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+    {
+        "type": "text",
+        "text": "Scaling Vision with Sparse Mixture of Experts ",
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+        "type": "text",
+        "text": "Carlos Riquelme ∗ Google Brain ",
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+    {
+        "type": "text",
+        "text": "Joan Puigcerver \\* Google Brain ",
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+    },
+    {
+        "type": "text",
+        "text": "Basil Mustafa \\* Google Brain ",
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+    },
+    {
+        "type": "text",
+        "text": "Maxim Neumann Google Brain ",
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+    },
+    {
+        "type": "text",
+        "text": "Rodolphe Jenatton Google Brain ",
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+    },
+    {
+        "type": "text",
+        "text": "André Susano Pinto Google Brain ",
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+    },
+    {
+        "type": "text",
+        "text": "Daniel Keysers Google Brain ",
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+    {
+        "type": "text",
+        "text": "Neil Houlsby Google Brain ",
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+        "text": "Sparsely-gated Mixture of Experts networks (MoEs) have demonstrated excellent scalability in Natural Language Processing. In Computer Vision, however, almost all performant networks are “dense”, that is, every input is processed by every parameter. We present a Vision MoE (V-MoE), a sparse version of the Vision Transformer, that is scalable and competitive with the largest dense networks. When applied to image recognition, V-MoE matches the performance of state-ofthe-art networks, while requiring as little as half of the compute at inference time. Further, we propose an extension to the routing algorithm that can prioritize subsets of each input across the entire batch, leading to adaptive per-image compute. This allows V-MoE to trade-off performance and compute smoothly at test-time. Finally, we demonstrate the potential of V-MoE to scale vision models, and train a 15B parameter model that attains $9 0 . 3 5 \\%$ on ImageNet. ",
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+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+        "text": "Deep learning historically shows that increasing network capacity and dataset size generally improves performance. In computer vision, large models pre-trained on large datasets often achieve the state of the art [57, 50, 36, 20, 3]. This approach has had even more success in Natural Language Processing (NLP), where large pre-trained models are ubiquitous, and perform very well on many tasks [48, 18]. Text Transformers [61] are the largest models to date, some with over 100B parameters [9]. However, training and serving such models is expensive [56, 46]. This is partially because these deep networks are typically “dense”– every example is processed using every parameter –thus, scale comes at high computational cost. In contrast, conditional computation [5] aims to increase model capacity while keeping the training and inference cost roughly constant by applying only a subset of parameters to each example. In NLP, sparse Mixture of Experts (MoEs) are gaining popularity [54, 39, 22], enabling training and inference with fewer resources while unlocking trillion parameter models. ",
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+        "text": "In this work, we explore conditional computation for vision at scale. We introduce the Vision MoE (V-MoE), a sparse variant of the recent Vision Transformer (ViT) architecture [20] for image classification. The V-MoE replaces a subset of the dense feedforward layers in ViT with sparse MoE layers, where each image patch is “routed” to a subset of “experts” (MLPs). Due to unique failure modes and non-differentiability, routing in deep sparse models is challenging. We explore various design choices, and present an effective recipe for the pre-training and transfer of V-MoE, notably outperforming their dense counterparts. We further show that V-MoE models are remarkably flexible. The performance vs. inference-cost trade-off of already trained models can be smoothly adjusted during inference by modulating the sparsity level with respect to the input and/or the model weights. Also, we open-source our implementation and a number of V-MoE models trained on ImageNet-21k.2 ",
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+        "image_caption": [
+            "Figure 1: Overview of the architecture. V-MoE is composed of $L$ ViT blocks. In some, we replace the MLP with a sparsely activated mixture of MLPs. Each MLP (the expert) is stored on a separate device, and processes a fixed number of tokens. The communication of these tokens between devices =  expert uses a capacity ratio C = 43 : the sparse MoE layer receives 12 tokens per device, but each is shown in this example, which depicts the case when $k = 1$ expert is selected per token. Here each expert has capacity for 16 ( $\\textstyle \\frac { 1 6 \\cdot 1 } { 1 2 } = \\frac { 4 } { 3 }$ ; see Section 2.4). Non-expert components of V-MoE such as routers, attention layers and normal MLP blocks are replicated identically across devices. "
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+        "type": "text",
+        "text": "With V-MoE, we can scale to model sizes of 15B parameters, the largest vision models to date. We match the performance of state-of-the-art dense models, while requiring fewer time to train. Alternatively, V-MoE can match the cost of ViT while achieving better performance. To help control this tradeoff, we propose Batch Prioritized Routing, a routing algorithm that repurposes model sparsity to skip the computation of some patches, reducing compute on uninformative image regions. ",
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+        "type": "text",
+        "text": "We summarize our main contributions as follows: ",
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+        "text": "Vision models at scale. We present the Vision Mixture of Experts, a distributed sparsely-activated Transformer model for vision. We train models with up to $2 4 \\mathrm { M o E }$ layers, 32 experts per layer, and almost 15B parameters. We show that these models can be stably trained, seamlessly used for transfer, and successfully fine-tuned with as few as 1 000 datapoints. Moreover, our largest model achieves $9 0 . 3 5 \\%$ test accuracy on ImageNet when fine-tuned. ",
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+        "text": "Performance and inference. We show V-MoEs strongly outperform their dense counterparts on upstream, few-shot and full fine-tuning metrics in absolute terms. Moreover, at inference time, the V-MoE models can be adjusted to either (i) match the largest dense model’s performance while using as little as half the compute, or actual runtime, or (ii) significantly outperform it at the same cost. ",
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+        "text": "Batch Prioritized Routing. We propose a new priority-based routing algorithm that allows V-MoEs to discard the least useful patches. Thus, we devote less compute to each image. In particular, we show V-MoEs match the performance of the dense models while saving $20 \\%$ of the training FLOPs. Analysis. We provide some visualization of the routing decisions, revealing patterns and conclusions which helped motivate design decisions and may further improve understanding in the field. ",
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+        "type": "text",
+        "text": "2 The Vision Mixture of Experts ",
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+        "text": "We first describe MoEs and sparse MoEs. We then present how we apply this methodology to vision, before explaining our design choices for the routing algorithm and the implementation of V-MoEs. ",
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+        "text": "2.1 Conditional Computation with MoEs ",
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+        "text": "Conditional computation aims at activating different subsets of a network for different inputs [5]. A mixture-of-experts model is a specific instantiation whereby different model “experts” are responsible for different regions of the input space [31]. ",
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+        "text": "We follow the setting of [54], who present for deep learning a mixture of experts layer with $E$ experts as $\\begin{array} { r } { \\mathrm { M o E } ( \\mathbf { x } ) = \\sum _ { i = 1 } ^ { E } g ( \\mathbf { x } ) _ { i } e _ { i } ( \\mathbf { x } ) } \\end{array}$ where $\\mathbf { x } \\in \\mathbb { R } ^ { D }$ is the input to the layer, $\\boldsymbol { e } _ { i } : \\mathbb { R } ^ { \\boldsymbol { \\bar { D } } } \\mapsto \\mathbb { R } ^ { D }$ the function computed by expert $i$ , and $g : \\mathbb { R } ^ { D } \\mapsto \\mathbb { R } ^ { E }$ is the “routing” function which prescribes the input-conditioned weight for the experts. Both $e _ { i }$ and $g$ are parameterized by neural networks. As defined, this is still a dense network. However, if $g$ is sparse, i.e., restricted to assign only $k \\ll E$ non-zero weights, then unused experts need not be computed. This unlocks super-linear scaling of the number of model parameters with respect to inference and training compute. ",
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+        "type": "text",
+        "text": "",
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+    {
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+        "text": "2.2 MoEs for Vision ",
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+        "text": "We explore the application of sparsity to vision in the context of the Vision Transformer (ViT) [20]. ViT has been shown to scale well in the transfer learning setting, attaining better accuracies than CNNs with less pre-training compute. ViT processes images as a sequence of patches. An input image is first divided into a grid of equal-sized patches. These are linearly projected to the Transformer’s [61] hidden size. After adding positional embeddings, the patch embeddings (tokens) are processed by a Transformer, which consists predominately of alternating self-attention and MLP layers. ",
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+        "text": "The MLPs have two layers and a GeLU [29] non-linearity: $\\mathrm { M L P } ( \\mathbf { x } ) = \\mathbf { W } _ { \\mathrm { 2 } } \\ \\sigma _ { \\mathrm { g e l u } } ( \\mathbf { W } _ { \\mathrm { 1 } } \\mathbf { x } )$ . For Vision MoE, we replace a subset of these with MoE layers, where each expert is an MLP; see Figure 1. The experts have the same architecture $e _ { i } ( { \\bf x } ) = \\mathrm { M L P } _ { \\theta _ { i } } ( { \\bf x } )$ but with different weights $\\theta _ { i } = \\left( \\mathbf { W } _ { 1 } ^ { i } , \\mathbf { W } _ { 2 } ^ { i } \\right)$ . =This follows a similar design pattern as the M4 machine translation model [39]. ",
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+        "type": "text",
+        "text": "2.3 Routing ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "For each MoE layer in V-MoE, we use the routing function $g ( \\mathbf { x } ) \\mathbf { \\Psi } = \\mathrm { T O P } _ { k }$ softmax $\\left( \\mathbf { W } \\mathbf { x } + \\epsilon \\right)$ , where $\\mathrm { T O P } _ { k }$ is an operation that sets all elements of the vector to zero except the elements with the largest $k$ values, and $\\epsilon$ is sampled independently $\\epsilon \\sim \\mathcal { N } ( 0 , \\frac { 1 } { E ^ { 2 } } )$ entry-wise. In practice, we use $k = 1$ or $k = 2$ . In the context of the Vision Transformer, $\\mathbf { x }$ =is a representation of an image token at some =layer of the network. Therefore, V-MoE routes patch representations, not entire images. ",
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+        "type": "text",
+        "text": "The difference between previous formulations [54] is that we apply $\\mathrm { T O P } _ { k }$ after the softmax over experts weights [39], instead of before. This allows us to train with $k = 1$ (otherwise gradients with respect to routings are zero almost everywhere) and also performs better for $k > 1$ (see Appendix A). ",
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+        "text": "Finally, we add a small amount of noise with standard deviation $\\frac { 1 } { E }$ to the activations $\\mathbf { W } \\mathbf { x }$ . We empirically found this performed well but that the setup was robust to this parameter. The noise typically altered routing decisions ${ \\sim } 1 5 \\%$ of the time in earlier layers, and ${ \\sim } 2 { - } 3 \\%$ in deeper layers. ",
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+    {
+        "type": "text",
+        "text": "2.4 Expert’s Buffer Capacity ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "During training, sparse models may favor only a small set of experts [26, 52]. This common failure mode can cause two problems. First, statistical inefficiency: in the limit of collapse to a single expert, the model is no more powerful than a dense model. Second, computational inefficiency: imbalanced assignment of items to experts may lead to a poor hardware utilization. ",
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+        "text": "To combat imbalance and simplify our implementation, we fix the buffer capacity of each expert (i.e. the number of tokens that each expert processes), and train our model with auxiliary losses that encourage load balancing. This is essentially the same approach as followed by [54, 39, 22]. In our case, we use slight variants of two of the auxiliary losses proposed in [54], as described in Appendix A. ",
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+        "text": "We define the buffer capacity of an expert $( B _ { e } )$ as a function of the number of images in the batch number of experts $( N )$ , the number of tokens per image $( E )$ , and the capacity ratio $( P )$ , the number of selected experts per token $( C )$ : $B _ { e } =$ round $\\textstyle \\left( { \\frac { k N P C } { E } } \\right)$ . $( k )$ , the total ",
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+        "type": "text",
+        "text": "If the router assigns more than $B _ { e }$ tokens to a given expert, only $B _ { e }$ of them are processed. The remaining tokens are not entirely ‘lost’ as their information is preserved by residual connections (the top diagram of Figure 1). Also, if $k > 1$ , several experts try to process each token. Tokens are never >fully discarded. If an expert is assigned fewer than $B _ { e }$ tokens, the rest of its buffer is zero-padded. ",
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+        "type": "text",
+        "text": "We use the capacity ratio to adjust the capacity of the experts. With $C > 1$ , a slack capacity is added to account for a potential routing imbalance. This is typically useful for fine-tuning when the new data might come from a very different distribution than during upstream training. With $C < 1$ , the router is forced to ignore some assignments. In Section 4 we propose a new algorithm that takes advantage of setting $C \\ll 1$ to discard the least useful tokens and save compute during inference. ",
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+        "type": "text",
+        "text": "3 Transfer Learning ",
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+        "text": "In this section, we first present training different variants of V-MoE on a large dataset (Section 3.2) in order to be used for Transfer Learning afterwards. The ability to easily adapt our massive models to new tasks, using a small amount of data from the new task, is extremely valuable: it allows to amortize the cost of pre-training across multiple tasks. We consider two different approaches to Transfer Learning: linear few-shot learning on fixed representations and full fine-tuning of the model. ",
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+    {
+        "type": "text",
+        "text": "3.1 Models ",
+        "text_level": 1,
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+        "text": "We build V-MoE on different variants of ViT [20]: ViT-S(mall), ViT-B(ase), ViT-L(arge) and ViTH(uge), the hyperparameters of which are described in Appendix B.5. There are three additional major design decisions that affect the cost (and potentially the quality) of our model: ",
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+    {
+        "type": "text",
+        "text": "Number of MoE layers. Following [39], we place the MoEs on every other layer (we refer to these as V-MoE Every-2). In addition, we experimented with using fewer MoE layers, by placing them on the last- $\\boldsymbol { n }$ even blocks (thus we dub these V-MoE Last-n). In Appendix E.1 we observe that, although using fewer MoE layers decreases the number of parameters of the model, it has typically little impact on quality and can speed-up the models significantly, since less communication overhead is incurred. ",
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+    {
+        "type": "text",
+        "text": "Number of selected experts $k$ : The cost of our model does not depend on the total number of experts but the number of selected ones per token. Concurrent works in NLP fix $k = 1$ [22] or $k = 2$ [54, 39]. In our case, we use by default $k = 2$ (see Figure 10 in Appendix B for the exploration of different values of $k$ ), while we found the total number of experts $E = 3 2$ to be the sweet spot in our setting. ",
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+    {
+        "type": "text",
+        "text": "Buffer capacity $C$ : As mentioned in Section 2.4, we use a fixed buffer capacity. While this is typically regarded as a downside or engineering difficulty to implement these models, we can adjust the capacity ratio to control different trade-offs. We can intentionally set it to a low ratio to save compute, using Batch Prioritized Routing (see Section 4). During upstream training, we set $C = 1 . 0 5$ by default to give a small amount of slack without increasing the cost noticeably. ",
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+    {
+        "type": "text",
+        "text": "Note that for a given trained model, the latter two— $k$ and $C$ —can be adjusted without further training, whereas the positioning and quantity of expert layers is effectively fixed to match pre-training. ",
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+    {
+        "type": "text",
+        "text": "3.2 Data ",
+        "text_level": 1,
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+        "text": "We pre-train our models on JFT-300M [57], a semi-automatically noisy-labeled dataset. It has $\\sim 3 0 5 \\mathrm { M }$ training and 50 000 validation images, organised in a hierarchy of 18 291 classes (average 1.89 labels per image). We deduplicate it with respect to all our validation/test sets as in previous efforts [36].3 ",
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+    {
+        "type": "text",
+        "text": "Our few-shot experiments on ImageNet (i.e. ILSVRC2012) use only 1, 5, or 10 shots per class to adapt the upstream model, evaluating the resulting model on the validation set. ",
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+        "text": "We also fine-tuned the pre-trained models on the full training set (ca. 1M images). We report performance in a similar regime for four other datasets in Appendix B.5. Lastly, we explore the ability to fine-tune our large models in the low-data regime by evaluating them on the Visual Task Adaptation Benchmark (VTAB) [69], a diverse suite of 19 tasks with only 1 000 data points per task. As well as natural image classification, VTAB includes specialized tasks (e.g. medical or satellite imagery) and structured tasks (e.g. counting or assessing rotation/distance). ",
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+        "text": "3.3 Upstream results ",
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+        "text": "JFT is a multilabel dataset, so we measure model performance via precision $@ 1$ (see Appendix B.6 for details). Note that as in previous works [20], hyperparameters were tuned for transfer performance, and JFT precision could be improved at the expense of downstream tasks e.g. by reducing weight decay. Figure 2a shows the quality of different V-MoE and ViT variants with respect to total training compute and time. It shows models that select $k = 2$ experts and place MoEs in the last $n$ even blocks $\\hslash = 5$ for V-MoE-H, $n = 2$ otherwise), but the best results are achieved by V-MoE-H/14 Every-2 (see Table 2, 14 is the patch size). L/16’s are trained for 7 or 14 epochs. See Appendix B.5 for all results. ",
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+        "img_path": "images/c9583ef8573f88b5ab52bb6cdc466f3a72778204fe31d121cb2429ff9c12592a.jpg",
+        "image_caption": [
+            "Figure 2: JFT-300M Precision $@ 1$ and ImageNet 5-shot accuracy. Colors represent different ViT variants, markers represent either standard ViT or V-MoEs on the last $n$ even blocks. The lines represent the Pareto frontier of ViT (dashed) and V-MoE (solid) variants. "
+        ],
+        "image_footnote": [],
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+    {
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+        "img_path": "images/40e9744cbeb03d4891d930be558dc399a68ae80031ccc224b2837d477dced43f.jpg",
+        "image_caption": [
+            "Figure 3: ImageNet Fine-Tuning Accuracy. Colors represent different VIT variants, markers represent either standard ViT or V-MoEs on the last $n$ even blocks. Lines show the Pareto frontier of VIT (dashed) and V-MoE (solid). "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "table",
+        "img_path": "images/f4640af04c8dd45ba58858379b5fca7caf4214ec75ce47e59bac0305765ba3b4.jpg",
+        "table_caption": [
+            "Table 1: VTAB. Scores and $9 5 \\%$ confidence intervals for ViT and V-MoE. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>ViT</td><td>V-MoE</td></tr><tr><td>L/16 76.3±0.5</td><td>77.2±0.4</td></tr><tr><td>H/14 77.6±0.2</td><td>77.8±0.4</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Expert models provide notable gains across all model sizes, for only a mild increase in FLOPs, establishing a new Pareto frontier (gray lines). Alternatively, we can match or improve performance of ViT models at lower cost (e.g. V-MoE-L/16 improves upon ViT-H/14). Similar conclusions hold for training time, which includes communication overhead of dispatching data across devices. ",
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+    {
+        "type": "text",
+        "text": "3.4 Linear few-shot results ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We evaluate the quality of the representations learned using few-shot linear transfer. Given training examples from the new dataset $\\{ ( X , Y ) _ { i } \\}$ , we use the pre-trained model $\\mathcal { M }$ to extract a fixed representation $\\mathcal { M } ( x _ { i } )$ of each image. We fit a linear regression model mapping $\\mathcal { M } ( x _ { i } )$ to the one-hot encoding of the target labels $Y _ { i }$ , following [20] (see [27, Chapter 5] for background). ",
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+    {
+        "type": "text",
+        "text": "Figure 2b shows that the upstream gains are preserved under 5-shot ImageNet evaluation, considering both compute and time; in other words, the quality of the representations learned by V-MoE also outperforms ViT models when looking at a new task. Table 2 further shows the results on $\\{ 1 , 1 0 \\}$ -shot for some selected models, and the full detailed results are available in Appendix B.5. ",
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+    {
+        "type": "text",
+        "text": "3.5 Full fine-tuning results ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The typically most performant approach for Transfer Learning [19] consists of replacing the upstream classification head with a new task-specific one and fine-tuning the whole model. Though one may expect that massive models like V-MoEs require special handling for fine-tuning, we broadly follow the standard fine-tuning protocol for Vision Transformers. We use the auxiliary loss during fine-tuning as well, although we observe that it is often not needed in this step, as the router is already well trained. We explore the two sets of tasks considered therein: ",
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+        "type": "image",
+        "img_path": "images/50ec35f7ecf28e9c0130e04ba518b414285354dcdb6207d834c1ac8587a7979a.jpg",
+        "image_caption": [
+            "Figure 4: White patches are discarded tokens in the first layer of experts, for different capacities, using Batch Prioritized Routing (Section 4.1) with a V-MoE-H/14. See Appendix D for more examples. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Full data. We follow the setup of [20], except that we apply a dropout rate of 0.1 on the expert MLPs (as done in [22]), and we halve the number of fine-tuning steps for all datasets other than ImageNet. Figure 3 shows the results on ImageNet (averaged over three runs). Here, V-MoE also performs better than dense counterparts, though we suspect the fine-tuning protocol could be further improved and tailored to the sparse models. See Table 8 for all details, including results on other datasets. ",
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+    {
+        "type": "text",
+        "text": "Low-data regime. On the VTAB benchmark, we use a similar setup and hyperparameter budget as [20] (but fine-tune with half the schedule length). Table 1 shows that, while performance is similar for V-MoE-H/14, experts provide significant gains at the ViT-L/16 level, indicating that despite the large size of these models, they can still be fine-tuned with small amounts of data and no further tricks. ",
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+    {
+        "type": "text",
+        "text": "3.6 Scaling up V-MoE ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Finally, we test how well V-MoE can scale vision models to a very large number of parameters, while continuing to improve performance. For this, we increase the size of the model and use a larger pre-training dataset: JFT-3B is a larger version of JFT-300M, it contains almost 3B images and is noisily annotated with 30k classes. Inspired by [68], we apply the changes detailed in Appendix B.3, and train a 48-block V-MoE model, with every-2 expert placement (32 experts and $k = 2$ ), resulting in a model with 14.7B parameters, which we denote by V-MoE-15B. ",
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+        "type": "text",
+        "text": "We successfully train V-MoE-15B, which is, as far as we are aware, the largest vision model to date. It has an impressive $8 2 . 7 8 \\%$ accuracy on 5-shot ImageNet and $9 0 . 3 5 \\%$ when fully fine-tuned, as shown in Appendix B.5, which also includes more details about the model. Training this model required $1 6 . 8 \\mathrm { k }$ TPUv3-core-days. To contextualize this result, the current state of the art on ImageNet is Meta Pseudo-Labelling (MPL) [49]. MPL trains an EfficientNet-based model on unlabelled JFT-300M using ImageNet pseudo-labelling, achieving $9 0 . 2 \\%$ while requiring $2 2 . 5 \\mathrm { k }$ TPUv3-core-days. ",
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+    {
+        "type": "text",
+        "text": "4 Skipping Tokens with Batch Prioritized Routing ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We present a new routing algorithm that allows the model to prioritize important tokens (corresp. patches). By simultaneously reducing the capacity of each expert, we can discard the least useful tokens. Intuitively, not every patch is equally important to classify a given image, e.g., most background patches can be dropped to let the model only focus on the ones with the relevant entities. ",
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+        "type": "text",
+        "text": "4.1 From Vanilla Routing to Batch Prioritized Routing ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "With the notation from Section 2, the routing function $\\mathbf { X } \\in \\mathbb { R } ^ { N \\cdot P \\times D }$ . A batch contains $N$ images composed of $P$ $g$ is applied row-wise to a batch of inputs tokens each; each row of $\\mathbf { X }$ corresponds to the $D$ -dimensional representation of a particular token of an image. Accordingly, $g ( \\mathbf { X } ) _ { t , i } \\in \\mathbb { R }$ denotes the routing weight for the $t$ -th token and the $i$ -th expert. ",
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+        "image_caption": [
+            "Figure 5: Reducing compute with priority routing. Performance vs. inference FLOPs for large models. V-MoEs with the original vanilla routing are represented by $\\bullet$ , while $\\mid$ shows V-MoEs where BPR and a mix of $C \\in \\{ 0 . 6 , 0 . 7 , 0 . 8 \\}$ and $k \\in \\{ 1 , 2 \\}$ are used to reduce compute. ViT models shown as $\\mathbf { x }$ . "
+        ],
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+        "image_caption": [
+            "Figure 6: Priority routing works where vanilla fails. Performance vs. inference capacity ratio for a V-MoE-H/14 model with $k = 2$ . Even for large $C$ ’s BPR outperforms vanilla; at low $C$ the difference is stark. BPR is competitive with dense by processing only $1 5 { - } 3 0 \\%$ of the tokens. "
+        ],
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+    {
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+    {
+        "type": "text",
+        "text": "In all routing algorithms considered, for $i < j$ , every TOP- $i$ assignment has priority over any TOP- $j$ <assignment. The router first tries to dispatch all $i ^ { \\mathrm { { t h } } }$ expert choices before assigning any $j ^ { \\mathrm { t h } }$ choice4. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Given the TOP- $\\cdot i$ position, the default—or vanilla—routing, as used in [54, 39, 22], assigns tokens to experts as follows. It sequentially goes over the rows of $g ( \\mathbf { X } )$ and assigns each token to its TOP- $i$ expert when the expert’s buffer is not full. As a result, priority is given to tokens depending on the rank of their corresponding row. While images in a batch are randomly ordered, tokens within an image follow a pre-defined fixed order. The algorithm is detailed in Algorithm 1 of Appendix C. ",
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+    {
+        "type": "text",
+        "text": "Batch Prioritized Routing (BPR). To favour the “most important” tokens, we propose to compute a priority score $s ( \\mathbf { x } )$ on each token, and sort $g ( \\mathbf { X } )$ accordingly before proceeding with the allocation. We sort tokens based on their maximum routing weight, formally $s ( \\mathbf { \\bar { X } } ) _ { t } = \\operatorname* { m a x } _ { i } g ( \\mathbf { X } ) _ { t , i }$ . The sum of TOP- $k$ weights, i.e. $s ( \\mathbf { X } ) _ { t } = \\sum _ { i } g ( \\mathbf { X } ) _ { t , i }$ =, worked equally well. These two simple approaches =outperformed other options we explored, e.g., directly parameterising and learning the function $s$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We reuse the router outputs as a proxy for the priority of allocation. Our experiments show this preserves the performant predictive behaviour of the model, even though the router outputs primarily encode how well tokens and experts can be paired, not the token’s “importance” for the final classification task. Figure 4 visualizes token prioritisation with Batch Prioritized Routing for increasingly small capacities. Since all tokens across all images in the batch $\\mathbf { X }$ compete with each other, different images may receive different amounts of compute. We summarize BPR in Algorithm 2, in Appendix C. ",
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+    {
+        "type": "text",
+        "text": "4.2 Skip tokens with low capacity $C$ ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Batch Prioritized Routing opens the door to reducing the buffer size by smartly selecting which tokens to favor. This can have a dramatic impact in the computational cost of the overall sparse model. We discuss now inference and training results with $C$ defined in Section 2.4 in the regime $C \\ll 1$ . ",
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+    {
+        "type": "text",
+        "text": "At inference time. Prioritized routing is agnostic to how the model was originally trained. Figure 6 shows the effect of reducing compute at inference time by using BPR versus vanilla routing, on a V-MoE-H/14 model trained using vanilla routing. The difference in performance between both methods is remarkable —especially for $C \\leq 0 . 5$ , where the model truly starts fully dropping tokens, as $k = 2$ . Also, BPR allows the model to be competitive with the dense one even at quite low capacities. =As shown in Figure 5 for V-MoE-L/16 and V-MoE-H/14, Batch Prioritized Routing and low $C$ allow V-MoE to smoothly trade-off performance and FLOPS at inference time, quite a unique model feature. More concretely, Table 10 shows V-MoE models can beat the dense VIT-H performance by using less than half the FLOPs and less than $60 \\%$ of the runtime. Conversely, we can match the inference FLOPs cost and preserve a one-point accuracy gain in ImageNet/5shot and almost three-point in JFT precision at one (Table 11). Dense models generally require less runtime for the same amount of FLOPs due to the data transfer involved in the V-MoE implementation. ",
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+    {
+        "type": "image",
+        "img_path": "images/640deb97f79b2375225dd1090af5f85b4d407bb4a860255dc9270e4de0ca5a73.jpg",
+        "image_caption": [
+            "Figure 7: Deeper routing decisions correlate with image classes. We show $4 ~ \\mathrm { M o E }$ layers of a V-MoE-H/14. The $x$ -axis corresponds to the 32 experts in a layer. The $y$ -axis are the 1 000 ImageNet classes; orderings for both axes are different across plots. For each pair (expert $e$ , class $c$ ) we show the average routing weight for the tokens corresponding to all images with class $c$ for that particular expert $e$ . Figure 29 includes all the remaining layers; see Appendix E.2 for details. "
+        ],
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "At training time. Batch Prioritized Routing can also be leveraged during training. In Appendix C we show how expert models with max-weight routing can match the dense performance while saving around $20 \\%$ of the total training FLOPs, and strongly outperform vanilla with a similar FLOP budget. ",
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+    {
+        "type": "text",
+        "text": "5 Model Analysis ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Although large-scale sparse MoEs have led to strong performance [22, 39, 54], little is known and understood about how the internals of those complex models work. We argue that such exploratory experiments can inform the design of new algorithms. In this section, we provide the first such analysis at this scale, which guided the development of the algorithms presented in the paper. ",
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+    {
+        "type": "text",
+        "text": "Specialized experts. Intuitively, routers should learn to distribute images across experts based on their similarity. For instance, if the model had three experts, and the task mainly involved three categories—say animals, cars, and buildings—one would expect an expert to specialize in each of those. We test this intuition, with some obvious caveats: (a) experts are placed at several network depths, (b) $k$ experts are combined, and (c) routing happens at the token rather than the image level. ",
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+    {
+        "type": "text",
+        "text": "Figure 7 illustrates how many images of a given ImageNet class use each expert. The plots were produced by running a fine-tuned V-MoE-H Every-2 model. Interestingly, we saw similar patterns with the upstream model without fine-tuning. Experts specialize in discriminating between small sets of classes (those primarily routed through the expert). In earlier MoE layers we do not observe this. Experts may instead focus on aspects common to all classes (background, basic shapes, colours) - for example, Figure 30 (Appendix E) shows correlations with patch location in earlier layers. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The value of routers. After training a sparse MoE, it is natural to study the usefulness of the learned routers, in the light of several pitfalls. For example, the routers may just act as a load balancer if experts end up learning very similar functions, or the routers may simply choose poor assignments. In Appendix E.1, we replace, after training, one router at a time with a uniformly random router. The models are robust to early routing changes while more sensitive to the decisions in the last layers. ",
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+    {
+        "type": "text",
+        "text": "Routing weights distributions. We analyse the router outputs in Appendix E.3, and observe the distribution of selected weights varies wildly across different mixture of experts layers. ",
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+    {
+        "type": "text",
+        "text": "Changing $k$ at inference time. We have observed expert models are remarkably flexible. Somewhat surprisingly, sparse models are fairly robust to mismatches between their training and inference configurations. In Appendix E.4, we explore the effect of training with some original value of $k$ while applying the model at inference time with a different $\\boldsymbol { k } ^ { \\prime } \\neq \\boldsymbol { k }$ . This can be handy to control (decrease ≠or increase) the amount of FLOPs per input in a particular production system. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "6 Related work ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Conditional Computation. To grow the number of model parameters without proportionally increasing the computational cost, conditional computation [5, 15, 12] only activates some relevant parts of the model in an input-dependent fashion, like in decision trees [7]. In deep learning, the activation of portions of the model can use stochastic neurons [6] or reinforcement learning [4, 17, 53]. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Mixture of Experts. MoEs [31, 34, 10, 66] combine the outputs of sub-models known as experts via a router in an input-dependent way. MoEs have successfully used this form of conditional computation in a range of applications [23, 30, 58, 55, 67]. An input can select either all experts [21] or only a sparse mixture thereof as in recent massive language models [54, 39, 22]. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "MoEs for Language. MoEs have recently scaled language models up to trillions of parameters. Our approach is inspired by [54] who proposed a top- $k$ gating in LSTMs, with auxiliary losses ensuring the expert balance [26]. [39] further scaled up this approach for transformers, showing strong gains for neural machine translation. With over one trillion parameters and one expert per input, [22] sped up pre-training compared to a dense baseline [50] while showing gains thanks to transfer and distillation. [40] alternatively enforced a balanced routing by solving a linear assignment problem. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "MoEs for Vision. For computer vision, previous work on MoEs [21, 2, 25, 1, 63, 47, 64] focused on architectures whose scale is considerably smaller than that of both language models and our model. In DeepMoE [63], the “experts” are the channels of convolutional layers that are adaptively selected by a multi-headed sparse gate. This is similar to [64] where the kernels of convolutional layers are activated on a per-example basis. Other approaches use shallow MoEs, learning a single router, either disjointly [25] or jointly [2], together with CNNs playing the role of experts. [1] further have a cost-aware procedure to bias the assignments of inputs across the experts. Unlike shallow MoEs, we operate with up to several tens of routing decisions per token along the depth of the model. Scaling up routing depth was marked as a major challenge in [51], which we successfully tackle in our work. ",
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+    {
+        "type": "text",
+        "text": "7 Conclusions ",
+        "text_level": 1,
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+    },
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+        "text": "We have employed sparse conditional computation to train some of the largest vision models to date, showing significant improvements in representation learning and transfer learning. Alongside V-MoE, we have proposed Batch Prioritized Routing, which allows successful repurposing of model sparsity to introduce sparsity with respect to the inputs. This can be done without further adapting the model, allowing the re-use of trained models with sparse conditional computation. ",
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+    {
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+        "text": "This has interesting connotations for recent work in NLP using sparse models; recent analysis shows model sparsity is the most promising way to reduce model $\\mathrm { C O } _ { 2 }$ emissions [46] and that $90 \\%$ of the footprint stems from inference costs — we present an algorithm which takes the most efficient models and makes them even more efficient without any further model adaptation. ",
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+    {
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+        "text": "This is just the beginning of conditional computation at scale for vision; extensions include scaling up the expert count, reducing dependency on data and improving transfer of the representations produced by sparse models. Directions relating to heterogeneous expert architectures and conditional variable-length routes should also be fruitful. We expect increasing importance of sparse model scaling, especially in data rich domains such as large scale multimodal or video modeling. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
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+        "text": "We thank Alex Kolesnikov, Lucas Beyer and Xiaohua Zhai for providing continuous help and details about scaling ViT models; Alexey Dosovitskiy, who provided some of the pre-trained ViT models; Ilya Tolstikhin, who suggested placing experts only in the last layers; Josip Djolonga for his early review of the manuscript; Dmitry Lepikhin for providing details about the original GShard implementation; Barret Zoph and Liam Fedus for insightful comments and feedback; James Bradbury, Blake Hechtman and the rest of JAX and TPU team who helped us running our models efficiently, and many others from Google Brain for their support. ",
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
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parse/train/SJa9iHgAZ/SJa9iHgAZ.md

@@ -0,0 +1,274 @@
+# RESIDUAL CONNECTIONS ENCOURAGE ITERATIVE INFERENCE
+
+Stanisław Jastrz˛ebsk $^ { 1 , 2 , * }$ , Devansh Arpit $^ { 2 , * }$ , Nicolas Ballas3, Vikas Verma5, Tong Che2 & Yoshua Bengio2,6
+
+1 Jagiellonian University, Cracow, Poland   
+2 MILA, Université de Montréal, Canada   
+3 Facebook, Montreal, Canada   
+4 University of Bonn, Bonn, Germany   
+5 Aalto University, Finland   
+6 CIFAR Senior Fellow   
+∗ Equal Contribution
+
+# ABSTRACT
+
+Residual networks (Resnets) have become a prominent architecture in deep learning. However, a comprehensive understanding of Resnets is still a topic of ongoing research. A recent view argues that Resnets perform iterative refinement of features. We attempt to further expose properties of this aspect. To this end, we study Resnets both analytically and empirically. We formalize the notion of iterative refinement in Resnets by showing that residual connections naturally encourage features of residual blocks to move along the negative gradient of loss as we go from one block to the next. In addition, our empirical analysis suggests that Resnets are able to perform both representation learning and iterative refinement. In general, a Resnet block tends to concentrate representation learning behavior in the first few layers while higher layers perform iterative refinement of features. Finally we observe that sharing residual layers naively leads to representation explosion and counterintuitively, overfitting, and we show that simple existing strategies can help alleviating this problem.
+
+# 1 INTRODUCTION
+
+Traditionally, deep neural network architectures (e.g. VGG Simonyan & Zisserman (2014), AlexNet Krizhevsky et al. (2012), etc.) have been compositional in nature, meaning a hidden layer applies an affine transformation followed by non-linearity, with a different transformation at each layer. However, a major problem with deep architectures has been that of vanishing and exploding gradients. To address this problem, solutions like better activations (ReLU Nair & Hinton (2010)), weight initialization methods Glorot & Bengio (2010); He et al. (2015) and normalization methods Ioffe & Szegedy (2015); Arpit et al. (2016) have been proposed. Nonetheless, training compositional networks deeper than $1 5 - 2 0$ layers remains a challenging task.
+
+Recently, residual networks (Resnets He et al. (2016a)) were introduced to tackle these issues and are considered a breakthrough in deep learning because of their ability to learn very deep networks and achieve state-of-the-art performance. Besides this, performance of Resnets are generally found to remain largely unaffected by removing individual residual blocks or shuffling adjacent blocks Veit et al. (2016). These attributes of Resnets stem from the fact that residual blocks transform representations additively instead of compositionally (like traditional deep networks). This additive framework along with the aforementioned attributes has given rise to two school of thoughts about Resnets– the ensemble view where they are thought to learn an exponential ensemble of shallower models Veit et al. (2016), and the unrolled iterative estimation view Liao & Poggio (2016); Greff et al. (2016), where Resnet layers are thought to iteratively refine representations instead of learning new ones. While the success of Resnets may be attributed partly to both these views, our work takes steps towards achieving a deeper understanding of Resnets in terms of its iterative feature refinement perspective. Our contributions are as follows:
+
+1. We study Resnets analytically and provide a formal view of iterative feature refinement using Taylor’s expansion, showing that for any loss function, a residual block naturally encourages representations to move along the negative gradient of the loss with respect to hidden representations. Each residual block is therefore encouraged to take a gradient step in order to minimize the loss in the hidden representation space. We empirically confirm this by measuring the cosine between the output of a residual block and the gradient of loss with respect to the hidden representations prior to the application of the residual block.
+
+2. We empirically observe that Resnet blocks can perform both hierarchical representation learning (where each block discovers a different representation) and iterative feature refinement (where each block improves slightly but keeps the semantics of the representation of the previous layer). Specifically in Resnets, lower residual blocks learn to perform representation learning, meaning that they change representations significantly and removing these blocks can sometimes drastically hurt prediction performance. The higher blocks on the other hand essentially learn to perform iterative inference– minimizing the loss function by moving the hidden representation along the negative gradient direction. In the presence of shortcut connections1, representation learning is dominantly performed by the shortcut connection layer and most of residual blocks tend to perform iterative feature refinement.
+
+3. The iterative refinement view suggests that deep networks can potentially leverage intensive parameter sharing for the layer performing iterative inference. But sharing large number of residual blocks without loss of performance has not been successfully achieved yet. Towards this end we study two ways of reusing residual blocks: 1. Sharing residual blocks during training; 2. Unrolling a residual block for more steps that it was trained to unroll. We find that training Resnet with naively shared blocks leads to bad performance. We expose reasons for this failure and investigate a preliminary fix for this problem.
+
+# 2 BACKGROUND AND RELATED WORK
+
+# Residual Networks and their analysis:
+
+Recently, several papers have investigated the behavior of Resnets (He et al., 2016a). In (Veit et al., 2016; Littwin & Wolf, 2016), authors argue that Resnets are an ensemble of relatively shallow networks. This is based on the unraveled view of Resnets where there exist an exponential number of paths between the input and prediction layer. Further, observations that shuffling and dropping of residual blocks do not affect performance significantly also support this claim. Other works discuss the possibility that residual networks are approximating recurrent networks (Liao & Poggio, 2016; Greff et al., 2016). This view is in part supported by the observation that the mathematical formulation of Resnets bares similarity to LSTM (Hochreiter & Schmidhuber, 1997), and that successive layers cooperate and preserve the feature identity. Resnets have also been studied from the perspective of boosting theory Huang et al. (2017). In this work the authors propose to learn Resnets in a layerwise manner using a local classifier.
+
+Our work has critical differences compared with the aforementioned studies. Most importantly we focus on a precise definition of iterative inference. In particular, we show that a residual block approximate a gradient descent step in the activation space. Our work can also be seen as relating the gap between the boosting and iterative inference interpretations since having a residual block whose output is aligned with negative gradient of loss is similar to how gradient boosting models work.
+
+# Iterative refinement and weight sharing:
+
+Humans frequently perform predictions with iterative refinement based on the level of difficulty of the task at hand. A leading hypothesis regarding the nature of information processing that happens in the visual cortex is that it performs fast feedforward inference (Thorpe et al., 1996) for easy stimuli or when quick response time is needed, and performs iterative refinement of prediction for complex stimuli (Vanmarcke et al., 2016). The latter is thought to be done by lateral connections within individual layers in the brain that iteratively act upon the current state of the layer to update it. This mechanism allows the brain to make fine grained predictions on complex tasks. A characteristic attribute of this mechanism is the recursive application of the lateral connections which can be thought of as shared weights in a recurrent model. The above views suggest that it is desirable to have deep network models that perform parameter sharing in order to make the iterative inference view complete.
+
+# 3 ITERATIVE INFERENCE IN RESNETS
+
+Our goal in this section is to formalize the notion of iterative inference in Resnets. We study the properties of representations that residual blocks tend to learn, as a result of being additive in nature, in contrast to traditional compositional networks. Specifically, we consider Resnet architectures (see figure 1) where the first hidden layer is a convolution layer, which is followed by $L$ residual blocks which may or may not have shortcut connections in between residual blocks.
+
+A residual block applied on a representation $\mathbf { h } _ { i }$ transforms the representation as,
+
+$$
+\mathbf { h } _ { i + 1 } = \mathbf { h } _ { i } + F _ { i } ( \mathbf { h } _ { i } )
+$$
+
+Consider $L$ such residual blocks stacked on top of each other followed by a loss function. Then, we can Taylor expand any given loss function $\mathcal { L }$ recursively as,
+
+$$
+\begin{array} { r l } & { \mathcal { L } ( \mathbf { h } _ { L } ) = \mathcal { L } ( \mathbf { h } _ { L - 1 } + F _ { L - 1 } ( \mathbf { h } _ { L - 1 } ) ) } \\ & { \qquad = \mathcal { L } ( \mathbf { h } _ { L - 1 } ) + F _ { L - 1 } ( \mathbf { h } _ { L - 1 } ) . \frac { \partial \mathcal { L } ( \mathbf { h } _ { L - 1 } ) } { \partial \mathbf { h } _ { L - 1 } } } \\ & { \qquad + \mathcal { O } ( F _ { L - 1 } ^ { 2 } ( \mathbf { h } _ { L - 1 } ) ) } \end{array}
+$$
+
+![](images/30c059da734b3d2c02faa59a50374494f97750a57f83d86bbe0194d629b12f90.jpg)  
+Figure 1: A typical residual network architecture.
+
+Here we have Taylor expanded the loss function around $\mathbf { h } _ { L - 1 }$ . We can similarly expand the loss function recursively around $\mathbf { h } _ { L - 2 }$ and so on until $\mathbf { h } _ { i }$ and get,
+
+$$
+\mathcal { L } ( \mathbf { h } _ { L } ) = \mathcal { L } ( \mathbf { h } _ { i } ) + \sum _ { j = i } ^ { L - 1 } F _ { j } ( \mathbf { h } _ { j } ) . \frac { \partial \mathcal { L } ( \mathbf { h } _ { j } ) } { \partial \mathbf { h } _ { j } } + \mathcal { O } ( F _ { j } ^ { 2 } ( \mathbf { h } _ { j } ) )
+$$
+
+Notice we have explicitly only written the first order terms of each expansion. The rest of the terms are absorbed in the higher order terms $\mathcal { O } ( . )$ . Further, the first order term is a good approximation when the magnitude of $F _ { j }$ is small enough. In other cases, the higher order terms come into effect as well.
+
+Thus in part, the loss equivalently minimizes the dot product between $F ( \mathbf { h } _ { i } )$ and $\frac { \partial \mathcal { L } ( \mathbf { h } _ { i } ) } { \partial \mathbf { h } _ { i } }$ , which can be achieved by making $F ( \mathbf { h } _ { i } )$ point in the opposite half space to that of $\frac { \partial \mathcal { L } ( \mathbf { h } _ { i } ) } { \partial \mathbf { h } _ { i } }$ . In other words, $\mathbf { h } _ { i } + F ( \mathbf { h } _ { i } )$ approximately moves $\mathbf { h } _ { i }$ in the same half space as that of − ∂L(hi) . The overall training criteria can then be seen as approximately minimizing the dot product between these 2 terms along a path in the $\mathbf { h }$ space between $\mathbf { h } _ { i }$ and $\mathbf { h } _ { L }$ such that loss gradually reduces as we take steps from $\mathbf { h } _ { i }$ to $\mathbf { h } _ { L }$ . The above analysis is justified in practice, as Resnets’ top layers output $F _ { j }$ has small magnitude (Greff et al., 2016), which we also report in Fig. 2.
+
+Given our analysis we formalize iterative inference in Resnets as moving down the energy (loss) surface. It is also worth noting the resemblance of the function of a residual block to stochastic gradient descent. We make a more formal argument in the appendix.
+
+# 4 EMPIRICAL ANALYSIS
+
+Experiments are performed on CIFAR-10 (Krizhevsky & Hinton, 2009) and CIFAR-100 (see appendix) using the original Resnet architecture He et al. (2016b) and two other architectures that we introduce for the purpose of our analysis (described below). Our main goal is to validate that residual networks perform iterative refinement as discussed above, showing its various consequences. Specifically, we set out to empirically answer the following questions:
+
+![](images/71e298d1b076e4b12047251edf34dc58e2a1c373f3e912f67fd5b14882e5001a.jpg)  
+Figure 2: Average ratio of $\ell ^ { 2 }$ norm of output of residual block to the norm of the input of residual block for (left to right) original Resnet, single representation Resnet, avg-pooling Resnet, and wideResnet on CIFAR-10. (Train and validation curves are overlapping.)
+
+![](images/87b38cb14e7916bd262b494e479290beb8391b11457aece9f86c7741977ef81e.jpg)  
+Figure 3: Final prediction accuracy when individual residual blocks are dropped for (left to right) original Resnet, single representation Resnet, avg-pooling Resnet, and wideResnet on CIFAR-10.
+
+![](images/0ec713edda21a6d1f7e5a31027d4d2dfd72b09deebe8676ef252a106245389c0.jpg)  
+Figure 4: Average cos loss between residual block $F ( \mathbf { h } _ { i } )$ and $\frac { \partial \mathcal { L } ( \mathbf { h } _ { i } ) } { \partial \mathbf { h } _ { i } }$ for (left to right) original Resnet, single representation Resnet, avg-pooling Resnet, and wideResnet on CIFAR-10.
+
+![](images/c0c51db0ffe5a1c77726680f0afd4cddf8fa251638820c2aa980f194b59cbedc.jpg)  
+Figure 5: Prediction accuracy when plugging classifier after hidden states in the last stage of Resnets(if any) during training for (left to right) original Resnet, single representation Resnet, avgpooling Resnet, and wideResnet on CIFAR-10. (Blue to red spectrum denotes lower to higher residual blocks)
+
+• Do residual blocks in Resnets behave similarly to each other or is there a distinction between blocks that perform iterative refinement vs. representation learning?   
+• Is the cosine between $\frac { \partial \mathcal { L } ( \mathbf { h } _ { i } ) } { \partial \mathbf { h } _ { i } }$ and $F _ { i } ( \mathbf { h } _ { i } )$ negative in residual networks?   
+• What kind of samples do residual blocks target?   
+• What happens when layers are shared in Resnets?
+
+Resnet architectures: We use the following four architectures for our analysis:
+
+1. Original Resnet-110 architecture: This is the same architecture as used in He et al. (2016b) starting with a $3 \times 3$ convolution layer with 16 filters followed by 54 residual blocks in three different stages (of 18 blocks each with 16, 32 and 64 filters respectively) each separated by a shortcut connections ( ${ \bf \Phi } _ { 1 } \times { \bf \Phi } _ { 1 }$ convolution layers that allow change in the hidden space dimensionality) inserted after the $1 8 ^ { t h }$ and $3 6 ^ { t h }$ residual blocks such that the 3 stages have hidden space of height-width $3 2 \times 3 2$ , $1 6 \times 1 6$ and $8 \times 8$ . The model has a total of 1, 742, 762 parameters.
+
+2. Single representation Resnet: This architecture starts with a $3 \times 3$ convolution layer with 100 filters. This is followed by 10 residual blocks such that all hidden representations have the same height and width of $3 2 \times 3 2$ and 100 filters are used in all the convolution layers in residual blocks as well.
+
+3. Avg-pooling Resnet: This architecture repeats the residual blocks of the single representation Resnet (described above) three times such that there is a $2 \times 2$ average pooling layer after each set of 10 residual blocks that reduces the height and width after each stage by half. Also, in contrast to single representation architecture, it uses 150 filters in all convolution layers. This is followed by the classification block as in the single representation Resnet. It has 12, 201, 310 parameters. We call this architecture the avg-pooling architecture. We also ran experiments with max pooling instead of average pooling but do not report results because they were similar except that max pool acts more non-linearly compared with average pooling, and hence the metrics from max pooling are more similar to those from original Resnet.
+
+4. Wide Resnet: This architecture starts with a $3 \times 3$ convolution layer followed by 3 stages of four residual blocks with 160, 320 and 640 number of filters respectively, and $3 \times 3$ kernel size in all convolution layers. This model has a total of 45,732,842 parameters.
+
+Experimental details: For all architectures, we use He-normal weight initialization as suggested in He et al. (2015), and biases are initialized to 0. For residual blocks, we use BatchNorm ReLU Conv BatchNorm ReLU Conv as suggested in He et al. (2016b). The classifier is composed of the following elements: BatchNorm ReLU AveragePool(8,8) Flatten Fully-Connected-Layer(#classes) Softmax. This model has 1, 829, 210 parameters. For all experiments for single representation and pooling Resnet architectures, we use SGD with momentum 0.9 and train for 200 epochs and 100 epochs (respectively) with learning rate 0.1 until epoch 40, 0.02 until 60, 0.004 until 80 and 0.0008 afterwards. For the original Resnet we use SGD with momentum 0.9 and train for 300 epochs with learning rate 0.1 until epoch 80, 0.01 until 120, 0.001 until 200, 0.00001 until 240 and 0.000011 afterwards. We use data augmentation (horizontal flipping and translation) during training of all architectures. For the wide Resnet architecture, we train the model with with learning rate 0.1 until epoch 60 and 0.02 until 100 epochs.
+
+Note: All experiments on CIFAR-100 are reported in the appendix. In addition, we also record the metrics reported in sections 4.1 and 4.2 as a function of epochs (shown in the appendix due to space limitations). The conclusions are similar to what is reported below.
+
+# 4.1 COSINE LOSS OF RESIDUAL BLOCKS
+
+In this experiment we directly validate our theoretical prediction about Resnets minimizing the dot product between gradient of loss and block output. To this end compute the cosine loss $\bar { \Gamma _ { i } } ( \mathbf { h } _ { i } ) . \frac { \partial \mathcal { L } ( \mathbf { h } _ { i } ) } { \partial \mathbf { h } _ { i } }$   
+A negative cosine loss and small $F _ { i } ( . )$ together suggest that $F _ { i } ( . )$ is refining $\begin{array} { r } { \overline { { { \| { \cal F } _ { i } ( { \bf h } _ { i } ) \| _ { 2 } } \| \frac { \partial { \mathcal { L } } ( { \bf h } _ { i } ) } { \partial { \bf h } _ { i } } \| _ { 2 } } } } \end{array}$   
+features by moving them in the half space of $- \frac { \partial \mathcal { L } ( \mathbf { h } _ { i } ) } { \partial \mathbf { h } _ { i } }$ , thus reducing the loss value for the corresponding data samples. Figure 4 shows the cosine loss for CIFAR-10 on train and validation sets. These figures show that cosine loss is consistently negative for all residual blocks but especially for the higher residual blocks. Also, notice for deeper architectures (original Resnet and pooling Resnet), the higher blocks achieve more negative cosine loss and are thus more iterative in nature. Further, since the higher residual blocks make smaller changes to representation (figure 2), the first order Taylor’s term becomes dominant and hence these blocks effectively move samples in the half space of the negative cosine loss thus reducing loss value of prediction. This result formalizes the sense in which residual blocks perform iterative refinement of features– move representations in the half space of $- \frac { \partial \mathcal { L } ( \mathbf { h } _ { i } ) } { \partial \mathbf { h } _ { i } }$ .
+
+# 4.2 REPRESENTATION LEARNING VS. FEATURE REFINEMENT
+
+In this section, we are interested in investigating the behavior of residual layers in terms of representation learning vs. refinement of features. To this end, we perform the following experiments.
+
+1. $\ell ^ { 2 }$ ratio $\| F _ { i } ( \mathbf { h } _ { i } ) \| _ { 2 } / \| \mathbf { h } _ { i } \| _ { 2 }$ : A residual block $F _ { i } ( . )$ transforms representation as $\mathbf { h } _ { i + 1 } \mathbf { \Psi } = \mathbf { h } _ { i } \mathbf { \Psi } +$ $F _ { i } ( \mathbf { h } _ { i } )$ . For every such block in a Resnet, we measure the $\ell ^ { 2 }$ ratio of $\| F _ { i } ( \mathbf h _ { i } ) \| _ { 2 } / \| \mathbf h _ { i } \| _ { 2 }$ averaged across samples. This ratio directly shows how significantly $F _ { i } ( . )$ changes the representation $\mathbf { h } _ { i }$ ; a large change can be argued to be a necessary condition for layer to perform representation learning. Figure 2 shows the $\ell ^ { 2 }$ ratio for CIFAR-10 on train and validation sets. For single representation
+
+Resnet and pooling Resnet, the first few residual blocks (especially the first residual block) changes representations significantly (up to twice the norm of the original representation), while the rest of the higher blocks are relatively much less significant and this effect is monotonic as we go to higher blocks. However this effect is not as drastic in the original Resnet and wide Resnet architectures which have two $1 \times 1$ (shortcut) convolution layers, thus adding up to a total of 3 convolution layers in the main path of the residual network (notice there exists only one convolution layer in the main path for the other two architectures). This suggests that residual blocks in general tend to learn to refine features but in the case when the network lacks enough compositional layers in the main path, lower residual blocks are forced to change representations significantly, as a proxy for the absence of compositional layers. Additionally, small $\ell ^ { \frac { \mathtt { A } } { 2 } }$ ratio justifies first order approximation used to derive our main result in Sec. 3.
+
+2. Effect of dropping residual layer on accuracy: We drop individual residual blocks from trained Resnets and make predictions using the rest of network on validation set. This analysis shows the significance of individual residual blocks towards the final accuracy that is achieved using all the residual blocks. Note, dropping individual residual blocks is possible because adjacent blocks operate in the same feature space. Figure 3 shows the result of dropping individual residual blocks. As one would expect given above analysis, dropping the first few residual layers (especially the first) for single representation Resnet and pooling Resnet leads to catastrophic performance drop while dropping most of the higher residual layers have minimal effect on performance. On the other hand, performance drops are not drastic for the original Resnet and wide Resnet architecture, which is in agreement with the observations in $\ell ^ { 2 }$ ratio experiments above.
+
+In another set of experiments, we measure validation accuracy after individual residual block during the training process. This set of experiments is achieved by plugging the classifier right after each residual block in the last stage of hidden representation (i.e., after the last shortcut connection, if any). This is shown in figure 5. The figures show that accuracy increases very gradually when adding more residual blocks in the last stage of all architectures.
+
+# 4.3 BORDERLINE EXAMPLES
+
+In this section we investigate which samples get correctly classified after the application of a residual block. Individual residual blocks in general lead to small improvements in performance. Intuitively, since these layers move representations minimally (as shown by previous analysis), the samples that lead to these minor accuracy jump should be near the decision boundary but getting misclassified by a slight margin. To confirm this intuition, we focus on borderline examples, defined as examples that require less than $1 0 \%$ probability change to flip prediction to, or from the correct class. We measure loss, accuracy and entropy over borderline examples over last 5 blocks of the network using the network final classifier. Experiment is performed on CIFAR-10 using Resnet-110 architecture.
+
+Fig 6 shows evolution of loss and accuracy on three groups of examples: borderline examples, already correctly classified and the whole dataset. While overall accuracy and loss remains similar across the top residual blocks, we observe that a significant chunk of borderline examples gets corrected by the immediate next residual block. This exposes the qualitative nature of examples that these feature refinement layers focus on, which is further reinforced by the fact that entropy decreases for all considered subsets. We also note that while train loss drops uniformly across layers, test sets loss increases after last block. Correcting this phenomenon could lead to improved generalization in Resnets, which we leave for future work.
+
+# 4.4 UNROLLING RESIDUAL NETWORK
+
+A fundamental requirement for a procedure to be truly iterative is to apply the same function. In this section we explore what happens when we unroll the last block of a trained residual network for more steps than it was trained for. Our main goal is to investigate if iterative inference generalizes to more steps than it was trained on. We focus on the same model as discussed in previous section, Resnet-110, and unroll the last residual block for 20 extra steps. Naively unrolling the network leads to activation explosion (we observe similar behavior in Sec. 4.5). To control for that effect, we added a scaling factor on the output of the last residual blocks. We hypothesize that controlling the scale limits the drift of the activation through the unrolled layer, i.e. they remains in a given neighbourhood on which the network is well behaved. Similarly to Sec. 4.3 we track evolution of loss and accuracy on three groups of examples: borderline examples, already correctly classified and the whole dataset. Experiments are repeated 4 times, and results are averaged.
+
+![](images/45ce3cb6c90773f203329525ca411c67d6772b3e60ca0f93709282eb77bc1563.jpg)  
+Figure 6: Accuracy, loss and entropy for last 5 blocks of Resnet-110. Performance on bordeline examples improves at the expense of performance (loss) of already correctly classified points (correct). This happens because last block output is encouraged by training to be negatively correlated (around $- 0 . 1$ cosine) with gradient of the loss.
+
+![](images/1cf6fb56177565aa740e3c99c5040589577c27df90aab5e834e0229f328a5f23.jpg)  
+Figure 7: Accuracy, loss and entropy for Resnet-110 with last block unrolled for 20 additional steps (with appropriate scaling). Borderline examples are corrected and overall performance accuracy improves. Note different scales for train and test. Curves are averaged over 4 runs.
+
+We first investigate how unrolling blocks impact loss and accuracy. Loss on train set improved uniformly from 0.0012 to 0.001, while it increased on test set. There are on average 51 borderline examples in test $\mathrm { s e t } ^ { 2 }$ , on which performance is improved from $4 3 \%$ to $5 3 \%$ , which yields slight improvement in accuracy on test set. Next we shift our attention to cosine loss. We observe that cosine loss remains negative on the first two steps without rescaling, and all steps after scaling. Figure 7 shows evolution of loss and accuracy on the three groups of examples: borderline examples, already correctly classified and the whole dataset. Cosine loss and $\ell ^ { 2 }$ ratio for each block are reported in Appendix E.
+
+To summarize, unrolling residual network to more steps than it was trained on improves both loss on train set, and maintains (in given neighbourhood) negative cosine loss on both train and test set.
+
+# 4.5 SHARING RESIDUAL LAYERS
+
+Our results suggest that top residual blocks should be shareable, because they perform similar iterative refinement. We consider a shared version of Resnet-110 model, where in each stage we share all the residual blocks from the $5 ^ { t h }$ block. All shared Resnets in this section have therefore a similar number of parameters as Resnet-38. Contrary to (Liao & Poggio, 2016) we observe that naively sharing the higher (iterative refinement) residual blocks of a Resnets in general leads to bad performance3 (especially for deeper Resnets).
+
+First, we compare the unshared and shared version of Resnet-110. The shared version uses approximately 3 times less parameters. In Fig. 8, we report the train and validation performances of the Resnet-110. We observe that naively sharing parameters of the top residual blocks leads both to overfitting (given similar training accuracy, the shared Resnet-110 has significantly lower validation performances) and underfitting (worse training accuracy than Resnet-110). We also compared our shared model with a Resnet-38 that has a similar number of parameters and observe worse validation performances, while achieving similar training accuracy.
+
+![](images/bfbb0a7746950c10206f50512c34dd98c7f540cf633119c7a63f023e3d29f4fe.jpg)  
+Figure 8: Resnet-110 with naively shared top 13 layers of each block compared with unshared Resnet-38. Left plot present training and validation curves, shared Resnet-110 heavily overfits. In the right plot we track gradient norm ratio between first block in first and last stage of resnet $\begin{array} { r } { r = | | \frac { \hat { \partial } L } { \partial h _ { 1 } } | | / \frac { \partial L } { \partial h _ { 1 + 2 n } } | | \big ) } \end{array}$ L1 ||/ ∂L∂h1+2n ||). Significantly larger ratio in the naive sharing model suggests, that the overfitting is caused by early layers dominating learning. Metrics are tracked on train (solid line) and validation data (dashed line)
+
+![](images/317e694fb4391b1b7f026c109e8bd64f8f99052c805e6808e33e71ff5988a165.jpg)  
+Figure 9: Ablation study of different strategies to remedy sharing leading to overfitting phenomenon in Residual Networks. Left figure shows effect on training and test accuracy. Right figure studies norm explosion. All components are important, but it is most crucial to unshare BN statistics.
+
+We notice that sharing layers make the layer activations explode during the forward propagation at initialization due to the repeated application of the same operation (Fig 8, right). Consequently, the norm of the gradients also explodes at initialization (Fig. 8, center).
+
+To address this issue we introduce a variant of recurrent batch normalization (Cooijmans et al., 2016), which proposes to initialize $\gamma$ to 0.1 and unshare statistics for every step. On top of this strategy, we also unshare $\gamma$ and $\beta$ parameters. Tab. 1 shows that using our strategy alleviates explosion problem and leads to small improvement over baseline with similar number of parameters. We also perform an ablation to study, see Figure. 9 (left), which show that all additions to naive strategy are necessary and drastically reduce the initial activation explosion. Finally, we observe a similar trend for cosine loss, intermediate accuracy, and $\ell ^ { 2 }$ ratio for the shared Resnet as for the unshared Resnet discussed in the previous Sections. Full results are reported in Appendix D.
+
+Unshared Batch Normalization strategy therefore mitigates this exploding activation problem. This problem, leading to exploding gradient in our case, appears frequently in recurrent neural network. This suggests that future unrolled Resnets should use insights from research on recurrent networks optimization, including careful initialization (Henaff et al., 2016) and parametrization changes (Hochreiter & Schmidhuber, 1997).
+
+<table><tr><td>Model</td><td>CIFAR10</td><td>CIFAR100</td><td>Parameters</td></tr><tr><td>Resnet-32</td><td>1.53/7.14</td><td>12.62/30.08</td><td>467k-473k</td></tr><tr><td>Resnet-38</td><td>1.20/6.99</td><td>10.04/29.66</td><td>565k-571k</td></tr><tr><td>Resnet-110-UBN</td><td>0.63/6.62</td><td>7.75/29.94</td><td>570k-576k</td></tr><tr><td>Resnet-146-UBN</td><td>0.68/6.82</td><td>7.21/29.49</td><td>573k-579k</td></tr><tr><td>Resnet-182-UBN</td><td>0.48/6.97</td><td>6.42 /29.33</td><td>576k-581k</td></tr><tr><td>Resnet-56</td><td>0.58/6.53</td><td>5.19/28.99</td><td>857k-863k</td></tr><tr><td>Resnet-110</td><td>0.22/6.13</td><td>1.26 /27.54</td><td>1734k-1740k</td></tr></table>
+
+Table 1: Train and test error of Resnet sharing top layers blocks (while using unshared both statistics and $\beta , \gamma$ in Batch Normalization) denoted as UBN (Unshared Batch Normalization) compared to baseline Resnet of varying depth. Training Resnet with unrolled layers can bring additional gain of $0 . 3 \%$ , while adding marginal amount of extra parameters. Runs are repeated 4 times.
+
+# 5 CONCLUSION
+
+Our main contribution is formalizing the view of iterative refinement in Resnets and showing analytically that residual blocks naturally encourage representations to move in the half space of negative loss gradient, thus implementing a gradient descent in the activation space (each block reduces loss and improves accuracy). We validate theory experimentally on a wide range of Resnet architectures.
+
+We further explored two forms of sharing blocks in Resnet. We show that Resnet can be unrolled to more steps than it was trained on. Next, we found that counterintuitively training residual blocks with shared blocks leads to overfitting. While we propose a variant of batch normalization to mitigate it, we leave further investigation of this phenomena for future work. We hope that our developed formal view, and practical results, will aid analysis of other models employing iterative inference and residual connections.
+
+# ACKNOWLEDGEMENTS
+
+We acknowledge the computing resources provided by ComputeCanada and CalculQuebec. SJ was supported by Grant No. DI 2014/016644 from Ministry of Science and Higher Education, Poland. DA was supported by IVADO.
+
+# REFERENCES
+
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+Tim Cooijmans, Nicolas Ballas, César Laurent, Çaglar Gülçehre, and Aaron Courville. Recurrent ˘ batch normalization. arXiv preprint arXiv:1603.09025, 2016.   
+X. Glorot and Y. Bengio. Understanding the difficulty of training deep feedforward neural networks. In Aistats, 2010.   
+K. Greff, R. Srivastava, and J. Schmidhuber. Highway and residual networks learn unrolled iterative estimation. arXiV, 2016.   
+K. He, X. Zhang, S. Ren, and J. Sun. Delving deep into rectifiers: Surpassing human-level performance on imagenet classification. In ICCV, 2015.   
+K. He, X. Zhang, S. Ren, and J. Sun. Deep residual learning for image recognition. In CVPR, 2016a.   
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+M. Henaff, A. Szlam, and Y. LeCun. Recurrent orthogonal networks and long-memory tasks. In ICML, 2016.   
+S. Hochreiter and J. Schmidhuber. Long short-term memory. Neural computation, 1997.
+
+Furong Huang, Jordan Ash, John Langford, and Robert Schapire. Learning deep resnet blocks sequentially using boosting theory. arXiv preprint arXiv:1706.04964, 2017.
+
+S. Ioffe and C. Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. In ICML, 2015.
+
+A. Krizhevsky and G. Hinton. Learning multiple layers of features from tiny images. 2009.
+
+A. Krizhevsky, I. Sutskever, and G. Hinton. Imagenet classification with deep convolutional neural networks. In NIPS, 2012.
+
+Q. Liao and T. Poggio. Bridging the gaps between residual learning, recurrent neural networks and visual cortex. arXiV, 2016.
+
+E. Littwin and L. Wolf. The loss surface of residual networks: Ensembles and the role of batch normalization. arXiV, 2016.
+
+V. Nair and G. Hinton. Rectified linear units improve restricted boltzmann machines. In ICML, 2010.
+
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+
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+
+S. Vanmarcke, F. Calders, and F. Wagemans. The time-course of ultrarapid categorization: The influence of scene congruency and top-down processing. i-Perception, 2016.
+
+A. Veit, M. Wilber, and S. Belongie. Residual networks are exponential ensembles of relatively shallow networks. arXiV, 2016.
+
+# Appendices
+
+A FURTHER ANALYSIS
+
+A.1 A SIDE-EFFECT OF MOVING IN THE HALF SPACE OF $- \frac { \partial \mathcal { L } ( \mathbf { h } _ { o } ) } { \partial \mathbf { h } _ { o } }$
+
+Let $\mathbf { h } _ { o } = \mathbf { W } \mathbf { x } + \mathbf { b }$ be the output of the first layer (convolution) of a ResNet. In this analysis we show that if $\mathbf { h } _ { o }$ moves in the half space of $- \frac { \partial \mathcal { L } ( \dot { \bf h } _ { o } ) } { \partial { \bf h } _ { o } }$ , then it is equivalent to updating the parameters of the convolution layer using a gradient update step. To see this, consider the change in $\mathbf { h } _ { o }$ from updating parameters using gradient descent with step size $\eta$ . This is given by,
+
+$$
+\begin{array} { r l } & { \Delta \mathbf { h } _ { o } = ( \mathbf { W } - \eta \frac { \partial \mathcal { L } } { \partial \mathbf { W } } ) \mathbf { x } + ( \mathbf { b } - \eta \frac { \partial \mathcal { L } } { \partial \mathbf { b } } ) - ( \mathbf { W } \mathbf { x } + \mathbf { b } ) } \\ & { \quad \quad = - \eta \frac { \partial \mathcal { L } } { \partial \mathbf { W } } \mathbf { x } - \eta \frac { \partial \mathcal { L } } { \partial \mathbf { b } } } \\ & { \quad \quad = - \eta \frac { \partial \mathcal { L } } { \partial \mathbf { h } _ { o } } \left( \frac { \partial \mathbf { h } _ { o } } { \partial \mathbf { W } } \mathbf { x } + \frac { \partial \mathbf { h } _ { o } } { \partial \mathbf { b } } \right) } \\ & { \quad \quad = - \eta \frac { \partial \mathcal { L } } { \partial \mathbf { h } _ { o } } \left( \| \mathbf { x } \| ^ { 2 } + 1 \right) } \\ & { \quad \quad \propto - \frac { \partial \mathcal { L } } { \partial \mathbf { h } _ { o } } } \end{array}
+$$
+
+Thus, moving $\mathbf { h } _ { o }$ in the half space of ∂L∂h has the same effect as that achieved by updating the parameters W, b using gradient descent. Although we found this insight interesting, we don’t build upon it in this paper. We leave this as a future work.
+
+# B ANALYSIS ON CIFAR-100
+
+Here we report the experiments as done in sections 4.2 and 4.1, for CIFAR-100 dataset. The plots are shown in figures 10, 11 and 12. The conclusions are same as reported in the main text for CIFAR-10.
+
+# C ANALYSIS OF INTERMEDIATE METRICS ON CIFAR-10 AND CIFAR-100
+
+Here we plot the accuracy, cosine loss and $\ell ^ { 2 }$ ratio metrics corresponding to each individual residual block on validation during the training process for CIFAR-10 (figures 13, 14, 5) and CIFAR-100 (figures 15, 16, 17). These plots are recorded only for the residual blocks in the last space for each architecture (this is because otherwise the dimensions of the output of the residual block and the classifier will not match). In the case of cosine loss after individual residual block, this set of experiments is achieved by plugging the classifier right after each hidden representation and measuring the cosine between the gradient w.r.t. hidden representation and the corresponding residual block’s output.
+
+We find that the accuracy after individual residual blocks increases gradually as we move from from lower to higher residua blocks. Cosine loss on the other hand consistently remains negative for all architectures. Finally $\ell ^ { 2 }$ ratio tends to increase for residual blocks as training progresses.
+
+# D ITERATIVE INFERENCE IN SHARED RESNET
+
+In this section we extend results from Sec. 4.5. We report cosine loss, intermediate accuracy, and $\ell ^ { 2 }$ ratio for naively shared Resnet in Fig. 19, and with unshared batch normalization in Fig. ??.
+
+![](images/1b17acb195696f29ae3640e938149cc123114da395f8c230d9ab97cb13e31f33.jpg)  
+Figure 10: Average cos loss between residual block $F ( \mathbf { h } _ { i } )$ and $\frac { \partial \mathcal { L } ( \mathbf { h } _ { i } ) } { \partial \mathbf { h } _ { i } }$ for (left to right) original Resnet, single representation Resnet, avg-pooling Resnet, and wideResnet on CIFAR-100.
+
+![](images/6c3cb1928f9b9760fc6b0a476c1dbfe114c61337c0a10818f9d2531121cc3492.jpg)  
+Figure 11: Average ratio of $\ell ^ { 2 }$ norm of output of residual block to the norm of the input of residual block for (left to right) original Resnet, single representation Resnet, avg-pooling Resnet, and wideResnet on CIFAR-100. (Train and validation curves are overlapping.)
+
+![](images/6fa9571f937f9a10fa428290fe15ddf9157886ca0638d0c70792a5dfa9dbd224.jpg)  
+Figure 12: Final prediction accuracy when individual residual blocks are dropped for (left to right) original Resnet, single representation Resnet, avg-pooling Resnet, and wideResnet on CIFAR-100.
+
+![](images/35b4afd2a7979b9304c3d47ec0ba7f16e0dbb19db8dffad74d7b420d54dffd86.jpg)  
+Figure 13: Average cos loss between residual block $F ( \mathbf { h } _ { i } )$ and $\frac { \partial \mathcal { L } ( \mathbf { h } _ { i } ) } { \partial \mathbf { h } _ { i } }$ during training for (left to right) original Resnet, single representation Resnet, avg-pooling Resnet, and wideResnet on CIFAR10. (Blue to red spectrum denotes lower to higher residual blocks)
+
+![](images/47804a663459cbe85a632d724d137c87e5c163c0da670b05c469de4e12d1d468.jpg)  
+Figure 14: Average ratio of $\ell ^ { 2 }$ norm of output of residual block to the norm of the input of residual block during training for (left to right) original Resnet, single representation Resnet, avg-pooling Resnet, and wideResnet on CIFAR-10. (Blue to red spectrum denotes lower to higher residual blocks)
+
+![](images/b779c3d0cc4cfdca2c6f72fe04e29662322b5aa8270237d0879be36375e249eb.jpg)  
+Figure 15: Average cos loss between residual block $F ( \mathbf { h } _ { i } )$ and $\frac { \partial \mathcal { L } ( \mathbf { h } _ { i } ) } { \partial \mathbf { h } _ { i } }$ during training for (left to right) original Resnet, single representation Resnet, avg-pooling Resnet, and wideResnet on CIFAR100. (Blue to red spectrum denotes lower to higher residual blocks)
+
+![](images/be96d4ee5e6092d0abc3923a29ede16bae2a0e2f69dfda342c24dce27448fdc7.jpg)  
+Figure 16: Average ratio of $\ell ^ { 2 }$ norm of output of residual block to the norm of the input of residual block during training for (left to right) original Resnet, single representation Resnet, avg-pooling Resnet, and wideResnet on CIFAR-100. (Blue to red spectrum denotes lower to higher residual blocks)
+
+![](images/0e1c34c0292272d61a67bcf11c4d7dee673f157548679b261a2c86de4c4c0365.jpg)  
+Figure 17: Prediction accuracy when plugging classifier after hidden states in the last stage of Resnets(if any) during training for (left to right) original Resnet, single representation Resnet, avgpooling Resnet, and wideResnet on CIFAR-100. (Blue to red spectrum denotes lower to higher residual blocks)
+
+![](images/6971a298000f2b641ebb3b1777ff1c7001887d45ceeab449029d48e8d485df97.jpg)  
+Figure 18: Cosine loss, $\ell ^ { 2 }$ ratio, and intermediate accuracy for shared Resnet-110 with unshared Batch Normalization (described in Sec. 4.5). Each curve represents different block in Resnet. Red is closest to output.
+
+![](images/d0f2347f07953ddb059bbfaad26f0e9816ebf0fbb936fa7d0fd52f4002719440.jpg)  
+Figure 19: Cosine loss, $\ell ^ { 2 }$ ratio, and intermediate accuracy for naively shared Resnet-110. Each curve represents different block in Resnet. Red is closest to output.
+
+![](images/e116cb21a9e9e5bf5578ddf325e9427b0d83225b4a27e53806f38c6c3d213ffe.jpg)  
+Figure 20: First figure show that cosine loss in Resnet-110 after unrolling generalizes to more steps than it was trained on. Second plot shows evolution of $\ell ^ { 2 }$ ratio for Resnet-110. Third plot reports cosine loss Resnet-110 with scaled version of final block, as considered in Sec. 4.4. Rightmost plots reports $\ell ^ { 2 }$ ratio for scaled Resnet-110. Vertical line in plots indicates number of steps network was trained on.
+
+# E UNROLLING RESIDUAL NETWORKS
+
+In this section we report additional results for unrolling residual network. Figure 20 shows evolution of cosine loss an $\ell ^ { 2 }$ ratio for Resnet-110 with unrolled last block for 20 additional steps.

parse/train/SJa9iHgAZ/SJa9iHgAZ_content_list.json

@@ -0,0 +1,1434 @@
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+    {
+        "type": "text",
+        "text": "RESIDUAL CONNECTIONS ENCOURAGE ITERATIVE INFERENCE ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Stanisław Jastrz˛ebsk $^ { 1 , 2 , * }$ , Devansh Arpit $^ { 2 , * }$ , Nicolas Ballas3, Vikas Verma5, Tong Che2 & Yoshua Bengio2,6 ",
+        "bbox": [
+            184,
+            169,
+            714,
+            199
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Jagiellonian University, Cracow, Poland   \n2 MILA, Université de Montréal, Canada   \n3 Facebook, Montreal, Canada   \n4 University of Bonn, Bonn, Germany   \n5 Aalto University, Finland   \n6 CIFAR Senior Fellow   \n∗ Equal Contribution ",
+        "bbox": [
+            183,
+            212,
+            457,
+            313
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            349,
+            544,
+            364
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Residual networks (Resnets) have become a prominent architecture in deep learning. However, a comprehensive understanding of Resnets is still a topic of ongoing research. A recent view argues that Resnets perform iterative refinement of features. We attempt to further expose properties of this aspect. To this end, we study Resnets both analytically and empirically. We formalize the notion of iterative refinement in Resnets by showing that residual connections naturally encourage features of residual blocks to move along the negative gradient of loss as we go from one block to the next. In addition, our empirical analysis suggests that Resnets are able to perform both representation learning and iterative refinement. In general, a Resnet block tends to concentrate representation learning behavior in the first few layers while higher layers perform iterative refinement of features. Finally we observe that sharing residual layers naively leads to representation explosion and counterintuitively, overfitting, and we show that simple existing strategies can help alleviating this problem. ",
+        "bbox": [
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+            386,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            617,
+            334,
+            633
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Traditionally, deep neural network architectures (e.g. VGG Simonyan & Zisserman (2014), AlexNet Krizhevsky et al. (2012), etc.) have been compositional in nature, meaning a hidden layer applies an affine transformation followed by non-linearity, with a different transformation at each layer. However, a major problem with deep architectures has been that of vanishing and exploding gradients. To address this problem, solutions like better activations (ReLU Nair & Hinton (2010)), weight initialization methods Glorot & Bengio (2010); He et al. (2015) and normalization methods Ioffe & Szegedy (2015); Arpit et al. (2016) have been proposed. Nonetheless, training compositional networks deeper than $1 5 - 2 0$ layers remains a challenging task. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recently, residual networks (Resnets He et al. (2016a)) were introduced to tackle these issues and are considered a breakthrough in deep learning because of their ability to learn very deep networks and achieve state-of-the-art performance. Besides this, performance of Resnets are generally found to remain largely unaffected by removing individual residual blocks or shuffling adjacent blocks Veit et al. (2016). These attributes of Resnets stem from the fact that residual blocks transform representations additively instead of compositionally (like traditional deep networks). This additive framework along with the aforementioned attributes has given rise to two school of thoughts about Resnets– the ensemble view where they are thought to learn an exponential ensemble of shallower models Veit et al. (2016), and the unrolled iterative estimation view Liao & Poggio (2016); Greff et al. (2016), where Resnet layers are thought to iteratively refine representations instead of learning new ones. While the success of Resnets may be attributed partly to both these views, our work takes steps towards achieving a deeper understanding of Resnets in terms of its iterative feature refinement perspective. Our contributions are as follows: ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1. We study Resnets analytically and provide a formal view of iterative feature refinement using Taylor’s expansion, showing that for any loss function, a residual block naturally encourages representations to move along the negative gradient of the loss with respect to hidden representations. Each residual block is therefore encouraged to take a gradient step in order to minimize the loss in the hidden representation space. We empirically confirm this by measuring the cosine between the output of a residual block and the gradient of loss with respect to the hidden representations prior to the application of the residual block. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2. We empirically observe that Resnet blocks can perform both hierarchical representation learning (where each block discovers a different representation) and iterative feature refinement (where each block improves slightly but keeps the semantics of the representation of the previous layer). Specifically in Resnets, lower residual blocks learn to perform representation learning, meaning that they change representations significantly and removing these blocks can sometimes drastically hurt prediction performance. The higher blocks on the other hand essentially learn to perform iterative inference– minimizing the loss function by moving the hidden representation along the negative gradient direction. In the presence of shortcut connections1, representation learning is dominantly performed by the shortcut connection layer and most of residual blocks tend to perform iterative feature refinement. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3. The iterative refinement view suggests that deep networks can potentially leverage intensive parameter sharing for the layer performing iterative inference. But sharing large number of residual blocks without loss of performance has not been successfully achieved yet. Towards this end we study two ways of reusing residual blocks: 1. Sharing residual blocks during training; 2. Unrolling a residual block for more steps that it was trained to unroll. We find that training Resnet with naively shared blocks leads to bad performance. We expose reasons for this failure and investigate a preliminary fix for this problem. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 BACKGROUND AND RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            507,
+            511,
+            522
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Residual Networks and their analysis: ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            537,
+            437,
+            553
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Recently, several papers have investigated the behavior of Resnets (He et al., 2016a). In (Veit et al., 2016; Littwin & Wolf, 2016), authors argue that Resnets are an ensemble of relatively shallow networks. This is based on the unraveled view of Resnets where there exist an exponential number of paths between the input and prediction layer. Further, observations that shuffling and dropping of residual blocks do not affect performance significantly also support this claim. Other works discuss the possibility that residual networks are approximating recurrent networks (Liao & Poggio, 2016; Greff et al., 2016). This view is in part supported by the observation that the mathematical formulation of Resnets bares similarity to LSTM (Hochreiter & Schmidhuber, 1997), and that successive layers cooperate and preserve the feature identity. Resnets have also been studied from the perspective of boosting theory Huang et al. (2017). In this work the authors propose to learn Resnets in a layerwise manner using a local classifier. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our work has critical differences compared with the aforementioned studies. Most importantly we focus on a precise definition of iterative inference. In particular, we show that a residual block approximate a gradient descent step in the activation space. Our work can also be seen as relating the gap between the boosting and iterative inference interpretations since having a residual block whose output is aligned with negative gradient of loss is similar to how gradient boosting models work. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Iterative refinement and weight sharing: ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            810,
+            454,
+            824
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Humans frequently perform predictions with iterative refinement based on the level of difficulty of the task at hand. A leading hypothesis regarding the nature of information processing that happens in the visual cortex is that it performs fast feedforward inference (Thorpe et al., 1996) for easy stimuli or when quick response time is needed, and performs iterative refinement of prediction for complex stimuli (Vanmarcke et al., 2016). The latter is thought to be done by lateral connections within individual layers in the brain that iteratively act upon the current state of the layer to update it. This mechanism allows the brain to make fine grained predictions on complex tasks. A characteristic attribute of this mechanism is the recursive application of the lateral connections which can be thought of as shared weights in a recurrent model. The above views suggest that it is desirable to have deep network models that perform parameter sharing in order to make the iterative inference view complete. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 ITERATIVE INFERENCE IN RESNETS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            222,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our goal in this section is to formalize the notion of iterative inference in Resnets. We study the properties of representations that residual blocks tend to learn, as a result of being additive in nature, in contrast to traditional compositional networks. Specifically, we consider Resnet architectures (see figure 1) where the first hidden layer is a convolution layer, which is followed by $L$ residual blocks which may or may not have shortcut connections in between residual blocks. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A residual block applied on a representation $\\mathbf { h } _ { i }$ transforms the representation as, ",
+        "bbox": [
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+            329,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5dd65aa5e9b63c4c665223b5cbda12a146229788178f710d821cb3a5e476102a.jpg",
+        "text": "$$\n\\mathbf { h } _ { i + 1 } = \\mathbf { h } _ { i } + F _ { i } ( \\mathbf { h } _ { i } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Consider $L$ such residual blocks stacked on top of each other followed by a loss function. Then, we can Taylor expand any given loss function $\\mathcal { L }$ recursively as, ",
+        "bbox": [
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+            390,
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b055d438d0677cbc5be456e0097e3342095257af904e332842f2b44d96e33ace.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathcal { L } ( \\mathbf { h } _ { L } ) = \\mathcal { L } ( \\mathbf { h } _ { L - 1 } + F _ { L - 1 } ( \\mathbf { h } _ { L - 1 } ) ) } \\\\ & { \\qquad = \\mathcal { L } ( \\mathbf { h } _ { L - 1 } ) + F _ { L - 1 } ( \\mathbf { h } _ { L - 1 } ) . \\frac { \\partial \\mathcal { L } ( \\mathbf { h } _ { L - 1 } ) } { \\partial \\mathbf { h } _ { L - 1 } } } \\\\ & { \\qquad + \\mathcal { O } ( F _ { L - 1 } ^ { 2 } ( \\mathbf { h } _ { L - 1 } ) ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/30c059da734b3d2c02faa59a50374494f97750a57f83d86bbe0194d629b12f90.jpg",
+        "image_caption": [
+            "Figure 1: A typical residual network architecture. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            643,
+            335,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Here we have Taylor expanded the loss function around $\\mathbf { h } _ { L - 1 }$ . We can similarly expand the loss function recursively around $\\mathbf { h } _ { L - 2 }$ and so on until $\\mathbf { h } _ { i }$ and get, ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/702e5c254d7690832c351f454f8b15bb53034608e4420db777485535ca9b52df.jpg",
+        "text": "$$\n\\mathcal { L } ( \\mathbf { h } _ { L } ) = \\mathcal { L } ( \\mathbf { h } _ { i } ) + \\sum _ { j = i } ^ { L - 1 } F _ { j } ( \\mathbf { h } _ { j } ) . \\frac { \\partial \\mathcal { L } ( \\mathbf { h } _ { j } ) } { \\partial \\mathbf { h } _ { j } } + \\mathcal { O } ( F _ { j } ^ { 2 } ( \\mathbf { h } _ { j } ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Notice we have explicitly only written the first order terms of each expansion. The rest of the terms are absorbed in the higher order terms $\\mathcal { O } ( . )$ . Further, the first order term is a good approximation when the magnitude of $F _ { j }$ is small enough. In other cases, the higher order terms come into effect as well. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Thus in part, the loss equivalently minimizes the dot product between $F ( \\mathbf { h } _ { i } )$ and $\\frac { \\partial \\mathcal { L } ( \\mathbf { h } _ { i } ) } { \\partial \\mathbf { h } _ { i } }$ , which can be achieved by making $F ( \\mathbf { h } _ { i } )$ point in the opposite half space to that of $\\frac { \\partial \\mathcal { L } ( \\mathbf { h } _ { i } ) } { \\partial \\mathbf { h } _ { i } }$ . In other words, $\\mathbf { h } _ { i } + F ( \\mathbf { h } _ { i } )$ approximately moves $\\mathbf { h } _ { i }$ in the same half space as that of − ∂L(hi) . The overall training criteria can then be seen as approximately minimizing the dot product between these 2 terms along a path in the $\\mathbf { h }$ space between $\\mathbf { h } _ { i }$ and $\\mathbf { h } _ { L }$ such that loss gradually reduces as we take steps from $\\mathbf { h } _ { i }$ to $\\mathbf { h } _ { L }$ . The above analysis is justified in practice, as Resnets’ top layers output $F _ { j }$ has small magnitude (Greff et al., 2016), which we also report in Fig. 2. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Given our analysis we formalize iterative inference in Resnets as moving down the energy (loss) surface. It is also worth noting the resemblance of the function of a residual block to stochastic gradient descent. We make a more formal argument in the appendix. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4 EMPIRICAL ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Experiments are performed on CIFAR-10 (Krizhevsky & Hinton, 2009) and CIFAR-100 (see appendix) using the original Resnet architecture He et al. (2016b) and two other architectures that we introduce for the purpose of our analysis (described below). Our main goal is to validate that residual networks perform iterative refinement as discussed above, showing its various consequences. Specifically, we set out to empirically answer the following questions: ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/71e298d1b076e4b12047251edf34dc58e2a1c373f3e912f67fd5b14882e5001a.jpg",
+        "image_caption": [
+            "Figure 2: Average ratio of $\\ell ^ { 2 }$ norm of output of residual block to the norm of the input of residual block for (left to right) original Resnet, single representation Resnet, avg-pooling Resnet, and wideResnet on CIFAR-10. (Train and validation curves are overlapping.) "
+        ],
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+        "img_path": "images/87b38cb14e7916bd262b494e479290beb8391b11457aece9f86c7741977ef81e.jpg",
+        "image_caption": [
+            "Figure 3: Final prediction accuracy when individual residual blocks are dropped for (left to right) original Resnet, single representation Resnet, avg-pooling Resnet, and wideResnet on CIFAR-10. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/0ec713edda21a6d1f7e5a31027d4d2dfd72b09deebe8676ef252a106245389c0.jpg",
+        "image_caption": [
+            "Figure 4: Average cos loss between residual block $F ( \\mathbf { h } _ { i } )$ and $\\frac { \\partial \\mathcal { L } ( \\mathbf { h } _ { i } ) } { \\partial \\mathbf { h } _ { i } }$ for (left to right) original Resnet, single representation Resnet, avg-pooling Resnet, and wideResnet on CIFAR-10. "
+        ],
+        "image_footnote": [],
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/c0c51db0ffe5a1c77726680f0afd4cddf8fa251638820c2aa980f194b59cbedc.jpg",
+        "image_caption": [
+            "Figure 5: Prediction accuracy when plugging classifier after hidden states in the last stage of Resnets(if any) during training for (left to right) original Resnet, single representation Resnet, avgpooling Resnet, and wideResnet on CIFAR-10. (Blue to red spectrum denotes lower to higher residual blocks) "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 3
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+    {
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+    {
+        "type": "text",
+        "text": "• Do residual blocks in Resnets behave similarly to each other or is there a distinction between blocks that perform iterative refinement vs. representation learning?   \n• Is the cosine between $\\frac { \\partial \\mathcal { L } ( \\mathbf { h } _ { i } ) } { \\partial \\mathbf { h } _ { i } }$ and $F _ { i } ( \\mathbf { h } _ { i } )$ negative in residual networks?   \n• What kind of samples do residual blocks target?   \n• What happens when layers are shared in Resnets? ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Resnet architectures: We use the following four architectures for our analysis: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "1. Original Resnet-110 architecture: This is the same architecture as used in He et al. (2016b) starting with a $3 \\times 3$ convolution layer with 16 filters followed by 54 residual blocks in three different stages (of 18 blocks each with 16, 32 and 64 filters respectively) each separated by a shortcut connections ( ${ \\bf \\Phi } _ { 1 } \\times { \\bf \\Phi } _ { 1 }$ convolution layers that allow change in the hidden space dimensionality) inserted after the $1 8 ^ { t h }$ and $3 6 ^ { t h }$ residual blocks such that the 3 stages have hidden space of height-width $3 2 \\times 3 2$ , $1 6 \\times 1 6$ and $8 \\times 8$ . The model has a total of 1, 742, 762 parameters. ",
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+    },
+    {
+        "type": "text",
+        "text": "2. Single representation Resnet: This architecture starts with a $3 \\times 3$ convolution layer with 100 filters. This is followed by 10 residual blocks such that all hidden representations have the same height and width of $3 2 \\times 3 2$ and 100 filters are used in all the convolution layers in residual blocks as well. ",
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "3. Avg-pooling Resnet: This architecture repeats the residual blocks of the single representation Resnet (described above) three times such that there is a $2 \\times 2$ average pooling layer after each set of 10 residual blocks that reduces the height and width after each stage by half. Also, in contrast to single representation architecture, it uses 150 filters in all convolution layers. This is followed by the classification block as in the single representation Resnet. It has 12, 201, 310 parameters. We call this architecture the avg-pooling architecture. We also ran experiments with max pooling instead of average pooling but do not report results because they were similar except that max pool acts more non-linearly compared with average pooling, and hence the metrics from max pooling are more similar to those from original Resnet. ",
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+    {
+        "type": "text",
+        "text": "4. Wide Resnet: This architecture starts with a $3 \\times 3$ convolution layer followed by 3 stages of four residual blocks with 160, 320 and 640 number of filters respectively, and $3 \\times 3$ kernel size in all convolution layers. This model has a total of 45,732,842 parameters. ",
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+        "type": "text",
+        "text": "Experimental details: For all architectures, we use He-normal weight initialization as suggested in He et al. (2015), and biases are initialized to 0. For residual blocks, we use BatchNorm ReLU Conv BatchNorm ReLU Conv as suggested in He et al. (2016b). The classifier is composed of the following elements: BatchNorm ReLU AveragePool(8,8) Flatten Fully-Connected-Layer(#classes) Softmax. This model has 1, 829, 210 parameters. For all experiments for single representation and pooling Resnet architectures, we use SGD with momentum 0.9 and train for 200 epochs and 100 epochs (respectively) with learning rate 0.1 until epoch 40, 0.02 until 60, 0.004 until 80 and 0.0008 afterwards. For the original Resnet we use SGD with momentum 0.9 and train for 300 epochs with learning rate 0.1 until epoch 80, 0.01 until 120, 0.001 until 200, 0.00001 until 240 and 0.000011 afterwards. We use data augmentation (horizontal flipping and translation) during training of all architectures. For the wide Resnet architecture, we train the model with with learning rate 0.1 until epoch 60 and 0.02 until 100 epochs. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Note: All experiments on CIFAR-100 are reported in the appendix. In addition, we also record the metrics reported in sections 4.1 and 4.2 as a function of epochs (shown in the appendix due to space limitations). The conclusions are similar to what is reported below. ",
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+    {
+        "type": "text",
+        "text": "4.1 COSINE LOSS OF RESIDUAL BLOCKS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this experiment we directly validate our theoretical prediction about Resnets minimizing the dot product between gradient of loss and block output. To this end compute the cosine loss $\\bar { \\Gamma _ { i } } ( \\mathbf { h } _ { i } ) . \\frac { \\partial \\mathcal { L } ( \\mathbf { h } _ { i } ) } { \\partial \\mathbf { h } _ { i } }$   \nA negative cosine loss and small $F _ { i } ( . )$ together suggest that $F _ { i } ( . )$ is refining $\\begin{array} { r } { \\overline { { { \\| { \\cal F } _ { i } ( { \\bf h } _ { i } ) \\| _ { 2 } } \\| \\frac { \\partial { \\mathcal { L } } ( { \\bf h } _ { i } ) } { \\partial { \\bf h } _ { i } } \\| _ { 2 } } } } \\end{array}$   \nfeatures by moving them in the half space of $- \\frac { \\partial \\mathcal { L } ( \\mathbf { h } _ { i } ) } { \\partial \\mathbf { h } _ { i } }$ , thus reducing the loss value for the corresponding data samples. Figure 4 shows the cosine loss for CIFAR-10 on train and validation sets. These figures show that cosine loss is consistently negative for all residual blocks but especially for the higher residual blocks. Also, notice for deeper architectures (original Resnet and pooling Resnet), the higher blocks achieve more negative cosine loss and are thus more iterative in nature. Further, since the higher residual blocks make smaller changes to representation (figure 2), the first order Taylor’s term becomes dominant and hence these blocks effectively move samples in the half space of the negative cosine loss thus reducing loss value of prediction. This result formalizes the sense in which residual blocks perform iterative refinement of features– move representations in the half space of $- \\frac { \\partial \\mathcal { L } ( \\mathbf { h } _ { i } ) } { \\partial \\mathbf { h } _ { i } }$ . ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.2 REPRESENTATION LEARNING VS. FEATURE REFINEMENT ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "In this section, we are interested in investigating the behavior of residual layers in terms of representation learning vs. refinement of features. To this end, we perform the following experiments. ",
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+        "text": "1. $\\ell ^ { 2 }$ ratio $\\| F _ { i } ( \\mathbf { h } _ { i } ) \\| _ { 2 } / \\| \\mathbf { h } _ { i } \\| _ { 2 }$ : A residual block $F _ { i } ( . )$ transforms representation as $\\mathbf { h } _ { i + 1 } \\mathbf { \\Psi } = \\mathbf { h } _ { i } \\mathbf { \\Psi } +$ $F _ { i } ( \\mathbf { h } _ { i } )$ . For every such block in a Resnet, we measure the $\\ell ^ { 2 }$ ratio of $\\| F _ { i } ( \\mathbf h _ { i } ) \\| _ { 2 } / \\| \\mathbf h _ { i } \\| _ { 2 }$ averaged across samples. This ratio directly shows how significantly $F _ { i } ( . )$ changes the representation $\\mathbf { h } _ { i }$ ; a large change can be argued to be a necessary condition for layer to perform representation learning. Figure 2 shows the $\\ell ^ { 2 }$ ratio for CIFAR-10 on train and validation sets. For single representation ",
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+        "type": "text",
+        "text": "Resnet and pooling Resnet, the first few residual blocks (especially the first residual block) changes representations significantly (up to twice the norm of the original representation), while the rest of the higher blocks are relatively much less significant and this effect is monotonic as we go to higher blocks. However this effect is not as drastic in the original Resnet and wide Resnet architectures which have two $1 \\times 1$ (shortcut) convolution layers, thus adding up to a total of 3 convolution layers in the main path of the residual network (notice there exists only one convolution layer in the main path for the other two architectures). This suggests that residual blocks in general tend to learn to refine features but in the case when the network lacks enough compositional layers in the main path, lower residual blocks are forced to change representations significantly, as a proxy for the absence of compositional layers. Additionally, small $\\ell ^ { \\frac { \\mathtt { A } } { 2 } }$ ratio justifies first order approximation used to derive our main result in Sec. 3. ",
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+        "type": "text",
+        "text": "2. Effect of dropping residual layer on accuracy: We drop individual residual blocks from trained Resnets and make predictions using the rest of network on validation set. This analysis shows the significance of individual residual blocks towards the final accuracy that is achieved using all the residual blocks. Note, dropping individual residual blocks is possible because adjacent blocks operate in the same feature space. Figure 3 shows the result of dropping individual residual blocks. As one would expect given above analysis, dropping the first few residual layers (especially the first) for single representation Resnet and pooling Resnet leads to catastrophic performance drop while dropping most of the higher residual layers have minimal effect on performance. On the other hand, performance drops are not drastic for the original Resnet and wide Resnet architecture, which is in agreement with the observations in $\\ell ^ { 2 }$ ratio experiments above. ",
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+    {
+        "type": "text",
+        "text": "In another set of experiments, we measure validation accuracy after individual residual block during the training process. This set of experiments is achieved by plugging the classifier right after each residual block in the last stage of hidden representation (i.e., after the last shortcut connection, if any). This is shown in figure 5. The figures show that accuracy increases very gradually when adding more residual blocks in the last stage of all architectures. ",
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+    {
+        "type": "text",
+        "text": "4.3 BORDERLINE EXAMPLES",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section we investigate which samples get correctly classified after the application of a residual block. Individual residual blocks in general lead to small improvements in performance. Intuitively, since these layers move representations minimally (as shown by previous analysis), the samples that lead to these minor accuracy jump should be near the decision boundary but getting misclassified by a slight margin. To confirm this intuition, we focus on borderline examples, defined as examples that require less than $1 0 \\%$ probability change to flip prediction to, or from the correct class. We measure loss, accuracy and entropy over borderline examples over last 5 blocks of the network using the network final classifier. Experiment is performed on CIFAR-10 using Resnet-110 architecture. ",
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+        "type": "text",
+        "text": "Fig 6 shows evolution of loss and accuracy on three groups of examples: borderline examples, already correctly classified and the whole dataset. While overall accuracy and loss remains similar across the top residual blocks, we observe that a significant chunk of borderline examples gets corrected by the immediate next residual block. This exposes the qualitative nature of examples that these feature refinement layers focus on, which is further reinforced by the fact that entropy decreases for all considered subsets. We also note that while train loss drops uniformly across layers, test sets loss increases after last block. Correcting this phenomenon could lead to improved generalization in Resnets, which we leave for future work. ",
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+    {
+        "type": "text",
+        "text": "4.4 UNROLLING RESIDUAL NETWORK ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A fundamental requirement for a procedure to be truly iterative is to apply the same function. In this section we explore what happens when we unroll the last block of a trained residual network for more steps than it was trained for. Our main goal is to investigate if iterative inference generalizes to more steps than it was trained on. We focus on the same model as discussed in previous section, Resnet-110, and unroll the last residual block for 20 extra steps. Naively unrolling the network leads to activation explosion (we observe similar behavior in Sec. 4.5). To control for that effect, we added a scaling factor on the output of the last residual blocks. We hypothesize that controlling the scale limits the drift of the activation through the unrolled layer, i.e. they remains in a given neighbourhood on which the network is well behaved. Similarly to Sec. 4.3 we track evolution of loss and accuracy on three groups of examples: borderline examples, already correctly classified and the whole dataset. Experiments are repeated 4 times, and results are averaged. ",
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+    {
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+        "img_path": "images/45ce3cb6c90773f203329525ca411c67d6772b3e60ca0f93709282eb77bc1563.jpg",
+        "image_caption": [
+            "Figure 6: Accuracy, loss and entropy for last 5 blocks of Resnet-110. Performance on bordeline examples improves at the expense of performance (loss) of already correctly classified points (correct). This happens because last block output is encouraged by training to be negatively correlated (around $- 0 . 1$ cosine) with gradient of the loss. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/1cf6fb56177565aa740e3c99c5040589577c27df90aab5e834e0229f328a5f23.jpg",
+        "image_caption": [
+            "Figure 7: Accuracy, loss and entropy for Resnet-110 with last block unrolled for 20 additional steps (with appropriate scaling). Borderline examples are corrected and overall performance accuracy improves. Note different scales for train and test. Curves are averaged over 4 runs. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "We first investigate how unrolling blocks impact loss and accuracy. Loss on train set improved uniformly from 0.0012 to 0.001, while it increased on test set. There are on average 51 borderline examples in test $\\mathrm { s e t } ^ { 2 }$ , on which performance is improved from $4 3 \\%$ to $5 3 \\%$ , which yields slight improvement in accuracy on test set. Next we shift our attention to cosine loss. We observe that cosine loss remains negative on the first two steps without rescaling, and all steps after scaling. Figure 7 shows evolution of loss and accuracy on the three groups of examples: borderline examples, already correctly classified and the whole dataset. Cosine loss and $\\ell ^ { 2 }$ ratio for each block are reported in Appendix E. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To summarize, unrolling residual network to more steps than it was trained on improves both loss on train set, and maintains (in given neighbourhood) negative cosine loss on both train and test set. ",
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+    {
+        "type": "text",
+        "text": "4.5 SHARING RESIDUAL LAYERS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "Our results suggest that top residual blocks should be shareable, because they perform similar iterative refinement. We consider a shared version of Resnet-110 model, where in each stage we share all the residual blocks from the $5 ^ { t h }$ block. All shared Resnets in this section have therefore a similar number of parameters as Resnet-38. Contrary to (Liao & Poggio, 2016) we observe that naively sharing the higher (iterative refinement) residual blocks of a Resnets in general leads to bad performance3 (especially for deeper Resnets). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "First, we compare the unshared and shared version of Resnet-110. The shared version uses approximately 3 times less parameters. In Fig. 8, we report the train and validation performances of the Resnet-110. We observe that naively sharing parameters of the top residual blocks leads both to overfitting (given similar training accuracy, the shared Resnet-110 has significantly lower validation performances) and underfitting (worse training accuracy than Resnet-110). We also compared our shared model with a Resnet-38 that has a similar number of parameters and observe worse validation performances, while achieving similar training accuracy. ",
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+    {
+        "type": "image",
+        "img_path": "images/bfbb0a7746950c10206f50512c34dd98c7f540cf633119c7a63f023e3d29f4fe.jpg",
+        "image_caption": [
+            "Figure 8: Resnet-110 with naively shared top 13 layers of each block compared with unshared Resnet-38. Left plot present training and validation curves, shared Resnet-110 heavily overfits. In the right plot we track gradient norm ratio between first block in first and last stage of resnet $\\begin{array} { r } { r = | | \\frac { \\hat { \\partial } L } { \\partial h _ { 1 } } | | / \\frac { \\partial L } { \\partial h _ { 1 + 2 n } } | | \\big ) } \\end{array}$ L1 ||/ ∂L∂h1+2n ||). Significantly larger ratio in the naive sharing model suggests, that the overfitting is caused by early layers dominating learning. Metrics are tracked on train (solid line) and validation data (dashed line) "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/317e694fb4391b1b7f026c109e8bd64f8f99052c805e6808e33e71ff5988a165.jpg",
+        "image_caption": [
+            "Figure 9: Ablation study of different strategies to remedy sharing leading to overfitting phenomenon in Residual Networks. Left figure shows effect on training and test accuracy. Right figure studies norm explosion. All components are important, but it is most crucial to unshare BN statistics. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We notice that sharing layers make the layer activations explode during the forward propagation at initialization due to the repeated application of the same operation (Fig 8, right). Consequently, the norm of the gradients also explodes at initialization (Fig. 8, center). ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To address this issue we introduce a variant of recurrent batch normalization (Cooijmans et al., 2016), which proposes to initialize $\\gamma$ to 0.1 and unshare statistics for every step. On top of this strategy, we also unshare $\\gamma$ and $\\beta$ parameters. Tab. 1 shows that using our strategy alleviates explosion problem and leads to small improvement over baseline with similar number of parameters. We also perform an ablation to study, see Figure. 9 (left), which show that all additions to naive strategy are necessary and drastically reduce the initial activation explosion. Finally, we observe a similar trend for cosine loss, intermediate accuracy, and $\\ell ^ { 2 }$ ratio for the shared Resnet as for the unshared Resnet discussed in the previous Sections. Full results are reported in Appendix D. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Unshared Batch Normalization strategy therefore mitigates this exploding activation problem. This problem, leading to exploding gradient in our case, appears frequently in recurrent neural network. This suggests that future unrolled Resnets should use insights from research on recurrent networks optimization, including careful initialization (Henaff et al., 2016) and parametrization changes (Hochreiter & Schmidhuber, 1997). ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/ddba59987491ef2561c82aab6500a1b67b4dd9e3b280e1c2d6e7aa6647279627.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>CIFAR10</td><td>CIFAR100</td><td>Parameters</td></tr><tr><td>Resnet-32</td><td>1.53/7.14</td><td>12.62/30.08</td><td>467k-473k</td></tr><tr><td>Resnet-38</td><td>1.20/6.99</td><td>10.04/29.66</td><td>565k-571k</td></tr><tr><td>Resnet-110-UBN</td><td>0.63/6.62</td><td>7.75/29.94</td><td>570k-576k</td></tr><tr><td>Resnet-146-UBN</td><td>0.68/6.82</td><td>7.21/29.49</td><td>573k-579k</td></tr><tr><td>Resnet-182-UBN</td><td>0.48/6.97</td><td>6.42 /29.33</td><td>576k-581k</td></tr><tr><td>Resnet-56</td><td>0.58/6.53</td><td>5.19/28.99</td><td>857k-863k</td></tr><tr><td>Resnet-110</td><td>0.22/6.13</td><td>1.26 /27.54</td><td>1734k-1740k</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Train and test error of Resnet sharing top layers blocks (while using unshared both statistics and $\\beta , \\gamma$ in Batch Normalization) denoted as UBN (Unshared Batch Normalization) compared to baseline Resnet of varying depth. Training Resnet with unrolled layers can bring additional gain of $0 . 3 \\%$ , while adding marginal amount of extra parameters. Runs are repeated 4 times. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            315,
+            318,
+            332
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Our main contribution is formalizing the view of iterative refinement in Resnets and showing analytically that residual blocks naturally encourage representations to move in the half space of negative loss gradient, thus implementing a gradient descent in the activation space (each block reduces loss and improves accuracy). We validate theory experimentally on a wide range of Resnet architectures. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We further explored two forms of sharing blocks in Resnet. We show that Resnet can be unrolled to more steps than it was trained on. Next, we found that counterintuitively training residual blocks with shared blocks leads to overfitting. While we propose a variant of batch normalization to mitigate it, we leave further investigation of this phenomena for future work. We hope that our developed formal view, and practical results, will aid analysis of other models employing iterative inference and residual connections. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            515,
+            367,
+            530
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We acknowledge the computing resources provided by ComputeCanada and CalculQuebec. SJ was supported by Grant No. DI 2014/016644 from Ministry of Science and Higher Education, Poland. DA was supported by IVADO. ",
+        "bbox": [
+            176,
+            546,
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+            588
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            611,
+            287,
+            625
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "D. Arpit, Y. Zhou, B. U Kota, and V. Govindaraju. Normalization propagation: A parametric technique for removing internal covariate shift in deep networks. ICML, 2016.   \nTim Cooijmans, Nicolas Ballas, César Laurent, Çaglar Gülçehre, and Aaron Courville. Recurrent ˘ batch normalization. arXiv preprint arXiv:1603.09025, 2016.   \nX. Glorot and Y. Bengio. Understanding the difficulty of training deep feedforward neural networks. In Aistats, 2010.   \nK. Greff, R. Srivastava, and J. Schmidhuber. Highway and residual networks learn unrolled iterative estimation. arXiV, 2016.   \nK. He, X. Zhang, S. Ren, and J. Sun. Delving deep into rectifiers: Surpassing human-level performance on imagenet classification. In ICCV, 2015.   \nK. He, X. Zhang, S. Ren, and J. Sun. Deep residual learning for image recognition. In CVPR, 2016a.   \nK. He, X. Zhang, S. Ren, and J. Sun. Identity mappings in deep residual networks. In ECCV, 2016b.   \nM. Henaff, A. Szlam, and Y. LeCun. Recurrent orthogonal networks and long-memory tasks. In ICML, 2016.   \nS. Hochreiter and J. Schmidhuber. Long short-term memory. Neural computation, 1997. ",
+        "bbox": [
+            169,
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+        ],
+        "page_idx": 8
+    },
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Appendices ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            343,
+            127
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A FURTHER ANALYSIS ",
+        "bbox": [
+            176,
+            148,
+            382,
+            166
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.1 A SIDE-EFFECT OF MOVING IN THE HALF SPACE OF $- \\frac { \\partial \\mathcal { L } ( \\mathbf { h } _ { o } ) } { \\partial \\mathbf { h } _ { o } }$ ",
+        "bbox": [
+            174,
+            181,
+            632,
+            202
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Let $\\mathbf { h } _ { o } = \\mathbf { W } \\mathbf { x } + \\mathbf { b }$ be the output of the first layer (convolution) of a ResNet. In this analysis we show that if $\\mathbf { h } _ { o }$ moves in the half space of $- \\frac { \\partial \\mathcal { L } ( \\dot { \\bf h } _ { o } ) } { \\partial { \\bf h } _ { o } }$ , then it is equivalent to updating the parameters of the convolution layer using a gradient update step. To see this, consider the change in $\\mathbf { h } _ { o }$ from updating parameters using gradient descent with step size $\\eta$ . This is given by, ",
+        "bbox": [
+            176,
+            214,
+            823,
+            276
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/00b49fa65fafc32521c2e616a5406fd7937229b5db97dc72f2fa0f6c9e639696.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\Delta \\mathbf { h } _ { o } = ( \\mathbf { W } - \\eta \\frac { \\partial \\mathcal { L } } { \\partial \\mathbf { W } } ) \\mathbf { x } + ( \\mathbf { b } - \\eta \\frac { \\partial \\mathcal { L } } { \\partial \\mathbf { b } } ) - ( \\mathbf { W } \\mathbf { x } + \\mathbf { b } ) } \\\\ & { \\quad \\quad = - \\eta \\frac { \\partial \\mathcal { L } } { \\partial \\mathbf { W } } \\mathbf { x } - \\eta \\frac { \\partial \\mathcal { L } } { \\partial \\mathbf { b } } } \\\\ & { \\quad \\quad = - \\eta \\frac { \\partial \\mathcal { L } } { \\partial \\mathbf { h } _ { o } } \\left( \\frac { \\partial \\mathbf { h } _ { o } } { \\partial \\mathbf { W } } \\mathbf { x } + \\frac { \\partial \\mathbf { h } _ { o } } { \\partial \\mathbf { b } } \\right) } \\\\ & { \\quad \\quad = - \\eta \\frac { \\partial \\mathcal { L } } { \\partial \\mathbf { h } _ { o } } \\left( \\| \\mathbf { x } \\| ^ { 2 } + 1 \\right) } \\\\ & { \\quad \\quad \\propto - \\frac { \\partial \\mathcal { L } } { \\partial \\mathbf { h } _ { o } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            323,
+            287,
+            673,
+            452
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Thus, moving $\\mathbf { h } _ { o }$ in the half space of ∂L∂h has the same effect as that achieved by updating the parameters W, b using gradient descent. Although we found this insight interesting, we don’t build upon it in this paper. We leave this as a future work. ",
+        "bbox": [
+            174,
+            462,
+            825,
+            507
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "B ANALYSIS ON CIFAR-100 ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            534,
+            429,
+            550
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Here we report the experiments as done in sections 4.2 and 4.1, for CIFAR-100 dataset. The plots are shown in figures 10, 11 and 12. The conclusions are same as reported in the main text for CIFAR-10. ",
+        "bbox": [
+            173,
+            569,
+            825,
+            598
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "C ANALYSIS OF INTERMEDIATE METRICS ON CIFAR-10 AND CIFAR-100 ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            637,
+            803,
+            654
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Here we plot the accuracy, cosine loss and $\\ell ^ { 2 }$ ratio metrics corresponding to each individual residual block on validation during the training process for CIFAR-10 (figures 13, 14, 5) and CIFAR-100 (figures 15, 16, 17). These plots are recorded only for the residual blocks in the last space for each architecture (this is because otherwise the dimensions of the output of the residual block and the classifier will not match). In the case of cosine loss after individual residual block, this set of experiments is achieved by plugging the classifier right after each hidden representation and measuring the cosine between the gradient w.r.t. hidden representation and the corresponding residual block’s output. ",
+        "bbox": [
+            173,
+            671,
+            825,
+            785
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We find that the accuracy after individual residual blocks increases gradually as we move from from lower to higher residua blocks. Cosine loss on the other hand consistently remains negative for all architectures. Finally $\\ell ^ { 2 }$ ratio tends to increase for residual blocks as training progresses. ",
+        "bbox": [
+            174,
+            791,
+            823,
+            834
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "D ITERATIVE INFERENCE IN SHARED RESNET ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            859,
+            568,
+            876
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In this section we extend results from Sec. 4.5. We report cosine loss, intermediate accuracy, and $\\ell ^ { 2 }$ ratio for naively shared Resnet in Fig. 19, and with unshared batch normalization in Fig. ??. ",
+        "bbox": [
+            173,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/1b17acb195696f29ae3640e938149cc123114da395f8c230d9ab97cb13e31f33.jpg",
+        "image_caption": [
+            "Figure 10: Average cos loss between residual block $F ( \\mathbf { h } _ { i } )$ and $\\frac { \\partial \\mathcal { L } ( \\mathbf { h } _ { i } ) } { \\partial \\mathbf { h } _ { i } }$ for (left to right) original Resnet, single representation Resnet, avg-pooling Resnet, and wideResnet on CIFAR-100. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            151,
+            821,
+            209
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/6c3cb1928f9b9760fc6b0a476c1dbfe114c61337c0a10818f9d2531121cc3492.jpg",
+        "image_caption": [
+            "Figure 11: Average ratio of $\\ell ^ { 2 }$ norm of output of residual block to the norm of the input of residual block for (left to right) original Resnet, single representation Resnet, avg-pooling Resnet, and wideResnet on CIFAR-100. (Train and validation curves are overlapping.) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            271,
+            823,
+            330
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/6fa9571f937f9a10fa428290fe15ddf9157886ca0638d0c70792a5dfa9dbd224.jpg",
+        "image_caption": [
+            "Figure 12: Final prediction accuracy when individual residual blocks are dropped for (left to right) original Resnet, single representation Resnet, avg-pooling Resnet, and wideResnet on CIFAR-100. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            398,
+            823,
+            462
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/35b4afd2a7979b9304c3d47ec0ba7f16e0dbb19db8dffad74d7b420d54dffd86.jpg",
+        "image_caption": [
+            "Figure 13: Average cos loss between residual block $F ( \\mathbf { h } _ { i } )$ and $\\frac { \\partial \\mathcal { L } ( \\mathbf { h } _ { i } ) } { \\partial \\mathbf { h } _ { i } }$ during training for (left to right) original Resnet, single representation Resnet, avg-pooling Resnet, and wideResnet on CIFAR10. (Blue to red spectrum denotes lower to higher residual blocks) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            609,
+            820,
+            667
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/47804a663459cbe85a632d724d137c87e5c163c0da670b05c469de4e12d1d468.jpg",
+        "image_caption": [
+            "Figure 14: Average ratio of $\\ell ^ { 2 }$ norm of output of residual block to the norm of the input of residual block during training for (left to right) original Resnet, single representation Resnet, avg-pooling Resnet, and wideResnet on CIFAR-10. (Blue to red spectrum denotes lower to higher residual blocks) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            743,
+            821,
+            803
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/b779c3d0cc4cfdca2c6f72fe04e29662322b5aa8270237d0879be36375e249eb.jpg",
+        "image_caption": [
+            "Figure 15: Average cos loss between residual block $F ( \\mathbf { h } _ { i } )$ and $\\frac { \\partial \\mathcal { L } ( \\mathbf { h } _ { i } ) } { \\partial \\mathbf { h } _ { i } }$ during training for (left to right) original Resnet, single representation Resnet, avg-pooling Resnet, and wideResnet on CIFAR100. (Blue to red spectrum denotes lower to higher residual blocks) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            145,
+            821,
+            204
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/be96d4ee5e6092d0abc3923a29ede16bae2a0e2f69dfda342c24dce27448fdc7.jpg",
+        "image_caption": [
+            "Figure 16: Average ratio of $\\ell ^ { 2 }$ norm of output of residual block to the norm of the input of residual block during training for (left to right) original Resnet, single representation Resnet, avg-pooling Resnet, and wideResnet on CIFAR-100. (Blue to red spectrum denotes lower to higher residual blocks) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            275,
+            821,
+            339
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/0e1c34c0292272d61a67bcf11c4d7dee673f157548679b261a2c86de4c4c0365.jpg",
+        "image_caption": [
+            "Figure 17: Prediction accuracy when plugging classifier after hidden states in the last stage of Resnets(if any) during training for (left to right) original Resnet, single representation Resnet, avgpooling Resnet, and wideResnet on CIFAR-100. (Blue to red spectrum denotes lower to higher residual blocks) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            425,
+            821,
+            484
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/6971a298000f2b641ebb3b1777ff1c7001887d45ceeab449029d48e8d485df97.jpg",
+        "image_caption": [
+            "Figure 18: Cosine loss, $\\ell ^ { 2 }$ ratio, and intermediate accuracy for shared Resnet-110 with unshared Batch Normalization (described in Sec. 4.5). Each curve represents different block in Resnet. Red is closest to output. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            205,
+            651,
+            790,
+            821
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/d0f2347f07953ddb059bbfaad26f0e9816ebf0fbb936fa7d0fd52f4002719440.jpg",
+        "image_caption": [
+            "Figure 19: Cosine loss, $\\ell ^ { 2 }$ ratio, and intermediate accuracy for naively shared Resnet-110. Each curve represents different block in Resnet. Red is closest to output. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            205,
+            101,
+            790,
+            276
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/e116cb21a9e9e5bf5578ddf325e9427b0d83225b4a27e53806f38c6c3d213ffe.jpg",
+        "image_caption": [
+            "Figure 20: First figure show that cosine loss in Resnet-110 after unrolling generalizes to more steps than it was trained on. Second plot shows evolution of $\\ell ^ { 2 }$ ratio for Resnet-110. Third plot reports cosine loss Resnet-110 with scaled version of final block, as considered in Sec. 4.4. Rightmost plots reports $\\ell ^ { 2 }$ ratio for scaled Resnet-110. Vertical line in plots indicates number of steps network was trained on. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            204,
+            332,
+            794,
+            460
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "E UNROLLING RESIDUAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            566,
+            496,
+            582
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In this section we report additional results for unrolling residual network. Figure 20 shows evolution of cosine loss an $\\ell ^ { 2 }$ ratio for Resnet-110 with unrolled last block for 20 additional steps. ",
+        "bbox": [
+            173,
+            598,
+            825,
+            626
+        ],
+        "page_idx": 13
+    }
+]

parse/train/SJgVHkrYDH/SJgVHkrYDH.md

@@ -0,0 +1,496 @@
+# LEARNING TO RETRIEVE REASONING PATHS OVER WIKIPEDIA GRAPH FOR QUESTION ANSWERING
+
+Akari Asai∗†, Kazuma Hashimoto‡, Hannaneh Hajishirzi†§, Richard Socher‡ & Caiming Xiong‡
+
+†University of Washington ‡Salesforce Research §Allen Institute for Artificial Intelligence   
+{akari,hannaneh}@cs.washington.edu   
+{k.hashimoto,rsocher,cxiong}@salesforce.com
+
+# ABSTRACT
+
+Answering questions that require multi-hop reasoning at web-scale necessitates retrieving multiple evidence documents, one of which often has little lexical or semantic relationship to the question. This paper introduces a new graphbased recurrent retrieval approach that learns to retrieve reasoning paths over the Wikipedia graph to answer multi-hop open-domain questions. Our retriever model trains a recurrent neural network that learns to sequentially retrieve evidence paragraphs in the reasoning path by conditioning on the previously retrieved documents. Our reader model ranks the reasoning paths and extracts the answer span included in the best reasoning path. Experimental results show state-of-the-art results in three open-domain QA datasets, showcasing the effectiveness and robustness of our method. Notably, our method achieves significant improvement in HotpotQA, outperforming the previous best model by more than 14 points.1
+
+# 1 INTRODUCTION
+
+Open-domain Question Answering (QA) is the task of answering a question given a large collection of text documents (e.g., Wikipedia). Most state-of-the-art approaches for open-domain QA (Chen et al., 2017; Wang et al., 2018a; Lee et al., 2018; Yang et al., 2019) leverage non-parameterized models (e.g., TF-IDF or BM25) to retrieve a fixed set of documents, where an answer span is extracted by a neural reading comprehension model. Despite the success of these pipeline methods in singlehop QA, whose questions can be answered based on a single paragraph, they often fail to retrieve the required evidence for answering multi-hop questions, e.g., the question in Figure 1. Multi-hop QA (Yang et al., 2018) usually requires finding more than one evidence document, one of which often consists of little lexical overlap or semantic relationship to the original question. However, retrieving a fixed list of documents independently does not capture relationships between evidence documents through bridge entities that are required for multi-hop reasoning.
+
+Recent open-domain QA methods learn end-to-end models to jointly retrieve and read documents (Seo et al., 2019; Lee et al., 2019). These methods, however, face challenges for entity-centric questions since compressing the necessary information into an embedding space does not capture lexical information in entities. Cognitive Graph (Ding et al., 2019) incorporates entity links between documents for multi-hop QA to extend the list of retrieved documents. This method, however, compiles a fixed list of documents independently and expects the reader to find the reasoning paths.
+
+In this paper, we introduce a new recurrent graph-based retrieval method that learns to retrieve evidence documents as reasoning paths for answering complex questions. Our method sequentially retrieves each evidence document, given the history of previously retrieved documents to form several reasoning paths in a graph of entities. Our method then leverages an existing reading comprehension model to answer questions by ranking the retrieved reasoning paths. The strong interplay between the retriever model and reader model enables our entire method to answer complex questions by exploring more accurate reasoning paths compared to other methods.
+
+![](images/5341787e714e2d214f06fea1dee98532372593e3609cbf71531d27fd1d839f5c.jpg)  
+Figure 1: An example of open-domain multi-hop question from HotpotQA. Paragraph 2 is unlikely to be retrieved using TF-IDF retrievers due to little lexical overlap to the given question.
+
+To be more specific, our method (sketched in Figure 2) constructs the Wikipedia paragraph graph using Wikipedia hyperlinks and document structures to model the relationships between paragraphs. Our retriever trains a recurrent neural network to score reasoning paths in this graph by maximizing the likelihood of selecting a correct evidence paragraph at each step and fine-tuning paragraph BERT encodings. Our reader model is a multi-task learner to score each reasoning path according to its likelihood of containing and extracting the correct answer phrase. We leverage data augmentation and negative example mining for robust training of both models.
+
+Our experimental results show that our method achieves the state-of-the-art results on HotpotQA full wiki and HotpotQA distractor settings (Yang et al., 2018), outperforming the previous stateof-the-art methods by more than 14 points absolute gain on the full wiki setting. We also evaluate our approach on SQuAD Open (Chen et al., 2017) and Natural Questions Open (Lee et al., 2019) without changing any architectural designs, achieving better or comparable to the state of the art, which suggests that our method is robust across different datasets. Additionally, our framework provides interpretable insights into the underlying entity relationships used for multi-hop reasoning.
+
+# 2 RELATED WORK
+
+Neural open-domain question answering Most current open-domain QA methods use a pipeline approach that includes a retriever and reader. Chen et al. (2017) incorporate a TF-IDF-based retriever with a state-of-the-art neural reading comprehension model. The subsequent work improves the heuristic retriever by re-ranking retrieved documents (Wang et al., 2018a;b; Lee et al., 2018; Lin et al., 2018). The performance of these methods is still bounded by the performance of the initial retrieval process. In multi-hop QA, non-parameterized retrievers face the challenge of retrieving all the relevant documents, one or some of which are lexically distant from the question. Recently, Lee et al. (2019) and Seo et al. (2019) introduce fully trainable models that retrieve a few candidates directly from large-scale Wikipedia collections. All these methods find evidence documents independently without the knowledge of previously selected documents or relationships between documents. This would result in failing to conduct multi-hop retrieval.
+
+Retrievers guided by entity links Most relevant to our work are recent studies that attempt to use entity links for multi-hop open-domain QA. Cognitive Graph (Ding et al., 2019) retrieves evidence documents offline, and trains a reading comprehension model to jointly predict possible answer spans and next-hop spans to extend the reasoning chain. Instead, we train our retriever to find reasoning paths directly. Concurrent with our work, Entity-centric IR (Godbole et al., 2019) uses entity linking for multi-hop retrieval. Unlike our method, this method does not learn to retrieve reasoning paths sequentially, nor study the interplay between retriever and reader. Moreover, while the previous approaches require a system to encode all possible nodes, our beam search decoding process only encodes the nodes on the reasoning paths, which significantly reduces the computational costs. PullNet (Sun et al., 2019) learns to retrieve question-aware sub-graphs from text corpora and knowledge bases (e.g., Freebase), while we focus on open-domain QA solely based on text.
+
+Multi-step (iterative) retrievers Similar to our recurrent retriever, multi-step retrievers explore multiple evidence documents iteratively. Multi-step reasoner (Das et al., 2019) repeats the retrieval process for a fixed number of steps, interacting with a reading comprehension model by reformulating the query in a latent space to enhance retrieval performance. Feldman & El-Yaniv (2019) also propose a query reformulation mechanism with a focus on multi-hop open-domain QA. Most recently, Qi et al. (2019) introduce GoldEn Retriever, which reads and generates search queries for two steps to search documents for HotpotQA full wiki. These methods do not use the graph structure of the documents during the iterative retrieval process. In addition, all of these multi-step retrieval methods do not accommodate arbitrary steps of reasoning and the termination condition is hard-coded. In contrast, our method leverages the Wikipedia graph to retrieve documents that are lexically or semantically distant to questions, and is adaptive to any reasoning path lengths, which leads to significant improvement over the previous work in HotpotQA and SQuAD Open.
+
+![](images/a022e60ad47bb8c69547f226d08e1ee9f406355a07b186ccc4cf766bef6162fb.jpg)  
+Figure 2: Overview of our framework.
+
+# 3 OPEN-DOMAIN QUESTION ANSWERING OVER WIKIPEDIA GRAPH
+
+Overview This paper introduces a new graph-based recurrent retrieval method (Section 3.1) that learns to find evidence documents as reasoning paths for answering complex questions. We then extend an existing reading comprehension model (Section 3.2) to answer questions given a collection of reasoning paths. Our method uses a strong interplay between retrieving and reading steps such that the retrieval method learns to retrieve a set of reasoning paths to narrow down the search space for our reader model, for robust pipeline process. Figure 2 sketches the overview of our QA model.
+
+We use Wikipedia for open-domain QA, where each article is divided into paragraphs, resulting in millions of paragraphs in total. Each paragraph $p$ is considered as our retrieval target. Given a question $q$ , our framework aims at deriving its answer $a$ by retrieving and reading reasoning paths, each of which is represented with a sequence of paragraphs: $E = [ p _ { i } , \dotsc , p _ { k } ]$ . We formulate the task by decomposing the objective into the retriever objective $S _ { \mathrm { r e t r } } ( q , E )$ that selects reasoning paths $E$ relevant to the question, and the reader objective $S _ { \mathrm { r e a d } } ( q , E , a )$ that finds the answer $a$ in $E$ :
+
+$$
+\operatorname * { a r g m a x } _ { E , a } S ( q , E , a ) \mathrm { s . t . } S ( q , E , a ) = S _ { \mathrm { r e t r } } ( q , E ) + S _ { \mathrm { r e a d } } ( q , E , a ) .
+$$
+
+# 3.1 LEARNING TO RETRIEVE REASONING PATHS
+
+Our method learns to retrieve reasoning paths across a graph structure. Evidence paragraphs for a complex question do not necessarily have lexical overlaps with the question, but one of them is likely to be retrieved, and its entity mentions and the question often entail another paragraph (e.g., Figure 1). To perform such multi-hop reasoning, we first construct a graph of paragraphs, covering all the Wikipedia paragraphs. Each node of the Wikipedia graph $\mathcal { G }$ represents a single paragraph $p _ { i }$ .
+
+Constructing the Wikipedia graph Hyperlinks are commonly used to construct relationships between articles on the web, usually maintained by article writers, and are thus useful knowledge resources. Wikipedia consists of its internal hyperlinks to connect articles. We use the hyperlinks to construct the direct edges in $\mathcal { G }$ . We also consider symmetric within-document links, allowing a paragraph to hop to other paragraphs in the same article. The Wikipedia graph $\mathcal { G }$ is densely connected and covers a wide range of topics that provide useful evidence for open-domain questions. This graph is constructed offline and is reused throughout training and inference for any question.
+
+# 3.1.1 THE GRAPH-BASED RECURRENT RETRIEVER
+
+General formulation with a recurrent retriever We use a Recurrent Neural Network (RNN) to model the reasoning paths for the question $q$ . At the $t { \cdot }$ -th time step $( t \geq 1 )$ ) our model selects a paragraph $p _ { i }$ among candidate paragraphs $\mathbf { C } _ { t }$ given the current hidden state $h _ { t }$ of the RNN. The initial hidden state $h _ { 1 }$ is independent of any questions or paragraphs, and based on a parameterized vector. We use BERT’s [CLS] token representation (Devlin et al., 2019) to independently encode each candidate paragraph $p _ { i }$ along with $q$ .2 We then compute the probability $P ( \boldsymbol { p } _ { i } | h _ { t } )$ that $p _ { i }$ is selected. The RNN selection procedure captures relationships between paragraphs in the reasoning path by conditioning on the selection history. The process is terminated when [EOE], the end-ofevidence symbol, is selected, to allow it to capture reasoning paths with arbitrary length given each question. More specifically, the process of selecting $p _ { i }$ at the $t$ -th step is formulated as follows:
+
+$$
+\begin{array} { r l } & { w _ { i } = \mathrm { B E R T } _ { [ \mathrm { C L S } ] } ( q , p _ { i } ) \in \mathbb { R } ^ { d } , } \\ & { P ( p _ { i } | h _ { t } ) = \sigma ( w _ { i } \cdot h _ { t } + b ) , } \\ & { \quad \quad h _ { t + 1 } = \mathrm { R N N } ( h _ { t } , w _ { i } ) \in \mathbb { R } ^ { d } , } \end{array}
+$$
+
+where $b \in \mathbb { R } ^ { 1 }$ is a bias term. Motivated by Salimans & Kingma (2016), we normalize the RNN states to control the scale of logits in Equation (3) and allow the model to learn multiple reasoning paths. The details of Equation (4) are described in Appendix A.1. The next candidate set $\mathbf { C } _ { t + 1 }$ is constructed to include paragraphs that are linked from the selected paragraph $p _ { i }$ in the graph. To allow our model to flexibly retrieve multiple paragraphs within $\mathbf { C } _ { t }$ , we also add $K$ -best paragraphs other than $p _ { i }$ (from $\mathbf { C } _ { t } .$ ) to $\mathbf { C } _ { t + 1 }$ , based on the probabilities. We typically set $K = 1$ in this paper.
+
+Beam search for candidate paragraphs It is computationally expensive to compute Equation (2) over millions of the possible paragraphs. Moreover, a fully trainable retriever often performs poorly for entity-centric questions such as SQuAD, since it does not explicitly maintain lexical information (Lee et al., 2019). To navigate our retriever in the large-scale graph effectively, we initialize candidate paragraphs with a TF-IDF-based retrieval and guide the search over the Wikipedia graph. In particular, the initial candidate set $\mathbf { C } _ { 1 }$ includes $F$ paragraphs with the highest TF-IDF scores with respect to the question. We expand $\mathbf { C } _ { t }$ $t \geq 2 ,$ ) by appending the [EOE] symbol. We additionally use a beam search to explore paths in the directed graph. We define the score of a reasoning path $E = [ p _ { i } , \dotsc , p _ { k } ]$ by multiplying the probabilities of selecting the paragraphs: $P ( p _ { i } | h _ { 1 } ) \ldots P ( p _ { k } | h _ { | E | } )$ . The beam search outputs the top $B$ reasoning paths $\mathbf { E } = \{ E _ { 1 } , \dots , E _ { B } \}$ with the highest scores to pass to the reader model i.e., $S ( q , E , a ) = S _ { \mathrm { r e a d } } ( q , E , a )$ for $E \in \mathbf { E }$ .
+
+In terms of the computational cost, the number of the paragraphs processed by Equation (2) is bounded by $\begin{array} { r } { \mathcal { O } ( | \mathbf { C } _ { 1 } | + B \sum _ { t \geq 2 } | \overline { { \mathbf { C } _ { t } } } | ) } \end{array}$ , where $B$ is the beam size and $\overline { { | \mathbf { C } _ { t } | } }$ is the average size of $\mathbf { C } _ { t }$ over the $B$ hypothesises.
+
+# 3.1.2 TRAINING OF THE GRAPH-BASED RECURRENT RETRIEVER
+
+Data augmentation We train our retriever in a supervised fashion using evidence paragraphs annotated for each question. For multi-hop QA, we have multiple paragraphs for each question, and single paragraph for single-hop QA. We first derive a ground-truth reasoning path $g = [ p _ { 1 } , \dotsc , p _ { | g | } ]$ using the available annotated data in each dataset. $p _ { | { g \vert } }$ is set to [EOE] for the termination condition. To relax and stabilize the training process, we augment the training data with additional reasoning paths – not necessarily the shortest paths – that can derive the answer. In particular, we add a new training path $g _ { r } = [ \bar { p _ { r } } , p _ { 1 } , \dotsc , p _ { | g | } ]$ by adding a paragraph $p _ { r } \in \mathbf { C _ { 1 } }$ that has a high TF-IDF score and is linked to the first paragraph $p _ { 1 }$ in the ground-truth path $g$ . Adding these new training paths helps at the test time when the first paragraph in the reasoning path does not necessarily appear among the paragraphs that initialize the Wikipedia search using the heuristic TF-IDF retrieval.
+
+Negative examples for robustness Our graph-based recurrent retriever needs to be trained to discriminate between relevant and irrelevant paragraphs at each step. We therefore use negative examples along with the ground-truth paragraphs; to be more specific, we use two types of negative examples: (1) TF-IDF-based and (2) hyperlink-based ones. For single-hop QA, we only use the type (1). For multi-hop QA, we use both types, and the type (2) is especially important to prevent our retriever from being distracted by reasoning paths without correct answer spans. We typically set the number of the negative examples to 50.
+
+2Appendix A.2 discusses the motivation, and Appendix C.4 shows results with an alternative approach.
+
+Loss function For the sequential prediction task, we estimate $P ( \boldsymbol { p } _ { i } | h _ { t } )$ independently in Equation (3) and use the binary cross-entropy loss to maximize probability values of all the possible paths. Note that using the widely-used cross-entropy loss with the softmax normalization over $\mathbf { C } _ { t }$ is not desirable here; maximizing the probabilities of $g$ and $g _ { r }$ contradict with each other. More specifically, the loss function of $g$ at the $t$ -th step is defined as follows:
+
+$$
+L _ { \mathrm { r e t r } } ( p _ { t } , h _ { t } ) = - \log P ( p _ { t } | h _ { t } ) - \sum _ { \tilde { p } \in \tilde { \mathbf { C } } _ { t } } \log \left( 1 - P ( \tilde { p } | h _ { t } ) \right) ,
+$$
+
+where $\tilde { \mathbf { C } } _ { t }$ is a set of the negative examples described above, and includes [EOE] for $t < | g |$ . We exclude $p _ { r }$ from $\tilde { \mathbf { C } } _ { 1 }$ for the sake of our multi-path learning. The loss is also defined with respect to $g _ { r }$ in the same way. All the model parameters, including those in BERT, are jointly optimized.
+
+# 3.2 READING AND ANSWERING GIVEN REASONING PATHS
+
+Our reader model first verifies each reasoning path in $\mathbf { E }$ , and finally outputs an answer span $a$ from the most plausible reasoning path. This interplay is effective in making our framework robust; this is further discussed in Appendix A.3. We model the reader as a multi-task learning of (1) reading comprehension, that extracts an answer span from a reasoning path $E$ using a standard approach (Seo et al., 2017; Xiong et al., 2017; Devlin et al., 2019), and (2) reasoning path re-ranking, that re-ranks the retrieved reasoning paths by computing the probability that the path includes the answer.
+
+For the reading comprehension task, we use BERT (Devlin et al., 2019), where the input is the concatenation of the question text and the text of all the paragraphs in $E$ . This lets our reader to fully leverage the self-attention mechanism across the concatenated paragraphs in the retrieved reasoning paths; this paragraph interaction is crucial for multi-hop reasoning (Wang et al., 2019a).
+
+We share the same model for re-ranking, and use the BERT’s [CLS] representation to estimate the probability of selecting $E$ to answer the question:
+
+$$
+\begin{array} { r } { P ( E | q ) = \sigma ( w _ { n } \cdot u _ { E } ) \mathrm { s . t . } u _ { E } = \mathrm { B E R T } _ { [ \mathrm { C L S } ] } ( q , E ) \in \mathbb { R } ^ { D } , } \end{array}
+$$
+
+where $w _ { n } \in \mathbb { R } ^ { D }$ is a weight vector. At the inference time, we select the best evidence $E _ { b e s t } \in \mathbf { E }$ by $P ( E | q )$ , and output the answer span by $S _ { \mathrm { r e a d } }$ :
+
+$$
+E _ { b e s t } = \underset { E \in \mathbf { E } } { \arg \operatorname* { m a x } } P ( E | q ) , ~ S _ { \mathrm { r e a d } } = \underset { i , j , ~ i \leq j } { \arg \operatorname* { m a x } } P _ { i } ^ { s t a r t } P _ { j } ^ { e n d } ,
+$$
+
+where $P _ { i } ^ { s t a r t } , P _ { j } ^ { e n d }$ denote the probability that the $i$ -th and $j$ -th tokens in $E _ { b e s t }$ are the start and end positions, respectively, of the answer span, and are calculated by following Devlin et al. (2019).
+
+Training examples To train the multi-task reader model, we use the ground-truth evidence paragraphs used for training our retriever. It is known to be effective in open-domain QA to use distantly supervised examples, which are not originally associated with the questions but include expected answer strings (Chen et al., 2017; Wang et al., 2018a; Hu et al., 2019). These distantly supervised examples are also effective to simulate the inference time process. Therefore, we combine distantly supervised examples from a TF-IDF retriever with the original supervised examples. Following the procedures in Chen et al. (2017), we add up to one distantly supervised example for each supervised example. We set the answer span as the string that matches $a$ and appears first.
+
+To train our reader model to discriminate between relevant and irrelevant reasoning paths, we augment the original training data with additional negative examples to simulate incomplete evidence. In particular, we add paragraphs that appear to be relevant to the given question but actually do not contain the answer. For multi-hop QA, we select one ground-truth paragraph including the answer span, and swap it with one of the TF-IDF top ranked paragraphs. For single-hop QA, we simply replace the single ground-truth paragraph with TF-IDF-based negative examples which do not include the expected answer string. For the distorted evidence $\tilde { E }$ , we aim at minimizing $P ( \tilde { E } | q )$ .
+
+Multi-task loss function The objective is the sum of cross entropy losses for the span prediction and re-ranking tasks. The loss for the question $q$ and its evidence candidate $E$ is as follows:
+
+$$
+L _ { \mathrm { r e a d } } = L _ { \mathrm { s p a n } } + L _ { \mathrm { n o \_ a n s w e r } } = ( - \log P _ { y ^ { s t a r t } } ^ { s t a r t } - \log P _ { y ^ { e n d } } ^ { e n d } ) - \log P ^ { r } ,
+$$
+
+where $y ^ { s t a r t }$ and $y ^ { e n d }$ are the ground-truth start and end indices, respectively. $L _ { \mathrm { n o \_ a n s w e r } }$ corresponds to the loss of the re-ranking model, to discriminate the distorted reasoning paths with no answers. $P ^ { r }$ is $P ( E | q )$ if $E$ is the ground-truth evidence; otherwise $P ^ { r } = 1 - P ( E | q )$ . We mask the span losses for negative examples, in order to avoid unexpected effects to the span predictions.
+
+# 4 EXPERIMENTS
+
+# 4.1 EXPERIMENTAL SETUP
+
+We evaluate our method in three open-domain Wikipedia-sourced datasets: HotpotQA, SQuAD Open and Natural Questions Open. We target all the English Wikipedia paragraphs for SQuAD Open and Natural Questions Open, and the first paragraph (introductory paragraph) of each article for HotpotQA following previous studies. More details can be found in Appendix B.
+
+HotpotQA HotpotQA (Yang et al., 2018) is a human-annotated large-scale multi-hop QA dataset. Each answer can be extracted from a collection of 10 paragraphs in the distractor setting, and from the entire Wikipedia in the full wiki setting. Two evidence paragraphs are associated with each question for training. Our primary target is the full wiki setting due to its open-domain scenario, and we use the distractor setting to evaluate how well our method works in a closed scenario where the two evidence paragraphs are always included. The dataset also provides annotations to evaluate the prediction of supporting sentences, and we adapt our retriever to the supporting fact prediction. Note that this subtask is specific to HotpotQA. More details are described in Appendix A.5.
+
+SQuAD Open SQuAD Open (Chen et al., 2017) is composed of questions from the original SQuAD dataset (Rajpurkar et al., 2016). This is a single-hop QA task, and a single paragraph is associated with each question in the training data.
+
+Natural Questions Open Natural Questions Open (Lee et al., 2019) is composed of questions from the Natural Questions dataset (Kwiatkowski et al., 2019),3 which is based on Google Search queries independently from the existing articles. A single paragraph is associated with each question, but our preliminary analysis showed that some questions benefit from multi-hop reasoning.
+
+Metrics We report standard $F l$ and $E M$ scores for HotpotQA and SQuAD Open, and EM score for Natural Questions Open to evaluate the overall QA accuracy to find the correct answers. For HotpotQA, we also report Supporting Fact F1 $( S P F I )$ and Supporting Fact EM (SP EM) to evaluate the sentence-level supporting fact retrieval accuracy. To evaluate the paragraph-level retrieval accuracy for the multi-hop reasoning, we use the following metrics: Answer Recall (AR), which evaluates the recall of the answer string among top paragraphs (Wang et al., 2018a; Das et al., 2019), Paragraph Recall $( P R )$ , which evaluates if at least one of the ground-truth paragraphs is included among the retrieved paragraphs, and Paragraph Exact Match $( P E M )$ , which evaluates if both of the ground-truth paragraphs for multi-hop reasoning are included among the retrieved paragraphs.
+
+Evidence Corpus and the Wikipedia graph We use English Wikipedia as the evidence corpus and do not use other data such as Google search snippets or external structured knowledge bases. We use the several versions of Wikipedia dumps for the three datasets (See Appendix B.5). To construct the Wikipedia graph, the hyperlinks are automatically extracted from the raw HTML source files. Directed edges are added between a paragraph $p _ { i }$ and all of the paragraphs included in the target article. The constructed graph consists of $3 2 . 7 \mathbf { M }$ nodes and 205.4M edges. For HotpotQA we only use the introductory paragraphs in the graph that includes about 5.2M nodes and 23.4M edges.
+
+Implementation details We use the pre-trained BERT models (Devlin et al., 2019) using the uncased base configuration ( $d = 7 6 8$ ) for our retriever and the whole word masking uncased large (wwm) configuration ( $d = 1 0 2 4 ^ { \prime }$ ) for our readers. We follow Chen et al. (2017) for the TF-IDF-based retrieval model and use the same hyper-parameters. We tuned the most important hyper-parameters, $F$ , the number of the initial TF-IDF-based paragraphs, and $B$ , the beam size, by mainly using the HotpotQA development set (the effects of increasing $F$ are shown in Figure 5 in Appendix C.3 along with the results with $B = 1$ ). If not specified, we set $B = 8$ for all the datasets, $F = 5 0 0$ for HotpotQA full wiki and SQuAD Open, and $F = 1 0 0$ for Natural Questions Open.
+
+<table><tr><td></td><td colspan="4">full wiki</td><td colspan="4">distractor</td></tr><tr><td></td><td colspan="2">QA</td><td colspan="2">SP</td><td colspan="2">QA</td><td colspan="2">SP</td></tr><tr><td>Models</td><td>F1</td><td>EM</td><td>F1</td><td>EM</td><td>F1</td><td>EM</td><td>F1</td><td>EM</td></tr><tr><td>Semantic Retrieval (Nie et al., 2019)</td><td>58.8</td><td>46.5</td><td>71.5</td><td>39.9</td><td>一</td><td></td><td></td><td>1</td></tr><tr><td>GoldEn Retriever (Qi et al.,2019)</td><td>49.8</td><td>1</td><td>64.6</td><td>一</td><td>1</td><td></td><td></td><td></td></tr><tr><td>Cognitive Graph (Ding et al., 2019)</td><td>49.4</td><td>37.6</td><td>58.5</td><td>23.1</td><td>一</td><td></td><td></td><td></td></tr><tr><td>DecompRC (Min et al.,2019c)</td><td>43.3</td><td></td><td></td><td></td><td>70.6</td><td></td><td></td><td></td></tr><tr><td>MUPPET (Feldman &amp; El-Yaniv,2019)</td><td>40.4</td><td>31.1</td><td>47.7</td><td>17.0</td><td>1</td><td></td><td>1</td><td></td></tr><tr><td>DFGN (Xiao et al., 2019)</td><td>1</td><td>1</td><td>1</td><td>1</td><td>69.2</td><td>55.4</td><td></td><td></td></tr><tr><td>QFE (Nishida et al., 2019)</td><td>1</td><td>1</td><td>1</td><td>一</td><td>68.7</td><td>53.7</td><td>84.7</td><td>58.8</td></tr><tr><td>Baseline (Yang et al., 2018)</td><td>34.4</td><td>24.7</td><td>41.0</td><td>5.3</td><td>58.3</td><td>44.4</td><td>66.7</td><td>22.0</td></tr><tr><td>Transformer-XH(Zhao et al., 2020)</td><td>62.4</td><td>50.2</td><td>71.6</td><td>42.2</td><td>1</td><td></td><td></td><td></td></tr><tr><td>Ours (Reader:BERT wwm)</td><td>73.3</td><td>60.5</td><td>76.1</td><td>49.3</td><td>81.2</td><td>68.0</td><td>1</td><td>1</td></tr><tr><td>Ours (Reader: BERT base)</td><td>65.8</td><td>52.7</td><td>75.0</td><td>47.9</td><td>73.3</td><td>59.4</td><td>85.2 84.6</td><td>58.6 57.4</td></tr></table>
+
+Table 1: HotpotQA development set results: QA and SP (supporting fact prediction) results on HotpotQA’s full wiki and distractor settings. “–” denotes no results are available.
+
+# 4.2 OVERALL RESULTS
+
+Table 1 compares our method with previous published methods on the HotpotQA development set. Our method significantly outperforms all the previous results across the evaluation metrics under both the full wiki and distractor settings. Notably, our method achieves $1 4 . 5 \mathrm { F } 1$ and $1 4 . 0 \mathrm { E M }$ gains compared to state-of-the-art Semantic Retrieval (Nie et al., 2019) and 10.9 F1 gains over the concurrent Transformer-XH model (Zhao et al., 2020) on full wiki. We can see that our method, even with the BERT base configuration for our reader, significantly outperforms all the previous QA scores. Moreover, our method shows significant improvement in predicting supporting facts in the full wiki setting. We compare the performance of our approach to other models on the HotpotQA full wiki official hidden test set in Table 2. We outperform all the published and unpublished models including up-to-date work (marked with $\clubsuit$ ) by large margins in terms of QA performance.
+
+On SQuAD Open, our model outperforms the concurrent state-of-the-art model (Wang et al., 2019b) by 2.9 F1 and 3.5 EM scores as shown in Table 3. Due to the fewer lexical overlap between questions and paragraphs on Natural Questions, pipelined approaches using term-based retrievers often face difficulties finding associated articles. Nevertheless, our approach matches the performance of the best end-to-end retriever (ORQA), as shown in Table 4. In addition to its competitive performance, our retriever can be handled on a single GPU machine, while a fully end-to-end retriever in general requires industry-scale computational resources for training (Seo et al., 2019). More results on these two datasets are discussed in Appendix D.
+
+# 4.3 PERFORMANCE OF REASONING PATH RETRIEVAL
+
+We compare our retriever with competitive retrieval methods for HotpotQA full wiki, with $F = 2 0$
+
+TF-IDF (Chen et al., 2017), the widely used retrieval method that scores paragraphs according to the TF-IDF scores of the question-paragraph pairs. We simply select the top-2 paragraphs.
+
+Re-rank (Nogueira & Cho, 2019) that learns to retrieve paragraphs by fine-tuning BERT to re-rank the top $F$ TF-IDF paragraphs. We select the top-2 paragraphs after re-ranking.
+
+Re-rank 2hop which extends Re-rank to accommodate two-hop reasoning. It first adds paragraphs linked from the top TF-IDF paragraphs. It then uses the same BERT model to select the paragraphs.
+
+Entity-centric IR is our re-implementation of Godbole et al. (2019) that is related to Re-rank 2hop, but instead of simply selecting the top two paragraphs, they re-rank the possible combinations of the paragraphs that are linked to each other.
+
+Cognitive Graph (Ding et al., 2019) that uses the provided prediction results of the Cognitive Graph model on the HotpotQA development dataset.
+
+Semantic Retrieval (Nie et al., 2019) that uses the provided prediction results of the state-of-the-art Semantic Retrieval model on the HotpotQA development dataset.
+
+Retrieval results Table 5 shows that our recurrent retriever yields $8 . 8 \mathrm { ~ P ~ }$ EM and 9.1 AR, leading to the improvement of 10.3 QA EM over Semantic Retrieval. The significant improvement from
+
+Table 2: HotpotQA full wiki test set results: official leaderboard results (on November 6, 2019) on the hidden test set of the HotpotQA full wiki setting. Work marked with ♣ appeared after September 25.   
+
+<table><tr><td colspan="2">Models QA</td><td colspan="2">SP</td></tr><tr><td>(*:anonymous)</td><td>F1 EM</td><td>F1</td><td>EM</td></tr><tr><td>Semantic Retrieval</td><td>57.3 45.3</td><td>70.8</td><td>38.7</td></tr><tr><td>GoldEn Retriever</td><td>48.6 37.9</td><td>64.2</td><td>30.7</td></tr><tr><td>Cognitive Graph</td><td>48.9 37.1</td><td>57.7</td><td>22.8</td></tr><tr><td>Entity-centric IR</td><td>46.3 35.4</td><td>43.2</td><td>0.06</td></tr><tr><td>MUPPET</td><td>40.3 30.6</td><td>47.3</td><td>16.7</td></tr><tr><td>DecompRC</td><td>40.7 30.0</td><td>1</td><td>1</td></tr><tr><td>QFE</td><td>38.1 28.7</td><td>44.4</td><td>14.2</td></tr><tr><td>Baseline</td><td>32.9 24.0</td><td>37.7</td><td>3.9</td></tr><tr><td>HGN*</td><td>69.2</td><td>56.7 76.4</td><td>50.0</td></tr><tr><td>MIR+EPS+BERT**</td><td>64.8 52.9</td><td>72.0</td><td>42.8</td></tr><tr><td>Transformer-XH*</td><td>60.8</td><td>49.0 70.0</td><td>41.7</td></tr><tr><td>Ours</td><td>73.0</td><td>60.0 76.4</td><td>49.1</td></tr></table>
+
+Table 3: SQuAD Open results: we report F1 and EM scores on the test set of SQuAD Open, following previous work.   
+
+<table><tr><td>Models multi-passage (Wang et al.,2019b)</td><td>F1</td><td>EM</td></tr><tr><td>ORQA (Lee et al., 2019) BM25+BERT (Lee et al.,2019) Weaver (Raison et al.,2018) RE (Hu et al.,2019) MUPPET (Feldman &amp; El-Yaniv,2019) BERTserini (Yang et al., 2019) DENSPI-hybrid (Seo et al.,2019) MINIMAL (Min et al.,2018) Multi-step Reasoner (Das et al., 2019) Paragraph Ranker (Lee et al., 2018) R(Wang et al.,2018a)</td><td>60.9 1 1 1 50.2 46.2 46.1 44.4 42.5 39.2 1 37.5</td><td>53.0 20.2 33.2 42.3 41.9 39.3 38.6 36.2 34.7 31.9 30.2 29.1</td></tr><tr><td>DrQA (Chen et al., 2017) Ours</td><td>1 63.8</td><td>29.3 56.5</td></tr></table>
+
+Table 4: Natural Questions Open results: we report EM scores on the test and development sets of Natural Questions Open, following previous work.   
+
+<table><tr><td rowspan="2">Models</td><td colspan="2">EM</td></tr><tr><td>Dev</td><td>Test</td></tr><tr><td>ORQA (Lee et al., 2019)</td><td>31.3</td><td>33.3</td></tr><tr><td>Hard EM (Min et al.,2019a)</td><td>28.8</td><td>28.1</td></tr><tr><td>BERT + BM 25 (Lee et al.,2019)</td><td>24.8</td><td>26.5</td></tr><tr><td>Ours</td><td>31.7</td><td>32.6</td></tr></table>
+
+Table 5: Retrieval evaluation: Comparing our retrieval method with other methods across Answer Recall, Paragraph Recall, Paragraph EM, and QA EM metrics.   
+
+<table><tr><td>Models</td><td>AR</td><td>PR</td><td>P EM</td><td>EM</td></tr><tr><td>Ours (F= 20)</td><td>87.0</td><td>93.3</td><td>72.7</td><td>56.8</td></tr><tr><td>TF-IDF</td><td>39.7</td><td>66.9</td><td>10.0</td><td>18.2</td></tr><tr><td>Re-rank</td><td>55.1</td><td>85.9</td><td>29.6</td><td>35.7</td></tr><tr><td>Re-rank 2hop</td><td>56.0</td><td>70.1</td><td>26.1</td><td>38.8</td></tr><tr><td>Entity-centric IR</td><td>63.4</td><td>87.3</td><td>34.9</td><td>42.0</td></tr><tr><td>Cognitive Graph</td><td>76.0</td><td>87.6</td><td>57.8</td><td>37.6</td></tr><tr><td>Semantic Retrieval</td><td>77.9</td><td>93.2</td><td>63.9</td><td>46.5</td></tr></table>
+
+Re-rank2hop to Entity-centric IR demonstrates that exploring entity links from the initially retrieved documents helps to retrieve the paragraphs with fewer lexical overlaps. On the other hand, comparing our retriever with Entity-centric IR and Semantic Retrieval shows the importance of learning to sequentially retrieve reasoning paths in the Wikipedia graph. It should be noted that our method with $F = 2 0$ outperforms all the QA EM scores in Table 1.
+
+# 4.4 ANALYSIS
+
+We conduct detailed analysis of our framework on the HotpotQA full wiki development set.
+
+Ablation study of our framework To study the effectiveness of our modeling choices, we compare the performance of variants of our framework. We ablate the retriever with 1) No recurrent module, which removes the recurrence from our retriever, and computes the probability of each paragraph to be included in reasoning paths independently and selects the path with the highest joint probability path on the graph; 2) No beam search, which uses a greedy search $B = 1$ ) in our recurrent retriever; 3) No link-based negative examples, which trains the retriever model without adding hyperlink-based negative examples besides TF-IDF-based negative examples.
+
+We ablate the reader model with 1) No reasoning path re-ranking, which outputs the answer only with the best reasoning path from the retriever model, and 2) No negative examples, which trains the model only with the gold paragraphs, removing $L _ { \mathrm { n o \_ a n s w e r } }$ from $L _ { \mathrm { r e a d } }$ . During inference,“No negative examples” reads all the paths and outputs an answer with the highest answer probability.
+
+Table 6: Ablation study: evaluating different variants of our model on HotpotQA full wiki.   
+
+<table><tr><td>Settings (F = 100)</td><td>F1</td><td>EM</td></tr><tr><td>full retriever, no recurrent module retriever, no beam search retriever, no link-based negatives</td><td>72.4 52.5 68.7 64.1</td><td>59.5 42.1 56.2 52.6</td></tr><tr><td>reader, no reasoning path re-ranking reader, no negative examples</td><td>70.1 53.7</td><td>57.4 43.3</td></tr></table>
+
+Table 7: Performance with different link structures: comparing our results on the Hotpot QA full wiki development set when we use an off-the-shelf entity linking system instead of the Wikipedia hyperlinks.   
+
+<table><tr><td rowspan=1 colspan=1>Settings (F = 100)</td><td rowspan=1 colspan=1>F1</td><td rowspan=1 colspan=1>EM</td></tr><tr><td rowspan=1 colspan=1>with hyperlinks with entity linking system</td><td rowspan=1 colspan=1>72.470.1</td><td rowspan=1 colspan=1>59.557.3</td></tr></table>
+
+Table 8: Performance with different reasoning path length: comparing the performance with different path length on HotpotQA full wiki. $L$ -step retrieval sets the number of the reasoning steps to a fixed number.   
+
+<table><tr><td rowspan=1 colspan=3>Settings (F = 100)     F1</td><td rowspan=1 colspan=1>EM</td></tr><tr><td rowspan=1 colspan=2>Adaptive retrieval</td><td rowspan=1 colspan=1>72.4</td><td rowspan=1 colspan=1>59.5</td></tr><tr><td rowspan=1 colspan=1>L-step retrieval</td><td rowspan=1 colspan=1>L=1L=2L=3L=4</td><td rowspan=1 colspan=1>45.871.470.166.3</td><td rowspan=1 colspan=1>35.558.557.753.9</td></tr></table>
+
+Table 9: Statistics of the reasoning paths: the average length and the distribution of length of the reasoning paths selected by our retriever and reader for HotpotQA full wiki. Avg. EM represents QA EM performance.   
+
+<table><tr><td>(F=100)</td><td>Retriever</td><td>Reader</td><td>EM</td></tr><tr><td>Avg.#ofL</td><td>1.96</td><td>2.21</td><td>with L</td></tr><tr><td>1</td><td>539</td><td>403</td><td>31.2</td></tr><tr><td>2</td><td>6,639</td><td>5,655</td><td>60.0</td></tr><tr><td>3</td><td>227</td><td>1,347</td><td>63.0</td></tr></table>
+
+Ablation results Table 6 shows that removing any of the listed components gives notable performance drop. The most critical component in our retriever model is the recurrent module, dropping the EM by 17.4 points. As shown in Figure 1, multi-step retrieval often relies on information mentioned in another paragraph. Therefore, without conditioning on the previous time steps, the model fails to retrieve the complete evidence. Training without hyperlink-based negative examples results in the second largest performance drop, indicating that the model can be easily distracted by reasoning paths without a correct answer and the importance of negative sampling for training. Replacing the beam search with the greedy search gives a performance drop of about 4 points on EM, which demonstrates that being aware of the graph structure is helpful in finding the best reasoning paths.
+
+Performance drop by removing the reasoning path re-ranking indicates the importance of verifying the reasoning paths in our reader. Not using negative examples to train the reader degrades EM more than 16 points, due to the over-confident predictions as discussed in Clark & Gardner (2018).
+
+The performance with an off-the-shelf entity linking system Although the existence of the hyperlinks is not special on the web, one question is how well our method works without the Wikipedia hyperlinks. We evaluate our method on the development set of HotpotQA full wiki with an off-theshelf entity linking system (Ferragina & Scaiella, 2011) to construct the document graph in our method. More details about this experimental setup can be found in Appendix B.7. Table 7 shows that our approach with the entity linking system shows only $2 . 3 \ \mathrm { F 1 }$ and $2 . 2 \ : \mathrm { E M }$ lower scores than those with the hyperlinks, still achieving the state of the art. This suggests that our approach is not restricted to the existence of the hyperlink information, and using hyperlinks is promising.
+
+The effectiveness of arbitrary-step retrieval The existing iterative retrieval methods fix the number of reasoning steps (Qi et al., 2019; Das et al., 2019; Godbole et al., 2019; Feldman & El-Yaniv, 2019), while our approach accommodates arbitrary steps of reasoning. We also evaluate our method by fixing the length of the reasoning path $( L = \{ 1 , 2 , 3 , 4 \} )$ ). Table 8 shows that out adaptive retrieval performs the best, although the length of all the annotated reasoning paths in HotpotQA is two. As discussed in Min et al. (2019b), we also observe that some questions are answerable based on a single paragraph, where our model flexibly selects a single paragraph and then terminates retrieval.
+
+The effectiveness of the interplay between retriever and reader Table 6 shows that the interplay between our retriever and reader models is effective. To understand this, we investigate the length of reasoning paths selected by our retriever and reader, and their final QA performance. Table 9 shows that the average length selected by our reader is notably longer than that by our retriever. Table 9 also presents the EM scores averaged over the questions with certain length of reasoning paths $( { \cal L } = \{ 1 , 2 , 3 \}$ ). We observe that our framework performs the best when it selects the reasoning paths with $L = 3$ , showing 63.0 EM score. Based on these observations, we expect the retriever favors a shorter path, while the reader tends to select a longer and more convincing multi-hop reasoning path to derive an answer string.
+
+![](images/2fab0de0d7ea4ea2c84b83905bb731f86e9b84a148e117a4942926d8a1522819.jpg)  
+Figure 3: Reasoning examples by our model (two paragraphs connected by a dotted line) and Re-rank (the bottom two paragraphs). Highlighted text denotes a bridge entity, and blue-underlined text represents hyperlinks.
+
+![](images/c8edbf87b7a401b990a6b1ccdeeb34aa6ffaded0ec76ec6dca1c3b92f5a620b7.jpg)  
+Figure 4: Reasoning examples by our retriever (the bottom paragraph) and our reader (two paragraphs connected by a dotted line). Highlighted text denotes a bridge entity, and blue-underlined text represents hyperlinks.
+
+Qualitative examples of retrieved reasoning paths Finally, we show two examples from HotpotQA full wiki, and Appendix C.5 presents more qualitative examples. In Figure 3, our approach successfully retrieves the correct reasoning path and answers correctly, while Re-rank fails. The top two paragraphs next to the graph are the introductory paragraphs of the two entities on the reasoning path, and the paragraph at the bottom shows the wrong paragraph selected by Re-rank. The “Millwall F.C.” has fewer lexical overlaps and the bridge entity “Millwall” is not stated in the given question. Thus, Re-rank chooses a wrong paragraph with high lexical overlaps to the given question. In Figure 4, we compare the reasoning paths ranked highest by our retriever and reader. Although the gold path is included among the top 8 paths selected by the beam search, our retriever model selects a wrong paragraph as the best reasoning path. By re-ranking the reasoning paths, the reader eventually selects the correct reasoning path (“2017-18 Wigan Athletic F.C. season” “EFL Cup”). This example shows the effectiveness of the strong interplay of our retriever and reader.
+
+# 5 CONCLUSION
+
+This paper introduces a new graph-based recurrent retrieval approach, which retrieves reasoning paths over the Wikipedia graph to answer multi-hop open-domain questions. Our retriever model learns to sequentially retrieve evidence paragraphs to form the reasoning path. Subsequently, our reader model re-ranks the reasoning paths, and it determines the final answer as the one extracted from the best reasoning path. Our experimental results significantly advance the state of the art on HotpotQA by more than 14 points absolute gain on the full wiki setting. Our approach also achieves the state-of-the-art performance on SQuAD Open and Natural Questions Open without any architectural changes, demonstrating the robustness of our method. Our method provides insights into the underlying entity relationships, and the discrete reasoning paths are helpful in interpreting our framework’s reasoning process. Future work involves end-to-end training of our graph-based recurrent retriever and reader for improving upon our current two-stage training.
+
+# ACKNOWLEDGMENTS
+
+We acknowledge grants from ONR N00014-18-1-2826, DARPA N66001-19-2-403, NSF (IIS1616112, IIS1252835), and Samsung GRO. We thank Sewon Min, David Wadden, Yizhong Wang, Akhilesh Gotmare, Tong Niu, and UW NLP group and Salesforce research members for their insightful discussions. We would also like to show our gratitude to Melvin Gruesbeck for providing us with the artistic figures presented in this paper. We thank the anonymous reviewers for their helpful and thoughtful comments. Akari Asai is supported by The Nakajima Foundation Fellowship.
+
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+
+# APPENDIX
+
+# A DETAILS ABOUT MODELING
+
+A.1 A NORMALIZED RNN
+
+We decompose Equation (4) as follows:
+
+$$
+a _ { t + 1 } = W _ { r } [ h _ { t } ; w _ { i } ] + b _ { r } , \quad h _ { t + 1 } = \frac { \alpha } { \left\| a _ { t + 1 } \right\| } a _ { t + 1 } ,
+$$
+
+where $W _ { r } \in \mathbb { R } ^ { d \times 2 d }$ is a weight matrix, $b _ { r } \in \mathbb { R } ^ { d }$ is a bias vector, and $\alpha \in \mathbb { R } ^ { 1 }$ is a scalar parameter (initialized with 1.0). We set the global initial state $a _ { 1 }$ to a parameterized vector $s \in \mathbb { R } ^ { d }$ , and we also parameterize an [EOE] vector $w _ { [ \mathrm { E O E } ] } \in \mathbb { R } ^ { d }$ for the [EOE] symbol. The use of $w _ { i }$ for both the input and output layers is inspired by Inan et al. (2017); Press & Wolf (2017). In addition, we align the norm of $w _ { \mathrm { [ E O E ] } }$ with those of $w _ { i }$ , by applying layer normalization (Ba et al., 2016) of the last layer in BERT because $w _ { \mathrm { [ E O E ] } }$ is used along with the BERT outputs. Without the layer normalization, the $L 2$ -norms of $w _ { i }$ and $w _ { \mathrm { [ E O E ] } }$ can be quite different, and the model can easily discriminate between them by the difference of the norms.
+
+# A.2 QUESTION-PARAGRAPH ENCODING IN OUR RETRIEVER COMPONENT
+
+Equation (2) shows that we compute each paragraph representation $w _ { i }$ conditioned on the question $q$ . An alternative approach is separately encoding the paragraphs and the question, to directly retrieve paragraphs (Lee et al., 2019; Seo et al., 2019; Das et al., 2019). However, due to the lack of explicit interactions between the paragraphs and the question, such a neural retriever using questionindependent paragraph encodings suffers from compressing the necessary information into fixeddimensional vectors, resulting in low performance on entity-centric questions (Lee et al., 2019). It has been shown that attention-based paragraph-question interactions improve the retrieval accuracy if the retrieval scale is tractable (Wang et al., 2018a; Lee et al., 2018). There is a trade-off between the scalability and the accuracy, and this work aims at striking the balance by jointly using the lexical matching retrieval and the graphs, followed by the rich question-paragraph encodings.
+
+A question-independent variant We can also formulate our retriever model by using a questionindependent approach. There are only two simple modifications. First, we reformulate Equation (2) as follows:
+
+$$
+w _ { i } = \mathrm { B E R T } _ { [ \mathrm { C L S } ] } ( p _ { i } ) ,
+$$
+
+where we no longer input the question $q$ together with the paragraphs. Next, we condition the initial RNN state $h _ { 1 }$ on the question information. More specifically, we compute $h _ { 1 }$ by using Equation (4) as follows:
+
+$$
+\begin{array} { l } { { w _ { q } = \mathrm { B E R T } _ { [ \mathrm { C L S } ] } ( q ) , } } \\ { { h _ { 1 } = \mathrm { R N N } ( h _ { 1 } ^ { \prime } , w _ { q } ) , } } \end{array}
+$$
+
+where $w _ { q }$ is computed by using the same BERT encoder as in Equation (10), and $h _ { 1 } ^ { \prime }$ is the original $h _ { 1 }$ used in our question-dependent approach as described in Appendix A.1. The remaining parts are exactly the same, and we can perform the reasoning path retrieval in the same manner.
+
+# A.3 WHY IS THE INTERPLAY IMPORTANT?
+
+Our retriever model learns to predict plausibility of the reasoning paths by capturing the paragraph interactions through the BERT’s [CLS] representations, after independently encoding the paragraphs along with the question; this makes our retriever scalable to the open-domain scenario. By contrast, our reader jointly learns to predict the plausibility and answer the question, and moreover, fully leverages the self-attention mechanism across the concatenated paragraphs in the retrieved reasoning paths; this paragraph interaction is crucial for multi-hop reasoning (Wang et al., 2019a). In summary, our retriever is scalable, but the top-1 prediction is not always enough to fully capture multi-hop reasoning to answer the question. Therefore, the additional re-ranking process mitigates the uncertainty and makes our framework more robust.
+
+# A.4 HANDLING YES-NO QUESTIONS IN OUR READER COMPONENT
+
+In the HotpotQA dataset, we need to handle yes-no questions as well as extracting answer spans from the paragraphs. We treat the two special types of the answers, yes and no, by extending the re-ranking model in Equation (6). In particular, we extend the binary classification to a multi-class classification task, where the positive “answerable” class is decomposed into the following three classes: span, yes, and no. If the probability of “yes” or “no” is the largest among the three classes, our reader directly outputs the label as the answer, without any span extraction. Otherwise, our reader uses the span extraction model to output the answer.
+
+# A.5 SUPPORTING FACT PREDICTION IN HOTPOTQA
+
+We adapt our recurrent retriever to the subtask of the supporting fact prediction in HotpotQA (Yang et al., 2018). The task is outputting sentences which support to answer the question. Such supporting sentences are annotated for the two ground-truth paragraphs in the training data. Since our framework outputs the most plausible reasoning path $E$ along with the answer, we can add an additional step to select supporting facts (sentences) from the paragraphs in $E$ . We train our recurrent retriever by using the training examples for the supporting fact prediction task, where the model parameters are not shared with those of our paragraph retriever. We replace the question-paragraph encoding in Equation (2) with question-answer-sentence encoding for the task, where a question string is concatenated with its answer string. The answer string is the ground-truth one during the training time. We then maximize the probability of selecting the ground-truth sequence of the supporting fact sentences, while setting the other sentences as negative examples. At test time, we use the best reasoning path and its predicted answer string from our retriever and reader models to finally output the supporting facts for each question. The supporting fact prediction task is performed after finalizing the reasoning path and the answer for each question, and hence this additional task does not affect the QA accuracy.
+
+# B DETAILS ABOUT EXPERIMENTS
+
+# B.1 DATASET DETAILS OF HOTPOTQA, SQUAD OPEN AND NATURAL QUESTIONS OPEN
+
+HotpotQA The HotpotQA training, development, and test datasets contain 90,564, 7,405 and 7,405 questions, respectively. To train our retriever model for the distractor setting, we use the distractor training data, where only the original ten paragraphs are associated with each question. The retriever model trained with this setting is also used in our ablation study as “retriever, no linkbased negatives” in Table 6. For the full wiki setting, we train our retriever model with the data augmentation technique and the additional negative examples described in Section 3.1.2. We use the same reader model, for both the settings, trained with the augmented additional references and the negative examples described in Section 3.2.
+
+SQuAD Open and Natural Questions Open For SQuAD Open, we use the original training set (containing 78,713 questions) as our training data, and the original development set (containing 10,570 questions) as our test data. For Natural Questions Open, we follow the dataset splits provided by Min et al. (2019a), and the training, development and test datasets contain 79,168, 8,757 and 3,610, respectively. For both the SQuAD Open and Natural Questions Open, we train our reader on the original examples with the augmented additional negative examples and the distantly supervised examples described in Section 3.2.
+
+# B.2 DERIVING GROUND-TRUTH REASONING PATHS
+
+Section 3.1.2 describes our training strategy for our recurrent retriever. We apply the data augmentation technique to HotpotQA and Natural Questions to consider multi-hop reasoning. To derive the ground-truth reasoning path $g$ , we use the ground-truth evidence paragraphs associated with the questions in the training data for each dataset. For SQuAD and Natural Questions Open, each training example has only single paragraph $p$ , and thus it is trivial to derive $g$ as $[ p , [ \mathrm { E O E } ] ]$ . For the multi-hop case, HotpotQA, we have two ground-truth paragraphs $p _ { 1 } , p _ { 2 }$ for each question. Assuming that $p _ { 2 }$ includes the answer string, we set $g = [ p _ { 1 } , \bar { p } _ { 2 }$ , $[ \mathrm { E O E } ] ]$ .
+
+# B.3 DETAILS ABOUT NEGATIVE EXAMPLES FOR OUR READER MODEL IN SQUAD OPEN AND NATURAL QUESTIONS OPEN
+
+To train our reader model for SQuAD Open, in addition to the TF-IDF top-ranked paragraphs, we add two types of additional negative examples: (i) paragraphs, which do not include the answer string, from the originally annotated articles, and (ii) “unanswerable” questions from SQuAD 2.0 (Rajpurkar et al., 2018). For Natural Questions Open, we add negative examples of the type (i).
+
+# B.4 TRAINING SETTINGS
+
+To use the pre-trained BERT models, we used the public code base, pytorch-transformers,4 written in PyTorch.5 For optimization, we used the code base’s implementation of the Adam optimizer (Kingma & Ba, 2015), with a weight-decay coefficient of 0.01 for non-bias parameters. A warm-up strategy in the code base was also used, with a warm-up rate of 0.1. Most of the settings follow the default settings. To train our recurrent retriever, we set the learning rate to $3 \cdot 1 0 ^ { - 5 }$ , and the maximum number of the training epochs to three. The mini-batch size is four; a mini-batch example consists of a question with its corresponding paragraphs. To train our reader model, we set the learning rate to $3 \cdot 1 0 ^ { - 5 }$ , and the maximum number of training epochs to two. Empirically we observe better performance with a larger batch size as discussed in previous work (Liu et al., 2019; Ott et al., 2018), and thus we set the mini-batch size to 120. A mini-batch example consists of a question with its evidence paragraphs. We will release our code to follow our experiments.
+
+# B.5 THE WIKIPEDIA DUMPS FOR EACH DATASET
+
+For HotpotQA full wiki, we use the pre-processed English Wikipedia dump from October, 2017, provided by the HotpotQA authors.6 For Natural Questions Open, we use the English Wikipedia dump from December 20, 2018, following Lee et al. (2019) and Min et al. (2019a). For SQuAD Open, we use the Wikipedia dump provided by Chen et al. (2017).
+
+Although using a single dump for different open-domain QA datasets is a common practice (Chen et al., 2017; Wang et al., 2018a; Lee et al., 2018), this potentially causes inconsistent or even unfair evaluation across different experimental settings, due to the temporal inconsistency of the Wikipedia articles. More concretely, every Wikipedia article is editable and and as a result, a fact can be rephrased or could be removed. For instance, a question from the SQuAD development set, “Where does Kenya rank on the CPI scale?” is originally paired with a paragraph from the article of Kenya. Based on a single sentence “Kenya ranks low on Transparency International’s Corruption Perception Index (CPI)” from the paragraph, an annotated answer span is “low.” However, this sentence has been rewritten as “Kenya has a high degree of corruption according to Transparency International’s Corruption Perception Index (CPI)” in a later version of the same article.7 This is problematic considering the major evaluation metrics based on string matching.
+
+Another problem exists especially in Natural Questions Open. The dataset contains real Google search queries, and some of them reflect temporal trends at the time when the queries were executed. If a query is related to a TV show broadcasted in 2018, we can hardly expect to extract the answer from a dump in 2017.
+
+Like this, although Wikipedia is a useful knowledge source for open-domain QA research, its rapidly evolving nature should be considered more carefully for the reproducibility. We will make all of the data including pre-processed Wikipedia articles for each experiment available for future research.
+
+B.6 DETAILS ABOUT INITIAL CANDIDATES $C _ { 1 }$ SELECTION
+
+To retrieve the initial candidates $C _ { 1 }$ for each question, we use a TF-IDF based retriever with the bi-gram hashing (Chen et al., 2017). For HotpotQA full wiki, we retrieve top $F$ introductory paragraphs, for each question, from a corpus including all the introductory paragraphs. For SQuAD Open and Natural Questions Open, we first retrieve 50 Wikipedia articles through the same TF-IDF retriever, and further run another TF-IDF-based paragraph retriever (Clark & Gardner, 2018; Min et al., 2019a) to retrieve $F$ paragraphs in total.
+
+# B.7 DETAILS ABOUT ENTITY LINKING EXPERIMENT
+
+We experiment with a variant of our approach, where we incorporate an entity linking system with our framework, in place of the Wikipedia hyperlinks. In this experiment, we first retrieve seed paragraphs using TF-IDF $F = 1 0 0$ ), and run an off-the-shelf entity linker (TagMe by Ferragina & Scaiella (2011)) over the paragraphs. If the entity linker detects some entities, we retrieve their corresponding Wikipedia articles, and add edges from the seed paragraphs to the entity-linked paragraphs. Once we build the graph, then we re-run all of the experiments while the other components are exactly the same. We use the TagMe official Python wrapper.8
+
+# C ADDITIONAL RESULTS ON HOTPOTQA
+
+# C.1 UPPER-BOUND OF OUR RETRIEVAL MODULE
+
+For scalability and computational efficiency, we bootstrap our retrieval module with TF-IDF retrieval; we first retrieval $F$ paragraphs using TF-IDF with the method described in Section B.6 and initialize $C _ { 1 }$ with these TF-IDF paragraphs. Although we expand our candidate paragraphs at each time step using the Wikipedia graph, if our method failed to retrieve paragraphs a few-hops away from the answer paragraphs, it is likely to fail to reach the answer paragraphs. To estimate the paragraph EM upper-bound, we have checked if two gold paragraphs are included in the top 20 TF-IDF paragraphs and their hyperlinked paragraphs in the HotpotQA full wiki setting. We found that for $7 5 . 4 \%$ of the questions, all of the gold paragraphs are included in the collections of the TF-IDF paragraphs and the hyperlinked paragraphs. Also, it should be noted when we only consider the TF-IDF retrieval results, the upper-bound drops to $3 5 . 1 \%$ , which suggests that the TF-IDF-based retrieval cannot effectively discover the paragraphs multi-hop away due to the few lexical overlap. When we increase the number of $F$ to 100 and 500, the upper-bound reaches $8 4 . 1 \%$ and $8 9 . 2 \%$ , respectively.
+
+C.2 PER-CATEGORY QUESTION ANSWERING AND RETRIEVAL PERFORMANCE ON HOTPOTQA FULL WIKI
+
+In HotpotQA, there are two types of questions, bridge and comparison. While comparison-type questions explicitly mention the two entities related to the given questions, in bridge-type questions, the bridge entities are rarely explicitly stated. This makes it hard for a retrieval system to discover the paragraphs entailed by the bridge entities only.
+
+We evaluate the question answering and paragraph retrieval performance for each of the two question types. We compare the PR, P EM and QA EM for each of the two categories with two state-of-theart models, Cognitive Graph (Ding et al., 2019) and Semantic Retrieval (Nie et al., 2019). Here, we set our initial TF-IDF number $F$ to 500. Table 10 shows that our retriever yields $1 6 . 5 \mathrm { ~ P ~ }$ EM
+
+<table><tr><td></td><td colspan="3">Total (7,405)</td><td colspan="3">Bridge (5.918)</td><td colspan="3">Comp (1,487)</td></tr><tr><td>Models</td><td>PR</td><td>PEM</td><td>EM</td><td>PR</td><td>PEM</td><td>EM</td><td>PR</td><td>PEM</td><td>EM</td></tr><tr><td>Ours_</td><td>94.3</td><td>75.7</td><td>60.5</td><td>93.9</td><td>73.7</td><td>57.8</td><td>98.7</td><td>83.5</td><td>70.5</td></tr><tr><td>Cognitive Graph (Ding et al., 2019)</td><td>87.6</td><td>57.8</td><td>-37.5</td><td>84.8</td><td>51.8</td><td>-36.1</td><td>98.6</td><td>81.6</td><td>53.7</td></tr><tr><td>Semantic Retrieval (Nie et al.,2019)</td><td>93.2</td><td>63.9</td><td>46.5</td><td>91.6</td><td>57.2</td><td>42.7</td><td>99.7</td><td>90.6</td><td>61.7</td></tr></table>
+
+Table 10: Retrieval evaluation: Comparing our retrieval method with other methods across Answer Recall, Paragraph Recall, Paragraph EM, and QA EM metrics.
+
+![](images/6f898a3f08d4a52be317f5fb6fded715c90a3b2b814f49b909f439c573030551.jpg)  
+Figure 5: Robustness to the increase of $F$ . We compare the F1 scores of our model, our model without a beam search and Re-rank with different number of $F$ .
+
+gain and 15.1 EM gain over Semantic Retrieval for the challenging bridge-type questions. For the comparison-type questions, our method achieves almost 10 point higher QA EM than Semantic Retrieval. We observed that some of the comparison-type questions can be answered based on single paragraph, and thus our model selects only one paragraph for some of these comparisontype questions, resulting in lower P EM scores on the comparison-type questions. We show several examples of the questions where we can answer based on single paragraph in Section C.5.
+
+# C.3 ON THE ROBUSTNESS TO THE INCREASE OF THE PARAGRAPHS
+
+As we discussed in 3.1.1, we aim at significantly reducing the search space and thus scaling the number of initial TF-IDF candidates. Increasing the number of the initial retrieved paragraphs often improves the recall of the evidence paragraphs of the datasets. On the other hand, increasing the candidate paragraphs introduces additional noises, may distract models, and eventually hurt the performance (Kratzwald & Feuerriegel, 2018). We compare the performance of three different approaches: (i) ours, (ii) ours (greedy, without reasoning path re-ranking), and (iii) $R e$ -rank. We increase the number of the TF-IDF-based retrieved paragraphs from 10 to 500 (For Re-rank, we compare the performance up to 200 paragraphs). Figure 5 clearly shows that our approach is robust towards the increase of the initial candidate paragraphs, and thus can constantly yield performance gains with more candidate paragraphs. Our approach with the greedy search also shows performance improvements; however, after a certain number, the greedy approach stops improving the performance. Re-rank starts suffering from the noises caused by many distracting paragraphs included in the initial candidate paragraphs at $F = 2 0 0$ .
+
+# C.4 RESULTS OF QUESTION-INDEPENDENT PARAGRAPH ENCODING FOR OUR RETRIEVER
+
+To show the importance of the question-paragraph encoding in our retriever model, we conduct an experiment on the development set of HotpotQA, by replacing it with the question-independent encoding described in Appendix A.2. For a fair comparison, we use the same initial TF-IDF-based retrieval (only for the full wiki setting), hyperlink-based Wikipedia graph, beam search, and reader model (BERT wwm). We train the alternative model without using the data augmentation technique (described in Section 3.1.2) for quick experiments.
+
+Table 11: Effects of the question-dependent paragraph encoding: Comparing our retriever model with and without the query-dependent encoding. For our question-dependent approach, the full wiki results correspond to “retriever, no link-based negatives” in Table 6, and the distractor results correspond to “Ours (Reader: BERT wwm)” Table 1, to make the results comparable.   
+
+<table><tr><td></td><td>full wiki (F = 100)</td><td colspan="2">distractor</td></tr><tr><td>Encoding method</td><td>QA F1 QA EM</td><td>QA F1</td><td>QA EM</td></tr><tr><td>Question-dependent (our main model)</td><td>64.1</td><td>81.2</td><td>68.0</td></tr><tr><td>Question-independent</td><td>52.6 47.3 37.8</td><td>80.0</td><td>66.4</td></tr></table>
+
+Table 11 shows the results in both the full wiki and distractor settings. As seen in this table, the QA F1 and EM performance significantly deteriorates on the full wiki setting, which demonstrates the importance of the question-dependent encoding for complex and entity-centric open-domain question answering.
+
+We can also see that the performance drop on the distractor setting is much smaller than that on the full wiki setting. This is due to its closed nature; for each question, we are given only ten paragraphs and the two gold paragraphs are always included, which significantly narrows the searching space down and makes the retrieval task much easier than that in the full wiki setting. Therefore, our recurrent retriever model is likely to discover the gold reasoning paths by the beam search, and our reader model can select the gold paths by the robust re-ranking approach. To verify this hypothesis, we checked the P EM score as a retrieval accuracy in the distractor setting. If we only consider the top-1 path from the beam search, the P EM score of the question-independent model is $12 \%$ lower than that of our question-dependent model. However, if we consider all the reasoning paths produced by the beam search, the coverage of the gold paths is almost the same. As a result, our reader model can perform similarly with both the question-dependent/independent approaches. This additionally shows the robustness of our re-ranking approach.
+
+# C.5 MORE QUALITATIVE ANALYSIS ON THE REASONING PATH ON HOTPOTQA FULL WIKI
+
+In this section, we conduct more qualitative analysis on the reasoning paths predicted by our model. Explicitly retrieving plausible reasoning paths and re-ranking the paths provide us interpretable insights into the underlying entity relationships used for multi-hop reasoning.
+
+As shown in Table 9, our model flexibly selects one or more paragraphs for each question. To understand these behaviors, we conduct qualitative analysis on these examples whose reasoning paths are shorter or longer than the original gold reasoning paths.
+
+Reasoning path only with single paragraph First, we show two examples (one is a bridge-type question and the other is a comparison-type question), where our retriever selects single paragraph and terminates without selecting any additional paragraphs.
+
+The bridge-type question in Table 12 shows that, while originally this question requires a system to read two paragraphs, Before I Go to Sleep (film) and Nicole Kidman, our retriever and reader eventually choose Nicole Kidman only. The second paragraph has a lot of lexical overlaps to the given question, and thus, a system may not need to read both of the paragraphs to answer.
+
+The comparison-type question in Table 12 also shows that even comparison-type questions do not always require two paragraphs to answer the questions, and our model only selects one paragraph necessary to answer the given example question. In this example, the question has large lexical overlap with one of the ground-truth paragraph (The Bears and I), resulting in allowing our model to answer the question based on the single paragraph.
+
+Min et al. (2019b) also observed that some of the questions do not necessarily require multi-hop reasoning, while HotpotQA is designed to require multi-hop reasoning (Yang et al., 2018). In that sense, we can say that our method automatical detects potentially single-hop questions.
+
+Reasoning path with three paragraphs All of the HotpotQA questions are authored by annotators who are shown two relevant paragraphs, and thus, originally the length of ground-truth reasoning paths is always two. On the other hand, as our model accommodates arbitrary steps of reasoning, it often selects reasoning paths longer than the original annotations as shown in Table 9. When our model selects a longer reasoning path for a HotpotQA question, does it contain paragraphs that provide additional evidence? We show an example in Table 13, so as to answer this question. Our model selects an additional paragraph, Blue Jeans (Lana Del Rey song) at the first step, and then selects the two annotated gold paragraphs. This first paragraph is strongly relevant to the given question, but does not contain the answer. This additional evidence might help the reader to find the correct bridge entity (“Back to December”).
+
+Table 12: Two examples of the questions that our model retrieves a reasoning path with only one paragraph. We partly remove sentences irrelevant to the questions. Words in red correspond to the answer strings.   
+
+<table><tr><td>Q[bridge]: Before I Go to Sleep stars an Australian actress, producer and occasional what?</td></tr><tr><td>Before IGo to Sleep (film): Before IGo to Sleep is a 2O14 mystery psychological thriller film writen and directed by Rowan Joff and based on the 2O11 novel of the same name by S. J. Watson. An international co-production between the United Kingdom, the United States, France,and Sweden, the film stars Nicole Kidman , Mark Strong, Colin Firth,and Anne-Marie Duff.</td></tr><tr><td>Nicole Kidman: Nicole Mary Kidman, , is an Australian actress, producer and occasional singer. She is the recipient of several awards, including an Academy Award, two Primetime Emmy Awards,a BAFTA Award, three Golden Globe Awards,and the Silver Bear for Best</td></tr><tr><td>Actress. Annotated reasoning path Before IGo to Sleep (film) →Nicole Kidman</td></tr><tr><td>Predicted reasoning path: Nicole Kidman</td></tr><tr><td>Q[comparison]: In between The Bears and I and Oceans which was released on July 31, 1974,by Buena Vista Distribution?</td></tr><tr><td>The Bears and I: The Bears and I is a 1974 American drama film directed by Bernard McEveety and written by John Whedon. The film stars Patrick Wayne, Chief Dan George, Andrew Duggan, Michael Ansara and Robert Pine. The film was released on July 31,1974, by Buena Vista Distribution.</td></tr><tr><td>Oceans (film): Oceans is a 2Oo9 French nature documentary film directed, produced,</td></tr><tr><td>co-written,and narrated by Jacques Perrin,with Jacques Cluzaud as co-director. Annotated reasoning path: The Bears and I, Oceans (film) Predicted reasoning path: The Bears and I</td></tr></table>
+
+# C.6 QUALITATIVE ANALYSIS ON THE REASONING PATH ON HOTPOTQA DISTRACTOR
+
+Although the main focus in this paper is on open-domain QA, we show the state-of-the-art performance on the HotpotQA distractor setting as well with the exactly same architecture. We conduct qualitative analysis to understand our model’s behavior in the closed setting. In this setting, the two ground-truth paragraphs are always given for each question.
+
+Table 14 shows two examples from the HotpotQA distractor setting. In the first example, P1 and P2 are its corresponding ground-truth paragraphs. At the first time step, our retriever does not expect that P2 is related to the evidence to answer the question, as the retriever is not aware of the bridge entity, “Pasek & Paul”. If we simply adopt the Re-rank strategy, P3 with the second highest probability is selected, resulting in a wrong paragraph selection. In our framework, our retriever is conditioned on the previous retrieval history and thus, at the second time step, it chooses the correct paragraph, P2, lowering the probability of P3. This clearly shows the effectiveness of our multi-step retrieval method in the closed setting as well. At the third step, our model stops the prediction by outputting [EOS]. In 588 examples $( 7 . 9 \% )$ of the entire distractor development dataset, the paragraph selection by our graph-based recurrent retriever differs from the top-2 strategy.
+
+Table 13: An example question where our model predicts reasoning paths of the length of three. Our model expects that the question is answerable based on the last paragraph of the annotated path.   
+
+<table><tr><td>Q: Yoann Lemoine,a French video director, has created music videos for Lana Del Rey, Katy Perry,and an orchestral country pop ballad by which top pop artist?</td></tr><tr><td>Yoann Lemoine: Yoann Lemoine (born 16 March 1983) is a French music video director, graphic designer and singer-songwriter. His most notable works include his music video direction for Katy Perry&#x27;s &quot;Teenage Dream”, Taylor Swift&#x27;s single “Back to December Lana Del Rey&#x27;s “Born to Die” and Mystery Jets’“Dreaming of Another World&quot;.</td></tr><tr><td>Back to December: “Back to December isa song written and recorded by American singer/songwriter Taylor Swift for her third studio album“Speak Now&quot; (2O1O). “Back to December&quot; is considered an orchestral country pop ballad and its lyrics are a remorseful plea for forgiveness for breaking up with a former lover.</td></tr><tr><td>Blue Jeans (Lana Del Rey song): “Blue Jeans&quot; is a song by American singer-songwriter Lana Del Rey for her second studio album “Born to Die&quot; (2012). Produced by Emile Haynie, the song was writen by Del Rey,Haynie,and Dan Heath. Charting across Europe and Asia, “Blue Jeans”reached the top 1O in Belgium,Poland,and Israel. The second was shot and directed by Yoann Lemoine,featuring film noir elements and crocodiles.</td></tr><tr><td>Annotated reasoning path: Yoann Lemoin-→ Back to December Predicted reasoning path: Blue Jeans (Lana Del Rey song) → Yoann Lemoin -→Back to December</td></tr></table>
+
+Table 14: Two examples from the HotpotQA distractor development set. Highlighted text shows the bridge entities for multi-hop reasoning, and also the words in red denote the predicted answer.   
+
+<table><tr><td rowspan=1 colspan=6>Q:Which songwriting duo composed music for &quot;La La Land&quot;,and created lyrics for ”AChristmas Story: The Musica?</td></tr><tr><td rowspan=1 colspan=3>P1: A Christmas Story: The Musical is a musical version of the film &quot;A ChristmasStory ... The musical has music and lyrics written byPasek &amp; Pauland the bookby Joseph Robinette.</td><td rowspan=1 colspan=1>0.98</td><td rowspan=1 colspan=1>0.00</td><td rowspan=1 colspan=1>0.00</td></tr><tr><td rowspan=1 colspan=3> P2: Benj Pasek and Justin Paul, known together as Pasek and Paul,are an Americansongwriting duo and composing team for musical theater, films,and television..they won both the Golden Globe and Academy Award for Best Original Song forthe song &quot;City of Stars”.</td><td rowspan=1 colspan=1>0.08</td><td rowspan=1 colspan=1>0.89?</td><td rowspan=1 colspan=1>0.00</td></tr><tr><td rowspan=1 colspan=3>P3: La La Land”is a song recorded by American singer Demi Lovato. It was writ-ten by Lovato,Joe Jonas,Nick Jonas and Kevin Jonas and produced by the JonasBrothers alongside John Fields,for Lovato&#x27;s debut studio album,&quot;Dont Forget&quot;(2008).</td><td rowspan=1 colspan=1>0.12</td><td rowspan=1 colspan=1>0.00</td><td rowspan=1 colspan=1>0.00</td></tr><tr><td rowspan=1 colspan=6>Q:Alexander Kerensky was defeated and destroyed by the Bolsheviks in the course of a civilwar that ended when ?</td></tr><tr><td rowspan=5 colspan=3>P1: The Socialist Revolutionary Party,or Party of Socialists-Revolutionaries sery&quot;)was a major political party in early 2Oth century Russia and a key player in theRussian Revolution．.. The anti-Bolshevik faction of this party,known as theRight SRs,which remained loyal to the Provisional Government leader AlexanderKerensky was defeated and destroyed by the Bolsheviks in the course ofthe Rus-sian Civil Warand subsequent persecution.P2:The Russian Civil War (November 1917 October 1922) was a multi-party warin the former Russian Empire immediately after the Russian Revolutions of 1917,as many factions vied to determine Russias political future.P3: Alexander Fyodorovich Kerensky was a Russian lawyer and key political fig-ure in the Russian Revolution of 1917.</td><td rowspan=4 colspan=1>0.950.00</td><td rowspan=4 colspan=1>0.000.87√</td><td rowspan=3 colspan=1>0.00</td></tr><tr><td rowspan=1 colspan=1>Russian R</td></tr><tr><td rowspan=1 colspan=2>Right SRs,whi</td></tr><tr><td rowspan=1 colspan=1>0.00</td></tr><tr><td rowspan=1 colspan=1>0.08</td><td rowspan=1 colspan=1>0.09</td><td rowspan=1 colspan=1>0.00</td></tr></table>
+
+Table 15: Statistics of the reasoning paths for SQuAD Open and Natural Questions Open: the average length and the distribution of length of the reasoning paths selected by our retriever and reader for SQuAD Open and Natural Questions Open.   
+
+<table><tr><td rowspan="2"></td><td colspan="2">SQuAD Open</td><td colspan="2">Natural Questions Open</td></tr><tr><td>Retriever 1.00</td><td>Reader 1.08</td><td>Retriever 1.23</td><td>Reader 1.54</td></tr><tr><td>1</td><td>10,570</td><td>9,759</td><td>6,719</td><td>4,047</td></tr><tr><td>2</td><td>0</td><td>811</td><td>2,038</td><td>4,702</td></tr><tr><td>3</td><td>0</td><td>0</td><td>0</td><td>8</td></tr></table>
+
+We present another example, where only the graph-based recurrent retrieval model succeeds in finding the correct paragraph pair, (P1, P2). The second question in Table 14 shows that at the first time step our retriever successfully selects P1, but does not pay attention to P2 at all, as the retriever is not aware of the bridge entity, “the Russian Civil War”. Again, once it is conditioned on P1, which includes the bridge entity, it can select P2 at the second time step. Like this, we can see how our model successfully learns to model relationships between paragraphs for multi-hop reasoning.
+
+# D ADDITIONAL RESULTS ON SQUAD OPEN AND NATURAL QUESTIONS OPEN
+
+Although the main focus of this work is on multi-hop open-domain QA, our framework shows competitive performance on the two open-domain QA datasets, SQuAD Open and Natural Questions Open. Both of the two dataets are originally created by assigning a single ground-truth paragraph for each question, and in that sense, our framework is not specific to multi-hop reasoning tasks. In this section, we further analyze our experimental results on the two datasets.
+
+SQuAD Open Table 15 shows statistics of the lengths of the selected reasoning paths on our SQuAD Open experiment. This table is analogous to Table 9 on our HotpotQA experiments. We can clearly see that our recurrent retriever always outputs a single paragraph for each question, if we only use the top-1 predictions. This is because our retriever model for this dataset is trained with the single-paragraph annotations. Our beam search can find longer reasoning paths, and as a result, the re-ranking process in our reader model somtimes selects the reasoning paths including two paragraphs. The trend is consistent with that in Table 9. However, the effects of selecting more than one paragraph do not have a big impact; we observed only $0 . 1 \%$ F1/EM improvement over our method with restricting the path length to one (based on the same experiment with $L = 1$ in Table 8). Considering that SQuAD is a single-hop QA dataset, the result matches our intuition.
+
+Natural Questions Open Table 15 also shows the results on Natural Questions Open, where we see the same trend again. Thanks to the ground-truth path augmentation technique, our recurrent retriever model prefers longer reasoning paths than those on SQuAD Open. We observed $1 \%$ EM improvement over the $L = 1$ baseline on Natural Questions Open, and next we show an example to discuss why our reasoning path approach can be effective on this dataset.
+
+Table 16 shows one example where our model finds a multi-hop reasoning path effectively in Natural Questions Open (development set). The question “who sang the original version of killing me so” has relatively fewer lexical overlap with the originally annotated paragraph (Killing Me Softly with His Song (V) in Table 16). Moreover, there are several entities named as “killing me softly” in Wikipedia, because many artists cover the song. To answer this question correctly, our retriever first selects Roberta Flack (I), and then hops to the originally annotated paragraph, Killing Me Softly with His Song (V). Our reader further verifies this reasoning path and extracts the correct answer from Killing Me Softly with His Song (V). This example shows that even without gold
+
+<table><tr><td>Q: who sang the original version of killing me softly</td></tr><tr><td>Roberta Flack (I): Roberta Cleopatra Flack (born February 10,1937) is an American singer. She is known for her No. 1 singles &quot;The First Time Ever I Saw Your Face&quot;, &quot;Killing Me Softly with His Song&#x27;</td></tr><tr><td>Killing Me Softly with His Song (V), The song was written in collaboration with Lori Lieberman, who recorded the song in late 1971. In 1973 it became a number - one hit in the US and Canada for Roberta Flack, Many artists have covered the song...</td></tr><tr><td>Annotated reasoning path: Killing Me Softly with His Song (V) Predicted reasoning Path: Roberta Flack (I) →Killing Me Softly with His Song (V)</td></tr></table>
+
+Table 16: An example from Natural Questions Open. The bold text represents titles and paragraph indices (e.g., (I) denotes that the paragraph is an introductory paragraph). The highlighted phrase represents a bridge entity and the text in red represents an answer span.
+
+reasoning paths annotations, our model trained on the augmented examples learns to retrieve multihop reasoning paths from the entire Wikipedia.
+
+These detailed experimental results on the two other open-domain QA datasets demonstrate that our framework learns to retrieve reasoning paths flexibly with evidence sufficient to answer a given question, according to each dataset’s nature.

parse/train/SyX0IeWAW/SyX0IeWAW.md

@@ -0,0 +1,223 @@
+# META LEARNING SHARED HIERARCHIES
+
+Kevin Frans Henry M. Gunn High School Work done as an intern at OpenAI kevinfrans2@gmail.com
+
+John Schulman OpenAI
+
+Jonathan Ho, Xi Chen, Pieter Abbeel UC Berkeley, Department of Electrical Engineering and Computer Science
+
+# ABSTRACT
+
+We develop a metalearning approach for learning hierarchically structured policies, improving sample efficiency on unseen tasks through the use of shared primitives—policies that are executed for large numbers of timesteps. Specifically, a set of primitives are shared within a distribution of tasks, and are switched between by task-specific policies. We provide a concrete metric for measuring the strength of such hierarchies, leading to an optimization problem for quickly reaching high reward on unseen tasks. We then present an algorithm to solve this problem end-to-end through the use of any off-the-shelf reinforcement learning method, by repeatedly sampling new tasks and resetting task-specific policies. We successfully discover1 meaningful motor primitives for the directional movement of four-legged robots, solely by interacting with distributions of mazes. We also demonstrate the transferability of primitives to solve long-timescale sparse-reward obstacle courses, and we enable 3D humanoid robots to robustly walk and crawl with the same policy.
+
+# 1 INTRODUCTION
+
+Humans encounter a wide variety of tasks throughout their lives and utilize prior knowledge to master new tasks quickly. In contrast, reinforcement learning algorithms are typically used to solve each task independently and from scratch, and they require far more experience than humans. While a large body of research seeks to improve the sample efficiency of reinforcement learning algorithms, there is a limit to learning speed in the absence of prior knowledge.
+
+We consider the setting where agents solve distributions of related tasks, with the goal of learning new tasks quickly. One challenge is that while we want to share information between the different tasks, these tasks have different optimal policies, so it’s suboptimal to learn a single shared policy for all tasks. Addressing this challenge, we propose a model containing a set of shared sub-policies (i.e., motor primitives), which are switched between by task-specific master policies. This design is closely related to the options framework (Sutton et al., 1999; Bacon et al., 2016), but applied to the setting of a task distribution. We propose a method for the end-to-end training of sub-policies that allow for quick learning on new tasks, handled solely by learning a master policy.
+
+Our contributions are as follows.
+
+• We formulate an optimization problem that answers the question of what is a good hierarchy?—the problem is to find a set of low-level motor primitives that enable the high-level master policy to be learned quickly. We propose an optimization algorithm that tractably and approximately solves the optimization problem we posed. The main novelty is in how we repeatedly reset the master policy, which allows us to adapt the sub-policies for fast learning.
+
+We will henceforth refer to our proposed method—including the hierarchical architecture and optimization algorithm—as MLSH, for metalearning shared hierarchies.
+
+We validate our approach on a wide range of environments, including 2D continuous movement, gridworld navigation, and 3D physics tasks involving the directional movement of robots. In the 3D environments, we enable humanoid robots to both walk and crawl with the same policy; and 4-legged robots to discover directional movement primitives to solve a distribution of mazes as well as sparse-reward obstacle courses. Our experiments show that our method is capable of learning meaningful sub-policies solely through interaction with a distributions of tasks, outperforming previously proposed algorithms. We also display that our method is efficient enough to learn in complex physics environments with long time horizons, and robust enough to transfer sub-policies towards otherwise unsolvable sparse-reward tasks.
+
+# 2 RELATED WORK
+
+Previous work in hierarchical reinforcement learning seeks to speed up the learning process by recombining a set of temporally extended primitives—the most well-known formulation is Options (Sutton et al., 1999). While the earliest work assumed that these options are given, more recent work seeks to learn them automatically (Vezhnevets et al., 2016; Daniel et al., 2016). Heess et al. (2016) discovers primitives by training over a set of simple tasks. Florensa et al. (2017) learns a master policy, where sub-policies are defined according to information-maximizing statistics. Bacon et al. (2016) introduces end-to-end learning of hierarchy through the options framework. Henderson et al. (2017) extends the options framework to include reward options. Several methods (Dayan & Hinton, 1993; Vezhnevets et al., 2017; Ghazanfari & Taylor, 2017) aim to learn a decomposition of complicated tasks into sub-goals. These prior works are mostly focused on the single-task setting and don’t account for the multi-task structure as part of the algorithm. Other past works (Thomas & Barto, 2011; Thomas, 2011; Thomas & Barto, 2012) have simultaneously learned modules that are used in conjunction to solve tasks, but do not incorporate temporal abstraction. On the other hand, our work takes advantage of the multi-task setting as a way to learn temporally extended primitives.
+
+There has also been work in metalearning, where information from past experiences is used to learn quickly on specific tasks. Andrychowicz et al. (2016) proposes the use of a recurrent LSTM network to generate parameter updates. Duan et al. (2016) and Wang et al. (2016) aim to use recurrent networks as the entire learning process, giving the network the same inputs a traditional RL method would receive. Mishra et al. (2017) tackles a similar problem, utilizing temporal convolutions rather than recurrency. Finn et al. (2017) accounts for fine-tuning of a shared policy, by optimizing through a second gradient step. While the prior work on metalearning optimizes to learn as much as possible in a small number of gradient updates, MLSH (our method) optimizes to learn quickly over a large number of policy gradient updates in the RL setting—a regime not yet explored by prior work.
+
+# 3 PROBLEM STATEMENT
+
+First, we will formally define the optimization problem we would like to solve, in which we have a distribution over tasks, and we would like to find parameters that enable an agent to learn quickly on tasks sampled from this distribution.
+
+Let $S$ and $A$ denote the state space and action space, respectively. A Markov Decision Process (MDP) is defined by the transition function $P ( s ^ { \prime } , \bar { r } | s , a )$ , where $( s ^ { \prime } , \bar { r } )$ are the next state and reward, and $( s , a )$ are the state and action.
+
+Let $P _ { M }$ denote a distribution over MDPs $M$ with the same state-action space $( S , A )$ . An agent is a function mapping from a multi-episode history $\left( s _ { 0 } , a _ { 0 } , r _ { 0 } , s _ { 1 } , a _ { 2 } , r _ { 2 } , \dots s _ { t - 1 } \right)$ to the next action $a _ { t }$ . Specifically, an agent consists of a reinforcement learning algorithm which iteratively updates a parameter vector $( \phi , \theta )$ that defines a stochastic policy $\pi _ { \phi , \theta } ( a | s )$ . $\phi$ parameters are shared between all tasks and held fixed at test time. $\theta$ is learned from scratch (from a zero or random initialization) per-task, and encodes the state of the learning process on that task. In the setting we consider, first an MDP $M$ is sampled from $P _ { M }$ , then an agent is incarnated with the shared parameters $\phi$ , along with randomly-initialized $\theta$ parameters. During an agent’s $T$ -step interaction with the sampled MDP $M$ , the agent iteratively updates its $\theta$ parameters.
+
+![](images/e7efbd7a9f86cbc819ac0db5e5cc053b78029791ed7fa5a23d741e22e77c3324.jpg)  
+Figure 1: Structure of a hierarchical sub-policy agent. $\theta$ represents the master policy, which selects a sub-policy to be active. In the diagram, $\phi _ { 3 }$ is the active sub-policy, and actions are taken according to its output.
+
+In other words, $\phi$ represents a set of parameters that is shared between tasks, and $\theta$ represents a set of per-task parameters, which is updated as the agent learns about the current task $M$ . An agent interacts with the task for $T$ timesteps, over multiple episodes, and receives total return $R =$ $r _ { 0 } + r _ { 1 } + . . . + r _ { T - 1 }$ . The meta-learning objective is to optimize the expected return during an agent’s entire lifetime, over the sampled tasks.
+
+$$
+\mathrm { m a x i m i z e } _ { \phi } E _ { M \sim P _ { M } , t = 0 \ldots T - 1 } [ R ]
+$$
+
+This objective tries to find a shared parameter vector $\phi$ that ensures that, when faced with a new MDP, the agent achieves high $T$ time-step returns by simply adapting $\theta$ while in this new MDP.
+
+While there are various possible architectures incorporating shared parameters $\phi$ and per-task parameters $\theta$ , we propose an architecture that is motivated by the ideas of hierarchical reinforcement learning. Specifically, the shared parameter vector $\phi$ consists of a set of subvectors $\phi _ { 1 } , \phi _ { 2 } , \ldots , \phi _ { K }$ , where each subvector $\phi _ { k }$ defines a sub-policy $\pi _ { \phi _ { k } } ( a | s )$ . The parameter $\theta$ is a separate neural network that switches between the sub-policies. That is, $\theta$ parametrizes a stochastic policy, called the master policy whose action is to choose the index $k \in \{ 1 , 2 , \ldots , K \}$ . Furthermore, as in some other hierarchical policy architectures (e.g. options (Sutton et al., 1999)), the master policy chooses actions at a slower timescale than the sub-policies $\phi _ { k }$ . In this work, the master policy samples actions at a fixed frequency of $N$ timesteps, i.e., at $t = 0 , N , 2 N , \ldots .$
+
+This architecture is illustrated in Figure 1. By discovering a strong set of sub-policies $\phi$ , learning on new tasks can be handled solely by updating the master policy $\theta$ . Furthermore, since the master policy chooses actions only every $N$ time steps, it sees a learning problem with a horizon that is only $1 / N$ times as long. Hence, it can adapt quickly to a new MDP $M$ , which is required by the learning objective (Equation (1)).
+
+# 4 ALGORITHM
+
+We would like to iteratively learn a set of sub-policies that allow newly incarnated agents to achieve maximum reward over $T$ -step interactions in a distribution of tasks.
+
+An optimal set of sub-policies must be fine-tuned enough to achieve high performance. At the same time, they must be robust enough to work on wide ranges of tasks. Optimal sets of sub-policies must also be diversely structured such that master policies can be learned quickly. We present an update scheme of sub-policy parameters $\phi$ leading naturally to these qualities.
+
+# 4.1 POLICY UPDATE IN MLSH
+
+In this section, we will describe the MLSH (metalearning shared hierarchies) algorithm for learning sub-policy parameters $\phi$ . Starting from a random initialization, the algorithm (Algorithm 1) iteratively performs update steps which can be broken into two main components: a warmup period to optimize master policy parameters $\theta$ , along with a joint update period where both $\theta$ and $\phi$ are optimized.
+
+# Algorithm 1 Meta Learning Shared Hierarchies
+
+<table><tr><td>Initialize Φ</td></tr><tr><td>repeat Initialize 0</td></tr><tr><td>Sample task M ~ PM</td></tr><tr><td>for w = O,1,.W (warmup period) do</td></tr><tr><td>Collect D timesteps of experience using T𝜙,θ</td></tr><tr><td>Update θ to maximize expected return from 1/N timescale viewpoint</td></tr><tr><td>end for for u = O,1,...U (joint update period) do</td></tr><tr><td>Collect D timesteps of experience using ,θ</td></tr><tr><td>Update θ to maximize expected return from 1/N timescale viewpoint</td></tr><tr><td>Update  to maximize expected return from full timescale viewpoint</td></tr><tr><td>end for</td></tr><tr><td>until convergence</td></tr></table>
+
+From a high-level view, an MLSH update is structured as follows. We first sample a task $M$ from the distribution $P _ { M }$ . We then initialize an agent, using a previous set of sub-policies, parameterized by $\phi$ , and a master policy with randomly-initialized parameters $\theta$ . We then run a warmup period to optimize $\theta$ . At this point, our agent contains of a set of general sub-policies $\phi$ , as well as a master policy $\theta$ fine-tuned to the task at hand. We enter the joint update period, where both $\theta$ and $\phi$ are updated. Finally, we sample a new task, reset $\theta$ , and repeat.
+
+The warmup period for optimizing the master policy $\theta$ is defined as follows. We assume a constant set of sub-policies as parameterized by $\phi$ . From the sampled task, we record $D$ timesteps of experience using $\pi _ { \phi , \theta } ( a | s )$ . We view this experience from the perspective of the master policy, as in Figure 2. Specifically, we consider the selection of a sub-policy as a single action. The next $N$ timesteps, along with corresponding state changes and rewards, are viewed as a single environment transition. We then update $\theta$ towards maximizing reward, using the collected experience along with an arbitrary reinforcement learning algorithm (for example DQN, A3C, TRPO, PPO) (Mnih et al., 2015; 2016; Schulman et al., 2015; 2017). We repeat this prodecure $W$ times.
+
+Next, we will define a joint update period where both sub-policies $\phi$ and master policy $\theta$ are updated. For $U$ iterations, we collect experience and optimize $\theta$ as defined in the warmup period. Additionally, we reuse the same experience, but viewed from the perspective of the sub-policies. We treat the master policy as an extension of the environment. Specifically, we consider the master policy’s decision as a discrete portion of the environment’s observation. For each $N$ -timestep slice of experience, we only update the parameters of the sub-policy that had been activated by the master policy.
+
+![](images/0f586eac4e1bf551f1fa11420d53959ba42edf5f5cbd6b831580d373c5a0ca98.jpg)  
+Figure 2: Unrolled structure for a master policy action lasting $N \ = \ 3$ timesteps. Left: When training the master policy, the update only depends on the master policy’s action and total reward (blue region), treating the individual actions and rewards as part of the environment transition (red region). Right: When training sub-policies, the update considers the master policy’s action as part of the observation (blue region), ignoring actions in other timesteps (red region)
+
+# 5 RATIONALE
+
+We will now provide intuition for why this framework leads to a set of sub-policies $\phi$ which allow agents to quickly reach high reward when learning $\theta$ on a new task. In metalearning methods, it is common to optimize for reward over an entire inner loop (in the case of MLSH, training $\theta$ for $T$ iterations). However, we instead choose to optimize $\phi$ towards maximizing reward within a single episode. Our argument relies on the assumption that the warmup period of $\theta$ will learn an optimal master policy, given a set of fixed sub-polices $\phi$ . As such, the optimal $\phi$ at $\theta _ { \mathrm { f i n a l } }$ is equivalent to the optimal $\phi$ for training $\theta$ from scratch. While this assumption is at some times false, such as when a gradient update overshoots the optimal $\theta$ policy, we empirically find the assumption accurate enough for training purposes.
+
+Next, we consider the inclusion of a warmup period. It is important that $\phi$ only be updated when $\theta$ is at a near-optimal level. A motivating example for this is a navigation task containing two possible destinations, as well as two sub-policies. If $\theta$ is random, the optimal sub-policies both lead the agent to the midpoint of the destinations. If $\theta$ contains information on the correct destination, the optimal sub-policies consist of one leading to the first destination, and the other to the second.
+
+Finally, we will address the reasoning behind limiting the update period to $U$ iterations. As we update the sub-policy parameters $\phi$ while reusing master policy parameters $\theta$ , we are assuming that re-training $\theta$ will result in roughly the same master policy. However, as $\phi$ changes, this assumption holds less weight. We therefore stop and re-train $\theta$ once a threshold of $U$ iterations has passed.
+
+# 6 EXPERIMENTS
+
+We hypothesize that meaningful sub-policies can be learned by operating over distributions of tasks, in an efficient enough manner to handle complex physics domains. We also hypothesize that subpolicies can be transferred to complicated tasks outside the training distribution. In the following section, we present a series of experiments designed to test the performance of our method, through comparison to baselines and past methods with hierarchy.
+
+# 6.1 EXPERIMENTAL SETUP
+
+We present a series of environments containing both shared and task-specific information. We examine two curves: the overall learning on the entire distribution $( \phi )$ , as well as the learning on a sampled individual task $\mathbf { \eta } ^ { ( \theta ) }$ . For overall training, we compare to a baseline of a shared policy trained jointly across all tasks from the distribution. We also compare to running MLSH without a warmup period.
+
+In the sampled individual task experiments, our MLSH agent utilizes sub-policies $( \phi )$ previously trained on the entire distribution, and only updates the master policy $\mathbf { \eta } ^ { ( \theta ) }$ towards the new task. To test the importance of the sub-policy structure, we compare against fine-tuning a single policy that has been optimized across all tasks. We also compare against training a new single policy from scratch, to test if the learned sub-policies are useful.
+
+For both master and sub-policies, we use 2 layer MLPs with a hidden size of 64. Master policy actions are sampled through a softmax distribution. We train both master and sub-policies using policy gradient methods, specifically PPO (Schulman et al., 2017). For collecting experience, we compute a batchsize of $D { = } 2 0 0 0$ timesteps. We use a much larger learning rate for $\theta$ (0.01) than for $\phi$ (0.0003), since $\phi$ parameters should remain relatively consistent throughout a single warmup and joint-update period.
+
+While the base MLSH algorithm is sequential, we run all experiments in a parallel multi-core setup for faster wall-clock training time. We split 120 cores into 10 groups of 12 cores, where a group represents a single MLSH learner which uses 12 cores to to collect experience in parallel. All groups sample individual tasks from the task distribution, and only $\phi$ parameters are shared. Viewed as a whole, we are optimizing a shared set of $\phi$ parameters towards 10 sampled tasks in parallel.
+
+To prevent periods where the $\phi$ parameters are receiving no gradients, we stagger the warmup periods of each group, so a new group enters warmup as soon as another group leaves. Once a group has finished both its warmup and joint-update period, a new task is sampled along with a new random initialization of $\theta$ , both of which are shared within all cores in the group. Warmup and joint-update lengths for individual environment distributions will be described in the following section. As a general rule, a good warmup duration represents the amount of gradient updates required to approach convergence of $\theta$ .
+
+![](images/5755c998dc06436b1ff9c95df68e96e1e92aa123b95d5381c5a89bfe99740278.jpg)  
+Figure 3: Sampled tasks from 2D moving bandits. Small green dot represents the agent, while blue and yellow dots represent potential goal points. Right: Blue/red arrows correspond to movements when taking sub-policies 1 and 2 respectively.
+
+6.2 CAN MEANINGFUL SUB-POLICIES BE LEARNED OVER A DISTRIBUTION OF TASKS, AND DO THEY OUTPERFORM A SHARED POLICY?
+
+Our motivating problem is a 2D moving bandits task (Figure 3), in which an agent is placed in a world and shown the positions of two randomly placed points. The agent may take discrete actions to move in the four cardinal directions, or opt to stay still. One of the two points is marked as correct, although the agent does not receive information on which one it is. The agent receives a reward of 1 if it is within a certain distance of the correct point, and a reward of 0 otherwise. Each episode lasts 50 timesteps, and master policy actions last for 10. We use two sub-policies, a warmup duration of 9, and a joint-update duration of 1.
+
+![](images/952cbd17c94a1496ff9e5b04287e2e96a3459fb1b3b09810f19a07abc7807b1b.jpg)  
+Figure 4: Learning curves for 2D Moving Bandits and Four Rooms.
+
+After training, MLSH learns sub-policies corresponding to movement towards each potential goal point. Training a master policy is faster than training a single policy from scratch, as we are tasked only with discovering the correct goal, rather than also learning primitive movement. Learning a shared policy, on the other hand, results in an agent that always moves towards a certain goal point, ignoring the other and thereby cutting expected reward by half. We additionally compare to an $\mathtt { R L } ^ { 2 }$ policy (Duan et al., 2016), which encounters the same problem as the shared policy and ignores one of the goal points.
+
+![](images/50c559a48b1e887e60b394612e7b32c45118f8db9f44720f43daadc41a744612.jpg)  
+Figure 5: Top: Ant Twowalk. Ant must maneuver towards red goal point, either towards the top or towards the right. Bottom Left: Walking. Humanoid must move horizontally while maintaining an upright stance. Bottom Right: Crawling. Humanoid must move horizontally while a height-limiting obstacle is present.
+
+We perform several ablation tests within the 2D moving bandits task. Removing the warmup period results in an MLSH agent which at first has both sub-policies moving to the same goal point, but gradually shifts one sub-policy towards the other point. Running the master policy on the same timescale as the sub-policies results in similar behavior to simply learning a shared policy, showing that the temporal extension of sub-policies is key. Finally, we run a hyperparameter comparison to test the influence of the sub-policy count and warmup duration.
+
+# 6.3 HOW DOES MLSH COMPARE TO PAST METHODS IN THE HIERARCHICAL DOMAIN?
+
+To compare to past methods, we consider the four-rooms domain described in Sutton et al. (1999) and expanded in Option Critic (Bacon et al., 2016). The agent starts at a specific spot in the gridworld, and is randomly assigned a goal position. A reward of 1 is awarded for being in the goal state. Episodes last for 100 timesteps, and master policy actions last for 25. We utilize four sub-policies, a warmup time of 20, and a joint-update time of 30.
+
+First, we repeatedly train MLSH and Option Critic on many random goals in the four-rooms domain, until reward stops improving. Then, we sample an unseen goal position and fine-tune. We compare against baselines of training a single policy from scratch, using PPO against MLSH, and Actor Critic against Option Critic. In Figure 4, while Option Critic performs similarly to its baseline, we can see MLSH reach high reward faster than the PPO baseline. It is worth noting that when fine-tuning, the PPO baseline naturally reaches more stable reward than Actor Critic, so we do not compare MLSH and Option Critic directly.
+
+# 6.4 IS THE MLSH FRAMEWORK SAMPLE-EFFICIENT ENOUGH TO LEARN DIVERSE SUB-POLICIES IN PHYSICS ENVIRONMENTS?
+
+To test the scalability of the MLSH algorithm, we present a series of physics-based tasks which we describe below, all which are simulated through Mujoco (Todorov et al., 2012). Diverse subpolicies are naturally discovered, as shown in Figure 5 and Figure 6. Episodes last 1000 timesteps, and master policy actions last 200. We use a warmup time of 20, and a joint-update time of 40.
+
+In the Twowalk tasks, we would like to examine if simulated robots can learn directional movement primitives. We test performance on a standard simulated four-legged ant, and use a sub-policy count of two. A destination point is placed in either the top edge of the world or the right edge of the world. Reward is given based on negative distance to this destination point.
+
+![](images/2a64ad4a78b48864dcc90ca1a8b516b9d3d78be1248e62d9351e77313e190412.jpg)  
+Figure 6: Top: Distribution of mazes. Red blocks are impassable tiles, and green blocks represent the goal. Bottom: Sub-policies learned from mazes to move up, right, and down.
+
+In addition, we would like to determine if diverse sub-policies can be automatically discovered solely through interaction with the environment. We present a task where Ant robots must move to destination points in a set of mazes (Figure 6). Without human supervision, Ant robots are able to learn directional movement sub-policies in three directions, and use them in combination to solve the mazes.
+
+In the Walk/Crawl task, we would like to determine if Humanoid robots can learn a variety of movement styles. Out of two possible locomotion objectives, one is randomly selected. In the first objective, the agent must move forwards while maintaining an upright stance. This was designed with a walking behavior in mind. In the second objective, the agent must move backwards underneath an obstacle limiting vertical height. This was designed to encourage a crawling behavior.
+
+Additionally, we test the transfer capabilities of sub-policies trained in the Walk/Crawl task by introducing an unseen combination task. The Humanoid agent must first walk forwards until a certain distance, at which point it must switch movements, turn around, and crawl backwards under an obstacle.
+
+<table><tr><td colspan="2">Reward on Walk/Crawl combination task</td></tr><tr><td>MLSHTransfer Shared Policy Transfer Single Policy</td><td>14333 6055 -643</td></tr></table>
+
+On both Twowalk and Walk/Crawl tasks, MLSH significantly outperforms baselines, displaying scalability into complex physics domains. Ant robots learn temporally-extended directional movement primitives that lead to efficient exploration of mazes. In addition, we successfully discover diverse Humanoid sub-policies for both walking and crawling.
+
+6.5 CAN SUB-POLICIES BE USED TO LEARN IN AN OTHERWISE UNSOLVABLE SPARSE PHYSICS ENVIRONMENT?
+
+Finally, we present a complex task that is unsolvable with naive PPO. The agent controls an Ant robot which has been placed into an obstacle course. The agent must navigate from the bottom-left corner to the top-right corner, to receive a reward of 1. In all other cases, the agent receives a reward of 0. Along the way, there are obstacles such as walls and a chasing enemy. We periodically reset the joints of the Ant robot to prevent it from falling over. An episode lasts for 2000 timesteps, and master policy actions last 200. To solve this task, we use sub-policies learned in the Ant Twowalk tasks. We then fine-tune the master policy on the obstacle course task.
+
+![](images/c45bb033ea0100b1e9b88a1ffab54662266208886df4ab8dd0b513236601577e.jpg)  
+Figure 7: Learning curves for Twowalk and Walk/Crawl tasks
+
+![](images/6ee2391f0ebd27ee68dc375118bbb0d74cc560b512c4cfe6455cc3b3dba69bb9.jpg)  
+Figure 8: Ant Obstacle course task. Agent must navigate to the green square in the top right corner. Entering the red circle causes an enemy to attack the agent, knocking it back.
+
+In the sparse reward setting, naive PPO cannot learn, as exploration over the space of primitive action sequences is unlikely to result in reward signal. On the other hand, MLSH allows for exploration over the space of sub-policies, where it is easier to discover a sequence that leads to reward.
+
+<table><tr><td colspan="2">Reward on Ant Obstacle task</td></tr><tr><td>MLSHTransfer Single Policy</td><td>193 0</td></tr></table>
+
+# 7 DISCUSSION
+
+In this work, we formulate an approach for the end-to-end metalearning of hierarchical policies. We present a model for representing shared information as a set of sub-policies. We then provide a framework for training these models over distributions of environments. Even though we do not optimize towards the true objective, we achieve significant speedups in learning. In addition, we naturally discover diverse sub-policies without the need for hand engineering.
+
+# 7.1 FUTURE WORK
+
+As there is no gradient signal being passed between the master and sub-policies, the MLSH model utilizes hard one-hot communication, as opposed to methods such as Gumbel-Softmax (Jang et al., 2016). This lack of a gradient also allows MLSH to be learning-method agnostic. While we used policy gradients in our experiments, it is entirely feasible to have the master or sub-policies be trained with evolution (Eigen) or Q-learning (Watkins & Dayan, 1992).
+
+From another point of view, our training framework can be seen as a method of joint optimization over two sets of parameters. This framework can be applied to other scenarios than learning subpolicies. For example, distributions of tasks with similar observation distributions but different reward functions could be solved with a shared observational network, while learning independent policies.
+
+This work draws inspiration from the domains of both hierarchical reinforcement learning and metalearning, the intersection at which architecture space has yet to be explored. For example, the set of sub-policies could be condensed into a single neural network, which receives a continuous vector from the master policy. If sample efficiency issues are addressed, several approximations in the MLSH method could be removed for a more unbiased estimator – such as training $\phi$ to maximize reward on the entire $T$ -timesteps, rather than on a single episode. We believe this work opens up many directions in training agents that can quickly adapt to new tasks.
+
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+Alexander Sasha Vezhnevets, Simon Osindero, Tom Schaul, Nicolas Heess, Max Jaderberg, David Silver, and Koray Kavukcuoglu. Feudal networks for hierarchical reinforcement learning. arXiv preprint 1703.01161, 2017.
+
+Jane X. Wang, Zeb Kurth-Nelson, Dhruva Tirumala, Hubert Soyer, Joel Z. Leibo, Remi Munos, ´ Charles Blundell, Dharshan Kumaran, and Matthew Botvinick. Learning to reinforcement learn. arXiv preprint 1611.05763, 2016.
+
+Christopher JCH Watkins and Peter Dayan. Q-learning. Machine learning, 8(3-4):279–292, 1992.

parse/train/SyX0IeWAW/SyX0IeWAW_content_list.json

@@ -0,0 +1,1271 @@
+[
+    {
+        "type": "text",
+        "text": "META LEARNING SHARED HIERARCHIES ",
+        "text_level": 1,
+        "bbox": [
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+            98,
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+            121
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Kevin Frans Henry M. Gunn High School Work done as an intern at OpenAI kevinfrans2@gmail.com ",
+        "bbox": [
+            184,
+            145,
+            410,
+            200
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "John Schulman OpenAI ",
+        "bbox": [
+            184,
+            222,
+            294,
+            250
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Jonathan Ho, Xi Chen, Pieter Abbeel UC Berkeley, Department of Electrical Engineering and Computer Science ",
+        "bbox": [
+            553,
+            145,
+            813,
+            188
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            287,
+            544,
+            303
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We develop a metalearning approach for learning hierarchically structured policies, improving sample efficiency on unseen tasks through the use of shared primitives—policies that are executed for large numbers of timesteps. Specifically, a set of primitives are shared within a distribution of tasks, and are switched between by task-specific policies. We provide a concrete metric for measuring the strength of such hierarchies, leading to an optimization problem for quickly reaching high reward on unseen tasks. We then present an algorithm to solve this problem end-to-end through the use of any off-the-shelf reinforcement learning method, by repeatedly sampling new tasks and resetting task-specific policies. We successfully discover1 meaningful motor primitives for the directional movement of four-legged robots, solely by interacting with distributions of mazes. We also demonstrate the transferability of primitives to solve long-timescale sparse-reward obstacle courses, and we enable 3D humanoid robots to robustly walk and crawl with the same policy. ",
+        "bbox": [
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+            320,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            546,
+            336,
+            563
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Humans encounter a wide variety of tasks throughout their lives and utilize prior knowledge to master new tasks quickly. In contrast, reinforcement learning algorithms are typically used to solve each task independently and from scratch, and they require far more experience than humans. While a large body of research seeks to improve the sample efficiency of reinforcement learning algorithms, there is a limit to learning speed in the absence of prior knowledge. ",
+        "bbox": [
+            174,
+            580,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We consider the setting where agents solve distributions of related tasks, with the goal of learning new tasks quickly. One challenge is that while we want to share information between the different tasks, these tasks have different optimal policies, so it’s suboptimal to learn a single shared policy for all tasks. Addressing this challenge, we propose a model containing a set of shared sub-policies (i.e., motor primitives), which are switched between by task-specific master policies. This design is closely related to the options framework (Sutton et al., 1999; Bacon et al., 2016), but applied to the setting of a task distribution. We propose a method for the end-to-end training of sub-policies that allow for quick learning on new tasks, handled solely by learning a master policy. ",
+        "bbox": [
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+            656,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Our contributions are as follows. ",
+        "bbox": [
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+            775,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "• We formulate an optimization problem that answers the question of what is a good hierarchy?—the problem is to find a set of low-level motor primitives that enable the high-level master policy to be learned quickly. We propose an optimization algorithm that tractably and approximately solves the optimization problem we posed. The main novelty is in how we repeatedly reset the master policy, which allows us to adapt the sub-policies for fast learning. ",
+        "bbox": [
+            214,
+            803,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We will henceforth refer to our proposed method—including the hierarchical architecture and optimization algorithm—as MLSH, for metalearning shared hierarchies. ",
+        "bbox": [
+            173,
+            103,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We validate our approach on a wide range of environments, including 2D continuous movement, gridworld navigation, and 3D physics tasks involving the directional movement of robots. In the 3D environments, we enable humanoid robots to both walk and crawl with the same policy; and 4-legged robots to discover directional movement primitives to solve a distribution of mazes as well as sparse-reward obstacle courses. Our experiments show that our method is capable of learning meaningful sub-policies solely through interaction with a distributions of tasks, outperforming previously proposed algorithms. We also display that our method is efficient enough to learn in complex physics environments with long time horizons, and robust enough to transfer sub-policies towards otherwise unsolvable sparse-reward tasks. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            286,
+            344,
+            303
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Previous work in hierarchical reinforcement learning seeks to speed up the learning process by recombining a set of temporally extended primitives—the most well-known formulation is Options (Sutton et al., 1999). While the earliest work assumed that these options are given, more recent work seeks to learn them automatically (Vezhnevets et al., 2016; Daniel et al., 2016). Heess et al. (2016) discovers primitives by training over a set of simple tasks. Florensa et al. (2017) learns a master policy, where sub-policies are defined according to information-maximizing statistics. Bacon et al. (2016) introduces end-to-end learning of hierarchy through the options framework. Henderson et al. (2017) extends the options framework to include reward options. Several methods (Dayan & Hinton, 1993; Vezhnevets et al., 2017; Ghazanfari & Taylor, 2017) aim to learn a decomposition of complicated tasks into sub-goals. These prior works are mostly focused on the single-task setting and don’t account for the multi-task structure as part of the algorithm. Other past works (Thomas & Barto, 2011; Thomas, 2011; Thomas & Barto, 2012) have simultaneously learned modules that are used in conjunction to solve tasks, but do not incorporate temporal abstraction. On the other hand, our work takes advantage of the multi-task setting as a way to learn temporally extended primitives. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "There has also been work in metalearning, where information from past experiences is used to learn quickly on specific tasks. Andrychowicz et al. (2016) proposes the use of a recurrent LSTM network to generate parameter updates. Duan et al. (2016) and Wang et al. (2016) aim to use recurrent networks as the entire learning process, giving the network the same inputs a traditional RL method would receive. Mishra et al. (2017) tackles a similar problem, utilizing temporal convolutions rather than recurrency. Finn et al. (2017) accounts for fine-tuning of a shared policy, by optimizing through a second gradient step. While the prior work on metalearning optimizes to learn as much as possible in a small number of gradient updates, MLSH (our method) optimizes to learn quickly over a large number of policy gradient updates in the RL setting—a regime not yet explored by prior work. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3 PROBLEM STATEMENT ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            667,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "First, we will formally define the optimization problem we would like to solve, in which we have a distribution over tasks, and we would like to find parameters that enable an agent to learn quickly on tasks sampled from this distribution. ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Let $S$ and $A$ denote the state space and action space, respectively. A Markov Decision Process (MDP) is defined by the transition function $P ( s ^ { \\prime } , \\bar { r } | s , a )$ , where $( s ^ { \\prime } , \\bar { r } )$ are the next state and reward, and $( s , a )$ are the state and action. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Let $P _ { M }$ denote a distribution over MDPs $M$ with the same state-action space $( S , A )$ . An agent is a function mapping from a multi-episode history $\\left( s _ { 0 } , a _ { 0 } , r _ { 0 } , s _ { 1 } , a _ { 2 } , r _ { 2 } , \\dots s _ { t - 1 } \\right)$ to the next action $a _ { t }$ . Specifically, an agent consists of a reinforcement learning algorithm which iteratively updates a parameter vector $( \\phi , \\theta )$ that defines a stochastic policy $\\pi _ { \\phi , \\theta } ( a | s )$ . $\\phi$ parameters are shared between all tasks and held fixed at test time. $\\theta$ is learned from scratch (from a zero or random initialization) per-task, and encodes the state of the learning process on that task. In the setting we consider, first an MDP $M$ is sampled from $P _ { M }$ , then an agent is incarnated with the shared parameters $\\phi$ , along with randomly-initialized $\\theta$ parameters. During an agent’s $T$ -step interaction with the sampled MDP $M$ , the agent iteratively updates its $\\theta$ parameters. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/e7efbd7a9f86cbc819ac0db5e5cc053b78029791ed7fa5a23d741e22e77c3324.jpg",
+        "image_caption": [
+            "Figure 1: Structure of a hierarchical sub-policy agent. $\\theta$ represents the master policy, which selects a sub-policy to be active. In the diagram, $\\phi _ { 3 }$ is the active sub-policy, and actions are taken according to its output. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            339,
+            107,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In other words, $\\phi$ represents a set of parameters that is shared between tasks, and $\\theta$ represents a set of per-task parameters, which is updated as the agent learns about the current task $M$ . An agent interacts with the task for $T$ timesteps, over multiple episodes, and receives total return $R =$ $r _ { 0 } + r _ { 1 } + . . . + r _ { T - 1 }$ . The meta-learning objective is to optimize the expected return during an agent’s entire lifetime, over the sampled tasks. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/51179ca76e4665e7ffad3b15b5f10de7efce4dd3a8ccbd76a1a21ef62d4cacce.jpg",
+        "text": "$$\n\\mathrm { m a x i m i z e } _ { \\phi } E _ { M \\sim P _ { M } , t = 0 \\ldots T - 1 } [ R ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            385,
+            410,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "This objective tries to find a shared parameter vector $\\phi$ that ensures that, when faced with a new MDP, the agent achieves high $T$ time-step returns by simply adapting $\\theta$ while in this new MDP. ",
+        "bbox": [
+            173,
+            431,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "While there are various possible architectures incorporating shared parameters $\\phi$ and per-task parameters $\\theta$ , we propose an architecture that is motivated by the ideas of hierarchical reinforcement learning. Specifically, the shared parameter vector $\\phi$ consists of a set of subvectors $\\phi _ { 1 } , \\phi _ { 2 } , \\ldots , \\phi _ { K }$ , where each subvector $\\phi _ { k }$ defines a sub-policy $\\pi _ { \\phi _ { k } } ( a | s )$ . The parameter $\\theta$ is a separate neural network that switches between the sub-policies. That is, $\\theta$ parametrizes a stochastic policy, called the master policy whose action is to choose the index $k \\in \\{ 1 , 2 , \\ldots , K \\}$ . Furthermore, as in some other hierarchical policy architectures (e.g. options (Sutton et al., 1999)), the master policy chooses actions at a slower timescale than the sub-policies $\\phi _ { k }$ . In this work, the master policy samples actions at a fixed frequency of $N$ timesteps, i.e., at $t = 0 , N , 2 N , \\ldots .$ ",
+        "bbox": [
+            173,
+            467,
+            825,
+            593
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "This architecture is illustrated in Figure 1. By discovering a strong set of sub-policies $\\phi$ , learning on new tasks can be handled solely by updating the master policy $\\theta$ . Furthermore, since the master policy chooses actions only every $N$ time steps, it sees a learning problem with a horizon that is only $1 / N$ times as long. Hence, it can adapt quickly to a new MDP $M$ , which is required by the learning objective (Equation (1)). ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4 ALGORITHM ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            689,
+            312,
+            704
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We would like to iteratively learn a set of sub-policies that allow newly incarnated agents to achieve maximum reward over $T$ -step interactions in a distribution of tasks. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "An optimal set of sub-policies must be fine-tuned enough to achieve high performance. At the same time, they must be robust enough to work on wide ranges of tasks. Optimal sets of sub-policies must also be diversely structured such that master policies can be learned quickly. We present an update scheme of sub-policy parameters $\\phi$ leading naturally to these qualities. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4.1 POLICY UPDATE IN MLSH ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            828,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section, we will describe the MLSH (metalearning shared hierarchies) algorithm for learning sub-policy parameters $\\phi$ . Starting from a random initialization, the algorithm (Algorithm 1) iteratively performs update steps which can be broken into two main components: a warmup period to optimize master policy parameters $\\theta$ , along with a joint update period where both $\\theta$ and $\\phi$ are optimized. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 Meta Learning Shared Hierarchies ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            488,
+            117
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "table",
+        "img_path": "images/2b69c408c58a19730eb581e713419511c0b4763a19eac6ec54f7e982a8161595.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Initialize Φ</td></tr><tr><td>repeat Initialize 0</td></tr><tr><td>Sample task M ~ PM</td></tr><tr><td>for w = O,1,.W (warmup period) do</td></tr><tr><td>Collect D timesteps of experience using T𝜙,θ</td></tr><tr><td>Update θ to maximize expected return from 1/N timescale viewpoint</td></tr><tr><td>end for for u = O,1,...U (joint update period) do</td></tr><tr><td>Collect D timesteps of experience using ,θ</td></tr><tr><td>Update θ to maximize expected return from 1/N timescale viewpoint</td></tr><tr><td>Update  to maximize expected return from full timescale viewpoint</td></tr><tr><td>end for</td></tr><tr><td>until convergence</td></tr></table>",
+        "bbox": [
+            178,
+            121,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "From a high-level view, an MLSH update is structured as follows. We first sample a task $M$ from the distribution $P _ { M }$ . We then initialize an agent, using a previous set of sub-policies, parameterized by $\\phi$ , and a master policy with randomly-initialized parameters $\\theta$ . We then run a warmup period to optimize $\\theta$ . At this point, our agent contains of a set of general sub-policies $\\phi$ , as well as a master policy $\\theta$ fine-tuned to the task at hand. We enter the joint update period, where both $\\theta$ and $\\phi$ are updated. Finally, we sample a new task, reset $\\theta$ , and repeat. ",
+        "bbox": [
+            173,
+            354,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The warmup period for optimizing the master policy $\\theta$ is defined as follows. We assume a constant set of sub-policies as parameterized by $\\phi$ . From the sampled task, we record $D$ timesteps of experience using $\\pi _ { \\phi , \\theta } ( a | s )$ . We view this experience from the perspective of the master policy, as in Figure 2. Specifically, we consider the selection of a sub-policy as a single action. The next $N$ timesteps, along with corresponding state changes and rewards, are viewed as a single environment transition. We then update $\\theta$ towards maximizing reward, using the collected experience along with an arbitrary reinforcement learning algorithm (for example DQN, A3C, TRPO, PPO) (Mnih et al., 2015; 2016; Schulman et al., 2015; 2017). We repeat this prodecure $W$ times. ",
+        "bbox": [
+            173,
+            445,
+            825,
+            558
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Next, we will define a joint update period where both sub-policies $\\phi$ and master policy $\\theta$ are updated. For $U$ iterations, we collect experience and optimize $\\theta$ as defined in the warmup period. Additionally, we reuse the same experience, but viewed from the perspective of the sub-policies. We treat the master policy as an extension of the environment. Specifically, we consider the master policy’s decision as a discrete portion of the environment’s observation. For each $N$ -timestep slice of experience, we only update the parameters of the sub-policy that had been activated by the master policy. ",
+        "bbox": [
+            173,
+            564,
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+            661
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/0f586eac4e1bf551f1fa11420d53959ba42edf5f5cbd6b831580d373c5a0ca98.jpg",
+        "image_caption": [
+            "Figure 2: Unrolled structure for a master policy action lasting $N \\ = \\ 3$ timesteps. Left: When training the master policy, the update only depends on the master policy’s action and total reward (blue region), treating the individual actions and rewards as part of the environment transition (red region). Right: When training sub-policies, the update considers the master policy’s action as part of the observation (blue region), ignoring actions in other timesteps (red region) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            253,
+            684,
+            740,
+            813
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "5 RATIONALE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            303,
+            117
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We will now provide intuition for why this framework leads to a set of sub-policies $\\phi$ which allow agents to quickly reach high reward when learning $\\theta$ on a new task. In metalearning methods, it is common to optimize for reward over an entire inner loop (in the case of MLSH, training $\\theta$ for $T$ iterations). However, we instead choose to optimize $\\phi$ towards maximizing reward within a single episode. Our argument relies on the assumption that the warmup period of $\\theta$ will learn an optimal master policy, given a set of fixed sub-polices $\\phi$ . As such, the optimal $\\phi$ at $\\theta _ { \\mathrm { f i n a l } }$ is equivalent to the optimal $\\phi$ for training $\\theta$ from scratch. While this assumption is at some times false, such as when a gradient update overshoots the optimal $\\theta$ policy, we empirically find the assumption accurate enough for training purposes. ",
+        "bbox": [
+            174,
+            132,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Next, we consider the inclusion of a warmup period. It is important that $\\phi$ only be updated when $\\theta$ is at a near-optimal level. A motivating example for this is a navigation task containing two possible destinations, as well as two sub-policies. If $\\theta$ is random, the optimal sub-policies both lead the agent to the midpoint of the destinations. If $\\theta$ contains information on the correct destination, the optimal sub-policies consist of one leading to the first destination, and the other to the second. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Finally, we will address the reasoning behind limiting the update period to $U$ iterations. As we update the sub-policy parameters $\\phi$ while reusing master policy parameters $\\theta$ , we are assuming that re-training $\\theta$ will result in roughly the same master policy. However, as $\\phi$ changes, this assumption holds less weight. We therefore stop and re-train $\\theta$ once a threshold of $U$ iterations has passed. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "6 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            417,
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+            433
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We hypothesize that meaningful sub-policies can be learned by operating over distributions of tasks, in an efficient enough manner to handle complex physics domains. We also hypothesize that subpolicies can be transferred to complicated tasks outside the training distribution. In the following section, we present a series of experiments designed to test the performance of our method, through comparison to baselines and past methods with hierarchy. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "6.1 EXPERIMENTAL SETUP ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            535,
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+            549
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We present a series of environments containing both shared and task-specific information. We examine two curves: the overall learning on the entire distribution $( \\phi )$ , as well as the learning on a sampled individual task $\\mathbf { \\eta } ^ { ( \\theta ) }$ . For overall training, we compare to a baseline of a shared policy trained jointly across all tasks from the distribution. We also compare to running MLSH without a warmup period. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the sampled individual task experiments, our MLSH agent utilizes sub-policies $( \\phi )$ previously trained on the entire distribution, and only updates the master policy $\\mathbf { \\eta } ^ { ( \\theta ) }$ towards the new task. To test the importance of the sub-policy structure, we compare against fine-tuning a single policy that has been optimized across all tasks. We also compare against training a new single policy from scratch, to test if the learned sub-policies are useful. ",
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+    {
+        "type": "text",
+        "text": "For both master and sub-policies, we use 2 layer MLPs with a hidden size of 64. Master policy actions are sampled through a softmax distribution. We train both master and sub-policies using policy gradient methods, specifically PPO (Schulman et al., 2017). For collecting experience, we compute a batchsize of $D { = } 2 0 0 0$ timesteps. We use a much larger learning rate for $\\theta$ (0.01) than for $\\phi$ (0.0003), since $\\phi$ parameters should remain relatively consistent throughout a single warmup and joint-update period. ",
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+        "type": "text",
+        "text": "While the base MLSH algorithm is sequential, we run all experiments in a parallel multi-core setup for faster wall-clock training time. We split 120 cores into 10 groups of 12 cores, where a group represents a single MLSH learner which uses 12 cores to to collect experience in parallel. All groups sample individual tasks from the task distribution, and only $\\phi$ parameters are shared. Viewed as a whole, we are optimizing a shared set of $\\phi$ parameters towards 10 sampled tasks in parallel. ",
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+        "type": "text",
+        "text": "To prevent periods where the $\\phi$ parameters are receiving no gradients, we stagger the warmup periods of each group, so a new group enters warmup as soon as another group leaves. Once a group has finished both its warmup and joint-update period, a new task is sampled along with a new random initialization of $\\theta$ , both of which are shared within all cores in the group. Warmup and joint-update lengths for individual environment distributions will be described in the following section. As a general rule, a good warmup duration represents the amount of gradient updates required to approach convergence of $\\theta$ . ",
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+        "image_caption": [
+            "Figure 3: Sampled tasks from 2D moving bandits. Small green dot represents the agent, while blue and yellow dots represent potential goal points. Right: Blue/red arrows correspond to movements when taking sub-policies 1 and 2 respectively. "
+        ],
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+        "type": "text",
+        "text": "6.2 CAN MEANINGFUL SUB-POLICIES BE LEARNED OVER A DISTRIBUTION OF TASKS, AND DO THEY OUTPERFORM A SHARED POLICY? ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our motivating problem is a 2D moving bandits task (Figure 3), in which an agent is placed in a world and shown the positions of two randomly placed points. The agent may take discrete actions to move in the four cardinal directions, or opt to stay still. One of the two points is marked as correct, although the agent does not receive information on which one it is. The agent receives a reward of 1 if it is within a certain distance of the correct point, and a reward of 0 otherwise. Each episode lasts 50 timesteps, and master policy actions last for 10. We use two sub-policies, a warmup duration of 9, and a joint-update duration of 1. ",
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+        "img_path": "images/952cbd17c94a1496ff9e5b04287e2e96a3459fb1b3b09810f19a07abc7807b1b.jpg",
+        "image_caption": [
+            "Figure 4: Learning curves for 2D Moving Bandits and Four Rooms. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "After training, MLSH learns sub-policies corresponding to movement towards each potential goal point. Training a master policy is faster than training a single policy from scratch, as we are tasked only with discovering the correct goal, rather than also learning primitive movement. Learning a shared policy, on the other hand, results in an agent that always moves towards a certain goal point, ignoring the other and thereby cutting expected reward by half. We additionally compare to an $\\mathtt { R L } ^ { 2 }$ policy (Duan et al., 2016), which encounters the same problem as the shared policy and ignores one of the goal points. ",
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+        "image_caption": [
+            "Figure 5: Top: Ant Twowalk. Ant must maneuver towards red goal point, either towards the top or towards the right. Bottom Left: Walking. Humanoid must move horizontally while maintaining an upright stance. Bottom Right: Crawling. Humanoid must move horizontally while a height-limiting obstacle is present. "
+        ],
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+    {
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "We perform several ablation tests within the 2D moving bandits task. Removing the warmup period results in an MLSH agent which at first has both sub-policies moving to the same goal point, but gradually shifts one sub-policy towards the other point. Running the master policy on the same timescale as the sub-policies results in similar behavior to simply learning a shared policy, showing that the temporal extension of sub-policies is key. Finally, we run a hyperparameter comparison to test the influence of the sub-policy count and warmup duration. ",
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+    {
+        "type": "text",
+        "text": "6.3 HOW DOES MLSH COMPARE TO PAST METHODS IN THE HIERARCHICAL DOMAIN? ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To compare to past methods, we consider the four-rooms domain described in Sutton et al. (1999) and expanded in Option Critic (Bacon et al., 2016). The agent starts at a specific spot in the gridworld, and is randomly assigned a goal position. A reward of 1 is awarded for being in the goal state. Episodes last for 100 timesteps, and master policy actions last for 25. We utilize four sub-policies, a warmup time of 20, and a joint-update time of 30. ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "First, we repeatedly train MLSH and Option Critic on many random goals in the four-rooms domain, until reward stops improving. Then, we sample an unseen goal position and fine-tune. We compare against baselines of training a single policy from scratch, using PPO against MLSH, and Actor Critic against Option Critic. In Figure 4, while Option Critic performs similarly to its baseline, we can see MLSH reach high reward faster than the PPO baseline. It is worth noting that when fine-tuning, the PPO baseline naturally reaches more stable reward than Actor Critic, so we do not compare MLSH and Option Critic directly. ",
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+    {
+        "type": "text",
+        "text": "6.4 IS THE MLSH FRAMEWORK SAMPLE-EFFICIENT ENOUGH TO LEARN DIVERSE SUB-POLICIES IN PHYSICS ENVIRONMENTS? ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To test the scalability of the MLSH algorithm, we present a series of physics-based tasks which we describe below, all which are simulated through Mujoco (Todorov et al., 2012). Diverse subpolicies are naturally discovered, as shown in Figure 5 and Figure 6. Episodes last 1000 timesteps, and master policy actions last 200. We use a warmup time of 20, and a joint-update time of 40. ",
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "In the Twowalk tasks, we would like to examine if simulated robots can learn directional movement primitives. We test performance on a standard simulated four-legged ant, and use a sub-policy count of two. A destination point is placed in either the top edge of the world or the right edge of the world. Reward is given based on negative distance to this destination point. ",
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+    {
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+        "img_path": "images/2a64ad4a78b48864dcc90ca1a8b516b9d3d78be1248e62d9351e77313e190412.jpg",
+        "image_caption": [
+            "Figure 6: Top: Distribution of mazes. Red blocks are impassable tiles, and green blocks represent the goal. Bottom: Sub-policies learned from mazes to move up, right, and down. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "In addition, we would like to determine if diverse sub-policies can be automatically discovered solely through interaction with the environment. We present a task where Ant robots must move to destination points in a set of mazes (Figure 6). Without human supervision, Ant robots are able to learn directional movement sub-policies in three directions, and use them in combination to solve the mazes. ",
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "In the Walk/Crawl task, we would like to determine if Humanoid robots can learn a variety of movement styles. Out of two possible locomotion objectives, one is randomly selected. In the first objective, the agent must move forwards while maintaining an upright stance. This was designed with a walking behavior in mind. In the second objective, the agent must move backwards underneath an obstacle limiting vertical height. This was designed to encourage a crawling behavior. ",
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "Additionally, we test the transfer capabilities of sub-policies trained in the Walk/Crawl task by introducing an unseen combination task. The Humanoid agent must first walk forwards until a certain distance, at which point it must switch movements, turn around, and crawl backwards under an obstacle. ",
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+    {
+        "type": "table",
+        "img_path": "images/4e5f814def5500512d6421614287cca5caa8fc6ecb947f64a5b8a5235555148b.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Reward on Walk/Crawl combination task</td></tr><tr><td>MLSHTransfer Shared Policy Transfer Single Policy</td><td>14333 6055 -643</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "On both Twowalk and Walk/Crawl tasks, MLSH significantly outperforms baselines, displaying scalability into complex physics domains. Ant robots learn temporally-extended directional movement primitives that lead to efficient exploration of mazes. In addition, we successfully discover diverse Humanoid sub-policies for both walking and crawling. ",
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "6.5 CAN SUB-POLICIES BE USED TO LEARN IN AN OTHERWISE UNSOLVABLE SPARSE PHYSICS ENVIRONMENT? ",
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "Finally, we present a complex task that is unsolvable with naive PPO. The agent controls an Ant robot which has been placed into an obstacle course. The agent must navigate from the bottom-left corner to the top-right corner, to receive a reward of 1. In all other cases, the agent receives a reward of 0. Along the way, there are obstacles such as walls and a chasing enemy. We periodically reset the joints of the Ant robot to prevent it from falling over. An episode lasts for 2000 timesteps, and master policy actions last 200. To solve this task, we use sub-policies learned in the Ant Twowalk tasks. We then fine-tune the master policy on the obstacle course task. ",
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+    {
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+        "img_path": "images/c45bb033ea0100b1e9b88a1ffab54662266208886df4ab8dd0b513236601577e.jpg",
+        "image_caption": [
+            "Figure 7: Learning curves for Twowalk and Walk/Crawl tasks "
+        ],
+        "image_footnote": [],
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+        "page_idx": 8
+    },
+    {
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+        "img_path": "images/6ee2391f0ebd27ee68dc375118bbb0d74cc560b512c4cfe6455cc3b3dba69bb9.jpg",
+        "image_caption": [
+            "Figure 8: Ant Obstacle course task. Agent must navigate to the green square in the top right corner. Entering the red circle causes an enemy to attack the agent, knocking it back. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
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+    {
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "In the sparse reward setting, naive PPO cannot learn, as exploration over the space of primitive action sequences is unlikely to result in reward signal. On the other hand, MLSH allows for exploration over the space of sub-policies, where it is easier to discover a sequence that leads to reward. ",
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+        "img_path": "images/eda2d50b73426dfdf2a6ebecb63aca7cbd8b6602fd23b9f95e7af28b93e67b4f.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Reward on Ant Obstacle task</td></tr><tr><td>MLSHTransfer Single Policy</td><td>193 0</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "7 DISCUSSION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this work, we formulate an approach for the end-to-end metalearning of hierarchical policies. We present a model for representing shared information as a set of sub-policies. We then provide a framework for training these models over distributions of environments. Even though we do not optimize towards the true objective, we achieve significant speedups in learning. In addition, we naturally discover diverse sub-policies without the need for hand engineering. ",
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+    {
+        "type": "text",
+        "text": "7.1 FUTURE WORK ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As there is no gradient signal being passed between the master and sub-policies, the MLSH model utilizes hard one-hot communication, as opposed to methods such as Gumbel-Softmax (Jang et al., 2016). This lack of a gradient also allows MLSH to be learning-method agnostic. While we used policy gradients in our experiments, it is entirely feasible to have the master or sub-policies be trained with evolution (Eigen) or Q-learning (Watkins & Dayan, 1992). ",
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+    {
+        "type": "text",
+        "text": "From another point of view, our training framework can be seen as a method of joint optimization over two sets of parameters. This framework can be applied to other scenarios than learning subpolicies. For example, distributions of tasks with similar observation distributions but different reward functions could be solved with a shared observational network, while learning independent policies. ",
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+    },
+    {
+        "type": "text",
+        "text": "This work draws inspiration from the domains of both hierarchical reinforcement learning and metalearning, the intersection at which architecture space has yet to be explored. For example, the set of sub-policies could be condensed into a single neural network, which receives a continuous vector from the master policy. If sample efficiency issues are addressed, several approximations in the MLSH method could be removed for a more unbiased estimator – such as training $\\phi$ to maximize reward on the entire $T$ -timesteps, rather than on a single episode. We believe this work opens up many directions in training agents that can quickly adapt to new tasks. ",
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+# DECOUPLING REPRESENTATION LEARNING FROM REINFORCEMENT LEARNING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+In an effort to overcome limitations of reward-driven feature learning in deep reinforcement learning (RL) from images, we propose decoupling representation learning from policy learning. To this end, we introduce a new unsupervised learning (UL) task, called Augmented Temporal Contrast (ATC), which trains a convolutional encoder to associate pairs of observations separated by a short time difference, under image augmentations and using a contrastive loss. In online RL experiments, we show that training the encoder exclusively using ATC matches or outperforms end-to-end RL in most environments. Additionally, we benchmark several leading UL algorithms by pre-training encoders on expert demonstrations and using them, with weights frozen, in RL agents; we find that agents using ATC-trained encoders outperform all others. We also train multi-task encoders on data from multiple environments and show generalization to different downstream RL tasks. Finally, we ablate components of ATC, and introduce a new data augmentation to enable replay of (compressed) latent images from pre-trained encoders when RL requires augmentation. Our experiments span visually diverse RL benchmarks in DeepMind Control, DeepMind Lab, and Atari, and our complete code is available at hiddenurl.
+
+# 1 INTRODUCTION
+
+Ever since the first fully-learned approach succeeded at playing Atari games from screen images (Mnih et al., 2015), standard practice in deep reinforcement learning (RL) has been to learn visual features and a control policy jointly, end-to-end. Several such deep RL algorithms have matured (Hessel et al., 2018; Schulman et al., 2017; Mnih et al., 2016; Haarnoja et al., 2018) and have been successfully applied to domains ranging from real-world (Levine et al., 2016; Kalashnikov et al., 2018) and simulated robotics (Lee et al., 2019; Laskin et al., 2020a; Hafner et al., 2020) to sophisticated video games (Berner et al., 2019; Jaderberg et al., 2019), and even high-fidelity driving simulators (Dosovitskiy et al., 2017). While the simplicity of end-to-end methods is appealing, relying on the reward function to learn visual features can be severely limiting. For example, it leaves features difficult to acquire under sparse rewards, and it can narrow their utility to a single task. Although our intent is broader than to focus on either sparse-reward or multi-task settings, they arise naturally in our studies. We investigate how to learn visual representations which are agnostic to rewards, without degrading the control policy.
+
+A number of recent works have significantly improved RL performance by introducing auxiliary losses, which are unsupervised tasks that provide feature-learning signal to the convolution neural network (CNN) encoder, additionally to the RL loss (Jaderberg et al., 2017; van den Oord et al., 2018; Laskin et al., 2020b; Guo et al., 2020; Schwarzer et al., 2020). Meanwhile, in the field of computer vision, recent efforts in unsupervised and self-supervised learning (Chen et al., 2020; Grill et al., 2020; He et al., 2019) have demonstrated that powerful feature extractors can be learned without labels, as evidenced by their usefulness for downstream tasks such as ImageNet classification. Together, these advances suggest that visual features for RL could possibly be learned entirely without rewards, which would grant greater flexibility to improve overall learning performance. To our knowledge, however, no single unsupervised learning (UL) task has been shown adequate for this purpose in general vision-based environments.
+
+In this paper, we demonstrate the first decoupling of representation learning from reinforcement learning that performs as well as or better than end-to-end RL. We update the encoder weights using only UL and train a control policy independently, on the (compressed) latent images. This capability stands in contrast to previous state-of-the-art methods, which have trained the UL and RL objectives jointly, or Laskin et al. (2020b), which observed diminished performance with decoupled encoders.
+
+Our main enabling contribution is a new unsupervised task tailored to reinforcement learning, which we call Augmented Temporal Contrast (ATC). ATC requires a model to associate observations from nearby time steps within the same trajectory (Anand et al., 2019). Observations are encoded via a convolutional neural network (shared with the RL agent) into a small latent space, where the InfoNCE loss is applied (van den Oord et al., 2018). Within each randomly sampled training batch, the positive observation, $o _ { t + k }$ , for every anchor, $o _ { t }$ , serves as negative for all other anchors. For regularization, observations undergo stochastic data augmentation (Laskin et al., 2020b) prior to encoding, namely random shift (Kostrikov et al., 2020), and a momentum encoder (He et al., 2020; Laskin et al., 2020b) is used to process the positives. A learned predictor layer further processes the anchor code (Grill et al., 2020; Chen et al., 2020) prior to contrasting. In summary, our algorithm is a novel combination of elements that enables generic learning of the structure of observations and transitions in MDPs without requiring rewards or actions as input.
+
+We include extensive experimental studies establishing the effectiveness of our algorithm in a visually diverse range of common RL environments: DeepMind Control Suite (DMControl; Tassa et al. 2018), DeepMind Lab (DMLab; Beattie et al. 2016), and Atari (Bellemare et al., 2013). Our experiments span discrete and continuous control, 2D and 3D visuals, and both on-policy and off policy RL algorithms. Complete code for all of our experiments is available at hiddenurl. Our empirical contributions are summarized as follows:
+
+Online RL with UL: We find that the convolutional encoder trained solely with the unsupervised ATC objective can fully replace the end-to-end RL encoder without degrading policy performance. ATC achieves nearly equal or greater performance in all DMControl and DMLab environments tested and in 5 of the 8 Atari games tested. In the other 3 Atari games, using ATC as an auxiliary loss or for weight initialization still brings improvements over end-to-end RL.
+
+Encoder Pre-Training Benchmarks: We pre-train the convolutional encoder to convergence on expert demonstrations, and evaluate it by training an RL agent using the encoder with weights frozen. We find that ATC matches or outperforms all prior UL algorithms as tested across all domains, demonstrating that ATC is a state-of-the-art UL algorithm for RL.
+
+Multi-Task Encoders: An encoder is trained on demonstrations from multiple environments, and is evaluated, with weights frozen, in separate downstream RL agents. A single encoder trained on four DMControl environments generalizes successfully, performing equal or better than end-to-end RL in four held-out environments. Similar attempts to generalize across eight diverse Atari games result in mixed performance, confirming some limited feature sharing among games.
+
+Ablations and Encoder Analysis: Components of ATC are ablated, showing their individual effects. Additionally, data augmentation is shown to be necessary in DMControl during RL even when using a frozen encoder. We introduce a new augmentation, subpixel random shift, which matches performance while augmenting the latent images, unlocking computation and memory benefits.
+
+# 2 RELATED WORK
+
+Several recent works have used unsupervised/self-supervised representation learning methods to improve performance in RL. The UNREAL agent (Jaderberg et al., 2017) introduced unsupervised auxiliary tasks to deep RL, including the Pixel Control task, a Q-learning method requiring predictions of screen changes in discrete control environments, which has become a standard in DMLab (Hessel et al., 2019). CPC (van den Oord et al., 2018) applied contrastive losses over multiple time steps as an auxiliary task for the convolutional and recurrent layers of RL agents, and it has been extended with future action-conditioning (Guo et al., 2018). Recently, PBL (Guo et al., 2020) surpassed these methods with an auxiliary loss of forward and backward predictions in the recurrent latent space using partial agent histories. Where the trend is of increasing sophistication in auxiliary recurrent architectures, our algorithm is markedly simpler, requiring only observations, and yet it proves sufficient in partially observed settings (POMDPs).
+
+ST-DIM (Anand et al., 2019) introduced various temporal, contrastive losses, including ones that operate on “local” features from an intermediate layer within the encoder, without data augmentation. CURL (Laskin et al., 2020b) introduced an augmented, contrastive auxiliary task similar to ours, including a momentum encoder but without temporal contrast. Mazoure et al. (2020) provided extensive analysis pertaining to InfoNCE losses on functions of successive time steps in MDPs, including local features in their auxiliary loss (DRIML) similar to ST-DIM, and finally conducted experiments using global temporal contrast of augmented observations in the Procgen (Cobbe et al., 2019) environment. Most recently, MPR (Schwarzer et al., 2020) combined data augmentation with multi-step, convolutional forward modeling and a similarity loss to improve DQN agents in the Atari 100k benchmark. Hafner et al. (2019; 2020); Lee et al. (2019) proposed to leverage world-modeling in a latent-space for continuous control. A small number of model-free methods have attempted to decouple encoder training from the RL loss as ablations, but have met reduced performance relative to end-to-end RL (Laskin et al., 2020b; Lee et al., 2020). None have previously been shown effective in as diverse a collection of RL environments as ours (Bellemare et al., 2013; Tassa et al., 2018; Beattie et al., 2016).
+
+Finn et al. (2016); Ha & Schmidhuber (2018) are example works which pretrained encoder features in advance using image reconstruction losses such as the VAE (Kingma & Welling, 2013). Devin et al. (2018); Kipf et al. (2019) pretrained object-centric representations, the latter learning a forward model by way of contrastive losses; Yan et al. (2020) introduced a similar technique to learn encoders supporting manipulation of deformable objects by traditional control methods. MERLIN (Wayne et al., 2018) trained a convolutional encoder and sophisticated memory module online, detached from the RL agent, which learned read-only accesses to memory. It used reconstruction and one-step latent-prediction losses and achieved high performance in DMLab-like environments with extreme partial observability. Our loss function may benefit those settings, as it outperforms similar reconstruction losses in our experiments. Decoupling unsupervised pretraining from downstream tasks is common in computer vision (Henaff et al., 2019; He et al., 2019; Chen et al., 2020) and has ´ favorable properties of providing task agnostic features which can be used for training smaller taskspecific networks, yielding significant gains in computational efficiency over end-to-end methods.
+
+# 3 AUGMENTED TEMPORAL CONTRAST
+
+Our unsupervised learning task, Augmented Temporal Contrast (ATC), requires a model to associate an observation, $o _ { t }$ , with one from a specified, near-future time step, $o _ { t + k }$ . Within each training batch, we apply stochastic data augmentation to the observations (Laskin et al., 2020b), namely random shift (Kostrikov et al., 2020), which is simple to implement and provides highly effective regularization in most cases. The augmented observations are encoded into a small latent space where a contrastive loss is applied. This task encourages the learned encoder to extract meaningful elements of the structure of the MDP from observations.
+
+Our architecture for ATC consists of four learned components - (i) a convolutional encoder, $f _ { \theta }$ , which processes the anchor observation, $o _ { t }$ , into the latent image $z _ { t } ~ = ~ f _ { \theta } ( \operatorname { A U G } ( o _ { t } ) )$ , (ii) a linear global compressor, $g _ { \phi }$ to produce a small latent code vector $c _ { t } = g _ { \phi } ( z _ { t } )$ , (iii) a residual predictor MLP, $h _ { \psi }$ , which acts as an implicit forward model to advance the code $p _ { t } ~ = ~ h _ { \psi } ( c _ { t } ) + c _ { t }$ and (iv) a contrastive transformation matrix, $W$ . To process the positive observation, $o _ { t + k }$ into the target code $\bar { c } _ { t \pm k } = \bar { g _ { \bar { \phi } } } ( f _ { \bar { \theta } } ( \operatorname { A U G } ( o _ { t + k } ) )$ , we use a momentum encoder (He et al., 2019) parameterized as a slowly moving average of the weights from the learned encoder and compressor layer:
+
+![](images/baabe3a71bc43acb562a542e60773e7f84f3eea98027ab6fe84d45f9966eb5b7.jpg)  
+Figure 1: Augmented Temporal Contrast— augmented observations are processed through a learned encoder $f _ { \theta }$ , compressor, $g _ { \phi }$ and residual predictor $h _ { \psi }$ , and are associated through a contrastive loss with a positive example from $k$ time steps later, processed through a momentum encoder.
+
+$$
+\bar { \theta }  ( 1 - \tau ) \bar { \theta } + \tau \theta ; \qquad \bar { \phi }  ( 1 - \tau ) \bar { \phi } + \tau \phi .
+$$
+
+The complete architecture is shown in Figure 1. The convolutional encoder, $f _ { \theta }$ , alone is shared with the RL agent.
+
+We employ the InfoNCE loss (Gutmann & Hyvarinen, 2010; van den Oord et al., 2018) using log- ¨ its computed bilinearly, as $l ~ = ~ p _ { t } W \bar { c } _ { t + k }$ . In our implementation, every anchor in the training batch utilizes the positives corresponding to all other anchors as its negative examples. Denoting an observation indexed from dataset $\mathcal { O }$ as $o _ { i }$ , and its positive as $o _ { i + }$ , the logits can be written as $l _ { i , j + } = p _ { i } W \bar { c } _ { j + }$ ; our loss function in practice is:
+
+$$
+\mathcal { L } ^ { A T C } = - \mathbb { E } _ { \mathcal { O } } \left[ \log \frac { \exp l _ { i , i + } } { \sum _ { o _ { j } \in \mathcal { O } } \exp l _ { i , j + } } \right] .
+$$
+
+# 4 EXPERIMENTS
+
+# 4.1 EVALUATION ENVIRONMENTS AND ALGORITHMS
+
+We evaluate ATC on three standard, visually diverse RL benchmarks - the DeepMind control suite (DMControl; Tassa et al. 2018), Atari games in the Arcade Learning Environment (Bellemare et al., 2013), and DeepMind Lab (DMLab; Beattie et al. 2016). Atari requires discrete control in arcadestyle games. DMControl is comprised of continuous control robotic locomotion and manipulation tasks. In contrast, DMLab requries the RL agent to reason in more visually complex 3D maze environments with partial observability.
+
+We use ATC to enhance both on-policy and off-policy RL algorithms. For DMControl, we use RADSAC (Laskin et al., 2020a; Haarnoja et al., 2018) with the augmentation of Kostrikov et al. (2020), which randomly shifts the image in each coordinate (by up to 4 pixels), replicating edge pixel values as necessary to restore the original image size. A difference from prior work is that we use more downsampling in our convolutional network, by using strides $( 2 , 2 , 2 , 1 )$ instead of $( 2 , 1 , 1 , 1 )$ to reduce the convolution output image by $2 5 \mathrm { x }$ .1 For both Atari and DMLab, we use PPO (Schulman et al., 2017). In Atari, we use feed-forward agents, sticky actions, and no end-of-life boundaries for RL episodes. In DMLab we used recurrent, LSTM agents receiving only a single time-step image input, the four-layer convolution encoder from Jaderberg et al. (2019), and we tuned the entropy bonus for each level. In the online setting, the ATC loss is trained using small replay buffer of recent experiences.
+
+We include all our own baselines for fair comparison and provide complete settings in an appendix. Unless otherwise noted, each curve represents a minimum of 3 random seeds. The bold lines show the average, and the lightly shaded area around each curve represents the maximum extent of the best and worst seeds at each checkpoint.
+
+# 4.2 ONLINE RL WITH ATC
+
+DMControl In the online setting, we found ATC to be capable of training the encoder by itself (i.e., with encoder fully detached from any RL gradient update), achieving essentially equal or better scores versus end-to-end RL in all six environments we tested, Figure 2. In CARTPOLE-SWINGUPSPARSE, where rewards are only received once the pole reaches vertical, ATC training enabled the agent to master the task significantly faster. The encoder is trained with one update for every RL update to the policy, using the same batch size, except in CHEETAH-RUN, which required twice the ATC updates.
+
+DMLab We experimented with two kinds of levels in DMLab: EXPLORE GOAL LOCATIONS, which requires repeatedly navigating a maze whose layout is randomized every episode, and LASERTAG THREE OPPONENTS, which requires fast reflexes to pursue and tag enemies at a distance. We found ATC capable of training fully detached encoders while achieving equal or better performance than end-to-end RL. Results are shown in Figure 3. Both environments exhibit sparsity which is greater in the “large” version than the “small” version, which our algorithm addresses, discussed next.
+
+![](images/3bc961b4e4d3021562434fb49dfc796b984802350f35775720c5fb85ddde2e02.jpg)  
+Figure 2: Online encoder training by ATC, fully detached from RL training, performs as well as end-to-end RL in DMControl, and better in sparse-reward environments (environment steps shown, see appendix for action repeats). Each curve is 10 random seeds.
+
+In EXPLORE, the goal object is rarely seen, especially early on, making its appearance difficult to learn. We therefore introduced prioritized sampling for ATC , with priorities corresponding to empirical absolute returns: $p \propto 1 + R _ { a b s }$ , where $\begin{array} { r } { R _ { a b s } ^ { - } = \sum _ { t = 0 } ^ { n } \gamma ^ { t } | r _ { t } | } \end{array}$ , to train more frequently on more informative scenes.2 Whereas uniform-ATC performs slightly below RL, uniform-ATC outperforms RL and nearly matches using ATC (uniform) as an auxiliary task. By considering the encoder as a stand-alone feature extractor separate from the policy, no importance sampling correction is required.
+
+In LASERTAG, enemies are often seen, but the reward of tagging one is rarely achieved by the random agent. ATC learns the relevant features anyway, boosting performance while the RL-only agent remains at zero average score. We found that increasing the rate of UL training to do twice as many updates3 further improved the score to match the ATC-auxiliary agent, showing flexibility to address the representation-learning bottleneck when opponents are dispersed.
+
+![](images/529869cdc0639395a9c3ea1832e5de7f047624be22923583fb22ece1a3bdf01b.jpg)  
+Figure 3: Online encoder training by ATC, fully detached from the RL agent, performs as well or better than end-to-end RL in DMLab (1 agent step $= 4$ environment steps, the standard action repeat). Prioritized ATC replay (EXPLORE) or increased ATC training (LASERTAG) addresses sparsities to nearly match performance of RL with ATC as an auxiliary loss $( \mathrm { R L + A T C }$ ). Each curve is 3 random seeds.
+
+Atari We tested a diverse subset of eight Atari games, shown in Figure 4. We found detachedencoder training to work as well as end-to-end RL in five games, but performance suffered in BREAKOUT and SPACE INVADERS in particular. Using ATC as an auxiliary task, however, improves performance in these games and others. We found it helpful to anneal the amount of UL training over the course of RL in Atari (details in an appendix). Notably, we found several games, including SPACE INVADERS, to benefit from using ATC only to initialize encoder weights, done using an initial $1 0 0 \mathrm { k }$ transitions gathered with a uniform random policy. Some of our remaining experiments provide more insights into the challenges of this domain.
+
+# 4.3 ENCODER PRE-TRAINING BENCHMARKS
+
+To benchmark the effectiveness of different UL algorithms for RL, we propose a new evaluation methodology that is similar to how UL pre-training techniques are measured in computer vision (see e.g. Chen et al. (2020); Grill et al. (2020)): (i) collect a data set composed of expert demonstrations from each environment; (ii) pre-train the CNN encoder with that data offline using UL; (iii) evaluate by using RL to learn a control policy while keeping the encoder weights frozen. This procedure isolates the asymptotic performance of each UL algorithm for RL. For convenience, we drew expert demonstrations from partially-trained RL agents, and every UL algorithm trained on the same data set for each environment. Our RL agents used the same post-encoder architectures as in the online experiments. Further details about pre-training by each algorithm are provided in an appendix.
+
+![](images/2cff74d17a83f427f4f8908036f45a1088f8557b69b4c13b3cf4d4bcf449ff73.jpg)  
+Figure 4: Online encoder training using ATC, fully detached from the RL agent, works well in 5 of 8 games tested (1 agent step $= 4$ environment steps, the standard action repeat). 6 of 8 games benefit significantly from using ATC as an auxiliary loss or for weight initialization. Each curve is 8 random seeds.
+
+DMControl We compare ATC against two competing algorithms: Augmented Contrast (AC), from CURL (Laskin et al., 2020b), which uses the same observation for the anchor and the positive, and a VAE (Kingma & Welling, 2013), for which we found better performance by introducing a time delay to the target observation (VAE-T). We found ATC to match or outperform the other algorithms, in all four test environments, as shown in Figure 5. Further, ATC is the only one to match or outperform the reference end-to-end RL across all cases.
+
+![](images/f309dd47f7083677e9f184338a22014aa84f7b0a85c672257f0eb8412dcdd8f0.jpg)  
+Figure 5: RL in DMControl, using encoders pre-trained on expert demonstrations using UL, with weights frozen—across all domains, ATC outperforms prior methods and the end-to-end RL reference. Each curve is a mininum of 4 random seeds.
+
+DMLab We compare against both Pixel Control (Jaderberg et al., 2017; Hessel et al., 2019) and CPC (van den Oord et al., 2018), which have been shown to bring strong benefits in DMLab. While all algorithms perform similarly well in EXPLORE, ATC performs significantly better in LASERTAG, Figure 6. Our algorithm is simpler than Pixel Control and CPC in the sense that it uses neither actions, deconvolution, nor recurrence.
+
+Atari We compare against Pixel Control, VAE-T, and a basic inverse model which predicts actions between pairs of observations. We also compare against Spatio-Temporal Deep InfoMax (ST-DIM), which uses temporal contrastive losses with “local” features from an intermediate convolution layer to ensure attention to the whole screen; it was shown to produce detailed game-state knowledge when applied to individual frames (Anand et al., 2019). Of the four games shown in Figure 7, ATC is the only UL algorithm to match the end-to-end RL reference in GRAVITAR and BREAKOUT, and it performs best in SPACE INVADERS.
+
+![](images/3154101adf4b820cc0467150274febe9fb7d21036dc97115152835c455f59101.jpg)  
+Figure 6: RL in DMLab, using pre-trained encoders with weights frozen–in LASERTAG especially, ATC outperforms leading prior UL algorithms.
+
+![](images/121138f0c210cac9a3c4cfd52b0433a919852489cef2654c02ed829c55b70b77.jpg)  
+Figure 7: RL in Atari, using pre-trained encoders with weights frozen—ATC outperforms several leading, prior UL algorithms and exceeds the end-to-end RL reference in 3 of the 4 games tested.
+
+# 4.4 MULTI-TASK ENCODERS
+
+In the offline setting, we conducted initial explorations into the capability of ATC to learn multi-task encoders, simply by pre-training on demonstrations from multiple environments. We evaluate the encoder by using it, with frozen weights, in separate RL agents learning each downstream task.
+
+DMControl Figure 8 shows our results in DMControl, where we pretrained using only the four environments in the top row. Although the encoder was never trained on the HOPPER, PENDULUM, nor FINGER domains, the multi-task encoder supports efficient RL in them. PENDULUM-SWINGUP and CARTPOLE-SWINGUP-SPARSE stand out as challenging environments which benefited from cross-domain and cross-task pre-training, respectively. The pretraining was remarkably efficient, requiring only 20,000 updates to the encoder.
+
+![](images/a1a082ebecffa5389e63c77f1ace713a46a8b37259167a313d56f09aba23a920.jpg)  
+Figure 8: Separate RL agents using a single encoder with weights frozen after pre-training on expert demonstrations from the four top environments. The encoder generalizes to four new environments, bottom row, where sparse reward tasks especially benefit from the transfer. Each curve is minimum 4 random seeds.
+
+Atari Atari proved a more challenging domain for learning multi-task encoders. Learning all eight games together in Figure 11, in the appendix, resulted in diminished performance relative to single-game pretraining in three of the eight. The decrease was partially alleviated by widening the encoder with twice as many filters per layer, indicating that representation capacity is a limiting factor. To test generalization, we conducted a seven-game pre-training experiment where we test the encoder on the held-out game. Most games suffered diminished performance (although still perform significantly higher than a frozen random encoder), confirming the limited extent to which visual features transfer across these games.
+
+![](images/01ef7105d2680afb75ad5c5c4eed915c080de929140a61da789373eff80f213c.jpg)  
+Figure 9: BREAKOUT benefits from contrasting against negatives from several neighboring time steps.
+
+![](images/539e70000e4ecf18d2eba89d8be66353cc7947a0d1d8196f6ef9391f02b75278.jpg)  
+Figure 10: An example scene from BREAKOUT, where a lowperformance UL encoder (without shift) focuses on the paddle. Introducing random shift and sequence data makes the highperformance UL encoder (full ATC) focus near the ball, as does the encoder from a fully-trained, end-to-end RL agent.
+
+# 4.5 ABLATIONS AND ENCODER ANALYSIS
+
+Random Shift in ATC In offline experiments, we discovered random shift augmentations to be helpful in all domains. To our knowledge, this is the first application of random shift to 3D visual environments as in DMLab. In Atari, we found performance in GRAVITAR to suffer from random shift, but reducing the probability of applying random shift to each observation from 1.0 to 0.1 alleviated the effect while still bringing benefits in other games, so we used this setting in our main experiments. Results are shown in Figure 12 in an appendix.
+
+Random Shift in RL In DMControl, we found the best results when using random shift during RL, even when training with a frozen encoder. This is evidence that the augmentation regularizes not only the representation but also the policy, which first processes the latent image into a 50- dimensional vector. To unlock computation and memory benefits of replaying only the latent images for the RL agent, we attempted to apply data augmentation to the latent image. But we found the smallest possible random shifts to be too extreme. Instead, we introduce a new augmentation, subpixel random shift, which linearly interpolates among neighboring pixels. As shown in Figure 13 in the appendix, this augmentation restores performance when applied to the latent images, allowing a pre-trained encoder to be entirely bypassed during policy training updates.
+
+Temporal Contrast on Sequences In BREAKOUT alone, we discovered that composing the UL training batch of trajectory segments, rather than individual transitions, gave a significant benefit. Treating all elements of the training batch independently provides “hard” negatives, since the encoder must distinguish between neighboring time steps. This setting had no effect in the other Atari games tested, and we found equal or better performance using individual transitions in DMControl and DMLab. Figure 9 further shows that using a similarity loss (Grill et al., 2020) does not capture the benefit.
+
+Encoder Analysis We analyzed the learned encoders in BREAKOUT to further study this ablation effect. Similar to Zagoruyko & Komodakis (2016), we compute spatial attention maps by mean-pooling the absolute values of the activations along the channel dimension and follow with a 2-dimensional spatial softmax. Figure 10 shows the attention of four different encoders on the displayed scene. The poorly performing UL encoder heavily utilizes the paddle to distinguish the observation. The UL encoder trained with random shift and sequence data, however, focuses near the ball, as does the fully-trained RL encoder. (The random encoder mostly highlights the bricks, which are less relevant for control.) In an appendix, we include other example encoder analyses from Atari and DMLab which show ATC-trained encoders attending only to key objects on the game screen, while RL-trained encoders additionally attend to potentially distracting features such as game score.
+
+# 5 CONCLUSION
+
+Reward-free representation learning from images provides flexibility and insights for improving deep RL agents. We have shown a broad range of cases where our new unsupervised learning algorithm can fully replace RL for training convolutional encoders while maintaining or improving online performance. In a small number of environments–a few of the Atari games–including the RL loss for encoder training still surpasses our UL-only method, leaving opportunities for further improvements in UL for RL.
+
+Our preliminary efforts to use actions as inputs (into the predictor MLP) or as prediction outputs (inverse loss) with ATC did not immediately yield strong improvements. We experimented only with random shift, but other augmentations may be useful, as well. In multi-task encoder training, our technique avoids any need for sophisticated reward-balancing (Hessel et al., 2019), but more advanced training methods may still help when the required features are in conflict, as in Atari, or if they otherwise impact our loss function unequally. On the theoretical side, it may be helpful to analyze the effects of domain shift on the policy when a detached representation is learned online.
+
+One obvious application of our offline methodology would be in the batch RL setting, where the agent learns from a fixed data set. Our offline experiments showed that a relatively small number of transitions are sufficient to learn rich representations by UL, and the lower limit could be further explored. Overall, we hope that our algorithm and experiments spur further developments leveraging unsupervised learning for reinforcement learning.
+
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+
+A APPENDIX
+
+# A.1 ALGORITHMS
+
+# Algorithm 1 Online RL with decoupled ATC encoder (steps distinct from end-to-end RL in blue)
+
+Require: θAT C, φπ . ATC model parameters (encoder $f _ { \theta }$ thru contrast $W$ ), policy parameters   
+1: $\mathcal { S }  \{ \}$ $\triangleright$ replay buffer of observations   
+2: $\bar { \theta } _ { A T C }  \theta _ { A T C }$ . initialize momentum encoder (conv and linear only)   
+3: repeat   
+4: Sample environment and policy, through encoder:   
+5: for 1 to m do . a minibatch   
+6: $\begin{array} { l } { a \sim { \pi } ( \cdot | f _ { \theta } ( s ) ; \phi ) , { s } ^ { \prime } \sim T ( s , a ) , r \sim R ( s , a , { s } ^ { \prime } ) } \\ { \quad S  S \cup \{ s \} } \\ { \quad s  s ^ { \prime } } \end{array}$   
+7: store observations (delete oldest if full)   
+8:   
+9: end for   
+10: Update policy by given RL formula: $\triangleright$ on- or off-policy   
+11: for 1 to n do $\triangleright$ given number RL updates per minibatch   
+12: $\phi _ { \pi }  \phi _ { \pi } + R L ( s , a , s ^ { \prime } , r ; \phi _ { \pi } )$ . stop gradient into encoder   
+13: end for   
+14: Update encoder (and contrastive model) by ATC:   
+15: for 1 to p do   
+16: 17: $\begin{array} { r l } & { s , s _ { + } \sim s } \\ & { \underline { { \theta } } _ { A T C }  \theta _ { A T C } - \lambda _ { A T C } \nabla _ { \theta _ { A T C } } \mathcal { L } ^ { A T C } ( s , s _ { + } ) } \end{array}$ sample observations: anchors and positives $\triangleright$ ATC gradient update   
+18: $\bar { \theta } _ { A T C }  ( 1 - \tau ) \bar { \theta } _ { A T C } + \tau \theta _ { A T C }$ . update momentum encoder (conv and linear only)   
+19: end for   
+20: until converged   
+21: return Encoder $f _ { \theta }$ and policy $\pi _ { \phi }$
+
+# A.2 ADDITIONAL FIGURES
+
+![](images/5543152dd76206baac60db6f57cb863ab7342d883bbae0c2215217cd33cfa063.jpg)  
+Figure 11: RL using multi-task encoders (all with weights frozen) for eight Atari games gives mixed performance, partially improved by increased network capacity (8-game-wide). Training on 7 games and testing on the held-out one yields diminished but non-zero performance, showing some limited feature transfer between games.
+
+In subpixel random shift, new pixels are a linearly weighted average of the four nearest pixels to a randomly chosen coordinate location. We used uniformly random horizontal and vertical shifts, and tested maximum displacements in $( \pm ) \left\{ 0 . 1 , 0 . 2 5 , 0 . 5 , 0 . 7 5 , 1 . 0 \right\}$ pixels (with “edge” mode padding $\pm 1 )$ . We found 0.5 to work well in all tested domains, restoring the performance of raw image augmentation but eliminating convolutions entirely from the RL training updates.
+
+![](images/4beb61e4ab5090b07268c86aa788662703783a664d7d871ba041b834c06ceeac.jpg)  
+Figure 12: Random shift augmentation helps in some Atari games and hurts in others, but applying with probability 0.1 is a performant middle ground. DMLab benefits from random shift. (Offline pre-training.)
+
+![](images/29ac0be5110940df74d1d5c3a4f0e6e4139c5451731cb9c5df2f23a86dd1c6d8.jpg)  
+Figure 13: Even after pre-training encoders for DMControl using random shift, RL requires augmentation— our subpixel augmentation acts on the (compressed) latent image, permitting its use in the replay buffer.
+
+![](images/1cce8da9c9fb4f83d8b7ebde417428125f913540be195ef011cceec933f3c869.jpg)  
+Figure 14: Attention map in BREAKOUT which shows the RL-trained encoder focusing on game score, whereas UL ATC encoder focuses properly on the paddle and ball.
+
+![](images/730bece2e848d8228104749b4de1d034becb59d3c9360f699590d057e7854254.jpg)  
+Figure 15: Attention map in LASERTAG. UL encoder with pixel control focuses on the score, while UL encoder with the proposed ATC focuses properly on the coin similar to RL-trained encoder.
+
+![](images/081300ec7a0548d8be5ff478ad26357c10b99de16be803312ba29ed20f5121ec.jpg)  
+Figure 16: Attention map in the LASERTAG which shows that UL encoders focus properly on the enemy similar to RL-trained encoder.
+
+Table 1: DMControl, RAD-SAC Hyperparameters.   
+
+<table><tr><td>HYPERPARAMETER</td><td>VALUE</td></tr><tr><td>OBSERVATION RENDERING</td><td>(84,84),RGB</td></tr><tr><td>RANDOM SHIFT PAD</td><td>±4</td></tr><tr><td>REPLAY BUFFER SIZE</td><td>1e5</td></tr><tr><td>INITIAL STEPS</td><td>1e4</td></tr><tr><td>STACKED FRAMES</td><td>3</td></tr><tr><td>ACTIONREPEAT</td><td>2(FINGER,WALKER)</td></tr><tr><td></td><td>8 (CARTPOLE)</td></tr><tr><td></td><td>4 (REST)</td></tr><tr><td>OPTIMIZER</td><td>ADAM</td></tr><tr><td>(β1,β2)→(fθ,Tψ,Q)</td><td>(.9,.999)</td></tr><tr><td>(β1,β)→(a) LEARNING RATE(fθ,πψ,QΦ)</td><td>(.5,.999)</td></tr><tr><td></td><td>2e-4(CHEETAH)</td></tr><tr><td>LEARNING RATE (α)</td><td>1e-3 (REST)</td></tr><tr><td>BATCH SIZE</td><td>1e-4</td></tr><tr><td></td><td>512(CHEETAH,PENDULUM) 256 (REST)</td></tr><tr><td>Q FUNCTION EMA T</td><td>0.01</td></tr><tr><td>CRITIC TARGET UPDATE FREQ</td><td>2</td></tr><tr><td>CONVOLUTION FILTERS</td><td>[32,32,32,32]</td></tr><tr><td>CONVOLUTION STRIDES</td><td>[2,2,2,1]</td></tr><tr><td>CONVOLUTION FILTER SIZE</td><td>3</td></tr><tr><td>ENCODER EMAT</td><td>0.05</td></tr><tr><td>LATENT DIMENSION</td><td>50</td></tr><tr><td>HIDDEN UNITS (MLP)</td><td>[1024,1024]</td></tr><tr><td>DISCOUNT </td><td>.99</td></tr><tr><td>INITIAL TEMPERATURE</td><td>0.1</td></tr><tr><td></td><td></td></tr></table>
+
+Table 2: Atari, PPO Hyperparameters.   
+
+<table><tr><td>HYPERPARAMETER</td><td>VALUE</td></tr><tr><td>OBSERVATION RENDERING</td><td>(84,84), GREY</td></tr><tr><td>STACKED FRAMES</td><td>4</td></tr><tr><td>ACTIONREPEAT</td><td>4</td></tr><tr><td>OPTIMIZER</td><td>ADAM</td></tr><tr><td>LEARNING RATE</td><td>2.5e-4</td></tr><tr><td>PARALLEL ENVIRONMENTS</td><td>16</td></tr><tr><td>SAMPLING INTERVAL</td><td>128</td></tr><tr><td>LIKELIHOOD RATIO CLIP, ∈</td><td>0.1</td></tr><tr><td>PPOEPOCHS</td><td>4</td></tr><tr><td>PPOMINIBATCHES</td><td>4</td></tr><tr><td>CONVOLUTION FILTERS</td><td>[32, 64, 64]</td></tr><tr><td>CONVOLUTION FILTER SIZES</td><td>[8,4,3]</td></tr><tr><td>CONVOLUTION STRIDES</td><td>[4,2,1]</td></tr><tr><td>HIDDEN UNITS (MLP)</td><td>[512]</td></tr><tr><td>DISCOUNT γ</td><td>.99</td></tr><tr><td>GENERALIZED ADVANTAGE ESTIMATION入</td><td>0.95</td></tr><tr><td>LEARNINGRATE ANNEALING</td><td>LINEAR</td></tr><tr><td>ENTROPY BONUS COEFFICIENT</td><td>0.01</td></tr><tr><td>EPISODIC LIVES</td><td>FALSE</td></tr><tr><td>REPEAT ACTIONPROBABILITY</td><td>0.25</td></tr><tr><td>REWARD CLIPPING</td><td>±1</td></tr><tr><td>VALUELOSS COEFFICIENT</td><td>1.0</td></tr></table>
+
+Table 3: DMLab, PPO Hyperparameters.   
+
+<table><tr><td>HYPERPARAMETER</td><td>VALUE</td></tr><tr><td>OBSERVATION RENDERING</td><td>(72,96),RGB</td></tr><tr><td>STACKED FRAMES</td><td>1</td></tr><tr><td>ACTION REPEAT</td><td>4</td></tr><tr><td>OPTIMIZER</td><td>ADAM</td></tr><tr><td>LEARNING RATE</td><td>2.5e-4</td></tr><tr><td>PARALLEL ENVIRONMENTS</td><td>16</td></tr><tr><td>SAMPLING INTERVAL</td><td>128</td></tr><tr><td>LIKELIHOOD RATIO CLIP, E</td><td>0.1</td></tr><tr><td>PPO EPOCHS</td><td>1</td></tr><tr><td>PPOMINIBATCHES</td><td>2</td></tr><tr><td>CONVOLUTION FILTERS</td><td>[32,64,64,64]</td></tr><tr><td>CONVOLUTION FILTER SIZES</td><td>[8,4,3,3]</td></tr><tr><td>CONVOLUTION STRIDES</td><td>[4,2,1, 1]</td></tr><tr><td>HIDDEN UNITS (LSTM)</td><td>[256]</td></tr><tr><td>SKIP CONNECTIONS</td><td>CONV3,4; LSTM</td></tr><tr><td>DISCOUNT </td><td>.99</td></tr><tr><td>GENERALIZED ADVANTAGE ESTIMATION 入</td><td>0.97</td></tr><tr><td>LEARNING RATE ANNEALING</td><td>NONE</td></tr><tr><td>ENTROPY BONUS COEFFICIENT</td><td>0.01 (EXPLORE)</td></tr><tr><td></td><td>0.0003 (LASERTAG)</td></tr><tr><td>VALUE LOSS COEFFICIENT</td><td>0.5</td></tr></table>
+
+# A.4 ONLINE ATC SETTINGS
+
+Table 4: Common ATC Hyperparameters.   
+
+<table><tr><td>HYPERPARAMETER</td><td>VALUE</td></tr><tr><td>RANDOM SHIFT PAD</td><td>±4</td></tr><tr><td>LEARNING RATE</td><td>1e-3</td></tr><tr><td>LEARNING RATE ANNEALING</td><td>COSINE</td></tr><tr><td>TARGETUPDATE INTERVAL</td><td>1</td></tr><tr><td>TARGET UPDATE T PREDICTOR HIDDEN SIZES,h</td><td>0.01</td></tr><tr><td></td><td>[512]</td></tr><tr><td>REPLAY BUFFER SIZE</td><td>1e5</td></tr></table>
+
+Table 5: DMControl ATC Hyperparameters.   
+
+<table><tr><td>HYPERPARAMETER</td><td>VALUE</td></tr><tr><td>RANDOMSHIFTPROBABILITY</td><td>1</td></tr><tr><td>BATCH SIZE</td><td>ASRL (INDIVIDUAL OBSERVATIONS)</td></tr><tr><td>TEMPORAL SHIFT,k</td><td>1</td></tr><tr><td>MIN AGENT STEPS TO UL</td><td>1e4</td></tr><tr><td>MIN AGENT STEPS TO RL</td><td>1e4</td></tr><tr><td>UL UPDATE SCHEDULE</td><td>ASRL</td></tr><tr><td></td><td>(2X CHEETAH)</td></tr><tr><td>LATENT SIZE</td><td>128</td></tr></table>
+
+Table 6: Atari ATC Hyperparameters.   
+
+<table><tr><td>HYPERPARAMETER</td><td>VALUE</td></tr><tr><td>RANDOM SHIFTPROBABILITY</td><td>0.1</td></tr><tr><td>BATCH SIZE</td><td>512 (32 TRAJECTORIES OF 16 TIME STEPS)</td></tr><tr><td>TEMPORAL SHIFT,k</td><td>3</td></tr><tr><td>MIN AGENT STEPS TO UL</td><td>5e4</td></tr><tr><td>MIN AGENT STEPS TO RL</td><td>1e5</td></tr><tr><td>ULUPDATE SCHEDULE</td><td>ANNEALED QUADRATICALLY FROM6 PER SAMPLER ITERATION (1e4 ONCE AT 1e5 STEPS FOR WEIGHT INITIALIZATION)</td></tr><tr><td>LATENT SIZE</td><td>256</td></tr></table>
+
+Table 7: DMLab ATC Hyperparameters.   
+
+<table><tr><td>HYPERPARAMETER</td><td>VALUE</td></tr><tr><td>RANDOMSHIFTPROBABILITY</td><td>1</td></tr><tr><td>BATCH SIZE</td><td>512 (INDIVIDUAL OBSERVATIONS)</td></tr><tr><td>TEMPORAL SHIFT,k</td><td>3</td></tr><tr><td>MIN AGENT STEPS TO UL</td><td>5e4</td></tr><tr><td>MIN AGENT STEPS TO RL</td><td>1e5</td></tr><tr><td>ULUPDATE SCHEDULE LATENT SIZE</td><td>2PER SAMPLERITERATION 256</td></tr></table>
+
+# A.5 OFFLINE PRE-TRAINING DETAILS
+
+We conducted coarse hyperparameter sweeps to tune each competing UL algorithm. In all cases, the best setting is the one shown in our comparisons.
+
+When our VAEs include a time difference between input and reconstruction observations, we include one hidden layer with action additionally input between the encoder and decoder. We tried both 1.0 and 0.1 KL-divergence weight in the VAE loss, and found 0.1 to perform better in both DMControl and Atari.
+
+DMControl For the VAE, we experimented with 0 and 1 time step difference between input and reconstruction target observations and training for either 1e4 or 5e4 updates. The best settings were 1-step temporal, and 5e4 updates, with batch size 128. ATC used 1-step temporal, 5e4 updates (although this can be significantly decreased), and batch size 256 (including CHEETAH). The pretraining data set consisted of the first 5e4 transitions from a RAD-SAC agent learning each task, including 5e3 random actions. Within this span, CARTPOLE and BALL IN CUP learned completely, but WALKER and CHEETAH reached average returns of 514 and 630, respectively (collected without the compressive convolution).
+
+DMLab For Pixel Control, we used the settings from Hessel et al. (2019) (see the appendix therein), except we used only empirical returns, computed offline (without bootstrapping). For CPC, we tried training batch shapes, $b a t c h \times t i m e$ in (64, 8), (32, 16), (16, 32), and found the setting with rollouts of length 16 to be best. We contrasted all elements of the batch against each other, rather than only forward constrasts. In all cases we also used 16 steps to warmup the LSTM. For all algorithms we tried learning rates $3 \mathrm { e } { - 4 }$ and 1e−3 and both 5e4 and 1.5e5 updates. For ATC and CPC, the lower learning rate and higher number of updates helped in LASERTAG especially. The pretraining data was 125e3 samples from partially trained RL agents receiving average returns of 127 and 6 in EXPLORE GOAL LOCATIONS SMALL and LASERTAG THREE OPPONENTS SMALL, respectively.
+
+Atari For the VAE, we experimented with 0, 1, and 3 time step difference between input and reconstruction target, and found 3 to work best. For ST-DIM we experimented with 1, 3, and 4 time steps differences, and batch sizes from 64 to 256, learning rates 1e−3 and 5e−4. Likewise, 3-step delay worked best. For the inverse model, we tried 1- and 3-step predictions, with 1-step working better overall, and found random shift augmentation to help. For pixel control, we used the settings in Jaderberg et al. (2017), again with full empirical returns. We ran each algorithm for up to 1e5 updates, although final ATC results used 5e4 updates. We ran each RL agent with and without observation normalization on the latent image and observed no difference in performance. Pretraining data was 125e3 samples sourced from the replay buffer of DQN agents trained for 15e6 steps with epsilon-greedy $\epsilon = 0 . 1$ . Evaluation scores were:
+
+Table 8: Atari Pre-Training Data Source Agents.   
+
+<table><tr><td>GAME</td><td>EVALUATION SCORE</td></tr><tr><td>ALIEN</td><td>1,800</td></tr><tr><td>BREAKOUT</td><td>279</td></tr><tr><td>FROSTBITE</td><td>1,400</td></tr><tr><td>GRAVITAR</td><td>390</td></tr><tr><td>PONG</td><td>18</td></tr><tr><td>QBERT</td><td>8,800</td></tr><tr><td>SEAQUEST</td><td>11,000</td></tr><tr><td>SPACE INVADERS</td><td>1,200</td></tr></table>

parse/train/_SKUm2AJpvN/_SKUm2AJpvN_content_list.json

@@ -0,0 +1,1657 @@
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+    {
+        "type": "text",
+        "text": "DECOUPLING REPRESENTATION LEARNING FROM REINFORCEMENT LEARNING ",
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+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
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+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In an effort to overcome limitations of reward-driven feature learning in deep reinforcement learning (RL) from images, we propose decoupling representation learning from policy learning. To this end, we introduce a new unsupervised learning (UL) task, called Augmented Temporal Contrast (ATC), which trains a convolutional encoder to associate pairs of observations separated by a short time difference, under image augmentations and using a contrastive loss. In online RL experiments, we show that training the encoder exclusively using ATC matches or outperforms end-to-end RL in most environments. Additionally, we benchmark several leading UL algorithms by pre-training encoders on expert demonstrations and using them, with weights frozen, in RL agents; we find that agents using ATC-trained encoders outperform all others. We also train multi-task encoders on data from multiple environments and show generalization to different downstream RL tasks. Finally, we ablate components of ATC, and introduce a new data augmentation to enable replay of (compressed) latent images from pre-trained encoders when RL requires augmentation. Our experiments span visually diverse RL benchmarks in DeepMind Control, DeepMind Lab, and Atari, and our complete code is available at hiddenurl. ",
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+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ever since the first fully-learned approach succeeded at playing Atari games from screen images (Mnih et al., 2015), standard practice in deep reinforcement learning (RL) has been to learn visual features and a control policy jointly, end-to-end. Several such deep RL algorithms have matured (Hessel et al., 2018; Schulman et al., 2017; Mnih et al., 2016; Haarnoja et al., 2018) and have been successfully applied to domains ranging from real-world (Levine et al., 2016; Kalashnikov et al., 2018) and simulated robotics (Lee et al., 2019; Laskin et al., 2020a; Hafner et al., 2020) to sophisticated video games (Berner et al., 2019; Jaderberg et al., 2019), and even high-fidelity driving simulators (Dosovitskiy et al., 2017). While the simplicity of end-to-end methods is appealing, relying on the reward function to learn visual features can be severely limiting. For example, it leaves features difficult to acquire under sparse rewards, and it can narrow their utility to a single task. Although our intent is broader than to focus on either sparse-reward or multi-task settings, they arise naturally in our studies. We investigate how to learn visual representations which are agnostic to rewards, without degrading the control policy. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A number of recent works have significantly improved RL performance by introducing auxiliary losses, which are unsupervised tasks that provide feature-learning signal to the convolution neural network (CNN) encoder, additionally to the RL loss (Jaderberg et al., 2017; van den Oord et al., 2018; Laskin et al., 2020b; Guo et al., 2020; Schwarzer et al., 2020). Meanwhile, in the field of computer vision, recent efforts in unsupervised and self-supervised learning (Chen et al., 2020; Grill et al., 2020; He et al., 2019) have demonstrated that powerful feature extractors can be learned without labels, as evidenced by their usefulness for downstream tasks such as ImageNet classification. Together, these advances suggest that visual features for RL could possibly be learned entirely without rewards, which would grant greater flexibility to improve overall learning performance. To our knowledge, however, no single unsupervised learning (UL) task has been shown adequate for this purpose in general vision-based environments. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this paper, we demonstrate the first decoupling of representation learning from reinforcement learning that performs as well as or better than end-to-end RL. We update the encoder weights using only UL and train a control policy independently, on the (compressed) latent images. This capability stands in contrast to previous state-of-the-art methods, which have trained the UL and RL objectives jointly, or Laskin et al. (2020b), which observed diminished performance with decoupled encoders. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Our main enabling contribution is a new unsupervised task tailored to reinforcement learning, which we call Augmented Temporal Contrast (ATC). ATC requires a model to associate observations from nearby time steps within the same trajectory (Anand et al., 2019). Observations are encoded via a convolutional neural network (shared with the RL agent) into a small latent space, where the InfoNCE loss is applied (van den Oord et al., 2018). Within each randomly sampled training batch, the positive observation, $o _ { t + k }$ , for every anchor, $o _ { t }$ , serves as negative for all other anchors. For regularization, observations undergo stochastic data augmentation (Laskin et al., 2020b) prior to encoding, namely random shift (Kostrikov et al., 2020), and a momentum encoder (He et al., 2020; Laskin et al., 2020b) is used to process the positives. A learned predictor layer further processes the anchor code (Grill et al., 2020; Chen et al., 2020) prior to contrasting. In summary, our algorithm is a novel combination of elements that enables generic learning of the structure of observations and transitions in MDPs without requiring rewards or actions as input. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We include extensive experimental studies establishing the effectiveness of our algorithm in a visually diverse range of common RL environments: DeepMind Control Suite (DMControl; Tassa et al. 2018), DeepMind Lab (DMLab; Beattie et al. 2016), and Atari (Bellemare et al., 2013). Our experiments span discrete and continuous control, 2D and 3D visuals, and both on-policy and off policy RL algorithms. Complete code for all of our experiments is available at hiddenurl. Our empirical contributions are summarized as follows: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Online RL with UL: We find that the convolutional encoder trained solely with the unsupervised ATC objective can fully replace the end-to-end RL encoder without degrading policy performance. ATC achieves nearly equal or greater performance in all DMControl and DMLab environments tested and in 5 of the 8 Atari games tested. In the other 3 Atari games, using ATC as an auxiliary loss or for weight initialization still brings improvements over end-to-end RL. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Encoder Pre-Training Benchmarks: We pre-train the convolutional encoder to convergence on expert demonstrations, and evaluate it by training an RL agent using the encoder with weights frozen. We find that ATC matches or outperforms all prior UL algorithms as tested across all domains, demonstrating that ATC is a state-of-the-art UL algorithm for RL. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Multi-Task Encoders: An encoder is trained on demonstrations from multiple environments, and is evaluated, with weights frozen, in separate downstream RL agents. A single encoder trained on four DMControl environments generalizes successfully, performing equal or better than end-to-end RL in four held-out environments. Similar attempts to generalize across eight diverse Atari games result in mixed performance, confirming some limited feature sharing among games. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Ablations and Encoder Analysis: Components of ATC are ablated, showing their individual effects. Additionally, data augmentation is shown to be necessary in DMControl during RL even when using a frozen encoder. We introduce a new augmentation, subpixel random shift, which matches performance while augmenting the latent images, unlocking computation and memory benefits. ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Several recent works have used unsupervised/self-supervised representation learning methods to improve performance in RL. The UNREAL agent (Jaderberg et al., 2017) introduced unsupervised auxiliary tasks to deep RL, including the Pixel Control task, a Q-learning method requiring predictions of screen changes in discrete control environments, which has become a standard in DMLab (Hessel et al., 2019). CPC (van den Oord et al., 2018) applied contrastive losses over multiple time steps as an auxiliary task for the convolutional and recurrent layers of RL agents, and it has been extended with future action-conditioning (Guo et al., 2018). Recently, PBL (Guo et al., 2020) surpassed these methods with an auxiliary loss of forward and backward predictions in the recurrent latent space using partial agent histories. Where the trend is of increasing sophistication in auxiliary recurrent architectures, our algorithm is markedly simpler, requiring only observations, and yet it proves sufficient in partially observed settings (POMDPs). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "ST-DIM (Anand et al., 2019) introduced various temporal, contrastive losses, including ones that operate on “local” features from an intermediate layer within the encoder, without data augmentation. CURL (Laskin et al., 2020b) introduced an augmented, contrastive auxiliary task similar to ours, including a momentum encoder but without temporal contrast. Mazoure et al. (2020) provided extensive analysis pertaining to InfoNCE losses on functions of successive time steps in MDPs, including local features in their auxiliary loss (DRIML) similar to ST-DIM, and finally conducted experiments using global temporal contrast of augmented observations in the Procgen (Cobbe et al., 2019) environment. Most recently, MPR (Schwarzer et al., 2020) combined data augmentation with multi-step, convolutional forward modeling and a similarity loss to improve DQN agents in the Atari 100k benchmark. Hafner et al. (2019; 2020); Lee et al. (2019) proposed to leverage world-modeling in a latent-space for continuous control. A small number of model-free methods have attempted to decouple encoder training from the RL loss as ablations, but have met reduced performance relative to end-to-end RL (Laskin et al., 2020b; Lee et al., 2020). None have previously been shown effective in as diverse a collection of RL environments as ours (Bellemare et al., 2013; Tassa et al., 2018; Beattie et al., 2016). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Finn et al. (2016); Ha & Schmidhuber (2018) are example works which pretrained encoder features in advance using image reconstruction losses such as the VAE (Kingma & Welling, 2013). Devin et al. (2018); Kipf et al. (2019) pretrained object-centric representations, the latter learning a forward model by way of contrastive losses; Yan et al. (2020) introduced a similar technique to learn encoders supporting manipulation of deformable objects by traditional control methods. MERLIN (Wayne et al., 2018) trained a convolutional encoder and sophisticated memory module online, detached from the RL agent, which learned read-only accesses to memory. It used reconstruction and one-step latent-prediction losses and achieved high performance in DMLab-like environments with extreme partial observability. Our loss function may benefit those settings, as it outperforms similar reconstruction losses in our experiments. Decoupling unsupervised pretraining from downstream tasks is common in computer vision (Henaff et al., 2019; He et al., 2019; Chen et al., 2020) and has ´ favorable properties of providing task agnostic features which can be used for training smaller taskspecific networks, yielding significant gains in computational efficiency over end-to-end methods. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 AUGMENTED TEMPORAL CONTRAST ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Our unsupervised learning task, Augmented Temporal Contrast (ATC), requires a model to associate an observation, $o _ { t }$ , with one from a specified, near-future time step, $o _ { t + k }$ . Within each training batch, we apply stochastic data augmentation to the observations (Laskin et al., 2020b), namely random shift (Kostrikov et al., 2020), which is simple to implement and provides highly effective regularization in most cases. The augmented observations are encoded into a small latent space where a contrastive loss is applied. This task encourages the learned encoder to extract meaningful elements of the structure of the MDP from observations. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our architecture for ATC consists of four learned components - (i) a convolutional encoder, $f _ { \\theta }$ , which processes the anchor observation, $o _ { t }$ , into the latent image $z _ { t } ~ = ~ f _ { \\theta } ( \\operatorname { A U G } ( o _ { t } ) )$ , (ii) a linear global compressor, $g _ { \\phi }$ to produce a small latent code vector $c _ { t } = g _ { \\phi } ( z _ { t } )$ , (iii) a residual predictor MLP, $h _ { \\psi }$ , which acts as an implicit forward model to advance the code $p _ { t } ~ = ~ h _ { \\psi } ( c _ { t } ) + c _ { t }$ and (iv) a contrastive transformation matrix, $W$ . To process the positive observation, $o _ { t + k }$ into the target code $\\bar { c } _ { t \\pm k } = \\bar { g _ { \\bar { \\phi } } } ( f _ { \\bar { \\theta } } ( \\operatorname { A U G } ( o _ { t + k } ) )$ , we use a momentum encoder (He et al., 2019) parameterized as a slowly moving average of the weights from the learned encoder and compressor layer: ",
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+    {
+        "type": "image",
+        "img_path": "images/baabe3a71bc43acb562a542e60773e7f84f3eea98027ab6fe84d45f9966eb5b7.jpg",
+        "image_caption": [
+            "Figure 1: Augmented Temporal Contrast— augmented observations are processed through a learned encoder $f _ { \\theta }$ , compressor, $g _ { \\phi }$ and residual predictor $h _ { \\psi }$ , and are associated through a contrastive loss with a positive example from $k$ time steps later, processed through a momentum encoder. "
+        ],
+        "image_footnote": [],
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/3b6612cbc946510bb0915acb04f3fb36e270572ed5386acdac47d595d9b5b1a1.jpg",
+        "text": "$$\n\\bar { \\theta }  ( 1 - \\tau ) \\bar { \\theta } + \\tau \\theta ; \\qquad \\bar { \\phi }  ( 1 - \\tau ) \\bar { \\phi } + \\tau \\phi .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The complete architecture is shown in Figure 1. The convolutional encoder, $f _ { \\theta }$ , alone is shared with the RL agent. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We employ the InfoNCE loss (Gutmann & Hyvarinen, 2010; van den Oord et al., 2018) using log- ¨ its computed bilinearly, as $l ~ = ~ p _ { t } W \\bar { c } _ { t + k }$ . In our implementation, every anchor in the training batch utilizes the positives corresponding to all other anchors as its negative examples. Denoting an observation indexed from dataset $\\mathcal { O }$ as $o _ { i }$ , and its positive as $o _ { i + }$ , the logits can be written as $l _ { i , j + } = p _ { i } W \\bar { c } _ { j + }$ ; our loss function in practice is: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a0cc31920aac899d135126d97b0917624c752fbb7f575ee845ab388f0c14d674.jpg",
+        "text": "$$\n\\mathcal { L } ^ { A T C } = - \\mathbb { E } _ { \\mathcal { O } } \\left[ \\log \\frac { \\exp l _ { i , i + } } { \\sum _ { o _ { j } \\in \\mathcal { O } } \\exp l _ { i , j + } } \\right] .\n$$",
+        "text_format": "latex",
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.1 EVALUATION ENVIRONMENTS AND ALGORITHMS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We evaluate ATC on three standard, visually diverse RL benchmarks - the DeepMind control suite (DMControl; Tassa et al. 2018), Atari games in the Arcade Learning Environment (Bellemare et al., 2013), and DeepMind Lab (DMLab; Beattie et al. 2016). Atari requires discrete control in arcadestyle games. DMControl is comprised of continuous control robotic locomotion and manipulation tasks. In contrast, DMLab requries the RL agent to reason in more visually complex 3D maze environments with partial observability. ",
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+    {
+        "type": "text",
+        "text": "We use ATC to enhance both on-policy and off-policy RL algorithms. For DMControl, we use RADSAC (Laskin et al., 2020a; Haarnoja et al., 2018) with the augmentation of Kostrikov et al. (2020), which randomly shifts the image in each coordinate (by up to 4 pixels), replicating edge pixel values as necessary to restore the original image size. A difference from prior work is that we use more downsampling in our convolutional network, by using strides $( 2 , 2 , 2 , 1 )$ instead of $( 2 , 1 , 1 , 1 )$ to reduce the convolution output image by $2 5 \\mathrm { x }$ .1 For both Atari and DMLab, we use PPO (Schulman et al., 2017). In Atari, we use feed-forward agents, sticky actions, and no end-of-life boundaries for RL episodes. In DMLab we used recurrent, LSTM agents receiving only a single time-step image input, the four-layer convolution encoder from Jaderberg et al. (2019), and we tuned the entropy bonus for each level. In the online setting, the ATC loss is trained using small replay buffer of recent experiences. ",
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+    {
+        "type": "text",
+        "text": "We include all our own baselines for fair comparison and provide complete settings in an appendix. Unless otherwise noted, each curve represents a minimum of 3 random seeds. The bold lines show the average, and the lightly shaded area around each curve represents the maximum extent of the best and worst seeds at each checkpoint. ",
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+    {
+        "type": "text",
+        "text": "4.2 ONLINE RL WITH ATC ",
+        "text_level": 1,
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+    {
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+        "text": "DMControl In the online setting, we found ATC to be capable of training the encoder by itself (i.e., with encoder fully detached from any RL gradient update), achieving essentially equal or better scores versus end-to-end RL in all six environments we tested, Figure 2. In CARTPOLE-SWINGUPSPARSE, where rewards are only received once the pole reaches vertical, ATC training enabled the agent to master the task significantly faster. The encoder is trained with one update for every RL update to the policy, using the same batch size, except in CHEETAH-RUN, which required twice the ATC updates. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "DMLab We experimented with two kinds of levels in DMLab: EXPLORE GOAL LOCATIONS, which requires repeatedly navigating a maze whose layout is randomized every episode, and LASERTAG THREE OPPONENTS, which requires fast reflexes to pursue and tag enemies at a distance. We found ATC capable of training fully detached encoders while achieving equal or better performance than end-to-end RL. Results are shown in Figure 3. Both environments exhibit sparsity which is greater in the “large” version than the “small” version, which our algorithm addresses, discussed next. ",
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+    {
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+        "img_path": "images/3bc961b4e4d3021562434fb49dfc796b984802350f35775720c5fb85ddde2e02.jpg",
+        "image_caption": [
+            "Figure 2: Online encoder training by ATC, fully detached from RL training, performs as well as end-to-end RL in DMControl, and better in sparse-reward environments (environment steps shown, see appendix for action repeats). Each curve is 10 random seeds. "
+        ],
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+        "text": "In EXPLORE, the goal object is rarely seen, especially early on, making its appearance difficult to learn. We therefore introduced prioritized sampling for ATC , with priorities corresponding to empirical absolute returns: $p \\propto 1 + R _ { a b s }$ , where $\\begin{array} { r } { R _ { a b s } ^ { - } = \\sum _ { t = 0 } ^ { n } \\gamma ^ { t } | r _ { t } | } \\end{array}$ , to train more frequently on more informative scenes.2 Whereas uniform-ATC performs slightly below RL, uniform-ATC outperforms RL and nearly matches using ATC (uniform) as an auxiliary task. By considering the encoder as a stand-alone feature extractor separate from the policy, no importance sampling correction is required. ",
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+        "type": "text",
+        "text": "In LASERTAG, enemies are often seen, but the reward of tagging one is rarely achieved by the random agent. ATC learns the relevant features anyway, boosting performance while the RL-only agent remains at zero average score. We found that increasing the rate of UL training to do twice as many updates3 further improved the score to match the ATC-auxiliary agent, showing flexibility to address the representation-learning bottleneck when opponents are dispersed. ",
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+        "image_caption": [
+            "Figure 3: Online encoder training by ATC, fully detached from the RL agent, performs as well or better than end-to-end RL in DMLab (1 agent step $= 4$ environment steps, the standard action repeat). Prioritized ATC replay (EXPLORE) or increased ATC training (LASERTAG) addresses sparsities to nearly match performance of RL with ATC as an auxiliary loss $( \\mathrm { R L + A T C }$ ). Each curve is 3 random seeds. "
+        ],
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+        "type": "text",
+        "text": "Atari We tested a diverse subset of eight Atari games, shown in Figure 4. We found detachedencoder training to work as well as end-to-end RL in five games, but performance suffered in BREAKOUT and SPACE INVADERS in particular. Using ATC as an auxiliary task, however, improves performance in these games and others. We found it helpful to anneal the amount of UL training over the course of RL in Atari (details in an appendix). Notably, we found several games, including SPACE INVADERS, to benefit from using ATC only to initialize encoder weights, done using an initial $1 0 0 \\mathrm { k }$ transitions gathered with a uniform random policy. Some of our remaining experiments provide more insights into the challenges of this domain. ",
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+    {
+        "type": "text",
+        "text": "4.3 ENCODER PRE-TRAINING BENCHMARKS",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To benchmark the effectiveness of different UL algorithms for RL, we propose a new evaluation methodology that is similar to how UL pre-training techniques are measured in computer vision (see e.g. Chen et al. (2020); Grill et al. (2020)): (i) collect a data set composed of expert demonstrations from each environment; (ii) pre-train the CNN encoder with that data offline using UL; (iii) evaluate by using RL to learn a control policy while keeping the encoder weights frozen. This procedure isolates the asymptotic performance of each UL algorithm for RL. For convenience, we drew expert demonstrations from partially-trained RL agents, and every UL algorithm trained on the same data set for each environment. Our RL agents used the same post-encoder architectures as in the online experiments. Further details about pre-training by each algorithm are provided in an appendix. ",
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+        "image_caption": [
+            "Figure 4: Online encoder training using ATC, fully detached from the RL agent, works well in 5 of 8 games tested (1 agent step $= 4$ environment steps, the standard action repeat). 6 of 8 games benefit significantly from using ATC as an auxiliary loss or for weight initialization. Each curve is 8 random seeds. "
+        ],
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+    {
+        "type": "text",
+        "text": "DMControl We compare ATC against two competing algorithms: Augmented Contrast (AC), from CURL (Laskin et al., 2020b), which uses the same observation for the anchor and the positive, and a VAE (Kingma & Welling, 2013), for which we found better performance by introducing a time delay to the target observation (VAE-T). We found ATC to match or outperform the other algorithms, in all four test environments, as shown in Figure 5. Further, ATC is the only one to match or outperform the reference end-to-end RL across all cases. ",
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+        "image_caption": [
+            "Figure 5: RL in DMControl, using encoders pre-trained on expert demonstrations using UL, with weights frozen—across all domains, ATC outperforms prior methods and the end-to-end RL reference. Each curve is a mininum of 4 random seeds. "
+        ],
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+    {
+        "type": "text",
+        "text": "DMLab We compare against both Pixel Control (Jaderberg et al., 2017; Hessel et al., 2019) and CPC (van den Oord et al., 2018), which have been shown to bring strong benefits in DMLab. While all algorithms perform similarly well in EXPLORE, ATC performs significantly better in LASERTAG, Figure 6. Our algorithm is simpler than Pixel Control and CPC in the sense that it uses neither actions, deconvolution, nor recurrence. ",
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+    {
+        "type": "text",
+        "text": "Atari We compare against Pixel Control, VAE-T, and a basic inverse model which predicts actions between pairs of observations. We also compare against Spatio-Temporal Deep InfoMax (ST-DIM), which uses temporal contrastive losses with “local” features from an intermediate convolution layer to ensure attention to the whole screen; it was shown to produce detailed game-state knowledge when applied to individual frames (Anand et al., 2019). Of the four games shown in Figure 7, ATC is the only UL algorithm to match the end-to-end RL reference in GRAVITAR and BREAKOUT, and it performs best in SPACE INVADERS. ",
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+        "image_caption": [
+            "Figure 6: RL in DMLab, using pre-trained encoders with weights frozen–in LASERTAG especially, ATC outperforms leading prior UL algorithms. "
+        ],
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+    {
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+        "image_caption": [
+            "Figure 7: RL in Atari, using pre-trained encoders with weights frozen—ATC outperforms several leading, prior UL algorithms and exceeds the end-to-end RL reference in 3 of the 4 games tested. "
+        ],
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+    {
+        "type": "text",
+        "text": "4.4 MULTI-TASK ENCODERS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In the offline setting, we conducted initial explorations into the capability of ATC to learn multi-task encoders, simply by pre-training on demonstrations from multiple environments. We evaluate the encoder by using it, with frozen weights, in separate RL agents learning each downstream task. ",
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+    {
+        "type": "text",
+        "text": "DMControl Figure 8 shows our results in DMControl, where we pretrained using only the four environments in the top row. Although the encoder was never trained on the HOPPER, PENDULUM, nor FINGER domains, the multi-task encoder supports efficient RL in them. PENDULUM-SWINGUP and CARTPOLE-SWINGUP-SPARSE stand out as challenging environments which benefited from cross-domain and cross-task pre-training, respectively. The pretraining was remarkably efficient, requiring only 20,000 updates to the encoder. ",
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+        "img_path": "images/a1a082ebecffa5389e63c77f1ace713a46a8b37259167a313d56f09aba23a920.jpg",
+        "image_caption": [
+            "Figure 8: Separate RL agents using a single encoder with weights frozen after pre-training on expert demonstrations from the four top environments. The encoder generalizes to four new environments, bottom row, where sparse reward tasks especially benefit from the transfer. Each curve is minimum 4 random seeds. "
+        ],
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+    {
+        "type": "text",
+        "text": "Atari Atari proved a more challenging domain for learning multi-task encoders. Learning all eight games together in Figure 11, in the appendix, resulted in diminished performance relative to single-game pretraining in three of the eight. The decrease was partially alleviated by widening the encoder with twice as many filters per layer, indicating that representation capacity is a limiting factor. To test generalization, we conducted a seven-game pre-training experiment where we test the encoder on the held-out game. Most games suffered diminished performance (although still perform significantly higher than a frozen random encoder), confirming the limited extent to which visual features transfer across these games. ",
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+        "img_path": "images/01ef7105d2680afb75ad5c5c4eed915c080de929140a61da789373eff80f213c.jpg",
+        "image_caption": [
+            "Figure 9: BREAKOUT benefits from contrasting against negatives from several neighboring time steps. "
+        ],
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+        "img_path": "images/539e70000e4ecf18d2eba89d8be66353cc7947a0d1d8196f6ef9391f02b75278.jpg",
+        "image_caption": [
+            "Figure 10: An example scene from BREAKOUT, where a lowperformance UL encoder (without shift) focuses on the paddle. Introducing random shift and sequence data makes the highperformance UL encoder (full ATC) focus near the ball, as does the encoder from a fully-trained, end-to-end RL agent. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "4.5 ABLATIONS AND ENCODER ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Random Shift in ATC In offline experiments, we discovered random shift augmentations to be helpful in all domains. To our knowledge, this is the first application of random shift to 3D visual environments as in DMLab. In Atari, we found performance in GRAVITAR to suffer from random shift, but reducing the probability of applying random shift to each observation from 1.0 to 0.1 alleviated the effect while still bringing benefits in other games, so we used this setting in our main experiments. Results are shown in Figure 12 in an appendix. ",
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+    {
+        "type": "text",
+        "text": "Random Shift in RL In DMControl, we found the best results when using random shift during RL, even when training with a frozen encoder. This is evidence that the augmentation regularizes not only the representation but also the policy, which first processes the latent image into a 50- dimensional vector. To unlock computation and memory benefits of replaying only the latent images for the RL agent, we attempted to apply data augmentation to the latent image. But we found the smallest possible random shifts to be too extreme. Instead, we introduce a new augmentation, subpixel random shift, which linearly interpolates among neighboring pixels. As shown in Figure 13 in the appendix, this augmentation restores performance when applied to the latent images, allowing a pre-trained encoder to be entirely bypassed during policy training updates. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Temporal Contrast on Sequences In BREAKOUT alone, we discovered that composing the UL training batch of trajectory segments, rather than individual transitions, gave a significant benefit. Treating all elements of the training batch independently provides “hard” negatives, since the encoder must distinguish between neighboring time steps. This setting had no effect in the other Atari games tested, and we found equal or better performance using individual transitions in DMControl and DMLab. Figure 9 further shows that using a similarity loss (Grill et al., 2020) does not capture the benefit. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Encoder Analysis We analyzed the learned encoders in BREAKOUT to further study this ablation effect. Similar to Zagoruyko & Komodakis (2016), we compute spatial attention maps by mean-pooling the absolute values of the activations along the channel dimension and follow with a 2-dimensional spatial softmax. Figure 10 shows the attention of four different encoders on the displayed scene. The poorly performing UL encoder heavily utilizes the paddle to distinguish the observation. The UL encoder trained with random shift and sequence data, however, focuses near the ball, as does the fully-trained RL encoder. (The random encoder mostly highlights the bricks, which are less relevant for control.) In an appendix, we include other example encoder analyses from Atari and DMLab which show ATC-trained encoders attending only to key objects on the game screen, while RL-trained encoders additionally attend to potentially distracting features such as game score. ",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Reward-free representation learning from images provides flexibility and insights for improving deep RL agents. We have shown a broad range of cases where our new unsupervised learning algorithm can fully replace RL for training convolutional encoders while maintaining or improving online performance. In a small number of environments–a few of the Atari games–including the RL loss for encoder training still surpasses our UL-only method, leaving opportunities for further improvements in UL for RL. ",
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+    {
+        "type": "text",
+        "text": "",
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "Our preliminary efforts to use actions as inputs (into the predictor MLP) or as prediction outputs (inverse loss) with ATC did not immediately yield strong improvements. We experimented only with random shift, but other augmentations may be useful, as well. In multi-task encoder training, our technique avoids any need for sophisticated reward-balancing (Hessel et al., 2019), but more advanced training methods may still help when the required features are in conflict, as in Atari, or if they otherwise impact our loss function unequally. On the theoretical side, it may be helpful to analyze the effects of domain shift on the policy when a detached representation is learned online. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "One obvious application of our offline methodology would be in the batch RL setting, where the agent learns from a fixed data set. Our offline experiments showed that a relatively small number of transitions are sufficient to learn rich representations by UL, and the lower limit could be further explored. Overall, we hope that our algorithm and experiments spur further developments leveraging unsupervised learning for reinforcement learning. ",
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+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "bbox": [
+            176,
+            103,
+            299,
+            117
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.1 ALGORITHMS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            133,
+            315,
+            148
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 Online RL with decoupled ATC encoder (steps distinct from end-to-end RL in blue) ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            166,
+            728,
+            194
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Require: θAT C, φπ . ATC model parameters (encoder $f _ { \\theta }$ thru contrast $W$ ), policy parameters   \n1: $\\mathcal { S }  \\{ \\}$ $\\triangleright$ replay buffer of observations   \n2: $\\bar { \\theta } _ { A T C }  \\theta _ { A T C }$ . initialize momentum encoder (conv and linear only)   \n3: repeat   \n4: Sample environment and policy, through encoder:   \n5: for 1 to m do . a minibatch   \n6: $\\begin{array} { l } { a \\sim { \\pi } ( \\cdot | f _ { \\theta } ( s ) ; \\phi ) , { s } ^ { \\prime } \\sim T ( s , a ) , r \\sim R ( s , a , { s } ^ { \\prime } ) } \\\\ { \\quad S  S \\cup \\{ s \\} } \\\\ { \\quad s  s ^ { \\prime } } \\end{array}$   \n7: store observations (delete oldest if full)   \n8:   \n9: end for   \n10: Update policy by given RL formula: $\\triangleright$ on- or off-policy   \n11: for 1 to n do $\\triangleright$ given number RL updates per minibatch   \n12: $\\phi _ { \\pi }  \\phi _ { \\pi } + R L ( s , a , s ^ { \\prime } , r ; \\phi _ { \\pi } )$ . stop gradient into encoder   \n13: end for   \n14: Update encoder (and contrastive model) by ATC:   \n15: for 1 to p do   \n16: 17: $\\begin{array} { r l } & { s , s _ { + } \\sim s } \\\\ & { \\underline { { \\theta } } _ { A T C }  \\theta _ { A T C } - \\lambda _ { A T C } \\nabla _ { \\theta _ { A T C } } \\mathcal { L } ^ { A T C } ( s , s _ { + } ) } \\end{array}$ sample observations: anchors and positives $\\triangleright$ ATC gradient update   \n18: $\\bar { \\theta } _ { A T C }  ( 1 - \\tau ) \\bar { \\theta } _ { A T C } + \\tau \\theta _ { A T C }$ . update momentum encoder (conv and linear only)   \n19: end for   \n20: until converged   \n21: return Encoder $f _ { \\theta }$ and policy $\\pi _ { \\phi }$ ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.2 ADDITIONAL FIGURES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            535,
+            377,
+            549
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/5543152dd76206baac60db6f57cb863ab7342d883bbae0c2215217cd33cfa063.jpg",
+        "image_caption": [
+            "Figure 11: RL using multi-task encoders (all with weights frozen) for eight Atari games gives mixed performance, partially improved by increased network capacity (8-game-wide). Training on 7 games and testing on the held-out one yields diminished but non-zero performance, showing some limited feature transfer between games. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            196,
+            569,
+            795,
+            739
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In subpixel random shift, new pixels are a linearly weighted average of the four nearest pixels to a randomly chosen coordinate location. We used uniformly random horizontal and vertical shifts, and tested maximum displacements in $( \\pm ) \\left\\{ 0 . 1 , 0 . 2 5 , 0 . 5 , 0 . 7 5 , 1 . 0 \\right\\}$ pixels (with “edge” mode padding $\\pm 1 )$ . We found 0.5 to work well in all tested domains, restoring the performance of raw image augmentation but eliminating convolutions entirely from the RL training updates. ",
+        "bbox": [
+            173,
+            827,
+            825,
+            897
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/4beb61e4ab5090b07268c86aa788662703783a664d7d871ba041b834c06ceeac.jpg",
+        "image_caption": [
+            "Figure 12: Random shift augmentation helps in some Atari games and hurts in others, but applying with probability 0.1 is a performant middle ground. DMLab benefits from random shift. (Offline pre-training.) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            194,
+            108,
+            799,
+            219
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/29ac0be5110940df74d1d5c3a4f0e6e4139c5451731cb9c5df2f23a86dd1c6d8.jpg",
+        "image_caption": [
+            "Figure 13: Even after pre-training encoders for DMControl using random shift, RL requires augmentation— our subpixel augmentation acts on the (compressed) latent image, permitting its use in the replay buffer. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            194,
+            282,
+            799,
+            382
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/1cce8da9c9fb4f83d8b7ebde417428125f913540be195ef011cceec933f3c869.jpg",
+        "image_caption": [
+            "Figure 14: Attention map in BREAKOUT which shows the RL-trained encoder focusing on game score, whereas UL ATC encoder focuses properly on the paddle and ball. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            189,
+            443,
+            797,
+            559
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/730bece2e848d8228104749b4de1d034becb59d3c9360f699590d057e7854254.jpg",
+        "image_caption": [
+            "Figure 15: Attention map in LASERTAG. UL encoder with pixel control focuses on the score, while UL encoder with the proposed ATC focuses properly on the coin similar to RL-trained encoder. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            196,
+            623,
+            803,
+            709
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/081300ec7a0548d8be5ff478ad26357c10b99de16be803312ba29ed20f5121ec.jpg",
+        "image_caption": [
+            "Figure 16: Attention map in the LASERTAG which shows that UL encoders focus properly on the enemy similar to RL-trained encoder. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            192,
+            781,
+            803,
+            868
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/6354325d20234acc183776724d31de98783f09d5ce38a9bcba2ffa3c123ec094.jpg",
+        "table_caption": [
+            "Table 1: DMControl, RAD-SAC Hyperparameters. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>HYPERPARAMETER</td><td>VALUE</td></tr><tr><td>OBSERVATION RENDERING</td><td>(84,84),RGB</td></tr><tr><td>RANDOM SHIFT PAD</td><td>±4</td></tr><tr><td>REPLAY BUFFER SIZE</td><td>1e5</td></tr><tr><td>INITIAL STEPS</td><td>1e4</td></tr><tr><td>STACKED FRAMES</td><td>3</td></tr><tr><td>ACTIONREPEAT</td><td>2(FINGER,WALKER)</td></tr><tr><td></td><td>8 (CARTPOLE)</td></tr><tr><td></td><td>4 (REST)</td></tr><tr><td>OPTIMIZER</td><td>ADAM</td></tr><tr><td>(β1,β2)→(fθ,Tψ,Q)</td><td>(.9,.999)</td></tr><tr><td>(β1,β)→(a) LEARNING RATE(fθ,πψ,QΦ)</td><td>(.5,.999)</td></tr><tr><td></td><td>2e-4(CHEETAH)</td></tr><tr><td>LEARNING RATE (α)</td><td>1e-3 (REST)</td></tr><tr><td>BATCH SIZE</td><td>1e-4</td></tr><tr><td></td><td>512(CHEETAH,PENDULUM) 256 (REST)</td></tr><tr><td>Q FUNCTION EMA T</td><td>0.01</td></tr><tr><td>CRITIC TARGET UPDATE FREQ</td><td>2</td></tr><tr><td>CONVOLUTION FILTERS</td><td>[32,32,32,32]</td></tr><tr><td>CONVOLUTION STRIDES</td><td>[2,2,2,1]</td></tr><tr><td>CONVOLUTION FILTER SIZE</td><td>3</td></tr><tr><td>ENCODER EMAT</td><td>0.05</td></tr><tr><td>LATENT DIMENSION</td><td>50</td></tr><tr><td>HIDDEN UNITS (MLP)</td><td>[1024,1024]</td></tr><tr><td>DISCOUNT </td><td>.99</td></tr><tr><td>INITIAL TEMPERATURE</td><td>0.1</td></tr><tr><td></td><td></td></tr></table>",
+        "bbox": [
+            297,
+            171,
+            696,
+            529
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/fd386492f80ce756558184732871bab012ab084fe00b344b2f79d155cd9c119c.jpg",
+        "table_caption": [
+            "Table 2: Atari, PPO Hyperparameters. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>HYPERPARAMETER</td><td>VALUE</td></tr><tr><td>OBSERVATION RENDERING</td><td>(84,84), GREY</td></tr><tr><td>STACKED FRAMES</td><td>4</td></tr><tr><td>ACTIONREPEAT</td><td>4</td></tr><tr><td>OPTIMIZER</td><td>ADAM</td></tr><tr><td>LEARNING RATE</td><td>2.5e-4</td></tr><tr><td>PARALLEL ENVIRONMENTS</td><td>16</td></tr><tr><td>SAMPLING INTERVAL</td><td>128</td></tr><tr><td>LIKELIHOOD RATIO CLIP, ∈</td><td>0.1</td></tr><tr><td>PPOEPOCHS</td><td>4</td></tr><tr><td>PPOMINIBATCHES</td><td>4</td></tr><tr><td>CONVOLUTION FILTERS</td><td>[32, 64, 64]</td></tr><tr><td>CONVOLUTION FILTER SIZES</td><td>[8,4,3]</td></tr><tr><td>CONVOLUTION STRIDES</td><td>[4,2,1]</td></tr><tr><td>HIDDEN UNITS (MLP)</td><td>[512]</td></tr><tr><td>DISCOUNT γ</td><td>.99</td></tr><tr><td>GENERALIZED ADVANTAGE ESTIMATION入</td><td>0.95</td></tr><tr><td>LEARNINGRATE ANNEALING</td><td>LINEAR</td></tr><tr><td>ENTROPY BONUS COEFFICIENT</td><td>0.01</td></tr><tr><td>EPISODIC LIVES</td><td>FALSE</td></tr><tr><td>REPEAT ACTIONPROBABILITY</td><td>0.25</td></tr><tr><td>REWARD CLIPPING</td><td>±1</td></tr><tr><td>VALUELOSS COEFFICIENT</td><td>1.0</td></tr></table>",
+        "bbox": [
+            297,
+            582,
+            699,
+            890
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/f8720bcdcf121b1e6f10dd22ce0d5b8a60fc9215c47b1af587225a2b26a4aa3a.jpg",
+        "table_caption": [
+            "Table 3: DMLab, PPO Hyperparameters. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>HYPERPARAMETER</td><td>VALUE</td></tr><tr><td>OBSERVATION RENDERING</td><td>(72,96),RGB</td></tr><tr><td>STACKED FRAMES</td><td>1</td></tr><tr><td>ACTION REPEAT</td><td>4</td></tr><tr><td>OPTIMIZER</td><td>ADAM</td></tr><tr><td>LEARNING RATE</td><td>2.5e-4</td></tr><tr><td>PARALLEL ENVIRONMENTS</td><td>16</td></tr><tr><td>SAMPLING INTERVAL</td><td>128</td></tr><tr><td>LIKELIHOOD RATIO CLIP, E</td><td>0.1</td></tr><tr><td>PPO EPOCHS</td><td>1</td></tr><tr><td>PPOMINIBATCHES</td><td>2</td></tr><tr><td>CONVOLUTION FILTERS</td><td>[32,64,64,64]</td></tr><tr><td>CONVOLUTION FILTER SIZES</td><td>[8,4,3,3]</td></tr><tr><td>CONVOLUTION STRIDES</td><td>[4,2,1, 1]</td></tr><tr><td>HIDDEN UNITS (LSTM)</td><td>[256]</td></tr><tr><td>SKIP CONNECTIONS</td><td>CONV3,4; LSTM</td></tr><tr><td>DISCOUNT </td><td>.99</td></tr><tr><td>GENERALIZED ADVANTAGE ESTIMATION 入</td><td>0.97</td></tr><tr><td>LEARNING RATE ANNEALING</td><td>NONE</td></tr><tr><td>ENTROPY BONUS COEFFICIENT</td><td>0.01 (EXPLORE)</td></tr><tr><td></td><td>0.0003 (LASERTAG)</td></tr><tr><td>VALUE LOSS COEFFICIENT</td><td>0.5</td></tr></table>",
+        "bbox": [
+            281,
+            383,
+            714,
+            679
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "A.4 ONLINE ATC SETTINGS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            385,
+            117
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "table",
+        "img_path": "images/dbb1e9311c2fc28345cc69b9baca0e75bf9727b26b2620b763cc884c22548f64.jpg",
+        "table_caption": [
+            "Table 4: Common ATC Hyperparameters. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>HYPERPARAMETER</td><td>VALUE</td></tr><tr><td>RANDOM SHIFT PAD</td><td>±4</td></tr><tr><td>LEARNING RATE</td><td>1e-3</td></tr><tr><td>LEARNING RATE ANNEALING</td><td>COSINE</td></tr><tr><td>TARGETUPDATE INTERVAL</td><td>1</td></tr><tr><td>TARGET UPDATE T PREDICTOR HIDDEN SIZES,h</td><td>0.01</td></tr><tr><td></td><td>[512]</td></tr><tr><td>REPLAY BUFFER SIZE</td><td>1e5</td></tr></table>",
+        "bbox": [
+            356,
+            171,
+            637,
+            290
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "table",
+        "img_path": "images/8a3aa29ba963e1da67e66a3ccf1307446fd907f35f4e5a8ae2ec870343697f9b.jpg",
+        "table_caption": [
+            "Table 5: DMControl ATC Hyperparameters. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>HYPERPARAMETER</td><td>VALUE</td></tr><tr><td>RANDOMSHIFTPROBABILITY</td><td>1</td></tr><tr><td>BATCH SIZE</td><td>ASRL (INDIVIDUAL OBSERVATIONS)</td></tr><tr><td>TEMPORAL SHIFT,k</td><td>1</td></tr><tr><td>MIN AGENT STEPS TO UL</td><td>1e4</td></tr><tr><td>MIN AGENT STEPS TO RL</td><td>1e4</td></tr><tr><td>UL UPDATE SCHEDULE</td><td>ASRL</td></tr><tr><td></td><td>(2X CHEETAH)</td></tr><tr><td>LATENT SIZE</td><td>128</td></tr></table>",
+        "bbox": [
+            267,
+            344,
+            725,
+            474
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "table",
+        "img_path": "images/e165c970cb841fc5cfc8506e327f5fd0b81a53589c0c9affdd96ae68afa3fa53.jpg",
+        "table_caption": [
+            "Table 6: Atari ATC Hyperparameters. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>HYPERPARAMETER</td><td>VALUE</td></tr><tr><td>RANDOM SHIFTPROBABILITY</td><td>0.1</td></tr><tr><td>BATCH SIZE</td><td>512 (32 TRAJECTORIES OF 16 TIME STEPS)</td></tr><tr><td>TEMPORAL SHIFT,k</td><td>3</td></tr><tr><td>MIN AGENT STEPS TO UL</td><td>5e4</td></tr><tr><td>MIN AGENT STEPS TO RL</td><td>1e5</td></tr><tr><td>ULUPDATE SCHEDULE</td><td>ANNEALED QUADRATICALLY FROM6 PER SAMPLER ITERATION (1e4 ONCE AT 1e5 STEPS FOR WEIGHT INITIALIZATION)</td></tr><tr><td>LATENT SIZE</td><td>256</td></tr></table>",
+        "bbox": [
+            186,
+            531,
+            805,
+            660
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "table",
+        "img_path": "images/ca8b3fe8e196c6472ea52706968ea3b0a9d1393822ead8b4005d4016a45e5672.jpg",
+        "table_caption": [
+            "Table 7: DMLab ATC Hyperparameters. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>HYPERPARAMETER</td><td>VALUE</td></tr><tr><td>RANDOMSHIFTPROBABILITY</td><td>1</td></tr><tr><td>BATCH SIZE</td><td>512 (INDIVIDUAL OBSERVATIONS)</td></tr><tr><td>TEMPORAL SHIFT,k</td><td>3</td></tr><tr><td>MIN AGENT STEPS TO UL</td><td>5e4</td></tr><tr><td>MIN AGENT STEPS TO RL</td><td>1e5</td></tr><tr><td>ULUPDATE SCHEDULE LATENT SIZE</td><td>2PER SAMPLERITERATION 256</td></tr></table>",
+        "bbox": [
+            276,
+            715,
+            718,
+            834
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "A.5 OFFLINE PRE-TRAINING DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            103,
+            452,
+            117
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "We conducted coarse hyperparameter sweeps to tune each competing UL algorithm. In all cases, the best setting is the one shown in our comparisons. ",
+        "bbox": [
+            171,
+            130,
+            823,
+            157
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "When our VAEs include a time difference between input and reconstruction observations, we include one hidden layer with action additionally input between the encoder and decoder. We tried both 1.0 and 0.1 KL-divergence weight in the VAE loss, and found 0.1 to perform better in both DMControl and Atari. ",
+        "bbox": [
+            174,
+            165,
+            825,
+            219
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "DMControl For the VAE, we experimented with 0 and 1 time step difference between input and reconstruction target observations and training for either 1e4 or 5e4 updates. The best settings were 1-step temporal, and 5e4 updates, with batch size 128. ATC used 1-step temporal, 5e4 updates (although this can be significantly decreased), and batch size 256 (including CHEETAH). The pretraining data set consisted of the first 5e4 transitions from a RAD-SAC agent learning each task, including 5e3 random actions. Within this span, CARTPOLE and BALL IN CUP learned completely, but WALKER and CHEETAH reached average returns of 514 and 630, respectively (collected without the compressive convolution). ",
+        "bbox": [
+            173,
+            236,
+            825,
+            348
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "DMLab For Pixel Control, we used the settings from Hessel et al. (2019) (see the appendix therein), except we used only empirical returns, computed offline (without bootstrapping). For CPC, we tried training batch shapes, $b a t c h \\times t i m e$ in (64, 8), (32, 16), (16, 32), and found the setting with rollouts of length 16 to be best. We contrasted all elements of the batch against each other, rather than only forward constrasts. In all cases we also used 16 steps to warmup the LSTM. For all algorithms we tried learning rates $3 \\mathrm { e } { - 4 }$ and 1e−3 and both 5e4 and 1.5e5 updates. For ATC and CPC, the lower learning rate and higher number of updates helped in LASERTAG especially. The pretraining data was 125e3 samples from partially trained RL agents receiving average returns of 127 and 6 in EXPLORE GOAL LOCATIONS SMALL and LASERTAG THREE OPPONENTS SMALL, respectively. ",
+        "bbox": [
+            173,
+            362,
+            825,
+            502
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Atari For the VAE, we experimented with 0, 1, and 3 time step difference between input and reconstruction target, and found 3 to work best. For ST-DIM we experimented with 1, 3, and 4 time steps differences, and batch sizes from 64 to 256, learning rates 1e−3 and 5e−4. Likewise, 3-step delay worked best. For the inverse model, we tried 1- and 3-step predictions, with 1-step working better overall, and found random shift augmentation to help. For pixel control, we used the settings in Jaderberg et al. (2017), again with full empirical returns. We ran each algorithm for up to 1e5 updates, although final ATC results used 5e4 updates. We ran each RL agent with and without observation normalization on the latent image and observed no difference in performance. Pretraining data was 125e3 samples sourced from the replay buffer of DQN agents trained for 15e6 steps with epsilon-greedy $\\epsilon = 0 . 1$ . Evaluation scores were: ",
+        "bbox": [
+            173,
+            517,
+            825,
+            656
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "table",
+        "img_path": "images/c1f121d7053ad6d46a9cab0d716131c16dde12a472f199eddcfc85bd816b80e2.jpg",
+        "table_caption": [
+            "Table 8: Atari Pre-Training Data Source Agents. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>GAME</td><td>EVALUATION SCORE</td></tr><tr><td>ALIEN</td><td>1,800</td></tr><tr><td>BREAKOUT</td><td>279</td></tr><tr><td>FROSTBITE</td><td>1,400</td></tr><tr><td>GRAVITAR</td><td>390</td></tr><tr><td>PONG</td><td>18</td></tr><tr><td>QBERT</td><td>8,800</td></tr><tr><td>SEAQUEST</td><td>11,000</td></tr><tr><td>SPACE INVADERS</td><td>1,200</td></tr></table>",
+        "bbox": [
+            357,
+            707,
+            637,
+            838
+        ],
+        "page_idx": 17
+    }
+]

parse/train/jlchsFOLfeF/jlchsFOLfeF.md

@@ -0,0 +1,278 @@
+# Invariance Principle Meets Information Bottleneck for Out-of-Distribution Generalization
+
+Kartik Ahuja†
+
+Ethan Caballero∗ †
+
+Dinghuai Zhang∗ †
+
+Jean-Christophe Gagnon-Audet †
+
+Yoshua Bengio †
+
+Ioannis Mitliagkas†
+
+Irina Rish†
+
+# Abstract
+
+The invariance principle from causality is at the heart of notable approaches such as invariant risk minimization (IRM) that seek to address out-of-distribution (OOD) generalization failures. Despite the promising theory, invariance principle-based approaches fail in common classification tasks, where invariant (causal) features capture all the information about the label. Are these failures due to the methods failing to capture the invariance? Or is the invariance principle itself insufficient? To answer these questions, we revisit the fundamental assumptions in linear regression tasks, where invariance-based approaches were shown to provably generalize OOD. In contrast to the linear regression tasks, we show that for linear classification tasks we need much stronger restrictions on the distribution shifts, or otherwise OOD generalization is impossible. Furthermore, even with appropriate restrictions on distribution shifts in place, we show that the invariance principle alone is insufficient. We prove that a form of the information bottleneck constraint along with invariance helps address key failures when invariant features capture all the information about the label and also retains the existing success when they do not. We propose an approach that incorporates both of these principles and demonstrate its effectiveness in several experiments.
+
+# 1 Introduction
+
+Recent years have witnessed an explosion of examples showing deep learning models are prone to exploiting shortcuts (spurious features) (Geirhos et al., 2020; Pezeshki et al., 2020) which make them fail to generalize out-of-distribution (OOD). In Beery et al. (2018), a convolutional neural network was trained to classify camels from cows; however, it was found that the model relied on the background color (e.g., green pastures for cows) and not on the properties of the animals (e.g., shape). These examples become very concerning when they occur in real-life applications (e.g., COVID-19 detection (DeGrave et al., 2020)).
+
+To address these out-of-distribution generalization failures, invariant risk minimization (Arjovsky et al., 2019) and several other works were proposed (Ahuja et al., 2020; Pezeshki et al., 2020; Krueger et al., 2020; Robey et al., 2021; Zhang et al., 2021). The invariance principle from causality (Peters et al., 2015; Pearl, 1995) is at the heart of these works. The principle distinguishes predictors that only rely on the causes of the label from those that do not. The optimal predictor that only focuses on the causes is invariant and min-max optimal (Rojas-Carulla et al., 2018; Koyama and Yamaguchi, 2020; Ahuja et al., 2021) under many distribution shifts but the same is not true for other predictors.
+
+Our contributions. Despite the promising theory, invariance principle-based approaches fail in settings (Aubin et al., 2021) where invariant features capture all information about the label contained in the input. A particular example is image classification (e.g., cow vs. camel) (Beery et al., 2018) where the label is a deterministic function of the invariant features (e.g., shape of the animal), and does not depend on the spurious features (e.g., background). To understand such failures, we revisit the fundamental assumptions in linear regression tasks, where invariance-based approaches were shown to provably generalize OOD. We show that, in contrast to the linear regression tasks, OOD generalization is significantly harder for linear classification tasks; we need much stronger restrictions in the form of support overlap assumptions3 on the distribution shifts, or otherwise it is not possible to guarantee OOD generalization under interventions on variables other than the target class. We then proceed to show that, even under the right assumptions on distribution shifts, the invariance principle is insufficient. However, we establish that information bottleneck (IB) constraints (Tishby et al., 2000), together with the invariance principle, provably works in both settings – when invariant features completely capture the information about the label and also when they do not. (Table 1 summarizes our theoretical results presented later). We propose an approach that combines both these principles and demonstrate its effectiveness on linear unit tests (Aubin et al., 2021) and on different real datasets.
+
+<table><tr><td>Task</td><td>Invariant features capture label info</td><td>Support overlap invariant features</td><td>Support overlap spurious features</td><td>OOD generalization guarantee (εtr→εall) ERM IRM IB-ERM</td><td></td><td>IB-IRM</td></tr><tr><td rowspan="5">Linear Classification</td><td>Full/Partial</td><td>No</td><td>Yes/No</td><td></td><td>Impossible for any algorithm to generalize OOD [Thm2]</td><td rowspan="5">[Thm3,4]</td></tr><tr><td>Full</td><td>Yes</td><td>No</td><td>区 X</td><td>√</td></tr><tr><td>Partial</td><td>Yes</td><td>No</td><td>X X</td><td>? X</td></tr><tr><td>Full</td><td>Yes</td><td>Yes</td><td>√</td><td>√ √</td></tr><tr><td>Partial</td><td>Yes</td><td>Yes</td><td>√ X √</td><td>? √</td></tr><tr><td>Linear</td><td>Full</td><td>No</td><td>No</td><td>√</td><td>× √</td><td></td></tr><tr><td>Regression</td><td>Partial</td><td>No</td><td>No</td><td>区</td><td>X</td><td>√ √ [Thm4]</td></tr></table>
+
+Table 1: Summary of the new and existing results (Arjovsky et al., 2019; Rosenfeld et al., 2021). IB-ERM (IRM): information bottleneck - empirical (invariant) risk minimization ERM (IRM).
+
+# 2 OOD generalization and invariance: background & failures
+
+Background. We consider a supervised training data $D$ gathered from a set of training environments $\mathcal { E } _ { t r }$ : $D = \{ D ^ { e } \} _ { e \in \mathcal { E } _ { t r } }$ , where $\bar { D ^ { e } } = \{ x _ { i } ^ { e } , y _ { i } ^ { e } \} _ { i = 1 } ^ { n ^ { e } }$ is the dataset from environment $e \in \mathcal { E } _ { t r }$ and $n ^ { e }$ is the number of instances in environment $e$ . $x _ { i } ^ { e } \in \mathbb { R } ^ { d }$ and $y _ { i } ^ { e } \in \mathcal { V } \subseteq \mathbb { R } ^ { k }$ correspond to the input feature value and the label for $i ^ { t h }$ instance respectively. Each $( x _ { i } ^ { e } , y _ { i } ^ { e } )$ is an i.i.d. draw from $\mathbb { P } ^ { e }$ , where $\mathbb { P } ^ { e }$ is the joint distribution of the input feature and the label in environment $e$ . Let $\mathcal { X } ^ { e }$ be the support of the input feature values in the environment $e$ . The goal of OOD generalization is to use training data $D$ to construct a predictor $f : \mathbb { R } ^ { d }  \mathbb { R } ^ { k }$ that performs well across many unseen environments in ${ \mathcal { E } } _ { a l l }$ , where $\mathcal { E } _ { a l l } \supset \mathcal { E } _ { t r }$ . Define the risk of $f$ in environment $e$ as $R ^ { e } ( f ) \doteq \mathbb { E } \bigl [ \ell ( f ( X ^ { e } ) , Y ^ { e } ) \bigr ]$ , where for example $\ell$ can be 0-1 loss, logistic loss, square loss, $( X ^ { e } , Y ^ { e } ) \sim { \mathbb { P } } ^ { e }$ , and the expectation $\mathbb { E }$ is w.r.t. $\mathbb { P } ^ { e }$ . Formally stated, our goal is to use the data from training environments $\mathcal { E } _ { t r }$ to find $f : \mathbb { R } ^ { d }  \mathcal { V }$ to minimize
+
+$$
+\operatorname* { m i n } _ { f } \operatorname* { m a x } _ { e \in { \mathscr { E } } _ { a l l } } R ^ { e } ( f ) .
+$$
+
+So far we did not state any restrictions on ${ \mathcal { E } } _ { a l l }$ . Consider binary classification: without any restrictions on ${ \mathcal { E } } _ { a l l }$ , no method can reduce the above objective ( $\ell$ is 0-1 loss) to below one. Suppose a method outputs $f ^ { * }$ ; if $\exists e \in \mathcal { E } _ { a l l } \ \backslash \mathcal { E } _ { t r }$ with labels based on $1 - f ^ { * }$ , then it achieves an error of one. Some assumptions on ${ \mathcal { E } } _ { a l l }$ are thus necessary. Consider how ${ \mathcal { E } } _ { a l l }$ is restricted using invariance for linear regressions (Arjovsky et al., 2019).
+
+Assumption 1. Linear regression structural equation model (SEM). In each $e \in \mathcal { E } _ { a l l }$
+
+$$
+\begin{array} { r l } & { Y ^ { e } \gets w _ { \mathrm { i n v } } ^ { * } \cdot Z _ { \mathrm { i n v } } ^ { e } + \epsilon ^ { e } , \quad Z _ { \mathrm { i n v } } ^ { e } \perp \epsilon ^ { e } , \quad \mathbb { E } [ \epsilon ^ { e } ] = 0 , \mathbb { E } \big [ | \epsilon ^ { e } | ^ { 2 } \big ] \leq \sigma _ { \mathrm { s u p } } ^ { 2 } } \\ & { X ^ { e } \gets S ( Z _ { \mathrm { i n v } } ^ { e } , Z _ { \mathrm { s p u } } ^ { e } ) } \end{array}
+$$
+
+where $\boldsymbol { w _ { \mathrm { i n v } } ^ { * } } \in \mathbb { R } ^ { m }$ , $Z _ { \mathsf { i n v } } ^ { e } \in \mathbb { R } ^ { m }$ , $Z _ { \mathsf { s p u } } \in \mathbb { R } ^ { o }$ , $S \in \mathbb { R } ^ { d \times ( m + o ) }$ , $S$ is invertible $( m + o = d )$ . We focus on invertible $S$ but several results extend to non-invertible $S$ as well (see Appendix).
+
+Assumption 1 states how $Y ^ { e }$ and $X ^ { e }$ are generated from latent invariant features $Z _ { \mathrm { i n v } } ^ { e \mathrm { ~ 4 ~ } }$ , latent spurious features $Z _ { \mathsf { s p u } } ^ { e }$ and noise $\epsilon ^ { e }$ . The relationship between label and invariant features is invariant, i.e., $w _ { \mathrm { i n v } } ^ { \ast }$ is fixed across all environments. However, the distributions of $Z _ { \mathrm { i n v } } ^ { e }$ , $Z _ { \mathsf { s p u } } ^ { e }$ , and $\epsilon ^ { e }$ are allowed to change arbitrarily across all the environments. Suppose is identity. If we regress only on the invariant features $Z _ { \mathrm { i n v } } ^ { e }$ , then the optimal solution is $w _ { \mathrm { i n v } } ^ { \ast }$ , which is independent of the environment, and the error it achieves is bounded above by the variance of $\epsilon ^ { e } ( \sigma _ { \mathsf { s u p } } ^ { 2 } )$ . If we regress on the entire $Z ^ { e }$ and the optimal predictor places a non-zero weight on $Z _ { \mathsf { s p u } } ^ { e }$ sup(e.g., $Z _ { \mathsf { s p u } } ^ { e } \gets Y ^ { e } + \zeta ^ { e } )$ , then this predictor fails to solve equation (1) ( $\exists e \in { \mathcal { E } } _ { a l l }$ , $Z _ { \mathsf { s p u } } ^ { e } \to \infty$ , error $ \infty$ , see Appendix for details). Also, not only regressing on $Z _ { \mathrm { i n v } } ^ { e }$ is better than on $Z ^ { e }$ , it can be shown that it is optimal, i.e., it solves equation (1) under Assumption 1 and achieves a value of $\sigma _ { \mathsf { s u p } } ^ { 2 }$ for the objective in equation (1).
+
+Invariant predictor. Define a linear representation map $\Phi : \mathbb { R } ^ { r \times d }$ (that transforms $X ^ { e }$ as $\Phi ( X ^ { e } ) ) ,$   
+and searc classifitations $\boldsymbol { w } : \mathbb { R } ^ { k \times r }$ on the represeis invariant (in tion ssum $w \cdot \Phi ( X ^ { e } ) )$ $\Phi$ ${ \mathbb E } [ Y ^ { e } | \bar { \Phi } ( X ^ { e } ) ]$ $\bar { \Phi } ( X ^ { e } ) = Z _ { \mathrm { i n v } } ^ { e }$ $\mathbb { E } [ Y ^ { e } | \Phi ( X ^ { e } ) ]$ $\Phi$   
+$w \cdot \Phi$ across the set of training environments $\mathcal { E } _ { t r }$ if there is a predictor $w$ that simultaneously achieves   
+the minimum risk, i.e., $w \in \arg \operatorname* { m i n } _ { \tilde { w } } R ^ { e } ( \tilde { w } \cdot \Phi ) , \forall e \in \mathcal { E } _ { t r }$ . The main objective of IRM is stated as
+
+$$
+\operatorname* { m i n } _ { w \in \mathbb { R } ^ { k \times r } , \Phi \in \mathbb { R } ^ { r \times d } } \frac { 1 } { | \mathcal { E } _ { t r } | } \sum _ { e \in \mathcal { E } _ { t r } } R ^ { e } ( w \cdot \Phi ) \quad \mathrm { s . t . } w \in \arg \operatorname* { m i n } _ { \tilde { w } \in \mathbb { R } ^ { k \times r } } R ^ { e } ( \tilde { w } \cdot \Phi ) , \ \forall e \in \mathcal { E } _ { t r } .
+$$
+
+Observe that if we drop the constraints in the above which search only over invariant predictors, then we get the standard empirical risk minimization (ERM) (Vapnik, 1992) (assuming all the training environments occur with equal probability). In all our theorems, we use 0-1 loss for binary classification $\mathcal { V } = \{ 0 , 1 \}$ and square loss for regression $\mathcal { V } = \mathbb { R }$ . For binary classification, the output of the predictor is given as $\mathsf { I } ( w \cdot \Phi ( X ^ { e } ) )$ , where $\mathsf { I } ( \cdot )$ is the indicator function that takes 1 if the input is $\geq 0$ and 0 otherwise, and the risk is $R ^ { e } ( w \cdot \Phi ) = \mathbb { E } \big [ | | ( w \cdot \Phi ( X ^ { e } ) ) - Y ^ { e } | \big ]$ . For regression, the output of the predictor is $w \cdot \Phi ( X ^ { e } )$ and the corresponding risk is $R ^ { e } ( w \cdot \Phi ) = \mathbb { E } \big [ ( w \cdot \Phi ( X ^ { e } ) - Y ^ { e } ) ^ { 2 } \big ]$ . We now present the main OOD generalization result from Arjovsky et al. (2019) for linear regressions.
+
+Theorem 1. (Informal) If Assumption $^ { l }$ is satisfied, $\mathsf { R a n k } [ \Phi ] > 0$ , $| \mathcal { E } _ { t r } | > 2 d ,$ , and $\mathcal { E } _ { t r }$ lie in a linear general position (a mild condition on the data in $\mathcal { E } _ { t r }$ , defined in the Appendix), then each solution to equation (3) achieves OOD generalization (solves equation (1), $\ b e \in \mathcal { E } _ { a l l }$ with $r i s k > \sigma _ { \mathsf { s u p } } ^ { 2 } ,$ .
+
+Despite the above guarantees, IRM has been shown to fail in several cases including linear SEMs in (Aubin et al., 2021). We take a closer look at these failures next.
+
+Understanding the failures: fully informative invariant features vs. partially informative invariant features (FIIF vs. PIIF). We define properties salient to the datasets/SEMs used in the OOD generalization literature. Each $e \in \mathcal { E } _ { a l l }$ , the distribution $( X ^ { e } , Y ^ { e } ) \sim { \mathbb { P } } ^ { e }$ satisfies the following properties. a) $\exists$ a map $\Phi ^ { * }$ (linear or not), which we call an invariant feature map, such that $\mathbb { E } \big [ Y ^ { e } \big | \Phi ^ { * } \big ( X ^ { e } \big ) \big ]$ is the same for all $e \in \mathcal { E } _ { a l l }$ and $Y ^ { e } \not \downarrow \Phi ^ { * } ( X ^ { e } )$ . These conditions ensure $\Phi ^ { * }$ maps to features that have a finite predictive power and have the same optimal predictor across ${ \mathcal { E } } _ { a l l }$ . For the SEM in Assumption 1, $\Phi ^ { * }$ maps to $Z _ { \mathrm { i n v } } ^ { e }$ . b) $\exists$ a map $\Psi ^ { * }$ (linear or not), which we call spurious feature map, such that $\mathbb { E } \big [ Y ^ { e } \big | \Psi ^ { * } \big ( X ^ { e } \big ) \big ]$ is not the same for all $e \in \mathcal { E } _ { a l l }$ and $Y ^ { e } \not \vdash \Psi ^ { * } ( X ^ { e } )$ for some environments. $\Psi ^ { * }$ often creates a hindrance in learning predictors that only rely on $\Phi ^ { * }$ . Note that $\Psi ^ { * }$ should not be a transformation of some $\Phi ^ { * }$ . For the SEM in Assumption 1, suppose $Z _ { \mathsf { s p u } } ^ { e }$ is anti-causally related to $Y ^ { e }$ , then $\Psi ^ { * }$ maps to $Z _ { \mathsf { s p u } } ^ { e }$ (See Appendix for an example).
+
+In the colored MNIST (CMNIST) dataset (Arjovsky et al., 2019), the digits are colored in such a way that in the training domain, color is highly predictive of the digit label but this correlation being spurious breaks down at test time. Suppose the invariant feature map $\Phi ^ { * }$ extracts the uncolored digit and the spurious feature map $\Psi ^ { * }$ extracts the background color. Ahuja et al. (2021) studied two variations of the colored MNIST dataset, which differed in the way final labels are generated from original MNIST labels (corrupted with noise or not). They showed that the IRM exhibits good OOD generalization $5 0 \%$ improvement over ERM) in anti-causal-CMNIST (AC-CMNIST, original data from Arjovsky et al. (2019)) but is no different from ERM and fails in covariate shift-CMNIST (CSCMNIST). In AC-CMNIST, the invariant features $\Phi ^ { * } ( X ^ { e } )$ (uncolored digit) are partially informative about the label, i.e., $Y \not \perp X ^ { e } | \Phi ^ { * } ( X ^ { e } )$ , and color contains information about label not contained in the uncolored digit. On the other hand in CS-CMNIST, invariant features are fully informative about the label, i.e., $Y \perp X ^ { e } | \Phi ^ { * } ( X ^ { e } )$ , i.e., they contains all the information about the label that is contained in input $X ^ { e }$ . Most human labelled datasets have fully informative invariant features; the labels (digit value) only depend on the invariant features (uncolored digit) and spurious features (color of the digit) do not affect the label. 5 In the rare case, when the humans are asked to label images in which the object being labelled itself is blurred, humans can rely on spurious features such as the background making such a data representative of PIIF setting. In Table 2, we divide the different datasets used in the literature based on informativeness of the invariant features. We observe that when the invariant features are fully informative, both IRM and ERM fail but only in classification tasks and not in regression tasks (Ahuja et al., 2021); this is consistent with the linear regression result in Theorem 1, where IRM succeeds regardless of whether $Y ^ { e } \perp X ^ { e } | Z _ { \mathsf { i n v } } ^ { e }$ holds or not. Motivated by this observation, we take a closer look at the classification tasks where invariant features are fully informative.
+
+<table><tr><td>Fully informative invariant features (FIF) ∀e ∈εau,Ye ⊥ Xe|Φ*(Xe)</td><td>Partially informative invariant features (PIIF) e∈εau Ye / Xe|Φ*(Xe)</td></tr><tr><td>Task:classification Example2/2S,CS-CMNIST</td><td>Task:classification or regression</td></tr><tr><td>SEM in Assumption 2</td><td>Example 1/1S,Example 3/3S,AC-CMNIST</td></tr><tr><td>ERM and IRMfail</td><td>SEM in Rosenfeld et al. (2021)</td></tr><tr><td>Theorem 3,4 (This paper)</td><td>ERMfails,IRM succeeds sometimes Theorem9,5.1 (Arjovsky et al.,2019;Rosenfeld etal.,2021)</td></tr></table>
+
+Table 2: Categorization of OOD evaluation datasets and SEMs. Example 1/1S, 2/2S, 3/3S from (Aubin et al., 2021), AC-CMNIST(Arjovsky et al., 2019), CS-CMNIST(Ahuja et al., 2021).
+
+# 3 OOD generalization theory for linear classification tasks
+
+A two-dimensional example with fully informative invariant features. We start with a 2D classification example (based on Nagarajan et al. (2021)), which can be understood as a simplified version of the CS-CMNIST dataset (Ahuja et al., 2021), Example 2/2S of Aubin et al. (2021), where both IRM and ERM fail. The example goes as follows. In each training environment $e \in \mathcal { E } _ { t r }$
+
+$Y ^ { e } \gets | \Big ( X _ { \mathsf { i n v } } ^ { e } - \frac { 1 } { 2 } \Big )$ , where $X _ { \mathrm { i n v } } ^ { e } \in \{ 0 , 1 \}$ is Bernoulli $\left( { \frac { 1 } { 2 } } \right)$ , $X _ { \mathsf { s p u } } ^ { e }  X _ { \mathsf { i n v } } ^ { e } \oplus W ^ { e }$ , where $W ^ { e } \in \{ 0 , 1 \}$ is Bernoulli $\left( 1 - p ^ { e } \right)$ with selection bias $p ^ { e } > \frac { 1 } { 2 }$ where Bernoulli $( a )$ takes value 1 with probability $a$ and 0 otherwise. Each training environment is characterized by the probability $p ^ { e }$ . Following Assumption 1, we assume that the labelling function does not change from $\mathcal { E } _ { t r }$ to ${ \mathcal { E } } _ { a l l }$ , thus the relation between the label and the invariant features does not change. Assume that the distribution of $X _ { \mathrm { i n v } } ^ { e }$ and $X _ { \mathsf { s p u } } ^ { e }$ can change arbitrarily. See Figure 1a) for a pictorial representation of this example illustrating the gist of the problem: there are many classifiers with the same error on $\mathcal { E } _ { t r }$ while only the one identical to the labelling function $\vert ( X _ { \mathrm { i n v } } ^ { e } - \frac { 1 } { 2 } )$ generalizes correctly OOD. Define a classifier $\begin{array} { r } { \mathsf { I } \big ( w _ { \mathsf { i n v } } x _ { \mathsf { i n v } } + w _ { \mathsf { s p u } } x _ { \mathsf { s p u } } - \frac { 1 } { 2 } \big ( w _ { \mathsf { i n v } } + w _ { \mathsf { s p u } } \big ) \big ) } \end{array}$ . Define a set of classifiers $\mathcal { S } = \{ ( w _ { \mathsf { i n v } } , w _ { \mathsf { s p u } } )$ s.t. $w _ { \mathsf { i n v } } > | w _ { \mathsf { s p u } } | \}$ . Observe that all the classifiers in $s$ achieve a zero classification error on the training environments. However, only classifiers for which $w _ { \mathsf { s p u } } = 0$ solve the OOD generalization (eq. (1)). With $\Phi$ as the identity, it can be shown that all the classifiers $s$ form an invariant predictor (satisfy the constraint in equation (3) over all the training environments when $\ell$ is the 0-1 loss). Observe that increasing the number of training environments to infinity does not address the problem, unlike with the linear regression result discussed in Theorem 1 (Arjovsky et al., 2019), where it was shown that if the number of environments increases linearly in the dimension of the data, then the solution to IRM also solves the OOD generalization (eq. (1)). 6 We use the above example to construct general SEMs for linear classification when the invariant features are fully informative. We follow the structure of the SEM from Assumption 1 in our construction.
+
+![](images/16c42a9b566c62313af5b6128943b376915fe626ce3a411fbef1120c46d26f93.jpg)  
+Figure 1: a) 2D classification example illustrating multiple invariant predictors: Most of these predictors rely on spurious features and each of them achieve zero error across all $\mathcal { E } _ { t r }$ , b) illustration of the impossibility result. If latent invariant features in the training environments are separable, then there are multiple equally good candidates that could have generated the data, and the algorithm cannot distinguish between these.
+
+# Assumption 2. Linear classification structural equation model (FIIF). In each $e \in \mathcal { E } _ { a l l }$
+
+$$
+\begin{array} { r l } & { Y ^ { e } \gets \mathsf { I } \big ( w _ { \mathsf { i n v } } ^ { * } \cdot Z _ { \mathsf { i n v } } ^ { e } \big ) \oplus N ^ { e } , \quad N ^ { e } \sim \mathsf { B e r n o u l i } ( q ) , q < \frac { 1 } { 2 } , \quad N ^ { e } \perp ( Z _ { \mathsf { i n v } } ^ { e } , Z _ { \mathsf { s p u } } ^ { e } ) , } \\ & { X ^ { e } \gets S \big ( Z _ { \mathsf { i n v } } ^ { e } , Z _ { \mathsf { s p u } } ^ { e } \big ) , } \end{array}
+$$
+
+where $\boldsymbol { w _ { \mathrm { i n v } } ^ { * } } \in \mathbb { R } ^ { m }$ with $\| w _ { \mathrm { i n v } } ^ { * } \| = 1$ is the labelling hyperplane, $Z _ { \mathrm { i n v } } ^ { e } \in \mathbb { R } ^ { m }$ , $Z _ { \mathsf { s p u } } ^ { e } \in \mathbb { R } ^ { o }$ , $N ^ { e }$ is binary noise with identical distribution across environments, $\oplus$ is the XOR operator, $S$ is invertible.
+
+If noise level $q$ is zero, then the above SEM covers linearly separable problems. See Figure 2a) for the directed acyclic graph (DAG) corresponding to this SEM. From the DAG observe that $Y ^ { e } \perp X ^ { e } | Z _ { \mathsf { i n v } } ^ { e }$ which implies that the invariant features are fully informative. Contrast this with a DAG that follows Assumption 1 shown in Figure 2b), where $Y ^ { e } \downarrow X ^ { e } | Z _ { \mathrm { i n v } } ^ { e }$ and thus the invariant features are not fully informative. If can change arb ${ \mathcal { E } } _ { a l l }$ follows the SEM in Assumption 2 and suppose the distribution of ly, then it can be shown that only a classifier identical to the labellin $Z _ { \mathrm { i n v } } ^ { e } , Z _ { \mathsf { s p u } } ^ { e }$ $\mathsf { I } ( w _ { \mathsf { i n v } } ^ { \ast } \cdot Z _ { \mathsf { i n v } } ^ { e } )$ can solve the OOD generalization (eq. (1)); such a classifier achieves an error of $q$ (noise level) in all the environments. As a result, if for a classifier we can find $e \in \mathcal { E } _ { a l l }$ that follows Assumption 2 where the error is greater than $q$ , then such a classifier does not solve equation (1). Now we ask – what are the minimal conditions on training environments $\mathcal { E } _ { t r }$ to achieve OOD generalization when ${ \mathcal { E } } _ { a l l }$ follow Assumption 2? To achieve OOD generalization for linear regressions, in Theorem 1, it was required that the number of training environments grows linearly in the dimension of the data. However, there was no restriction on the support of the latent invariant and latent spurious features, and they were allowed to change arbitrarily from train to test (for further discussion on this, see the Appendix). Can we continue to work with similar assumptions for the SEM in Assumption 2 and solve the OOD generalization (eq. (1))? We state some assumptions and notations to answer that. Define the support of the invariant (spurious) features $Z _ { \mathrm { i n v } } ^ { e } ( Z _ { \mathsf { s p u } } ^ { e } )$ in environment $e$ as $\mathcal { Z } _ { \mathrm { i n v } } ^ { e } ( \mathcal { Z } _ { \mathsf { s p u } } ^ { e } )$ .
+
+Assumption 3. Bounded invariant features. $\cup _ { e \in \mathcal { E } _ { t r } } \mathcal { Z } _ { \mathfrak { i n v } } ^ { e }$ is a bounded set.7
+
+Assumption 4. Bounded spurious features. $\cup _ { e \in { \mathcal E } _ { t r } } { \mathcal Z } _ { \mathsf { s p u } } ^ { e }$ is a bounded set.
+
+Assumption 5. Invariant feature support overlap. $\forall e \in \mathcal { E } _ { a l l } , \mathcal { Z } _ { \mathsf { i n v } } ^ { e } \subseteq \cup _ { e ^ { \prime } \in \mathcal { E } _ { t r } } \mathcal { Z } _ { \mathsf { i n v } } ^ { e ^ { \prime } }$   
+Assumption 6. Spurious feature support overlap. $\forall e \in \mathcal { E } _ { a l l } , \mathcal { Z } _ { \mathsf { s p u } } ^ { e } \subseteq \cup _ { e ^ { \prime } \in \mathcal { E } _ { t r } } \mathcal { Z } _ { \mathsf { s p u } } ^ { e ^ { \prime } }$
+
+Assumption 5 (6) states that the support of the invariant (spurious) features for unseen environments is the same as the union of the support over the training environments. It is important to note that support overlap does not imply that the distribution over the invariant features does not change. We now define a margin that measures how much the is training support of invariant features $Z _ { \mathrm { i n v } } ^ { e }$ separated by the labelling hyperplane $w _ { \mathrm { i n v } } ^ { \ast }$ . Define Inv-Margin $\begin{array} { r l } { } & { = \operatorname* { m i n } _ { z \in \cup _ { e \in \varepsilon _ { t r } } \mathcal { Z } _ { \mathrm { i n v } } ^ { e } } \mathsf { s g n } \big ( w _ { \mathsf { i n v } } ^ { * } \cdot z \big ) \big ( w _ { \mathsf { i n v } } ^ { * } \cdot z \big ) } \end{array}$ . This margin only coincides with the standard margin in support vector machines when the noise level $q$ is 0 (linearly separable) and $S$ is identity. If Inv-Margin $> 0$ , then the labelling hyperplane $w _ { \mathrm { i n v } } ^ { \ast }$ separates the support into two halves (see Figure 1b)).
+
+Assumption 7. Strictly separable invariant features. Inv-Margin $> 0$
+
+Next, we show the importance of support overlap for invariant features.
+
+Theorem 2. Impossibility of guaranteed OOD generalization for linear classification. Suppose each $e \in \mathcal { E } _ { a l l }$ follows Assumption 2. If for all the training environments $\mathcal { E } _ { t r }$ , the latent invariant features are bounded and strictly separable, i.e., Assumption 3 and 7 hold, then every deterministic algorithm fails to solve the OOD generalization (eq. (1)), i.e., for the output of every algorithm $\exists e \in \mathcal { E } _ { a l l }$ in which the error exceeds the minimum required value $q$ (noise level).
+
+The proofs to all the theorems are in the Appendix. We provide a high-level intuiton as to why invariant feature support overlap is crucial to the impossibility result. In Figure 1b), we show that if the support of latent invariant features are strictly separated by the labelling hyperplane $w _ { \mathrm { i n v } } ^ { \ast }$ , then we can find another valid hyperplane $w _ { \mathrm { i n v } } ^ { + }$ that is equally likely to have generated the same data. There is no algorithm that can distinguish between $w _ { \mathrm { i n v } } ^ { \ast }$ and $w _ { \mathrm { i n v } } ^ { + }$ . As a result, if we use data from the region where the hyperplanes disagree (yellow region Figure 1b)), then the algorithm fails.
+
+Significance of Theorem 2. We showed that without the support overlap assumption on the invariant features, OOD generalization is impossible for linear classification tasks. This is in contrast to linear regression in Theorem 1 (Arjovsky et al., 2019), where even in the absence of the support overlap assumption, guaranteed OOD generalization was possible. Applying the above Theorem 2 to the 2D case (eq. (4)) implies that we cannot assume that the support of invariant latent features can change, or else that case is also impossible to solve.
+
+Next, we ask what further assumptions are minimally needed to be able to solve the OOD generalization (eq. (1)). Each classifier can be written as $\bar { w } \cdot \dot { X ^ { e } } = \bar { w } \cdot S ( Z _ { \mathrm { i n v } } ^ { e } , Z _ { \mathrm { s p u } } ^ { e } ) = \tilde { w } _ { \mathrm { i n v } } \cdot Z _ { \mathrm { i n v } } ^ { e } + \tilde { w } _ { \mathrm { s p u } } Z _ { \mathrm { s p u } } ^ { e }$ . If $\tilde { w } _ { \mathsf { s p u } } \neq 0$ , then the classifier $\bar { w }$ is said to rely on spurious features.
+
+Theorem 3. Sufficiency and Insufficiency of ERM and IRM. Suppose each $\textit { e } \in \mathcal { E } _ { a l l }$ follows Assumption 2. Assume that a) the invariant features are strictly separable, bounded, and satisfy support overlap, $b$ ) the spurious features are bounded (Assumptions 3-5, 7 hold).
+
+• Sufficiency: If the spurious features satisfy support overlap (Assumption 6 holds), then both ERM and IRM solve the OOD generalization problem (eq. (1)). Also, there exist solutions to ERM and IRM solutions that rely on the spurious features and still achieve OOD generalization.
+
+• Insufficiency: If spurious features do not satisfy support overlap, then both ERM and IRM fail at solving the OOD generalization problem (eq. (1)). Also, there exist no such classifiers that rely on spurious features and also achieve OOD generalization.
+
+Significance of Theorem 3. From the first part, we learn that if the support overlap is satisfied for both the invariant features and the spurious features, then either ERM or IRM can solve the OOD generalization (eq. (1)). Interestingly, in this case we can have classifiers that rely on the spurious features and yet solve the OOD generalization (eq. (1)). For the 2D case (eq. (4)) this case implies that the entire set $s$ solves the OOD generalization (eq. (1)). From the second part, we learn that if support overlap holds for invariant features but not for spurious features, then the ideal OOD optimal predictors rely only on the invariant features. In this case, methods like ERM and IRM continue to rely on spurious features and fail at OOD generalization. For the above 2D case (eq. (4)) this implies that only the predictors that rely only on $X _ { \mathrm { i n v } } ^ { e }$ in the set $s$ solve the OOD generalization (eq. (1)).
+
+To summarize, we looked at SEMs for classification tasks when invariant features are fully informative, and find that the support overlap assumption over invariant features is necessary. Even in the presence of support overlap for invariant features, we showed that ERM and IRM can easily fail if the support overlap is violated for spurious features. This raises a natural question – Can we even solve the case with the support overlap assumption only on the invariant features? We will now show that the information bottleneck principle can help tackle these cases.
+
+# 4 Information bottleneck principle meets invariance principle
+
+Why the information bottleneck? The information bottleneck principle prescribes to learn a representation that compresses the input $X$ as much as possible while preserving all the relevant information about the target label $Y$ (Tishby et al., 2000). Mutual information $I ( { \bar { X } } ; \Phi ( X ) )$ is used to measure information compression. If representation $\Phi ( X )$ is a deterministic transformation of $X$ , then in principle we can use the entropy of $\Phi ( X )$ to measure compression (Kirsch et al., 2020). Let us revisit the 2D case (eq. (4)) and apply this principle to it. Following the second part of Theorem 3, where ERM and IRM failed, assume that invariant features satisfy the support overlap assumption, but make no such assumption for the spurious features. Consider three choices for $\Phi$ : identity (selects both features), selects invariant feature only, selects spurious feature only. The entropy of $\dot { H } ( \Phi ( X ^ { e } ) )$ when $\Phi$ is the identity is $H ( p ^ { e } ) + \log ( 2 )$ , where $\bar { H ( p ^ { e } ) }$ is the Shannon entropy in Bernoulli $( p ^ { e } )$ . If $\Phi$ selects the invariant/spurious features only, then $H ( \Phi ( X ^ { e } ) ) = \log ( 2 )$ . Among all three choices, the one that has the least entropy and also achieves zero error is the representation that focuses on the invariant feature. We could find the OOD optimal predictor in this example just by using information bottleneck. Does it mean the invariance principle isn’t needed? We answer this next.
+
+![](images/d7e9c47c2763b3601fbf394b12401d20b00afacedae56e0a8a224df7da6e7244.jpg)  
+Figure 2: Comparison of the DAG from Assumption 2 (fully informative invariant features) vs. DAGs from Rosenfeld et al. (2021); Arjovsky et al. (2019) (partially informative invariant features).
+
+Whand sider a simple classification SEM. In each , where all the random variables involve $e \in \mathcal { E } _ { t r }$ , i $Y ^ { e } \gets X _ { \mathsf { i n v } } ^ { 1 , e } \oplus X _ { \mathsf { i n v } } ^ { 2 , e } \oplus N ^ { e }$ $X _ { \mathsf { s p u } } ^ { e }  Y ^ { e } \oplus V ^ { e }$ $N ^ { e } , V ^ { e }$ are Berthen in li with parameters predictions based $q$ (in cal across are bette $\mathcal { E } _ { t r }$ ), a $c ^ { e }$ (varies across predictions ba $\mathcal { E } _ { t r }$ ) re on If . $c ^ { e } < q$ , $\mathcal { E } _ { t r }$ $X _ { \mathsf { s p u } } ^ { e }$ $\bar { X } _ { \mathfrak { i n v } } ^ { 1 , e } , X _ { \mathfrak { i n v } } ^ { 2 , e }$ $X _ { \mathrm { i n v } } ^ { 1 , e } , X _ { \mathrm { i n v } } ^ { 2 , e }$ formation band not on are uniform Bernoulli, then these features have a higher entropy than neck would bar using . Invariance constrai $X _ { \mathsf { i n v } } ^ { 1 , e } , X _ { \mathsf { i n v } } ^ { 2 , e }$ . Instead, we want the modge the model to focus on $X _ { \mathsf { s p u } } ^ { e }$ . In this case, $X _ { \mathsf { i n v } } ^ { 1 , e }$ $X _ { \mathsf { i n v } } ^ { 2 , e }$ $X _ { \mathsf { s p u } } ^ { e }$ $X _ { \mathsf { i n v } } ^ { 1 , e }$ $X _ { \mathsf { i n v } } ^ { 2 , e }$ example, observe that invariant features are partially informative unlike the 2D case (eq. (4)).
+
+Why invariance and information bottleneck? We have illustrated through simple examples when the information bottleneck is needed but not invariance and vice-versa. We now provide a simple example where both these constraints are needed at the same time. This example combines the 2D case (eq. (4)) and the example we highlighted in the paragraph above: $Y ^ { e } \gets X _ { \mathsf { i n v } } ^ { e } \oplus N ^ { e }$ , $X _ { \mathsf { s p u } } ^ { 1 , e } \gets X _ { \mathsf { i n v } } ^ { e } \oplus W ^ { e }$ , and $X _ { \mathsf { s p u } } ^ { 2 , e } \gets Y ^ { e } \oplus V ^ { e }$ . In this case, the invariance constraint does not allow representations that use information bottleneck c $X _ { \mathsf { s p u } } ^ { 2 , e }$ but does not prohibit representations that rely oints on top ensure that representations that only use $X _ { \mathsf { s p u } } ^ { 1 , e }$ . However,re used. We $X _ { \mathrm { i n v } } ^ { e }$ now describe an objective 8 that combines both these principles:
+
+$$
+\operatorname* { m i n } _ { w , \Phi } \sum _ { e \in { \mathcal E } _ { t r } } h ^ { e } \big ( w \cdot \Phi \big ) \quad \mathrm { s . t . } \ \frac { 1 } { | { \mathcal E } _ { t r } | } \sum _ { e \in { \mathcal E } _ { t r } } R ^ { e } \big ( w \cdot \Phi \big ) \leq r ^ { \mathrm { t h } } , \ w \in \arg \operatorname* { m i n } _ { \tilde { w } \in \mathbb R ^ { k \times r } } R ^ { e } \big ( \tilde { w } \cdot \Phi \big ) , \forall e \in { \mathcal E } _ { t r } ,
+$$
+
+where $h ^ { e }$ in the above is a lower bounded differential entropy defined below and $r ^ { \mathrm { t h } }$ is the threshold on the average risk. Typical information bottleneck based optimization in neural networks involves minimization of the entropy of the representation output from a certain hidden layer. For both analytical convenience and also because the above setup is a linear model, we work with the simplest form of bottleneck which directly minimizes the entropy of the output layer. Recall the definition of differential entropy of a random variable $X$ $\quad , h ( X ) = - \mathbf { \bar { \mathbb { E } } } _ { X } [ \log d \mathbb { P } _ { X } ]$ and $d \mathbb { P } _ { X }$ is the Radon-Nikodym derivative of $\mathbb { P } _ { X }$ with respect to Lebesgue measure. Because in general differential entropy has no lower bound, we add a small independent noise term $\zeta$ (Kirsch et al., 2020) to the classifier to ensure that the entropy is bounded below. We call the above optimization information bottleneck based invariant risk minimization (IB-IRM). In summary, among all the highly predictive invariant predictors we pick the ones that have the least entropy. If we drop the invariance constraint from the above optimization, we get information bottleneck based empirical risk minimization (IB-ERM). In the above formulation and following result, we assume that $X ^ { e }$ are continuous random variables; the results continue to hold for discrete $X ^ { e }$ as well (See Appendix for details).
+
+# Theorem 4. IB-IRM and IB-ERM vs. IRM and ERM
+
+8Results extend to alternate objective with information bottleneck constraints and average risk as objective.
+
+• Fully informative invariant features (FIIF). Suppose each $e \in \mathcal { E } _ { a l l }$ follows Assumption 2. Assume that the invariant features are strictly separable, bounded, and satisfy support overlap (Assumptions 3,5 and 7 hold). Also, for each $e \in \mathcal { E } _ { t r }$ $Z _ { \mathsf { s p u } } ^ { e }  A Z _ { \mathsf { i n v } } ^ { e } + W ^ { e }$ , where $A \in \mathbb { R } ^ { o \times m }$ , $W ^ { e } \in \mathbb { R } ^ { o }$ is continuous, bounded, and zero mean noise. Each solution to $I B$ -IRM (eq. (6), with $\ell$ as 0-1 loss, and $r ^ { \mathrm { t h } } = q ,$ ), and IB-ERM solves the OOD generalization (eq. (1)) but ERM and IRM (eq.(3)) fail.
+
+• Partially informative invariant features $( P I I F )$ . Suppose each $e \in \mathcal { E } _ { a l l }$ follows Assumption 1 and $\exists \ e \in { \mathcal { E } } _ { t r }$ such that $\mathbb { E } [ \epsilon ^ { e } Z _ { \mathsf { s p u } } ^ { e } ] \neq 0$ . If $| \mathcal { E } _ { t r } | > 2 d$ and the set $\mathcal { E } _ { t r }$ lies in a linear general position (a mild condition defined in the Appendix), then each solution to IB-IRM (eq. (6), with $\ell$ as square loss, $\sigma _ { \epsilon } ^ { 2 } < r ^ { \mathsf { t h } } \le \sigma _ { Y } ^ { 2 }$ , where $\sigma _ { Y } ^ { 2 }$ and $\sigma _ { \epsilon } ^ { 2 }$ are the variance in the label and noise across $\mathcal { E } _ { t r }$ ) and IRM (eq.(3)) solves OOD generalization (eq. (1)) but IB-ERM and ERM fail.
+
+Significance of Theorem 4 and remarks. In the first part (FIIF), IB-ERM and IB-IRM succeed without assuming support overlap for the spurious features, which was crucial for success of ERM and IRM in Theorem 3. This establishes that support overlap of spurious features is not a necessary condition. Observe that when invariant features are fully informative, IB-ERM and IB-IRM succeed, but when invariant features are partially informative IB-IRM and IRM succeed. In real data settings, we do not know if the invariant features are fully or partially informative. Since IB-IRM is the only common winner in both the settings, it would be pragmatic to use it in the absence of domain knowledge about the informativeness of the invariant features. In the paragraph preceding the objective in equation (6), we discussed examples where both the IB and IRM constraints were needed at the same time. In the Appendix, we generalize that example and show that if we change the assumptions in linear classification SEM in Assumption 2 such that the invariant features are partially informative, then we see the joint benefit of IB and IRM constraints. At this point, it is also worth pointing to a result in Rosenfeld et al. (2021), which focused on linear classification SEMs (DAG shown in Figure 2c) with partially informative invariant features. Under the assumption of complete support overlap for spurious and invariant features, authors showed IRM succeeds.
+
+# 4.1 Proposed approach
+
+We take the three terms from the optimization in equation (6) and create a weighted combination as
+
+$$
+\sum _ { \epsilon } \Big ( R ^ { e } ( \Phi ) + \lambda \| \nabla _ { w , w = 1 , 0 } R ^ { e } ( w \cdot \Phi ) \| ^ { 2 } + \nu h ^ { e } ( \Phi ) \Big ) \leq \sum _ { \epsilon } \Big ( R ^ { e } ( \Phi ) + \lambda \| \nabla _ { w , w = 1 , 0 } R ^ { e } ( w \cdot \Phi ) \| ^ { 2 } + \nu h ( \Phi ) \Big ) .
+$$
+
+In the LHS above, the first term corresponds to the risks across environments, the second term approximates invariance constraint (follows the IRMv1 objective (Arjovsky et al., 2019)), and the third term is the entropy of the classifier in each environment.
+
+In the RHS, $h ( \Phi )$ is the entropy of $\Phi$ unconditional on the environment (the entropy on the left-hand side is entropy conditional on the environment assuming all the environments are equally likely). Optimizing over differential entropy is not easy, and thus we resort to minimizing an upper bound of it (Kirsch et al., 2020). We use the standard result that among all continuous random variables with the same variance, Gaussian has the maximum differential entropy. Since the entropy of Gaussian increases with its variance, we use the variance of $\Phi$ instead of the differential entropy (For further details, see the Appendix). Our final objective is given as
+
+$$
+\sum _ { e } \Big ( R ^ { e } ( \Phi ) + \lambda \| \nabla _ { w , w = 1 . 0 } R ^ { e } ( w \cdot \Phi ) \| ^ { 2 } + \gamma \mathsf { V a r } ( \Phi ) \Big ) .
+$$
+
+# On the behavior of gradient descent with and without informa
+
+![](images/8a3cf0e7bca3d1fa872223a8320e95f17f3937c66e0aca73f9415c50e7db929d.jpg)  
+Figure 3: Comparing convergence of $\frac { \vert w _ { \mathsf { s p u } } \vert } { \sqrt { w _ { \mathsf { s p u } } ^ { 2 } + w _ { \mathsf { i n v } } ^ { 2 } } }$ (metric from Nagarajan et al. (2021)) for average selection bias $p = 0 . 9$ .
+
+tion bottleneck. In the entire discussion so far, we have focused on ensuring that the set of optimal solutions to the desired objective (IB-IRM, IB-ERM, etc.) correspond to the solutions of the OOD generalization problem (eq. (1)). In some simple cases, such as the 2D case (eq. (4)), it can be shown that gradient descent is biased towards selecting the ideal classifier (Soudry et al., 2018; Nagarajan et al., 2021). Even though gradient descent can eventually learn the ideal classifier that only relies on the invariant features, training is frustratingly slow as was shown by Nagarajan et al. (2021). In the next theorem, we characterize the impact of using IB penalty $( \mathsf { V a r } ( \Phi ) )$ in the 2D example (eq. (4)). We compare the methods in terms of | wspu(t)winv(t) |, which was the metric used in Nagarajan et al. (2021); $w _ { \mathsf { s p u } } ( t )$ and $w _ { \mathsf { i n v } } ( t )$ are the weights for the spurious feature and the invariant feature at time $t$ of training (assuming training happens with continuous time gradient descent).
+
+Theorem 5. Impact of $\pmb { I B }$ on learning speed. Suppose each $\textit { e } \in \mathcal { E } _ { t r }$ follows the $2 D$ case from equation (4). Set $\lambda = 0$ , $\gamma > 0$ in equation (7) to get the $I B$ -ERM objective with $\ell$ as exponential loss. Continuous-time gradient descent on this IB-ERM objective achieves $| \frac { w _ { \mathsf { s p u } } ( t ) } { w _ { \mathsf { i n v } } ( t ) } | \leq \epsilon$ in time less than $\frac { W _ { 0 } ( \frac { 1 } { 2 \gamma } ) } { 2 ( 1 - p ) \epsilon }$ $W _ { 0 } ( \cdot )$ denotes the principal branch of the Lambert $W$ function), while in the same time the ratio for ERM $\begin{array} { r } { \vert \frac { w _ { \mathrm { s p u } } ( t ) } { w _ { \mathrm { i n v } } ( t ) } \vert \ge \ln ( \frac { 1 + 2 p } { 3 - 2 p } ) / \ln \left( 1 + \frac { W _ { 0 } ( \frac { 1 } { 2 \gamma } ) } { 2 ( 1 - p ) \epsilon } \right) } \end{array}$ W0( 12γ )2(1−p) 
, where p = 1|Etr | Pe∈Etr pe .
+
+$| \frac { w _ { \mathsf { s p u } } ( t ) } { w _ { \mathsf { i n v } } ( t ) } |$ converges to zero for both methods, but it converges much faster for IB-ERM (for $p =$ $0 . 9 , \epsilon = 0 . 0 0 1 , \gamma = 0 . 5 8$ , the ratio for IB-ERM is $| \frac { w _ { \mathsf { s p u } } ( t ) } { w _ { \mathsf { i n v } } ( t ) } | \leq 0 . 0 0 1$ and ratio for ERM is $| \frac { w _ { \mathsf { s p u } } ( t ) } { w _ { \mathsf { i n v } } ( t ) } | \geq$ 0.09). In the above theorem, we analyzed the impact of information bottleneck only. The convergence analysis for both the penalties jointly comes with its own challenges, and we hope to explore this in future work. However, we carried out experiments with gradient descent on all the objectives for the 2D example (eq. (4)). See Figure 3 for the comparisons.
+
+# 5 Experiments
+
+Methods, datasets & metrics. We compare our approaches – information bottleneck based ERM (IBERM) and information bottleneck based IRM (IB-IRM) with ERM and IRM. We also compare with an Oracle model trained on data where spurious features are permuted to remove spurious correlations. We use all the datasets in Table 2, Terra Incognita dataset (Beery et al., 2018), and COCO (Ahmed et al., 2021). We follow the same protocol for tuning hyperparameters from Aubin et al. (2021); Arjovsky et al. (2019) for their respective datasets (see the Appendix for more details). As is reported in literature, for Example 2/2S, Example 3/3S we use classification error and for AC-CMNIST, CS-CMNIST, Terra Incognita, and COCO we use accuracy. For Example 1/1S, we use mean square error (MSE). The code for experiments can be found at https://github.com/ahujak/IB-IRM.
+
+Summary of results. In Table 3, we provide a comparison of methods for different examples in linear unit tests (Aubin et al., 2021) for three and six training environments. In Table 4, we provide a comparison of the methods for different CMNIST datasets, Terra Incognita and COCO dataset. Based on our Theorem 4, we do not expect ERM and IB-ERM to do well on Example 1/1S, Example 3/3S and AC-CMNIST as these datasets fall in the PIIF category, i.e, the invariant features are partially informative. On these examples, we find that IRM and IB-IRM do better than ERM and IB-ERM (for Example 3/3S when there are three environments all methods perform poorly). Based on our Theorem 4, we do not expect IRM and ERM to do well on Example 2/2S, CS-CMNIST, Terra Incognita and COCO dataset,9 as these datasets fall in the FIIF category, i.e., the invariant features are fully informative. On these FIIF examples, we find that IB-ERM always performs well (close to oracle), and in some cases IB-IRM also performs well. Our experiments confirm that IB penalty has a crucial role to play in FIIF settings and IRMv1 penalty has a crucial role to play in PIIF settings (to further this claim, we provide an ablation study in the Appendix). On Example 1/1S, AC-CMNIST, we find that IB-IRM is able to extract the benefit of IRMv1 penalty. On CS-CMNIST and Example 2/2S we find that IB-IRM is able to extract the benefit of IB penalty. In settings such as COCO dataset, where IB-IRM does not perform as well as IB-ERM, better hyperparameter tuning strategies should be able to help IB-IRM adapt and put a higher weight on IB penalty. Overall, we can conclude that IB-ERM improves over ERM (significantly in FIIF and marginally in PIIF settings), and IB-IRM improves over IRM (improves in FIIF settings and retains advantages in PIIF settings).
+
+Remark. As we move from three to six environments, we observe that MSE in Example 1/1S exhibits a larger variance. This is because of the way data is generated, the new environments that are sampled have labels that have a higher noise level (we follow the same procedure as in Aubin et al. (2021)).
+
+# 6 Extensions, limitations, and future work
+
+Extension to non-linear models and multi-class classification. In this work our theoretical analysis focused on linear models. Consider the map $X  S ( Z _ { \mathsf { i n v } } , Z _ { \mathsf { s p u } } )$ in Assumption 2. Suppose $S$ is non-linear and bijective. We can divide the learning task into two parts a) invert $S$ to obtain $Z _ { \mathrm { i n v } }$ , $Z _ { \mathsf { s p u } }$ and b) learn a linear model that only relies on the invariant features $Z _ { \mathrm { i n v } }$ to predict the label $Y$ . For part b), we can rely on the approaches proposed in this work. For part a), we need to leverage advancements in the field of non-linear ICA (Khemakhem et al., 2020). The current state-of-the-art to solve part a) requires strong structural assumptions on the dependence between all the components of $Z _ { \mathrm { i n v } }$ , $Z _ { \mathsf { s p u } }$ (Lu et al., 2021). Therefore, solving part a) and part b) in conjunction with minimal assumptions forms an exciting future work. In the entire work, the discussion was focused on binary classification tasks and regression tasks. For multi-class classification settings, we consider natural extension of the SEM in Assumption 2 (See the Appendix) and our main results continue to hold.
+
+<table><tr><td></td><td>#Envs</td><td>ERM</td><td>IB-ERM</td><td>IRM</td><td>IB-IRM</td><td>Oracle</td></tr><tr><td>Example1</td><td>3</td><td>13.36 ±1.49</td><td>12.96 ±1.30</td><td>11.15± 0.71</td><td>11.68 ± 0.90</td><td>10.42±0.16</td></tr><tr><td>Example1s</td><td>3</td><td>13.33 ± 1.49</td><td>12.92 ± 1.30</td><td>11.07 ± 0.68</td><td>11.74 ± 1.03</td><td>10.45±0.19</td></tr><tr><td>Example2</td><td>3</td><td>0.42 ± 0.01</td><td>0.00±0.00</td><td>0.45 ± 0.00</td><td>0.00 ±0.00</td><td>0.00 ±0.00</td></tr><tr><td>Example2s</td><td>3</td><td>0.45 ± 0.01</td><td>0.00 ± 0.01</td><td>0.45 ± 0.01</td><td>0.06 ± 0.12</td><td>0.00 ± 0.00</td></tr><tr><td>Example3</td><td>3</td><td>0.48 ± 0.07</td><td>0.49 ± 0.06</td><td>0.48 ± 0.07</td><td>0.48 ± 0.07</td><td>0.01 ± 0.00</td></tr><tr><td>Example3s</td><td>3</td><td>0.49 ± 0.06</td><td>0.49 ± 0.06</td><td>0.49 ± 0.07</td><td>0.49 ± 0.07</td><td>0.01 ±0.00</td></tr><tr><td>Example1</td><td>6</td><td>33.74 ± 60.18</td><td>32.03 ± 57.05</td><td>23.04 ± 40.64</td><td>25.66 ± 45.96</td><td>22.21±39.25</td></tr><tr><td>Example1s</td><td>6</td><td>33.62 ± 59.80</td><td>31.92 ± 56.70</td><td>22.92 ± 40.60</td><td>25.60 ± 45.62</td><td>22.13±38.93</td></tr><tr><td>Example2</td><td>6</td><td>0.37 ± 0.06</td><td>0.02 ± 0.05</td><td>0.46 ± 0.01</td><td>0.43 ± 0.11</td><td>0.00±0.00</td></tr><tr><td>Example2s</td><td>6</td><td>0.46 ± 0.01</td><td>0.02 ± 0.06</td><td>0.46 ± 0.01</td><td>0.45 ± 0.10</td><td>0.00±0.00</td></tr><tr><td>Example3</td><td>6</td><td>0.33 ± 0.18</td><td>0.26 ± 0.20</td><td>0.14 ± 0.18</td><td>0.19 ± 0.19</td><td>0.01±0.00</td></tr><tr><td>Example3s</td><td>6</td><td>0.36 ±0.19</td><td>0.27 ± 0.20</td><td>0.14 ± 0.18</td><td>0.19 ± 0.19</td><td>0.01±0.00</td></tr></table>
+
+Table 3: Comparisons on linear unit tests in terms of mean square error (regression) and classification error (classification). “#Envs” means the number of training environments.
+
+<table><tr><td></td><td>ERM</td><td>IB-ERM</td><td>IRM</td><td>IB-IRM</td></tr><tr><td>CS-CMNIST</td><td>60.27 ± 1.21</td><td>71.80 ± 0.69</td><td>61.49 ± 1.45</td><td>71.79 ± 0.70</td></tr><tr><td>AC-CMNIST</td><td>16.84 ± 0.82</td><td>50.24 ± 0.47</td><td>66.98 ± 1.65</td><td>67.67 ± 1.78</td></tr><tr><td>Terra Incognita</td><td>49.80 ± 4.40</td><td>56.40 ± 2.10</td><td>54.60 ± 1.30</td><td>54.10 ± 2.00</td></tr><tr><td>COCO</td><td>22.70 ± 1.04</td><td>31.66 ± 2.39</td><td>18.47 ± 10.20</td><td>25.10 ± 1.03</td></tr></table>
+
+Table 4: Classification accuracy percentage on colored MNISTs, Terra Incognita and COCO dataset.
+
+On the choice for IB penalty and IRMv1 penalty. We use the approximation for entropy (in equation (7)) described in Kirsch et al. (2020). The approximation (even though an upper bound) serves as an effective proxy for the true information bottleneck as shown in the experiments in Kirsch et al. (2020) (e.g., see their experiment on Imagenette dataset). Also, our experiments validate this approximation even in moderately high dimensions, as an example in CS-CMNIST, the dimension of the layer at which bottleneck constraints are applied is 256. Developing tighter approximations for information bottleneck in high dimensions and analyzing their impact on OOD generalization is an important future work. In recent works (Rosenfeld et al., 2021; Kamath et al., 2021; Gulrajani and Lopez-Paz, 2021), there has been criticism of different aspects of IRM, e.g., failure of IRMv1 penalty in non-linear models, the tuning of IRMv1 penalty, etc. Since we use IRMv1 penalty in our proposed loss, these criticisms apply to our objective as well. Other approximations of invariance have been proposed in the literature (Koyama and Yamaguchi, 2020; Ahuja et al., 2020; Chang et al., 2020). Exploring their benefits together with information bottleneck is a fruitful future work. Before concluding, we want to remark that we have already discussed the closest related works. However, we also provide a detailed discussion of the broader related literature in the Appendix.
+
+# 7 Conclusion
+
+In this work, we revisited the fundamental assumptions for OOD generalization for settings when invariant features capture all the information about the label. We showed how linear classification tasks are different and need much stronger assumptions than linear regression tasks. We provide a sharp characterization of performance of ERM and IRM under different assumptions on support overlap of invariant and spurious features. We showed that support overlap of invariant features is necessary or otherwise OOD generalization is impossible. However, ERM and IRM seem to fail even in the absence of support overlap of spurious features. We prove that a form of the information bottleneck constraint along with invariance goes a long way in overcoming the failures while retaining the existing provable guarantees.
+
+# Acknowledgements
+
+We thank Reyhane Askari Hemmat, Adam Ibrahim, Alexia Jolicoeur-Martineau, Divyat Mahajan, Ryan D’Orazio, Nicolas Loizou, Manuela Girotti, and Charles Guille-Escuret for the feedback. Kartik Ahuja would also like to thank Karthikeyan Shanmugam for discussions pertaining to the related works.
+
+# Funding disclosure
+
+We would like to thank Samsung Electronics Co., Ldt. for funding this research. Kartik Ahuja acknowledges the support provided by IVADO postdoctoral fellowship funding program. Yoshua Bengio acknowledges the support from CIFAR and IBM. Ioannis Mitliagkas acknowledges support from an NSERC Discovery grant (RGPIN-2019-06512), a Samsung grant, Canada CIFAR AI chair and MSR collaborative research grant. Irina Rish acknowledges the support from Canada CIFAR AI Chair Program and from the Canada Excellence Research Chairs Program. We thank Compute Canada for providing computational resources.
+
+# References
+
+Ahmed, F., Bengio, Y., van Seijen, H., and Courville, A. (2021). Systematic generalisation with group invariant predictions. In International Conference on Learning Representations.   
+Ahuja, K., Shanmugam, K., Varshney, K., and Dhurandhar, A. (2020). Invariant risk minimization games. In International Conference on Machine Learning, pages 145–155. PMLR.   
+Ahuja, K., Wang, J., Dhurandhar, A., Shanmugam, K., and Varshney, K. R. (2021). Empirical or invariant risk minimization? a sample complexity perspective. In International Conference on Learning Representations.   
+Arjovsky, M., Bottou, L., Gulrajani, I., and Lopez-Paz, D. (2019). Invariant risk minimization. arXiv preprint arXiv:1907.02893.   
+Aubin, B., Słowik, A., Arjovsky, M., Bottou, L., and Lopez-Paz, D. (2021). Linear unit-tests for invariance discovery. arXiv preprint arXiv:2102.10867.   
+Beery, S., Van Horn, G., and Perona, P. (2018). Recognition in terra incognita. In Proceedings of the European Conference on Computer Vision, pages 456–473.   
+Chang, S., Zhang, Y., Yu, M., and Jaakkola, T. S. (2020). Invariant rationalization. In International Conference on Machine Learning, 2020.   
+DeGrave, A. J., Janizek, J. D., and Lee, S.-I. (2020). AI for radiographic COVID-19 detection selects shortcuts over signal. medRxiv.   
+Geirhos, R., Jacobsen, J.-H., Michaelis, C., Zemel, R., Brendel, W., Bethge, M., and Wichmann, F. A. (2020). Shortcut learning in deep neural networks. Nature Machine Intelligence, 2(11):665–673.   
+Gulrajani, I. and Lopez-Paz, D. (2021). In search of lost domain generalization. In International Conference on Learning Representations.   
+Kamath, P., Tangella, A., Sutherland, D. J., and Srebro, N. (2021). Does invariant risk minimization capture invariance? arXiv preprint arXiv:2101.01134.   
+Khemakhem, I., Kingma, D., Monti, R., and Hyvarinen, A. (2020). Variational autoencoders and nonlinear ica: A unifying framework. In International Conference on Artificial Intelligence and Statistics, pages 2207–2217. PMLR.   
+Kirsch, A., Lyle, C., and Gal, Y. (2020). Unpacking information bottlenecks: Unifying informationtheoretic objectives in deep learning. arXiv preprint arXiv:2003.12537.   
+Koyama, M. and Yamaguchi, S. (2020). Out-of-distribution generalization with maximal invariant predictor. arXiv preprint arXiv:2008.01883.   
+Krueger, D., Caballero, E., Jacobsen, J.-H., Zhang, A., Binas, J., Zhang, D., Priol, R. L., and Courville, A. (2020). Out-of-distribution generalization via risk extrapolation (rex). arXiv preprint arXiv:2003.00688.   
+Lu, C., Wu, Y., Hernández-Lobato, J. M., and Schölkopf, B. (2021). Nonlinear invariant risk minimization: A causal approach. arXiv preprint arXiv:2102.12353.   
+Nagarajan, V., Andreassen, A., and Neyshabur, B. (2021). Understanding the failure modes of out-of-distribution generalization. In International Conference on Learning Representations.   
+Pearl, J. (1995). Causal diagrams for empirical research. Biometrika, 82(4):669–688.   
+Peters, J., Bühlmann, P., and Meinshausen, N. (2015). Causal inference using invariant prediction: identification and confidence intervals. arXiv preprint arXiv:1501.01332.   
+Pezeshki, M., Kaba, S.-O., Bengio, Y., Courville, A., Precup, D., and Lajoie, G. (2020). Gradient starvation: A learning proclivity in neural networks. arXiv preprint arXiv:2011.09468.   
+Robey, A., Pappas, G. J., and Hassani, H. (2021). Model-based domain generalization. arXiv preprint arXiv:2102.11436.   
+Rojas-Carulla, M., Schölkopf, B., Turner, R., and Peters, J. (2018). Invariant models for causal transfer learning. The Journal of Machine Learning Research, 19(1):1309–1342.   
+Rosenfeld, E., Ravikumar, P. K., and Risteski, A. (2021). The risks of invariant risk minimization. In International Conference on Learning Representations.   
+Soudry, D., Hoffer, E., Nacson, M. S., Gunasekar, S., and Srebro, N. (2018). The implicit bias of gradient descent on separable data. The Journal of Machine Learning Research, 19(1):2822–2878.   
+Tishby, N., Pereira, F. C., and Bialek, W. (2000). The information bottleneck method. arXiv preprint physics/0004057.   
+Vapnik, V. (1992). Principles of risk minimization for learning theory. In Advances in neural information processing systems, pages 831–838.   
+Zhang, D., Ahuja, K., Xu, Y., Wang, Y., and Courville, A. C. (2021). Can subnetwork structure be the key to out-of-distribution generalization? In ICML.
+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] See Section 2-5 and the additional details such as the proofs in the supplementary material.   
+(b) Did you describe the limitations of your work? [Yes] See Section 4.1 and Section 6.   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Section A.1 in the Appendix in the supplementary material.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Section 2-4.   
+(b) Did you include complete proofs of all theoretical results? [Yes] See the Appendix in the Supplementary Material.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] See https://github.com/ahujak/IB-IRM
+
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section A.2 in the Appendix in the supplementary material.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] See Section A.2 in the Appendix in the supplementary material.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Section A.2 in the Appendix in the supplementary material.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] We use the codes from following github repositories https://github.com/ facebookresearch/DomainBed, https://github.com/facebookresearch/ InvariantRiskMinimization and https://github.com/facebookresearch/ InvarianceUnitTests and we have cited the creators in the Section A.2 in the Appendix in the supplementary material.   
+(b) Did you mention the license of the assets? [Yes] All the repositories mentioned above use MIT license. We have mentioned this in Section A.2 in the Appendix in the supplementary material.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] We have included code for our experiments in the supplementary material.   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/train/jlchsFOLfeF/jlchsFOLfeF_content_list.json

@@ -0,0 +1,1286 @@
+[
+    {
+        "type": "text",
+        "text": "Invariance Principle Meets Information Bottleneck for Out-of-Distribution Generalization ",
+        "text_level": 1,
+        "bbox": [
+            191,
+            122,
+            810,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Kartik Ahuja† ",
+        "bbox": [
+            240,
+            220,
+            341,
+            236
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ethan Caballero∗ † ",
+        "bbox": [
+            418,
+            220,
+            547,
+            236
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Dinghuai Zhang∗ † ",
+        "bbox": [
+            622,
+            220,
+            756,
+            237
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Jean-Christophe Gagnon-Audet † ",
+        "bbox": [
+            202,
+            256,
+            434,
+            272
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Yoshua Bengio † ",
+        "bbox": [
+            506,
+            257,
+            619,
+            272
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ioannis Mitliagkas† ",
+        "bbox": [
+            658,
+            256,
+            794,
+            272
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Irina Rish† ",
+        "bbox": [
+            462,
+            292,
+            539,
+            308
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            343,
+            535,
+            359
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The invariance principle from causality is at the heart of notable approaches such as invariant risk minimization (IRM) that seek to address out-of-distribution (OOD) generalization failures. Despite the promising theory, invariance principle-based approaches fail in common classification tasks, where invariant (causal) features capture all the information about the label. Are these failures due to the methods failing to capture the invariance? Or is the invariance principle itself insufficient? To answer these questions, we revisit the fundamental assumptions in linear regression tasks, where invariance-based approaches were shown to provably generalize OOD. In contrast to the linear regression tasks, we show that for linear classification tasks we need much stronger restrictions on the distribution shifts, or otherwise OOD generalization is impossible. Furthermore, even with appropriate restrictions on distribution shifts in place, we show that the invariance principle alone is insufficient. We prove that a form of the information bottleneck constraint along with invariance helps address key failures when invariant features capture all the information about the label and also retains the existing success when they do not. We propose an approach that incorporates both of these principles and demonstrate its effectiveness in several experiments. ",
+        "bbox": [
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+            375,
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+            608
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+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            633,
+            310,
+            651
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recent years have witnessed an explosion of examples showing deep learning models are prone to exploiting shortcuts (spurious features) (Geirhos et al., 2020; Pezeshki et al., 2020) which make them fail to generalize out-of-distribution (OOD). In Beery et al. (2018), a convolutional neural network was trained to classify camels from cows; however, it was found that the model relied on the background color (e.g., green pastures for cows) and not on the properties of the animals (e.g., shape). These examples become very concerning when they occur in real-life applications (e.g., COVID-19 detection (DeGrave et al., 2020)). ",
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+        "text": "To address these out-of-distribution generalization failures, invariant risk minimization (Arjovsky et al., 2019) and several other works were proposed (Ahuja et al., 2020; Pezeshki et al., 2020; Krueger et al., 2020; Robey et al., 2021; Zhang et al., 2021). The invariance principle from causality (Peters et al., 2015; Pearl, 1995) is at the heart of these works. The principle distinguishes predictors that only rely on the causes of the label from those that do not. The optimal predictor that only focuses on the causes is invariant and min-max optimal (Rojas-Carulla et al., 2018; Koyama and Yamaguchi, 2020; Ahuja et al., 2021) under many distribution shifts but the same is not true for other predictors. ",
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+        "text": "Our contributions. Despite the promising theory, invariance principle-based approaches fail in settings (Aubin et al., 2021) where invariant features capture all information about the label contained in the input. A particular example is image classification (e.g., cow vs. camel) (Beery et al., 2018) where the label is a deterministic function of the invariant features (e.g., shape of the animal), and does not depend on the spurious features (e.g., background). To understand such failures, we revisit the fundamental assumptions in linear regression tasks, where invariance-based approaches were shown to provably generalize OOD. We show that, in contrast to the linear regression tasks, OOD generalization is significantly harder for linear classification tasks; we need much stronger restrictions in the form of support overlap assumptions3 on the distribution shifts, or otherwise it is not possible to guarantee OOD generalization under interventions on variables other than the target class. We then proceed to show that, even under the right assumptions on distribution shifts, the invariance principle is insufficient. However, we establish that information bottleneck (IB) constraints (Tishby et al., 2000), together with the invariance principle, provably works in both settings – when invariant features completely capture the information about the label and also when they do not. (Table 1 summarizes our theoretical results presented later). We propose an approach that combines both these principles and demonstrate its effectiveness on linear unit tests (Aubin et al., 2021) and on different real datasets. ",
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+        "table_footnote": [
+            "Table 1: Summary of the new and existing results (Arjovsky et al., 2019; Rosenfeld et al., 2021). IB-ERM (IRM): information bottleneck - empirical (invariant) risk minimization ERM (IRM). "
+        ],
+        "table_body": "<table><tr><td>Task</td><td>Invariant features capture label info</td><td>Support overlap invariant features</td><td>Support overlap spurious features</td><td>OOD generalization guarantee (εtr→εall) ERM IRM IB-ERM</td><td></td><td>IB-IRM</td></tr><tr><td rowspan=\"5\">Linear Classification</td><td>Full/Partial</td><td>No</td><td>Yes/No</td><td></td><td>Impossible for any algorithm to generalize OOD [Thm2]</td><td rowspan=\"5\">[Thm3,4]</td></tr><tr><td>Full</td><td>Yes</td><td>No</td><td>区 X</td><td>√</td></tr><tr><td>Partial</td><td>Yes</td><td>No</td><td>X X</td><td>? X</td></tr><tr><td>Full</td><td>Yes</td><td>Yes</td><td>√</td><td>√ √</td></tr><tr><td>Partial</td><td>Yes</td><td>Yes</td><td>√ X √</td><td>? √</td></tr><tr><td>Linear</td><td>Full</td><td>No</td><td>No</td><td>√</td><td>× √</td><td></td></tr><tr><td>Regression</td><td>Partial</td><td>No</td><td>No</td><td>区</td><td>X</td><td>√ √ [Thm4]</td></tr></table>",
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+        "type": "text",
+        "text": "2 OOD generalization and invariance: background & failures ",
+        "text_level": 1,
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+        "text": "Background. We consider a supervised training data $D$ gathered from a set of training environments $\\mathcal { E } _ { t r }$ : $D = \\{ D ^ { e } \\} _ { e \\in \\mathcal { E } _ { t r } }$ , where $\\bar { D ^ { e } } = \\{ x _ { i } ^ { e } , y _ { i } ^ { e } \\} _ { i = 1 } ^ { n ^ { e } }$ is the dataset from environment $e \\in \\mathcal { E } _ { t r }$ and $n ^ { e }$ is the number of instances in environment $e$ . $x _ { i } ^ { e } \\in \\mathbb { R } ^ { d }$ and $y _ { i } ^ { e } \\in \\mathcal { V } \\subseteq \\mathbb { R } ^ { k }$ correspond to the input feature value and the label for $i ^ { t h }$ instance respectively. Each $( x _ { i } ^ { e } , y _ { i } ^ { e } )$ is an i.i.d. draw from $\\mathbb { P } ^ { e }$ , where $\\mathbb { P } ^ { e }$ is the joint distribution of the input feature and the label in environment $e$ . Let $\\mathcal { X } ^ { e }$ be the support of the input feature values in the environment $e$ . The goal of OOD generalization is to use training data $D$ to construct a predictor $f : \\mathbb { R } ^ { d }  \\mathbb { R } ^ { k }$ that performs well across many unseen environments in ${ \\mathcal { E } } _ { a l l }$ , where $\\mathcal { E } _ { a l l } \\supset \\mathcal { E } _ { t r }$ . Define the risk of $f$ in environment $e$ as $R ^ { e } ( f ) \\doteq \\mathbb { E } \\bigl [ \\ell ( f ( X ^ { e } ) , Y ^ { e } ) \\bigr ]$ , where for example $\\ell$ can be 0-1 loss, logistic loss, square loss, $( X ^ { e } , Y ^ { e } ) \\sim { \\mathbb { P } } ^ { e }$ , and the expectation $\\mathbb { E }$ is w.r.t. $\\mathbb { P } ^ { e }$ . Formally stated, our goal is to use the data from training environments $\\mathcal { E } _ { t r }$ to find $f : \\mathbb { R } ^ { d }  \\mathcal { V }$ to minimize ",
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+        "type": "equation",
+        "img_path": "images/82247be238930160b96568cbc9aca673eb0d046f9fef292a5f231e2e482b05fd.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { f } \\operatorname* { m a x } _ { e \\in { \\mathscr { E } } _ { a l l } } R ^ { e } ( f ) .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "So far we did not state any restrictions on ${ \\mathcal { E } } _ { a l l }$ . Consider binary classification: without any restrictions on ${ \\mathcal { E } } _ { a l l }$ , no method can reduce the above objective ( $\\ell$ is 0-1 loss) to below one. Suppose a method outputs $f ^ { * }$ ; if $\\exists e \\in \\mathcal { E } _ { a l l } \\ \\backslash \\mathcal { E } _ { t r }$ with labels based on $1 - f ^ { * }$ , then it achieves an error of one. Some assumptions on ${ \\mathcal { E } } _ { a l l }$ are thus necessary. Consider how ${ \\mathcal { E } } _ { a l l }$ is restricted using invariance for linear regressions (Arjovsky et al., 2019). ",
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+        "type": "text",
+        "text": "Assumption 1. Linear regression structural equation model (SEM). In each $e \\in \\mathcal { E } _ { a l l }$ ",
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+        "type": "equation",
+        "img_path": "images/bfdd42655287578461fbf6504c939bb8e458cf08851bfb035e28af6a79f64286.jpg",
+        "text": "$$\n\\begin{array} { r l } & { Y ^ { e } \\gets w _ { \\mathrm { i n v } } ^ { * } \\cdot Z _ { \\mathrm { i n v } } ^ { e } + \\epsilon ^ { e } , \\quad Z _ { \\mathrm { i n v } } ^ { e } \\perp \\epsilon ^ { e } , \\quad \\mathbb { E } [ \\epsilon ^ { e } ] = 0 , \\mathbb { E } \\big [ | \\epsilon ^ { e } | ^ { 2 } \\big ] \\leq \\sigma _ { \\mathrm { s u p } } ^ { 2 } } \\\\ & { X ^ { e } \\gets S ( Z _ { \\mathrm { i n v } } ^ { e } , Z _ { \\mathrm { s p u } } ^ { e } ) } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where $\\boldsymbol { w _ { \\mathrm { i n v } } ^ { * } } \\in \\mathbb { R } ^ { m }$ , $Z _ { \\mathsf { i n v } } ^ { e } \\in \\mathbb { R } ^ { m }$ , $Z _ { \\mathsf { s p u } } \\in \\mathbb { R } ^ { o }$ , $S \\in \\mathbb { R } ^ { d \\times ( m + o ) }$ , $S$ is invertible $( m + o = d )$ . We focus on invertible $S$ but several results extend to non-invertible $S$ as well (see Appendix). ",
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+        "type": "text",
+        "text": "Assumption 1 states how $Y ^ { e }$ and $X ^ { e }$ are generated from latent invariant features $Z _ { \\mathrm { i n v } } ^ { e \\mathrm { ~ 4 ~ } }$ , latent spurious features $Z _ { \\mathsf { s p u } } ^ { e }$ and noise $\\epsilon ^ { e }$ . The relationship between label and invariant features is invariant, i.e., $w _ { \\mathrm { i n v } } ^ { \\ast }$ is fixed across all environments. However, the distributions of $Z _ { \\mathrm { i n v } } ^ { e }$ , $Z _ { \\mathsf { s p u } } ^ { e }$ , and $\\epsilon ^ { e }$ are allowed to change arbitrarily across all the environments. Suppose is identity. If we regress only on the invariant features $Z _ { \\mathrm { i n v } } ^ { e }$ , then the optimal solution is $w _ { \\mathrm { i n v } } ^ { \\ast }$ , which is independent of the environment, and the error it achieves is bounded above by the variance of $\\epsilon ^ { e } ( \\sigma _ { \\mathsf { s u p } } ^ { 2 } )$ . If we regress on the entire $Z ^ { e }$ and the optimal predictor places a non-zero weight on $Z _ { \\mathsf { s p u } } ^ { e }$ sup(e.g., $Z _ { \\mathsf { s p u } } ^ { e } \\gets Y ^ { e } + \\zeta ^ { e } )$ , then this predictor fails to solve equation (1) ( $\\exists e \\in { \\mathcal { E } } _ { a l l }$ , $Z _ { \\mathsf { s p u } } ^ { e } \\to \\infty$ , error $ \\infty$ , see Appendix for details). Also, not only regressing on $Z _ { \\mathrm { i n v } } ^ { e }$ is better than on $Z ^ { e }$ , it can be shown that it is optimal, i.e., it solves equation (1) under Assumption 1 and achieves a value of $\\sigma _ { \\mathsf { s u p } } ^ { 2 }$ for the objective in equation (1). ",
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+    {
+        "type": "text",
+        "text": "Invariant predictor. Define a linear representation map $\\Phi : \\mathbb { R } ^ { r \\times d }$ (that transforms $X ^ { e }$ as $\\Phi ( X ^ { e } ) ) ,$   \nand searc classifitations $\\boldsymbol { w } : \\mathbb { R } ^ { k \\times r }$ on the represeis invariant (in tion ssum $w \\cdot \\Phi ( X ^ { e } ) )$ $\\Phi$ ${ \\mathbb E } [ Y ^ { e } | \\bar { \\Phi } ( X ^ { e } ) ]$ $\\bar { \\Phi } ( X ^ { e } ) = Z _ { \\mathrm { i n v } } ^ { e }$ $\\mathbb { E } [ Y ^ { e } | \\Phi ( X ^ { e } ) ]$ $\\Phi$   \n$w \\cdot \\Phi$ across the set of training environments $\\mathcal { E } _ { t r }$ if there is a predictor $w$ that simultaneously achieves   \nthe minimum risk, i.e., $w \\in \\arg \\operatorname* { m i n } _ { \\tilde { w } } R ^ { e } ( \\tilde { w } \\cdot \\Phi ) , \\forall e \\in \\mathcal { E } _ { t r }$ . The main objective of IRM is stated as ",
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+    {
+        "type": "equation",
+        "img_path": "images/b913c53ad184ff951ca63d5befca877bddba13960bb439a0899b7718ed057f54.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { w \\in \\mathbb { R } ^ { k \\times r } , \\Phi \\in \\mathbb { R } ^ { r \\times d } } \\frac { 1 } { | \\mathcal { E } _ { t r } | } \\sum _ { e \\in \\mathcal { E } _ { t r } } R ^ { e } ( w \\cdot \\Phi ) \\quad \\mathrm { s . t . } w \\in \\arg \\operatorname* { m i n } _ { \\tilde { w } \\in \\mathbb { R } ^ { k \\times r } } R ^ { e } ( \\tilde { w } \\cdot \\Phi ) , \\ \\forall e \\in \\mathcal { E } _ { t r } .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Observe that if we drop the constraints in the above which search only over invariant predictors, then we get the standard empirical risk minimization (ERM) (Vapnik, 1992) (assuming all the training environments occur with equal probability). In all our theorems, we use 0-1 loss for binary classification $\\mathcal { V } = \\{ 0 , 1 \\}$ and square loss for regression $\\mathcal { V } = \\mathbb { R }$ . For binary classification, the output of the predictor is given as $\\mathsf { I } ( w \\cdot \\Phi ( X ^ { e } ) )$ , where $\\mathsf { I } ( \\cdot )$ is the indicator function that takes 1 if the input is $\\geq 0$ and 0 otherwise, and the risk is $R ^ { e } ( w \\cdot \\Phi ) = \\mathbb { E } \\big [ | | ( w \\cdot \\Phi ( X ^ { e } ) ) - Y ^ { e } | \\big ]$ . For regression, the output of the predictor is $w \\cdot \\Phi ( X ^ { e } )$ and the corresponding risk is $R ^ { e } ( w \\cdot \\Phi ) = \\mathbb { E } \\big [ ( w \\cdot \\Phi ( X ^ { e } ) - Y ^ { e } ) ^ { 2 } \\big ]$ . We now present the main OOD generalization result from Arjovsky et al. (2019) for linear regressions. ",
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+        "type": "text",
+        "text": "Theorem 1. (Informal) If Assumption $^ { l }$ is satisfied, $\\mathsf { R a n k } [ \\Phi ] > 0$ , $| \\mathcal { E } _ { t r } | > 2 d ,$ , and $\\mathcal { E } _ { t r }$ lie in a linear general position (a mild condition on the data in $\\mathcal { E } _ { t r }$ , defined in the Appendix), then each solution to equation (3) achieves OOD generalization (solves equation (1), $\\ b e \\in \\mathcal { E } _ { a l l }$ with $r i s k > \\sigma _ { \\mathsf { s u p } } ^ { 2 } ,$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Despite the above guarantees, IRM has been shown to fail in several cases including linear SEMs in (Aubin et al., 2021). We take a closer look at these failures next. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Understanding the failures: fully informative invariant features vs. partially informative invariant features (FIIF vs. PIIF). We define properties salient to the datasets/SEMs used in the OOD generalization literature. Each $e \\in \\mathcal { E } _ { a l l }$ , the distribution $( X ^ { e } , Y ^ { e } ) \\sim { \\mathbb { P } } ^ { e }$ satisfies the following properties. a) $\\exists$ a map $\\Phi ^ { * }$ (linear or not), which we call an invariant feature map, such that $\\mathbb { E } \\big [ Y ^ { e } \\big | \\Phi ^ { * } \\big ( X ^ { e } \\big ) \\big ]$ is the same for all $e \\in \\mathcal { E } _ { a l l }$ and $Y ^ { e } \\not \\downarrow \\Phi ^ { * } ( X ^ { e } )$ . These conditions ensure $\\Phi ^ { * }$ maps to features that have a finite predictive power and have the same optimal predictor across ${ \\mathcal { E } } _ { a l l }$ . For the SEM in Assumption 1, $\\Phi ^ { * }$ maps to $Z _ { \\mathrm { i n v } } ^ { e }$ . b) $\\exists$ a map $\\Psi ^ { * }$ (linear or not), which we call spurious feature map, such that $\\mathbb { E } \\big [ Y ^ { e } \\big | \\Psi ^ { * } \\big ( X ^ { e } \\big ) \\big ]$ is not the same for all $e \\in \\mathcal { E } _ { a l l }$ and $Y ^ { e } \\not \\vdash \\Psi ^ { * } ( X ^ { e } )$ for some environments. $\\Psi ^ { * }$ often creates a hindrance in learning predictors that only rely on $\\Phi ^ { * }$ . Note that $\\Psi ^ { * }$ should not be a transformation of some $\\Phi ^ { * }$ . For the SEM in Assumption 1, suppose $Z _ { \\mathsf { s p u } } ^ { e }$ is anti-causally related to $Y ^ { e }$ , then $\\Psi ^ { * }$ maps to $Z _ { \\mathsf { s p u } } ^ { e }$ (See Appendix for an example). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the colored MNIST (CMNIST) dataset (Arjovsky et al., 2019), the digits are colored in such a way that in the training domain, color is highly predictive of the digit label but this correlation being spurious breaks down at test time. Suppose the invariant feature map $\\Phi ^ { * }$ extracts the uncolored digit and the spurious feature map $\\Psi ^ { * }$ extracts the background color. Ahuja et al. (2021) studied two variations of the colored MNIST dataset, which differed in the way final labels are generated from original MNIST labels (corrupted with noise or not). They showed that the IRM exhibits good OOD generalization $5 0 \\%$ improvement over ERM) in anti-causal-CMNIST (AC-CMNIST, original data from Arjovsky et al. (2019)) but is no different from ERM and fails in covariate shift-CMNIST (CSCMNIST). In AC-CMNIST, the invariant features $\\Phi ^ { * } ( X ^ { e } )$ (uncolored digit) are partially informative about the label, i.e., $Y \\not \\perp X ^ { e } | \\Phi ^ { * } ( X ^ { e } )$ , and color contains information about label not contained in the uncolored digit. On the other hand in CS-CMNIST, invariant features are fully informative about the label, i.e., $Y \\perp X ^ { e } | \\Phi ^ { * } ( X ^ { e } )$ , i.e., they contains all the information about the label that is contained in input $X ^ { e }$ . Most human labelled datasets have fully informative invariant features; the labels (digit value) only depend on the invariant features (uncolored digit) and spurious features (color of the digit) do not affect the label. 5 In the rare case, when the humans are asked to label images in which the object being labelled itself is blurred, humans can rely on spurious features such as the background making such a data representative of PIIF setting. In Table 2, we divide the different datasets used in the literature based on informativeness of the invariant features. We observe that when the invariant features are fully informative, both IRM and ERM fail but only in classification tasks and not in regression tasks (Ahuja et al., 2021); this is consistent with the linear regression result in Theorem 1, where IRM succeeds regardless of whether $Y ^ { e } \\perp X ^ { e } | Z _ { \\mathsf { i n v } } ^ { e }$ holds or not. Motivated by this observation, we take a closer look at the classification tasks where invariant features are fully informative. ",
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+    {
+        "type": "table",
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+        "table_caption": [],
+        "table_footnote": [
+            "Table 2: Categorization of OOD evaluation datasets and SEMs. Example 1/1S, 2/2S, 3/3S from (Aubin et al., 2021), AC-CMNIST(Arjovsky et al., 2019), CS-CMNIST(Ahuja et al., 2021). "
+        ],
+        "table_body": "<table><tr><td>Fully informative invariant features (FIF) ∀e ∈εau,Ye ⊥ Xe|Φ*(Xe)</td><td>Partially informative invariant features (PIIF) e∈εau Ye / Xe|Φ*(Xe)</td></tr><tr><td>Task:classification Example2/2S,CS-CMNIST</td><td>Task:classification or regression</td></tr><tr><td>SEM in Assumption 2</td><td>Example 1/1S,Example 3/3S,AC-CMNIST</td></tr><tr><td>ERM and IRMfail</td><td>SEM in Rosenfeld et al. (2021)</td></tr><tr><td>Theorem 3,4 (This paper)</td><td>ERMfails,IRM succeeds sometimes Theorem9,5.1 (Arjovsky et al.,2019;Rosenfeld etal.,2021)</td></tr></table>",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "3 OOD generalization theory for linear classification tasks ",
+        "text_level": 1,
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+        "text": "A two-dimensional example with fully informative invariant features. We start with a 2D classification example (based on Nagarajan et al. (2021)), which can be understood as a simplified version of the CS-CMNIST dataset (Ahuja et al., 2021), Example 2/2S of Aubin et al. (2021), where both IRM and ERM fail. The example goes as follows. In each training environment $e \\in \\mathcal { E } _ { t r }$ ",
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+        "type": "text",
+        "text": "$Y ^ { e } \\gets | \\Big ( X _ { \\mathsf { i n v } } ^ { e } - \\frac { 1 } { 2 } \\Big )$ , where $X _ { \\mathrm { i n v } } ^ { e } \\in \\{ 0 , 1 \\}$ is Bernoulli $\\left( { \\frac { 1 } { 2 } } \\right)$ , $X _ { \\mathsf { s p u } } ^ { e }  X _ { \\mathsf { i n v } } ^ { e } \\oplus W ^ { e }$ , where $W ^ { e } \\in \\{ 0 , 1 \\}$ is Bernoulli $\\left( 1 - p ^ { e } \\right)$ with selection bias $p ^ { e } > \\frac { 1 } { 2 }$ where Bernoulli $( a )$ takes value 1 with probability $a$ and 0 otherwise. Each training environment is characterized by the probability $p ^ { e }$ . Following Assumption 1, we assume that the labelling function does not change from $\\mathcal { E } _ { t r }$ to ${ \\mathcal { E } } _ { a l l }$ , thus the relation between the label and the invariant features does not change. Assume that the distribution of $X _ { \\mathrm { i n v } } ^ { e }$ and $X _ { \\mathsf { s p u } } ^ { e }$ can change arbitrarily. See Figure 1a) for a pictorial representation of this example illustrating the gist of the problem: there are many classifiers with the same error on $\\mathcal { E } _ { t r }$ while only the one identical to the labelling function $\\vert ( X _ { \\mathrm { i n v } } ^ { e } - \\frac { 1 } { 2 } )$ generalizes correctly OOD. Define a classifier $\\begin{array} { r } { \\mathsf { I } \\big ( w _ { \\mathsf { i n v } } x _ { \\mathsf { i n v } } + w _ { \\mathsf { s p u } } x _ { \\mathsf { s p u } } - \\frac { 1 } { 2 } \\big ( w _ { \\mathsf { i n v } } + w _ { \\mathsf { s p u } } \\big ) \\big ) } \\end{array}$ . Define a set of classifiers $\\mathcal { S } = \\{ ( w _ { \\mathsf { i n v } } , w _ { \\mathsf { s p u } } )$ s.t. $w _ { \\mathsf { i n v } } > | w _ { \\mathsf { s p u } } | \\}$ . Observe that all the classifiers in $s$ achieve a zero classification error on the training environments. However, only classifiers for which $w _ { \\mathsf { s p u } } = 0$ solve the OOD generalization (eq. (1)). With $\\Phi$ as the identity, it can be shown that all the classifiers $s$ form an invariant predictor (satisfy the constraint in equation (3) over all the training environments when $\\ell$ is the 0-1 loss). Observe that increasing the number of training environments to infinity does not address the problem, unlike with the linear regression result discussed in Theorem 1 (Arjovsky et al., 2019), where it was shown that if the number of environments increases linearly in the dimension of the data, then the solution to IRM also solves the OOD generalization (eq. (1)). 6 We use the above example to construct general SEMs for linear classification when the invariant features are fully informative. We follow the structure of the SEM from Assumption 1 in our construction. ",
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+            "Figure 1: a) 2D classification example illustrating multiple invariant predictors: Most of these predictors rely on spurious features and each of them achieve zero error across all $\\mathcal { E } _ { t r }$ , b) illustration of the impossibility result. If latent invariant features in the training environments are separable, then there are multiple equally good candidates that could have generated the data, and the algorithm cannot distinguish between these. "
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+        "type": "text",
+        "text": "Assumption 2. Linear classification structural equation model (FIIF). In each $e \\in \\mathcal { E } _ { a l l }$ ",
+        "text_level": 1,
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+        "img_path": "images/b532d04e66db61709084dc8211b4bf438a8846ea365f0e28930f0f2183e7ba08.jpg",
+        "text": "$$\n\\begin{array} { r l } & { Y ^ { e } \\gets \\mathsf { I } \\big ( w _ { \\mathsf { i n v } } ^ { * } \\cdot Z _ { \\mathsf { i n v } } ^ { e } \\big ) \\oplus N ^ { e } , \\quad N ^ { e } \\sim \\mathsf { B e r n o u l i } ( q ) , q < \\frac { 1 } { 2 } , \\quad N ^ { e } \\perp ( Z _ { \\mathsf { i n v } } ^ { e } , Z _ { \\mathsf { s p u } } ^ { e } ) , } \\\\ & { X ^ { e } \\gets S \\big ( Z _ { \\mathsf { i n v } } ^ { e } , Z _ { \\mathsf { s p u } } ^ { e } \\big ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\boldsymbol { w _ { \\mathrm { i n v } } ^ { * } } \\in \\mathbb { R } ^ { m }$ with $\\| w _ { \\mathrm { i n v } } ^ { * } \\| = 1$ is the labelling hyperplane, $Z _ { \\mathrm { i n v } } ^ { e } \\in \\mathbb { R } ^ { m }$ , $Z _ { \\mathsf { s p u } } ^ { e } \\in \\mathbb { R } ^ { o }$ , $N ^ { e }$ is binary noise with identical distribution across environments, $\\oplus$ is the XOR operator, $S$ is invertible. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "If noise level $q$ is zero, then the above SEM covers linearly separable problems. See Figure 2a) for the directed acyclic graph (DAG) corresponding to this SEM. From the DAG observe that $Y ^ { e } \\perp X ^ { e } | Z _ { \\mathsf { i n v } } ^ { e }$ which implies that the invariant features are fully informative. Contrast this with a DAG that follows Assumption 1 shown in Figure 2b), where $Y ^ { e } \\downarrow X ^ { e } | Z _ { \\mathrm { i n v } } ^ { e }$ and thus the invariant features are not fully informative. If can change arb ${ \\mathcal { E } } _ { a l l }$ follows the SEM in Assumption 2 and suppose the distribution of ly, then it can be shown that only a classifier identical to the labellin $Z _ { \\mathrm { i n v } } ^ { e } , Z _ { \\mathsf { s p u } } ^ { e }$ $\\mathsf { I } ( w _ { \\mathsf { i n v } } ^ { \\ast } \\cdot Z _ { \\mathsf { i n v } } ^ { e } )$ can solve the OOD generalization (eq. (1)); such a classifier achieves an error of $q$ (noise level) in all the environments. As a result, if for a classifier we can find $e \\in \\mathcal { E } _ { a l l }$ that follows Assumption 2 where the error is greater than $q$ , then such a classifier does not solve equation (1). Now we ask – what are the minimal conditions on training environments $\\mathcal { E } _ { t r }$ to achieve OOD generalization when ${ \\mathcal { E } } _ { a l l }$ follow Assumption 2? To achieve OOD generalization for linear regressions, in Theorem 1, it was required that the number of training environments grows linearly in the dimension of the data. However, there was no restriction on the support of the latent invariant and latent spurious features, and they were allowed to change arbitrarily from train to test (for further discussion on this, see the Appendix). Can we continue to work with similar assumptions for the SEM in Assumption 2 and solve the OOD generalization (eq. (1))? We state some assumptions and notations to answer that. Define the support of the invariant (spurious) features $Z _ { \\mathrm { i n v } } ^ { e } ( Z _ { \\mathsf { s p u } } ^ { e } )$ in environment $e$ as $\\mathcal { Z } _ { \\mathrm { i n v } } ^ { e } ( \\mathcal { Z } _ { \\mathsf { s p u } } ^ { e } )$ . ",
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+        "type": "text",
+        "text": "Assumption 3. Bounded invariant features. $\\cup _ { e \\in \\mathcal { E } _ { t r } } \\mathcal { Z } _ { \\mathfrak { i n v } } ^ { e }$ is a bounded set.7 ",
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+        "type": "text",
+        "text": "Assumption 4. Bounded spurious features. $\\cup _ { e \\in { \\mathcal E } _ { t r } } { \\mathcal Z } _ { \\mathsf { s p u } } ^ { e }$ is a bounded set. ",
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+        "type": "text",
+        "text": "Assumption 5. Invariant feature support overlap. $\\forall e \\in \\mathcal { E } _ { a l l } , \\mathcal { Z } _ { \\mathsf { i n v } } ^ { e } \\subseteq \\cup _ { e ^ { \\prime } \\in \\mathcal { E } _ { t r } } \\mathcal { Z } _ { \\mathsf { i n v } } ^ { e ^ { \\prime } }$   \nAssumption 6. Spurious feature support overlap. $\\forall e \\in \\mathcal { E } _ { a l l } , \\mathcal { Z } _ { \\mathsf { s p u } } ^ { e } \\subseteq \\cup _ { e ^ { \\prime } \\in \\mathcal { E } _ { t r } } \\mathcal { Z } _ { \\mathsf { s p u } } ^ { e ^ { \\prime } }$ ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Assumption 5 (6) states that the support of the invariant (spurious) features for unseen environments is the same as the union of the support over the training environments. It is important to note that support overlap does not imply that the distribution over the invariant features does not change. We now define a margin that measures how much the is training support of invariant features $Z _ { \\mathrm { i n v } } ^ { e }$ separated by the labelling hyperplane $w _ { \\mathrm { i n v } } ^ { \\ast }$ . Define Inv-Margin $\\begin{array} { r l } { } & { = \\operatorname* { m i n } _ { z \\in \\cup _ { e \\in \\varepsilon _ { t r } } \\mathcal { Z } _ { \\mathrm { i n v } } ^ { e } } \\mathsf { s g n } \\big ( w _ { \\mathsf { i n v } } ^ { * } \\cdot z \\big ) \\big ( w _ { \\mathsf { i n v } } ^ { * } \\cdot z \\big ) } \\end{array}$ . This margin only coincides with the standard margin in support vector machines when the noise level $q$ is 0 (linearly separable) and $S$ is identity. If Inv-Margin $> 0$ , then the labelling hyperplane $w _ { \\mathrm { i n v } } ^ { \\ast }$ separates the support into two halves (see Figure 1b)). ",
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+        "type": "text",
+        "text": "Assumption 7. Strictly separable invariant features. Inv-Margin $> 0$ ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Next, we show the importance of support overlap for invariant features. ",
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+        "type": "text",
+        "text": "Theorem 2. Impossibility of guaranteed OOD generalization for linear classification. Suppose each $e \\in \\mathcal { E } _ { a l l }$ follows Assumption 2. If for all the training environments $\\mathcal { E } _ { t r }$ , the latent invariant features are bounded and strictly separable, i.e., Assumption 3 and 7 hold, then every deterministic algorithm fails to solve the OOD generalization (eq. (1)), i.e., for the output of every algorithm $\\exists e \\in \\mathcal { E } _ { a l l }$ in which the error exceeds the minimum required value $q$ (noise level). ",
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+        "type": "text",
+        "text": "The proofs to all the theorems are in the Appendix. We provide a high-level intuiton as to why invariant feature support overlap is crucial to the impossibility result. In Figure 1b), we show that if the support of latent invariant features are strictly separated by the labelling hyperplane $w _ { \\mathrm { i n v } } ^ { \\ast }$ , then we can find another valid hyperplane $w _ { \\mathrm { i n v } } ^ { + }$ that is equally likely to have generated the same data. There is no algorithm that can distinguish between $w _ { \\mathrm { i n v } } ^ { \\ast }$ and $w _ { \\mathrm { i n v } } ^ { + }$ . As a result, if we use data from the region where the hyperplanes disagree (yellow region Figure 1b)), then the algorithm fails. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Significance of Theorem 2. We showed that without the support overlap assumption on the invariant features, OOD generalization is impossible for linear classification tasks. This is in contrast to linear regression in Theorem 1 (Arjovsky et al., 2019), where even in the absence of the support overlap assumption, guaranteed OOD generalization was possible. Applying the above Theorem 2 to the 2D case (eq. (4)) implies that we cannot assume that the support of invariant latent features can change, or else that case is also impossible to solve. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Next, we ask what further assumptions are minimally needed to be able to solve the OOD generalization (eq. (1)). Each classifier can be written as $\\bar { w } \\cdot \\dot { X ^ { e } } = \\bar { w } \\cdot S ( Z _ { \\mathrm { i n v } } ^ { e } , Z _ { \\mathrm { s p u } } ^ { e } ) = \\tilde { w } _ { \\mathrm { i n v } } \\cdot Z _ { \\mathrm { i n v } } ^ { e } + \\tilde { w } _ { \\mathrm { s p u } } Z _ { \\mathrm { s p u } } ^ { e }$ . If $\\tilde { w } _ { \\mathsf { s p u } } \\neq 0$ , then the classifier $\\bar { w }$ is said to rely on spurious features. ",
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+    {
+        "type": "text",
+        "text": "Theorem 3. Sufficiency and Insufficiency of ERM and IRM. Suppose each $\\textit { e } \\in \\mathcal { E } _ { a l l }$ follows Assumption 2. Assume that a) the invariant features are strictly separable, bounded, and satisfy support overlap, $b$ ) the spurious features are bounded (Assumptions 3-5, 7 hold). ",
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+        "type": "text",
+        "text": "• Sufficiency: If the spurious features satisfy support overlap (Assumption 6 holds), then both ERM and IRM solve the OOD generalization problem (eq. (1)). Also, there exist solutions to ERM and IRM solutions that rely on the spurious features and still achieve OOD generalization. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "• Insufficiency: If spurious features do not satisfy support overlap, then both ERM and IRM fail at solving the OOD generalization problem (eq. (1)). Also, there exist no such classifiers that rely on spurious features and also achieve OOD generalization. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Significance of Theorem 3. From the first part, we learn that if the support overlap is satisfied for both the invariant features and the spurious features, then either ERM or IRM can solve the OOD generalization (eq. (1)). Interestingly, in this case we can have classifiers that rely on the spurious features and yet solve the OOD generalization (eq. (1)). For the 2D case (eq. (4)) this case implies that the entire set $s$ solves the OOD generalization (eq. (1)). From the second part, we learn that if support overlap holds for invariant features but not for spurious features, then the ideal OOD optimal predictors rely only on the invariant features. In this case, methods like ERM and IRM continue to rely on spurious features and fail at OOD generalization. For the above 2D case (eq. (4)) this implies that only the predictors that rely only on $X _ { \\mathrm { i n v } } ^ { e }$ in the set $s$ solve the OOD generalization (eq. (1)). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To summarize, we looked at SEMs for classification tasks when invariant features are fully informative, and find that the support overlap assumption over invariant features is necessary. Even in the presence of support overlap for invariant features, we showed that ERM and IRM can easily fail if the support overlap is violated for spurious features. This raises a natural question – Can we even solve the case with the support overlap assumption only on the invariant features? We will now show that the information bottleneck principle can help tackle these cases. ",
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+    {
+        "type": "text",
+        "text": "4 Information bottleneck principle meets invariance principle ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "Why the information bottleneck? The information bottleneck principle prescribes to learn a representation that compresses the input $X$ as much as possible while preserving all the relevant information about the target label $Y$ (Tishby et al., 2000). Mutual information $I ( { \\bar { X } } ; \\Phi ( X ) )$ is used to measure information compression. If representation $\\Phi ( X )$ is a deterministic transformation of $X$ , then in principle we can use the entropy of $\\Phi ( X )$ to measure compression (Kirsch et al., 2020). Let us revisit the 2D case (eq. (4)) and apply this principle to it. Following the second part of Theorem 3, where ERM and IRM failed, assume that invariant features satisfy the support overlap assumption, but make no such assumption for the spurious features. Consider three choices for $\\Phi$ : identity (selects both features), selects invariant feature only, selects spurious feature only. The entropy of $\\dot { H } ( \\Phi ( X ^ { e } ) )$ when $\\Phi$ is the identity is $H ( p ^ { e } ) + \\log ( 2 )$ , where $\\bar { H ( p ^ { e } ) }$ is the Shannon entropy in Bernoulli $( p ^ { e } )$ . If $\\Phi$ selects the invariant/spurious features only, then $H ( \\Phi ( X ^ { e } ) ) = \\log ( 2 )$ . Among all three choices, the one that has the least entropy and also achieves zero error is the representation that focuses on the invariant feature. We could find the OOD optimal predictor in this example just by using information bottleneck. Does it mean the invariance principle isn’t needed? We answer this next. ",
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+        "img_path": "images/d7e9c47c2763b3601fbf394b12401d20b00afacedae56e0a8a224df7da6e7244.jpg",
+        "image_caption": [
+            "Figure 2: Comparison of the DAG from Assumption 2 (fully informative invariant features) vs. DAGs from Rosenfeld et al. (2021); Arjovsky et al. (2019) (partially informative invariant features). "
+        ],
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+    {
+        "type": "text",
+        "text": "Whand sider a simple classification SEM. In each , where all the random variables involve $e \\in \\mathcal { E } _ { t r }$ , i $Y ^ { e } \\gets X _ { \\mathsf { i n v } } ^ { 1 , e } \\oplus X _ { \\mathsf { i n v } } ^ { 2 , e } \\oplus N ^ { e }$ $X _ { \\mathsf { s p u } } ^ { e }  Y ^ { e } \\oplus V ^ { e }$ $N ^ { e } , V ^ { e }$ are Berthen in li with parameters predictions based $q$ (in cal across are bette $\\mathcal { E } _ { t r }$ ), a $c ^ { e }$ (varies across predictions ba $\\mathcal { E } _ { t r }$ ) re on If . $c ^ { e } < q$ , $\\mathcal { E } _ { t r }$ $X _ { \\mathsf { s p u } } ^ { e }$ $\\bar { X } _ { \\mathfrak { i n v } } ^ { 1 , e } , X _ { \\mathfrak { i n v } } ^ { 2 , e }$ $X _ { \\mathrm { i n v } } ^ { 1 , e } , X _ { \\mathrm { i n v } } ^ { 2 , e }$ formation band not on are uniform Bernoulli, then these features have a higher entropy than neck would bar using . Invariance constrai $X _ { \\mathsf { i n v } } ^ { 1 , e } , X _ { \\mathsf { i n v } } ^ { 2 , e }$ . Instead, we want the modge the model to focus on $X _ { \\mathsf { s p u } } ^ { e }$ . In this case, $X _ { \\mathsf { i n v } } ^ { 1 , e }$ $X _ { \\mathsf { i n v } } ^ { 2 , e }$ $X _ { \\mathsf { s p u } } ^ { e }$ $X _ { \\mathsf { i n v } } ^ { 1 , e }$ $X _ { \\mathsf { i n v } } ^ { 2 , e }$ example, observe that invariant features are partially informative unlike the 2D case (eq. (4)). ",
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+    {
+        "type": "text",
+        "text": "Why invariance and information bottleneck? We have illustrated through simple examples when the information bottleneck is needed but not invariance and vice-versa. We now provide a simple example where both these constraints are needed at the same time. This example combines the 2D case (eq. (4)) and the example we highlighted in the paragraph above: $Y ^ { e } \\gets X _ { \\mathsf { i n v } } ^ { e } \\oplus N ^ { e }$ , $X _ { \\mathsf { s p u } } ^ { 1 , e } \\gets X _ { \\mathsf { i n v } } ^ { e } \\oplus W ^ { e }$ , and $X _ { \\mathsf { s p u } } ^ { 2 , e } \\gets Y ^ { e } \\oplus V ^ { e }$ . In this case, the invariance constraint does not allow representations that use information bottleneck c $X _ { \\mathsf { s p u } } ^ { 2 , e }$ but does not prohibit representations that rely oints on top ensure that representations that only use $X _ { \\mathsf { s p u } } ^ { 1 , e }$ . However,re used. We $X _ { \\mathrm { i n v } } ^ { e }$ now describe an objective 8 that combines both these principles: ",
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+        "img_path": "images/64dc4c3e94f9e25e20968d6df854a736e075f5b0347d75988a10b7e68c7b66eb.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { w , \\Phi } \\sum _ { e \\in { \\mathcal E } _ { t r } } h ^ { e } \\big ( w \\cdot \\Phi \\big ) \\quad \\mathrm { s . t . } \\ \\frac { 1 } { | { \\mathcal E } _ { t r } | } \\sum _ { e \\in { \\mathcal E } _ { t r } } R ^ { e } \\big ( w \\cdot \\Phi \\big ) \\leq r ^ { \\mathrm { t h } } , \\ w \\in \\arg \\operatorname* { m i n } _ { \\tilde { w } \\in \\mathbb R ^ { k \\times r } } R ^ { e } \\big ( \\tilde { w } \\cdot \\Phi \\big ) , \\forall e \\in { \\mathcal E } _ { t r } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $h ^ { e }$ in the above is a lower bounded differential entropy defined below and $r ^ { \\mathrm { t h } }$ is the threshold on the average risk. Typical information bottleneck based optimization in neural networks involves minimization of the entropy of the representation output from a certain hidden layer. For both analytical convenience and also because the above setup is a linear model, we work with the simplest form of bottleneck which directly minimizes the entropy of the output layer. Recall the definition of differential entropy of a random variable $X$ $\\quad , h ( X ) = - \\mathbf { \\bar { \\mathbb { E } } } _ { X } [ \\log d \\mathbb { P } _ { X } ]$ and $d \\mathbb { P } _ { X }$ is the Radon-Nikodym derivative of $\\mathbb { P } _ { X }$ with respect to Lebesgue measure. Because in general differential entropy has no lower bound, we add a small independent noise term $\\zeta$ (Kirsch et al., 2020) to the classifier to ensure that the entropy is bounded below. We call the above optimization information bottleneck based invariant risk minimization (IB-IRM). In summary, among all the highly predictive invariant predictors we pick the ones that have the least entropy. If we drop the invariance constraint from the above optimization, we get information bottleneck based empirical risk minimization (IB-ERM). In the above formulation and following result, we assume that $X ^ { e }$ are continuous random variables; the results continue to hold for discrete $X ^ { e }$ as well (See Appendix for details). ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "Theorem 4. IB-IRM and IB-ERM vs. IRM and ERM ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "8Results extend to alternate objective with information bottleneck constraints and average risk as objective. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "• Fully informative invariant features (FIIF). Suppose each $e \\in \\mathcal { E } _ { a l l }$ follows Assumption 2. Assume that the invariant features are strictly separable, bounded, and satisfy support overlap (Assumptions 3,5 and 7 hold). Also, for each $e \\in \\mathcal { E } _ { t r }$ $Z _ { \\mathsf { s p u } } ^ { e }  A Z _ { \\mathsf { i n v } } ^ { e } + W ^ { e }$ , where $A \\in \\mathbb { R } ^ { o \\times m }$ , $W ^ { e } \\in \\mathbb { R } ^ { o }$ is continuous, bounded, and zero mean noise. Each solution to $I B$ -IRM (eq. (6), with $\\ell$ as 0-1 loss, and $r ^ { \\mathrm { t h } } = q ,$ ), and IB-ERM solves the OOD generalization (eq. (1)) but ERM and IRM (eq.(3)) fail. ",
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+    },
+    {
+        "type": "text",
+        "text": "• Partially informative invariant features $( P I I F )$ . Suppose each $e \\in \\mathcal { E } _ { a l l }$ follows Assumption 1 and $\\exists \\ e \\in { \\mathcal { E } } _ { t r }$ such that $\\mathbb { E } [ \\epsilon ^ { e } Z _ { \\mathsf { s p u } } ^ { e } ] \\neq 0$ . If $| \\mathcal { E } _ { t r } | > 2 d$ and the set $\\mathcal { E } _ { t r }$ lies in a linear general position (a mild condition defined in the Appendix), then each solution to IB-IRM (eq. (6), with $\\ell$ as square loss, $\\sigma _ { \\epsilon } ^ { 2 } < r ^ { \\mathsf { t h } } \\le \\sigma _ { Y } ^ { 2 }$ , where $\\sigma _ { Y } ^ { 2 }$ and $\\sigma _ { \\epsilon } ^ { 2 }$ are the variance in the label and noise across $\\mathcal { E } _ { t r }$ ) and IRM (eq.(3)) solves OOD generalization (eq. (1)) but IB-ERM and ERM fail. ",
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+        "type": "text",
+        "text": "Significance of Theorem 4 and remarks. In the first part (FIIF), IB-ERM and IB-IRM succeed without assuming support overlap for the spurious features, which was crucial for success of ERM and IRM in Theorem 3. This establishes that support overlap of spurious features is not a necessary condition. Observe that when invariant features are fully informative, IB-ERM and IB-IRM succeed, but when invariant features are partially informative IB-IRM and IRM succeed. In real data settings, we do not know if the invariant features are fully or partially informative. Since IB-IRM is the only common winner in both the settings, it would be pragmatic to use it in the absence of domain knowledge about the informativeness of the invariant features. In the paragraph preceding the objective in equation (6), we discussed examples where both the IB and IRM constraints were needed at the same time. In the Appendix, we generalize that example and show that if we change the assumptions in linear classification SEM in Assumption 2 such that the invariant features are partially informative, then we see the joint benefit of IB and IRM constraints. At this point, it is also worth pointing to a result in Rosenfeld et al. (2021), which focused on linear classification SEMs (DAG shown in Figure 2c) with partially informative invariant features. Under the assumption of complete support overlap for spurious and invariant features, authors showed IRM succeeds. ",
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+    {
+        "type": "text",
+        "text": "4.1 Proposed approach ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We take the three terms from the optimization in equation (6) and create a weighted combination as ",
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+    {
+        "type": "equation",
+        "img_path": "images/64fe9b87c4434d040a316d05f0d84125c069d9f2ea7795f2d226e9fc107677b4.jpg",
+        "text": "$$\n\\sum _ { \\epsilon } \\Big ( R ^ { e } ( \\Phi ) + \\lambda \\| \\nabla _ { w , w = 1 , 0 } R ^ { e } ( w \\cdot \\Phi ) \\| ^ { 2 } + \\nu h ^ { e } ( \\Phi ) \\Big ) \\leq \\sum _ { \\epsilon } \\Big ( R ^ { e } ( \\Phi ) + \\lambda \\| \\nabla _ { w , w = 1 , 0 } R ^ { e } ( w \\cdot \\Phi ) \\| ^ { 2 } + \\nu h ( \\Phi ) \\Big ) .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "In the LHS above, the first term corresponds to the risks across environments, the second term approximates invariance constraint (follows the IRMv1 objective (Arjovsky et al., 2019)), and the third term is the entropy of the classifier in each environment. ",
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+        "type": "text",
+        "text": "In the RHS, $h ( \\Phi )$ is the entropy of $\\Phi$ unconditional on the environment (the entropy on the left-hand side is entropy conditional on the environment assuming all the environments are equally likely). Optimizing over differential entropy is not easy, and thus we resort to minimizing an upper bound of it (Kirsch et al., 2020). We use the standard result that among all continuous random variables with the same variance, Gaussian has the maximum differential entropy. Since the entropy of Gaussian increases with its variance, we use the variance of $\\Phi$ instead of the differential entropy (For further details, see the Appendix). Our final objective is given as ",
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+    {
+        "type": "equation",
+        "img_path": "images/6b591483f320a5fd2e306fe30e65a6bf3a1071917ee245a75fa42982bce60ea9.jpg",
+        "text": "$$\n\\sum _ { e } \\Big ( R ^ { e } ( \\Phi ) + \\lambda \\| \\nabla _ { w , w = 1 . 0 } R ^ { e } ( w \\cdot \\Phi ) \\| ^ { 2 } + \\gamma \\mathsf { V a r } ( \\Phi ) \\Big ) .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "On the behavior of gradient descent with and without informa",
+        "text_level": 1,
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+    {
+        "type": "image",
+        "img_path": "images/8a3cf0e7bca3d1fa872223a8320e95f17f3937c66e0aca73f9415c50e7db929d.jpg",
+        "image_caption": [
+            "Figure 3: Comparing convergence of $\\frac { \\vert w _ { \\mathsf { s p u } } \\vert } { \\sqrt { w _ { \\mathsf { s p u } } ^ { 2 } + w _ { \\mathsf { i n v } } ^ { 2 } } }$ (metric from Nagarajan et al. (2021)) for average selection bias $p = 0 . 9$ . "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "tion bottleneck. In the entire discussion so far, we have focused on ensuring that the set of optimal solutions to the desired objective (IB-IRM, IB-ERM, etc.) correspond to the solutions of the OOD generalization problem (eq. (1)). In some simple cases, such as the 2D case (eq. (4)), it can be shown that gradient descent is biased towards selecting the ideal classifier (Soudry et al., 2018; Nagarajan et al., 2021). Even though gradient descent can eventually learn the ideal classifier that only relies on the invariant features, training is frustratingly slow as was shown by Nagarajan et al. (2021). In the next theorem, we characterize the impact of using IB penalty $( \\mathsf { V a r } ( \\Phi ) )$ in the 2D example (eq. (4)). We compare the methods in terms of | wspu(t)winv(t) |, which was the metric used in Nagarajan et al. (2021); $w _ { \\mathsf { s p u } } ( t )$ and $w _ { \\mathsf { i n v } } ( t )$ are the weights for the spurious feature and the invariant feature at time $t$ of training (assuming training happens with continuous time gradient descent). ",
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+    {
+        "type": "text",
+        "text": "Theorem 5. Impact of $\\pmb { I B }$ on learning speed. Suppose each $\\textit { e } \\in \\mathcal { E } _ { t r }$ follows the $2 D$ case from equation (4). Set $\\lambda = 0$ , $\\gamma > 0$ in equation (7) to get the $I B$ -ERM objective with $\\ell$ as exponential loss. Continuous-time gradient descent on this IB-ERM objective achieves $| \\frac { w _ { \\mathsf { s p u } } ( t ) } { w _ { \\mathsf { i n v } } ( t ) } | \\leq \\epsilon$ in time less than $\\frac { W _ { 0 } ( \\frac { 1 } { 2 \\gamma } ) } { 2 ( 1 - p ) \\epsilon }$ $W _ { 0 } ( \\cdot )$ denotes the principal branch of the Lambert $W$ function), while in the same time the ratio for ERM $\\begin{array} { r } { \\vert \\frac { w _ { \\mathrm { s p u } } ( t ) } { w _ { \\mathrm { i n v } } ( t ) } \\vert \\ge \\ln ( \\frac { 1 + 2 p } { 3 - 2 p } ) / \\ln \\left( 1 + \\frac { W _ { 0 } ( \\frac { 1 } { 2 \\gamma } ) } { 2 ( 1 - p ) \\epsilon } \\right) } \\end{array}$ W0( 12γ )2(1−p)\u000f \u0001, where p = 1|Etr | Pe∈Etr pe . ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "$| \\frac { w _ { \\mathsf { s p u } } ( t ) } { w _ { \\mathsf { i n v } } ( t ) } |$ converges to zero for both methods, but it converges much faster for IB-ERM (for $p =$ $0 . 9 , \\epsilon = 0 . 0 0 1 , \\gamma = 0 . 5 8$ , the ratio for IB-ERM is $| \\frac { w _ { \\mathsf { s p u } } ( t ) } { w _ { \\mathsf { i n v } } ( t ) } | \\leq 0 . 0 0 1$ and ratio for ERM is $| \\frac { w _ { \\mathsf { s p u } } ( t ) } { w _ { \\mathsf { i n v } } ( t ) } | \\geq$ 0.09). In the above theorem, we analyzed the impact of information bottleneck only. The convergence analysis for both the penalties jointly comes with its own challenges, and we hope to explore this in future work. However, we carried out experiments with gradient descent on all the objectives for the 2D example (eq. (4)). See Figure 3 for the comparisons. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Methods, datasets & metrics. We compare our approaches – information bottleneck based ERM (IBERM) and information bottleneck based IRM (IB-IRM) with ERM and IRM. We also compare with an Oracle model trained on data where spurious features are permuted to remove spurious correlations. We use all the datasets in Table 2, Terra Incognita dataset (Beery et al., 2018), and COCO (Ahmed et al., 2021). We follow the same protocol for tuning hyperparameters from Aubin et al. (2021); Arjovsky et al. (2019) for their respective datasets (see the Appendix for more details). As is reported in literature, for Example 2/2S, Example 3/3S we use classification error and for AC-CMNIST, CS-CMNIST, Terra Incognita, and COCO we use accuracy. For Example 1/1S, we use mean square error (MSE). The code for experiments can be found at https://github.com/ahujak/IB-IRM. ",
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+    {
+        "type": "text",
+        "text": "Summary of results. In Table 3, we provide a comparison of methods for different examples in linear unit tests (Aubin et al., 2021) for three and six training environments. In Table 4, we provide a comparison of the methods for different CMNIST datasets, Terra Incognita and COCO dataset. Based on our Theorem 4, we do not expect ERM and IB-ERM to do well on Example 1/1S, Example 3/3S and AC-CMNIST as these datasets fall in the PIIF category, i.e, the invariant features are partially informative. On these examples, we find that IRM and IB-IRM do better than ERM and IB-ERM (for Example 3/3S when there are three environments all methods perform poorly). Based on our Theorem 4, we do not expect IRM and ERM to do well on Example 2/2S, CS-CMNIST, Terra Incognita and COCO dataset,9 as these datasets fall in the FIIF category, i.e., the invariant features are fully informative. On these FIIF examples, we find that IB-ERM always performs well (close to oracle), and in some cases IB-IRM also performs well. Our experiments confirm that IB penalty has a crucial role to play in FIIF settings and IRMv1 penalty has a crucial role to play in PIIF settings (to further this claim, we provide an ablation study in the Appendix). On Example 1/1S, AC-CMNIST, we find that IB-IRM is able to extract the benefit of IRMv1 penalty. On CS-CMNIST and Example 2/2S we find that IB-IRM is able to extract the benefit of IB penalty. In settings such as COCO dataset, where IB-IRM does not perform as well as IB-ERM, better hyperparameter tuning strategies should be able to help IB-IRM adapt and put a higher weight on IB penalty. Overall, we can conclude that IB-ERM improves over ERM (significantly in FIIF and marginally in PIIF settings), and IB-IRM improves over IRM (improves in FIIF settings and retains advantages in PIIF settings). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Remark. As we move from three to six environments, we observe that MSE in Example 1/1S exhibits a larger variance. This is because of the way data is generated, the new environments that are sampled have labels that have a higher noise level (we follow the same procedure as in Aubin et al. (2021)). ",
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+    {
+        "type": "text",
+        "text": "6 Extensions, limitations, and future work ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Extension to non-linear models and multi-class classification. In this work our theoretical analysis focused on linear models. Consider the map $X  S ( Z _ { \\mathsf { i n v } } , Z _ { \\mathsf { s p u } } )$ in Assumption 2. Suppose $S$ is non-linear and bijective. We can divide the learning task into two parts a) invert $S$ to obtain $Z _ { \\mathrm { i n v } }$ , $Z _ { \\mathsf { s p u } }$ and b) learn a linear model that only relies on the invariant features $Z _ { \\mathrm { i n v } }$ to predict the label $Y$ . For part b), we can rely on the approaches proposed in this work. For part a), we need to leverage advancements in the field of non-linear ICA (Khemakhem et al., 2020). The current state-of-the-art to solve part a) requires strong structural assumptions on the dependence between all the components of $Z _ { \\mathrm { i n v } }$ , $Z _ { \\mathsf { s p u } }$ (Lu et al., 2021). Therefore, solving part a) and part b) in conjunction with minimal assumptions forms an exciting future work. In the entire work, the discussion was focused on binary classification tasks and regression tasks. For multi-class classification settings, we consider natural extension of the SEM in Assumption 2 (See the Appendix) and our main results continue to hold. ",
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+    {
+        "type": "table",
+        "img_path": "images/88e356c422b6d677fd02faef845186a0a65441b956430aee8cef6b7f1ac8dd1d.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 3: Comparisons on linear unit tests in terms of mean square error (regression) and classification error (classification). “#Envs” means the number of training environments. "
+        ],
+        "table_body": "<table><tr><td></td><td>#Envs</td><td>ERM</td><td>IB-ERM</td><td>IRM</td><td>IB-IRM</td><td>Oracle</td></tr><tr><td>Example1</td><td>3</td><td>13.36 ±1.49</td><td>12.96 ±1.30</td><td>11.15± 0.71</td><td>11.68 ± 0.90</td><td>10.42±0.16</td></tr><tr><td>Example1s</td><td>3</td><td>13.33 ± 1.49</td><td>12.92 ± 1.30</td><td>11.07 ± 0.68</td><td>11.74 ± 1.03</td><td>10.45±0.19</td></tr><tr><td>Example2</td><td>3</td><td>0.42 ± 0.01</td><td>0.00±0.00</td><td>0.45 ± 0.00</td><td>0.00 ±0.00</td><td>0.00 ±0.00</td></tr><tr><td>Example2s</td><td>3</td><td>0.45 ± 0.01</td><td>0.00 ± 0.01</td><td>0.45 ± 0.01</td><td>0.06 ± 0.12</td><td>0.00 ± 0.00</td></tr><tr><td>Example3</td><td>3</td><td>0.48 ± 0.07</td><td>0.49 ± 0.06</td><td>0.48 ± 0.07</td><td>0.48 ± 0.07</td><td>0.01 ± 0.00</td></tr><tr><td>Example3s</td><td>3</td><td>0.49 ± 0.06</td><td>0.49 ± 0.06</td><td>0.49 ± 0.07</td><td>0.49 ± 0.07</td><td>0.01 ±0.00</td></tr><tr><td>Example1</td><td>6</td><td>33.74 ± 60.18</td><td>32.03 ± 57.05</td><td>23.04 ± 40.64</td><td>25.66 ± 45.96</td><td>22.21±39.25</td></tr><tr><td>Example1s</td><td>6</td><td>33.62 ± 59.80</td><td>31.92 ± 56.70</td><td>22.92 ± 40.60</td><td>25.60 ± 45.62</td><td>22.13±38.93</td></tr><tr><td>Example2</td><td>6</td><td>0.37 ± 0.06</td><td>0.02 ± 0.05</td><td>0.46 ± 0.01</td><td>0.43 ± 0.11</td><td>0.00±0.00</td></tr><tr><td>Example2s</td><td>6</td><td>0.46 ± 0.01</td><td>0.02 ± 0.06</td><td>0.46 ± 0.01</td><td>0.45 ± 0.10</td><td>0.00±0.00</td></tr><tr><td>Example3</td><td>6</td><td>0.33 ± 0.18</td><td>0.26 ± 0.20</td><td>0.14 ± 0.18</td><td>0.19 ± 0.19</td><td>0.01±0.00</td></tr><tr><td>Example3s</td><td>6</td><td>0.36 ±0.19</td><td>0.27 ± 0.20</td><td>0.14 ± 0.18</td><td>0.19 ± 0.19</td><td>0.01±0.00</td></tr></table>",
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+    {
+        "type": "table",
+        "img_path": "images/e9034b0e33dae062049a2a6c57860b82487d904f334acf43705faa81987dcd7d.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 4: Classification accuracy percentage on colored MNISTs, Terra Incognita and COCO dataset. "
+        ],
+        "table_body": "<table><tr><td></td><td>ERM</td><td>IB-ERM</td><td>IRM</td><td>IB-IRM</td></tr><tr><td>CS-CMNIST</td><td>60.27 ± 1.21</td><td>71.80 ± 0.69</td><td>61.49 ± 1.45</td><td>71.79 ± 0.70</td></tr><tr><td>AC-CMNIST</td><td>16.84 ± 0.82</td><td>50.24 ± 0.47</td><td>66.98 ± 1.65</td><td>67.67 ± 1.78</td></tr><tr><td>Terra Incognita</td><td>49.80 ± 4.40</td><td>56.40 ± 2.10</td><td>54.60 ± 1.30</td><td>54.10 ± 2.00</td></tr><tr><td>COCO</td><td>22.70 ± 1.04</td><td>31.66 ± 2.39</td><td>18.47 ± 10.20</td><td>25.10 ± 1.03</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "On the choice for IB penalty and IRMv1 penalty. We use the approximation for entropy (in equation (7)) described in Kirsch et al. (2020). The approximation (even though an upper bound) serves as an effective proxy for the true information bottleneck as shown in the experiments in Kirsch et al. (2020) (e.g., see their experiment on Imagenette dataset). Also, our experiments validate this approximation even in moderately high dimensions, as an example in CS-CMNIST, the dimension of the layer at which bottleneck constraints are applied is 256. Developing tighter approximations for information bottleneck in high dimensions and analyzing their impact on OOD generalization is an important future work. In recent works (Rosenfeld et al., 2021; Kamath et al., 2021; Gulrajani and Lopez-Paz, 2021), there has been criticism of different aspects of IRM, e.g., failure of IRMv1 penalty in non-linear models, the tuning of IRMv1 penalty, etc. Since we use IRMv1 penalty in our proposed loss, these criticisms apply to our objective as well. Other approximations of invariance have been proposed in the literature (Koyama and Yamaguchi, 2020; Ahuja et al., 2020; Chang et al., 2020). Exploring their benefits together with information bottleneck is a fruitful future work. Before concluding, we want to remark that we have already discussed the closest related works. However, we also provide a detailed discussion of the broader related literature in the Appendix. ",
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+    {
+        "type": "text",
+        "text": "7 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this work, we revisited the fundamental assumptions for OOD generalization for settings when invariant features capture all the information about the label. We showed how linear classification tasks are different and need much stronger assumptions than linear regression tasks. We provide a sharp characterization of performance of ERM and IRM under different assumptions on support overlap of invariant and spurious features. We showed that support overlap of invariant features is necessary or otherwise OOD generalization is impossible. However, ERM and IRM seem to fail even in the absence of support overlap of spurious features. We prove that a form of the information bottleneck constraint along with invariance goes a long way in overcoming the failures while retaining the existing provable guarantees. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgements ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We thank Reyhane Askari Hemmat, Adam Ibrahim, Alexia Jolicoeur-Martineau, Divyat Mahajan, Ryan D’Orazio, Nicolas Loizou, Manuela Girotti, and Charles Guille-Escuret for the feedback. Kartik Ahuja would also like to thank Karthikeyan Shanmugam for discussions pertaining to the related works. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Funding disclosure ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We would like to thank Samsung Electronics Co., Ldt. for funding this research. Kartik Ahuja acknowledges the support provided by IVADO postdoctoral fellowship funding program. Yoshua Bengio acknowledges the support from CIFAR and IBM. Ioannis Mitliagkas acknowledges support from an NSERC Discovery grant (RGPIN-2019-06512), a Samsung grant, Canada CIFAR AI chair and MSR collaborative research grant. Irina Rish acknowledges the support from Canada CIFAR AI Chair Program and from the Canada Excellence Research Chairs Program. We thank Compute Canada for providing computational resources. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            343,
+            266,
+            359
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+        "text": "Ahmed, F., Bengio, Y., van Seijen, H., and Courville, A. (2021). Systematic generalisation with group invariant predictions. In International Conference on Learning Representations.   \nAhuja, K., Shanmugam, K., Varshney, K., and Dhurandhar, A. (2020). Invariant risk minimization games. In International Conference on Machine Learning, pages 145–155. PMLR.   \nAhuja, K., Wang, J., Dhurandhar, A., Shanmugam, K., and Varshney, K. R. (2021). Empirical or invariant risk minimization? a sample complexity perspective. In International Conference on Learning Representations.   \nArjovsky, M., Bottou, L., Gulrajani, I., and Lopez-Paz, D. (2019). Invariant risk minimization. arXiv preprint arXiv:1907.02893.   \nAubin, B., Słowik, A., Arjovsky, M., Bottou, L., and Lopez-Paz, D. (2021). Linear unit-tests for invariance discovery. arXiv preprint arXiv:2102.10867.   \nBeery, S., Van Horn, G., and Perona, P. (2018). Recognition in terra incognita. In Proceedings of the European Conference on Computer Vision, pages 456–473.   \nChang, S., Zhang, Y., Yu, M., and Jaakkola, T. S. (2020). Invariant rationalization. In International Conference on Machine Learning, 2020.   \nDeGrave, A. J., Janizek, J. D., and Lee, S.-I. (2020). AI for radiographic COVID-19 detection selects shortcuts over signal. medRxiv.   \nGeirhos, R., Jacobsen, J.-H., Michaelis, C., Zemel, R., Brendel, W., Bethge, M., and Wichmann, F. A. (2020). Shortcut learning in deep neural networks. Nature Machine Intelligence, 2(11):665–673.   \nGulrajani, I. and Lopez-Paz, D. (2021). In search of lost domain generalization. In International Conference on Learning Representations.   \nKamath, P., Tangella, A., Sutherland, D. J., and Srebro, N. (2021). Does invariant risk minimization capture invariance? arXiv preprint arXiv:2101.01134.   \nKhemakhem, I., Kingma, D., Monti, R., and Hyvarinen, A. (2020). Variational autoencoders and nonlinear ica: A unifying framework. In International Conference on Artificial Intelligence and Statistics, pages 2207–2217. PMLR.   \nKirsch, A., Lyle, C., and Gal, Y. (2020). Unpacking information bottlenecks: Unifying informationtheoretic objectives in deep learning. arXiv preprint arXiv:2003.12537.   \nKoyama, M. and Yamaguchi, S. (2020). Out-of-distribution generalization with maximal invariant predictor. arXiv preprint arXiv:2008.01883.   \nKrueger, D., Caballero, E., Jacobsen, J.-H., Zhang, A., Binas, J., Zhang, D., Priol, R. L., and Courville, A. (2020). Out-of-distribution generalization via risk extrapolation (rex). arXiv preprint arXiv:2003.00688.   \nLu, C., Wu, Y., Hernández-Lobato, J. M., and Schölkopf, B. (2021). Nonlinear invariant risk minimization: A causal approach. arXiv preprint arXiv:2102.12353.   \nNagarajan, V., Andreassen, A., and Neyshabur, B. (2021). Understanding the failure modes of out-of-distribution generalization. In International Conference on Learning Representations.   \nPearl, J. (1995). Causal diagrams for empirical research. Biometrika, 82(4):669–688.   \nPeters, J., Bühlmann, P., and Meinshausen, N. (2015). Causal inference using invariant prediction: identification and confidence intervals. arXiv preprint arXiv:1501.01332.   \nPezeshki, M., Kaba, S.-O., Bengio, Y., Courville, A., Precup, D., and Lajoie, G. (2020). Gradient starvation: A learning proclivity in neural networks. arXiv preprint arXiv:2011.09468.   \nRobey, A., Pappas, G. J., and Hassani, H. (2021). Model-based domain generalization. arXiv preprint arXiv:2102.11436.   \nRojas-Carulla, M., Schölkopf, B., Turner, R., and Peters, J. (2018). Invariant models for causal transfer learning. The Journal of Machine Learning Research, 19(1):1309–1342.   \nRosenfeld, E., Ravikumar, P. K., and Risteski, A. (2021). The risks of invariant risk minimization. In International Conference on Learning Representations.   \nSoudry, D., Hoffer, E., Nacson, M. S., Gunasekar, S., and Srebro, N. (2018). The implicit bias of gradient descent on separable data. The Journal of Machine Learning Research, 19(1):2822–2878.   \nTishby, N., Pereira, F. C., and Bialek, W. (2000). The information bottleneck method. arXiv preprint physics/0004057.   \nVapnik, V. (1992). Principles of risk minimization for learning theory. In Advances in neural information processing systems, pages 831–838.   \nZhang, D., Ahuja, K., Xu, Y., Wang, Y., and Courville, A. C. (2021). Can subnetwork structure be the key to out-of-distribution generalization? In ICML. ",
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parse/train/rk8wKk-R-/rk8wKk-R-.md

@@ -0,0 +1,336 @@
+# CONVOLUTIONAL SEQUENCE MODELING REVISITED
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+This paper revisits the problem of sequence modeling using convolutional architectures. Although both convolutional and recurrent architectures have a long history in sequence prediction, the current “default” mindset in much of the deep learning community is that generic sequence modeling is best handled using recurrent networks. The goal of this paper is to question this assumption. Specifically, we consider a simple generic temporal convolution network (TCN), which adopts features from modern ConvNet architectures such as a dilations and residual connections. We show that on a variety of sequence modeling tasks, including many frequently used as benchmarks for evaluating recurrent networks, the TCN outperforms baseline RNN methods (LSTMs, GRUs, and vanilla RNNs) and sometimes even highly specialized approaches. We further show that the potential “infinite memory” advantage that RNNs have over TCNs is largely absent in practice: TCNs indeed exhibit longer effective history sizes than their recurrent counterparts. As a whole, we argue that it may be time to (re)consider ConvNets as the default “go to” architecture for sequence modeling.
+
+# 1 INTRODUCTION
+
+Since the re-emergence of neural networks to the forefront of machine learning, two types of network architectures have played a pivotal role: the convolutional network, often used for vision and higher-dimensional input data; and the recurrent network, typically used for modeling sequential data. These two types of architectures have become so ingrained in modern deep learning that they can be viewed as constituting the “pillars” of deep learning approaches. This paper looks at the problem of sequence modeling, predicting how a sequence will evolve over time. This is a key problem in domains spanning audio, language modeling, music processing, time series forecasting, and many others. Although exceptions certainly exist in some domains, the current “default” thinking in the deep learning community is that these sequential tasks are best handled by some type of recurrent network. Our aim is to revisit this default thinking, and specifically ask whether modern convolutional architectures are in fact just as powerful for sequence modeling.
+
+Before making the main claims of our paper, some history of convolutional and recurrent models for sequence modeling is useful. In the early history of neural networks, convolutional models were specifically proposed as a means of handling sequence data, the idea being that one could slide a 1-D convolutional filter over the data (and stack such layers together) to predict future elements of a sequence from past ones (Hinton, 1989; LeCun et al., 1995). Thus, the idea of using convolutional models for sequence modeling goes back to the beginning of convolutional architectures themselves. However, these models were subsequently largely abandoned for many sequence modeling tasks in favor of recurrent networks (Elman, 1990). The reasoning for this appears straightforward: while convolutional architectures have a limited ability to look back in time (i.e., their receptive field is limited by the size and layers of the filters), recurrent networks have no such limitation. Because recurrent networks propagate forward a hidden state, they are theoretically capable of infinite memory, the ability to make predictions based upon data that occurred arbitrarily long ago in the sequence. This possibility seems to be realized even moreso for the now-standard architectures of Long ShortTerm Memory networks (LSTMs) (Hochreiter & Schmidhuber, 1997), or recent incarnations such as the Gated Recurrent Unit (GRU) (Cho et al., 2014); these architectures aim to avoid the “vanishing gradient” challenge of traditional RNNs and appear to provide a means to actually realize this infinite memory.
+
+Given the substantial limitations of convolutional architectures at the time that RNNs/LSTMs were initially proposed (when deep convolutional architectures were difficult to train, and strategies such as dilated convolutions had not reached widespread use), it is no surprise that CNNs fell out of favor to RNNs. While there have been a few notable examples in recent years of CNNs applied to sequence modeling (e.g., the WaveNet (Oord et al., 2016a) and PixelCNN (Oord et al., 2016b) architectures), the general “folk wisdom” of sequence modeling prevails, that the first avenue of attack for these problems should be some form of recurrent network.
+
+The fundamental aim of this paper is to revisit this folk wisdom, and thereby make a counterclaim. We argue that with the tools of modern convolutional architectures at our disposal (namely the ability to train very deep networks via residual connections and other similar mechanisms, plus the ability to increase receptive field size via dilations), in fact convolutional architectures typically outperform recurrent architectures on sequence modeling tasks, especially (and perhaps somewhat surprisingly) on domains where a long effective history length is needed to make proper predictions.
+
+This paper consists of two main contributions. First, we describe a generic, baseline temporal convolutional network (TCN) architecture, combining best practices in the design of modern convolutional architectures, including residual layers and dilation. We emphasize that we are not claiming to invent the practice of applying convolutional architectures to sequence prediction, and indeed the TCN architecture here mirrors closely architectures such as WaveNet (in fact TCN is notably simpler in some respects). We do, however, want to propose a generic modern form of convolutional sequence prediction for subsequent experimentation. Second, and more importantly, we extensively evaluate the TCN model versus alternative approaches on a wide variety of sequence modeling tasks, spanning many domains and datasets that have typically been the purview of recurrent models, including word- and character-level language modeling, polyphonic music prediction, and other baseline tasks commonly used to evaluate recurrent architectures. Although our baseline TCN can be outperformed by specialized (and typically highly-tuned) RNNs in some cases, for the majority of problems the TCN performs best, with minimal tuning on the architecture or the optimization. This paper also analyzes empirically the myth of “infinite memory” in RNNs, and shows that in practice, TCNs of similar size and complexity may actually demonstrate longer effective history sizes. Our chief claim in this paper is thus an empirical one: rather than presuming that RNNs will be the default best method for sequence modeling tasks, it may be time to (re)consider ConvNets as the “go-to” approach when facing a new dataset or task in sequence modeling.
+
+# 2 RELATED WORK
+
+In this section we highlight some of the key innovations in the history of recurrent and convolutional architectures for sequence prediction.
+
+Recurrent networks broadly refer to networks that maintain a vector of hidden activations, which are kept over time by propagating them through the network. The intuitive appeal of this approach is that the hidden state can act as a sort of “memory” of everything that has been seen so far in a sequence, without the need for keeping an explicit history. Unfortunately, such memory comes at a cost, and it is well-known that the na¨ıve RNN architecture is difficult to train due to the exploding/vanishing gradient problem (Bengio et al., 1994).
+
+A number of solutions have been proposed to address this issue. More than twenty years ago, Hochreiter & Schmidhuber (1997) introduced the now-ubiquitous Long Short-Term Memory (LSTM) which uses a set of gates to explicitly maintain memory cells that are propagated forward in time. Other solutions or refinements include a simplified variant of LSTM, the Gated Recurrent Unit (GRU) (Cho et al., 2014), peephole connections (Gers et al., 2002), Clockwork RNN (Koutnik et al., 2014) and recent works such as MI-RNN (Wu et al., 2016) and the Dilated RNN (Chang et al., 2017). Alternatively, several regularization techniques have been proposed to better train LSTMs, such as those based upon the properties of the RNN dynamical system (Pascanu et al., 2013); more recently, strategies such as Zoneout (Krueger et al., 2017) and AWD-LSTM (Merity et al., 2017) were also introduced to regularize LSTM in various ways, and have achieved exceptional results in the field of language modeling.
+
+While it is frequently criticized as a seemingly “ad-hoc” architecture, LSTMs have proven to be extremely robust and is very hard to improve upon by other recurrent architectures, at least for general problems. Jozefowicz et al. (2015) concluded that if there were “architectures much better than the LSTM”, then they were “not trivial to find”. However, while they evaluated a variety of recurrent architectures with different combinations of components via an evolutionary search, they did not consider architectures that were fundamentally different from the recurrent ones.
+
+The history of convolutional architectures for time series is comparatively shorter, as they soon fell out of favor compared to recurrent architectures for these tasks, though are also seeing a resurgence in recent years. Waibel et al. (1989) and Bottou et al. (1990) studied the usage of time-delay networks (TDNNs) for sequences, one of the earliest local-connection-based networks in this domain. LeCun et al. (1995) then proposed and examined the usage of CNNs on time-series data, pointing out that the same kind of feature extraction used in images could work well on sequence modeling with convolutional filters. Recent years have seen a re-emergence of convolutional models for sequence data. Perhaps most notably, the WaveNet (Oord et al., 2016a) applied a stacked convolutional architecture to model audio signals, using a combination of dilations (Yu & Koltun, 2015), skip connections, gating, and conditioning on context stacks; the WaveNet mode was additionally applied to a few other contexts, such as financial applications (Borovykh et al., 2017). Non-dilated gated convolutions have also been applied in the context of language modeling (Dauphin et al., 2017). And finally, convolutional models have seen a recent adoption in sequence to sequence modeling and machine translations applications, such as the ByteNet (Kalchbrenner et al., 2016) and ConvS2S architectures (Gehring et al., 2017).
+
+Despite these successes, the general consensus of the deep learning community seems to be that RNNs (here meaning all RNNs including LSTM and its variants) are better suited to sequence modeling for two apparent reasons: 1) as discussed before, RNNs are theoretically capable of infinite memory; and 2) RNN models are inherently suitable for sequential inputs of varying length, whereas CNNs seem to be more appropriate in domains with fixed-size inputs (e.g., vision).
+
+With this as the context, this paper reconsiders convolutional sequence modeling in general, first introducing a simple general-purpose convolutional sequence modeling architecture that can be applied in all the same scenarios as an RNN (the architecture acts as a “drop-in” replacement for RNNs of any kind). We then extensively evaluate the performance of the architecture on tasks from different domains, focusing on domains and settings that have been used explicitly as applications and benchmarks for RNNs in the recent past. With regard to the specific architectures mentioned above (e.g. WaveNet, ByteNet, gated convolutional language models), the primary goal here is to describe a simple, application-independent architecture that avoids much of the extra specialized components of these architectures (gating, complex residuals, context stacks, or the encoder-decoder architectures of seq2seq models), and keeps only the “standard” convolutional components from most image architectures, with the restriction that the convolutions be causal. In several cases we specifically compare the architecture with and without additional components (e.g., gating elements), and highlight that it does not seem to substantially improve performance of the architecture across domains. Thus, the primary goal of this paper is to provide a baseline architecture for convolutional sequence prediction tasks, and to evaluate the performance of this model across multiple domains.
+
+# 3 CONVOLUTIONAL SEQUENCE MODELING
+
+In this section, we propose a generic architecture for convolutional sequence prediction, and generally refer to it as Temporal Convolution Networks (TCNs). We emphasize that we adopt this term not as a label for a truly new architecture, but as a simple descriptive term for this and similar architectures. The distinguishing characteristics of the TCN are that: 1) the convolutions in the architecture are causal, meaning that there is no information “leakage” between future and past; 2) the architecture can take a sequence of any length and map it to an output sequence of the same length, just as with an RNN. Beyond this, we emphasize how to build very long effective history sizes (i.e., the ability for the networks to look very far into the past to make a prediction) using a combination of very deep networks (augmented with residual layers) and dilated convolutions.
+
+# 3.1 THE SEQUENCE MODELING TASK
+
+Before defining the network structure, we highlight the nature of the sequence modeling task. We suppose that we are given a sequence of inputs $x _ { 0 } , \ldots , x _ { T }$ , and we wish to predict some correspond
+
+![](images/e4db16452e482b763cd61552fcde4b5c943fbde3fd808c44865ddb9b473e207d.jpg)  
+Figure 1: A simple causal convolution with filter size 3.
+
+ing outputs $y _ { 0 } , \ldots , y _ { T }$ at each time. The key constraint is that to predict the output $y _ { t }$ for some time $t$ , we are constrained to only use those inputs that have been previously observed: $x _ { 0 } , \ldots , x _ { t }$ . Formally, a sequence modeling network is any function $f : \mathcal { X } ^ { T + \bar { 1 } }  \mathcal { Y } ^ { T + \bar { 1 } }$ that produces this mapping
+
+$$
+\hat { y } _ { 0 } , \dots , \hat { y } _ { T } = f ( x _ { 0 } , \dots , x _ { T } )
+$$
+
+if it satisfies the causal constraint that $y _ { t }$ depends only on $x _ { 0 } , \ldots , x _ { t }$ , and not on any “future” inputs $x _ { t + 1 } , \dots , x _ { T }$ . The goal of learning in the sequence modeling setting is to find the network $f$ minimizing some expected loss between the actual outputs and predictions $L ( y _ { 0 } , \dots , y _ { T } , f ( x _ { 0 } , \dots , x _ { T } ) )$ where the sequences and outputs are drawn according to some distribution.
+
+This formalism encompasses many settings such as auto-regressive prediction (where we try to predict some signal given its past) by setting the target output to be simply the input shifted by one time step. It does not, however, directly capture domains such as machine translation, or sequenceto-sequence prediction in general, since in these cases the entire input sequence (including “future” states) can be used to predict each output (though the techniques can naturally be extended to work in such settings).
+
+# 3.2 CAUSAL CONVOLUTIONS AND THE TCN
+
+As mentioned above, the TCN is based upon two principles: the fact that the network produces an output of the same length as the input, and the fact that there can be no leakage from the future into the past. To accomplish the first point, the TCN uses a 1D fully-convolutional network (FCN) architecture (Long et al., 2015), where each hidden layer is the same length as the input layer, and zero padding of length (kernel size − 1) is added to keep subsequent layers the same length as previous ones. To achieve the second point, the TCN uses causal convolutions, convolutions where a subsequent output at time $t$ is convolved only with elements from time $t$ and before in the previous layer.1 Graphically, the network is shown in Figure 1. Put in a simple manner:
+
+$$
+\mathrm { T C N } = \mathrm { 1 D F C N } + \mathrm { c a u s a l ~ c o n v o l u t i o n s }
+$$
+
+It is worth emphasizing that this is essentially the same architecture as the time delay neural network proposed nearly 30 years ago by Waibel et al. (1989), with the sole tweak of zero padding to ensure equal sizes of all layers.
+
+However, a major disadvantage of this “na¨ıve” approach is that in order to achieve a long effective history size, we need an extremely deep network or very large filters, neither of which were particularly feasible when the methods were first introduced. Thus, in the following sections, we describe how techniques from modern convolutional architectures can be integrated into the TCN to allow for both very deep networks and very long effective history.
+
+![](images/3d3384271f9fbd194ae3ecf347bb7e1e6b27e790b9063a1b747424be7bcaf5f9.jpg)  
+Figure 2: A dilated causal convolution with dilation factors $d = 1 , 2 , 4$ and filter size $k = 3$ . The receptive field is able to cover all values from the input sequence.
+
+# 3.3 DILATED CONVOLUTIONS
+
+Through convolutional filters, as previously addressed, a simple causal convolution is only able to look back at a history with size linear in the depth of the network. This makes it challenging to apply the aforementioned causal convolution on sequence tasks, especially those requiring longer history. Our solution here, used previously for example in audio synthesis by Oord et al. (2016a), is to employ dilated convolutions (Yu & Koltun, 2015) that enable an exponentially large receptive field. More formally, for a 1-D sequence input $\mathbf { x } \in \mathbb { R } ^ { n }$ and a filter $f : \bar { \{ 0 , \dots , k - 1 \bar { \} } }  \mathbb { R }$ , the dilated convolution operation $F$ on element $s$ of the sequence is defined as
+
+$$
+F ( s ) = ( \mathbf { x } * _ { d } f ) ( s ) = \sum _ { i = 0 } ^ { k - 1 } f ( i ) \cdot \mathbf { x } _ { s + d \cdot i }
+$$
+
+where $d$ is the dilation factor and $k$ is the filter size. Dilation is thus equivalent to introducing a fixed step between every two adjacent filter taps. When taking $d = 1$ , for example, a dilated convolution is trivially a normal convolution operation. Using larger dilations enables an output at the top level to represent a wider range of inputs, thus effectively expanding the receptive field of a ConvNet.
+
+This gives us two ways to increase the receptive field of the TCN: by choosing larger filter sizes $k$ , and by increasing the dilation factor $d$ , where the effective history of one such layer is $( k - 1 ) d$ . As is common when using dilated convolutions, we increase $d$ exponentially with the depth of the network (i.e., $d = O ( 2 ^ { i } )$ at level $i$ of the network). This ensures that there is some filter that hits each input within the effective history, while also allowing for an extremely large effective history using deep networks. We provide an illustration in Figure 2. Using filter size $k = 3$ and dilation factor $d = 1 , 2 , 4$ , the receptive field is able to cover all values from the input sequence.
+
+# 3.4 RESIDUAL CONNECTIONS
+
+Proposed by He et al. (2016), residual functions have proven to be especially useful in effectively training deep networks. In a residual network, each residual block contains a branch leading out to a series of transformations $\mathcal { F }$ , whose outputs are added to the input $\mathbf { x }$ of the block:
+
+$$
+o = \mathrm { A c t i v a t i o n } ( \mathbf { x } + \mathcal { F } ( \mathbf { x } ) )
+$$
+
+This effectively allows for the layers to learn modifications to the identity mapping rather than the entire transformation, which has been repeatedly shown to benefit very deep networks.
+
+As the TCN’s receptive field depends on the network depth $n$ as well as filter size $k$ and dilation factor $d$ , stabilization of deeper and larger TCNs becomes important. For example, in a case where the prediction could depend on a history of size $2 ^ { 1 2 }$ and a high-dimensional input sequence, a network of up to 12 layers could be needed. Each layer, more specifically, consists of multiple filters for feature extraction. In our design of the generic TCN model, we therefore employed a generic residual module in place of a convolutional layer.
+
+The residual block for our baseline TCN is shown in figure 3a. Within a residual block, the TCN has 2 layers of dilated causal convolution and non-linearity, for which we used the rectified linear unit
+
+(a) TCN residual block. An 1x1 convolution is added when residual input and output have different dimensions.
+
+![](images/1813e828cfa0df05d2cac09fee75934b751463303c74b3ec805bc1f1e7e53827.jpg)  
+Figure 3: A visualization of the TCN residual block
+
+(b) An example of residual connection in TCN. The blue lines are filters in the residual function, and the green lines are identity mappings.
+
+(ReLU) (Nair & Hinton, 2010). For normalization, we applied Weight Normalization (Salimans & Kingma, 2016) to the filters in the dilated convolution (where we note that the filters are essentially vectors of size $k \times 1 \AA$ ). In addition, a 2-D dropout (Srivastava et al., 2014) layer was added after each dilated convolution for regularization: at each training step, a whole channel (in the width dimension) is zeroed out.
+
+However, whereas in standard ResNet the input is passed in and added directly to the output of the residual function, in TCN (and ConvNet in general) the input and output could have different widths. Therefore in our TCN, when the input-output widths disagree, we use an additional 1x1 convolution to ensure that element-wise addition $\oplus$ receives tensors of the same shape (see Figure 3a, 3b).
+
+Note that many further optimizations (e.g., gating, skip connections, context stacking as in audio generation using WaveNet) are possible in a TCN than what we described here. However, in this paper, we aim to present a generic, general-purpose TCN, to which additional twists can be added as needed. As we are going to show in Section 4, this general-purpose architecture is already able to outperform recurrent units like LSTM on a number of tasks by a good margin.
+
+# 3.5 ADVANTAGES OF TCN SEQUENCE MODELING
+
+There are several key advantages to a TCN model with the ingredients that we described above.
+
+• Parallelism. Unlike in RNNs where the predictions for later timesteps must wait for their predecessors to complete, in a convolutional architecture these computations can be done in parallel since the same filter is used in each layer. Therefore, in training and evaluation, a (possibly long) input sequence can be processed as a whole in TCN, instead of serially as in RNN, which depends on the length of the sequence and could be less efficient.
+
+• Flexible receptive field size. With a TCN, we can change its receptive field size in multiple ways. For instance, stacking more dilated (causal) convolutional layers, using larger dilation factors, or increasing the filter size are all viable options (with possibly different interpretations). TCN is thus easy to tune and adapt to different domains, since we now can directly control the size of the model’s memory.
+
+• Stable gradients. Unlike recurrent architectures, TCN has a backpropagation path that is different from the temporal direction of the sequence. This enables it to avoid the problem of exploding/vanishing gradients, which is a major issue for RNNs (and which led to the development of LSTM, GRU, HF-RNN, etc.).
+
+• Low memory requirement for training. In a task where the input sequence is long, a structure such as LSTM can easily use up a lot of memory to store the partial results for backpropagation (e.g., the results for each gate of the cell). However, in TCN, the backpropagation path only depends on the network depth and the filters are shared in each layer, which means that in practice, as model size or sequence length gets large, TCN is likely to use less memory than RNNs.
+
+# 3.6 DISADVANTAGES OF TCN SEQUENCE MODELING
+
+We also summarize two disadvantages of using TCN instead of RNNs.
+
+• Data storage in evaluation. In evaluation/testing, RNNs only need to maintain a hidden state and take in a current input $x _ { t }$ in order to generate a prediction. In other words, a “summary” of the entire history is provided by the fixed-length set of vectors $h _ { t }$ , which means that the actual observed sequence can be discarded (and indeed, the hidden state can be used as a kind of encoder for all the observed history). In contrast, the TCN still needs to take in a sequence with non-trivial length (precisely the effective history length) in order to predict, thus possibly requiring more memory during evaluation. • Potential parameter change for a transfer of domain. Different domains can have different requirements on the amount of history the model needs to memorize. Therefore, when transferring a model from a domain where only little memory is needed (i.e., small $k$ and $d$ ) to a domain where much larger memory is required (i.e., much larger $k$ and $d$ ), TCN may perform poorly for not having a sufficiently large receptive field.
+
+We want to emphasize, though, that we believe the notable lack of “infinite memory” for a TCN is decidedly not a practical disadvantage, since, as we show in Section 4, the TCN method actually outperforms RNNs in terms of the ability to deal with long temporal dependencies.
+
+# 4 EXPERIMENTS
+
+In this section, we conduct a series of experiments using the baseline TCN (described in section 3) and generic RNNs (namely LSTMs, GRUs, and vanilla RNNs). These experiments cover tasks and datasets from various domains, aiming to test different aspects of a model’s ability to learn sequence modeling. In several cases, specialized RNN models, or methods with particular forms of regularization can indeed vastly outperform both generic RNNs and the TCN on particular problems, which we highlight when applicable. But as a general-purpose architecture, we believe the experiments make a compelling case for the TCN as the “first attempt” approach for many sequential problems.
+
+All experiments reported in this section used the same TCN architecture, just varying the depth of the network and occasionally the kernel size. We use an exponential dilation $d = 2 ^ { n }$ for layer $n$ in the network, and the Adam optimizer (Kingma & Ba, 2015) with learning rate 0.002 for TCN (unless otherwise noted). We also empirically find that gradient clipping helped training convergence of TCN, and we pick the maximum norm to clip from [0.3, 1]. When training recurrent models, we use a simple grid search to find a good set of hyperparameters (in particular, optimizer, recurrent drop $p \in [ 0 . 0 5 , 0 . 5 ]$ , the learning rate, gradient clipping, and initial forget-gate bias), while keeping the network around the same size as TCN. No other optimizations, such as gating mechanism (see Appendix D), or highway network, were added to TCN or the RNNs. The hyperparameters we use for TCN on different tasks are reported in Table 2 in Appendix B. In addition, we conduct a series controlled experiments to investigate the effects of filter size and residual function on the TCN’s performance. These results can be found in Appendix C.
+
+# 4.1 TASKS AND RESULTS SUMMARY
+
+In this section we highlight the general performance of generic TCNs vs generic LSTMs for a variety of domains from the sequential modeling literature. A complete description of each task, as well as references to some prior works that evaluated them, is given in Appendix A. In brief, the tasks we consider are: the adding problem, sequential MNIST, permuted MNIST (P-MNIST), the copy memory task, the Nottingham and JSB Chorales polyphonic music tasks, Penn Treebank (PTB), Wikitext-103 and LAMBADA word-level language modeling, as well as PTB and text8 characterlevel language modeling.
+
+Table 1: Complete comparison of the TCN to regularized recurrent architectures in various tasks.   
+
+<table><tr><td rowspan="2">Sequential Tasks</td><td rowspan="2">Model Size (≈)</td><td colspan="4">Models</td></tr><tr><td>LSTM</td><td>GRU</td><td>RNN</td><td>TCN (ours)</td></tr><tr><td>Seq. MNIST (accuracy)</td><td>70K</td><td>87.2</td><td>96.2</td><td>21.5</td><td>99.0</td></tr><tr><td>P-Seq. MNIST (accuracy)</td><td>70K</td><td>85.7</td><td>87.3</td><td>25.3</td><td>97.2</td></tr><tr><td>The Adding Problem T=600 (loss)</td><td>70K</td><td>0.164</td><td>5.3e-5</td><td>0.177</td><td>5.8e-5</td></tr><tr><td>Copy Memory T=1000 (loss)</td><td>16K</td><td>0.0204</td><td>0.0197</td><td>1</td><td>3.5e-5</td></tr><tr><td>Music JSB Chorales (loss)</td><td>300K</td><td>8.45</td><td>8.43</td><td>8.91</td><td>8.10</td></tr><tr><td>Music Nottingham (loss)</td><td>1M</td><td>3.29</td><td>3.46</td><td>-</td><td>3.07</td></tr><tr><td>Word-level PTB (ppl)</td><td>13M</td><td>84.77</td><td>92.48</td><td>114.50</td><td>90.17</td></tr><tr><td>Word-level Wiki-103 (ppl)</td><td>-</td><td>48.4 (large)</td><td>-</td><td>1</td><td>45.19</td></tr><tr><td>Word-level LAMBADA (ppl)</td><td>-</td><td>4186</td><td>-</td><td>14725</td><td>1279</td></tr><tr><td>Char-level PTB (bpc)</td><td>3M</td><td>1.41</td><td>1.42</td><td>1.52</td><td>1.35</td></tr><tr><td>Char-level text8 (bpc)</td><td>5M</td><td>1.52</td><td>1.56</td><td>1.69</td><td>1.45</td></tr></table>
+
+![](images/ea6a8e859d355bb1c75732fd3c99b5619813b7328366f413036a0d184b56a648.jpg)  
+Figure 4: Results of TCN vs. recurrent architectures on the adding problem
+
+A summary comparison of TCNs to the standard RNN architectures (LSTM, GRU, and vanilla RNN) is shown in Table 1. We will highlight many of these results below, and want to emphasize that for several tasks the baseline RNN architectures are still far from the state of the art (see Table 4), but in total the results make a strong case that the TCN architecture, as a generic sequence modeling framework, is often superior to generic RNN approaches. We now consider several of these experiments in detail, generally distinguishing between the “recurrent benchmark” tasks designed to show the limitations of networks for sequence modeling (adding problem, sequential & permuted MNIST, copy memory), and the “applied” tasks (polyphonic music and language modeling).
+
+# 4.2 BASELINE RECURRENT TASKS
+
+We first compare the results of the TCN architecture to those of RNNs on the toy baseline tasks that have been frequently used to evaluate sequential modeling (Hochreiter & Schmidhuber, 1997; Martens & Sutskever, 2011; Pascanu et al., 2013; Le et al., 2015; Cooijmans et al., 2016; Zhang et al., 2016; Krueger et al., 2017; Wisdom et al., 2016; Arjovsky et al., 2016).
+
+The Adding Problem. Convergence results for the adding problem, for problem sizes $T =$ 200, 400, 600, are shown in Figure 4; all models were chosen to have roughly 70K parameters. In all three cases, TCNs quickly converged to a virtually perfect solution (i.e., an MSE loss very close to 0). LSTMs and vanilla RNNs performed significantly worse, while on this task GRUs also performed quite well, even though their convergence was slightly slower than TCNs.
+
+Sequential MNIST and P-MNIST. Results on sequential and permuted MNIST, run over 10 epochs, are shown in Figures 5a and 5b; all models were picked to have roughly 70K parameters. For both problems, TCNs substantially outperform the alternative architectures, both in terms of convergence time and final performance level on the task. For the permuted sequential MNIST, TCNs outperform state of the art results using recurrent nets $( 9 5 . 9 \% )$ with Zoneout+Recurrent BatchNorm (Cooijmans et al., 2016; Krueger et al., 2017), a highly optimized method for regularizing RNNs.
+
+![](images/404df7deb85ffa4ae5c1945673693b1c2d442023841918eb5df6debfac1e9251.jpg)  
+Figure 5: Results of TCN vs. recurrent architectures on the Sequential MNIST and P-MNIST
+
+![](images/4cc3f571686b4edc1e41610e5d0966fcd5f75e40a815be1d639a5bcb8bd1de0b.jpg)  
+Figure 6: Result of TCN vs. recurrent architectures on the Copy Memory Task, for different $T$
+
+Copy Memory Task. Finally, Figure 6 shows the results of the different methods (with roughly the same size) on the copy memory task. Again, the TCNs quickly converge to correct answers, while the LSTM and GRU simply converge to the same loss as predicting all zeros. In this case we also compare to the recently-proposed EURNN (Jing et al., 2017), which was highlighted to perform well on this task. While both perform well for sequence length $T = 5 0 0$ , the TCN again has a clear advantage for $T = 1 0 0 0$ and $T = 2 0 0 0$ (in terms of both loss and convergence).
+
+# 4.3 RESULTS ON POLYPHONIC MUSIC AND LANGUAGE MODELING
+
+Next, we compare the results of the TCN architecture to recurrent architectures on 6 different real datasets in polyphonic music as well as word- and character-level language modeling. These are areas where sequence modeling has been used most frequently. As domains where there is considerable practical interests, there have also been many specialized RNNs developed for these tasks (e.g., Zhang et al. (2016); Ha et al. (2017); Krueger et al. (2017); Grave et al. (2016); Greff et al. (2017); Merity et al. (2017)). We mention some of these comparisons when useful, but the primary goal here is to compare the generic TCN model to other generic RNN architectures, so we focus mainly on these comparisons.
+
+Polyphonic Music. On the Nottingham and JSB Chorales datasets, the TCN with virtually no tuning is again able to beat the other models by a considerable margin (see Table 1), and even outperforms some improved recurrent models for this task such as HF-RNN (Boulanger-Lewandowski et al., 2012) and Diagonal RNN (Subakan & Smaragdis, 2017). Note however that other models such as the Deep Belief Net LSTM (Vohra et al., 2015) perform substantially better on this task; we believe this is likely due to the fact that the datasets involved in polyphonic music are relatively small, and thus the right regularization method or generative modeling procedure can improve performance significantly. This is largely orthogonal to the RNN/TCN distinction, as a similar variant of TCN may well be possible.
+
+Word-level Language Modeling. Language modeling remains one of the primary applications of recurrent networks in general, where many recent works have been focusing on optimizing the usage of LSTMs (see Krueger et al. (2017); Merity et al. (2017)). In our implementation, we follow standard practices such as tying the weights of encoder and decoder layers for both TCN and RNNs (Press & Wolf, 2016), which significantly reduces the number of parameters in the model. When training the language modeling tasks, we use SGD optimizer with annealing learning rate (by a factor of 0.5) for TCN and RNNs.
+
+Results on word-level language modeling are reported in Table 1. With a fine-tuned LSTM (i.e., with recurrent and embedding dropout, etc.), we find LSTM can outperform TCN in perplexity on the Penn TreeBank (PTB) dataset, where the TCN model still beats both GRU and vanilla RNN. On the much larger Wikitext-103 corpus, however, without performing much hyperparameter search (due to lengthy training process), we still observe that TCN outperforms the state of the art LSTM results (48.4 in perplexity) by Grave et al. (2016) (without continuous cache pointer; see Table 4). The same superiority is observed on the LAMBADA test (Paperno et al., 2016), where TCN achieves a much lower perplexity than its recurrent counterparts in predicting the last word based on a very long context (see Appendix A). We will further analyze this in section 4.4.
+
+Character-level Language Modeling. The results of applying TCN and alternative models on PTB and text8 data for character-level language modeling are shown in Table 1, with performance measured in bits per character (bpc). While beaten by the state of the art (see Table 4), the generic TCN outperforms regularized LSTM and GRU as well as methods such as Norm-stabilized LSTM (Krueger & Memisevic, 2015). Moreover, we note that using a filter size of $k \leq 4$ works better than larger filter sizes in character-level language modeling, which suggests that capturing short history is more important than longer dependencies in these tasks.
+
+# 4.4 MEMORY SIZE OF TCN AND RNNS
+
+Finally, one of the important reasons why RNNs have been preferred over CNNs for general sequence modeling is that theoretically, recurrent architectures are capable of an infinite memory. We therefore attempt to study here how much memory TCN and LSTM/GRU are able to actually “backtrack”, via the copy memory task and the LAMBADA language modeling task.
+
+The copy memory task is a simple but perfect task to examine a model’s ability to pick up its memory from a (possibly) distant past (by varying the value of sequence length $T$ ). However, different from the setting in Section 4.2, in order to compare the results for different sequence lengths, here we only report the accuracy on the last $I O$ elements of the output sequence. We used a model size of 10K for both TCN and RNNs.
+
+![](images/294b6d1f2397e8ec063d0643ac527f33c030df7f7c912acd3fc3b3fa37f37eaa.jpg)  
+Figure 7: Accuracy on the copy memory task for varying sequence length $T$ .
+
+The results are shown in Figure 7. TCNs consistently converge to $100 \%$ accuracy for all sequence lengths, whereas it is increasingly challenging for recurrent models to memorize as $T$ grows (with accuracy converging to that of a random guess). LSTM’s accuracy quickly falls below $20 \%$ for $T \geq 5 0$ , which suggests that instead of infinite memory, LSTMs are only good at recalling a short history instead.
+
+This observation is also backed up by the experiments of TCN on the LAMBADA dataset, which is specifically designed to test a model’s textual understanding in a broader discourse. The objective of LAMBADA dataset is to predict the last word of the target sentence given a sufficiently long context (see Appendix A for more details). Most of the existing models fail to guess accurately on this task. As shown in Table 1, TCN outperforms LSTMs by a significant margin in perplexity on LAMBADA (with a smaller network and virtually no tuning).
+
+These results indicate that TCNs, despite their apparent finite history, in practice maintain a longer effective history than their recurrent counterparts. We would like to emphasize that this empirical observation does not contradict the good results that prior works have achieved using LSTM, such as in language modeling on PTB. In fact, the very success of $n$ -gram models (Brown et al., 1992) suggested that language modeling might not need a very long memory, a conclusion also reached by prior works such as Dauphin et al. (2017).
+
+# 5 DISCUSSION
+
+In this work, we revisited the topic of modeling sequence predictions using convolutional architectures. We introduced the key components of the TCN and analyzed some vital advantages and disadvantages of using TCN for sequence predictions instead of RNNs. Further, we compared our generic TCN model to the recurrent architectures on a set of experiments that span a wide range of domains and datasets. Through these experiments, we have shown that TCN with minimal tuning can outperform LSTM/GRU of the same model size (and with standard regularizations) in most of the tasks. Further experiments on the copy memory task and LAMBADA task revealed that TCNs actually has a better capability for long-term memory than the comparable recurrent architectures, which are commonly believed to have unlimited memory.
+
+It is still important to note that, however, we only presented a generic architecture here, with components all coming from standard modern convolutional networks (e.g., normalization, dropout, residual network). And indeed, on specific problems, the TCN model can still be beaten by some specialized RNNs that adopt carefully designed optimization strategies. Nevertheless, we believe the experiment results in Section 4 might be a good signal that instead of considering RNNs as the “default” methodology for sequence modeling, convolutional networks too, can be a promising and powerful toolkit in studying time-series data.
+
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+
+# A DESCRIPTION OF BENCHMARK TASKS
+
+The Adding Problem: In this task, each input consists of a length- $\mathbf { \nabla } \cdot n$ sequence of depth 2, with all values randomly chosen in [0, 1], and the second dimension being all zeros expect for two elements that are marked by 1. The objective is to sum the two random values whose second dimensions are marked by 1. Simply predicting the sum to be 1 should give an MSE of about 0.1767. First introduced by Hochreiter & Schmidhuber (1997), the addition problem have been consistently used as a pathological test for evaluating sequential models (Pascanu et al., 2013; Le et al., 2015; Zhang et al., 2016; Arjovsky et al., 2016).
+
+Sequential MNIST & P-MNIST: Sequential MNIST is frequently used to test a recurrent network’s ability to combine its information from a long memory context in order to make classification prediction (Le et al., 2015; Zhang et al., 2016; Cooijmans et al., 2016; Krueger et al., 2017; Jing et al., 2017). In this task, MNIST (Lecun et al., 1998) images are presented to the model as a $7 8 4 \times 1$ sequence for digit classification In a more challenging setting, we also permuted the order of the sequence by a random (fixed) order and tested the TCN on this permuted MNIST (P-MNIST) task.
+
+Copy Memory Task: In copy memory task, each input sequence has length $T + 2 0$ . The first 10 values are chosen randomly from digit [1-8] with the rest being all zeros, except for the last 11 entries which are marked by 9 (the first $" 9 "$ is a delimiter). The goal of this task is to generate an output of same length that is zero everywhere, except the last 10 values after the delimiter, where the model is expected to repeat the same 10 values at the start of the input. This was used by prior works such as Arjovsky et al. (2016); Wisdom et al. (2016); Jing et al. (2017); but we also extended the sequence lengths to up to $T = 2 0 0 0$ .
+
+JSB Chorales: JSB Chorales dataset (Allan & Williams, 2005) is a polyphonic music dataset consisting of the entire corpus of 382 four-part harmonized chorales by J. S. Bach. In a polyphonic music dataset, each input is a sequence of elements having 88 dimensions, representing the 88 keys on a piano. Therefore, each element $x _ { t }$ is a chord written in as binary vector, in which a “1” indicates a key pressed.
+
+Nottingham: Nottingham dataset 2 is a collection of 1200 British and American folk tunes. Nottingham is a much larger dataset than JSB Chorales. Along with JSB Chorales, Nottingham has been used in a number of works that investigated recurrent models’ applicability in polyphonic music (Greff et al., 2017; Chung et al., 2014), and the performance for both tasks are measured in terms of negative log-likelihood (NLL) loss.
+
+PennTreebank: We evaluated TCN on the PennTreebank (PTB) dataset (Marcus et al., 1993), for both character-level and word-level language modeling. When used as a character-level language corpus, PTB contains 5059K characters for training, 396K for validation and 446K for testing, with an alphabet size of 50. When used as a word-level language corpus, PTB contains 888K words for training, 70K for validation and 79K for testing, with vocabulary size 10000. This is a highly studied dataset in the field of language modeling (Miyamoto & Cho, 2016; Krueger et al., 2017; Merity et al., 2017), with exceptional results have been achieved by some highly optimized RNNs.
+
+Wikitext-103: Wikitext-103 (Merity et al., 2016) is almost 110 times as large as PTB, featuring a vocabulary size of about 268K. The dataset contains 28K Wikipedia articles (about 103 million words) for training, 60 articles (about 218K words) for validation and 60 articles (246K words) for testing. This is a more representative (and realistic) dataset than PTB as it contains a much larger vocabulary, including many rare vocabularies.
+
+LAMBADA: Introduced by Paperno et al. (2016), LAMBADA (LA nguage Modeling Boadened to Account for Discourse Aspects) is a dataset consisting of 10K passages extracted from novels, with on average 4.6 sentences as context, and 1 target sentence whose last word is to be predicted. This dataset was built so that human can guess naturally and perfectly when given the context, but would fail to do so when only given the target sentence. Therefore, LAMBADA is a very challenging dataset that evaluates a model’s textual understanding and ability to keep track of information in the broader discourse. Here is an example of a test in the LAMBADA dataset, where the last word “miscarriage” is to be predicted (which is not in the context):
+
+Context: “Yes, I thought I was going to lose the baby.”“I was scared too.” he stated, sincerity flooding his eyes. “You were?”“Yes, of course. Why do you even ask?”“This baby wasn’t exactly planned for.”
+
+Target Sentence: “Do you honestly think that I would want you to have a ”
+
+Target Word: miscarriage
+
+This dataset was evaluated in prior works such as Paperno et al. (2016); Grave et al. (2016). In general, better results on LAMBADA indicate that a model is better at capturing information from longer and broader context. The training data for LAMBADA is the full text of 2,662 novels with more than 200M words 3, and the vocabulary size is about 93K.
+
+text8: We also used tex $8 ^ { 4 }$ dataset for character level language modeling (Mikolov et al., 2012). Compared to PTB, text8 is about 20 times as large, with about 100 million characters from Wikipedia (90M for training, 5M for validation and 5M for testing). The corpus contains 27 unique alphabets.
+
+# B HYPERPARAMETERS SETTINGS
+
+# B.1 HYPERPARAMETERS FOR TCN
+
+In this supplementary section, we report in a table (see Table 2) the hyperparameters we used when applying the generic TCN model on the different tasks/datasets. The most important factor for picking parameters is to make sure that the TCN has a sufficiently large receptive field by choosing $k$ and $n$ that can cover the amount of context needed for the task.
+
+Table 2: TCN parameter settings for experiments in Section. 4   
+
+<table><tr><td colspan="7">TCN SETTINGS</td></tr><tr><td>Dataset/Task</td><td>Subtask</td><td>k n</td><td>Hidden</td><td>Dropout</td><td>Grad Clip</td><td>Note</td></tr><tr><td rowspan="3">The Adding Problem</td><td>T=200</td><td>6</td><td>7 27</td><td></td><td></td><td></td></tr><tr><td>T= 400</td><td>7</td><td>7 27</td><td>0.0</td><td>N/A</td><td></td></tr><tr><td>T= 600</td><td>8</td><td>8 24</td><td></td><td></td><td></td></tr><tr><td rowspan="2">Seq. MNIST</td><td></td><td>7</td><td>8</td><td>25</td><td>0.0</td><td>N/A</td></tr><tr><td></td><td>6</td><td>8 20</td><td></td><td></td><td></td></tr><tr><td>Permuted MNIST</td><td></td><td>7 6</td><td>8 8</td><td>25 0.0 20</td><td>N/A</td><td></td></tr><tr><td rowspan="3">Copy Memory Task</td><td>T=500</td><td>6</td><td>9</td><td>10</td><td></td><td></td></tr><tr><td>T= 1000</td><td>8</td><td>8</td><td>10 0.05</td><td>1.0</td><td>RMSprop 5e-4</td></tr><tr><td>T= 2000</td><td>8</td><td>9 10</td><td></td><td></td><td></td></tr><tr><td>Music JSB Chorales</td><td>1</td><td>3</td><td>2</td><td>150</td><td>0.5 0.4</td><td></td></tr><tr><td>Music Nottingham</td><td>1</td><td>6</td><td>4</td><td>150</td><td>0.2 0.4</td><td></td></tr><tr><td rowspan="3">Word-level LM</td><td>PTB</td><td>3</td><td>4 600</td><td></td><td></td><td>Embed. size 600</td></tr><tr><td>Wiki-103</td><td>3</td><td>5 1000</td><td>0.4</td><td>0.3</td><td>Embed. size 400</td></tr><tr><td>LAMBADA</td><td>4</td><td>5 500</td><td></td><td></td><td>Embed. size 500</td></tr><tr><td rowspan="2">Char-level LM</td><td>PTB</td><td>3</td><td>3</td><td>450</td><td></td><td></td></tr><tr><td>text8</td><td>2</td><td>5</td><td>0.1 520</td><td>0.15</td><td>Embed. size 100</td></tr></table>
+
+As previously mentioned in Section 4, the number of hidden units was chosen based on $k$ and $n$ such that the model size is approximately at the same level as the recurrent models. In the table above, a gradient clip of N/A means no gradient clipping was applied. However, in larger tasks, we empirically found that adding a gradient clip value (we randomly picked from [0.2, 1]) helps the training convergence.
+
+# B.2 HYPERPARAMETERS FOR LSTM/GRU
+
+We also report the parameter setting for LSTM in Table 3. These values are picked from hyperparameter search for LSTMs that have up to 3 layers, and the optimizers are chosen from {SGD, Adam, RMSprop, Adagrad}.
+
+GRU hyperparameters were chosen in a similar fashion, but with more hidden units to keep the total model size approximately the same (since a GRU cell is smaller).
+
+# B.3 COMPARE TO THE STATE OF THE ART RESULTS
+
+As previously noted, TCN can still be outperformed by optimized RNNs in some of the tasks, whose results are summarized in Table 4 below. The same TCN architecture is used across all tasks.
+
+Note that the size of the SoTA model may be different from the size of the TCN.
+
+Table 3: LSTM parameter settings for experiments in Section 4.   
+
+<table><tr><td colspan="8">LSTM SETTINGS(KEY PARAMETERS)</td></tr><tr><td>Dataset/Task</td><td>Subtask</td><td>n</td><td>Hidden</td><td>Dropout</td><td>Grad Clip</td><td>Bias</td><td>Note</td></tr><tr><td rowspan="3">The Adding Problem</td><td>T= 200</td><td>2</td><td>77</td><td></td><td>50</td><td>5.0</td><td>SGD 1e-3</td></tr><tr><td>T= 400</td><td>2</td><td>77</td><td>0.0</td><td>50</td><td>10.0</td><td>Adam 2e-3</td></tr><tr><td>T= 600</td><td>1</td><td>130</td><td></td><td>5</td><td>1.0</td><td>1</td></tr><tr><td>Seq. MNIST</td><td>:</td><td>1</td><td>130</td><td>0.0</td><td>1</td><td>1.0</td><td>RMSprop 1e-3</td></tr><tr><td>Permuted MNIST</td><td>1</td><td>1</td><td>130</td><td>0.0</td><td>1</td><td>10.0</td><td>RMSprop 1e-3</td></tr><tr><td rowspan="3">Copy Memory Task</td><td>T= 500</td><td>1</td><td>50</td><td></td><td>0.25</td><td></td><td></td></tr><tr><td>T=1000</td><td>1</td><td>50</td><td>0.05</td><td>1</td><td>1</td><td>RMSprop/Adam</td></tr><tr><td>T= 2000</td><td>3</td><td>28</td><td></td><td>1</td><td></td><td></td></tr><tr><td>Music JSB Chorales</td><td>1</td><td>2</td><td>200</td><td>0.2</td><td>1</td><td>10.0</td><td>SGD/Adam</td></tr><tr><td>Music Nottingham</td><td>-</td><td>3</td><td>280</td><td>0.1</td><td>0.5</td><td>-</td><td>Adam 4e-3</td></tr><tr><td rowspan="3">Word-level LM</td><td></td><td>1</td><td>500</td><td></td><td>1</td><td>-</td><td></td></tr><tr><td>PTB</td><td>3</td><td>700</td><td>0.4</td><td>0.3</td><td>1.0</td><td>SGD 30,Emb. 700, etc.</td></tr><tr><td>Wiki-103</td><td>-</td><td>-</td><td>·</td><td>·</td><td>·</td><td>Grave et al. (2016)</td></tr><tr><td></td><td>LAMBADA</td><td>-</td><td>-</td><td>-</td><td>1</td><td>-</td><td>Grave et al. (2016)</td></tr><tr><td rowspan="2">Char-level LM</td><td>PTB</td><td></td><td>600</td><td>0.1</td><td>0.5</td><td>-</td><td>Emb. size 120</td></tr><tr><td>text8</td><td>21</td><td>1024</td><td>0.15</td><td>0.5</td><td>1</td><td>Adam 1e-2</td></tr></table>
+
+Table 4: State of the art (SoTA) results for tasks in Section 4.   
+
+<table><tr><td rowspan=1 colspan=6>TCN VS. SoTA RESULTS</td></tr><tr><td rowspan=1 colspan=1>Task</td><td rowspan=1 colspan=1>TCN Result</td><td rowspan=1 colspan=1>Size</td><td rowspan=1 colspan=1>SoTA</td><td rowspan=1 colspan=1>Size</td><td rowspan=1 colspan=1>Model</td></tr><tr><td rowspan=1 colspan=1>Seq. MNIST (acc.)</td><td rowspan=1 colspan=1>99.0</td><td rowspan=1 colspan=1>21K</td><td rowspan=1 colspan=1>99.0</td><td rowspan=1 colspan=1>21K</td><td rowspan=1 colspan=1>Dilated GRU (Chang et al., 2017)</td></tr><tr><td rowspan=1 colspan=1>P-MNIST (acc.)</td><td rowspan=1 colspan=1>97.2</td><td rowspan=1 colspan=1>42K</td><td rowspan=1 colspan=1>95.9</td><td rowspan=1 colspan=1>42K</td><td rowspan=1 colspan=1> Zoneout (Krueger et al., 2017)</td></tr><tr><td rowspan=1 colspan=1>Adding Prob. 600 (loss)</td><td rowspan=1 colspan=1>5.8e-5</td><td rowspan=1 colspan=1>70K</td><td rowspan=1 colspan=1>5.3e-5</td><td rowspan=1 colspan=1>70K</td><td rowspan=1 colspan=1>Regularized GRU</td></tr><tr><td rowspan=1 colspan=1>Copy Memory 1000 (loss)</td><td rowspan=1 colspan=1>3.5e-5</td><td rowspan=1 colspan=1>70K</td><td rowspan=1 colspan=1>0.011</td><td rowspan=1 colspan=1>70K</td><td rowspan=1 colspan=1>EURNN (Jing et al., 2017)</td></tr><tr><td rowspan=1 colspan=1>JSB Chorales (loss)</td><td rowspan=1 colspan=1>8.10</td><td rowspan=1 colspan=1>300K</td><td rowspan=1 colspan=1>3.47</td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1>DBN+LSTM(Vohra et al., 2015)</td></tr><tr><td rowspan=1 colspan=1>Nottingham (loss)</td><td rowspan=1 colspan=1>3.07</td><td rowspan=1 colspan=1>1M</td><td rowspan=1 colspan=1>1.32</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>DBN+LSTM (Vohra et al., 2015)</td></tr><tr><td rowspan=1 colspan=1>Word PTB (ppl)</td><td rowspan=1 colspan=1>90.17</td><td rowspan=1 colspan=1>13M</td><td rowspan=1 colspan=1>52.8</td><td rowspan=1 colspan=1>24M</td><td rowspan=1 colspan=1>AWD-LSTM + Cont.Cache(Merity et al., 2017)</td></tr><tr><td rowspan=1 colspan=1>Word Wiki-103 (ppl)</td><td rowspan=1 colspan=1>45.19</td><td rowspan=1 colspan=1>148M</td><td rowspan=1 colspan=1>40.4</td><td rowspan=1 colspan=1>&gt;300M</td><td rowspan=1 colspan=1>Neural Cache Model (Large)(Grave et al., 2016)</td></tr><tr><td rowspan=1 colspan=1>Word LAMBADA (ppl)</td><td rowspan=1 colspan=1>1279</td><td rowspan=1 colspan=1>56M</td><td rowspan=1 colspan=1>138</td><td rowspan=1 colspan=1>&gt;100M</td><td rowspan=1 colspan=1>Neural Cache Model (Large)(Grave et al., 2016)</td></tr><tr><td rowspan=1 colspan=1>Char PTB (bpc)</td><td rowspan=1 colspan=1>1.35</td><td rowspan=1 colspan=1>3M</td><td rowspan=1 colspan=1>1.22</td><td rowspan=1 colspan=1>14M</td><td rowspan=1 colspan=1>2-LayerNorm HyperLSTM(Ha et al., 2017)</td></tr><tr><td rowspan=1 colspan=1>Char text8 (bpc)</td><td rowspan=1 colspan=1>1.45</td><td rowspan=1 colspan=1>4.6M</td><td rowspan=1 colspan=1>1.29</td><td rowspan=1 colspan=1>&gt;12M</td><td rowspan=1 colspan=1>HM-LSTM (Chung et al., 2016)</td></tr></table>
+
+![](images/5e61b9b3ee15df3f95a75c0ba19ebb15a423e433d11b444e7c29d44ad108a8d4.jpg)  
+Figure 8: Controlled experiments that examine different components of the TCN model
+
+In this section we briefly study, via controlled experiments, the effect of filter size and residual block on the TCN’s ability to model different sequential tasks. Figure 8 shows the results of this ablative analysis. We kept the model size and the depth of the networks exactly the same within each experiment so that dilation factor is controlled. We conducted the experiment on three very different tasks: the copy memory task, permuted MNIST (P-MNIST), as well as word-level PTB language modeling.
+
+Through these experiments, we empirically confirm that both filter sizes and residuals play important roles in TCN’s capability of modeling potentially long dependencies. In both the copy memory and the permuted MNIST task, we observed faster convergence and better result for larger filter sizes (e.g. in the copy memory task, a filter size $k \geq 3$ led to only suboptimal convergence). In word-level PTB, we find a filter size of $k = 3$ works best. This is not a complete surprise, since a size- $k$ filter on the inputs is analogous to a $k$ -gram model in language modeling.
+
+Results of control experiments on the residual function are shown in Figure 8d, 8e and 8f. In all three scenarios, we observe that the residual stabilizes the training by bringing a faster convergence as well as better final results, compared to TCN with the same model size but no residual block.
+
+# D EXPERIMENTS: GATING MECHANISM ON TCN
+
+One component that has shown to be effective in adapting a TCN to language modeling is the gating mechanism within the residual block, which was used in works such as Dauphin et al. (2017). In this section, we empirically evaluate the effects of adding gated units to TCN.
+
+We replace the ReLU within the TCN residual block with a gating mechanism, represented by an elementwise product between two convolutional layers, with one of them also passing through a sigmoid function $\sigma ( x ) ^ { 5 }$ . Prior works such as Dauphin et al. (2017) has used similar gating to control the path through which information flows in the network, and achieved great performance on language modeling tasks.
+
+Table 5: TCN with Gating Mechanism within Residual Block.   
+
+<table><tr><td rowspan=1 colspan=3>RELU TCN VS.GATED TCN RESULTS</td></tr><tr><td rowspan=1 colspan=1>Task</td><td rowspan=1 colspan=1>TCN</td><td rowspan=1 colspan=1>TCN + Gating</td></tr><tr><td rowspan=1 colspan=1>Seq. MNIST (acc.)</td><td rowspan=1 colspan=1>99.0</td><td rowspan=1 colspan=1>99.0</td></tr><tr><td rowspan=1 colspan=1>P-MNIST (acc.)</td><td rowspan=1 colspan=1>97.2</td><td rowspan=1 colspan=1>96.9</td></tr><tr><td rowspan=1 colspan=1>Adding Prob. 600 (loss)</td><td rowspan=1 colspan=1>5.8e-5</td><td rowspan=1 colspan=1>5.6e-5</td></tr><tr><td rowspan=1 colspan=1>CopyMemory 11000 (loss)</td><td rowspan=1 colspan=1>3.5e-5</td><td rowspan=1 colspan=1>0.00508</td></tr><tr><td rowspan=1 colspan=1>JSB Chorales (loss)</td><td rowspan=1 colspan=1>8.10</td><td rowspan=1 colspan=1>8.13</td></tr><tr><td rowspan=1 colspan=1>Nottingham (loss)</td><td rowspan=1 colspan=1>3.07</td><td rowspan=1 colspan=1>3.12</td></tr><tr><td rowspan=1 colspan=1>Word PTB (ppl)</td><td rowspan=1 colspan=1>90.17</td><td rowspan=1 colspan=1>88.91</td></tr><tr><td rowspan=1 colspan=1>Char PTB (bpc)</td><td rowspan=1 colspan=1>1.35</td><td rowspan=1 colspan=1>1.343</td></tr><tr><td rowspan=1 colspan=1>Char text8 (bpc)</td><td rowspan=1 colspan=1>1.45</td><td rowspan=1 colspan=1>1.48</td></tr></table>
+
+Through these comparisons, we notice that gating components do further improve the TCN results on certain language modeling datasets like PTB, which agrees with prior works. However, we do not observe such benefits to exist in general on sequence prediction tasks, such as on polyphonic music datasets, and those simpler benchmark tasks requiring more long-term memories. For example, on the copy memory task with $T = 1 0 0 0$ , we find that gating mechanism deteriorates the convergence of TCN to a suboptimal result that is only slightly better than random guess.

parse/train/rk8wKk-R-/rk8wKk-R-_content_list.json

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+    {
+        "type": "text",
+        "text": "CONVOLUTIONAL SEQUENCE MODELING REVISITED ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            148,
+            398,
+            176
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            212,
+            544,
+            228
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "This paper revisits the problem of sequence modeling using convolutional architectures. Although both convolutional and recurrent architectures have a long history in sequence prediction, the current “default” mindset in much of the deep learning community is that generic sequence modeling is best handled using recurrent networks. The goal of this paper is to question this assumption. Specifically, we consider a simple generic temporal convolution network (TCN), which adopts features from modern ConvNet architectures such as a dilations and residual connections. We show that on a variety of sequence modeling tasks, including many frequently used as benchmarks for evaluating recurrent networks, the TCN outperforms baseline RNN methods (LSTMs, GRUs, and vanilla RNNs) and sometimes even highly specialized approaches. We further show that the potential “infinite memory” advantage that RNNs have over TCNs is largely absent in practice: TCNs indeed exhibit longer effective history sizes than their recurrent counterparts. As a whole, we argue that it may be time to (re)consider ConvNets as the default “go to” architecture for sequence modeling. ",
+        "bbox": [
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+            247,
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+            455
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            492,
+            334,
+            508
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Since the re-emergence of neural networks to the forefront of machine learning, two types of network architectures have played a pivotal role: the convolutional network, often used for vision and higher-dimensional input data; and the recurrent network, typically used for modeling sequential data. These two types of architectures have become so ingrained in modern deep learning that they can be viewed as constituting the “pillars” of deep learning approaches. This paper looks at the problem of sequence modeling, predicting how a sequence will evolve over time. This is a key problem in domains spanning audio, language modeling, music processing, time series forecasting, and many others. Although exceptions certainly exist in some domains, the current “default” thinking in the deep learning community is that these sequential tasks are best handled by some type of recurrent network. Our aim is to revisit this default thinking, and specifically ask whether modern convolutional architectures are in fact just as powerful for sequence modeling. ",
+        "bbox": [
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+            529,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Before making the main claims of our paper, some history of convolutional and recurrent models for sequence modeling is useful. In the early history of neural networks, convolutional models were specifically proposed as a means of handling sequence data, the idea being that one could slide a 1-D convolutional filter over the data (and stack such layers together) to predict future elements of a sequence from past ones (Hinton, 1989; LeCun et al., 1995). Thus, the idea of using convolutional models for sequence modeling goes back to the beginning of convolutional architectures themselves. However, these models were subsequently largely abandoned for many sequence modeling tasks in favor of recurrent networks (Elman, 1990). The reasoning for this appears straightforward: while convolutional architectures have a limited ability to look back in time (i.e., their receptive field is limited by the size and layers of the filters), recurrent networks have no such limitation. Because recurrent networks propagate forward a hidden state, they are theoretically capable of infinite memory, the ability to make predictions based upon data that occurred arbitrarily long ago in the sequence. This possibility seems to be realized even moreso for the now-standard architectures of Long ShortTerm Memory networks (LSTMs) (Hochreiter & Schmidhuber, 1997), or recent incarnations such as the Gated Recurrent Unit (GRU) (Cho et al., 2014); these architectures aim to avoid the “vanishing gradient” challenge of traditional RNNs and appear to provide a means to actually realize this infinite memory. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Given the substantial limitations of convolutional architectures at the time that RNNs/LSTMs were initially proposed (when deep convolutional architectures were difficult to train, and strategies such as dilated convolutions had not reached widespread use), it is no surprise that CNNs fell out of favor to RNNs. While there have been a few notable examples in recent years of CNNs applied to sequence modeling (e.g., the WaveNet (Oord et al., 2016a) and PixelCNN (Oord et al., 2016b) architectures), the general “folk wisdom” of sequence modeling prevails, that the first avenue of attack for these problems should be some form of recurrent network. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The fundamental aim of this paper is to revisit this folk wisdom, and thereby make a counterclaim. We argue that with the tools of modern convolutional architectures at our disposal (namely the ability to train very deep networks via residual connections and other similar mechanisms, plus the ability to increase receptive field size via dilations), in fact convolutional architectures typically outperform recurrent architectures on sequence modeling tasks, especially (and perhaps somewhat surprisingly) on domains where a long effective history length is needed to make proper predictions. ",
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+    {
+        "type": "text",
+        "text": "This paper consists of two main contributions. First, we describe a generic, baseline temporal convolutional network (TCN) architecture, combining best practices in the design of modern convolutional architectures, including residual layers and dilation. We emphasize that we are not claiming to invent the practice of applying convolutional architectures to sequence prediction, and indeed the TCN architecture here mirrors closely architectures such as WaveNet (in fact TCN is notably simpler in some respects). We do, however, want to propose a generic modern form of convolutional sequence prediction for subsequent experimentation. Second, and more importantly, we extensively evaluate the TCN model versus alternative approaches on a wide variety of sequence modeling tasks, spanning many domains and datasets that have typically been the purview of recurrent models, including word- and character-level language modeling, polyphonic music prediction, and other baseline tasks commonly used to evaluate recurrent architectures. Although our baseline TCN can be outperformed by specialized (and typically highly-tuned) RNNs in some cases, for the majority of problems the TCN performs best, with minimal tuning on the architecture or the optimization. This paper also analyzes empirically the myth of “infinite memory” in RNNs, and shows that in practice, TCNs of similar size and complexity may actually demonstrate longer effective history sizes. Our chief claim in this paper is thus an empirical one: rather than presuming that RNNs will be the default best method for sequence modeling tasks, it may be time to (re)consider ConvNets as the “go-to” approach when facing a new dataset or task in sequence modeling. ",
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+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+        "text": "In this section we highlight some of the key innovations in the history of recurrent and convolutional architectures for sequence prediction. ",
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+        "type": "text",
+        "text": "Recurrent networks broadly refer to networks that maintain a vector of hidden activations, which are kept over time by propagating them through the network. The intuitive appeal of this approach is that the hidden state can act as a sort of “memory” of everything that has been seen so far in a sequence, without the need for keeping an explicit history. Unfortunately, such memory comes at a cost, and it is well-known that the na¨ıve RNN architecture is difficult to train due to the exploding/vanishing gradient problem (Bengio et al., 1994). ",
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+        "type": "text",
+        "text": "A number of solutions have been proposed to address this issue. More than twenty years ago, Hochreiter & Schmidhuber (1997) introduced the now-ubiquitous Long Short-Term Memory (LSTM) which uses a set of gates to explicitly maintain memory cells that are propagated forward in time. Other solutions or refinements include a simplified variant of LSTM, the Gated Recurrent Unit (GRU) (Cho et al., 2014), peephole connections (Gers et al., 2002), Clockwork RNN (Koutnik et al., 2014) and recent works such as MI-RNN (Wu et al., 2016) and the Dilated RNN (Chang et al., 2017). Alternatively, several regularization techniques have been proposed to better train LSTMs, such as those based upon the properties of the RNN dynamical system (Pascanu et al., 2013); more recently, strategies such as Zoneout (Krueger et al., 2017) and AWD-LSTM (Merity et al., 2017) were also introduced to regularize LSTM in various ways, and have achieved exceptional results in the field of language modeling. ",
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+    {
+        "type": "text",
+        "text": "While it is frequently criticized as a seemingly “ad-hoc” architecture, LSTMs have proven to be extremely robust and is very hard to improve upon by other recurrent architectures, at least for general problems. Jozefowicz et al. (2015) concluded that if there were “architectures much better than the LSTM”, then they were “not trivial to find”. However, while they evaluated a variety of recurrent architectures with different combinations of components via an evolutionary search, they did not consider architectures that were fundamentally different from the recurrent ones. ",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "The history of convolutional architectures for time series is comparatively shorter, as they soon fell out of favor compared to recurrent architectures for these tasks, though are also seeing a resurgence in recent years. Waibel et al. (1989) and Bottou et al. (1990) studied the usage of time-delay networks (TDNNs) for sequences, one of the earliest local-connection-based networks in this domain. LeCun et al. (1995) then proposed and examined the usage of CNNs on time-series data, pointing out that the same kind of feature extraction used in images could work well on sequence modeling with convolutional filters. Recent years have seen a re-emergence of convolutional models for sequence data. Perhaps most notably, the WaveNet (Oord et al., 2016a) applied a stacked convolutional architecture to model audio signals, using a combination of dilations (Yu & Koltun, 2015), skip connections, gating, and conditioning on context stacks; the WaveNet mode was additionally applied to a few other contexts, such as financial applications (Borovykh et al., 2017). Non-dilated gated convolutions have also been applied in the context of language modeling (Dauphin et al., 2017). And finally, convolutional models have seen a recent adoption in sequence to sequence modeling and machine translations applications, such as the ByteNet (Kalchbrenner et al., 2016) and ConvS2S architectures (Gehring et al., 2017). ",
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+        "type": "text",
+        "text": "Despite these successes, the general consensus of the deep learning community seems to be that RNNs (here meaning all RNNs including LSTM and its variants) are better suited to sequence modeling for two apparent reasons: 1) as discussed before, RNNs are theoretically capable of infinite memory; and 2) RNN models are inherently suitable for sequential inputs of varying length, whereas CNNs seem to be more appropriate in domains with fixed-size inputs (e.g., vision). ",
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+    {
+        "type": "text",
+        "text": "With this as the context, this paper reconsiders convolutional sequence modeling in general, first introducing a simple general-purpose convolutional sequence modeling architecture that can be applied in all the same scenarios as an RNN (the architecture acts as a “drop-in” replacement for RNNs of any kind). We then extensively evaluate the performance of the architecture on tasks from different domains, focusing on domains and settings that have been used explicitly as applications and benchmarks for RNNs in the recent past. With regard to the specific architectures mentioned above (e.g. WaveNet, ByteNet, gated convolutional language models), the primary goal here is to describe a simple, application-independent architecture that avoids much of the extra specialized components of these architectures (gating, complex residuals, context stacks, or the encoder-decoder architectures of seq2seq models), and keeps only the “standard” convolutional components from most image architectures, with the restriction that the convolutions be causal. In several cases we specifically compare the architecture with and without additional components (e.g., gating elements), and highlight that it does not seem to substantially improve performance of the architecture across domains. Thus, the primary goal of this paper is to provide a baseline architecture for convolutional sequence prediction tasks, and to evaluate the performance of this model across multiple domains. ",
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+    {
+        "type": "text",
+        "text": "3 CONVOLUTIONAL SEQUENCE MODELING ",
+        "text_level": 1,
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+        "text": "In this section, we propose a generic architecture for convolutional sequence prediction, and generally refer to it as Temporal Convolution Networks (TCNs). We emphasize that we adopt this term not as a label for a truly new architecture, but as a simple descriptive term for this and similar architectures. The distinguishing characteristics of the TCN are that: 1) the convolutions in the architecture are causal, meaning that there is no information “leakage” between future and past; 2) the architecture can take a sequence of any length and map it to an output sequence of the same length, just as with an RNN. Beyond this, we emphasize how to build very long effective history sizes (i.e., the ability for the networks to look very far into the past to make a prediction) using a combination of very deep networks (augmented with residual layers) and dilated convolutions. ",
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+        "type": "text",
+        "text": "3.1 THE SEQUENCE MODELING TASK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Before defining the network structure, we highlight the nature of the sequence modeling task. We suppose that we are given a sequence of inputs $x _ { 0 } , \\ldots , x _ { T }$ , and we wish to predict some correspond",
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+        "img_path": "images/e4db16452e482b763cd61552fcde4b5c943fbde3fd808c44865ddb9b473e207d.jpg",
+        "image_caption": [
+            "Figure 1: A simple causal convolution with filter size 3. "
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+        "type": "text",
+        "text": "ing outputs $y _ { 0 } , \\ldots , y _ { T }$ at each time. The key constraint is that to predict the output $y _ { t }$ for some time $t$ , we are constrained to only use those inputs that have been previously observed: $x _ { 0 } , \\ldots , x _ { t }$ . Formally, a sequence modeling network is any function $f : \\mathcal { X } ^ { T + \\bar { 1 } }  \\mathcal { Y } ^ { T + \\bar { 1 } }$ that produces this mapping ",
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+        "type": "equation",
+        "img_path": "images/3e15ca49e4386d69f8cfa21ce83287ba4a20ad4563252e97e510139c1ffba7ab.jpg",
+        "text": "$$\n\\hat { y } _ { 0 } , \\dots , \\hat { y } _ { T } = f ( x _ { 0 } , \\dots , x _ { T } )\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "if it satisfies the causal constraint that $y _ { t }$ depends only on $x _ { 0 } , \\ldots , x _ { t }$ , and not on any “future” inputs $x _ { t + 1 } , \\dots , x _ { T }$ . The goal of learning in the sequence modeling setting is to find the network $f$ minimizing some expected loss between the actual outputs and predictions $L ( y _ { 0 } , \\dots , y _ { T } , f ( x _ { 0 } , \\dots , x _ { T } ) )$ where the sequences and outputs are drawn according to some distribution. ",
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+        "type": "text",
+        "text": "This formalism encompasses many settings such as auto-regressive prediction (where we try to predict some signal given its past) by setting the target output to be simply the input shifted by one time step. It does not, however, directly capture domains such as machine translation, or sequenceto-sequence prediction in general, since in these cases the entire input sequence (including “future” states) can be used to predict each output (though the techniques can naturally be extended to work in such settings). ",
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+    {
+        "type": "text",
+        "text": "3.2 CAUSAL CONVOLUTIONS AND THE TCN ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "As mentioned above, the TCN is based upon two principles: the fact that the network produces an output of the same length as the input, and the fact that there can be no leakage from the future into the past. To accomplish the first point, the TCN uses a 1D fully-convolutional network (FCN) architecture (Long et al., 2015), where each hidden layer is the same length as the input layer, and zero padding of length (kernel size − 1) is added to keep subsequent layers the same length as previous ones. To achieve the second point, the TCN uses causal convolutions, convolutions where a subsequent output at time $t$ is convolved only with elements from time $t$ and before in the previous layer.1 Graphically, the network is shown in Figure 1. Put in a simple manner: ",
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+    {
+        "type": "equation",
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+        "text": "$$\n\\mathrm { T C N } = \\mathrm { 1 D F C N } + \\mathrm { c a u s a l ~ c o n v o l u t i o n s }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "It is worth emphasizing that this is essentially the same architecture as the time delay neural network proposed nearly 30 years ago by Waibel et al. (1989), with the sole tweak of zero padding to ensure equal sizes of all layers. ",
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+        "type": "text",
+        "text": "However, a major disadvantage of this “na¨ıve” approach is that in order to achieve a long effective history size, we need an extremely deep network or very large filters, neither of which were particularly feasible when the methods were first introduced. Thus, in the following sections, we describe how techniques from modern convolutional architectures can be integrated into the TCN to allow for both very deep networks and very long effective history. ",
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+    {
+        "type": "image",
+        "img_path": "images/3d3384271f9fbd194ae3ecf347bb7e1e6b27e790b9063a1b747424be7bcaf5f9.jpg",
+        "image_caption": [
+            "Figure 2: A dilated causal convolution with dilation factors $d = 1 , 2 , 4$ and filter size $k = 3$ . The receptive field is able to cover all values from the input sequence. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "3.3 DILATED CONVOLUTIONS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Through convolutional filters, as previously addressed, a simple causal convolution is only able to look back at a history with size linear in the depth of the network. This makes it challenging to apply the aforementioned causal convolution on sequence tasks, especially those requiring longer history. Our solution here, used previously for example in audio synthesis by Oord et al. (2016a), is to employ dilated convolutions (Yu & Koltun, 2015) that enable an exponentially large receptive field. More formally, for a 1-D sequence input $\\mathbf { x } \\in \\mathbb { R } ^ { n }$ and a filter $f : \\bar { \\{ 0 , \\dots , k - 1 \\bar { \\} } }  \\mathbb { R }$ , the dilated convolution operation $F$ on element $s$ of the sequence is defined as ",
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+        "img_path": "images/c2f18d1bcf810be91cf4f6415c0e25ca6867b5ddba3cb28385e6dd7d5bc0ee2e.jpg",
+        "text": "$$\nF ( s ) = ( \\mathbf { x } * _ { d } f ) ( s ) = \\sum _ { i = 0 } ^ { k - 1 } f ( i ) \\cdot \\mathbf { x } _ { s + d \\cdot i }\n$$",
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+    {
+        "type": "text",
+        "text": "where $d$ is the dilation factor and $k$ is the filter size. Dilation is thus equivalent to introducing a fixed step between every two adjacent filter taps. When taking $d = 1$ , for example, a dilated convolution is trivially a normal convolution operation. Using larger dilations enables an output at the top level to represent a wider range of inputs, thus effectively expanding the receptive field of a ConvNet. ",
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+        "type": "text",
+        "text": "This gives us two ways to increase the receptive field of the TCN: by choosing larger filter sizes $k$ , and by increasing the dilation factor $d$ , where the effective history of one such layer is $( k - 1 ) d$ . As is common when using dilated convolutions, we increase $d$ exponentially with the depth of the network (i.e., $d = O ( 2 ^ { i } )$ at level $i$ of the network). This ensures that there is some filter that hits each input within the effective history, while also allowing for an extremely large effective history using deep networks. We provide an illustration in Figure 2. Using filter size $k = 3$ and dilation factor $d = 1 , 2 , 4$ , the receptive field is able to cover all values from the input sequence. ",
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+        "type": "text",
+        "text": "3.4 RESIDUAL CONNECTIONS ",
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+        "type": "text",
+        "text": "Proposed by He et al. (2016), residual functions have proven to be especially useful in effectively training deep networks. In a residual network, each residual block contains a branch leading out to a series of transformations $\\mathcal { F }$ , whose outputs are added to the input $\\mathbf { x }$ of the block: ",
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+        "img_path": "images/a14b11eeb99c1c7768b94eec8e763486c2f41521fee1459687c6c0433c4fd694.jpg",
+        "text": "$$\no = \\mathrm { A c t i v a t i o n } ( \\mathbf { x } + \\mathcal { F } ( \\mathbf { x } ) )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "This effectively allows for the layers to learn modifications to the identity mapping rather than the entire transformation, which has been repeatedly shown to benefit very deep networks. ",
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+    {
+        "type": "text",
+        "text": "As the TCN’s receptive field depends on the network depth $n$ as well as filter size $k$ and dilation factor $d$ , stabilization of deeper and larger TCNs becomes important. For example, in a case where the prediction could depend on a history of size $2 ^ { 1 2 }$ and a high-dimensional input sequence, a network of up to 12 layers could be needed. Each layer, more specifically, consists of multiple filters for feature extraction. In our design of the generic TCN model, we therefore employed a generic residual module in place of a convolutional layer. ",
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+    {
+        "type": "text",
+        "text": "The residual block for our baseline TCN is shown in figure 3a. Within a residual block, the TCN has 2 layers of dilated causal convolution and non-linearity, for which we used the rectified linear unit ",
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+        "type": "text",
+        "text": "(a) TCN residual block. An 1x1 convolution is added when residual input and output have different dimensions. ",
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+    {
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+        "img_path": "images/1813e828cfa0df05d2cac09fee75934b751463303c74b3ec805bc1f1e7e53827.jpg",
+        "image_caption": [
+            "Figure 3: A visualization of the TCN residual block "
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+        "type": "text",
+        "text": "(b) An example of residual connection in TCN. The blue lines are filters in the residual function, and the green lines are identity mappings. ",
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+    {
+        "type": "text",
+        "text": "(ReLU) (Nair & Hinton, 2010). For normalization, we applied Weight Normalization (Salimans & Kingma, 2016) to the filters in the dilated convolution (where we note that the filters are essentially vectors of size $k \\times 1 \\AA$ ). In addition, a 2-D dropout (Srivastava et al., 2014) layer was added after each dilated convolution for regularization: at each training step, a whole channel (in the width dimension) is zeroed out. ",
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+        "type": "text",
+        "text": "However, whereas in standard ResNet the input is passed in and added directly to the output of the residual function, in TCN (and ConvNet in general) the input and output could have different widths. Therefore in our TCN, when the input-output widths disagree, we use an additional 1x1 convolution to ensure that element-wise addition $\\oplus$ receives tensors of the same shape (see Figure 3a, 3b). ",
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+    {
+        "type": "text",
+        "text": "Note that many further optimizations (e.g., gating, skip connections, context stacking as in audio generation using WaveNet) are possible in a TCN than what we described here. However, in this paper, we aim to present a generic, general-purpose TCN, to which additional twists can be added as needed. As we are going to show in Section 4, this general-purpose architecture is already able to outperform recurrent units like LSTM on a number of tasks by a good margin. ",
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+        "type": "text",
+        "text": "3.5 ADVANTAGES OF TCN SEQUENCE MODELING ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "There are several key advantages to a TCN model with the ingredients that we described above. ",
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+        "type": "text",
+        "text": "• Parallelism. Unlike in RNNs where the predictions for later timesteps must wait for their predecessors to complete, in a convolutional architecture these computations can be done in parallel since the same filter is used in each layer. Therefore, in training and evaluation, a (possibly long) input sequence can be processed as a whole in TCN, instead of serially as in RNN, which depends on the length of the sequence and could be less efficient. ",
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+        "type": "text",
+        "text": "• Flexible receptive field size. With a TCN, we can change its receptive field size in multiple ways. For instance, stacking more dilated (causal) convolutional layers, using larger dilation factors, or increasing the filter size are all viable options (with possibly different interpretations). TCN is thus easy to tune and adapt to different domains, since we now can directly control the size of the model’s memory. ",
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+        "type": "text",
+        "text": "• Stable gradients. Unlike recurrent architectures, TCN has a backpropagation path that is different from the temporal direction of the sequence. This enables it to avoid the problem of exploding/vanishing gradients, which is a major issue for RNNs (and which led to the development of LSTM, GRU, HF-RNN, etc.). ",
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+        "type": "text",
+        "text": "• Low memory requirement for training. In a task where the input sequence is long, a structure such as LSTM can easily use up a lot of memory to store the partial results for backpropagation (e.g., the results for each gate of the cell). However, in TCN, the backpropagation path only depends on the network depth and the filters are shared in each layer, which means that in practice, as model size or sequence length gets large, TCN is likely to use less memory than RNNs. ",
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+    {
+        "type": "text",
+        "text": "3.6 DISADVANTAGES OF TCN SEQUENCE MODELING ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We also summarize two disadvantages of using TCN instead of RNNs. ",
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+        "type": "text",
+        "text": "• Data storage in evaluation. In evaluation/testing, RNNs only need to maintain a hidden state and take in a current input $x _ { t }$ in order to generate a prediction. In other words, a “summary” of the entire history is provided by the fixed-length set of vectors $h _ { t }$ , which means that the actual observed sequence can be discarded (and indeed, the hidden state can be used as a kind of encoder for all the observed history). In contrast, the TCN still needs to take in a sequence with non-trivial length (precisely the effective history length) in order to predict, thus possibly requiring more memory during evaluation. • Potential parameter change for a transfer of domain. Different domains can have different requirements on the amount of history the model needs to memorize. Therefore, when transferring a model from a domain where only little memory is needed (i.e., small $k$ and $d$ ) to a domain where much larger memory is required (i.e., much larger $k$ and $d$ ), TCN may perform poorly for not having a sufficiently large receptive field. ",
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+        "text": "We want to emphasize, though, that we believe the notable lack of “infinite memory” for a TCN is decidedly not a practical disadvantage, since, as we show in Section 4, the TCN method actually outperforms RNNs in terms of the ability to deal with long temporal dependencies. ",
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+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we conduct a series of experiments using the baseline TCN (described in section 3) and generic RNNs (namely LSTMs, GRUs, and vanilla RNNs). These experiments cover tasks and datasets from various domains, aiming to test different aspects of a model’s ability to learn sequence modeling. In several cases, specialized RNN models, or methods with particular forms of regularization can indeed vastly outperform both generic RNNs and the TCN on particular problems, which we highlight when applicable. But as a general-purpose architecture, we believe the experiments make a compelling case for the TCN as the “first attempt” approach for many sequential problems. ",
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+        "type": "text",
+        "text": "All experiments reported in this section used the same TCN architecture, just varying the depth of the network and occasionally the kernel size. We use an exponential dilation $d = 2 ^ { n }$ for layer $n$ in the network, and the Adam optimizer (Kingma & Ba, 2015) with learning rate 0.002 for TCN (unless otherwise noted). We also empirically find that gradient clipping helped training convergence of TCN, and we pick the maximum norm to clip from [0.3, 1]. When training recurrent models, we use a simple grid search to find a good set of hyperparameters (in particular, optimizer, recurrent drop $p \\in [ 0 . 0 5 , 0 . 5 ]$ , the learning rate, gradient clipping, and initial forget-gate bias), while keeping the network around the same size as TCN. No other optimizations, such as gating mechanism (see Appendix D), or highway network, were added to TCN or the RNNs. The hyperparameters we use for TCN on different tasks are reported in Table 2 in Appendix B. In addition, we conduct a series controlled experiments to investigate the effects of filter size and residual function on the TCN’s performance. These results can be found in Appendix C. ",
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+    {
+        "type": "text",
+        "text": "4.1 TASKS AND RESULTS SUMMARY ",
+        "text_level": 1,
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+        "text": "In this section we highlight the general performance of generic TCNs vs generic LSTMs for a variety of domains from the sequential modeling literature. A complete description of each task, as well as references to some prior works that evaluated them, is given in Appendix A. In brief, the tasks we consider are: the adding problem, sequential MNIST, permuted MNIST (P-MNIST), the copy memory task, the Nottingham and JSB Chorales polyphonic music tasks, Penn Treebank (PTB), Wikitext-103 and LAMBADA word-level language modeling, as well as PTB and text8 characterlevel language modeling. ",
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+        "table_caption": [
+            "Table 1: Complete comparison of the TCN to regularized recurrent architectures in various tasks. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Sequential Tasks</td><td rowspan=\"2\">Model Size (≈)</td><td colspan=\"4\">Models</td></tr><tr><td>LSTM</td><td>GRU</td><td>RNN</td><td>TCN (ours)</td></tr><tr><td>Seq. MNIST (accuracy)</td><td>70K</td><td>87.2</td><td>96.2</td><td>21.5</td><td>99.0</td></tr><tr><td>P-Seq. MNIST (accuracy)</td><td>70K</td><td>85.7</td><td>87.3</td><td>25.3</td><td>97.2</td></tr><tr><td>The Adding Problem T=600 (loss)</td><td>70K</td><td>0.164</td><td>5.3e-5</td><td>0.177</td><td>5.8e-5</td></tr><tr><td>Copy Memory T=1000 (loss)</td><td>16K</td><td>0.0204</td><td>0.0197</td><td>1</td><td>3.5e-5</td></tr><tr><td>Music JSB Chorales (loss)</td><td>300K</td><td>8.45</td><td>8.43</td><td>8.91</td><td>8.10</td></tr><tr><td>Music Nottingham (loss)</td><td>1M</td><td>3.29</td><td>3.46</td><td>-</td><td>3.07</td></tr><tr><td>Word-level PTB (ppl)</td><td>13M</td><td>84.77</td><td>92.48</td><td>114.50</td><td>90.17</td></tr><tr><td>Word-level Wiki-103 (ppl)</td><td>-</td><td>48.4 (large)</td><td>-</td><td>1</td><td>45.19</td></tr><tr><td>Word-level LAMBADA (ppl)</td><td>-</td><td>4186</td><td>-</td><td>14725</td><td>1279</td></tr><tr><td>Char-level PTB (bpc)</td><td>3M</td><td>1.41</td><td>1.42</td><td>1.52</td><td>1.35</td></tr><tr><td>Char-level text8 (bpc)</td><td>5M</td><td>1.52</td><td>1.56</td><td>1.69</td><td>1.45</td></tr></table>",
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+        "image_caption": [
+            "Figure 4: Results of TCN vs. recurrent architectures on the adding problem "
+        ],
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+    {
+        "type": "text",
+        "text": "A summary comparison of TCNs to the standard RNN architectures (LSTM, GRU, and vanilla RNN) is shown in Table 1. We will highlight many of these results below, and want to emphasize that for several tasks the baseline RNN architectures are still far from the state of the art (see Table 4), but in total the results make a strong case that the TCN architecture, as a generic sequence modeling framework, is often superior to generic RNN approaches. We now consider several of these experiments in detail, generally distinguishing between the “recurrent benchmark” tasks designed to show the limitations of networks for sequence modeling (adding problem, sequential & permuted MNIST, copy memory), and the “applied” tasks (polyphonic music and language modeling). ",
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+    {
+        "type": "text",
+        "text": "4.2 BASELINE RECURRENT TASKS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We first compare the results of the TCN architecture to those of RNNs on the toy baseline tasks that have been frequently used to evaluate sequential modeling (Hochreiter & Schmidhuber, 1997; Martens & Sutskever, 2011; Pascanu et al., 2013; Le et al., 2015; Cooijmans et al., 2016; Zhang et al., 2016; Krueger et al., 2017; Wisdom et al., 2016; Arjovsky et al., 2016). ",
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+    {
+        "type": "text",
+        "text": "The Adding Problem. Convergence results for the adding problem, for problem sizes $T =$ 200, 400, 600, are shown in Figure 4; all models were chosen to have roughly 70K parameters. In all three cases, TCNs quickly converged to a virtually perfect solution (i.e., an MSE loss very close to 0). LSTMs and vanilla RNNs performed significantly worse, while on this task GRUs also performed quite well, even though their convergence was slightly slower than TCNs. ",
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+    {
+        "type": "text",
+        "text": "Sequential MNIST and P-MNIST. Results on sequential and permuted MNIST, run over 10 epochs, are shown in Figures 5a and 5b; all models were picked to have roughly 70K parameters. For both problems, TCNs substantially outperform the alternative architectures, both in terms of convergence time and final performance level on the task. For the permuted sequential MNIST, TCNs outperform state of the art results using recurrent nets $( 9 5 . 9 \\% )$ with Zoneout+Recurrent BatchNorm (Cooijmans et al., 2016; Krueger et al., 2017), a highly optimized method for regularizing RNNs. ",
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+        "img_path": "images/404df7deb85ffa4ae5c1945673693b1c2d442023841918eb5df6debfac1e9251.jpg",
+        "image_caption": [
+            "Figure 5: Results of TCN vs. recurrent architectures on the Sequential MNIST and P-MNIST "
+        ],
+        "image_footnote": [],
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+    {
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+        "image_caption": [
+            "Figure 6: Result of TCN vs. recurrent architectures on the Copy Memory Task, for different $T$ "
+        ],
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+    {
+        "type": "text",
+        "text": "Copy Memory Task. Finally, Figure 6 shows the results of the different methods (with roughly the same size) on the copy memory task. Again, the TCNs quickly converge to correct answers, while the LSTM and GRU simply converge to the same loss as predicting all zeros. In this case we also compare to the recently-proposed EURNN (Jing et al., 2017), which was highlighted to perform well on this task. While both perform well for sequence length $T = 5 0 0$ , the TCN again has a clear advantage for $T = 1 0 0 0$ and $T = 2 0 0 0$ (in terms of both loss and convergence). ",
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+    {
+        "type": "text",
+        "text": "4.3 RESULTS ON POLYPHONIC MUSIC AND LANGUAGE MODELING ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Next, we compare the results of the TCN architecture to recurrent architectures on 6 different real datasets in polyphonic music as well as word- and character-level language modeling. These are areas where sequence modeling has been used most frequently. As domains where there is considerable practical interests, there have also been many specialized RNNs developed for these tasks (e.g., Zhang et al. (2016); Ha et al. (2017); Krueger et al. (2017); Grave et al. (2016); Greff et al. (2017); Merity et al. (2017)). We mention some of these comparisons when useful, but the primary goal here is to compare the generic TCN model to other generic RNN architectures, so we focus mainly on these comparisons. ",
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+    {
+        "type": "text",
+        "text": "Polyphonic Music. On the Nottingham and JSB Chorales datasets, the TCN with virtually no tuning is again able to beat the other models by a considerable margin (see Table 1), and even outperforms some improved recurrent models for this task such as HF-RNN (Boulanger-Lewandowski et al., 2012) and Diagonal RNN (Subakan & Smaragdis, 2017). Note however that other models such as the Deep Belief Net LSTM (Vohra et al., 2015) perform substantially better on this task; we believe this is likely due to the fact that the datasets involved in polyphonic music are relatively small, and thus the right regularization method or generative modeling procedure can improve performance significantly. This is largely orthogonal to the RNN/TCN distinction, as a similar variant of TCN may well be possible. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Word-level Language Modeling. Language modeling remains one of the primary applications of recurrent networks in general, where many recent works have been focusing on optimizing the usage of LSTMs (see Krueger et al. (2017); Merity et al. (2017)). In our implementation, we follow standard practices such as tying the weights of encoder and decoder layers for both TCN and RNNs (Press & Wolf, 2016), which significantly reduces the number of parameters in the model. When training the language modeling tasks, we use SGD optimizer with annealing learning rate (by a factor of 0.5) for TCN and RNNs. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Results on word-level language modeling are reported in Table 1. With a fine-tuned LSTM (i.e., with recurrent and embedding dropout, etc.), we find LSTM can outperform TCN in perplexity on the Penn TreeBank (PTB) dataset, where the TCN model still beats both GRU and vanilla RNN. On the much larger Wikitext-103 corpus, however, without performing much hyperparameter search (due to lengthy training process), we still observe that TCN outperforms the state of the art LSTM results (48.4 in perplexity) by Grave et al. (2016) (without continuous cache pointer; see Table 4). The same superiority is observed on the LAMBADA test (Paperno et al., 2016), where TCN achieves a much lower perplexity than its recurrent counterparts in predicting the last word based on a very long context (see Appendix A). We will further analyze this in section 4.4. ",
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+    {
+        "type": "text",
+        "text": "Character-level Language Modeling. The results of applying TCN and alternative models on PTB and text8 data for character-level language modeling are shown in Table 1, with performance measured in bits per character (bpc). While beaten by the state of the art (see Table 4), the generic TCN outperforms regularized LSTM and GRU as well as methods such as Norm-stabilized LSTM (Krueger & Memisevic, 2015). Moreover, we note that using a filter size of $k \\leq 4$ works better than larger filter sizes in character-level language modeling, which suggests that capturing short history is more important than longer dependencies in these tasks. ",
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+    {
+        "type": "text",
+        "text": "4.4 MEMORY SIZE OF TCN AND RNNS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Finally, one of the important reasons why RNNs have been preferred over CNNs for general sequence modeling is that theoretically, recurrent architectures are capable of an infinite memory. We therefore attempt to study here how much memory TCN and LSTM/GRU are able to actually “backtrack”, via the copy memory task and the LAMBADA language modeling task. ",
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+    {
+        "type": "text",
+        "text": "The copy memory task is a simple but perfect task to examine a model’s ability to pick up its memory from a (possibly) distant past (by varying the value of sequence length $T$ ). However, different from the setting in Section 4.2, in order to compare the results for different sequence lengths, here we only report the accuracy on the last $I O$ elements of the output sequence. We used a model size of 10K for both TCN and RNNs. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/294b6d1f2397e8ec063d0643ac527f33c030df7f7c912acd3fc3b3fa37f37eaa.jpg",
+        "image_caption": [
+            "Figure 7: Accuracy on the copy memory task for varying sequence length $T$ . "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "The results are shown in Figure 7. TCNs consistently converge to $100 \\%$ accuracy for all sequence lengths, whereas it is increasingly challenging for recurrent models to memorize as $T$ grows (with accuracy converging to that of a random guess). LSTM’s accuracy quickly falls below $20 \\%$ for $T \\geq 5 0$ , which suggests that instead of infinite memory, LSTMs are only good at recalling a short history instead. ",
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+    },
+    {
+        "type": "text",
+        "text": "This observation is also backed up by the experiments of TCN on the LAMBADA dataset, which is specifically designed to test a model’s textual understanding in a broader discourse. The objective of LAMBADA dataset is to predict the last word of the target sentence given a sufficiently long context (see Appendix A for more details). Most of the existing models fail to guess accurately on this task. As shown in Table 1, TCN outperforms LSTMs by a significant margin in perplexity on LAMBADA (with a smaller network and virtually no tuning). ",
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+        "type": "text",
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+        "text": "",
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "These results indicate that TCNs, despite their apparent finite history, in practice maintain a longer effective history than their recurrent counterparts. We would like to emphasize that this empirical observation does not contradict the good results that prior works have achieved using LSTM, such as in language modeling on PTB. In fact, the very success of $n$ -gram models (Brown et al., 1992) suggested that language modeling might not need a very long memory, a conclusion also reached by prior works such as Dauphin et al. (2017). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In this work, we revisited the topic of modeling sequence predictions using convolutional architectures. We introduced the key components of the TCN and analyzed some vital advantages and disadvantages of using TCN for sequence predictions instead of RNNs. Further, we compared our generic TCN model to the recurrent architectures on a set of experiments that span a wide range of domains and datasets. Through these experiments, we have shown that TCN with minimal tuning can outperform LSTM/GRU of the same model size (and with standard regularizations) in most of the tasks. Further experiments on the copy memory task and LAMBADA task revealed that TCNs actually has a better capability for long-term memory than the comparable recurrent architectures, which are commonly believed to have unlimited memory. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "It is still important to note that, however, we only presented a generic architecture here, with components all coming from standard modern convolutional networks (e.g., normalization, dropout, residual network). And indeed, on specific problems, the TCN model can still be beaten by some specialized RNNs that adopt carefully designed optimization strategies. Nevertheless, we believe the experiment results in Section 4 might be a good signal that instead of considering RNNs as the “default” methodology for sequence modeling, convolutional networks too, can be a promising and powerful toolkit in studying time-series data. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            565,
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+        ],
+        "page_idx": 10
+    },
+    {
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Computational linguistics, 19(2):313–330, 1993.   \nMartens, J. and Sutskever, I. Learning recurrent neural networks with hessian-free optimization. In International Conference on Machine Learning (ICML-11), pp. 1033–1040, 2011.   \nMerity, S., Xiong, C., Bradbury, J., and Socher, R. Pointer sentinel mixture models. arXiv preprint arXiv:1609.07843, 2016.   \nMerity, S., Keskar, N. S., and Socher, R. Regularizing and Optimizing LSTM Language Models. arXiv preprint arXiv:1708.02182, 2017.   \nMikolov, T., Sutskever, I., Deoras, A., Le, H.-S., Kombrink, S., and Cernocky, J. Subword language modeling with neural networks. preprint, 2012.   \nMiyamoto, Y. and Cho, K. Gated word-character recurrent language model. arXiv preprint arXiv:1606.01700, 2016.   \nNair, V. and Hinton, G. E. Rectified linear units improve restricted boltzmann machines. 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Using the output embedding to improve language models. arXiv preprint arXiv:1608.05859, 2016.   \nSalimans, T. and Kingma, D. P. Weight normalization: A simple reparameterization to accelerate training of deep neural networks. In Advances in Neural Information Processing Systems, pp. 901–909. 2016.   \nSrivastava, N., Hinton, G. E., Krizhevsky, A., Sutskever, I., and Salakhutdinov, R. Dropout: a simple way to prevent neural networks from overfitting. Journal of machine learning research, 15(1): 1929–1958, 2014.   \nSubakan, Y. C. and Smaragdis, P. Diagonal rnns in symbolic music modeling. arXiv preprint arXiv:1704.05420, 2017.   \nVohra, R., Goel, K., and Sahoo, J. Modeling temporal dependencies in data using a dbn-lstm. In Data Science and Advanced Analytics (DSAA), 2015. 36678 2015. IEEE International Conference on, pp. 1–4. IEEE, 2015.   \nWaibel, A., Hanazawa, T., Hinton, G., Shikano, K., and Lang, K. J. Phoneme recognition using time-delay neural networks. IEEE transactions on acoustics, speech, and signal processing, 37 (3):328–339, 1989.   \nWisdom, S., Powers, T., Hershey, J., Le Roux, J., and Atlas, L. Full-capacity unitary recurrent neural networks. In Advances in Neural Information Processing Systems, pp. 4880–4888, 2016.   \nWu, Y., Zhang, S., Zhang, Y., Bengio, Y., and Salakhutdinov, R. R. On multiplicative integration with recurrent neural networks. In Advances in Neural Information Processing Systems, pp. 2856– 2864, 2016.   \nYu, F. and Koltun, V. Multi-scale context aggregation by dilated convolutions. In International Conference on Learning Representations, 2015.   \nZhang, S., Wu, Y., Che, T., Lin, Z., Memisevic, R., Salakhutdinov, R. R., and Bengio, Y. Architectural complexity measures of recurrent neural networks. In Advances in Neural Information Processing Systems, pp. 1822–1830. 2016. ",
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+        "type": "text",
+        "text": "A DESCRIPTION OF BENCHMARK TASKS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The Adding Problem: In this task, each input consists of a length- $\\mathbf { \\nabla } \\cdot n$ sequence of depth 2, with all values randomly chosen in [0, 1], and the second dimension being all zeros expect for two elements that are marked by 1. The objective is to sum the two random values whose second dimensions are marked by 1. Simply predicting the sum to be 1 should give an MSE of about 0.1767. First introduced by Hochreiter & Schmidhuber (1997), the addition problem have been consistently used as a pathological test for evaluating sequential models (Pascanu et al., 2013; Le et al., 2015; Zhang et al., 2016; Arjovsky et al., 2016). ",
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+        "type": "text",
+        "text": "Sequential MNIST & P-MNIST: Sequential MNIST is frequently used to test a recurrent network’s ability to combine its information from a long memory context in order to make classification prediction (Le et al., 2015; Zhang et al., 2016; Cooijmans et al., 2016; Krueger et al., 2017; Jing et al., 2017). In this task, MNIST (Lecun et al., 1998) images are presented to the model as a $7 8 4 \\times 1$ sequence for digit classification In a more challenging setting, we also permuted the order of the sequence by a random (fixed) order and tested the TCN on this permuted MNIST (P-MNIST) task. ",
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+        "type": "text",
+        "text": "Copy Memory Task: In copy memory task, each input sequence has length $T + 2 0$ . The first 10 values are chosen randomly from digit [1-8] with the rest being all zeros, except for the last 11 entries which are marked by 9 (the first $\" 9 \"$ is a delimiter). The goal of this task is to generate an output of same length that is zero everywhere, except the last 10 values after the delimiter, where the model is expected to repeat the same 10 values at the start of the input. This was used by prior works such as Arjovsky et al. (2016); Wisdom et al. (2016); Jing et al. (2017); but we also extended the sequence lengths to up to $T = 2 0 0 0$ . ",
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+    {
+        "type": "text",
+        "text": "JSB Chorales: JSB Chorales dataset (Allan & Williams, 2005) is a polyphonic music dataset consisting of the entire corpus of 382 four-part harmonized chorales by J. S. Bach. In a polyphonic music dataset, each input is a sequence of elements having 88 dimensions, representing the 88 keys on a piano. Therefore, each element $x _ { t }$ is a chord written in as binary vector, in which a “1” indicates a key pressed. ",
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+        "type": "text",
+        "text": "Nottingham: Nottingham dataset 2 is a collection of 1200 British and American folk tunes. Nottingham is a much larger dataset than JSB Chorales. Along with JSB Chorales, Nottingham has been used in a number of works that investigated recurrent models’ applicability in polyphonic music (Greff et al., 2017; Chung et al., 2014), and the performance for both tasks are measured in terms of negative log-likelihood (NLL) loss. ",
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+        "type": "text",
+        "text": "PennTreebank: We evaluated TCN on the PennTreebank (PTB) dataset (Marcus et al., 1993), for both character-level and word-level language modeling. When used as a character-level language corpus, PTB contains 5059K characters for training, 396K for validation and 446K for testing, with an alphabet size of 50. When used as a word-level language corpus, PTB contains 888K words for training, 70K for validation and 79K for testing, with vocabulary size 10000. This is a highly studied dataset in the field of language modeling (Miyamoto & Cho, 2016; Krueger et al., 2017; Merity et al., 2017), with exceptional results have been achieved by some highly optimized RNNs. ",
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+    {
+        "type": "text",
+        "text": "Wikitext-103: Wikitext-103 (Merity et al., 2016) is almost 110 times as large as PTB, featuring a vocabulary size of about 268K. The dataset contains 28K Wikipedia articles (about 103 million words) for training, 60 articles (about 218K words) for validation and 60 articles (246K words) for testing. This is a more representative (and realistic) dataset than PTB as it contains a much larger vocabulary, including many rare vocabularies. ",
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+    {
+        "type": "text",
+        "text": "LAMBADA: Introduced by Paperno et al. (2016), LAMBADA (LA nguage Modeling Boadened to Account for Discourse Aspects) is a dataset consisting of 10K passages extracted from novels, with on average 4.6 sentences as context, and 1 target sentence whose last word is to be predicted. This dataset was built so that human can guess naturally and perfectly when given the context, but would fail to do so when only given the target sentence. Therefore, LAMBADA is a very challenging dataset that evaluates a model’s textual understanding and ability to keep track of information in the broader discourse. Here is an example of a test in the LAMBADA dataset, where the last word “miscarriage” is to be predicted (which is not in the context): ",
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+        "text": "Context: “Yes, I thought I was going to lose the baby.”“I was scared too.” he stated, sincerity flooding his eyes. “You were?”“Yes, of course. Why do you even ask?”“This baby wasn’t exactly planned for.” ",
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+    {
+        "type": "text",
+        "text": "Target Sentence: “Do you honestly think that I would want you to have a ” ",
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+    {
+        "type": "text",
+        "text": "Target Word: miscarriage ",
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+    {
+        "type": "text",
+        "text": "This dataset was evaluated in prior works such as Paperno et al. (2016); Grave et al. (2016). In general, better results on LAMBADA indicate that a model is better at capturing information from longer and broader context. The training data for LAMBADA is the full text of 2,662 novels with more than 200M words 3, and the vocabulary size is about 93K. ",
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+    {
+        "type": "text",
+        "text": "text8: We also used tex $8 ^ { 4 }$ dataset for character level language modeling (Mikolov et al., 2012). Compared to PTB, text8 is about 20 times as large, with about 100 million characters from Wikipedia (90M for training, 5M for validation and 5M for testing). The corpus contains 27 unique alphabets. ",
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+    {
+        "type": "text",
+        "text": "B HYPERPARAMETERS SETTINGS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "B.1 HYPERPARAMETERS FOR TCN ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this supplementary section, we report in a table (see Table 2) the hyperparameters we used when applying the generic TCN model on the different tasks/datasets. The most important factor for picking parameters is to make sure that the TCN has a sufficiently large receptive field by choosing $k$ and $n$ that can cover the amount of context needed for the task. ",
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+        "table_caption": [
+            "Table 2: TCN parameter settings for experiments in Section. 4 "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"7\">TCN SETTINGS</td></tr><tr><td>Dataset/Task</td><td>Subtask</td><td>k n</td><td>Hidden</td><td>Dropout</td><td>Grad Clip</td><td>Note</td></tr><tr><td rowspan=\"3\">The Adding Problem</td><td>T=200</td><td>6</td><td>7 27</td><td></td><td></td><td></td></tr><tr><td>T= 400</td><td>7</td><td>7 27</td><td>0.0</td><td>N/A</td><td></td></tr><tr><td>T= 600</td><td>8</td><td>8 24</td><td></td><td></td><td></td></tr><tr><td rowspan=\"2\">Seq. MNIST</td><td></td><td>7</td><td>8</td><td>25</td><td>0.0</td><td>N/A</td></tr><tr><td></td><td>6</td><td>8 20</td><td></td><td></td><td></td></tr><tr><td>Permuted MNIST</td><td></td><td>7 6</td><td>8 8</td><td>25 0.0 20</td><td>N/A</td><td></td></tr><tr><td rowspan=\"3\">Copy Memory Task</td><td>T=500</td><td>6</td><td>9</td><td>10</td><td></td><td></td></tr><tr><td>T= 1000</td><td>8</td><td>8</td><td>10 0.05</td><td>1.0</td><td>RMSprop 5e-4</td></tr><tr><td>T= 2000</td><td>8</td><td>9 10</td><td></td><td></td><td></td></tr><tr><td>Music JSB Chorales</td><td>1</td><td>3</td><td>2</td><td>150</td><td>0.5 0.4</td><td></td></tr><tr><td>Music Nottingham</td><td>1</td><td>6</td><td>4</td><td>150</td><td>0.2 0.4</td><td></td></tr><tr><td rowspan=\"3\">Word-level LM</td><td>PTB</td><td>3</td><td>4 600</td><td></td><td></td><td>Embed. size 600</td></tr><tr><td>Wiki-103</td><td>3</td><td>5 1000</td><td>0.4</td><td>0.3</td><td>Embed. size 400</td></tr><tr><td>LAMBADA</td><td>4</td><td>5 500</td><td></td><td></td><td>Embed. size 500</td></tr><tr><td rowspan=\"2\">Char-level LM</td><td>PTB</td><td>3</td><td>3</td><td>450</td><td></td><td></td></tr><tr><td>text8</td><td>2</td><td>5</td><td>0.1 520</td><td>0.15</td><td>Embed. size 100</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "As previously mentioned in Section 4, the number of hidden units was chosen based on $k$ and $n$ such that the model size is approximately at the same level as the recurrent models. In the table above, a gradient clip of N/A means no gradient clipping was applied. However, in larger tasks, we empirically found that adding a gradient clip value (we randomly picked from [0.2, 1]) helps the training convergence. ",
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+    {
+        "type": "text",
+        "text": "B.2 HYPERPARAMETERS FOR LSTM/GRU ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "We also report the parameter setting for LSTM in Table 3. These values are picked from hyperparameter search for LSTMs that have up to 3 layers, and the optimizers are chosen from {SGD, Adam, RMSprop, Adagrad}. ",
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+    {
+        "type": "text",
+        "text": "GRU hyperparameters were chosen in a similar fashion, but with more hidden units to keep the total model size approximately the same (since a GRU cell is smaller). ",
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+    {
+        "type": "text",
+        "text": "B.3 COMPARE TO THE STATE OF THE ART RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "As previously noted, TCN can still be outperformed by optimized RNNs in some of the tasks, whose results are summarized in Table 4 below. The same TCN architecture is used across all tasks. ",
+        "bbox": [
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+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Note that the size of the SoTA model may be different from the size of the TCN. ",
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+        "page_idx": 16
+    },
+    {
+        "type": "table",
+        "img_path": "images/73d40644ba73244a6c32276d6302ef92fc2e7e143437b73c5e720fe7870168fc.jpg",
+        "table_caption": [
+            "Table 3: LSTM parameter settings for experiments in Section 4. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"8\">LSTM SETTINGS(KEY PARAMETERS)</td></tr><tr><td>Dataset/Task</td><td>Subtask</td><td>n</td><td>Hidden</td><td>Dropout</td><td>Grad Clip</td><td>Bias</td><td>Note</td></tr><tr><td rowspan=\"3\">The Adding Problem</td><td>T= 200</td><td>2</td><td>77</td><td></td><td>50</td><td>5.0</td><td>SGD 1e-3</td></tr><tr><td>T= 400</td><td>2</td><td>77</td><td>0.0</td><td>50</td><td>10.0</td><td>Adam 2e-3</td></tr><tr><td>T= 600</td><td>1</td><td>130</td><td></td><td>5</td><td>1.0</td><td>1</td></tr><tr><td>Seq. MNIST</td><td>:</td><td>1</td><td>130</td><td>0.0</td><td>1</td><td>1.0</td><td>RMSprop 1e-3</td></tr><tr><td>Permuted MNIST</td><td>1</td><td>1</td><td>130</td><td>0.0</td><td>1</td><td>10.0</td><td>RMSprop 1e-3</td></tr><tr><td rowspan=\"3\">Copy Memory Task</td><td>T= 500</td><td>1</td><td>50</td><td></td><td>0.25</td><td></td><td></td></tr><tr><td>T=1000</td><td>1</td><td>50</td><td>0.05</td><td>1</td><td>1</td><td>RMSprop/Adam</td></tr><tr><td>T= 2000</td><td>3</td><td>28</td><td></td><td>1</td><td></td><td></td></tr><tr><td>Music JSB Chorales</td><td>1</td><td>2</td><td>200</td><td>0.2</td><td>1</td><td>10.0</td><td>SGD/Adam</td></tr><tr><td>Music Nottingham</td><td>-</td><td>3</td><td>280</td><td>0.1</td><td>0.5</td><td>-</td><td>Adam 4e-3</td></tr><tr><td rowspan=\"3\">Word-level LM</td><td></td><td>1</td><td>500</td><td></td><td>1</td><td>-</td><td></td></tr><tr><td>PTB</td><td>3</td><td>700</td><td>0.4</td><td>0.3</td><td>1.0</td><td>SGD 30,Emb. 700, etc.</td></tr><tr><td>Wiki-103</td><td>-</td><td>-</td><td>·</td><td>·</td><td>·</td><td>Grave et al. (2016)</td></tr><tr><td></td><td>LAMBADA</td><td>-</td><td>-</td><td>-</td><td>1</td><td>-</td><td>Grave et al. (2016)</td></tr><tr><td rowspan=\"2\">Char-level LM</td><td>PTB</td><td></td><td>600</td><td>0.1</td><td>0.5</td><td>-</td><td>Emb. size 120</td></tr><tr><td>text8</td><td>21</td><td>1024</td><td>0.15</td><td>0.5</td><td>1</td><td>Adam 1e-2</td></tr></table>",
+        "bbox": [
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+            469
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "table",
+        "img_path": "images/278d38dcdbaf08f00e3539a54c3a34cbb2dc66d4ae477cbd1ad7a96a3913491a.jpg",
+        "table_caption": [
+            "Table 4: State of the art (SoTA) results for tasks in Section 4. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=6>TCN VS. SoTA RESULTS</td></tr><tr><td rowspan=1 colspan=1>Task</td><td rowspan=1 colspan=1>TCN Result</td><td rowspan=1 colspan=1>Size</td><td rowspan=1 colspan=1>SoTA</td><td rowspan=1 colspan=1>Size</td><td rowspan=1 colspan=1>Model</td></tr><tr><td rowspan=1 colspan=1>Seq. MNIST (acc.)</td><td rowspan=1 colspan=1>99.0</td><td rowspan=1 colspan=1>21K</td><td rowspan=1 colspan=1>99.0</td><td rowspan=1 colspan=1>21K</td><td rowspan=1 colspan=1>Dilated GRU (Chang et al., 2017)</td></tr><tr><td rowspan=1 colspan=1>P-MNIST (acc.)</td><td rowspan=1 colspan=1>97.2</td><td rowspan=1 colspan=1>42K</td><td rowspan=1 colspan=1>95.9</td><td rowspan=1 colspan=1>42K</td><td rowspan=1 colspan=1> Zoneout (Krueger et al., 2017)</td></tr><tr><td rowspan=1 colspan=1>Adding Prob. 600 (loss)</td><td rowspan=1 colspan=1>5.8e-5</td><td rowspan=1 colspan=1>70K</td><td rowspan=1 colspan=1>5.3e-5</td><td rowspan=1 colspan=1>70K</td><td rowspan=1 colspan=1>Regularized GRU</td></tr><tr><td rowspan=1 colspan=1>Copy Memory 1000 (loss)</td><td rowspan=1 colspan=1>3.5e-5</td><td rowspan=1 colspan=1>70K</td><td rowspan=1 colspan=1>0.011</td><td rowspan=1 colspan=1>70K</td><td rowspan=1 colspan=1>EURNN (Jing et al., 2017)</td></tr><tr><td rowspan=1 colspan=1>JSB Chorales (loss)</td><td rowspan=1 colspan=1>8.10</td><td rowspan=1 colspan=1>300K</td><td rowspan=1 colspan=1>3.47</td><td rowspan=1 colspan=1>：</td><td rowspan=1 colspan=1>DBN+LSTM(Vohra et al., 2015)</td></tr><tr><td rowspan=1 colspan=1>Nottingham (loss)</td><td rowspan=1 colspan=1>3.07</td><td rowspan=1 colspan=1>1M</td><td rowspan=1 colspan=1>1.32</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>DBN+LSTM (Vohra et al., 2015)</td></tr><tr><td rowspan=1 colspan=1>Word PTB (ppl)</td><td rowspan=1 colspan=1>90.17</td><td rowspan=1 colspan=1>13M</td><td rowspan=1 colspan=1>52.8</td><td rowspan=1 colspan=1>24M</td><td rowspan=1 colspan=1>AWD-LSTM + Cont.Cache(Merity et al., 2017)</td></tr><tr><td rowspan=1 colspan=1>Word Wiki-103 (ppl)</td><td rowspan=1 colspan=1>45.19</td><td rowspan=1 colspan=1>148M</td><td rowspan=1 colspan=1>40.4</td><td rowspan=1 colspan=1>&gt;300M</td><td rowspan=1 colspan=1>Neural Cache Model (Large)(Grave et al., 2016)</td></tr><tr><td rowspan=1 colspan=1>Word LAMBADA (ppl)</td><td rowspan=1 colspan=1>1279</td><td rowspan=1 colspan=1>56M</td><td rowspan=1 colspan=1>138</td><td rowspan=1 colspan=1>&gt;100M</td><td rowspan=1 colspan=1>Neural Cache Model (Large)(Grave et al., 2016)</td></tr><tr><td rowspan=1 colspan=1>Char PTB (bpc)</td><td rowspan=1 colspan=1>1.35</td><td rowspan=1 colspan=1>3M</td><td rowspan=1 colspan=1>1.22</td><td rowspan=1 colspan=1>14M</td><td rowspan=1 colspan=1>2-LayerNorm HyperLSTM(Ha et al., 2017)</td></tr><tr><td rowspan=1 colspan=1>Char text8 (bpc)</td><td rowspan=1 colspan=1>1.45</td><td rowspan=1 colspan=1>4.6M</td><td rowspan=1 colspan=1>1.29</td><td rowspan=1 colspan=1>&gt;12M</td><td rowspan=1 colspan=1>HM-LSTM (Chung et al., 2016)</td></tr></table>",
+        "bbox": [
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+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/5e61b9b3ee15df3f95a75c0ba19ebb15a423e433d11b444e7c29d44ad108a8d4.jpg",
+        "image_caption": [
+            "Figure 8: Controlled experiments that examine different components of the TCN model "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            133,
+            821,
+            438
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "In this section we briefly study, via controlled experiments, the effect of filter size and residual block on the TCN’s ability to model different sequential tasks. Figure 8 shows the results of this ablative analysis. We kept the model size and the depth of the networks exactly the same within each experiment so that dilation factor is controlled. We conducted the experiment on three very different tasks: the copy memory task, permuted MNIST (P-MNIST), as well as word-level PTB language modeling. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "Through these experiments, we empirically confirm that both filter sizes and residuals play important roles in TCN’s capability of modeling potentially long dependencies. In both the copy memory and the permuted MNIST task, we observed faster convergence and better result for larger filter sizes (e.g. in the copy memory task, a filter size $k \\geq 3$ led to only suboptimal convergence). In word-level PTB, we find a filter size of $k = 3$ works best. This is not a complete surprise, since a size- $k$ filter on the inputs is analogous to a $k$ -gram model in language modeling. ",
+        "bbox": [
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+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "Results of control experiments on the residual function are shown in Figure 8d, 8e and 8f. In all three scenarios, we observe that the residual stabilizes the training by bringing a faster convergence as well as better final results, compared to TCN with the same model size but no residual block. ",
+        "bbox": [
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+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "D EXPERIMENTS: GATING MECHANISM ON TCN ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            102,
+            599,
+            118
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "One component that has shown to be effective in adapting a TCN to language modeling is the gating mechanism within the residual block, which was used in works such as Dauphin et al. (2017). In this section, we empirically evaluate the effects of adding gated units to TCN. ",
+        "bbox": [
+            173,
+            133,
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+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "We replace the ReLU within the TCN residual block with a gating mechanism, represented by an elementwise product between two convolutional layers, with one of them also passing through a sigmoid function $\\sigma ( x ) ^ { 5 }$ . Prior works such as Dauphin et al. (2017) has used similar gating to control the path through which information flows in the network, and achieved great performance on language modeling tasks. ",
+        "bbox": [
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+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "table",
+        "img_path": "images/5f6f9c7e069590d701d2fd81496ef2a036bcc6446052a5d148a33ffb57b1e7e2.jpg",
+        "table_caption": [
+            "Table 5: TCN with Gating Mechanism within Residual Block. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=3>RELU TCN VS.GATED TCN RESULTS</td></tr><tr><td rowspan=1 colspan=1>Task</td><td rowspan=1 colspan=1>TCN</td><td rowspan=1 colspan=1>TCN + Gating</td></tr><tr><td rowspan=1 colspan=1>Seq. MNIST (acc.)</td><td rowspan=1 colspan=1>99.0</td><td rowspan=1 colspan=1>99.0</td></tr><tr><td rowspan=1 colspan=1>P-MNIST (acc.)</td><td rowspan=1 colspan=1>97.2</td><td rowspan=1 colspan=1>96.9</td></tr><tr><td rowspan=1 colspan=1>Adding Prob. 600 (loss)</td><td rowspan=1 colspan=1>5.8e-5</td><td rowspan=1 colspan=1>5.6e-5</td></tr><tr><td rowspan=1 colspan=1>CopyMemory 11000 (loss)</td><td rowspan=1 colspan=1>3.5e-5</td><td rowspan=1 colspan=1>0.00508</td></tr><tr><td rowspan=1 colspan=1>JSB Chorales (loss)</td><td rowspan=1 colspan=1>8.10</td><td rowspan=1 colspan=1>8.13</td></tr><tr><td rowspan=1 colspan=1>Nottingham (loss)</td><td rowspan=1 colspan=1>3.07</td><td rowspan=1 colspan=1>3.12</td></tr><tr><td rowspan=1 colspan=1>Word PTB (ppl)</td><td rowspan=1 colspan=1>90.17</td><td rowspan=1 colspan=1>88.91</td></tr><tr><td rowspan=1 colspan=1>Char PTB (bpc)</td><td rowspan=1 colspan=1>1.35</td><td rowspan=1 colspan=1>1.343</td></tr><tr><td rowspan=1 colspan=1>Char text8 (bpc)</td><td rowspan=1 colspan=1>1.45</td><td rowspan=1 colspan=1>1.48</td></tr></table>",
+        "bbox": [
+            325,
+            291,
+            671,
+            513
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "text",
+        "text": "Through these comparisons, we notice that gating components do further improve the TCN results on certain language modeling datasets like PTB, which agrees with prior works. However, we do not observe such benefits to exist in general on sequence prediction tasks, such as on polyphonic music datasets, and those simpler benchmark tasks requiring more long-term memories. For example, on the copy memory task with $T = 1 0 0 0$ , we find that gating mechanism deteriorates the convergence of TCN to a suboptimal result that is only slightly better than random guess. ",
+        "bbox": [
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+        ],
+        "page_idx": 19
+    }
+]

parse/train/rkhlb8lCZ/rkhlb8lCZ_content_list.json

@@ -0,0 +1,1554 @@
+[
+    {
+        "type": "text",
+        "text": "WAVELET POOLING FOR CONVOLUTIONAL NEURAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Travis Williams Department of Electrical Engineering North Carolina A&T State University Greensboro, NC 27410, USA tlwilli3@aggies.ncat.edu ",
+        "bbox": [
+            183,
+            170,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Robert Li   \nDepartment of Electrical Engineering   \nNorth Carolina A&T State University   \nGreensboro, NC 27410, USA   \neeli@ncat.edu ",
+        "bbox": [
+            506,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            276,
+            544,
+            291
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Convolutional Neural Networks continuously advance the progress of 2D and 3D image and object classification. The steadfast usage of this algorithm requires constant evaluation and upgrading of foundational concepts to maintain progress. Network regularization techniques typically focus on convolutional layer operations, while leaving pooling layer operations without suitable options. We introduce Wavelet Pooling as another alternative to traditional neighborhood pooling. This method decomposes features into a second level decomposition, and discards the first-level subbands to reduce feature dimensions. This method addresses the overfitting problem encountered by max pooling, while reducing features in a more structurally compact manner than pooling via neighborhood regions. Experimental results on four benchmark classification datasets demonstrate our proposed method outperforms or performs comparatively with methods like max, mean, mixed, and stochastic pooling. ",
+        "bbox": [
+            233,
+            313,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            532,
+            336,
+            547
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Convolutional Neural Networks (CNNs) have become the standard-bearer in image and object classification (Nielsen, 2015). Due to the layer structures conforming to the shape of the inputs, CNNs consistently classify images, objects, videos, etc. at a higher accuracy rate than vector-based deep learning techniques (Nielsen, 2015). The strength of this algorithm motivates researchers to constantly evaluate and upgrade foundational concepts to continue growth and progress. The key components of CNN, the convolutional layer and pooling layer, consistently undergo modifications and innovations to elevate accuracy and efficiency of CNNs beyond previous benchmarks. ",
+        "bbox": [
+            174,
+            568,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Pooling has roots in predecessors to CNN such as Neocognitron, which manual subsampling by the user occurs (Fukushima, 1979), and Cresceptron, which introduces the first max pooling operation in deep learning (Weng et al., 1992). Pooling subsamples the results of the convolutional layers, gradually reducing spatial dimensions of the data throughout the network. The benefits of this operation are to reduce parameters, increase computational efficiency, and regulate overfitting (Boureau et al., 2010). ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Methods of pooling vary, with the most popular form being max pooling, and secondarily, average pooling (Nielsen, 2015; Lee et al., 2016). These forms of pooling are deterministic, efficient, and simple, but have weaknesses hindering the potential for optimal network learning (Lee et al., 2016; Yu et al., 2014). Other pooling operations, notably mixed pooling and stochastic pooling, use probabilistic approaches to correct some of the issues of the prior methods (Yu et al., 2014; Zeiler & Fergus, 2013). ",
+        "bbox": [
+            174,
+            763,
+            825,
+            847
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "However, one commonality all these pooling operations employ a neighborhood approach to subsampling, reminiscent of nearest neighbor interpolation in image processing. Neighborhood interpolation techniques perform fast, with simplicity and efficiency, but introduce artifacts such as edge halos, blurring, and aliasing (Parker et al., 1983). Minimizing discontinuities in the data are critical to aiding in network regularization, and increasing classification accuracy. ",
+        "bbox": [
+            174,
+            854,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We propose a wavelet pooling algorithm that uses a second-level wavelet decomposition to subsample features. Our approach forgoes the nearest neighbor interpolation method in favor of an organic, subband method that more accurately represents the feature contents with less artifacts. We compare our proposed pooling method to max, mean, mixed, and stochastic pooling to verify its validity, and ability to produce near equal or superior results. We test these methods on benchmark image classification datasets such as Mixed National Institute of Standards and Technology (MNIST) (Lecun et al., 1998), Canadian Institute for Advanced Research (CIFAR-10) (Krizhevsky, 2009), Street House View Numbers (SHVN) (Netzer et al., 2011), and Karolinska Directed Emotional Faces (KDEF) (Lundqvist et al., 1998). We perform all simulations in MATLAB R2016b. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The rest of this paper organizes as follows: Section 2 gives the background, Section 3 describes the proposed methods, Section 4 discusses the experimental results, and Section 5 gives the summary and conclusion. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 BACKGROUND ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            297,
+            326,
+            315
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Pooling is another term for subsampling. In this layer, the dimensions of the output of the convolutional layer are condensed. The dimensionality reduction happens by summarizing a region into one neuron value, and this occurs until all neurons have been affected. The two most popular forms of pooling are max pooling and average pooling (Nielsen, 2015; Lee et al., 2016). Max pooling involves taking the maximum value of a region $R _ { i j }$ and selecting it for the condensed feature map. Average pooling involves calculating the average value of a region and selecting it for the condensed feature map. The max pooling function is expressed as: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/96256f98bb18add71d7231184ea243c76406ffbd6209827174c687418e25efcd.jpg",
+        "text": "$$\na _ { k i j } = \\operatorname* { m a x } _ { ( p , q ) \\in R _ { i j } } ( a _ { k p q } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            423,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "While average pooling is shown by the following equation: ",
+        "bbox": [
+            173,
+            484,
+            562,
+            500
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/aabc723f17606442abbbe8411354a5d08dfd3b6d2e69b740ed5e1bc85abc11fb.jpg",
+        "text": "$$\na _ { k i j } = \\frac { 1 } { | R _ { i j } | } \\sum _ { ( p , q ) \\in R _ { i j } } a _ { k p q }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            408,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Where $a _ { k i j }$ is the output activation of the $k ^ { t h }$ feature map at $( i , j ) , a _ { k p q }$ is the input activation at $( p , q )$ within $R _ { i j }$ , and $| R _ { i j } |$ is the size of the pooling region. An illustration of both of these pooling methods is expressed in Figure 1 (Williams & Li, 2016): ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "table",
+        "img_path": "images/ae2b5d15b50e837d11f60d04adaee5d22ccbc44a25ff8c80e4d73036d941c506.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=8>Convolution Output</td></tr><tr><td rowspan=1 colspan=1>55</td><td rowspan=1 colspan=1>20</td><td rowspan=1 colspan=1>87</td><td rowspan=1 colspan=1>46</td><td rowspan=1 colspan=1>32</td><td rowspan=1 colspan=1>59</td><td rowspan=1 colspan=1>76</td><td rowspan=1 colspan=1>58</td></tr><tr><td rowspan=1 colspan=1>58</td><td rowspan=1 colspan=1>25</td><td rowspan=1 colspan=1>61</td><td rowspan=1 colspan=1>29</td><td rowspan=1 colspan=1>34</td><td rowspan=1 colspan=1>79</td><td rowspan=1 colspan=1>94</td><td rowspan=1 colspan=1>15</td></tr><tr><td rowspan=1 colspan=1>29</td><td rowspan=1 colspan=1>35</td><td rowspan=1 colspan=1>71</td><td rowspan=1 colspan=1>63</td><td rowspan=1 colspan=1>94</td><td rowspan=1 colspan=1>73</td><td rowspan=1 colspan=1>12</td><td rowspan=1 colspan=1>17</td></tr><tr><td rowspan=1 colspan=1>58</td><td rowspan=1 colspan=1>22</td><td rowspan=1 colspan=1>41</td><td rowspan=1 colspan=1>98</td><td rowspan=1 colspan=1>32</td><td rowspan=1 colspan=1>45</td><td rowspan=1 colspan=1>16</td><td rowspan=1 colspan=1>76</td></tr><tr><td rowspan=1 colspan=1>92</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>16</td><td rowspan=1 colspan=1>。</td><td rowspan=1 colspan=1>46</td><td rowspan=1 colspan=1>84</td><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>7</td></tr><tr><td rowspan=1 colspan=1>38</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>86</td><td rowspan=1 colspan=1>97</td><td rowspan=1 colspan=1>64</td><td rowspan=1 colspan=1>13</td><td rowspan=1 colspan=1>15</td><td rowspan=1 colspan=1>48</td></tr><tr><td rowspan=1 colspan=1>32</td><td rowspan=1 colspan=1>19</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>42</td><td rowspan=1 colspan=1>45</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>56</td></tr><tr><td rowspan=1 colspan=1>6</td><td rowspan=1 colspan=1>。</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>8</td><td rowspan=1 colspan=1>66</td><td rowspan=1 colspan=1>44</td><td rowspan=1 colspan=1>72</td><td rowspan=1 colspan=1>8</td></tr></table>",
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+        "table_caption": [],
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+        "table_body": "<table><tr><td rowspan=1 colspan=4>Max Pooling</td></tr><tr><td rowspan=1 colspan=1>58</td><td rowspan=1 colspan=1>87</td><td rowspan=1 colspan=1>79</td><td rowspan=1 colspan=1>94</td></tr><tr><td rowspan=1 colspan=1>58</td><td rowspan=1 colspan=1>98</td><td rowspan=1 colspan=1>94</td><td rowspan=1 colspan=1>76</td></tr><tr><td rowspan=1 colspan=1>92</td><td rowspan=1 colspan=1>97</td><td rowspan=1 colspan=1>84</td><td rowspan=1 colspan=1>48</td></tr><tr><td rowspan=1 colspan=1>32</td><td rowspan=1 colspan=1>42</td><td rowspan=1 colspan=1>66</td><td rowspan=1 colspan=1>72</td></tr></table>",
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+        "table_caption": [
+            "Mean Pooling ",
+            "Figure 1: Example of Max & Average Pooling with Stride of 2 "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>40</td><td rowspan=1 colspan=1>56</td><td rowspan=1 colspan=1>51</td><td rowspan=1 colspan=1>61</td></tr><tr><td rowspan=1 colspan=1>36</td><td rowspan=1 colspan=1>68</td><td rowspan=1 colspan=1>61</td><td rowspan=1 colspan=1>30</td></tr><tr><td rowspan=1 colspan=1>35</td><td rowspan=1 colspan=1>50</td><td rowspan=1 colspan=1>52</td><td rowspan=1 colspan=1>19</td></tr><tr><td rowspan=1 colspan=1>14</td><td rowspan=1 colspan=1>16</td><td rowspan=1 colspan=1>39</td><td rowspan=1 colspan=1>34</td></tr></table>",
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+        "type": "text",
+        "text": "While max and average pooling both are effective, simple methods, they also have shortcomings. Max pooling, depending on the data, can erase details from an image (Yu et al., 2014; Zeiler & Fergus, 2013). This happens if the main details have less intensity than the insignificant details. In addition, max pooling commonly overfits training data (Yu et al., 2014; Zeiler & Fergus, 2013). Average pooling, depending on the data, can dilute pertinent details from an image. The averaging of data with values much lower than significant details causes this action (Yu et al., 2014; Zeiler & Fergus, 2013). Figure 2 illustrates these shortcomings using the toy image example: ",
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+    {
+        "type": "text",
+        "text": "To combat these issues, researchers have created probabilistic pooling methods. Mixed pooling combines max and average pooling by randomly selecting one method over the other during training (Yu et al., 2014). There is no set way to perform mixed pooling. This method is applied arbitrarily in three different ways (1) for all features within a layer, (2) mixed between features within a layer, or (3) mixed between regions for different features within a layer (Lee et al., 2016; Yu et al., 2014). Mixed pooling is shown in the following equation: ",
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+        "type": "image",
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+        "image_caption": [
+            "Figure 2: Shortcomings of Max & Average Pooling using Toy Image "
+        ],
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+    {
+        "type": "equation",
+        "img_path": "images/810abb892685aa03d5c0bcbf82849277ab61731c7397b6db601c69d034565553.jpg",
+        "text": "$$\na _ { k i j } = \\lambda \\cdot \\operatorname* { m a x } _ { ( p , q ) \\in R _ { i j } } ( a _ { k p q } ) + ( 1 - \\lambda ) \\cdot \\frac { 1 } { | R _ { i j } | } \\sum _ { ( p , q ) \\in R _ { i j } } a _ { k p q }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\lambda$ is a random value 0 or 1, indicating max or average pooling for a particular region/feature/layer. ",
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+    {
+        "type": "text",
+        "text": "Another probabilistic pooling method, called stochastic pooling, improves upon max pooling by randomly sampling from neighborhood regions based on the probability values of each activation (Zeiler & Fergus, 2013). These probabilities $p$ for each region are calculated by normalizing the activations within the region: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0d2e08ff045c90c3ebad7d96be890d1c61b60368eb4af50d1604645af5509ab6.jpg",
+        "text": "$$\np _ { p q } = \\frac { a _ { p q } } { \\sum _ { ( p , q ) \\in R _ { i j } } a _ { p q } }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The pooled activation is sampled from a multinomial distribution based on $p$ to pick a location $l$ within the region (Zeiler & Fergus, 2013). The process is captured in the following equation (Zeiler & Fergus, 2013): ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/143890786b56eaed5381e8725788170ecc0758ccc388218cf344eb93c79a8973.jpg",
+        "text": "$$\na _ { k i j } = a _ { l } \\quad w h e r e \\quad l \\sim P ( p _ { 1 } , . . . , p _ { | R _ { i j } | } )\n$$",
+        "text_format": "latex",
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/2ca0fb375b792ba9a9b71febe50ad09430c488f886e3669eda98dc3d1a51f74a.jpg",
+        "image_caption": [
+            "Figure 3 displays a visual example of stochastic pooling on a $3 { \\bf x } 3$ region: ",
+            "Figure 3: Stochastic Pooling Example "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In Figure 3, a region of activations are shown, and in the adjacent region, their corresponding probabilities based on Equation 4. In any given region, the activations with the highest probabilities have the higher chance of selection. However, any activation can be chosen. In this example, the stochastic pooling method selects the midrange activation with a probability of $13 \\%$ . By being based off probability, and not deterministic, stochastic pooling avoids the shortcomings of max and average pooling, while enjoying some of the advantages of max pooling (Zeiler & Fergus, 2013). ",
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+    {
+        "type": "text",
+        "text": "3 PROPOSED METHOD ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The previously highlighted pooling methods use neighborhoods to subsample, almost identical to nearest neighbor interpolation. Previous studies explore the possibilities of wavelets in image interpolation versus traditional methods (Dumic et al., 2007). Our proposed pooling method uses wavelets to reduce the dimensions of the feature maps. We propose using the wavelet transform to minimize artifacts resulting from neighborhood reduction (Parker et al., 1983). We postulate that our approach, which discards the first-order subbands, more organically captures the data compression. This organic reduction therefore lessens the creation of jagged edges and other artifacts that may impede correct image classification. ",
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+    {
+        "type": "text",
+        "text": "3.1 FORWARD PROPAGATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The proposed wavelet pooling scheme pools features by performing a 2nd order decomposition in the wavelet domain according to the fast wavelet transform (FWT) (Mallat, 1989; Nason & Silverman, 1995; Strang & Nguyen, 1996; Burrus et al., 1998), which is a more efficient implementation of the two-dimensional discrete wavelet transform (DWT) as follows (Chui, 1992; Strang & Strela, 1995; Rieder et al., 1994): ",
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+    {
+        "type": "equation",
+        "img_path": "images/0f49fa3a1145ee2c146cf489e4b6df00272270ec28477ceadafad307de0314a9.jpg",
+        "text": "$$\nW _ { \\varphi } [ j + 1 , k ] = h _ { \\varphi } [ - n ] * W _ { \\varphi } [ j , n ] | _ { n = 2 k , k \\le 0 }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "equation",
+        "img_path": "images/0609908e4062e191e19ac12b04bce7ee58ef380833c98786aa052d88a140f59a.jpg",
+        "text": "$$\nW _ { \\psi } [ j + 1 , k ] = h _ { \\psi } [ - n ] * W _ { \\psi } [ j , n ] | _ { n = 2 k , k \\le 0 }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where $\\varphi$ is the approximation function, and $\\psi$ is the detail function, $W _ { \\varphi }$ , $W _ { \\psi }$ are called approximation and detail coefficients. $h _ { \\varphi } [ - n ]$ and $h _ { \\psi } [ - n ]$ are the time reversed scaling and wavelet vectors, (n) represents the sample in the vector, while (j) denotes the resolution level. When using the FWT on images, we apply it twice (once on the rows, then again on the columns). By doing this in combination, we obtain our detail subbands (LH, HL, HH) at each decomposition level, and our approximation subband (LL) for the highest decomposition level. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "After performing the 2nd order decomposition, we reconstruct the image features, but only using the 2nd order wavelet subbands. This method pools the image features by a factor of 2 using the inverse FWT (IFWT) (Mallat, 1989; Nason & Silverman, 1995; Strang & Nguyen, 1996; Burrus et al., 1998), which is based off of the inverse DWT (IDWT) (Chui, 1992; Strang & Strela, 1995; Rieder et al., 1994): ",
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+    {
+        "type": "equation",
+        "img_path": "images/cdb4cf23e293b7194ec0346dab26fe5f6e4d83f8b059debeabadc5d152bffe68.jpg",
+        "text": "$$\nW _ { \\varphi } [ j , k ] = h _ { \\varphi } [ - n ] * W _ { \\varphi } [ j + 1 , n ] + h _ { \\psi } [ - n ] * W _ { \\psi } [ j + 1 , n ] | _ { n = { \\frac { k } { 2 } } , k \\leq 0 }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/4f3d158ff784108d752855ef0f3ac75a0287d53a4f9a1c33a8ce754c00e1ff07.jpg",
+        "image_caption": [
+            "Figure 4 gives an illustration of the algorithm for the forward propagation of wavelet pooling: ",
+            "Figure 4: Wavelet Pooling Forward Propagation Algorithm "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "3.2 BACKPROPAGATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The proposed wavelet pooling algorithm performs backpropagation by reversing the process of its forward propagation. First, the image feature being back propagated undergoes $1 ^ { s t }$ order wavelet decomposition. After decomposition, the detail coefficient subbands upsample by a factor of 2 to create a new $1 ^ { s t }$ level decomposition. The initial decomposition then becomes the $2 ^ { n d }$ level decomposition. Finally, this new $2 ^ { n d }$ order wavelet decomposition reconstructs the image feature for further backpropagation using the IDWT. Figure 5 details the backpropagation algorithm of wavelet pooling: ",
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+    },
+    {
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+        "img_path": "images/67febc9a06058a1f18b93c877fafd23b66b72930b4b25034992e16feaa33623b.jpg",
+        "image_caption": [
+            "Figure 5: Wavelet Pooling Backpropagation Algorithm "
+        ],
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+    {
+        "type": "text",
+        "text": "4 RESULTS AND DISCUSSION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "All CNN experiments use MatConvNet (Vedaldi & Lenc, 2015). All training uses stochastic gradient descent (Bottou, 2010). For our proposed method, the wavelet basis is the Haar wavelet, mainly for its even, square subbands. All experiments are run on a 64-bit operating system, with an Intel Core i7-6800k CPU $\\textcircled { a } ~ 3 . 4 0 ~ \\mathrm { G H z }$ processor, with 64.0 GB of RAM. We utilize two GeForce Titan X Pascal GPUs with 12 GB of video memory for all training. All CNN structures except for MNIST use a network loosely based on Zeilers network (Zeiler & Fergus, 2013). We repeat the experiments with Dropout (Srivastava, 2013) and replace Local Response Normalization (Krizhevsky, 2009) with Batch Normalization (Ioffe & Szegedy, 2015) for CIFAR-10 and SHVN (Dropout only) to examine how these regularization techniques change the pooling results. To test the effectiveness of each pooling method on each dataset, we solely pool with that method for all pooling layers in that network. All pooling methods use a $2 \\mathbf { x } 2$ window for an even comparison to the proposed method. Figure 6 gives a selection of each of the datasets. ",
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+    {
+        "type": "text",
+        "text": "4.1 MNIST ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The network architecture is based on the example MNIST structure from MatConvNet, with batch normalization inserted. All other parameters are the same. Figure 7 shows our network structure for the MNIST experiments: ",
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+    {
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+        "image_caption": [
+            "Figure 6: Selection of Image Datasets "
+        ],
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+    {
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+        "img_path": "images/95c98f5bb8cd108027f9a420c1a864090ebfbba981f7d03348eb869f3d9d69e0.jpg",
+        "image_caption": [
+            "Figure 7: CNN MNIST Structure Block Diagram "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "The input training data and test data come from the MNIST database of handwritten digits. The full training set of 60,000 images is used, as well as the full testing set of 10,000 images. Table 1 shows our proposed method outperforms all methods. Given the small number of epochs, max pooling is the only method to start to overfit the data during training. Mixed and stochastic pooling show a rocky trajectory, but do not overfit. Average and wavelet pooling show a smoother descent in learning and error reduction. Figure 8 shows the energy of each method per epoch. ",
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+    {
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+        "img_path": "images/deb898ce72595fd7b606ef3ff92190e8d12d0e163a108fe312e27b2c0706e6b5.jpg",
+        "image_caption": [
+            "Figure 8: MNIST Pooling Method Energy Performance of Training & Validation Sets "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Table 1 shows the accuracy of each method: ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/a6b912bae6eddb62cd4dc830a9acdc22854093ed9e55f20b874153f9c85bfcfb.jpg",
+        "table_caption": [
+            "Table 1: MNIST Performance of Pooling Methods "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Average</td><td rowspan=1 colspan=1>Max</td><td rowspan=1 colspan=1>Mixed</td><td rowspan=1 colspan=1>Stochastic</td><td rowspan=1 colspan=1>Wavelet</td></tr><tr><td rowspan=1 colspan=1>Accuracy (%)</td><td rowspan=1 colspan=1>98.72</td><td rowspan=1 colspan=1>98.80</td><td rowspan=1 colspan=1>98.86</td><td rowspan=1 colspan=1>98.90</td><td rowspan=1 colspan=1>99.01</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.2 CIFAR-10 ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We run two sets of experiments with the pooling methods. The first is a regular network structure with no dropout layers. We use this network to observe each pooling method without extra regularization. The second uses dropout and batch normalization, and performs over 30 more epochs to observe the effects of these changes. Figure 9 shows our network structure for the CIFAR-10 experiments: ",
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+    {
+        "type": "text",
+        "text": "The input training and test data come from the CIFAR-10 dataset. The full training set of 50,000 images is used, as well as the full testing set of 10,000 images. For both cases, with no dropout, and with dropout, Table 2 and Table 3 show our proposed method has the second highest accuracy. Max pooling overfits fairly quickly, while wavelet pooling resists overfitting. The change in learning rate prevents our method from overfitting, and it continues to show a slower propensity for learning. Mixed and stochastic pooling maintain a consistent progression of learning, and their validation sets trend at a similar, but better rate than our proposed method. Average pooling shows the smoothest descent in learning and error reduction, especially in the validation set. Figure 10 shows the energy of each method per epoch. ",
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/eab1af088fdeeed9fe5e1fdc17353c47fe3cad910a704148ae6af01cf8847a1f.jpg",
+        "image_caption": [
+            "Figure 9: CNN CIFAR-10 Structure Block Diagram "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/8343d05229f091b0ccaf3b71410317e193a25d016e7d738194f7228e11c02b31.jpg",
+        "image_caption": [
+            "Figure 10: CIFAR-10 Pooling Method Energy Performance of Training & Validation Sets "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Tables 2 and 3 show the accuracy of each method: ",
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/9b67fda20f7fd98f822b1c94a378949b1df1a54d39ab44e670aa5f1fb50ade77.jpg",
+        "table_caption": [
+            "Table 2: CIFAR-10 Performance of Pooling Methods "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Average</td><td rowspan=1 colspan=1>Max</td><td rowspan=1 colspan=1>Mixed</td><td rowspan=1 colspan=1>Stochastic</td><td rowspan=1 colspan=1>Wavelet</td></tr><tr><td rowspan=1 colspan=1>Accuracy (%)</td><td rowspan=1 colspan=1>76.51</td><td rowspan=1 colspan=1>71.42</td><td rowspan=1 colspan=1>73.77</td><td rowspan=1 colspan=1>73.03</td><td rowspan=1 colspan=1>74.42</td></tr></table>",
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+        "page_idx": 6
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+    {
+        "type": "table",
+        "img_path": "images/3663e2c55b5220b2a5b231c028d227ebab76bf02e4771d5f6ebb2f76b6edf696.jpg",
+        "table_caption": [
+            "Table 3: CIFAR-10 Performance of Pooling Methods $^ +$ Dropout "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Average</td><td>Max</td><td>Mixed</td><td>Stochastic</td><td>Wavelet</td></tr><tr><td>Accuracy (%)</td><td>81.15</td><td>80.30</td><td>79.21</td><td>80.09</td><td>80.28</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "4.3 SHVN ",
+        "text_level": 1,
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We run two sets of experiments with the pooling methods. The first is a regular network structure with no dropout layers. We use this network to observe each pooling method without extra regularization. The second uses dropout to observe the effects of this change. Figure 11 shows our network structure for the SHVN experiments: ",
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+    },
+    {
+        "type": "text",
+        "text": "The input training and test data come from the SHVN dataset. For the case with no dropout, we use 55,000 images from the training set. For the case with dropout, we use the full training set of 73,257 images, a validation set of 30,000 images we extract from the extra training set of 531,131 images, as well as the full testing set of 26,032 images. For both cases, with no dropout, and with dropout, ",
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/b3a9dba3655fb309e5a32d14191f972a1e36fd28e5668c13d8d56ef140140af0.jpg",
+        "image_caption": [
+            "Figure 11: CNN SHVN Structure Block Diagram "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 4 and Table 5 show our proposed method has the second lowest accuracy. Max and wavelet pooling both slightly overfit the data. Our method follows the path of max pooling, but performs slightly better in maintaining some stability. Mixed, stochastic, and average pooling maintain a slow progression of learning, and their validation sets trend at near identical rates. Figure 12 shows the energy of each method per epoch. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/5cc6b495708b04e274769b4f7b6e0755d6376f1b03d688956c266d1d52014244.jpg",
+        "image_caption": [
+            "Figure 12: SHVN Pooling Method Energy Performance of Training & Validation Sets "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Tables 4 and 5 shows the accuracy of each method: ",
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/53751ae4e66d146973948d5021db925d706c8715255920cfd0ec8c748cc2b02d.jpg",
+        "table_caption": [
+            "Table 4: SHVN Performance of Pooling Methods "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Average</td><td>Max</td><td>Mixed</td><td>Stochastic</td><td>Wavelet</td></tr><tr><td>Accuracy (%)</td><td>89.83</td><td>88.09</td><td>89.25</td><td>89.97</td><td>88.51</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/cb7d4fc25e63379f0f91246871650b2691f667af38d5fcc9c82a8705bbc9c2f1.jpg",
+        "table_caption": [
+            "Table 5: SHVN Performance of Pooling Methods $^ +$ Dropout "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Average</td><td rowspan=1 colspan=1>Max</td><td rowspan=1 colspan=1>Mixed</td><td rowspan=1 colspan=1>Stochastic</td><td rowspan=1 colspan=1>Wavelet</td></tr><tr><td rowspan=1 colspan=1>Accuracy (%)</td><td rowspan=1 colspan=1>92.80</td><td rowspan=1 colspan=1>92.18</td><td rowspan=1 colspan=1>92.13</td><td rowspan=1 colspan=1>91.04</td><td rowspan=1 colspan=1>91.10</td></tr></table>",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.4 KDEF ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We run one set of experiments with the pooling methods that includes dropout. Figure 13 shows our network structure for the KDEF experiments: ",
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The input training and test data come from the KDEF dataset. This dataset contains 4,900 images of 35 people displaying seven basic emotions (afraid, angry, disgusted, happy, neutral, sad, and surprised) using facial expressions. They display emotions at five poses (full left and right profiles, half left and right profiles, and straight). ",
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+    {
+        "type": "image",
+        "img_path": "images/5313b03d57e276b0ac92f5d863296f151505f94fa4cade18ea3fc474dc79355c.jpg",
+        "image_caption": [
+            "Figure 13: CNN KDEF Structure Block Diagram "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This dataset contains a few errors that we fix (missing or corrupted images, uncropped images, etc.). All of the missing images are at angles of -90, -45, 45, or 90 degrees. We fix the missing and corrupt images by mirroring their counterparts in MATLAB and adding them back to the dataset. We manually crop the images that need to match the dimensions set by the creators $( 7 6 2 \\mathrm { ~ x ~ } 5 6 2 )$ . KDEF does not designate a training or test data set. We shuffle the data and separate 3,900 images as training data, and 1,000 images as test data. We resize the images to $1 2 8 \\mathrm { x } 1 2 8$ because of memory and time constraints. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The dropout layers regulate the network and maintain stability in spite of some pooling methods known to overfit. Table 6 shows our proposed method has the second highest accuracy. Max pooling eventually overfits, while wavelet pooling resists overfitting. Average and mixed pooling resist overfitting, but are unstable for most of the learning. Stochastic pooling maintains a consistent progression of learning. Wavelet pooling also follows a smoother, consistent progression of learning. Figure 14 shows the energy of each method per epoch. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/537316cbfa6f3889562dfa12841eb7528f683ecceedd841395fefc3334bf1b8a.jpg",
+        "image_caption": [
+            "Figure 14: KDEF Pooling Method Energy Performance of Training & Validation Sets "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            191,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 6 shows the accuracy of each method: ",
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/f8d2ffaa5167e2f0a69554989e9664cb5181ccb13a4eff3c44a8999f636923be.jpg",
+        "table_caption": [
+            "Table 6: KDEF Performance of Pooling Methods $^ +$ Dropout "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Average</td><td>Max</td><td>Mixed</td><td>Stochastic</td><td>Wavelet</td></tr><tr><td>Accuracy (%)</td><td>76.5</td><td>75.6</td><td>72.6</td><td>72.7</td><td>75.9</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.5 COMPUTATIONAL COMPLEXITY ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Our construction and implementation of wavelet pooling is not efficient. We present this proposed methods as a proof-of-concept, to show its potential and validity, and also to be open to massive improvements. The main area of improvement is computational efficiency. As a proof-of-concept, the code written to implement this method is not at its peak form. Additionally, we did not have the time, space, or resources to optimize the code. We view the accuracy results and novelty as a starting point to spawn improvements, both from our own research as well as other researchers. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We calculate efficiency in terms of mathematical operations (multiplications, additions, logical, etc.) that each method utilizes to complete its algorithm. For max pooling, we calculate operations based on the worst-case scenarios for each neighborhood in finding the maximum value. For average pooling, we calculate the number of additions and division for each neighborhood. Mixed pooling is the mean value of both average and max pooling. We calculate operations for stochastic pooling by counting the number of mathematical operations as well as the random selection of the values based on probability (Roulette Wheel Selection). For wavelet pooling, we calculate the number of operations for each subband at each level, in both decomposition and reconstruction. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Table 7 shows the number of mathematical operations for one image in forward propagation. This table shows that for all methods, average pooling has the least number of computations, followed by mixed pooling, with max pooling not far behind. Stochastic pooling is the least computationally efficient pooling method out of the neighborhood-based methods. It uses about $3 \\mathbf { x }$ more mathematical operations than average pooling, the most computationally efficient. ",
+        "bbox": [
+            174,
+            257,
+            823,
+            327
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "However, wavelet pooling by far is the least computationally efficient method, using 54 to $2 1 3 \\mathrm { x }$ more mathematical operations than average pooling. This is partially due to the implementation of the subband coding, which did not implement multidimensional decomposition and reconstruction. ",
+        "bbox": [
+            174,
+            334,
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+            376
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "table",
+        "img_path": "images/045767e0bc9a93197dcfa2aa53bbc019abf39bcbff5bf7fb526bf90c0063a048.jpg",
+        "table_caption": [
+            "Table 7: Number of Mathematical Operations for Each Method According to Dataset "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>MNIST</td><td rowspan=1 colspan=1>CIFAR-10</td><td rowspan=1 colspan=1>SHVN</td><td rowspan=1 colspan=1>KDEF</td></tr><tr><td rowspan=1 colspan=1>Max</td><td rowspan=1 colspan=1>6.2K</td><td rowspan=1 colspan=1>13K</td><td rowspan=1 colspan=1>26K</td><td rowspan=1 colspan=1>50K</td></tr><tr><td rowspan=1 colspan=1>Avg</td><td rowspan=1 colspan=1>3.5K</td><td rowspan=1 colspan=1>7.4K</td><td rowspan=1 colspan=1>15K</td><td rowspan=1 colspan=1>29K</td></tr><tr><td rowspan=1 colspan=1>Mix</td><td rowspan=1 colspan=1>4.8K</td><td rowspan=1 colspan=1>10K</td><td rowspan=1 colspan=1>20K</td><td rowspan=1 colspan=1>40K</td></tr><tr><td rowspan=1 colspan=1>Stoch</td><td rowspan=1 colspan=1>10.6K</td><td rowspan=1 colspan=1>22K</td><td rowspan=1 colspan=1>45K</td><td rowspan=1 colspan=1>86K</td></tr><tr><td rowspan=1 colspan=1>Wav</td><td rowspan=1 colspan=1>110K</td><td rowspan=1 colspan=1>405K</td><td rowspan=1 colspan=1>810K</td><td rowspan=1 colspan=1>6.2M</td></tr></table>",
+        "bbox": [
+            326,
+            395,
+            669,
+            484
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Nonetheless, by implementing our method through good coding practices (vectorization, architecture, etc.), GPUs, and an improved FTW algorithm, this method can prove to be a viable option. There exists a few improvements to the FTW algorithm that utilize multidimensional wavelets (Karlsson & Vetterli, 1988; Weeks & Bayoumi, 1998), lifting (Valens, 1999), parallelization Holmstrom (1995), as well as other methods that boast of improving the efficiency in speed and memory ¨ (Oliver & Malumbres, 2008; Khoromskij & Miao, 2014; Kopenkov, 2008) ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We prove wavelet pooling has potential to equal or eclipse some of the traditional methods currently utilized in CNNs. Our proposed method outperforms all others in the MNIST dataset, outperforms all but one in the CIFAR-10 and KDEF datasets, and performs within respectable ranges of the pooling methods that outdo it in the SHVN dataset. The addition of dropout and batch normalization show our proposed methods response to network regularization. Like the non-dropout cases, it outperforms all but one in both the CIFAR- $1 0 ~ \\&$ KDEF datasets, and performs within respectable ranges of the pooling methods that outdo it in the SHVN dataset. Our results confirm previous studies proving that no one pooling method is superior, but some perform better than others depending on the dataset and network structure Boureau et al. (2010); Lee et al. (2016). Furthermore, many networks alternate between different pooling methods to maximize the effectiveness of each method. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Future work and improvements in this area could be to vary the wavelet basis to explore which basis performs best for the pooling. Altering the upsampling and downsampling factors in the decomposition and reconstruction can lead to better image feature reductions outside of the $2 \\mathbf { x } 2$ scale. Retention of the subbands we discard for the backpropagation could lead to higher accuracies and fewer errors. Improving the method of FTW we use could greatly increase computational efficiency. Finally, analyzing the structural similarity (SSIM) of wavelet pooling versus other methods could further prove the vitality of using our approach. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            104,
+            328,
+            117
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "This research is supported by the Title III HBGI PhD Fellowship grant from the U.S. Department of Education. ",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
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+# DATASET DISTILLATION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Model distillation aims to distill the knowledge of a complex model into a simpler one. In this paper, we consider an alternative formulation called dataset distillation: we keep the model fixed and instead attempt to distill the knowledge from a large training dataset into a small one. The idea is to synthesize a small number of data points that do not need to come from the correct data distribution, but will, when given to the learning algorithm as training data, approximate the model trained on the original data. For example, we show that it is possible to compress 60, 000 MNIST training images into just 10 synthetic distilled images (one per class) and achieve close to the original performance, given a fixed network initialization. We evaluate our method in various initialization settings. Experiments on multiple datasets, MNIST, CIFAR10, PASCAL-VOC, and CUB-200, demonstrate the advantage of our approach compared to alternative methods. Finally, we include a real-world application of dataset distillation to the continual learning setting: we show that storing distilled images as episodic memory of previous tasks can alleviate forgetting more effectively than real images.
+
+# 1 INTRODUCTION
+
+Hinton et al. (2015) proposed network distillation as a way to transfer the knowledge from an ensemble of many separately-trained networks into a single, typically compact network, performing a type of model compression. In this paper, we are considering a related but orthogonal task: rather than distilling the model, we propose to distill the dataset. Unlike network distillation, we keep the model fixed but encapsulate the knowledge of the entire training dataset, which typically contains thousands to millions of images, into a small number of synthetic training images. We show that we can go as low as one synthetic image per category, training the same model to reach surprisingly good performance on these synthetic images. For example, in Figure 1a, we compress 60, 000 training images of MNIST digit dataset into only 10 synthetic images (one per category), given a fixed network initialization. Training the standard LENET (LeCun et al., 1998) on these 10 images yields test-time MNIST recognition performance of $9 4 \%$ , compared to $9 9 \%$ for the original dataset. For networks with unknown random weights, 100 synthetic images train to $8 9 \%$ . We name our method Dataset Distillation and these images distilled images.
+
+But why is dataset distillation interesting? First, there is the purely scientific question of how much data is encoded in a given training set and how compressible it is? Second, we wish to know whether it is possible to “load up" a given network with an entire dataset-worth of knowledge by a handful of images. This is in contrast to traditional training that often requires tens of thousands of data samples. Finally, on the practical side, dataset distillation enables applications that require compressing data with its task. We demonstrate that under the continual learning setting, storing distilled images as memory of past task and data can alleviate catastrophic forgetting (McCloskey and Cohen, 1989).
+
+A key question is whether it is even possible to compress a dataset into a small set of synthetic data samples. For example, is it possible to train an image classification model on synthetic images that are not on the manifold of natural images? Conventional wisdom would suggest that the answer is no, as the synthetic training data may not follow the same distribution of the real test data. Yet, in this work, we show that this is indeed possible.
+
+We present an optimization algorithm for synthesizing a small number of synthetic data samples not only capturing much of the original training data but also tailored explicitly for fast model training with only a few data point. To achieve our goal, we first derive the network weights as a differentiable function of our synthetic training data. Given this connection, instead of optimizing the network weights for a particular training objective, we optimize the pixel values of our distilled images. However, this formulation requires access to the initial weights of the network. To relax this assumption, we develop a method for generating distilled images for randomly initialized networks. To further boost performance, we propose an iterative version, where the same distilled images are reused over multiple gradient descent steps so that the knowledge can be fully transferred into the model. Finally, we study a simple linear model, deriving a lower bound on the size of distilled data required to achieve the same performance as training on the full dataset.
+
+![](images/a3349f9c1635457a4bda7e72a8f70ee5582b7484123d58a89342b2223f0666cc.jpg)  
+Figure 1: We distill the knowledge of tens of thousands of images into a few synthetic training images called distilled images. On MNIST, 100 distilled images can train a standard LENET with a random initialization to $8 9 \%$ test accuracy, compared to $9 9 \%$ when fully trained. On CIFAR10, 100 distilled images can train a network with a random initialization to $4 1 \%$ test accuracy, compared to $8 0 \%$ when fully trained. In Section 3.6, we show that these distilled images can efficiently store knowledge of previous tasks for continual learning.
+
+We demonstrate that a handful of distilled images can be used to train a model with a fixed initialization to achieve surprisingly high performance. For networks pre-trained on other tasks, our method can find distilled images for fast model fine-tuning. We test our method on several initialization settings: fixed initialization, random initialization, fixed pre-trained weights, and random pre-trained weights. Extensive experiments on four publicly available datasets, MNIST, CIFAR10, PASCAL-VOC, and CUB-200, show that our approach often outperforms existing methods. Finally, we demonstrate that for continual learning methods that store limited-size past data samples as episodic memory (Lopez-Paz and Ranzato, 2017; Kirkpatrick et al., 2017), storing our distilled data instead is much more effective. Our distilled images contain richer information about the past data and tasks, and we show experimental evidence on standard continual learning benchmarks. Our code, data, and models will be available upon publication.
+
+# 2 RELATED WORK
+
+Knowledge distillation. The main inspiration for this paper is network distillation (Hinton et al., 2015), a widely used technique in ensemble learning (Radosavovic et al., 2018) and model compression (Ba and Caruana, 2014; Romero et al., 2015; Howard et al., 2017). While network distillation aims to distill the knowledge of multiple networks into a single model, our goal is to compress the knowledge of an entire dataset into a few synthetic data. Our method is also related to the theoretical concept of teaching dimension, which specifies the minimal size of data needed to teach a target model to a learner (Shinohara and Miyano, 1991; Goldman and Kearns, 1995). However, methods (Zhu, 2013; 2015) inspired by this concept require the existence of target models, which our method does not.
+
+Dataset pruning, core-set construction, and instance selection. Another way to distill knowledge is to summarize the entire dataset by a small subset, either by only using the “valuable” data for model training (Angelova et al., 2005; Felzenszwalb et al., 2010; Lapedriza et al., 2013) or by only labeling the “valuable” data via active learning (Cohn et al., 1996; Tong and Koller, 2001). Similarly, core-set construction (Tsang et al., 2005; Har-Peled and Kushal, 2007; Bachem et al., 2017; Sener and Savarese, 2018) and instance selection (Olvera-López et al., 2010) methods aim to select a subset of the entire training data, such that models trained on the subset will perform as well as the model trained on the full dataset. For example, solutions to many classical linear learning algorithms, e.g., Perceptron (Rosenblatt, 1957) and SVMs (Hearst et al., 1998), are weighted sums of subsets of training examples, which can be viewed as core-sets. However, algorithms constructing these subsets require many more training examples per category than we do, in part because their “valuable” images have to be real, whereas our distilled images are exempt from this constraint.
+
+Gradient-based hyperparameter optimization. Our work bears similarity with gradient-based hyperparameter optimization techniques, which compute the gradient of hyperparameter w.r.t. the final validation loss by reversing the entire training procedure (Bengio, 2000; Domke, 2012; Maclaurin et al., 2015; Pedregosa, 2016). We also backpropagate errors through optimization steps. However, we use only training set data and focus more heavily on learning synthetic training data rather than tuning hyperparameters. To our knowledge, this direction has only been slightly touched on previously (Maclaurin et al., 2015). We explore it in greater depth and demonstrate the idea of dataset distillation in various settings. More crucially, our distilled images work well across random initialization weights, not possible by prior work.
+
+Understanding datasets. Researchers have presented various approaches for understanding and visualizing learned models (Zeiler and Fergus, 2014; Zhou et al., 2015; Mahendran and Vedaldi, 2015; Bau et al., 2017; Koh and Liang, 2017). Unlike these approaches, we are interested in understanding the intrinsic properties of the training data rather than a specific trained model. Analyzing training datasets has, in the past, been mainly focused on the investigation of bias in datasets (Ponce et al., 2006; Torralba and Efros, 2011). For example, Torralba and Efros (2011) proposed to quantify the “value” of dataset samples using cross-dataset generalization. Our method offers a different perspective for understanding datasets by distilling full datasets into a few synthetic samples.
+
+# 3 FORMULATION
+
+Given a model and a dataset, we aim to obtain a new, much-reduced synthetic dataset which performs almost as well as the original dataset. We first present our main optimization algorithm for training a network with a fixed initialization with one gradient descent (GD) step (Section 3.1). In Section 3.2, we derive the resolution to a more challenging case, where initial weights are random rather than fixed. In Section 3.3, we further study a linear network case to help readers understand both the properties and limitations of our method. We also discuss the distribution of initial weights with which our method can work well. In Section 3.4, we extend our approach to reuse the same distilled images over 2, 000 gradient descent steps and largely improve the performance. Finally, Section 3.5 discusses dataset distillation for different initialization distributions. Finally, in Section 3.6, we show that our distilled images can be used as effective episodic memory for continual learning tasks.
+
+Consider a training dataset $\mathbf { x } = \{ x _ { i } \} _ { i = 1 } ^ { N }$ , we parameterize our neural network as $\theta$ and denote $\ell ( x _ { i } , \theta )$ as the loss function that represents the loss of this network on a data point $x _ { i }$ . Our task is to find the minimizer of the empirical error over entire training data:
+
+$$
+\theta ^ { * } = \underset { \theta } { \arg \operatorname* { m i n } } \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \ell ( x _ { i } , \theta ) \triangleq \underset { \theta } { \arg \operatorname* { m i n } } \ell ( \mathbf { x } , \theta ) ,
+$$
+
+where for notation simplicity we overload the $\ell ( \cdot )$ notation so that $\ell ( \mathbf { x } , \theta )$ represents the average error of $\theta$ over the entire dataset. We make the mild assumption that $\ell$ is twice-differentiable, which holds true for the majority of modern machine learning models and tasks.
+
+# 3.1 OPTIMIZING DISTILLED DATA
+
+Standard training usually applies minibatch stochastic gradient descent or its variants. At each step $t$ , a minibatch of training data $\mathbf { x } _ { t } = \{ x _ { t , j } \} _ { j = 1 } ^ { n }$ is sampled to update the current parameters as
+
+$$
+\theta _ { t + 1 } = \theta _ { t } - \eta \nabla _ { \theta _ { t } } \ell ( \mathbf { x } _ { t } , \theta _ { t } ) ,
+$$
+
+where is the learning rate. Such a training process often takes tens of thousands or even millions of update steps to converge. Instead, we learn a tiny set of synthetic distilled training data $\tilde { \mathbf { x } } = \{ \tilde { x } _ { i } \} _ { i = 1 } ^ { M }$ with and a corresponding learning rate $\tilde { \eta }$ so that a single GD step such as
+
+$$
+\theta _ { 1 } = \theta _ { 0 } - \tilde { \eta } \nabla _ { \theta _ { 0 } } \ell ( \tilde { \mathbf { x } } , \theta _ { 0 } )
+$$
+
+# Algorithm 1 Dataset Distillation
+
+Input: $p ( \theta _ { 0 } )$ : distribution of initial weights; $M$ : the number of distilled data   
+Input: $\alpha$ : step size; $n$ : batch size; $T$ : the number of optimization iterations; $\tilde { \eta } _ { 0 }$ : initial value for $\tilde { \eta }$   
+1: Initialize $\hat { \tilde { \mathbf { x } } } = \{ \tilde { x } _ { i } \} _ { i = 1 } ^ { M }$ either from $\mathcal { N } ( 0 , I )$ or from real training images. Initialize $\tilde { \eta }  \tilde { \eta } _ { 0 }$   
+2: for each training step $t = 1$ to $T$ do   
+3: Get a minibatch of real training data $\mathbf { x } _ { t } = \{ x _ { t , j } \} _ { j = 1 } ^ { n }$   
+4: Sample a batch of initial weights $\theta _ { 0 } ^ { ( j ) } \sim p ( \theta _ { 0 } )$   
+5: for each sampled $\theta _ { 0 } ^ { ( j ) }$ do   
+6: Compute updated parameter with GD: $\theta _ { 1 } ^ { ( j ) } = \theta _ { 0 } ^ { ( j ) } - \tilde { \eta } \nabla _ { \theta _ { 0 } ^ { ( j ) } } \ell ( \tilde { \mathbf { x } } , \theta _ { 0 } ^ { ( j ) } )$   
+7: Evaluate the objective function on real training data: $\mathscr { L } ^ { ( j ) } = \ell ( \mathbf { x } _ { t } , \boldsymbol { \theta } _ { 1 } ^ { ( j ) } )$   
+8: end for   
+9: Update $\begin{array} { r } { \tilde { \mathbf { x } }  \tilde { \mathbf { x } } - \alpha \nabla _ { \tilde { \mathbf { x } } } \sum _ { j } \mathcal { L } ^ { ( j ) } , \mathrm { a n d } \tilde { \eta }  \tilde { \eta } - \alpha \nabla _ { \tilde { \eta } } \sum _ { j } \mathcal { L } ^ { ( j ) } } \end{array}$
+
+10: end for Output: distilled data $\tilde { \bf x }$ and optimized learning rate $\tilde { \eta }$
+
+using these learned synthetic data $\tilde { \bf x }$ can greatly boost the performance on the real test set. Given an initial $\theta _ { 0 }$ , we obtain these synthetic data $\tilde { \mathbf { x } }$ and learning rate $\tilde { \eta }$ by minimizing the objective below $\mathcal { L }$
+
+$$
+\begin{array} { r l } & { \tilde { \mathbf { x } } ^ { * } , \tilde { \eta } ^ { * } = \underset { \tilde { \mathbf { x } } , \tilde { \eta } } { \arg \operatorname* { m i n } } \mathcal { L } ( \tilde { \mathbf { x } } , \tilde { \eta } ; \theta _ { 0 } ) } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad } \end{array}
+$$
+
+where we derive the new weights $\theta _ { 1 }$ as a function of distilled data $\tilde { \bf x }$ and learning rate $\tilde { \eta }$ using Equation 2 and then evaluate the new weights over all the real training data $\mathbf { x }$ . The loss $\mathcal { L } ( \tilde { \mathbf { x } } , \tilde { \eta } ; \theta _ { 0 } )$ is differentiable w.r.t. $\tilde { \bf x }$ and $\tilde { \eta }$ , and can thus be optimized using standard gradient-based methods. In many classification tasks, the data $\mathbf { x }$ may contain discrete parts, e.g., class labels in data-label pairs. For such cases, we fix the discrete parts rather than learn them.
+
+# 3.2 DISTILLATION FOR RANDOM INITIALIZATIONS
+
+Unfortunately, the above distilled data is optimized for a given initialization, and does not generalize well to other initializations, as it encodes the information of both the training dataset $\mathbf { x }$ and a particular network initialization $\theta _ { 0 }$ . To address this issue, we turn to calculate a small number of distilled data that can work for networks with random initializations from a specific distribution. We formulate the optimization problem as follows:
+
+$$
+\tilde { \mathbf { x } } ^ { * } , \tilde { \eta } ^ { * } = \underset { \tilde { \mathbf { x } } , \tilde { \eta } } { \arg \operatorname* { m i n } } \mathbb { E } _ { \theta _ { 0 } \sim p ( \theta _ { 0 } ) } \mathcal { L } ( \tilde { \mathbf { x } } , \tilde { \eta } ; \theta _ { 0 } ) ,
+$$
+
+where the network initialization $\theta _ { 0 }$ is randomly sampled from a distribution $p ( \theta _ { 0 } )$ . During our optimization, the distilled data are optimized to work well for randomly initialized networks. In practice, we observe that the final distilled data generalize well to unseen initializations. In addition, these distilled images often look quite informative, encoding the discriminative features of each category (e.g., in Figure 2). Algorithm 1 illustrates our main method.
+
+As the optimization (Equation 4) is highly non-linear and complex, the initialization of $\tilde { \bf x }$ plays a critical role in the final performance. We experiment with different initialization strategies and observe that using random real images as initialization often produces better distilled images compared to random initialization, e.g., $\mathcal { N } ( 0 , I )$ .
+
+For a compact set distilled data to be properly learned, it turns out having only one GD step is far from sufficient. Next, we derive a lower bound on the size of distilled data needed for a simple model with arbitrary initial $\theta _ { 0 }$ in one GD step, and discuss its implications on our algorithm.
+
+# 3.3 ANALYSIS OF A SIMPLE LINEAR CASE
+
+This section studies our formulation in a simple linear regression problem with quadratic loss. We derive a lower bound of the size of distilled data needed to achieve the same performance as training on the full dataset for arbitrary initialization with one GD step. Consider a dataset $\mathbf { x }$ containing $N$ data-target pairs $\{ ( d _ { i } , t _ { i } ) \} _ { i = 1 } ^ { N }$ , where $d _ { i } \in \mathbb { R } ^ { D }$ and $t _ { i } \in \mathbb { R }$ , which we represent as two matrices: an $N \times D$ data matrix $\mathbf { d }$ and an $N \times 1$ target matrix $\mathbf { t }$ . Given the mean squared error metric and a $D \times 1$ weight matrix $\theta$ , we have
+
+$$
+\ell ( { \mathbf x } , \theta ) = \ell ( ( { \mathbf d } , { \mathbf t } ) , \theta ) = \frac { 1 } { 2 N } \| { \mathbf d } \theta - { \mathbf t } \| ^ { 2 } .
+$$
+
+We aim to learn $M$ synthetic data-target pairs $\tilde { \mathbf { x } } = ( \tilde { \mathbf { d } } , \tilde { \mathbf { t } } )$ , where $\tilde { \mathbf { d } }$ is an $M \times D$ matrix, $\tilde { \mathbf { t } }$ an $M \times 1$ matrix $M \ll N ,$ ), and $\tilde { \eta }$ the learning rate, to minimize $\ell ( \mathbf { x } , \theta _ { 0 } - \tilde { \eta } \nabla _ { \theta _ { 0 } } \ell ( \tilde { \mathbf { x } } , \theta _ { 0 } ) )$ . The updated weight matrix after one GD step with these distilled data is
+
+$$
+\begin{array} { r l } & { \theta _ { 1 } = \theta _ { 0 } - \tilde { \eta } \nabla _ { \theta _ { 0 } } \ell ( \tilde { \mathbf { x } } , \theta _ { 0 } ) } \\ & { \quad = \theta _ { 0 } - \frac { \tilde { \eta } } { M } \tilde { \mathbf { d } } ^ { T } ( \tilde { \mathbf { d } } \theta _ { 0 } - \tilde { \mathbf { t } } ) } \\ & { \quad = ( \mathbf { I } - \frac { \tilde { \eta } } { M } \tilde { \mathbf { d } } ^ { T } \tilde { \mathbf { d } } ) \theta _ { 0 } + \frac { \tilde { \eta } } { M } \tilde { \mathbf { d } } ^ { T } \tilde { \mathbf { t } } . } \end{array}
+$$
+
+For the quadratic loss, there always exists distilled data $\tilde { \mathbf { x } }$ that can achieve the same performance as training on the full dataset $\mathbf { x }$ (i.e., attaining the global minimum) for any initialization $\theta _ { 0 }$ . For example, given any global minimum solution $\theta ^ { * }$ , we can choose $\tilde { \mathbf { d } } = N \cdot \mathbf { I }$ and $\tilde { \mathbf { t } } = N \cdot \boldsymbol { \theta } ^ { * }$ . But how small can the size of the distilled data be? For such models, the global minimum is attained at any $\theta ^ { * }$ satisfying $\mathbf { d } ^ { T } \mathbf { d } \theta ^ { * } = \mathbf { d } ^ { T } \mathbf { t }$ . Substituting Equation 6 in the condition above, we have
+
+$$
+\mathbf { d } ^ { T } \mathbf { d } ( \mathbf { I } - \frac { \tilde { \eta } } { M } \tilde { \mathbf { d } } ^ { T } \tilde { \mathbf { d } } ) \theta _ { 0 } + \frac { \tilde { \eta } } { M } \mathbf { d } ^ { T } \mathbf { d } \tilde { \mathbf { d } } ^ { T } \tilde { \mathbf { t } } = \mathbf { d } ^ { T } \mathbf { t } .
+$$
+
+Here we make the mild assumption that the feature columns of the data matrix $\mathbf { d }$ are independent (i.e., ${ \bf d } ^ { T } { \bf d }$ has full rank). For a $\bar { \bf x } = ( \tilde { \bf d } , \tilde { \bf t } )$ to satisfy the above equation for any $\theta _ { 0 }$ , we must have
+
+$$
+\mathbf { I } - \frac { \widetilde { \eta } } { M } \widetilde { \mathbf { d } } ^ { T } \widetilde { \mathbf { d } } = \mathbf { 0 } ,
+$$
+
+which implies that $\tilde { \mathbf { d } } ^ { T } \tilde { \mathbf { d } }$ has full rank and $M \geq D$
+
+Discussion. The analysis above only considers a simple case but suggests that any small number of distilled data fail to generalize to arbitrary initial $\theta _ { 0 }$ . This is intuitively expected as the optimization target $\ell ( \mathbf { x } , \theta _ { 1 } ) = \ell ( \bar { \mathbf { x } } , \theta _ { 0 } - \tilde { \eta } \nabla _ { \theta _ { 0 } } \ell ( \tilde { \mathbf { x } } , \theta _ { 0 } ) )$ depends on the local behavior of $\ell ( { \mathbf { x } } , \cdot )$ around $\theta _ { 0 }$ (e.g., gradient magnitude), which can be drastically different across various initializations $\theta _ { 0 }$ . The lower bound $M \geq D$ is a quite restricting one, considering that real datasets often have thousands to even hundreds of thousands of dimensions (e.g., images). This analysis motivates us to avoid the limitation of using one GD step by extending to multiple steps in the next section.
+
+# 3.4 MULTIPLE GRADIENT DESCENT STEPS
+
+We extend Algorithm 1 to more than one gradient descent steps by changing Line 6 to multiple sequential GD steps on the same batch of distilled data, i.e., each step $i$ performs
+
+$$
+\theta _ { i + 1 } = \theta _ { i } - \tilde { \eta } _ { i } \nabla _ { \theta _ { i } } \ell ( \tilde { \mathbf { x } } , \theta _ { i } ) ,
+$$
+
+and changing Line 9 to backpropagate through all steps. We do not share the same learning rates across steps as later steps often require lower learning rates.
+
+Naively computing gradients is memory and computationally intensive. Therefore, we exploit a recent technique called back-gradient optimization, which allows for significantly faster gradient calculation in reverse-mode differentiation (Domke, 2012; Maclaurin et al., 2015). Specifically, back-gradient optimization formulates the necessary second-order terms into efficient Hessian-vector products (Pearlmutter, 1994), which can be easily calculated with modern automatic differentiation systems such as PyTorch (Paszke et al., 2017).
+
+# 3.5 DISTRIBUTION OF INITIAL WEIGHTS
+
+There is freedom in choosing the distribution of initial weights $p ( \theta _ { 0 } )$ . In this work, we explore the following four practical choices in the experiments:
+
+(a) MNIST. These distilled images can train unknown random initializations to $8 8 . 5 1 \% \pm 1 . 1 1 \%$ test accuracy in 2000 GD steps.
+
+![](images/0a9d824eb361ad8280563fe1315d8bbfe0b9ac40de2724c177b8ff475188136c.jpg)  
+(b) CIFAR10. These distilled images can train unknown random initializations to $4 1 . 2 3 \% \pm 0 . 8 8 \%$ test accuracy in $5 0 \mathrm { G D }$ steps.
+
+Figure 2: Distilled images trained for random initialization a batch of 100 distilled images (ten per class). Only 30 of 100 distilled images are shown here. Please see the appendix for the full result.
+
+• Random initialization: Distribution over random initial weights, e.g., He Initialization (He et al., 2015) and Xavier Initialization (Glorot and Bengio, 2010) for neural networks. • Fixed initialization: A particular fixed network initialized by the method above. • Random pre-trained weights: Distribution over models pre-trained on other tasks or datasets, e.g., ALEXNET (Krizhevsky et al., 2012) networks trained on ImageNet (Deng et al., 2009). • Fixed pre-trained weights: A particular fixed network pre-trained on other tasks and datasets.
+
+Distillation with pre-trained weights. Such learned distilled data essentially fine-tune weights pre-trained on one dataset to perform well for a new dataset, thus bridging the gap between the two domains. Domain mismatch and dataset bias represent a challenging problem in machine learning (Torralba and Efros, 2011; Daume III, 2007; Saenko et al., 2010). In this work, we characterize the domain mismatch via distilled data. In Section 4.1.2, we show that a small number of distilled images are sufficient to quickly adapt convolutional neural network (CNN) models to new datasets and tasks.
+
+# 3.6 APPLICATION TO CONTINUAL LEARNING
+
+To guard against domain shift, several continual learning methods store a subset of training samples in a small memory buffer, and restrict future updates to maintain reasonable performance on these stored samples (Rebuffi et al., 2017; Kirkpatrick et al., 2017; Lopez-Paz and Ranzato, 2017; Nguyen et al., 2018). As our distilled images contain rich information about the past training data and task, they could naturally serve as a compressed memory of the past.
+
+To test this, we modify a recent continual learning method called Gradient Episodic Memory (GEM) (Lopez-Paz and Ranzato, 2017). GEM enforces inequality constraints such that the new model, after being trained on the new data and task, should perform at least as well as the old model on the previously stored data and tasks. Here, we store our distilled data for each task instead of randomly drawn training samples as used in GEM. We use the distilled data to construct inequality constraints, and solve the optimization using quadratic programming, same as in GEM. As shown in Section 4.2, our method compares favorably against several baselines that rely on real images.
+
+# 4 EXPERIMENTS
+
+In this section, we report experiments of regular image classifications on MNIST (LeCun, 1998) and CIFAR10 (Krizhevsky and Hinton, 2009), adaptation from ImageNet (Deng et al., 2009) to PASCAL-VOC (Everingham et al., 2010) and CUB-200 (Wah et al., 2011), and continual learning on permuted MNIST and CIFAR100.
+
+Baselines. For each experiment, in addition to baselines specific to the setting, we generally compare our method against baselines trained with data derived or selected from real training images:
+
+• Random real images: We randomly sample the same number of real images per category.   
+• Optimized real images: We sample different sets of random real images as above, and choose the top $2 0 \%$ best performing sets.   
+• $k$ -means $^ { + + }$ : We apply $k$ -means $^ { + + }$ (Arthur and Vassilvitskii, 2007) clustering to each category, and extract the cluster centroids.
+
+![](images/1b5e9553885791db954a1e14c31f2b9ac8a5ecec65fe56c457d907cedfc3c31e.jpg)
+
+Figure 3: Distillation performance with varying numbers of GD steps and a fixed number of distilled images.   
+
+<table><tr><td rowspan=3 colspan=1></td><td rowspan=1 colspan=2>Ours</td><td rowspan=1 colspan=6>Baselines</td></tr><tr><td rowspan=2 colspan=1>Fixed init.</td><td rowspan=2 colspan=1>Random init.</td><td rowspan=1 colspan=4>Used as training data in CNN</td><td rowspan=1 colspan=2>Used in KNN classification</td></tr><tr><td rowspan=1 colspan=1>Random real</td><td rowspan=1 colspan=1>Optimized real</td><td rowspan=1 colspan=1>k-means++</td><td rowspan=1 colspan=1>Average real</td><td rowspan=1 colspan=1>Random real</td><td rowspan=1 colspan=1>k-means++</td></tr><tr><td rowspan=1 colspan=1>MNIST</td><td rowspan=1 colspan=1>94.4</td><td rowspan=1 colspan=1>88.5± 1.1</td><td rowspan=1 colspan=1>82.8± 1.8</td><td rowspan=1 colspan=1>83.8± 2.1</td><td rowspan=1 colspan=1>86.7± 1.4</td><td rowspan=1 colspan=1>77.7 ± 2.7</td><td rowspan=1 colspan=1>71.5 ± 2.1</td><td rowspan=1 colspan=1>92.4±0.2</td></tr><tr><td rowspan=1 colspan=1>CIFAR10</td><td rowspan=1 colspan=1>45.2</td><td rowspan=1 colspan=1>41.2±0.9</td><td rowspan=1 colspan=1>24.8 ± 1.5</td><td rowspan=1 colspan=1>24.9 ± 1.4</td><td rowspan=1 colspan=1>26.7± 1.8</td><td rowspan=1 colspan=1>22.8±0.8</td><td rowspan=1 colspan=1>18.8± 1.3</td><td rowspan=1 colspan=1>29.4± 0.4</td></tr></table>
+
+Table 1: Comparison between our method and various baselines. All methods use ten images per category (100 in total), except for the average real images baseline, which reuses the same images in different GD steps. For MNIST, our method uses $2 0 0 0 \mathrm { G D }$ steps, and baselines use the best among #steps $\in \{ 1 , 1 0 0 , 5 0 0 , 1 0 0 0 , 2 0 0 0 \}$ . For CIFAR10, our method uses $5 0 \mathrm { G D }$ steps, and baselines use the best among #steps $\in \{ 1 , 5 , 1 0 , 2 0 , 5 0 0 \}$ . In addition, we include a K-nearest neighbors (KNN) baseline, and report best results among all combinations of distance metric $\in \{ l _ { 1 } , l _ { 2 } \}$ and one or three neighbors.
+
+• Average real images: We compute the average image for each category.
+
+Please see the appendix for more details about training and baselines, and additional results.
+
+# 4.1 DATASET DISTILLATION
+
+We first present experimental results on training classifiers either from scratch or adapting from pre-trained weights. For MNIST, the distilled images are trained with LENET (LeCun et al., 1998), which achieves about $9 9 \%$ test accuracy if conventionally trained. For CIFAR10, we use a network architecture (Krizhevsky, 2012) that achieves around $8 0 \%$ test accuracy if conventionally trained. For ImageNet adaptations, we use an ALEXNET (Krizhevsky et al., 2012). We use 2000 GD steps for MNIST and $5 0 \mathrm { G D }$ steps for CIFAR10. For random initializations and random pre-trained weights, we report means and standard deviations over 200 held-out models, unless otherwise stated.
+
+For baselines, we perform each evaluation on 200 held-out models using all possible combinations of learning rate $\in \{$ distilled learning rates $\tilde { \eta } ^ { * }$ , 1e-3, 3e-3, 1e-2, 3e-2, 1e-1, 3e-1} and several choices of numbers of training GD steps (see table captions for details), and report results with the best performing combination.
+
+# 4.1.1 DISTILLATION WITH WEIGHTS SAMPLED FRO NETWORK INITIALIZATION
+
+Fixed initialization. With access to initial network weights, distilled images can directly train a fixed network to reach high performance. Experiment results show that just 10 distilled images (one per class) can boost the performance of a LENET with an initial accuracy $8 . 2 5 \%$ to a final accuracy of $9 3 . 8 2 \%$ on MNIST in $2 0 0 0 \mathrm { G D }$ steps. Using 100 distilled images (ten per class) can raise the final accuracy can be raised to $9 4 . 4 1 \%$ , as shown in the first column of Table 1. Similarly, 100 distilled images can train a network with an initial accuracy $1 0 . 7 5 \%$ to test accuracy of $4 5 . 1 5 \%$ on CIFAR10 in $5 0 \mathrm { G D }$ steps.
+
+Random initialization. Figure 2 distilled images trained with randomly sampled initializations using Xavier Initialization (Glorot and Bengio, 2010). While the resulting average test accuracy from these images are not as high as those for fixed initialization, these distilled images crucially do not require a specific initial point, and thus could potentially generalize to a much wider range of starting points. In Section 4.2 below, we present preliminary results of achieving nontrivial gains from applying such distilled images to classifier networks during a continual learning training process.
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Oursw/fixedpre-trained</td><td rowspan=1 colspan=1>Oursw/randompre-trained</td><td rowspan=1 colspan=1>Random real</td><td rowspan=1 colspan=1>Optimized real</td><td rowspan=1 colspan=1>k-means++</td><td rowspan=1 colspan=1>Average real</td><td rowspan=1 colspan=1>Few-shotadaptationMotiian et al. (2017)</td><td rowspan=1 colspan=1>No adaptation</td><td rowspan=1 colspan=1>Train on fulltarget dataset</td></tr><tr><td rowspan=1 colspan=1>M→u</td><td rowspan=1 colspan=1>97.9</td><td rowspan=1 colspan=1>95.4±1.8</td><td rowspan=1 colspan=1>94.9 ±0.8</td><td rowspan=1 colspan=1>95.2± 0.7</td><td rowspan=1 colspan=1>94.8±0.7</td><td rowspan=1 colspan=1>93.9±0.8</td><td rowspan=1 colspan=1>96.7±0.5</td><td rowspan=1 colspan=1>90.4±3.0</td><td rowspan=1 colspan=1>97.3±0.3</td></tr><tr><td rowspan=1 colspan=1>U→M</td><td rowspan=1 colspan=1>93.2</td><td rowspan=1 colspan=1>92.7 ±1.4</td><td rowspan=1 colspan=1>87.1 ± 2.9</td><td rowspan=1 colspan=1>87.6 ± 2.1</td><td rowspan=1 colspan=1>88.0± 2.2</td><td rowspan=1 colspan=1>78.4 ± 5.0</td><td rowspan=1 colspan=1>89.2 ± 2.4</td><td rowspan=1 colspan=1>67.5±3.9</td><td rowspan=1 colspan=1>98.6± 0.5</td></tr><tr><td rowspan=1 colspan=1>S→M</td><td rowspan=1 colspan=1>96.2</td><td rowspan=1 colspan=1>85.2± 4.7</td><td rowspan=1 colspan=1>84.6 ± 2.1</td><td rowspan=1 colspan=1>85.2 ± 1.2</td><td rowspan=1 colspan=1>86.5± 1.2</td><td rowspan=1 colspan=1>74.9 ± 2.6</td><td rowspan=1 colspan=1>74.0 ± 1.5</td><td rowspan=1 colspan=1>51.6 ± 2.8</td><td rowspan=1 colspan=1>98.6±0.5</td></tr></table>
+
+Table 2: Adapting models among MNIST $( \mathcal { M } )$ , USPS $( \mathcal { U } )$ , and SVHN $( S )$ using 100 distilled images. Our method outperforms few-shot domain adaptation (Motiian et al., 2017) and other baselines in most settings. Due to computation limitations, the 100 distilled images are split into 10 minibatches applied in 10 sequential GD steps, and the entire set of 100 distilled images is iterated through 3 times $\mathrm { 3 0 G D }$ steps in total). For baselines, we train the model using the same number of images with $\{ 1 , 3 , 5 \}$ times and report the best result.
+
+Table 3: Adapting an ALEXNET pre-trained on ImageNet to PASCAL-VOC and CUB-200. We use one distilled image per category, repeatedly applied via three GD steps. Our method significantly outperforms the baselines. For baselines, we train the model with $\{ 1 , 3 , 5 \}$ GD steps and report the best. Results are over 10 runs.   
+
+<table><tr><td rowspan=1 colspan=1>Target dataset</td><td rowspan=1 colspan=1>Ours</td><td rowspan=1 colspan=1>Random real</td><td rowspan=1 colspan=1>Optimized real</td><td rowspan=1 colspan=1>Average real</td><td rowspan=1 colspan=1>Fine-tune on fulltarget dataset</td></tr><tr><td rowspan=1 colspan=1>PASCAL-VOC</td><td rowspan=1 colspan=1>70.75</td><td rowspan=1 colspan=1>19.41 ± 3.73</td><td rowspan=1 colspan=1>23.82± 3.66</td><td rowspan=1 colspan=1>9.94</td><td rowspan=1 colspan=1>75.57±0.18</td></tr><tr><td rowspan=1 colspan=1>CUB-200</td><td rowspan=1 colspan=1>38.76</td><td rowspan=1 colspan=1>7.11 ± 0.66</td><td rowspan=1 colspan=1>7.23± 0.78</td><td rowspan=1 colspan=1>2.88</td><td rowspan=1 colspan=1>41.21 ± 0.51</td></tr></table>
+
+Multiple gradient descent steps. Section 3.3 has shown theoretical limitations of using only one step in a simple linear case. In Figure 3, we empirically verify for deep networks that using multiple steps drastically outperforms the single step method, given the same number of distilled images. Table 1 summarizes the results of our method and all baselines. Our method with both fixed and random initializations outperforms all the baselines on CIFAR10 and most of the baselines on MNIST.
+
+# 4.1.2 DISTILLATION WITH PRE-TRAINED INITIAL WEIGHTS
+
+Next, we show the extended setting of our algorithm discussed in Section 3.5, where the weights are not randomly initialized but pre-trained on a particular dataset. In this section, for random initial weights, we train the distilled images on 2000 pre-trained models and evaluate them on 200 unseen models.
+
+Fixed and random pre-trained weights on digits. As shown in Section 3.5, we can optimize distilled images to quickly fine-tune pre-trained models on a new dataset. Table 2 shows that our method is more effective than various baselines on adaptation between three digits datasets: MNIST, USPS (Hull, 1994), and SVHN (Netzer et al., 2011). We also compare our method against a stateof-the-art few-shot domain adaptation method (Motiian et al., 2017). Although our method uses the entire training set to compute the distilled images, both methods use the same number of images to distill the knowledge of target dataset. Prior work (Motiian et al., 2017) is outperformed by our method with fixed pre-trained weights on all the tasks, and by our method with random pre-trained weights on two of the three tasks. This result shows that our distilled images effectively compress the information of target datasets.
+
+Fixed pre-trained ALEXNET to PASCAL-VOC and CUB-200. In Table 3, we adapt a widely used ALEXNET model pre-trained on ImageNet to image classification on PASCAL-VOC and CUB-200 datasets. Given only one distilled image per category, our method outperforms various baselines significantly. Our method is on par with fine-tuning on the full datasets with thousands of images.
+
+# 4.2 APPLICATION TO CONTINUAL LEARNING
+
+We modify Gradient Episodic Memory (GEM) (Lopez-Paz and Ranzato, 2017) to store distilled data for each task rather than real training images. Experiments in Lopez-Paz and Ranzato (2017) use large memory buffers, up to $2 5 \%$ of the training set. Instead, we focus on a more realistic scenario where the buffer is rather small $( \leq 1 \%$ of the training set). Following the experiment settings and architecture choices from Lopez-Paz and Ranzato (2017), we consider two continual learning tasks:
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Permuted MNIST</td><td rowspan=1 colspan=1>CIFAR100</td></tr><tr><td rowspan=3 colspan=1>Memory size per task = 10</td><td rowspan=1 colspan=1>iCaRL (Rebuffi et al., 2017)</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>42.4</td></tr><tr><td rowspan=1 colspan=1>GEM (Lopez-Paz and Ranzato,2017)</td><td rowspan=1 colspan=1>67.4</td><td rowspan=1 colspan=1>43.8</td></tr><tr><td rowspan=1 colspan=1>GEM + Ours</td><td rowspan=1 colspan=1>75.6</td><td rowspan=1 colspan=1>52.8</td></tr><tr><td rowspan=2 colspan=1>Memory size per task = 40</td><td rowspan=1 colspan=1>iCaRL</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>45.8</td></tr><tr><td rowspan=1 colspan=1>GEM</td><td rowspan=1 colspan=1>75.3</td><td rowspan=1 colspan=1>51.6</td></tr><tr><td rowspan=2 colspan=1>Memory size per task = 50</td><td rowspan=1 colspan=1>iCaRL</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>46.9</td></tr><tr><td rowspan=1 colspan=1>GEM</td><td rowspan=1 colspan=1>75.8</td><td rowspan=1 colspan=1>52.4</td></tr><tr><td rowspan=1 colspan=1>No memory buffer</td><td rowspan=1 colspan=1>EWC (Kirkpatrick et al., 2017)</td><td rowspan=1 colspan=1>63.5</td><td rowspan=1 colspan=1>45.6</td></tr></table>
+
+Table 4: Continual learning results. Distilled images are trained with random Xavier Initialization distribution.   
+For permuted MNIST, they are trained with 2000 GD steps. For CIFAR100, they are trained for 200 GD steps.
+
+• Permuted MNIST: 20 classification tasks each formed by using a different permutation to arrange pixels from MNIST images. Each task contains 1, 000 training images. The classifier used has 2 hidden layers each with 100 neurons.
+
+• CIFAR100: 20 classification tasks formed by splitting the 100 classes into 20 equal subsets of 5 classes. Each task contains 2, 500 training images. The classifier used is RESNET18 (He et al., 2016).
+
+Table 4 shows that using distilled data drastically improves final overall accuracy on all tasks, and reduces buffer size by up to $5 \times$ compared to the original GEM that uses real images. We only report the basic iCaRL (Rebuffi et al., 2017) setting on CIFAR100 because it requires similar input distributions across all tasks, and it is unclear how to properly inject distilled images into its specialized examplar selection procedure.
+
+The appendix details the hyper-parameters tested for each continual learning algorithm.
+
+# 5 DISCUSSION AND LIMITATIONS
+
+In this paper, we have presented dataset distillation for compressing the knowledge of entire training data into a few synthetic training images. We demonstrate how to train a network to reach surprisingly good performance with only a small number of distilled images. Finally, the distilled images can efficiently store the memory of previous tasks in the continual learning setting.
+
+Many challenges remain for knowledge distillation of data. Although our method generalizes well to random initializations, it is still limited to a particular network architecture. Since loss surfaces for different architectures might be drastically different, a more flexible method of applying the distilled data may overcome this difficulty. Another limitation is the increasing computation and memory requirements for finding the distilled data as the number of images and steps increases. To compress large-scale datasets such as ImageNet, we may need first-order gradient approximations to make the optimization computationally feasible.
+
+Nonetheless, we are encouraged by the findings in this paper on the possibilities of training large models with a few distilled data, leading to potential applications such as accelerating network evaluation in neural architecture search (Zoph and Le, 2017). We believe that the ideas developed in this work might give new insights into the quantity and type of data that deep networks are able to process, and hopefully inspire others to think along this direction.
+
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+M. McCloskey and N. J. Cohen. Catastrophic interference in connectionist networks: The sequential learning problem. In Psychology of learning and motivation, volume 24, pages 109–165. 1989.   
+S. Motiian, Q. Jones, S. Iranmanesh, and G. Doretto. Few-shot adversarial domain adaptation. In NIPS, 2017.   
+Y. Netzer, T. Wang, A. Coates, A. Bissacco, B. Wu, and A. Y. Ng. Reading digits in natural images with unsupervised feature learning. In NIPS workshop, 2011.   
+C. V. Nguyen, Y. Li, T. D. Bui, and R. E. Turner. Variational continual learning. In ICLR, 2018.   
+J. A. Olvera-López, J. A. Carrasco-Ochoa, J. F. Martínez-Trinidad, and J. Kittler. A review of instance selection methods. Artificial Intelligence Review, 34(2):133–143, 2010.   
+A. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer. Automatic differentiation in pytorch. In ICLR Workshop, 2017.   
+B. A. Pearlmutter. Fast exact multiplication by the hessian. Neural computation, 6(1):147–160, 1994.   
+F. Pedregosa. Hyperparameter optimization with approximate gradient. In ICML, 2016.   
+J. Ponce, T. L. Berg, M. Everingham, D. A. Forsyth, M. Hebert, S. Lazebnik, M. Marszalek, C. Schmid, B. C. Russell, A. Torralba, et al. Dataset issues in object recognition. In Toward category-level object recognition. 2006.   
+I. Radosavovic, P. Dollár, R. Girshick, G. Gkioxari, and K. He. Data distillation: Towards omni-supervised learning. In CVPR, 2018.   
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+
+# S-1 EXPERIMENT DETAILS
+
+In our experiments, we disable dropout layers in the networks due to the randomness and computational cost they introduce in distillation. Moreover, we initialize the distilled learning rates with a constant between 0.001 and 0.02 depending on the task, and use the Adam solver (Kingma and Ba, 2015) with a learning rate of 0.001. For random initialization and random pre-trained weights, we sample 4 to 16 initial weights in each optimization step. We run all the experiments on NVIDIA 1080 Ti, 2080 Ti, Titan $\mathrm { X p }$ , and V100 GPUs. We use one GPU for fixed initial weights and up to four GPUs for random initial weights. Each training typically takes 1 to 6 hours.
+
+Below we describe the details of our baselines using real training images.
+
+• Random real images: We randomly sample the same number of real images per category. We evaluate the performance over 10 randomly sampled sets. Optimized real images: We sample 50 sets of real images using the procedure above, pick 10 sets that achieve the best performance on 20 held-out models and 1024 randomly chosen training images, and evaluate the performance of these 10 sets.   
+• $k$ -means++: For each category, we use $k$ -means $^ { + + }$ (Arthur and Vassilvitskii, 2007) clustering to extract the same number of cluster centroids as the number of distilled images in our method. We evaluate the method over 10 runs.   
+• Average real images: We compute the average image of all the images in each category, which is repeated to match the same total number of images. We evaluate the model only once because average images are deterministic.
+
+To enforce our optimized learning rate to be positive, we apply softplus to a scalar trained parameter. For continual learning experiment on CIFAR10 dataset, to compare with GEM (Lopez-Paz and Ranzato, 2017), we replace the Batch normalization (Wu and He, 2018) with Group normalization (Ioffe and Szegedy, 2015) in RESNET18 (He et al., 2016), as it is difficult to run back-gradient optimization through batch norm running statistics. For a fair comparison, we use the same architecture for our method and other baselines.
+
+For dataset distillation experiments with pre-trained initial weights, distilled images are initialized with $\mathcal { N } ( 0 , 1 )$ at the beginning of training. For other experiments, distilled images are initialized with random real samples, unless otherwise stated.
+
+# S-1.1 CONTINUAL LEARNING EXPERIMENT DETAILS
+
+For the compared continual learning methods, we report the best report from the following combinations of hyper-parameters:
+
+• GEM: – $\gamma \in \{ 0 , 0 . 1 , 0 . 2 , 0 . 3 , 0 . 4 , 0 . 5 , 0 . 6 , 0 . 7 , 0 . 8 , 0 . 9 , 1 \} .$ – learning rate $= 0 . 1$ .
+
+• iCARL: – regularization $\in \{ 0 . 0 0 1 , 0 . 0 0 3 , 0 . 0 1 , 0 . 0 3 , 0 . 1 , 0 . 3 , 1 . 0 \} .$ – learning rate $= 0 . 1$ .
+
+# S-2 ADDITIONAL EXPERIMENTS RESULTS
+
+• Figures 4 and 5 show distilled images trained for random initializations on MNIST and CIFAR10.   
+• Figures 6, 7, and 8 show distilled images trained for adapting random pre-trained models on digits datasets including MNIST, USPS, and SVHN.
+
+![](images/4209657f2c33c19e198b09c8cbc393418f574a497c85796f01fc2729127a09f8.jpg)  
+Figure 4: Dataset distillation for random initializations on MNIST. This batch of 100 distilled images are repeatedly applied in $2 0 0 0 \mathrm { G D }$ steps.. These images train average test accuracy to $8 8 . 5 1 \% \pm 1 . 1 1 \%$ .
+
+![](images/24b32ec4cd0387ba959edbec8c823bf0fbad4161bbd7ac0bd20add99a88af9e4.jpg)  
+Figure 5: Dataset distillation for random initializations on CIFAR10. This batch of 100 distilled images are repeatedly applied in $5 0 \mathrm { G D }$ steps. These images train average test accuracy to $4 1 . 2 3 \% \pm 0 . 8 8 \%$ .
+
+![](images/b949921ddccd833f8d916579ad504ea9b666f832339d3f6930c379883ef0585f.jpg)  
+Figure 6: Dataset distillation for adapting random pretrained models from USPS to MNIST. 100 distilled images are split into $1 0 \mathrm { G D }$ steps, shown as 10 rows here. Top row is the earliest GD step, and bottom row is the last. The 10 steps are iterated over three times to finish adaptation, leading to a total of $3 0 \mathrm { G D }$ steps. These images train average test accuracy on 200 held out models from $6 7 . 5 4 \% \pm 3 . 9 1 \%$ to $9 2 . 7 4 \% \pm 1 . 3 8 \%$ .
+
+![](images/d296fc0060d9496141a27d0d5f986acbdf8b184d2aed4c093b3a2067bde87cca.jpg)  
+Figure 7: Dataset distillation for adapting random pretrained models from MNIST to USPS. 100 distilled images are split into $1 0 \mathrm { G D }$ steps, shown as 10 rows here. Top row is the earliest GD step, and bottom row is the last. The 10 steps are iterated over three times to finish adaptation, leading to a total of $3 0 \mathrm { G D }$ steps. These images train average test accuracy on 200 held out models from $9 0 . 4 3 \% \pm 2 . 9 7 \%$ to $9 5 . 3 8 \% \pm 1 . 8 1 \%$ .
+
+![](images/095b3f814a2782b7b4960b8fd1b03c044205c281667dffaf5a13394b53e2190b.jpg)  
+Figure 8: Dataset distillation for adapting random pretrained models from SVHN to MNIST. 100 distilled images are split into $1 0 \mathrm { G D }$ steps, shown as 10 rows here. Top row is the earliest GD step, and bottom row is the last. The 10 steps are iterated over three times to finish adaptation, leading to a total of $3 0 \mathrm { G D }$ steps. These images train average test accuracy on 200 held out models from $5 1 . 6 4 \% \pm 2 . 7 7 \%$ to $8 5 . 2 1 \% \pm 4 . 7 3 \%$ .

parse/train/ryxO3gBtPB/ryxO3gBtPB_content_list.json

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+[
+    {
+        "type": "text",
+        "text": "DATASET DISTILLATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
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+            398,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            210,
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+            224
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Model distillation aims to distill the knowledge of a complex model into a simpler one. In this paper, we consider an alternative formulation called dataset distillation: we keep the model fixed and instead attempt to distill the knowledge from a large training dataset into a small one. The idea is to synthesize a small number of data points that do not need to come from the correct data distribution, but will, when given to the learning algorithm as training data, approximate the model trained on the original data. For example, we show that it is possible to compress 60, 000 MNIST training images into just 10 synthetic distilled images (one per class) and achieve close to the original performance, given a fixed network initialization. We evaluate our method in various initialization settings. Experiments on multiple datasets, MNIST, CIFAR10, PASCAL-VOC, and CUB-200, demonstrate the advantage of our approach compared to alternative methods. Finally, we include a real-world application of dataset distillation to the continual learning setting: we show that storing distilled images as episodic memory of previous tasks can alleviate forgetting more effectively than real images. ",
+        "bbox": [
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+            242,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            481,
+            334,
+            496
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Hinton et al. (2015) proposed network distillation as a way to transfer the knowledge from an ensemble of many separately-trained networks into a single, typically compact network, performing a type of model compression. In this paper, we are considering a related but orthogonal task: rather than distilling the model, we propose to distill the dataset. Unlike network distillation, we keep the model fixed but encapsulate the knowledge of the entire training dataset, which typically contains thousands to millions of images, into a small number of synthetic training images. We show that we can go as low as one synthetic image per category, training the same model to reach surprisingly good performance on these synthetic images. For example, in Figure 1a, we compress 60, 000 training images of MNIST digit dataset into only 10 synthetic images (one per category), given a fixed network initialization. Training the standard LENET (LeCun et al., 1998) on these 10 images yields test-time MNIST recognition performance of $9 4 \\%$ , compared to $9 9 \\%$ for the original dataset. For networks with unknown random weights, 100 synthetic images train to $8 9 \\%$ . We name our method Dataset Distillation and these images distilled images. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "But why is dataset distillation interesting? First, there is the purely scientific question of how much data is encoded in a given training set and how compressible it is? Second, we wish to know whether it is possible to “load up\" a given network with an entire dataset-worth of knowledge by a handful of images. This is in contrast to traditional training that often requires tens of thousands of data samples. Finally, on the practical side, dataset distillation enables applications that require compressing data with its task. We demonstrate that under the continual learning setting, storing distilled images as memory of past task and data can alleviate catastrophic forgetting (McCloskey and Cohen, 1989). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A key question is whether it is even possible to compress a dataset into a small set of synthetic data samples. For example, is it possible to train an image classification model on synthetic images that are not on the manifold of natural images? Conventional wisdom would suggest that the answer is no, as the synthetic training data may not follow the same distribution of the real test data. Yet, in this work, we show that this is indeed possible. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We present an optimization algorithm for synthesizing a small number of synthetic data samples not only capturing much of the original training data but also tailored explicitly for fast model training with only a few data point. To achieve our goal, we first derive the network weights as a differentiable function of our synthetic training data. Given this connection, instead of optimizing the network weights for a particular training objective, we optimize the pixel values of our distilled images. However, this formulation requires access to the initial weights of the network. To relax this assumption, we develop a method for generating distilled images for randomly initialized networks. To further boost performance, we propose an iterative version, where the same distilled images are reused over multiple gradient descent steps so that the knowledge can be fully transferred into the model. Finally, we study a simple linear model, deriving a lower bound on the size of distilled data required to achieve the same performance as training on the full dataset. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/a3349f9c1635457a4bda7e72a8f70ee5582b7484123d58a89342b2223f0666cc.jpg",
+        "image_caption": [
+            "Figure 1: We distill the knowledge of tens of thousands of images into a few synthetic training images called distilled images. On MNIST, 100 distilled images can train a standard LENET with a random initialization to $8 9 \\%$ test accuracy, compared to $9 9 \\%$ when fully trained. On CIFAR10, 100 distilled images can train a network with a random initialization to $4 1 \\%$ test accuracy, compared to $8 0 \\%$ when fully trained. In Section 3.6, we show that these distilled images can efficiently store knowledge of previous tasks for continual learning. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            202,
+            89,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            380,
+            825,
+            491
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We demonstrate that a handful of distilled images can be used to train a model with a fixed initialization to achieve surprisingly high performance. For networks pre-trained on other tasks, our method can find distilled images for fast model fine-tuning. We test our method on several initialization settings: fixed initialization, random initialization, fixed pre-trained weights, and random pre-trained weights. Extensive experiments on four publicly available datasets, MNIST, CIFAR10, PASCAL-VOC, and CUB-200, show that our approach often outperforms existing methods. Finally, we demonstrate that for continual learning methods that store limited-size past data samples as episodic memory (Lopez-Paz and Ranzato, 2017; Kirkpatrick et al., 2017), storing our distilled data instead is much more effective. Our distilled images contain richer information about the past data and tasks, and we show experimental evidence on standard continual learning benchmarks. Our code, data, and models will be available upon publication. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            674,
+            339,
+            690
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Knowledge distillation. The main inspiration for this paper is network distillation (Hinton et al., 2015), a widely used technique in ensemble learning (Radosavovic et al., 2018) and model compression (Ba and Caruana, 2014; Romero et al., 2015; Howard et al., 2017). While network distillation aims to distill the knowledge of multiple networks into a single model, our goal is to compress the knowledge of an entire dataset into a few synthetic data. Our method is also related to the theoretical concept of teaching dimension, which specifies the minimal size of data needed to teach a target model to a learner (Shinohara and Miyano, 1991; Goldman and Kearns, 1995). However, methods (Zhu, 2013; 2015) inspired by this concept require the existence of target models, which our method does not. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Dataset pruning, core-set construction, and instance selection. Another way to distill knowledge is to summarize the entire dataset by a small subset, either by only using the “valuable” data for model training (Angelova et al., 2005; Felzenszwalb et al., 2010; Lapedriza et al., 2013) or by only labeling the “valuable” data via active learning (Cohn et al., 1996; Tong and Koller, 2001). Similarly, core-set construction (Tsang et al., 2005; Har-Peled and Kushal, 2007; Bachem et al., 2017; Sener and Savarese, 2018) and instance selection (Olvera-López et al., 2010) methods aim to select a subset of the entire training data, such that models trained on the subset will perform as well as the model trained on the full dataset. For example, solutions to many classical linear learning algorithms, e.g., Perceptron (Rosenblatt, 1957) and SVMs (Hearst et al., 1998), are weighted sums of subsets of training examples, which can be viewed as core-sets. However, algorithms constructing these subsets require many more training examples per category than we do, in part because their “valuable” images have to be real, whereas our distilled images are exempt from this constraint. ",
+        "bbox": [
+            174,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
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+            188
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Gradient-based hyperparameter optimization. Our work bears similarity with gradient-based hyperparameter optimization techniques, which compute the gradient of hyperparameter w.r.t. the final validation loss by reversing the entire training procedure (Bengio, 2000; Domke, 2012; Maclaurin et al., 2015; Pedregosa, 2016). We also backpropagate errors through optimization steps. However, we use only training set data and focus more heavily on learning synthetic training data rather than tuning hyperparameters. To our knowledge, this direction has only been slightly touched on previously (Maclaurin et al., 2015). We explore it in greater depth and demonstrate the idea of dataset distillation in various settings. More crucially, our distilled images work well across random initialization weights, not possible by prior work. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Understanding datasets. Researchers have presented various approaches for understanding and visualizing learned models (Zeiler and Fergus, 2014; Zhou et al., 2015; Mahendran and Vedaldi, 2015; Bau et al., 2017; Koh and Liang, 2017). Unlike these approaches, we are interested in understanding the intrinsic properties of the training data rather than a specific trained model. Analyzing training datasets has, in the past, been mainly focused on the investigation of bias in datasets (Ponce et al., 2006; Torralba and Efros, 2011). For example, Torralba and Efros (2011) proposed to quantify the “value” of dataset samples using cross-dataset generalization. Our method offers a different perspective for understanding datasets by distilling full datasets into a few synthetic samples. ",
+        "bbox": [
+            174,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 FORMULATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            457,
+            331,
+            473
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Given a model and a dataset, we aim to obtain a new, much-reduced synthetic dataset which performs almost as well as the original dataset. We first present our main optimization algorithm for training a network with a fixed initialization with one gradient descent (GD) step (Section 3.1). In Section 3.2, we derive the resolution to a more challenging case, where initial weights are random rather than fixed. In Section 3.3, we further study a linear network case to help readers understand both the properties and limitations of our method. We also discuss the distribution of initial weights with which our method can work well. In Section 3.4, we extend our approach to reuse the same distilled images over 2, 000 gradient descent steps and largely improve the performance. Finally, Section 3.5 discusses dataset distillation for different initialization distributions. Finally, in Section 3.6, we show that our distilled images can be used as effective episodic memory for continual learning tasks. ",
+        "bbox": [
+            174,
+            488,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Consider a training dataset $\\mathbf { x } = \\{ x _ { i } \\} _ { i = 1 } ^ { N }$ , we parameterize our neural network as $\\theta$ and denote $\\ell ( x _ { i } , \\theta )$ as the loss function that represents the loss of this network on a data point $x _ { i }$ . Our task is to find the minimizer of the empirical error over entire training data: ",
+        "bbox": [
+            174,
+            633,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6fc7f25027a062fdadbb802b5b9b72fc1913009da9f0decffa389e09a03d389d.jpg",
+        "text": "$$\n\\theta ^ { * } = \\underset { \\theta } { \\arg \\operatorname* { m i n } } \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\ell ( x _ { i } , \\theta ) \\triangleq \\underset { \\theta } { \\arg \\operatorname* { m i n } } \\ell ( \\mathbf { x } , \\theta ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            338,
+            679,
+            658,
+            722
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where for notation simplicity we overload the $\\ell ( \\cdot )$ notation so that $\\ell ( \\mathbf { x } , \\theta )$ represents the average error of $\\theta$ over the entire dataset. We make the mild assumption that $\\ell$ is twice-differentiable, which holds true for the majority of modern machine learning models and tasks. ",
+        "bbox": [
+            174,
+            724,
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+            767
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 OPTIMIZING DISTILLED DATA ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            784,
+            419,
+            797
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Standard training usually applies minibatch stochastic gradient descent or its variants. At each step $t$ , a minibatch of training data $\\mathbf { x } _ { t } = \\{ x _ { t , j } \\} _ { j = 1 } ^ { n }$ is sampled to update the current parameters as ",
+        "bbox": [
+            173,
+            809,
+            825,
+            839
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5eb7a074ee2abe3683ef2053d243894d69af7508bf2deb780949de5f5e381590.jpg",
+        "text": "$$\n\\theta _ { t + 1 } = \\theta _ { t } - \\eta \\nabla _ { \\theta _ { t } } \\ell ( \\mathbf { x } _ { t } , \\theta _ { t } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            406,
+            843,
+            588,
+            859
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where is the learning rate. Such a training process often takes tens of thousands or even millions of update steps to converge. Instead, we learn a tiny set of synthetic distilled training data $\\tilde { \\mathbf { x } } = \\{ \\tilde { x } _ { i } \\} _ { i = 1 } ^ { M }$ with and a corresponding learning rate $\\tilde { \\eta }$ so that a single GD step such as ",
+        "bbox": [
+            174,
+            863,
+            825,
+            905
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a5f61ac2eb4472d573380be11a3bf9c3a611a647691030d6bdfe76da55a2356d.jpg",
+        "text": "$$\n\\theta _ { 1 } = \\theta _ { 0 } - \\tilde { \\eta } \\nabla _ { \\theta _ { 0 } } \\ell ( \\tilde { \\mathbf { x } } , \\theta _ { 0 } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            418,
+            909,
+            580,
+            925
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 Dataset Distillation ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Input: $p ( \\theta _ { 0 } )$ : distribution of initial weights; $M$ : the number of distilled data   \nInput: $\\alpha$ : step size; $n$ : batch size; $T$ : the number of optimization iterations; $\\tilde { \\eta } _ { 0 }$ : initial value for $\\tilde { \\eta }$   \n1: Initialize $\\hat { \\tilde { \\mathbf { x } } } = \\{ \\tilde { x } _ { i } \\} _ { i = 1 } ^ { M }$ either from $\\mathcal { N } ( 0 , I )$ or from real training images. Initialize $\\tilde { \\eta }  \\tilde { \\eta } _ { 0 }$   \n2: for each training step $t = 1$ to $T$ do   \n3: Get a minibatch of real training data $\\mathbf { x } _ { t } = \\{ x _ { t , j } \\} _ { j = 1 } ^ { n }$   \n4: Sample a batch of initial weights $\\theta _ { 0 } ^ { ( j ) } \\sim p ( \\theta _ { 0 } )$   \n5: for each sampled $\\theta _ { 0 } ^ { ( j ) }$ do   \n6: Compute updated parameter with GD: $\\theta _ { 1 } ^ { ( j ) } = \\theta _ { 0 } ^ { ( j ) } - \\tilde { \\eta } \\nabla _ { \\theta _ { 0 } ^ { ( j ) } } \\ell ( \\tilde { \\mathbf { x } } , \\theta _ { 0 } ^ { ( j ) } )$   \n7: Evaluate the objective function on real training data: $\\mathscr { L } ^ { ( j ) } = \\ell ( \\mathbf { x } _ { t } , \\boldsymbol { \\theta } _ { 1 } ^ { ( j ) } )$   \n8: end for   \n9: Update $\\begin{array} { r } { \\tilde { \\mathbf { x } }  \\tilde { \\mathbf { x } } - \\alpha \\nabla _ { \\tilde { \\mathbf { x } } } \\sum _ { j } \\mathcal { L } ^ { ( j ) } , \\mathrm { a n d } \\tilde { \\eta }  \\tilde { \\eta } - \\alpha \\nabla _ { \\tilde { \\eta } } \\sum _ { j } \\mathcal { L } ^ { ( j ) } } \\end{array}$ ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "10: end for Output: distilled data $\\tilde { \\bf x }$ and optimized learning rate $\\tilde { \\eta }$ ",
+        "bbox": [
+            174,
+            284,
+            500,
+            308
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "using these learned synthetic data $\\tilde { \\bf x }$ can greatly boost the performance on the real test set. Given an initial $\\theta _ { 0 }$ , we obtain these synthetic data $\\tilde { \\mathbf { x } }$ and learning rate $\\tilde { \\eta }$ by minimizing the objective below $\\mathcal { L }$ ",
+        "bbox": [
+            173,
+            330,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d0f32eb75872ba2d9fd5220d541a2de5ff5eadaa92d7b9e089ee2e7c6c80662d.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\tilde { \\mathbf { x } } ^ { * } , \\tilde { \\eta } ^ { * } = \\underset { \\tilde { \\mathbf { x } } , \\tilde { \\eta } } { \\arg \\operatorname* { m i n } } \\mathcal { L } ( \\tilde { \\mathbf { x } } , \\tilde { \\eta } ; \\theta _ { 0 } ) } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            354,
+            362,
+            640,
+            449
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where we derive the new weights $\\theta _ { 1 }$ as a function of distilled data $\\tilde { \\bf x }$ and learning rate $\\tilde { \\eta }$ using Equation 2 and then evaluate the new weights over all the real training data $\\mathbf { x }$ . The loss $\\mathcal { L } ( \\tilde { \\mathbf { x } } , \\tilde { \\eta } ; \\theta _ { 0 } )$ is differentiable w.r.t. $\\tilde { \\bf x }$ and $\\tilde { \\eta }$ , and can thus be optimized using standard gradient-based methods. In many classification tasks, the data $\\mathbf { x }$ may contain discrete parts, e.g., class labels in data-label pairs. For such cases, we fix the discrete parts rather than learn them. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 DISTILLATION FOR RANDOM INITIALIZATIONS ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+            554
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Unfortunately, the above distilled data is optimized for a given initialization, and does not generalize well to other initializations, as it encodes the information of both the training dataset $\\mathbf { x }$ and a particular network initialization $\\theta _ { 0 }$ . To address this issue, we turn to calculate a small number of distilled data that can work for networks with random initializations from a specific distribution. We formulate the optimization problem as follows: ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0f46a6fb4a98c4f65cb20b3f2e7e5b1c368ccfc69f68d678dc064fc6dfb374cc.jpg",
+        "text": "$$\n\\tilde { \\mathbf { x } } ^ { * } , \\tilde { \\eta } ^ { * } = \\underset { \\tilde { \\mathbf { x } } , \\tilde { \\eta } } { \\arg \\operatorname* { m i n } } \\mathbb { E } _ { \\theta _ { 0 } \\sim p ( \\theta _ { 0 } ) } \\mathcal { L } ( \\tilde { \\mathbf { x } } , \\tilde { \\eta } ; \\theta _ { 0 } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            367,
+            640,
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+            666
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where the network initialization $\\theta _ { 0 }$ is randomly sampled from a distribution $p ( \\theta _ { 0 } )$ . During our optimization, the distilled data are optimized to work well for randomly initialized networks. In practice, we observe that the final distilled data generalize well to unseen initializations. In addition, these distilled images often look quite informative, encoding the discriminative features of each category (e.g., in Figure 2). Algorithm 1 illustrates our main method. ",
+        "bbox": [
+            174,
+            670,
+            825,
+            741
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As the optimization (Equation 4) is highly non-linear and complex, the initialization of $\\tilde { \\bf x }$ plays a critical role in the final performance. We experiment with different initialization strategies and observe that using random real images as initialization often produces better distilled images compared to random initialization, e.g., $\\mathcal { N } ( 0 , I )$ . ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For a compact set distilled data to be properly learned, it turns out having only one GD step is far from sufficient. Next, we derive a lower bound on the size of distilled data needed for a simple model with arbitrary initial $\\theta _ { 0 }$ in one GD step, and discuss its implications on our algorithm. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3 ANALYSIS OF A SIMPLE LINEAR CASE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            869,
+            475,
+            883
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This section studies our formulation in a simple linear regression problem with quadratic loss. We derive a lower bound of the size of distilled data needed to achieve the same performance as training on the full dataset for arbitrary initialization with one GD step. Consider a dataset $\\mathbf { x }$ containing $N$ data-target pairs $\\{ ( d _ { i } , t _ { i } ) \\} _ { i = 1 } ^ { N }$ , where $d _ { i } \\in \\mathbb { R } ^ { D }$ and $t _ { i } \\in \\mathbb { R }$ , which we represent as two matrices: an $N \\times D$ data matrix $\\mathbf { d }$ and an $N \\times 1$ target matrix $\\mathbf { t }$ . Given the mean squared error metric and a $D \\times 1$ weight matrix $\\theta$ , we have ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4b634e49d9f5d416d2816a5c635aa951c07d5e5d6f7b13b886d25de943a1a280.jpg",
+        "text": "$$\n\\ell ( { \\mathbf x } , \\theta ) = \\ell ( ( { \\mathbf d } , { \\mathbf t } ) , \\theta ) = \\frac { 1 } { 2 N } \\| { \\mathbf d } \\theta - { \\mathbf t } \\| ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            366,
+            165,
+            632,
+            194
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We aim to learn $M$ synthetic data-target pairs $\\tilde { \\mathbf { x } } = ( \\tilde { \\mathbf { d } } , \\tilde { \\mathbf { t } } )$ , where $\\tilde { \\mathbf { d } }$ is an $M \\times D$ matrix, $\\tilde { \\mathbf { t } }$ an $M \\times 1$ matrix $M \\ll N ,$ ), and $\\tilde { \\eta }$ the learning rate, to minimize $\\ell ( \\mathbf { x } , \\theta _ { 0 } - \\tilde { \\eta } \\nabla _ { \\theta _ { 0 } } \\ell ( \\tilde { \\mathbf { x } } , \\theta _ { 0 } ) )$ . The updated weight matrix after one GD step with these distilled data is ",
+        "bbox": [
+            174,
+            200,
+            825,
+            244
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c22467c9be65b89d2faa67be3ecbc64469f8857e7e3d35cb8e9336fcf5bbb136.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\theta _ { 1 } = \\theta _ { 0 } - \\tilde { \\eta } \\nabla _ { \\theta _ { 0 } } \\ell ( \\tilde { \\mathbf { x } } , \\theta _ { 0 } ) } \\\\ & { \\quad = \\theta _ { 0 } - \\frac { \\tilde { \\eta } } { M } \\tilde { \\mathbf { d } } ^ { T } ( \\tilde { \\mathbf { d } } \\theta _ { 0 } - \\tilde { \\mathbf { t } } ) } \\\\ & { \\quad = ( \\mathbf { I } - \\frac { \\tilde { \\eta } } { M } \\tilde { \\mathbf { d } } ^ { T } \\tilde { \\mathbf { d } } ) \\theta _ { 0 } + \\frac { \\tilde { \\eta } } { M } \\tilde { \\mathbf { d } } ^ { T } \\tilde { \\mathbf { t } } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            385,
+            248,
+            611,
+            330
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For the quadratic loss, there always exists distilled data $\\tilde { \\mathbf { x } }$ that can achieve the same performance as training on the full dataset $\\mathbf { x }$ (i.e., attaining the global minimum) for any initialization $\\theta _ { 0 }$ . For example, given any global minimum solution $\\theta ^ { * }$ , we can choose $\\tilde { \\mathbf { d } } = N \\cdot \\mathbf { I }$ and $\\tilde { \\mathbf { t } } = N \\cdot \\boldsymbol { \\theta } ^ { * }$ . But how small can the size of the distilled data be? For such models, the global minimum is attained at any $\\theta ^ { * }$ satisfying $\\mathbf { d } ^ { T } \\mathbf { d } \\theta ^ { * } = \\mathbf { d } ^ { T } \\mathbf { t }$ . Substituting Equation 6 in the condition above, we have ",
+        "bbox": [
+            173,
+            332,
+            825,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a36b802abca8fa439f7cb2f52b9acd8a9c194d3278bc0ec64c9964f18ad53616.jpg",
+        "text": "$$\n\\mathbf { d } ^ { T } \\mathbf { d } ( \\mathbf { I } - \\frac { \\tilde { \\eta } } { M } \\tilde { \\mathbf { d } } ^ { T } \\tilde { \\mathbf { d } } ) \\theta _ { 0 } + \\frac { \\tilde { \\eta } } { M } \\mathbf { d } ^ { T } \\mathbf { d } \\tilde { \\mathbf { d } } ^ { T } \\tilde { \\mathbf { t } } = \\mathbf { d } ^ { T } \\mathbf { t } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            349,
+            409,
+            648,
+            439
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Here we make the mild assumption that the feature columns of the data matrix $\\mathbf { d }$ are independent (i.e., ${ \\bf d } ^ { T } { \\bf d }$ has full rank). For a $\\bar { \\bf x } = ( \\tilde { \\bf d } , \\tilde { \\bf t } )$ to satisfy the above equation for any $\\theta _ { 0 }$ , we must have ",
+        "bbox": [
+            173,
+            444,
+            826,
+            474
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/06e780a2790e091669522bd4c15fda339dbe40eacac0c3715765501fc0364abc.jpg",
+        "text": "$$\n\\mathbf { I } - \\frac { \\widetilde { \\eta } } { M } \\widetilde { \\mathbf { d } } ^ { T } \\widetilde { \\mathbf { d } } = \\mathbf { 0 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            437,
+            478,
+            558,
+            510
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "which implies that $\\tilde { \\mathbf { d } } ^ { T } \\tilde { \\mathbf { d } }$ has full rank and $M \\geq D$ ",
+        "bbox": [
+            173,
+            515,
+            504,
+            531
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Discussion. The analysis above only considers a simple case but suggests that any small number of distilled data fail to generalize to arbitrary initial $\\theta _ { 0 }$ . This is intuitively expected as the optimization target $\\ell ( \\mathbf { x } , \\theta _ { 1 } ) = \\ell ( \\bar { \\mathbf { x } } , \\theta _ { 0 } - \\tilde { \\eta } \\nabla _ { \\theta _ { 0 } } \\ell ( \\tilde { \\mathbf { x } } , \\theta _ { 0 } ) )$ depends on the local behavior of $\\ell ( { \\mathbf { x } } , \\cdot )$ around $\\theta _ { 0 }$ (e.g., gradient magnitude), which can be drastically different across various initializations $\\theta _ { 0 }$ . The lower bound $M \\geq D$ is a quite restricting one, considering that real datasets often have thousands to even hundreds of thousands of dimensions (e.g., images). This analysis motivates us to avoid the limitation of using one GD step by extending to multiple steps in the next section. ",
+        "bbox": [
+            173,
+            537,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.4 MULTIPLE GRADIENT DESCENT STEPS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            652,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We extend Algorithm 1 to more than one gradient descent steps by changing Line 6 to multiple sequential GD steps on the same batch of distilled data, i.e., each step $i$ performs ",
+        "bbox": [
+            171,
+            678,
+            823,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/81c541387be5bdc4f33bcc227da6e05bf73b49d37a5f3a834d290838e665a4b8.jpg",
+        "text": "$$\n\\theta _ { i + 1 } = \\theta _ { i } - \\tilde { \\eta } _ { i } \\nabla _ { \\theta _ { i } } \\ell ( \\tilde { \\mathbf { x } } , \\theta _ { i } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            408,
+            712,
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+            729
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "and changing Line 9 to backpropagate through all steps. We do not share the same learning rates across steps as later steps often require lower learning rates. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Naively computing gradients is memory and computationally intensive. Therefore, we exploit a recent technique called back-gradient optimization, which allows for significantly faster gradient calculation in reverse-mode differentiation (Domke, 2012; Maclaurin et al., 2015). Specifically, back-gradient optimization formulates the necessary second-order terms into efficient Hessian-vector products (Pearlmutter, 1994), which can be easily calculated with modern automatic differentiation systems such as PyTorch (Paszke et al., 2017). ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.5 DISTRIBUTION OF INITIAL WEIGHTS ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            869,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "There is freedom in choosing the distribution of initial weights $p ( \\theta _ { 0 } )$ . In this work, we explore the following four practical choices in the experiments: ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "(a) MNIST. These distilled images can train unknown random initializations to $8 8 . 5 1 \\% \\pm 1 . 1 1 \\%$ test accuracy in 2000 GD steps. ",
+        "bbox": [
+            173,
+            169,
+            491,
+            207
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/0a9d824eb361ad8280563fe1315d8bbfe0b9ac40de2724c177b8ff475188136c.jpg",
+        "image_caption": [
+            "(b) CIFAR10. These distilled images can train unknown random initializations to $4 1 . 2 3 \\% \\pm 0 . 8 8 \\%$ test accuracy in $5 0 \\mathrm { G D }$ steps. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Figure 2: Distilled images trained for random initialization a batch of 100 distilled images (ten per class). Only 30 of 100 distilled images are shown here. Please see the appendix for the full result. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "• Random initialization: Distribution over random initial weights, e.g., He Initialization (He et al., 2015) and Xavier Initialization (Glorot and Bengio, 2010) for neural networks. • Fixed initialization: A particular fixed network initialized by the method above. • Random pre-trained weights: Distribution over models pre-trained on other tasks or datasets, e.g., ALEXNET (Krizhevsky et al., 2012) networks trained on ImageNet (Deng et al., 2009). • Fixed pre-trained weights: A particular fixed network pre-trained on other tasks and datasets. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Distillation with pre-trained weights. Such learned distilled data essentially fine-tune weights pre-trained on one dataset to perform well for a new dataset, thus bridging the gap between the two domains. Domain mismatch and dataset bias represent a challenging problem in machine learning (Torralba and Efros, 2011; Daume III, 2007; Saenko et al., 2010). In this work, we characterize the domain mismatch via distilled data. In Section 4.1.2, we show that a small number of distilled images are sufficient to quickly adapt convolutional neural network (CNN) models to new datasets and tasks. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.6 APPLICATION TO CONTINUAL LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            491,
+            495,
+            503
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To guard against domain shift, several continual learning methods store a subset of training samples in a small memory buffer, and restrict future updates to maintain reasonable performance on these stored samples (Rebuffi et al., 2017; Kirkpatrick et al., 2017; Lopez-Paz and Ranzato, 2017; Nguyen et al., 2018). As our distilled images contain rich information about the past training data and task, they could naturally serve as a compressed memory of the past. ",
+        "bbox": [
+            174,
+            516,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To test this, we modify a recent continual learning method called Gradient Episodic Memory (GEM) (Lopez-Paz and Ranzato, 2017). GEM enforces inequality constraints such that the new model, after being trained on the new data and task, should perform at least as well as the old model on the previously stored data and tasks. Here, we store our distilled data for each task instead of randomly drawn training samples as used in GEM. We use the distilled data to construct inequality constraints, and solve the optimization using quadratic programming, same as in GEM. As shown in Section 4.2, our method compares favorably against several baselines that rely on real images. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            712,
+            326,
+            727
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this section, we report experiments of regular image classifications on MNIST (LeCun, 1998) and CIFAR10 (Krizhevsky and Hinton, 2009), adaptation from ImageNet (Deng et al., 2009) to PASCAL-VOC (Everingham et al., 2010) and CUB-200 (Wah et al., 2011), and continual learning on permuted MNIST and CIFAR100. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Baselines. For each experiment, in addition to baselines specific to the setting, we generally compare our method against baselines trained with data derived or selected from real training images: ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "• Random real images: We randomly sample the same number of real images per category.   \n• Optimized real images: We sample different sets of random real images as above, and choose the top $2 0 \\%$ best performing sets.   \n• $k$ -means $^ { + + }$ : We apply $k$ -means $^ { + + }$ (Arthur and Vassilvitskii, 2007) clustering to each category, and extract the cluster centroids. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/1b5e9553885791db954a1e14c31f2b9ac8a5ecec65fe56c457d907cedfc3c31e.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "table",
+        "img_path": "images/3c29393b4dd57dbea3f199701fa0304ce910795d567c3c07975c726775c1d8fb.jpg",
+        "table_caption": [
+            "Figure 3: Distillation performance with varying numbers of GD steps and a fixed number of distilled images. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=3 colspan=1></td><td rowspan=1 colspan=2>Ours</td><td rowspan=1 colspan=6>Baselines</td></tr><tr><td rowspan=2 colspan=1>Fixed init.</td><td rowspan=2 colspan=1>Random init.</td><td rowspan=1 colspan=4>Used as training data in CNN</td><td rowspan=1 colspan=2>Used in KNN classification</td></tr><tr><td rowspan=1 colspan=1>Random real</td><td rowspan=1 colspan=1>Optimized real</td><td rowspan=1 colspan=1>k-means++</td><td rowspan=1 colspan=1>Average real</td><td rowspan=1 colspan=1>Random real</td><td rowspan=1 colspan=1>k-means++</td></tr><tr><td rowspan=1 colspan=1>MNIST</td><td rowspan=1 colspan=1>94.4</td><td rowspan=1 colspan=1>88.5± 1.1</td><td rowspan=1 colspan=1>82.8± 1.8</td><td rowspan=1 colspan=1>83.8± 2.1</td><td rowspan=1 colspan=1>86.7± 1.4</td><td rowspan=1 colspan=1>77.7 ± 2.7</td><td rowspan=1 colspan=1>71.5 ± 2.1</td><td rowspan=1 colspan=1>92.4±0.2</td></tr><tr><td rowspan=1 colspan=1>CIFAR10</td><td rowspan=1 colspan=1>45.2</td><td rowspan=1 colspan=1>41.2±0.9</td><td rowspan=1 colspan=1>24.8 ± 1.5</td><td rowspan=1 colspan=1>24.9 ± 1.4</td><td rowspan=1 colspan=1>26.7± 1.8</td><td rowspan=1 colspan=1>22.8±0.8</td><td rowspan=1 colspan=1>18.8± 1.3</td><td rowspan=1 colspan=1>29.4± 0.4</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Comparison between our method and various baselines. All methods use ten images per category (100 in total), except for the average real images baseline, which reuses the same images in different GD steps. For MNIST, our method uses $2 0 0 0 \\mathrm { G D }$ steps, and baselines use the best among #steps $\\in \\{ 1 , 1 0 0 , 5 0 0 , 1 0 0 0 , 2 0 0 0 \\}$ . For CIFAR10, our method uses $5 0 \\mathrm { G D }$ steps, and baselines use the best among #steps $\\in \\{ 1 , 5 , 1 0 , 2 0 , 5 0 0 \\}$ . In addition, we include a K-nearest neighbors (KNN) baseline, and report best results among all combinations of distance metric $\\in \\{ l _ { 1 } , l _ { 2 } \\}$ and one or three neighbors. ",
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+    },
+    {
+        "type": "text",
+        "text": "• Average real images: We compute the average image for each category. ",
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Please see the appendix for more details about training and baselines, and additional results. ",
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+    },
+    {
+        "type": "text",
+        "text": "4.1 DATASET DISTILLATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We first present experimental results on training classifiers either from scratch or adapting from pre-trained weights. For MNIST, the distilled images are trained with LENET (LeCun et al., 1998), which achieves about $9 9 \\%$ test accuracy if conventionally trained. For CIFAR10, we use a network architecture (Krizhevsky, 2012) that achieves around $8 0 \\%$ test accuracy if conventionally trained. For ImageNet adaptations, we use an ALEXNET (Krizhevsky et al., 2012). We use 2000 GD steps for MNIST and $5 0 \\mathrm { G D }$ steps for CIFAR10. For random initializations and random pre-trained weights, we report means and standard deviations over 200 held-out models, unless otherwise stated. ",
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+    },
+    {
+        "type": "text",
+        "text": "For baselines, we perform each evaluation on 200 held-out models using all possible combinations of learning rate $\\in \\{$ distilled learning rates $\\tilde { \\eta } ^ { * }$ , 1e-3, 3e-3, 1e-2, 3e-2, 1e-1, 3e-1} and several choices of numbers of training GD steps (see table captions for details), and report results with the best performing combination. ",
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+    },
+    {
+        "type": "text",
+        "text": "4.1.1 DISTILLATION WITH WEIGHTS SAMPLED FRO NETWORK INITIALIZATION",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Fixed initialization. With access to initial network weights, distilled images can directly train a fixed network to reach high performance. Experiment results show that just 10 distilled images (one per class) can boost the performance of a LENET with an initial accuracy $8 . 2 5 \\%$ to a final accuracy of $9 3 . 8 2 \\%$ on MNIST in $2 0 0 0 \\mathrm { G D }$ steps. Using 100 distilled images (ten per class) can raise the final accuracy can be raised to $9 4 . 4 1 \\%$ , as shown in the first column of Table 1. Similarly, 100 distilled images can train a network with an initial accuracy $1 0 . 7 5 \\%$ to test accuracy of $4 5 . 1 5 \\%$ on CIFAR10 in $5 0 \\mathrm { G D }$ steps. ",
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+    {
+        "type": "text",
+        "text": "Random initialization. Figure 2 distilled images trained with randomly sampled initializations using Xavier Initialization (Glorot and Bengio, 2010). While the resulting average test accuracy from these images are not as high as those for fixed initialization, these distilled images crucially do not require a specific initial point, and thus could potentially generalize to a much wider range of starting points. In Section 4.2 below, we present preliminary results of achieving nontrivial gains from applying such distilled images to classifier networks during a continual learning training process. ",
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+    {
+        "type": "table",
+        "img_path": "images/097057089a7eeefd7636aa1d4989a017fe7107a25fcd89eda6cf9b1644f72145.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Oursw/fixedpre-trained</td><td rowspan=1 colspan=1>Oursw/randompre-trained</td><td rowspan=1 colspan=1>Random real</td><td rowspan=1 colspan=1>Optimized real</td><td rowspan=1 colspan=1>k-means++</td><td rowspan=1 colspan=1>Average real</td><td rowspan=1 colspan=1>Few-shotadaptationMotiian et al. (2017)</td><td rowspan=1 colspan=1>No adaptation</td><td rowspan=1 colspan=1>Train on fulltarget dataset</td></tr><tr><td rowspan=1 colspan=1>M→u</td><td rowspan=1 colspan=1>97.9</td><td rowspan=1 colspan=1>95.4±1.8</td><td rowspan=1 colspan=1>94.9 ±0.8</td><td rowspan=1 colspan=1>95.2± 0.7</td><td rowspan=1 colspan=1>94.8±0.7</td><td rowspan=1 colspan=1>93.9±0.8</td><td rowspan=1 colspan=1>96.7±0.5</td><td rowspan=1 colspan=1>90.4±3.0</td><td rowspan=1 colspan=1>97.3±0.3</td></tr><tr><td rowspan=1 colspan=1>U→M</td><td rowspan=1 colspan=1>93.2</td><td rowspan=1 colspan=1>92.7 ±1.4</td><td rowspan=1 colspan=1>87.1 ± 2.9</td><td rowspan=1 colspan=1>87.6 ± 2.1</td><td rowspan=1 colspan=1>88.0± 2.2</td><td rowspan=1 colspan=1>78.4 ± 5.0</td><td rowspan=1 colspan=1>89.2 ± 2.4</td><td rowspan=1 colspan=1>67.5±3.9</td><td rowspan=1 colspan=1>98.6± 0.5</td></tr><tr><td rowspan=1 colspan=1>S→M</td><td rowspan=1 colspan=1>96.2</td><td rowspan=1 colspan=1>85.2± 4.7</td><td rowspan=1 colspan=1>84.6 ± 2.1</td><td rowspan=1 colspan=1>85.2 ± 1.2</td><td rowspan=1 colspan=1>86.5± 1.2</td><td rowspan=1 colspan=1>74.9 ± 2.6</td><td rowspan=1 colspan=1>74.0 ± 1.5</td><td rowspan=1 colspan=1>51.6 ± 2.8</td><td rowspan=1 colspan=1>98.6±0.5</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 2: Adapting models among MNIST $( \\mathcal { M } )$ , USPS $( \\mathcal { U } )$ , and SVHN $( S )$ using 100 distilled images. Our method outperforms few-shot domain adaptation (Motiian et al., 2017) and other baselines in most settings. Due to computation limitations, the 100 distilled images are split into 10 minibatches applied in 10 sequential GD steps, and the entire set of 100 distilled images is iterated through 3 times $\\mathrm { 3 0 G D }$ steps in total). For baselines, we train the model using the same number of images with $\\{ 1 , 3 , 5 \\}$ times and report the best result. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/bd9d752318ee7eeaf8cef47467ab7567fb189d01405a6b87283d517b5c93f495.jpg",
+        "table_caption": [
+            "Table 3: Adapting an ALEXNET pre-trained on ImageNet to PASCAL-VOC and CUB-200. We use one distilled image per category, repeatedly applied via three GD steps. Our method significantly outperforms the baselines. For baselines, we train the model with $\\{ 1 , 3 , 5 \\}$ GD steps and report the best. Results are over 10 runs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Target dataset</td><td rowspan=1 colspan=1>Ours</td><td rowspan=1 colspan=1>Random real</td><td rowspan=1 colspan=1>Optimized real</td><td rowspan=1 colspan=1>Average real</td><td rowspan=1 colspan=1>Fine-tune on fulltarget dataset</td></tr><tr><td rowspan=1 colspan=1>PASCAL-VOC</td><td rowspan=1 colspan=1>70.75</td><td rowspan=1 colspan=1>19.41 ± 3.73</td><td rowspan=1 colspan=1>23.82± 3.66</td><td rowspan=1 colspan=1>9.94</td><td rowspan=1 colspan=1>75.57±0.18</td></tr><tr><td rowspan=1 colspan=1>CUB-200</td><td rowspan=1 colspan=1>38.76</td><td rowspan=1 colspan=1>7.11 ± 0.66</td><td rowspan=1 colspan=1>7.23± 0.78</td><td rowspan=1 colspan=1>2.88</td><td rowspan=1 colspan=1>41.21 ± 0.51</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Multiple gradient descent steps. Section 3.3 has shown theoretical limitations of using only one step in a simple linear case. In Figure 3, we empirically verify for deep networks that using multiple steps drastically outperforms the single step method, given the same number of distilled images. Table 1 summarizes the results of our method and all baselines. Our method with both fixed and random initializations outperforms all the baselines on CIFAR10 and most of the baselines on MNIST. ",
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+    },
+    {
+        "type": "text",
+        "text": "4.1.2 DISTILLATION WITH PRE-TRAINED INITIAL WEIGHTS",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Next, we show the extended setting of our algorithm discussed in Section 3.5, where the weights are not randomly initialized but pre-trained on a particular dataset. In this section, for random initial weights, we train the distilled images on 2000 pre-trained models and evaluate them on 200 unseen models. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Fixed and random pre-trained weights on digits. As shown in Section 3.5, we can optimize distilled images to quickly fine-tune pre-trained models on a new dataset. Table 2 shows that our method is more effective than various baselines on adaptation between three digits datasets: MNIST, USPS (Hull, 1994), and SVHN (Netzer et al., 2011). We also compare our method against a stateof-the-art few-shot domain adaptation method (Motiian et al., 2017). Although our method uses the entire training set to compute the distilled images, both methods use the same number of images to distill the knowledge of target dataset. Prior work (Motiian et al., 2017) is outperformed by our method with fixed pre-trained weights on all the tasks, and by our method with random pre-trained weights on two of the three tasks. This result shows that our distilled images effectively compress the information of target datasets. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Fixed pre-trained ALEXNET to PASCAL-VOC and CUB-200. In Table 3, we adapt a widely used ALEXNET model pre-trained on ImageNet to image classification on PASCAL-VOC and CUB-200 datasets. Given only one distilled image per category, our method outperforms various baselines significantly. Our method is on par with fine-tuning on the full datasets with thousands of images. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.2 APPLICATION TO CONTINUAL LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            824,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We modify Gradient Episodic Memory (GEM) (Lopez-Paz and Ranzato, 2017) to store distilled data for each task rather than real training images. Experiments in Lopez-Paz and Ranzato (2017) use large memory buffers, up to $2 5 \\%$ of the training set. Instead, we focus on a more realistic scenario where the buffer is rather small $( \\leq 1 \\%$ of the training set). Following the experiment settings and architecture choices from Lopez-Paz and Ranzato (2017), we consider two continual learning tasks: ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/6a61a4e9a732e104741b079f7c1c4bb9b4bc489058191cbd9eac96653a075e2b.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Permuted MNIST</td><td rowspan=1 colspan=1>CIFAR100</td></tr><tr><td rowspan=3 colspan=1>Memory size per task = 10</td><td rowspan=1 colspan=1>iCaRL (Rebuffi et al., 2017)</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>42.4</td></tr><tr><td rowspan=1 colspan=1>GEM (Lopez-Paz and Ranzato,2017)</td><td rowspan=1 colspan=1>67.4</td><td rowspan=1 colspan=1>43.8</td></tr><tr><td rowspan=1 colspan=1>GEM + Ours</td><td rowspan=1 colspan=1>75.6</td><td rowspan=1 colspan=1>52.8</td></tr><tr><td rowspan=2 colspan=1>Memory size per task = 40</td><td rowspan=1 colspan=1>iCaRL</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>45.8</td></tr><tr><td rowspan=1 colspan=1>GEM</td><td rowspan=1 colspan=1>75.3</td><td rowspan=1 colspan=1>51.6</td></tr><tr><td rowspan=2 colspan=1>Memory size per task = 50</td><td rowspan=1 colspan=1>iCaRL</td><td rowspan=1 colspan=1>二</td><td rowspan=1 colspan=1>46.9</td></tr><tr><td rowspan=1 colspan=1>GEM</td><td rowspan=1 colspan=1>75.8</td><td rowspan=1 colspan=1>52.4</td></tr><tr><td rowspan=1 colspan=1>No memory buffer</td><td rowspan=1 colspan=1>EWC (Kirkpatrick et al., 2017)</td><td rowspan=1 colspan=1>63.5</td><td rowspan=1 colspan=1>45.6</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 4: Continual learning results. Distilled images are trained with random Xavier Initialization distribution.   \nFor permuted MNIST, they are trained with 2000 GD steps. For CIFAR100, they are trained for 200 GD steps. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "• Permuted MNIST: 20 classification tasks each formed by using a different permutation to arrange pixels from MNIST images. Each task contains 1, 000 training images. The classifier used has 2 hidden layers each with 100 neurons. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "• CIFAR100: 20 classification tasks formed by splitting the 100 classes into 20 equal subsets of 5 classes. Each task contains 2, 500 training images. The classifier used is RESNET18 (He et al., 2016). ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 4 shows that using distilled data drastically improves final overall accuracy on all tasks, and reduces buffer size by up to $5 \\times$ compared to the original GEM that uses real images. We only report the basic iCaRL (Rebuffi et al., 2017) setting on CIFAR100 because it requires similar input distributions across all tasks, and it is unclear how to properly inject distilled images into its specialized examplar selection procedure. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The appendix details the hyper-parameters tested for each continual learning algorithm. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 DISCUSSION AND LIMITATIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            513,
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+            530
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we have presented dataset distillation for compressing the knowledge of entire training data into a few synthetic training images. We demonstrate how to train a network to reach surprisingly good performance with only a small number of distilled images. Finally, the distilled images can efficiently store the memory of previous tasks in the continual learning setting. ",
+        "bbox": [
+            174,
+            545,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Many challenges remain for knowledge distillation of data. Although our method generalizes well to random initializations, it is still limited to a particular network architecture. Since loss surfaces for different architectures might be drastically different, a more flexible method of applying the distilled data may overcome this difficulty. Another limitation is the increasing computation and memory requirements for finding the distilled data as the number of images and steps increases. To compress large-scale datasets such as ImageNet, we may need first-order gradient approximations to make the optimization computationally feasible. ",
+        "bbox": [
+            174,
+            608,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Nonetheless, we are encouraged by the findings in this paper on the possibilities of training large models with a few distilled data, leading to potential applications such as accelerating network evaluation in neural architecture search (Zoph and Le, 2017). We believe that the ideas developed in this work might give new insights into the quantity and type of data that deep networks are able to process, and hopefully inspire others to think along this direction. ",
+        "bbox": [
+            174,
+            713,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
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In ACL, 2007.   \nJ. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei. Imagenet: A large-scale hierarchical image database. In CVPR, 2009.   \nJ. Domke. Generic methods for optimization-based modeling. In Artificial Intelligence and Statistics, pages 318–326, 2012.   \nM. Everingham, L. Van Gool, C. K. Williams, J. Winn, and A. Zisserman. The pascal visual object classes (voc) challenge. IJCV, 88(2):303–338, 2010.   \nP. F. Felzenszwalb, R. B. Girshick, D. McAllester, and D. Ramanan. Object detection with discriminatively trained part-based models. PAMI, 32(9):1627–1645, 2010.   \nX. Glorot and Y. Bengio. Understanding the difficulty of training deep feedforward neural networks. In International conference on artificial intelligence and statistics, 2010.   \nS. A. Goldman and M. J. Kearns. On the complexity of teaching. Journal of Computer and System Sciences, 50 (1):20–31, 1995.   \nS. Har-Peled and A. Kushal. Smaller coresets for k-median and k-means clustering. Discrete & Computational Geometry, 37(1):3–19, 2007.   \nK. He, X. Zhang, S. Ren, and J. Sun. Delving deep into rectifiers: Surpassing human-level performance on imagenet classification. In ICCV, 2015.   \nK. 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.   \nM. A. Hearst, S. T. Dumais, E. Osuna, J. Platt, and B. Scholkopf. Support vector machines. IEEE Intelligent Systems and their applications, 13(4):18–28, 1998.   \nG. Hinton, O. Vinyals, and J. Dean. Distilling the knowledge in a neural network. In NIPS Deep Learning and Representation Learning Workshop, 2015.   \nA. 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. In CVPR, 2017.   \nJ. J. Hull. A database for handwritten text recognition research. PAMI, 16(5):550–554, 1994.   \nS. Ioffe and C. Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. In ICLR, 2015.   \nD. Kingma and J. Ba. Adam: A method for stochastic optimization. In ICLR, 2015.   \nJ. Kirkpatrick, R. Pascanu, N. Rabinowitz, J. Veness, G. Desjardins, A. A. Rusu, K. Milan, J. Quan, T. Ramalho, A. Grabska-Barwinska, et al. Overcoming catastrophic forgetting in neural networks. PNAS, 114(13): 3521–3526, 2017.   \nP. W. Koh and P. Liang. Understanding black-box predictions via influence functions. In ICML, 2017.   \nA. Krizhevsky. cuda-convnet: High-performance $\\mathrm { c } { + } { + } I$ /cuda implementation of convolutional neural networks. https://github.com/akrizhevsky/cuda-convnet2, 2012.   \nA. Krizhevsky and G. Hinton. Learning multiple layers of features from tiny images. Technical report, Citeseer, 2009.   \nA. Krizhevsky, I. Sutskever, and G. E. Hinton. Imagenet classification with deep convolutional neural networks. In NIPS, 2012.   \nA. Lapedriza, H. Pirsiavash, Z. Bylinskii, and A. Torralba. Are all training examples equally valuable? arXiv preprint arXiv:1311.6510, 2013.   \nY. LeCun. The mnist database of handwritten digits. http://yann. lecun. com/exdb/mnist/, 1998.   \nY. LeCun, L. Bottou, Y. Bengio, and P. Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998.   \nD. Lopez-Paz and M. Ranzato. Gradient episodic memory for continual learning. In NIPS, 2017.   \nD. Maclaurin, D. Duvenaud, and R. Adams. Gradient-based hyperparameter optimization through reversible learning. In ICML, 2015.   \nA. Mahendran and A. Vedaldi. Understanding deep image representations by inverting them. In CVPR, 2015.   \nM. McCloskey and N. J. Cohen. Catastrophic interference in connectionist networks: The sequential learning problem. In Psychology of learning and motivation, volume 24, pages 109–165. 1989.   \nS. Motiian, Q. Jones, S. Iranmanesh, and G. Doretto. Few-shot adversarial domain adaptation. In NIPS, 2017.   \nY. Netzer, T. Wang, A. Coates, A. Bissacco, B. Wu, and A. Y. Ng. Reading digits in natural images with unsupervised feature learning. In NIPS workshop, 2011.   \nC. V. Nguyen, Y. Li, T. D. Bui, and R. E. Turner. Variational continual learning. In ICLR, 2018.   \nJ. A. Olvera-López, J. A. Carrasco-Ochoa, J. F. Martínez-Trinidad, and J. Kittler. A review of instance selection methods. Artificial Intelligence Review, 34(2):133–143, 2010.   \nA. Paszke, S. Gross, S. Chintala, G. Chanan, E. Yang, Z. DeVito, Z. Lin, A. Desmaison, L. Antiga, and A. Lerer. Automatic differentiation in pytorch. In ICLR Workshop, 2017.   \nB. A. Pearlmutter. Fast exact multiplication by the hessian. Neural computation, 6(1):147–160, 1994.   \nF. Pedregosa. Hyperparameter optimization with approximate gradient. In ICML, 2016.   \nJ. Ponce, T. L. Berg, M. Everingham, D. A. Forsyth, M. Hebert, S. Lazebnik, M. Marszalek, C. Schmid, B. C. Russell, A. Torralba, et al. Dataset issues in object recognition. In Toward category-level object recognition. 2006.   \nI. Radosavovic, P. Dollár, R. Girshick, G. Gkioxari, and K. He. Data distillation: Towards omni-supervised learning. In CVPR, 2018.   \nS.-A. Rebuffi, A. Kolesnikov, G. Sperl, and C. H. Lampert. icarl: Incremental classifier and representation learning. In CVPR, 2017.   \nA. Romero, N. Ballas, S. E. Kahou, A. Chassang, C. Gatta, and Y. Bengio. Fitnets: Hints for thin deep nets. In ICLR, 2015.   \nF. Rosenblatt. The perceptron, a perceiving and recognizing automaton Project Para. Cornell Aeronautical Laboratory, 1957.   \nK. Saenko, B. Kulis, M. Fritz, and T. Darrell. Adapting visual category models to new domains. In ECCV, 2010.   \nO. Sener and S. Savarese. Active learning for convolutional neural networks: A core-set approach. In ICLR, 2018.   \nA. Shinohara and S. Miyano. Teachability in computational learning. New Generation Computing, 8(4):337–347, 1991.   \nS. Tong and D. Koller. Support vector machine active learning with applications to text classification. JMLR, 2 (Nov):45–66, 2001.   \nA. Torralba and A. A. Efros. Unbiased look at dataset bias. In CVPR. IEEE, 2011.   \nI. W. Tsang, J. T. Kwok, and P.-M. Cheung. Core vector machines: Fast svm training on very large data sets. JMLR, 6(Apr):363–392, 2005.   \nC. Wah, S. Branson, P. Welinder, P. Perona, and S. Belongie. The Caltech-UCSD Birds-200-2011 Dataset. Technical Report CNS-TR-2011-001, California Institute of Technology, 2011.   \nY. Wu and K. He. Group normalization. In ECCV, 2018.   \nM. D. Zeiler and R. Fergus. Visualizing and understanding convolutional networks. In ECCV, 2014.   \nB. Zhou, A. Khosla, A. Lapedriza, A. Oliva, and A. Torralba. Object detectors emerge in deep scene cnns. In ICLR, 2015.   \nX. Zhu. Machine teaching for bayesian learners in the exponential family. In NIPS, 2013.   \nX. Zhu. Machine teaching: An inverse problem to machine learning and an approach toward optimal education. In AAAI, 2015.   \nB. Zoph and Q. V. Le. Neural architecture search with reinforcement learning. In ICLR, 2017. ",
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+        "type": "text",
+        "text": "S-1 EXPERIMENT DETAILS ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "In our experiments, we disable dropout layers in the networks due to the randomness and computational cost they introduce in distillation. Moreover, we initialize the distilled learning rates with a constant between 0.001 and 0.02 depending on the task, and use the Adam solver (Kingma and Ba, 2015) with a learning rate of 0.001. For random initialization and random pre-trained weights, we sample 4 to 16 initial weights in each optimization step. We run all the experiments on NVIDIA 1080 Ti, 2080 Ti, Titan $\\mathrm { X p }$ , and V100 GPUs. We use one GPU for fixed initial weights and up to four GPUs for random initial weights. Each training typically takes 1 to 6 hours. ",
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+        "type": "text",
+        "text": "Below we describe the details of our baselines using real training images. ",
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+        "text": "• Random real images: We randomly sample the same number of real images per category. We evaluate the performance over 10 randomly sampled sets. Optimized real images: We sample 50 sets of real images using the procedure above, pick 10 sets that achieve the best performance on 20 held-out models and 1024 randomly chosen training images, and evaluate the performance of these 10 sets.   \n• $k$ -means++: For each category, we use $k$ -means $^ { + + }$ (Arthur and Vassilvitskii, 2007) clustering to extract the same number of cluster centroids as the number of distilled images in our method. We evaluate the method over 10 runs.   \n• Average real images: We compute the average image of all the images in each category, which is repeated to match the same total number of images. We evaluate the model only once because average images are deterministic. ",
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+        "type": "text",
+        "text": "To enforce our optimized learning rate to be positive, we apply softplus to a scalar trained parameter. For continual learning experiment on CIFAR10 dataset, to compare with GEM (Lopez-Paz and Ranzato, 2017), we replace the Batch normalization (Wu and He, 2018) with Group normalization (Ioffe and Szegedy, 2015) in RESNET18 (He et al., 2016), as it is difficult to run back-gradient optimization through batch norm running statistics. For a fair comparison, we use the same architecture for our method and other baselines. ",
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+        "text": "For dataset distillation experiments with pre-trained initial weights, distilled images are initialized with $\\mathcal { N } ( 0 , 1 )$ at the beginning of training. For other experiments, distilled images are initialized with random real samples, unless otherwise stated. ",
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+        "text": "S-1.1 CONTINUAL LEARNING EXPERIMENT DETAILS ",
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+        "text": "For the compared continual learning methods, we report the best report from the following combinations of hyper-parameters: ",
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+        "text": "• GEM: – $\\gamma \\in \\{ 0 , 0 . 1 , 0 . 2 , 0 . 3 , 0 . 4 , 0 . 5 , 0 . 6 , 0 . 7 , 0 . 8 , 0 . 9 , 1 \\} .$ – learning rate $= 0 . 1$ . ",
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+        "text": "• iCARL: – regularization $\\in \\{ 0 . 0 0 1 , 0 . 0 0 3 , 0 . 0 1 , 0 . 0 3 , 0 . 1 , 0 . 3 , 1 . 0 \\} .$ – learning rate $= 0 . 1$ . ",
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+        "type": "text",
+        "text": "S-2 ADDITIONAL EXPERIMENTS RESULTS ",
+        "text_level": 1,
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+        "text": "• Figures 4 and 5 show distilled images trained for random initializations on MNIST and CIFAR10.   \n• Figures 6, 7, and 8 show distilled images trained for adapting random pre-trained models on digits datasets including MNIST, USPS, and SVHN. ",
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+            "Figure 4: Dataset distillation for random initializations on MNIST. This batch of 100 distilled images are repeatedly applied in $2 0 0 0 \\mathrm { G D }$ steps.. These images train average test accuracy to $8 8 . 5 1 \\% \\pm 1 . 1 1 \\%$ . "
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+            "Figure 5: Dataset distillation for random initializations on CIFAR10. This batch of 100 distilled images are repeatedly applied in $5 0 \\mathrm { G D }$ steps. These images train average test accuracy to $4 1 . 2 3 \\% \\pm 0 . 8 8 \\%$ . "
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+            "Figure 6: Dataset distillation for adapting random pretrained models from USPS to MNIST. 100 distilled images are split into $1 0 \\mathrm { G D }$ steps, shown as 10 rows here. Top row is the earliest GD step, and bottom row is the last. The 10 steps are iterated over three times to finish adaptation, leading to a total of $3 0 \\mathrm { G D }$ steps. These images train average test accuracy on 200 held out models from $6 7 . 5 4 \\% \\pm 3 . 9 1 \\%$ to $9 2 . 7 4 \\% \\pm 1 . 3 8 \\%$ . "
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+            "Figure 7: Dataset distillation for adapting random pretrained models from MNIST to USPS. 100 distilled images are split into $1 0 \\mathrm { G D }$ steps, shown as 10 rows here. Top row is the earliest GD step, and bottom row is the last. The 10 steps are iterated over three times to finish adaptation, leading to a total of $3 0 \\mathrm { G D }$ steps. These images train average test accuracy on 200 held out models from $9 0 . 4 3 \\% \\pm 2 . 9 7 \\%$ to $9 5 . 3 8 \\% \\pm 1 . 8 1 \\%$ . "
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+            "Figure 8: Dataset distillation for adapting random pretrained models from SVHN to MNIST. 100 distilled images are split into $1 0 \\mathrm { G D }$ steps, shown as 10 rows here. Top row is the earliest GD step, and bottom row is the last. The 10 steps are iterated over three times to finish adaptation, leading to a total of $3 0 \\mathrm { G D }$ steps. These images train average test accuracy on 200 held out models from $5 1 . 6 4 \\% \\pm 2 . 7 7 \\%$ to $8 5 . 2 1 \\% \\pm 4 . 7 3 \\%$ . "
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+        "text": "Distributed Learning with Strategic Users: A Repeated Game Approach ",
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+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
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+        "text": "Abstract ",
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+        "text": "1 We consider a distributed learning setting where strategic users are incentivized, by   \n2 a cost-sensitive fusion center, to train a learning model based on local data. The   \n3 users are not obliged to provide their true gradient updates and the fusion center   \n4 is not capable of validating the authenticity of reported updates. Thus motivated,   \n5 we formulate the interactions between the fusion center and the users as repeated   \n6 games, manifesting an under-explored interplay between machine learning and   \n7 game theory. We then develop an incentive mechanism for the fusion center based   \n8 on a joint gradient estimation and user action classification scheme, and study its   \n9 impact on the convergence performance of distributed learning. Further, we devise   \n10 an adaptive zero-determinant (ZD) strategy, thereby generalizing the celebrated ZD   \n11 strategy to the repeated games with time-varying stochastic errors. Theoretical and   \n12 empirical analysis show that the fusion center can incentivize the strategic users to   \n13 cooperate and report informative gradient updates, thus ensuring the convergence. ",
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+        "text": "14 1 Introduction ",
+        "text_level": 1,
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+        "text": "15 Distributed machine learning is becoming increasingly important in large-scale problems with data  \n16 intensive applications [18, 21, 25, 37]. Notably, federated learning has emerged as an attractive   \n17 distributed computing paradigm that aims to learn an accurate model without collecting data from the   \n18 owners and storing it in the cloud: The training data is kept locally on the computing devices which   \n19 participate in the model training and report gradient updates (or its variants) based on local data [19].   \n20 In this work, we study a distributed learning scheme in which privacy-aware users train a global model   \n21 with a fusion center. We consider the users to be rational, self-interested and risk-neutral. The users   \n22 are not compelled to contribute their resources unconditionally, unless they are sufficiently rewarded,   \n23 and the system may reach a noncooperative Nash equilibrium where the users do not participate in   \n24 training. This departs from conventional distributed learning schemes where the agents directly follow   \n25 the lead of the fusion center $( \\mathrm { F C } ) ^ { 1 }$ and send their gradients. Since the users are strategic, a paramount   \n26 objective for the FC is to design an effective reward mechanism to incentivize self-interested users to   \n27 provide informative gradient updates. The repeated game enriches the distributed learning framework   \n28 with the idea of many agents interacting within a common uncertain environment, and this framework   \n29 provides a new perspective to specify how agents can strategically choose the learning updates how   \n30 the resulting changes impact the performance of the learning efforts.   \n31 Challenges and Contributions. There are a number of challenges in distributed learning with   \n32 strategic users. First, the users are not obliged to entirely dedicate their resources and they may not   \n33 fulfill their roles in the training of the algorithm if it were not for their own interest. Secondly, the   \n34 FC cannot directly validate data driven gradient updates due to their stochastic nature. The quality   \n35 of the updates can vary over time and across the users since each user can control his own dataset.   \n36 The interactions among users and the FC are repeated, and each user is capable of devising intricate   \n37 strategies based on the past interactions. From a game-theoretic perspective, the fusion center’s ability   \n38 to reciprocate against non-cooperative user actions is significantly restricted since she cannot directly   \n39 observe the user actions. Finally, the FC is not allowed to impose penalties on the users and positive   \n40 rewards are the only options at her disposal to incentivize user participation. The work proposed here   \n41 is, to the best of our knowledge, the first distributed learning framework to consider these challenges.   \n42 In this study, we model the interactions (in terms of gradient reporting and reward) between the   \n43 FC and the users as repeated games, which intertwine with the updates in distributed learning. We   \n44 propose a reward mechanism for the fusion center, based on an adaptive zero-determinant strategy,   \n45 thereby generalizing the celebrated ZD strategy to the repeated games with time-varying stochastic   \n46 errors. To tackle the challenge that the FC cannot directly verify the received reported gradients,   \n47 we devise a gradient estimation and user action classification. Our findings demonstrate that, by   \n48 employing adaptive ZD strategies, the FC can incentivize the strategic users to cooperate and report   \n49 informative gradient updates, thus ensuing the convergence of distributed learning. ",
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+        "image_caption": [
+            "Figure 1: The fusion center (FC) trains the learning model with strategic users who are not obliged to report their gradients. (a) The objective of the FC is to incentivize users to cooperate by giving rewards so as to learn the model. (b) If the user is cooperative, he reports a privacy-preserved version of his gradient signal. Otherwise, the user is defective and sends an arbitrary uninformative signal. (c) The FC and the user each choose to cooperate or defect with respective payoffs as shown. "
+        ],
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+        "text": "Detailed discussion on related work is relegated to Appendix A, due to space limitation. ",
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+        "text": "2 Distributed Learning with Strategic Users as Repeated Games ",
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+        "text": "52 We consider a distributed learning setting with $K$ strategic users $\\mathcal { K } = \\{ 1 , \\ldots , K \\}$ and a fusion center   \n53 (FC), and the optimization problem is given as follows: ",
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+        "text": "$$\n\\operatorname* { m i n } _ { \\theta \\in \\mathbb { R } ^ { n } } F ( \\theta ) : = \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } \\mathbb { E } _ { Z _ { k } \\sim \\mathcal { D } } \\big [ \\mathcal { L } ( \\theta ; Z _ { k } ) \\big ] ,\n$$",
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+        "text": "54 where $\\mathcal { L } ( \\cdot )$ is the loss function. In each iteration, each user gets a mini-batch of $s$ i.i.d. sam  \n55 ples from an unknown distribution $\\mathcal { D }$ , and computes the stochastic gradient signal as $X _ { k , t } : =$   \n56 1s Psi=1 ∇θ L\u0000θt; zik,t\u0001, where zik,t is the ith sampled data of user k at time t.   \n57 Stage Game Formulation: Actions and Payoffs. The action and the reported signal of user $\\mathbf { k }$ in   \n58 iteration $t$ are denoted with $B _ { k , t } \\in \\{ c , d \\}$ and $Y _ { k , t }$ , respectively. As depicted in Fig. 1, a user is   \n59 cooperative $( B _ { k , t } = c )$ if he is sending the privacy-preserved version of his gradient $X _ { k , t }$ . Otherwise,   \n60 the user is defective and sends a noise signal $\\Upsilon _ { k , t } \\sim \\mathcal { N } ( 0 , \\Xi _ { t } )$ independent of $X _ { k , t }$ : ",
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+        "text": "$$\nY _ { k , t } = { \\left\\{ \\begin{array} { l l } { X _ { k , t } + N _ { k , t } , } & { { \\mathrm { i f ~ } } B _ { k , t } = c { \\mathrm { ~ ( c o o p e r a t i v e ) } } ; } \\\\ { \\Upsilon _ { k , t } , } & { { \\mathrm { i f ~ } } B _ { k , t } = d { \\mathrm { ~ ( d e f e c t i v e ) } } . } \\end{array} \\right. }\n$$",
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+        "text": "61 Remark 1. Note that $N _ { k , t }$ is independent of $X _ { k , t }$ and $N _ { k , t } \\sim \\mathcal { N } ( \\vec { 0 } , \\nu _ { t } ^ { 2 } \\mathbf { I } )$ . If $\\begin{array} { r } { \\| \\nabla _ { \\theta } \\mathcal { L } ( \\theta ; z ) \\| _ { 2 } \\leq \\ell . } \\end{array}$ for all   \n62 $\\theta$ and $z$ , then this privacy-preservation mechanism enjoys $\\epsilon _ { t }$ -differential privacy, with $\\epsilon _ { t } = \\ell ^ { 2 } / s ^ { 2 } \\nu _ { t } ^ { 2 }$   \n63 for mini-batch size s. The details are provided in Appendix.   \n64 The payoff structure of a single interplay between the fusion center and a user is depicted in Fig 1b.   \n65 In iteration $t$ , when a user cooperates, he provides an information gain $R$ to the FC at his privacy   \n66 cost $V _ { \\mathrm { U } } R$ with $0 < V _ { \\mathrm { U } } \\le 1$ . When a user defects, he does not provide any information gain and does   \n67 not incur any privacy cost. The FC may distribute rewards at the end of each iteration to incentivize   \n68 the users. We denote the action of the FC toward user $k$ as $A _ { k , t } \\in \\{ C , D \\}$ . The FC is cooperative   \n69 $( A _ { k , t } { = } C )$ if she makes a payment $r$ to the user at her cost $r V _ { \\mathrm { F C } }$ with $0 < V _ { \\mathrm { F C } } \\leq 1$ . The FC is defective   \n70 $( A _ { k , t } = D )$ , if she does not make any payment to the user. The factor $V _ { \\mathrm { F C } }$ captures the difference in the   \n71 valuation of the reward between the FC and the user; for instance, the reward can be a coupon which   \n72 may be redeemed in the future. Denote the FC’s payoff vector by ${ \\bf S } _ { \\mathrm { F C } } = [ R - r V _ { \\mathrm { F C } } , - r \\bar { V } _ { \\mathrm { F C } } , R , 0 ]$   \n73 and that of the users by $\\mathbf { S } _ { \\mathrm { U } } = [ r - V _ { \\mathrm { U } } R , r , - V _ { \\mathrm { U } } R , \\mathbf { \\bar { 0 } } ]$ . In this paper, we only analyze the case where   \n74 $R { > } r V _ { \\mathrm { F C } }$ and $r > V _ { \\mathrm { U } } R$ . Otherwise, the FC or users do not have any incentive to cooperate.   \n75 The FC cannot observe the actions of the users and her realized payoffs. We assume that users do   \n76 not communicate or collude with each other. They cannot observe the actions of other users and the   \n77 actions of the FC toward other users. Next, we will discuss how to devise effective strategies for the   \n78 FC to incentivize cooperative user action for the repeated game in a cost-effective manner.   \n79 Repeated Games between Users and Fusion Center. A salient feature of $2 \\times 2$ repeated games   \n80 is that players with longer memories of the history of the game have no advantage over those with   \n81 shorter ones when each stage game is identically repeated infinite times [31]. Thus, without loss of   \n82 generality, we assume the user strategies only depend on the outcomes of the last round. Let $q _ { 1 } , q _ { 2 } , q _ { 3 }$   \n83 and $q _ { 4 }$ denote the probabilities of cooperation for the user conditioned on the joint action pair of   \n84 the previous iteration, that is $\\left( A _ { k , t - 1 } , B _ { k , t - 1 } \\right)$ , in the order of $( C , c ) , ( C , d ) , ( \\tilde { D _ { , } c } )$ and $( D , d )$ . The   \n85 user’s strategy vector is defined as $\\mathbf { q } = [ q _ { 1 } , q _ { 2 } , q _ { 3 } , q _ { 4 } ]$ .   \n86 Analogous to the user strategies, let $p _ { 1 } , p _ { 2 } , p _ { 3 }$ and $p _ { 4 }$ denote the probabilities of cooperation for   \n87 the FC conditioned on $\\left( A _ { k , t - 1 } , B _ { k , t } \\right)$ , in the order of $( C , c ) , ( C , d ) , ( D , c )$ and $( D , d )$ . The fusion   \n88 center’s strategy vector is defined as $\\mathbf { p } = [ p _ { 1 } , p _ { 2 } , p _ { 3 } , p _ { 4 } ]$ . The joint action pair of the user and the   \n89 FC is considered as the state of the game in iteration $t$ : $\\left( A _ { k , t } , B _ { k , t } \\right)$ . The strategy vectors $\\mathbf { p }$ and $\\mathbf { q }$   \n90 imply a Markov state transition matrix as follows: ",
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+        "text": "$$\n\\Omega = \\left[ \\begin{array} { c c c c } { q _ { 1 } p _ { 1 } } & { ( 1 - q _ { 1 } ) p _ { 2 } } & { q _ { 1 } ( 1 - p _ { 1 } ) } & { ( 1 - q _ { 1 } ) ( 1 - p _ { 2 } ) } \\\\ { q _ { 2 } p _ { 1 } } & { ( 1 - q _ { 2 } ) p _ { 2 } } & { q _ { 2 } ( 1 - p _ { 1 } ) } & { ( 1 - q _ { 2 } ) ( 1 - p _ { 2 } ) } \\\\ { q _ { 3 } p _ { 3 } } & { ( 1 - q _ { 3 } ) p _ { 4 } } & { q _ { 3 } ( 1 - p _ { 3 } ) } & { ( 1 - q _ { 3 } ) ( 1 - p _ { 4 } ) } \\\\ { q _ { 4 } p _ { 3 } } & { ( 1 - q _ { 4 } ) p _ { 4 } } & { q _ { 4 } ( 1 - p _ { 3 } ) } & { ( 1 - q _ { 4 } ) ( 1 - p _ { 4 } ) } \\end{array} \\right] .\n$$",
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+        "text": "91 Let $\\Lambda ^ { * }$ be the stationary vector of the transition matrix $\\overline { { \\Omega } }$ , i.e., $\\Lambda ^ { * } = \\Lambda ^ { * } \\overline { { \\Omega } }$ . We can find the expected   \n92 payoffs of the FC and the user in the stationary state as $s _ { \\mathrm { F C } } ^ { * } = \\Lambda ^ { * } \\mathbf { S } _ { \\mathrm { F C } } ^ { \\top }$ and $s _ { \\mathrm { U } } ^ { * } = \\Lambda ^ { * } \\mathbf { S } _ { \\mathrm { U } } ^ { \\top }$ . The FC sets   \n93 her strategy $\\mathbf { p }$ satisfying, for some real values $\\varphi _ { 0 } , \\varphi _ { 1 }$ and $\\varphi _ { 2 }$ , the equation ",
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+        "type": "equation",
+        "img_path": "images/06137764ace26981eb18ad96fd37aa85bb005b1532ff8170d347e113cb395ba9.jpg",
+        "text": "$$\n\\left[ p _ { 1 } - 1 , \\ p _ { 2 } - 1 , \\ p _ { 3 } , \\ p _ { 4 } \\right] = \\varphi _ { 0 } { \\bf S } _ { \\mathrm { F C } } + \\varphi _ { 1 } { \\bf S } _ { \\mathrm { U } } + \\varphi _ { 2 } { \\bf 1 } .\n$$",
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+        "text": "4 This class of strategies are called zero-determinant (ZD) strategies, which enforce a linear relation between the expected payoffs, given by 5 $\\varphi _ { 0 } s _ { \\mathrm { F C } } ^ { * } + \\varphi _ { 1 } s _ { \\mathrm { U } } ^ { * } + \\varphi _ { 2 } = 0$ , regardless of the user strategy [31]. ",
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+        "text": "96 Remark 2. The ZD strategy is a powerful tool to incentivize the users cooperation for the $F C$   \n97 because she can unilaterally set $s _ { \\mathrm { U } } ^ { * }$ or establish an extortionate linear relation between $s _ { \\mathrm { U } } ^ { * }$ and $s _ { \\mathrm { F C } } ^ { * }$ .   \n98 Against such an $F C$ strategy, the user’s best response which maximizes his payoff is full cooperation,   \n99 $\\mathbf { q } ^ { * } = [ 1 1 1 1 ]$ . The details are provided in Appendix $C$ .   \n100 Against the FC who is equipped with the ZD strategy, the user can increase his expected payoff only   \n101 by cooperating more often, and consequently his best response is full cooperation. Assuming that   \n102 there are sufficiently many participating users, the FC has the absolute leverage against any single   \n103 user who tries to negotiate with her. Nevertheless, the FC cannot directly employ the ZD strategy   \n104 since she cannot observe the true actions of the users. In the next section, we will study the use of ZD   \n105 strategy can be extended in the scope of distributed learning. ",
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+        "type": "text",
+        "text": "106 3 Distributed Stochastic Gradient Descent with Strategic Users ",
+        "text_level": 1,
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+        "text": "107 For the ease of exposition, in this paper we focus on an interesting variant of the classical stochastic   \n108 gradient descent algorithm using the gradient signals reported by strategic users (SGD-SU). In each   \n109 iteration, the FC collects the reported gradients of the users and update the model as follows: ",
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+        "text": "$$\n\\begin{array} { r } { \\theta _ { t } = \\theta _ { t - 1 } - \\eta _ { t } \\cdot \\widehat { m } _ { t } \\big ( \\mathbf { Y } _ { t } \\big ) , } \\end{array}\n$$",
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+        "table_caption": [
+            "Algorithm 1: Stochastic Gradient Descent with Strategic Users (SGD-SU) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"3\">1 fort=1,2,...,T-1do Fusion Center: broadcast the current iterate 0t-1 to all the users</td></tr><tr><td>2 3</td><td rowspan=\"2\">forall k ∈{1,2,...,K}do</td><td rowspan=\"2\">Xk,t+Nk,t cooperative action,</td></tr><tr><td>4</td></tr><tr><td></td><td colspan=\"2\">User k: compute the gradient Xk,t and Yk,t ← Yk,t defective action, Fusion Center: form the gradient estimate mt(Yt) ← K(A1Ωt-1)qT 1 K Yk,t</td></tr><tr><td>5 6</td><td colspan=\"2\">k=1 update model parameter0t ← 0t-1 - ntmt(Yt)</td></tr><tr><td>7</td><td colspan=\"2\">C (cooperative） if Ytmt&gt;|/mt|l2 classify the users Bk,t (mt, Y𝑘,t) ←</td></tr><tr><td></td><td colspan=\"2\">(7) d (defective) else</td></tr><tr><td>8 9</td><td colspan=\"2\">compute the detection and false alarm probabilities using(8)and (11) compute the adaptive strategies (9) and distribute the rewards accordingly</td></tr></table>",
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+        "text": "110 where $\\mathbf { Y _ { t } } = [ Y _ { 1 , t } \\ldots Y _ { K , t } ]$ , $\\eta _ { t }$ is the step size and $\\widehat { m } _ { t }$ is the gradient estimator. The FC cannot directly b111 observe user actions and verify the reported gradients. This gives rise to two coupled challenges: ",
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+        "text": "• The gradient estimator $\\widehat { m } _ { t }$ should be resilient against the uninformative reports of defective users. • Although the ZD strategies are powerful tools to incentivize user cooperation, the FC cannot directly employ a ZD strategy because she cannot observe the users’ actions. ",
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+        "text": "115 To tackle these difficulties, we will first introduce a gradient estimation and user classification scheme   \n116 and discuss the impact of user action classification errors on the dynamics of repeated games. As   \n117 outlined in Algorithm 1. we will develop adaptive FC strategies which generalize the classical ZD   \n118 strategies to the repeated games with time-varying stochastic errors. ",
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+    {
+        "type": "text",
+        "text": "119 3.1 Joint Gradient Estimation and User Action Classification ",
+        "text_level": 1,
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+        "text": "120 The stochastic gradients can be decomposed as $X _ { k , t } = m _ { t } + W _ { k , t }$ where $m _ { t } : = \\nabla _ { \\theta } F ( \\theta _ { t } )$ is the   \n121 population gradient and $W _ { k , t }$ is the zero-mean noise term [30]. The unknown parameter $m _ { t }$ is the   \n122 mean of the reported gradient $Y _ { k , t }$ when the user is cooperative $( B _ { k , t } = c )$ ). The defective users   \n123 send zero-mean random noise as their reported gradients. The FC needs to classify the reported   \n124 gradients and obtain an estimate of $m _ { t }$ for the SGD-SU update in (5). These two problems are   \n125 coupled with each other, and the joint scheme is, therefore, comprised of a gradient estimator $\\widehat { m } _ { t }$ ,   \n126 and a classification rule $\\widehat { B } _ { k , t }$ . To tackle this difficult problem, we first investigate gradient estimation.   \n127 Let $\\Lambda _ { 1 }$ be the initial state distribution of the games between the users and the FC. A modified   \n128 empirical mean based gradient estimator can be employed as follows: ",
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+        "text": "$$\n\\begin{array} { r } { \\widehat { m } _ { t } ( { \\mathbf { Y } _ { t } } ) : = \\frac { 1 } { K ( \\Lambda _ { 1 } \\Omega ^ { t - 1 } ) { \\mathbf { q } ^ { \\top } } } \\sum _ { k = 1 } ^ { K } Y _ { k , t } . } \\end{array}\n$$",
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+        "text": "129 It is easy to verify that $\\widehat { m } _ { t } ( \\cdot )$ is an unbiased estimator if the FC is able to employ her strategies $\\mathbf { p }$   \n130 bwithout any errors and the state distribution of the repeated games are governed by the state transition   \n131 matrix $\\Omega$ as in (3) without any perturbations. ",
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+        "text": "132 Using the gradient estimator $\\widehat { m } _ { t } ( \\cdot )$ , the FC can form the user action classification rule as ",
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+        "img_path": "images/1532dc4c310e7ad20c8183f9818d77623d3316e120e276983b701f300749d75d.jpg",
+        "text": "$$\n\\widehat { B } _ { k , t } \\left( \\widehat { m } _ { t } ( \\mathbf { Y } _ { t } ) , Y _ { k , t } \\right) = \\left\\{ \\begin{array} { l l } { \\widehat { c } } & { \\mathrm { i f } \\quad Y _ { k , t } ^ { \\top } \\widehat { m } _ { t } > \\frac { 1 } { 2 } \\| \\widehat { m } _ { t } \\| ^ { 2 } , } \\\\ { \\widehat { d } } & { \\mathrm { e l s e } ; } \\end{array} \\right.\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "133 where $\\hat { d }$ (or $\\hat { c }$ ) is the defective (or cooperative) label. The noise in the stochastic gradients, $W _ { k , t }$ ,   \n134 can be approximated as a zero mean Gaussian r.v. [17, 22, 26, 36]. Recall from (2) that cooperative   \n135 users send the privacy-preserved versions of their gradient. This implies $Y _ { k , t } \\sim { \\mathcal { N } } ( m _ { t } , \\Sigma _ { t } )$ , given   \n136 $B _ { k , t } = c$ , where $\\Sigma _ { t } : = \\mathrm { c o v } [ W _ { k , t } ] + \\nu _ { t } ^ { 2 } \\mathrm { I }$ . Thus, the detection and false alarm probabilities of the   \n137 classifier, denoted by $\\Phi _ { t }$ and $\\Psi _ { t }$ respectively, can be found as ",
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+        "img_path": "images/aa8695e649d5885dfd2774d338360f54c1a6a64cfbb7911ab2278fefa39f72f0.jpg",
+        "text": "$$\n\\begin{array} { r } { \\Phi _ { t } = 1 - \\mathcal { Q } \\left( \\frac { m _ { t } ^ { \\top } \\widehat { m } _ { t } - \\frac { 1 } { 2 } \\Vert \\widehat { m } _ { t } \\Vert ^ { 2 } } { \\sqrt { \\widehat { m } _ { t } ^ { \\top } \\Sigma _ { t } \\widehat { m } _ { t } } } \\right) \\quad \\mathrm { a n d } \\quad \\Psi _ { t } = \\mathcal { Q } \\left( \\frac { \\frac { 1 } { 2 } \\Vert \\widehat { m } _ { t } \\Vert ^ { 2 } } { \\sqrt { \\widehat { m } _ { t } ^ { \\top } \\Xi _ { t } \\widehat { m } _ { t } } } \\right) . } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "38 Remark 3. The linear classifier (7) is an effective tool under the homoscedasticity assumption. If   \n9 that is violated, the FC can employ different classifiers. The details are provided in Appendix for the   \n0 Classifier Design. ",
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+        "text": "In the next subsection, we discuss how the FC can devise her strategies building on the joint gradient estimation and user action classification scheme. ",
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+        "type": "text",
+        "text": "3.2 Adaptive Strategies for Fusion Center ",
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+        "text": "144 Although the ZD strategies, $\\mathbf { p }$ , provide the FC an efficient and powerful mechanism to encourage   \n145 the user’s cooperation; the FC cannot directly use $\\mathbf { p }$ since they are conditioned on the user’s ac  \n146 tion, $B _ { k , t }$ , which is not observable to her. Alternatively, the FC can use the classification results   \n147 after carefully adapting her strategies to mitigate the adverse effects of inevitable classification   \n148 errors. Let $\\pi _ { t , 1 } , \\pi _ { t , 2 } , \\pi _ { t , 3 }$ and $\\pi _ { t , 4 }$ denote the probabilities of cooperation for the FC conditioned   \n149 on $( A _ { k , t - 1 } , \\widehat { B } _ { k , t } )$ , in the order of $( C , \\hat { c } ) , ( C , \\hat { d } ) , ( D , \\hat { c } )$ and $( D , { \\hat { d } } )$ . These are referred to as adaptive   \n150 strategies and the FC sets these probabilities satisfying the following system of equations: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1840b63bfb04789ac7bcb55620fc4236c91dd369fa5a1c2ae9a81612554ddb93.jpg",
+        "text": "$$\n\\begin{array} { r l } & { p _ { 1 } = \\pi _ { t , 1 } \\Phi _ { t } + \\pi _ { t , 2 } ( 1 - \\Phi _ { t } ) , \\quad p _ { 2 } = \\pi _ { t , 1 } \\Psi _ { t } + \\pi _ { t , 2 } ( 1 - \\Psi _ { t } ) , } \\\\ & { p _ { 3 } = \\pi _ { t , 3 } \\Phi _ { t } + \\pi _ { t , 4 } ( 1 - \\Phi _ { t } ) , \\quad p _ { 4 } = \\pi _ { t , 3 } \\Psi _ { t } + \\pi _ { t , 4 } ( 1 - \\Psi _ { t } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            318,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Suppose 151 $\\frac { \\Phi _ { t } } { \\Psi _ { t } } \\geq \\frac { p _ { 1 } } { p _ { 2 } }$ and $\\frac { \\Phi _ { t } } { \\Psi _ { t } } \\geq \\frac { p _ { 3 } } { p _ { 4 } }$ p3 . Then the unique solution to the system above is given by ",
+        "bbox": [
+            142,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/74f9b1165a77a0192d9a1df0ae23ec0d8e0367a64f436a76cbb87950c92e830b.jpg",
+        "text": "$$\n\\begin{array} { c c } { \\pi _ { t , 1 } = \\displaystyle \\frac { p _ { 1 } ( 1 - \\Psi _ { t } ) - p _ { 2 } ( 1 - \\Phi _ { t } ) } { \\Phi _ { t } - \\Psi _ { t } } , } & { \\pi _ { t , 2 } = \\displaystyle \\frac { p _ { 2 } \\Phi _ { t } - p _ { 1 } \\Psi _ { t } } { \\Phi _ { t } - \\Psi _ { t } } , } \\\\ { \\pi _ { t , 3 } = \\displaystyle \\frac { p _ { 3 } ( 1 - \\Psi _ { t } ) - p _ { 4 } ( 1 - \\Phi _ { t } ) } { \\Phi _ { t } - \\Psi _ { t } } , } & { \\pi _ { t , 4 } = \\displaystyle \\frac { p _ { 4 } \\Phi _ { t } - p _ { 3 } \\Psi _ { t } } { \\Phi _ { t } - \\Psi _ { t } } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "152 Remark 4. If the FC directly employed the $Z D$ strategies without any adaptation, i.e., she cooperates   \n153 with probability $p _ { i }$ conditioned on classification output; the repeated games may not converge to   \n154 the stationary state $\\Lambda ^ { * }$ and a linear relation between the expected payoffs (4) may not be enforced   \n155 because the classification errors yield an additive disturbance on the state transition matrix as follows ",
+        "bbox": [
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+            462,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5594e463f2fd1ed6e6796d63268acdb64394b054a6597ca4d00bb8ff313accf2.jpg",
+        "text": "$$\n\\Omega - ( p _ { 1 } - p _ { 2 } ) \\left\\{ { \\bf q } ^ { \\top } [ 1 - \\Phi _ { t } ~ 0 ~ 1 - \\Phi _ { t } ~ 0 ] + ( { \\bf 1 } - { \\bf q } ) ^ { \\top } [ 0 ~ \\Psi _ { t } ~ 0 ~ \\Psi _ { t } ] \\right\\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            274,
+            527,
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+            547
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Adaptive strategies (9) cancel out this adverse disturbance on the dynamics of the repeated game ",
+        "bbox": [
+            158,
+            549,
+            807,
+            564
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "158 In the absence of classification errors ( $\\Phi _ { t } = 1$ and $\\Psi _ { t } = 0$ ), the adaptive strategies reduce to the ZD   \n159 strategies, i.e., $\\pi _ { t } = \\mathbf { p }$ . Classification errors force the FC to be more retaliatory than dictated by the   \n160 ZD strategy $\\mathbf { p }$ , i.e., $\\pi _ { t , 1 } > p _ { 1 }$ , $\\pi _ { t , 3 } > p _ { 3 }$ , $\\pi _ { t , 2 } < p _ { 2 }$ and $\\pi _ { t , 4 } < p _ { 4 }$ . In general, detection and false alarm   \n161 probabilities, $\\Phi _ { t }$ and $\\Psi _ { t }$ , are time-varying; thus the adaptive strategies also change over time. ",
+        "bbox": [
+            140,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "162 3.3 The Impact of Estimation Errors on Repeated Game Dynamics ",
+        "text_level": 1,
+        "bbox": [
+            150,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "163 The proposed adaptive strategies (9) requires the knowledge of detection probability, $\\Phi _ { t }$ . However, 164 the FC cannot exactly compute $\\Phi _ { t }$ using (8) since she does not have the knowledge of $m _ { t }$ . Instead, she can form her estimate 165 $\\hat { \\Phi } _ { t }$ using $\\widehat { m } _ { t }$ : ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/633d8d2e080c253199b188872b2239bc58676ed14cd636ac429aecb00054c06a.jpg",
+        "text": "$$\n\\widehat { \\Phi } _ { t } = 1 - \\mathcal { Q } \\left( \\frac { \\frac { 1 } { 2 } \\| \\widehat { m } _ { t } \\| ^ { 2 } } { \\sqrt { \\widehat { m } _ { t } ^ { \\top } \\Sigma _ { t } \\widehat { m } _ { t } } } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            405,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "166 Due to the inevitable gradient estimation errors, in general, we have $\\widehat \\Phi _ { t } \\neq \\Phi _ { t }$ . As a result, the FC   \n167 cannot exactly employ the adaptive FC strategies dictated by Eq. 9. With several steps of variable   \n168 substitutions, this yields an additive perturbation on the state transition matrix as follows: ",
+        "bbox": [
+            142,
+            772,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e3987109cc8b8b314f79d098776351668970335c9d6e3382b6a3fc2778343906.jpg",
+        "text": "$$\n\\widetilde { \\Omega } _ { t } = \\Omega + V _ { t } \\Omega ^ { \\bot } \\mathrm { ~ w i t h ~ } V _ { t } : = \\frac { \\widehat { \\Phi } _ { t } - \\Phi _ { t } } { \\widehat { \\Phi } _ { t } - \\Psi _ { t } } \\mathrm { ~ a n d ~ } \\Omega ^ { \\bot } : = ( p _ { 1 } - p _ { 2 } ) \\mathbf { q } ^ { \\top } [ - 1 \\mathrm { ~ 0 ~ 1 ~ } 0 ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            819,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "169 Let $\\tilde { \\Lambda } _ { t }$ be the probability distribution over the state space of the games $\\{ C c , C d , D c , D d \\}$ at the start   \n170 of iteration $t$ . According to (12), the state distributions follow the transition rule such that ",
+        "bbox": [
+            137,
+            859,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c66effef7b71e7d6676f607e1b27f6b245f15c5a22049e4036111fd68871c980.jpg",
+        "text": "$$\n\\widetilde { \\Lambda } _ { t + 1 } = \\widetilde { \\Lambda } _ { t } \\widetilde { \\Omega } _ { t } = \\widetilde { \\Lambda } _ { t } \\left( \\Omega + V _ { t } \\Omega ^ { \\perp } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            385,
+            892,
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+            914
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "171 Note that $\\Lambda _ { t }$ can be considered as the state distribution of the repeated games in the absence of   \n172 perturbations on the state transition matrix. For the FC, $\\Lambda _ { t }$ is the designed state distribution in which   \n173 the ZD strategy dominates against any user strategy. ",
+        "bbox": [
+            137,
+            90,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Next, we study the time-varying perturbation terms. Using (8) and (11),174 $V _ { t }$ can be found $\\mathrm { a s } ^ { 2 }$ ",
+        "bbox": [
+            138,
+            138,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/37a98a2b45b384c9d2a856c5e635ae27aa0a2e4a33fa404bcfc2fa51a7ef12bb.jpg",
+        "text": "$$\n\\begin{array} { r } { \\gamma _ { t } = \\frac { \\widehat { \\Phi } _ { t } - \\Phi _ { t } } { \\widehat { \\Phi } _ { t } - \\Psi _ { t } } = \\frac { Q ( \\frac { \\widehat { m } _ { t } ^ { \\top } ( m \\epsilon - \\frac { 1 } { 2 } \\widehat { m } \\widehat { u } _ { t } ) } { \\sqrt { \\widehat { m } _ { t } ^ { \\top } \\Sigma _ { t } \\widehat { m } _ { t } } } ) - Q ( \\frac { 1 } { \\sqrt { \\widehat { m } _ { t } ^ { \\top } \\Sigma _ { t } \\widehat { m } _ { t } } } ) } { 1 - Q ( \\frac { \\frac { 1 } { 2 } \\| \\widehat { m } _ { t } \\| ^ { 2 } } { \\sqrt { \\widehat { m } _ { t } ^ { \\top } \\Sigma _ { t } \\widehat { m } _ { t } } } ) - Q ( \\frac { \\frac { 1 } { 2 } \\| \\widehat { m } _ { t } \\| ^ { 2 } } { \\sqrt { \\widehat { m } _ { t } ^ { \\top } \\Sigma _ { t } \\widehat { m } _ { t } } } ) } = \\frac { Q ( \\frac { \\widehat { m } _ { t } ( \\widehat { m } \\epsilon - \\widehat { m } \\widehat { u } _ { t } ) } { \\sqrt { \\operatorname { R a y } ( \\Sigma _ { t } , \\widehat { m } _ { t } ) } } + \\frac { 1 } { 2 } \\| \\widehat { m } _ { t } \\| } { 1 - Q ( \\frac { \\frac { 1 } { 2 } \\| \\widehat { m } _ { t } \\| } { \\sqrt { \\operatorname { R a y } ( \\Sigma _ { t } , \\widehat { m } _ { t } ) } } ) - Q ( \\frac { \\frac { 1 } { 2 } \\| \\widehat { m } _ { t } \\| } { \\sqrt { \\operatorname { R a y } ( \\Xi _ { t } , \\widehat { m } _ { t } ) } } ) } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            161,
+            826,
+            232
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "175 In the presence of these perturbations, to establish stability guarantees on the dynamics of the repeated   \n176 games, we impose the following assumption on the norm of the gradient estimator: ",
+        "bbox": [
+            147,
+            238,
+            826,
+            267
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Assumption 1. We assume tha177 $^ { \\prime } \\Vert \\widehat { m } _ { t } \\Vert \\geq \\operatorname* { m a x } \\big \\{ 2 \\sqrt { \\mathrm { R a y } ( \\widehat { m } _ { t } , \\Sigma _ { t } ) } , 2 \\sqrt { \\mathrm { R a y } ( \\widehat { m } _ { t } , \\Xi _ { t } ) } , \\sqrt { \\widehat { m } _ { t } ^ { \\top } ( m _ { t } - \\widehat { m } _ { t } ) } \\big \\} .$ ",
+        "bbox": [
+            150,
+            271,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "178 Note that these conditions are primarily associated to the accuracy of the linear classifier (7) which   \n179 operates effectively when the mean vectors of the classes are sufficiently separated. The following   \n180 result indicates that, due to the perturbations on the state transition matrix, the real state distribution   \n181 $\\tilde { \\Lambda } _ { t }$ is a noisy version of $\\Lambda _ { t }$ .   \n82 Lemma 1. Let $\\Lambda _ { 1 }$ denote the initial state distributions of the games between the FC and the users.   \n183 Under Assumption $I$ , we have that ",
+        "bbox": [
+            140,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            150,
+            363,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/aeb8facbe553b770a44c0bbdc379e5d84a63063b2102ff014a82209c971ada0a.jpg",
+        "text": "$$\n\\tilde { \\Lambda } _ { t } = \\Lambda _ { t } + \\Lambda _ { 1 } \\sum _ { i = 1 } ^ { t - 1 } V _ { i } \\Omega ^ { i - 1 } \\Omega ^ { \\perp } \\Omega ^ { t - 1 - i } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "184 This noise on the state distributions will manifest as a novel bias term in the gradient estimation. In   \n185 the next subsection, we will provide the convergence analysis of SGD-SU which will mainly focus   \n186 on the characterization of this bias term. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "7 3.4 Convergence Results ",
+        "text_level": 1,
+        "bbox": [
+            158,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "188 In this section, we provide the convergence guarantee for SGD-SU (5). Let $\\mathcal { F } _ { t }$ denote the $\\sigma$ -algebra,   \n189 generated by $\\{ \\theta _ { 1 } , \\mathbf { Y } _ { i } , i < t \\}$ . In particular, $\\mathcal { F } _ { t }$ should be interpreted as the history of SGD-SU up to   \n190 iteration $t$ , just before $\\mathbf { Y } _ { t }$ is generated. Thus, conditioning on $\\mathcal { F } _ { t }$ can be thought of as conditioning   \n191 on $\\{ \\theta _ { 1 } , \\widetilde { \\Lambda } _ { 1 } , \\mathbf { \\check { Y } } _ { 1 } , \\dots , \\theta _ { t - 1 } , \\widetilde { \\Lambda } _ { t - 1 } , \\mathbf { Y } _ { t - 1 } , \\theta _ { t } , \\widetilde { \\Lambda } _ { t } \\}$ . For convenience, denote $\\mathbb { E } _ { t } [ \\cdot ] : = \\mathbb { E } _ { t } [ \\cdot | \\mathcal { F } _ { t } ]$ . Observe   \n192 that, we can decompose the gradient estimator $\\widehat { m } _ { t }$ as follows: ",
+        "bbox": [
+            140,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a42a61f3056f8e47962794ffa03921cb0fee828adaf4630be105c563a64cf73a.jpg",
+        "text": "$$\n\\widehat { m } _ { t } ( \\cdot ) = m _ { t } ( 1 + \\zeta _ { t } ) + \\mathcal { E } _ { t } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "193 where $\\zeta _ { t }$ is the estimation bias term due to the perturbations on the state transition matrix, given by ",
+        "bbox": [
+            142,
+            635,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7743fe96e25fee78a6e246cd2feea45e71d876f9cbb44656836f3517d9817570.jpg",
+        "text": "$$\n\\begin{array} { r } { \\zeta _ { t } = \\frac { 1 } { m _ { t } } \\left( \\mathbb { E } _ { t } [ \\widehat { m } _ { t } ] - m _ { t } \\right) = \\frac { \\sum _ { k = 1 } ^ { K } \\mathrm { P } \\left( B _ { k , t } = c | \\mathcal { F } _ { t } \\right) } { K \\left( \\Lambda _ { t } \\mathbf { q } ^ { \\top } \\right) } - 1 } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            326,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "194 and $\\mathcal { E } _ { t }$ is the estimation noise term, given by $\\mathcal { E } _ { t } = \\widehat { m } _ { t } - \\mathbb { E } _ { t } [ \\widehat { m } _ { t } ]$ . Conditioned on $\\mathcal { F } _ { t }$ , the probability of a user taking the cooperative action, in iteration 195 $t$ , is given by $\\mathrm { P } ( B _ { k , t } = c | \\mathcal { F } _ { t } ) = \\widetilde { \\Lambda } _ { t } \\mathbf { q } ^ { \\top }$ . The bias term, 196 $\\zeta _ { t }$ , can be found as follows: ",
+        "bbox": [
+            140,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/66724c1f164b1cfd3f1de652638c3cbaae9eded5b7a587cf7a9d78527c9ed0b0.jpg",
+        "text": "$$\n\\zeta _ { t } = \\frac { \\widetilde { \\Lambda } _ { t } \\mathbf { q } ^ { \\intercal } } { \\Lambda _ { t } \\mathbf { q } ^ { \\intercal } } - 1 .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            444,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "97 From Lemma 1 and (15), it is clear that the perturbations on the state transition matrix (12), directly   \n198 translates into a bias in the gradient estimation rule. ",
+        "bbox": [
+            151,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To establish convergence guarantees for the SGD-SU in (5), 199 $\\Lambda _ { t } \\mathbf { q } ^ { \\top }$ and $\\widetilde { \\Lambda } _ { t } \\mathbf { q } ^ { \\top }$ must meet the following 00 criteria during the course of the algorithm: ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "202 The first condition $\\Lambda _ { t } \\mathbf { q } ^ { \\top } \\geq 0 . 5$ is very mild in the sense that it merely requires that the probability   \n203 of user cooperation dictated by the memory-1 strategies $\\mathbf { p }$ and $\\mathbf { q }$ $\\mathrm { ~ \\textit ~ { ~ 1 ~ } ~ } \\times \\mathrm { ~ 4 ~ }$ vectors), in the absence   \n204 of perturbations, is larger than 0.5. The second condition $\\widetilde { \\Lambda } _ { t } \\mathbf { q } ^ { \\top } > 0$ states that, in the presence of   \n205 perturbations, the probability of user cooperation is always positive3. ",
+        "bbox": [
+            140,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "206 By Assumption 2, there exists a positive constant $H _ { T }$ such that ",
+        "bbox": [
+            145,
+            155,
+            588,
+            171
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3b6fe5d3cea76537ebfe4cf5df0e7f2c9ff61a4ea5ed18098ad8ffe9b892373c.jpg",
+        "text": "$$\n0 < | \\zeta _ { t } | < H _ { T } < 1 , \\forall t \\in \\{ 1 , \\ldots , T \\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            370,
+            174,
+            625,
+            191
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Further, we have the following lemma characterizing the properties of estimation noise. ",
+        "bbox": [
+            160,
+            194,
+            750,
+            209
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "08 Lemma 2. Conditioned on $\\mathcal { F } _ { t } ,$ , the estimation noise in iteration $t$ , denoted $\\mathcal { E } _ { t }$ , is a zero-mean random   \n09 vector with the mean square error given by ",
+        "bbox": [
+            151,
+            212,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8c2e785b59a64995b48921b2c7b97163990e088f892926e90dd356531cd89791.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbb { E } _ { t } [ \\| \\mathcal { E } _ { t } \\| ^ { 2 } ] = \\frac { 1 } { K \\left( \\Lambda _ { t } \\mathbf { q } ^ { \\top } \\right) } \\left( ( \\zeta _ { t } + 1 ) \\mathrm { t r } \\left( \\Sigma _ { t } - \\Xi _ { t } \\right) + \\frac { 1 } { \\Lambda _ { t } \\mathbf { q } ^ { \\top } } \\mathrm { t r } \\left( \\Xi _ { t } \\right) \\right) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "210 By (16) and (17), we have that ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/73f8793b7e9f4c91e68f094ed341b68ca90f3443bb1b96630aa3df69df924922.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbb { E } _ { t } \\left[ \\Vert \\mathcal { E } _ { t } \\Vert ^ { 2 } \\right] \\leq \\frac { E _ { T } } { K } \\mathrm { ~ w i t h ~ } E _ { T } : = \\frac { 1 } { \\Lambda _ { t } \\mathbf { q } ^ { \\top } } \\Bigg [ \\big ( H _ { T } + 1 \\big ) \\mathrm { t r } \\big ( \\Sigma _ { t } - \\Xi _ { t } \\big ) + \\frac { 1 } { \\Lambda _ { t } \\mathbf { q } ^ { \\top } } \\mathrm { t r } \\big ( \\Xi _ { t } \\big ) \\Bigg ] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "11 We impose the following assumption on the objective function, which is standard for performance   \n12 analysis of stochastic gradient-based methods [3, 28]. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "213 Assumption 3. The objective function $F$ and the SGD-SU satisfy the following: ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "214 ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "(i) $F$ is $L -$ smooth, that is, $F$ is differentiable and its gradient is $L$ −Lipschitz: ",
+        "bbox": [
+            202,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ebd6b7a08886a5ed70a8b2668516a8693231a8f1bb594a5c30dd2cb401613f35.jpg",
+        "text": "$$\n\\lVert \\nabla F ( { \\boldsymbol { \\theta } } ) - \\nabla F ( { \\boldsymbol { \\theta } } ^ { \\prime } ) \\rVert \\leq L \\lVert { \\boldsymbol { \\theta } } - { \\boldsymbol { \\theta } } \\rVert , \\ \\forall { \\boldsymbol { \\theta } } , { \\boldsymbol { \\theta } } ^ { \\prime } \\in \\mathbb { R } ^ { n } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "(ii) The sequence of iterates $\\{ \\theta _ { t } \\}$ is contained in an open set over which $F$ is bounded below by a scalar $F _ { \\mathrm { i n f } }$ . ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Our next result describes the behavior of the sequence of gradients of $F$ when fixed step sizes are employed. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "19 Theorem 1. Under Assumptions 2 and 3, suppose that the SGD-SU (5) is run for $T$ iterations with $a$ fixed stepsize 0 $\\bar { \\beta }$ satisfying ",
+        "bbox": [
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+            506,
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f83162874a8fd0ae230993b5f4a9e75248a25db0ea5724412dba658129c3322e.jpg",
+        "text": "$$\n0 < \\bar { \\beta } \\leq \\frac { 1 } { L ( 1 + H _ { T } ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "221 Then, the SGD algorithm with strategic users satisfies that ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f6f76f445fd3d82f32495275d888759837de4f207bff8852ccc29fcd77f6fb7c.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbb { E } \\left[ \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\left. \\nabla F ( \\theta _ { t } ) \\right. ^ { 2 } \\right] \\leq \\frac { L E _ { T } } { K ( 1 - H _ { T } ) } + \\frac { 2 ( F ( \\theta _ { 1 } ) - F _ { \\mathrm { i n f } } ) } { \\bar { \\beta } T ( 1 - H _ { T } ) } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "222 Theorem 1 illustrates the impact of the perturbations on the state transition matrix (12) on the   \n223 convergence rate of SGD-SU. When $H _ { T }$ is close to 0, SGD-SU performs similar to the basic   \n224 minibatch SGD. On the other hand, if $H _ { T }$ is close to 1, the optimality gap may be large. Our next   \n225 result will characterize the gradient estimation bias term $\\zeta _ { t }$ . First, we have the following assumption   \n226 on the state transition matrix $\\Omega$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Assumption 4. The state transition matrix $\\Omega$ can be diagonalized as $\\Omega = \\Gamma \\mathcal { U } \\Gamma ^ { - 1 }$ with $\\mathcal { U }$ has the eigenvalues of $\\Omega$ in descending order of magnitude: $1 \\geq | u _ { 2 } | \\geq | u _ { 3 } | \\geq | u _ { 4 } | \\geq 0 .$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Denote the element of 229 $\\Gamma ^ { - 1 }$ in the $i ^ { \\mathrm { t h } }$ row and $j ^ { \\mathrm { t h } }$ column as $\\Gamma _ { i j } ^ { - 1 }$ . Denote the four rows of $\\Gamma ^ { - 1 }$ by 230 $\\vec { \\gamma } _ { 1 } , \\dots , \\vec { \\gamma } _ { 4 }$ . Next, we define $\\delta$ as ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/7af7b71a7605db7e9496a9bf2cbcde62840d4b0ab3716603ba3555340ddf053a.jpg",
+        "text": "$$\n\\delta : = \\Bigg ( \\operatorname* { m a x } _ { j \\in \\{ 2 , 3 , 4 \\} } \\left| \\Gamma _ { 3 j } - \\Gamma _ { 1 j } \\right| \\Bigg ) \\Bigg ( \\operatorname* { m a x } _ { j \\in \\{ 2 , 3 , 4 \\} } \\left| \\vec { \\gamma } _ { j } \\mathbf { q } ^ { \\top } \\right| ^ { 2 } \\Bigg ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "231 Further, the first order Taylor approximation of the scalar variable $V _ { t }$ can be found as follows: ",
+        "bbox": [
+            150,
+            803,
+            781,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ddee0bb4b3b641ca3dd7016944245d4659777618a6d6c13ea5c0d72bbb39f5a7.jpg",
+        "text": "$$\nV _ { t } = \\frac { m _ { t } ^ { \\top } ( \\widehat { m } _ { t } - m _ { t } ) } { \\| m _ { t } \\| ^ { 2 } } h _ { t } ( m _ { t } ) \\mathrm { ~ w i t h ~ } h _ { t } ( m _ { t } ) : = \\frac { \\frac { \\| m _ { t } \\| } { \\sqrt { 2 \\pi \\mathrm { R a y } ( \\Sigma _ { t } , m _ { t } ) } } \\mathrm { e x p } \\left( - \\frac { 1 } { 8 } \\frac { \\| m _ { t } \\| ^ { 2 } } { \\mathrm { R a y } ( \\Sigma _ { t } , m _ { t } ) } \\right) } { 1 - \\mathcal { Q } \\left( \\frac { \\| m _ { t } \\| } { 2 \\sqrt { \\mathrm { R a y } ( \\Sigma _ { t } , m _ { t } ) } } \\right) - \\mathcal { Q } \\left( \\frac { \\| m _ { t } \\| } { 2 \\sqrt { \\mathrm { R a y } ( \\Xi _ { t } , m _ { t } ) } } \\right) } \\mathrm { . }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "3A sufficient condition for this requirement is that user strategies are forgiving in nature, i.e., $q _ { 1 } , q _ { 2 } , q _ { 3 } , q _ { 4 } > 0$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Define 232 $h _ { t } ^ { \\operatorname* { m a x } } : = \\operatorname* { m a x } _ { i \\in \\{ 1 , \\dots , t \\} } h _ { i } ( m _ { i } )$ . Our next result indicates that, the estimation bias term $\\zeta _ { t }$ can 233 be found in terms of the past gradient estimation errors. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "234 Theorem 2. Under Assumptions $^ { l }$ , 2 and 4, the gradient estimation bias term $\\zeta _ { t }$ , can be found as ",
+        "bbox": [
+            148,
+            123,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e5d1201c638953c453fbfda8774be7e0c2aea9f3dad9c588f201936c22a586b0.jpg",
+        "text": "$$\n\\zeta _ { t } = ( p _ { 1 } - p _ { 2 } ) \\sum _ { i = 1 } ^ { t - 1 } \\frac { \\Lambda _ { i } \\mathbf { q } ^ { \\top } } { \\Lambda _ { t } \\mathbf { q } ^ { \\top } } \\frac { m _ { i } ^ { \\top } \\mathcal { E } _ { i } } { \\| m _ { i } \\| ^ { 2 } } h _ { i } ( m _ { i } ) \\Delta _ { i , t }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            339,
+            143,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "235 with ",
+        "bbox": [
+            140,
+            185,
+            205,
+            199
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4d9213fd96be43406b55954bbea33c4faa0ddeae6d3c3968b519951d85304cac.jpg",
+        "text": "$$\n| \\Delta _ { i , t } | \\leq \\delta | u _ { 2 } | ^ { t - 1 - i } + \\delta ^ { 2 } h _ { t - 1 } ^ { \\operatorname* { m a x } } | u _ { 2 } | ^ { t - 2 - i } ( t - i - 1 ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            326,
+            195,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "236 Further, for some $0 < \\eta < 1$ we have ",
+        "bbox": [
+            142,
+            217,
+            418,
+            232
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bdf131b0ce41db55bf09ed366108b0018dfd9fb98a9758ad6d6133d59fe11f0d.jpg",
+        "text": "$$\n\\mathrm { P } \\left( | \\zeta _ { t } | < \\eta | \\alpha _ { 1 } , \\dots , \\alpha _ { t - 1 } \\right) > 1 - \\frac { \\sum _ { i = 1 } ^ { t - 1 } \\alpha _ { i } ^ { 2 } } { K \\eta ^ { 2 } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            352,
+            238,
+            642,
+            273
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "237 with ",
+        "bbox": [
+            140,
+            280,
+            205,
+            294
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7a5761b15647df382b73ccb140a7c6bb9c97e39e270b736cc5f5f3ce23fc18fa.jpg",
+        "text": "$$\n\\alpha _ { i } ^ { 2 } = \\frac { 2 \\left| ( \\nu _ { i } ^ { 2 } - \\xi _ { i } ^ { 2 } ) + \\frac { m _ { i } ^ { \\top } \\Sigma _ { i } m _ { i } } { \\Vert m _ { i } \\Vert ^ { 2 } } \\right| + \\frac { \\xi _ { i } ^ { 2 } } { \\Lambda _ { i } \\mathbf { q } ^ { \\top } } } { \\Vert m _ { i } \\Vert ^ { 2 } \\left( \\Lambda _ { i } \\mathbf { q } ^ { \\top } \\right) } \\left[ \\frac { \\Lambda _ { i } \\mathbf { q } ^ { \\top } } { \\Lambda _ { t } \\mathbf { q } ^ { \\top } } \\right] ^ { 2 } h _ { i } ^ { 2 } \\Delta _ { i , t } ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            297,
+            681,
+            343
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "238 Note that Eq. (21) indicates that, the estimation bias term $\\zeta _ { t }$ can be expanded in terms of past gradient   \n239 estimation errors. We prove that the absolute values of the coefficients, $| \\Delta _ { i , t } |$ ’s, are bounded as ",
+        "bbox": [
+            137,
+            354,
+            825,
+            385
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d57df4fe47b78de15dca50d019723c64f1d264a102b8c98fdc0657ea07ebd862.jpg",
+        "text": "$$\n| \\Delta _ { i , t } | \\leq \\delta | u _ { 2 } | ^ { t - 1 - i } + \\delta ^ { 2 } h _ { t - 1 } ^ { \\operatorname* { m a x } } | u _ { 2 } | ^ { t - 2 - i } ( t - i - 1 ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            326,
+            391,
+            668,
+            410
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "240 where $u _ { 2 }$ is the eigenvalue of $\\Omega$ with the second highest absolute value. Since $\\Omega$ is a row stochastic   \n241 matrix, $| u _ { 2 } | \\le 1$ . When $| u _ { 2 } |$ is strictly less than 1, $\\Delta _ { i , t }$ ’s decay fast as $t - i$ grows. This can also be   \n242 interpreted as the impact of past gradient estimation errors fade away quickly. Using this result, in   \n243 Eq.(22), we derive a high probability upper bound on the estimation bias term $\\zeta _ { t }$ . ",
+        "bbox": [
+            140,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "244 4 Experiments ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            491,
+            312,
+            507
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "245 In this section, we evaluate the performance of SGD-SU (5) using real-life datasets. All the results in   \n246 the preceding section assert convergence for the SG method (5) under the assumption that the FC can   \n247 access $\\Sigma _ { t }$ and $\\Xi _ { t }$ . In a real-life machine learning setting with strategic users, this information may   \n248 not be available to the FC. For convenience, define $\\widehat { \\mathcal { K } } _ { t } ^ { c }$ and $\\widehat { \\mathcal { K } } _ { t } ^ { d }$ as the sets of users who are classified   \n249 as cooperative $( \\hat { c } )$ and defective $( \\hat { d } )$ at iteration $t$ . Based on the user action classification, the FC can   \n250 form her estimates for the covariance matrices under the cooperative and defective actions as follows: ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cb86ed9decbae1464042897f436a2e3f2f39545e93e637e19208d0e4ce0465e5.jpg",
+        "text": "$$\n\\widehat { \\Sigma } _ { t } = \\frac { 1 } { \\vert \\widehat { K } _ { t } ^ { \\mathrm { c } } \\vert } \\sum _ { k \\in \\widehat { K } _ { t } ^ { \\mathrm { c } } } \\left( Y _ { k , t } - \\bar { Y } _ { t } ^ { \\mathrm { c } } \\right) \\left( Y _ { k , t } - \\bar { Y } _ { t } ^ { \\mathrm { c } } \\right) ^ { \\top } \\mathrm { ~ a n d ~ } \\widehat { \\Xi } _ { t } = \\frac { 1 } { \\vert \\widehat { K } _ { t } ^ { \\mathrm { d } } \\vert } \\sum _ { k \\in \\widehat { K } _ { t } ^ { \\mathrm { d } } } \\left( Y _ { k , t } - \\bar { Y } _ { t } ^ { \\mathrm { d } } \\right) \\left( Y _ { k , t } - \\bar { Y } _ { t } ^ { \\mathrm { d } } \\right) ^ { \\top } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            200,
+            609,
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+            648
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "where 251 $\\begin{array} { r } { \\bar { Y } _ { t } ^ { \\mathrm { c } } = \\frac { 1 } { | \\hat { K } _ { t } ^ { \\mathrm { c } } | } \\sum _ { k \\in \\hat { K } _ { t } ^ { \\mathrm { c } } } Y _ { k , t } } \\end{array}$ and $\\begin{array} { r } { \\bar { Y } _ { t } ^ { \\mathrm { d } } = \\frac { 1 } { | \\hat { K } _ { t } ^ { \\mathrm { d } } | } \\sum _ { k \\in \\hat { K } _ { t } ^ { \\mathrm { c } } } Y _ { k , t } } \\end{array}$ ",
+        "bbox": [
+            142,
+            654,
+            558,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "252 In our first set of experiments, we consider a binary logistic classification problem and use the KDD  \n253 Cup 04 dataset [6]. The goal of binary logistic classification experiments is to learn a classification   \n254 rule that differentiates between two types of particles generated in high energy collider experiments   \n255 based on 78 attributes [6]. In our second set of experiments, we consider a neural network trained on   \n256 the MNIST dataset. The number of users is chosen as $K = 5 0$ and mini-batch size is $s = 1 0$ . In the   \n257 experiments, we have tested the performance of two different ZD strategies, namely equalizer and   \n258 extortion[31].   \n259 For the logistic classification problem, Fig. 4a and 4b, depict the optimality gap under four different   \n260 user strategies: $\\mathbf { q } = [ 0 . 9 \\ 0 . { \\overset { \\cdot } { 1 } } 5 \\ 0 . 9 \\ 0 . 1 5 ]$ (stubborn), $\\mathbf { q } ^ { - } = [ 0 . 9 \\ 0 . 9 \\ 0 . 1 5 \\ 0 . 1 5 ]$ (tit-for-tat, ), ${ \\bf q } =$   \n261 $[ 0 . 9 0 . 1 5 0 . 1 5 0 . 9 ]$ (win-stay-lose-switch) and $\\mathbf { q } = [ 0 . 9 0 . 9 0 . 9 0 . 9 0 . 9 ]$ ] (full cooperation). For the full   \n262 cooperation, coin toss, tit-for-tat and stubborn user strategies, SGSU converges quickly. For Pavlov   \n263 user strategies, SGSU can eventually approach, albeit more slowly than other cases. Fig 4c and 4d   \n264 illustrate the probability of user cooperation, $\\widetilde { \\Lambda } _ { t } \\mathbf { q } ^ { \\top }$ , across different user strategies. The experimental   \n265 results validate Lemma 1 and the empirical user cooperation probabilities match the theoretical except   \n266 when the users are Pavlov. Unsurprisingly, when the users follow full cooperation (or coin toss)   \n267 strategy, they cooperate with probability 0.9 (or 0.5) regardless of the actual states of the repeated   \n268 games. For the cases with stubborn and tit-for-tat users, the games quickly converge to the steady   \n269 state distribution. Interestingly, for the cases with Pavlov users, the probability of user cooperation   \n270 decreases over time. This is associated to the performance of the linear classifier. For the image   \n271 classification problem, Fig 4e-h depict the training loss and testing accuracy across iterations for   \n272 different FC and user strategies. In all experiments, SGSU converges in the presence of strategic   \n273 users. Further details regarding the Experimental results are relegated to Appendix. ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/7159baeb09837f87e9b893545a3aeb956911b3ca3da5511cfadf1f4917dee7f8.jpg",
+        "image_caption": [
+            "Figure 2: Stochastic Descent Algorithm with Strategic Users "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "274 5 Future Directions ",
+        "text_level": 1,
+        "bbox": [
+            147,
+            483,
+            354,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this work, we study a distributed learning framework where strategic users train a learning model with a fusion center. The main objective of the FC is to encourage users to be cooperative by distributing rewards. Based on this, we devise a reward mechanism for the FC based on the ZDstrategies. Further, we examine the performance of SGD algorithm in the presence of strategic users. Our findings reveal that the algorithm has provable convergence and our empirical results verify our theoretical analysis. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "281 We are also working on the development of robust estimation tools in distributed learning with   \n282 strategic users. The geometric median is a reliable estimation technique when the collected data   \n283 contain outliers of large magnitude [10, 14, 24, 27]: ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
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+        "text": "$$\n\\operatorname { M e d } ( \\mathbf { Y } _ { t } ) : = \\arg \\operatorname* { m i n } _ { y \\in \\mathbb { R } ^ { n } } \\sum _ { k = 1 } ^ { K } \\| y - Y _ { k , t } \\| _ { 2 } .\n$$",
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+        "text": "284 The FC can use Med as a robust gradient estimator, especially when the variance of the uninformative   \n285 signals, $\\xi _ { t } ^ { 2 }$ , reported by the defective users, is very high. The geometric median (24) can be computed   \n286 by the Weiszfeld’s algorithm [34, 35], which is a special case of iteratively reweighted least squares.   \n287 In contrast, with the knowledge of $\\mathbf { q }$ , the modified sample mean estimator (6) allows the FC to trade   \n288 robustness for overall tractability of the algorithm with reduced computational complexity.   \n289 The linear classifier is vulnerable to vanishing gradients as the stochastic gradient descent algorithm   \n290 with strategic users (SGD-SU) converges to $\\theta ^ { * }$ . This can be addressed by modifying the classifier   \n291 to incorporate the information contained in the norm of the reported gradients. Furthermore, we   \n292 discuss how to extend the convergence guarantee for SGSU to allow heterogeneous user strategies.   \n293 The details are presented in Appendix. ",
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+        "text": "References ",
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