It is common practice in deep learning to use overparameterized networks and train for as long as possible; there are numerous studies that show, both theoretically and empirically, that such practices surprisingly do not unduly harm the generalization performance of the classifier. In this paper, we empirically study this phenomenon in the setting of adversarially trained deep networks, which are trained to minimize the loss under worst-case adversarial perturbations. We find that overfitting to the training set does in fact harm robust performance to a very large degree in adversarially robust training across multiple datasets (SVHN, CIFAR-10, CIFAR-100, and ImageNet) and perturbation models ($\ell_\infty$ and $\ell_2$). Based upon this observed effect, we show that the performance gains of virtually all recent algorithmic improvements upon adversarial training can be matched by simply using early stopping. We also show that effects such as the double descent curve do still occur in adversarially trained models, yet fail to explain the observed overfitting. Finally, we study several classical and modern deep learning remedies for overfitting, including regularization and data augmentation, and find that no approach in isolation improves significantly upon the gains achieved by early stopping. All code for reproducing the experiments as well as pretrained model weights and training logs can be found at https://github.com/locuslab/robust_overfitting.
Pennsylvania State University（宾夕法尼亚州立大学）；The University of Hong Kong（香港大学）；University of California, Los Angeles（加州大学洛杉矶分校）
"Benign overfitting", where classifiers memorize noisy training data yet still achieve a good generalization performance, has drawn great attention in the machine learning community. To explain this surprising phenomenon, a series of works have provided theoretical justification in over-parameterized linear regression, classification, and kernel methods. However, it is not clear if benign overfitting still occurs in the presence of adversarial examples, i.e., examples with tiny and intentional perturbations to fool the classifiers. In this paper, we show that benign overfitting indeed occurs i…
Purdue University（普渡大学）；University of Nebraska-Lincoln（内布拉斯加大学林肯分校）
Adversarially robust learning aims to design algorithms that are robust to small adversarial perturbations on input variables. Beyond the existing studies on the predictive performance to adversarial samples, our goal is to understand statistical properties of adversarially robust estimates and analyze adversarial risk in the setup of linear regression models. By discovering the statistical minimax rate of convergence of adversarially robust estimators, we emphasize the importance of incorporating model information, e.g., sparsity, in adversarially robust learning. Further, we reveal an explic…
Toyota Technological Institute at Chicago（芝加哥丰田技术研究院）；Purdue University（普渡大学）
We present a minimax optimal learner for the problem of learning predictors robust to adversarial examples at test-time. Interestingly, we find that this requires new algorithmic ideas and approaches to adversarially robust learning. In particular, we show, in a strong negative sense, the suboptimality of the robust learner proposed by Montasser, Hanneke, and Srebro (2019) and a broader family of learners we identify as local learners. Our results are enabled by adopting a global perspective, specifically, through a key technical contribution: the global one-inclusion graph, which may be of in…
Huazhong University of Science and Technology（华中科技大学）；Peking University（北京大学）；Carnegie Mellon University（卡内基梅隆大学）；Cornell University（康奈尔大学）
Neural network robustness has recently been highlighted by the existence of adversarial examples. Many previous works show that the learned networks do not perform well on perturbed test data, and significantly more labeled data is required to achieve adversarially robust generalization. In this paper, we theoretically and empirically show that with just more unlabeled data, we can learn a model with better adversarially robust generalization. The key insight of our results is based on a risk decomposition theorem, in which the expected robust risk is separated into two parts: the stability pa…
Max Planck Institute for Informatics（马克斯·普朗克信息学研究所）；Saarland Informatics Campus（萨尔兰信息学园区）；University of Tübingen（图宾根大学）
Adversarial training (AT) has become the de-facto standard to obtain models robust against adversarial examples. However, AT exhibits severe robust overfitting: cross-entropy loss on adversarial examples, so-called robust loss, decreases continuously on training examples, while eventually increasing on test examples. In practice, this leads to poor robust generalization, i.e., adversarial robustness does not generalize well to new examples. In this paper, we study the relationship between robust generalization and flatness of the robust loss landscape in weight space, i.e., whether robust loss…
Toyota Technological Institute at Chicago（芝加哥丰田技术研究院）
We study the problem of learning predictors that are robust to adversarial examples with respect to an unknown perturbation set, relying instead on interaction with an adversarial attacker or access to attack oracles, examining different models for such interactions. We obtain upper bounds on the sample complexity and upper and lower bounds on the number of required interactions, or number of successful attacks, in different interaction models, in terms of the VC and Littlestone dimensions of the hypothesis class of predictors, and without any assumptions on the perturbation set.
Adversarial robustness has become a fundamental requirement in modern machine learning applications. Yet, there has been surprisingly little statistical understanding so far. In this paper, we provide the first result of the optimal minimax guarantees for the excess risk for adversarially robust classification, under Gaussian mixture model proposed by \cite{schmidt2018adversarially}. The results are stated in terms of the Adversarial Signal-to-Noise Ratio (AdvSNR), which generalizes a similar notion for standard linear classification to the adversarial setting. For the Gaussian mixtures with A…
Batch Normalization (BatchNorm) is effective for improving the performance and accelerating the training of deep neural networks. However, it has also shown to be a cause of adversarial vulnerability, i.e., networks without it are more robust to adversarial attacks. In this paper, we investigate how BatchNorm causes this vulnerability and proposed new normalization that is robust to adversarial attacks. We first observe that adversarial images tend to shift the distribution of BatchNorm input, and this shift makes train-time estimated population statistics inaccurate. We hypothesize that these…
In this paper we propose a new family of algorithms, ATENT, for training adversarially robust deep neural networks. We formulate a new loss function that is equipped with an additional entropic regularization. Our loss function considers the contribution of adversarial samples that are drawn from a specially designed distribution in the data space that assigns high probability to points with high loss and in the immediate neighborhood of training samples. Our proposed algorithms optimize this loss to seek adversarially robust valleys of the loss landscape. Our approach achieves competitive (or…
Knowledge distillation is effective for producing small, high-performance neural networks for classification, but these small networks are vulnerable to adversarial attacks. This paper studies how adversarial robustness transfers from teacher to student during knowledge distillation. We find that a large amount of robustness may be inherited by the student even when distilled on only clean images. Second, we introduce Adversarially Robust Distillation (ARD) for distilling robustness onto student networks. In addition to producing small models with high test accuracy like conventional distillat…
The Pennsylvania State University（宾夕法尼亚州立大学）；Michigan State University（密歇根州立大学）；Sony AI（索尼人工智能研究院）
Recent studies demonstrated that the adversarially robust learning under $\ell_\infty$ attack is harder to generalize to different domains than standard domain adaptation. How to transfer robustness across different domains has been a key question in domain adaptation field. To investigate the fundamental difficulty behind adversarially robust domain adaptation (or robustness transfer), we propose to analyze a key complexity measure that controls the cross-domain generalization: the adversarial Rademacher complexity over {\em symmetric difference hypothesis space} $\mathcal{H} Δ\mathcal{H}$. F…
School of Computing and Information Systems, The University of Melbourne（墨尔本大学计算与信息系统学院）；Key Lab. of Machine Perception, School of Artificial Intelligence, Peking University（北京大学人工智能学院机器感知重点实验室）；Institute for Artificial Intelligence, Peking University（北京大学人工智能研究院）；University of California, Los Angeles（加州大学洛杉矶分校）；School of Computer Science, Fudan University（复旦大学计算机科学学院）
Deep neural networks (DNNs) are known to be vulnerable to adversarial attacks. A range of defense methods have been proposed to train adversarially robust DNNs, among which adversarial training has demonstrated promising results. However, despite preliminary understandings developed for adversarial training, it is still not clear, from the architectural perspective, what configurations can lead to more robust DNNs. In this paper, we address this gap via a comprehensive investigation on the impact of network width and depth on the robustness of adversarially trained DNNs. Specifically, we make …
SenseTime（商汤科技）；Zhongguancun Laboratory（中关村实验室）；Hefei Comprehensive National Science Center（合肥综合性国家科学中心）；JD Explore Academy（京东探索研究院）；Beihang University（北京航空航天大学）；Chinese Academy of Sciences（中国科学院）；Zhejiang University（浙江大学）
Adversarial training has been demonstrated to be one of the most effective remedies for defending adversarial examples, yet it often suffers from the huge robustness generalization gap on unseen testing adversaries, deemed as the adversarially robust generalization problem. Despite the preliminary understandings devoted to adversarially robust generalization, little is known from the architectural perspective. To bridge the gap, this paper for the first time systematically investigated the relationship between adversarially robust generalization and architectural design. Inparticular, we compr…
University of California, San Diego（加州大学圣地亚哥分校）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）
We consider the sample complexity of learning with adversarial robustness. Most prior theoretical results for this problem have considered a setting where different classes in the data are close together or overlapping. Motivated by some real applications, we consider, in contrast, the well-separated case where there exists a classifier with perfect accuracy and robustness, and show that the sample complexity narrates an entirely different story. Specifically, for linear classifiers, we show a large class of well-separated distributions where the expected robust loss of any algorithm is at lea…
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Weierstrass Institute（魏尔斯特拉斯研究所）；InstaDeep Ltd.（InstaDeep公司）
We propose a scalable robust learning algorithm combining kernel smoothing and robust optimization. Our method is motivated by the convex analysis perspective of distributionally robust optimization based on probability metrics, such as the Wasserstein distance and the maximum mean discrepancy. We adapt the integral operator using supremal convolution in convex analysis to form a novel function majorant used for enforcing robustness. Our method is simple in form and applies to general loss functions and machine learning models. Exploiting a connection with optimal transport, we prove theoretic…
Standard adversarial training approaches suffer from robust overfitting where the robust accuracy decreases when models are adversarially trained for too long. The origin of this problem is still unclear and conflicting explanations have been reported, i.e., memorization effects induced by large loss data or because of small loss data and growing differences in loss distribution of training samples as the adversarial training progresses. Consequently, several mitigation approaches including early stopping, temporal ensembling and weight perturbations on small loss data have been proposed to mi…
The effects of adversarial training on semantic segmentation networks has not been thoroughly explored. While previous work has shown that adversarially-trained image classifiers can be used to perform image synthesis, we have yet to understand how best to leverage an adversarially-trained segmentation network to do the same. Using a simple optimizer, we demonstrate that adversarially-trained semantic segmentation networks can be used to perform image inpainting and generation. Our experiments demonstrate that adversarially-trained segmentation networks are more robust and indeed exhibit perce…
Adversarial training is one of the most effective defenses against adversarial attacks. Previous works suggest that overfitting is a dominant phenomenon in adversarial training leading to a large generalization gap between test and train accuracy in neural networks. In this work, we show that the observed generalization gap is closely related to the choice of the activation function. In particular, we show that using activation functions with low (exact or approximate) curvature values has a regularization effect that significantly reduces both the standard and robust generalization gaps in ad…
Recent works have shown the effectiveness of randomized smoothing as a scalable technique for building neural network-based classifiers that are provably robust to $\ell_2$-norm adversarial perturbations. In this paper, we employ adversarial training to improve the performance of randomized smoothing. We design an adapted attack for smoothed classifiers, and we show how this attack can be used in an adversarial training setting to boost the provable robustness of smoothed classifiers. We demonstrate through extensive experimentation that our method consistently outperforms all existing provabl…
University of Maryland（马里兰大学）；Cornell University（康奈尔大学）
Transfer learning, in which a network is trained on one task and re-purposed on another, is often used to produce neural network classifiers when data is scarce or full-scale training is too costly. When the goal is to produce a model that is not only accurate but also adversarially robust, data scarcity and computational limitations become even more cumbersome. We consider robust transfer learning, in which we transfer not only performance but also robustness from a source model to a target domain. We start by observing that robust networks contain robust feature extractors. By training class…
McGill University（麦吉尔大学）；University of Minnesota（明尼苏达大学）
In this work we study input gradient regularization of deep neural networks, and demonstrate that such regularization leads to generalization proofs and improved adversarial robustness. The proof of generalization does not overcome the curse of dimensionality, but it is independent of the number of layers in the networks. The adversarial robustness regularization combines adversarial training, which we show to be equivalent to Total Variation regularization, with Lipschitz regularization. We demonstrate empirically that the regularized models are more robust, and that gradient norms of images …
In this paper, we explore the effect of architecture completeness on adversarial robustness. We train models with different architectures on CIFAR-10 and MNIST dataset. For each model, we vary different number of layers and different number of nodes in the layer. For every architecture candidate, we use Fast Gradient Sign Method (FGSM) to generate untargeted adversarial attacks and use adversarial training to defend against those attacks. For each architecture candidate, we report pre-attack, post-attack and post-defense accuracy for the model as well as the architecture parameters and the imp…
Technion – Israel Institute of Technology（以色列理工学院）；Intel – Artificial Intelligence Products Group (AIPG)（英特尔人工智能产品事业部）
Even though deep learning has shown unmatched performance on various tasks, neural networks have been shown to be vulnerable to small adversarial perturbations of the input that lead to significant performance degradation. In this work we extend the idea of adding white Gaussian noise to the network weights and activations during adversarial training (PNI) to the injection of colored noise for defense against common white-box and black-box attacks. We show that our approach outperforms PNI and various previous approaches in terms of adversarial accuracy on CIFAR-10 and CIFAR-100 datasets. In a…
While deep learning has led to remarkable results on a number of challenging problems, researchers have discovered a vulnerability of neural networks in adversarial settings, where small but carefully chosen perturbations to the input can make the models produce extremely inaccurate outputs. This makes these models particularly unsuitable for safety-critical application domains (e.g. self-driving cars) where robustness is extremely important. Recent work has shown that augmenting training with adversarially generated data provides some degree of robustness against test-time attacks. In this pa…
Recent works have theoretically and empirically shown that deep neural networks (DNNs) have an inherent vulnerability to small perturbations. Applying the Deep k-Nearest Neighbors (DkNN) classifier, we observe a dramatically increasing robustness-accuracy trade-off as the layer goes deeper. In this work, we propose a Deep Adversarially-Enhanced k-Nearest Neighbors (DAEkNN) method which achieves higher robustness than DkNN and mitigates the robustness-accuracy trade-off in deep layers through two key elements. First, DAEkNN is based on an adversarially trained model. Second, DAEkNN makes predic…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Machine learning (ML) robustness and domain generalization are fundamentally correlated: they essentially concern data distribution shifts under adversarial and natural settings, respectively. On one hand, recent studies show that more robust (adversarially trained) models are more generalizable. On the other hand, there is a lack of theoretical understanding of their fundamental connections. In this paper, we explore the relationship between regularization and domain transferability considering different factors such as norm regularization and data augmentations (DA). We propose a general the…
Training machine learning models that are robust against adversarial inputs poses seemingly insurmountable challenges. To better understand adversarial robustness, we consider the underlying problem of learning robust representations. We develop a notion of representation vulnerability that captures the maximum change of mutual information between the input and output distributions, under the worst-case input perturbation. Then, we prove a theorem that establishes a lower bound on the minimum adversarial risk that can be achieved for any downstream classifier based on its representation vulner…
Neural networks are vulnerable to adversarial attacks -- small visually imperceptible crafted noise which when added to the input drastically changes the output. The most effective method of defending against these adversarial attacks is to use the methodology of adversarial training. We analyze the adversarially trained robust models to study their vulnerability against adversarial attacks at the level of the latent layers. Our analysis reveals that contrary to the input layer which is robust to adversarial attack, the latent layer of these robust models are highly susceptible to adversarial …
Convolutional neural networks (CNNs) have achieved beyond human-level accuracy in the image classification task and are widely deployed in real-world environments. However, CNNs show vulnerability to adversarial perturbations that are well-designed noises aiming to mislead the classification models. In order to defend against the adversarial perturbations, adversarially trained GAN (ATGAN) is proposed to improve the adversarial robustness generalization of the state-of-the-art CNNs trained by adversarial training. ATGAN incorporates adversarial training into standard GAN training procedure to …
University of California Santa Barbara（加州大学圣巴巴拉分校）
Machine learning models are vulnerable to adversarial attacks that can often cause misclassification by introducing small but well designed perturbations. In this paper, we explore, in the setting of classical composite hypothesis testing, a defense strategy based on the generalized likelihood ratio test (GLRT), which jointly estimates the class of interest and the adversarial perturbation. We evaluate the GLRT approach for the special case of binary hypothesis testing in white Gaussian noise under $\ell_{\infty}$ norm-bounded adversarial perturbations, a setting for which a minimax strategy o…
Recently, some works found an interesting phenomenon that adversarially robust classifiers can generate good images comparable to generative models. We investigate this phenomenon from an energy perspective and provide a novel explanation. We reformulate adversarial example generation, adversarial training, and image generation in terms of an energy function. We find that adversarial training contributes to obtaining an energy function that is flat and has low energy around the real data, which is the key for generative capability. Based on our new understanding, we further propose a better ad…
University of Tübingen（图宾根大学）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；ETH Zürich（苏黎世联邦理工学院）
Algorithmic recourse seeks to provide actionable recommendations for individuals to overcome unfavorable classification outcomes from automated decision-making systems. Recourse recommendations should ideally be robust to reasonably small uncertainty in the features of the individual seeking recourse. In this work, we formulate the adversarially robust recourse problem and show that recourse methods that offer minimally costly recourse fail to be robust. We then present methods for generating adversarially robust recourse for linear and for differentiable classifiers. Finally, we show that reg…
Many machine learning models are vulnerable to adversarial attacks; for example, adding adversarial perturbations that are imperceptible to humans can often make machine learning models produce wrong predictions with high confidence. Moreover, although we may obtain robust models on the training dataset via adversarial training, in some problems the learned models cannot generalize well to the test data. In this paper, we focus on $\ell_\infty$ attacks, and study the adversarially robust generalization problem through the lens of Rademacher complexity. For binary linear classifiers, we prove t…
Adversarial Training has proved to be an effective training paradigm to enforce robustness against adversarial examples in modern neural network architectures. Despite many efforts, explanations of the foundational principles underpinning the effectiveness of Adversarial Training are limited and far from being widely accepted by the Deep Learning community. In this paper, we describe surprising properties of adversarially-trained models, shedding light on mechanisms through which robustness against adversarial attacks is implemented. Moreover, we highlight limitations and failure modes affecti…
Deep neural networks are known to be overconfident when applied to out-of-distribution (OOD) inputs which clearly do not belong to any class. This is a problem in safety-critical applications since a reliable assessment of the uncertainty of a classifier is a key property, allowing the system to trigger human intervention or to transfer into a safe state. In this paper, we aim for certifiable worst case guarantees for OOD detection by enforcing not only low confidence at the OOD point but also in an $l_\infty$-ball around it. For this purpose, we use interval bound propagation (IBP) to upper b…
South China Agricultural University（华南农业大学）；Pazhou Lab（琶洲实验室）
Since the Lipschitz properties of convolutional neural networks (CNNs) are widely considered to be related to adversarial robustness, we theoretically characterize the $\ell_1$ norm and $\ell_\infty$ norm of 2D multi-channel convolutional layers and provide efficient methods to compute the exact $\ell_1$ norm and $\ell_\infty$ norm. Based on our theorem, we propose a novel regularization method termed norm decay, which can effectively reduce the norms of convolutional layers and fully-connected layers. Experiments show that norm-regularization methods, including norm decay, weight decay, and s…
Increasingly more similarities between human vision and convolutional neural networks (CNNs) have been revealed in the past few years. Yet, vanilla CNNs often fall short in generalizing to adversarial or out-of-distribution (OOD) examples which humans demonstrate superior performance. Adversarial training is a leading learning algorithm for improving the robustness of CNNs on adversarial and OOD data; however, little is known about the properties, specifically the shape bias and internal features learned inside adversarially-robust CNNs. In this paper, we perform a thorough, systematic study t…
Adversarial training is widely used to improve the robustness of deep neural networks to adversarial attack. However, adversarial training is prone to overfitting, and the cause is far from clear. This work sheds light on the mechanisms underlying overfitting through analyzing the loss landscape w.r.t. the input. We find that robust overfitting results from standard training, specifically the minimization of the clean loss, and can be mitigated by regularization of the loss gradients. Moreover, we find that robust overfitting turns severer during adversarial training partially because the grad…
Guardian of the Ensembles: Introducing Pairwise Adversarially Robust Loss for Resisting Adversarial Attacks in DNN Ensembles
Centre for Computational and Data Sciences, IIT Kharagpur（印度理工学院克勒格布尔分校计算与数据科学中心）；Computer Science and Engineering Department, IIT Kharagpur（印度理工学院克勒格布尔分校计算机科学与工程系）；Center for Cyber Security, New York University Abu Dhabi（纽约大学阿布扎比分校网络安全中心）；Department of Artificial Intelligence, IIT Kharagpur（印度理工学院克勒格布尔分校人工智能系）；Computer Science and Engineering Department, IIT Patna（印度理工学院巴特那分校计算机科学与工程系）
Object detection is an important vision task and has emerged as an indispensable component in many vision system, rendering its robustness as an increasingly important performance factor for practical applications. While object detection models have been demonstrated to be vulnerable against adversarial attacks by many recent works, very few efforts have been devoted to improving their robustness. In this work, we take an initial attempt towards this direction. We first revisit and systematically analyze object detectors and many recently developed attacks from the perspective of model robustn…
In recent years, the ML community has seen surges of interest in both adversarially robust learning and implicit layers, but connections between these two areas have seldom been explored. In this work, we combine innovations from these areas to tackle the problem of N-k security-constrained optimal power flow (SCOPF). N-k SCOPF is a core problem for the operation of electrical grids, and aims to schedule power generation in a manner that is robust to potentially k simultaneous equipment outages. Inspired by methods in adversarially robust training, we frame N-k SCOPF as a minimax optimization …
University of Edinburgh（爱丁堡大学）；Samsung AI Center, Cambridge（三星剑桥人工智能中心）
Stochastic Neural Networks (SNNs) that inject noise into their hidden layers have recently been shown to achieve strong robustness against adversarial attacks. However, existing SNNs are usually heuristically motivated, and often rely on adversarial training, which is computationally costly. We propose a new SNN that achieves state-of-the-art performance without relying on adversarial training, and enjoys solid theoretical justification. Specifically, while existing SNNs inject learned or hand-tuned isotropic noise, our SNN learns an anisotropic noise distribution to optimize a learning-theore…
