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+    {
+        "type": "text",
+        "text": "CLDA: Contrastive Learning for Semi-Supervised Domain Adaptation ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Ankit Singh ",
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+        "text": "Department of Computer Science Indian Institute of Technology, Madras singh.ankit@cse.iitm.ac.in ",
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Unsupervised Domain Adaptation (UDA) aims to align the labeled source distribution with the unlabeled target distribution to obtain domain invariant predictive models. However, the application of well-known UDA approaches does not generalize well in Semi-Supervised Domain Adaptation (SSDA) scenarios where few labeled samples from the target domain are available. This paper proposes a simple Contrastive Learning framework for semi-supervised Domain Adaptation (CLDA) that attempts to bridge the intra-domain gap between the labeled and unlabeled target distributions and the inter-domain gap between source and unlabeled target distribution in SSDA. We suggest employing class-wise contrastive learning to reduce the inter-domain gap and instance-level contrastive alignment between the original(input image) and strongly augmented unlabeled target images to minimize the intra-domain discrepancy. We have empirically shown that both of these modules complement each other to achieve superior performance. Experiments on three well-known domain adaptation benchmark datasets, namely DomainNet, Office-Home, and Office31, demonstrate the effectiveness of our approach. CLDA achieves state-of-the-art results on all the above datasets. ",
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+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Deep Convolutional networks [30, 52] have shown impressive performance in various computer vision tasks, e.g., image classification [19, 22] and action recognition [48, 23, 57, 32]. However, there is an inherent problem of generalizability with deep-learning models, i.e., models trained on one dataset(source domain) does not perform well on another domain. This loss of generalization is due to the presence of domain shift [11, 55] across the dataset. Recent works [46, 29] have shown that the presence of few labeled data from the target domain can significantly boost the performance of the convolutional neural network(CNN) based models. This observation led to the formulation of Semi-Supervised Domain Adaption (SSDA), which is a variant of Unsupervised Domain Adaptation where we have access to a few labeled samples from the target domain. ",
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+    {
+        "type": "text",
+        "text": "Unsupervised domain adaptation methods [42, 12, 36, 51, 35] try to transfer knowledge from the label rich source domain to the unlabeled target domain. Many such existing domain adaptation approaches [42, 12, 51] align the features of the source distribution with the target distribution without considering the category of the samples. These class-agnostic methods fail to generate discriminative features when aligning global distributions. Recently, owing to the success of contrastive approaches [6, 18, 39], in self-representation learning, some recent works [26, 28] have turned to instance-based contrastive approaches to reduce discrepancies across domains. ",
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+    {
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+        "text": "[46] reveals that the direct application of the well-known UDA approaches in Semi-Supervised Domain Adaptation yields sub-optimal performance. [29] has shown that supervision from labeled source and target samples can only ensure the partial cross-domain feature alignment. This creates aligned and unaligned sub-distributions of the target domain, causing intra-domain discrepancy apart from inter-domain discrepancy in SSDA. ",
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+        "img_path": "images/6b685103502be02c0b22dad148a39834efc5e2349ac431e7ecf24318a048e14e.jpg",
+        "image_caption": [
+            "Figure 1: Conceptual description of CLDA approach. (a) Intial distribution of samples from both domain .(b) Instance Contrastive Alignment ensures unlabeled target samples move into the low entropy area forming robust clusters (c) Inter-Domain Contrastive Alignment minimizes the distance between the clusters of same class from both domain (d) The clusters of both domain are well aligned and samples are far away from decision boundary. "
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+        "type": "text",
+        "text": "In this work, we propose CLDA, a simple single-stage novel contrastive learning framework to address the aforementioned problem. Our framework contains two significant components to learn domain agnostic representation. First, Inter-Domain Contrastive Alignment reduces the discrepancy between centroids of the same class from the source and the target domain while increasing the distance between the class centroids of different classes from both source and target domain. This ensures clusters of the same class from both domains are near each other in latent space than the clusters of the other classes from both domains. ",
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+        "text": "Second, inspired by the success of self-representation learning in semi-supervised settings [17, 6, 49], we propose to use Instance Contrastive Alignment to reduce the intra-domain discrepancy. In this, we first generate the augmented views of the unlabeled target images using image augmentation methods. Alignment of the features of the original and augmented images of the unlabeled samples from the target domain ensures that they are closer to each other in latent space. The alignment between two variants of the same image ensures that the classifier boundary lies in the low-density regions assuring that the feature representations of two variants of the unlabeled target images are similar, which helps to generate better clusters for the target domain. ",
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+        "text": "In summary, our key contributions are as follows. 1) We propose a novel, simple single-stage training framework for Semi-supervised Domain Adaptation. 2)We propose using alignment at class centroids and instance levels to reduce inter and intra domain discrepancies present in SSDA. 3)We evaluate the effectiveness of different augmentation approaches, for instance-based contrastive alignment in the SSDA setting. 4)We evaluate our approach over three well-known Domain Adaptation datasets (DomainNet, Office-Home, and Office31) to gain insights. Our approach achieves the state of the art results across multiple datasets showing its effectiveness. We perform extensive ablation experiments highlighting the role of different components of our framework. ",
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+    {
+        "type": "text",
+        "text": "2 Related Works ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "2.1 Unsupervised Domain Adaptation ",
+        "text_level": 1,
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+        "text": "Unsupervised Domain Adaptation (UDA) [14] is a well-studied problem, and most UDA algorithms reduce the domain gap by matching the features of the sources and target domain [16, 4, 24, 36, 51, 27]. Feature-based alignment methods reduce the global divergence [16, 51] between source and target distribution. Adversarial learning [12, 5, 34, 35, 42, 41] based approaches have shown impressive performance in reducing the divergence between source and target domains. It involves training the model to generate features to deceive the domain classifier, invariantly making the generated features domain agnostic. Recently, Image translation methods [20, 21, 38] have been explored in UDA where an image from the target domain is translated to the source domain to be treated as an image from the source domain to overcome the divergence present across domains. ",
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+    {
+        "type": "text",
+        "text": "Despite remarkable progress in UDA, [46] shows the UDA approaches do not perform well in the SSDA setting, which we consider in this work. ",
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+    {
+        "type": "text",
+        "text": "2.2 Semi-Supervised Learning ",
+        "text_level": 1,
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+        "text": "Semi-Supervised Learning(SSL) aims to leverage the vast amount of unlabeled data with limited labeled data to improve classifier performance. The main difference between SSL and SSDA is that SSL uses data sampled from the same distribution while SSDA deals with data sampled from two domains with inherent domain discrepancy. The current line of work in SSL [50, 3, 31, 10] follows consistency-based approaches to reduce the intra-domain gap. Mean teacher [53] uses two copies of the same model (student model and teacher model) to ensure consistency across augmented views of the images. Weights of the teacher model are updated as the exponential moving average of the weights of the student model. Mix-Match [3] and ReMixMatch [2] use interpolation between labeled and unlabeled data to generate perturbed features. Recently introduced FixMatch [50] achieves impressive performance using the confident pseudo labels of the unlabeled samples and treating them as labels for the strongly perturbed samples. However, direct application of SSL in the SSDA setting yields sub-optimal performance as the presumption in the SSL is that distributions of labeled and unlabeled data are identical, which is not the case in SSDA. ",
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+    {
+        "type": "text",
+        "text": "2.3 Contrastive Learning ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Contrastive Learning(CL) has shown impressive performance in self-representation learning [6, 1, 18, 54, 39]. Most contrastive learning methods align the representations of the positive pair (similar images) to be close to each other while making negative pairs apart. In semantic segmentation, [33] uses patch-wise contrastive learning to reduce the domain divergence by aligning the similar patches across domains. In domain adaptation, contrastive learning [28, 26] has been applied for alignment at the instance level to learn domain agnostic representations. [26, 28] use samples from the same class as positive pairs, and samples from different classes are counted as negative pairs. [26] modifies Maximum Mean Discrepancy (MMD) [16] loss to be used as a contrastive loss. In contrast to [28, 26], our work proposes to use contrastive learning in SSDA setting both at the class and instance level (across perturbed samples of the same image) to learn the semantic structure of the data better. ",
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+    {
+        "type": "text",
+        "text": "2.4 Semi-Supervised Domain Adaptation ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Semi-Supervised Domain Adaptation (SSDA) aims to reduce the discrepancy between the source and target distribution in the presence of limited labeled target samples. [46] first proposed to align the source and target distributions using adversarial training. [29] shows the presence of intra domain discrepancy in the target distribution and introduces a framework to mitigate it. [25] uses consistency alongside multiple adversarial strategies on top of MME [46]. [9] introduced the meta-learning framework for Semi-Supervised Domain Adaptation. [58] breaks down the SSDA problem into two subproblems, namely, SSL in the target domain and UDA problem across the source and target domains, and learn the optimal weights of the network using co-training. [37] proposed to use pretraining of the feature extractor and consistency across perturbed samples as a simple yet effective strategy for SSDA. [44] introduces a framework for SSDA consisting of a shared feature extractor and two classifiers with opposite purposes, which are trained in an alternative fashion; where one classifier tries to cluster the target samples while the other scatter the source samples, so that target features are well aligned with source domain features. Most of the above approaches are based on adversarial training, while our work proposes to use contrastive learning-based feature alignment at the class level and the instance level to reduce discrepancy across domains. ",
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+    {
+        "type": "text",
+        "text": "3 Methodology ",
+        "text_level": 1,
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+        "text": "In this section, we present our novel Semi-Supervised Domain Adaptation approach to learn domain agnostic representation. We will first introduce the background and notations used in our work and then describe our approach and its components in detail. ",
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+        "img_path": "images/e51ffd9d13deb32f523a10afdc3af7280418f9b75dd1bbf0946ee7f544d92b65.jpg",
+        "image_caption": [
+            "Figure 2: Outline of our CLDA Framework Our approach consists of aligning the outputs of the neural network at two levels. At the instance level, we try to maximize the similarity between features of unlabeled target images and strongly augmented unlabeled target images using Instance Contrastive Alignment. At the class level, we pass the images from both domains through the network, where we assign the labels to features of unlabeled target images and compute the centroids of each class of the target domain. Similarly, we compute the centroids for source domain features using their class labels. Finally, we maximize the similarity between centroids of the same class across domains by employing Inter-Domain Contrastive Alignment. We also used cross-entropy loss on the labeled source and target images, apart from the above components in our framework. "
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+    {
+        "type": "text",
+        "text": "3.1 Problem Formulation ",
+        "text_level": 1,
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+        "text": "In Semi-Supervised Domain Adaptation, we have datasets sampled from two domains. The source dataset contains labeled images $\\mathcal { D } _ { s } = \\{ ( x _ { i } ^ { s } , y _ { i } ^ { s } ) \\} _ { i = 1 } ^ { N _ { s } } \\subset \\mathcal { R } ^ { d } \\times \\mathbf { \\bar { \\mathcal { V } } }$ sampled from some distribution $P _ { S } ( X , Y )$ . Besides that, we have two sets of data sampled from target domain distribution $P _ { T } ( X , Y )$ . We denote the labeled set of images sampled from the target domain as $\\mathcal { D } _ { l t } = \\{ ( x _ { i } ^ { l t } , y _ { i } ^ { l t } ) \\} _ { i = 1 } ^ { N _ { l t } }$ . The unlabeled set sampled from target domain $\\mathcal { D } _ { t } = \\{ ( x _ { i } ^ { t } ) \\} _ { i = 1 } ^ { N _ { t } }$ contains large number of images $( N _ { t } \\gg N _ { l t } )$ without any corresponding labels associated with them. We also denote the labeled data from both domains as $\\mathcal { D } _ { l } = \\mathcal { D } _ { s } \\cup \\mathcal { D } _ { l t }$ . Labels $y _ { i } ^ { s }$ and $y _ { i } ^ { l t }$ of the samples from source and labeled target set correspond to one of the categories of the dataset having $K$ different classes/categories $i . e$ $\\bar { Y = \\{ 1 , 2 , . . . \\bar { K } \\} }$ . Our goal is to learn a task specific classifier using $D _ { s } , D _ { l t }$ and $D _ { t }$ to accurately predict labels on test data from target domain. ",
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+        "text": "3.2 Supervised Training ",
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+        "text": "Labeled source and target samples are passed through the CNN-based feature extractor $\\mathcal { G } ( . )$ to obtain corresponding features, which are then passed through task-specific classifier $\\mathcal F ( . )$ to minimize the well-known cross-entropy loss on the labeled images from both source and target domains. ",
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+        "img_path": "images/6ad462806f90acc7495b4054ba58388eaf979f97f17cd95276ad58b5043df9d4.jpg",
+        "text": "$$\n\\mathcal { L } _ { s u p } = - \\sum _ { k = 1 } ^ { K } ( y ^ { i } ) _ { k } \\log ( \\mathcal { F } ( \\mathcal { G } ( ( x _ { l } ^ { i } ) ) _ { k }\n$$",
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+        "type": "text",
+        "text": "3.3 Inter-Domain Contrastive Alignment ",
+        "text_level": 1,
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+        "text": "Our method is based on the observation that the samples from the same category across domains must cluster in the latent space. However, this is observed only for the source domain due to the availability of the labels. Samples from the target domain do not align to form clusters due to the domain shift between the target and the source distributions. This discrepancy between the cluster of the same category across domains is reduced by aligning the centroids of each class of source and target domain. [6, 17] have shown that having a separate projection space is beneficial for contrastive training. Instead of using a separate projection, we have used the outputs from the task-specific classifier as features to align the clusters across the domain. ",
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+        "text": "We represent the centroid of the images from the source domain belonging to class $k$ as the mean of their features, which can be written as ",
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+        "type": "equation",
+        "img_path": "images/7a592ec5fa5a41cd2cee8867897545cacdbb35c75cf4f7f125ba4946b73f5655.jpg",
+        "text": "$$\nC _ { k } ^ { s } = \\frac { \\displaystyle \\sum _ { i = 1 } ^ { i = B } \\mathbb { 1 } _ { \\{ y _ { i } ^ { s } = k \\} } \\mathcal { F } ( \\mathcal { G } ( x _ { i } ^ { s } ) ) } { \\displaystyle \\sum _ { i = 1 } ^ { i = B } \\mathbb { 1 } _ { \\{ y _ { i } ^ { s } = k \\} } }\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where $B$ is the size of batch. We maintain a memory bank $\\displaystyle C ^ { s } = [ C _ { 1 } ^ { s } , C _ { 2 } ^ { s } , . . . . C _ { K } ^ { s } ] )$ to store the centroids of each class from source domain. We use exponential moving average to update these centroid values during the training ",
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+        "type": "equation",
+        "img_path": "images/108d56bcfbc3761ab0da004d1aaeef3c428782f5c30c58a6b410722dd9560192.jpg",
+        "text": "$$\nC _ { k } ^ { s } = \\rho ( C _ { k } ^ { s } ) _ { s t e p } + ( 1 - \\rho ) ( C _ { k } ^ { s } ) _ { s t e p - 1 }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\rho$ is a momentum term, and $( C _ { k } ^ { s } ) _ { s t e p }$ and $( C _ { k } ^ { s } ) _ { s t e p - 1 }$ are the centroid values of class $k$ at the current and previous step, respectively. ",
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+        "type": "text",
+        "text": "We also need to cluster the unlabeled target samples for Inter-Domain Contrastive Alignment. The pseudo labels obtained from the task specific classifier as shown in Eq (3) is used as the class labels for the corresponding unlabeled target samples. ",
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+        "type": "equation",
+        "img_path": "images/5b08c952c1b05ac646d9446c37e36e22fc4eb113b72b0a8cbdac3cbab42e6a64.jpg",
+        "text": "$$\n\\hat { y _ { i } ^ { t } } = a r g m a x ( ( \\mathcal { F } ( \\mathcal { G } ( x _ { i } ^ { t } ) ) )\n$$",
+        "text_format": "latex",
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+        "text": "Similar to the source domain , we also calculate the separate cluster centroid $C _ { k } ^ { t }$ for each of the class $k$ of the target samples present in the minibatch as per the Eq (2) where unlabeled target images replace the images from the source domain with their corresponding pseudo label. The model is then trained to maximize the similarity between the cluster representation of each class $k$ from the source and the target domain. $C _ { k } ^ { s }$ and $C _ { k } ^ { t }$ form the positive pair while the remaining cluster centroids from both domains form the negative pairs. The remaining clusters from both domains are pushed apart in the latent space. This is achieved through employing a modified NT-Xent (normalized temperature-scaled cross-entropy) contrastive loss [6, 39, 49, 33] for domain adaptation given by ",
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+        "text": "$$\n\\mathcal { L } _ { c l u } ( C _ { i } ^ { t } , C _ { i } ^ { s } ) = - \\log \\frac { h \\bigl ( C _ { i } ^ { t } , C _ { i } ^ { s } \\bigr ) } { h \\bigl ( C _ { i } ^ { t } , C _ { i } ^ { s } \\bigr ) + \\underset { t \\in \\{ s , t \\} } { \\overset { K } { \\sum } } \\mathbb { 1 } _ { \\{ r \\neq i \\} } h \\bigl ( C _ { i } ^ { t } , C _ { r } ^ { q } \\bigr ) }\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where $\\begin{array} { r } { h ( \\mathbf { u } , \\mathbf { v } ) = \\exp \\big ( \\frac { \\mathbf { u } ^ { \\top } \\mathbf { v } } { \\| \\mathbf { u } \\| _ { 2 } \\| \\mathbf { v } \\| _ { 2 } } / \\tau \\big ) } \\end{array}$ measures the exponential of cosine similarity , $\\mathbb { 1 }$ is an indicator function and $\\tau$ is the temperature hyperparameter. ",
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+        "type": "text",
+        "text": "3.4 Instance Contrastive Alignment ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Recent works on contrastive learning [18, 39, 6] show encouraging results in single domain settings. [28] extends contrastive learning into multi-domain settings. Inspired by such success, we employ Instance Contrastive Learning to form stable and correct cluster cores in the target domain. ",
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+    {
+        "type": "text",
+        "text": "To perform contrastive alignment at the instance level, we first generate a strongly augmented version of the unlabeled target image i.e $\\tilde { x _ { i } ^ { t } } = \\psi ( x _ { i } ^ { t } )$ where $\\psi ( . )$ is the strong augmentation function [8]. Next, we employ the NT-Xent loss [6, 39] as defined in Eq (5) to ensure that these two variants of the same image are closer to each other in the latent space while the rest of the images in minibatch of size $B$ are pushed apart. This idea stems from the cluster assumption in an ideal classifier, which states the decision boundary should lie in the low-density region, ensuring consistent prediction for different augmented variants of the same image. ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "equation",
+        "img_path": "images/5320eef9102ca9a5bb8ea5c51f7df47cb24149b14f752a16ffd72633e169dc34.jpg",
+        "text": "$$\n\\mathcal { L } _ { i n s } ( \\tilde { x } _ { i } ^ { t } , x _ { i } ^ { t } ) = - \\log \\frac { h \\big ( \\mathcal { F } ( \\mathcal { G } ( \\tilde { x } _ { i } ^ { t } ) , \\mathcal { F } ( \\mathcal { G } ( x _ { i } ^ { t } ) ) ) } { \\displaystyle \\sum _ { r = 1 } ^ { B } h \\big ( \\mathcal { F } ( \\mathcal { G } ( \\tilde { x } _ { i } ^ { t } ) ) , \\mathcal { F } ( \\mathcal { G } ( x _ { r } ^ { t } ) ) \\big ) + \\displaystyle \\sum _ { r = 1 } ^ { B } \\mathbb { 1 } _ { \\{ r \\neq i \\} } h \\big ( \\mathcal { F } ( \\mathcal { G } ( \\tilde { x } _ { i } ^ { t } ) ) , \\mathcal { F } ( \\mathcal { G } ( \\tilde { x } _ { r } ^ { t } ) ) \\big ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In SSDA, [29] has shown that target distribution gets divided into aligned and unaligned subdistribution in the presence of very few labeled target data. Thus, aligning the unaligned subdistribution can lead to improved performance, while perturbing the aligned sub-distribution can result in a negative transfer. Therefore, we only propagate the gradients for strongly augmented images to avoid perturbing the aligned sub-distribution in the target domain. ",
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+    {
+        "type": "text",
+        "text": "[6] shows stronger augmentation in contrastive learning leads to improved performance. Consistent prediction across the input and strongly augmented unlabeled images in Instance Contrastive Alignment forces the unaligned target sub-distribution to move away from the low-density region towards aligned distribution. This ensures better clustering in the unlabeled target distribution, which is validated by improved accuracy as shown in Table 5 after employing Instance Contrastive Alignment with Inter-Domain Contrastive Alignment. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Both of the components of the CLDA framework are necessary for the improved performance, as shown in Table 5 . Instance Contrastive Alignment ensures that unlabeled target samples are consistent and are in the high-density region. However, it does not assure alignment between source and unlabeled target samples. Inter-Domain Contrastive Alignment reduces the discrepancy between unlabeled target samples and source domain but unlabeled target samples closer to the decision boundary might get pushed towards the wrong classes resulting in negative transfer. Thus, combining both components results in a much better alignment of the unlabeled target samples towards the source domain, leading to improved performance of the framework. ",
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+    },
+    {
+        "type": "text",
+        "text": "3.5 Overall framework and training objective ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The overall training objective employs supervised loss, Inter-Domain Contrastive Alignment and Instance Contrastive Alignment which can be formulated as follows: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/3005ea8b332501d03f3b20f9050675d2ac0aa4a5b09393e6c2489fe02f7e0a88.jpg",
+        "text": "$$\n\\mathcal { L } _ { t o t } = \\mathcal { L } _ { s u p } + \\alpha * \\mathcal { L } _ { c l u } + \\beta * \\mathcal { L } _ { i n s }\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "We train the model in our framework by employing overall training loss described as in (6). ",
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+    },
+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "4.1 Experimental Setup ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We evaluate the effectiveness of our approach on three different domain adaptation datasets: DomainNet [43], Office-Home [56] and Office31 [45]. DomainNet [43] is a large-scale domain adaptation dataset with 345 classes across 6 domains. Following MME [46], we use a subset of the dataset containing 126 categories across four domains: Real(R), Clipart(C), Sketch(S), and Painting(P). The performance on DomainNet is evaluated using 7 different combinations out of possible 12 combinations. Office-Home [56] is another widely used domain adaptation benchmark dataset with 65 classes across four domains: Art(Ar), Product $( \\mathrm { P r } )$ , Clipart(Cl), and Real (Rl). We perform experiments on all possible combinations of 4 domains. Office31 [45] is a relatively smaller dataset containing just 31 categories of data across three domains- Amazon(A), Dslr(D), Webcam(W). Following prior work [46, 29], we evaluate our approach on two combinations for the office31 dataset. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For the fair comparison, we use the data-splits (train, validation, and test splits) released by [46] on Github 1. We use the same settings for the benchmark datasets as in the prior work [46, 29], including the number of labeled samples in the target domain, which are consistent across all experiments. ",
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+    {
+        "type": "text",
+        "text": "4.2 Implementation Details ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Similar to the previous works on SSDA [46, 29, 9], we use Resnet34 and Alexnet as the backbone networks in our paper. We only used VGG for Office31 due to its higher memory requirements. The feature generator model is initialized with ImageNet weights, and the classifier is randomly initialized and has the same architecture as in [46, 29, 9]. All our experiments are performed using Pytorch [40].We use an identical set of hyperparameters $\\langle \\alpha = 4$ , $\\beta = 1$ ) across all our experiments other than minibatch size. All the hyperparameters values are decided using validation performance on Product to Art experiments on the Office-Home dataset. We have set $\\tau = 5$ in our experiments. Each minibatch of size $B$ contains an equal number of source and labeled target examples, while the number of unlabeled target samples is $\\mu \\times B$ . We study the effect of $\\mu$ in section 4.5. Resnet34 experiments are performed with minibatch size, $B = 3 2$ and Alexnet models are trained with $B = 2 4$ We use $\\mu = 4$ for all our experiments. We use SGD optimizer with a momentum of 0.9 and an initial learning rate of 0.01 with cosine learning rate decay for all our experiments. Weight decay is set to 0.0005 for all our models. Other details of the experiments are included in the supplementary. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/e492adbf1b627dc56901542b95751e5968717b6c73a7b68b65ce2529d528f809.jpg",
+        "table_caption": [
+            "Table 1: Performance Comparison in Office-Home. Numbers show top-1 accuracy values for different domain adaptation scenarios under 3-shot setting using Alexnet and Resnet34 as backbone networks. We have highlighted the best method for each transfer task. CLDA surpasses all the baseline methods in most adaptation scenarios. Our Proposed framework achieves the best average performance among all compared methods. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Net</td><td>Method</td><td>R1→C1</td><td>R1→Pr</td><td>Rl→Ar</td><td>Pr-→Rl</td><td>Pr→Cl</td><td>Pr→Ar</td><td>Ar→Pl</td><td>Ar-→C1</td><td>Ar-→Rl</td><td>CI-→Rl</td><td>Cl→Ar</td><td>C1-→Pr</td><td>Mean</td></tr><tr><td rowspan=\"10\">Alexnet</td><td>S+T</td><td>44.6</td><td>66.7</td><td>47.7</td><td>57.8</td><td>44.4</td><td>36.1</td><td>57.6</td><td>38.8</td><td>57.0</td><td>54.3</td><td>37.5</td><td>57.9</td><td>50.0</td></tr><tr><td>DANN</td><td>47.2</td><td>66.7</td><td>46.6</td><td>58.1</td><td>44.4</td><td>36.1</td><td>57.2</td><td>39.8</td><td>56.6</td><td>54.3</td><td>38.6</td><td>57.9</td><td>50.3</td></tr><tr><td>ADR</td><td>37.8</td><td>63.5</td><td>45.4</td><td>53.5</td><td>32.5</td><td>32.2</td><td>49.5</td><td>31.8</td><td>53.4</td><td>49.7</td><td>34.2</td><td>50.4</td><td>44.5</td></tr><tr><td>CDAN</td><td>36.1</td><td>62.3</td><td>42.2</td><td>52.7</td><td>28.0</td><td>27.8</td><td>48.7</td><td>28.0</td><td>51.3</td><td>41.0</td><td>26.8</td><td>49.9</td><td>41.2</td></tr><tr><td>ENT</td><td>44.9</td><td>70.4</td><td>47.1</td><td>60.3</td><td>41.2</td><td>34.6</td><td>60.7</td><td>37.8</td><td>60.5</td><td>58.0</td><td>31.8</td><td>63.4</td><td>50.9</td></tr><tr><td>MME</td><td>51.2</td><td>73.0</td><td>50.3</td><td>61.6</td><td>47.2</td><td>40.7</td><td>63.9</td><td>43.8</td><td>61.4</td><td>59.9</td><td>44.7</td><td>64.7</td><td>55.2</td></tr><tr><td>Meta-MME</td><td>50.3</td><td>-</td><td>-</td><td>-</td><td>48.3</td><td>40.3</td><td>:</td><td>44.5</td><td>-</td><td>-</td><td>44.5</td><td>-</td><td>-</td></tr><tr><td>BiAT</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>56.4</td></tr><tr><td>APE CLDA(ours)</td><td>51.9</td><td>74.6</td><td>51.2</td><td>61.6</td><td>47.9</td><td>42.1</td><td>65.5</td><td>44.5</td><td>60.9</td><td>58.1</td><td>44.3</td><td>64.8</td><td>55.6</td></tr><tr><td></td><td>51.5</td><td>74.1</td><td>54.3</td><td>67.0</td><td>47.9</td><td>47.0</td><td>65.8</td><td>47.4</td><td>66.6</td><td>64.1</td><td>46.8</td><td>67.5</td><td>58.3</td></tr><tr><td rowspan=\"7\">Resnet34</td><td>S+T</td><td>55.7</td><td>80.8</td><td>67.8</td><td>73.1</td><td>53.8</td><td>63.5</td><td>73.1</td><td>54.0</td><td>74.2</td><td>68.3</td><td>57.6</td><td>72.3</td><td>66.2</td></tr><tr><td>DANN</td><td>57.3</td><td>75.5</td><td>65.2</td><td>69.2</td><td>51.8</td><td>56.6</td><td>68.3</td><td>54.7</td><td>73.8</td><td>67.1</td><td>55.1</td><td>67.5</td><td>63.5</td></tr><tr><td>ENT</td><td>62.6</td><td>85.7</td><td>70.2</td><td>79.9</td><td>60.5</td><td>63.9</td><td>79.5</td><td>61.3</td><td>79.1</td><td>76.4</td><td>64.7</td><td>79.1</td><td>71.9</td></tr><tr><td>MME</td><td>64.6</td><td>85.5</td><td>71.3</td><td>80.1</td><td>64.6</td><td>65.5</td><td>79.0</td><td>63.6</td><td>79.7</td><td>76.6</td><td>67.2</td><td>79.3</td><td>73.1</td></tr><tr><td>Meta-MME</td><td>65.2</td><td>-</td><td>-</td><td>-</td><td>64.5</td><td>66.7</td><td>-</td><td>63.3</td><td>-</td><td>-</td><td>67.5</td><td>-</td><td>-</td></tr><tr><td>APE</td><td>66.4</td><td>86.2</td><td>73.4</td><td>82.0</td><td>65.2</td><td>66.1</td><td>81.1</td><td>63.9</td><td>80.2</td><td>76.8</td><td>66.6</td><td>79.9</td><td>74.0</td></tr><tr><td>CLDA (ours)</td><td>66.0</td><td>87.6</td><td>76.7</td><td>82.2</td><td>63.9</td><td>72.4</td><td>81.4</td><td>63.4</td><td>81.3</td><td>80.3</td><td>70.5</td><td>80.9</td><td>75.5</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.3 Baselines ",
+        "text_level": 1,
+        "bbox": [
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+            532,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We compare our CLDA framework with previous state-of-the-art SSDA approaches : MME [46], APE [29], BiAT [25] , UODA [44], Meta-MME [9] and ENT [15] using the performance reported by these papers. papers. We also included the results from adversarial based baseline methods: DANN [13], ADR [47] and CDAN [35] as reported in [46]. We also provide the $\\mathbf { s } { + } \\mathbf { T }$ results where the model is trained using all the labeled samples across domains. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.4 Results ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            648,
+            263,
+            664
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 1- 3 show top-1 accuracies and mean accuracies for different combination of domain adaptation scenarios for all three datasets in comparison with baseline SSDA methods. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Office-Home. Table 1 contains the results of the Office-Home dataset for 3-shot setting with Alexnet and Resnet34 as backbone networks. Results for the 1-shot adaptation scenarios are included in the supplementary. Our method consistently performs better than the baseline approaches and achieves $5 8 . 3 \\%$ and $7 5 . 5 \\%$ mean accuracy with Alexnet and Resnet34, respectively. Our approach surpasses the state-of-the-art SSDA approaches in most of the adaptation tasks. In some domain adaptation cases, such as $\\mathrm { C l }  \\mathrm { R l }$ , $\\mathbf { R } 1  \\mathbf { A r }$ and $\\mathrm { P r }  \\mathrm { A r }$ , we exceeded APE by more than $3 \\%$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "DomainNet: Our CLDA approach surpasses the performance of existing SSDA baselines as shown in Table 2. Using Alexnet backbone, our method improves over BiAT by $5 . 2 \\%$ and $4 . 9 \\%$ in 1-shot and 3-shot settings, respectively. We obtain similarly improved performance when we switch the neural backbone from Alexnet to Resnet34. With Resnet34 as the backbone, we gain $4 . 3 \\%$ and $3 . 6 \\%$ over APE in 1-shot and 3-shot settings, respectively. Similar to the Office-Home, our approach surpasses the well-known domain adaptation benchmarks methods in most domain adaptation tasks of the DomainNet dataset. Such consistent improved performance shows that our approach reduces both inter and intra domain discrepancy prevalent in SSDA. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/b07128146fdfa0e187d3c8fb2b36a656e6b052e0fb3ef039fb5c2da52221fc8b.jpg",
+        "table_caption": [
+            "Table 2: Performance Comparison in DomainNet. Numbers show Top-1 accuracy values for different domain adaptation scenarios under 1-shot and 3-shot settings using Alexnet and Resnet34 as backbone networks. CLDA achieves better performance than all the baseline methods in most of the domain adaptation tasks. We have highlighted the best approach for each domain adaptation task. Our Proposed framework achieves the best average performance among all compared methods. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Net</td><td rowspan=\"2\">Method</td><td colspan=\"2\">R→C</td><td colspan=\"2\">R→P</td><td colspan=\"2\">P→C</td><td colspan=\"2\">C→S</td><td colspan=\"2\">S→P</td><td colspan=\"2\">R→S</td><td colspan=\"2\">P→R</td><td colspan=\"2\">Mean 1-shot3-shot</td></tr><tr><td>1-shot3-shot</td><td></td><td>1-shot</td><td>3-shot</td><td></td><td>1-shot3-shot</td><td>1-shot</td><td>3-shot</td><td>1-shot</td><td>3-shot</td><td>1-shot3-shot</td><td></td><td>1-shot3-shot</td><td></td><td></td><td></td></tr><tr><td rowspan=\"9\">Alexnet</td><td>S+T</td><td>43.3</td><td>47.1</td><td>42.4</td><td>45.0 43.8</td><td>40.1 39.1</td><td>44.9</td><td>33.6</td><td>36.4</td><td>35.7</td><td>38.4</td><td>29.1</td><td>33.3</td><td>55.8</td><td>58.7</td><td>40.0 40.4</td><td>43.4</td></tr><tr><td>DANN</td><td>43.3</td><td>46.1</td><td>41.6</td><td></td><td></td><td>41.0</td><td>35.9</td><td>36.5</td><td>36.9</td><td>38.9</td><td>32.5</td><td>33.4</td><td>53.5</td><td>57.3</td><td></td><td>42.4</td></tr><tr><td>ADR</td><td>43.1</td><td>46.2</td><td>41.4</td><td>44.4</td><td>39.3</td><td>43.6</td><td>32.8</td><td>36.4</td><td>33.1</td><td>38.9</td><td>29.1</td><td>32.4</td><td>55.9</td><td>57.3</td><td>39.2</td><td>42.7</td></tr><tr><td>CDAN</td><td>46.3</td><td>46.8</td><td>45.7</td><td>45.0</td><td>38.3</td><td>42.3</td><td>27.5</td><td>29.5</td><td>30.2</td><td>33.7</td><td>28.8</td><td>31.3</td><td>56.7</td><td>58.7</td><td>39.1</td><td>41.0</td></tr><tr><td>ENT</td><td>37.0</td><td>45.5</td><td>35.6</td><td>42.6</td><td>26.8</td><td>40.4</td><td>18.9</td><td>31.1</td><td>15.1</td><td>29.6</td><td>18.0</td><td>29.6</td><td>52.2</td><td>60.0</td><td>29.1</td><td>39.8</td></tr><tr><td>MME</td><td>48.9</td><td>55.6</td><td>48.0</td><td>49.0</td><td>46.7</td><td>51.7</td><td>36.3</td><td>39.4</td><td>39.4</td><td>43.0</td><td>33.3</td><td>37.9</td><td>56.8</td><td>60.7</td><td>44.2</td><td>48.2</td></tr><tr><td>Meta-MME</td><td>1</td><td>56.4</td><td>-</td><td>50.2</td><td></td><td>51.9</td><td>，</td><td>39.6</td><td>，</td><td>43.7</td><td>-</td><td>38.7</td><td>1</td><td>60.7</td><td>-</td><td>48.8</td></tr><tr><td>BiAT</td><td>54.2</td><td>58.6</td><td>49.2</td><td>50.6</td><td>44.0</td><td>52.0</td><td>37.7</td><td>41.9</td><td>39.6</td><td>42.1</td><td>37.2</td><td>42.0</td><td>56.9</td><td>58.8</td><td>45.5</td><td>49.4</td></tr><tr><td>APE CLDA (ours)</td><td>47.7 56.3</td><td>54.6 59.9</td><td>49.0 56.0</td><td>50.5</td><td>46.9</td><td>52.1 54.6</td><td>38.5 42.5</td><td>42.6</td><td>38.5</td><td>42.2</td><td>33.8 38.0</td><td>38.7</td><td>57.5</td><td>61.4</td><td>44.6</td><td>48.9</td></tr><tr><td></td><td></td><td></td><td></td><td>57.2</td><td>50.8</td><td></td><td></td><td>47.3</td><td>46.8</td><td></td><td>51.4</td><td>42.7</td><td></td><td>64.4</td><td>67.0</td><td>50.7</td><td>54.3</td></tr><tr><td rowspan=\"10\"></td><td>S+T</td><td>55.6</td><td>60.0</td><td>60.6</td><td>62.2</td><td>56.8</td><td>59.4</td><td>50.8</td><td>55.0</td><td>56.0</td><td>59.5</td><td>46.3</td><td>50.1</td><td>71.8</td><td>73.9</td><td>56.9</td><td>60.0</td></tr><tr><td>DANN</td><td>58.2</td><td>59.8</td><td>61.4</td><td>62.8</td><td>56.3</td><td>59.6</td><td>52.8</td><td>55.4</td><td>57.4</td><td>59.9</td><td>52.2</td><td>54.9</td><td>70.3</td><td>72.2</td><td>58.4</td><td>60.7</td></tr><tr><td>ADR</td><td>57.1</td><td>60.7</td><td>61.3</td><td>61.9</td><td>57.0</td><td>60.7</td><td>51.0</td><td>54.4</td><td>56.0</td><td>59.9</td><td>49.0</td><td>51.1</td><td>72.0</td><td>74.2</td><td>57.6</td><td>60.4</td></tr><tr><td>CDAN</td><td>65.0</td><td>69.0</td><td>64.9</td><td>67.3</td><td>63.7</td><td>68.4</td><td>53.1</td><td>57.8</td><td>63.4</td><td>65.3</td><td>54.5</td><td>59.0</td><td>73.2</td><td>78.5</td><td>62.5</td><td>66.5</td></tr><tr><td>ENT</td><td>65.2</td><td>71.0</td><td>65.9</td><td>69.2</td><td>65.4</td><td>71.1</td><td>54.6</td><td>60.0</td><td>59.7</td><td>62.1</td><td>52.1</td><td>61.1</td><td>75.0</td><td>78.6</td><td>62.6</td><td>67.6</td></tr><tr><td>MME</td><td>70.0</td><td>72.2</td><td>67.7</td><td>69.7</td><td>69.0</td><td>71.7</td><td>56.3</td><td>61.8</td><td>64.8</td><td>66.8</td><td>61.0</td><td>61.9</td><td>76.1</td><td>78.5</td><td>66.4</td><td>68.9</td></tr><tr><td>UODA</td><td>72.7</td><td>75.4</td><td>70.3</td><td>71.5</td><td>69.8</td><td>73.2</td><td>60.5</td><td>64.1</td><td>66.4</td><td>69.4</td><td>62.7</td><td>64.2</td><td>77.3</td><td>80.8</td><td>68.5</td><td>71.2</td></tr><tr><td>Meta-MME</td><td>1</td><td>73.5</td><td>1</td><td>70.3</td><td>1</td><td>72.8</td><td>-</td><td>62.8</td><td>1</td><td>68.0</td><td>-</td><td>63.8</td><td>-</td><td>79.2</td><td>-</td><td>70.1</td></tr><tr><td>BiAT</td><td>73.0</td><td>74.9</td><td>68.0</td><td>68.8</td><td>71.6</td><td>74.6</td><td>57.9</td><td>61.5</td><td>63.9</td><td>67.5</td><td>58.5</td><td>62.1</td><td>77.0</td><td>78.6</td><td>67.1</td><td>69.7</td></tr><tr><td>APE</td><td>70.4</td><td>76.6 77.7</td><td>70.8 75.1</td><td>72.1 75.7</td><td>72.9</td><td>76.7 76.4</td><td>56.7 63.7</td><td>63.1 69.7</td><td>64.5 70.2</td><td>66.1 73.7</td><td>63.0 67.1</td><td>67.8</td><td>76.6</td><td>79.4 82.9</td><td>67.6 71.9</td><td>71.7</td></tr><tr><td></td><td>CLDA (ours)</td><td>76.1</td><td></td><td></td><td></td><td>71.0</td><td></td><td></td><td></td><td></td><td></td><td>71.1</td><td>80.1</td><td></td><td></td><td>75.3</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/dd1369bf12b986d21a391bd7d4ccdb2af9511edb91769f1c16ac4b8bc1d50c72.jpg",
+        "table_caption": [
+            "Table 3: Performance Comparison in Office31. Numbers show Top-1 accuracy values for different domain adaptation scenarios under 1-shot and 3-shot settings using Alexnet and VGG as backbone networks. CLDA outperforms all the baseline approaches in both scenarios. We have highlighted the superior method on each domain adaptation task. Our Proposed framework achieves the best mean accuracy among all baseline methods. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"5\">Alexnet</td><td colspan=\"6\">VGG</td></tr><tr><td></td><td colspan=\"2\">W→A</td><td colspan=\"2\">D→A</td><td colspan=\"2\">Mean</td><td colspan=\"2\">W→A</td><td colspan=\"2\">D→A</td><td colspan=\"2\">Mean</td></tr><tr><td>Method</td><td>1-shot</td><td>3-shot</td><td>1-shot</td><td>3-shot</td><td>1-shot</td><td>3-shot</td><td>1-shot</td><td>3-shot</td><td>1-shot</td><td>3-shot</td><td>1-shot</td><td>3-shot</td></tr><tr><td>S+T</td><td>50.4</td><td>61.2</td><td>50.0</td><td>62.4</td><td>50.2</td><td>61.8</td><td>169.2</td><td>73.2</td><td>68.2</td><td>73.3</td><td>68.7</td><td>73.25</td></tr><tr><td>DANN</td><td>57.0</td><td>64.4</td><td>54.5</td><td>65.2</td><td>55.8</td><td>64.8</td><td>69.3</td><td>75.4</td><td>70.4</td><td>74.6</td><td>69.85</td><td>75.0</td></tr><tr><td>ADR</td><td>50.2</td><td>61.2</td><td>50.9</td><td>61.4</td><td>50.6</td><td>61.3</td><td>69.7</td><td>73.3</td><td>69.2</td><td>74.1</td><td>69.45</td><td>73.7</td></tr><tr><td>CDAN</td><td>50.4</td><td>60.3</td><td>48.5</td><td>61.4</td><td>49.5</td><td>60.8</td><td>65.9</td><td>74.4</td><td>64.4</td><td>71.4</td><td>65.15</td><td>72.9</td></tr><tr><td>ENT</td><td>50.7</td><td>64.0</td><td>50.0</td><td>66.2</td><td>50.4</td><td>65.1</td><td>69.1</td><td>75.4</td><td>72.1</td><td>75.1</td><td>70.6</td><td>75.25</td></tr><tr><td>MME</td><td>57.2</td><td>67.3</td><td>55.8</td><td>67.8</td><td>56.5</td><td>67.6</td><td>73.1</td><td>76.3</td><td>73.6</td><td>77.6</td><td>73.35</td><td>76.95</td></tr><tr><td>BiAT</td><td>57.9</td><td>68.2</td><td>54.6</td><td>68.5</td><td>56.3</td><td>68.4</td><td>-</td><td>1</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>APE</td><td>1</td><td>67.6</td><td>-</td><td>69.0</td><td>-</td><td>68.3</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>CLDA</td><td>64.6</td><td>70.5</td><td>62.7</td><td>72.5</td><td>63.6</td><td>71.5</td><td>76.2</td><td>78.6</td><td>75.1</td><td>76.7</td><td>75.6</td><td>77.6</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Office31: Similar to other datasets, our proposed method with Alexnet and VGG as neural backbone achieves the best performance in both domain adaption scenarios for office31 as shown in Table 3. Using Alexnet backbone, we beat the APE [29] by $3 . 2 \\%$ in 3-shot and BiAT by $7 . 3 \\%$ in 1-shot settings. We observe similar gains over all the baselines methods with VGG as the neural network backbone. This shows the efficacy of our proposed approach irrespective of the used backbone. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.5 Ablation Studies ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We perform extensive ablation experiments to analyze our CLDA framework and the effects of the different components and hyperparameters. We perform these experiments on the 3-shot $\\mathrm { P r }  \\mathrm { A r }$ domain adaptation task of the Office-Home dataset using Resnet34 unless specified otherwise. ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Effectiveness of Individual Modules: Our CLDA framework is composed of two modules: InterDomain Contrastive Alignment and Instance Contrastive Alignment. We investigate the significance of each component of our framework by dropping the other during training. We observe that the test accuracy drops from $7 2 . 4 \\%$ to $6 8 . 3 \\%$ when only Inter-Domain Contrastive Alignment is used, and it drops to $6 7 . 7 \\%$ when Instance Contrastive Alignment is used alone as shown in Table 5(a). Though individual modules do not yield high performance on their own but once combined, they surpass their individual performance by a margin of around $4 \\%$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/2abf7ba35fef703ef49820e43a123faacc5e07006a05c4339d02362b5664ac47.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Augmentation</td><td>Test Accuracy(Pr→Ar)</td><td>Test Accuracy(Rl→Ar)</td></tr><tr><td>Horizontal Flipping (Hflip)</td><td>68.1</td><td>73.4</td></tr><tr><td>Hflip + Color Jitter</td><td>67.6</td><td>74.9</td></tr><tr><td>Hflip+ Color Jitter+ Grayscale</td><td>70.2</td><td>76.2</td></tr><tr><td>Rand Augment (RA) [8]</td><td>71.1</td><td>74.6</td></tr><tr><td>RA + Grayscale</td><td>72.4</td><td>76.7</td></tr><tr><td>Auto Augment [7]</td><td>69.9</td><td>75.3</td></tr></table>",
+        "bbox": [
+            240,
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/c559af2063faac46d7120f29aecc53ce7e351334606d0acc8b6b5191dcde3732.jpg",
+        "image_caption": [
+            "Table 4: Effect of Strong Augmentations Numbers show the test accuracy on 3-shot domain adaptation tasks of the Office-Home dataset with Resnet34 with different augmentation policies. ",
+            "Accuracy vs Ratio of unlabeled to labeled data "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+            263,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/985e3697944763a1d34ecb7f6cdc7302257c7d45fd5a260543d6e9fc35ea2c15.jpg",
+        "image_caption": [
+            "Accuracy vs Weight of Instance Contrastive Alignment "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            186,
+            265,
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/ce7daee479da15ce20a942f1f3f10d535fc557418e4ecdea10f11f78515c6b41.jpg",
+        "image_caption": [
+            "Accuracy vs Weight of Inter-Domain Contrastive Alignment ",
+            "Figure 3: Effect of different hyperparameters on 3-shot $\\mathbf { P r }  \\mathbf { A r }$ (Product to Art) data adaptation scenario on the Office-Home using Resnet34. (a) Effect of varying the weight of Instance Contrastive Alignment on validation and test Accuracy (b) Effect of varying weight of Inter-Domain Contrastive Alignment on validation and test Accuracy (c) Effect of $\\mu$ , ratio of unlabeled target to labeled target data on validation and test accuracy. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            390,
+            267,
+            604,
+            339
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            454,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Effect of Different Hyperparameters: We analyze the importance of different hyperparameters used in our approach. We observe that the weight of Instance Contrastive Alignment affects the performance of our approach as the test accuracy drops from $7 2 . 4 \\%$ to $7 0 . 7 \\%$ when we set $\\alpha$ to 1 instead of its optimal value of 4 as shown in figure 3. We also notice that increasing $\\beta$ led to a reduction of the validation and test performance. We also look into the effect of $\\mu$ , which is the ratio of unlabeled to labeled data in a minibatch. We observe that an increasing value of $\\mu$ increases the performance till $\\mu = 4$ , after which it starts to drop, as shown in figure 3. ",
+        "bbox": [
+            173,
+            517,
+            825,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Importance of Instance Contrastive Alignment: Instance Contrastive Alignment ensures similar representation across different variants of the unlabeled target images. This consistency is also ensured by other well-known SSL approaches like FixMatch [50]. We perform an ablation experiment replacing Instance Contrastive Alignment with FixMatch. We also compare with L1 and L2 loss to have a fair analysis. As shown in Table 5 (b) Instance Contrastive Alignment helps to achieve superior performance in comparison with other consistency-based approaches. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/998f7f90ed00b73df61128f42bf2602973c5e9a096abf52c1628e4458c9a5260.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "(a) Ablation Study on the effectiveness of Individual components of the CLDA framework on $\\mathrm { P r }  \\mathrm { A r }$ adaptation task of the OfficeHome dataset using Resnet34. "
+        ],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Approach</td><td rowspan=1 colspan=1>Test Accuracy</td></tr><tr><td rowspan=1 colspan=1>CLDA w/o Instance ContrastiveCLDA w/o Inter-Domain Contrastive</td><td rowspan=1 colspan=1>68.367.7</td></tr><tr><td rowspan=1 colspan=1>CLDA (ours)</td><td rowspan=1 colspan=1>72.4</td></tr></table>",
+        "bbox": [
+            181,
+            750,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/cf3bba68d151be47b1303fca72ebfd3d477f7415261db69b0681f508d84214c1.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "(b) Ablation Study on other consistency based approaches on $\\mathrm { P r }  \\mathrm { A r }$ domain adaptation task of the OfficeHome using Resnet34. "
+        ],
+        "table_body": "<table><tr><td>Approach</td><td>Test Accuracy</td></tr><tr><td>Fix-Match</td><td>70.8</td></tr><tr><td>L1 loss</td><td>69.4</td></tr><tr><td>L2 loss</td><td>69.3</td></tr><tr><td>CLDA (ours)</td><td>72.4</td></tr></table>",
+        "bbox": [
+            596,
+            741,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 5: Experiments to understand the significance of individual components of our framework. ",
+        "bbox": [
+            178,
+            892,
+            813,
+            909
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/97c110de42a3e2ad41cbc5d37ffa0148386eca4a8affdb0710c054c62c84e00c.jpg",
+        "table_caption": [
+            "Table 6: Ablation study to understand the effect of outliers in target domain. Numbers show the test accuracy of 1-shot domain adaptation tasks of the Office-Home dataset with Resnet34. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Experiments</td><td>O samples mislabeled</td><td>8 samples mislabeled (~ 12%)</td><td>16 samples mislabeled (~ 25%)</td></tr><tr><td>Pr→Ar</td><td>66.2</td><td>66.0</td><td>65.7</td></tr><tr><td>Rl→Ar</td><td>72.6</td><td>72.05</td><td>71.56</td></tr></table>",
+        "bbox": [
+            240,
+            88,
+            759,
+            127
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Effect of Other Clustering Techniques: Inter-Domain Contrastive Alignment requires pseudo labels for the unlabeled target data for clustering. In this ablation experiment, we replace our approach of using the model’s prediction as a pseudo label with K-means clustering, which we invoke after every 50 steps and use the generated centroids for the next 50 steps to obtain pseudo-class labels for unlabeled target data. We observe a drop in performance (from $7 2 . 4 \\%$ to $\\bar { 7 1 } . 2 \\%$ ) when using K-means to obtain the pseudo label for unlabeled target images. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Effect of Augmentation Policy: We look into different augmentation policies for the Instance Contrastive Alignment. As suggested in [6], a stronger augmentation policy for contrastive learning increases the performance of the model. We find that RandAugment [8] with Grayscale augmentation policy gives better results over other augmentation policies. The influence of the strong augmentation can be observed from $\\sim 4 \\%$ improvement in the performance when the augmentation policy is switched from horizontal flipping to RandAugment with Grayscale. Table 4 contains the test accuracy of different augmentation policies on 3-shot $\\mathrm { P r }  \\mathrm { A r }$ and $\\mathrm { R l }  \\mathrm { P r }$ domain adaption tasks of the Office-Home dataset with Resnet34. ",
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+    {
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+        "text": "Effect of Noisy-Labeled Target Samples: In SSDA, we have few labeled samples from the target domain; however, the presence of noisy-labeled target samples can have an adverse effect on the performance. To understand the effect of noisy-labeled target samples on the framework, we conducted experiments on the 1-shot $\\mathrm { P r }  \\mathrm { A r }$ and $\\mathbf { R } 1  \\mathbf { A r }$ domain adaptation scenarios of the Office-Home dataset with Resnet34, where we mislabeled some previously labeled target samples as shown in Table 6. We observe a small decrease in performance of our framework ( from $6 6 . \\hat { 2 \\% }$ to $6 5 . 7 \\%$ for $\\mathrm { P r }  \\mathrm { A r }$ and from $7 2 . 6 \\%$ to $7 1 . 5 6 \\%$ for $\\mathbb { R } 1 \\to \\mathrm { A r }$ ) when mislabeled target samples increase from $0 \\%$ to $\\sim 2 5 \\%$ in both domain adaptation scenarios showing the robustness of our framework. ",
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+    {
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+        "text": "5 Conclusion ",
+        "text_level": 1,
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+    {
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+        "text": "In this work, we present a novel single-stage contrastive learning framework for semi-supervised domain adaptation. The framework consists of Inter-Domain Contrastive Alignment and InstanceContrastive Alignment, where the former maximizes the similarity between centroids of the same class from both domains and later maximizes the similarity between augmented views of the unlabeled target images. We show that both of the components of the framework are necessary for improved performance. We demonstrate the effectiveness of our approach on three standard domain adaptation benchmark datasets, outperforming the well-known SSDA methods. ",
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+    {
+        "type": "text",
+        "text": "6 Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
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+        "text": "The work is supported by Half-Time Research Assistantship (HTRA) grants from the Ministry of Education, India. We would also like to thank Saurav Chakraborty and Athira Nambiar for their valuable suggestions and feedback to improve the work. ",
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
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+        "text": "[1] Philip Bachman, R Devon Hjelm, and William Buchwalter. “Learning Representations by Maximizing Mutual Information Across Views”. In: Neural Information Processing Systems. 2019.   \n[2] David Berthelot, Nicholas Carlini, Ekin D. Cubuk, Alex Kurakin, Kihyuk Sohn, Han Zhang, and Colin Raffel. “ReMixMatch: Semi-Supervised Learning with Distribution Matching and Augmentation Anchoring”. In: International Conference on Learning Representations. 2020.   \n[3] David Berthelot, Nicholas Carlini, Ian Goodfellow, Nicolas Papernot, Avital Oliver, and Colin Raffel. “MixMatch: A Holistic Approach to Semi-Supervised Learning”. In: Neural Information Processing Systems. 2019.   \n[4] Konstantinos Bousmalis, George Trigeorgis, Nathan Silberman, Dilip Krishnan, and Dumitru Erhan. “Domain Separation Networks”. In: Neural Information Processing Systems. 2016.   \n[5] Chao Chen, Zhihong Chen, Boyuan Jiang, and Xinyu Jin. “Joint Domain Alignment and Discriminative Feature Learning for Unsupervised Deep Domain Adaptation”. In: AAAI Conference on Artificial Intelligence. Vol. 33. 01. 2019, pp. 3296–3303.   \n[6] Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey E. Hinton. “A Simple Framework for Contrastive Learning of Visual Representations”. In: International Conference on Machine Learning. PMLR. 2020, pp. 1597–1607.   \n[7] Ekin Dogus Cubuk, Barret Zoph, Dandelion Mané, Vijay Vasudevan, and Quoc V. Le. “AutoAugment: Learning Augmentation Strategies From Data”. In: IEEE Conference on Computer Vision and Pattern Recognition (2019), pp. 113–123.   \n[8] Ekin Dogus Cubuk, Barret Zoph, Jonathon Shlens, and Quoc V. Le. “Randaugment: Practical automated data augmentation with a reduced search space”. In: IEEE Conference on Computer Vision and Pattern Recognition Workshops (2020), pp. 3008–3017.   \n[9] Timothy Hospedales Da Li. “Online Meta-Learning for Multi-Source and Semi-Supervised Domain Adaptation”. In: European Conference on Computer Vision. 2020.   \n[10] Zihang Dai, Zhilin Yang, Fan Yang, William W. Cohen, and Ruslan Salakhutdinov. “Good Semi-supervised Learning That Requires a Bad GAN”. In: Neural Information Processing Systems abs/1705.09783 (2017).   \n[11] Jeff Donahue, Yangqing Jia, Oriol Vinyals, Judy Hoffman, Ning Zhang, Eric Tzeng, and Trevor Darrell. “DeCAF: A Deep Convolutional Activation Feature for Generic Visual Recognition”. In: International Conference on Machine Learning. PMLR. 2014.   \n[12] Yaroslav Ganin and Victor Lempitsky. “Unsupervised Domain Adaptation by Backpropagation”. In: International Conference on Machine Learning. PMLR. 2015, pp. 1180–1189.   \n[13] Yaroslav Ganin, Evgeniya Ustinova, Hana Ajakan, Pascal Germain, Hugo Larochelle, François Laviolette, Mario Marchand, and Victor Lempitsky. “Domain-Adversarial Training of Neural Networks”. In: The Journal of Machine Learning Research 17.1 (2016), pp. 2096–2030.   \n[14] Raghuraman Gopalan, Ruonan Li, and Rama Chellappa. “Domain adaptation for object recognition: An unsupervised approach”. In: IEEE International Conference on Computer Vision (2011), pp. 999–1006.   \n[15] Yves Grandvalet and Yoshua Bengio. “Semi-supervised learning by entropy minimization”. In: Neural Information Processing Systems. 2004, pp. 529–536.   \n[16] Arthur Gretton, Karsten M Borgwardt, Malte J Rasch, Bernhard Schölkopf, and Alexander Smola. “A Kernel Two-Sample Test”. In: The Journal of Machine Learning Research 13.1 (2012), pp. 723–773.   \n[17] Jean-Bastien Grill, Florian Strub, Florent Altch’e, Corentin Tallec, Pierre H. Richemond, Elena Buchatskaya, Carl Doersch, Bernardo Avila Pires, Zhaohan Daniel Guo, Mohammad Gheshlaghi Azar, et al. “Bootstrap Your Own Latent: A New Approach to Self-Supervised Learning”. In: ArXiv abs/2006.07733 (2020).   \n[18] Kaiming He, Haoqi Fan, Yuxin Wu, Saining Xie, and Ross B. Girshick. “Momentum Contrast for Unsupervised Visual Representation Learning”. In: IEEE Conference on Computer Vision and Pattern Recognition (2020), pp. 9726–9735.   \n[19] Kaiming Heand, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. “Deep Residual Learning for Image Recognition”. In: IEEE Conference on Computer Vision and Pattern Recognition. 2016, pp. 770–778.   \n[20] Judy Hoffman, Eric Tzeng, Taesung Park, Jun-Yan Zhu, Phillip Isola, Kate Saenko, Alexei A. Efros, and Trevor Darrell. “CyCADA: Cycle-Consistent Adversarial Domain Adaptation”. In: International Conference on Machine Learning. PMLR. 2018.   \n[21] Lanqing Hu, Meina Kan, Shiguang Shan, and Xilin Chen. “Duplex Generative Adversarial Network for Unsupervised Domain Adaptation”. In: IEEE Conference on Computer Vision and Pattern Recognition (2018), pp. 1498–1507.   \n[22] Gao Huang, Zhuang Liu, Laurens Van Der Maaten, and Kilian Q. Weinberger. “Densely Connected Convolutional Networks”. In: IEEE Conference on Computer Vision and Pattern Recognition (2017), pp. 2261–2269.   \n[23] Shuiwang Ji, Wei Xu, Ming Yang, and Kai Yu. “3D Convolutional Neural Networks for Human Action Recognition”. In: IEEE Transactions on Pattern Analysis and Machine Intelligence 35 (2013), pp. 221–231.   \n[24] Jian Shen and Yanru Qu and Weinan Zhang and Yong Yu. “Wasserstein Distance Guided Representation Learning for Domain Adaptation”. In: AAAI Conference on Artificial Intelligence. 2018.   \n[25] Pin Jiang, Aming Wu, Yahong Han, Yunfeng Shao, Meiyu Qi, and Bingshuai Li. “Bidirectional Adversarial Training for Semi-Supervised Domain Adaptation”. In: International Joint Conferences on Artificial Intelligence. 2020.   \n[26] Guoliang Kang, Lu Jiang, Yi Yang, and Alexander G Hauptmann. “Contrastive Adaptation Network for Unsupervised Domain Adaptation”. In: IEEE Conference on Computer Vision and Pattern Recognition (2019), pp. 4888–4897.   \n[27] Guoliang Kang, Liang Zheng, Yan Yan, and Yi Yang. “Deep Adversarial Attention Alignment for Unsupervised Domain Adaptation: the Benefit of Target Expectation Maximization”. In: European Conference on Computer Vision. 2018.   \n[28] Donghyun Kim, Kuniaki Saito, Tae-Hyun Oh, Bryan A. Plummer, Stan Sclaroff, and Kate Saenko. “Cross-domain Self-supervised Learning for Domain Adaptation with Few Source Labels”. In: IEEE International Conference on Computer Vision abs/2003.08264 (2021).   \n[29] Tae-Kyung Kim and Changick Kim. “Attract, Perturb, and Explore: Learning a Feature Alignment Network for Semi-supervised Domain Adaptation”. In: European Conference on Computer Vision. 2020.   \n[30] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E. Hinton. “ImageNet Classification with Deep Convolutional Neural Networks”. In: Neural Information Processing Systems 25 (2012), pp. 1097–1105.   \n[31] Chongxuan Li, Kun Xu, Zhu Jun, and Zhang Bo. “Triple Generative Adversarial Nets”. In: Neural Information Processing Systems abs/1703.02291 (2017).   \n[32] Ji Lin, Chuang Gan, and Song Han. “TSM: Temporal Shift Module for Efficient Video Understanding”. In: IEEE International Conference on Computer Vision (2019), pp. 7082– 7092.   \n[33] Weizhe Liu, David Ferstl, Samuel Schulter, Lukas Zebedin, Pascal Fua, and Christian Leistner. “Domain Adaptation for Semantic Segmentation via Patch-Wise Contrastive Learning”. In: arXiv preprint arXiv:2104.11056 (2021).   \n[34] Mingsheng Long, Yue Cao, Jianmin Wang, and Michael I. Jordan. “Learning Transferable Features with Deep Adaptation Networks”. In: International Conference on Machine Learning. PMLR. 2015, pp. 97–105.   \n[35] Mingsheng Long, Zhangjie Cao, Jianmin Wang, and Michael I. Jordan. “Conditional Adversarial Domain Adaptation”. In: Neural Information Processing Systems. 2018.   \n[36] Mingsheng Long, Han Zhu, Jianmin Wang, and Michael I. Jordan. “Unsupervised Domain Adaptation with Residual Transfer Networks”. In: Neural Information Processing Systems. 2016.   \n[37] Samarth Mishra, Kate Saenko, and Venkatesh Saligrama. “Surprisingly Simple SemiSupervised Domain Adaptation with Pretraining and Consistency”. In: ArXiv abs/2101.12727 (2021).   \n[38] Zak Murez, Soheil Kolouri, David Kriegman, Ravi Ramamoorthi, and Kyungnam Kim. “Image to Image Translation for Domain Adaptation”. In: IEEE Conference on Computer Vision and Pattern Recognition (2018), pp. 4500–4509.   \n[39] Aaron van den Oord, Yazhe Li, and Oriol Vinyals. “Representation Learning with Contrastive Predictive Coding”. In: ArXiv abs/1807.03748 (2018).   \n[40] Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Köpf, Edward Yang, Zach DeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala. “PyTorch: An Imperative Style, High-Performance Deep Learning Library”. In: Neural Information Processing Systems. 2019.   \n[41] Sujoy Paul, Yi-Hsuan Tsai, Samuel Schulter, Amit K Roy-Chowdhury, and Manmohan Chandraker. “Domain adaptive semantic segmentation using weak labels”. In: European Conference on Computer Vision. 2020, pp. 571–587.   \n[42] Zhongyi Pei, Zhangjie Cao, Mingsheng Long, and Jianmin Wang. “Multi-Adversarial Domain Adaptation”. In: AAAI Conference on Artificial Intelligence. 2018.   \n[43] Xingchao Peng, Qinxun Bai, Xide Xia, Zijun Huang, Kate Saenko, and Bo Wang. “Moment Matching for Multi-Source Domain Adaptation”. In: IEEE International Conference on Computer Vision. 2019, pp. 1406–1415.   \n[44] Can Qin, Lichen Wang, Qianqian Ma, Yu Yin, Huan Wang, and Yun Fu. “Contradictory Structure Learning for Semi-supervised Domain Adaptation”. In: Proceedings of the 2021 SIAM International Conference on Data Mining (SDM). SIAM. 2021, pp. 576–584.   \n[45] Kate Saenko, Brian Kulis, Mario Fritz, and Trevor Darrell. “Adapting Visual Category Models to New Domains”. In: European Conference on Computer Vision. 2010.   \n[46] Kuniaki Saito, Donghyun Kim, Stan Sclaroff, Trevor Darrell, and Kate Saenko. “SemiSupervised Domain Adaptation via Minimax Entropy”. In: IEEE International Conference on Computer Vision (2019), pp. 8049–8057.   \n[47] Kuniaki Saito, Yoshitaka Ushiku, Tatsuya Harada, and Kate Saenko. “Adversarial Dropout Regularization”. In: International Conference on Learning Representations. 2018.   \n[48] Karen Simonyan and Andrew Zisserman. “Two-Stream Convolutional Networks for Action Recognition in Videos”. In: Neural Information Processing Systems. 2014.   \n[49] Ankit Singh, Omprakash Chakraborty, Ashutosh Varshney, Rameswar panda, Rogerio Feris, Kate Saenko, and Abir Das. “Semi-Supervised Action Recognition With Temporal Contrastive Learning”. In: IEEE Conference on Computer Vision and Pattern Recognition. June 2021, pp. 10389–10399.   \n[50] Kihyuk Sohn, David Berthelot, Chun-Liang Li, Zizhao Zhang, Nicholas Carlini, Ekin D Cubuk, Alex Kurakin, Han Zhang, and Colin Raffel. “FixMatch: Simplifying Semi-Supervised Learning with Consistency and Confidence”. In: Neural Information Processing Systems (2020).   \n[51] Baochen Sun and Kate Saenko. “Deep CORAL: Correlation Alignment for Deep Domain Adaptation”. In: European Conference on Computer Vision Workshops. 2016.   \n[52] Christian Szegedy, Sergey Ioffe, Vincent Vanhoucke, and Alexander Amir Alemi. “Inceptionv4, Inception-ResNet and the Impact of Residual Connections on Learning”. In: AAAI Conference on Artificial Intelligence. 2017.   \n[53] Antti Tarvainen and Harri Valpola. “Mean teachers are better role models: Weight-averaged consistency targets improve semi-supervised deep learning results”. In: Neural Information Processing Systems. 2017.   \n[54] Yonglong Tian, Dilip Krishnan, and Phillip Isola. “Contrastive Multiview Coding”. In: European Conference on Computer Vision. 2020.   \n[55] Eric Tzeng, Judy Hoffman, Kate Saenko, and Trevor Darrell. “Adversarial Discriminative Domain Adaptation”. In: IEEE Conference on Computer Vision and Pattern Recognition (2017), pp. 2962–2971.   \n[56] Hemanth Venkateswara, Jose Eusebio, Shayok Chakraborty, and Sethuraman Panchanathan. “Deep Hashing Network for Unsupervised Domain Adaptation”. In: IEEE Conference on Computer Vision and Pattern Recognition (2017), pp. 5385–5394.   \n[57] Limin Wang, Yuanjun Xiong, Zhe Wang, Yu Qiao, Dahua Lin, Xiaoou Tang, and Luc Van Gool. “Temporal Segment Networks for Action Recognition in Videos”. In: IEEE Transactions on Pattern Analysis and Machine Intelligence 41 (2019), pp. 2740–2755.   \n[58] Luyu Yang, Yan Wang, Mingfei Gao, Abhinav Shrivastava, Kilian Q. Weinberger, Wei-Lun Chao, and Ser-Nam Lim. “Deep Co-Training with Task Decomposition for Semi-Supervised Domain Adaptation”. In: IEEE International Conference on Computer Vision. 2021. ",
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parse/train/HkmaTz-0W/HkmaTz-0W.md

@@ -0,0 +1,334 @@
+# VISUALIZING THE LOSS LANDSCAPE OF NEURAL NETS
+
+# Anonymous authors
+
+Paper under double-blind review
+
+# ABSTRACT
+
+Neural network training relies on our ability to find “good” minimizers of highly non-convex loss functions. It is well known that certain network architecture designs (e.g., skip connections) produce loss functions that train easier, and wellchosen training parameters (batch size, learning rate, optimizer) produce minimizers that generalize better. However, the reasons for these differences, and their effect on the underlying loss landscape, is not well understood.
+
+In this paper, we explore the structure of neural loss functions, and the effect of loss landscapes on generalization, using a range of visualization methods. First, we introduce a simple “filter normalization” method that helps us visualize loss function curvature, and make meaningful side-by-side comparisons between loss functions. Then, using a variety of visualizations, we explore how network architecture effects the loss landscape, and how training parameters affect the shape of minimizers.
+
+# 1 INTRODUCTION
+
+Training neural networks requires minimizing a high-dimensional non-convex loss function – a task that is hard in theory, but sometimes easy in practice. Despite the NP-hardness of training general neural loss functions (Blum & Rivest, 1989), simple gradient methods often find global minimizers (parameter configurations with zero or near-zero training loss), even when data and labels are randomized before training (Zhang et al., 2017). However, this good behavior is not universal; the trainability of neural nets is highly dependent on network architecture design choices, the choice of optimizer, variable initialization, and a variety of other considerations. Unfortunately, the effect of each of these choices on the structure of the underlying loss surface is unclear. Because of the prohibitive cost of loss function evaluations (which requires looping over all the data points in the training set), studies in this field have remained predominantly theoretical.
+
+![](images/a14706c29d892ca8c4fe95f8306eb80e5ef1b4694e7c755e61d5a5f55c9ef4dd.jpg)  
+Figure 1: The loss surfaces of ResNet-56 with/without skip connections. The vertical axis is logarithmic to show dynamic range. The proposed filter normalization scheme is used to enable comparisons of sharpness/flatness between the two figures.
+
+Our goal is to use high-resolution visualizations to provide an empirical characterization of neural loss functions, and to explore how different network architecture choices affect the loss landscape. Furthermore, we explore how the non-convex structure of neural loss functions relates to their trainability, and how the geometry of neural minimizers (i.e., their sharpness/flatness, and their surrounding landscape), affects their generalization properties.
+
+To do this in a meaningful way, we propose a simple “filter normalization” scheme that enables us to do side-by-side comparisons of different minima found by different methods. We then use visualizations to explore sharpness/flatness of minimizers found by different methods, as well as the effect of network architecture choices (use of skip connections, number of filters, network depth) on the loss landscape. Out goal is to understand how differences in loss function geometry effect the generalization of neural nets.
+
+# 1.1 CONTRIBUTIONS
+
+In this article, we study methods for producing meaningful loss function visualizations. Then, using these visualization methods, we explore how loss landscape geometry effects generalization error and trainability. More specifically, we address the following issues:
+
+• We reveal faults in a number of visualization methods for loss functions, and show that simple visualization strategies fail to accurately capture the local geometry (sharpness or flatness) of loss function minimizers. We present a simple visualization method based on “filter normalization” that enables side-by-side comparisons of different minimizers. The sharpness of minimizers correlates well with generalization error when this visualization is used, even when making sharpness comparisons across disparate network architectures and training methods.   
+• We observe that, when networks become sufficiently deep, neural loss landscapes suddenly transition from being nearly convex to being highly chaotic. This transition from convex to chaotic behavior, which seem to have been previously unnoticed, coincides with a dramatic drop in generalization error, and ultimately to a lack of trainability. We show that skip connections promote flat minimizers and prevent the transition to chaotic behavior, which helps explain why skip connections are necessary for training extremely deep networks.   
+• We study the visualization of SGD optimization trajectories. We explain the difficulties that arise when visualizing these trajectories, and show that optimization trajectories lie in an extremely low dimensional space. This low dimensionality can be explained by the presence of large nearly convex regions in the loss landscape, such as those observed in our 2-dimensional visualizations.
+
+# 2 THEORETICAL BACKGROUND & RELATED WORK
+
+Visualizations have the potential to help us answer several important questions about why neural networks work. In particular, why are we able to minimize highly non-convex neural loss functions? And why do the resulting minima generalize?
+
+Because of the difficultly of visualizing loss functions, most studies of loss landscapes are largely theoretical in nature. A number of authors have studied our ability to minimize neural loss functions. Using random matrix theory and spin glass theory, several authors have shown that local minima are of low objective value (Dauphin et al., 2014; Choromanska et al., 2015). It can also be shown that local minima are global minima, provided one assumes linear neurons (Hardt & Ma, 2017), very wide layers (Nguyen & Hein, 2017), or full rank weight matrices (Yun et al., 2017). These assumptions have been relaxed by Kawaguchi (2016) and Lu & Kawaguchi (2017), although some assumptions (e.g., of the loss functions) are still required. Soudry & Hoffer (2017); Freeman & Bruna (2017); Xie et al. (2017) also analyzed shallow networks with one or two hidden layers under mild conditions.
+
+Another approach is to show that we can expect good minimizers, not simply because of the endogenous properties of neural networks, but because of the optimizers. For restricted network classes such as those with one hidden layer, with some extra assumptions on the sample distribution, globally optimal or near-optimal solutions can be found by common optimization methods (Soltanolkotabi et al., 2017; Li & Yuan, 2017; Tian, 2017). For networks with specific structures, Safran & Shamir (2016) and Haeffele & Vidal (2017) show there likely exists a monotonically decreasing path from an initialization to a global minimum. Swirszcz et al. (2017) show counterexamples that achieve “bad” local minima for toy problems.
+
+Also of interest is work on assessing the sharpness/flatness of local minima. Hochreiter & Schmidhuber (1997) defined “flatness” as the size of the connected region around the minimum where the training loss remains low. Keskar et al. (2017) propose $\epsilon$ -sharpness, which looks at the maximum loss in a bounded neighborhood of a minimum. Flatness can also be defined using the local curvature of the loss function at a critical point. Keskar et al. (2017) suggests that this information is encoded in the eigenvalues of the Hessian. However, Dinh et al. (2017) show that these quantitative measure of sharpness are problematic because they are not invariant to symmetries in the network, and are thus not sufficient to determine its generalization ability. This issue was addressed in Chaudhari et al. (2017), who used local entropy as a measure of sharpness. This measure is invariant to the simple transformation used by Dinh et al. (2017), but difficult to quantify for large networks.
+
+Theoretical results make some restrictive assumptions such as the independence of the input samples, or restrictions on non-linearities and loss functions. For this reason, visualizations play a key role in verifying the validity of theoretical assumptions, and understanding loss function behavior in real-world systems. In the next section, we briefly review methods that have been used for this purpose.
+
+# 3 THE BASICS OF LOSS FUNCTION VISUALIZATION
+
+Neural networks are trained on a corpus of feature vectors (e.g., images) $\{ x _ { i } \}$ and accompanying labels $\{ y _ { i } \}$ by minimizing a loss of the form
+
+$$
+L ( \theta ) = \frac { 1 } { m } \sum _ { i = 1 } ^ { m } \ell ( x _ { i } , y _ { i } ; \theta )
+$$
+
+where $\theta$ denotes the parameters (weights) of the neural network, the function $\ell ( x _ { i } , y _ { i } ; \theta )$ measures how well the neural network with parameters $\theta$ predicts the label of a data sample, and $m$ is the number of data samples.
+
+Neural nets contain many parameters, and so their loss functions live in a very high-dimensional space. Unfortunately, visualizations are only possible using low-dimensional 1D (line) or 2D (surface) plots. Several methods exist for closing this dimensionality gap.
+
+1-Dimensional Linear Interpolation One simple and lightweight way to plot loss functions is to choose two sets of parameters $\theta _ { 1 }$ and $\theta _ { 2 }$ , and plot the values of the loss function along the line connecting these two points. We can parameterize this line by choosing a scalar parameter $\alpha$ , and defining the weighted average
+
+$$
+\theta _ { \alpha } = ( 1 - \alpha ) \theta _ { 1 } + \alpha \theta _ { 2 } .
+$$
+
+Finally, we plot the function $f ( \alpha ) = L ( \theta _ { \alpha } )$ . This strategy was taken by Goodfellow et al. (2015), who studied the loss surface along the line between a (random) initial guess, and a nearby minimizer obtained by stochastic gradient descent. This method has been widely used to study the “sharpness” and “flatness��� of different minima, and the dependence of sharpness on batch-size (Keskar et al., 2017; Dinh et al., 2017). Smith & Topin (2017) use the same 1D interpolation technique to show different minima and the “peaks” between them, while Im et al. (2016) plot the line between minima obtained via different optimizers.
+
+The 1D linear interpolation method suffers from several weaknesses. First, it is difficult to visualize non-convexities using 1D plots. Indeed, the authors of (Goodfellow et al., 2015) found that loss functions appear to lack local minima along the minimization trajectory. We will see later, using 2D methods, that some loss functions have extreme non-convexities, and that these non-convexities correlate with the difference in generalization between different network architectures. Second, this method does not consider batch normalization or invariance symmetries in the network. For this reason, the visual sharpness comparisons produced by 1D interpolation plots may be misleading; this issue will be explored in depth in Section 5.
+
+2D Contour Plots To use this approach, one chooses a center point $\theta ^ { * }$ in the graph, and chooses two direction vectors, $\delta$ and $\eta$ . One then plots a function of the form $f ( \alpha ) = L ( { \bar { \theta } } ^ { * } { \bar { + } } \alpha \delta )$ in the 1D (line) case, or
+
+$$
+f ( \alpha , \beta ) = L ( \theta ^ { * } + \alpha \delta + \beta \eta )
+$$
+
+in the 2D (surface) case. This approach was used in (Goodfellow et al., 2015) to explore the trajectories of different minimization methods. It was also used in (Im et al., 2016) to show that different optimization algorithms find different local minima within the 2D projected space.
+
+Because of the computational burden of 2D plotting, these methods generally result in low-resolution plots of small regions that have not captured the complex non-convexity of loss surfaces. Below, we use high-resolution visualizations over large slices of weight space to visualize how network design affects non-convex structure.
+
+# 4 PROPOSED VISUALIZATION: FILTER-WISE NORMALIZATION
+
+This study relies heavily on plots of the form (1) produced using random direction vectors, $\delta$ and $\eta$ , each sampled from a random Gaussian distribution with appropriate scaling (described below).
+
+While the “random directions” approach to plotting is simple, it cannot be used to compare the geometry of two different minimizers or two different networks. This is because of the scale invariance in network weights. When ReLU non-linearities are used, the network remains unchanged if we (for example) multiply the weights in one layer of a network by 10, and divide the next layer by 10. This invariance is even more prominent when batch normalization is used. In this case, the size (i.e., norm) of a filter is irrelevant because the output of each layer is re-scaled during batch normalization. For this reason, a network’s behavior remains unchanged if we re-scale the weights.
+
+Scale invariance prevents us from making meaningful comparisons between plots, unless special precautions are taken. A neural network with large weights may appear to have a smooth and slowly varying loss function; perturbing the weights by one unit will have very little effect on network performance if the weights live on a scale much larger than one. However, if the weights are much smaller than one, then that same one unit perturbation may have a catastrophic effect, making the loss function appear quite sensitive to weight perturbations. Keep in mind that neural nets are scale invariant; if the small-parameter and large-parameter networks in this example are equivalent (because one is simply a re-scaling of the other), then any apparent differences in the loss function are merely an artifact of scale invariance. This scale invariance was exploited by Dinh et al. (2017) to build pairs of equivalent networks that have different apparent sharpness.
+
+To remove this scaling effect, we plot loss functions using filter-wise normalized directions. To obtain such directions for a network with parameters $\theta$ , we begin by producing a random Gaussian direction vector $d$ with dimensions compatible with $\theta$ . Then we normalize each filter in $d$ to have the same norm of the corresponding filter in $\theta$ . In other words, we make the replacement
+
+$$
+d _ { i }  { \frac { d _ { i } } { \Vert d _ { i } \Vert } } \Vert \theta _ { i } \Vert ,
+$$
+
+where $d _ { i }$ represents the ith filter of $d$ (not the ith weight), and $\lVert \boldsymbol { \theta } _ { i } \rVert$ denotes the Frobenius norm of the ith filter of $\theta$ . Note that the filter-wise normalization is different from that of (Im et al., 2016), which normalize the direction without considering the norm of individual filters.
+
+The proposed scaling is an important factor when making meaningful plots of loss function geometry. We will explore the importance of proper scaling below as we explore the sharpness/flatness of different minimizers. In this context, we show that the sharpness of filter-normalized plots correlates with generalization error, while plots without filter normalization can be very misleading.
+
+# 5 THE SHARP VS FLAT DILEMMA
+
+Section 4 introduces the concept of filter normalization, and provides an intuitive justification for its use. In this section, we address the issue of whether sharp minimizers generalize better than flat minimizers. In doing so, we will see that the sharpness of minimizers correlates well with generalization error when filter normalization is used. This enables side-by-side comparisons between plots. In contrast, the sharpness of non-filter normalized plots may appear distorted and unpredictable.
+
+It is widely thought that small-batch SGD produces “flat” minimizers that generalize better, while large batch sizes produce “sharp” minima with poor generalization (Chaudhari et al., 2017; Keskar et al., 2017; Hochreiter & Schmidhuber, 1997). This claim is disputed though, with Dinh et al. (2017); Kawaguchi et al. (2017) arguing that generalization is not directly related to the curvature of loss surfaces, and some authors proposing specialized training methods that achieve good performance with large batch sizes (Hoffer et al., 2017; Goyal et al., 2017; De et al., 2017).
+
+Here, we explore the difference between sharp and flat minimizers. We begin by discussing difficulties that arise when performing such a visualization, and how proper normalization can prevent such plots from producing distorted results.
+
+We train a CIFAR-10 classifier using a 9-layer VGG network (Simonyan & Zisserman, 2015) with Batch Normalization (Ioffe & Szegedy, 2015). We use two batch sizes: a large batch size of 8192 ( $1 6 . 4 \%$ of the training data of CIFAR-10), and a small batch size of 128. Let $\theta _ { s }$ and $\theta _ { l }$ indicate the solutions obtained by running SGD using small and large batch sizes, respectively1. Using the linear interpolation approach (Goodfellow et al., 2015), we plot the loss values on both training and testing data sets of CIFAR-10, along a direction containing the two solutions, i.e., $f ( \alpha ) = L ( \theta _ { s } + \alpha ( \theta _ { l } - \theta _ { s } ) )$ .
+
+![](images/ef50b7737be38ec876e0cdb95e3e3af393016f78c52f3f6208f9ac227293847e.jpg)  
+Figure 2: 1D linear interpolation of solutions obtained by small-batch and large-batch methods for VGG9. The blue lines are loss values and the red lines are accuracies. The solid lines are training curves and the dashed lines are for testing. Small batch is at abscissa 0, and large batch is at abscissa 1.
+
+Table 1: Test errors of VGG-9 on CIFAR-10 with different optimization algorithms and hyperparameters.   
+
+<table><tr><td rowspan="3"></td><td rowspan="3"></td><td colspan="2">SGD</td><td colspan="2">Adam</td></tr><tr><td>bs=128</td><td>bs=8192</td><td>bs=128</td><td>bs=8192</td></tr><tr><td>VGG-9</td><td>WD= 0</td><td>7.37</td><td>11.07</td><td>7.44</td><td>10.91</td></tr><tr><td></td><td>WD = 5e-4</td><td>6.00</td><td>10.19</td><td>7.80</td><td>9.52</td></tr></table>
+
+Similar to Keskar et al. (2017), we also superimpose the classification accuracy in red. This plot is shown in Figure 2.
+
+Figures 2(a) and 2(b) show linear interpolation plots with $\theta _ { s }$ at $\mathbf { X }$ -axis location 0, and $\theta _ { l }$ at location $1 ^ { 2 }$ . As observed by Keskar et al. (2017), we can clearly see that the small-batch solution is quite wide, while the large-batch solution is sharp. However, this sharpness balance can be flipped simply by turning on weight decay (Krogh & Hertz, 1992). Figures 2(c) and 2(d) show results of the same experiment, except this time with a non-zero weight decay parameter. This time, the large batch minimizer is considerably flatter than the sharp small batch minimizer. However, we see from Table 1 that small batches generalize better in all 4 experiments; there is no apparent correlation between sharpness and generalization. We will see that these side-by-side sharpness comparisons are extremely misleading, and fail to capture the endogenous properties of the minima.
+
+The apparent differences in sharpness in Figure 2 can be explained by examining the weights of each minimizer. Histograms of the networks weights are shown for each experiment in Figure 3. We see that, when a large batch is used with zero weight decay, the resulting weights tends to be smaller than in the small batch case. We reverse this effect by adding weight decay; in this case the large batch minimizer has much larger weights than the small batch minimizer. This difference in scale occurs for a simple reason: A smaller batch size results in more weight updates per epoch than a large batch size, and so the shrinking effect of weight decay (which imposes a penalty on the norm of the weights) is more pronounced.
+
+Figure 2 in not visualizing the endogenous sharpness of minimizers, but rather just the (irrelevant) weight scaling. The scaling of weights in these networks is irrelevant because batch normalization re-scales the outputs to have unit variance. However, small weights still appear more sensitive to perturbations, and produce sharper looking minimizers.
+
+Filter normalized plots We repeat the experiment in Figure 2, but this time we plot the loss function near each minimizer separately using random filter-normalized directions. This removes the apparent differences in geometry caused by the scaling depicted in Figure 3. The results, presented in Figure 4, still show differences in sharpness between small batch and large batch minima, however these differences are much more subtle than it would appear in the un-normalized plots.
+
+We also visualize these results using two random directions and contour plots. As shown in Figure 5, the weights obtained with small batch size and non-zero weight decay have wider contours than the sharper large batch minimizers. Similar for Resnet-56 appear in Figure 12 of the Appendix.
+
+![](images/c5a7d4f64b3d10940e5b0e8fcf8a2ad96ebd92066bfd9cc81dc1eeb9a61973a2.jpg)  
+Figure 3: Histogram of weights. With zero weight decay, small-batch methods produce large weights. With non-zero weight decay, small-batch methods produce smaller weights.
+
+![](images/0089ebcfa1e4c7c29fcc8cf3c9b44b137d8e12613e49aba8c463103b971b4e17.jpg)  
+Figure 4: The shape of minima obtained using different optimization algorithms, with varying batch size and weight decay. The title of each subfigure contains the optimizer, batch size, and test error. The first row has no weight decay and the second row uses weight decay 5e-4.
+
+![](images/19fc60a8c5a0284a9dbe8f9ce56c82c214fd602dbeff6b22bd17998217d2f207.jpg)  
+Figure 5: 2D visualization of solutions obtained by SGD with small-batch and large-batch. Similar to Figure 4, the first row uses zero weight decay and the second row sets weight decay to 5e-4.
+
+Generalization and Flatness Using the filter-normalized plots in Figures 4 and 5, we can make side-by-side comparisons between minimizers, and we see that now sharpness correlates well with generalization error. Large batches produced visually sharper minima (although not dramatically so) with higher test error. Interestingly, the Adam optimizer attained larger test error than SGD, and, as predicted, the corresponding minima are visually sharper. Results of a similar experiment using ResNet-56 are presented in the Appendix (Figure 12).
+
+# 6 WHAT MAKES NEURAL NETWORKS TRAINABLE? INSIGHTS ON THE (NON) CONVEXITY STRUCTURE OF LOSS SURFACES
+
+Our ability to find global minimizers to neural loss functions is not universal; it seems that some neural architectures are easier to minimize than others. For example, using skip connections, He et al. (2016) were able to train extremely deep architectures, while comparable architectures without skip connections are not trainable. Furthermore, our ability to train seems to depend strongly on the initial parameters from which training starts.
+
+Using visualization methods, we do an empirical study of neural architectures to explore why the non-convexity of loss functions seems to be problematic in some situations, but not in others. We aim to provide insight into the following questions: Do loss functions have significant non-convexity at all? If prominent non-convexities exist, why are they not problematic in all situations? Why are some architectures easy to train, and why are results so sensitive to the initialization? We will see that different architectures have extreme differences in non-convexity structure that answer these questions, and that these differences correlate with generalization error.
+
+# 6.1 EXPERIMENTAL SETUP
+
+To understand the effects of network architecture on non-convexity, we trained a number of networks, and plotted the landscape around the obtained minimizers using the filter-normalized random direction method described in Section 4.
+
+We consider three classes of neural networks:
+
+• Residual networks that are optimized for performance on CIFAR (He et al., 2016). We consider ResNet-20, ResNet-56, and ResNet-110, where each name is labeled with the number of convolutional layers it has. “VGG-like” networks that do not contain shortcut/skip connections. We produced these networks simply by removing the skip connections from the CIFAR-optimized ResNets. We call these networks ResNet-20-noshort, ResNet-56-noshort, and ResNet-110-noshort. Note that these networks do not all perform well on the CIFAR-10 task. We use them purely for experimental purposes to explore the effect of shortcut connections. “Wide” ResNets that have been optimized for ImageNet rather than CIFAR. These networks have more filters per layer than the CIFAR optimized networks, and also have different numbers of layers. These models include ResNet-18, ResNet-34, and ResNet-50.
+
+All models are trained on the CIFAR-10 dataset using SGD with Nesterov momentum, batch-size 128, and 0.0005 weight decay for 300 epochs. The learning rate was initialized at 0.1, and decreased by a factor of 10 at epochs 150, 225 and 275. Deeper experimental VGG-like networks (e.g., ResNet-56-noshort, as described below) required a smaller initial learning rate of 0.01.
+
+High resolution 2D plots of the minimizers for different neural networks are shown in Figure 6. Results are shown as contour plots rather than surface plots because this makes it extremely easy to see non-convex structures and evaluate sharpness. For surface plots of ResNet-56, see Figure 1. Note that the center of each plot corresponds to the minimizer, and the two axes parameterize two random directions with filter-wise normalization as in (1). We make several observations below about how architecture effects the loss landscape. We also provide loss and error values for these networks in Table 2, and convergence curves in Figure 14 of the Appendix.
+
+# 6.2 THE EFFECT OF NETWORK DEPTH
+
+From Figure 6, we see that network depth has a dramatic effect on the loss surfaces of neural networks when skip connections are not used. The network ResNet-20-noshort has a fairly benign landscape dominated by a region with convex contours in the center, and no dramatic non-convexity. This isn’t too surprising: the original VGG networks for ImageNet had 19 layers and could be trained effectively (Simonyan & Zisserman, 2015).
+
+However, as network depth increases, the loss surface of the VGG-like nets spontaneously transitions from (nearly) convex to chaotic. ResNet-56-noshort has dramatic non-convexities and large regions where the gradient directions (which are normal to the contours depicted in the plots) do not point towards the minimizer at the center. Also, the loss function becomes extremely large as we move in some directions. ResNet-110-noshort displays even more dramatic non-convexities, and becomes extremely steep as we move in all directions shown in the plot. Furthermore, note that the minimizers at the center of the deep VGG-like nets seem to be fairly sharp. In the case of ResNet-56-noshort, the minimizer is also fairly ill-conditioned, as the contours near the minimizer have significant eccentricity.
+
+![](images/09e1fa6d53f9c16d0e4f2124e26f3227824bd8d411eafd3b74ff9db658ad4338.jpg)  
+Figure 6: 2D visualization of the solutions of different networks.
+
+Table 2: Loss values and errors for different architectures.   
+
+<table><tr><td></td><td>Filters</td><td>Training Loss</td><td>Training Error</td><td>Test Error</td></tr><tr><td rowspan="2">ResNet-20 ResNet-20-noshort</td><td rowspan="2">16</td><td>0.017</td><td>0.286</td><td>7.37</td></tr><tr><td>0.025</td><td>0.560</td><td>8.18</td></tr><tr><td>ResNet-56</td><td rowspan="2">16</td><td>0.004</td><td>0.052</td><td>5.89</td></tr><tr><td>ResNet-56-noshort</td><td>0.024</td><td>0.704</td><td>10.83</td></tr><tr><td>ResNet-110</td><td rowspan="2">16</td><td>0.002</td><td>0.042</td><td>5.79</td></tr><tr><td>ResNet-110-noshort</td><td>0.258</td><td>8.732</td><td>16.44</td></tr><tr><td>ResNet-18</td><td rowspan="2">64 64</td><td>0.002</td><td>0.026</td><td>5.42</td></tr><tr><td>ResNet-34</td><td>0.001</td><td>0.014</td><td>4.73</td></tr><tr><td>ResNet-50</td><td>64</td><td>0.001</td><td>0.006</td><td>4.55</td></tr></table>
+
+# 6.3 SHORTCUT CONNECTIONS TO THE RESCUE
+
+Shortcut connections have a dramatic effect of the geometry of the loss functions. In Figure 6, we see that residual connections prevent the transition to chaotic behavior as depth increases. In fact, the width and shape of the 0.1-level contour is almost identical for the 20- and 110-layer networks.
+
+Interestingly, the effect of skip connections seems to be most important for deep networks. For the more shallow networks (ResNet-20 and ResNet-20-noshort), the effect of skip connections is fairly unnoticeable. However residual connections prevent the explosion of non-convexity that occurs when networks get deep. This effect seems to apply to other kinds of skip connections as well; Figure 13 of the Appendix shows the loss landscape of DenseNet (Huang et al., 2017), which shows no noticeable non-convexity.
+
+# 6.4 WIDE MODELS VS THIN MODELS
+
+To see the effect of the number of convolutional filters per layer, we compare the narrow CIFARoptimized ResNets (ResNet-20/56/110) with wider ResNets (ResNet-18/34/50) that have more filters and were optimized for ImageNet. From Figure 6, we see that the wider models have loss landscapes with no noticeable chaotic behavior. Increased network width resulted in flat minima and wide regions of apparent convexity.
+
+This effect is also validated by Figure 7, in which we plot the landscape of ResNet-56, but we multiple the number of filter per layer by $k = 2 , 4$ , and 8. We see that increased width prevents chaotic behavior, and skip connections dramatically widen minimizers. Finally, note that sharpness correlates extremely well with test error.
+
+![](images/c55cb4af6dfc31fb4a6c5b394992c8d9c6370fc38dc358befd09e109f60dd5bd.jpg)  
+Figure 7: Wide-ResNet-56 (WRN-56) on CIFAR-10 both with shortcut connections (top) and without (bottom). The label $k = 2$ means twice as many filters per layer, $k = 4$ means 4 times, etc. Test error is reported below each figure.
+
+# 6.5 IMPLICATIONS FOR NETWORK INITIALIZATION
+
+One of the most interesting observations seen in Figure 6 is that loss landscapes for all the networks considered seem to be partitioned into a well-defined region of low loss value and convex contours, surrounded by a well-defined region of high loss value and non-convex contours.
+
+This partitioning of chaotic and convex regions may explain the importance of good initialization strategies, and also the easy training behavior of “good” architectures. When using normalized random initialization strategies such as those proposed by Glorot & Bengio (2010), typical neural networks attain an initial loss value less than 2.5. The well behaved loss landscapes in Figure 6 (ResNets, and shallow VGG-like nets) are dominated by large, flat, nearly convex attractors that rise to a loss value of 4 or greater. For such landscapes, a random initialization will likely lie in the “well- behaved” loss region, and the optimization algorithm might never “see” the pathological non-convexities that occur on the high loss chaotic plateaus.
+
+Chaotic loss landscapes (ResNet-56-noshort and ResNet-110-noshort) have shallower regions of convexity that rise to lower loss values. For sufficiently deep networks with shallow enough attractors, the initial iterate will likely lie in the chaotic region where the gradients are uninformative. In our experiments, SGD was unable to train a 156 layer network without skip connections (even with very low learning rates), which adds weight to this hypothesis.
+
+# 6.6 LANDSCAPE GEOMETRY AFFECTS GENERALIZATION
+
+Table 2 displays the training and test error for the networks depicted in Figure 6. Both Figures 6 and 7 show that landscape geometry has a dramatic effect on generalization. First, note that visually flatter minimizers consistently correspond to lower test error, which further strengthens our assertion that filter normalization is a natural way to visualize loss function geometry.
+
+Second, we notice that chaotic landscapes (deep networks without skip connections) result in worse training and test error, while more convex landscapes have lower error values. In fact, the most convex landscapes (the wide ResNets in the bottom row of Figure 6) generalize the best of all the networks. This latter class of networks show no noticeable chaotic behavior at all.
+
+# 7 VISUALIZING OPTIMIZATION PATHS
+
+Finally, we explore methods for visualizing the trajectories of different optimizers. For this application, random directions are ineffective. We will provide a theoretical explanation for why random directions fail, and explore methods for effectively plotting trajectories on top of loss function contours.
+
+Several authors have observed that random direction fail to capture the variation in optimization trajectories, including Gallagher & Downs (2003); Lorch (2016); Lipton (2016); Liao & Poggio (2017). Several failed visualizations are depicted in Figure 8. In Figure 8(a), we see the iterates of SGD projected onto the plane defined by two random directions. Almost none of the motion is captured (notice the super-zoomed-in axes and the seemingly random walk). This problem was noticed by Goodfellow et al. (2015), who then visualized trajectories using one direction that points from initialization to solution, and one random direction. This approach is shown in Figure 8(b). As seen in Figure 8(c), the random axis captures almost no variation, leading to the (misleading) appearance of a straight line path.
+
+![](images/6b58a26902081c8eab5b05e4d0561fdea81fbc2a651f28634198198c59d961af.jpg)  
+Figure 8: Ineffective visualizations of optimizer trajectories. These visualizations suffer from the orthogonality of random directions in high dimensions.
+
+# 7.1 WHY RANDOM DIRECTIONS FAIL: LOW DIMENSIONAL OPTIMIZATION TRAJECTORIES
+
+It is well known that two random vectors in a high dimensional space will be nearly orthogonal with high probability. In fact, the expected cosine similarity between Gaussian random vectors in $n$ dimensions is roughly $\sqrt { 2 / ( \pi n ) }$ (Goldstein & Studer (2016), Lemma 5).
+
+This is problematic when optimization trajectories lie in extremely low dimensional spaces. In this case, a randomly chosen vector will lie orthogonal to the low-rank space containing the optimization path, and a projection onto a random direction will capture almost no variation. Figure 8(b) suggests that optimization trajectories are low dimensional because the random direction captures orders of magnitude less variation than the vector that points along the optimization path. Below, we use PCA directions to directly validate this low dimensionality, and also to produce effective visualizations.
+
+# 7.2 EFFECTIVE TRAJECTORY PLOTTING USING PCA DIRECTIONS
+
+To capture variation in trajectories, we need to use non-random (and carefully chosen) directions. Here, we suggest an approach based on PCA that allows us to measure how much variation we’ve captured; we also provide plots of these trajectories along the contours of the loss surface.
+
+Let $\theta _ { i }$ denote model parameters at epoch $i$ and the final solution as $\theta _ { n }$ . Given $m$ training epochs, we can apply PCA to the matrix $M = [ \theta _ { 0 } - \theta _ { n } ; \cdot \cdot \cdot ; \theta _ { n - 1 } - \theta _ { n } ]$ , and then select the two most explanatory directions. Optimizer trajectories (blue dots) and loss surfaces along PCA directions are shown in Figure 9. Epochs where the learning rate was decreased are shown as red dots. On each axis, we measure the amount of variation in the descent path captured by that PCA direction.
+
+We see some interesting behavior in these plots. At early stages of training, the paths tend to move perpendicular to the contours of the loss surface, i.e., along the gradient directions as one would expect from non-stochastic gradient descent. The stochasticity becomes fairly pronounced in several plots during the later stages of training. This is particularly true of the plots that use weight decay and small batches (which leads to more gradient noise, and a more radical departure from deterministic gradient directions). When weight decay and small batches are used, we see the path turn nearly parallel to the contours and “orbit” the solution when the stepsize is large. When the stepsize is dropped (at the red dot), the effective noise in the system decreases, and we see a kink in the path as the trajectory falls into the nearest local minimizer.
+
+Finally, we can directly observe that the descent path is very low dimensional: between $40 \%$ and $90 \%$ of the variation in the descent paths lies in a space of only 2 dimensions. The optimization trajectories in Figure 9 appear to be dominated by movement in the direction of a nearby attractor. This low dimensionality is compatible with the observations in Section 6.5, where we observed that non-chaotic landscapes are dominated by wide, flat minimizers.
+
+![](images/8d33544243937b60993d712bc2c60f7cc045d69afe806c2a339b6f5d0aca7c66.jpg)  
+Figure 9: Projected learning trajectories use normalized PCA directions for VGG-9. The left plot in each subfigure uses batch size 128, and the right one uses batch size 8192.
+
+# 8 CONCLUSION
+
+In this paper, we presented a new, more accurate visualization technique that provided insights into the consequences of a variety of choices facing the neural network practitioner, including network architecture, optimizer selection, and batch size.
+
+Neural networks have advanced dramatically in recent years, largely on the back of anecdotal knowledge and theoretical results with complex assumptions. For progress to continue to be made, a more general understanding of the structure of neural networks is needed. Our hope is that effective visualization, when coupled with continued advances in theory, can result in faster training, simpler models, and better generalization.
+
+# REFERENCES
+
+Avrim Blum and Ronald L Rivest. Training a 3-node neural network is np-complete. In Advances in neural information processing systems, pp. 494–501, 1989.
+
+Pratik Chaudhari, Anna Choromanska, Stefano Soatto, and Yann LeCun. Entropy-sgd: Biasing gradient descent into wide valleys. ICLR, 2017.
+
+Anna Choromanska, Mikael Henaff, Michael Mathieu, Gerard Ben Arous, and Yann LeCun. The ´ loss surfaces of multilayer networks. In AISTATS, 2015.
+
+Yann N Dauphin, Razvan Pascanu, Caglar Gulcehre, Kyunghyun Cho, Surya Ganguli, and Yoshua Bengio. Identifying and attacking the saddle point problem in high-dimensional non-convex optimization. In Advances in neural information processing systems, pp. 2933–2941, 2014.
+
+Soham De, Abhay Yadav, David Jacobs, and Tom Goldstein. Automated inference with adaptive batches. In Artificial Intelligence and Statistics, pp. 1504–1513, 2017.
+
+Laurent Dinh, Razvan Pascanu, Samy Bengio, and Yoshua Bengio. Sharp minima can generalize for deep nets. ICLR, 2017.
+
+C Daniel Freeman and Joan Bruna. Topology and geometry of half-rectified network optimization. ICLR, 2017.
+
+Marcus Gallagher and Tom Downs. Visualization of learning in multilayer perceptron networks using principal component analysis. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 33(1):28–34, 2003.
+
+Xavier Glorot and Yoshua Bengio. Understanding the difficulty of training deep feedforward neural networks. In Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, pp. 249–256, 2010.
+
+Tom Goldstein and Christoph Studer. Phasemax: Convex phase retrieval via basis pursuit. arXiv preprint arXiv:1610.07531, 2016.
+
+Ian J Goodfellow, Oriol Vinyals, and Andrew M Saxe. Qualitatively characterizing neural network optimization problems. ICLR, 2015.
+
+Priya Goyal, Piotr Dollar, Ross Girshick, Pieter Noordhuis, Lukasz Wesolowski, Aapo Kyrola, ´ Andrew Tulloch, Yangqing Jia, and Kaiming He. Accurate, large minibatch sgd: Training imagenet in 1 hour. arXiv preprint arXiv:1706.02677, 2017.
+
+Benjamin D Haeffele and Rene Vidal. Global optimality in neural network training. In ´ Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 7331–7339, 2017.
+
+Moritz Hardt and Tengyu Ma. Identity matters in deep learning. ICLR, 2017.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep Residual Learning for Image Recognition. In CVPR, 2016.
+
+Sepp Hochreiter and Jurgen Schmidhuber. Flat minima. ¨ Neural Computation, 9(1):1–42, 1997.
+
+Elad Hoffer, Itay Hubara, and Daniel Soudry. Train longer, generalize better: closing the generaliza tion gap in large batch training of neural networks. NIPS, 2017.
+
+Gao Huang, Zhuang Liu, Kilian Q Weinberger, and Laurens van der Maaten. Densely connected convolutional networks. CVPR, 2017.
+
+Daniel Jiwoong Im, Michael Tao, and Kristin Branson. An empirical analysis of deep network loss surfaces. arXiv preprint arXiv:1612.04010, 2016.
+
+Sergey Ioffe and Christian Szegedy. Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift. 2015.
+
+Kenji Kawaguchi. Deep learning without poor local minima. NIPS, 2016.
+
+Kenji Kawaguchi, Leslie Pack Kaelbling, and Yoshua Bengio. Generalization in deep learning. arXiv preprint arXiv:1710.05468, 2017.
+
+Nitish Shirish Keskar, Dheevatsa Mudigere, Jorge Nocedal, Mikhail Smelyanskiy, and Ping Tak Peter Tang. On large-batch training for deep learning: Generalization gap and sharp minima. ICLR, 2017.
+
+Anders Krogh and John A Hertz. A simple weight decay can improve generalization. In NIPS, 1992.
+
+Yuanzhi Li and Yang Yuan. Convergence analysis of two-layer neural networks with relu activation. arXiv preprint arXiv:1705.09886, 2017.
+
+Qianli Liao and Tomaso Poggio. Theory of deep learning ii: Landscape of the empirical risk in deep learning. arXiv preprint arXiv:1703.09833, 2017.
+
+Zachary C Lipton. Stuck in a what? adventures in weight space. ICLR Workshop, 2016.
+
+Eliana Lorch. Visualizing deep network training trajectories with pca. ICML WorkshopWorkshop on Visualization for Deep Learning, 2016.
+
+Haihao Lu and Kenji Kawaguchi. Depth creates no bad local minima. arXiv preprint arXiv:1702.08580, 2017.
+
+Quynh Nguyen and Matthias Hein. The loss surface of deep and wide neural networks. International Conference on Machine Learning, 2017.
+
+Itay Safran and Ohad Shamir. On the quality of the initial basin in overspecified neural networks. In International Conference on Machine Learning, pp. 774–782, 2016.
+
+Karen Simonyan and Andrew Zisserman. Very Deep Convolutional Networks for Large-Scale Image Recognition. In ICLR, 2015.
+
+Leslie N Smith and Nicholay Topin. Exploring loss function topology with cyclical learning rates. arXiv preprint arXiv:1702.04283, 2017.
+
+Mahdi Soltanolkotabi, Adel Javanmard, and Jason D Lee. Theoretical insights into the optimization landscape of over-parameterized shallow neural networks. arXiv preprint arXiv:1707.04926, 2017.
+
+Daniel Soudry and Elad Hoffer. Exponentially vanishing sub-optimal local minima in multilayer neural networks. arXiv preprint arXiv:1702.05777, 2017.
+
+Grzegorz Swirszcz, Wojciech Marian Czarnecki, and Razvan Pascanu. Local minima in training of neural networks. stat, 1050:17, 2017.
+
+Yuandong Tian. An analytical formula of population gradient for two-layered relu network and its applications in convergence and critical point analysis. ICML, 2017.
+
+Bo Xie, Yingyu Liang, and Le Song. Diverse neural network learns true target functions. In Artificial Intelligence and Statistics, pp. 1216–1224, 2017.
+
+Chulhee Yun, Suvrit Sra, and Ali Jadbabaie. Global optimality conditions for deep neural networks. arXiv preprint arXiv:1707.02444, 2017.
+
+Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning requires rethinking generalization. ICLR, 2017.
+
+# 9 APPENDIX
+
+![](images/610175cbc64cf1ae69b262234cde47d9ff25b3b42f6d1c8fb9c0e9c8696c1bab.jpg)  
+Figure 10: 1D linear interpolation of solutions obtained by small-batch and large-batch methods for ResNet56. The blue lines are loss values and the red lines are error. The solid lines are training curves and the dashed lines are for testing.
+
+![](images/5dc79c487760afd5e52819863f3727b33d7f93699648a1573eb732d6e3055567.jpg)  
+Figure 11: The shape of minima obtained via different optimization algorithms for ResNet-56, with varying batch size and weight decay. Similar to Figure 4, the first row uses zero weight decay and the second row uses 5e-4 weight decay.
+
+Generalization error for each plot is shown in Table 3.
+
+![](images/51c9dc79a1510514ec1b5270885d6f3c0f59cca940263448f32e6e8e3acc82d5.jpg)  
+Figure 12: 2D visualization of solutions of ResNet-56 obtained by SGD/Adam with small-batch and large-batch. Similar to Figure 11, the first row uses zero weight decay and the second row sets weight decay to 5e-4.
+
+Table 3: Test error for ResNet-56 with different optimization algorithms and batch-size/weight-decay parameters.   
+
+<table><tr><td rowspan="3"></td><td colspan="2">SGD</td><td colspan="2">Adam</td></tr><tr><td>bs=128</td><td>bs=4096</td><td>bs=128</td><td>bs=4096</td></tr><tr><td>WD= 0</td><td>8.26</td><td>13.93</td><td>9.55</td><td>14.30</td></tr><tr><td>WD = 5e-4</td><td>5.89</td><td>10.59</td><td>7.67</td><td>12.36</td></tr></table>
+
+![](images/82ab00c5f26ae528c7ab8fa745db3229313bdbdc5c56807e9c7afd9f1c347d5d.jpg)  
+Figure 13: The loss landscape for DenseNet-121 trained on CIFAR-10. The final training error is 0.002 and the testing error is 4.37
+
+![](images/0baeb9b3d7877532ae268f66b858c5530fc075aaa8c9a1edf0c0f6e79f301065.jpg)  
+Figure 14: Convergence curves for different architectures. The first row is for training loss and the second row are training and testing error curves.
+
+![](images/0c310c541da47bed32b8513c30c1792a62f50b9bcacdcb51b6f31d18970657bd.jpg)  
+Figure 15: Training and testing loss curves for VGG-9. Dashed lines are for testing, solid for training.

parse/train/HkmaTz-0W/HkmaTz-0W_content_list.json

@@ -0,0 +1,1823 @@
+[
+    {
+        "type": "text",
+        "text": "VISUALIZING THE LOSS LANDSCAPE OF NEURAL NETS ",
+        "text_level": 1,
+        "bbox": [
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+            101,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors ",
+        "text_level": 1,
+        "bbox": [
+            184,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Paper under double-blind review ",
+        "bbox": [
+            184,
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+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
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+            544,
+            250
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Neural network training relies on our ability to find “good” minimizers of highly non-convex loss functions. It is well known that certain network architecture designs (e.g., skip connections) produce loss functions that train easier, and wellchosen training parameters (batch size, learning rate, optimizer) produce minimizers that generalize better. However, the reasons for these differences, and their effect on the underlying loss landscape, is not well understood. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we explore the structure of neural loss functions, and the effect of loss landscapes on generalization, using a range of visualization methods. First, we introduce a simple “filter normalization” method that helps us visualize loss function curvature, and make meaningful side-by-side comparisons between loss functions. Then, using a variety of visualizations, we explore how network architecture effects the loss landscape, and how training parameters affect the shape of minimizers. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Training neural networks requires minimizing a high-dimensional non-convex loss function – a task that is hard in theory, but sometimes easy in practice. Despite the NP-hardness of training general neural loss functions (Blum & Rivest, 1989), simple gradient methods often find global minimizers (parameter configurations with zero or near-zero training loss), even when data and labels are randomized before training (Zhang et al., 2017). However, this good behavior is not universal; the trainability of neural nets is highly dependent on network architecture design choices, the choice of optimizer, variable initialization, and a variety of other considerations. Unfortunately, the effect of each of these choices on the structure of the underlying loss surface is unclear. Because of the prohibitive cost of loss function evaluations (which requires looping over all the data points in the training set), studies in this field have remained predominantly theoretical. ",
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+    {
+        "type": "image",
+        "img_path": "images/a14706c29d892ca8c4fe95f8306eb80e5ef1b4694e7c755e61d5a5f55c9ef4dd.jpg",
+        "image_caption": [
+            "Figure 1: The loss surfaces of ResNet-56 with/without skip connections. The vertical axis is logarithmic to show dynamic range. The proposed filter normalization scheme is used to enable comparisons of sharpness/flatness between the two figures. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our goal is to use high-resolution visualizations to provide an empirical characterization of neural loss functions, and to explore how different network architecture choices affect the loss landscape. Furthermore, we explore how the non-convex structure of neural loss functions relates to their trainability, and how the geometry of neural minimizers (i.e., their sharpness/flatness, and their surrounding landscape), affects their generalization properties. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To do this in a meaningful way, we propose a simple “filter normalization” scheme that enables us to do side-by-side comparisons of different minima found by different methods. We then use visualizations to explore sharpness/flatness of minimizers found by different methods, as well as the effect of network architecture choices (use of skip connections, number of filters, network depth) on the loss landscape. Out goal is to understand how differences in loss function geometry effect the generalization of neural nets. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1.1 CONTRIBUTIONS ",
+        "text_level": 1,
+        "bbox": [
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+            319,
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+            333
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this article, we study methods for producing meaningful loss function visualizations. Then, using these visualization methods, we explore how loss landscape geometry effects generalization error and trainability. More specifically, we address the following issues: ",
+        "bbox": [
+            176,
+            345,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We reveal faults in a number of visualization methods for loss functions, and show that simple visualization strategies fail to accurately capture the local geometry (sharpness or flatness) of loss function minimizers. We present a simple visualization method based on “filter normalization” that enables side-by-side comparisons of different minimizers. The sharpness of minimizers correlates well with generalization error when this visualization is used, even when making sharpness comparisons across disparate network architectures and training methods.   \n• We observe that, when networks become sufficiently deep, neural loss landscapes suddenly transition from being nearly convex to being highly chaotic. This transition from convex to chaotic behavior, which seem to have been previously unnoticed, coincides with a dramatic drop in generalization error, and ultimately to a lack of trainability. We show that skip connections promote flat minimizers and prevent the transition to chaotic behavior, which helps explain why skip connections are necessary for training extremely deep networks.   \n• We study the visualization of SGD optimization trajectories. We explain the difficulties that arise when visualizing these trajectories, and show that optimization trajectories lie in an extremely low dimensional space. This low dimensionality can be explained by the presence of large nearly convex regions in the loss landscape, such as those observed in our 2-dimensional visualizations. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 THEORETICAL BACKGROUND & RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            717,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Visualizations have the potential to help us answer several important questions about why neural networks work. In particular, why are we able to minimize highly non-convex neural loss functions? And why do the resulting minima generalize? ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Because of the difficultly of visualizing loss functions, most studies of loss landscapes are largely theoretical in nature. A number of authors have studied our ability to minimize neural loss functions. Using random matrix theory and spin glass theory, several authors have shown that local minima are of low objective value (Dauphin et al., 2014; Choromanska et al., 2015). It can also be shown that local minima are global minima, provided one assumes linear neurons (Hardt & Ma, 2017), very wide layers (Nguyen & Hein, 2017), or full rank weight matrices (Yun et al., 2017). These assumptions have been relaxed by Kawaguchi (2016) and Lu & Kawaguchi (2017), although some assumptions (e.g., of the loss functions) are still required. Soudry & Hoffer (2017); Freeman & Bruna (2017); Xie et al. (2017) also analyzed shallow networks with one or two hidden layers under mild conditions. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Another approach is to show that we can expect good minimizers, not simply because of the endogenous properties of neural networks, but because of the optimizers. For restricted network classes such as those with one hidden layer, with some extra assumptions on the sample distribution, globally optimal or near-optimal solutions can be found by common optimization methods (Soltanolkotabi et al., 2017; Li & Yuan, 2017; Tian, 2017). For networks with specific structures, Safran & Shamir (2016) and Haeffele & Vidal (2017) show there likely exists a monotonically decreasing path from an initialization to a global minimum. Swirszcz et al. (2017) show counterexamples that achieve “bad” local minima for toy problems. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Also of interest is work on assessing the sharpness/flatness of local minima. Hochreiter & Schmidhuber (1997) defined “flatness” as the size of the connected region around the minimum where the training loss remains low. Keskar et al. (2017) propose $\\epsilon$ -sharpness, which looks at the maximum loss in a bounded neighborhood of a minimum. Flatness can also be defined using the local curvature of the loss function at a critical point. Keskar et al. (2017) suggests that this information is encoded in the eigenvalues of the Hessian. However, Dinh et al. (2017) show that these quantitative measure of sharpness are problematic because they are not invariant to symmetries in the network, and are thus not sufficient to determine its generalization ability. This issue was addressed in Chaudhari et al. (2017), who used local entropy as a measure of sharpness. This measure is invariant to the simple transformation used by Dinh et al. (2017), but difficult to quantify for large networks. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Theoretical results make some restrictive assumptions such as the independence of the input samples, or restrictions on non-linearities and loss functions. For this reason, visualizations play a key role in verifying the validity of theoretical assumptions, and understanding loss function behavior in real-world systems. In the next section, we briefly review methods that have been used for this purpose. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 THE BASICS OF LOSS FUNCTION VISUALIZATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Neural networks are trained on a corpus of feature vectors (e.g., images) $\\{ x _ { i } \\}$ and accompanying labels $\\{ y _ { i } \\}$ by minimizing a loss of the form ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9c7d8143ead33d52d704a82f72fb5b5a162ece24fd132eee72414648a71300ec.jpg",
+        "text": "$$\nL ( \\theta ) = \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } \\ell ( x _ { i } , y _ { i } ; \\theta )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            411,
+            523,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\theta$ denotes the parameters (weights) of the neural network, the function $\\ell ( x _ { i } , y _ { i } ; \\theta )$ measures how well the neural network with parameters $\\theta$ predicts the label of a data sample, and $m$ is the number of data samples. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Neural nets contain many parameters, and so their loss functions live in a very high-dimensional space. Unfortunately, visualizations are only possible using low-dimensional 1D (line) or 2D (surface) plots. Several methods exist for closing this dimensionality gap. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "1-Dimensional Linear Interpolation One simple and lightweight way to plot loss functions is to choose two sets of parameters $\\theta _ { 1 }$ and $\\theta _ { 2 }$ , and plot the values of the loss function along the line connecting these two points. We can parameterize this line by choosing a scalar parameter $\\alpha$ , and defining the weighted average ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/eeb817ecfe5ec80266ec0e8e367d173ddcf9fc7b429cc82cdf7c300e37418cf2.jpg",
+        "text": "$$\n\\theta _ { \\alpha } = ( 1 - \\alpha ) \\theta _ { 1 } + \\alpha \\theta _ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Finally, we plot the function $f ( \\alpha ) = L ( \\theta _ { \\alpha } )$ . This strategy was taken by Goodfellow et al. (2015), who studied the loss surface along the line between a (random) initial guess, and a nearby minimizer obtained by stochastic gradient descent. This method has been widely used to study the “sharpness” and “flatness” of different minima, and the dependence of sharpness on batch-size (Keskar et al., 2017; Dinh et al., 2017). Smith & Topin (2017) use the same 1D interpolation technique to show different minima and the “peaks” between them, while Im et al. (2016) plot the line between minima obtained via different optimizers. ",
+        "bbox": [
+            173,
+            748,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The 1D linear interpolation method suffers from several weaknesses. First, it is difficult to visualize non-convexities using 1D plots. Indeed, the authors of (Goodfellow et al., 2015) found that loss functions appear to lack local minima along the minimization trajectory. We will see later, using 2D methods, that some loss functions have extreme non-convexities, and that these non-convexities correlate with the difference in generalization between different network architectures. Second, this method does not consider batch normalization or invariance symmetries in the network. For this reason, the visual sharpness comparisons produced by 1D interpolation plots may be misleading; this issue will be explored in depth in Section 5. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2D Contour Plots To use this approach, one chooses a center point $\\theta ^ { * }$ in the graph, and chooses two direction vectors, $\\delta$ and $\\eta$ . One then plots a function of the form $f ( \\alpha ) = L ( { \\bar { \\theta } } ^ { * } { \\bar { + } } \\alpha \\delta )$ in the 1D (line) case, or ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/20bc9ed5441644e55fb317349eb32512411915f9154c7aa2145b501d1bc6b60a.jpg",
+        "text": "$$\nf ( \\alpha , \\beta ) = L ( \\theta ^ { * } + \\alpha \\delta + \\beta \\eta )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            401,
+            222,
+            596,
+            239
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "in the 2D (surface) case. This approach was used in (Goodfellow et al., 2015) to explore the trajectories of different minimization methods. It was also used in (Im et al., 2016) to show that different optimization algorithms find different local minima within the 2D projected space. ",
+        "bbox": [
+            173,
+            251,
+            825,
+            294
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Because of the computational burden of 2D plotting, these methods generally result in low-resolution plots of small regions that have not captured the complex non-convexity of loss surfaces. Below, we use high-resolution visualizations over large slices of weight space to visualize how network design affects non-convex structure. ",
+        "bbox": [
+            174,
+            300,
+            825,
+            357
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 PROPOSED VISUALIZATION: FILTER-WISE NORMALIZATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            383,
+            707,
+            400
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This study relies heavily on plots of the form (1) produced using random direction vectors, $\\delta$ and $\\eta$ , each sampled from a random Gaussian distribution with appropriate scaling (described below). ",
+        "bbox": [
+            174,
+            419,
+            823,
+            446
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "While the “random directions” approach to plotting is simple, it cannot be used to compare the geometry of two different minimizers or two different networks. This is because of the scale invariance in network weights. When ReLU non-linearities are used, the network remains unchanged if we (for example) multiply the weights in one layer of a network by 10, and divide the next layer by 10. This invariance is even more prominent when batch normalization is used. In this case, the size (i.e., norm) of a filter is irrelevant because the output of each layer is re-scaled during batch normalization. For this reason, a network’s behavior remains unchanged if we re-scale the weights. ",
+        "bbox": [
+            174,
+            454,
+            825,
+            551
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Scale invariance prevents us from making meaningful comparisons between plots, unless special precautions are taken. A neural network with large weights may appear to have a smooth and slowly varying loss function; perturbing the weights by one unit will have very little effect on network performance if the weights live on a scale much larger than one. However, if the weights are much smaller than one, then that same one unit perturbation may have a catastrophic effect, making the loss function appear quite sensitive to weight perturbations. Keep in mind that neural nets are scale invariant; if the small-parameter and large-parameter networks in this example are equivalent (because one is simply a re-scaling of the other), then any apparent differences in the loss function are merely an artifact of scale invariance. This scale invariance was exploited by Dinh et al. (2017) to build pairs of equivalent networks that have different apparent sharpness. ",
+        "bbox": [
+            174,
+            558,
+            825,
+            698
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To remove this scaling effect, we plot loss functions using filter-wise normalized directions. To obtain such directions for a network with parameters $\\theta$ , we begin by producing a random Gaussian direction vector $d$ with dimensions compatible with $\\theta$ . Then we normalize each filter in $d$ to have the same norm of the corresponding filter in $\\theta$ . In other words, we make the replacement ",
+        "bbox": [
+            174,
+            704,
+            825,
+            761
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bc1721a5ea12224aa682ec7f2836c96faf11a61f21a111ed34be92a392c889b6.jpg",
+        "text": "$$\nd _ { i }  { \\frac { d _ { i } } { \\Vert d _ { i } \\Vert } } \\Vert \\theta _ { i } \\Vert ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            442,
+            772,
+            553,
+            806
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $d _ { i }$ represents the ith filter of $d$ (not the ith weight), and $\\lVert \\boldsymbol { \\theta } _ { i } \\rVert$ denotes the Frobenius norm of the ith filter of $\\theta$ . Note that the filter-wise normalization is different from that of (Im et al., 2016), which normalize the direction without considering the norm of individual filters. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The proposed scaling is an important factor when making meaningful plots of loss function geometry. We will explore the importance of proper scaling below as we explore the sharpness/flatness of different minimizers. In this context, we show that the sharpness of filter-normalized plots correlates with generalization error, while plots without filter normalization can be very misleading. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "5 THE SHARP VS FLAT DILEMMA ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            459,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Section 4 introduces the concept of filter normalization, and provides an intuitive justification for its use. In this section, we address the issue of whether sharp minimizers generalize better than flat minimizers. In doing so, we will see that the sharpness of minimizers correlates well with generalization error when filter normalization is used. This enables side-by-side comparisons between plots. In contrast, the sharpness of non-filter normalized plots may appear distorted and unpredictable. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "It is widely thought that small-batch SGD produces “flat” minimizers that generalize better, while large batch sizes produce “sharp” minima with poor generalization (Chaudhari et al., 2017; Keskar et al., 2017; Hochreiter & Schmidhuber, 1997). This claim is disputed though, with Dinh et al. (2017); Kawaguchi et al. (2017) arguing that generalization is not directly related to the curvature of loss surfaces, and some authors proposing specialized training methods that achieve good performance with large batch sizes (Hoffer et al., 2017; Goyal et al., 2017; De et al., 2017). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Here, we explore the difference between sharp and flat minimizers. We begin by discussing difficulties that arise when performing such a visualization, and how proper normalization can prevent such plots from producing distorted results. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We train a CIFAR-10 classifier using a 9-layer VGG network (Simonyan & Zisserman, 2015) with Batch Normalization (Ioffe & Szegedy, 2015). We use two batch sizes: a large batch size of 8192 ( $1 6 . 4 \\%$ of the training data of CIFAR-10), and a small batch size of 128. Let $\\theta _ { s }$ and $\\theta _ { l }$ indicate the solutions obtained by running SGD using small and large batch sizes, respectively1. Using the linear interpolation approach (Goodfellow et al., 2015), we plot the loss values on both training and testing data sets of CIFAR-10, along a direction containing the two solutions, i.e., $f ( \\alpha ) = L ( \\theta _ { s } + \\alpha ( \\theta _ { l } - \\theta _ { s } ) )$ . ",
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/ef50b7737be38ec876e0cdb95e3e3af393016f78c52f3f6208f9ac227293847e.jpg",
+        "image_caption": [
+            "Figure 2: 1D linear interpolation of solutions obtained by small-batch and large-batch methods for VGG9. The blue lines are loss values and the red lines are accuracies. The solid lines are training curves and the dashed lines are for testing. Small batch is at abscissa 0, and large batch is at abscissa 1. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/98c197d8db6f14ab8dac05d572f1b918604d116a07327fac2ead42e5745aae02.jpg",
+        "table_caption": [
+            "Table 1: Test errors of VGG-9 on CIFAR-10 with different optimization algorithms and hyperparameters. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"3\"></td><td rowspan=\"3\"></td><td colspan=\"2\">SGD</td><td colspan=\"2\">Adam</td></tr><tr><td>bs=128</td><td>bs=8192</td><td>bs=128</td><td>bs=8192</td></tr><tr><td>VGG-9</td><td>WD= 0</td><td>7.37</td><td>11.07</td><td>7.44</td><td>10.91</td></tr><tr><td></td><td>WD = 5e-4</td><td>6.00</td><td>10.19</td><td>7.80</td><td>9.52</td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Similar to Keskar et al. (2017), we also superimpose the classification accuracy in red. This plot is shown in Figure 2. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Figures 2(a) and 2(b) show linear interpolation plots with $\\theta _ { s }$ at $\\mathbf { X }$ -axis location 0, and $\\theta _ { l }$ at location $1 ^ { 2 }$ . As observed by Keskar et al. (2017), we can clearly see that the small-batch solution is quite wide, while the large-batch solution is sharp. However, this sharpness balance can be flipped simply by turning on weight decay (Krogh & Hertz, 1992). Figures 2(c) and 2(d) show results of the same experiment, except this time with a non-zero weight decay parameter. This time, the large batch minimizer is considerably flatter than the sharp small batch minimizer. However, we see from Table 1 that small batches generalize better in all 4 experiments; there is no apparent correlation between sharpness and generalization. We will see that these side-by-side sharpness comparisons are extremely misleading, and fail to capture the endogenous properties of the minima. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The apparent differences in sharpness in Figure 2 can be explained by examining the weights of each minimizer. Histograms of the networks weights are shown for each experiment in Figure 3. We see that, when a large batch is used with zero weight decay, the resulting weights tends to be smaller than in the small batch case. We reverse this effect by adding weight decay; in this case the large batch minimizer has much larger weights than the small batch minimizer. This difference in scale occurs for a simple reason: A smaller batch size results in more weight updates per epoch than a large batch size, and so the shrinking effect of weight decay (which imposes a penalty on the norm of the weights) is more pronounced. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Figure 2 in not visualizing the endogenous sharpness of minimizers, but rather just the (irrelevant) weight scaling. The scaling of weights in these networks is irrelevant because batch normalization re-scales the outputs to have unit variance. However, small weights still appear more sensitive to perturbations, and produce sharper looking minimizers. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Filter normalized plots We repeat the experiment in Figure 2, but this time we plot the loss function near each minimizer separately using random filter-normalized directions. This removes the apparent differences in geometry caused by the scaling depicted in Figure 3. The results, presented in Figure 4, still show differences in sharpness between small batch and large batch minima, however these differences are much more subtle than it would appear in the un-normalized plots. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We also visualize these results using two random directions and contour plots. As shown in Figure 5, the weights obtained with small batch size and non-zero weight decay have wider contours than the sharper large batch minimizers. Similar for Resnet-56 appear in Figure 12 of the Appendix. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/c5a7d4f64b3d10940e5b0e8fcf8a2ad96ebd92066bfd9cc81dc1eeb9a61973a2.jpg",
+        "image_caption": [
+            "Figure 3: Histogram of weights. With zero weight decay, small-batch methods produce large weights. With non-zero weight decay, small-batch methods produce smaller weights. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/0089ebcfa1e4c7c29fcc8cf3c9b44b137d8e12613e49aba8c463103b971b4e17.jpg",
+        "image_caption": [
+            "Figure 4: The shape of minima obtained using different optimization algorithms, with varying batch size and weight decay. The title of each subfigure contains the optimizer, batch size, and test error. The first row has no weight decay and the second row uses weight decay 5e-4. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/19fc60a8c5a0284a9dbe8f9ce56c82c214fd602dbeff6b22bd17998217d2f207.jpg",
+        "image_caption": [
+            "Figure 5: 2D visualization of solutions obtained by SGD with small-batch and large-batch. Similar to Figure 4, the first row uses zero weight decay and the second row sets weight decay to 5e-4. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Generalization and Flatness Using the filter-normalized plots in Figures 4 and 5, we can make side-by-side comparisons between minimizers, and we see that now sharpness correlates well with generalization error. Large batches produced visually sharper minima (although not dramatically so) with higher test error. Interestingly, the Adam optimizer attained larger test error than SGD, and, as predicted, the corresponding minima are visually sharper. Results of a similar experiment using ResNet-56 are presented in the Appendix (Figure 12). ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "6 WHAT MAKES NEURAL NETWORKS TRAINABLE? INSIGHTS ON THE (NON) CONVEXITY STRUCTURE OF LOSS SURFACES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Our ability to find global minimizers to neural loss functions is not universal; it seems that some neural architectures are easier to minimize than others. For example, using skip connections, He et al. (2016) were able to train extremely deep architectures, while comparable architectures without skip connections are not trainable. Furthermore, our ability to train seems to depend strongly on the initial parameters from which training starts. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Using visualization methods, we do an empirical study of neural architectures to explore why the non-convexity of loss functions seems to be problematic in some situations, but not in others. We aim to provide insight into the following questions: Do loss functions have significant non-convexity at all? If prominent non-convexities exist, why are they not problematic in all situations? Why are some architectures easy to train, and why are results so sensitive to the initialization? We will see that different architectures have extreme differences in non-convexity structure that answer these questions, and that these differences correlate with generalization error. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6.1 EXPERIMENTAL SETUP ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To understand the effects of network architecture on non-convexity, we trained a number of networks, and plotted the landscape around the obtained minimizers using the filter-normalized random direction method described in Section 4. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We consider three classes of neural networks: ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "• Residual networks that are optimized for performance on CIFAR (He et al., 2016). We consider ResNet-20, ResNet-56, and ResNet-110, where each name is labeled with the number of convolutional layers it has. “VGG-like” networks that do not contain shortcut/skip connections. We produced these networks simply by removing the skip connections from the CIFAR-optimized ResNets. We call these networks ResNet-20-noshort, ResNet-56-noshort, and ResNet-110-noshort. Note that these networks do not all perform well on the CIFAR-10 task. We use them purely for experimental purposes to explore the effect of shortcut connections. “Wide” ResNets that have been optimized for ImageNet rather than CIFAR. These networks have more filters per layer than the CIFAR optimized networks, and also have different numbers of layers. These models include ResNet-18, ResNet-34, and ResNet-50. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "All models are trained on the CIFAR-10 dataset using SGD with Nesterov momentum, batch-size 128, and 0.0005 weight decay for 300 epochs. The learning rate was initialized at 0.1, and decreased by a factor of 10 at epochs 150, 225 and 275. Deeper experimental VGG-like networks (e.g., ResNet-56-noshort, as described below) required a smaller initial learning rate of 0.01. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "High resolution 2D plots of the minimizers for different neural networks are shown in Figure 6. Results are shown as contour plots rather than surface plots because this makes it extremely easy to see non-convex structures and evaluate sharpness. For surface plots of ResNet-56, see Figure 1. Note that the center of each plot corresponds to the minimizer, and the two axes parameterize two random directions with filter-wise normalization as in (1). We make several observations below about how architecture effects the loss landscape. We also provide loss and error values for these networks in Table 2, and convergence curves in Figure 14 of the Appendix. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6.2 THE EFFECT OF NETWORK DEPTH ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "From Figure 6, we see that network depth has a dramatic effect on the loss surfaces of neural networks when skip connections are not used. The network ResNet-20-noshort has a fairly benign landscape dominated by a region with convex contours in the center, and no dramatic non-convexity. This isn’t too surprising: the original VGG networks for ImageNet had 19 layers and could be trained effectively (Simonyan & Zisserman, 2015). ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "However, as network depth increases, the loss surface of the VGG-like nets spontaneously transitions from (nearly) convex to chaotic. ResNet-56-noshort has dramatic non-convexities and large regions where the gradient directions (which are normal to the contours depicted in the plots) do not point towards the minimizer at the center. Also, the loss function becomes extremely large as we move in some directions. ResNet-110-noshort displays even more dramatic non-convexities, and becomes extremely steep as we move in all directions shown in the plot. Furthermore, note that the minimizers at the center of the deep VGG-like nets seem to be fairly sharp. In the case of ResNet-56-noshort, the minimizer is also fairly ill-conditioned, as the contours near the minimizer have significant eccentricity. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/09e1fa6d53f9c16d0e4f2124e26f3227824bd8d411eafd3b74ff9db658ad4338.jpg",
+        "image_caption": [
+            "Figure 6: 2D visualization of the solutions of different networks. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/40226d86d4ee5767c94ded31a90966ad5bd948e604b7e23e2165d0c6465e84b3.jpg",
+        "table_caption": [
+            "Table 2: Loss values and errors for different architectures. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Filters</td><td>Training Loss</td><td>Training Error</td><td>Test Error</td></tr><tr><td rowspan=\"2\">ResNet-20 ResNet-20-noshort</td><td rowspan=\"2\">16</td><td>0.017</td><td>0.286</td><td>7.37</td></tr><tr><td>0.025</td><td>0.560</td><td>8.18</td></tr><tr><td>ResNet-56</td><td rowspan=\"2\">16</td><td>0.004</td><td>0.052</td><td>5.89</td></tr><tr><td>ResNet-56-noshort</td><td>0.024</td><td>0.704</td><td>10.83</td></tr><tr><td>ResNet-110</td><td rowspan=\"2\">16</td><td>0.002</td><td>0.042</td><td>5.79</td></tr><tr><td>ResNet-110-noshort</td><td>0.258</td><td>8.732</td><td>16.44</td></tr><tr><td>ResNet-18</td><td rowspan=\"2\">64 64</td><td>0.002</td><td>0.026</td><td>5.42</td></tr><tr><td>ResNet-34</td><td>0.001</td><td>0.014</td><td>4.73</td></tr><tr><td>ResNet-50</td><td>64</td><td>0.001</td><td>0.006</td><td>4.55</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6.3 SHORTCUT CONNECTIONS TO THE RESCUE ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Shortcut connections have a dramatic effect of the geometry of the loss functions. In Figure 6, we see that residual connections prevent the transition to chaotic behavior as depth increases. In fact, the width and shape of the 0.1-level contour is almost identical for the 20- and 110-layer networks. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Interestingly, the effect of skip connections seems to be most important for deep networks. For the more shallow networks (ResNet-20 and ResNet-20-noshort), the effect of skip connections is fairly unnoticeable. However residual connections prevent the explosion of non-convexity that occurs when networks get deep. This effect seems to apply to other kinds of skip connections as well; Figure 13 of the Appendix shows the loss landscape of DenseNet (Huang et al., 2017), which shows no noticeable non-convexity. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6.4 WIDE MODELS VS THIN MODELS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "To see the effect of the number of convolutional filters per layer, we compare the narrow CIFARoptimized ResNets (ResNet-20/56/110) with wider ResNets (ResNet-18/34/50) that have more filters and were optimized for ImageNet. From Figure 6, we see that the wider models have loss landscapes with no noticeable chaotic behavior. Increased network width resulted in flat minima and wide regions of apparent convexity. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This effect is also validated by Figure 7, in which we plot the landscape of ResNet-56, but we multiple the number of filter per layer by $k = 2 , 4$ , and 8. We see that increased width prevents chaotic behavior, and skip connections dramatically widen minimizers. Finally, note that sharpness correlates extremely well with test error. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/c55cb4af6dfc31fb4a6c5b394992c8d9c6370fc38dc358befd09e109f60dd5bd.jpg",
+        "image_caption": [
+            "Figure 7: Wide-ResNet-56 (WRN-56) on CIFAR-10 both with shortcut connections (top) and without (bottom). The label $k = 2$ means twice as many filters per layer, $k = 4$ means 4 times, etc. Test error is reported below each figure. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            158,
+            285,
+            839,
+            500
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6.5 IMPLICATIONS FOR NETWORK INITIALIZATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            587,
+            537,
+            599
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "One of the most interesting observations seen in Figure 6 is that loss landscapes for all the networks considered seem to be partitioned into a well-defined region of low loss value and convex contours, surrounded by a well-defined region of high loss value and non-convex contours. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This partitioning of chaotic and convex regions may explain the importance of good initialization strategies, and also the easy training behavior of “good” architectures. When using normalized random initialization strategies such as those proposed by Glorot & Bengio (2010), typical neural networks attain an initial loss value less than 2.5. The well behaved loss landscapes in Figure 6 (ResNets, and shallow VGG-like nets) are dominated by large, flat, nearly convex attractors that rise to a loss value of 4 or greater. For such landscapes, a random initialization will likely lie in the “well- behaved” loss region, and the optimization algorithm might never “see” the pathological non-convexities that occur on the high loss chaotic plateaus. ",
+        "bbox": [
+            174,
+            661,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Chaotic loss landscapes (ResNet-56-noshort and ResNet-110-noshort) have shallower regions of convexity that rise to lower loss values. For sufficiently deep networks with shallow enough attractors, the initial iterate will likely lie in the chaotic region where the gradients are uninformative. In our experiments, SGD was unable to train a 156 layer network without skip connections (even with very low learning rates), which adds weight to this hypothesis. ",
+        "bbox": [
+            174,
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+            849
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6.6 LANDSCAPE GEOMETRY AFFECTS GENERALIZATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            869,
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+            882
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Table 2 displays the training and test error for the networks depicted in Figure 6. Both Figures 6 and 7 show that landscape geometry has a dramatic effect on generalization. First, note that visually flatter minimizers consistently correspond to lower test error, which further strengthens our assertion that filter normalization is a natural way to visualize loss function geometry. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
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+            132
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Second, we notice that chaotic landscapes (deep networks without skip connections) result in worse training and test error, while more convex landscapes have lower error values. In fact, the most convex landscapes (the wide ResNets in the bottom row of Figure 6) generalize the best of all the networks. This latter class of networks show no noticeable chaotic behavior at all. ",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "7 VISUALIZING OPTIMIZATION PATHS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Finally, we explore methods for visualizing the trajectories of different optimizers. For this application, random directions are ineffective. We will provide a theoretical explanation for why random directions fail, and explore methods for effectively plotting trajectories on top of loss function contours. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Several authors have observed that random direction fail to capture the variation in optimization trajectories, including Gallagher & Downs (2003); Lorch (2016); Lipton (2016); Liao & Poggio (2017). Several failed visualizations are depicted in Figure 8. In Figure 8(a), we see the iterates of SGD projected onto the plane defined by two random directions. Almost none of the motion is captured (notice the super-zoomed-in axes and the seemingly random walk). This problem was noticed by Goodfellow et al. (2015), who then visualized trajectories using one direction that points from initialization to solution, and one random direction. This approach is shown in Figure 8(b). As seen in Figure 8(c), the random axis captures almost no variation, leading to the (misleading) appearance of a straight line path. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/6b58a26902081c8eab5b05e4d0561fdea81fbc2a651f28634198198c59d961af.jpg",
+        "image_caption": [
+            "Figure 8: Ineffective visualizations of optimizer trajectories. These visualizations suffer from the orthogonality of random directions in high dimensions. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "7.1 WHY RANDOM DIRECTIONS FAIL: LOW DIMENSIONAL OPTIMIZATION TRAJECTORIES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "It is well known that two random vectors in a high dimensional space will be nearly orthogonal with high probability. In fact, the expected cosine similarity between Gaussian random vectors in $n$ dimensions is roughly $\\sqrt { 2 / ( \\pi n ) }$ (Goldstein & Studer (2016), Lemma 5). ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "This is problematic when optimization trajectories lie in extremely low dimensional spaces. In this case, a randomly chosen vector will lie orthogonal to the low-rank space containing the optimization path, and a projection onto a random direction will capture almost no variation. Figure 8(b) suggests that optimization trajectories are low dimensional because the random direction captures orders of magnitude less variation than the vector that points along the optimization path. Below, we use PCA directions to directly validate this low dimensionality, and also to produce effective visualizations. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "7.2 EFFECTIVE TRAJECTORY PLOTTING USING PCA DIRECTIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "To capture variation in trajectories, we need to use non-random (and carefully chosen) directions. Here, we suggest an approach based on PCA that allows us to measure how much variation we’ve captured; we also provide plots of these trajectories along the contours of the loss surface. ",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Let $\\theta _ { i }$ denote model parameters at epoch $i$ and the final solution as $\\theta _ { n }$ . Given $m$ training epochs, we can apply PCA to the matrix $M = [ \\theta _ { 0 } - \\theta _ { n } ; \\cdot \\cdot \\cdot ; \\theta _ { n - 1 } - \\theta _ { n } ]$ , and then select the two most explanatory directions. Optimizer trajectories (blue dots) and loss surfaces along PCA directions are shown in Figure 9. Epochs where the learning rate was decreased are shown as red dots. On each axis, we measure the amount of variation in the descent path captured by that PCA direction. ",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            176,
+            103,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We see some interesting behavior in these plots. At early stages of training, the paths tend to move perpendicular to the contours of the loss surface, i.e., along the gradient directions as one would expect from non-stochastic gradient descent. The stochasticity becomes fairly pronounced in several plots during the later stages of training. This is particularly true of the plots that use weight decay and small batches (which leads to more gradient noise, and a more radical departure from deterministic gradient directions). When weight decay and small batches are used, we see the path turn nearly parallel to the contours and “orbit” the solution when the stepsize is large. When the stepsize is dropped (at the red dot), the effective noise in the system decreases, and we see a kink in the path as the trajectory falls into the nearest local minimizer. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Finally, we can directly observe that the descent path is very low dimensional: between $40 \\%$ and $90 \\%$ of the variation in the descent paths lies in a space of only 2 dimensions. The optimization trajectories in Figure 9 appear to be dominated by movement in the direction of a nearby attractor. This low dimensionality is compatible with the observations in Section 6.5, where we observed that non-chaotic landscapes are dominated by wide, flat minimizers. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/8d33544243937b60993d712bc2c60f7cc045d69afe806c2a339b6f5d0aca7c66.jpg",
+        "image_caption": [
+            "Figure 9: Projected learning trajectories use normalized PCA directions for VGG-9. The left plot in each subfigure uses batch size 128, and the right one uses batch size 8192. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "8 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            671,
+            318,
+            686
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we presented a new, more accurate visualization technique that provided insights into the consequences of a variety of choices facing the neural network practitioner, including network architecture, optimizer selection, and batch size. ",
+        "bbox": [
+            176,
+            703,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Neural networks have advanced dramatically in recent years, largely on the back of anecdotal knowledge and theoretical results with complex assumptions. For progress to continue to be made, a more general understanding of the structure of neural networks is needed. Our hope is that effective visualization, when coupled with continued advances in theory, can result in faster training, simpler models, and better generalization. ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            287,
+            118
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Avrim Blum and Ronald L Rivest. Training a 3-node neural network is np-complete. In Advances in neural information processing systems, pp. 494–501, 1989. ",
+        "bbox": [
+            176,
+            126,
+            823,
+            155
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Pratik Chaudhari, Anna Choromanska, Stefano Soatto, and Yann LeCun. Entropy-sgd: Biasing gradient descent into wide valleys. ICLR, 2017. ",
+        "bbox": [
+            174,
+            162,
+            823,
+            193
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Anna Choromanska, Mikael Henaff, Michael Mathieu, Gerard Ben Arous, and Yann LeCun. The ´ loss surfaces of multilayer networks. In AISTATS, 2015. ",
+        "bbox": [
+            173,
+            200,
+            823,
+            231
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Yann N Dauphin, Razvan Pascanu, Caglar Gulcehre, Kyunghyun Cho, Surya Ganguli, and Yoshua Bengio. Identifying and attacking the saddle point problem in high-dimensional non-convex optimization. In Advances in neural information processing systems, pp. 2933–2941, 2014. ",
+        "bbox": [
+            178,
+            238,
+            823,
+            282
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Soham De, Abhay Yadav, David Jacobs, and Tom Goldstein. Automated inference with adaptive batches. In Artificial Intelligence and Statistics, pp. 1504–1513, 2017. ",
+        "bbox": [
+            171,
+            290,
+            823,
+            320
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Laurent Dinh, Razvan Pascanu, Samy Bengio, and Yoshua Bengio. Sharp minima can generalize for deep nets. ICLR, 2017. ",
+        "bbox": [
+            173,
+            328,
+            823,
+            357
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C Daniel Freeman and Joan Bruna. Topology and geometry of half-rectified network optimization. ICLR, 2017. ",
+        "bbox": [
+            174,
+            366,
+            825,
+            395
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Marcus Gallagher and Tom Downs. Visualization of learning in multilayer perceptron networks using principal component analysis. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 33(1):28–34, 2003. ",
+        "bbox": [
+            174,
+            404,
+            825,
+            446
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Xavier Glorot and Yoshua Bengio. Understanding the difficulty of training deep feedforward neural networks. In Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, pp. 249–256, 2010. ",
+        "bbox": [
+            176,
+            455,
+            823,
+            498
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Tom Goldstein and Christoph Studer. Phasemax: Convex phase retrieval via basis pursuit. arXiv preprint arXiv:1610.07531, 2016. ",
+        "bbox": [
+            174,
+            507,
+            821,
+            536
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Ian J Goodfellow, Oriol Vinyals, and Andrew M Saxe. Qualitatively characterizing neural network optimization problems. ICLR, 2015. ",
+        "bbox": [
+            173,
+            545,
+            821,
+            574
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Priya Goyal, Piotr Dollar, Ross Girshick, Pieter Noordhuis, Lukasz Wesolowski, Aapo Kyrola, ´ Andrew Tulloch, Yangqing Jia, and Kaiming He. Accurate, large minibatch sgd: Training imagenet in 1 hour. arXiv preprint arXiv:1706.02677, 2017. ",
+        "bbox": [
+            173,
+            582,
+            825,
+            626
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Benjamin D Haeffele and Rene Vidal. Global optimality in neural network training. In ´ Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 7331–7339, 2017. ",
+        "bbox": [
+            174,
+            633,
+            821,
+            664
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Moritz Hardt and Tengyu Ma. Identity matters in deep learning. ICLR, 2017. ",
+        "bbox": [
+            174,
+            671,
+            681,
+            688
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep Residual Learning for Image Recognition. In CVPR, 2016. ",
+        "bbox": [
+            173,
+            695,
+            820,
+            726
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Sepp Hochreiter and Jurgen Schmidhuber. Flat minima. ¨ Neural Computation, 9(1):1–42, 1997. ",
+        "bbox": [
+            174,
+            733,
+            797,
+            750
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Elad Hoffer, Itay Hubara, and Daniel Soudry. Train longer, generalize better: closing the generaliza tion gap in large batch training of neural networks. NIPS, 2017. ",
+        "bbox": [
+            174,
+            757,
+            820,
+            786
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Gao Huang, Zhuang Liu, Kilian Q Weinberger, and Laurens van der Maaten. Densely connected convolutional networks. CVPR, 2017. ",
+        "bbox": [
+            174,
+            795,
+            823,
+            824
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Daniel Jiwoong Im, Michael Tao, and Kristin Branson. An empirical analysis of deep network loss surfaces. arXiv preprint arXiv:1612.04010, 2016. ",
+        "bbox": [
+            174,
+            833,
+            821,
+            863
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Sergey Ioffe and Christian Szegedy. Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift. 2015. ",
+        "bbox": [
+            176,
+            871,
+            825,
+            900
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Kenji Kawaguchi. Deep learning without poor local minima. NIPS, 2016. ",
+        "bbox": [
+            174,
+            909,
+            660,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Kenji Kawaguchi, Leslie Pack Kaelbling, and Yoshua Bengio. Generalization in deep learning. arXiv preprint arXiv:1710.05468, 2017. ",
+        "bbox": [
+            173,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Nitish Shirish Keskar, Dheevatsa Mudigere, Jorge Nocedal, Mikhail Smelyanskiy, and Ping Tak Peter Tang. On large-batch training for deep learning: Generalization gap and sharp minima. ICLR, 2017. ",
+        "bbox": [
+            176,
+            140,
+            823,
+            184
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Anders Krogh and John A Hertz. A simple weight decay can improve generalization. In NIPS, 1992. ",
+        "bbox": [
+            173,
+            193,
+            825,
+            208
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Yuanzhi Li and Yang Yuan. Convergence analysis of two-layer neural networks with relu activation. arXiv preprint arXiv:1705.09886, 2017. ",
+        "bbox": [
+            173,
+            217,
+            825,
+            246
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Qianli Liao and Tomaso Poggio. Theory of deep learning ii: Landscape of the empirical risk in deep learning. arXiv preprint arXiv:1703.09833, 2017. ",
+        "bbox": [
+            174,
+            253,
+            825,
+            284
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Zachary C Lipton. Stuck in a what? adventures in weight space. ICLR Workshop, 2016. ",
+        "bbox": [
+            171,
+            291,
+            751,
+            308
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Eliana Lorch. Visualizing deep network training trajectories with pca. ICML WorkshopWorkshop on Visualization for Deep Learning, 2016. ",
+        "bbox": [
+            173,
+            315,
+            823,
+            344
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Haihao Lu and Kenji Kawaguchi. Depth creates no bad local minima. arXiv preprint arXiv:1702.08580, 2017. ",
+        "bbox": [
+            171,
+            353,
+            825,
+            382
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Quynh Nguyen and Matthias Hein. The loss surface of deep and wide neural networks. International Conference on Machine Learning, 2017. ",
+        "bbox": [
+            173,
+            391,
+            825,
+            420
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Itay Safran and Ohad Shamir. On the quality of the initial basin in overspecified neural networks. In International Conference on Machine Learning, pp. 774–782, 2016. ",
+        "bbox": [
+            171,
+            429,
+            825,
+            458
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Karen Simonyan and Andrew Zisserman. Very Deep Convolutional Networks for Large-Scale Image Recognition. In ICLR, 2015. ",
+        "bbox": [
+            171,
+            467,
+            823,
+            496
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Leslie N Smith and Nicholay Topin. Exploring loss function topology with cyclical learning rates. arXiv preprint arXiv:1702.04283, 2017. ",
+        "bbox": [
+            171,
+            503,
+            825,
+            534
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Mahdi Soltanolkotabi, Adel Javanmard, and Jason D Lee. Theoretical insights into the optimization landscape of over-parameterized shallow neural networks. arXiv preprint arXiv:1707.04926, 2017. ",
+        "bbox": [
+            171,
+            541,
+            825,
+            571
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Daniel Soudry and Elad Hoffer. Exponentially vanishing sub-optimal local minima in multilayer neural networks. arXiv preprint arXiv:1702.05777, 2017. ",
+        "bbox": [
+            169,
+            579,
+            823,
+            609
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Grzegorz Swirszcz, Wojciech Marian Czarnecki, and Razvan Pascanu. Local minima in training of neural networks. stat, 1050:17, 2017. ",
+        "bbox": [
+            171,
+            617,
+            825,
+            647
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Yuandong Tian. An analytical formula of population gradient for two-layered relu network and its applications in convergence and critical point analysis. ICML, 2017. ",
+        "bbox": [
+            171,
+            655,
+            823,
+            685
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Bo Xie, Yingyu Liang, and Le Song. Diverse neural network learns true target functions. In Artificial Intelligence and Statistics, pp. 1216–1224, 2017. ",
+        "bbox": [
+            171,
+            693,
+            825,
+            723
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Chulhee Yun, Suvrit Sra, and Ali Jadbabaie. Global optimality conditions for deep neural networks. arXiv preprint arXiv:1707.02444, 2017. ",
+        "bbox": [
+            171,
+            731,
+            825,
+            760
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning requires rethinking generalization. ICLR, 2017. ",
+        "bbox": [
+            173,
+            768,
+            823,
+            797
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "9 APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            294,
+            117
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/610175cbc64cf1ae69b262234cde47d9ff25b3b42f6d1c8fb9c0e9c8696c1bab.jpg",
+        "image_caption": [
+            "Figure 10: 1D linear interpolation of solutions obtained by small-batch and large-batch methods for ResNet56. The blue lines are loss values and the red lines are error. The solid lines are training curves and the dashed lines are for testing. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            236,
+            170,
+            759,
+            469
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/5dc79c487760afd5e52819863f3727b33d7f93699648a1573eb732d6e3055567.jpg",
+        "image_caption": [
+            "Figure 11: The shape of minima obtained via different optimization algorithms for ResNet-56, with varying batch size and weight decay. Similar to Figure 4, the first row uses zero weight decay and the second row uses 5e-4 weight decay. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            169,
+            554,
+            852,
+            756
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Generalization error for each plot is shown in Table 3. ",
+        "bbox": [
+            173,
+            830,
+            529,
+            844
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/51c9dc79a1510514ec1b5270885d6f3c0f59cca940263448f32e6e8e3acc82d5.jpg",
+        "image_caption": [
+            "Figure 12: 2D visualization of solutions of ResNet-56 obtained by SGD/Adam with small-batch and large-batch. Similar to Figure 11, the first row uses zero weight decay and the second row sets weight decay to 5e-4. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            169,
+            147,
+            852,
+            362
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/f889b340b32ecb02fe5d2760ff8bde83eee6ebad92bc7f2a54de6e9e607681d4.jpg",
+        "table_caption": [
+            "Table 3: Test error for ResNet-56 with different optimization algorithms and batch-size/weight-decay parameters. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"3\"></td><td colspan=\"2\">SGD</td><td colspan=\"2\">Adam</td></tr><tr><td>bs=128</td><td>bs=4096</td><td>bs=128</td><td>bs=4096</td></tr><tr><td>WD= 0</td><td>8.26</td><td>13.93</td><td>9.55</td><td>14.30</td></tr><tr><td>WD = 5e-4</td><td>5.89</td><td>10.59</td><td>7.67</td><td>12.36</td></tr></table>",
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+    },
+    {
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+        "image_caption": [
+            "Figure 13: The loss landscape for DenseNet-121 trained on CIFAR-10. The final training error is 0.002 and the testing error is 4.37 "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            370,
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+        "image_caption": [
+            "Figure 14: Convergence curves for different architectures. The first row is for training loss and the second row are training and testing error curves. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            172,
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+        "page_idx": 16
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+        "image_caption": [
+            "Figure 15: Training and testing loss curves for VGG-9. Dashed lines are for testing, solid for training. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            207,
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+    }
+]

parse/train/S1FQEfZA-/S1FQEfZA-.md

@@ -0,0 +1,230 @@
+# A CLASSIFICATION–BASED PERSPECTIVE ON GAN DISTRIBUTIONS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+A fundamental, and still largely unanswered, question in the context of Generative Adversarial Networks (GANs) is whether GANs are actually able to capture the key characteristics of the datasets they are trained on. The current approaches to examining this issue require significant human supervision, such as visual inspection of sampled images, and often offer only fairly limited scalability. In this paper, we propose new techniques that employ classification–based perspective to evaluate synthetic GAN distributions and their capability to accurately reflect the essential properties of the training data. These techniques require only minimal human supervision and can easily be scaled and adapted to evaluate a variety of state-of-the-art GANs on large, popular datasets. They also indicate that GANs have significant problems in reproducing the more distributional properties of the training dataset. In particular, the diversity of such synthetic data is orders of magnitude smaller than that of the original data.
+
+# 1 INTRODUCTION
+
+Generative Adversarial Networks (GANs) (Goodfellow et al., 2014) have garnered a significant amount of attention due to their ability to learn generative models of multiple natural image datasets (Radford et al., 2015; Denton et al., 2015; Zhang et al., 2016; Zhu et al., 2017). Since their conception, a fundamental question regarding GANs is to what extent they truly learn the underlying data distribution. This is a key issue for multiple reasons. From a scientific perspective, understanding the capabilities of common GANs can shed light on what precisely the adversarial training setup allows the GAN to learn. From an engineering standpoint, it is important to grasp the power and limitations of the GAN framework when applying it in concrete applications. Due to the broad potential applicability of GANs, researchers have investigated this question in a variety of ways.
+
+When we evaluate the quality of a GAN, an obvious first check is to establish that the generated samples lie in the support of the true distribution. In the case of images, this corresponds to checking if the generated samples look realistic. Indeed, visual inspection of generated images is currently the most common way of assessing the quality of a given GAN. Individual humans can performs this task quickly and reliably, and various GANs have achieved impressive results for generating realistic-looking images of faces and indoor scenes (Salimans et al., 2016; Denton et al., 2015).
+
+Once we have established that GANs produce realistic-looking images, the next concern is that the GAN might simply be memorizing the training dataset. While this hypothesis cannot be ruled out entirely, there is evidence that GANs perform at least some non-trivial modeling of the unknown distribution. Previous studies show that interpolations in the latent space of the generator produce novel and meaningful image variations (Radford et al., 2015), and that there is a clear disparity between generated samples and their nearest neighbors in the true dataset (Arora & Zhang, 2017).
+
+Taken together, these results provide evidence that GANs could constitute successful distribution learning algorithms, which motivates studying their distributions in more detail. The direct approach is to compare the probability density assigned by the generator with estimates of the true distribution (Wu et al., 2016). However, in the context of GANs and high-dimensional image distributions, this is complicated by two factors. First, GANs do not naturally provide probability estimates for their samples. Second, estimating the probability density of the true distribution is a challenging problem itself (the adversarial training framework specifically avoids this issue). Hence prior work has only investigated the probability density of GANs on simple datasets such as MNIST (Wu et al., 2016).
+
+Since reliably computing probability densities in high dimensions is challenging, we can instead study the behavior of GANs in low-dimensional problems such as two-dimensional Gaussian mixtures. Here, a common failure of GANs is mode collapse, wherein the generator assigns a disproportionately large mass to a subset of modes from the true distribution (Goodfellow, 2016). This raises concerns about a lack of diversity in the synthetic GAN distributions, and recent work shows that the learned distributions of two common GANs indeed have (moderately) low support size for the CelebA dataset (Arora & Zhang, 2017). However, the approach of Arora & Zhang (2017) heavily relies on a human annotator in order to identify duplicates. Hence it does not easily scale to comparing many variants of GANs or asking more fine-grained questions than collision statistics. Overall, our understanding of synthetic GAN distributions remains blurry, largely due to the lack of versatile tools for a quantitative evaluation of GANs in realistic settings. The focus of this work is precisly to address this question:
+
+# Can we develop principled and quantitative approaches to study synthetic GAN distributions?
+
+To this end, we propose two new evaluation techniques for synthetic GAN distributions. Our methods are inspired by the idea of comparing moments of distributions, which is at the heart of many methods in classical statistics. Although simple moments of high-dimensional distributions are often not semantically meaningful, we can extend this idea to distributions of realistic images by leveraging image statistics identified using convolutional neural networks. In particular, we train image classifiers in order to construct test functions corresponding to semantically meaningful properties of the distributions. An important feature of our approach is that it requires only light human supervision and can easily be scaled to evaluating many GANs and large synthetic datasets.
+
+Using our new evaluation techniques, we study five state-of-the-art GANs on the CelebA and LSUN datasets, arguably the two most common testbeds for advanced GANs. We find that most of the GANs significantly distort the relative frequency of even basic image attributes, such as the hair style of a person or the type of room in an indoor scene. This clearly indicates a mismatch between the true and synthetic distributions. Moreover, we conduct experiments to explore the diversity of GAN distributions. We use synthetic GAN data to train image classifiers and find that these have significantly lower accuracy than classifiers trained on the true data set. This points towards a lack of diversity in the GAN data, and again towards a discrepancy between the true and synthetic distributions. In fact, our additional examinations show that the diversity in GANs is only comparable to a subset of the true data that is $1 0 0 \times$ smaller.
+
+# 2 UNDERSTANDING GANS THROUGH THE LENS OF CLASSIFICATION
+
+When comparing two distributions, a common first test is to compute low-order moments such as the mean and the variance. If the distributions are simple enough, these quantities provide a good understanding for how similar they are. Moreover, low-order moments have a precise definition and are usually quick to compute. On the other hand, low-order moments can also be misleading for more complicated, high-dimensional distributions. As a concrete example, consider a generative model of digits (such as MNIST). If a generator produces digits that are shifted by a significant amount yet otherwise perfect, we will probably still consider this as a good approximation of the true distribution. However, the expectation (mean moment) of the generator distribution can be very different from the expectation of the true data distribution. This raises the question of what other properties of high-dimensional image distributions are easy to test yet semantically meaningful.
+
+In the next two subsections, we describe two concrete approaches to evaluate synthetic GAN data that are easy to compute yet capture relevant information about the distribution. The common theme is that we employ convolutional neural networks in order to capture properties of the distributions that are hard to describe in a mathematically precise way, but usually well-defined for a human (e.g., what fraction of the images shows a smiling person?). Automating the process of annotating images with such high-level information will allow us to study various aspects of synthetic GAN data.
+
+# 2.1 QUANTIFYING MODE COLLAPSE
+
+Mode collapse refers to the tendency of the generator to concentrate a large probability mass on a few modes of the true distribution. While there is ample evidence for the presence of mode-collapse in GANs (Goodfellow, 2016; Arora & Zhang, 2017; Metz et al., 2016), elegant visualizations of this phenomena are somewhat restricted to toy problems on low-dimensional distributions (Goodfellow,
+
+2016; Metz et al., 2016). For image datasets, it is common to rely on human annotators and derived heuristics (see Section 2.3). While these methods have their merits, they are restrictive both in the scale and granularity of testing. Here we propose a classification-based tool to assess how good GANs are at assigning the right mass across broad concepts/modes. To do so, we use a trained classifier as an expert “annotator” that labels important features in synthetic data, and then analyze the resulting distribution. Specifically, our goal is to investigate if a GAN trained on a well-balanced dataset (i.e., contains equal number of samples from each class) can learn to reproduce this balanced structure. Let $D = ( \dot { X } , Y ) = \{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { N }$ represent a dataset of size $N$ with $C$ classes, where $( x _ { i } , y _ { i } )$ denote an image-label pair drawn from true data. If the dataset $D$ is balanced, it contains $N / C$ images per class. The procedure for computing class distribution in synthetic data is:
+
+1. Train an annotator (a multi-class classifier) using the dataset $D$ .   
+2. Train an unconditional GAN on the images $X$ from dataset $D$ , without using class labels.   
+3. Create a synthetic dataset by sampling $N$ images from a GAN and labeling them using the annotator from Step 1.
+
+The annotated data generated via the above procedure can provide insight into the GAN’s class distribution at the scale of the entire dataset. Moreover, we can vary the granularity of mode analysis by choosing richer classification tasks, i.e., more challenging classes or a larger number of them. In Section 3.3, we use this technique to visualize mode collapse in several state-of-the-art GANs on the CelebA and LSUN datasets. All the studied GANs show significant mode collapse and the effect becomes more pronounced when the granularity of the annotator is increased (larger number of classes). We also investigate the temporal aspect of the GAN setup and find that the dominant mode varies widely over the course of training. Our approach also enables us to benchmark and compare GANs on different datasets based on the extent of mode collapse in the learned distributions.
+
+# 2.2 MEASURING DIVERSITY
+
+Our above method for inspecting distribution of modes in synthetic data provides a coarse look at the statistics of the underlying distribution. While the resulting quantities are semantically meaningful, they capture only simple notions of diversity. To get a more holistic view on the sample diversity in the synthetic distribution, we now describe a second classification-based approach for evaluating GAN distributions. The main question that motivates it is: Can GANs recover the key aspects of real data to enable training a good classifier? We believe that this is an interesting measure of sample diversity for two reasons. First, classification of high-dimensional image data is a challenging problem, so a good training dataset will require a sufficiently diverse sample from the distribution. Second, augmenting data for classification problems is one of the proposed use cases of GANs (e.g., see the recent work of Shrivastava et al. (2017)).
+
+If GANs are truly able to capture the quality and diversity of the underlying data distribution, we expect almost no gap between classifiers trained on true data and synthetic data from a GAN. A generic method to produce data from GANs for classification is to train separate GANs for each class in the dataset $D$ .1 Samples from these class-wise GANs can then be pooled together to get a labeled synthetic dataset. Note that the labels are trivially determined based on the class modeled by the particular GAN from which a sample is drawn. We perform the following steps to assess the classification performance of synthetic data vs. true data:
+
+1. Train a classifier on the true data $D$ (from Section 2.1) as a benchmark for comparison. 2. Train $C$ separate unconditional GANs, one per class in dataset D. 3. Generate a balanced synthetic labeled dataset of size $_ \mathrm { N }$ by consolidating an equal number of samples drawn from each of these $C$ GANs. The labels obtained by aggregating samples from per-class GANs are designated as “default” labels for the synthetic dataset. Note that by design, both true and synthetic datasets have $N$ samples, with $N / C$ examples per class. 4. Use synthetic labeled data from Step 3 to train classifier with the same architecture as Step 1.
+
+Comparing the classifiers from Steps 1 and 4 can then shed light on the disparity between the two distributions. Radford et al. (2015) conducted an experiment similar to Step 2 on the MNIST dataset using a conditional GAN. They found that samples from their DCGAN performed comparably to true data on nearest neighbor classification. We obtained similar good results on MNIST, which could be due to the efficacy of GANs in learning the MNIST distribution or due to the ease of getting good accuracy on MNIST even with a small training set (Rolnick et al., 2017). To clarify this question, we restrict our analysis to more complex datasets, specifically CelebA and LSUN.
+
+We evaluate the two following properties in our classification task:
+
+(i) How well can the GANs recover nuances of the decision boundary, which is reflected by how easily the classifier can fit the training data?   
+(ii) How does the diversity of synthetic data compare to that of true data when measured by classification accuracy on a hold-out set of true data?
+
+We observe that all the studied GANs have very low diversity in this metric. In particular, the accuracy achieved by a classifier trained on GAN data is comparable only to the accuracy of a classifier trained on a $1 0 0 \times$ (or more) subsampled version of the true dataset. Even if we draw more samples from the GANs to produce a training set several times larger than the true dataset, there is no improvement in performance. Looking at the classification accuracy gives us a way to compare different models on a potential downstream application of GANs. Interestingly, we find that visual quality of samples does not necessarily correlate with good classification performance.
+
+# 2.3 RELATED WORK
+
+In GAN literature, it is common to investigate performance using metrics that involve human supervision. Arora & Zhang (2017) proposed a measure based on manually counting duplicates in GAN samples as a heuristic for the support or diversity of the learned distribution. In Wu et al. (2016), manual classification of a small sample (100 images) of GAN generated MNIST images is used as a test for the GAN is missing certain modes. Such annotator-based metrics have clear advantages in identifying relevant failure-modes of synthetic samples, which explains why visual inspection (eyeballing) is still the most popular approach to assess GAN samples.
+
+There have also been various attempts to build good metrics for GANs that are not based on manual heuristics. Parzen window estimation can be used to approximate the log-likelihood of the distribution, though it is known to work poorly for high-dimensional data (Theis et al., 2016). Wu et al. (2016) develop a method to get a better estimate for log-likelihood using annealed importance sampling. Salimans et al. (2016) propose a metric known as Inception Score, where the entropy in the labels predicted by a pre-trained Inception network is used to assess the diversity in GAN samples.
+
+# 3 EXPERIMENTS
+
+In the following sub-sections we describe the setup and results for our classification-based GAN benchmarks. Additional details can be found in Section 5 in the Appendix.
+
+# 3.1 DATASETS
+
+GANs have shown promise in generating realistic samples, resulting in efforts to apply them to a broad spectrum of datasets. However, the Large-scale CelebFaces Attributes (CelebA) (Liu et al., 2015) and Large-Scale Scene Understanding (LSUN) (Yu et al., 2015) datasets remain the most popular and canonical ones in developing and evaluating GAN variants. Conveniently, these datasets also have rich annotations, making them particularly suited for our classification–based evaluations. Details on the setup for classification tasks for these datasets are given in the Appendix (Section 5).
+
+# 3.2 MODELS
+
+Using our framework, we perform a comparative study of several popular variants of GANs:
+
+1. Deep Convolutional GAN (DCGAN): Convolutional GAN trained using a Jensen–Shannon divergence–based objective (Goodfellow et al., 2014; Radford et al., 2015).   
+2. Wasserstein GAN (WGAN): GAN that uses a Wasserstein distance–based objective (Arjovsky et al., 2017).   
+3. Adversarially Learned Inference (ALI): GAN that uses joint adversarial training of generative and inference networks (Dumoulin et al., 2017).   
+4. Boundary Equilibrium GAN (BEGAN): Auto-encoder style GAN trained using Wasserstein distance objective (Berthelot et al., 2017).
+
+![](images/62ced381625d83d2930cddcf355b5368dc556dd7a50f2dae5311314091417820.jpg)  
+Figure 1: Visualizations of mode collapse in the synthetic, GAN-generated data produced after training on our chosen subsets of CelebA and LSUN datasets. Left panel shows the relative distribution of classes in samples drawn from synthetic datasets extracted at the end of the training process, and compares is to the true data distribution (leftmost plots). On the right, shown is the evolution of analogous class distribution for different GANs over the course of training. BEGAN did not converge on the LSUN tasks and hence is excluded from the corresponding analysis.
+
+5. Improved GAN (ImGAN): GAN that uses semi-supervised learning (labels are part of GAN training), with various other architectural and procedural improvements (Salimans et al., 2016).
+
+All the aforementioned GANs are unconditional, however, ImGAN has access to class labels as a part of the semi-supervised training process. We use standard implementations for each of these models, details of which are provided in the Appendix (Section 5). We also used the prescribed hyper-parameter settings for each GAN, including number of iterations we train them for. Our analysis is based on $6 4 \times 6 4$ samples, which is a size at which GAN generated samples tend to be of high quality. We also use visual inspection to ascertain that the perceptual quality of GAN samples in our experiments is comparable to those reported in previous studies. We demonstrate sample images in Figures 2 and 3 in the Appendix. BEGAN did not converge in our experiments on the LSUN dataset and hence is excluded from the corresponding analysis.
+
+In our study, we use two types of classification models:
+
+1. ResNet: 32-Layer Residual network He et al. (2016). Here, we choose a ResNet as it is a standard classifier in vision and yields high accuracy on various datasets, making it a reliable baseline.   
+2. Linear Model: This is a network with one-fully connected layer between the input and output (no hidden layers) with a softmax non-linearity. If the dimensions of input $x$ and output $\hat { y }$ , are $D$ and $C$ (number of classes) respectively, then linear models implement the function $\bar { y } = \bar { \sigma ( W ^ { T } x + b ) }$ ,
+
+where $W$ is a $D \times C$ matrix, $b$ is a $C \times 1$ vector and $\sigma ( \cdot )$ is the softmax function. Due to it’s simplicity, this model will serve as a useful baseline in some of our experiments.
+
+We always train the classifiers to convergence, with decaying learning rate and no data augmentation.
+
+# 3.3 EXAMINATION OF MODE COLLAPSE
+
+Experimental results for quantifying mode collapse through classification tasks, described in Section 2.1, are presented below. Table 2 in the Appendix gives details on datasets (subsets of CelebA and LSUN) used in our analysis, such as size $( N )$ , number of classes $( C )$ , and accuracy of the annotator, i.e., a classifier pre-trained on true data, which is then used to label the synthetic, GANgenerated data. Figure 1 presents class distribution in synthetic data, as specified by these annotators. The left panel compares the relative distribution of modes in true data (uniform) with that in various GAN-generated datasets. Each of these datasets is created by drawing $N$ samples from the GAN after it was trained on the corresponding true dataset. The right panel illustrates the evolution of class distributions in various GANs over the course of training2.
+
+Results: These visualization lead to the following findings:
+
+• All GANs seem to suffer from significant mode-collapse. This becomes more apparent when the annotator granularity is increased, by considering a larger set of classes. For instance, one should compare the relatively balanced class distributions in the 3-class LSUN task to the near-absence of some modes in the 5-class task. • Mode collapse is prevalent in GANs throughout the training process, and does not seem to recede over time. Instead the dominant mode(s) often fluctuate wildly over the course of the training. • For each task, often there is a common set of modes onto which the distint GANs exhibit collapse.
+
+In addition to viewing our method as an approach to analyze the mode collapse, we can also use it as a benchmark for GAN comparison. From this perspective, we can observe that, on CelebA, DCGAN and ALI learn somewhat balanced distributions, while WGAN, BEGAN and Improved GAN show prominent mode collapse. This is in contrast to the results obtained LSUN, where, for example, WGAN exhibit relatively small mode collapse, while ALI suffers from significant mode collapse even on the simple 3-class task. This highlights the general challenge in real world applications of GANs: they often perform well on the datasets they were designed for (e.g. ALI on CelebA and WGAN on LSUN), but extension to new datasets is not straightforward. Temporal analysis of mode-collapse shows that there is wide variation in the dominant mode for WGAN and Improved GAN, whereas for BEGAN, the same mode(s) often dominates the entire training process.
+
+# 3.4 DIVERSITY EXPERIMENTS
+
+Using the procedure outlined in Section 2.2, we perform a quantitative assessment of sample diversity in GANs on the CelebA and LSUN datasets. We restrict our experiments to binary classification as we find they have sufficient complexity to highlight the disparity between true and synthetic data. Selected results for classification-based evaluation of GANs are presented in Table 1. A more extensive study is presented in Table 3, and Figures 4 and 5 in the Appendix (Section 5).
+
+As a preliminary check, we inspect the quality of our labeled GAN datasets. For this, we use high-accuracy annotators from Section 2.1 to predict labels for GAN generated data and measure consistency between the predicted and default labels (label correctness). We also inspect confidence scores, defined as the softmax probabilities for predicted class, of the annotator. The motivation behind these metrics is that if the classifier can correctly and with high-confidence predict labels for labeled GAN samples, then it is likely that they are convincing examples of that class, and hence of good “quality”. Empirical results for label agreement and annotator confidence of GAN generated datasets are shown in Tables 1 and 3, and Figure 4. We also report an equivalent Inception Score (Salimans et al., 2016), similar to that described in Section 2.3. Using the Inception network to get the label distribution may not be meaningful for face or scene images. Instead, we compute the Inception Score using the label distribution predicted from the annotator networks. Score is computed as $e x p ( \mathbb { E } _ { x } [ { \bf K L } ( p ( y | x ) ) | | p ( y ) ] )$ , where $y$ refers to label predictions from the annotators 3.
+
+<table><tr><td rowspan=5 colspan=1>Task</td><td rowspan=1 colspan=13>Classification Performance</td></tr><tr><td rowspan=4 colspan=1>Data Source</td><td rowspan=4 colspan=2>LabelCorrectness(%)</td><td rowspan=4 colspan=2>Inception Score(μ±σ)</td><td rowspan=1 colspan=8>Accuracy (%)</td></tr><tr><td rowspan=1 colspan=7>Linear model</td><td rowspan=1 colspan=1>ResNet</td></tr><tr><td rowspan=1 colspan=5>↑1</td><td rowspan=1 colspan=2>↑10</td><td rowspan=1 colspan=1>↑</td></tr><tr><td rowspan=1 colspan=3>Train</td><td rowspan=1 colspan=2>Test</td><td rowspan=1 colspan=1>Train</td><td rowspan=1 colspan=1>Test</td><td rowspan=1 colspan=1>Test</td></tr><tr><td rowspan=8 colspan=1>CelebASmiling (Y/N)# Images: 156160</td><td rowspan=3 colspan=1>TrueTrue↓64True↓256True、↓512True↓1024</td><td rowspan=3 colspan=2>92.4</td><td rowspan=3 colspan=2>1.69 ± 0.0074</td><td rowspan=2 colspan=3>85.787.691.593.7</td><td rowspan=3 colspan=2>85.685.082.480.076.2</td><td rowspan=3 colspan=1></td><td rowspan=3 colspan=1></td><td rowspan=1 colspan=1>92.4</td></tr><tr><td rowspan=2 colspan=1>87.882.177.871.2</td></tr><tr><td rowspan=1 colspan=3>95.0</td></tr><tr><td rowspan=5 colspan=1>DCGANWGANALIBEGANImproved GAN</td><td rowspan=5 colspan=2>96.198.293.393.598.4</td><td rowspan=5 colspan=2>1.67 ± 0.00281.68 ± 0.00311.71 ± 0.00271.74 ± 0.00281.88 ± 0.0021</td><td rowspan=2 colspan=3>100.096.8</td><td rowspan=2 colspan=2>77.183.4</td><td rowspan=2 colspan=1>100.096.8</td><td rowspan=2 colspan=1>77.183.5</td><td rowspan=2 colspan=1>63.365.3</td></tr><tr><td rowspan=1 colspan=2>27</td></tr><tr><td rowspan=1 colspan=3>94.5</td><td rowspan=3 colspan=2>80.169.570.2</td><td rowspan=3 colspan=1>95.098.5100.0</td><td rowspan=3 colspan=1>82.469.670.1</td><td rowspan=3 colspan=1>55.864.161.6</td></tr><tr><td rowspan=1 colspan=3>98.5</td></tr><tr><td rowspan=1 colspan=3>100.0</td></tr><tr><td rowspan=6 colspan=1>LSUNBedroom/Kitchen# Images: 200000</td><td rowspan=2 colspan=1>TrueTrue↓512True↓1024True↓2048True4096</td><td rowspan=2 colspan=2>98.2</td><td rowspan=2 colspan=2>1.94 ± 0.0217</td><td rowspan=1 colspan=3>64.764.765.298.7</td><td rowspan=2 colspan=2>64.164.064.056.255.1</td><td rowspan=2 colspan=1></td><td rowspan=2 colspan=1></td><td rowspan=2 colspan=1>99.176.466.956.555.1</td></tr><tr><td rowspan=1 colspan=3>100.0</td><td rowspan=1 colspan=1>.0</td><td rowspan=1 colspan=1>55.1</td></tr><tr><td rowspan=4 colspan=1>DCGANWGANALIImproved GAN</td><td rowspan=4 colspan=2>92.787.880.484.2</td><td rowspan=1 colspan=2>1.85± 0.0036</td><td rowspan=1 colspan=2>90</td><td rowspan=1 colspan=2>90.8</td><td rowspan=1 colspan=2>56.5</td><td rowspan=1 colspan=1>91.2</td><td rowspan=1 colspan=1>56.3</td><td rowspan=4 colspan=1>51.255.750.551.2</td></tr><tr><td rowspan=2 colspan=1>87.880.4</td><td rowspan=1 colspan=2>1.70 ± 0.0023</td><td rowspan=1 colspan=3>86.2</td><td rowspan=1 colspan=1>58</td><td rowspan=1 colspan=1>96.3</td><td rowspan=2 colspan=1>54.150.8</td></tr><tr><td rowspan=2 colspan=2>1.62 ± 0.00261.68 ± 0.0030</td><td rowspan=1 colspan=3>80.7</td><td rowspan=1 colspan=2>49.7</td><td rowspan=1 colspan=1>81.7</td></tr><tr><td rowspan=1 colspan=3>91.6</td><td rowspan=1 colspan=2>55.9</td><td rowspan=1 colspan=1>90.8</td><td rowspan=1 colspan=1>56.5</td></tr></table>
+
+Table 1: Select results from the comparative study on classification performance of true data vs. GANs on the CelebA and LSUN datasets. Label correctness measures the agreement between default labels for the synthetic datasets, and those predicted by the annotator, a classifier trained on true data. Shown alongside are the equivalent inception scores computed using labels predicted by the annotator (rather than an Inception Network). Training and test accuracies for a linear model on the various true and synthetic datasets are reported. Also presented are the corresponding accuracies for this classifier trained on down-sampled true data $\left( \downarrow _ { M } \right)$ and oversampled synthetic data $( \uparrow _ { L } )$ . Test accuracy for ResNets trained on these datasets is also shown (training accuracy was always $1 0 0 \%$ ), though it is noticeable that deep networks suffer from issues when trained on synthetic datasets.
+
+Next, we train classifiers using the true and labeled GAN-generated datasets and study their performance in terms of accuracy on a hold-out set of true data. ResNets (and other deep variants) yield good classification performance on true data, but suffer from severe overfitting on the synthetic data, leading to poor test accuracy. This already indicates a possible problem with GANs and the diversity of the data they generate. But to highlight this problem better and avoid the issues that stem from overfitting, we also look for a classifier which does not always overfit on the synthetic data. We, however, observed that even training simple networks, such as one fully connected layer with few hidden units, led to overfitting on synthetic data. Hence, we resorted to a very basic linear model described in Section 3.2. Tables 1 and 3 shows results from binary classification experiments using linear models, with the training and test accuracies of the classifier on various datasets.
+
+Finally, to get a better understanding of the underlying ”diversity” of synthetic datasets, we train linear models using down-sampled versions of true data (no augmentation), and compare this to the performance of synthetic data, as shown in Tables 1 and 3. Down-sampling the data by a factor of $M$ , denoted as $\downarrow _ { M }$ implies selecting a random $N / M$ subset of the data $D$ . Visualizations of how GAN classification performance compares with (down-sampled) true data are in Figure 5 in the Appendix. A natural argument in the defense of GANs is that we can oversample them, i.e. generate datasets much larger than the size of training data. Results for linear models trained using a 10-fold oversampling of GANs (drawing $1 0 N$ samples), denoted by $\uparrow _ { 1 0 }$ , are show in Tables 1 and 3.
+
+Results: The major findings from these experiments are:
+
+• Based on Tables 1 and 3, and Figure 4, we see strong agreement between annotator labels and true labels for synthetic data, on par with the scores for the test set of true data. It is thus apparent that the GAN images are of high-quality, as expected based on the visual inspection. These scores are lower for LSUN than CelebA, potentially due to lower quality of generated LSUN images. From these results, we can get a broad understanding of how good GANs are at producing convincing/representative samples from different classes across datasets. This also shows that simple classification-based benchmarks can highlight relevant properties of synthetic datasets.
+
+• The equivalent inception score is not very informative and is similar for the true (hold-out set) and synthetic datasets. This is not surprising given the simple nature of our binary classification task and the fact that the true and synthetic datasets have almost a uniform distribution over labels. • It is evident from Table 1 that there is a large performance gap between true and synthetic data on classification tasks. Inspection of training accuracies shows that linear models are able to nearly fit the synthetic datasets, but are grossly underfitting on true data. Given the high scores of synthetic data on the previous experiments to assess dataset ‘quality’ (Tables 1 and 3, and Figure 4), it is likely that the poor classification performance is more indicative of lack of ‘diversity’. Comparing GAN performance to that of down-sampled true data reveals that the learned distribution, which was trained on datasets that have around hundred thousand data points exhibits diversity that is on par with what only mere couple of hundreds of true data samples constitute! This shows that, at least from the point of view of classification, the diversity of the GAN generated data is severely lacking. • Oversampling GANs by 10-fold to produce larger datasets does not improve classification performance. The disparity between true and synthetic data remains nearly unchanged even after this significant oversampling, further highlighting the lack of diversity in GANs.
+
+In terms of the conclusions of relative performance of various GANs, we observe that WGAN and ALI (on CelebA) perform better than the other GANs. While BEGAN samples have good perceptual quality (see Figure 2), they consistently perform badly on our classification tasks. On the other hand, WGAN samples have relatively poor visual quality but seem to outperform other GANs in classification tasks. This is a strong indicator of the need to consider other metrics, such as the ones proposed in this paper, in addition to visual inspection to study GANs. For LSUN, the gap between true and synthetic data is much larger, with the classifiers getting near random performance on all the synthetic datasets. Note that these classifiers get poor test accuracy on LSUN but are not overfitting on the training data. In this case, we speculate the lower performance could be due to both lower quality and diversity of LSUN samples.
+
+In summary, our key experimental finding is that even simple classification–based tests can hold tremendous potential to shed insight on the learned distribution in GANs. This not only helps us to get a deeper understanding of many of the underlying issues, but also provides with a more quantitative and rigorous platform on which to compare different GANs. Our techniques could, in principle, be also applied to assess other generative models such as Variational Auto-Encoders (VAEs) Kingma & Welling (2014). However, VAEs have significant problems in generating realistic samples on the datasets used in our analysis in the first place – see Arora & Zhang (2017).
+
+# 4 CONCLUSIONS
+
+In this paper, we put forth techniques for examining the ability of GANs to capture key characteristics of the training data, through the lens of classification. Our tools are scalable, quantitative and automatic (no need for visual inspection of images). They thus are capable of studying state-ofthe-art GANs on realistic, large-scale image datasets. Further, they serve as a mean to perform a nuanced comparison of GANs and to identify their relative merits, including properties that cannot be discerned from mere visual inspection.
+
+We then use the developed techniques to perform empirical studies on popular GANs on the CelebA and LSUN datasets. Our examination shows that mode collapse is indeed a prevalent issue for GANs. Also, we observe that synthetic GAN-generated datasets have significantly reduced diversity, at least when examined from a classification perspective. In fact, the diversity of such synthetic data is often few orders of magnitude smaller than that of the true data. Furthermore, this gap in diversity does not seem to be bridged by simply producing much larger datasets by oversampling GANs. Finally, we also notice that good perceptual quality of samples does not necessarily correlate – and might sometime even anti-correlate – with distribution diversity. These findings suggest that we need to go beyond the visual inspection–based evaluations and look for more quantitative tools for assessing quality of GANs, such as the ones presented in this paper.
+
+# REFERENCES
+
+Martin Arjovsky, Soumith Chintala, and Leon Bottou. Wasserstein generative adversarial networks. ´ In International Conference on Machine Learning, pp. 214–223, 2017.
+
+Sanjeev Arora and Yi Zhang. Do GANs actually learn the distribution? An empirical study. arXiv preprint arXiv:1706.08224, 2017.
+
+David Berthelot, Tom Schumm, and Luke Metz. BEGAN: Boundary equilibrium generative adversarial networks. arXiv preprint arXiv:1703.10717, 2017.
+
+Emily L Denton, Soumith Chintala, Rob Fergus, et al. Deep generative image models using a laplacian pyramid of adversarial networks. In Advances in neural information processing systems, pp. 1486–1494, 2015.
+
+Vincent Dumoulin, Ishmael Belghazi, Ben Poole, Alex Lamb, Martin Arjovsky, Olivier Mastropietro, and Aaron Courville. Adversarially learned inference. In International Conference on Learning Representations, Apr 2017. URL https://arxiv.org/abs/1606.00704.
+
+Ian Goodfellow. Nips 2016 tutorial: Generative adversarial networks. arXiv preprint arXiv:1701.00160, 2016.
+
+Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in neural information processing systems, pp. 2672–2680, 2014.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770–778, 2016.
+
+Diederik P Kingma and Max Welling. Auto-encoding variational bayes. In International Conference on Learning Representations, Apr 2014. URL https://arxiv.org/abs/1312.6114.
+
+Ziwei Liu, Ping Luo, Xiaogang Wang, and Xiaoou Tang. Deep learning face attributes in the wild. In Proceedings of International Conference on Computer Vision (ICCV), 2015.
+
+Luke Metz, Ben Poole, David Pfau, and Jascha Sohl-Dickstein. Unrolled generative adversarial networks. arXiv preprint arXiv:1611.02163, 2016.
+
+Alec Radford, Luke Metz, and Soumith Chintala. Unsupervised representation learning with deep convolutional generative adversarial networks. arXiv preprint arXiv:1511.06434, 2015.
+
+David Rolnick, Andreas Veit, Serge Belongie, and Nir Shavit. Deep learning is robust to massive label noise. arXiv preprint arXiv:1705.10694, 2017.
+
+Tim Salimans, Ian Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, and Xi Chen. Improved techniques for training GANs. In Advances in Neural Information Processing Systems, pp. 2234–2242, 2016.
+
+Ashish Shrivastava, Tomas Pfister, Oncel Tuzel, Joshua Susskind, Wenda Wang, and Russell Webb. Learning from simulated and unsupervised images through adversarial training. In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), July 2017.
+
+L. Theis, A. van den Oord, and M. Bethge. A note on the evaluation of generative models. In International Conference on Learning Representations, Apr 2016. URL http://arxiv.org/ abs/1511.01844.
+
+Yuhuai Wu, Yuri Burda, Ruslan Salakhutdinov, and Roger Grosse. On the quantitative analysis of decoder-based generative models. arXiv preprint arXiv:1611.04273, 2016.
+
+Fisher Yu, Yinda Zhang, Shuran Song, Ari Seff, and Jianxiong Xiao. LSUN: Construction of a large-scale image dataset using deep learning with humans in the loop. arXiv preprint arXiv:1506.03365, 2015.
+
+Han Zhang, Tao Xu, Hongsheng Li, Shaoting Zhang, Xiaolei Huang, Xiaogang Wang, and Dimitris Metaxas. Stackgan: Text to photo-realistic image synthesis with stacked generative adversarial networks. arXiv preprint arXiv:1612.03242, 2016.
+
+Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A Efros. Unpaired image-to-image translation using cycle-consistent adversarial networks. arXiv preprint arXiv:1703.10593, 2017.
+
+# 5 APPENDIX
+
+# 5.1 EXPERIMENTAL SETUP
+
+# 5.1.1 DATASETS FOR CLASSIFICATION TASKS
+
+To assess GAN performance from the perspective of classification, we construct a set of classification tasks on the CelebA and LSUN datasets. In the case of the LSUN dataset, images are annotated with scene category labels, which makes it straightforward to use this data for binary and multiclass classification. On the other hand, each image in the CelebA dataset is labeled with 40 binary attributes. As a result, a single image has multiple associated attribute labels. Here, we construct classification tasks can by considering binary combinations of an attribute(s) (examples are shown in Figure 2). Attributes used in our experiments were chosen such that the resulting dataset was large, and classifiers trained on true data got high-accuracy so as to be good annotators for the synthetic data. Details on datasets used in our classification tasks, such as training set size $( N )$ , number of classes $( C )$ , and accuracy of the annotator, i.e., a classifier pre-trained on true data which is used to label the synthetic GAN-generated data, are provided in Table 2.
+
+<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=1>N</td><td rowspan=1 colspan=1>C</td><td rowspan=1 colspan=1>Annotator&#x27;s Accuracy (%)</td></tr><tr><td rowspan=1 colspan=1>CelebA: Makeup, Smiling</td><td rowspan=1 colspan=1>102,436</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>90.9, 92.4</td></tr><tr><td rowspan=1 colspan=1>CelebA: Male, Mouth Open</td><td rowspan=1 colspan=1>115,660</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>97.9, 93.5</td></tr><tr><td rowspan=1 colspan=1>CelebA: Bangs, Smiling</td><td rowspan=1 colspan=1>45,196</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>93.9,92.4</td></tr><tr><td rowspan=1 colspan=1>LSUN: Bedroom, Kitchen, Classroom</td><td rowspan=1 colspan=1>150,000</td><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>98.7</td></tr><tr><td rowspan=1 colspan=1>LSUN: Bedroom, Conference Room, Dining Room,Kitchen,Living Room</td><td rowspan=1 colspan=1>250,000</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>93.7</td></tr></table>
+
+Table 2: Details of CelebA and LSUN subsets used for the studies in Section 3.3. Here, we use a classifier trained on true data as an annotator that let’s us infer label distribution for the synthetic, GAN-generated data. $N$ is the size of the training set and $C$ is the number of classes in the true and synthetic datasets. Annotator’s accuracy refers to the accuracy of the classifier on a test set of true data. For CelebA, we use a combination of attribute-wise binary classifiers as annotators due their higher accuracy compared to a single classifier trained jointly on all the four classes.
+
+# 5.1.2 MODELS
+
+Benchmarks were performed on standard implementations - • DCGAN: https://github.com/carpedm20/DCGAN-tensorflow • WGAN: https://github.com/martinarjovsky/WassersteinGAN • ALI: https://github.com/IshmaelBelghazi/ALI • BEGAN :https://github.com/carpedm20/BEGAN-tensorflow • Improved GAN: https://github.com/openai/improved-gan • ResNet Classifier: Variation of the standard TensorFlow ResNet https://github.com/ tensorflow/models/blob/master/research/resnet/resnet_model.py
+
+# 5.2 ADDITION EXPERIMENTAL RESULTS
+
+# 5.2.1 SAMPLE QUALITY
+
+For each of our benchmark experiments, we ascertain that the visual quality of samples produced by the GANs is comparable to that reported in prior work. Examples of random samples drawn for multi-class datasets from both true and synthetic data are shown in Figure 2 for the CelebA dataset, and in Figure!3 for the LSUN dataset.
+
+# 5.2.2 MODE COLLAPSE EXPERIMENTS
+
+In the studies to observe mode collapse in GANs described in Sections 2.1 and 3.3, we use a pretrained classifier as an annotator to obtain the class distribution for datasets generated from unconditional GANs. Figure 4 shows histograms of annotator confidence for the datasets used for benchmarking listed in Table 2. As can be seen in these figures, the annotator confidence for the synthetic data is comparable to that on the hold-out set of true data. Thus, it seems likely that the
+
+![](images/ffd1eadad3e04387b4653a31d59a532c5b3dd825d3c778538887a71a3f4fdf6e.jpg)  
+Figure 2: Illustration of datasets from CelebA used in proposed classification-based benchmarks to evaluate GANs. Shown alongside are images sampled from various unconditional GANs trained on this dataset. Labels for the GAN samples are obtained using a pre-trained classifier as an annotator.
+
+GAN generated samples are of good quality and are truly representative examples of their respective classes, as expected based on visual inspection.
+
+# 5.2.3 DIVERSITY EXPERIMENTS
+
+Table 3 presents an extension of the comparative study of classification performance of true and GAN generated data provided in Table 1. Visualizations of how test accuracies of a linear model classifier trained on GAN data compares with one trained on true data is shown in Figure 5. For each task, the bold curve shows test accuracy of a classifier trained on true data as a function of true dataset size. A down-sampling factor of $M$ corresponds to training the classifier on a random $N / M$ subset of true data. The dashed curves show test accuracy of classifiers trained on GAN datasets, obtained by drawing $N$ samples from GANs at the culmination of the training process. Based on these visualizations, it is apparent that GANs have comparable classification performance to a subset of training data that is more than a $1 0 0 \mathbf { x }$ smaller. Thus, from the perspective of classification, GANs have diversity on par with a few hundred true data samples.
+
+![](images/e5b8ba0ba9dfbdfd0a8afcc1a9e78e8a86380ad9f5d7993dbf0217cebfd010d4.jpg)  
+(b) 5-class dataset from LSUN for Bedroom, Conference Room, Dining Room, Kitchen, Living Room.
+
+![](images/2ebb51f850f33b6ea62b998998d116ae8a6c587d4bac7ba0cb1ceccba4d70787.jpg)  
+Figure 3: Illustration of datasets from LSUN used in proposed classification-based benchmarks to evaluate GANs. Shown alongside are images sampled from various unconditional GANs trained on this dataset. Labels for the GAN samples are obtained using a pre-trained classifier as an annotator.   
+Figure 4: Histograms of annotator confidence (softmax probability) during label prediction on true data (test set) and synthetic data for tasks on the CelebA and LSUN datasets (see Section 3.4).
+
+![](images/bcdf6de7631c3992c2acb9241b93e99085cff09b8f2d85b8369478a356137e12.jpg)  
+Figure 5: Illustration of the classification performance of true data compared with GAN-generated synthetic datasets based on experiments described in Section 3.4. Classification is performed using a basic linear model, described in Section 3.2, and performance is reported in terms of accuracy on a hold-out set of true data. In the plots, the bold curve captures the classification performance of models trained on true data vs the size of the true dataset (maximum size is $N$ ). Dashed lines represent performance of classifiers trained on various GAN-generated datasets of size $N$ . These plots indicate that GAN samples have ”diversity” comparable to a small subset (few hundred samples) of true data. Here the notion of diversity is one that is relevant for classification tasks.
+
+<table><tr><td rowspan=5 colspan=1>Task</td><td rowspan=1 colspan=8>Classification Performance</td></tr><tr><td rowspan=4 colspan=1>Data Source</td><td rowspan=4 colspan=1>LabelCorrectness(%)</td><td rowspan=4 colspan=1>Inception Score(μ±σ)</td><td rowspan=1 colspan=5>Accuracy (%)</td></tr><tr><td rowspan=1 colspan=4>Linear model</td><td rowspan=1 colspan=1>ResNet</td></tr><tr><td rowspan=1 colspan=2>↑1</td><td rowspan=1 colspan=2>↑10</td><td rowspan=1 colspan=1>↑1</td></tr><tr><td rowspan=1 colspan=1>Train</td><td rowspan=1 colspan=1>Test</td><td rowspan=1 colspan=1>Train</td><td rowspan=1 colspan=1>Test</td><td rowspan=1 colspan=1>Test</td></tr><tr><td rowspan=6 colspan=1>CelebAMale (Y/N)# Images: 136522</td><td rowspan=2 colspan=1>TrueTrue↓64True↓256True↓512True↓1024</td><td rowspan=2 colspan=1>97.9</td><td rowspan=2 colspan=1>1.98 ± 0.0033</td><td rowspan=2 colspan=1>88.189.691.696.3100.0</td><td rowspan=2 colspan=1>88.888.786.983.883.1</td><td rowspan=2 colspan=1></td><td rowspan=2 colspan=1></td><td rowspan=1 colspan=1>97.992.9</td></tr><tr><td rowspan=1 colspan=1>89.882.681.4</td></tr><tr><td rowspan=4 colspan=1>DCGANWGANALIBEGANImproved GAN</td><td rowspan=4 colspan=1>98.298.399.299.399.8</td><td rowspan=4 colspan=1>1.97± 0.00131.97 ± 0.00131.99 ± 0.00081.99 ± 0.00061.99 ± 0.0004</td><td rowspan=4 colspan=1>100.096.795.897.9100.0</td><td rowspan=4 colspan=1>79.284.086.778.075.6</td><td rowspan=4 colspan=1>100.096.795.898.0100.0</td><td rowspan=1 colspan=1>79.6</td><td rowspan=1 colspan=1>56.4</td></tr><tr><td rowspan=3 colspan=1>83.986.778.271.0</td><td rowspan=1 colspan=1>50.0</td></tr><tr><td rowspan=1 colspan=1>58.9</td></tr><tr><td rowspan=1 colspan=1>55.471.7</td></tr><tr><td rowspan=4 colspan=1>CelebASmiling (Y/N)# Images: 156160</td><td rowspan=2 colspan=1>TrueTrue64True↓256True↓512True↓1024</td><td rowspan=2 colspan=1>92.4</td><td rowspan=2 colspan=1>1.69 ± 0.0074</td><td rowspan=2 colspan=1>85.787.691.593.795.0</td><td rowspan=2 colspan=1>85.685.082.480.076.2</td><td rowspan=2 colspan=1></td><td rowspan=2 colspan=1></td><td rowspan=1 colspan=1>92.487.8</td></tr><tr><td rowspan=1 colspan=1>82.177.871.2</td></tr><tr><td rowspan=2 colspan=1>DCGANWGANALIBEGANImproved GAN</td><td rowspan=2 colspan=1>96.198.293.393.598.4</td><td rowspan=1 colspan=1>1.67± 0.00281.68 ± 0.00311.71 ± 0.0027</td><td rowspan=1 colspan=1>100.096.894.5</td><td rowspan=1 colspan=1>77.183.480.1</td><td rowspan=1 colspan=1>100.096.895.0</td><td rowspan=1 colspan=1>77.183.582.4</td><td rowspan=1 colspan=1>63.365.355.8</td></tr><tr><td rowspan=1 colspan=1>1.74 ± 0.00281.88 ± 0.0021</td><td rowspan=1 colspan=1>98.5100.0</td><td rowspan=1 colspan=1>69.570.2</td><td rowspan=1 colspan=1>98.5100.0</td><td rowspan=1 colspan=1>69.670.1</td><td rowspan=1 colspan=1>64.161.6</td></tr><tr><td rowspan=2 colspan=1>CelebABlack Hair (Y/N)#Images: 77812</td><td rowspan=1 colspan=1>TrueTrue↓64True↓256True↓512True↓1024</td><td rowspan=1 colspan=1>84.5</td><td rowspan=1 colspan=1>1.68 ± 0.0112</td><td rowspan=1 colspan=1>76.479.786.389100.0</td><td rowspan=1 colspan=1>76.575.472.668.765.4</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>84.580.075.873.972.7</td></tr><tr><td rowspan=1 colspan=1>DCGANWGANALIBEGANImproved GAN</td><td rowspan=1 colspan=1>86.776.079.487.686.7</td><td rowspan=1 colspan=1>1.68 ± 0.00401.60 ± 0.00551.63 ± 0.00281.74 ± 0.00281.64 ± 0.0045</td><td rowspan=1 colspan=1>100.094.494.994.1100.0</td><td rowspan=1 colspan=1>70.973.771.067.670.3</td><td rowspan=1 colspan=1>100.094.394.994.1100.0</td><td rowspan=1 colspan=1>70.573.470.267.769.1</td><td rowspan=1 colspan=1>53.458.555.767.270.2</td></tr><tr><td rowspan=3 colspan=1>LSUNBedroom/Kitchen#Images: 200000</td><td rowspan=1 colspan=1>TrueTrue↓512True↓1024True↓2048True4096</td><td rowspan=1 colspan=1>98.2</td><td rowspan=1 colspan=1>1.94 ± 0.0217</td><td rowspan=1 colspan=1>64.764.765.298.7100.0</td><td rowspan=1 colspan=1>64.164.064.056.255.1</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>99.176.466.956.555.1</td></tr><tr><td rowspan=2 colspan=1>DCGANWGANALIImproved GAN</td><td rowspan=2 colspan=1>92.787.880.484.2</td><td rowspan=1 colspan=1>1.85± 0.00361.70 ± 0.0023</td><td rowspan=1 colspan=1>90.886.2</td><td rowspan=1 colspan=1>56.558.2</td><td rowspan=1 colspan=1>91.296.3</td><td rowspan=1 colspan=1>56.354.1</td><td rowspan=1 colspan=1>31.255.7</td></tr><tr><td rowspan=1 colspan=1>1.62 ± 0.00261.68 ± 0.0030</td><td rowspan=1 colspan=1>80.791.6</td><td rowspan=1 colspan=1>49.755.9</td><td rowspan=1 colspan=1>81.790.8</td><td rowspan=1 colspan=1>50.856.5</td><td rowspan=1 colspan=1>50.551.2</td></tr></table>
+
+Table 3: Detailed version of the comparative study of the classification performance of true data and GANs on the CelebA and LSUN datasets shown in Table 1, based on experiments described in Section 3.4. Label correctness measures the agreement between default labels for the synthetic datasets, and those predicted by the annotator, a classifier trained on the true data. Shown alongside are the equivalent inception scores computed using labels predicted by the annotator (instead of the Inception Network). Training and test accuracies for a linear model classifier on the various true and synthetic datasets are reported. Also presented are the corresponding accuracies for a linear model trained on down-sampled true data $\left( \downarrow _ { M } \right)$ and oversampled synthetic data $( \uparrow _ { L } )$ . Test accuracy for ResNets trained on these datasets is also shown (training accuracy was always $1 0 0 \%$ ), though it is noticeable that deep networks suffer from issues when trained on synthetic datasets.

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+        "text": "A CLASSIFICATION–BASED PERSPECTIVE ON GAN DISTRIBUTIONS ",
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+        "text": "Anonymous authors Paper under double-blind review ",
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+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+        "text": "A fundamental, and still largely unanswered, question in the context of Generative Adversarial Networks (GANs) is whether GANs are actually able to capture the key characteristics of the datasets they are trained on. The current approaches to examining this issue require significant human supervision, such as visual inspection of sampled images, and often offer only fairly limited scalability. In this paper, we propose new techniques that employ classification–based perspective to evaluate synthetic GAN distributions and their capability to accurately reflect the essential properties of the training data. These techniques require only minimal human supervision and can easily be scaled and adapted to evaluate a variety of state-of-the-art GANs on large, popular datasets. They also indicate that GANs have significant problems in reproducing the more distributional properties of the training dataset. In particular, the diversity of such synthetic data is orders of magnitude smaller than that of the original data. ",
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+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+    },
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+        "text": "Generative Adversarial Networks (GANs) (Goodfellow et al., 2014) have garnered a significant amount of attention due to their ability to learn generative models of multiple natural image datasets (Radford et al., 2015; Denton et al., 2015; Zhang et al., 2016; Zhu et al., 2017). Since their conception, a fundamental question regarding GANs is to what extent they truly learn the underlying data distribution. This is a key issue for multiple reasons. From a scientific perspective, understanding the capabilities of common GANs can shed light on what precisely the adversarial training setup allows the GAN to learn. From an engineering standpoint, it is important to grasp the power and limitations of the GAN framework when applying it in concrete applications. Due to the broad potential applicability of GANs, researchers have investigated this question in a variety of ways. ",
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+    {
+        "type": "text",
+        "text": "When we evaluate the quality of a GAN, an obvious first check is to establish that the generated samples lie in the support of the true distribution. In the case of images, this corresponds to checking if the generated samples look realistic. Indeed, visual inspection of generated images is currently the most common way of assessing the quality of a given GAN. Individual humans can performs this task quickly and reliably, and various GANs have achieved impressive results for generating realistic-looking images of faces and indoor scenes (Salimans et al., 2016; Denton et al., 2015). ",
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "Once we have established that GANs produce realistic-looking images, the next concern is that the GAN might simply be memorizing the training dataset. While this hypothesis cannot be ruled out entirely, there is evidence that GANs perform at least some non-trivial modeling of the unknown distribution. Previous studies show that interpolations in the latent space of the generator produce novel and meaningful image variations (Radford et al., 2015), and that there is a clear disparity between generated samples and their nearest neighbors in the true dataset (Arora & Zhang, 2017). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Taken together, these results provide evidence that GANs could constitute successful distribution learning algorithms, which motivates studying their distributions in more detail. The direct approach is to compare the probability density assigned by the generator with estimates of the true distribution (Wu et al., 2016). However, in the context of GANs and high-dimensional image distributions, this is complicated by two factors. First, GANs do not naturally provide probability estimates for their samples. Second, estimating the probability density of the true distribution is a challenging problem itself (the adversarial training framework specifically avoids this issue). Hence prior work has only investigated the probability density of GANs on simple datasets such as MNIST (Wu et al., 2016). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Since reliably computing probability densities in high dimensions is challenging, we can instead study the behavior of GANs in low-dimensional problems such as two-dimensional Gaussian mixtures. Here, a common failure of GANs is mode collapse, wherein the generator assigns a disproportionately large mass to a subset of modes from the true distribution (Goodfellow, 2016). This raises concerns about a lack of diversity in the synthetic GAN distributions, and recent work shows that the learned distributions of two common GANs indeed have (moderately) low support size for the CelebA dataset (Arora & Zhang, 2017). However, the approach of Arora & Zhang (2017) heavily relies on a human annotator in order to identify duplicates. Hence it does not easily scale to comparing many variants of GANs or asking more fine-grained questions than collision statistics. Overall, our understanding of synthetic GAN distributions remains blurry, largely due to the lack of versatile tools for a quantitative evaluation of GANs in realistic settings. The focus of this work is precisly to address this question: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Can we develop principled and quantitative approaches to study synthetic GAN distributions? ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To this end, we propose two new evaluation techniques for synthetic GAN distributions. Our methods are inspired by the idea of comparing moments of distributions, which is at the heart of many methods in classical statistics. Although simple moments of high-dimensional distributions are often not semantically meaningful, we can extend this idea to distributions of realistic images by leveraging image statistics identified using convolutional neural networks. In particular, we train image classifiers in order to construct test functions corresponding to semantically meaningful properties of the distributions. An important feature of our approach is that it requires only light human supervision and can easily be scaled to evaluating many GANs and large synthetic datasets. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Using our new evaluation techniques, we study five state-of-the-art GANs on the CelebA and LSUN datasets, arguably the two most common testbeds for advanced GANs. We find that most of the GANs significantly distort the relative frequency of even basic image attributes, such as the hair style of a person or the type of room in an indoor scene. This clearly indicates a mismatch between the true and synthetic distributions. Moreover, we conduct experiments to explore the diversity of GAN distributions. We use synthetic GAN data to train image classifiers and find that these have significantly lower accuracy than classifiers trained on the true data set. This points towards a lack of diversity in the GAN data, and again towards a discrepancy between the true and synthetic distributions. In fact, our additional examinations show that the diversity in GANs is only comparable to a subset of the true data that is $1 0 0 \\times$ smaller. ",
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+    {
+        "type": "text",
+        "text": "2 UNDERSTANDING GANS THROUGH THE LENS OF CLASSIFICATION",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "When comparing two distributions, a common first test is to compute low-order moments such as the mean and the variance. If the distributions are simple enough, these quantities provide a good understanding for how similar they are. Moreover, low-order moments have a precise definition and are usually quick to compute. On the other hand, low-order moments can also be misleading for more complicated, high-dimensional distributions. As a concrete example, consider a generative model of digits (such as MNIST). If a generator produces digits that are shifted by a significant amount yet otherwise perfect, we will probably still consider this as a good approximation of the true distribution. However, the expectation (mean moment) of the generator distribution can be very different from the expectation of the true data distribution. This raises the question of what other properties of high-dimensional image distributions are easy to test yet semantically meaningful. ",
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+    {
+        "type": "text",
+        "text": "In the next two subsections, we describe two concrete approaches to evaluate synthetic GAN data that are easy to compute yet capture relevant information about the distribution. The common theme is that we employ convolutional neural networks in order to capture properties of the distributions that are hard to describe in a mathematically precise way, but usually well-defined for a human (e.g., what fraction of the images shows a smiling person?). Automating the process of annotating images with such high-level information will allow us to study various aspects of synthetic GAN data. ",
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+    {
+        "type": "text",
+        "text": "2.1 QUANTIFYING MODE COLLAPSE",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Mode collapse refers to the tendency of the generator to concentrate a large probability mass on a few modes of the true distribution. While there is ample evidence for the presence of mode-collapse in GANs (Goodfellow, 2016; Arora & Zhang, 2017; Metz et al., 2016), elegant visualizations of this phenomena are somewhat restricted to toy problems on low-dimensional distributions (Goodfellow, ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2016; Metz et al., 2016). For image datasets, it is common to rely on human annotators and derived heuristics (see Section 2.3). While these methods have their merits, they are restrictive both in the scale and granularity of testing. Here we propose a classification-based tool to assess how good GANs are at assigning the right mass across broad concepts/modes. To do so, we use a trained classifier as an expert “annotator” that labels important features in synthetic data, and then analyze the resulting distribution. Specifically, our goal is to investigate if a GAN trained on a well-balanced dataset (i.e., contains equal number of samples from each class) can learn to reproduce this balanced structure. Let $D = ( \\dot { X } , Y ) = \\{ ( x _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { N }$ represent a dataset of size $N$ with $C$ classes, where $( x _ { i } , y _ { i } )$ denote an image-label pair drawn from true data. If the dataset $D$ is balanced, it contains $N / C$ images per class. The procedure for computing class distribution in synthetic data is: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "1. Train an annotator (a multi-class classifier) using the dataset $D$ .   \n2. Train an unconditional GAN on the images $X$ from dataset $D$ , without using class labels.   \n3. Create a synthetic dataset by sampling $N$ images from a GAN and labeling them using the annotator from Step 1. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The annotated data generated via the above procedure can provide insight into the GAN’s class distribution at the scale of the entire dataset. Moreover, we can vary the granularity of mode analysis by choosing richer classification tasks, i.e., more challenging classes or a larger number of them. In Section 3.3, we use this technique to visualize mode collapse in several state-of-the-art GANs on the CelebA and LSUN datasets. All the studied GANs show significant mode collapse and the effect becomes more pronounced when the granularity of the annotator is increased (larger number of classes). We also investigate the temporal aspect of the GAN setup and find that the dominant mode varies widely over the course of training. Our approach also enables us to benchmark and compare GANs on different datasets based on the extent of mode collapse in the learned distributions. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 MEASURING DIVERSITY ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our above method for inspecting distribution of modes in synthetic data provides a coarse look at the statistics of the underlying distribution. While the resulting quantities are semantically meaningful, they capture only simple notions of diversity. To get a more holistic view on the sample diversity in the synthetic distribution, we now describe a second classification-based approach for evaluating GAN distributions. The main question that motivates it is: Can GANs recover the key aspects of real data to enable training a good classifier? We believe that this is an interesting measure of sample diversity for two reasons. First, classification of high-dimensional image data is a challenging problem, so a good training dataset will require a sufficiently diverse sample from the distribution. Second, augmenting data for classification problems is one of the proposed use cases of GANs (e.g., see the recent work of Shrivastava et al. (2017)). ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "If GANs are truly able to capture the quality and diversity of the underlying data distribution, we expect almost no gap between classifiers trained on true data and synthetic data from a GAN. A generic method to produce data from GANs for classification is to train separate GANs for each class in the dataset $D$ .1 Samples from these class-wise GANs can then be pooled together to get a labeled synthetic dataset. Note that the labels are trivially determined based on the class modeled by the particular GAN from which a sample is drawn. We perform the following steps to assess the classification performance of synthetic data vs. true data: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "1. Train a classifier on the true data $D$ (from Section 2.1) as a benchmark for comparison. 2. Train $C$ separate unconditional GANs, one per class in dataset D. 3. Generate a balanced synthetic labeled dataset of size $_ \\mathrm { N }$ by consolidating an equal number of samples drawn from each of these $C$ GANs. The labels obtained by aggregating samples from per-class GANs are designated as “default” labels for the synthetic dataset. Note that by design, both true and synthetic datasets have $N$ samples, with $N / C$ examples per class. 4. Use synthetic labeled data from Step 3 to train classifier with the same architecture as Step 1. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Comparing the classifiers from Steps 1 and 4 can then shed light on the disparity between the two distributions. Radford et al. (2015) conducted an experiment similar to Step 2 on the MNIST dataset using a conditional GAN. They found that samples from their DCGAN performed comparably to true data on nearest neighbor classification. We obtained similar good results on MNIST, which could be due to the efficacy of GANs in learning the MNIST distribution or due to the ease of getting good accuracy on MNIST even with a small training set (Rolnick et al., 2017). To clarify this question, we restrict our analysis to more complex datasets, specifically CelebA and LSUN. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We evaluate the two following properties in our classification task: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "(i) How well can the GANs recover nuances of the decision boundary, which is reflected by how easily the classifier can fit the training data?   \n(ii) How does the diversity of synthetic data compare to that of true data when measured by classification accuracy on a hold-out set of true data? ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We observe that all the studied GANs have very low diversity in this metric. In particular, the accuracy achieved by a classifier trained on GAN data is comparable only to the accuracy of a classifier trained on a $1 0 0 \\times$ (or more) subsampled version of the true dataset. Even if we draw more samples from the GANs to produce a training set several times larger than the true dataset, there is no improvement in performance. Looking at the classification accuracy gives us a way to compare different models on a potential downstream application of GANs. Interestingly, we find that visual quality of samples does not necessarily correlate with good classification performance. ",
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "2.3 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In GAN literature, it is common to investigate performance using metrics that involve human supervision. Arora & Zhang (2017) proposed a measure based on manually counting duplicates in GAN samples as a heuristic for the support or diversity of the learned distribution. In Wu et al. (2016), manual classification of a small sample (100 images) of GAN generated MNIST images is used as a test for the GAN is missing certain modes. Such annotator-based metrics have clear advantages in identifying relevant failure-modes of synthetic samples, which explains why visual inspection (eyeballing) is still the most popular approach to assess GAN samples. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "There have also been various attempts to build good metrics for GANs that are not based on manual heuristics. Parzen window estimation can be used to approximate the log-likelihood of the distribution, though it is known to work poorly for high-dimensional data (Theis et al., 2016). Wu et al. (2016) develop a method to get a better estimate for log-likelihood using annealed importance sampling. Salimans et al. (2016) propose a metric known as Inception Score, where the entropy in the labels predicted by a pre-trained Inception network is used to assess the diversity in GAN samples. ",
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+    {
+        "type": "text",
+        "text": "3 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In the following sub-sections we describe the setup and results for our classification-based GAN benchmarks. Additional details can be found in Section 5 in the Appendix. ",
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+    {
+        "type": "text",
+        "text": "3.1 DATASETS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "GANs have shown promise in generating realistic samples, resulting in efforts to apply them to a broad spectrum of datasets. However, the Large-scale CelebFaces Attributes (CelebA) (Liu et al., 2015) and Large-Scale Scene Understanding (LSUN) (Yu et al., 2015) datasets remain the most popular and canonical ones in developing and evaluating GAN variants. Conveniently, these datasets also have rich annotations, making them particularly suited for our classification–based evaluations. Details on the setup for classification tasks for these datasets are given in the Appendix (Section 5). ",
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+    {
+        "type": "text",
+        "text": "3.2 MODELS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Using our framework, we perform a comparative study of several popular variants of GANs: ",
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+    {
+        "type": "text",
+        "text": "1. Deep Convolutional GAN (DCGAN): Convolutional GAN trained using a Jensen–Shannon divergence–based objective (Goodfellow et al., 2014; Radford et al., 2015).   \n2. Wasserstein GAN (WGAN): GAN that uses a Wasserstein distance–based objective (Arjovsky et al., 2017).   \n3. Adversarially Learned Inference (ALI): GAN that uses joint adversarial training of generative and inference networks (Dumoulin et al., 2017).   \n4. Boundary Equilibrium GAN (BEGAN): Auto-encoder style GAN trained using Wasserstein distance objective (Berthelot et al., 2017). ",
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/62ced381625d83d2930cddcf355b5368dc556dd7a50f2dae5311314091417820.jpg",
+        "image_caption": [
+            "Figure 1: Visualizations of mode collapse in the synthetic, GAN-generated data produced after training on our chosen subsets of CelebA and LSUN datasets. Left panel shows the relative distribution of classes in samples drawn from synthetic datasets extracted at the end of the training process, and compares is to the true data distribution (leftmost plots). On the right, shown is the evolution of analogous class distribution for different GANs over the course of training. BEGAN did not converge on the LSUN tasks and hence is excluded from the corresponding analysis. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "5. Improved GAN (ImGAN): GAN that uses semi-supervised learning (labels are part of GAN training), with various other architectural and procedural improvements (Salimans et al., 2016). ",
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+    {
+        "type": "text",
+        "text": "All the aforementioned GANs are unconditional, however, ImGAN has access to class labels as a part of the semi-supervised training process. We use standard implementations for each of these models, details of which are provided in the Appendix (Section 5). We also used the prescribed hyper-parameter settings for each GAN, including number of iterations we train them for. Our analysis is based on $6 4 \\times 6 4$ samples, which is a size at which GAN generated samples tend to be of high quality. We also use visual inspection to ascertain that the perceptual quality of GAN samples in our experiments is comparable to those reported in previous studies. We demonstrate sample images in Figures 2 and 3 in the Appendix. BEGAN did not converge in our experiments on the LSUN dataset and hence is excluded from the corresponding analysis. ",
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "In our study, we use two types of classification models: ",
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+    {
+        "type": "text",
+        "text": "1. ResNet: 32-Layer Residual network He et al. (2016). Here, we choose a ResNet as it is a standard classifier in vision and yields high accuracy on various datasets, making it a reliable baseline.   \n2. Linear Model: This is a network with one-fully connected layer between the input and output (no hidden layers) with a softmax non-linearity. If the dimensions of input $x$ and output $\\hat { y }$ , are $D$ and $C$ (number of classes) respectively, then linear models implement the function $\\bar { y } = \\bar { \\sigma ( W ^ { T } x + b ) }$ , ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $W$ is a $D \\times C$ matrix, $b$ is a $C \\times 1$ vector and $\\sigma ( \\cdot )$ is the softmax function. Due to it’s simplicity, this model will serve as a useful baseline in some of our experiments. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We always train the classifiers to convergence, with decaying learning rate and no data augmentation. ",
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+    {
+        "type": "text",
+        "text": "3.3 EXAMINATION OF MODE COLLAPSE ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Experimental results for quantifying mode collapse through classification tasks, described in Section 2.1, are presented below. Table 2 in the Appendix gives details on datasets (subsets of CelebA and LSUN) used in our analysis, such as size $( N )$ , number of classes $( C )$ , and accuracy of the annotator, i.e., a classifier pre-trained on true data, which is then used to label the synthetic, GANgenerated data. Figure 1 presents class distribution in synthetic data, as specified by these annotators. The left panel compares the relative distribution of modes in true data (uniform) with that in various GAN-generated datasets. Each of these datasets is created by drawing $N$ samples from the GAN after it was trained on the corresponding true dataset. The right panel illustrates the evolution of class distributions in various GANs over the course of training2. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Results: These visualization lead to the following findings: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "• All GANs seem to suffer from significant mode-collapse. This becomes more apparent when the annotator granularity is increased, by considering a larger set of classes. For instance, one should compare the relatively balanced class distributions in the 3-class LSUN task to the near-absence of some modes in the 5-class task. • Mode collapse is prevalent in GANs throughout the training process, and does not seem to recede over time. Instead the dominant mode(s) often fluctuate wildly over the course of the training. • For each task, often there is a common set of modes onto which the distint GANs exhibit collapse. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In addition to viewing our method as an approach to analyze the mode collapse, we can also use it as a benchmark for GAN comparison. From this perspective, we can observe that, on CelebA, DCGAN and ALI learn somewhat balanced distributions, while WGAN, BEGAN and Improved GAN show prominent mode collapse. This is in contrast to the results obtained LSUN, where, for example, WGAN exhibit relatively small mode collapse, while ALI suffers from significant mode collapse even on the simple 3-class task. This highlights the general challenge in real world applications of GANs: they often perform well on the datasets they were designed for (e.g. ALI on CelebA and WGAN on LSUN), but extension to new datasets is not straightforward. Temporal analysis of mode-collapse shows that there is wide variation in the dominant mode for WGAN and Improved GAN, whereas for BEGAN, the same mode(s) often dominates the entire training process. ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.4 DIVERSITY EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "text": "Using the procedure outlined in Section 2.2, we perform a quantitative assessment of sample diversity in GANs on the CelebA and LSUN datasets. We restrict our experiments to binary classification as we find they have sufficient complexity to highlight the disparity between true and synthetic data. Selected results for classification-based evaluation of GANs are presented in Table 1. A more extensive study is presented in Table 3, and Figures 4 and 5 in the Appendix (Section 5). ",
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+        "text": "As a preliminary check, we inspect the quality of our labeled GAN datasets. For this, we use high-accuracy annotators from Section 2.1 to predict labels for GAN generated data and measure consistency between the predicted and default labels (label correctness). We also inspect confidence scores, defined as the softmax probabilities for predicted class, of the annotator. The motivation behind these metrics is that if the classifier can correctly and with high-confidence predict labels for labeled GAN samples, then it is likely that they are convincing examples of that class, and hence of good “quality”. Empirical results for label agreement and annotator confidence of GAN generated datasets are shown in Tables 1 and 3, and Figure 4. We also report an equivalent Inception Score (Salimans et al., 2016), similar to that described in Section 2.3. Using the Inception network to get the label distribution may not be meaningful for face or scene images. Instead, we compute the Inception Score using the label distribution predicted from the annotator networks. Score is computed as $e x p ( \\mathbb { E } _ { x } [ { \\bf K L } ( p ( y | x ) ) | | p ( y ) ] )$ , where $y$ refers to label predictions from the annotators 3. ",
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+        "table_body": "<table><tr><td rowspan=5 colspan=1>Task</td><td rowspan=1 colspan=13>Classification Performance</td></tr><tr><td rowspan=4 colspan=1>Data Source</td><td rowspan=4 colspan=2>LabelCorrectness(%)</td><td rowspan=4 colspan=2>Inception Score(μ±σ)</td><td rowspan=1 colspan=8>Accuracy (%)</td></tr><tr><td rowspan=1 colspan=7>Linear model</td><td rowspan=1 colspan=1>ResNet</td></tr><tr><td rowspan=1 colspan=5>↑1</td><td rowspan=1 colspan=2>↑10</td><td rowspan=1 colspan=1>↑</td></tr><tr><td rowspan=1 colspan=3>Train</td><td rowspan=1 colspan=2>Test</td><td rowspan=1 colspan=1>Train</td><td rowspan=1 colspan=1>Test</td><td rowspan=1 colspan=1>Test</td></tr><tr><td rowspan=8 colspan=1>CelebASmiling (Y/N)# Images: 156160</td><td rowspan=3 colspan=1>TrueTrue↓64True↓256True、↓512True↓1024</td><td rowspan=3 colspan=2>92.4</td><td rowspan=3 colspan=2>1.69 ± 0.0074</td><td rowspan=2 colspan=3>85.787.691.593.7</td><td rowspan=3 colspan=2>85.685.082.480.076.2</td><td rowspan=3 colspan=1></td><td rowspan=3 colspan=1></td><td rowspan=1 colspan=1>92.4</td></tr><tr><td rowspan=2 colspan=1>87.882.177.871.2</td></tr><tr><td rowspan=1 colspan=3>95.0</td></tr><tr><td rowspan=5 colspan=1>DCGANWGANALIBEGANImproved GAN</td><td rowspan=5 colspan=2>96.198.293.393.598.4</td><td rowspan=5 colspan=2>1.67 ± 0.00281.68 ± 0.00311.71 ± 0.00271.74 ± 0.00281.88 ± 0.0021</td><td rowspan=2 colspan=3>100.096.8</td><td rowspan=2 colspan=2>77.183.4</td><td rowspan=2 colspan=1>100.096.8</td><td rowspan=2 colspan=1>77.183.5</td><td rowspan=2 colspan=1>63.365.3</td></tr><tr><td rowspan=1 colspan=2>27</td></tr><tr><td rowspan=1 colspan=3>94.5</td><td rowspan=3 colspan=2>80.169.570.2</td><td rowspan=3 colspan=1>95.098.5100.0</td><td rowspan=3 colspan=1>82.469.670.1</td><td rowspan=3 colspan=1>55.864.161.6</td></tr><tr><td rowspan=1 colspan=3>98.5</td></tr><tr><td rowspan=1 colspan=3>100.0</td></tr><tr><td rowspan=6 colspan=1>LSUNBedroom/Kitchen# Images: 200000</td><td rowspan=2 colspan=1>TrueTrue↓512True↓1024True↓2048True4096</td><td rowspan=2 colspan=2>98.2</td><td rowspan=2 colspan=2>1.94 ± 0.0217</td><td rowspan=1 colspan=3>64.764.765.298.7</td><td rowspan=2 colspan=2>64.164.064.056.255.1</td><td rowspan=2 colspan=1></td><td rowspan=2 colspan=1></td><td rowspan=2 colspan=1>99.176.466.956.555.1</td></tr><tr><td rowspan=1 colspan=3>100.0</td><td rowspan=1 colspan=1>.0</td><td rowspan=1 colspan=1>55.1</td></tr><tr><td rowspan=4 colspan=1>DCGANWGANALIImproved GAN</td><td rowspan=4 colspan=2>92.787.880.484.2</td><td rowspan=1 colspan=2>1.85± 0.0036</td><td rowspan=1 colspan=2>90</td><td rowspan=1 colspan=2>90.8</td><td rowspan=1 colspan=2>56.5</td><td rowspan=1 colspan=1>91.2</td><td rowspan=1 colspan=1>56.3</td><td rowspan=4 colspan=1>51.255.750.551.2</td></tr><tr><td rowspan=2 colspan=1>87.880.4</td><td rowspan=1 colspan=2>1.70 ± 0.0023</td><td rowspan=1 colspan=3>86.2</td><td rowspan=1 colspan=1>58</td><td rowspan=1 colspan=1>96.3</td><td rowspan=2 colspan=1>54.150.8</td></tr><tr><td rowspan=2 colspan=2>1.62 ± 0.00261.68 ± 0.0030</td><td rowspan=1 colspan=3>80.7</td><td rowspan=1 colspan=2>49.7</td><td rowspan=1 colspan=1>81.7</td></tr><tr><td rowspan=1 colspan=3>91.6</td><td rowspan=1 colspan=2>55.9</td><td rowspan=1 colspan=1>90.8</td><td rowspan=1 colspan=1>56.5</td></tr></table>",
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+        "type": "text",
+        "text": "Table 1: Select results from the comparative study on classification performance of true data vs. GANs on the CelebA and LSUN datasets. Label correctness measures the agreement between default labels for the synthetic datasets, and those predicted by the annotator, a classifier trained on true data. Shown alongside are the equivalent inception scores computed using labels predicted by the annotator (rather than an Inception Network). Training and test accuracies for a linear model on the various true and synthetic datasets are reported. Also presented are the corresponding accuracies for this classifier trained on down-sampled true data $\\left( \\downarrow _ { M } \\right)$ and oversampled synthetic data $( \\uparrow _ { L } )$ . Test accuracy for ResNets trained on these datasets is also shown (training accuracy was always $1 0 0 \\%$ ), though it is noticeable that deep networks suffer from issues when trained on synthetic datasets. ",
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+        "type": "text",
+        "text": "Next, we train classifiers using the true and labeled GAN-generated datasets and study their performance in terms of accuracy on a hold-out set of true data. ResNets (and other deep variants) yield good classification performance on true data, but suffer from severe overfitting on the synthetic data, leading to poor test accuracy. This already indicates a possible problem with GANs and the diversity of the data they generate. But to highlight this problem better and avoid the issues that stem from overfitting, we also look for a classifier which does not always overfit on the synthetic data. We, however, observed that even training simple networks, such as one fully connected layer with few hidden units, led to overfitting on synthetic data. Hence, we resorted to a very basic linear model described in Section 3.2. Tables 1 and 3 shows results from binary classification experiments using linear models, with the training and test accuracies of the classifier on various datasets. ",
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+        "text": "Finally, to get a better understanding of the underlying ”diversity” of synthetic datasets, we train linear models using down-sampled versions of true data (no augmentation), and compare this to the performance of synthetic data, as shown in Tables 1 and 3. Down-sampling the data by a factor of $M$ , denoted as $\\downarrow _ { M }$ implies selecting a random $N / M$ subset of the data $D$ . Visualizations of how GAN classification performance compares with (down-sampled) true data are in Figure 5 in the Appendix. A natural argument in the defense of GANs is that we can oversample them, i.e. generate datasets much larger than the size of training data. Results for linear models trained using a 10-fold oversampling of GANs (drawing $1 0 N$ samples), denoted by $\\uparrow _ { 1 0 }$ , are show in Tables 1 and 3. ",
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+        "type": "text",
+        "text": "Results: The major findings from these experiments are: ",
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+        "text": "• Based on Tables 1 and 3, and Figure 4, we see strong agreement between annotator labels and true labels for synthetic data, on par with the scores for the test set of true data. It is thus apparent that the GAN images are of high-quality, as expected based on the visual inspection. These scores are lower for LSUN than CelebA, potentially due to lower quality of generated LSUN images. From these results, we can get a broad understanding of how good GANs are at producing convincing/representative samples from different classes across datasets. This also shows that simple classification-based benchmarks can highlight relevant properties of synthetic datasets. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "• The equivalent inception score is not very informative and is similar for the true (hold-out set) and synthetic datasets. This is not surprising given the simple nature of our binary classification task and the fact that the true and synthetic datasets have almost a uniform distribution over labels. • It is evident from Table 1 that there is a large performance gap between true and synthetic data on classification tasks. Inspection of training accuracies shows that linear models are able to nearly fit the synthetic datasets, but are grossly underfitting on true data. Given the high scores of synthetic data on the previous experiments to assess dataset ‘quality’ (Tables 1 and 3, and Figure 4), it is likely that the poor classification performance is more indicative of lack of ‘diversity’. Comparing GAN performance to that of down-sampled true data reveals that the learned distribution, which was trained on datasets that have around hundred thousand data points exhibits diversity that is on par with what only mere couple of hundreds of true data samples constitute! This shows that, at least from the point of view of classification, the diversity of the GAN generated data is severely lacking. • Oversampling GANs by 10-fold to produce larger datasets does not improve classification performance. The disparity between true and synthetic data remains nearly unchanged even after this significant oversampling, further highlighting the lack of diversity in GANs. ",
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+    },
+    {
+        "type": "text",
+        "text": "In terms of the conclusions of relative performance of various GANs, we observe that WGAN and ALI (on CelebA) perform better than the other GANs. While BEGAN samples have good perceptual quality (see Figure 2), they consistently perform badly on our classification tasks. On the other hand, WGAN samples have relatively poor visual quality but seem to outperform other GANs in classification tasks. This is a strong indicator of the need to consider other metrics, such as the ones proposed in this paper, in addition to visual inspection to study GANs. For LSUN, the gap between true and synthetic data is much larger, with the classifiers getting near random performance on all the synthetic datasets. Note that these classifiers get poor test accuracy on LSUN but are not overfitting on the training data. In this case, we speculate the lower performance could be due to both lower quality and diversity of LSUN samples. ",
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+    {
+        "type": "text",
+        "text": "In summary, our key experimental finding is that even simple classification–based tests can hold tremendous potential to shed insight on the learned distribution in GANs. This not only helps us to get a deeper understanding of many of the underlying issues, but also provides with a more quantitative and rigorous platform on which to compare different GANs. Our techniques could, in principle, be also applied to assess other generative models such as Variational Auto-Encoders (VAEs) Kingma & Welling (2014). However, VAEs have significant problems in generating realistic samples on the datasets used in our analysis in the first place – see Arora & Zhang (2017). ",
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+    {
+        "type": "text",
+        "text": "4 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this paper, we put forth techniques for examining the ability of GANs to capture key characteristics of the training data, through the lens of classification. Our tools are scalable, quantitative and automatic (no need for visual inspection of images). They thus are capable of studying state-ofthe-art GANs on realistic, large-scale image datasets. Further, they serve as a mean to perform a nuanced comparison of GANs and to identify their relative merits, including properties that cannot be discerned from mere visual inspection. ",
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+    },
+    {
+        "type": "text",
+        "text": "We then use the developed techniques to perform empirical studies on popular GANs on the CelebA and LSUN datasets. Our examination shows that mode collapse is indeed a prevalent issue for GANs. Also, we observe that synthetic GAN-generated datasets have significantly reduced diversity, at least when examined from a classification perspective. In fact, the diversity of such synthetic data is often few orders of magnitude smaller than that of the true data. Furthermore, this gap in diversity does not seem to be bridged by simply producing much larger datasets by oversampling GANs. Finally, we also notice that good perceptual quality of samples does not necessarily correlate – and might sometime even anti-correlate – with distribution diversity. These findings suggest that we need to go beyond the visual inspection–based evaluations and look for more quantitative tools for assessing quality of GANs, such as the ones presented in this paper. ",
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            287,
+            118
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Martin Arjovsky, Soumith Chintala, and Leon Bottou. Wasserstein generative adversarial networks. ´ In International Conference on Machine Learning, pp. 214–223, 2017. ",
+        "bbox": [
+            171,
+            128,
+            823,
+            159
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Sanjeev Arora and Yi Zhang. Do GANs actually learn the distribution? An empirical study. arXiv preprint arXiv:1706.08224, 2017. ",
+        "bbox": [
+            171,
+            166,
+            823,
+            195
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "David Berthelot, Tom Schumm, and Luke Metz. BEGAN: Boundary equilibrium generative adversarial networks. arXiv preprint arXiv:1703.10717, 2017. ",
+        "bbox": [
+            169,
+            205,
+            823,
+            234
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Emily L Denton, Soumith Chintala, Rob Fergus, et al. Deep generative image models using a laplacian pyramid of adversarial networks. In Advances in neural information processing systems, pp. 1486–1494, 2015. ",
+        "bbox": [
+            171,
+            243,
+            823,
+            286
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Vincent Dumoulin, Ishmael Belghazi, Ben Poole, Alex Lamb, Martin Arjovsky, Olivier Mastropietro, and Aaron Courville. Adversarially learned inference. In International Conference on Learning Representations, Apr 2017. URL https://arxiv.org/abs/1606.00704. ",
+        "bbox": [
+            173,
+            296,
+            821,
+            340
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Ian Goodfellow. Nips 2016 tutorial: Generative adversarial networks. arXiv preprint arXiv:1701.00160, 2016. ",
+        "bbox": [
+            171,
+            348,
+            825,
+            378
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in neural information processing systems, pp. 2672–2680, 2014. ",
+        "bbox": [
+            173,
+            387,
+            821,
+            431
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770–778, 2016. ",
+        "bbox": [
+            173,
+            439,
+            821,
+            483
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Diederik P Kingma and Max Welling. Auto-encoding variational bayes. In International Conference on Learning Representations, Apr 2014. URL https://arxiv.org/abs/1312.6114. ",
+        "bbox": [
+            169,
+            492,
+            823,
+            522
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Ziwei Liu, Ping Luo, Xiaogang Wang, and Xiaoou Tang. Deep learning face attributes in the wild. In Proceedings of International Conference on Computer Vision (ICCV), 2015. ",
+        "bbox": [
+            173,
+            530,
+            818,
+            560
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Luke Metz, Ben Poole, David Pfau, and Jascha Sohl-Dickstein. Unrolled generative adversarial networks. arXiv preprint arXiv:1611.02163, 2016. ",
+        "bbox": [
+            173,
+            569,
+            820,
+            599
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Alec Radford, Luke Metz, and Soumith Chintala. Unsupervised representation learning with deep convolutional generative adversarial networks. arXiv preprint arXiv:1511.06434, 2015. ",
+        "bbox": [
+            173,
+            608,
+            823,
+            637
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "David Rolnick, Andreas Veit, Serge Belongie, and Nir Shavit. Deep learning is robust to massive label noise. arXiv preprint arXiv:1705.10694, 2017. ",
+        "bbox": [
+            173,
+            646,
+            823,
+            676
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Tim Salimans, Ian Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, and Xi Chen. Improved techniques for training GANs. In Advances in Neural Information Processing Systems, pp. 2234–2242, 2016. ",
+        "bbox": [
+            173,
+            685,
+            825,
+            728
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Ashish Shrivastava, Tomas Pfister, Oncel Tuzel, Joshua Susskind, Wenda Wang, and Russell Webb. Learning from simulated and unsupervised images through adversarial training. In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), July 2017. ",
+        "bbox": [
+            176,
+            737,
+            823,
+            781
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "L. Theis, A. van den Oord, and M. Bethge. A note on the evaluation of generative models. In International Conference on Learning Representations, Apr 2016. URL http://arxiv.org/ abs/1511.01844. ",
+        "bbox": [
+            174,
+            790,
+            823,
+            833
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Yuhuai Wu, Yuri Burda, Ruslan Salakhutdinov, and Roger Grosse. On the quantitative analysis of decoder-based generative models. arXiv preprint arXiv:1611.04273, 2016. ",
+        "bbox": [
+            169,
+            843,
+            823,
+            872
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Fisher Yu, Yinda Zhang, Shuran Song, Ari Seff, and Jianxiong Xiao. LSUN: Construction of a large-scale image dataset using deep learning with humans in the loop. arXiv preprint arXiv:1506.03365, 2015. ",
+        "bbox": [
+            174,
+            881,
+            825,
+            924
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Han Zhang, Tao Xu, Hongsheng Li, Shaoting Zhang, Xiaolei Huang, Xiaogang Wang, and Dimitris Metaxas. Stackgan: Text to photo-realistic image synthesis with stacked generative adversarial networks. arXiv preprint arXiv:1612.03242, 2016. ",
+        "bbox": [
+            176,
+            103,
+            823,
+            146
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A Efros. Unpaired image-to-image translation using cycle-consistent adversarial networks. arXiv preprint arXiv:1703.10593, 2017. ",
+        "bbox": [
+            173,
+            155,
+            823,
+            184
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "5 APPENDIX ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "5.1 EXPERIMENTAL SETUP ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "5.1.1 DATASETS FOR CLASSIFICATION TASKS ",
+        "text_level": 1,
+        "bbox": [
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+            154,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "To assess GAN performance from the perspective of classification, we construct a set of classification tasks on the CelebA and LSUN datasets. In the case of the LSUN dataset, images are annotated with scene category labels, which makes it straightforward to use this data for binary and multiclass classification. On the other hand, each image in the CelebA dataset is labeled with 40 binary attributes. As a result, a single image has multiple associated attribute labels. Here, we construct classification tasks can by considering binary combinations of an attribute(s) (examples are shown in Figure 2). Attributes used in our experiments were chosen such that the resulting dataset was large, and classifiers trained on true data got high-accuracy so as to be good annotators for the synthetic data. Details on datasets used in our classification tasks, such as training set size $( N )$ , number of classes $( C )$ , and accuracy of the annotator, i.e., a classifier pre-trained on true data which is used to label the synthetic GAN-generated data, are provided in Table 2. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/f9f605a42758dec1c5e59700c1fed878b0d2b505eba596b80596ddccac6404f8.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Dataset</td><td rowspan=1 colspan=1>N</td><td rowspan=1 colspan=1>C</td><td rowspan=1 colspan=1>Annotator&#x27;s Accuracy (%)</td></tr><tr><td rowspan=1 colspan=1>CelebA: Makeup, Smiling</td><td rowspan=1 colspan=1>102,436</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>90.9, 92.4</td></tr><tr><td rowspan=1 colspan=1>CelebA: Male, Mouth Open</td><td rowspan=1 colspan=1>115,660</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>97.9, 93.5</td></tr><tr><td rowspan=1 colspan=1>CelebA: Bangs, Smiling</td><td rowspan=1 colspan=1>45,196</td><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>93.9,92.4</td></tr><tr><td rowspan=1 colspan=1>LSUN: Bedroom, Kitchen, Classroom</td><td rowspan=1 colspan=1>150,000</td><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>98.7</td></tr><tr><td rowspan=1 colspan=1>LSUN: Bedroom, Conference Room, Dining Room,Kitchen,Living Room</td><td rowspan=1 colspan=1>250,000</td><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>93.7</td></tr></table>",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Table 2: Details of CelebA and LSUN subsets used for the studies in Section 3.3. Here, we use a classifier trained on true data as an annotator that let’s us infer label distribution for the synthetic, GAN-generated data. $N$ is the size of the training set and $C$ is the number of classes in the true and synthetic datasets. Annotator’s accuracy refers to the accuracy of the classifier on a test set of true data. For CelebA, we use a combination of attribute-wise binary classifiers as annotators due their higher accuracy compared to a single classifier trained jointly on all the four classes. ",
+        "bbox": [
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+            455,
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+            540
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "5.1.2 MODELS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            558,
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+            573
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Benchmarks were performed on standard implementations - • DCGAN: https://github.com/carpedm20/DCGAN-tensorflow • WGAN: https://github.com/martinarjovsky/WassersteinGAN • ALI: https://github.com/IshmaelBelghazi/ALI • BEGAN :https://github.com/carpedm20/BEGAN-tensorflow • Improved GAN: https://github.com/openai/improved-gan • ResNet Classifier: Variation of the standard TensorFlow ResNet https://github.com/ tensorflow/models/blob/master/research/resnet/resnet_model.py ",
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+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "5.2 ADDITION EXPERIMENTAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            710,
+            468,
+            724
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "5.2.1 SAMPLE QUALITY",
+        "text_level": 1,
+        "bbox": [
+            176,
+            734,
+            354,
+            750
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "For each of our benchmark experiments, we ascertain that the visual quality of samples produced by the GANs is comparable to that reported in prior work. Examples of random samples drawn for multi-class datasets from both true and synthetic data are shown in Figure 2 for the CelebA dataset, and in Figure!3 for the LSUN dataset. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "5.2.2 MODE COLLAPSE EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            829,
+            455,
+            844
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In the studies to observe mode collapse in GANs described in Sections 2.1 and 3.3, we use a pretrained classifier as an annotator to obtain the class distribution for datasets generated from unconditional GANs. Figure 4 shows histograms of annotator confidence for the datasets used for benchmarking listed in Table 2. As can be seen in these figures, the annotator confidence for the synthetic data is comparable to that on the hold-out set of true data. Thus, it seems likely that the ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/ffd1eadad3e04387b4653a31d59a532c5b3dd825d3c778538887a71a3f4fdf6e.jpg",
+        "image_caption": [
+            "Figure 2: Illustration of datasets from CelebA used in proposed classification-based benchmarks to evaluate GANs. Shown alongside are images sampled from various unconditional GANs trained on this dataset. Labels for the GAN samples are obtained using a pre-trained classifier as an annotator. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 11
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+    {
+        "type": "text",
+        "text": "GAN generated samples are of good quality and are truly representative examples of their respective classes, as expected based on visual inspection. ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "5.2.3 DIVERSITY EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            814,
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+        "page_idx": 11
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+    {
+        "type": "text",
+        "text": "Table 3 presents an extension of the comparative study of classification performance of true and GAN generated data provided in Table 1. Visualizations of how test accuracies of a linear model classifier trained on GAN data compares with one trained on true data is shown in Figure 5. For each task, the bold curve shows test accuracy of a classifier trained on true data as a function of true dataset size. A down-sampling factor of $M$ corresponds to training the classifier on a random $N / M$ subset of true data. The dashed curves show test accuracy of classifiers trained on GAN datasets, obtained by drawing $N$ samples from GANs at the culmination of the training process. Based on these visualizations, it is apparent that GANs have comparable classification performance to a subset of training data that is more than a $1 0 0 \\mathbf { x }$ smaller. Thus, from the perspective of classification, GANs have diversity on par with a few hundred true data samples. ",
+        "bbox": [
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+        "page_idx": 11
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+    {
+        "type": "image",
+        "img_path": "images/e5b8ba0ba9dfbdfd0a8afcc1a9e78e8a86380ad9f5d7993dbf0217cebfd010d4.jpg",
+        "image_caption": [
+            "(b) 5-class dataset from LSUN for Bedroom, Conference Room, Dining Room, Kitchen, Living Room. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            99,
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+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/2ebb51f850f33b6ea62b998998d116ae8a6c587d4bac7ba0cb1ceccba4d70787.jpg",
+        "image_caption": [
+            "Figure 3: Illustration of datasets from LSUN used in proposed classification-based benchmarks to evaluate GANs. Shown alongside are images sampled from various unconditional GANs trained on this dataset. Labels for the GAN samples are obtained using a pre-trained classifier as an annotator. ",
+            "Figure 4: Histograms of annotator confidence (softmax probability) during label prediction on true data (test set) and synthetic data for tasks on the CelebA and LSUN datasets (see Section 3.4). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            627,
+            813,
+            852
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/bcdf6de7631c3992c2acb9241b93e99085cff09b8f2d85b8369478a356137e12.jpg",
+        "image_caption": [
+            "Figure 5: Illustration of the classification performance of true data compared with GAN-generated synthetic datasets based on experiments described in Section 3.4. Classification is performed using a basic linear model, described in Section 3.2, and performance is reported in terms of accuracy on a hold-out set of true data. In the plots, the bold curve captures the classification performance of models trained on true data vs the size of the true dataset (maximum size is $N$ ). Dashed lines represent performance of classifiers trained on various GAN-generated datasets of size $N$ . These plots indicate that GAN samples have ”diversity” comparable to a small subset (few hundred samples) of true data. Here the notion of diversity is one that is relevant for classification tasks. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            176,
+            816,
+            526
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/05a7df5055e01b2f771d77a55cb4b5c4dd395cbd6af90140f1e5a143bb68cf1d.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=5 colspan=1>Task</td><td rowspan=1 colspan=8>Classification Performance</td></tr><tr><td rowspan=4 colspan=1>Data Source</td><td rowspan=4 colspan=1>LabelCorrectness(%)</td><td rowspan=4 colspan=1>Inception Score(μ±σ)</td><td rowspan=1 colspan=5>Accuracy (%)</td></tr><tr><td rowspan=1 colspan=4>Linear model</td><td rowspan=1 colspan=1>ResNet</td></tr><tr><td rowspan=1 colspan=2>↑1</td><td rowspan=1 colspan=2>↑10</td><td rowspan=1 colspan=1>↑1</td></tr><tr><td rowspan=1 colspan=1>Train</td><td rowspan=1 colspan=1>Test</td><td rowspan=1 colspan=1>Train</td><td rowspan=1 colspan=1>Test</td><td rowspan=1 colspan=1>Test</td></tr><tr><td rowspan=6 colspan=1>CelebAMale (Y/N)# Images: 136522</td><td rowspan=2 colspan=1>TrueTrue↓64True↓256True↓512True↓1024</td><td rowspan=2 colspan=1>97.9</td><td rowspan=2 colspan=1>1.98 ± 0.0033</td><td rowspan=2 colspan=1>88.189.691.696.3100.0</td><td rowspan=2 colspan=1>88.888.786.983.883.1</td><td rowspan=2 colspan=1></td><td rowspan=2 colspan=1></td><td rowspan=1 colspan=1>97.992.9</td></tr><tr><td rowspan=1 colspan=1>89.882.681.4</td></tr><tr><td rowspan=4 colspan=1>DCGANWGANALIBEGANImproved GAN</td><td rowspan=4 colspan=1>98.298.399.299.399.8</td><td rowspan=4 colspan=1>1.97± 0.00131.97 ± 0.00131.99 ± 0.00081.99 ± 0.00061.99 ± 0.0004</td><td rowspan=4 colspan=1>100.096.795.897.9100.0</td><td rowspan=4 colspan=1>79.284.086.778.075.6</td><td rowspan=4 colspan=1>100.096.795.898.0100.0</td><td rowspan=1 colspan=1>79.6</td><td rowspan=1 colspan=1>56.4</td></tr><tr><td rowspan=3 colspan=1>83.986.778.271.0</td><td rowspan=1 colspan=1>50.0</td></tr><tr><td rowspan=1 colspan=1>58.9</td></tr><tr><td rowspan=1 colspan=1>55.471.7</td></tr><tr><td rowspan=4 colspan=1>CelebASmiling (Y/N)# Images: 156160</td><td rowspan=2 colspan=1>TrueTrue64True↓256True↓512True↓1024</td><td rowspan=2 colspan=1>92.4</td><td rowspan=2 colspan=1>1.69 ± 0.0074</td><td rowspan=2 colspan=1>85.787.691.593.795.0</td><td rowspan=2 colspan=1>85.685.082.480.076.2</td><td rowspan=2 colspan=1></td><td rowspan=2 colspan=1></td><td rowspan=1 colspan=1>92.487.8</td></tr><tr><td rowspan=1 colspan=1>82.177.871.2</td></tr><tr><td rowspan=2 colspan=1>DCGANWGANALIBEGANImproved GAN</td><td rowspan=2 colspan=1>96.198.293.393.598.4</td><td rowspan=1 colspan=1>1.67± 0.00281.68 ± 0.00311.71 ± 0.0027</td><td rowspan=1 colspan=1>100.096.894.5</td><td rowspan=1 colspan=1>77.183.480.1</td><td rowspan=1 colspan=1>100.096.895.0</td><td rowspan=1 colspan=1>77.183.582.4</td><td rowspan=1 colspan=1>63.365.355.8</td></tr><tr><td rowspan=1 colspan=1>1.74 ± 0.00281.88 ± 0.0021</td><td rowspan=1 colspan=1>98.5100.0</td><td rowspan=1 colspan=1>69.570.2</td><td rowspan=1 colspan=1>98.5100.0</td><td rowspan=1 colspan=1>69.670.1</td><td rowspan=1 colspan=1>64.161.6</td></tr><tr><td rowspan=2 colspan=1>CelebABlack Hair (Y/N)#Images: 77812</td><td rowspan=1 colspan=1>TrueTrue↓64True↓256True↓512True↓1024</td><td rowspan=1 colspan=1>84.5</td><td rowspan=1 colspan=1>1.68 ± 0.0112</td><td rowspan=1 colspan=1>76.479.786.389100.0</td><td rowspan=1 colspan=1>76.575.472.668.765.4</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>84.580.075.873.972.7</td></tr><tr><td rowspan=1 colspan=1>DCGANWGANALIBEGANImproved GAN</td><td rowspan=1 colspan=1>86.776.079.487.686.7</td><td rowspan=1 colspan=1>1.68 ± 0.00401.60 ± 0.00551.63 ± 0.00281.74 ± 0.00281.64 ± 0.0045</td><td rowspan=1 colspan=1>100.094.494.994.1100.0</td><td rowspan=1 colspan=1>70.973.771.067.670.3</td><td rowspan=1 colspan=1>100.094.394.994.1100.0</td><td rowspan=1 colspan=1>70.573.470.267.769.1</td><td rowspan=1 colspan=1>53.458.555.767.270.2</td></tr><tr><td rowspan=3 colspan=1>LSUNBedroom/Kitchen#Images: 200000</td><td rowspan=1 colspan=1>TrueTrue↓512True↓1024True↓2048True4096</td><td rowspan=1 colspan=1>98.2</td><td rowspan=1 colspan=1>1.94 ± 0.0217</td><td rowspan=1 colspan=1>64.764.765.298.7100.0</td><td rowspan=1 colspan=1>64.164.064.056.255.1</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>99.176.466.956.555.1</td></tr><tr><td rowspan=2 colspan=1>DCGANWGANALIImproved GAN</td><td rowspan=2 colspan=1>92.787.880.484.2</td><td rowspan=1 colspan=1>1.85± 0.00361.70 ± 0.0023</td><td rowspan=1 colspan=1>90.886.2</td><td rowspan=1 colspan=1>56.558.2</td><td rowspan=1 colspan=1>91.296.3</td><td rowspan=1 colspan=1>56.354.1</td><td rowspan=1 colspan=1>31.255.7</td></tr><tr><td rowspan=1 colspan=1>1.62 ± 0.00261.68 ± 0.0030</td><td rowspan=1 colspan=1>80.791.6</td><td rowspan=1 colspan=1>49.755.9</td><td rowspan=1 colspan=1>81.790.8</td><td rowspan=1 colspan=1>50.856.5</td><td rowspan=1 colspan=1>50.551.2</td></tr></table>",
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Table 3: Detailed version of the comparative study of the classification performance of true data and GANs on the CelebA and LSUN datasets shown in Table 1, based on experiments described in Section 3.4. Label correctness measures the agreement between default labels for the synthetic datasets, and those predicted by the annotator, a classifier trained on the true data. Shown alongside are the equivalent inception scores computed using labels predicted by the annotator (instead of the Inception Network). Training and test accuracies for a linear model classifier on the various true and synthetic datasets are reported. Also presented are the corresponding accuracies for a linear model trained on down-sampled true data $\\left( \\downarrow _ { M } \\right)$ and oversampled synthetic data $( \\uparrow _ { L } )$ . Test accuracy for ResNets trained on these datasets is also shown (training accuracy was always $1 0 0 \\%$ ), though it is noticeable that deep networks suffer from issues when trained on synthetic datasets. ",
+        "bbox": [
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+        "page_idx": 14
+    }
+]

parse/train/_61Qh8tULj_/_61Qh8tULj_.md

@@ -0,0 +1,286 @@
+# Conflict-Averse Gradient Descent for Multi-task Learning
+
+†Bo Liu, †Xingchao Liu, ‡Xiaojie Jin, †,§Peter Stone, †Qiang Liu †The University of Texas at Austin, §Sony AI, $^ \ddag$ Bytedance Research {bliu,xcliu,pstone,lqiang}@cs.utexas.edu, xjjin0731@gmail.com
+
+# Abstract
+
+The goal of multi-task learning is to enable more efficient learning than single task learning by sharing model structures for a diverse set of tasks. A standard multi-task learning objective is to minimize the average loss across all tasks. While straightforward, using this objective often results in much worse final performance for each task than learning them independently. A major challenge in optimizing a multi-task model is the conflicting gradients, where gradients of different task objectives are not well aligned so that following the average gradient direction can be detrimental to specific tasks’ performance. Previous work has proposed several heuristics to manipulate the task gradients for mitigating this problem. But most of them lack convergence guarantee and/or could converge to any Pareto-stationary point. In this paper, we introduce Conflict-Averse Gradient descent (CAGrad) which minimizes the average loss function, while leveraging the worst local improvement of individual tasks to regularize the algorithm trajectory. CAGrad balances the objectives automatically and still provably converges to a minimum over the average loss. It includes the regular gradient descent (GD) and the multiple gradient descent algorithm (MGDA) in the multi-objective optimization (MOO) literature as special cases. On a series of challenging multi-task supervised learning and reinforcement learning tasks, CAGrad achieves improved performance over prior state-of-the-art multi-objective gradient manipulation methods. Code is available at https://github.com/Cranial-XIX/CAGrad.
+
+# 1 Introduction
+
+Multi-task learning (MTL) refers to learning a single model that can tackle multiple different tasks [11, 28, 44, 38]. By sharing parameters across tasks, MTL methods learn more efficiently with an overall smaller model size compared to learning with separate models [38, 40, 25]. Moreover, it has been shown that MTL could in principle improve the quality of the learned representation and therefore benefit individual tasks [35, 43, 34]. For example, an early MTL result by [2] demonstrated that training a neural network to recognize doors could be improved by simultaneously training it to recognize doorknobs.
+
+However, learning multiple tasks simultaneously can be a challenging optimization problem because it involves multiple objectives [38]. The most popular MTL objective in practice is the average loss over all tasks. Even when this average loss is exactly the true objective (as opposed to only caring about a single task as in the door/doorknob example), directly optimizing the average loss could lead to undesirable performance, e.g. the optimizer struggles to make progress so the learning performance significantly deteriorates. A known cause of this phenomenon is the conflicting gradients [41]: gradients from different tasks 1) may have varying scales with the largest gradient dominating the update, and 2) may point in different directions so that directly optimizing the average loss can be quite detrimental to a specific task’s performance.
+
+![](images/14d2ae6187fa423d10a6e6e4e172d11a36640fdd1044a7d8913cb1c05d44c78d.jpg)  
+Figure 1: The optimization challenges faced by gradient descent (GD) and existing gradient manipulation methods like the multiple gradient descent algorithm (MGDA) [6] and PCGrad [41]. MGDA, PCGrad and CAGrad are applied on top of the Adam optimizer [16]. For each methods, we repeat 3 runs of optimization from different initial points (labeled with $\bullet$ ). Each optimization trajectory is colored from red to yellow. GD with Adam gets stuck on two of the initial points because the gradient of one task overshadows that of the other task, causing the algorithm to jump back and forth between the walls of a steep valley without making progress along the floor of the valley. MGDA and PCGrad stop optimization as soon as they reach the Pareto set.
+
+To address this problem, previous work either adaptively re-weights the objectives of each task based on heuristics [3, 15] or seeks a better update vector [30, 41] by manipulating the task gradients. However, existing work often lacks convergence guarantees or only provably converges to any point on the Pareto set of the objectives. This means the final convergence point of these methods may largely depend on the initial model parameters. As a result, it is possible that these methods over-optimize one objective while overlooking the others (See Fig. 1).
+
+Motivated by the limitation of current methods, we introduce Conflict-Averse Gradient descent (CAGrad), which reduces the conflict among gradients and still provably converges to a minimum of the average loss. The idea of CAGrad is simple: it looks for an update vector that maximizes the worst local improvement of any objective in a neighborhood of the average gradient. In this way, CAGrad automatically balances different objectives and smoothly converges to an optimal point of the average loss. Specifically, we show that vanilla gradient descent (GD) and the multiple gradient descent algorithm (MGDA) are special cases of CAGrad (See Sec. 3.1). We demonstrate that CAGrad can improve over prior state-of-the-art gradient manipulation methods on a series of challenging multi-task supervised, semi-supervised, and reinforcement learning problems.
+
+# 2 Background
+
+In this section, we first introduce the problem setup of multi-task learning (MTL). Then we analyze the optimization challenge of MTL and discuss the limitation of prior gradient manipulation methods.
+
+# 2.1 Multi-task Learning and its Challenge
+
+In multi-task learning (MTL), we are given $K \geq 2$ different tasks, each of which is associated with a loss function $L _ { i } ( \theta )$ for a shared set of parameters $\theta$ . The goal is to find an optimal $\theta \in \mathbb { R } ^ { m }$ that achieves low losses across all tasks. In practice, a standard objective for MTL is minimizing the average loss over all tasks:
+
+$$
+\theta ^ { * } = \underset { \theta \in \mathbb { R } ^ { m } } { \arg \operatorname* { m i n } } \left\{ L _ { 0 } ( \theta ) \triangleq \frac { 1 } { K } \sum _ { i = 1 } ^ { K } L _ { i } ( \theta ) \right\} .
+$$
+
+Unfortunately, directly optimizing (1) using gradient descent may significantly compromise the optimization of individual losses in practice. A major source of this phenomenon is known as the conflicting gradients [41].
+
+Optimization Challenge: Conflicting Gradients Denote by $g _ { i } = \nabla L _ { i } ( \theta )$ the gradient of task $i$ , and $\begin{array} { r } { g _ { 0 } = \nabla L _ { 0 } ( \theta ) = \frac { 1 } { K } \sum _ { i } ^ { K } g _ { i } } \end{array}$ the averaged gradient. With learning rate $\alpha \in \mathbb { R } ^ { + }$ , $\theta  \theta - \alpha g _ { 0 }$ is the steepest descent update that appears to be the most natural update to follow when optimizing (1). However, $g _ { 0 }$ may conflict with individual gradients, i.e. $\exists \ i$ , $\langle g _ { i } , g _ { 0 } \rangle < 0$ . When this conflict is large, following $g _ { 0 }$ will decrease the performance on task $i$ . As observed by [41] and illustrated in Fig. 1, when $\theta$ is near a steep “valley", where a specific task’s gradient dominates the update, manipulating the direction and magnitude of $g _ { 0 }$ often leads to better optimization.
+
+# 2.2 Prior Attempts and Convergence Issues
+
+Several methods have been proposed to manipulate the task gradients to form a new update vector and have shown improved performance on MTL. Sener et al. apply the multiple-gradient descent algorithm (MGDA) [6] for MTL, which directly optimizes towards the Pareto set [30]. Chen et al. dynamically re-weight each $L _ { i }$ using a pre-defined heuristic [3]. More recently, PCGrad identifies conflicting gradients as the motivation behind manipulating the gradients and projects each task gradient to the normal plane of others to reduce the conflict [41]. While all these methods have shown success at improving the learning performance of MTL, they manipulate the gradient without respecting the original objective (1). Therefore, these methods could in principle converge to any point in the Pareto set (See Fig. 1 and Sec. 3.2). We provide the detailed algorithms of MGDA and PCGrad in Appendix A.1 and A.2, and a visualization of the update vector by each method in Fig. 2.
+
+# 3 Method
+
+We introduce our main algorithm, Conflict-Averse Gradient descent in Sec. 3.1, and then show theoretical analysis in Sec. 3.2.
+
+# 3.1 Conflict-Averse Gradient Descent
+
+Assume we update $\theta$ by $\theta ^ { \prime }  \theta - \alpha d$ , where $\alpha$ is a step size and $d$ an update vector. We want to choose $d$ to decrease not only the average loss $L _ { 0 }$ , but also every individual loss. To do so, we consider the minimum decrease rate across the losses,
+
+$$
+R ( \theta , d ) = \operatorname* { m a x } _ { i \in [ K ] } \left\{ { \frac { 1 } { \alpha } } \left( L _ { i } ( \theta - \alpha d ) - L _ { i } ( \theta ) \right) \right\} \approx - \operatorname* { m i n } _ { i \in [ K ] } \langle g _ { i } , d \rangle ,
+$$
+
+where we use the first-order Taylor approximation assuming $\alpha$ is small. If $R ( \theta , d ) < 0$ , it means that all losses are decreased with the update given a sufficiently small $\alpha$ . Therefore, $R ( \theta , d )$ can be regarded as a measurement of conflict among objectives.
+
+With the above measurement, our algorithm finds an update vector that minimizes such conflict to mitigate the optimization challenge while still converging to an optimum of the main objective $L _ { 0 } ( \theta )$ . To this end, we introduce Conflict-Averse Gradient descent (CAGrad), which on each optimization step determines the update $d$ by solving the following optimization problem:
+
+$$
+\operatorname* { m a x } _ { d \in \mathbb { R } ^ { m } } \operatorname* { m i n } _ { i \in [ K ] } \langle g _ { i } , d \rangle \quad \mathrm { s . t . } \quad \left\| d - g _ { 0 } \right\| \leq c \left\| g _ { 0 } \right\| ,
+$$
+
+Here, $c \in [ 0 , 1 )$ is a pre-specified hyper-parameter that controls the convergence rate (See Sec. 3.2). The optimization problem (3) looks for the best update vector within a local ball centered at the averaged gradient $g _ { 0 }$ , which also minimizes the conflict in losses measured by (2). Since we focus on MTL and choose the average loss as the main objective, $g _ { 0 }$ is the average gradient. However, CAGrad also applies when $g _ { 0 }$ is the gradient of some other user-specified objective. We leave exploring this possibility as a future direction.
+
+Dual Objective The optimization problem (3) involves decision variable $d$ that has the same dimension as the number of parameters in $\theta$ , which could be millions for a deep neural network. It is not practical to directly solve for $d$ on every optimization step. However, the dual problem of Eq. (3), as we will derive in the following, only involves solving for a decision variable $w \in \mathbb { R } ^ { K }$ , which can be efficiently found using standard optimization libraries [7]. Specifically, first note that $\begin{array} { r } { \operatorname* { m i n } _ { i } \langle g _ { i } , d \rangle = \operatorname* { m i n } _ { w \in \mathcal { W } } \langle \sum _ { i } w _ { i } \bar { g } _ { i } , d \rangle } \end{array}$ , where $w = ( w _ { 1 } , \dots , w _ { K } ) \in \mathbb { R } ^ { K }$ and $\mathcal { W }$ denotes the probability simplex, i.e. $\begin{array} { r } { \mathcal { W } = \{ w : \sum _ { i } w _ { i } = 1 } \end{array}$ and $w _ { i } ~ \geq ~ 0 \}$ . Denote $\begin{array} { r } { g _ { w } \ = \ \sum _ { i } w _ { i } g _ { i } } \end{array}$ and $\phi = c ^ { 2 } \left\| g _ { 0 } \right\| ^ { 2 }$ . The Lagrangian of the objective in Eq. (3) is
+
+$$
+\operatorname* { m a x } _ { d \in \mathbb { R } ^ { m } } \operatorname* { m i n } _ { \lambda \geq 0 , w \in \mathcal { W } } g _ { w } ^ { \top } d - \lambda ( \left. g _ { 0 } - d \right. ^ { 2 } - \phi ) / 2 .
+$$
+
+Since the objective for $d$ is concave with linear constraints, by switching the min and max, we reach the dual form without changing the solution by Slater’s condition:
+
+$$
+\operatorname* { m i n } _ { \lambda \geq 0 , w \in \mathcal { W } } \operatorname* { m a x } _ { d \in \mathbb { R } ^ { m } } g _ { w } ^ { \top } d - \lambda \left. g _ { 0 } - d \right. ^ { 2 } / 2 + \lambda \phi / 2 .
+$$
+
+<table><tr><td>Input: Initial model parameter vector 0o,differentiable loss functions {Li}=1,a constant c ∈ [0,1) and learning rate α ∈ R+.</td></tr><tr><td>repeat</td></tr><tr><td>At the t-th optimization step,define go = k∑i-1 VLi(θt-1) and  =c² /ol²2. 1K Solve</td></tr><tr><td>K min, F(w) := gg +√Φ|lgull，where gω = 1 M wiVLi(0t-1).</td></tr><tr><td>K wEW i=1</td></tr><tr><td></td></tr><tr><td>Update 0t = 0t-1-α (go+ i 1/2 9w). 1igwll</td></tr><tr><td>until convergence</td></tr><tr><td></td></tr></table>
+
+We end up with the following optimization problem w.r.t. $w$ after several steps of calculus,
+
+$$
+\boldsymbol { w } ^ { * } = \underset { \boldsymbol { w } \in \mathcal { W } } { \arg \operatorname* { m i n } } \ : g _ { \boldsymbol { w } } ^ { \top } \boldsymbol { g } _ { 0 } + \sqrt { \phi } \left\| \boldsymbol { g } _ { \boldsymbol { w } } \right\| ,
+$$
+
+where the optimal $\lambda ^ { * } = \| g _ { w ^ { * } } \| / \phi ^ { 1 / 2 }$ and the optimal update $d ^ { * } = g _ { 0 } + g _ { w ^ { * } } / \lambda ^ { * }$ . The detailed derivation is provided in Appendix A.3 and the entire CAGrad algorithm is summarized in Alg. 1. The dimension of $w$ equals to the number of objectives $K$ , which usually ranges from 2 to tens and is much smaller than the number of parameters in a neural network. Therefore, in practice, we solve the dual objective to perform the update of CAGrad.
+
+Remark In Alg. 1, when $c = 0$ , CAGrad recovers the typical gradient descent with $d = g _ { 0 }$ . On the other hand, when $c \to \infty$ , then minimizing $F ( w )$ is equivalent to $\operatorname* { m i n } _ { w } \left\| g _ { w } \right\|$ . This coincides with the multiple gradient descent algorithm (MGDA) [6], which uses the minimum norm vector in the convex hull of the individual gradients as the update direction (see Fig. 2; second column). MGDA is a gradient-based multi-objective optimization designed to converge to an arbitrary point on the Pareto set, that is, it leaves all the points on the Pareto set as fixed points (and hence can not control which specific point it will converge to). It is different from our method which targets to minimize $L _ { 0 }$ while using gradient conflict to regularize the optimization trajectory. As we will analyze in the following section, to guarantee that CAGrad converges to an optimum of $L _ { 0 } ( \theta )$ , we have to ensure $0 \leq c < 1$
+
+# 3.2 Convergence Analysis
+
+In this section we first formally introduce the related Pareto concepts and then analyze CAGrad’s convergence property. Particularly, in Alg. 1, when $c < 1$ , CAGrad is guaranteed to converge to a minimum point of the average loss $L _ { 0 }$ .
+
+Pareto Concepts Unlike single task learning where any two parameter vectors $\theta _ { 1 }$ and $\theta _ { 2 }$ can be ordered in the sense that either $L ( \theta _ { 1 } ) \leq L ( \mathsf { \bar { \theta } } _ { 2 } )$ or $L ( \theta _ { 1 } ) \ge L \bar { ( \theta _ { 2 } ) }$ holds, MTL could have two parameter vectors where one performs better for task $i$ and the other performs better for task $j \neq i$ To this end, we need the notion of Pareto-optimality [13].
+
+Definition 3.1 (Pareto optimal and stationary points). Let $\pmb { L } ( \theta ) = \{ L _ { i } ( \theta ) \colon i \in [ K ] \}$ be a set of differentiable loss functions from $\mathbb { R } ^ { m }$ to $\mathbb { R }$ . For two points $\theta , \theta ^ { \prime } \in \mathbb { R } ^ { m }$ , we say that $\theta$ is Pareto dominated by $\theta ^ { \prime }$ , denoted by $\mathbf { } \cdot \mathbf { L } ( \theta ^ { \prime } ) \prec L ( \theta )$ , if $L _ { i } ( \theta ^ { \prime } ) \leq L _ { i } ( \theta )$ for all $i \in [ K ]$ and $\pmb { L } ( \theta ^ { \prime } ) \neq \pmb { L } ( \theta )$ . A point $\theta \in \mathbb { R } ^ { m }$ is said to be Pareto-optimal if there exists no $\theta ^ { \prime } \in \mathbb { R } ^ { m }$ such that $\mathbf { \dot { L } } ( \theta ^ { \prime } ) \prec \dot { \mathbf { L } } ( \theta )$ . The set of all Pareto-optimal points is called the Pareto set. $A$ point $\theta$ is called Pareto-stationary if we have $\mathrm { m i n } _ { w \in \mathcal { W } } \| g _ { w } ( \theta ) \| = 0$ , where $\begin{array} { r } { g _ { w } ( \theta ) = \sum _ { i = 1 } ^ { K } w _ { i } \nabla L _ { i } ( \theta ) } \end{array}$ , and $\mathcal { W }$ is the probability simplex on $[ K ]$ .
+
+Similar to the case of single-objective differentiable optimization, a local Pareto optimal point $\theta$ must be Pareto stationary (see e.g., [6]).
+
+Theorem 3.2 (Convergence of CAGrad). Assume the individual loss functions $L _ { 0 } , L _ { 1 } , \dots , L _ { K }$ are differentiable on $\mathbb { R } ^ { m }$ and their gradients $\nabla L _ { i } ( \theta )$ are all $H$ -Lipschitz, i.e. $\lVert \nabla L _ { i } ( x ) - \nabla L _ { i } ( y ) \rVert \leq$ $H \parallel x - y \parallel$ for $i = 0 , 1 , \ldots , K$ where $0 \leq H \leq \infty$ . Assume $L _ { 0 } ^ { * } = \operatorname* { i n f } _ { \theta \in \mathbb { R } ^ { m } } \dot { L } _ { 0 } ( \theta ) > - \infty$ .
+
+With a fixed step size $\alpha$ satisfying $0 < \alpha \leq 1 / H$ , we have for the CAGrad in Alg. 1:
+
+![](images/5c9c3173a60de5bc0f95c0dd8cfdd3adf5f1660683b1fd79cd70859bb92f45e3.jpg)  
+Figure 2: The combined update vector $d$ (in red) of a two-task learning problem with gradient descent (GD), multiple gradient descent algorithm (MGDA), PCGrad and Conflict-Averse Gradient descent (CAGrad). The two task-specific gradients are labeled $g _ { 1 }$ and $g _ { 2 }$ . MGDA’s objective is given in its primal form (See Appendix A.1). For PCGrad, each gradient is first projected onto the normal plane of the other (the dashed arrows). Then the final update vector is the average of the two projected gradients. CAGrad finds the best update vector within a ball around the average gradient that maximizes the worse local improvement between task 1 and task 2.
+
+1) For any $c \geq 1$ , all the fixed points of CAGrad are Pareto-stationary points of $( L _ { 0 } , L _ { 1 } , \ldots , L _ { K } )$
+
+2) In particular, if we take $0 \leq c < 1$ , then CAGrad satisfies
+
+$$
+\sum _ { t = 0 } ^ { T } \left. \nabla L _ { 0 } ( \theta _ { t } ) \right. ^ { 2 } \leq \frac { 2 ( L _ { 0 } ( \theta _ { 0 } ) - L _ { 0 } ^ { * } ) } { \alpha ( 1 - c ^ { 2 } ) } .
+$$
+
+This means that the algorithm converges to a stationary point of $\nabla L _ { 0 }$ if we take $0 \leq c < 1$ . The proof is in Appendix A.3. As we discuss earlier, unlike our method, MGDA is designed to converge to an arbitrary point on the Pareto set, without explicit control of which point it will converges to. Another algorithm with similar property is PCGrad [41], which is a gradient-based algorithm that mitigates the conflicting gradients problem by removing the conflicting components of each gradient with respect to the other gradients before averaging them to form the final update; see Fig. 2, third column for the illustration. Similar to MGDA, as shown in [41], PCGrad also converges to an arbitrary Pareto point without explicit control of which point it will arrive at.
+
+# 3.3 Practical Speedup
+
+A typical drawback of methods that manipulate gradients is the computation overhead. For computing the optimal update vector, a method usually requires $K$ back-propagations to find all individual gradients $g _ { i }$ , in addition to the time required for optimization. This can be prohibitive for the scenario with many tasks. To this end, we propose to only sample a subset of tasks $S \subseteq [ K ]$ , compute their corresponding gradients $\{ g _ { i } \mid i \in { \overline { { S } } } \}$ and the averaged gradient $g _ { 0 }$ . Then we optimize $d$ in:
+
+$$
+\operatorname* { m a x } _ { d \in \mathbb { R } ^ { m } } \operatorname* { m i n } \left( \langle \frac { K g _ { 0 } - \sum _ { i \in S } g _ { i } } { K - | S | } , d \rangle , \ : \ : \operatorname* { m i n } _ { i \in S } \langle g _ { i } , d \rangle \right) \mathrm { s . t . } \| d - g _ { 0 } \| \leq c \| g _ { 0 } \|
+$$
+
+Remark Note that the convergence guarantee in Thm. 3.2 still holds for Eq. 4 as the constraint does not change (See Appendix A.3). The time complexity is $\mathcal { O } ( ( \vert S \vert N + T )$ , where $N$ denotes the time for one pass of back-propagation and $T$ denotes the optimization time. For few-task learning $K < 1 0$ ), usually $T \ll N$ . When $S = [ K ]$ , we recover the full CAGrad algorithm.
+
+# 4 Related Work
+
+Multi-task Learning Due to its benefit with regards to data and computational efficiency, multi-task learning (MTL) has broad applications in vision, language, and robotics [11, 28, 22, 44, 38]. A number of MTL-friendly architectures have been proposed using task-specific modules [25, 11], attentionbased mechanisms [21] or activating different paths along the deep networks to tackle MTL [27, 40]. Apart from designing new architectures, another branch of methods focus on decomposing a large problem into smaller local problems that could be quickly learned by smaller models [29, 26, 37, 8]. Then a unified policy is learned from the smaller models using knowledge distillation [12].
+
+MTL Optimization In this work, we focus on the optimization challenge of MTL [38]. Gradient manipulation methods are designed specifically to balance the learning of each task. The simplest form of gradient manipulation is to re-weight the task losses based on specific criteria, e.g., uncertainty [15], gradient norm [3], or difficulty [9]. These methods are mostly heuristics and their performance can be unstable. Recently, two methods [30, 41] that manipulate the gradients to find a better local update vector have become popular. Sener et al [30] view MTL as a multi-objective optimization problem, and use multiple gradient descent algorithm for optimization. PCGrad [41] identifies a major optimization challenge for MTL, the conflicting gradients, and proposes to project each task gradient to the normal plane of other task gradients before combining them together to form the final update vector. Though yielding good empirical performance, both methods can only guarantee convergence to a Pareto-stationary point, but not knowing where it exactly converges to. More recently, GradDrop [4] randomly drops out task gradients based on how much they conflict. IMTLG [20] seeks an update vector that has equal projections on each task gradient. RotoGrad [14] separately scales and rotates task gradients to mitigate optimization conflict.
+
+Our method, CAGrad, also manipulates the gradient to find a better optimization trajectory. Like other MTL optimization techniques, CAGrad is model-agnostic. However, unlike prior methods, CAGrad converges to the optimal point in theory and achieves better empirical performance on both toy multi-objective optimization tasks and real-world applications.
+
+# 5 Experiment
+
+We conduct experiments to answer the following questions:
+
+Question (1) Do CAGrad, MGDA and PCGrad behave consistently with their theoretical properties in practice? (yes)
+
+Question (2) Does CAGrad recover GD and MGDA when varying the constant $c ?$ (yes)
+
+Question (3) How does CAGrad perform in both performance and computational efficiency compared to prior state-of-the-art methods, on challenging multi-task learning problems under the supervised, semi-supervised and reinforcement learning settings? (CAGrad improves over prior state-of-the-art methods under all settings)
+
+# 5.1 Convergence and Ablation over c
+
+To answer questions (1) and (2), we create a toy optimization example to evaluate the convergence of CAGrad compared to MGDA and PCGrad. On the same toy example, we ablate over the constant $c$ and show that CAGrad recovers GD and MGDA with proper $c$ values. Next, to test CAGrad on more complicated neural models, we perform the same set of experiments on the Multi-Fashion+MNIST benchmark [19] with a shrinked LeNet architecture [18] (in which each layer has a reduced number of neurons compared to the original LeNet). Please refer to Appendix B for more details.
+
+For the toy optimization example, we modify the toy example used by $\mathrm { Y u }$ et al. [41] and consider $\theta = ( \theta _ { 1 } , \bar { \theta _ { 2 } } ) \overset { \cdot } { \in } \mathbb { R } ^ { 2 }$ with the following individual loss functions:
+
+$$
+\begin{array} { r l } & { L _ { 1 } ( \theta ) = c _ { 1 } ( \theta ) f _ { 1 } ( \theta ) + c _ { 2 } ( \theta ) g _ { 1 } ( \theta ) \mathrm { ~ a n d ~ } L _ { 2 } ( \theta ) = c _ { 1 } ( \theta ) f _ { 2 } ( \theta ) + c _ { 2 } ( \theta ) g _ { 2 } ( \theta ) , \forall } \\ & { f _ { 1 } ( \theta ) = \log \big ( \operatorname* { m a x } ( | 0 . 5 ( - \theta _ { 1 } - 7 ) - \operatorname { t a n h } { ( - \theta _ { 2 } ) } | , \ 0 . 0 0 0 0 0 5 ) \big ) + 6 , } \\ & { f _ { 2 } ( \theta ) = \log \big ( \operatorname* { m a x } ( | 0 . 5 ( - \theta _ { 1 } + 3 ) - \operatorname { t a n h } { ( - \theta _ { 2 } ) } + 2 | , \ 0 . 0 0 0 0 0 5 ) \big ) + 6 , } \\ & { g _ { 1 } ( \theta ) = \big ( ( - \theta _ { 1 } + 7 ) ^ { 2 } + 0 . 1 * ( - \theta _ { 2 } - 8 ) ^ { 2 } \big ) / 1 0 - 2 0 , } \\ & { g _ { 2 } ( \theta ) = \big ( ( - \theta _ { 1 } - 7 ) ^ { 2 } + 0 . 1 * ( - \theta _ { 2 } - 8 ) ^ { 2 } \big ) / 1 0 - 2 0 , } \\ & { c _ { 1 } ( \theta ) = \operatorname* { m a x } ( \operatorname { t a n h } { ( 0 . 5 * \theta _ { 2 } ) } , \ 0 ) \mathrm { ~ a n d ~ } c _ { 2 } ( \theta ) = \operatorname* { m a x } ( \operatorname { t a n h } { ( - 0 . 5 * \theta _ { 2 } ) } , \ 0 ) . } \end{array}
+$$
+
+The average loss $L _ { 0 }$ and individual losses $L _ { 1 }$ and $L _ { 2 }$ are shown in Fig. 1. We then pick 5 initial parameter vectors $\theta _ { \mathrm { i n i t } } \in \{ ( - 8 . 5 , 7 . 5 ) , ( - 8 . 5 , 5 ) , ( 0 , 0 ) , ( 9 , 9 ) , ( 1 0 , - 8 ) \}$ and plot the corresponding optimization trajectories with different methods in Fig. 3. As shown in Fig. 3, GD gets stuck in 2 out of the 5 runs while other methods all converge to the Pareto set. MGDA and PCGrad converge to different Pareto-stationary points depending on $\theta _ { \mathrm { i n i t } }$ . CAGrad with $c = 0$ recovers GD and CAGrad with $c = 1 0$ approximates MGDA well (in theory it requires $c \to \infty$ to exactly recover MGDA).
+
+![](images/75d2f251d4cdaccd9c44f7ec6d7b1f0f9b38469e21607d25f96a255236d70ce3.jpg)  
+Figure 3: The left four plots are 5 runs of each algorithms from 5 different initial parameter vectors, where trajectories are colored from red to yellow. The right two plots are CAGrad’s results with a varying $\bar { c ^ { \prime } } \in \{ 0 , 0 . 2 , 0 . 5 , 0 . 8 , 1 0 \}$ .
+
+Next, we apply the same set of experiments on the multi-task classification benchmark MultiFashion+MNIST [19]. This benchmark consists of images that are generated by overlaying an image from FashionMNIST dataset [39] on top of another image from MNIST dataset [5]. The two images are positioned on the top-left and bottom-right separately. We consider a shrinked LeNet as our model, and train it with Adam [16] optimizer with a 0.001 learning rate for 50 epochs using a batch size of 256. Due to the highly non-convex nature of the neural network, we are not able to visualize the entire Pareto set. But we provide the final training losses of different methods over three independent runs in Fig. 4. As shown, CAGrad achieves the lowest average loss with $c = 0 . 2$ . In addition, PCGrad and MGDA focus on optimizing task 1 and task 2 separately. Lastly, CAGrad with $c = 0$ and $c = 1 0$ roughly recovers the final performance of GD and MGDA. By increasing $c _ { \cdot }$ , the model performance shifts from more GD-like to more MGDA-like, though due to the non-convex nature of neural networks, CAGrad with $0 \leq c < 1$ does not necessarily converge to the exact same point.
+
+![](images/bee5020abe60ded51f7f3f7db9c65b38616afcdb9d451be9bb97a872fad1223b.jpg)  
+Figure 4: The average and individual training losses on the Fashion-and-MNIST benchmark by running GD, MGDA, PCGrad and CAGrad with different $c$ values. GD gets stuck at the steep valley (the area with a cloud of dots), which other methods can pass. MGDA and PCGrad converge randomly on the Pareto set.
+
+# 5.2 Multi-task Supervised Learning
+
+To answer question (3) in the supervised learning setting, we follow the experiment setup from Yu et al. [41] and consider the NYU-v2 and CityScapes vision datasets. NYU-v2 contains 3 tasks: 13- class semantic segmentation, depth estimation, and surface normal prediction. CityScapes similarly contains 2 tasks: 7-class semantic segmentation and depth estimation. Here, we follow [41] and combine CAGrad with a state-of-the-art MTL method MTAN [21], which applies attention mechanism on top of the SegNet architecture [1]. We compare CAGrad with PCGrad, vanilla MTAN and CrossStitch [25], which is another MTL method that modifies the network architecture. MTAN originally experiments with equal loss weighting and two other dynamic loss weighting heuristics [15, 3]. For a fair comparison, all methods are applied under the equal weighting scheme and we use the same training setup from [3]. We search $\bar { c } \in \{ 0 . 1 , 0 . 2 , . . . \bar { 0 } . 9 \}$ with the best average training loss for CAGrad on both datasets (0.4 for NYU-v2 and 0.2 for Cityscapes). We perform a two-tailed, Student’s $t$ -test under equal sample sizes, unequal variance setup and mark the results that are significant with an $^ *$ . Following Maninis et al.[24], we also compute the average per-task performance drop of method $m$ with respect to the single-tasking baseline $b$ : 1K PKi=1(−1)li (Mm,i − Mb,i)/Mb,i where $l _ { i } = 1$ if a higher value is better for a criterion $M _ { i }$ on task $i$ and 0 otherwise. The single-tasking baseline (independent) refers to training individual tasks with a vanilla SegNet. Results are shown in Tab. 1 and Tab. 2.
+
+Given the single task performance, CAGrad performs better on the task that is overlooked by other methods (Surface Normal in NYU-v2 and Depth in CityScapes) and matches other methods’ performance on the rest of the tasks. We also provide the final test losses and the per-epoch training time of each method in Fig. 5 in Appendix B.2.
+
+Table 1: Multi-task learning results on NYU-v2 dataset. #P denotes the relative model size compared to the vanilla SegNet. Each experiment is repeated over 3 random seeds and the mean is reported. The best average result among all multi-task methods is marked in bold. MGDA, PCGrad, GradDrop and CAGrad are applied on the MTAN backbone. CAGrad has statistically significant improvement over baselines methods with an $^ *$ , tested with a $p$ -value of 0.1.   
+
+<table><tr><td rowspan="3">#P.</td><td rowspan="3">Method</td><td colspan="2">Segmentation</td><td colspan="2">Depth</td><td colspan="5">Surface Normal</td><td rowspan="3">△m%</td></tr><tr><td colspan="2">(Higher Better)</td><td colspan="2">(Lower Better)</td><td colspan="2">Angle Distance (Lower Better)</td><td colspan="2">Within t° (Higher Better)</td></tr><tr><td>mIoU</td><td>Pix Acc</td><td>Abs Err</td><td>Rel Err</td><td>Mean Median</td><td>11.25</td><td>22.5</td><td>30</td></tr><tr><td>3</td><td>Independent</td><td>38.30</td><td>63.76</td><td>0.6754</td><td>0.2780</td><td>25.01</td><td>19.21</td><td>30.14</td><td>57.20</td><td>69.15</td><td></td></tr><tr><td>~3</td><td>Cross-Stitch [25]</td><td>37.42</td><td>63.51</td><td>0.5487</td><td>0.2188</td><td>*28.85</td><td>*24.52</td><td>*22.75</td><td>*46.58</td><td>*59.56</td><td>6.96</td></tr><tr><td>1.77</td><td>MTAN [21]</td><td>39.29</td><td>65.33</td><td>0.5493</td><td>0.2263</td><td>*28.15</td><td>*23.96</td><td>*22.09</td><td>*47.50</td><td>*61.08</td><td>5.59</td></tr><tr><td>1.77</td><td>MGDA [30]</td><td>*30.47</td><td>*59.90</td><td>*0.6070</td><td>*0.2555</td><td>24.88</td><td>19.45</td><td>29.18</td><td>56.88</td><td>69.36</td><td>1.38</td></tr><tr><td>1.77</td><td>PCGrad [41]</td><td>38.06</td><td>64.64</td><td>0.5550</td><td>0.2325</td><td>*27.41</td><td>*22.80</td><td>*23.86</td><td>*49.83</td><td>*63.14</td><td>3.97</td></tr><tr><td>1.77</td><td>GradDrop [4]</td><td>39.39</td><td>65.12</td><td>0.5455</td><td>0.2279</td><td>*27.48</td><td>*22.96</td><td>*23.38</td><td>*49.44</td><td>*62.87</td><td>3.58</td></tr><tr><td>1.77</td><td>CAGrad (ours)</td><td>39.79</td><td>65.49</td><td>0.5486</td><td>0.2250</td><td>26.31</td><td>21.58</td><td>25.61</td><td>52.36</td><td>65.58</td><td>0.20</td></tr></table>
+
+Table 2: Multi-task learning results on CityScapes Challenge. Each experiment is repeated over 3 random seeds and the mean is reported. The best average result among all multi-task methods is marked in bold. PCGrad and CAGrad are applied on the MTAN backbone. CAGrad has statistically significant improvement over baselines methods with an $^ *$ , tested with a $p$ -value of 0.1.   
+
+<table><tr><td rowspan="2">#P.</td><td rowspan="2">Method</td><td colspan="2">Segmentation</td><td colspan="2">Depth</td><td rowspan="2">△m%</td></tr><tr><td>mIoU</td><td>(Higher Better) Pix Acc</td><td>(Lower Better) Abs Err</td><td>Rel Err</td></tr><tr><td>2</td><td>Independent</td><td>74.01</td><td>93.16</td><td>0.0125</td><td>27.77</td><td></td></tr><tr><td>~3</td><td>Cross-Stitch [25]</td><td>*73.08</td><td>*92.79</td><td>*0.0165</td><td>*118.5</td><td>90.02</td></tr><tr><td>1.77</td><td>MTAN [21]</td><td>75.18</td><td>93.49</td><td>*0.0155</td><td>*46.77</td><td>22.60</td></tr><tr><td>1.77</td><td>MGDA [30]</td><td>*68.84</td><td>*91.54</td><td>0.0309</td><td>33.50</td><td>44.14</td></tr><tr><td>1.77</td><td>PCGrad [41]</td><td>75.13</td><td>93.48</td><td>0.0154</td><td>42.07</td><td>18.29</td></tr><tr><td>1.77</td><td>GradDrop [4]</td><td>75.27</td><td>93.53</td><td>*0.0157</td><td>*47.54</td><td>23.73</td></tr><tr><td>1.77</td><td>CAGrad (ours)</td><td>75.16</td><td>93.48</td><td>0.0141</td><td>37.60</td><td>11.64</td></tr></table>
+
+# 5.3 Multi-task Reinforcement Learning
+
+To answer question (3) in the reinforcement learning (RL) setting, we apply CAGrad on the MT10 and MT50 benchmarks from the Meta-World environment [42]. In particular, MT10 and MT50 contains 10 and 50 robot manipulation tasks. Following [33], we use Soft Actor-Critic (SAC) [10] as the underlying RL training algorithm. We compare against Multi-task SAC (SAC with a shared model), Multi-headed SAC (SAC with a shared backbone and task-specific head), Multi-task SAC $^ +$ Task Encoder (SAC with a shared model and the input includes a task embedding) [42] and PCGrad [41]. We also compare with Soft Modularization [40] that routes different modules in a shared model to form different policies. Lastly, we also include a recent method (CARE) that considers language metadata and uses a mixture of expert encoder for MTL. We follow the same experiment setup from [33]. The results are shown in Tab. 3. CAGrad outperforms all baselines except for CARE which benefits from extra information from the metadata. We also apply the practical speedup in Sec. 3.3 and sub-sample 4 and 8 tasks for MT10 and MT50 (CAGrad-Fast). CAGrad-fast achieves comparable performance against the state-of-the-art method while achieving a $2 \mathbf { x }$ (MT10) and $5 \mathbf { x }$ (MT50) speedup over PCGrad. We provide a visualization of tasks from MT10 and MT50, and the comparison of computational efficiency in Appendix B.3.
+
+# 5.4 Semi-supervised Learning with Auxiliary Tasks
+
+Training with auxiliary tasks to improve the performance of a main task is another popular application of MTL. Here, we take semi-supervised learning as an instance. We combine different optimization
+
+<table><tr><td rowspan="2">Method</td><td>MetaworldMT10</td><td>MetaworldMT50</td></tr><tr><td>success (mean ± stderr)</td><td>success (mean ± stderr)</td></tr><tr><td>Multi-task SAC [42]</td><td>0.49 ±0.073</td><td>0.36 ±0.013</td></tr><tr><td>Multi-task SAC+ Task Encoder [42]</td><td>0.54 ±0.047</td><td>0.40 ±0.024</td></tr><tr><td>Multi-headed SAC [42]</td><td>0.61 ±0.036</td><td>0.45 ±0.064</td></tr><tr><td>PCGrad [41]</td><td>0.72 ±0.022</td><td>0.50 ±0.017</td></tr><tr><td>Soft Modularization [40]</td><td>0.73 ±0.043</td><td>0.50 ±0.035</td></tr><tr><td>CAGrad (ours)</td><td>0.83 ±0.045</td><td>0.52 ±0.023</td></tr><tr><td>CAGrad-Fast (ours)</td><td>0.82 ±0.039</td><td>0.50 ±0.016</td></tr><tr><td>CARE [33]</td><td>0.84 ±0.051</td><td>0.54 ±0.031</td></tr><tr><td>One SAC agent per task (upper bound)</td><td>0.90 ±0.032</td><td>0.74 ±0.041</td></tr></table>
+
+Table 3: Multi-task reinforcement learning results on the Metaworld benchmarks. Results are averaged over 10 independent runs and the best result is marked in bold.
+
+algorithms with Auxiliary Task Reweighting for Minimum-data Learning (ARML) [31], a state-ofthe-art semi-supervised learning algorithm. The loss function is composed of the main task and two auxiliary tasks:
+
+$$
+\begin{array} { r } { L _ { 0 } = L _ { C E } ( \theta ; D _ { l } ) + w _ { 1 } L _ { a u x } ^ { 1 } ( \theta ; D _ { u } ) + w _ { 2 } L _ { a u x } ^ { 2 } ( \theta ; D _ { u } ) , } \end{array}
+$$
+
+where are au $L _ { C E }$ is the main cross-entropy classification loss on the lay unsupervised learning losses on the unlabeled dataset d dataset . We use $D _ { l }$ , and  sam $L _ { a u x } ^ { 1 } , L _ { a u x } ^ { 2 }$ $D _ { u }$ $w _ { 1 }$ $w _ { 2 }$ from ARML, and use the CIFAR10 dataset [17], which contains 50,000 training images and 10,000 test images. $10 \%$ of the training images is held out as the validation set. We test PCGrad, MGDA and CAGrad with 500, 1000 and 2000 labeled images. The rest of the training set is used for auxiliary tasks. For all the methods, we use the same labeled dataset, the same learning rate and train them for 200 epochs with the Adam [16] optimizer. Please refer to Appendix B.4 for more experimental details. Results are shown in Tab. 4. With all the different number of labels, CAGrad yields the best averaged test accuracy. We observed that MGDA performs much worse than the ARML baseline, because it significantly overlooks the main classification task. We also compare different gradient manipulation methods on the same task with GradNorm [3], which dynamically adjusts $w _ { 1 }$ and $w _ { 2 }$ during training. The results and conclusions are similar to those for ARML.
+
+<table><tr><td>Method</td><td>500 labels</td><td>1000 labels</td><td>2000 labels</td></tr><tr><td>ARML [31]</td><td>67.05 ±0.16</td><td>73.22 ±0.26</td><td>81.35 ±0.36</td></tr><tr><td>ARML + PCGrad [41]</td><td>67.49 ±0.64</td><td>73.23 ±0.62</td><td>81.91 ±0.19</td></tr><tr><td>ARML +MGDA[30]</td><td>49.27 ±0.68</td><td>60.11 ±2.35</td><td>60.78 ±0.17</td></tr><tr><td>ARML +CAGrad (Ours)</td><td>68.25 ±0.37</td><td>74.37 ±0.42</td><td>82.81 ±0.48</td></tr><tr><td>GradNorm [3]</td><td>67.35 ±0.15</td><td>73.53 ±0.23</td><td>81.03 ±0.71</td></tr><tr><td>GradNorm + PCGrad [41]</td><td>67.83 ±0.19</td><td>73.91 ±0.09</td><td>82.72 ±0.19</td></tr><tr><td>GradNorm + MGDA [30]</td><td>36.99 ±2.11</td><td>57.94 ±0.92</td><td>59.12 ±0.63</td></tr><tr><td>GradNorm + CAGrad (Ours)</td><td>67.53 ±0.26</td><td>74.72 ±0.19</td><td>83.15 ±0.56</td></tr></table>
+
+Table 4: Semi-supervised Learning with auxiliary tasks on CIFAR10. We report the average test accuracy over 3 independent runs for each method and mark the best result in bold.
+
+# 6 Conclusion
+
+In this work, we introduce the Conflict-Averse Gradient descent (CAGrad) algorithm that explicitly optimizes the minimum decrease rate of any specific task’s loss while still provably converging to the optimum of the average loss. CAGrad generalizes the gradient descent and multiple gradient descent algorithm, and demonstrates improved performance across several challenging multi-task learning problems compared to the state-of-the-art methods. While we focus mainly on optimizing the average loss, an interesting future direction is to look at main objectives other than the average loss under the multi-task setting.
+
+# Acknowledgements
+
+The research was conducted in the statistical learning and AI group (SLAI) and the Learning Agents Research Group (LARG) in computer science at UT Austin. SLAI research is supported in part by CAREER-1846421, SenSE-2037267, EAGER-2041327, and Office of Navy Research, and NSF AI Institute for Foundations of Machine Learning (IFML). LARG research is supported in part by NSF (CPS-1739964, IIS-1724157, FAIN-2019844), ONR (N00014-18-2243), ARO (W911NF-19-2-0333), DARPA, Lockheed Martin, GM, Bosch, and UT Austin’s Good Systems grand challenge. Peter Stone serves as the Executive Director of Sony AI America and receives financial compensation for this work. The terms of this arrangement have been reviewed and approved by the University of Texas at Austin in accordance with its policy on objectivity in research. Xingchao Liu is supported in part by a funding from BP.
+
+# References
+
+[1] Vijay Badrinarayanan, Alex Kendall, and Roberto Cipolla. Segnet: A deep convolutional encoder-decoder architecture for image segmentation. IEEE transactions on pattern analysis and machine intelligence, 39(12):2481–2495, 2017.   
+[2] Rich Caruana. Multitask learning. Machine learning, 28(1):41–75, 1997.   
+[3] Zhao Chen, Vijay Badrinarayanan, Chen-Yu Lee, and Andrew Rabinovich. Gradnorm: Gradient normalization for adaptive loss balancing in deep multitask networks. In International Conference on Machine Learning, pages 794–803. PMLR, 2018. [4] Zhao Chen, Jiquan Ngiam, Yanping Huang, Thang Luong, Henrik Kretzschmar, Yuning Chai, and Dragomir Anguelov. Just pick a sign: Optimizing deep multitask models with gradient sign dropout. arXiv preprint arXiv:2010.06808, 2020.   
+[5] Li Deng. The mnist database of handwritten digit images for machine learning research. IEEE Signal Processing Magazine, 29(6):141–142, 2012.   
+[6] Jean-Antoine Désidéri. Multiple-gradient descent algorithm (mgda) for multiobjective optimization. Comptes Rendus Mathematique, 350(5-6):313–318, 2012. [7] Steven Diamond and Stephen Boyd. CVXPY: A Python-embedded modeling language for convex optimization. Journal of Machine Learning Research, 17(83):1–5, 2016. [8] Dibya Ghosh, Avi Singh, Aravind Rajeswaran, Vikash Kumar, and Sergey Levine. Divide-andconquer reinforcement learning. arXiv preprint arXiv:1711.09874, 2017.   
+[9] Michelle Guo, Albert Haque, De-An Huang, Serena Yeung, and Li Fei-Fei. Dynamic task prioritization for multitask learning. In Proceedings of the European Conference on Computer Vision (ECCV), pages 270–287, 2018.   
+[10] Tuomas Haarnoja, Aurick Zhou, Pieter Abbeel, and Sergey Levine. Soft actor-critic: Offpolicy maximum entropy deep reinforcement learning with a stochastic actor. In International Conference on Machine Learning, pages 1861–1870. PMLR, 2018.   
+[11] Kazuma Hashimoto, Caiming Xiong, Yoshimasa Tsuruoka, and Richard Socher. A joint manytask model: Growing a neural network for multiple nlp tasks. arXiv preprint arXiv:1611.01587, 2016.   
+[12] Geoffrey Hinton, Oriol Vinyals, and Jeff Dean. Distilling the knowledge in a neural network. arXiv preprint arXiv:1503.02531, 2015.   
+[13] Harold M Hochman and James D Rodgers. Pareto optimal redistribution. The American economic review, 59(4):542–557, 1969.   
+[14] Adrián Javaloy and Isabel Valera. Rotograd: Dynamic gradient homogenization for multi-task learning. arXiv preprint arXiv:2103.02631, 2021.   
+[15] Alex Kendall, Yarin Gal, and Roberto Cipolla. Multi-task learning using uncertainty to weigh losses for scene geometry and semantics. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 7482–7491, 2018.   
+[16] Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.   
+[17] Alex Krizhevsky, Geoffrey Hinton, et al. Learning multiple layers of features from tiny images. 2009.   
+[18] Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998.   
+[19] Xi Lin, Hui-Ling Zhen, Zhenhua Li, Qingfu Zhang, and Sam Kwong. Pareto multi-task learning. arXiv preprint arXiv:1912.12854, 2019.   
+[20] Liyang Liu, Yi Li, Zhanghui Kuang, Jing-Hao Xue, Yimin Chen, Wenming Yang, Qingmin Liao, and Wayne Zhang. Towards impartial multi-task learning. In International Conference on Learning Representations, 2020.   
+[21] Shikun Liu, Edward Johns, and Andrew J Davison. End-to-end multi-task learning with attention. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 1871–1880, 2019.   
+[22] Xingchao Liu, Xing Han, Na Zhang, and Qiang Liu. Certified monotonic neural networks. arXiv preprint arXiv:2011.10219, 2020.   
+[23] Debabrata Mahapatra and Vaibhav Rajan. Multi-task learning with user preferences: Gradient descent with controlled ascent in pareto optimization. In International Conference on Machine Learning, pages 6597–6607. PMLR, 2020.   
+[24] Kevis-Kokitsi Maninis, Ilija Radosavovic, and Iasonas Kokkinos. Attentive single-tasking of multiple tasks. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 1851–1860, 2019.   
+[25] Ishan Misra, Abhinav Shrivastava, Abhinav Gupta, and Martial Hebert. Cross-stitch networks for multi-task learning. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 3994–4003, 2016.   
+[26] Emilio Parisotto, Jimmy Lei Ba, and Ruslan Salakhutdinov. Actor-mimic: Deep multitask and transfer reinforcement learning. arXiv preprint arXiv:1511.06342, 2015.   
+[27] Clemens Rosenbaum, Tim Klinger, and Matthew Riemer. Routing networks: Adaptive selection of non-linear functions for multi-task learning. arXiv preprint arXiv:1711.01239, 2017.   
+[28] Sebastian Ruder. An overview of multi-task learning in deep neural networks. arXiv preprint arXiv:1706.05098, 2017.   
+[29] Andrei A Rusu, Sergio Gomez Colmenarejo, Caglar Gulcehre, Guillaume Desjardins, James Kirkpatrick, Razvan Pascanu, Volodymyr Mnih, Koray Kavukcuoglu, and Raia Hadsell. Policy distillation. arXiv preprint arXiv:1511.06295, 2015.   
+[30] Ozan Sener and Vladlen Koltun. Multi-task learning as multi-objective optimization. arXiv preprint arXiv:1810.04650, 2018.   
+[31] Baifeng Shi, Judy Hoffman, Kate Saenko, Trevor Darrell, and Huijuan Xu. Auxiliary task reweighting for minimum-data learning. Advances in Neural Information Processing Systems, 33, 2020.   
+[32] Shagun Sodhani and Amy Zhang. Mtrl - multi task rl algorithms. Github, 2021.   
+[33] Shagun Sodhani, Amy Zhang, and Joelle Pineau. Multi-task reinforcement learning with context-based representations. arXiv preprint arXiv:2102.06177, 2021.   
+[34] Charles Stein. Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. In Contribution to the Theory of Statistics, pages 197–206. University of California Press, 2020.   
+[35] Kevin Swersky, Jasper Snoek, and Ryan Prescott Adams. Multi-task bayesian optimization. 2013.   
+[36] Antti Tarvainen and Harri Valpola. Mean teachers are better role models: Weight-averaged consistency targets improve semi-supervised deep learning results. In Proceedings of the 31st International Conference on Neural Information Processing Systems, pages 1195–1204, 2017.   
+[37] Yee Whye Teh, Victor Bapst, Wojciech Marian Czarnecki, John Quan, James Kirkpatrick, Raia Hadsell, Nicolas Heess, and Razvan Pascanu. Distral: Robust multitask reinforcement learning. arXiv preprint arXiv:1707.04175, 2017.   
+[38] Simon Vandenhende, Stamatios Georgoulis, Wouter Van Gansbeke, Marc Proesmans, Dengxin Dai, and Luc Van Gool. Multi-task learning for dense prediction tasks: A survey. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2021.   
+[39] Han Xiao, Kashif Rasul, and Roland Vollgraf. Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv preprint arXiv:1708.07747, 2017.   
+[40] Ruihan Yang, Huazhe Xu, Yi Wu, and Xiaolong Wang. Multi-task reinforcement learning with soft modularization. arXiv preprint arXiv:2003.13661, 2020.   
+[41] Tianhe Yu, Saurabh Kumar, Abhishek Gupta, Sergey Levine, Karol Hausman, and Chelsea Finn. Gradient surgery for multi-task learning. arXiv preprint arXiv:2001.06782, 2020.   
+[42] Tianhe Yu, Deirdre Quillen, Zhanpeng He, Ryan Julian, Karol Hausman, Chelsea Finn, and Sergey Levine. Meta-world: A benchmark and evaluation for multi-task and meta reinforcement learning. In Conference on Robot Learning, pages 1094–1100. PMLR, 2020.   
+[43] Amir R Zamir, Alexander Sax, William Shen, Leonidas J Guibas, Jitendra Malik, and Silvio Savarese. Taskonomy: Disentangling task transfer learning. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 3712–3722, 2018.   
+[44] Yu Zhang and Qiang Yang. A survey on multi-task learning. IEEE Transactions on Knowledge and Data Engineering, 2021.
+
+# 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 Sec. 3.2 for the convergence analysis, Fig. 1 for the challenges faced by previous methods, and Sec. 5 for empirical evaluation of these challenges and the advantage of CAGrad.   
+(b) Did you describe the limitations of your work? [Yes] See Sec. 6. Currently we mainly focus on optimizing the average loss, which could be replaced by other main objectives.   
+(c) Did you discuss any potential negative societal impacts of your work? [N/A] Our method does not have potential negative societal impacts.   
+(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] The assumptions are stated in Thm. 3.2.   
+(b) Did you include complete proofs of all theoretical results? [Yes] The complete proof is included in Appendix A.3.
+
+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] We mention most of the details to reproduce the result in Sec. 5 and provide the rest of details of each experiment in Appendix.B. The code comes with the supplementary material.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Appendix.B and Sec. 5.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] For each experiment except for the toy (since there is no stochasticity), we run over multiple $\left( \geq 3 \right)$ seeds.   
+(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] We explicitly compare the computational efficiency in Fig. 5. More details on the resources are provided in the corresponding sections in Appendix.B.
+
+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] For most of the experiment, we follow the exact experiment setup and use the corresponding opensource code from previous works and have cited and compared against them.   
+(b) Did you mention the license of the assets? [Yes] All code and data are publicly available under MIT license   
+(c) Did you include any new assets either in the supplemental material or as a URL? [No] No new assets are introduced for our experiment. The only thing we modified is a shrinked LeNet, where the details are provided in Appendix.B.   
+(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] The data we use are publicly available data that has been used by a lot of prior research. There should be no personally identifiable information or offensive content.
+
+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] No human subjects involved.   
+(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/_61Qh8tULj_/_61Qh8tULj__content_list.json

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+        "type": "text",
+        "text": "Conflict-Averse Gradient Descent for Multi-task Learning ",
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+        "type": "text",
+        "text": "†Bo Liu, †Xingchao Liu, ‡Xiaojie Jin, †,§Peter Stone, †Qiang Liu †The University of Texas at Austin, §Sony AI, $^ \\ddag$ Bytedance Research {bliu,xcliu,pstone,lqiang}@cs.utexas.edu, xjjin0731@gmail.com ",
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+        "text": "Abstract ",
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+    },
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+        "text": "The goal of multi-task learning is to enable more efficient learning than single task learning by sharing model structures for a diverse set of tasks. A standard multi-task learning objective is to minimize the average loss across all tasks. While straightforward, using this objective often results in much worse final performance for each task than learning them independently. A major challenge in optimizing a multi-task model is the conflicting gradients, where gradients of different task objectives are not well aligned so that following the average gradient direction can be detrimental to specific tasks’ performance. Previous work has proposed several heuristics to manipulate the task gradients for mitigating this problem. But most of them lack convergence guarantee and/or could converge to any Pareto-stationary point. In this paper, we introduce Conflict-Averse Gradient descent (CAGrad) which minimizes the average loss function, while leveraging the worst local improvement of individual tasks to regularize the algorithm trajectory. CAGrad balances the objectives automatically and still provably converges to a minimum over the average loss. It includes the regular gradient descent (GD) and the multiple gradient descent algorithm (MGDA) in the multi-objective optimization (MOO) literature as special cases. On a series of challenging multi-task supervised learning and reinforcement learning tasks, CAGrad achieves improved performance over prior state-of-the-art multi-objective gradient manipulation methods. Code is available at https://github.com/Cranial-XIX/CAGrad. ",
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+        "text": "1 Introduction ",
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+        "text": "Multi-task learning (MTL) refers to learning a single model that can tackle multiple different tasks [11, 28, 44, 38]. By sharing parameters across tasks, MTL methods learn more efficiently with an overall smaller model size compared to learning with separate models [38, 40, 25]. Moreover, it has been shown that MTL could in principle improve the quality of the learned representation and therefore benefit individual tasks [35, 43, 34]. For example, an early MTL result by [2] demonstrated that training a neural network to recognize doors could be improved by simultaneously training it to recognize doorknobs. ",
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+        "text": "However, learning multiple tasks simultaneously can be a challenging optimization problem because it involves multiple objectives [38]. The most popular MTL objective in practice is the average loss over all tasks. Even when this average loss is exactly the true objective (as opposed to only caring about a single task as in the door/doorknob example), directly optimizing the average loss could lead to undesirable performance, e.g. the optimizer struggles to make progress so the learning performance significantly deteriorates. A known cause of this phenomenon is the conflicting gradients [41]: gradients from different tasks 1) may have varying scales with the largest gradient dominating the update, and 2) may point in different directions so that directly optimizing the average loss can be quite detrimental to a specific task’s performance. ",
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+        "image_caption": [
+            "Figure 1: The optimization challenges faced by gradient descent (GD) and existing gradient manipulation methods like the multiple gradient descent algorithm (MGDA) [6] and PCGrad [41]. MGDA, PCGrad and CAGrad are applied on top of the Adam optimizer [16]. For each methods, we repeat 3 runs of optimization from different initial points (labeled with $\\bullet$ ). Each optimization trajectory is colored from red to yellow. GD with Adam gets stuck on two of the initial points because the gradient of one task overshadows that of the other task, causing the algorithm to jump back and forth between the walls of a steep valley without making progress along the floor of the valley. MGDA and PCGrad stop optimization as soon as they reach the Pareto set. "
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+        "text": "To address this problem, previous work either adaptively re-weights the objectives of each task based on heuristics [3, 15] or seeks a better update vector [30, 41] by manipulating the task gradients. However, existing work often lacks convergence guarantees or only provably converges to any point on the Pareto set of the objectives. This means the final convergence point of these methods may largely depend on the initial model parameters. As a result, it is possible that these methods over-optimize one objective while overlooking the others (See Fig. 1). ",
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+        "text": "Motivated by the limitation of current methods, we introduce Conflict-Averse Gradient descent (CAGrad), which reduces the conflict among gradients and still provably converges to a minimum of the average loss. The idea of CAGrad is simple: it looks for an update vector that maximizes the worst local improvement of any objective in a neighborhood of the average gradient. In this way, CAGrad automatically balances different objectives and smoothly converges to an optimal point of the average loss. Specifically, we show that vanilla gradient descent (GD) and the multiple gradient descent algorithm (MGDA) are special cases of CAGrad (See Sec. 3.1). We demonstrate that CAGrad can improve over prior state-of-the-art gradient manipulation methods on a series of challenging multi-task supervised, semi-supervised, and reinforcement learning problems. ",
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+    {
+        "type": "text",
+        "text": "2 Background ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we first introduce the problem setup of multi-task learning (MTL). Then we analyze the optimization challenge of MTL and discuss the limitation of prior gradient manipulation methods. ",
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+        "type": "text",
+        "text": "2.1 Multi-task Learning and its Challenge ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In multi-task learning (MTL), we are given $K \\geq 2$ different tasks, each of which is associated with a loss function $L _ { i } ( \\theta )$ for a shared set of parameters $\\theta$ . The goal is to find an optimal $\\theta \\in \\mathbb { R } ^ { m }$ that achieves low losses across all tasks. In practice, a standard objective for MTL is minimizing the average loss over all tasks: ",
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+        "type": "equation",
+        "img_path": "images/889c361e1f527d61e1d650c6e00c51324425e5bcbd51d752464c23b62cbb07a3.jpg",
+        "text": "$$\n\\theta ^ { * } = \\underset { \\theta \\in \\mathbb { R } ^ { m } } { \\arg \\operatorname* { m i n } } \\left\\{ L _ { 0 } ( \\theta ) \\triangleq \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } L _ { i } ( \\theta ) \\right\\} .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Unfortunately, directly optimizing (1) using gradient descent may significantly compromise the optimization of individual losses in practice. A major source of this phenomenon is known as the conflicting gradients [41]. ",
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+    {
+        "type": "text",
+        "text": "Optimization Challenge: Conflicting Gradients Denote by $g _ { i } = \\nabla L _ { i } ( \\theta )$ the gradient of task $i$ , and $\\begin{array} { r } { g _ { 0 } = \\nabla L _ { 0 } ( \\theta ) = \\frac { 1 } { K } \\sum _ { i } ^ { K } g _ { i } } \\end{array}$ the averaged gradient. With learning rate $\\alpha \\in \\mathbb { R } ^ { + }$ , $\\theta  \\theta - \\alpha g _ { 0 }$ is the steepest descent update that appears to be the most natural update to follow when optimizing (1). However, $g _ { 0 }$ may conflict with individual gradients, i.e. $\\exists \\ i$ , $\\langle g _ { i } , g _ { 0 } \\rangle < 0$ . When this conflict is large, following $g _ { 0 }$ will decrease the performance on task $i$ . As observed by [41] and illustrated in Fig. 1, when $\\theta$ is near a steep “valley\", where a specific task’s gradient dominates the update, manipulating the direction and magnitude of $g _ { 0 }$ often leads to better optimization. ",
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+    {
+        "type": "text",
+        "text": "2.2 Prior Attempts and Convergence Issues ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Several methods have been proposed to manipulate the task gradients to form a new update vector and have shown improved performance on MTL. Sener et al. apply the multiple-gradient descent algorithm (MGDA) [6] for MTL, which directly optimizes towards the Pareto set [30]. Chen et al. dynamically re-weight each $L _ { i }$ using a pre-defined heuristic [3]. More recently, PCGrad identifies conflicting gradients as the motivation behind manipulating the gradients and projects each task gradient to the normal plane of others to reduce the conflict [41]. While all these methods have shown success at improving the learning performance of MTL, they manipulate the gradient without respecting the original objective (1). Therefore, these methods could in principle converge to any point in the Pareto set (See Fig. 1 and Sec. 3.2). We provide the detailed algorithms of MGDA and PCGrad in Appendix A.1 and A.2, and a visualization of the update vector by each method in Fig. 2. ",
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+        "type": "text",
+        "text": "3 Method ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We introduce our main algorithm, Conflict-Averse Gradient descent in Sec. 3.1, and then show theoretical analysis in Sec. 3.2. ",
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+        "type": "text",
+        "text": "3.1 Conflict-Averse Gradient Descent ",
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+        "type": "text",
+        "text": "Assume we update $\\theta$ by $\\theta ^ { \\prime }  \\theta - \\alpha d$ , where $\\alpha$ is a step size and $d$ an update vector. We want to choose $d$ to decrease not only the average loss $L _ { 0 }$ , but also every individual loss. To do so, we consider the minimum decrease rate across the losses, ",
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+        "img_path": "images/5fa1a63842e5aace12db9343eb021e68b73c9a9e847a4628ef2e468de1fd3b96.jpg",
+        "text": "$$\nR ( \\theta , d ) = \\operatorname* { m a x } _ { i \\in [ K ] } \\left\\{ { \\frac { 1 } { \\alpha } } \\left( L _ { i } ( \\theta - \\alpha d ) - L _ { i } ( \\theta ) \\right) \\right\\} \\approx - \\operatorname* { m i n } _ { i \\in [ K ] } \\langle g _ { i } , d \\rangle ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where we use the first-order Taylor approximation assuming $\\alpha$ is small. If $R ( \\theta , d ) < 0$ , it means that all losses are decreased with the update given a sufficiently small $\\alpha$ . Therefore, $R ( \\theta , d )$ can be regarded as a measurement of conflict among objectives. ",
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+        "type": "text",
+        "text": "With the above measurement, our algorithm finds an update vector that minimizes such conflict to mitigate the optimization challenge while still converging to an optimum of the main objective $L _ { 0 } ( \\theta )$ . To this end, we introduce Conflict-Averse Gradient descent (CAGrad), which on each optimization step determines the update $d$ by solving the following optimization problem: ",
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+        "img_path": "images/adfaf4eaa2fcef74c38ef4bcec42646e25a369f72db8a89fa9d92aa8de22a9d2.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { d \\in \\mathbb { R } ^ { m } } \\operatorname* { m i n } _ { i \\in [ K ] } \\langle g _ { i } , d \\rangle \\quad \\mathrm { s . t . } \\quad \\left\\| d - g _ { 0 } \\right\\| \\leq c \\left\\| g _ { 0 } \\right\\| ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Here, $c \\in [ 0 , 1 )$ is a pre-specified hyper-parameter that controls the convergence rate (See Sec. 3.2). The optimization problem (3) looks for the best update vector within a local ball centered at the averaged gradient $g _ { 0 }$ , which also minimizes the conflict in losses measured by (2). Since we focus on MTL and choose the average loss as the main objective, $g _ { 0 }$ is the average gradient. However, CAGrad also applies when $g _ { 0 }$ is the gradient of some other user-specified objective. We leave exploring this possibility as a future direction. ",
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+    {
+        "type": "text",
+        "text": "Dual Objective The optimization problem (3) involves decision variable $d$ that has the same dimension as the number of parameters in $\\theta$ , which could be millions for a deep neural network. It is not practical to directly solve for $d$ on every optimization step. However, the dual problem of Eq. (3), as we will derive in the following, only involves solving for a decision variable $w \\in \\mathbb { R } ^ { K }$ , which can be efficiently found using standard optimization libraries [7]. Specifically, first note that $\\begin{array} { r } { \\operatorname* { m i n } _ { i } \\langle g _ { i } , d \\rangle = \\operatorname* { m i n } _ { w \\in \\mathcal { W } } \\langle \\sum _ { i } w _ { i } \\bar { g } _ { i } , d \\rangle } \\end{array}$ , where $w = ( w _ { 1 } , \\dots , w _ { K } ) \\in \\mathbb { R } ^ { K }$ and $\\mathcal { W }$ denotes the probability simplex, i.e. $\\begin{array} { r } { \\mathcal { W } = \\{ w : \\sum _ { i } w _ { i } = 1 } \\end{array}$ and $w _ { i } ~ \\geq ~ 0 \\}$ . Denote $\\begin{array} { r } { g _ { w } \\ = \\ \\sum _ { i } w _ { i } g _ { i } } \\end{array}$ and $\\phi = c ^ { 2 } \\left\\| g _ { 0 } \\right\\| ^ { 2 }$ . The Lagrangian of the objective in Eq. (3) is ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/249471cc392a3bd1a1fd38e6bd87e15f6b3194218e730715b42a11c263411e31.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { d \\in \\mathbb { R } ^ { m } } \\operatorname* { m i n } _ { \\lambda \\geq 0 , w \\in \\mathcal { W } } g _ { w } ^ { \\top } d - \\lambda ( \\left. g _ { 0 } - d \\right. ^ { 2 } - \\phi ) / 2 .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Since the objective for $d$ is concave with linear constraints, by switching the min and max, we reach the dual form without changing the solution by Slater’s condition: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/d18fbd86da08bc545247a9432cc748f5e248533a2017b8897c4e9627008fbc26.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\lambda \\geq 0 , w \\in \\mathcal { W } } \\operatorname* { m a x } _ { d \\in \\mathbb { R } ^ { m } } g _ { w } ^ { \\top } d - \\lambda \\left. g _ { 0 } - d \\right. ^ { 2 } / 2 + \\lambda \\phi / 2 .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "table",
+        "img_path": "images/d8fa599614866baf4fa4f8bc66c07829dd06caa02ff9c914b65f62782fb2a1a8.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Input: Initial model parameter vector 0o,differentiable loss functions {Li}=1,a constant c ∈ [0,1) and learning rate α ∈ R+.</td></tr><tr><td>repeat</td></tr><tr><td>At the t-th optimization step,define go = k∑i-1 VLi(θt-1) and  =c² /ol²2. 1K Solve</td></tr><tr><td>K min, F(w) := gg +√Φ|lgull，where gω = 1 M wiVLi(0t-1).</td></tr><tr><td>K wEW i=1</td></tr><tr><td></td></tr><tr><td>Update 0t = 0t-1-α (go+ i 1/2 9w). 1igwll</td></tr><tr><td>until convergence</td></tr><tr><td></td></tr></table>",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We end up with the following optimization problem w.r.t. $w$ after several steps of calculus, ",
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/c587f0ba73a6c7d3e9cf88cb9993a883ee9e688ec81768c6fc9a13ea42b183e0.jpg",
+        "text": "$$\n\\boldsymbol { w } ^ { * } = \\underset { \\boldsymbol { w } \\in \\mathcal { W } } { \\arg \\operatorname* { m i n } } \\ : g _ { \\boldsymbol { w } } ^ { \\top } \\boldsymbol { g } _ { 0 } + \\sqrt { \\phi } \\left\\| \\boldsymbol { g } _ { \\boldsymbol { w } } \\right\\| ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            385,
+            315,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where the optimal $\\lambda ^ { * } = \\| g _ { w ^ { * } } \\| / \\phi ^ { 1 / 2 }$ and the optimal update $d ^ { * } = g _ { 0 } + g _ { w ^ { * } } / \\lambda ^ { * }$ . The detailed derivation is provided in Appendix A.3 and the entire CAGrad algorithm is summarized in Alg. 1. The dimension of $w$ equals to the number of objectives $K$ , which usually ranges from 2 to tens and is much smaller than the number of parameters in a neural network. Therefore, in practice, we solve the dual objective to perform the update of CAGrad. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Remark In Alg. 1, when $c = 0$ , CAGrad recovers the typical gradient descent with $d = g _ { 0 }$ . On the other hand, when $c \\to \\infty$ , then minimizing $F ( w )$ is equivalent to $\\operatorname* { m i n } _ { w } \\left\\| g _ { w } \\right\\|$ . This coincides with the multiple gradient descent algorithm (MGDA) [6], which uses the minimum norm vector in the convex hull of the individual gradients as the update direction (see Fig. 2; second column). MGDA is a gradient-based multi-objective optimization designed to converge to an arbitrary point on the Pareto set, that is, it leaves all the points on the Pareto set as fixed points (and hence can not control which specific point it will converge to). It is different from our method which targets to minimize $L _ { 0 }$ while using gradient conflict to regularize the optimization trajectory. As we will analyze in the following section, to guarantee that CAGrad converges to an optimum of $L _ { 0 } ( \\theta )$ , we have to ensure $0 \\leq c < 1$ ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 Convergence Analysis ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this section we first formally introduce the related Pareto concepts and then analyze CAGrad’s convergence property. Particularly, in Alg. 1, when $c < 1$ , CAGrad is guaranteed to converge to a minimum point of the average loss $L _ { 0 }$ . ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Pareto Concepts Unlike single task learning where any two parameter vectors $\\theta _ { 1 }$ and $\\theta _ { 2 }$ can be ordered in the sense that either $L ( \\theta _ { 1 } ) \\leq L ( \\mathsf { \\bar { \\theta } } _ { 2 } )$ or $L ( \\theta _ { 1 } ) \\ge L \\bar { ( \\theta _ { 2 } ) }$ holds, MTL could have two parameter vectors where one performs better for task $i$ and the other performs better for task $j \\neq i$ To this end, we need the notion of Pareto-optimality [13]. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Definition 3.1 (Pareto optimal and stationary points). Let $\\pmb { L } ( \\theta ) = \\{ L _ { i } ( \\theta ) \\colon i \\in [ K ] \\}$ be a set of differentiable loss functions from $\\mathbb { R } ^ { m }$ to $\\mathbb { R }$ . For two points $\\theta , \\theta ^ { \\prime } \\in \\mathbb { R } ^ { m }$ , we say that $\\theta$ is Pareto dominated by $\\theta ^ { \\prime }$ , denoted by $\\mathbf { } \\cdot \\mathbf { L } ( \\theta ^ { \\prime } ) \\prec L ( \\theta )$ , if $L _ { i } ( \\theta ^ { \\prime } ) \\leq L _ { i } ( \\theta )$ for all $i \\in [ K ]$ and $\\pmb { L } ( \\theta ^ { \\prime } ) \\neq \\pmb { L } ( \\theta )$ . A point $\\theta \\in \\mathbb { R } ^ { m }$ is said to be Pareto-optimal if there exists no $\\theta ^ { \\prime } \\in \\mathbb { R } ^ { m }$ such that $\\mathbf { \\dot { L } } ( \\theta ^ { \\prime } ) \\prec \\dot { \\mathbf { L } } ( \\theta )$ . The set of all Pareto-optimal points is called the Pareto set. $A$ point $\\theta$ is called Pareto-stationary if we have $\\mathrm { m i n } _ { w \\in \\mathcal { W } } \\| g _ { w } ( \\theta ) \\| = 0$ , where $\\begin{array} { r } { g _ { w } ( \\theta ) = \\sum _ { i = 1 } ^ { K } w _ { i } \\nabla L _ { i } ( \\theta ) } \\end{array}$ , and $\\mathcal { W }$ is the probability simplex on $[ K ]$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Similar to the case of single-objective differentiable optimization, a local Pareto optimal point $\\theta$ must be Pareto stationary (see e.g., [6]). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Theorem 3.2 (Convergence of CAGrad). Assume the individual loss functions $L _ { 0 } , L _ { 1 } , \\dots , L _ { K }$ are differentiable on $\\mathbb { R } ^ { m }$ and their gradients $\\nabla L _ { i } ( \\theta )$ are all $H$ -Lipschitz, i.e. $\\lVert \\nabla L _ { i } ( x ) - \\nabla L _ { i } ( y ) \\rVert \\leq$ $H \\parallel x - y \\parallel$ for $i = 0 , 1 , \\ldots , K$ where $0 \\leq H \\leq \\infty$ . Assume $L _ { 0 } ^ { * } = \\operatorname* { i n f } _ { \\theta \\in \\mathbb { R } ^ { m } } \\dot { L } _ { 0 } ( \\theta ) > - \\infty$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "With a fixed step size $\\alpha$ satisfying $0 < \\alpha \\leq 1 / H$ , we have for the CAGrad in Alg. 1: ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/5c9c3173a60de5bc0f95c0dd8cfdd3adf5f1660683b1fd79cd70859bb92f45e3.jpg",
+        "image_caption": [
+            "Figure 2: The combined update vector $d$ (in red) of a two-task learning problem with gradient descent (GD), multiple gradient descent algorithm (MGDA), PCGrad and Conflict-Averse Gradient descent (CAGrad). The two task-specific gradients are labeled $g _ { 1 }$ and $g _ { 2 }$ . MGDA’s objective is given in its primal form (See Appendix A.1). For PCGrad, each gradient is first projected onto the normal plane of the other (the dashed arrows). Then the final update vector is the average of the two projected gradients. CAGrad finds the best update vector within a ball around the average gradient that maximizes the worse local improvement between task 1 and task 2. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "1) For any $c \\geq 1$ , all the fixed points of CAGrad are Pareto-stationary points of $( L _ { 0 } , L _ { 1 } , \\ldots , L _ { K } )$ ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "2) In particular, if we take $0 \\leq c < 1$ , then CAGrad satisfies ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b00bc7b869378e60e2444cbe7d40e296e579e088de0c6b3e6be3801936505574.jpg",
+        "text": "$$\n\\sum _ { t = 0 } ^ { T } \\left. \\nabla L _ { 0 } ( \\theta _ { t } ) \\right. ^ { 2 } \\leq \\frac { 2 ( L _ { 0 } ( \\theta _ { 0 } ) - L _ { 0 } ^ { * } ) } { \\alpha ( 1 - c ^ { 2 } ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "This means that the algorithm converges to a stationary point of $\\nabla L _ { 0 }$ if we take $0 \\leq c < 1$ . The proof is in Appendix A.3. As we discuss earlier, unlike our method, MGDA is designed to converge to an arbitrary point on the Pareto set, without explicit control of which point it will converges to. Another algorithm with similar property is PCGrad [41], which is a gradient-based algorithm that mitigates the conflicting gradients problem by removing the conflicting components of each gradient with respect to the other gradients before averaging them to form the final update; see Fig. 2, third column for the illustration. Similar to MGDA, as shown in [41], PCGrad also converges to an arbitrary Pareto point without explicit control of which point it will arrive at. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 Practical Speedup ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "A typical drawback of methods that manipulate gradients is the computation overhead. For computing the optimal update vector, a method usually requires $K$ back-propagations to find all individual gradients $g _ { i }$ , in addition to the time required for optimization. This can be prohibitive for the scenario with many tasks. To this end, we propose to only sample a subset of tasks $S \\subseteq [ K ]$ , compute their corresponding gradients $\\{ g _ { i } \\mid i \\in { \\overline { { S } } } \\}$ and the averaged gradient $g _ { 0 }$ . Then we optimize $d$ in: ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "equation",
+        "img_path": "images/0ac10dde2ee514b0fda7a275ea05546939e1631abb1526d2ce1861fe4df1a6d2.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { d \\in \\mathbb { R } ^ { m } } \\operatorname* { m i n } \\left( \\langle \\frac { K g _ { 0 } - \\sum _ { i \\in S } g _ { i } } { K - | S | } , d \\rangle , \\ : \\ : \\operatorname* { m i n } _ { i \\in S } \\langle g _ { i } , d \\rangle \\right) \\mathrm { s . t . } \\| d - g _ { 0 } \\| \\leq c \\| g _ { 0 } \\|\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Remark Note that the convergence guarantee in Thm. 3.2 still holds for Eq. 4 as the constraint does not change (See Appendix A.3). The time complexity is $\\mathcal { O } ( ( \\vert S \\vert N + T )$ , where $N$ denotes the time for one pass of back-propagation and $T$ denotes the optimization time. For few-task learning $K < 1 0$ ), usually $T \\ll N$ . When $S = [ K ]$ , we recover the full CAGrad algorithm. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 Related Work ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Multi-task Learning Due to its benefit with regards to data and computational efficiency, multi-task learning (MTL) has broad applications in vision, language, and robotics [11, 28, 22, 44, 38]. A number of MTL-friendly architectures have been proposed using task-specific modules [25, 11], attentionbased mechanisms [21] or activating different paths along the deep networks to tackle MTL [27, 40]. Apart from designing new architectures, another branch of methods focus on decomposing a large problem into smaller local problems that could be quickly learned by smaller models [29, 26, 37, 8]. Then a unified policy is learned from the smaller models using knowledge distillation [12]. ",
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+    {
+        "type": "text",
+        "text": "MTL Optimization In this work, we focus on the optimization challenge of MTL [38]. Gradient manipulation methods are designed specifically to balance the learning of each task. The simplest form of gradient manipulation is to re-weight the task losses based on specific criteria, e.g., uncertainty [15], gradient norm [3], or difficulty [9]. These methods are mostly heuristics and their performance can be unstable. Recently, two methods [30, 41] that manipulate the gradients to find a better local update vector have become popular. Sener et al [30] view MTL as a multi-objective optimization problem, and use multiple gradient descent algorithm for optimization. PCGrad [41] identifies a major optimization challenge for MTL, the conflicting gradients, and proposes to project each task gradient to the normal plane of other task gradients before combining them together to form the final update vector. Though yielding good empirical performance, both methods can only guarantee convergence to a Pareto-stationary point, but not knowing where it exactly converges to. More recently, GradDrop [4] randomly drops out task gradients based on how much they conflict. IMTLG [20] seeks an update vector that has equal projections on each task gradient. RotoGrad [14] separately scales and rotates task gradients to mitigate optimization conflict. ",
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+    {
+        "type": "text",
+        "text": "Our method, CAGrad, also manipulates the gradient to find a better optimization trajectory. Like other MTL optimization techniques, CAGrad is model-agnostic. However, unlike prior methods, CAGrad converges to the optimal point in theory and achieves better empirical performance on both toy multi-objective optimization tasks and real-world applications. ",
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+    {
+        "type": "text",
+        "text": "5 Experiment ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We conduct experiments to answer the following questions: ",
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+    {
+        "type": "text",
+        "text": "Question (1) Do CAGrad, MGDA and PCGrad behave consistently with their theoretical properties in practice? (yes) ",
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+    {
+        "type": "text",
+        "text": "Question (2) Does CAGrad recover GD and MGDA when varying the constant $c ?$ (yes) ",
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+    {
+        "type": "text",
+        "text": "Question (3) How does CAGrad perform in both performance and computational efficiency compared to prior state-of-the-art methods, on challenging multi-task learning problems under the supervised, semi-supervised and reinforcement learning settings? (CAGrad improves over prior state-of-the-art methods under all settings) ",
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+    {
+        "type": "text",
+        "text": "5.1 Convergence and Ablation over c ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To answer questions (1) and (2), we create a toy optimization example to evaluate the convergence of CAGrad compared to MGDA and PCGrad. On the same toy example, we ablate over the constant $c$ and show that CAGrad recovers GD and MGDA with proper $c$ values. Next, to test CAGrad on more complicated neural models, we perform the same set of experiments on the Multi-Fashion+MNIST benchmark [19] with a shrinked LeNet architecture [18] (in which each layer has a reduced number of neurons compared to the original LeNet). Please refer to Appendix B for more details. ",
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+    {
+        "type": "text",
+        "text": "For the toy optimization example, we modify the toy example used by $\\mathrm { Y u }$ et al. [41] and consider $\\theta = ( \\theta _ { 1 } , \\bar { \\theta _ { 2 } } ) \\overset { \\cdot } { \\in } \\mathbb { R } ^ { 2 }$ with the following individual loss functions: ",
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+    {
+        "type": "equation",
+        "img_path": "images/d08e3254ebff6bc055148a96629a8ca033e870653a63ba79bada58df7e636f9f.jpg",
+        "text": "$$\n\\begin{array} { r l } & { L _ { 1 } ( \\theta ) = c _ { 1 } ( \\theta ) f _ { 1 } ( \\theta ) + c _ { 2 } ( \\theta ) g _ { 1 } ( \\theta ) \\mathrm { ~ a n d ~ } L _ { 2 } ( \\theta ) = c _ { 1 } ( \\theta ) f _ { 2 } ( \\theta ) + c _ { 2 } ( \\theta ) g _ { 2 } ( \\theta ) , \\forall } \\\\ & { f _ { 1 } ( \\theta ) = \\log \\big ( \\operatorname* { m a x } ( | 0 . 5 ( - \\theta _ { 1 } - 7 ) - \\operatorname { t a n h } { ( - \\theta _ { 2 } ) } | , \\ 0 . 0 0 0 0 0 5 ) \\big ) + 6 , } \\\\ & { f _ { 2 } ( \\theta ) = \\log \\big ( \\operatorname* { m a x } ( | 0 . 5 ( - \\theta _ { 1 } + 3 ) - \\operatorname { t a n h } { ( - \\theta _ { 2 } ) } + 2 | , \\ 0 . 0 0 0 0 0 5 ) \\big ) + 6 , } \\\\ & { g _ { 1 } ( \\theta ) = \\big ( ( - \\theta _ { 1 } + 7 ) ^ { 2 } + 0 . 1 * ( - \\theta _ { 2 } - 8 ) ^ { 2 } \\big ) / 1 0 - 2 0 , } \\\\ & { g _ { 2 } ( \\theta ) = \\big ( ( - \\theta _ { 1 } - 7 ) ^ { 2 } + 0 . 1 * ( - \\theta _ { 2 } - 8 ) ^ { 2 } \\big ) / 1 0 - 2 0 , } \\\\ & { c _ { 1 } ( \\theta ) = \\operatorname* { m a x } ( \\operatorname { t a n h } { ( 0 . 5 * \\theta _ { 2 } ) } , \\ 0 ) \\mathrm { ~ a n d ~ } c _ { 2 } ( \\theta ) = \\operatorname* { m a x } ( \\operatorname { t a n h } { ( - 0 . 5 * \\theta _ { 2 } ) } , \\ 0 ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The average loss $L _ { 0 }$ and individual losses $L _ { 1 }$ and $L _ { 2 }$ are shown in Fig. 1. We then pick 5 initial parameter vectors $\\theta _ { \\mathrm { i n i t } } \\in \\{ ( - 8 . 5 , 7 . 5 ) , ( - 8 . 5 , 5 ) , ( 0 , 0 ) , ( 9 , 9 ) , ( 1 0 , - 8 ) \\}$ and plot the corresponding optimization trajectories with different methods in Fig. 3. As shown in Fig. 3, GD gets stuck in 2 out of the 5 runs while other methods all converge to the Pareto set. MGDA and PCGrad converge to different Pareto-stationary points depending on $\\theta _ { \\mathrm { i n i t } }$ . CAGrad with $c = 0$ recovers GD and CAGrad with $c = 1 0$ approximates MGDA well (in theory it requires $c \\to \\infty$ to exactly recover MGDA). ",
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+    {
+        "type": "image",
+        "img_path": "images/75d2f251d4cdaccd9c44f7ec6d7b1f0f9b38469e21607d25f96a255236d70ce3.jpg",
+        "image_caption": [
+            "Figure 3: The left four plots are 5 runs of each algorithms from 5 different initial parameter vectors, where trajectories are colored from red to yellow. The right two plots are CAGrad’s results with a varying $\\bar { c ^ { \\prime } } \\in \\{ 0 , 0 . 2 , 0 . 5 , 0 . 8 , 1 0 \\}$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Next, we apply the same set of experiments on the multi-task classification benchmark MultiFashion+MNIST [19]. This benchmark consists of images that are generated by overlaying an image from FashionMNIST dataset [39] on top of another image from MNIST dataset [5]. The two images are positioned on the top-left and bottom-right separately. We consider a shrinked LeNet as our model, and train it with Adam [16] optimizer with a 0.001 learning rate for 50 epochs using a batch size of 256. Due to the highly non-convex nature of the neural network, we are not able to visualize the entire Pareto set. But we provide the final training losses of different methods over three independent runs in Fig. 4. As shown, CAGrad achieves the lowest average loss with $c = 0 . 2$ . In addition, PCGrad and MGDA focus on optimizing task 1 and task 2 separately. Lastly, CAGrad with $c = 0$ and $c = 1 0$ roughly recovers the final performance of GD and MGDA. By increasing $c _ { \\cdot }$ , the model performance shifts from more GD-like to more MGDA-like, though due to the non-convex nature of neural networks, CAGrad with $0 \\leq c < 1$ does not necessarily converge to the exact same point. ",
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+    {
+        "type": "image",
+        "img_path": "images/bee5020abe60ded51f7f3f7db9c65b38616afcdb9d451be9bb97a872fad1223b.jpg",
+        "image_caption": [
+            "Figure 4: The average and individual training losses on the Fashion-and-MNIST benchmark by running GD, MGDA, PCGrad and CAGrad with different $c$ values. GD gets stuck at the steep valley (the area with a cloud of dots), which other methods can pass. MGDA and PCGrad converge randomly on the Pareto set. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "5.2 Multi-task Supervised Learning ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To answer question (3) in the supervised learning setting, we follow the experiment setup from Yu et al. [41] and consider the NYU-v2 and CityScapes vision datasets. NYU-v2 contains 3 tasks: 13- class semantic segmentation, depth estimation, and surface normal prediction. CityScapes similarly contains 2 tasks: 7-class semantic segmentation and depth estimation. Here, we follow [41] and combine CAGrad with a state-of-the-art MTL method MTAN [21], which applies attention mechanism on top of the SegNet architecture [1]. We compare CAGrad with PCGrad, vanilla MTAN and CrossStitch [25], which is another MTL method that modifies the network architecture. MTAN originally experiments with equal loss weighting and two other dynamic loss weighting heuristics [15, 3]. For a fair comparison, all methods are applied under the equal weighting scheme and we use the same training setup from [3]. We search $\\bar { c } \\in \\{ 0 . 1 , 0 . 2 , . . . \\bar { 0 } . 9 \\}$ with the best average training loss for CAGrad on both datasets (0.4 for NYU-v2 and 0.2 for Cityscapes). We perform a two-tailed, Student’s $t$ -test under equal sample sizes, unequal variance setup and mark the results that are significant with an $^ *$ . Following Maninis et al.[24], we also compute the average per-task performance drop of method $m$ with respect to the single-tasking baseline $b$ : 1K PKi=1(−1)li (Mm,i − Mb,i)/Mb,i where $l _ { i } = 1$ if a higher value is better for a criterion $M _ { i }$ on task $i$ and 0 otherwise. The single-tasking baseline (independent) refers to training individual tasks with a vanilla SegNet. Results are shown in Tab. 1 and Tab. 2. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Given the single task performance, CAGrad performs better on the task that is overlooked by other methods (Surface Normal in NYU-v2 and Depth in CityScapes) and matches other methods’ performance on the rest of the tasks. We also provide the final test losses and the per-epoch training time of each method in Fig. 5 in Appendix B.2. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/1b4b4fc21e7177445c86b5f32798f8169b735cac85f90f474da59a13de4df985.jpg",
+        "table_caption": [
+            "Table 1: Multi-task learning results on NYU-v2 dataset. #P denotes the relative model size compared to the vanilla SegNet. Each experiment is repeated over 3 random seeds and the mean is reported. The best average result among all multi-task methods is marked in bold. MGDA, PCGrad, GradDrop and CAGrad are applied on the MTAN backbone. CAGrad has statistically significant improvement over baselines methods with an $^ *$ , tested with a $p$ -value of 0.1. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"3\">#P.</td><td rowspan=\"3\">Method</td><td colspan=\"2\">Segmentation</td><td colspan=\"2\">Depth</td><td colspan=\"5\">Surface Normal</td><td rowspan=\"3\">△m%</td></tr><tr><td colspan=\"2\">(Higher Better)</td><td colspan=\"2\">(Lower Better)</td><td colspan=\"2\">Angle Distance (Lower Better)</td><td colspan=\"2\">Within t° (Higher Better)</td></tr><tr><td>mIoU</td><td>Pix Acc</td><td>Abs Err</td><td>Rel Err</td><td>Mean Median</td><td>11.25</td><td>22.5</td><td>30</td></tr><tr><td>3</td><td>Independent</td><td>38.30</td><td>63.76</td><td>0.6754</td><td>0.2780</td><td>25.01</td><td>19.21</td><td>30.14</td><td>57.20</td><td>69.15</td><td></td></tr><tr><td>~3</td><td>Cross-Stitch [25]</td><td>37.42</td><td>63.51</td><td>0.5487</td><td>0.2188</td><td>*28.85</td><td>*24.52</td><td>*22.75</td><td>*46.58</td><td>*59.56</td><td>6.96</td></tr><tr><td>1.77</td><td>MTAN [21]</td><td>39.29</td><td>65.33</td><td>0.5493</td><td>0.2263</td><td>*28.15</td><td>*23.96</td><td>*22.09</td><td>*47.50</td><td>*61.08</td><td>5.59</td></tr><tr><td>1.77</td><td>MGDA [30]</td><td>*30.47</td><td>*59.90</td><td>*0.6070</td><td>*0.2555</td><td>24.88</td><td>19.45</td><td>29.18</td><td>56.88</td><td>69.36</td><td>1.38</td></tr><tr><td>1.77</td><td>PCGrad [41]</td><td>38.06</td><td>64.64</td><td>0.5550</td><td>0.2325</td><td>*27.41</td><td>*22.80</td><td>*23.86</td><td>*49.83</td><td>*63.14</td><td>3.97</td></tr><tr><td>1.77</td><td>GradDrop [4]</td><td>39.39</td><td>65.12</td><td>0.5455</td><td>0.2279</td><td>*27.48</td><td>*22.96</td><td>*23.38</td><td>*49.44</td><td>*62.87</td><td>3.58</td></tr><tr><td>1.77</td><td>CAGrad (ours)</td><td>39.79</td><td>65.49</td><td>0.5486</td><td>0.2250</td><td>26.31</td><td>21.58</td><td>25.61</td><td>52.36</td><td>65.58</td><td>0.20</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/b00bf52cba1934c26af0b2c54c5de722d4b0df277dca62cc44a4ec127c4e2cce.jpg",
+        "table_caption": [
+            "Table 2: Multi-task learning results on CityScapes Challenge. Each experiment is repeated over 3 random seeds and the mean is reported. The best average result among all multi-task methods is marked in bold. PCGrad and CAGrad are applied on the MTAN backbone. CAGrad has statistically significant improvement over baselines methods with an $^ *$ , tested with a $p$ -value of 0.1. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">#P.</td><td rowspan=\"2\">Method</td><td colspan=\"2\">Segmentation</td><td colspan=\"2\">Depth</td><td rowspan=\"2\">△m%</td></tr><tr><td>mIoU</td><td>(Higher Better) Pix Acc</td><td>(Lower Better) Abs Err</td><td>Rel Err</td></tr><tr><td>2</td><td>Independent</td><td>74.01</td><td>93.16</td><td>0.0125</td><td>27.77</td><td></td></tr><tr><td>~3</td><td>Cross-Stitch [25]</td><td>*73.08</td><td>*92.79</td><td>*0.0165</td><td>*118.5</td><td>90.02</td></tr><tr><td>1.77</td><td>MTAN [21]</td><td>75.18</td><td>93.49</td><td>*0.0155</td><td>*46.77</td><td>22.60</td></tr><tr><td>1.77</td><td>MGDA [30]</td><td>*68.84</td><td>*91.54</td><td>0.0309</td><td>33.50</td><td>44.14</td></tr><tr><td>1.77</td><td>PCGrad [41]</td><td>75.13</td><td>93.48</td><td>0.0154</td><td>42.07</td><td>18.29</td></tr><tr><td>1.77</td><td>GradDrop [4]</td><td>75.27</td><td>93.53</td><td>*0.0157</td><td>*47.54</td><td>23.73</td></tr><tr><td>1.77</td><td>CAGrad (ours)</td><td>75.16</td><td>93.48</td><td>0.0141</td><td>37.60</td><td>11.64</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "5.3 Multi-task Reinforcement Learning ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To answer question (3) in the reinforcement learning (RL) setting, we apply CAGrad on the MT10 and MT50 benchmarks from the Meta-World environment [42]. In particular, MT10 and MT50 contains 10 and 50 robot manipulation tasks. Following [33], we use Soft Actor-Critic (SAC) [10] as the underlying RL training algorithm. We compare against Multi-task SAC (SAC with a shared model), Multi-headed SAC (SAC with a shared backbone and task-specific head), Multi-task SAC $^ +$ Task Encoder (SAC with a shared model and the input includes a task embedding) [42] and PCGrad [41]. We also compare with Soft Modularization [40] that routes different modules in a shared model to form different policies. Lastly, we also include a recent method (CARE) that considers language metadata and uses a mixture of expert encoder for MTL. We follow the same experiment setup from [33]. The results are shown in Tab. 3. CAGrad outperforms all baselines except for CARE which benefits from extra information from the metadata. We also apply the practical speedup in Sec. 3.3 and sub-sample 4 and 8 tasks for MT10 and MT50 (CAGrad-Fast). CAGrad-fast achieves comparable performance against the state-of-the-art method while achieving a $2 \\mathbf { x }$ (MT10) and $5 \\mathbf { x }$ (MT50) speedup over PCGrad. We provide a visualization of tasks from MT10 and MT50, and the comparison of computational efficiency in Appendix B.3. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.4 Semi-supervised Learning with Auxiliary Tasks ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Training with auxiliary tasks to improve the performance of a main task is another popular application of MTL. Here, we take semi-supervised learning as an instance. We combine different optimization ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/95459a7dda17678dbbd5c8abc56480277bd94b92784a60bb74f26410edef5d56.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 3: Multi-task reinforcement learning results on the Metaworld benchmarks. Results are averaged over 10 independent runs and the best result is marked in bold. "
+        ],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td>MetaworldMT10</td><td>MetaworldMT50</td></tr><tr><td>success (mean ± stderr)</td><td>success (mean ± stderr)</td></tr><tr><td>Multi-task SAC [42]</td><td>0.49 ±0.073</td><td>0.36 ±0.013</td></tr><tr><td>Multi-task SAC+ Task Encoder [42]</td><td>0.54 ±0.047</td><td>0.40 ±0.024</td></tr><tr><td>Multi-headed SAC [42]</td><td>0.61 ±0.036</td><td>0.45 ±0.064</td></tr><tr><td>PCGrad [41]</td><td>0.72 ±0.022</td><td>0.50 ±0.017</td></tr><tr><td>Soft Modularization [40]</td><td>0.73 ±0.043</td><td>0.50 ±0.035</td></tr><tr><td>CAGrad (ours)</td><td>0.83 ±0.045</td><td>0.52 ±0.023</td></tr><tr><td>CAGrad-Fast (ours)</td><td>0.82 ±0.039</td><td>0.50 ±0.016</td></tr><tr><td>CARE [33]</td><td>0.84 ±0.051</td><td>0.54 ±0.031</td></tr><tr><td>One SAC agent per task (upper bound)</td><td>0.90 ±0.032</td><td>0.74 ±0.041</td></tr></table>",
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+        "text": "algorithms with Auxiliary Task Reweighting for Minimum-data Learning (ARML) [31], a state-ofthe-art semi-supervised learning algorithm. The loss function is composed of the main task and two auxiliary tasks: ",
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+        "text": "$$\n\\begin{array} { r } { L _ { 0 } = L _ { C E } ( \\theta ; D _ { l } ) + w _ { 1 } L _ { a u x } ^ { 1 } ( \\theta ; D _ { u } ) + w _ { 2 } L _ { a u x } ^ { 2 } ( \\theta ; D _ { u } ) , } \\end{array}\n$$",
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+        "text": "where are au $L _ { C E }$ is the main cross-entropy classification loss on the lay unsupervised learning losses on the unlabeled dataset d dataset . We use $D _ { l }$ , and  sam $L _ { a u x } ^ { 1 } , L _ { a u x } ^ { 2 }$ $D _ { u }$ $w _ { 1 }$ $w _ { 2 }$ from ARML, and use the CIFAR10 dataset [17], which contains 50,000 training images and 10,000 test images. $10 \\%$ of the training images is held out as the validation set. We test PCGrad, MGDA and CAGrad with 500, 1000 and 2000 labeled images. The rest of the training set is used for auxiliary tasks. For all the methods, we use the same labeled dataset, the same learning rate and train them for 200 epochs with the Adam [16] optimizer. Please refer to Appendix B.4 for more experimental details. Results are shown in Tab. 4. With all the different number of labels, CAGrad yields the best averaged test accuracy. We observed that MGDA performs much worse than the ARML baseline, because it significantly overlooks the main classification task. We also compare different gradient manipulation methods on the same task with GradNorm [3], which dynamically adjusts $w _ { 1 }$ and $w _ { 2 }$ during training. The results and conclusions are similar to those for ARML. ",
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+        "table_footnote": [
+            "Table 4: Semi-supervised Learning with auxiliary tasks on CIFAR10. We report the average test accuracy over 3 independent runs for each method and mark the best result in bold. "
+        ],
+        "table_body": "<table><tr><td>Method</td><td>500 labels</td><td>1000 labels</td><td>2000 labels</td></tr><tr><td>ARML [31]</td><td>67.05 ±0.16</td><td>73.22 ±0.26</td><td>81.35 ±0.36</td></tr><tr><td>ARML + PCGrad [41]</td><td>67.49 ±0.64</td><td>73.23 ±0.62</td><td>81.91 ±0.19</td></tr><tr><td>ARML +MGDA[30]</td><td>49.27 ±0.68</td><td>60.11 ±2.35</td><td>60.78 ±0.17</td></tr><tr><td>ARML +CAGrad (Ours)</td><td>68.25 ±0.37</td><td>74.37 ±0.42</td><td>82.81 ±0.48</td></tr><tr><td>GradNorm [3]</td><td>67.35 ±0.15</td><td>73.53 ±0.23</td><td>81.03 ±0.71</td></tr><tr><td>GradNorm + PCGrad [41]</td><td>67.83 ±0.19</td><td>73.91 ±0.09</td><td>82.72 ±0.19</td></tr><tr><td>GradNorm + MGDA [30]</td><td>36.99 ±2.11</td><td>57.94 ±0.92</td><td>59.12 ±0.63</td></tr><tr><td>GradNorm + CAGrad (Ours)</td><td>67.53 ±0.26</td><td>74.72 ±0.19</td><td>83.15 ±0.56</td></tr></table>",
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+        "type": "text",
+        "text": "6 Conclusion ",
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+        "text": "In this work, we introduce the Conflict-Averse Gradient descent (CAGrad) algorithm that explicitly optimizes the minimum decrease rate of any specific task’s loss while still provably converging to the optimum of the average loss. CAGrad generalizes the gradient descent and multiple gradient descent algorithm, and demonstrates improved performance across several challenging multi-task learning problems compared to the state-of-the-art methods. While we focus mainly on optimizing the average loss, an interesting future direction is to look at main objectives other than the average loss under the multi-task setting. ",
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+        "type": "text",
+        "text": "Acknowledgements ",
+        "text_level": 1,
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+        "text": "The research was conducted in the statistical learning and AI group (SLAI) and the Learning Agents Research Group (LARG) in computer science at UT Austin. SLAI research is supported in part by CAREER-1846421, SenSE-2037267, EAGER-2041327, and Office of Navy Research, and NSF AI Institute for Foundations of Machine Learning (IFML). LARG research is supported in part by NSF (CPS-1739964, IIS-1724157, FAIN-2019844), ONR (N00014-18-2243), ARO (W911NF-19-2-0333), DARPA, Lockheed Martin, GM, Bosch, and UT Austin’s Good Systems grand challenge. Peter Stone serves as the Executive Director of Sony AI America and receives financial compensation for this work. The terms of this arrangement have been reviewed and approved by the University of Texas at Austin in accordance with its policy on objectivity in research. Xingchao Liu is supported in part by a funding from BP. ",
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+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
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+            174,
+            280,
+            266,
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+        ],
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+    },
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+        "text": "[1] Vijay Badrinarayanan, Alex Kendall, and Roberto Cipolla. Segnet: A deep convolutional encoder-decoder architecture for image segmentation. IEEE transactions on pattern analysis and machine intelligence, 39(12):2481–2495, 2017.   \n[2] Rich Caruana. Multitask learning. Machine learning, 28(1):41–75, 1997.   \n[3] Zhao Chen, Vijay Badrinarayanan, Chen-Yu Lee, and Andrew Rabinovich. Gradnorm: Gradient normalization for adaptive loss balancing in deep multitask networks. In International Conference on Machine Learning, pages 794–803. PMLR, 2018. [4] Zhao Chen, Jiquan Ngiam, Yanping Huang, Thang Luong, Henrik Kretzschmar, Yuning Chai, and Dragomir Anguelov. Just pick a sign: Optimizing deep multitask models with gradient sign dropout. arXiv preprint arXiv:2010.06808, 2020.   \n[5] Li Deng. The mnist database of handwritten digit images for machine learning research. IEEE Signal Processing Magazine, 29(6):141–142, 2012.   \n[6] Jean-Antoine Désidéri. Multiple-gradient descent algorithm (mgda) for multiobjective optimization. Comptes Rendus Mathematique, 350(5-6):313–318, 2012. [7] Steven Diamond and Stephen Boyd. CVXPY: A Python-embedded modeling language for convex optimization. Journal of Machine Learning Research, 17(83):1–5, 2016. [8] Dibya Ghosh, Avi Singh, Aravind Rajeswaran, Vikash Kumar, and Sergey Levine. Divide-andconquer reinforcement learning. arXiv preprint arXiv:1711.09874, 2017.   \n[9] Michelle Guo, Albert Haque, De-An Huang, Serena Yeung, and Li Fei-Fei. Dynamic task prioritization for multitask learning. In Proceedings of the European Conference on Computer Vision (ECCV), pages 270–287, 2018.   \n[10] Tuomas Haarnoja, Aurick Zhou, Pieter Abbeel, and Sergey Levine. Soft actor-critic: Offpolicy maximum entropy deep reinforcement learning with a stochastic actor. In International Conference on Machine Learning, pages 1861–1870. PMLR, 2018.   \n[11] Kazuma Hashimoto, Caiming Xiong, Yoshimasa Tsuruoka, and Richard Socher. A joint manytask model: Growing a neural network for multiple nlp tasks. arXiv preprint arXiv:1611.01587, 2016.   \n[12] Geoffrey Hinton, Oriol Vinyals, and Jeff Dean. Distilling the knowledge in a neural network. arXiv preprint arXiv:1503.02531, 2015.   \n[13] Harold M Hochman and James D Rodgers. Pareto optimal redistribution. The American economic review, 59(4):542–557, 1969.   \n[14] Adrián Javaloy and Isabel Valera. Rotograd: Dynamic gradient homogenization for multi-task learning. arXiv preprint arXiv:2103.02631, 2021.   \n[15] Alex Kendall, Yarin Gal, and Roberto Cipolla. Multi-task learning using uncertainty to weigh losses for scene geometry and semantics. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 7482–7491, 2018.   \n[16] Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.   \n[17] Alex Krizhevsky, Geoffrey Hinton, et al. Learning multiple layers of features from tiny images. 2009.   \n[18] Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998.   \n[19] Xi Lin, Hui-Ling Zhen, Zhenhua Li, Qingfu Zhang, and Sam Kwong. Pareto multi-task learning. arXiv preprint arXiv:1912.12854, 2019.   \n[20] Liyang Liu, Yi Li, Zhanghui Kuang, Jing-Hao Xue, Yimin Chen, Wenming Yang, Qingmin Liao, and Wayne Zhang. Towards impartial multi-task learning. In International Conference on Learning Representations, 2020.   \n[21] Shikun Liu, Edward Johns, and Andrew J Davison. End-to-end multi-task learning with attention. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 1871–1880, 2019.   \n[22] Xingchao Liu, Xing Han, Na Zhang, and Qiang Liu. Certified monotonic neural networks. arXiv preprint arXiv:2011.10219, 2020.   \n[23] Debabrata Mahapatra and Vaibhav Rajan. Multi-task learning with user preferences: Gradient descent with controlled ascent in pareto optimization. In International Conference on Machine Learning, pages 6597–6607. PMLR, 2020.   \n[24] Kevis-Kokitsi Maninis, Ilija Radosavovic, and Iasonas Kokkinos. Attentive single-tasking of multiple tasks. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 1851–1860, 2019.   \n[25] Ishan Misra, Abhinav Shrivastava, Abhinav Gupta, and Martial Hebert. Cross-stitch networks for multi-task learning. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 3994–4003, 2016.   \n[26] Emilio Parisotto, Jimmy Lei Ba, and Ruslan Salakhutdinov. Actor-mimic: Deep multitask and transfer reinforcement learning. arXiv preprint arXiv:1511.06342, 2015.   \n[27] Clemens Rosenbaum, Tim Klinger, and Matthew Riemer. Routing networks: Adaptive selection of non-linear functions for multi-task learning. arXiv preprint arXiv:1711.01239, 2017.   \n[28] Sebastian Ruder. An overview of multi-task learning in deep neural networks. arXiv preprint arXiv:1706.05098, 2017.   \n[29] Andrei A Rusu, Sergio Gomez Colmenarejo, Caglar Gulcehre, Guillaume Desjardins, James Kirkpatrick, Razvan Pascanu, Volodymyr Mnih, Koray Kavukcuoglu, and Raia Hadsell. Policy distillation. arXiv preprint arXiv:1511.06295, 2015.   \n[30] Ozan Sener and Vladlen Koltun. Multi-task learning as multi-objective optimization. arXiv preprint arXiv:1810.04650, 2018.   \n[31] Baifeng Shi, Judy Hoffman, Kate Saenko, Trevor Darrell, and Huijuan Xu. Auxiliary task reweighting for minimum-data learning. Advances in Neural Information Processing Systems, 33, 2020.   \n[32] Shagun Sodhani and Amy Zhang. Mtrl - multi task rl algorithms. Github, 2021.   \n[33] Shagun Sodhani, Amy Zhang, and Joelle Pineau. Multi-task reinforcement learning with context-based representations. arXiv preprint arXiv:2102.06177, 2021.   \n[34] Charles Stein. Inadmissibility of the usual estimator for the mean of a multivariate normal distribution. In Contribution to the Theory of Statistics, pages 197–206. University of California Press, 2020.   \n[35] Kevin Swersky, Jasper Snoek, and Ryan Prescott Adams. Multi-task bayesian optimization. 2013.   \n[36] Antti Tarvainen and Harri Valpola. Mean teachers are better role models: Weight-averaged consistency targets improve semi-supervised deep learning results. In Proceedings of the 31st International Conference on Neural Information Processing Systems, pages 1195–1204, 2017.   \n[37] Yee Whye Teh, Victor Bapst, Wojciech Marian Czarnecki, John Quan, James Kirkpatrick, Raia Hadsell, Nicolas Heess, and Razvan Pascanu. Distral: Robust multitask reinforcement learning. arXiv preprint arXiv:1707.04175, 2017.   \n[38] Simon Vandenhende, Stamatios Georgoulis, Wouter Van Gansbeke, Marc Proesmans, Dengxin Dai, and Luc Van Gool. Multi-task learning for dense prediction tasks: A survey. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2021.   \n[39] Han Xiao, Kashif Rasul, and Roland Vollgraf. Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv preprint arXiv:1708.07747, 2017.   \n[40] Ruihan Yang, Huazhe Xu, Yi Wu, and Xiaolong Wang. Multi-task reinforcement learning with soft modularization. arXiv preprint arXiv:2003.13661, 2020.   \n[41] Tianhe Yu, Saurabh Kumar, Abhishek Gupta, Sergey Levine, Karol Hausman, and Chelsea Finn. Gradient surgery for multi-task learning. arXiv preprint arXiv:2001.06782, 2020.   \n[42] Tianhe Yu, Deirdre Quillen, Zhanpeng He, Ryan Julian, Karol Hausman, Chelsea Finn, and Sergey Levine. Meta-world: A benchmark and evaluation for multi-task and meta reinforcement learning. In Conference on Robot Learning, pages 1094–1100. PMLR, 2020.   \n[43] Amir R Zamir, Alexander Sax, William Shen, Leonidas J Guibas, Jitendra Malik, and Silvio Savarese. Taskonomy: Disentangling task transfer learning. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 3712–3722, 2018.   \n[44] Yu Zhang and Qiang Yang. A survey on multi-task learning. IEEE Transactions on Knowledge and Data Engineering, 2021. ",
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+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] See Sec. 3.2 for the convergence analysis, Fig. 1 for the challenges faced by previous methods, and Sec. 5 for empirical evaluation of these challenges and the advantage of CAGrad.   \n(b) Did you describe the limitations of your work? [Yes] See Sec. 6. Currently we mainly focus on optimizing the average loss, which could be replaced by other main objectives.   \n(c) Did you discuss any potential negative societal impacts of your work? [N/A] Our method does not have potential negative societal impacts.   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
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+        "text": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] The assumptions are stated in Thm. 3.2.   \n(b) Did you include complete proofs of all theoretical results? [Yes] The complete proof is included in Appendix A.3. ",
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+        "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] We mention most of the details to reproduce the result in Sec. 5 and provide the rest of details of each experiment in Appendix.B. The code comes with the supplementary material.   \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Appendix.B and Sec. 5.   \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] For each experiment except for the toy (since there is no stochasticity), we run over multiple $\\left( \\geq 3 \\right)$ seeds.   \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] We explicitly compare the computational efficiency in Fig. 5. More details on the resources are provided in the corresponding sections in Appendix.B. ",
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parse/train/wK2fDDJ5VcF/wK2fDDJ5VcF.md

@@ -0,0 +1,163 @@
+# Learning to Walk in Minutes Using Massively Parallel Deep Reinforcement Learning
+
+Nikita Rudin ETH Zurich and NVIDIA rudinn@ethz.ch
+
+David Hoeller ETH Zurich and NVIDIA dhoeller@ethz.ch
+
+Philipp Reist NVIDIA preist@nvidia.com
+
+Marco Hutter ETH Zurich mahutter@ethz.com
+
+Abstract: In this work, we present and study a training set-up that achieves fast policy generation for real-world robotic tasks by using massive parallelism on a single workstation GPU. We analyze and discuss the impact of different training algorithm components in the massively parallel regime on the final policy performance and training times. In addition, we present a novel game-inspired curriculum that is well suited for training with thousands of simulated robots in parallel. We evaluate the approach by training the quadrupedal robot ANYmal to walk on challenging terrain. The parallel approach allows training policies for flat terrain in under four minutes, and in twenty minutes for uneven terrain. This represents a speedup of multiple orders of magnitude compared to previous work. Finally, we transfer the policies to the real robot to validate the approach. We open-source our training code to help accelerate further research in the field of learned legged locomotion: https://leggedrobotics.github.io/legged_gym/.
+
+Keywords: Reinforcement Learning, Legged Robots, Sim-to-real
+
+![](images/b6480bd30beb6d1b9762af85125a32c3c8067d4548699e684f5c674891d50b92.jpg)  
+Figure 1: Thousands of robots learning to walk in simulation.
+
+# 1 Introduction
+
+Deep reinforcement learning (DRL) is proving to be a powerful tool for robotics. Tasks such as legged locomotion [1], manipulation [2], and navigation [3], have been solved using these new tools, and research continues to keep adding more and more challenging tasks to the list. The amount of data required to train a policy increases with the task complexity. For this reason, most work focuses on training in simulation before transferring to real robots. We have reached a point where multiple days or even weeks are needed to fully train an agent with current simulators. For example, OpenAI’s block reorientation task was trained for up to 14 days and their Rubik’s cube solving policy took several months to train [4]. The problem is exacerbated by the fact that deep reinforcement learning requires hyper-parameter tuning to obtain a suitable solution which requires sequentially rerunning time-consuming training. Reducing training times using massively parallel approaches such as presented here can therefore help improve the quality and time-to-deployment of DRL policies, as a training setup can be iterated on more often in the same time frame.
+
+In this paper, we examine the effects of massive parallelism for on-policy DRL algorithms and present considerations in how the standard RL formulation and the most commonly used hyperparameters should be adapted to learn efficiently in the highly parallel regime. Additionally, we present a novel game-inspired curriculum which automatically adapts the task difficulty to the performance of the policy. The proposed curriculum architecture is straightforward to implement, does not require tuning, and is well suited for the massively parallel regime. Common robotic simulators such as Mujoco [5], Bullet [6], or Raisim [7] feature efficient multi-body dynamics implementations. However, they have been developed to run on CPUs with only a reduced amount of parallelism. In this work, we use NVIDIA’s Isaac Gym simulation environment [8], which runs both the simulation and training on the GPU and is capable of simulating thousands of robots in parallel.
+
+The massively parallel training regime has been explored before [4, 9] in the context of distributed systems with a network of thousands of CPUs each running a separate instance of the simulation. The parallelization was achieved by averaging the gradients between the different workers without reducing the number of samples provided by each agent. This results in large batch sizes of millions of samples for each policy update which improves the learning dynamics, but does not optimize the overall training time. In parallel, recent works have aimed to increase the simulation throughput and reduce training times of standard DRL benchmark tasks. A framework combining parallel simulation with multi-GPU training [10] was proposed to achieve fast training using hundreds of parallel agents. In the context of visual navigation, large batch simulation has been used to increase the training throughput [11]. Furthermore, GPU accelerated physics simulation has been shown to significantly improve the training time of the Humanoid running task [12]. A differentiable simulator running on Google’s TPUs has also been shown to greatly accelerate the training of multiple tasks [13]. We build upon [10, 12] by pushing the parallelization further, optimizing the training algorithm, and applying the approach to a challenging real-world robotics task.
+
+Perceptive and dynamic locomotion for legged robots in unstructured environments is a demanding task that, until recently, had only been partially demonstrated with complex model-based approaches [14, 15]. Learning-based approaches are emerging as a promising alternative. For quadrupeds, DRL has been used to train blind policies robust to highly uneven ground [16] (12 hours of training). Perceptive locomotion over challenging terrain has been achieved by combining learning with optimal control techniques [17, 18] (82 and 88 hours of training) and recently, a fully learned approach has shown great robustness in this setting [19] (120 hours of training). Similarly, bipedal robots have also been trained to walk blindly on stairs [20] (training time not reported). With our approach we can train a perceptive policy in under 20 minutes on a single GPU, with the complexity of simto-real transfer to the hardware, which increases the performance and robustness requirements and provides clear validation of the overall approach. Training such behaviors in minutes opens up new exciting possibilities ranging from automatic tuning to customized training using scans of particular environments.
+
+# 2 Massively Parallel Reinforcement Learning
+
+Current (on-policy) reinforcement learning algorithms are divided into two parts: data collection and policy update. The policy update, which corresponds to back-propagation for neural networks, is easily performed in parallel on the GPU. Parallelizing data collection is not as straightforward. Each step consists of policy inference, simulation, reward, and observation calculation. Current popular pipelines have the simulation and reward/observation calculation computed on the CPU, making the GPU unsuitable for policy inference because of communication bottle-necks. Data transfer over PCIe is known to be the weakest link of GPU acceleration, and can be as much as 50 times slower than the GPU processing time alone [21]. Furthermore, with CPU data collection, a large amount of data must be sent to the GPU for each policy update, slowing down the overall process. Limited parallelization can be achieved by using multiple CPU cores and spawning many processes, each running the simulation for one agent. However, the number of agents is quickly limited by the number of cores and other issues such as memory usage. We explore the potential of massive parallelism with Isaac Gym’s end-to-end data collection and policy updates on the GPU, significantly reducing data copying and improving simulation throughput.
+
+# 2.1 Simulation Throughput
+
+The main factor affecting the total simulation throughput is the number of robots simulated in parallel. Modern GPUs can handle tens of thousands of parallel instructions. Similarly, IsaacGym’s PhysX engine can process thousands of robots in a single simulation and all other computations of our pipeline are vectorized to scale favorably with the number of robots. Using a single simulation with thousands of robots presents some new challenges. For example, a single common terrain mesh must be used, and it cannot be easily changed at each reset. We circumvent this problem by creating the whole mesh with all terrain types and levels tiled side by side. We change the terrain level of the robots by physically moving them on the mesh. In supplementary material, we show the computational time of different parts of the pipeline, examine how these times scale with the number of robots, and provide other techniques to optimize the simulation throughput.
+
+# 2.2 DRL Algorithm
+
+We build upon a custom implementation of the Proximal Policy Optimization (PPO) algorithm [22]. Our implementation is designed to perform every operation and store all the data on the GPU. In order to efficiently learn from thousands of robots in parallel, we perform some essential modifications to the algorithm and change some of the commonly used hyper-parameter values.
+
+# 2.2.1 Hyper-Parameters Modification
+
+In an on-policy algorithm such as PPO, a fixed policy collects a selected amount of data before doing the next policy update. This batch size, $B$ , is a crucial hyper-parameter for successful learning. With too little data, the gradients will be too noisy, and the algorithm will not learn effectively. With too much data, the samples become repetitive, and the algorithm cannot extract more information from them. These samples represent wasted simulation time and slow down the overall training. We have $B = n _ { r o b o t s } n _ { s t e p s }$ , where $ { n _ { s t e p s } }$ is the number of steps each robot takes per policy update and $n _ { r o b o t s }$ the number of robots simulated in parallel. Since we increase $n _ { r o b o t s }$ by a few orders of magnitude, we must choose a small $n _ { s t e p s }$ to keep $B$ reasonable and hence optimize training times, which is a setting that has not been extensively explored for on-policy reinforcement learning algorithms. It turns out that we can not choose $n _ { s t e p s }$ to be arbitrarily low. The algorithm requires trajectories with coherent temporal information to learn effectively. Even though, in theory, information of single steps could be used, we find that the algorithm fails to converge to the optimal solution below a certain threshold. This can be explained by the fact that we use Generalized Advantage Estimation (GAE) [23], which requires rewards from multiple time steps to be effective. For our task, we find that the algorithm struggles when we provide fewer than 25 consecutive steps, corresponding to $0 . 5 \mathrm { s }$ of simulated time. It is important to distinguish $ { n _ { s t e p s } }$ from the maximum episode length leading to a time-out and a reset, which we define as $2 0 \mathrm { s }$ . The environments are reset when they reach this maximum length and not after each iteration, meaning that a single episode can cover many policy updates. This limits the total number of robots training in parallel, and consequently, prohibits us from using the full computational capabilities of the GPU.
+
+The mini-batch size represents the size of the chunks in which the batch size is split to perform backpropagation. We find that having mini-batch sizes much larger than what is usually considered best practice is beneficial for our massively parallel use case. We use mini-batches of tens of thousands of samples and observe that it stabilizes the learning process without increasing the total training time.
+
+# 2.2.2 Reset Handling
+
+During training, the robots must be reset whenever they fall, and also after some time to keep them exploring new trajectories and terrains. The PPO algorithm includes a critic predicting an infinite horizon sum of future discounted rewards. Resets break this infinite horizon assumption and can lead to inferior critic performance if not handled carefully. Resets based on failure or reaching a goal are not a problem because the critic can predict them. However, a reset based on a time out can not be predicted (we do not provide episode time in the observations). The solution is to distinguish the two termination modes and augment the reward with the expected infinite sum of discounted future rewards in a time-out case. In other words, we bootstrap the target of the critic with its own prediction. This solution has been discussed in [24], but interestingly, this distinction is not part of the widely used Gym environment interface [25] and is ignored by popular implementations such as Stable-Baselines $[ 2 6 ] ^ { 1 }$ . After investigating multiple implementations, we conclude that this important detail is often avoided by assuming that the environments either never time out or only on the very last step of a batch collection. In our case, with few robot steps per batch, we can not make such an assumption since a meaningful episode length covers the collection of many batches. We modify the standard Gym interface to detect time-outs and implement the bootstrapping solution. In supplementary material, we show the effect of this solution on the total reward as well as the critic loss.
+
+![](images/9b48083b251353c43d2e71ff8968b79f99600e9e05461c3dd723d12ace8ceddd.jpg)  
+Figure 2: Terrain types used for training and testing in simulation. (a) Randomly rough terrain with variations of $0 . 1 \mathrm { m }$ . (b) Sloped terrain with an inclination of $2 5 \mathrm { d e g }$ . (c) Stairs with a width of $0 . 3 \mathrm { m }$ and height of $\mathrm { 0 . 2 m }$ . (d) Randomized, discrete obstacles with heights of up to $\pm 0 . 2 \mathrm { m }$ .
+
+# 3 Task Description
+
+A quadruped robot must learn to walk across challenging terrain, including uneven surfaces, slopes, stairs, and obstacles, while following base-heading and linear-velocity commands. We conduct most of the simulation and real-world deployment experiments on the ANYbotics ANYmal C robot. However, in simulation, we demonstrate the broader applicability of the approach by additionally training policies for ANYmal B, ANYmal C with an attached arm, and the Unitree A1 robots.
+
+# 3.1 Game-Inspired Curriculum
+
+The terrains are selected to be representative of real-world environments. We create five types of procedurally generated terrains presented in Fig. 2: flat, sloped, randomly rough, discrete obstacles, and stairs. The terrains are tiled squares with $8 \mathrm { m }$ sides. The robots start at the center of the terrain and are given randomized heading and velocity commands (kept constant for the duration of an episode) pushing them to walk across the terrain. Slopes and stairs are organized in pyramids to allow traversability in all directions.
+
+Previous works have shown the benefits of using an automated curriculum of task difficulty to learn complex locomotion policies [28, 29, 16]. Similarly, we find that it is essential to first train the policy on less challenging terrain before progressively increasing the complexity. We adopt a solution inspired by [16], but replace the particle filter approach with a new game-inspired automatic curriculum. All robots are assigned a terrain type and a level that represents the difficulty of that terrain. For stairs and randomized obstacles, we gradually increase the step height from $5 \mathrm { c m }$ to $2 0 \mathrm { c m }$ . Sloped terrain inclination is increased from 0 deg to 25 deg. If a robot manages to walk past the borders of its terrain, its level is increased, and at the next reset, it will start on more difficult terrain. However, if at the end of an episode it moved by less than half of the distance required by its target velocity, its level is reduced again. Robots solving the highest level are looped back to a randomly selected level to increase the diversity and avoid catastrophic forgetting. This approach has the advantage of training the robots at a level of difficulty tailored to their performance without requiring any external tuning. It adapts the difficulty level for each terrain type individually and provides us with visual and quantitative feedback on the progress of the training. When the robots have reached the final level and are evenly spread across all terrains due to looping back, we can conclude they have fully learned to solve the task.
+
+![](images/ecaf8198256af450c96e9b482c8d3d3a06909d2addf9a2530c761fb4193fd419.jpg)  
+Figure 3: 4000 robots progressing through the terrains with automatic curriculum, after 500 (top) and 1000 (bottom) policy updates. The robots start the training session on the first row (closest to the camera) and progressively reach harder terrains.
+
+The proposed curriculum structure is well suited for the massively parallel regime. With thousands of robots we can directly use their current progress in the curriculum as the distribution of the policy’s performance, and do not need learn it with a generator network [30]. Furthermore, our method doesn’t require tuning and is straightforward to implement in a parallel manner with nearzero processing cost. We remove the computational overhead of re-sampling and re-generating new terrains needed for the particle filter approach.
+
+Fig. 3 shows robots progressing through the terrains at two different stages of the training process. On complex terrain types, the robots require more training iterations to reach the highest levels. The distribution of robots after 500 iterations shows that while the policy is able to cross sloped terrains and to go down stairs, climbing stairs and traversing obstacles requires more training iterations. However, after 1000 iterations, the robots have reached the most challenging level for all terrain types and are spread across the map. We train for a total for 1500 iterations to let the policy converge to its highest performance.
+
+# 3.2 Observations, Actions, and Rewards
+
+The policy receives proprioceptive measurements of the robot as well as terrain information around the robot’s base. The observations are composed of: base linear and angular velocities, measurement of the gravity vector, joint positions and velocities, the previous actions selected by the policy, and finally, 108 measurements of the terrain sampled from a grid around the robot’s base. Each measurement is the distance from the terrain surface to the robot’s base height.
+
+The total reward is a weighted sum of nine terms, detailed in supplementary material. The main terms encourage the robot to follow the commanded velocities while avoiding undesired base velocities along other axes. In order to create a smoother, more natural motion, we also penalize joint torques, joint accelerations, joint target changes, and collisions. Contacts with the knees, shanks or between the feet and a vertical surface are considered collisions, while contacts with the base are considered crashes and lead to resets. Finally, we add an additional reward term encouraging the robot to take longer steps, which results in a more visually appealing behavior. We train a single policy with the same rewards for all terrains.
+
+The actions are interpreted as desired joint positions sent to the motors. There, a PD controller produces motor torques. In contrast to other works [16, 20], neither the reward function nor the action space has any gait-dependent elements.
+
+# 3.3 Sim-to-Real Additions
+
+In order to make the trained policies amenable for sim-to-real transfer, we randomize the friction of the ground, add noise to the observations and randomly push the robots during the episode to teach them a more stable stance. Each robot has a friction coefficient sampled uniformly in [0.5, 1.25]. The pushes happen every $1 0 \mathrm { s }$ . The robots’ base is accelerated up to $\pm 1 \mathrm { m } / \mathrm { s }$ in both $\mathbf { X }$ and y directions. The amount of noise is based on real data measured on the robot and is detailed in supplementary material.
+
+The ANYmal robot uses series elastic actuators with fairly complex dynamics, which are hard to model in simulation. For this reason and following the methodology of previous work [1], we use a neural network to compute torques from joint position commands. However, we simplify the inputs of the model. Instead of concatenating past measurements at fixed time steps and sending all of that information to a standard feed-forward network, we only provide the current measurements to an LSTM network. A potential drawback of this set-up is that the policy does not have the temporal information of the actuators as in previous work. We have experimented with various ways of providing that information through memory mechanisms for the policy but found that it does not improve the final performance.
+
+# 4 Results
+
+# 4.1 Effects of Massive Parallelism
+
+In this section, we study the effects of the number of parallel robots on the final performance of the policy. In order to use the total reward as a single representative metric, we have to remove the curriculum, otherwise a more performant policy sees its task difficulty increase and consequently a decrease in the total reward. As such, we simplify the task by reducing the maximum step size of stairs and obstacles and directly train robots on the full range of difficulties.
+
+We begin by setting a baseline with $n _ { r o b o t s } = 2 0 0 0 0$ and $n _ { s t e p s } = 5 0$ , resulting in a batch size of 1M samples. Using this very large batch size results in the best policy but at the cost of a relatively long training time.
+
+We then conduct experiments in which we increase the number of robots while keeping the batch size constant. As a result, the number of steps each robot takes per policy update decreases. In this case, the training time decreases with a higher number of robots, but the policy performance drops if that number is too high. We start from 128 robots corresponding to the level of parallelization of previous CPU implementations and increase that number up to 16384, which is close to the maximum amount of robots we could simulate on rough terrain with Isaac Gym running on a single workstation GPU.
+
+In Fig. 4, we compare these results with the baseline, which allows us to select the most favorable trade-off between policy performance and training time. We see two interesting effects at play. First, when the number of robots is too high, the performance drops sharply, which can be explained by the time horizon of each robot becoming too small. As expected, with larger batch sizes, the overall reward is higher, and the time horizon effect is shifted, meaning that we can use more robots before seeing the drop. On the other hand, below a certain threshold, we see a slow decrease in performance with fewer robots. We believe this is explained by the fact that the samples are very similar with many steps per robot because of the relatively small time steps between them. This means that for the same amount of samples, there is less diversity in the data. In other words, with a low number of robots, we are further from the standard assumption that the samples are independent and identically distributed, which seems to have a noticeable effect on the training process. In terms of training time, we see a nearly linear scaling up to 4000 robots, after which simulation throughput gains slow down. As such, we can conclude that increasing the number of robots is beneficial for both final performance and training time, but there is an upper limit on this number after which an on-policy algorithm cannot learn effectively. Increasing the batch size to values much larger than what is typically used in similar works seems highly beneficial. Unfortunately, it also scales the training time so it is a trade-off that must be balanced. From the third plot we can conclude that using 2048 to 4096 robots with a batch size of $\approx 1 0 0 k$ or $\approx 2 0 0 k$ provides the best trade-off for this specific task.
+
+![](images/b511ee0c2aca987f5fa8e39a0d14f3ee54098cf431491e0a271f167d3bdcff6f.jpg)  
+Figure 4: (a) Average and standard deviation (over 5 runs) of the total reward of an episode after 1500 policy updates for different number of robots and 3 different batch sizes. The ideal case of a batch size of 1M samples with 20000 robots is shown in red. (b) Total training time for the same experiments. (c) Reward dependency on total training time. Colors represent the number of robots, while shapes show the batch size (circles: 49152, crosses: 98304, triangles: 196608). Points in the upper left part of the graph (highlighted in green) represent the most desirable configuration.
+
+![](images/f15e005bc1e3d3db64d82f1f7796d15a227032608677590845862fc30d3d7da3.jpg)  
+Figure 5: Success rate of the tested policy on increasing terrain complexities. Robots start in the center of the terrain and are given a forward velocity command of $0 . 7 5 \mathrm { m } / \mathrm { s }$ , and a side velocity command randomized within $[ - 0 . 1 , 0 . 1 ] \mathrm { m } / \mathrm { s }$ . (a) Success rate for climbing stairs, descending stairs and traversing discrete obstacles. (b) Success rate for climbing and descending sloped terrains.
+
+![](images/4eab7c74f6cac548026b3ff28e7f10ab6832f65e03b74ce577a9f1ea90202882.jpg)  
+Figure 6: ANYmal C with a fixed arm, ANYmal B, A1 and Cassie in simulation.
+
+# 4.2 Simulation
+
+For our simulation and deployment experiments, we use a policy trained with 4096 robots and a batch size of 98304, which we train for 1500 policy updates in under 20 minutes2. We begin by measuring the performance of our trained policy in simulation. To that end, we perform robustness and traversability tests. For each terrain type, we command the robots to traverse the representative difficulty of the terrain at high forward velocity and measure the success rate. A success is defined as managing to cross the terrain while avoiding any contacts on the robot’s base. Fig. 5 shows the results for the different terrains. For stairs, we see a nearly $1 0 0 \%$ success rate for steps up to $\mathrm { 0 . 2 m }$ , which is the hardest stair difficulty we train on and close to the kinematic limits of our robot. Randomized obstacles seem to be more demanding, with the success rate decreasing steadily. We must note that in this case, the largest step is double the reported height since neighboring obstacles can have positive and negative heights. In the case of slopes, we can observe that after $2 5 \mathrm { d e g }$ the robots are not able to climb anymore but still learn to slide down with a moderate success rate.
+
+Given our relatively simple rewards and action space, the policy is free to adopt any gait and behavior. Interestingly, it always converges to a trotting gait, but there are often artifacts in the behavior, such as a dragging leg or unreasonably high or low base heights. After tuning of the reward weights, we can obtain a policy that respects all our constraints and can be transferred to the physical robot.
+
+To verify the generalizability of the approach, we train policies for multiple robots with the same set-up. We use the ANYmal C robot with a fixed robotic arm, which adds about $2 0 \%$ of additional weight, and the ANYmal B robot, which has comparable dimensions but modified kinematic and dynamic properties. In these two cases, we can retrain a policy without any modifications to the rewards or algorithm hyper-parameters and obtain a very similar performance. Next, we use the Unitree A1 robot, which has smaller dimensions, four times lower weight, and a different leg configuration. In this case, we remove the actuator model of the ANYdrive motors, reduce PD gains and the torque penalties, and change the default joint configurations. We can train a dynamic policy that learns to solve the same terrains even with the reduced size of the robot. Finally, we apply our approach to Agility Robotics’ bipedal robot Cassie. We find that an additional reward encouraging standing on a single foot is necessary to achieve a walking gait. With this addition, we are able to train the robot on the same terrains as its quadrupedal counterparts. Fig. 6 shows the different robots.
+
+![](images/f0dd8a9ddc78c32773c604659ba9f56cbd659b68c879ccbb91f00fe6ccb27b59.jpg)  
+Figure 7: Locomotion policy, trained in under $2 0 \mathrm { { m i n } }$ , deployed on the physical robot.
+
+# 4.3 Sim-to-real Transfer
+
+On the physical robot, our policy is fixed. We compute the observations from the robot’s sensors, feed them to the policy, and directly send the produced actions as target joint positions to the motors. We do not apply any additional filtering or constraint satisfaction checks. The terrain height measurements are queried from an elevation map that the robot is building from Lidar scans.
+
+Unfortunately, this height map is far from perfect, which results in a decrease in robustness between simulation and reality. We observe that these issues mainly occur at high velocities and therefore reduce the maximum linear velocity commands to $0 . 6 \mathrm { m } / \mathrm { s }$ for policies deployed on the hardware. The robot can walk up and down stairs and handles obstacles in a dynamic manner. We show samples of these experiments in Fig. 7 and in the supplementary video. To overcome issues with imperfect terrain mapping or state estimation drift, the authors of [19] implemented a teacher-student set-up, which provided outstanding robustness even in adverse conditions. As part of future work, we plan to merge the two approaches.
+
+# 5 Conclusion
+
+In this work, we demonstrated that a complex real-world robotics task can be trained in minutes with an on-policy deep reinforcement learning algorithm. Using an end-to-end GPU pipeline with thousands of robots simulated in parallel, combined with our proposed curriculum structure, we showed that the training time can be reduced by multiple orders of magnitude compared to previous work. We discussed multiple modifications to the learning algorithm and the standard hyper-parameters required to use the massively parallel regime effectively. Using our fast training pipeline, we performed many training runs, simplified the set-up, and kept only essential components. We showed that the task can be solved using simple observation and action spaces as well as relatively straightforward rewards without encouraging particular gaits or providing motion primitives.
+
+The purpose of this work is not to obtain the absolute best-performing policy with the highest robustness. For that use case, many other techniques can be incorporated into the pipeline. We aim to show that a policy can be trained in record time with our set-up while still being usable on the real hardware. We wish to shift other researchers’ perspective on the required training time for a real-world application, and hope that our work can serve as a reference for future research. We expect many other tasks to benefit from the massively parallel regime. By reducing the training time of these future robotic tasks, we can greatly accelerate the developments in this field.
+
+# Acknowledgments
+
+We would like to thank Mayank Mittal, Joonho Lee, Takahiro Miki, and Peter Werner for their valuable suggestions and help with hardware experiments as well as the Isaac Gym and PhysX teams for their continuous support.
+
+# References
+
+[1] J. Hwangbo, J. Lee, A. Dosovitskiy, D. Bellicoso, V. Tsounis, V. Koltun, and M. Hutter. Learning agile and dynamic motor skills for legged robots. Science Robotics, 4(26), 2019.   
+[2] S. Gu, E. Holly, T. Lillicrap, and S. Levine. Deep reinforcement learning for robotic manipulation with asynchronous off-policy updates. In IEEE International Conference on Robotics and Automation (ICRA), May 2017.   
+[3] G. Kahn, A. Villaflor, B. Ding, P. Abbeel, and S. Levine. Self-supervised deep reinforcement learning with generalized computation graphs for robot navigation. In IEEE International Conference on Robotics and Automation (ICRA), 2018.   
+[4] OpenAI, I. Akkaya, M. Andrychowicz, M. Chociej, M. Litwin, B. McGrew, A. Petron, A. Paino, M. Plappert, G. Powell, R. Ribas, J. Schneider, N. Tezak, J. Tworek, P. Welinder, L. Weng, Q. Yuan, W. Zaremba, and L. Zhang. Solving rubik’s cube with a robot hand, 2019.   
+[5] E. Todorov, T. Erez, and Y. Tassa. Mujoco: A physics engine for model-based control. In IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 2012.   
+[6] E. Coumans and Y. Bai. Pybullet, a python module for physics simulation for games, robotics and machine learning. http://pybullet.org, 2016–2021.   
+[7] J. Hwangbo, J. Lee, and M. Hutter. Per-contact iteration method for solving contact dynamics. IEEE Robotics and Automation Letters, 3(2), 2018. URL www.raisim.com.   
+[8] V. Makoviychuk, L. Wawrzyniak, Y. Guo, M. Lu, K. Storey, M. Macklin, D. Hoeller, N. Rudin, A. Allshire, A. Handa, and G. State. Isaac gym: High performance GPU based physics simulation for robot learning. In Conference on Neural Information Processing Systems (NeurIPS) Datasets and Benchmarks Track, 2021.   
+[9] N. Heess, D. TB, S. Sriram, J. Lemmon, J. Merel, G. Wayne, Y. Tassa, T. Erez, Z. Wang, S. M. A. Eslami, M. A. Riedmiller, and D. Silver. Emergence of locomotion behaviours in rich environments. CoRR, abs/1707.02286, 2017.   
+[10] A. Stooke and P. Abbeel. Accelerated methods for deep reinforcement learning. CoRR, abs/1803.02811, 2018.   
+[11] B. Shacklett, E. Wijmans, A. Petrenko, M. Savva, D. Batra, V. Koltun, and K. Fatahalian. Large batch simulation for deep reinforcement learning. In International Conference on Learning Representations (ICLR), 2021.   
+[12] J. Liang, V. Makoviychuk, A. Handa, N. Chentanez, M. Macklin, and D. Fox. Gpu-accelerated robotic simulation for distributed reinforcement learning. In Conference on Robot Learning (CoRL), 2018.   
+[13] C. D. Freeman, E. Frey, A. Raichuk, S. Girgin, I. Mordatch, and O. Bachem. Brax - a differentiable physics engine for large scale rigid body simulation. In 35th Conference on Neural Information Processing Systems (NeurIPS) Datasets and Benchmarks Track, 2021.   
+[14] A. Bouman, M. F. Ginting, N. Alatur, M. Palieri, D. D. Fan, T. Touma, T. Pailevanian, S.- K. Kim, K. Otsu, J. Burdick, and A.-a. Agha-Mohammadi. Autonomous spot: Long-range autonomous exploration of extreme environments with legged locomotion. In IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 2020.   
+[15] C. Gehring, P. Fankhauser, L. Isler, R. Diethelm, S. Bachmann, M. Potz, L. Gerstenberg, and M. Hutter. Anymal in the field: Solving industrial inspection of an offshore hvdc platform with a quadrupedal robot. In Field and Service Robotics, 2021.   
+[16] J. Lee, J. Hwangbo, L. Wellhausen, V. Koltun, and M. Hutter. Learning quadrupedal locomotion over challenging terrain. Science Robotics, 5(47), 2020.   
+[17] V. Tsounis, M. Alge, J. Lee, F. Farshidian, and M. Hutter. Deepgait: Planning and control of quadrupedal gaits using deep reinforcement learning. IEEE Robotics and Automation Letters, PP, 03 2020.   
+[18] S. Gangapurwala, M. Geisert, R. Orsolino, M. Fallon, and I. Havoutis. Real-time trajectory adaptation for quadrupedal locomotion using deep reinforcement learning. In IEEE International Conference on Robotics and Automation (ICRA), 2021.   
+[19] T. Miki, J. Lee, L. Wellhausen, V. Koltun, and M. Hutter. Wild anymal: Robust zero-shot perceptive locomotion. Submitted to Science Robotics, 2021.   
+[20] J. Siekmann, K. Green, J. Warila, A. Fern, and J. W. Hurst. Blind bipedal stair traversal via sim-to-real reinforcement learning. CoRR, abs/2105.08328, 2021.   
+[21] C. Gregg and K. Hazelwood. Where is the data? why you cannot debate cpu vs. gpu performance without the answer. In IEEE International Symposium on Performance Analysis of Systems and Software (ISPASS), 2011.   
+[22] J. Schulman, F. Wolski, P. Dhariwal, A. Radford, and O. Klimov. Proximal policy optimization algorithms. CoRR, abs/1707.06347, 2017.   
+[23] J. Schulman, P. Moritz, S. Levine, M. Jordan, and P. Abbeel. High-dimensional continuous control using generalized advantage estimation. In Proceedings of the International Conference on Learning Representations (ICLR), 2016.   
+[24] F. Pardo, A. Tavakoli, V. Levdik, and P. Kormushev. Time limits in reinforcement learning. CoRR, abs/1712.00378, 2017.   
+[25] G. Brockman, V. Cheung, L. Pettersson, J. Schneider, J. Schulman, J. Tang, and W. Zaremba. Openai gym, 2016.   
+[26] A. Hill, A. Raffin, M. Ernestus, A. Gleave, A. Kanervisto, R. Traore, P. Dhariwal, C. Hesse, O. Klimov, A. Nichol, M. Plappert, A. Radford, J. Schulman, S. Sidor, and Y. Wu. Stable baselines. https://github.com/hill-a/stable-baselines, 2018.   
+[27] J. Achiam. Spinning up in deep reinforcement learning, 2018. URL https://spinningup. openai.com/en/latest/.   
+[28] R. Wang, J. Lehman, J. Clune, and K. O. Stanley. Paired open-ended trailblazer (POET): endlessly generating increasingly complex and diverse learning environments and their solutions. CoRR, abs/1901.01753, 2019.   
+[29] Z. Xie, H. Y. Ling, N. H. Kim, and M. van de Panne. Allsteps: Curriculum-driven learning of stepping stone skills. Proceedings of ACM SIGGRAPH / Eurographics Symposium on Computer Animation, 2020.   
+[30] C. Florensa, D. Held, X. Geng, and P. Abbeel. Automatic goal generation for reinforcement learning agents. In Proceedings of the 35th International Conference on Machine Learning (ICML), volume 80 of Proceedings of Machine Learning Research, 2018.

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+        "text": "Learning to Walk in Minutes Using Massively Parallel Deep Reinforcement Learning ",
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+        "text": "Nikita Rudin ETH Zurich and NVIDIA rudinn@ethz.ch ",
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+        "text": "Abstract: In this work, we present and study a training set-up that achieves fast policy generation for real-world robotic tasks by using massive parallelism on a single workstation GPU. We analyze and discuss the impact of different training algorithm components in the massively parallel regime on the final policy performance and training times. In addition, we present a novel game-inspired curriculum that is well suited for training with thousands of simulated robots in parallel. We evaluate the approach by training the quadrupedal robot ANYmal to walk on challenging terrain. The parallel approach allows training policies for flat terrain in under four minutes, and in twenty minutes for uneven terrain. This represents a speedup of multiple orders of magnitude compared to previous work. Finally, we transfer the policies to the real robot to validate the approach. We open-source our training code to help accelerate further research in the field of learned legged locomotion: https://leggedrobotics.github.io/legged_gym/. ",
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+        "text": "Keywords: Reinforcement Learning, Legged Robots, Sim-to-real ",
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+        "image_caption": [
+            "Figure 1: Thousands of robots learning to walk in simulation. "
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+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+        "text": "Deep reinforcement learning (DRL) is proving to be a powerful tool for robotics. Tasks such as legged locomotion [1], manipulation [2], and navigation [3], have been solved using these new tools, and research continues to keep adding more and more challenging tasks to the list. The amount of data required to train a policy increases with the task complexity. For this reason, most work focuses on training in simulation before transferring to real robots. We have reached a point where multiple days or even weeks are needed to fully train an agent with current simulators. For example, OpenAI’s block reorientation task was trained for up to 14 days and their Rubik’s cube solving policy took several months to train [4]. The problem is exacerbated by the fact that deep reinforcement learning requires hyper-parameter tuning to obtain a suitable solution which requires sequentially rerunning time-consuming training. Reducing training times using massively parallel approaches such as presented here can therefore help improve the quality and time-to-deployment of DRL policies, as a training setup can be iterated on more often in the same time frame. ",
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+        "text": "In this paper, we examine the effects of massive parallelism for on-policy DRL algorithms and present considerations in how the standard RL formulation and the most commonly used hyperparameters should be adapted to learn efficiently in the highly parallel regime. Additionally, we present a novel game-inspired curriculum which automatically adapts the task difficulty to the performance of the policy. The proposed curriculum architecture is straightforward to implement, does not require tuning, and is well suited for the massively parallel regime. Common robotic simulators such as Mujoco [5], Bullet [6], or Raisim [7] feature efficient multi-body dynamics implementations. However, they have been developed to run on CPUs with only a reduced amount of parallelism. In this work, we use NVIDIA’s Isaac Gym simulation environment [8], which runs both the simulation and training on the GPU and is capable of simulating thousands of robots in parallel. ",
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+        "text": "The massively parallel training regime has been explored before [4, 9] in the context of distributed systems with a network of thousands of CPUs each running a separate instance of the simulation. The parallelization was achieved by averaging the gradients between the different workers without reducing the number of samples provided by each agent. This results in large batch sizes of millions of samples for each policy update which improves the learning dynamics, but does not optimize the overall training time. In parallel, recent works have aimed to increase the simulation throughput and reduce training times of standard DRL benchmark tasks. A framework combining parallel simulation with multi-GPU training [10] was proposed to achieve fast training using hundreds of parallel agents. In the context of visual navigation, large batch simulation has been used to increase the training throughput [11]. Furthermore, GPU accelerated physics simulation has been shown to significantly improve the training time of the Humanoid running task [12]. A differentiable simulator running on Google’s TPUs has also been shown to greatly accelerate the training of multiple tasks [13]. We build upon [10, 12] by pushing the parallelization further, optimizing the training algorithm, and applying the approach to a challenging real-world robotics task. ",
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+        "text": "Perceptive and dynamic locomotion for legged robots in unstructured environments is a demanding task that, until recently, had only been partially demonstrated with complex model-based approaches [14, 15]. Learning-based approaches are emerging as a promising alternative. For quadrupeds, DRL has been used to train blind policies robust to highly uneven ground [16] (12 hours of training). Perceptive locomotion over challenging terrain has been achieved by combining learning with optimal control techniques [17, 18] (82 and 88 hours of training) and recently, a fully learned approach has shown great robustness in this setting [19] (120 hours of training). Similarly, bipedal robots have also been trained to walk blindly on stairs [20] (training time not reported). With our approach we can train a perceptive policy in under 20 minutes on a single GPU, with the complexity of simto-real transfer to the hardware, which increases the performance and robustness requirements and provides clear validation of the overall approach. Training such behaviors in minutes opens up new exciting possibilities ranging from automatic tuning to customized training using scans of particular environments. ",
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+        "type": "text",
+        "text": "2 Massively Parallel Reinforcement Learning ",
+        "text_level": 1,
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+        "text": "Current (on-policy) reinforcement learning algorithms are divided into two parts: data collection and policy update. The policy update, which corresponds to back-propagation for neural networks, is easily performed in parallel on the GPU. Parallelizing data collection is not as straightforward. Each step consists of policy inference, simulation, reward, and observation calculation. Current popular pipelines have the simulation and reward/observation calculation computed on the CPU, making the GPU unsuitable for policy inference because of communication bottle-necks. Data transfer over PCIe is known to be the weakest link of GPU acceleration, and can be as much as 50 times slower than the GPU processing time alone [21]. Furthermore, with CPU data collection, a large amount of data must be sent to the GPU for each policy update, slowing down the overall process. Limited parallelization can be achieved by using multiple CPU cores and spawning many processes, each running the simulation for one agent. However, the number of agents is quickly limited by the number of cores and other issues such as memory usage. We explore the potential of massive parallelism with Isaac Gym’s end-to-end data collection and policy updates on the GPU, significantly reducing data copying and improving simulation throughput. ",
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+        "text": "2.1 Simulation Throughput ",
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+        "text": "The main factor affecting the total simulation throughput is the number of robots simulated in parallel. Modern GPUs can handle tens of thousands of parallel instructions. Similarly, IsaacGym’s PhysX engine can process thousands of robots in a single simulation and all other computations of our pipeline are vectorized to scale favorably with the number of robots. Using a single simulation with thousands of robots presents some new challenges. For example, a single common terrain mesh must be used, and it cannot be easily changed at each reset. We circumvent this problem by creating the whole mesh with all terrain types and levels tiled side by side. We change the terrain level of the robots by physically moving them on the mesh. In supplementary material, we show the computational time of different parts of the pipeline, examine how these times scale with the number of robots, and provide other techniques to optimize the simulation throughput. ",
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+        "text": "2.2 DRL Algorithm ",
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+        "text": "We build upon a custom implementation of the Proximal Policy Optimization (PPO) algorithm [22]. Our implementation is designed to perform every operation and store all the data on the GPU. In order to efficiently learn from thousands of robots in parallel, we perform some essential modifications to the algorithm and change some of the commonly used hyper-parameter values. ",
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+        "text": "2.2.1 Hyper-Parameters Modification ",
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+        "text": "In an on-policy algorithm such as PPO, a fixed policy collects a selected amount of data before doing the next policy update. This batch size, $B$ , is a crucial hyper-parameter for successful learning. With too little data, the gradients will be too noisy, and the algorithm will not learn effectively. With too much data, the samples become repetitive, and the algorithm cannot extract more information from them. These samples represent wasted simulation time and slow down the overall training. We have $B = n _ { r o b o t s } n _ { s t e p s }$ , where $ { n _ { s t e p s } }$ is the number of steps each robot takes per policy update and $n _ { r o b o t s }$ the number of robots simulated in parallel. Since we increase $n _ { r o b o t s }$ by a few orders of magnitude, we must choose a small $n _ { s t e p s }$ to keep $B$ reasonable and hence optimize training times, which is a setting that has not been extensively explored for on-policy reinforcement learning algorithms. It turns out that we can not choose $n _ { s t e p s }$ to be arbitrarily low. The algorithm requires trajectories with coherent temporal information to learn effectively. Even though, in theory, information of single steps could be used, we find that the algorithm fails to converge to the optimal solution below a certain threshold. This can be explained by the fact that we use Generalized Advantage Estimation (GAE) [23], which requires rewards from multiple time steps to be effective. For our task, we find that the algorithm struggles when we provide fewer than 25 consecutive steps, corresponding to $0 . 5 \\mathrm { s }$ of simulated time. It is important to distinguish $ { n _ { s t e p s } }$ from the maximum episode length leading to a time-out and a reset, which we define as $2 0 \\mathrm { s }$ . The environments are reset when they reach this maximum length and not after each iteration, meaning that a single episode can cover many policy updates. This limits the total number of robots training in parallel, and consequently, prohibits us from using the full computational capabilities of the GPU. ",
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+        "type": "text",
+        "text": "The mini-batch size represents the size of the chunks in which the batch size is split to perform backpropagation. We find that having mini-batch sizes much larger than what is usually considered best practice is beneficial for our massively parallel use case. We use mini-batches of tens of thousands of samples and observe that it stabilizes the learning process without increasing the total training time. ",
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+        "type": "text",
+        "text": "2.2.2 Reset Handling ",
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+        "type": "text",
+        "text": "During training, the robots must be reset whenever they fall, and also after some time to keep them exploring new trajectories and terrains. The PPO algorithm includes a critic predicting an infinite horizon sum of future discounted rewards. Resets break this infinite horizon assumption and can lead to inferior critic performance if not handled carefully. Resets based on failure or reaching a goal are not a problem because the critic can predict them. However, a reset based on a time out can not be predicted (we do not provide episode time in the observations). The solution is to distinguish the two termination modes and augment the reward with the expected infinite sum of discounted future rewards in a time-out case. In other words, we bootstrap the target of the critic with its own prediction. This solution has been discussed in [24], but interestingly, this distinction is not part of the widely used Gym environment interface [25] and is ignored by popular implementations such as Stable-Baselines $[ 2 6 ] ^ { 1 }$ . After investigating multiple implementations, we conclude that this important detail is often avoided by assuming that the environments either never time out or only on the very last step of a batch collection. In our case, with few robot steps per batch, we can not make such an assumption since a meaningful episode length covers the collection of many batches. We modify the standard Gym interface to detect time-outs and implement the bootstrapping solution. In supplementary material, we show the effect of this solution on the total reward as well as the critic loss. ",
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+        "image_caption": [
+            "Figure 2: Terrain types used for training and testing in simulation. (a) Randomly rough terrain with variations of $0 . 1 \\mathrm { m }$ . (b) Sloped terrain with an inclination of $2 5 \\mathrm { d e g }$ . (c) Stairs with a width of $0 . 3 \\mathrm { m }$ and height of $\\mathrm { 0 . 2 m }$ . (d) Randomized, discrete obstacles with heights of up to $\\pm 0 . 2 \\mathrm { m }$ . "
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+        "text": "3 Task Description ",
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+        "text": "A quadruped robot must learn to walk across challenging terrain, including uneven surfaces, slopes, stairs, and obstacles, while following base-heading and linear-velocity commands. We conduct most of the simulation and real-world deployment experiments on the ANYbotics ANYmal C robot. However, in simulation, we demonstrate the broader applicability of the approach by additionally training policies for ANYmal B, ANYmal C with an attached arm, and the Unitree A1 robots. ",
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+        "text": "3.1 Game-Inspired Curriculum ",
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+        "type": "text",
+        "text": "The terrains are selected to be representative of real-world environments. We create five types of procedurally generated terrains presented in Fig. 2: flat, sloped, randomly rough, discrete obstacles, and stairs. The terrains are tiled squares with $8 \\mathrm { m }$ sides. The robots start at the center of the terrain and are given randomized heading and velocity commands (kept constant for the duration of an episode) pushing them to walk across the terrain. Slopes and stairs are organized in pyramids to allow traversability in all directions. ",
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+        "text": "Previous works have shown the benefits of using an automated curriculum of task difficulty to learn complex locomotion policies [28, 29, 16]. Similarly, we find that it is essential to first train the policy on less challenging terrain before progressively increasing the complexity. We adopt a solution inspired by [16], but replace the particle filter approach with a new game-inspired automatic curriculum. All robots are assigned a terrain type and a level that represents the difficulty of that terrain. For stairs and randomized obstacles, we gradually increase the step height from $5 \\mathrm { c m }$ to $2 0 \\mathrm { c m }$ . Sloped terrain inclination is increased from 0 deg to 25 deg. If a robot manages to walk past the borders of its terrain, its level is increased, and at the next reset, it will start on more difficult terrain. However, if at the end of an episode it moved by less than half of the distance required by its target velocity, its level is reduced again. Robots solving the highest level are looped back to a randomly selected level to increase the diversity and avoid catastrophic forgetting. This approach has the advantage of training the robots at a level of difficulty tailored to their performance without requiring any external tuning. It adapts the difficulty level for each terrain type individually and provides us with visual and quantitative feedback on the progress of the training. When the robots have reached the final level and are evenly spread across all terrains due to looping back, we can conclude they have fully learned to solve the task. ",
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+        "img_path": "images/ecaf8198256af450c96e9b482c8d3d3a06909d2addf9a2530c761fb4193fd419.jpg",
+        "image_caption": [
+            "Figure 3: 4000 robots progressing through the terrains with automatic curriculum, after 500 (top) and 1000 (bottom) policy updates. The robots start the training session on the first row (closest to the camera) and progressively reach harder terrains. "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "The proposed curriculum structure is well suited for the massively parallel regime. With thousands of robots we can directly use their current progress in the curriculum as the distribution of the policy’s performance, and do not need learn it with a generator network [30]. Furthermore, our method doesn’t require tuning and is straightforward to implement in a parallel manner with nearzero processing cost. We remove the computational overhead of re-sampling and re-generating new terrains needed for the particle filter approach. ",
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+    {
+        "type": "text",
+        "text": "Fig. 3 shows robots progressing through the terrains at two different stages of the training process. On complex terrain types, the robots require more training iterations to reach the highest levels. The distribution of robots after 500 iterations shows that while the policy is able to cross sloped terrains and to go down stairs, climbing stairs and traversing obstacles requires more training iterations. However, after 1000 iterations, the robots have reached the most challenging level for all terrain types and are spread across the map. We train for a total for 1500 iterations to let the policy converge to its highest performance. ",
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+    {
+        "type": "text",
+        "text": "3.2 Observations, Actions, and Rewards ",
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+    {
+        "type": "text",
+        "text": "The policy receives proprioceptive measurements of the robot as well as terrain information around the robot’s base. The observations are composed of: base linear and angular velocities, measurement of the gravity vector, joint positions and velocities, the previous actions selected by the policy, and finally, 108 measurements of the terrain sampled from a grid around the robot’s base. Each measurement is the distance from the terrain surface to the robot’s base height. ",
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+        "text": "The total reward is a weighted sum of nine terms, detailed in supplementary material. The main terms encourage the robot to follow the commanded velocities while avoiding undesired base velocities along other axes. In order to create a smoother, more natural motion, we also penalize joint torques, joint accelerations, joint target changes, and collisions. Contacts with the knees, shanks or between the feet and a vertical surface are considered collisions, while contacts with the base are considered crashes and lead to resets. Finally, we add an additional reward term encouraging the robot to take longer steps, which results in a more visually appealing behavior. We train a single policy with the same rewards for all terrains. ",
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+        "type": "text",
+        "text": "The actions are interpreted as desired joint positions sent to the motors. There, a PD controller produces motor torques. In contrast to other works [16, 20], neither the reward function nor the action space has any gait-dependent elements. ",
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+        "type": "text",
+        "text": "3.3 Sim-to-Real Additions ",
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+    {
+        "type": "text",
+        "text": "In order to make the trained policies amenable for sim-to-real transfer, we randomize the friction of the ground, add noise to the observations and randomly push the robots during the episode to teach them a more stable stance. Each robot has a friction coefficient sampled uniformly in [0.5, 1.25]. The pushes happen every $1 0 \\mathrm { s }$ . The robots’ base is accelerated up to $\\pm 1 \\mathrm { m } / \\mathrm { s }$ in both $\\mathbf { X }$ and y directions. The amount of noise is based on real data measured on the robot and is detailed in supplementary material. ",
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+        "text": "The ANYmal robot uses series elastic actuators with fairly complex dynamics, which are hard to model in simulation. For this reason and following the methodology of previous work [1], we use a neural network to compute torques from joint position commands. However, we simplify the inputs of the model. Instead of concatenating past measurements at fixed time steps and sending all of that information to a standard feed-forward network, we only provide the current measurements to an LSTM network. A potential drawback of this set-up is that the policy does not have the temporal information of the actuators as in previous work. We have experimented with various ways of providing that information through memory mechanisms for the policy but found that it does not improve the final performance. ",
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+    {
+        "type": "text",
+        "text": "4 Results ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "4.1 Effects of Massive Parallelism ",
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+    {
+        "type": "text",
+        "text": "In this section, we study the effects of the number of parallel robots on the final performance of the policy. In order to use the total reward as a single representative metric, we have to remove the curriculum, otherwise a more performant policy sees its task difficulty increase and consequently a decrease in the total reward. As such, we simplify the task by reducing the maximum step size of stairs and obstacles and directly train robots on the full range of difficulties. ",
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+        "type": "text",
+        "text": "We begin by setting a baseline with $n _ { r o b o t s } = 2 0 0 0 0$ and $n _ { s t e p s } = 5 0$ , resulting in a batch size of 1M samples. Using this very large batch size results in the best policy but at the cost of a relatively long training time. ",
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+        "type": "text",
+        "text": "We then conduct experiments in which we increase the number of robots while keeping the batch size constant. As a result, the number of steps each robot takes per policy update decreases. In this case, the training time decreases with a higher number of robots, but the policy performance drops if that number is too high. We start from 128 robots corresponding to the level of parallelization of previous CPU implementations and increase that number up to 16384, which is close to the maximum amount of robots we could simulate on rough terrain with Isaac Gym running on a single workstation GPU. ",
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+        "type": "text",
+        "text": "In Fig. 4, we compare these results with the baseline, which allows us to select the most favorable trade-off between policy performance and training time. We see two interesting effects at play. First, when the number of robots is too high, the performance drops sharply, which can be explained by the time horizon of each robot becoming too small. As expected, with larger batch sizes, the overall reward is higher, and the time horizon effect is shifted, meaning that we can use more robots before seeing the drop. On the other hand, below a certain threshold, we see a slow decrease in performance with fewer robots. We believe this is explained by the fact that the samples are very similar with many steps per robot because of the relatively small time steps between them. This means that for the same amount of samples, there is less diversity in the data. In other words, with a low number of robots, we are further from the standard assumption that the samples are independent and identically distributed, which seems to have a noticeable effect on the training process. In terms of training time, we see a nearly linear scaling up to 4000 robots, after which simulation throughput gains slow down. As such, we can conclude that increasing the number of robots is beneficial for both final performance and training time, but there is an upper limit on this number after which an on-policy algorithm cannot learn effectively. Increasing the batch size to values much larger than what is typically used in similar works seems highly beneficial. Unfortunately, it also scales the training time so it is a trade-off that must be balanced. From the third plot we can conclude that using 2048 to 4096 robots with a batch size of $\\approx 1 0 0 k$ or $\\approx 2 0 0 k$ provides the best trade-off for this specific task. ",
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+            "Figure 4: (a) Average and standard deviation (over 5 runs) of the total reward of an episode after 1500 policy updates for different number of robots and 3 different batch sizes. The ideal case of a batch size of 1M samples with 20000 robots is shown in red. (b) Total training time for the same experiments. (c) Reward dependency on total training time. Colors represent the number of robots, while shapes show the batch size (circles: 49152, crosses: 98304, triangles: 196608). Points in the upper left part of the graph (highlighted in green) represent the most desirable configuration. "
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+            "Figure 5: Success rate of the tested policy on increasing terrain complexities. Robots start in the center of the terrain and are given a forward velocity command of $0 . 7 5 \\mathrm { m } / \\mathrm { s }$ , and a side velocity command randomized within $[ - 0 . 1 , 0 . 1 ] \\mathrm { m } / \\mathrm { s }$ . (a) Success rate for climbing stairs, descending stairs and traversing discrete obstacles. (b) Success rate for climbing and descending sloped terrains. "
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+        "image_caption": [
+            "Figure 6: ANYmal C with a fixed arm, ANYmal B, A1 and Cassie in simulation. "
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+        "text": "4.2 Simulation ",
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+        "text": "For our simulation and deployment experiments, we use a policy trained with 4096 robots and a batch size of 98304, which we train for 1500 policy updates in under 20 minutes2. We begin by measuring the performance of our trained policy in simulation. To that end, we perform robustness and traversability tests. For each terrain type, we command the robots to traverse the representative difficulty of the terrain at high forward velocity and measure the success rate. A success is defined as managing to cross the terrain while avoiding any contacts on the robot’s base. Fig. 5 shows the results for the different terrains. For stairs, we see a nearly $1 0 0 \\%$ success rate for steps up to $\\mathrm { 0 . 2 m }$ , which is the hardest stair difficulty we train on and close to the kinematic limits of our robot. Randomized obstacles seem to be more demanding, with the success rate decreasing steadily. We must note that in this case, the largest step is double the reported height since neighboring obstacles can have positive and negative heights. In the case of slopes, we can observe that after $2 5 \\mathrm { d e g }$ the robots are not able to climb anymore but still learn to slide down with a moderate success rate. ",
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+        "text": "Given our relatively simple rewards and action space, the policy is free to adopt any gait and behavior. Interestingly, it always converges to a trotting gait, but there are often artifacts in the behavior, such as a dragging leg or unreasonably high or low base heights. After tuning of the reward weights, we can obtain a policy that respects all our constraints and can be transferred to the physical robot. ",
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+        "type": "text",
+        "text": "To verify the generalizability of the approach, we train policies for multiple robots with the same set-up. We use the ANYmal C robot with a fixed robotic arm, which adds about $2 0 \\%$ of additional weight, and the ANYmal B robot, which has comparable dimensions but modified kinematic and dynamic properties. In these two cases, we can retrain a policy without any modifications to the rewards or algorithm hyper-parameters and obtain a very similar performance. Next, we use the Unitree A1 robot, which has smaller dimensions, four times lower weight, and a different leg configuration. In this case, we remove the actuator model of the ANYdrive motors, reduce PD gains and the torque penalties, and change the default joint configurations. We can train a dynamic policy that learns to solve the same terrains even with the reduced size of the robot. Finally, we apply our approach to Agility Robotics’ bipedal robot Cassie. We find that an additional reward encouraging standing on a single foot is necessary to achieve a walking gait. With this addition, we are able to train the robot on the same terrains as its quadrupedal counterparts. Fig. 6 shows the different robots. ",
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+        "image_caption": [
+            "Figure 7: Locomotion policy, trained in under $2 0 \\mathrm { { m i n } }$ , deployed on the physical robot. "
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+    {
+        "type": "text",
+        "text": "4.3 Sim-to-real Transfer ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "On the physical robot, our policy is fixed. We compute the observations from the robot’s sensors, feed them to the policy, and directly send the produced actions as target joint positions to the motors. We do not apply any additional filtering or constraint satisfaction checks. The terrain height measurements are queried from an elevation map that the robot is building from Lidar scans. ",
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+        "text": "Unfortunately, this height map is far from perfect, which results in a decrease in robustness between simulation and reality. We observe that these issues mainly occur at high velocities and therefore reduce the maximum linear velocity commands to $0 . 6 \\mathrm { m } / \\mathrm { s }$ for policies deployed on the hardware. The robot can walk up and down stairs and handles obstacles in a dynamic manner. We show samples of these experiments in Fig. 7 and in the supplementary video. To overcome issues with imperfect terrain mapping or state estimation drift, the authors of [19] implemented a teacher-student set-up, which provided outstanding robustness even in adverse conditions. As part of future work, we plan to merge the two approaches. ",
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+    {
+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this work, we demonstrated that a complex real-world robotics task can be trained in minutes with an on-policy deep reinforcement learning algorithm. Using an end-to-end GPU pipeline with thousands of robots simulated in parallel, combined with our proposed curriculum structure, we showed that the training time can be reduced by multiple orders of magnitude compared to previous work. We discussed multiple modifications to the learning algorithm and the standard hyper-parameters required to use the massively parallel regime effectively. Using our fast training pipeline, we performed many training runs, simplified the set-up, and kept only essential components. We showed that the task can be solved using simple observation and action spaces as well as relatively straightforward rewards without encouraging particular gaits or providing motion primitives. ",
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+    {
+        "type": "text",
+        "text": "The purpose of this work is not to obtain the absolute best-performing policy with the highest robustness. For that use case, many other techniques can be incorporated into the pipeline. We aim to show that a policy can be trained in record time with our set-up while still being usable on the real hardware. We wish to shift other researchers’ perspective on the required training time for a real-world application, and hope that our work can serve as a reference for future research. We expect many other tasks to benefit from the massively parallel regime. By reducing the training time of these future robotic tasks, we can greatly accelerate the developments in this field. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgments ",
+        "text_level": 1,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We would like to thank Mayank Mittal, Joonho Lee, Takahiro Miki, and Peter Werner for their valuable suggestions and help with hardware experiments as well as the Isaac Gym and PhysX teams for their continuous support. ",
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
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+        "text": "[1] J. Hwangbo, J. Lee, A. Dosovitskiy, D. Bellicoso, V. Tsounis, V. Koltun, and M. Hutter. Learning agile and dynamic motor skills for legged robots. Science Robotics, 4(26), 2019.   \n[2] S. Gu, E. Holly, T. Lillicrap, and S. Levine. Deep reinforcement learning for robotic manipulation with asynchronous off-policy updates. In IEEE International Conference on Robotics and Automation (ICRA), May 2017.   \n[3] G. Kahn, A. Villaflor, B. Ding, P. Abbeel, and S. Levine. Self-supervised deep reinforcement learning with generalized computation graphs for robot navigation. In IEEE International Conference on Robotics and Automation (ICRA), 2018.   \n[4] OpenAI, I. Akkaya, M. Andrychowicz, M. Chociej, M. Litwin, B. McGrew, A. Petron, A. Paino, M. Plappert, G. Powell, R. Ribas, J. Schneider, N. Tezak, J. Tworek, P. Welinder, L. Weng, Q. Yuan, W. Zaremba, and L. Zhang. Solving rubik’s cube with a robot hand, 2019.   \n[5] E. Todorov, T. Erez, and Y. Tassa. Mujoco: A physics engine for model-based control. In IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 2012.   \n[6] E. Coumans and Y. Bai. Pybullet, a python module for physics simulation for games, robotics and machine learning. http://pybullet.org, 2016–2021.   \n[7] J. Hwangbo, J. Lee, and M. Hutter. Per-contact iteration method for solving contact dynamics. IEEE Robotics and Automation Letters, 3(2), 2018. URL www.raisim.com.   \n[8] V. Makoviychuk, L. Wawrzyniak, Y. Guo, M. Lu, K. Storey, M. Macklin, D. Hoeller, N. Rudin, A. Allshire, A. Handa, and G. State. Isaac gym: High performance GPU based physics simulation for robot learning. In Conference on Neural Information Processing Systems (NeurIPS) Datasets and Benchmarks Track, 2021.   \n[9] N. Heess, D. TB, S. Sriram, J. Lemmon, J. Merel, G. Wayne, Y. Tassa, T. Erez, Z. Wang, S. M. A. Eslami, M. A. Riedmiller, and D. Silver. Emergence of locomotion behaviours in rich environments. CoRR, abs/1707.02286, 2017.   \n[10] A. Stooke and P. Abbeel. Accelerated methods for deep reinforcement learning. CoRR, abs/1803.02811, 2018.   \n[11] B. Shacklett, E. Wijmans, A. Petrenko, M. Savva, D. Batra, V. Koltun, and K. Fatahalian. Large batch simulation for deep reinforcement learning. In International Conference on Learning Representations (ICLR), 2021.   \n[12] J. Liang, V. Makoviychuk, A. Handa, N. Chentanez, M. Macklin, and D. Fox. Gpu-accelerated robotic simulation for distributed reinforcement learning. In Conference on Robot Learning (CoRL), 2018.   \n[13] C. D. Freeman, E. Frey, A. Raichuk, S. Girgin, I. Mordatch, and O. Bachem. Brax - a differentiable physics engine for large scale rigid body simulation. In 35th Conference on Neural Information Processing Systems (NeurIPS) Datasets and Benchmarks Track, 2021.   \n[14] A. Bouman, M. F. Ginting, N. Alatur, M. Palieri, D. D. Fan, T. Touma, T. Pailevanian, S.- K. Kim, K. Otsu, J. Burdick, and A.-a. Agha-Mohammadi. Autonomous spot: Long-range autonomous exploration of extreme environments with legged locomotion. In IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 2020.   \n[15] C. Gehring, P. Fankhauser, L. Isler, R. Diethelm, S. Bachmann, M. Potz, L. Gerstenberg, and M. Hutter. Anymal in the field: Solving industrial inspection of an offshore hvdc platform with a quadrupedal robot. In Field and Service Robotics, 2021.   \n[16] J. Lee, J. Hwangbo, L. Wellhausen, V. Koltun, and M. Hutter. Learning quadrupedal locomotion over challenging terrain. Science Robotics, 5(47), 2020.   \n[17] V. Tsounis, M. Alge, J. Lee, F. Farshidian, and M. Hutter. Deepgait: Planning and control of quadrupedal gaits using deep reinforcement learning. IEEE Robotics and Automation Letters, PP, 03 2020.   \n[18] S. Gangapurwala, M. Geisert, R. Orsolino, M. Fallon, and I. Havoutis. Real-time trajectory adaptation for quadrupedal locomotion using deep reinforcement learning. In IEEE International Conference on Robotics and Automation (ICRA), 2021.   \n[19] T. Miki, J. Lee, L. Wellhausen, V. Koltun, and M. Hutter. Wild anymal: Robust zero-shot perceptive locomotion. Submitted to Science Robotics, 2021.   \n[20] J. Siekmann, K. Green, J. Warila, A. Fern, and J. W. Hurst. Blind bipedal stair traversal via sim-to-real reinforcement learning. CoRR, abs/2105.08328, 2021.   \n[21] C. Gregg and K. Hazelwood. Where is the data? why you cannot debate cpu vs. gpu performance without the answer. In IEEE International Symposium on Performance Analysis of Systems and Software (ISPASS), 2011.   \n[22] J. Schulman, F. Wolski, P. Dhariwal, A. Radford, and O. Klimov. Proximal policy optimization algorithms. CoRR, abs/1707.06347, 2017.   \n[23] J. Schulman, P. Moritz, S. Levine, M. Jordan, and P. Abbeel. High-dimensional continuous control using generalized advantage estimation. In Proceedings of the International Conference on Learning Representations (ICLR), 2016.   \n[24] F. Pardo, A. Tavakoli, V. Levdik, and P. Kormushev. Time limits in reinforcement learning. CoRR, abs/1712.00378, 2017.   \n[25] G. Brockman, V. Cheung, L. Pettersson, J. Schneider, J. Schulman, J. Tang, and W. Zaremba. Openai gym, 2016.   \n[26] A. Hill, A. Raffin, M. Ernestus, A. Gleave, A. Kanervisto, R. Traore, P. Dhariwal, C. Hesse, O. Klimov, A. Nichol, M. Plappert, A. Radford, J. Schulman, S. Sidor, and Y. Wu. Stable baselines. https://github.com/hill-a/stable-baselines, 2018.   \n[27] J. Achiam. Spinning up in deep reinforcement learning, 2018. URL https://spinningup. openai.com/en/latest/.   \n[28] R. Wang, J. Lehman, J. Clune, and K. O. Stanley. Paired open-ended trailblazer (POET): endlessly generating increasingly complex and diverse learning environments and their solutions. CoRR, abs/1901.01753, 2019.   \n[29] Z. Xie, H. Y. Ling, N. H. Kim, and M. van de Panne. Allsteps: Curriculum-driven learning of stepping stone skills. Proceedings of ACM SIGGRAPH / Eurographics Symposium on Computer Animation, 2020.   \n[30] C. Florensa, D. Held, X. Geng, and P. Abbeel. Automatic goal generation for reinforcement learning agents. In Proceedings of the 35th International Conference on Machine Learning (ICML), volume 80 of Proceedings of Machine Learning Research, 2018. ",
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