jpg",
+        "image_caption": [
+            "Figure 3: (a) Our DVBF-LL model trained on pendulum image sequences. The upper plots show the latent space with coloring according to the ground truth with angles on the left and angular velocities on the right. The lower plots show regression results for predicting ground truth from the latent representation. The latent space plots show clearly that all information for representing the full state of a pendulum is encoded in each latent state. (b) DKF from Krishnan et al. (2015) trained on the same pendulum dataset. The latent space plot shows that DKF fails to learn velocities of the pendulum. It is therefore not able to capture all information for representing the full pendulum state. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "ground truth latent dynamical system. The experimental setup is described in the Supplementary Material. We published the code for DVBF and a link will be made available at https://brml. org/projects/dvbf. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.1 DYNAMIC PENDULUM ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "In order to test our algorithm on truly non-Markovian observations of a dynamical system, we simulated a dynamic torque-controlled pendulum governed by the differential equation ",
+        "bbox": [
+            173,
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a2c84e38c39e525dc1ff6ad2340e632908af077e0aa2ea41e9721ff6c3b0e4df.jpg",
+        "text": "$$\nm l ^ { 2 } { \\ddot { \\varphi } } ( t ) = - \\mu { \\dot { \\varphi } } ( t ) + m g l \\sin \\varphi ( t ) + u ( t ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "$m = l = 1 , \\mu = 0 . 5 , g = 9 . 8 1$ , via numerical integration, and then converted the ground-truth angle $\\varphi$ into an image observation in $\\mathcal { X }$ . The one-dimensional control corresponds to angle acceleration (which is proportional to joint torque). Angle and angular velocity fully describe the system. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Figure 3 shows the latent spaces for identical input data learned by DVBF-LL and DKF, respectively, colored with the ground truth in the top row. It should be noted that latent samples are shown, not means of posterior distributions. The state-space model was allowed to use three latent dimensions. As we can see in fig. 3a, DVBF-LL learned a two-dimensional manifold embedding, i.e., it encoded the angle in polar coordinates (thus circumventing the discontinuity of angles modulo $2 \\pi$ ). The bottom row shows ordinary least-squares regressions (OLS) underlining the performance: there exists a high correlation between latent states and ground-truth angle and angular velocity for DVBF-LL. On the contrary, fig. 3b verifies our prediction that DKF is equally capable of learning the angle, but extracts little to no information on angular velocity. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The OLS regression results shown in table 1 validate this observation.3 Predicting $\\sin ( \\varphi )$ and $\\cos ( \\varphi )$ , i.e., polar coordinates of the ground-truth angle $\\varphi$ , works almost equally well for DVBF-LL and DKF, with DVBF-LL slightly outperforming DKF. For predicting the ground truth velocity $\\dot { \\varphi }$ , DVBF-LL shows remarkable performance. DKF, instead, contains hardly any information, resulting in a very low goodness-of-fit score of $R ^ { 2 } = 0 . 0 3 5$ . ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/97bb943c76be0c1bc1610950d8d37237225c28f1948f27e3e996f91c92ed7272.jpg",
+        "table_caption": [
+            "Table 1: Results for pendulum OLS regressions of all latent states on respective dependent variable. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"3\">DVBF-LL</td><td colspan=\"2\">DKF</td></tr><tr><td></td><td></td><td>Log-Likelihood</td><td>R² Log-Likelihood</td><td>R²</td></tr><tr><td>Dependent</td><td>sin()</td><td>3990.8</td><td>0.961 0.982</td><td>1737.6 0.929</td></tr><tr><td>ground truth</td><td>cos()</td><td>7231.1</td><td>6614.2</td><td>0.979</td></tr><tr><td>variable</td><td>6</td><td>-11139 0.916</td><td>-20289</td><td>0.035</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/886df3e2f7dd268db086c2c03abc30a59759b198fe32da3389af3e840dbf5cc5.jpg",
+        "image_caption": [
+            "(b) Reconstructive latent walk. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/d58aeaeeefb189a121cf98cce5b00a9f7e03e587fd265f3883b8988c3b5125e1.jpg",
+        "image_caption": [
+            "(a) Generative latent walk. ",
+            "(c) Ground truth (top), reconstructions (middle), generative samples (bottom) from identical initial latent state. ",
+            "Figure 4: (a) Latent space walk in generative mode. (b) Latent space walk in filtering mode. (c) Ground truth and samples from recognition and generative model. The reconstruction sampling has access to observation sequence and performs filtering. The generative samples only get access to the observations once for creating the initial state while all subsequent samples are predicted from this single initial state. The red bar indicates the length of training sequences. Samples beyond show the generalization capabilities for sequences longer than during training. The complete sequence can be found in the Appendix in fig. 7. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "table",
+        "img_path": "images/0bda0edc9c640a14782193ac928d48fc81fbe31215bb5bcb175fdb6e40da574f.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=31>1            5               10               15              20                      40              45</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>`</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>.</td><td rowspan=1 colspan=1>,</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>,</td><td rowspan=1 colspan=1>,</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1>.</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>.</td><td rowspan=1 colspan=1>.</td><td rowspan=1 colspan=1>.</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>、</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>·</td><td rowspan=1 colspan=1>·</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>·</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>、</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>,</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1>、</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1>.</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>.</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>.</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>,</td><td rowspan=1 colspan=1>,</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>,</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Figure 4 shows that the strong relation between ground truth and latent state is beneficial for generative sampling. All plots show 100 time steps of a pendulum starting from the exact same latent state and not being actuated. The top row plots show a purely generative walk in the latent space on the left, and a walk in latent space that is corrected by filtering observations on the right. We can see that both follow a similar trajectory to an attractor. The generative model is more prone to noise when approaching the attractor. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The bottom plot shows the first 45 steps of the corresponding observations (top row), reconstructions (middle row), and generative samples (without correcting from observations). Interestingly, DVBF works very well even though the sequence is much longer than all training sequences (indicated by the red line). ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table (2) shows values of the lower bound to the marginal data likelihood (for DVBF-LL, this corresponds to eq. (11)). We see that DVBF-LL outperforms DKF in terms of compression, but only with a slight margin, which does not reflect the better generative sampling as Theis et al. (2015) argue. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/c9482434209e7dde8771c88440c90feb407a9a4f7b03d8c70dac223a41f3459b.jpg",
+        "image_caption": [
+            "Figure 5: (a) Two dimensions of 4D bouncing ball latent space. Ground truth x and y coordinates are combined into a regular $3 \\times 3$ checkerboard coloring. This checkerboard is correctly extracted by the embedding. (b) Remaining two latent dimensions. Same latent samples, colored with ball velocities in x and y direction (left and right image, respectively). The smooth, perpendicular coloring indicates that the ground truth value is stored in the latent dimension. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.2 BOUNCING BALL ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The bouncing ball experiment features a ball rolling within a bounding box in a plane. The system has a two-dimensional control input, added to the directed velocity of the ball. If the ball hits the wall, it bounces off, so that the true dynamics are highly dependent on the current position and velocity of the ball. The system’s state is four-dimensional, two dimensions each for position and velocity. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Consequently, we use a DVBF-LL with four latent dimensions. Figure 5 shows that DVBF again captures the entire system dynamics in the latent space. The checkerboard is quite a remarkable result: the ground truth position of the ball lies within the 2D unit square, the bounding box. In order to visualize how ground truth reappears in the learned latent states, we show the warping of the ground truth bounding box into the latent space. To this end, we partitioned (discretized) the ground truth unit square into a regular 3x3 checkerboard with respective coloring. We observed that DVBF learned to extract the 2D position from the 256 pixels, and aligned them in two dimensions of the latent space in strong correspondence to the physical system. The algorithm does the exact same pixel-to-2D inference that a human observer automatically does when looking at the image. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/0fee90b828d2c622647068fb4a7868c7ca217d5d2ed1f6ad3072d29a40f8c53b.jpg",
+        "image_caption": [
+            "Figure 6: Ground truth (top), reconstructions (middle), generative samples (bottom) from identical initial latent state for the two bouncing balls experiment. Red bar indicates length of training sequences. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.3 TWO BOUNCING BALLS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Another more complex environment4 features two balls in a bounding box. We used a 10-dimensional latent space to fully capture the position and velocity information of the balls. Reconstruction and generative samples are shown in fig. 6. Same as in the pendulum example we get a generative model with stable predictions beyond training data sequence length. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We have proposed Deep Variational Bayes Filters (DVBF), a new method to learn state space models from raw non-Markovian sequence data. DVBFs perform latent dynamic system identification, and subsequently overcome intractable inference. As DVBFs make use of stochastic gradient variational Bayes they naturally scale to large data sets. In a series of vision-based experiments we demonstrated that latent states can be recovered which identify the underlying physical quantities. The generative model showed stable long-term predictions far beyond the sequence length used during training. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            340,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Part of this work was conducted at Chair of Robotics and Embedded Systems, Department of Informatics, Technische Universität München, Germany, and supported by the TACMAN project, EC Grant agreement no. 610967, within the FP7 framework programme. ",
+        "bbox": [
+            176,
+            364,
+            821,
+            407
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We would like to thank Jost Tobias Springenberg, Adam Kosiorek, Moritz Münst, and anonymous reviewers for valuable input. ",
+        "bbox": [
+            173,
+            414,
+            823,
+            443
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            464,
+            287,
+            479
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Justin Bayer and Christian Osendorfer. Learning stochastic recurrent networks. arXiv preprint arXiv:1411.7610, 2014. ",
+        "bbox": [
+            173,
+            488,
+            825,
+            517
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Charles Blundell, Julien Cornebise, Koray Kavukcuoglu, and Daan Wierstra. Weight uncertainty in neural networks. arXiv preprint arXiv:1505.05424, 2015. ",
+        "bbox": [
+            171,
+            527,
+            825,
+            558
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Léon Bottou. Large-scale machine learning with stochastic gradient descent. In Proceedings of COMPSTAT’2010, pp. 177–186. Springer, 2010. ",
+        "bbox": [
+            171,
+            568,
+            825,
+            597
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Junyoung Chung, Kyle Kastner, Laurent Dinh, Kratarth Goel, Aaron C. Courville, and Yoshua Bengio. A recurrent latent variable model for sequential data. CoRR, abs/1506.02216, 2015. URL http://arxiv.org/abs/1506.02216. ",
+        "bbox": [
+            173,
+            607,
+            826,
+            651
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Marc Deisenroth and Carl E Rasmussen. Pilco: A model-based and data-efficient approach to policy search. In Proceedings of the 28th International Conference on machine learning (ICML-11), pp. 465–472, 2011. ",
+        "bbox": [
+            173,
+            661,
+            826,
+            704
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Zoubin Ghahramani and Geoffrey E Hinton. Parameter estimation for linear dynamical systems. Technical report, Technical Report CRG-TR-96-2, University of Toronto, Dept. of Computer Science, 1996. ",
+        "bbox": [
+            174,
+            715,
+            826,
+            757
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Zoubin Ghahramani and Geoffrey E Hinton. Variational learning for switching state-space models. Neural computation, 12(4):831–864, 2000. ",
+        "bbox": [
+            169,
+            768,
+            825,
+            797
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Alex Graves. Generating sequences with recurrent neural networks. arXiv preprint arXiv:1308.0850, 2013. ",
+        "bbox": [
+            173,
+            809,
+            823,
+            838
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Geoffrey E Hinton and Drew Van Camp. Keeping the neural networks simple by minimizing the description length of the weights. In Proceedings of the sixth annual conference on Computational learning theory, pp. 5–13. ACM, 1993. ",
+        "bbox": [
+            174,
+            849,
+            826,
+            891
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "4We used the script attached to Sutskever & Hinton (2007) for generating our datasets. ",
+        "bbox": [
+            197,
+            910,
+            705,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Antti Honkela, Tapani Raiko, Mikael Kuusela, Matti Tornio, and Juha Karhunen. Approximate riemannian conjugate gradient learning for fixed-form variational bayes. Journal of Machine Learning Research, 11(Nov):3235–3268, 2010. ",
+        "bbox": [
+            174,
+            103,
+            823,
+            146
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Matthew J Johnson, David Duvenaud, Alexander B Wiltschko, Sandeep R Datta, and Ryan P Adams. Structured VAEs: Composing probabilistic graphical models and variational autoencoders. arXiv preprint arXiv:1603.06277, 2016. ",
+        "bbox": [
+            174,
+            155,
+            823,
+            196
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Simon J Julier and Jeffrey K Uhlmann. New extension of the kalman filter to nonlinear systems. In AeroSense’97, pp. 182–193. International Society for Optics and Photonics, 1997. ",
+        "bbox": [
+            171,
+            207,
+            823,
+            236
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Rudolph E Kalman and Richard S Bucy. New results in linear filtering and prediction theory. Journal of basic engineering, 83(1):95–108, 1961. ",
+        "bbox": [
+            173,
+            244,
+            823,
+            273
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Diederik P Kingma and Max Welling. Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114, 2013. ",
+        "bbox": [
+            173,
+            281,
+            823,
+            310
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Jonathan Ko and Dieter Fox. Learning gp-bayesfilters via gaussian process latent variable models. Autonomous Robots, 30(1):3–23, 2011. ",
+        "bbox": [
+            171,
+            319,
+            823,
+            349
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Rahul G Krishnan, Uri Shalit, and David Sontag. Deep Kalman filters. arXiv preprint arXiv:1511.05121, 2015. ",
+        "bbox": [
+            173,
+            357,
+            823,
+            387
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Stephan Mandt, James McInerney, Farhan Abrol, Rajesh Ranganath, and David Blei. Variational tempering. In Proceedings of the 19th International Conference on Artificial Intelligence and Statistics, pp. 704–712, 2016. ",
+        "bbox": [
+            173,
+            395,
+            826,
+            438
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Kevin McGoff, Sayan Mukherjee, Natesh Pillai, et al. Statistical inference for dynamical systems: A review. Statistics Surveys, 9:209–252, 2015. ",
+        "bbox": [
+            173,
+            446,
+            821,
+            476
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Danilo J. Rezende, Shakir Mohamed, and Daan Wierstra. Stochastic backpropagation and approximate inference in deep generative models. In Tony Jebara and Eric P. Xing (eds.), Proceedings of the 31st International Conference on Machine Learning (ICML-14), pp. 1278–1286. JMLR Workshop and Conference Proceedings, 2014. URL http://jmlr.org/proceedings/ papers/v32/rezende14.pdf. ",
+        "bbox": [
+            173,
+            484,
+            826,
+            555
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Danilo Jimenez Rezende and Shakir Mohamed. Variational inference with normalizing flows. arXiv preprint arXiv:1505.05770, 2015. ",
+        "bbox": [
+            171,
+            564,
+            825,
+            593
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Ilya Sutskever and Geoffrey E. Hinton. Learning multilevel distributed representations for high-dimensional sequences. In Marina Meila and Xiaotong Shen (eds.), Proceedings of the Eleventh International Conference on Artificial Intelligence and Statistics (AISTATS-07), volume 2, pp. 548–555. Journal of Machine Learning Research - Proceedings Track, 2007. URL http://jmlr.csail.mit.edu/proceedings/papers/v2/sutskever07a/ sutskever07a.pdf. ",
+        "bbox": [
+            174,
+            602,
+            826,
+            686
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Leonid Kuvayev Rich Sutton. Model-based reinforcement learning with an approximate, learned model. In Proceedings of the ninth Yale workshop on adaptive and learning systems, pp. 101–105, 1996. ",
+        "bbox": [
+            173,
+            694,
+            826,
+            737
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Lucas Theis, Aäron van den Oord, and Matthias Bethge. A note on the evaluation of generative models. arXiv preprint arXiv:1511.01844, 2015. ",
+        "bbox": [
+            173,
+            746,
+            821,
+            775
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Harri Valpola and Juha Karhunen. An unsupervised ensemble learning method for nonlinear dynamic state-space models. Neural computation, 14(11):2647–2692, 2002. ",
+        "bbox": [
+            173,
+            784,
+            823,
+            813
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Manuel Watter, Jost Springenberg, Joschka Boedecker, and Martin Riedmiller. Embed to control: A locally linear latent dynamics model for control from raw images. In Advances in Neural Information Processing Systems, pp. 2728–2736, 2015. ",
+        "bbox": [
+            173,
+            821,
+            826,
+            864
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A SUPPLEMENTARY TO LOWER BOUND ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            519,
+            118
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.1 ANNEALED KL-DIVERGENCE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            135,
+            428,
+            150
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We used the analytical solution of the annealed KL-divergence in eq. (10) for optimization: ",
+        "bbox": [
+            168,
+            161,
+            769,
+            179
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/44b2f95f328037b96719d78948b36f215713eae436dc1023c3e1884a3dd7cce8.jpg",
+        "text": "$$\n\\begin{array} { r l r } & { } & { \\mathbb { E } _ { q _ { \\phi } } [ - \\ln q _ { \\phi } ( { \\bf w } _ { 1 : T } \\mid { \\bf x } _ { 1 : T } , { \\bf u } _ { 1 : T } ) + c _ { i } \\ln p ( { \\bf w } _ { 1 : T } ) ] = } \\\\ & { } & { c _ { i } \\frac { 1 } { 2 } \\ln ( 2 \\pi \\sigma _ { p } ^ { 2 } ) - \\frac { 1 } { 2 } \\ln ( 2 \\pi \\sigma _ { q } ^ { 2 } ) + c _ { i } \\frac { \\sigma _ { q } ^ { 2 } + ( \\mu _ { q } - \\mu _ { p } ) ^ { 2 } } { 2 \\sigma _ { p } ^ { 2 } } - \\frac { 1 } { 2 } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            207,
+            684,
+            267
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B SUPPLEMENTARY TO IMPLEMENTATION ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            285,
+            540,
+            301
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B.1 EXPERIMENTAL SETUP ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            318,
+            377,
+            333
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In all our experiments, we use sequences of 15 raw images of the respective system with $1 6 \\times 1 6$ pixels each, i.e., observation space $\\mathcal { X } \\subset \\mathbb { R } ^ { 2 5 6 }$ , as well as control inputs of varying dimension and interpretation depending on the experiment. We used training, validation and test sets with 500 sequences each. Control input sequences were drawn randomly (“motor babbling”). Additional details about the implementation can be found in the published code at https://brml.org/ projects/dvbf. ",
+        "bbox": [
+            173,
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+            430
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B.2 ADDITIONAL EXPERIMENT PLOTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+            464
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/e2917c69eb71d5423688cb5a5187f15f229e6017c11ecfbeb490972b4bf65861.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            486,
+            823,
+            518
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Figure 7: Ground truth and samples from recognition and generative model. Complete version of fig. 4 with all missing samples present. ",
+        "bbox": [
+            173,
+            530,
+            825,
+            559
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B.3 IMPLEMENTATION DETAILS FOR DVBF IN PENDULUM EXPERIMENT ",
+        "bbox": [
+            174,
+            606,
+            689,
+            622
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• Input: 15 timesteps of $1 6 ^ { 2 }$ observation dimensions and 1 action dimension   \n• Latent Space: 3 dimensions   \n• Observation Network $p ( \\mathbf { x } _ { t } | \\mathbf { z } _ { t } ) = \\mathcal { N } ( \\mathbf { x } _ { t } ; \\mu ( \\mathbf { z } _ { t } ) , \\boldsymbol { \\sigma } )$ : $1 2 8 \\mathrm { R e L U + 1 6 ^ { 2 } }$ identity output   \n• Recognition Model: $1 2 8 { \\mathrm { R e L U } } + 6$ identity output ",
+        "bbox": [
+            217,
+            627,
+            782,
+            709
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bebe4f1e24da8d062d155f65cb381e65e1a0c05e84feab2ab2b37326e3f11efd.jpg",
+        "text": "$$\n\\begin{array} { r } { q ( \\mathbf { w } _ { t } | \\mathbf { z } _ { t } , \\mathbf { x } _ { t + 1 } , \\mathbf { u } _ { t } ) = \\mathcal { N } ( \\mathbf { w } _ { t } ; \\boldsymbol { \\mu } , \\boldsymbol { \\sigma } ) , } \\\\ { ( \\boldsymbol { \\mu } , \\boldsymbol { \\sigma } ) = f ( \\mathbf { z } _ { t } , \\mathbf { x } _ { t + 1 } , \\mathbf { u } _ { t } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            411,
+            718,
+            645,
+            755
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "• Transition Network ${ \\pmb { \\alpha } } _ { t } ( { \\bf z } _ { t } )$ : 16 softmax output   \n• Initial Network $\\mathbf { w } _ { 1 } \\sim p ( \\mathbf { x } _ { 1 : T } )$ : Fast Dropout BiRNN with: $1 2 8 { \\mathrm { R e L U } } + 3$ identity output   \n• Initial Transition ${ \\bf z } _ { 1 } ( { \\bf w } _ { 1 } )$ : $1 2 8 { \\mathrm { R e L U } } + 3$ identity output   \n• Optimizer: adadelta, 0.1 step rate   \n• Inverse temperature: $c _ { 0 } = 0 . 0 1$ , updated every 250th gradient update, $T _ { A } = 1 0 ^ { 5 }$ iterations   \n• Batch-size: 500 ",
+        "bbox": [
+            215,
+            797,
+            823,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B.4 IMPLEMENTATION DETAILS FOR DVBF IN BOUNCING BALL EXPERIMENT ",
+        "bbox": [
+            173,
+            102,
+            730,
+            118
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• Input: 15 timesteps of $1 6 ^ { 2 }$ observation dimensions and 2 action dimension   \n• Latent Space: 4 dimensions   \n• Observation Network $p ( \\mathbf { x } _ { t } | \\mathbf { z } _ { t } ) = \\mathcal { N } ( \\mathbf { x } _ { t } ; \\mu ( \\mathbf { z } _ { t } ) , \\boldsymbol { \\sigma } )$ : $1 2 8 \\mathrm { R e L U + 1 6 ^ { 2 } }$ identity output   \n• Recognition Model: $1 2 8 { \\mathrm { R e L U } } + 8$ identity output ",
+        "bbox": [
+            215,
+            123,
+            781,
+            196
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9c872ff87c4c00ec9a1b2a6e2cba9244349ee3b2cd9efc085d2f5681b82ab3e8.jpg",
+        "text": "$$\n\\begin{array} { r } { q ( \\mathbf { w } _ { t } | \\mathbf { z } _ { t } , \\mathbf { x } _ { t + 1 } , \\mathbf { u } _ { t } ) = \\mathcal { N } ( \\mathbf { w } _ { t } ; \\boldsymbol { \\mu } , \\boldsymbol { \\sigma } ) , } \\\\ { ( \\boldsymbol { \\mu } , \\boldsymbol { \\sigma } ) = f ( \\mathbf { z } _ { t } , \\mathbf { x } _ { t + 1 } , \\mathbf { u } _ { t } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            411,
+            202,
+            643,
+            239
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• Transition Network ${ \\pmb { \\alpha } } _ { t } ( { \\bf z } _ { t } )$ : 16 softmax output   \n• Initial Network $\\mathbf { w } _ { 1 } \\sim p ( \\mathbf { x } _ { 1 : T } )$ : Fast Dropout BiRNN with: $1 2 8 { \\mathrm { R e L U } } + 4$ identity output   \n• Initial Transition ${ \\bf z } _ { 1 } ( { \\bf w } _ { 1 } )$ : $1 2 8 { \\mathrm { R e L U } } + 4$ identity output   \nOptimizer: adadelta, 0.1 step rate   \n• Inverse temperature: $c _ { 0 } = 0 . 0 1$ , updated every 250th gradient update, $T _ { A } = 1 0 ^ { 5 }$ iterations   \n• Batch-size: 500 ",
+        "bbox": [
+            215,
+            276,
+            823,
+            387
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B.5 IMPLEMENTATION DETAILS FOR DVBF IN TWO BOUNCING BALLS EXPERIMENT ",
+        "bbox": [
+            168,
+            402,
+            776,
+            417
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• Input: 15 timesteps of $2 0 ^ { 2 }$ observation dimensions and 2000 samples   \n• Latent Space: 10 dimensions   \n• Observation Network $p ( \\mathbf { x } _ { t } | \\mathbf { z } _ { t } ) = \\mathcal { N } ( \\mathbf { x } _ { t } ; \\mu ( \\mathbf { z } _ { t } ) , \\boldsymbol { \\sigma } )$ : $1 2 8 \\mathrm { R e L U + 2 0 ^ { 2 } }$ sigmoid output   \n• Recognition Model: $1 2 8 \\mathrm { R e L U } + 2 0 $ identity output ",
+        "bbox": [
+            215,
+            422,
+            784,
+            497
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/94ce0144818c1fa2d19b992da17f1675b3bec77353118582bbf330cba97cd557.jpg",
+        "text": "$$\n\\begin{array} { r } { q ( \\mathbf { w } _ { t } | \\mathbf { z } _ { t } , \\mathbf { x } _ { t + 1 } , \\mathbf { u } _ { t } ) = \\mathcal { N } ( \\mathbf { w } _ { t } ; \\boldsymbol { \\mu } , \\boldsymbol { \\sigma } ) , } \\\\ { ( \\boldsymbol { \\mu } , \\boldsymbol { \\sigma } ) = f ( \\mathbf { z } _ { t } , \\mathbf { x } _ { t + 1 } , \\mathbf { u } _ { t } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            411,
+            502,
+            643,
+            539
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• Transition Network ${ \\pmb { \\alpha } } _ { t } ( { \\bf z } _ { t } )$ : 64 softmax output   \n• Initial Network $\\mathbf { w } _ { 1 } \\sim p ( \\mathbf { x } _ { 1 : T } )$ : MLP with: $1 2 8 \\mathrm { R e L U } + 1 0$ identity output   \n• Initial Transition ${ \\bf z } _ { 1 } ( { \\bf w } _ { 1 } )$ : $1 2 8 \\mathrm { R e L U } + 1 0$ identity output   \n• Optimizer: adam, 0.001 step rate   \n• Inverse temperature: $c _ { 0 } = 0 . 0 1$ , updated every gradient update, $T _ { A } = 2 ~ 1 0 ^ { 5 }$ iterations   \n• Batch-size: 80 ",
+        "bbox": [
+            215,
+            575,
+            795,
+            685
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B.6 IMPLEMENTATION DETAILS FOR DKF IN PENDULUM EXPERIMENT ",
+        "bbox": [
+            174,
+            702,
+            679,
+            717
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• Input: 15 timesteps of $1 6 ^ { 2 }$ observation dimensions and 1 action dimension   \n• Latent Space: 3 dimensions   \n• Observation Network $p ( \\mathbf { x } _ { t } | \\mathbf { z } _ { t } ) = { \\mathcal { N } } ( \\mathbf { x } _ { t } ; \\mu ( \\mathbf { z } _ { t } ) , \\sigma ( \\mathbf { z } _ { t } ) ) \\colon 1 2 8 { \\mathrm { ~ S i g m o i d } } + 1 2 8 { \\mathrm { ~ S i g m o i d } } + 2 1 6 ^ { 2 }$ identity output   \n• Recognition Model: Fast Dropout BiRNN 128 Sigmoid $+ ~ 1 2 8$ Sigmoid $^ { + 3 }$ identity output   \n• Transition Network $p ( \\mathbf { z } _ { t } | \\mathbf { z } _ { t - 1 } , \\mathbf { u } _ { t - 1 } )$ : 128 Sigmoid $+ ~ 1 2 8$ Sigmoid $+ ~ 6$ output   \n• Optimizer: adam, 0.001 step rate   \n• Inverse temperature: $c _ { 0 } = 0 . 0 1$ , updated every $2 5 \\mathrm { t h }$ gradient update, $T _ { A } = 2 0 0 0$ iterations   \n• Batch-size: 500 ",
+        "bbox": [
+            215,
+            723,
+            826,
+            885
+        ],
+        "page_idx": 12
+    }
+]

parse/train/SJ3dBGZ0Z/SJ3dBGZ0Z.md

@@ -0,0 +1,266 @@
+# LSH SOFTMAX: SUB-LINEAR LEARNING AND INFERENCE OF THE SOFTMAX LAYER IN DEEP ARCHITECTURES
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Log-linear models models are widely used in machine learning, and in particular are ubiquitous in deep learning architectures in the form of the softmax. While exact inference and learning of these requires linear time, it can be done approximately in sub-linear time with strong concentrations guarantees. In this work, we present LSH Softmax, a method to perform sub-linear learning and inference of the softmax layer in the deep learning setting. Our method relies on the popular Locality-Sensitive Hashing to build a well-concentrated gradient estimator, using nearest neighbors and uniform samples. We also present an inference scheme in sub-linear time for LSH Softmax using the Gumbel distribution. On language modeling, we show that Recurrent Neural Networks trained with LSH Softmax perform on-par with computing the exact softmax while requiring sub-linear computations.
+
+# 1 INTRODUCTION
+
+Deep neural networks have achieved impressive successes in tasks spanning vision (He et al., 2016; Krizhevsky et al., 2012), language (Bahdanau et al., 2014), speech (Graves et al., 2013; Oord et al., 2016) and videos (Abu-El-Haija et al., 2016). While these models can vastly differ in architecture, activation functions, and presence of recurrence, they (almost) all share a common trait: the softmax layer. The softmax layer, or log-linear model, is a widely used model in machine learning and statistics that transforms a feature vector into a distribution over the output space, modeling log-probabilities as a linear function of the feature vector. For example, in object classification, the softmax layer at the end of a deep convolutional network transforms a feature vector into a probability distribution over classes for the image; in language modeling using recurrent neural networks, it maps the hidden state to a distribution over next words.
+
+While parameterizing for logits offers modeling flexibility, inference and learning have linear runtime in the number of classes. Indeed, both of these require computing the un-normalized probability for every class to compute the partition function and retrieve an actual probability distribution. Problems with large output spaces arise naturally in many areas like natural language processing (NLP), where the output space is a language’s vocabulary and can be on the order of hundreds of thousands of elements Jozefowicz et al. (2016); Jean et al. (2014). This can also occur in computer vision (Joulin et al., 2016) when attempting tag prediction on massive, weakly-labeled datasets such as Flickr100M (Thomee et al., 2015).
+
+Many solutions have been proposed to address this bottleneck, all revolving around two themes: approximation of the softmax probabilities or computation of exact probabilities for an approximate model. Canonical examples of the former are importance sampling (IS) or noise contrastive estimation (NCE; Gutmann & Hyvarinen (2012)). Instead of computing probabilities over the whole ¨ output space, these methods compute the softmax over a smaller, sampled vocabulary and re-weight the probabilities, providing an unbiased estimator. An illustration of the latter is Hierarchical Softmax (Morin & Bengio, 2005), where the output classes are first clustered such that you only need to compute the softmax over a smaller output space. While the former is an unbiased estimate, it comes with no concentration guarantees, and it is often more art than science to craft proposal distributions which will provide low-variance estimators. The latter, while efficient, requires carefully hand-crafted clustering of the output space, at the risk of making mistakes from which there is no recovery.
+
+More recently, estimators based on nearest neighbor search have been proposed for inference and learning in log-linear models (Mussmann & Ermon, 2016; Mussmann et al., 2017). These estimators hinge on Maximum Inner Product Search using Locality-Sensitive to retrieve the largest logits of the distribution and account for the tail with uniformly sampled classes. They boast strong theoretical guarantees and well-established concentration bounds. However, they were constrained to toy settings and not directly applicable to real-world, large-scale, machine learning. In this work, we build upon these estimators to make them amenable to deep learning practitioners, without losing any theoretical guarantees. We first show how they can be extended to be usable within training of deep learning models, then present our efficient implementation, adapted to deep learning hardware and frameworks. Finally, we show the applicability and efficiency of our method by evaluating on a real-world task: language modeling. We show significant perplexity gains against competing methods with significant speed-ups.
+
+Our contributions are as follows:
+
+• We present a new deep learning layer, LSH Softmax, an efficient replacement for the softmax layer based on Locality-Sensitive Hashing and the Gumbel distribution, for any deep learning architecture, with strong theoretical guarantees for sub-linear learning and inference. • We provide details for efficient implementation on deep learning hardware (GPUs) and modern deep learning frameworks (Abadi et al., 2016; Maclaurin et al.)). • Empirically, we show, on several datasets, that training and sampling from LSH Softmax performs similarly to an exact softmax while requiring significantly less FLOPS.
+
+# 2 BACKGROUND
+
+In this section, we first provide a quick overview of Neural Networks and the most popular classification layer, the softmax layer. We then present the Gumbel distribution (Gumbel & Lieblein, 1954) and introduce Locality-Sensitive Hashing (Indyk & Motwani, 1998), both of which our estimator is built upon for inference and learning. Notationally, $\mathcal { X }$ is the input space, e.g. $\mathcal { X } \triangleq \mathbf { R } ^ { d }$ and $\mathcal { V }$ is a discrete output space: $\mathcal { Y } \triangleq \{ 1 , \dots , C \}$ .
+
+# 2.1 NEURAL NETWORKS
+
+Feedforward Networks Neural networks models are built hierarchically by applying linear and non-linear transformations in alternating fashion. Formally, given input $x \in \mathcal { X }$ , an $m$ -layer neural network with $\sigma ( \cdot )$ non-linearity transforms $x$ into $h$ defined as:
+
+$$
+\begin{array} { r } { \boldsymbol { h } = \sigma ( { W _ { m } } \cdot \sigma ( \dots \cdot \sigma ( { W _ { 1 } } \cdot { x } + { b _ { 1 } } ) + \dots ) + { b _ { m } } ) . } \end{array}
+$$
+
+$\{ W _ { i } \} _ { i \le m }$ and $\{ b _ { i } \} _ { i \le m }$ are learned weights of the network. $\sigma ( \cdot )$ denotes an element-wise nonlinearity such as ReLU $( \operatorname* { m a x } ( \cdot , 0 ) )$ or sigmoid $( ( 1 + \exp ( - \cdot ) ) ^ { - 1 } )$ ).
+
+Recurrent Networks Recurrent Neural Networks (RNN) are an extension of the previous setting to arbitrarily long sequences by keeping an internal state $h _ { t }$ . Formally, given an input sequence $( x _ { 1 } , \dots , x _ { T } )$ , it can be written as a dynamical system of the form:
+
+$$
+h _ { 0 } = \mathbf { 0 } ; h _ { t } = \sigma ( U h _ { t - 1 } + V x _ { t } ) .
+$$
+
+where $U$ and $V$ are learnable weight matrices. In practice, this parametrization is not wellconditioned for optimization as it can be subject to vanishing or exploding gradients and in practice the Longer Short Term Memory (LSTM; Hochreiter & Schmidhuber (1997)) is preferred.
+
+In both cases, these outputs are then given as input to a softmax layer which produces a distribution over the output space $\mathcal { V }$ . In the rest of this work, we denote by $\phi$ the parameters of the neural network.
+
+# 2.2 SOFTMAX
+
+The softmax layer is the common name given to a log-linear model for multi-classification at the end of a neural network. Let us consider the multi-classification setting with inputs in $\mathcal { X }$ and outputs in $\mathcal { V }$ . Given a feature vector $\psi ( x )$ and $C$ weight vectors $\{ \theta _ { c } \} _ { c \leq C }$ , the softmax layer parameterizes the following distribution:
+
+$$
+p ( Y = c | x ; \theta ) \propto \exp ( \psi ( x ) ^ { T } \theta _ { c } )
+$$
+
+In the particular case of neural networks, $p ( y | x ; \theta , \phi ) \propto \exp ( h ^ { T } \theta _ { i } )$ . $\{ h ^ { T } \theta _ { i } \} _ { i \leq C }$ are called the logits. It is important to note that computing the distribution over the output space, for inference or learning, requires $O ( C )$ operations. For the rest of this work, $\theta$ denotes the parameters of the softmax whereas $\phi$ denotes the parameters of the neural network (producing the feature).
+
+# 2.3 GUMBEL DISTRIBUTION
+
+First introduced by Gumbel & Lieblein (1954), the Gumbel distribution is defined by the following cumulative distribution function: $p ( G < s ) = \exp ( - \exp ( - s ) )$ . More practically, one can sample from the Gumbel distribution by first sampling $U \sim \mathcal { U } [ 0 , 1 ]$ and returning $G = \dot { - } \log ( - \log ( \bar { U } ) )$ . This distribution is particularly useful as it casts sampling as optimization.
+
+Theorem 1 (Maddison et al. (2014)). Let $\{ y _ { i } \} _ { i \le C }$ be un-normalized probabilities (or logits) over $\mathcal { V }$ and let $\{ G _ { i } \} _ { i \leq C }$ be i.i.d Gumbel variables. Then:
+
+$$
+\arg \operatorname* { m a x } _ { i \leq C } \{ y _ { i } + G _ { i } \} \sim \mathrm { C a t e g o r i c a l } \left\{ \frac { 1 } { Z } e ^ { y _ { i } } \right\} _ { i \leq C }
+$$
+
+# 2.4 MIPS AND LOCALITY-SENSITIVE HASHING
+
+Nearest neighbor search is a task that arises in many fields, such as information retrieval. Given a fixed set of vectors $s$ and a distance, this task consists of, given any incoming query $q$ , returning the vectors closest to the query according to the specified distance. In this work, we will be interested in the Maximum Inner Product Search (MIPS) task. Let $\mathcal { S } = \{ s _ { 1 } , \ldots , s _ { N } \}$ be a subset of $\mathbf { R } ^ { d }$ . Given a query $q \in \mathbf { R } ^ { d }$ , MIPS aims at retrieving arg $\operatorname* { m a x } _ { s \in \mathcal { S } } q ^ { T } s$ .
+
+This requires $\Theta ( N )$ operations as one has to compute the dot-product of $q$ with all elements of $s$ . In the case where we assume that, for a given set $s$ , it is needed to retrieve the nearest neighbor for a large numbers of queries, we can achieve amortized sub-linear time.
+
+This problem is commonly addressed with space partitioning techniques, such as Locality-Sensitive Hashing (LSH; Indyk & Motwani (1998)). LSH leverages hashing to reduce the number of candidate vectors to evaluate, based on the idea that similar vectors will hash in the same bucket. We have the following result:
+
+Theorem 2 (Indyk & Motwani (1998)). Given a set $s$ of size $N$ , a similarity measure $d ( \cdot , \cdot )$ and $a$ family of hash functions $\mathcal { H }$ s.t. for $S > T$ and $p > q$ :
+
+$$
+\begin{array} { r } { \bullet \forall x , y \in S , d ( x , y ) \geq S \Rightarrow p \left[ h ( x ) = h ( y ) \right] \geq p . } \\ { \bullet \forall x , y \in S , d ( x , y ) \leq T \Rightarrow p \left[ h ( x ) = h ( y ) \right] \leq q . } \end{array}
+$$
+
+we can construct a data structure s.t. given an incoming query $q$ , a nearest neighbor can be retrieved, with high probability, in sub-linear time $O ( N ^ { \rho } \log N )$ with $\begin{array} { r } { \rho \triangleq \frac { \log p } { \log q } < 1 } \end{array}$ .
+
+Recent work builds on top of LSH to either reduce the number of tables (Lv et al., 2007), or utilize more expressive hash functions (Andoni et al., 2015). A common family of hash is the hyperplane hash, i.e. for $\boldsymbol { v } \sim \mathcal { N } ( 0 , I ) , \boldsymbol { h } _ { \boldsymbol { v } } ( \boldsymbol { x } ) = \mathrm { s i g n } \left( \boldsymbol { v } ^ { T } \boldsymbol { x } \right)$ , also called Signed Random Projections (Charikar, 2002). For the rest of this work, we denote $b$ the number of hashing bits (equivalently, the number of random vectors) per table, and $L$ the number of tables.
+
+# 3 LEARNING
+
+In this section, we show how we can apply Theorem 3.5 of (Mussmann et al., 2017) to enable sublinear learning of softmax parameters in the context of deep models, i.e. where both weights and inputs can change. This is crucial for real-world use.
+
+Deep learning models for both classification (Krizhevsky et al., 2012) and generation (Mikolov, 2012) are often trained with a maximum-likelihood objective. Formally, given a training pair $( x , y ) \in \mathcal { X } \times \mathcal { Y }$ , one aims at maximizing $\log p ( \boldsymbol { y } | \boldsymbol { x } ; \boldsymbol { \theta } , \boldsymbol { \phi } )$ , where $\theta \in \Theta$ and $\phi \in \Phi$ are respectively the parameters of the softmax and of the neural network. To optimize this model, the usual method is to use back-propagation (Rumelhart et al., 1988) to differentiate and then perform stochastic gradient descent (SGD; LeCun et al. (1998)) on $\theta$ and $\phi$ .
+
+Let’s denote by $f ( x ; \phi ) \triangleq h$ the feature vector given as input to the softmax. Given our notation, the objective is written as $\begin{array} { r } { \ell ( x , y , \boldsymbol { \theta } , \boldsymbol { \phi } ) \triangleq \log p ( y | x ; \boldsymbol { \theta } , \boldsymbol { \phi } ) = h ^ { T } \boldsymbol { \theta } _ { y } - \log \sum _ { i < C } \exp ( h ^ { T } \boldsymbol { \theta } _ { i } ) } \end{array}$ . For backpropagation, we need to compute the gradient of $\ell$ w.r.t to both $\theta$ and $h -$ the gradient w.r.t. $h$ is then passed down to compute the gradient w.r.t. $\phi$ .
+
+$$
+\begin{array} { r l } & { \nabla _ { h } \ell = \theta _ { y } - \mathbb { E } _ { i \sim p ( \cdot \vert x ; \theta , \phi ) } \left[ \theta _ { i } \right] } \\ & { \nabla _ { \theta _ { i } } \ell = \mathbf { 1 } _ { i = y } h - h \frac { \exp \left( h ^ { T } \theta _ { i } \right) } { Z _ { \theta } ( h ) } } \end{array}
+$$
+
+with $\begin{array} { r } { Z _ { \theta } ( h ) \ \triangleq \ \sum _ { i } \exp ( h ^ { T } \theta _ { i } ) } \end{array}$ . Computing these gradients clearly requires $O ( | \mathcal { V } | )$ operations. In practice, this constitutes a major bottleneck for large output spaces. Mussmann et al. (2017) shows how to compute expectation in in sub-linear time, with a well-concentrated estimator using an LSH structure. Intuitively, we can build a good estimate of the partition function by retrieving the largest logits (using LSH) and accounting for the tail with uniform samples. Applying this result, we can compute the expectations necessary to compute the softmax gradients in sub-linear time. This is described in Theorem 3.
+
+Theorem 3 (LSH Softmax for Learning). Let $h = f ( x ; \phi )$ be input to a softmax layer with parameters $\{ \theta _ { c } \} _ { c \leq C }$ and define $\ell ( x , y , \theta , \phi )$ as previously. Given $s$ , the $k$ -nearest neighbors of $h$ in $\{ \theta _ { c } \} _ { c \leq C }$ and $\tau$ , $l$ uniform samples from $\left\{ 1 , \dots , C \right\} - { \mathcal { S } }$ , let us define:
+
+$$
+\begin{array} { l } { \displaystyle \hat { Z } _ { \theta } ( h ) \triangleq \sum _ { i \in \mathcal { S } } \exp ( h ^ { T } \theta _ { i } ) + \frac { C - k } { l } \sum _ { i \in \mathcal { T } } \exp ( h ^ { T } \theta _ { i } ) } \\ { \displaystyle \quad \hat { g } _ { \theta _ { i } } \triangleq h \mathbf { 1 } _ { i = y } - \left( \mathbf { 1 } _ { i \in \mathcal { S } } + \frac { C - k } { l } \mathbf { 1 } _ { i \in \mathcal { T } } \right) h _ { t } \frac { \exp ( h ^ { T } \theta _ { i } ) } { \hat { Z } _ { \theta } ( h ) } } \\ { \displaystyle \quad \hat { g } _ { h } \triangleq \theta _ { y } - \frac { 1 } { \hat { Z } _ { \theta } ( h ) } \left[ \sum _ { i \in \mathcal { S } } \theta _ { i } \exp ( h ^ { T } \theta _ { i } ) + \frac { C - k } { l } \sum _ { i \in \mathcal { T } } \theta _ { i } \exp ( h ^ { T } \theta _ { i } ) \right] } \end{array}
+$$
+
+These estimators are well concentrated: i.e. for $\epsilon , \delta > 0 \quad$ , $\begin{array} { r } { i f k = l = O \left( n ^ { { \frac { 2 } { 3 } } } { \frac { 1 } { \epsilon } } { \sqrt { \frac { 1 } { \delta } } } \right) } \end{array}$ , then with probability greater than $1 - \delta$ :
+
+$$
+| Z _ { \theta } ( h ) - \hat { Z } _ { \theta } ( h ) | \leq \epsilon ; \forall i \leq C , | | \nabla _ { \theta _ { i } } \ell - \hat { g } _ { \theta _ { i } } | | \leq \epsilon ; | | \nabla _ { h } \ell - \hat { g } _ { h } | | \leq \epsilon
+$$
+
+While Theorem 3 provides computation of the gradients in sub-linear time, it is only usable in a setting where the weights $( \{ \theta _ { i } \} _ { i \leq C } )$ are not updated. Indeed, querying nearest neighbors in sublinear time assumes that an appropriate data structure (here LSH) was built in advance. However, when training deep models, we are required to update the weights at every training step. This necessitates online updating of the LSH structure. To maintain the sub-linear runtime, we perform these updates in a sparse manner. We describe in Algorithm 1 how this estimator can be used in a training loop, with weight updating and sparse LSH updates.
+
+Proposition 4. The softmax computations described in Algorithm 1 run in sub-linear time.
+
+# Algorithm 1 Fast Training of the Softmax layer
+
+Inputs: Dataset $\mathcal { D } = \{ ( x ^ { ( i ) } , y ^ { ( i ) } \} _ { i \leq N } \subset \mathcal { X } \times \mathcal { Y } , k , l , n _ { \mathrm { i t e r s } }$ number of training iterations.   
+Initialize $\theta$ and $\phi$   
+Initialize the MIPS structure with {θi}i≤|V|.   
+for $j \leq n _ { \mathrm { i t e r s } } { \bf d o }$ Sample an example $( x , y )$ from $\mathcal { D }$ . $\Delta \theta \gets 0$ Compute $h  f ( x ; \phi )$ Find $s$ , $k$ -nearest-neighbors of $\mathbf { h }$ using the MIPS. Define $\tau$ as $l$ indexes uniformly sampled from ${ \mathcal { V } } - { \mathcal { S } }$ . $\begin{array} { r l r } & { \hat { Z } _ { \theta } ( h ) \gets \sum _ { i \in S } \exp ( h ^ { T } \theta _ { i } ) + \frac { | \mathcal { V } | - k } { l } \sum _ { i \in \mathcal { T } } \exp ( h ^ { T } \theta _ { i } ) } & { \mathrm { ~ \mathbb { P } ~ P a r t ~ } } \\ & { \mathrm { O u t p u t ~ } \hat { \ell } = h ^ { T } \theta _ { y } - \log \hat { Z } _ { \theta } ( h ) } & \\ & { \Delta \theta _ { i } \gets \Delta \theta _ { i } + h \mathbf { 1 } _ { i = y } - \left( \mathbf { 1 } _ { i \in S } + \frac { | \mathcal { V } | - k } { l } \mathbf { 1 } _ { i \in \mathcal { T } } \right) h \frac { \exp ( h ^ { T } \theta _ { i } ) } { \hat { Z } _ { \theta } ( h ) } } \\ & { \hat { g } _ { h } \gets \theta _ { y } - \frac { 1 } { \hat { Z } _ { \theta _ { i } } ( h ) } \left[ \sum _ { i \in \mathcal { S } } \theta _ { i } \exp ( h ^ { T } \theta _ { i } ) + \frac { | \mathcal { V } | - k } { l } \sum _ { i \in \mathcal { T } } \theta _ { i } \exp ( h ^ { T } \theta _ { i } ) \right] } \end{array}$ ition function estimate Pass down $\hat { g } _ { \bf h }$ for back-propagation. Re-hash the updated vectors (at most $( k + l ) )$ into the right buckets.   
+end for
+
+Proof. The softmax computations can be split into three parts: retrieving nearest neighbors, computing the forward/backward passes, and rehashing updated vectors. With a sub-linear MIPS such as LSH, the first part is guaranteed to be sub-linear. For the second part, computing the partition function and the entire gradient estimator requires computing a finite number of sums over $O ( k + l ) = O ( n ^ { \frac { 2 } { 3 } } )$ terms, which is sub-linear. The third part consists of re-hashing updated vectors. Re-hashing a vector is a constant operation (consisting of $b \times L$ dot-products) and thus, given that only a sub-linear number of vectors are updated, re-hashing is sub-linear. □
+
+# 4 INFERENCE
+
+In the last section, we presented a method to speed-up training time based on an LSH data structure. In addition to these training time gains, LSH Softmax can be utilized for computational gains at inference time as well. While MAP inference can be easily derived from the MIPS structure, sampling from the conditional distribution is often required (e.g. to generate diverse sentences in language modeling or machine translation). These gains can be crucial for large-scale deployment. This is a direct application of (Mussmann et al., 2017) that once again leverages a MIPS structure and the Gumbel distribution. By lazily evaluating Gumbel noise, once can devise an inference scheme which allows to sample from log-linear models in sub-linear time.
+
+Theorem 5 (LSH Softmax for Inference). We reuse the same notations as the once in Theorem 3. We define $t \triangleq - \log ( - \log ( 1 - l / C ) )$ . Let $\{ G _ { i } \} _ { i \leq k }$ be $k$ samples from the Gumbel distribution. We then proceed to sample $m \sim$ Binomial $( C , l / \bar { C } )$ , and sample $\tau$ , m points from ${ \mathcal { V } } - { \mathcal { S } }$ with associated Gumbels $\{ G _ { i } ^ { \prime } \} _ { i \leq m }$ s.t. each $G _ { i } ^ { \prime }$ are larger than $t$ . Let us define:
+
+$$
+\begin{array} { r } { \hat { y } \stackrel { \triangle } { = } \mathrm { a r g } \operatorname* { m a x } \{ h ^ { T } \theta _ { i } + G _ { i } , i \in \mathcal { S } \} \bigcup \{ h ^ { T } \theta _ { i } + G _ { i } ^ { \prime } , i \in \mathcal { T } \} . } \end{array}
+$$
+
+Let $\epsilon , \delta > 0$ , we then have the two following results:
+
+1. For $k = l \geq \sqrt { \log \frac { 1 } { \delta } }$ , $\hat { y }$ is a sample from $p ( \boldsymbol { y } | \boldsymbol { x } ; \boldsymbol { \theta } , \phi )$ with probability greater than $1 - \delta$ .
+
+2. This inference scheme runs in sub-linear time.
+
+Proof. (Mussmann et al., 2017)
+
+We denote by $p ^ { \mathtt { G u m b e l } } ( \cdot | h ; \theta )$ the implicit distribution over $\mathcal { V }$ provided by this inference scheme. While we can sample from $p ^ { \mathtt { G u m b e 1 } }$ , we note that the likelihood is intractable. We also emphasize that this scheme can be utilized for any softmax model, regardless of the training method.
+
+# 5 EFFICIENT IMPLEMENTATION
+
+Recent successes of deep neural networks hinge on their efficient implementation on specialized hardware: Graphics Processor Units (GPU), which enables training of large models in reasonable time. Often, methods with theoretically faster runtime are dismissed by practitioners because of their incompatibility with the hardware, rendering them hard to implement efficiently and ultimately not widely used. In this section, we first detail how our method is indeed amenable to GPU implementation and can amount to wall-clock gains in practice, and explain why LSH Softmax is easy to implement in the context of modern deep learning frameworks who often provide a gradient computation API.
+
+GPU Implementation Standard LSH implementations consist of three steps:
+
+1. Hashing: Given a query $q \in \mathbf { R } ^ { d }$ , hash $q$ into $L$ tables i.e. computing $b \times L$ dot-product with (random) hyperplanes.   
+2. Look-up: Given $L$ signatures in $\{ 0 , 1 \} ^ { b }$ , retrieve candidates in each of the $L$ tables. Let us denote $C _ { q }$ the number of candidates retrieved.   
+3. Distances: Given those candidates $\{ x _ { 1 } , . . . , x _ { C _ { q } } \} \subset \mathbf { R } ^ { d }$ , compute the distances $\{ q ^ { T } x _ { i } \} _ { i \leq C _ { q } }$ and only return the closest one. It is also important to note that deep learning models are often trained using minibatch optimization;   
+let us describe how each of these steps can be computed efficiently and in the minibatch setting.
+
+The first step is amenable to the GPU setting; a batch of queries $\{ q _ { i } \} _ { i \leq m } \subset \mathbf { R } ^ { d }$ can be represented by $Q \in \mathbf { R } ^ { m \times d }$ . Given that the hyperplanes are similarly presented in matrix form i.e. $H \in { \mathbf { R } } ^ { d \times ( b \times L ) }$ , the hashing step is equivalent to $\mathrm { s i g n } \left( Q \cdot H \right) \in \{ \bar { 0 } , \bar { 1 } \} ^ { m \times ( b \times L ) }$ . This is the type of operations that GPUs excel at: matrix-multiply followed by element-wise function.
+
+The second step, while not as compatible with GPU, is still massively parallelizable using multithreading on CPU. Given the computed signatures, one can run parallelism at the query level (i.e. each thread retrieves candidates for a given query), rendering that step efficient. It also allows for more memory-efficient look-up such as (Lv et al., 2007).
+
+The last operation is, once again, very amenable to GPU. It simply consists of a gather (i.e. building a matrix with the appropriate indexes from the candidates) into a 3-d tensor. Indeed, after the previous step, the LSH structure returns $m$ lists of $s$ candidates, and the gather step returns the appropriate vectors from the vocabulary into a 3-d tensor of shape $\mathbf { R } ^ { m \times s \times d }$ . As the batched queries can be also seen as a 3-d tensor $\mathbf { R } ^ { m \times d \times 1 }$ , computing the exact distances then reduces to a batch matrix-multiply which is a very efficient operation on GPU.
+
+Software Implementation Another crucial point for practitioners is the ability to rely on frameworks automatically providing gradients, such as (Abadi et al., 2016; Maclaurin et al.), to implement deep learning models; this abstracts away the need to write down the exact gradients which can be both cumbersome and error-prone. An additional advantage of our estimator is that it can be effortlessly implemented in these frameworks. Indeed, given logits computed over the nearest-neighbors and the additional uniformly sampled indexes, one can compute the estimate of the partition function and thus an estimate of the loss. Computing the gradient estimators now reduces to differentiating this loss, which can be very simply done using the framework’s differentiation API.
+
+# 6 EXPERIMENTS
+
+After having presented our new layer LSH Softmax, we now proceed to show its applicability and efficiency in a real-world setting for deep learning practitioners, specifically towards language modeling. We first show that our method significantly outperforms approximate softmax baselines while performing within $2 0 \%$ of the performance of the exact softmax. We then provide a computational comparison. While we evaluate our method on NLP tasks, we want to emphasize that it is directly applicable to other domains, such as vision. However, public vision benchmark datasets with large output spaces require significantly more computational resources (e.g. 98 GPU nodes for 8 days for Flickr100M (Thomee et al., 2015)) which is outside the scope of this paper.
+
+# 6.1 LANGUAGE MODELING
+
+Language modeling is the task of, given a sequence of words $( w _ { 1 } , \dots , w _ { T } )$ in a vocabulary $\nu$ , estimating $\begin{array} { r } { p ( w _ { 1 } , \dots , w _ { T } ) = \prod _ { t \leq T } \overline { { p ( w _ { t } | w _ { < t } ) } } } \end{array}$ . Substantial work has been done to model these distributions using non-parametric $n$ -gram counts with additional smoothing techniques, but can fail to model long histories because of an exponential number of sequences. Recently, parametric models using RNNs have shown impressive success on these tasks (Mikolov, 2012). In this setting, large output spaces arise naturally, as the vocabulary size can range from $1 0 ^ { 4 }$ to $1 0 ^ { 6 }$ . We first describe our experimental protocol, and then report perplexity (ppl) of LSH Softmax against a set of baselines on this task for several datasets.
+
+Datasets We evaluate our method on three standard datasets for Language Modeling with varying number of characters and vocabulary size:
+
+• Penn TreeBank (PTB): We follow the pre-processing described by (Mikolov, 2012), which results in $9 2 9 k$ training tokens, $7 3 k$ validation and $8 2 k$ test tokens with a $1 0 k$ vocabulary size.   
+• Text8 is a dataset consisting of the first 100 millions characters of Wikipedia, and has a vocabulary size of $4 4 k$ . This dataset has been used recently in the context of language modeling (Xie et al., 2017). We use the $9 0 M$ first words for training and split the remaining between the validation and test set. Wikitext-2. First introduced in Merity et al. (2016), this is a selected corpus of Wikipedia articles. It has a vocabulary size of $3 3 k$ and contains $2 1 7 k$ tokens. As previously, we split between a training, validation and testing set.
+
+Baselines We evaluate the performance of models trained with (1) exact softmax i.e. computed over the entire output space, (2) Biased Importance Sampled softmax (BIS), as presented in (Jean et al., 2014), which consists of sub-sampling the vocabulary according to a proposal distribution based on unigram counts, and (3) Negative Sampling (NS), proposed in Mikolov et al. (2013), equivalent to (BIS) with a uniform distribution, (4) standard Importance Sampling (Jozefowicz et al., 2016) and (5) Noise-Contrastive Estimation (NCE; Gutmann & Hyvarinen (2012)). These baselines ¨ are what practitioners canonically use to circumvent the bottleneck of large output spaces.
+
+Implementation Details Our architecture is a 2-layer RNN with LSTM cells and 650 hidden units. Weights are initialized uniformly within $[ - 0 . 1 , 0 . 1 ]$ . Our models are trained using SGD using gradient clipping, with an initial learning rate of 20. This learning rate is annealed when the validation perplexity plateaus. Our models are trained for 40 epochs for PTB, 3 epochs for Text8, 25 epochs for Wikitext-2. With the notations of Theorem 3, for LSH Softmax, we choose $k = 1 0 \sqrt { | \nu | }$ and $l = \sqrt { | \nu | }$ . For the IS and NS baselines, we choose to sample $k + l$ classes from the output space for a fair comparison. We choose the number of bits per signature $b \triangleq \log _ { 2 } | \nu |$ and choose $L$ , number of tables, to have sufficient recall for the MIPS task.
+
+# 6.1.1 LEARNING
+
+We report perplexity for a fixed architecture but comparing different softmax evaluations; we present both learning curves and perplexity on each set. We report the perplexity of all trained models using the exact probabilities i.e. the full softmax. Perplexities are reported in Table 1 and learning curves in Figure 1. We see that LSH Softmax consistently outperforms the approximate baselines by a fair margin while performing a similar number of operations, showcasing the strength of this estimator. We also observe from the training curves that approximate methods’ performances tend to plateau, as IS and NS cannot target the proper classes to push down. In constrast, LSH Softmax does not.
+
+# 6.2 COMPUTATIONAL COMPARISON
+
+Having established that the proposed estimator performs very well on real-world tasks, we now proceed to evaluate the computation gains. It is important to note that for models with large output spaces, the softmax computation can amount to about $8 0 \%$ of the total computation (Joulin et al.,
+
+<table><tr><td>Method</td><td colspan="3">PTB</td><td colspan="3">Wikitext-2</td><td colspan="3">Text8</td></tr><tr><td></td><td>Train</td><td>Val</td><td>Test</td><td>Train</td><td>Val</td><td>Test</td><td>Train</td><td>Val</td><td>Test</td></tr><tr><td>Exact</td><td>29.67</td><td>83.52</td><td>79.80</td><td>38.05</td><td>101.88</td><td>95.06</td><td>164.68</td><td>151.92</td><td>189.67</td></tr><tr><td>BIS</td><td>48.76</td><td>133.26</td><td>135.51</td><td>65.57</td><td>214.9</td><td>205.65</td><td>1</td><td></td><td>1</td></tr><tr><td>NS</td><td>32.12</td><td>103.26</td><td>101.48</td><td>42.66</td><td>142.82</td><td>136.30</td><td>255.62</td><td>234.02</td><td>281.11</td></tr><tr><td>IS</td><td>1</td><td>1</td><td>114.33</td><td>1</td><td>1</td><td>128.38</td><td>1</td><td>1</td><td>205.94</td></tr><tr><td>NCE</td><td>1</td><td>1</td><td>115.30</td><td>1</td><td>1</td><td>122.04</td><td>1</td><td>1</td><td>386.87</td></tr><tr><td>Ours</td><td>25.68</td><td>97.45</td><td>92.91</td><td>63.60</td><td>124.51</td><td>115.11</td><td>206.20</td><td>178.86</td><td>224.42</td></tr></table>
+
+Table 1: LSH Softmax performs closest to the exact softmax and handily outperforms importance sampling based methods with no concentration guarantees.
+
+![](images/e4394f30c6264e9787a1aa710c32d67dcdcdbcdcab1d5313e67dc48ee8dce17f.jpg)  
+Figure 1: LSH Softmax converges faster than compared baselines on all three datasets. IS is not reported for Text8 as the results were order of magnitude worse than compared method.
+
+2016; Ji et al., 2015); we thus choose to only evaluate computational gains in the softmax layer. We evaluate our method in CPU, with a batch size of 1, to have an accurate estimation of the ratio of FLOPS. We report both speed-up and validation perplexity (ppl) relative difference with the exact softmax for LSH Softmax and NS. Note that NS requires the same number of operations as importance sampling (IS) but outperforms it in all tasks. Additionally, we show the speed-ups one can achieve on the One Billion Word dataset (Chelba et al., 2013), whose ppl was not evaluated due to computational constraints. We report the results in Table 2. We observe that, while faster, NS performs significantly worse than LSH Softmax. Furthermore, its performance deteriorates significantly when increasing the size of the output space, contrary to LSH Softmax which always performs in the same relative range.
+
+<table><tr><td>Method</td><td colspan="2">PTB</td><td colspan="2">Wikitext-2</td><td colspan="2">Text8</td><td>Billion Word</td></tr><tr><td></td><td>Speed-up</td><td>△ppl</td><td>Speed-up</td><td>△ppl</td><td>Speed-up</td><td>△ ppl</td><td>Speed-up</td></tr><tr><td>NS</td><td>2.8×</td><td>23.6%</td><td>3.7×</td><td>40.2%</td><td>3.1×</td><td>54.0%</td><td>5.7×</td></tr><tr><td>Ours</td><td>1.6×</td><td>16.7%</td><td>2.4×</td><td>22.2%</td><td>2.3×</td><td>17.8%</td><td>4.1×</td></tr></table>
+
+Table 2: LSH Softmax performs closest to the exact softmax and handily outperforms importance sampling based methods with no concentration guarantees.
+
+# 7 RELATED WORK
+
+In recent years, MIPS-based estimators for log-linear models have been explored in the literature. Vijayanarasimhan et al. (2014) propose retrieving the largest logits using LSH and estimating the Softmax using only those classes. Their method is encompassed in ours by simply setting $l$ to 0. However, we note that not accounting for the tail can lead to highly biased gradients. Indeed, Mussmann et al. (2017) show that, using only the top- $k$ largest values leads to significantly worse performance. In a similar direction, Spring & Shrivastava (2017b) propose using LSH at each layer and only retaining the largest activations which can be viewed as a form of adaptive dropout. This work differs with ours in two ways: first of all, their paper provides no theoretical guarantees and secondly, they focus on reducing memory footprint which is not the aim of our work. Finally, Spring & Shrivastava (2017a) proposed using the LSH structure as a proposal distribution to evaluate the Softmax. While unbiased and efficient, their method does not offer any concentration guarantees and the estimator can have arbitrarily bad variance.
+
+# 8 CONCLUSION
+
+In this work, we presented LSH Softmax, a softmax approximation layer for large output spaces with sub-linear learning and inference cost (in the number of states) and strong theoretical guarantees. We showcased both its applicability and efficiency by evaluating LSH on a common NLP task, language modeling. On several datasets for this task, we report perplexity closest to exact training among all baselines, as well as significant speed-ups. Our hope is that, for any architecture, this layer could be chosen in lieu of softmax, when the output space is sufficiently large to warrant the approximation. To that end, we plan to release source-code with the camera-ready version.
+
+# REFERENCES
+
+Mart´ın Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, et al. Tensorflow: Large-scale machine learning on heterogeneous distributed systems. arXiv preprint arXiv:1603.04467, 2016.   
+Sami Abu-El-Haija, Nisarg Kothari, Joonseok Lee, Paul Natsev, George Toderici, Balakrishnan Varadarajan, and Sudheendra Vijayanarasimhan. Youtube- $. 8 \mathrm { m }$ : A large-scale video classification benchmark. arXiv preprint arXiv:1609.08675, 2016.   
+Alexandr Andoni, Piotr Indyk, Thijs Laarhoven, Ilya Razenshteyn, and Ludwig Schmidt. Practical and optimal lsh for angular distance. In Advances in Neural Information Processing Systems, pp. 1225–1233, 2015.   
+Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. arXiv preprint arXiv:1409.0473, 2014.   
+Moses S Charikar. Similarity estimation techniques from rounding algorithms. In Proceedings of the thiryfourth annual ACM symposium on Theory of computing, pp. 380–388. ACM, 2002.   
+Ciprian Chelba, Tomas Mikolov, Mike Schuster, Qi Ge, Thorsten Brants, Phillipp Koehn, and Tony Robinson. One billion word benchmark for measuring progress in statistical language modeling. arXiv preprint arXiv:1312.3005, 2013.   
+Alex Graves, Abdel-rahman Mohamed, and Geoffrey Hinton. Speech recognition with deep recurrent neural networks. In Acoustics, speech and signal processing (icassp), 2013 ieee international conference on, pp. 6645–6649. IEEE, 2013.   
+Emil Julius Gumbel and Julius Lieblein. Statistical theory of extreme values and some practical applications: a series of lectures. 1954.   
+Michael U Gutmann and Aapo Hyvarinen. Noise-contrastive estimation of unnormalized statistical models, ¨ with applications to natural image statistics. Journal of Machine Learning Research, 13(Feb):307–361, 2012.   
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Computer Vision and Pattern Recognition (CVPR), 2016 IEEE Conference on, 2016.   
+Sepp Hochreiter and Jurgen Schmidhuber. Long short-term memory. ¨ Neural computation, 9(8):1735–1780, 1997.   
+Piotr Indyk and Rajeev Motwani. Approximate nearest neighbors: towards removing the curse of dimensionality. In Proceedings of the thirtieth annual ACM symposium on Theory of computing, pp. 604–613. ACM, 1998.   
+Sebastien Jean, Kyunghyun Cho, Roland Memisevic, and Yoshua Bengio. On using very large target vocabulary ´ for neural machine translation. arXiv preprint arXiv:1412.2007, 2014.
+
+Shihao Ji, SVN Vishwanathan, Nadathur Satish, Michael J Anderson, and Pradeep Dubey. Blackout: Speeding up recurrent neural network language models with very large vocabularies. arXiv preprint arXiv:1511.06909, 2015.
+
+Armand Joulin, Laurens van der Maaten, Allan Jabri, and Nicolas Vasilache. Learning visual features from large weakly supervised data. In European Conference on Computer Vision, pp. 67–84. Springer, 2016.
+
+Rafal Jozefowicz, Oriol Vinyals, Mike Schuster, Noam Shazeer, and Yonghui Wu. Exploring the limits of language modeling. arXiv preprint arXiv:1602.02410, 2016.
+
+Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In Advances in neural information processing systems, pp. 1097–1105, 2012.
+
+Yann LeCun, Leon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document ´ recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998.
+
+Qin Lv, William Josephson, Zhe Wang, Moses Charikar, and Kai Li. Multi-probe lsh: efficient indexing for high-dimensional similarity search. In Proceedings of the 33rd international conference on Very large data bases, pp. 950–961. VLDB Endowment, 2007.
+
+Dougal Maclaurin, David Duvenaud, Matthew Johnson, and Ryan P Adams. Autograd: Reverse-mode differentiation of native python, 2015. URL http://github. com/HIPS/autograd.
+
+Chris J Maddison, Daniel Tarlow, and Tom Minka. A\* sampling. In Advances in Neural Information Processing Systems, pp. 3086–3094, 2014.
+
+Stephen Merity, Caiming Xiong, James Bradbury, and Richard Socher. Pointer sentinel mixture models. arXiv preprint arXiv:1609.07843, 2016.
+
+Toma´s Mikolov. Statistical language models based on neural networks. ˇ Presentation at Google, Mountain View, 2nd April, 2012.
+
+Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg S Corrado, and Jeff Dean. Distributed representations of words and phrases and their compositionality. In Advances in neural information processing systems, pp. 3111–3119, 2013.
+
+Frederic Morin and Yoshua Bengio. Hierarchical probabilistic neural network language model. In Aistats, volume 5, pp. 246–252, 2005.
+
+Stephen Mussmann and Stefano Ermon. Learning and inference via maximum inner product search. In Proceedings of The 33rd International Conference on Machine Learning, pp. 2587–2596, 2016.
+
+Stephen Mussmann, Daniel Levy, and Stefano Ermon. Fast amortized inference and learning in log-linear models with randomly perturbed nearest neighbor search. arXiv preprint arXiv:1707.03372, 2017.
+
+Aaron van den Oord, Sander Dieleman, Heiga Zen, Karen Simonyan, Oriol Vinyals, Alex Graves, Nal Kalchbrenner, Andrew Senior, and Koray Kavukcuoglu. Wavenet: A generative model for raw audio. arXiv preprint arXiv:1609.03499, 2016.
+
+David E Rumelhart, Geoffrey E Hinton, Ronald J Williams, et al. Learning representations by back-propagating errors. Cognitive modeling, 5(3):1, 1988.
+
+Ryan Spring and Anshumali Shrivastava. A new unbiased and efficient class of lsh-based samplers and estimators for partition function computation in log-linear models. arXiv preprint arXiv:1703.05160, 2017a.
+
+Ryan Spring and Anshumali Shrivastava. Scalable and sustainable deep learning via randomized hashing. In Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 445–454. ACM, 2017b.
+
+Bart Thomee, David A Shamma, Gerald Friedland, Benjamin Elizalde, Karl Ni, Douglas Poland, Damian Borth, and Li-Jia Li. The new data and new challenges in multimedia research. arXiv preprint arXiv:1503.01817, 1(8), 2015.
+
+Sudheendra Vijayanarasimhan, Jonathon Shlens, Rajat Monga, and Jay Yagnik. Deep networks with large output spaces. arXiv preprint arXiv:1412.7479, 2014.
+
+Ziang Xie, Sida I Wang, Jiwei Li, Daniel Levy, Aiming Nie, Dan Jurafsky, and Andrew Y Ng. Data noising as ´ smoothing in neural network language models. arXiv preprint arXiv:1703.02573, 2017.

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+        "text": "Log-linear models models are widely used in machine learning, and in particular are ubiquitous in deep learning architectures in the form of the softmax. While exact inference and learning of these requires linear time, it can be done approximately in sub-linear time with strong concentrations guarantees. In this work, we present LSH Softmax, a method to perform sub-linear learning and inference of the softmax layer in the deep learning setting. Our method relies on the popular Locality-Sensitive Hashing to build a well-concentrated gradient estimator, using nearest neighbors and uniform samples. We also present an inference scheme in sub-linear time for LSH Softmax using the Gumbel distribution. On language modeling, we show that Recurrent Neural Networks trained with LSH Softmax perform on-par with computing the exact softmax while requiring sub-linear computations. ",
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+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "text": "Deep neural networks have achieved impressive successes in tasks spanning vision (He et al., 2016; Krizhevsky et al., 2012), language (Bahdanau et al., 2014), speech (Graves et al., 2013; Oord et al., 2016) and videos (Abu-El-Haija et al., 2016). While these models can vastly differ in architecture, activation functions, and presence of recurrence, they (almost) all share a common trait: the softmax layer. The softmax layer, or log-linear model, is a widely used model in machine learning and statistics that transforms a feature vector into a distribution over the output space, modeling log-probabilities as a linear function of the feature vector. For example, in object classification, the softmax layer at the end of a deep convolutional network transforms a feature vector into a probability distribution over classes for the image; in language modeling using recurrent neural networks, it maps the hidden state to a distribution over next words. ",
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+        "type": "text",
+        "text": "While parameterizing for logits offers modeling flexibility, inference and learning have linear runtime in the number of classes. Indeed, both of these require computing the un-normalized probability for every class to compute the partition function and retrieve an actual probability distribution. Problems with large output spaces arise naturally in many areas like natural language processing (NLP), where the output space is a language’s vocabulary and can be on the order of hundreds of thousands of elements Jozefowicz et al. (2016); Jean et al. (2014). This can also occur in computer vision (Joulin et al., 2016) when attempting tag prediction on massive, weakly-labeled datasets such as Flickr100M (Thomee et al., 2015). ",
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+        "type": "text",
+        "text": "Many solutions have been proposed to address this bottleneck, all revolving around two themes: approximation of the softmax probabilities or computation of exact probabilities for an approximate model. Canonical examples of the former are importance sampling (IS) or noise contrastive estimation (NCE; Gutmann & Hyvarinen (2012)). Instead of computing probabilities over the whole ¨ output space, these methods compute the softmax over a smaller, sampled vocabulary and re-weight the probabilities, providing an unbiased estimator. An illustration of the latter is Hierarchical Softmax (Morin & Bengio, 2005), where the output classes are first clustered such that you only need to compute the softmax over a smaller output space. While the former is an unbiased estimate, it comes with no concentration guarantees, and it is often more art than science to craft proposal distributions which will provide low-variance estimators. The latter, while efficient, requires carefully hand-crafted clustering of the output space, at the risk of making mistakes from which there is no recovery. ",
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "More recently, estimators based on nearest neighbor search have been proposed for inference and learning in log-linear models (Mussmann & Ermon, 2016; Mussmann et al., 2017). These estimators hinge on Maximum Inner Product Search using Locality-Sensitive to retrieve the largest logits of the distribution and account for the tail with uniformly sampled classes. They boast strong theoretical guarantees and well-established concentration bounds. However, they were constrained to toy settings and not directly applicable to real-world, large-scale, machine learning. In this work, we build upon these estimators to make them amenable to deep learning practitioners, without losing any theoretical guarantees. We first show how they can be extended to be usable within training of deep learning models, then present our efficient implementation, adapted to deep learning hardware and frameworks. Finally, we show the applicability and efficiency of our method by evaluating on a real-world task: language modeling. We show significant perplexity gains against competing methods with significant speed-ups. ",
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+    {
+        "type": "text",
+        "text": "Our contributions are as follows: ",
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+        "text": "• We present a new deep learning layer, LSH Softmax, an efficient replacement for the softmax layer based on Locality-Sensitive Hashing and the Gumbel distribution, for any deep learning architecture, with strong theoretical guarantees for sub-linear learning and inference. • We provide details for efficient implementation on deep learning hardware (GPUs) and modern deep learning frameworks (Abadi et al., 2016; Maclaurin et al.)). • Empirically, we show, on several datasets, that training and sampling from LSH Softmax performs similarly to an exact softmax while requiring significantly less FLOPS. ",
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+    {
+        "type": "text",
+        "text": "2 BACKGROUND ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this section, we first provide a quick overview of Neural Networks and the most popular classification layer, the softmax layer. We then present the Gumbel distribution (Gumbel & Lieblein, 1954) and introduce Locality-Sensitive Hashing (Indyk & Motwani, 1998), both of which our estimator is built upon for inference and learning. Notationally, $\\mathcal { X }$ is the input space, e.g. $\\mathcal { X } \\triangleq \\mathbf { R } ^ { d }$ and $\\mathcal { V }$ is a discrete output space: $\\mathcal { Y } \\triangleq \\{ 1 , \\dots , C \\}$ . ",
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+    {
+        "type": "text",
+        "text": "2.1 NEURAL NETWORKS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Feedforward Networks Neural networks models are built hierarchically by applying linear and non-linear transformations in alternating fashion. Formally, given input $x \\in \\mathcal { X }$ , an $m$ -layer neural network with $\\sigma ( \\cdot )$ non-linearity transforms $x$ into $h$ defined as: ",
+        "bbox": [
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+    {
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+        "img_path": "images/b1857c5fabbe9a98ed0e5de081a820cc3225ec929fbe867c99e85f5df5ad2153.jpg",
+        "text": "$$\n\\begin{array} { r } { \\boldsymbol { h } = \\sigma ( { W _ { m } } \\cdot \\sigma ( \\dots \\cdot \\sigma ( { W _ { 1 } } \\cdot { x } + { b _ { 1 } } ) + \\dots ) + { b _ { m } } ) . } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "$\\{ W _ { i } \\} _ { i \\le m }$ and $\\{ b _ { i } \\} _ { i \\le m }$ are learned weights of the network. $\\sigma ( \\cdot )$ denotes an element-wise nonlinearity such as ReLU $( \\operatorname* { m a x } ( \\cdot , 0 ) )$ or sigmoid $( ( 1 + \\exp ( - \\cdot ) ) ^ { - 1 } )$ ). ",
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+    {
+        "type": "text",
+        "text": "Recurrent Networks Recurrent Neural Networks (RNN) are an extension of the previous setting to arbitrarily long sequences by keeping an internal state $h _ { t }$ . Formally, given an input sequence $( x _ { 1 } , \\dots , x _ { T } )$ , it can be written as a dynamical system of the form: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/07c6ed687a5a4f6b864dc3fef9c17691c8c94b3e3d0122cff84fe2f723f9b0fd.jpg",
+        "text": "$$\nh _ { 0 } = \\mathbf { 0 } ; h _ { t } = \\sigma ( U h _ { t - 1 } + V x _ { t } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $U$ and $V$ are learnable weight matrices. In practice, this parametrization is not wellconditioned for optimization as it can be subject to vanishing or exploding gradients and in practice the Longer Short Term Memory (LSTM; Hochreiter & Schmidhuber (1997)) is preferred. ",
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+        "text": "In both cases, these outputs are then given as input to a softmax layer which produces a distribution over the output space $\\mathcal { V }$ . In the rest of this work, we denote by $\\phi$ the parameters of the neural network. ",
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+    {
+        "type": "text",
+        "text": "2.2 SOFTMAX ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The softmax layer is the common name given to a log-linear model for multi-classification at the end of a neural network. Let us consider the multi-classification setting with inputs in $\\mathcal { X }$ and outputs in $\\mathcal { V }$ . Given a feature vector $\\psi ( x )$ and $C$ weight vectors $\\{ \\theta _ { c } \\} _ { c \\leq C }$ , the softmax layer parameterizes the following distribution: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/8a0c9e1983e88e810fd076b4b3a891a748370769b1bafafe8c2aab2cfefaed16.jpg",
+        "text": "$$\np ( Y = c | x ; \\theta ) \\propto \\exp ( \\psi ( x ) ^ { T } \\theta _ { c } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the particular case of neural networks, $p ( y | x ; \\theta , \\phi ) \\propto \\exp ( h ^ { T } \\theta _ { i } )$ . $\\{ h ^ { T } \\theta _ { i } \\} _ { i \\leq C }$ are called the logits. It is important to note that computing the distribution over the output space, for inference or learning, requires $O ( C )$ operations. For the rest of this work, $\\theta$ denotes the parameters of the softmax whereas $\\phi$ denotes the parameters of the neural network (producing the feature). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2.3 GUMBEL DISTRIBUTION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "First introduced by Gumbel & Lieblein (1954), the Gumbel distribution is defined by the following cumulative distribution function: $p ( G < s ) = \\exp ( - \\exp ( - s ) )$ . More practically, one can sample from the Gumbel distribution by first sampling $U \\sim \\mathcal { U } [ 0 , 1 ]$ and returning $G = \\dot { - } \\log ( - \\log ( \\bar { U } ) )$ . This distribution is particularly useful as it casts sampling as optimization. ",
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+        "type": "text",
+        "text": "Theorem 1 (Maddison et al. (2014)). Let $\\{ y _ { i } \\} _ { i \\le C }$ be un-normalized probabilities (or logits) over $\\mathcal { V }$ and let $\\{ G _ { i } \\} _ { i \\leq C }$ be i.i.d Gumbel variables. Then: ",
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+        "img_path": "images/3608702dcb4f85e5b8094677b5cb25b3be2324d782d769551d55ffb93b05634f.jpg",
+        "text": "$$\n\\arg \\operatorname* { m a x } _ { i \\leq C } \\{ y _ { i } + G _ { i } \\} \\sim \\mathrm { C a t e g o r i c a l } \\left\\{ \\frac { 1 } { Z } e ^ { y _ { i } } \\right\\} _ { i \\leq C }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "2.4 MIPS AND LOCALITY-SENSITIVE HASHING ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Nearest neighbor search is a task that arises in many fields, such as information retrieval. Given a fixed set of vectors $s$ and a distance, this task consists of, given any incoming query $q$ , returning the vectors closest to the query according to the specified distance. In this work, we will be interested in the Maximum Inner Product Search (MIPS) task. Let $\\mathcal { S } = \\{ s _ { 1 } , \\ldots , s _ { N } \\}$ be a subset of $\\mathbf { R } ^ { d }$ . Given a query $q \\in \\mathbf { R } ^ { d }$ , MIPS aims at retrieving arg $\\operatorname* { m a x } _ { s \\in \\mathcal { S } } q ^ { T } s$ . ",
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+        "type": "text",
+        "text": "This requires $\\Theta ( N )$ operations as one has to compute the dot-product of $q$ with all elements of $s$ . In the case where we assume that, for a given set $s$ , it is needed to retrieve the nearest neighbor for a large numbers of queries, we can achieve amortized sub-linear time. ",
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+    {
+        "type": "text",
+        "text": "This problem is commonly addressed with space partitioning techniques, such as Locality-Sensitive Hashing (LSH; Indyk & Motwani (1998)). LSH leverages hashing to reduce the number of candidate vectors to evaluate, based on the idea that similar vectors will hash in the same bucket. We have the following result: ",
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+    {
+        "type": "text",
+        "text": "Theorem 2 (Indyk & Motwani (1998)). Given a set $s$ of size $N$ , a similarity measure $d ( \\cdot , \\cdot )$ and $a$ family of hash functions $\\mathcal { H }$ s.t. for $S > T$ and $p > q$ : ",
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+    {
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+        "img_path": "images/194ca0aaf119cecb4599379e9e9657f7895467917a64bd353ff82e86ea60ddfc.jpg",
+        "text": "$$\n\\begin{array} { r } { \\bullet \\forall x , y \\in S , d ( x , y ) \\geq S \\Rightarrow p \\left[ h ( x ) = h ( y ) \\right] \\geq p . } \\\\ { \\bullet \\forall x , y \\in S , d ( x , y ) \\leq T \\Rightarrow p \\left[ h ( x ) = h ( y ) \\right] \\leq q . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "we can construct a data structure s.t. given an incoming query $q$ , a nearest neighbor can be retrieved, with high probability, in sub-linear time $O ( N ^ { \\rho } \\log N )$ with $\\begin{array} { r } { \\rho \\triangleq \\frac { \\log p } { \\log q } < 1 } \\end{array}$ . ",
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+    },
+    {
+        "type": "text",
+        "text": "Recent work builds on top of LSH to either reduce the number of tables (Lv et al., 2007), or utilize more expressive hash functions (Andoni et al., 2015). A common family of hash is the hyperplane hash, i.e. for $\\boldsymbol { v } \\sim \\mathcal { N } ( 0 , I ) , \\boldsymbol { h } _ { \\boldsymbol { v } } ( \\boldsymbol { x } ) = \\mathrm { s i g n } \\left( \\boldsymbol { v } ^ { T } \\boldsymbol { x } \\right)$ , also called Signed Random Projections (Charikar, 2002). For the rest of this work, we denote $b$ the number of hashing bits (equivalently, the number of random vectors) per table, and $L$ the number of tables. ",
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+    {
+        "type": "text",
+        "text": "3 LEARNING ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we show how we can apply Theorem 3.5 of (Mussmann et al., 2017) to enable sublinear learning of softmax parameters in the context of deep models, i.e. where both weights and inputs can change. This is crucial for real-world use. ",
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+    {
+        "type": "text",
+        "text": "Deep learning models for both classification (Krizhevsky et al., 2012) and generation (Mikolov, 2012) are often trained with a maximum-likelihood objective. Formally, given a training pair $( x , y ) \\in \\mathcal { X } \\times \\mathcal { Y }$ , one aims at maximizing $\\log p ( \\boldsymbol { y } | \\boldsymbol { x } ; \\boldsymbol { \\theta } , \\boldsymbol { \\phi } )$ , where $\\theta \\in \\Theta$ and $\\phi \\in \\Phi$ are respectively the parameters of the softmax and of the neural network. To optimize this model, the usual method is to use back-propagation (Rumelhart et al., 1988) to differentiate and then perform stochastic gradient descent (SGD; LeCun et al. (1998)) on $\\theta$ and $\\phi$ . ",
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+    {
+        "type": "text",
+        "text": "Let’s denote by $f ( x ; \\phi ) \\triangleq h$ the feature vector given as input to the softmax. Given our notation, the objective is written as $\\begin{array} { r } { \\ell ( x , y , \\boldsymbol { \\theta } , \\boldsymbol { \\phi } ) \\triangleq \\log p ( y | x ; \\boldsymbol { \\theta } , \\boldsymbol { \\phi } ) = h ^ { T } \\boldsymbol { \\theta } _ { y } - \\log \\sum _ { i < C } \\exp ( h ^ { T } \\boldsymbol { \\theta } _ { i } ) } \\end{array}$ . For backpropagation, we need to compute the gradient of $\\ell$ w.r.t to both $\\theta$ and $h -$ the gradient w.r.t. $h$ is then passed down to compute the gradient w.r.t. $\\phi$ . ",
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+    {
+        "type": "equation",
+        "img_path": "images/6526206059f3f7fbc64a2693a0a38797482661c6a459c2e497a6c8ce17450f89.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\nabla _ { h } \\ell = \\theta _ { y } - \\mathbb { E } _ { i \\sim p ( \\cdot \\vert x ; \\theta , \\phi ) } \\left[ \\theta _ { i } \\right] } \\\\ & { \\nabla _ { \\theta _ { i } } \\ell = \\mathbf { 1 } _ { i = y } h - h \\frac { \\exp \\left( h ^ { T } \\theta _ { i } \\right) } { Z _ { \\theta } ( h ) } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "with $\\begin{array} { r } { Z _ { \\theta } ( h ) \\ \\triangleq \\ \\sum _ { i } \\exp ( h ^ { T } \\theta _ { i } ) } \\end{array}$ . Computing these gradients clearly requires $O ( | \\mathcal { V } | )$ operations. In practice, this constitutes a major bottleneck for large output spaces. Mussmann et al. (2017) shows how to compute expectation in in sub-linear time, with a well-concentrated estimator using an LSH structure. Intuitively, we can build a good estimate of the partition function by retrieving the largest logits (using LSH) and accounting for the tail with uniform samples. Applying this result, we can compute the expectations necessary to compute the softmax gradients in sub-linear time. This is described in Theorem 3. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Theorem 3 (LSH Softmax for Learning). Let $h = f ( x ; \\phi )$ be input to a softmax layer with parameters $\\{ \\theta _ { c } \\} _ { c \\leq C }$ and define $\\ell ( x , y , \\theta , \\phi )$ as previously. Given $s$ , the $k$ -nearest neighbors of $h$ in $\\{ \\theta _ { c } \\} _ { c \\leq C }$ and $\\tau$ , $l$ uniform samples from $\\left\\{ 1 , \\dots , C \\right\\} - { \\mathcal { S } }$ , let us define: ",
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+    {
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+        "img_path": "images/7e091c38b1d08161e2d9135fade35bc83f097904bd912d5bac6b36a60fb1ea88.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\hat { Z } _ { \\theta } ( h ) \\triangleq \\sum _ { i \\in \\mathcal { S } } \\exp ( h ^ { T } \\theta _ { i } ) + \\frac { C - k } { l } \\sum _ { i \\in \\mathcal { T } } \\exp ( h ^ { T } \\theta _ { i } ) } \\\\ { \\displaystyle \\quad \\hat { g } _ { \\theta _ { i } } \\triangleq h \\mathbf { 1 } _ { i = y } - \\left( \\mathbf { 1 } _ { i \\in \\mathcal { S } } + \\frac { C - k } { l } \\mathbf { 1 } _ { i \\in \\mathcal { T } } \\right) h _ { t } \\frac { \\exp ( h ^ { T } \\theta _ { i } ) } { \\hat { Z } _ { \\theta } ( h ) } } \\\\ { \\displaystyle \\quad \\hat { g } _ { h } \\triangleq \\theta _ { y } - \\frac { 1 } { \\hat { Z } _ { \\theta } ( h ) } \\left[ \\sum _ { i \\in \\mathcal { S } } \\theta _ { i } \\exp ( h ^ { T } \\theta _ { i } ) + \\frac { C - k } { l } \\sum _ { i \\in \\mathcal { T } } \\theta _ { i } \\exp ( h ^ { T } \\theta _ { i } ) \\right] } \\end{array}\n$$",
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+    {
+        "type": "text",
+        "text": "These estimators are well concentrated: i.e. for $\\epsilon , \\delta > 0 \\quad$ , $\\begin{array} { r } { i f k = l = O \\left( n ^ { { \\frac { 2 } { 3 } } } { \\frac { 1 } { \\epsilon } } { \\sqrt { \\frac { 1 } { \\delta } } } \\right) } \\end{array}$ , then with probability greater than $1 - \\delta$ : ",
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a047eeb2598619d2b4a4452168116398a54230122e1ea5049955e46189ded4d3.jpg",
+        "text": "$$\n| Z _ { \\theta } ( h ) - \\hat { Z } _ { \\theta } ( h ) | \\leq \\epsilon ; \\forall i \\leq C , | | \\nabla _ { \\theta _ { i } } \\ell - \\hat { g } _ { \\theta _ { i } } | | \\leq \\epsilon ; | | \\nabla _ { h } \\ell - \\hat { g } _ { h } | | \\leq \\epsilon\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "While Theorem 3 provides computation of the gradients in sub-linear time, it is only usable in a setting where the weights $( \\{ \\theta _ { i } \\} _ { i \\leq C } )$ are not updated. Indeed, querying nearest neighbors in sublinear time assumes that an appropriate data structure (here LSH) was built in advance. However, when training deep models, we are required to update the weights at every training step. This necessitates online updating of the LSH structure. To maintain the sub-linear runtime, we perform these updates in a sparse manner. We describe in Algorithm 1 how this estimator can be used in a training loop, with weight updating and sparse LSH updates. ",
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+    },
+    {
+        "type": "text",
+        "text": "Proposition 4. The softmax computations described in Algorithm 1 run in sub-linear time. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 Fast Training of the Softmax layer ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Inputs: Dataset $\\mathcal { D } = \\{ ( x ^ { ( i ) } , y ^ { ( i ) } \\} _ { i \\leq N } \\subset \\mathcal { X } \\times \\mathcal { Y } , k , l , n _ { \\mathrm { i t e r s } }$ number of training iterations.   \nInitialize $\\theta$ and $\\phi$   \nInitialize the MIPS structure with {θi}i≤|V|.   \nfor $j \\leq n _ { \\mathrm { i t e r s } } { \\bf d o }$ Sample an example $( x , y )$ from $\\mathcal { D }$ . $\\Delta \\theta \\gets 0$ Compute $h  f ( x ; \\phi )$ Find $s$ , $k$ -nearest-neighbors of $\\mathbf { h }$ using the MIPS. Define $\\tau$ as $l$ indexes uniformly sampled from ${ \\mathcal { V } } - { \\mathcal { S } }$ . $\\begin{array} { r l r } & { \\hat { Z } _ { \\theta } ( h ) \\gets \\sum _ { i \\in S } \\exp ( h ^ { T } \\theta _ { i } ) + \\frac { | \\mathcal { V } | - k } { l } \\sum _ { i \\in \\mathcal { T } } \\exp ( h ^ { T } \\theta _ { i } ) } & { \\mathrm { ~ \\mathbb { P } ~ P a r t ~ } } \\\\ & { \\mathrm { O u t p u t ~ } \\hat { \\ell } = h ^ { T } \\theta _ { y } - \\log \\hat { Z } _ { \\theta } ( h ) } & \\\\ & { \\Delta \\theta _ { i } \\gets \\Delta \\theta _ { i } + h \\mathbf { 1 } _ { i = y } - \\left( \\mathbf { 1 } _ { i \\in S } + \\frac { | \\mathcal { V } | - k } { l } \\mathbf { 1 } _ { i \\in \\mathcal { T } } \\right) h \\frac { \\exp ( h ^ { T } \\theta _ { i } ) } { \\hat { Z } _ { \\theta } ( h ) } } \\\\ & { \\hat { g } _ { h } \\gets \\theta _ { y } - \\frac { 1 } { \\hat { Z } _ { \\theta _ { i } } ( h ) } \\left[ \\sum _ { i \\in \\mathcal { S } } \\theta _ { i } \\exp ( h ^ { T } \\theta _ { i } ) + \\frac { | \\mathcal { V } | - k } { l } \\sum _ { i \\in \\mathcal { T } } \\theta _ { i } \\exp ( h ^ { T } \\theta _ { i } ) \\right] } \\end{array}$ ition function estimate Pass down $\\hat { g } _ { \\bf h }$ for back-propagation. Re-hash the updated vectors (at most $( k + l ) )$ into the right buckets.   \nend for ",
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+    {
+        "type": "text",
+        "text": "Proof. The softmax computations can be split into three parts: retrieving nearest neighbors, computing the forward/backward passes, and rehashing updated vectors. With a sub-linear MIPS such as LSH, the first part is guaranteed to be sub-linear. For the second part, computing the partition function and the entire gradient estimator requires computing a finite number of sums over $O ( k + l ) = O ( n ^ { \\frac { 2 } { 3 } } )$ terms, which is sub-linear. The third part consists of re-hashing updated vectors. Re-hashing a vector is a constant operation (consisting of $b \\times L$ dot-products) and thus, given that only a sub-linear number of vectors are updated, re-hashing is sub-linear. □ ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4 INFERENCE ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In the last section, we presented a method to speed-up training time based on an LSH data structure. In addition to these training time gains, LSH Softmax can be utilized for computational gains at inference time as well. While MAP inference can be easily derived from the MIPS structure, sampling from the conditional distribution is often required (e.g. to generate diverse sentences in language modeling or machine translation). These gains can be crucial for large-scale deployment. This is a direct application of (Mussmann et al., 2017) that once again leverages a MIPS structure and the Gumbel distribution. By lazily evaluating Gumbel noise, once can devise an inference scheme which allows to sample from log-linear models in sub-linear time. ",
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+        "type": "text",
+        "text": "Theorem 5 (LSH Softmax for Inference). We reuse the same notations as the once in Theorem 3. We define $t \\triangleq - \\log ( - \\log ( 1 - l / C ) )$ . Let $\\{ G _ { i } \\} _ { i \\leq k }$ be $k$ samples from the Gumbel distribution. We then proceed to sample $m \\sim$ Binomial $( C , l / \\bar { C } )$ , and sample $\\tau$ , m points from ${ \\mathcal { V } } - { \\mathcal { S } }$ with associated Gumbels $\\{ G _ { i } ^ { \\prime } \\} _ { i \\leq m }$ s.t. each $G _ { i } ^ { \\prime }$ are larger than $t$ . Let us define: ",
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d47be3fc3886be50d33ac0202065cfe8239e8ca41cfecbf015ce77331ca92160.jpg",
+        "text": "$$\n\\begin{array} { r } { \\hat { y } \\stackrel { \\triangle } { = } \\mathrm { a r g } \\operatorname* { m a x } \\{ h ^ { T } \\theta _ { i } + G _ { i } , i \\in \\mathcal { S } \\} \\bigcup \\{ h ^ { T } \\theta _ { i } + G _ { i } ^ { \\prime } , i \\in \\mathcal { T } \\} . } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Let $\\epsilon , \\delta > 0$ , we then have the two following results: ",
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+        "type": "text",
+        "text": "1. For $k = l \\geq \\sqrt { \\log \\frac { 1 } { \\delta } }$ , $\\hat { y }$ is a sample from $p ( \\boldsymbol { y } | \\boldsymbol { x } ; \\boldsymbol { \\theta } , \\phi )$ with probability greater than $1 - \\delta$ . ",
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+    },
+    {
+        "type": "text",
+        "text": "2. This inference scheme runs in sub-linear time. ",
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Proof. (Mussmann et al., 2017) ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We denote by $p ^ { \\mathtt { G u m b e l } } ( \\cdot | h ; \\theta )$ the implicit distribution over $\\mathcal { V }$ provided by this inference scheme. While we can sample from $p ^ { \\mathtt { G u m b e 1 } }$ , we note that the likelihood is intractable. We also emphasize that this scheme can be utilized for any softmax model, regardless of the training method. ",
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+    {
+        "type": "text",
+        "text": "5 EFFICIENT IMPLEMENTATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Recent successes of deep neural networks hinge on their efficient implementation on specialized hardware: Graphics Processor Units (GPU), which enables training of large models in reasonable time. Often, methods with theoretically faster runtime are dismissed by practitioners because of their incompatibility with the hardware, rendering them hard to implement efficiently and ultimately not widely used. In this section, we first detail how our method is indeed amenable to GPU implementation and can amount to wall-clock gains in practice, and explain why LSH Softmax is easy to implement in the context of modern deep learning frameworks who often provide a gradient computation API. ",
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+    {
+        "type": "text",
+        "text": "GPU Implementation Standard LSH implementations consist of three steps: ",
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+    {
+        "type": "text",
+        "text": "1. Hashing: Given a query $q \\in \\mathbf { R } ^ { d }$ , hash $q$ into $L$ tables i.e. computing $b \\times L$ dot-product with (random) hyperplanes.   \n2. Look-up: Given $L$ signatures in $\\{ 0 , 1 \\} ^ { b }$ , retrieve candidates in each of the $L$ tables. Let us denote $C _ { q }$ the number of candidates retrieved.   \n3. Distances: Given those candidates $\\{ x _ { 1 } , . . . , x _ { C _ { q } } \\} \\subset \\mathbf { R } ^ { d }$ , compute the distances $\\{ q ^ { T } x _ { i } \\} _ { i \\leq C _ { q } }$ and only return the closest one. It is also important to note that deep learning models are often trained using minibatch optimization;   \nlet us describe how each of these steps can be computed efficiently and in the minibatch setting. ",
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+    {
+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "The first step is amenable to the GPU setting; a batch of queries $\\{ q _ { i } \\} _ { i \\leq m } \\subset \\mathbf { R } ^ { d }$ can be represented by $Q \\in \\mathbf { R } ^ { m \\times d }$ . Given that the hyperplanes are similarly presented in matrix form i.e. $H \\in { \\mathbf { R } } ^ { d \\times ( b \\times L ) }$ , the hashing step is equivalent to $\\mathrm { s i g n } \\left( Q \\cdot H \\right) \\in \\{ \\bar { 0 } , \\bar { 1 } \\} ^ { m \\times ( b \\times L ) }$ . This is the type of operations that GPUs excel at: matrix-multiply followed by element-wise function. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The second step, while not as compatible with GPU, is still massively parallelizable using multithreading on CPU. Given the computed signatures, one can run parallelism at the query level (i.e. each thread retrieves candidates for a given query), rendering that step efficient. It also allows for more memory-efficient look-up such as (Lv et al., 2007). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The last operation is, once again, very amenable to GPU. It simply consists of a gather (i.e. building a matrix with the appropriate indexes from the candidates) into a 3-d tensor. Indeed, after the previous step, the LSH structure returns $m$ lists of $s$ candidates, and the gather step returns the appropriate vectors from the vocabulary into a 3-d tensor of shape $\\mathbf { R } ^ { m \\times s \\times d }$ . As the batched queries can be also seen as a 3-d tensor $\\mathbf { R } ^ { m \\times d \\times 1 }$ , computing the exact distances then reduces to a batch matrix-multiply which is a very efficient operation on GPU. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Software Implementation Another crucial point for practitioners is the ability to rely on frameworks automatically providing gradients, such as (Abadi et al., 2016; Maclaurin et al.), to implement deep learning models; this abstracts away the need to write down the exact gradients which can be both cumbersome and error-prone. An additional advantage of our estimator is that it can be effortlessly implemented in these frameworks. Indeed, given logits computed over the nearest-neighbors and the additional uniformly sampled indexes, one can compute the estimate of the partition function and thus an estimate of the loss. Computing the gradient estimators now reduces to differentiating this loss, which can be very simply done using the framework’s differentiation API. ",
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+    {
+        "type": "text",
+        "text": "6 EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "After having presented our new layer LSH Softmax, we now proceed to show its applicability and efficiency in a real-world setting for deep learning practitioners, specifically towards language modeling. We first show that our method significantly outperforms approximate softmax baselines while performing within $2 0 \\%$ of the performance of the exact softmax. We then provide a computational comparison. While we evaluate our method on NLP tasks, we want to emphasize that it is directly applicable to other domains, such as vision. However, public vision benchmark datasets with large output spaces require significantly more computational resources (e.g. 98 GPU nodes for 8 days for Flickr100M (Thomee et al., 2015)) which is outside the scope of this paper. ",
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+    {
+        "type": "text",
+        "text": "6.1 LANGUAGE MODELING ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Language modeling is the task of, given a sequence of words $( w _ { 1 } , \\dots , w _ { T } )$ in a vocabulary $\\nu$ , estimating $\\begin{array} { r } { p ( w _ { 1 } , \\dots , w _ { T } ) = \\prod _ { t \\leq T } \\overline { { p ( w _ { t } | w _ { < t } ) } } } \\end{array}$ . Substantial work has been done to model these distributions using non-parametric $n$ -gram counts with additional smoothing techniques, but can fail to model long histories because of an exponential number of sequences. Recently, parametric models using RNNs have shown impressive success on these tasks (Mikolov, 2012). In this setting, large output spaces arise naturally, as the vocabulary size can range from $1 0 ^ { 4 }$ to $1 0 ^ { 6 }$ . We first describe our experimental protocol, and then report perplexity (ppl) of LSH Softmax against a set of baselines on this task for several datasets. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Datasets We evaluate our method on three standard datasets for Language Modeling with varying number of characters and vocabulary size: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "• Penn TreeBank (PTB): We follow the pre-processing described by (Mikolov, 2012), which results in $9 2 9 k$ training tokens, $7 3 k$ validation and $8 2 k$ test tokens with a $1 0 k$ vocabulary size.   \n• Text8 is a dataset consisting of the first 100 millions characters of Wikipedia, and has a vocabulary size of $4 4 k$ . This dataset has been used recently in the context of language modeling (Xie et al., 2017). We use the $9 0 M$ first words for training and split the remaining between the validation and test set. Wikitext-2. First introduced in Merity et al. (2016), this is a selected corpus of Wikipedia articles. It has a vocabulary size of $3 3 k$ and contains $2 1 7 k$ tokens. As previously, we split between a training, validation and testing set. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Baselines We evaluate the performance of models trained with (1) exact softmax i.e. computed over the entire output space, (2) Biased Importance Sampled softmax (BIS), as presented in (Jean et al., 2014), which consists of sub-sampling the vocabulary according to a proposal distribution based on unigram counts, and (3) Negative Sampling (NS), proposed in Mikolov et al. (2013), equivalent to (BIS) with a uniform distribution, (4) standard Importance Sampling (Jozefowicz et al., 2016) and (5) Noise-Contrastive Estimation (NCE; Gutmann & Hyvarinen (2012)). These baselines ¨ are what practitioners canonically use to circumvent the bottleneck of large output spaces. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Implementation Details Our architecture is a 2-layer RNN with LSTM cells and 650 hidden units. Weights are initialized uniformly within $[ - 0 . 1 , 0 . 1 ]$ . Our models are trained using SGD using gradient clipping, with an initial learning rate of 20. This learning rate is annealed when the validation perplexity plateaus. Our models are trained for 40 epochs for PTB, 3 epochs for Text8, 25 epochs for Wikitext-2. With the notations of Theorem 3, for LSH Softmax, we choose $k = 1 0 \\sqrt { | \\nu | }$ and $l = \\sqrt { | \\nu | }$ . For the IS and NS baselines, we choose to sample $k + l$ classes from the output space for a fair comparison. We choose the number of bits per signature $b \\triangleq \\log _ { 2 } | \\nu |$ and choose $L$ , number of tables, to have sufficient recall for the MIPS task. ",
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+    {
+        "type": "text",
+        "text": "6.1.1 LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We report perplexity for a fixed architecture but comparing different softmax evaluations; we present both learning curves and perplexity on each set. We report the perplexity of all trained models using the exact probabilities i.e. the full softmax. Perplexities are reported in Table 1 and learning curves in Figure 1. We see that LSH Softmax consistently outperforms the approximate baselines by a fair margin while performing a similar number of operations, showcasing the strength of this estimator. We also observe from the training curves that approximate methods’ performances tend to plateau, as IS and NS cannot target the proper classes to push down. In constrast, LSH Softmax does not. ",
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+    {
+        "type": "text",
+        "text": "6.2 COMPUTATIONAL COMPARISON ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Having established that the proposed estimator performs very well on real-world tasks, we now proceed to evaluate the computation gains. It is important to note that for models with large output spaces, the softmax computation can amount to about $8 0 \\%$ of the total computation (Joulin et al., ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/f8a83c34c2d21d636dd4ac12e13dd8d0c15f0fc505c211cd3af449f5686f14da.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td colspan=\"3\">PTB</td><td colspan=\"3\">Wikitext-2</td><td colspan=\"3\">Text8</td></tr><tr><td></td><td>Train</td><td>Val</td><td>Test</td><td>Train</td><td>Val</td><td>Test</td><td>Train</td><td>Val</td><td>Test</td></tr><tr><td>Exact</td><td>29.67</td><td>83.52</td><td>79.80</td><td>38.05</td><td>101.88</td><td>95.06</td><td>164.68</td><td>151.92</td><td>189.67</td></tr><tr><td>BIS</td><td>48.76</td><td>133.26</td><td>135.51</td><td>65.57</td><td>214.9</td><td>205.65</td><td>1</td><td></td><td>1</td></tr><tr><td>NS</td><td>32.12</td><td>103.26</td><td>101.48</td><td>42.66</td><td>142.82</td><td>136.30</td><td>255.62</td><td>234.02</td><td>281.11</td></tr><tr><td>IS</td><td>1</td><td>1</td><td>114.33</td><td>1</td><td>1</td><td>128.38</td><td>1</td><td>1</td><td>205.94</td></tr><tr><td>NCE</td><td>1</td><td>1</td><td>115.30</td><td>1</td><td>1</td><td>122.04</td><td>1</td><td>1</td><td>386.87</td></tr><tr><td>Ours</td><td>25.68</td><td>97.45</td><td>92.91</td><td>63.60</td><td>124.51</td><td>115.11</td><td>206.20</td><td>178.86</td><td>224.42</td></tr></table>",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 1: LSH Softmax performs closest to the exact softmax and handily outperforms importance sampling based methods with no concentration guarantees. ",
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+    {
+        "type": "image",
+        "img_path": "images/e4394f30c6264e9787a1aa710c32d67dcdcdbcdcab1d5313e67dc48ee8dce17f.jpg",
+        "image_caption": [
+            "Figure 1: LSH Softmax converges faster than compared baselines on all three datasets. IS is not reported for Text8 as the results were order of magnitude worse than compared method. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "2016; Ji et al., 2015); we thus choose to only evaluate computational gains in the softmax layer. We evaluate our method in CPU, with a batch size of 1, to have an accurate estimation of the ratio of FLOPS. We report both speed-up and validation perplexity (ppl) relative difference with the exact softmax for LSH Softmax and NS. Note that NS requires the same number of operations as importance sampling (IS) but outperforms it in all tasks. Additionally, we show the speed-ups one can achieve on the One Billion Word dataset (Chelba et al., 2013), whose ppl was not evaluated due to computational constraints. We report the results in Table 2. We observe that, while faster, NS performs significantly worse than LSH Softmax. Furthermore, its performance deteriorates significantly when increasing the size of the output space, contrary to LSH Softmax which always performs in the same relative range. ",
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+        "type": "table",
+        "img_path": "images/9319b26b814a94ad8321c4b0daed6b511eb992e14dab427cd64213e22e549d33.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td colspan=\"2\">PTB</td><td colspan=\"2\">Wikitext-2</td><td colspan=\"2\">Text8</td><td>Billion Word</td></tr><tr><td></td><td>Speed-up</td><td>△ppl</td><td>Speed-up</td><td>△ppl</td><td>Speed-up</td><td>△ ppl</td><td>Speed-up</td></tr><tr><td>NS</td><td>2.8×</td><td>23.6%</td><td>3.7×</td><td>40.2%</td><td>3.1×</td><td>54.0%</td><td>5.7×</td></tr><tr><td>Ours</td><td>1.6×</td><td>16.7%</td><td>2.4×</td><td>22.2%</td><td>2.3×</td><td>17.8%</td><td>4.1×</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 2: LSH Softmax performs closest to the exact softmax and handily outperforms importance sampling based methods with no concentration guarantees. ",
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+    {
+        "type": "text",
+        "text": "7 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In recent years, MIPS-based estimators for log-linear models have been explored in the literature. Vijayanarasimhan et al. (2014) propose retrieving the largest logits using LSH and estimating the Softmax using only those classes. Their method is encompassed in ours by simply setting $l$ to 0. However, we note that not accounting for the tail can lead to highly biased gradients. Indeed, Mussmann et al. (2017) show that, using only the top- $k$ largest values leads to significantly worse performance. In a similar direction, Spring & Shrivastava (2017b) propose using LSH at each layer and only retaining the largest activations which can be viewed as a form of adaptive dropout. This work differs with ours in two ways: first of all, their paper provides no theoretical guarantees and secondly, they focus on reducing memory footprint which is not the aim of our work. Finally, Spring & Shrivastava (2017a) proposed using the LSH structure as a proposal distribution to evaluate the Softmax. While unbiased and efficient, their method does not offer any concentration guarantees and the estimator can have arbitrarily bad variance. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "8 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this work, we presented LSH Softmax, a softmax approximation layer for large output spaces with sub-linear learning and inference cost (in the number of states) and strong theoretical guarantees. We showcased both its applicability and efficiency by evaluating LSH on a common NLP task, language modeling. On several datasets for this task, we report perplexity closest to exact training among all baselines, as well as significant speed-ups. Our hope is that, for any architecture, this layer could be chosen in lieu of softmax, when the output space is sufficiently large to warrant the approximation. To that end, we plan to release source-code with the camera-ready version. ",
+        "bbox": [
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+            253,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            372,
+            285,
+            386
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Mart´ın Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S Corrado, Andy Davis, Jeffrey Dean, Matthieu Devin, et al. Tensorflow: Large-scale machine learning on heterogeneous distributed systems. arXiv preprint arXiv:1603.04467, 2016.   \nSami Abu-El-Haija, Nisarg Kothari, Joonseok Lee, Paul Natsev, George Toderici, Balakrishnan Varadarajan, and Sudheendra Vijayanarasimhan. Youtube- $. 8 \\mathrm { m }$ : A large-scale video classification benchmark. arXiv preprint arXiv:1609.08675, 2016.   \nAlexandr Andoni, Piotr Indyk, Thijs Laarhoven, Ilya Razenshteyn, and Ludwig Schmidt. Practical and optimal lsh for angular distance. In Advances in Neural Information Processing Systems, pp. 1225–1233, 2015.   \nDzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. arXiv preprint arXiv:1409.0473, 2014.   \nMoses S Charikar. Similarity estimation techniques from rounding algorithms. In Proceedings of the thiryfourth annual ACM symposium on Theory of computing, pp. 380–388. ACM, 2002.   \nCiprian Chelba, Tomas Mikolov, Mike Schuster, Qi Ge, Thorsten Brants, Phillipp Koehn, and Tony Robinson. One billion word benchmark for measuring progress in statistical language modeling. arXiv preprint arXiv:1312.3005, 2013.   \nAlex Graves, Abdel-rahman Mohamed, and Geoffrey Hinton. Speech recognition with deep recurrent neural networks. In Acoustics, speech and signal processing (icassp), 2013 ieee international conference on, pp. 6645–6649. IEEE, 2013.   \nEmil Julius Gumbel and Julius Lieblein. Statistical theory of extreme values and some practical applications: a series of lectures. 1954.   \nMichael U Gutmann and Aapo Hyvarinen. Noise-contrastive estimation of unnormalized statistical models, ¨ with applications to natural image statistics. Journal of Machine Learning Research, 13(Feb):307–361, 2012.   \nKaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Computer Vision and Pattern Recognition (CVPR), 2016 IEEE Conference on, 2016.   \nSepp Hochreiter and Jurgen Schmidhuber. Long short-term memory. ¨ Neural computation, 9(8):1735–1780, 1997.   \nPiotr Indyk and Rajeev Motwani. Approximate nearest neighbors: towards removing the curse of dimensionality. In Proceedings of the thirtieth annual ACM symposium on Theory of computing, pp. 604–613. ACM, 1998.   \nSebastien Jean, Kyunghyun Cho, Roland Memisevic, and Yoshua Bengio. On using very large target vocabulary ´ for neural machine translation. arXiv preprint arXiv:1412.2007, 2014. ",
+        "bbox": [
+            171,
+            387,
+            826,
+            934
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Shihao Ji, SVN Vishwanathan, Nadathur Satish, Michael J Anderson, and Pradeep Dubey. Blackout: Speeding up recurrent neural network language models with very large vocabularies. arXiv preprint arXiv:1511.06909, 2015. ",
+        "bbox": [
+            174,
+            104,
+            823,
+            143
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Armand Joulin, Laurens van der Maaten, Allan Jabri, and Nicolas Vasilache. Learning visual features from large weakly supervised data. In European Conference on Computer Vision, pp. 67–84. Springer, 2016. ",
+        "bbox": [
+            173,
+            151,
+            823,
+            179
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Rafal Jozefowicz, Oriol Vinyals, Mike Schuster, Noam Shazeer, and Yonghui Wu. Exploring the limits of language modeling. arXiv preprint arXiv:1602.02410, 2016. ",
+        "bbox": [
+            174,
+            188,
+            821,
+            214
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolutional neural networks. In Advances in neural information processing systems, pp. 1097–1105, 2012. ",
+        "bbox": [
+            174,
+            222,
+            820,
+            250
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Yann LeCun, Leon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document ´ recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998. ",
+        "bbox": [
+            173,
+            257,
+            823,
+            285
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Qin Lv, William Josephson, Zhe Wang, Moses Charikar, and Kai Li. Multi-probe lsh: efficient indexing for high-dimensional similarity search. In Proceedings of the 33rd international conference on Very large data bases, pp. 950–961. VLDB Endowment, 2007. ",
+        "bbox": [
+            176,
+            292,
+            823,
+            332
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Dougal Maclaurin, David Duvenaud, Matthew Johnson, and Ryan P Adams. Autograd: Reverse-mode differentiation of native python, 2015. URL http://github. com/HIPS/autograd. ",
+        "bbox": [
+            171,
+            340,
+            821,
+            367
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Chris J Maddison, Daniel Tarlow, and Tom Minka. A\\* sampling. In Advances in Neural Information Processing Systems, pp. 3086–3094, 2014. ",
+        "bbox": [
+            176,
+            376,
+            821,
+            402
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Stephen Merity, Caiming Xiong, James Bradbury, and Richard Socher. Pointer sentinel mixture models. arXiv preprint arXiv:1609.07843, 2016. ",
+        "bbox": [
+            174,
+            410,
+            825,
+            438
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Toma´s Mikolov. Statistical language models based on neural networks. ˇ Presentation at Google, Mountain View, 2nd April, 2012. ",
+        "bbox": [
+            173,
+            445,
+            823,
+            473
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg S Corrado, and Jeff Dean. Distributed representations of words and phrases and their compositionality. In Advances in neural information processing systems, pp. 3111–3119, 2013. ",
+        "bbox": [
+            173,
+            481,
+            823,
+            521
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Frederic Morin and Yoshua Bengio. Hierarchical probabilistic neural network language model. In Aistats, volume 5, pp. 246–252, 2005. ",
+        "bbox": [
+            171,
+            529,
+            823,
+            556
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Stephen Mussmann and Stefano Ermon. Learning and inference via maximum inner product search. In Proceedings of The 33rd International Conference on Machine Learning, pp. 2587–2596, 2016. ",
+        "bbox": [
+            173,
+            564,
+            823,
+            592
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Stephen Mussmann, Daniel Levy, and Stefano Ermon. Fast amortized inference and learning in log-linear models with randomly perturbed nearest neighbor search. arXiv preprint arXiv:1707.03372, 2017. ",
+        "bbox": [
+            173,
+            599,
+            823,
+            627
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Aaron van den Oord, Sander Dieleman, Heiga Zen, Karen Simonyan, Oriol Vinyals, Alex Graves, Nal Kalchbrenner, Andrew Senior, and Koray Kavukcuoglu. Wavenet: A generative model for raw audio. arXiv preprint arXiv:1609.03499, 2016. ",
+        "bbox": [
+            173,
+            635,
+            821,
+            674
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "David E Rumelhart, Geoffrey E Hinton, Ronald J Williams, et al. Learning representations by back-propagating errors. Cognitive modeling, 5(3):1, 1988. ",
+        "bbox": [
+            171,
+            683,
+            823,
+            710
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Ryan Spring and Anshumali Shrivastava. A new unbiased and efficient class of lsh-based samplers and estimators for partition function computation in log-linear models. arXiv preprint arXiv:1703.05160, 2017a. ",
+        "bbox": [
+            171,
+            718,
+            823,
+            744
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Ryan Spring and Anshumali Shrivastava. Scalable and sustainable deep learning via randomized hashing. In Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 445–454. ACM, 2017b. ",
+        "bbox": [
+            174,
+            753,
+            821,
+            792
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Bart Thomee, David A Shamma, Gerald Friedland, Benjamin Elizalde, Karl Ni, Douglas Poland, Damian Borth, and Li-Jia Li. The new data and new challenges in multimedia research. arXiv preprint arXiv:1503.01817, 1(8), 2015. ",
+        "bbox": [
+            173,
+            800,
+            823,
+            840
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Sudheendra Vijayanarasimhan, Jonathon Shlens, Rajat Monga, and Jay Yagnik. Deep networks with large output spaces. arXiv preprint arXiv:1412.7479, 2014. ",
+        "bbox": [
+            171,
+            848,
+            821,
+            876
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Ziang Xie, Sida I Wang, Jiwei Li, Daniel Levy, Aiming Nie, Dan Jurafsky, and Andrew Y Ng. Data noising as ´ smoothing in neural network language models. arXiv preprint arXiv:1703.02573, 2017. ",
+        "bbox": [
+            173,
+            883,
+            820,
+            911
+        ],
+        "page_idx": 9
+    }
+]

parse/train/_RnHyIeu5Y5/_RnHyIeu5Y5_content_list.json

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+    {
+        "type": "text",
+        "text": "ViTAE: Vision Transformer Advanced by Exploring Intrinsic Inductive Bias ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Yufei Xu1∗ Qiming Zhang1∗ Jing Zhang1 Dacheng Tao2,1 ",
+        "bbox": [
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+        ],
+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "1The University of Sydney, Australia, 2JD Explore Academy, China ",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "{yuxu7116,qzha2506}@uni.sydney.edu.au, jing.zhang1@sydney.edu.au, dacheng.tao@gmail.com ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Transformers have shown great potential in various computer vision tasks owing to their strong capability in modeling long-range dependency using the self-attention mechanism. Nevertheless, vision transformers treat an image as 1D sequence of visual tokens, lacking an intrinsic inductive bias (IB) in modeling local visual structures and dealing with scale variance. Alternatively, they require large-scale training data and longer training schedules to learn the IB implicitly. In this paper, we propose a new Vision Transformer Advanced by Exploring intrinsic IB from convolutions, i.e., ViTAE. Technically, ViTAE has several spatial pyramid reduction modules to downsample and embed the input image into tokens with rich multi-scale context by using multiple convolutions with different dilation rates. In this way, it acquires an intrinsic scale invariance IB and is able to learn robust feature representation for objects at various scales. Moreover, in each transformer layer, ViTAE has a convolution block in parallel to the multi-head selfattention module, whose features are fused and fed into the feed-forward network. Consequently, it has the intrinsic locality IB and is able to learn local features and global dependencies collaboratively. Experiments on ImageNet as well as downstream tasks prove the superiority of ViTAE over the baseline transformer and concurrent works. Source code and pretrained models will be available at code. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Transformers [79, 17, 40, 14, 46, 61] have shown a domination trend in NLP studies owing to their strong ability in modeling long-range dependencies by the self-attention mechanism [67, 81, 51]. Such success and good properties of transformers has inspired following many works that apply them in various computer vision tasks [19, 100, 97, 80, 7]. Among them, ViT [19] is the pioneering pure transformer model that embeds images into a sequence of visual tokens and models the global dependencies among them with stacked transformer blocks. Although it achieves promising performance on image classification, it requires large-scale training data and a longer training schedule. One important reason is that ViT ",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "image",
+        "img_path": "images/4515266c49c6aff7e7d656459cb5d8b9e3662e222790882fee6a157e2ad56504.jpg",
+        "image_caption": [
+            "Figure 1: Comparison of data and training efficiency of T2T-ViT-7 and ViTAE-T on ImageNet. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "lacks intrinsic inductive bias (IB) in modeling local visual structures (e.g., edges and corners) and dealing with objects at various scales like convolutions. Alternatively, ViT has to learn such IB implicitly from large-scale data. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Unlike vision transformers, Convolution Neural Networks (CNNs) naturally equip with the intrinsic IBs of scale-invariance and locality and still serve as prevalent backbones in vision tasks [26, 70, 62, 8, 96]. The success of CNNs inspires us to explore intrinsic IBs in vision transformers. We start by analyzing the above two IBs of CNNs, i.e., locality and scale-invariance. Convolution that computes local correlation among neighbor pixels is good at extracting local features such as edges and corners. Consequently, CNNs can provide plentiful low-level features at the shallow layers [94], which are then aggregated into high-level features progressively by a bulk of sequential convolutions [32, 68, 71]. Moreover, CNNs have a hierarchy structure to extract multi-scale features at different layers [68, 38, 26]. Besides, intra-layer convolutions can also learn features at different scales by varying their kernel sizes and dilation rates [25, 70, 8, 45, 96]. Consequently, scale-invariant feature representation can be obtained via intra- or inter-layer feature fusion. Nevertheless, CNNs are not well suited to model long-range dependencies2, which is the key advantage of transformers. An interesting question comes up: Can we improve vision transformers by leveraging the good properties of CNNs? Recently, DeiT [76] explores the idea of distilling knowledge from CNNs to transformers to facilitate training and improve the performance. However, it requires an off-the-shelf CNN model as the teacher and consumes extra training cost. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Different from DeiT, we explicitly introduce intrinsic IBs into vision transformers by re-designing the network structures in this paper. Current vision transformers always obtain tokens with singlescale context [19, 93, 80, 86, 47, 69, 77] and learn to adapt to objects at different scales from data. For example, T2T-ViT [93] improves ViT by delicately generating tokens in a soft split manner. Specifically, it uses a series of Tokens-to-Token transformation layers to aggregate single-scale neighboring contextual information and progressively structurizes the image to tokens. Motivated by the success of CNNs in dealing with scale variance, we explore a similar design in transformers, i.e., intra-layer convolutions with different receptive fields [70, 91], to embed multi-scale context into tokens. Such a design allows tokens to carry useful features of objects at various scales, thereby naturally having the intrinsic scale-invariance IB and explicitly facilitating transformers to learn scale-invariant features more efficiently from data. On the other hand, low-level local features are fundamental elements to generate high-level discriminative features. Although transformers can also learn such features at shallow layers from data, they are not skilled as convolutions by design. Recently, [89, 43, 21] stack convolutions and attention layers sequentially and demonstrate that locality is a reasonable compensation of global dependency. However, this serial structure ignores the global context during locality modeling (and vice versa). To avoid such a dilemma, we follow the “divide-and-conquer” idea and propose to model locality and long-range dependencies in parallel and then fuse the features to account for both. In this way, we empower transformers to learn local and long-range features within each block more effectively. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Technically, we propose a new Vision Transformers Advanced by Exploring Intrinsic Inductive Bias $( V i T A E )$ , which is a combination of two types of basic cells, i.e., reduction cell (RC) and normal cell (NC). RCs are used to downsample and embed the input images into tokens with rich multi-scale context while NCs aim to jointly model locality and global dependencies in the token sequence. Moreover, these two types of cells share a simple basic structure, i.e., paralleled attention module and convolutional layers followed by a feed-forward network (FFN). It is noteworthy that RC has an extra pyramid reduction module with atrous convolutions of different dilation rates to embed multi-scale context into tokens. Following the setting in [93], we stack three reduction cells to reduce the spatial resolution by $1 / 1 6$ and a series of NCs to learn discriminative features from data. ViTAE outperforms representative vision transformers in terms of data efficiency and training efficiency (see Figure 1), as well as classification accuracy and generalization on downstream tasks. ",
+        "bbox": [
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+            636,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our contributions are threefold. First, we explore two types of intrinsic IB in transformers, i.e., scale invariance and locality, and demonstrate the effectiveness of this idea in improving the feature learning ability of transformers. Second, we design a new transformer architecture named ViTAE based on two new reduction and normal cells to intrinsically incorporate the above two IBs. The proposed ViTAE embeds multi-scale context into tokens and learns both local and long-range features effectively. Third, ViTAE outperforms representative vision transformers regarding classification accuracy, data efficiency, training efficiency, and generalization on downstream tasks. ViTAE achieves $7 5 . 3 \\%$ and $8 2 . 0 \\%$ top-1 accuracy on ImageNet with 4.8M and 23.6M parameters, respectively. ",
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+    {
+        "type": "text",
+        "text": "",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "2.1 CNNs with intrinsic IB ",
+        "text_level": 1,
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+        "text": "CNNs have led to a series of breakthroughs in image classification [38, 94, 26, 95, 87] and downstream computer vision tasks. The convolution operations in CNNs extract local features from the neighbor pixels within the receptive field determined by the kernel size [42]. Following the intuition that local pixels are more likely to be correlated in images [41], CNNs have the intrinsic IB in modeling locality. In addition to the locality, another critical topic in visual tasks is scale-invariance, where multi-scale features are needed to represent the objects at different scales effectively [49, 90]. For example, to effectively learn features of large objects, a large receptive field is needed by either using large convolution kernels [90, 91] or a series of convolution layers in deeper architectures [26, 32, 68, 71]. To construct multi-scale feature representation, the classical idea is using image pyramid [8, 1, 55, 4, 39, 16], where features are hand-crafted or learned from a pyramid of images at different resolutions respectively [44, 8, 52, 63, 35, 3]. Accordingly, features from the small scale image mainly encode the large objects while features from the large scale image respond more to small objects. In addition to the above inter-layer fusion way, another way is to aggregate multi-scale context by using multiple convolutions with different receptive fields within a single layer, i.e., intra-layer fusion [96, 71, 70, 70, 72]. Either inter-layer fusion or intra-layer fusion empower CNNs an intrinsic IB in modeling scale-invariance. This paper introduces such an IB to vision transformers by following the intra-layer fusion idea and utilizing multiple convolutions with different dilation rates in the reduction cells to encode multi-scale context into each visual token. ",
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+    {
+        "type": "text",
+        "text": "2.2 Vision transformers with learned IB ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "ViT [19] is the pioneering work that applies a pure transformer to vision tasks and achieves promising results. However, since ViT lacks intrinsic inductive bias in modeling local visual structures, it indeed learns the IB from amounts of data implicitly. Following works along this direction are to simplify the model structures with fewer intrinsic IBs and directly learn them from large scale data [50, 74, 75, 22, 18, 20, 27] which have achieved promising results and been studied actively. Another direction is to leverage the intrinsic IB from CNNs to facilitate the training of vision transformers, e.g., using less training data or shorter training schedules. For example, DeiT [76] proposes to distill knowledge from CNNs to transformers during training. However, it requires an off-the-shelf CNN model as a teacher, introducing extra computation cost during training. Recently, some works try to introduce the intrinsic IB of CNNs into vision transformers explicitly [23, 58, 21, 43, 15, 89, 83, 92, 6, 47, 11]. For example, [43, 21, 83] stack convolutions and attention layers sequentially, resulting in a serial structure and modeling the locality and global dependency accordingly. [80, 28] design sequential stage-wise structures while [47, 33] apply attention within local windows. However, these serial structure may ignore the global context during locality modeling (and vice versa). [88] establishes connection across different scales at the cost of heavy computation. Instead, we follow the “divide-and-conquer” idea and propose to model locality and global dependencies simultaneously via a parallel structure within each transformer layer. Conformer [58], the most relevant concurrent work to us, employs a unit to explore inter-block interactions between parallel convolution and transformer blocks. In contrast, in ViTAE, the convolution and attention modules are designed to be complementary to each other within the transformer block. In addition, Conformer is not designed to have inherent scale invariance IB. ",
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+    {
+        "type": "text",
+        "text": "3 Methodology ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "3.1 Revisit vision transformer ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We first give a brief review of vision transformer in this part. To adapt transformers to vision tasks, ViT [19] first splits an image $x \\in R ^ { H \\times W \\times C }$ into tokens with a reduction ratio of $p$ (i.e., $x _ { t } \\in R ^ { ( ( H \\times W ) / p ^ { 2 } ) \\times D } )$ , where $H , W$ and $C$ denote the height, width, and channel dimensions of the input image, $D = C p ^ { 2 }$ denotes the token dimension. Then, an extra class token is concatenated to the visual tokens before adding position embeddings in an element-wise manner. The resulting tokens are fed into the following transformer layers. Each transformer layer is composed of two parts, i.e., a multi-head self-attention module (MHSA) and a feed forward network (FFN). ",
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+        "img_path": "images/12ff638562a0b78bbcef09633d771c21fc798831f54a00e148f38d5e89e3596b.jpg",
+        "image_caption": [
+            "Figure 2: The structure of the proposed ViTAE. It is constructed by stacking three RCs and several NCs. Both types of cells share a simple basic structure, i.e., an MHSA module and a parallel convolutional module followed by an FFN. In particular, RC has an extra pyramid reduction module using atrous convolutions with different dilation rates to embed multi-scale context into tokens. "
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+        "type": "text",
+        "text": "MHSA Multi-head self-attention extends single-head self-attention (SHSA) by using different projection matrices for each head. Specifically, the input tokens $x _ { t }$ are first projected to queries $( Q )$ , keys $( K )$ and values $( V )$ using projection matrices, i.e., $Q , K , V = x _ { t } W _ { Q } , x _ { t } Q _ { K } , x _ { t } Q _ { V }$ , where $W _ { Q / K / V } \\in R ^ { D \\times D }$ denotes the projection matrix for query, key, and value, respectively. Then, the self-attention operation is calculated as: ",
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+    {
+        "type": "equation",
+        "img_path": "images/67c1b6a153e357266ce63dcf66004997f007dc52c414762d6054a92d21ab5683.jpg",
+        "text": "$$\nA t t e n t i o n ( Q , K , V ) = s o f t m a x ( \\frac { Q K ^ { T } } { \\sqrt { D } } ) V .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "This SHSA module is repeated for $h$ times to formulate the MHSA module, where $h$ is the number of heads. The output features of the $h$ heads are concatenated along the channel dimension and formulate the output of the MHSA module. ",
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+        "type": "text",
+        "text": "FFN FFN is placed on top of the MHSA module and applied to each token identically and separately. It consists of two linear transformations with an activation function in between. Besides, a layer normalization [2] and a shortcut are added before and aside from the MHSA and FFN, respectively. ",
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+    {
+        "type": "text",
+        "text": "3.2 Overview architecture of ViTAE ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "ViTAE aims to introduce the intrinsic IB in CNNs to vision transformers. As shown in Figure 2, ViTAE is composed of two types of cells, i.e., RCs and NCs. RCs are responsible for embedding multi-scale context and local information into tokens, and NCs are used to further model the locality and long-range dependencies in the tokens. Taken an image $x \\in R ^ { H \\times W \\times C }$ as input, three RCs are used to gradually downsample $x$ by $4 \\times , 2 \\times$ , and $2 \\times$ , respectively. Thereby, the output tokens of the RCs are of size $[ H / 1 6 , W / 1 6 , D ]$ where $D$ is the token dimension (64 in our experiments). The output tokens of RCs are then flattened as $R ^ { H W / 2 5 6 \\times D }$ , concatenated with the class token, and added by the sinusoid position encoding. Next, the tokens are fed into the following NCs, which keep the length of the tokens. Finally, the prediction probability is obtained using a linear classification layer on the class token from the last NC. ",
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+    {
+        "type": "text",
+        "text": "3.3 Reduction cell ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Instead of directly splitting and flatten images into visual tokens based on a linear image patch embedding layer, we devise the reduction cell to embed multi-scale context and local information into visual tokens, which introduces the intrinsic scale-invariance and locality IBs from convolutions. Technically, RC has two parallel branches responsible for modeling locality and long-range dependency, respectively, followed by an FFN for feature transformation. We denote the input feature of the $i _ { t h } \\ : \\mathrm { R C }$ as $f _ { i } \\in \\dot { R } ^ { H _ { i } \\times W _ { i } \\times D _ { i } }$ . The input of the first RC is the image $x$ . In the global dependencies branch, $f _ { i }$ is firstly fed into a Pyramid Reduction Module (PRM) to extract multi-scale context, i.e., ",
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+    {
+        "type": "equation",
+        "img_path": "images/995f71351d9a0d0f5ed166388dc336a063cfcb324dbc2f6372f07fdf76e058d7.jpg",
+        "text": "$$\nf _ { i } ^ { m s } \\triangleq P R M _ { i } ( f _ { i } ) = C a t ( [ C o n v _ { i j } ( f _ { i } ; s _ { i j } , r _ { i } ) | s _ { i j } \\in S _ { i } , r _ { i } \\in  { \\mathcal { R } } ] ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $C o n v _ { i j } ( \\cdot )$ indicates the $j$ th convolutional layer in the PRM $( P R M _ { i } ( \\cdot ) )$ . It uses a dilation rate $s _ { i j }$ from the predefined dilation rate set $S _ { i }$ corresponding to the ith RC. Note that we use stride convolution to reduce the spatial dimension of features by a ratio $r _ { i }$ from the predefined reduction ratio set $\\mathcal { R }$ . The conv features are concatenated along the channel dimension, i.e., $f _ { i } ^ { m s } \\in$ $R ^ { ( W _ { i } / p ) \\times ( H _ { i } / p ) \\times ( | S _ { i } | D ) }$ , where $| { S _ { i } } |$ denotes the number of dilation rates in $S _ { i }$ . $f _ { i } ^ { m s }$ is then processed by an MHSA module to model long-range dependencies, i.e., ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/47bb46efbf400464cc18685ec76d7959be215ba831aad37913cc57da0f8d98b9.jpg",
+        "text": "$$\nf _ { i } ^ { g } = M H S A _ { i } ( I m g 2 S e q ( f _ { i } ^ { m s } ) ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $I m g 2 S e q ( \\cdot )$ is a simple reshape operation to flatten the feature map to a 1D sequence. In this way, $f _ { i } ^ { g }$ embeds the multi-scale context in each token. In addition, we use a Parallel Convolutional Module (PCM) to embed local context within the tokens, which are fused with $f _ { i } ^ { g }$ as follows: ",
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+    {
+        "type": "equation",
+        "img_path": "images/aab14cd318e095613617d3482aa2f81617b3140ca8a4a2e53d1fa0230ff83a82.jpg",
+        "text": "$$\nf _ { i } ^ { l g } = f _ { i } ^ { g } + { \\cal P } { \\cal C } M _ { i } ( f _ { i } ) .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Here, $P C M _ { i } ( \\cdot )$ represents the PCM, which is composed of three stacked convolution layers and an $I m g 2 S e q ( \\cdot )$ operation. It is noteworthy that the parallel convolution branch has the same spatial downsampling ratio as the PRM by using stride convolutions. In this way, the token features can carry both local and multi-scale context, implying that RC acquires the locality IB and scale-invariance IB by design. The fused tokens are then processed by the FFN, reshaped back to feature maps, and fed into the following RC or NC, i.e., ",
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+    {
+        "type": "equation",
+        "img_path": "images/716d8b1849d36a1ad308d5ce41f0df05844de9ab779da38fd00d2d340e97c83e.jpg",
+        "text": "$$\nf _ { i + 1 } = S e q 2 I m g ( F F N _ { i } ( f _ { i } ^ { l g } ) + f _ { i } ^ { l g } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where the $S e q 2 I m g ( \\cdot )$ is a simple reshape operation to reshape a token sequence back to feature maps. $F F N _ { i } ( \\cdot )$ represents the FFN in the ith RC. In our ViTAE, three RCs are stacked sequentially to gradually reduce the input image’s spatial dimension by $4 \\times , 2 \\times$ , and $2 \\times$ , respectively. The feature maps generated by the last RC are of a size of $[ H / 1 6 , W / 1 6 , D ]$ , which are then flattened into visual tokens and fed into the following NCs. ",
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+    {
+        "type": "text",
+        "text": "3.4 Normal cell ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "As shown in the bottom right part of Figure 2, NCs share a similar structure with the reduction cell except for the absence of the PRM. Due to the relatively small $( \\frac { 1 } { 1 6 } \\times )$ spatial size of feature maps after RCs, it is unnecessary to use PRM in NCs. Given $f _ { 3 }$ from the third RC, we first concatenate it with the class token $t _ { c l s }$ , and then add it to the positional encodings to get the input tokens $t$ for the following NCs. Here we ignore the subscript for clarity since all NCs have an identical architecture but different learnable weights. $t _ { c l s }$ is randomly initialized at the start of training and fixed during the inference. Similar to the RC, the tokens are fed into the MHSA module, i.e., $t _ { g } = M H S A ( t )$ . Meanwhile, they are reshaped to 2D feature maps and fed into the PCM, i.e., $t _ { l } = \\bar { I } m g 2 S e q ( P C M ( S e q 2 I m g ( t ) ) )$ . Note that the class token is discarded in PCM because it has no spatial connections with other visual tokens. To further reduce the parameters in NCs, we use group convolutions in PCM. The features from MHSA and PCM are then fused via element-wise sum, i.e., $t _ { l g } = t _ { g } + t _ { l }$ . Finally, $t _ { l g }$ are fed into the FFN to get the output features of NC, i.e., $t _ { n c } = F F N ( t _ { l g } ) \\overline { { + } } t _ { l g }$ . Similar to ViT [19], we apply layer normalization to the class token generated by the last NC and feed it to the classification head to get the final classification result. ",
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+    {
+        "type": "text",
+        "text": "3.5 Model details ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We use two variants of ViTAE in our experiments for a fair comparison of other models with similar model sizes. The details of them are summarized in Table 1. In the first RC, the default convolution kernel size is $7 \\times 7$ with a ",
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+    {
+        "type": "table",
+        "img_path": "images/b78024fbd1f12dcc881657db015afe79b126382b56ffdf33fae9095f44e9bcf1.jpg",
+        "table_caption": [
+            "Table 1: Model details of two variants of ViTAE. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Reduction Cell Dilation</td><td>Cells</td><td>Normal Cell Heads Embed Cells</td><td></td><td>Params Macs (M)</td><td>(G)</td></tr><tr><td>ViTAE-T</td><td>[1,2,3,4] √</td><td>3</td><td>4 256</td><td>7</td><td>4.8</td><td>1.5</td></tr><tr><td>ViTAE-S</td><td>[1,2,3,4] √</td><td>3</td><td>6</td><td>384 14</td><td>23.6</td><td>5.6</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "stride of 4 and dilation rates of $\\mathcal { S } _ { 1 } = [ 1 , 2 , 3 , 4 ]$ . In the following two RCs, the convolution kernel size is $3 \\times 3$ with a stride of 2 and dilation rates of $S _ { 2 } = [ 1 , 2 , 3 ]$ and $S _ { 3 } = [ 1 , 2 ]$ , respectively. Since the spatial dimension of tokens decreases, there is no need to use large kernels and dilation rates. PCM in both RCs and NCs comprises three convolutional layers with a kernel size of $3 \\times 3$ . ",
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+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "4.1 Implementation details ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We train and test the proposed ViTAE model on the standard ImageNet [38] dataset, which contains about 1.3 million images and covers 1k classes. Unless explicitly stated, the image size during training is set to $2 2 4 \\times 2 2 4$ . We use the AdamW [48] optimizer with the cosine learning rate scheduler and uses the data augmentation strategy exactly the same as T2T [93] for a fair comparison, regarding the training strategies and the size of models. We use a batch size of 512 for training all our models and set the initial learning rate to be 5e-4. The results of our models can be found in Table 2, where all the models are trained for 300 epochs on 8 V100 GPUs. The models are built on PyTorch [57] and TIMM [82]. ",
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+    {
+        "type": "text",
+        "text": "4.2 Comparison with the state-of-the-art ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We compare our ViTAE with both CNN models and vision transformers with similar model sizes in Table 2. Both Top-1/5 accuracy and real Top-1 accuracy on the ImageNet validation set are reported. We categorize the methods into CNN models, vision transformers with learned IB, and vision transformers with introduced intrinsic IB. Compared with CNN models, our ViTAE-T achieves a $7 5 . 3 \\%$ Top-1 accuracy, which is better than ResNet-18 with more parameters. The real Top-1 accuracy of the ViTAE model is $8 2 . 9 \\%$ , which is comparable to ResNet-50 that has four more times of parameters than ours. Similarly, our ViTAE-S achieves $8 2 . 0 \\%$ Top-1 accuracy with half of the parameters of ResNet-101 and ResNet-152, showing the superiority of learning both local and longrange features from specific structures with corresponding intrinsic IBs by design. Similar phenomena can also be observed when comparing ViTAE-T with MobileNetV1 [31] and MobileNetV2 [65], where ViTAE obtains better performance with fewer parameters. When compared with larger models which are searched according to NAS [73], our ViTAE-S achieves a similar performance when using $3 8 4 \\times 3 8 4$ images as input, which further shows the potential of vision transformers with intrinsic IB. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In addition, among the transformers with learned IB, ViT is the first pure transformer model for visual recognition. DeiT shares the same structure with ViT but uses different data augmentation and training strategies to facilitate the learning of transformers. DeiT⚗ denotes using an off-the-shelf CNN model as the teacher model to train DeiT, which introduces the intrinsic IB from CNN to transformer implicitly in a knowledge distillation manner, showing better performance than the vanilla ViT on the ImageNet dataset. It is exciting to see that our ViTAE-T with fewer parameters even outperforms the distilled model DeiT⚗, demonstrating the efficacy of introducing intrinsic IBs in transformers by design. Besides, compared with other transformers with explicit intrinsic IB, our ViTAE with fewer parameters also achieves comparable or better performance. For instance, ViTAE-T achieves comparable performance with LocalVit-T but has 1M fewer parameters, demonstrating the superiority of the proposed RCs and NCs in introducing intrinsic IBs. ",
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+    {
+        "type": "text",
+        "text": "4.3 Ablation study ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We use T2T-ViT [93] as our baseline model in the following ablation study of our ViTAE. As shown in Table 3, we investigate the hyper-parameter settings in RCs and NCs by isolating them separately. All the models are trained for 100 epochs on ImageNet and follow the same training setting and data augmentation strategy as described in Section 4.1. ",
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+    {
+        "type": "table",
+        "img_path": "images/e514264bc0b3536b852ece05e614d3ca959237e923c2f3d7e1d3c16e5426e296.jpg",
+        "table_caption": [
+            "Table 2: Comparison of ViTAE and SOTA methods on the ImageNet validation set. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Type Model</td><td rowspan=\"2\">Params (M)</td><td rowspan=\"2\">MACs (G)</td><td rowspan=\"2\">Input Size</td><td colspan=\"3\">ImageNet Real</td></tr><tr><td>Top-1</td><td>Top-5</td><td>Top-1</td></tr><tr><td rowspan=\"9\">CNN</td><td>ResNet-18 [26]</td><td>11.7</td><td>3.6</td><td>224</td><td>70.3</td><td>86.7</td><td>77.3</td></tr><tr><td>ResNet-50 [26]</td><td>25.6</td><td>7.6</td><td>224</td><td>76.7</td><td>93.3</td><td>82.5</td></tr><tr><td>ResNet-101 [26]</td><td>44.5</td><td>15.2</td><td>224</td><td>78.3</td><td>94.1</td><td>83.7</td></tr><tr><td>ResNet-152 [26]</td><td>60.2</td><td>22.6</td><td>224</td><td>78.9</td><td>94.4</td><td>84.1</td></tr><tr><td>EfficientNet-B0 [73]</td><td>5.3</td><td>0.8</td><td>224</td><td>77.1</td><td>93.3</td><td>83.5</td></tr><tr><td>EfficientNet-B4 [73]</td><td>19.3</td><td>8.4</td><td>380</td><td>82.9</td><td>96.4</td><td>88.0</td></tr><tr><td>MobileNetV1 [31]</td><td>4.3</td><td>0.6</td><td>224</td><td>72.3</td><td>1</td><td>-</td></tr><tr><td>MobileNetV2(1.4) [65]</td><td>6.9</td><td>0.6</td><td>224</td><td>74.7</td><td>-</td><td>-</td></tr><tr><td>RegNetY-600M[62]</td><td>6.1</td><td>1.2</td><td>224</td><td>75.5</td><td>-</td><td>-</td></tr><tr><td>RegNetY-4GF[62] RegNetY-8GF[62]</td><td>20.6 39.2</td><td>8.0 16.0</td><td>224</td><td>80.0</td><td>1</td><td>86.4</td></tr><tr><td></td><td></td><td></td><td></td><td>224</td><td>81.7</td><td>1</td><td>87.4</td></tr><tr><td rowspan=\"14\"></td><td>DeiT-T[76]</td><td>5.7</td><td>2.6</td><td>224</td><td>72.2</td><td>91.1</td><td>80.6</td></tr><tr><td>DeiT-T [76]</td><td>5.7</td><td>2.6</td><td>224</td><td>74.5</td><td>91.9</td><td>82.1</td></tr><tr><td>LocalViT-T[43]</td><td>5.9</td><td>2.6</td><td>224</td><td>74.8</td><td>92.6</td><td></td></tr><tr><td>LocalViT-T2T[43]</td><td>4.3</td><td>2.4</td><td>224</td><td>72.5</td><td>-</td><td>1 1</td></tr><tr><td>ConT-Ti [89]</td><td>5.8</td><td>1.6</td><td>224</td><td>74.9</td><td>-</td><td>-</td></tr><tr><td>PiT-Ti [29]</td><td>4.9</td><td>1.4</td><td>224</td><td>73.0</td><td>-</td><td>1</td></tr><tr><td>T2T-ViT-7 [93]</td><td>4.3</td><td>1.2</td><td>224</td><td>71.7</td><td>90.9</td><td>79.7</td></tr><tr><td>ViTAE-T</td><td>4.8</td><td>1.5</td><td>224</td><td>75.3</td><td>92.7</td><td>82.9</td></tr><tr><td>ViTAE-T ↑ 384</td><td>4.8</td><td>5.7</td><td>384</td><td>77.2</td><td>93.8</td><td>84.4</td></tr><tr><td>CeiT-T [92]</td><td>6.4</td><td>2.4</td><td>224</td><td>76.4</td><td>93.4</td><td>83.6</td></tr><tr><td>ConViT-Ti[15]</td><td>6.0</td><td>2.0</td><td>224</td><td>73.1</td><td>1</td><td>1</td></tr><tr><td>Cross ViT-Ti [6]</td><td>6.9</td><td>3.2</td><td>224</td><td>73.4</td><td>1</td><td>1</td></tr><tr><td>ViTAE-6M</td><td>6.5</td><td>2.0</td><td>224</td><td>77.9</td><td>94.1</td><td>84.9</td></tr><tr><td>PVT-T[80] LocalViT-PVT [43]</td><td>13.2</td><td>3.8</td><td>224</td><td>75.1</td><td>1</td><td></td></tr><tr><td rowspan=\"8\">PiT-XS [29] ConT-M [89] ViTAE-13M DeiT-S [76]</td><td>13.5</td><td>9.6</td><td>224</td><td></td><td>94.2</td><td>-</td></tr><tr><td>ConViT-Ti+ [15] 10.0</td><td>4.0</td><td>224</td><td>78.2 76.7</td><td></td><td>1</td></tr><tr><td></td><td>2.8</td><td></td><td>78.1</td><td>1</td><td>-</td></tr><tr><td>10.6 19.2</td><td>6.2</td><td>224 224</td><td>80.2</td><td>-</td><td>1</td></tr><tr><td>13.2</td><td>3.4</td><td>224</td><td>81.0</td><td>- 95.4</td><td>- 86.8</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>22.1 DeiT-S [76] 22.1</td><td>9.8 9.8</td><td>224 224</td><td>79.9 81.2</td><td>95.0 95.4</td><td>85.7 86.8</td></tr><tr><td>PVT-S[80]</td><td>7.6</td><td>224</td><td>79.8</td><td>-</td><td></td></tr><tr><td></td><td>24.5 23.5</td><td>5.2</td><td></td><td>81.3</td><td></td><td>1</td></tr><tr><td>Conformer-Ti [58] Swin-T[47]</td><td></td><td></td><td>224</td><td></td><td>-</td><td></td></tr><tr><td>CeiT-S [92]</td><td>29.0</td><td>9.0</td><td>224</td><td>81.3</td><td>-</td><td>1</td></tr><tr><td>CvT-13 [83]</td><td>24.2 20.0</td><td>9.0</td><td>224</td><td>82.0 81.6</td><td>95.9</td><td>87.3 86.7</td></tr><tr><td>ConViT-S[15]</td><td>27.0</td><td>9.0 10.8</td><td>224 224</td><td>81.3</td><td>1 1</td><td>1</td></tr><tr><td>Cross ViT-S [6]</td><td>26.7</td><td>11.2</td><td>224</td><td>81.0</td><td>1</td><td>1</td></tr><tr><td>PiT-S [29]</td><td>23.5</td><td>4.8</td><td>224</td><td>80.9</td><td></td><td></td></tr><tr><td>TNT-S [23]</td><td></td><td></td><td></td><td></td><td>-</td><td>1</td></tr><tr><td>Twins-PCPVT-S[10]</td><td>23.8</td><td>10.4</td><td>224</td><td>81.3</td><td>95.6</td><td>-</td></tr><tr><td></td><td>24.1</td><td>7.4</td><td>224</td><td>81.2</td><td>-</td><td>-</td></tr><tr><td>Twins-SVT-S [10]</td><td>24.0</td><td>5.6</td><td>224</td><td>81.7</td><td>-</td><td>1</td></tr><tr><td>T2T-ViT-14 [93]</td><td>21.5</td><td>5.2</td><td>224</td><td>81.5</td><td>95.7</td><td>86.8</td></tr><tr><td>ViTAE-S</td><td>23.6</td><td>5.6</td><td>224</td><td>82.0</td><td>95.9</td><td>87.0</td></tr><tr><td>ViTAE-S ↑ 384</td><td>23.6</td><td>20.2</td><td>384</td><td>83.0</td><td>96.2</td><td>87.5</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We use $\\checkmark$ and $\\times$ to denote whether or not the corresponding module is enabled during the experiments. If all columns under the RC and NC are marked $\\times$ as shown in the first row, the model becomes the standard T2T-ViT model. “Pre” indicates the output features of PCM and MHSA are fused before FFN while “Post” indicates a late fusion strategy correspondingly. “BN” indicates whether PCM uses BN after the convolutional layer or not. $\\mathit { \\Omega } ^ { 6 } \\times 3 \\mathit { \\Omega } ^ { 5 }$ in the first column denotes that the dilation rate set is the same in the three RCs. “ $[ 1 , 2 , 3 , 4 ]$ $\\downarrow ^ { \\circ }$ denotes using lower dilation rates in deeper RCs, i.e., $\\mathcal { S } _ { 1 } = [ 1 , 2 , 3 , 4 ]$ , $S _ { 2 } = [ 1 , 2 , 3 ]$ , $S _ { 3 } = [ 1 , 2 ]$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "As can be seen, using a pre-fusion strategy and BN achieves the best $6 9 . 9 \\%$ Top-1 accuracy among other settings. It is noteworthy that all the variants of NC outperform the vanilla T2T-ViT, implying the effectiveness of PCM, which introduces the intrinsic locality IB in transformers. It can also be observed that BN plays an important role in improving the model’s performance as it can help to alleviate the scale deviation between convolution’s and attention’s features. For the RC, we first investigate the impact of using different dilation rates in the PRM, as shown in the first column. As can be seen, using larger dilation rates (e.g., 4 or 5) does not deliver better performance. We suspect that larger dilation rates may lead to plain features in the deeper RCs due to the smaller resolution of feature maps. To ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/8cb6fb2c3865a2b27116e792990f87e65f07efdf975a9d033b0acebcd1d733eb.jpg",
+        "table_caption": [
+            "Table 3: Ablation Study of RCs and NCs in our ViTAE. “Pre” indicates the output features of PCM and MHSA are fused before FFN while “Post” indicates a late fusion strategy correspondingly. “BN” indicates whether PCM uses BN or not. “ $[ 1 , 2 , 3 , 4 ]$ $\\downarrow ^ { \\circ }$ denotes using smaller dilation rates in deeper RCs, i.e., $\\mathcal { S } _ { 1 } = [ 1 , 2 , 3 , 4 ]$ , $\\mathsf { \\bar { S } } _ { 2 } = [ 1 , 2 , 3 ]$ , $ { S _ { 3 } } = [ 1 , 2 ]$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Reduction Cell</td><td></td><td>Normal Cell</td><td rowspan=\"2\">Top-1</td></tr><tr><td>Dilation (S1 ~ S3) PCM</td><td>Pre</td><td>BN Post</td></tr><tr><td>× ×</td><td>×</td><td>× ×</td><td>68.7</td></tr><tr><td>× ×</td><td>√</td><td>× ×</td><td>69.1</td></tr><tr><td>× ×</td><td>×</td><td>√ ×</td><td>69.0</td></tr><tr><td>× ×</td><td>×</td><td>√ √</td><td>68.8</td></tr><tr><td>× ×</td><td>√</td><td>× √</td><td>69.9</td></tr><tr><td>[1,2]×3</td><td>× ×</td><td>×</td><td>× 69.5</td></tr><tr><td>[1,2,3]×3 ×</td><td>×</td><td>× ×</td><td>69.9</td></tr><tr><td>[1,2,3,4] × 3 ×</td><td>×</td><td>× ×</td><td>69.2</td></tr><tr><td>[1,2,3,4,5] × 3 ×</td><td>×</td><td>× ×</td><td>68.9</td></tr><tr><td>[1,2,3,4]↓ ×</td><td>×</td><td>× ×</td><td>69.8</td></tr><tr><td>[1,2,3,4]↓ √</td><td>×</td><td>× ×</td><td>71.7</td></tr><tr><td>[1,2,3,4]↓ √</td><td>√</td><td>× √</td><td>72.6</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "validate the hypothesis, we use smaller dilation rates in deeper RCs as denoted by $[ 1 , 2 , 3 , 4 ] \\downarrow$ . As can be seen, it achieves comparable performance as $[ 1 , 2 , 3 ] \\times$ . However, compared with $[ 1 , 2 , 3 , 4 ] \\downarrow$ , $[ 1 , 2 , 3 ] \\times$ increases the amount of parameters from 4.35M to $4 . 6 \\mathsf { M }$ . Therefore, we select $[ 1 , 2 , 3 , 4 ] \\downarrow$ as the default setting. In addition, after using PCM in the RC, it introduces the intrinsic locality IB, and the performance increases to $7 1 . 7 \\%$ Top-1 accuracy. Finally, the combination of RCs and NCs achieves the best accuracy at $7 2 . 6 \\%$ , demonstrating the complementarity between our RCs and NCs. ",
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+    },
+    {
+        "type": "text",
+        "text": "4.4 Data efficiency and training efficiency ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To validate the effectiveness of the introduced intrinsic IBs in improving data efficiency and training efficiency, we compare our ViTAE with T2T-ViT at different training settings: (a) training them using $20 \\%$ , $60 \\%$ , and $100 \\%$ ImageNet training set for equivalent 100 epochs on the full ImageNet training set, e.g., we employ 5 times epochs when using $20 \\%$ data for training compared with using $100 \\%$ data; and (b) training them using the full ImageNet training set for 100, 200, and 300 epochs respectively. The results are shown in Figure 1. As can be seen, ViTAE consistently outperforms the T2T-ViT baseline by a large margin in terms of both data efficiency and training efficiency. For example, ViTAE using only $20 \\%$ training data achieves comparable performance with T2T-ViT using all data. When $60 \\%$ training data are used, ViTAE significantly outperforms T2T-ViT using all data by about an absolute $3 \\%$ accuracy. It is also noteworthy that ViTAE trained for only 100 epochs has outperformed T2T-ViT trained for 300 epochs. After training ViTAE for 300 epochs, its performance is significantly boosted to $7 5 . 3 \\%$ Top-1 accuracy. With the proposed RCs and NCs, the transformer layers in our ViTAE only need to focus on modeling long-range dependencies, leaving the locality and multi-scale context modeling to its convolution counterparts, i.e., PCM and PRM. Such a “divide-and-conquer” strategy facilitates the training of vision transformers, making it possible to learn more efficiently with less training data and fewer training epochs. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To further validate the data efficiency of ViTAE model, we train the ViTAE model from scratch on the smaller datasets, i.e., Cifar10 and Cifar100. The results are summarized in Table 4. It can be viewed that with only $1 / 7$ number of epochs, the ViTAE-T model achieves better classification performance on Cifar10 dataset, with far fewer parameters (4.8M v.s. 86M), which further confirms ViTAE model’s data efficiency. ",
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "table",
+        "img_path": "images/63a31781efc1cf86aa1701ec0651b0baf9513eb3aeb90723d610223388fffb83.jpg",
+        "table_caption": [
+            "Table 4: Results of training from scratch on Cifar10/100. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Params (M)</td><td>Top-1 Acc</td><td>Epochs</td><td>Dataset</td></tr><tr><td>DeiT-B</td><td>86.0</td><td>97.5</td><td>7000</td><td>Cifar10</td></tr><tr><td>ViTAE-T</td><td>4.8</td><td>97.7</td><td>1000</td><td>Cifar10</td></tr><tr><td>ViTAE-T</td><td>4.8</td><td>85.0</td><td>1000</td><td>Cifar100</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.5 Generalization on downstream tasks ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            239,
+            465,
+            253
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/1811ce2c1001783f4b47fef070c93481a3eed9f0f8f8b509c00a30ea1da1a596.jpg",
+        "table_caption": [
+            "Table 5: Generalization of ViTAE and SOTA methods on different downstream tasks. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Params (M)</td><td>Cifar10</td><td>Cifar100</td><td>iNat19</td><td>Cars</td><td>Flowers</td><td>Pets</td></tr><tr><td>Grafit ResNet-50 [78]</td><td>25.6</td><td>-</td><td>=</td><td>75.9</td><td>92.5</td><td>98.2</td><td>-</td></tr><tr><td>EfficientNet-B5 [73]</td><td>30</td><td>98.1</td><td>91.1</td><td>-</td><td>-</td><td>98.5</td><td>-</td></tr><tr><td>ViT-B/16 [19]</td><td>86.5</td><td>98.1</td><td>87.1</td><td>-</td><td>1</td><td>89.5</td><td>93.8</td></tr><tr><td>ViT-L/16 [19]</td><td>304.3</td><td>97.9</td><td>86.4</td><td>-</td><td>-</td><td>89.7</td><td>93.6</td></tr><tr><td>DeiT-B [76]</td><td>86.6</td><td>99.1</td><td>90.8</td><td>77.7</td><td>92.1</td><td>98.4</td><td>-</td></tr><tr><td>T2T-ViT-14 [93]</td><td>21.5</td><td>98.3</td><td>88.4</td><td>-</td><td>-</td><td>-</td><td>1</td></tr><tr><td>ViTAE-T</td><td>4.8</td><td>97.3</td><td>86.0</td><td>73.3</td><td>89.5</td><td>97.5</td><td>92.6</td></tr><tr><td>ViTAE-S</td><td>23.6</td><td>98.8</td><td>90.8</td><td>76.0</td><td>91.4</td><td>97.8</td><td>94.2</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We further investigate the generalization of the proposed ViTAE models on downstream tasks by finetuning them on the training sets of several fine-grained classification tasks3, including Flowers [53], Cars [36], Pets [56], and iNaturalist19. We also fine-tune the proposed ViTAE models on Cifar10 [37] and Cifar100 [37]. The results are shown in Table 5. It can be seen that ViTAE achieves SOTA performance on most of the datasets using comparable or fewer parameters. These results demonstrate that the good generalization ability of our ViTAE. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.6 Visual inspection of ViTAE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            523,
+            400,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "To further analyze the property of our ViTAE, we first calculate the average attention distance of each layer in ViTAE-T and the baseline T2T-ViT-7 on the ImageNet test set, respectively. The results are shown in Figure 3. It can be observed that with the usage of PCM, which focuses on modeling locality, the transformer layers in the proposed NCs can better focus on modeling long-range dependencies, especially in shallow layers. In the deep layers, the average attention distances of ViTAE-T and T2T-ViT-7 are almost the same since modeling long-range dependencies is much more important. ",
+        "bbox": [
+            173,
+            549,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "These results confirm the effectiveness of the adopted “divide-and-conquer” idea in the proposed ViTAE, i.e., introducing the intrinsic locality IB from convolutions into vision transformers makes it possible that transformer layers only need to be responsible to long-range dependencies, since locality can be well modeled by convolutions in PCM. ",
+        "bbox": [
+            173,
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/0d27a138487be9062b7073116e5379b9d74b5e8122843b673cecf6fb02d9776d.jpg",
+        "image_caption": [
+            "Figure 3: The average per-layer attention distance of T2T-ViT-7 and our ViTAE-T. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            531,
+            648,
+            823,
+            750
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Besides, we apply Grad-CAM [66] on the MHSA’s output in the last NC to qualitatively inspect ViTAE. The visualization results are provided in Figure 4. Compared with the baseline T2T-ViT, our ViTAE covers the single or multiple targets in the images more precisely and attends less to the background. Moreover, ViTAE can better handle the scale variance issue as shown in Figure 4(b). Namely, it can precisely cover the birds no matter they are in small, middle, or large size. Such observations demonstrate that introducing the intrinsic IBs of locality and scale-invariance from convolutions to transformers helps ViTAE learn more discriminate features than the pure transformers. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/c40cbf192eae3a11c74a33bbbb56c6df92c409dc0cc54401ef7f9004a71cc003.jpg",
+        "image_caption": [
+            "Figure 4: Visual inspection of T2T-ViT and ViTAE using Grad-CAM [66]. (a) Images containing multiple or single objects and the heatmaps. (b) Images containing the same class of objects at different scales and the heatmaps (Best viewed in color). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "5 Limitation and discussion ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
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+        "type": "text",
+        "text": "In this paper, we explore two types of IBs and incorporate them into transformers through the proposed reduction and normal cells. With the collaboration of these two cells, our ViTAE model achieves impressive performance on the ImageNet with fast convergence and high data efficiency. Nevertheless, due to computational resource constraints, we have not scaled the ViTAE model and train it on largesize dataset, e.g., ImageNet-21K [38] and JFT-300M [30]. Although it remains unclear by now, we are optimistic about its scale property from the following preliminary evidence. As illustrated in Figure 2, our ViTAE model can be viewed as an intra-cell ensemble of complementary transformer layers and convolution layers owing to the skip connection and parallel structure. According to the attention distance analysis shown in Figure 3, the ensemble nature enables the transformer layers and convolution layers to focus on what they are good at, i.e., modeling long-range dependencies and locality. Therefore, ViTAE is very likely to learn better feature representation from large-scale data. Besides, we only study two typical IBs in this paper. More kinds of IBs such as constituting viewpoint invariance [64] can be explored in the future study. ",
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+        "text": "6 Conclusion ",
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+        "text": "In this paper, we re-design the transformer block by proposing two basic cells (reduction cells and normal cells) to incorporate two types of intrinsic inductive bias (IB) into transformers, i.e., locality and scale-invariance, resulting in a simple yet effective vision transformer architecture named ViTAE. Extensive experiments show that ViTAE outperforms representative vision transformers in various respects including classification accuracy, data efficiency, training efficiency, and generalization ability on downstream tasks. We plan to scale ViTAE to the large or huge model size and train it on large-size datasets in the future study. In addition, other kinds of IBs will also be investigated. We hope that this study will provide valuable insights to the following studies of introducing intrinsic IB into vision transformers and understanding the impact of intrinsic and learned IBs. ",
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+        "text": "Acknowledgement Dr. Jing Zhang is supported by the ARC project FL-170100117. ",
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+        "text": "References ",
+        "text_level": 1,
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+            174,
+            90,
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+        "type": "text",
+        "text": "processing. RCA engineer, 29(6):33–41, 1984. [2] J. L. Ba, J. R. Kiros, and G. E. Hinton. Layer normalization. arXiv preprint arXiv:1607.06450, 2016.   \n[3] H. Bay, T. Tuytelaars, and L. Van Gool. Surf: Speeded up robust features. In European conference on computer vision, pages 404–417. Springer, 2006. [4] P. J. Burt and E. H. Adelson. The laplacian pyramid as a compact image code. In Readings in computer vision, pages 671–679. Elsevier, 1987.   \n[5] Z. Cai and N. Vasconcelos. Cascade r-cnn: Delving into high quality object detection. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 6154–6162, 2018. [6] C.-F. Chen, Q. Fan, and R. Panda. Crossvit: Cross-attention multi-scale vision transformer for image classification. arXiv preprint arXiv:2103.14899, 2021. [7] H. Chen, Y. Wang, T. Guo, C. Xu, Y. Deng, Z. Liu, S. Ma, C. Xu, C. Xu, and W. Gao. Pre-trained image processing transformer. arXiv preprint arXiv:2012.00364, 2020. [8] L.-C. Chen, G. Papandreou, F. Schroff, and H. Adam. Rethinking atrous convolution for semantic image segmentation. arXiv preprint arXiv:1706.05587, 2017.   \n[9] X. Chen, S. Xie, and K. He. An empirical study of training self-supervised vision transformers. arXiv preprint arXiv:2104.02057, 2021.   \n[10] X. Chu, Z. Tian, Y. Wang, B. Zhang, H. Ren, X. Wei, H. Xia, and C. Shen. Twins: Revisiting spatial attention design in vision transformers. arXiv preprint arXiv:2104.13840, 2021.   \n[11] X. Chu, Z. Tian, B. Zhang, X. Wang, X. Wei, H. Xia, and C. Shen. Conditional positional encodings for vision transformers. arXiv preprint arXiv:2102.10882, 2021.   \n[12] M. Contributors. MMSegmentation: Openmmlab semantic segmentation toolbox and benchmark. https: //github.com/open-mmlab/mmsegmentation, 2020.   \n[13] M. Contributors. Openmmlab pose estimation toolbox and benchmark. https://github.com/ open-mmlab/mmpose, 2020.   \n[14] Z. Dai, Z. Yang, Y. Yang, J. Carbonell, Q. V. Le, and R. Salakhutdinov. Transformer-xl: Attentive language models beyond a fixed-length context. arXiv preprint arXiv:1901.02860, 2019.   \n[15] S. d’Ascoli, H. Touvron, M. Leavitt, A. Morcos, G. Biroli, and L. Sagun. Convit: Improving vision transformers with soft convolutional inductive biases. arXiv preprint arXiv:2103.10697, 2021.   \n[16] H. Demirel and G. Anbarjafari. Image resolution enhancement by using discrete and stationary wavelet decomposition. IEEE transactions on image processing, 20(5):1458–1460, 2010.   \n[17] J. Devlin, M.-W. Chang, K. Lee, and K. Toutanova. Bert: Pre-training of deep bidirectional transformers for language understanding. arXiv preprint arXiv:1810.04805, 2018.   \n[18] X. Ding, X. Zhang, J. Han, and G. Ding. Repmlp: Re-parameterizing convolutions into fully-connected layers for image recognition. arXiv preprint arXiv:2105.01883, 2021.   \n[19] A. Dosovitskiy, L. Beyer, A. Kolesnikov, D. Weissenborn, X. Zhai, T. Unterthiner, M. Dehghani, M. Minderer, G. Heigold, S. Gelly, et al. An image is worth 16x16 words: Transformers for image recognition at scale. arXiv preprint arXiv:2010.11929, 2020.   \n[20] A. El-Nouby, H. Touvron, M. Caron, P. Bojanowski, M. Douze, A. Joulin, I. Laptev, N. Neverova, G. Synnaeve, J. Verbeek, et al. Xcit: Cross-covariance image transformers. arXiv preprint arXiv:2106.09681, 2021.   \n[21] B. Graham, A. El-Nouby, H. Touvron, P. Stock, A. Joulin, H. Jégou, and M. Douze. Levit: a vision transformer in convnet’s clothing for faster inference. arXiv preprint arXiv:2104.01136, 2021.   \n[22] M.-H. Guo, Z.-N. Liu, T.-J. Mu, and S.-M. Hu. Beyond self-attention: External attention using two linear layers for visual tasks. arXiv preprint arXiv:2105.02358, 2021.   \n[23] K. Han, A. Xiao, E. Wu, J. Guo, C. Xu, and Y. Wang. Transformer in transformer. arXiv preprint arXiv:2103.00112, 2021.   \n[24] K. He, G. Gkioxari, P. Dollár, and R. Girshick. Mask r-cnn. In Proceedings of the IEEE international conference on computer vision, pages 2961–2969, 2017.   \n[25] K. He, X. Zhang, S. Ren, and J. Sun. Spatial pyramid pooling in deep convolutional networks for visual recognition. IEEE transactions on pattern analysis and machine intelligence, 37(9):1904–1916, 2015.   \n[26] K. He, X. Zhang, S. Ren, and J. Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770–778, 2016.   \n[27] L. He, Y. Dong, Y. Wang, D. Tao, and Z. Lin. Gauge equivariant transformer. In Thirty-Fifth Conference on Neural Information Processing Systems, 2021.   \n[28] B. Heo, S. Yun, D. Han, S. Chun, J. Choe, and S. J. Oh. Rethinking spatial dimensions of vision transformers. In International Conference on Computer Vision (ICCV), 2021.   \n[29] B. Heo, S. Yun, D. Han, S. Chun, J. Choe, and S. J. Oh. Rethinking spatial dimensions of vision transformers. arXiv preprint arXiv:2103.16302, 2021.   \n[30] G. Hinton, O. Vinyals, and J. Dean. Distilling the knowledge in a neural network. In NIPS Deep Learning and Representation Learning Workshop, 2015.   \n[31] A. G. Howard, M. Zhu, B. Chen, D. Kalenichenko, W. Wang, T. Weyand, M. Andreetto, and H. Adam. Mobilenets: Efficient convolutional neural networks for mobile vision applications. arXiv preprint arXiv:1704.04861, 2017.   \n[32] G. Huang, Z. Liu, L. Van Der Maaten, and K. Q. Weinberger. Densely connected convolutional networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 4700–4708, vision transformer. arXiv preprint arXiv:2106.03650, 2021.   \n[34] S. Ioffe and C. Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. In International conference on machine learning, pages 448–456. PMLR, 2015.   \n[35] Y. Ke and R. Sukthankar. Pca-sift: A more distinctive representation for local image descriptors. In Proceedings of the IEEE conference on computer vision and pattern recognition, volume 2, pages II–II. IEEE, 2004.   \n[36] J. Krause, M. Stark, J. Deng, and L. Fei-Fei. 3d object representations for fine-grained categorization. In 4th International IEEE Workshop on 3D Representation and Recognition (3dRR-13), Sydney, Australia, 2013.   \n[37] A. Krizhevsky, G. Hinton, et al. Learning multiple layers of features from tiny images. 2009.   \n[38] A. Krizhevsky, I. Sutskever, and G. E. Hinton. Imagenet classification with deep convolutional neural networks. Advances in neural information processing systems, 25:1097–1105, 2012.   \n[39] W.-S. Lai, J.-B. Huang, N. Ahuja, and M.-H. Yang. Deep laplacian pyramid networks for fast and accurate super-resolution. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 624–632, 2017.   \n[40] Z. Lan, M. Chen, S. Goodman, K. Gimpel, P. Sharma, and R. Soricut. Albert: A lite bert for self-supervised learning of language representations. arXiv preprint arXiv:1909.11942, 2019.   \n[41] Y. LeCun, Y. Bengio, et al. Convolutional networks for images, speech, and time series. The handbook of brain theory and neural networks, 3361(10):1995, 1995.   \n[42] Y. LeCun, Y. Bengio, and G. Hinton. Deep learning. nature, 521(7553):436–444, 2015.   \n[43] Y. Li, K. Zhang, J. Cao, R. Timofte, and L. Van Gool. Localvit: Bringing locality to vision transformers. arXiv preprint arXiv:2104.05707, 2021.   \n[44] G. Lin, C. Shen, A. Van Den Hengel, and I. Reid. Efficient piecewise training of deep structured models for semantic segmentation. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 3194–3203, 2016.   \n[45] T.-Y. Lin, P. Dollár, R. Girshick, K. He, B. Hariharan, and S. Belongie. Feature pyramid networks for object detection. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 2117–2125, 2017.   \n[46] Y. Liu, M. Ott, N. Goyal, J. Du, M. Joshi, D. Chen, O. Levy, M. Lewis, L. Zettlemoyer, and V. Stoyanov. Roberta: A robustly optimized bert pretraining approach. arXiv preprint arXiv:1907.11692, 2019.   \n[47] Z. Liu, Y. Lin, Y. Cao, H. Hu, Y. Wei, Z. Zhang, S. Lin, and B. Guo. Swin transformer: Hierarchical vision transformer using shifted windows. arXiv preprint arXiv:2103.14030, 2021.   \n[48] I. Loshchilov and F. Hutter. Decoupled weight decay regularization. In International Conference on Learning Representations, 2018.   \n[49] W. Luo, Y. Li, R. Urtasun, and R. S. Zemel. Understanding the effective receptive field in deep convolutional neural networks. In Proceedings of the 30th International Conference on Neural Information Processing Systems, volume 29, pages 4898–4906, 2016.   \n[50] L. Melas-Kyriazi. Do you even need attention? a stack of feed-forward layers does surprisingly well on imagenet. arXiv: Computer Vision and Pattern Recognition, 2021.   \n[51] H. Nam, J.-W. Ha, and J. Kim. Dual attention networks for multimodal reasoning and matching. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 299–307, 2017.   \n[52] P. C. Ng and S. Henikoff. Sift: Predicting amino acid changes that affect protein function. Nucleic acids research, 31(13):3812–3814, 2003.   \n[53] M.-E. Nilsback and A. Zisserman. Automated flower classification over a large number of classes. In Indian Conference on Computer Vision, Graphics and Image Processing, Dec 2008.   \n[54] S. W. Oh, J.-Y. Lee, N. Xu, and S. J. Kim. Video object segmentation using space-time memory networks. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 9226–9235, 2019.   \n[55] H. Olkkonen and P. Pesola. Gaussian pyramid wavelet transform for multiresolution analysis of images. Graphical Models and Image Processing, 58(4):394–398, 1996.   \n[56] O. M. Parkhi, A. Vedaldi, A. Zisserman, and C. V. Jawahar. Cats and dogs. In IEEE Conference on Computer Vision and Pattern Recognition, 2012.   \n[57] A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, A. Desmaison, A. Kopf, E. Yang, Z. DeVito, M. Raison, A. Tejani, S. Chilamkurthy, B. Steiner, L. Fang, J. Bai, and S. Chintala. Pytorch: An imperative style, high-performance deep learning library. In Advances in Neural Information Processing Systems, volume 32, pages 8026–8037, 2019.   \n[58] Z. Peng, W. Huang, S. Gu, L. Xie, Y. Wang, J. Jiao, and Q. Ye. Conformer: Local features coupling global representations for visual recognition. arXiv preprint arXiv:2105.03889, 2021.   \n[59] F. Perazzi, J. Pont-Tuset, B. McWilliams, L. Van Gool, M. Gross, and A. Sorkine-Hornung. A benchmark dataset and evaluation methodology for video object segmentation. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 724–732, 2016.   \n[60] J. Pont-Tuset, F. Perazzi, S. Caelles, P. Arbeláez, A. Sorkine-Hornung, and L. Van Gool. The 2017 davis challenge on video object segmentation. arXiv preprint arXiv:1704.00675, 2017.   \n[61] A. Radford, J. Wu, R. Child, D. Luan, D. Amodei, and I. Sutskever. Language models are unsupervised multitask learners. OpenAI blog, 1(8):9, 2019.   \n[62] I. Radosavovic, R. P. Kosaraju, R. Girshick, K. He, and P. Dollár. Designing network design spaces. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 10428–10436, 2020.   \n[63] E. Rublee, V. Rabaud, K. Konolige, and G. Bradski. Orb: An efficient alternative to sift or surf. In Proceedings of the IEEE international conference on computer vision, pages 2564–2571. Ieee, 2011.   \n[64] S. Sabour, N. Frosst, and G. E. Hinton. Dynamic routing between capsules. arXiv preprint arXiv:1710.09829, 2017.   \n[65] M. Sandler, A. Howard, M. Zhu, A. Zhmoginov, and L.-C. Chen. Mobilenetv2: Inverted residuals and linear bottlenecks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 4510–4520, 2018.   \n[66] R. R. Selvaraju, M. Cogswell, A. Das, R. Vedantam, D. Parikh, and D. Batra. Grad-cam: Visual explanations from deep networks via gradient-based localization. In Proceedings of the IEEE international conference on computer vision, pages 618–626, 2017.   \n[67] P. Shaw, J. Uszkoreit, and A. Vaswani. Self-attention with relative position representations. arXiv preprint arXiv:1803.02155, 2018.   \n[68] K. Simonyan and A. Zisserman. Very deep convolutional networks for large-scale image recognition. arXiv preprint arXiv:1409.1556, 2014.   \n[69] A. Srinivas, T.-Y. Lin, N. Parmar, J. Shlens, P. Abbeel, and A. Vaswani. Bottleneck transformers for visual recognition. arXiv preprint arXiv:2101.11605, 2021.   \n[70] C. Szegedy, S. Ioffe, V. Vanhoucke, and A. Alemi. Inception-v4, inception-resnet and the impact of residual connections on learning. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 31, 2017.   \n[71] C. Szegedy, W. Liu, Y. Jia, P. Sermanet, S. Reed, D. Anguelov, D. Erhan, V. Vanhoucke, and A. Rabinovich. Going deeper with convolutions. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 1–9, 2015.   \n[72] C. Szegedy, V. Vanhoucke, S. Ioffe, J. Shlens, and Z. Wojna. Rethinking the inception architecture for computer vision. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 2818–2826, 2016.   \n[73] M. Tan and Q. Le. Efficientnet: Rethinking model scaling for convolutional neural networks. In International Conference on Machine Learning, pages 6105–6114. PMLR, 2019.   \n[74] I. Tolstikhin, N. Houlsby, A. Kolesnikov, L. Beyer, X. Zhai, T. Unterthiner, J. Yung, D. Keysers, J. Uszkoreit, M. Lucic, and A. Dosovitskiy. Mlp-mixer: An all-mlp architecture for vision. arXiv preprint arXiv:2105.01601, 2021.   \n[75] H. Touvron, P. Bojanowski, M. Caron, M. Cord, A. El-Nouby, E. Grave, A. Joulin, G. Synnaeve, J. Verbeek, and H. Jégou. Resmlp: Feedforward networks for image classification with data-efficient training. arXiv preprint arXiv:2105.03404, 2021.   \n[76] H. Touvron, M. Cord, M. Douze, F. Massa, A. Sablayrolles, and H. Jégou. Training data-efficient image transformers & distillation through attention. arXiv preprint arXiv:2012.12877, 2020.   \n[77] H. Touvron, M. Cord, A. Sablayrolles, G. Synnaeve, and H. Jégou. Going deeper with image transformers. arXiv preprint arXiv:2103.17239, 2021.   \n[78] H. Touvron, A. Sablayrolles, M. Douze, M. Cord, and H. Jégou. Grafit: Learning fine-grained image representations with coarse labels. arXiv preprint arXiv:2011.12982, 2020.   \n[79] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin. Attention is all you need. In Proceedings of the 31st International Conference on Neural Information Processing Systems, volume 30, pages 5998–6008, 2017.   \n[80] W. Wang, E. Xie, X. Li, D.-P. Fan, K. Song, D. Liang, T. Lu, P. Luo, and L. Shao. Pyramid vision transformer: A versatile backbone for dense prediction without convolutions. arXiv preprint arXiv:2102.12122, 2021.   \n[81] X. Wang, R. Girshick, A. Gupta, and K. He. Non-local neural networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 7794–7803, 2018.   \n[82] R. Wightman. Pytorch image models. https://github.com/rwightman/pytorch-image-models, 2019.   \n[83] H. Wu, B. Xiao, N. Codella, M. Liu, X. Dai, L. Yuan, and L. Zhang. Cvt: Introducing convolutions to vision transformers. arXiv preprint arXiv:2103.15808, 2021.   \n[84] B. Xiao, H. Wu, and Y. Wei. Simple baselines for human pose estimation and tracking. In Proceedings of the European Conference on Computer Vision (ECCV), September 2018.   \n[85] T. Xiao, Y. Liu, B. Zhou, Y. Jiang, and J. Sun. Unified perceptual parsing for scene understanding. In Proceedings of the European Conference on Computer Vision (ECCV), pages 418–434, 2018.   \n[86] J. Xie, R. Zeng, Q. Wang, Z. Zhou, and P. Li. So-vit: Mind visual tokens for vision transformer. arXiv preprint arXiv:2104.10935, 2021.   \n[87] S. Xie, R. Girshick, P. Dollár, Z. Tu, and K. He. Aggregated residual transformations for deep neural networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 1492–1500, 2017.   \n[88] W. Xu, Y. Xu, T. Chang, and Z. Tu. Co-scale conv-attentional image transformers. In Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), pages 9981–9990, October 2021.   \n[89] H. Yan, Z. Li, W. Li, C. Wang, M. Wu, and C. Zhang. Contnet: Why not use convolution and transformer at the same time? arXiv preprint arXiv:2104.13497, 2021.   \n[90] F. Yu and V. Koltun. Multi-scale context aggregation by dilated convolutions. In ICLR 2016 : International Conference on Learning Representations 2016, 2016.   \n[91] F. Yu, V. Koltun, and T. Funkhouser. Dilated residual networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 472–480, 2017.   \n[92] K. Yuan, S. Guo, Z. Liu, A. Zhou, F. Yu, and W. Wu. Incorporating convolution designs into visual transformers. arXiv preprint arXiv:2103.11816, 2021.   \n[93] L. Yuan, Y. Chen, T. Wang, W. Yu, Y. Shi, Z. Jiang, F. E. Tay, J. Feng, and S. Yan. Tokens-to-token vit: Training vision transformers from scratch on imagenet. arXiv preprint arXiv:2101.11986, 2021.   \n[94] M. D. Zeiler and R. Fergus. Visualizing and understanding convolutional networks. In European conference on computer vision, pages 818–833. Springer, 2014.   \n[95] X. Zhang, X. Zhou, M. Lin, and J. Sun. Shufflenet: An extremely efficient convolutional neural network for mobile devices. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 6848–6856, 2018.   \n[96] H. Zhao, J. Shi, X. Qi, X. Wang, and J. Jia. Pyramid scene parsing network. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 2881–2890, 2017.   \n[97] S. Zheng, J. Lu, H. Zhao, X. Zhu, Z. Luo, Y. Wang, Y. Fu, J. Feng, T. Xiang, P. H. S. Torr, and L. Zhang. Rethinking semantic segmentation from a sequence-to-sequence perspective with transformers. arXiv preprint arXiv:2012.15840, 2020.   \n[98] B. Zhou, H. Zhao, X. Puig, S. Fidler, A. Barriuso, and A. Torralba. Scene parsing through ade20k dataset. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 633–641, 2017.   \n[99] B. Zhou, H. Zhao, X. Puig, T. Xiao, S. Fidler, A. Barriuso, and A. Torralba. Semantic understanding of scenes through the ade20k dataset. International Journal of Computer Vision, 127(3):302–321, 2019.   \n[100] X. Zhu, W. Su, L. Lu, B. Li, X. Wang, and J. Dai. Deformable detr: Deformable transformers for end-to-end object detection. In International Conference on Learning Representations, 2021. ",
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