Adversarial training and data augmentation with noise are widely adopted techniques to enhance the performance of neural networks. This paper investigates adversarial training and data augmentation with noise in the context of regularized regression in a reproducing kernel Hilbert space (RKHS). We establish the limiting formula for these techniques as the attack and noise size, as well as the regularization parameter, tend to zero. Based on this limiting formula, we analyze specific scenarios and demonstrate that, without appropriate regularization, these two methods may have larger generalization error and Lipschitz constant than standard kernel regression. However, by selecting the appropriate regularization parameter, these two methods can outperform standard kernel regression and achieve smaller generalization error and Lipschitz constant. These findings support the empirical observations that adversarial training can lead to overfitting, and appropriate regularization methods, such as early stopping, can alleviate this issue.
Although fast adversarial training has demonstrated both robustness and efficiency, the problem of "catastrophic overfitting" has been observed. This is a phenomenon in which, during single-step adversarial training, the robust accuracy against projected gradient descent (PGD) suddenly decreases to 0% after a few epochs, whereas the robust accuracy against fast gradient sign method (FGSM) increases to 100%. In this paper, we demonstrate that catastrophic overfitting is very closely related to the characteristic of single-step adversarial training which uses only adversarial examples with the m…
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…
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 robu…
The repeated community-wide reuse of test sets in popular benchmark problems raises doubts about the credibility of reported test-error rates. Verifying whether a learned model is overfitted to a test set is challenging as independent test sets drawn from the same data distribution are usually unavailable, while other test sets may introduce a distribution shift. We propose a new hypothesis test that uses only the original test data to detect overfitting. It utilizes a new unbiased error estimate that is based on adversarial examples generated from the test data and importance weighting. Overf…
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…
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…
The University of Sydney（悉尼大学）；Hong Kong Baptist University（香港浸会大学）；JD Explore Academy（京东探索研究院）；University of Science and Technology of China（中国科学技术大学）；Nanjing University of Science and Technology（南京理工大学）；The University of Melbourne（墨尔本大学）
Robust overfitting widely exists in adversarial training of deep networks. The exact underlying reasons for this are still not completely understood. Here, we explore the causes of robust overfitting by comparing the data distribution of \emph{non-overfit} (weak adversary) and \emph{overfitted} (strong adversary) adversarial training, and observe that the distribution of the adversarial data generated by weak adversary mainly contain small-loss data. However, the adversarial data generated by strong adversary is more diversely distributed on the large-loss data and the small-loss data. Given t…
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…
On the Impact of Hard Adversarial Instances on Overfitting in Adversarial Training
Hong Kong University of Science and Technology（香港科技大学）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；City University of Hong Kong（香港城市大学）；University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Adversarial training (AT) has proven to be one of the most effective ways to defend Deep Neural Networks (DNNs) against adversarial attacks. However, the phenomenon of robust overfitting, i.e., the robustness will drop sharply at a certain stage, always exists during AT. It is of great importance to decrease this robust generalization gap in order to obtain a robust model. In this paper, we present an in-depth study towards the robust overfitting from a new angle. We observe that consistency regularization, a popular technique in semi-supervised learning, has a similar goal as AT and can be us…
Provable Robust Overfitting Mitigation in Wasserstein Distributionally Robust Optimization
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）；Kaiyuan International Mathematical Sciences Institute（开元国际数学科学研究院）
Making deep neural networks robust to small adversarial noises has recently been sought in many applications. Adversarial training through iterative projected gradient descent (PGD) has been established as one of the mainstream ideas to achieve this goal. However, PGD is computationally demanding and often prohibitive in case of large datasets and models. For this reason, single-step PGD, also known as FGSM, has recently gained interest in the field. Unfortunately, FGSM-training leads to a phenomenon called ``catastrophic overfitting," which is a sudden drop in the adversarial accuracy under t…
Adversarial training suffers from the issue of robust overfitting, which seriously impairs its generalization performance. Data augmentation, which is effective at preventing overfitting in standard training, has been observed by many previous works to be ineffective in mitigating overfitting in adversarial training. This work proves that, contrary to previous findings, data augmentation alone can significantly boost accuracy and robustness in adversarial training. We find that the hardness and the diversity of data augmentation are important factors in combating robust overfitting. In general…
University of California San Diego（加利福尼亚大学圣地亚哥分校）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Our goal is to understand why the robustness drops after conducting adversarial training for too long. Although this phenomenon is commonly explained as overfitting, our analysis suggest that its primary cause is perturbation underfitting. We observe that after training for too long, FGSM-generated perturbations deteriorate into random noise. Intuitively, since no parameter updates are made to strengthen the perturbation generator, once this process collapses, it could be trapped in such local optima. Also, sophisticating this process could mostly avoid the robustness drop, which supports that…
Qatar Computing Research Institute, HBKU（哈马德·本·哈利法大学卡塔尔计算研究所）
Adversarial training has been empirically shown to be more prone to overfitting than standard training. The exact underlying reasons still need to be fully understood. In this paper, we identify one cause of overfitting related to current practices of generating adversarial samples from misclassified samples. To address this, we propose an alternative approach that leverages the misclassified samples to mitigate the overfitting problem. We show that our approach achieves better generalization while having comparable robustness to state-of-the-art adversarial training methods on a wide range of…
Benign Overfitting in Adversarial Training for Vision Transformers
Division of CEMSE, King Abdullah University of Science and Technology（科廷大学科学与工程学院）；School of Statistics, Renmin University of China（中国人民大学统计学院）；Department of Computer Science and Engineering, State University of New York at Buffalo（纽约州立大学布法罗分校计算机科学与工程学院）；School of Computer Science, University of Auckland（奥克兰大学计算机科学学院）
Preventing Catastrophic Overfitting in Fast Adversarial Training: A Bi-level Optimization Perspective
School of Artificial Intelligence, Xidian University（西安电子科技大学人工智能学院）；School of Computer Science and Technology, Xidian University（西安电子科技大学计算机科学与技术学院）；School of Engineering, Westlake University（西湖大学工程学院）
Electronics and Telecommunications Research Institute（电子与电信研究所）
Adversarial training deep neural networks often experience serious overfitting problem. Recently, it is explained that the overfitting happens because the sample complexity of training data is insufficient to generalize robustness. In traditional machine learning, one way to relieve overfitting from the lack of data is to use ensemble methods. However, adversarial training multiple networks is extremely expensive. Moreover, we found that there is a dilemma on choosing target model to generate adversarial examples. Optimizing attack to the members of ensemble will be suboptimal attack to the en…
Recent developments in the filed of Deep Learning have demonstrated that Deep Neural Networks(DNNs) are vulnerable to adversarial examples. Specifically, in image classification, an adversarial example can fool the well trained deep neural networks by adding barely imperceptible perturbations to clean images. Adversarial Training, one of the most direct and effective methods, minimizes the losses of perturbed-data to learn robust deep networks against adversarial attacks. It has been proven that using the fast gradient sign method (FGSM) can achieve Fast Adversarial Training. However, FGSM-bas…
Although fast adversarial training provides an efficient approach for building robust networks, it may suffer from a serious problem known as catastrophic overfitting (CO), where multi-step robust accuracy suddenly collapses to zero. In this paper, we for the first time decouple single-step adversarial examples into data-information and self-information, which reveals an interesting phenomenon called "self-fitting". Self-fitting, i.e., the network learns the self-information embedded in single-step perturbations, naturally leads to the occurrence of CO. When self-fitting occurs, the network ex…
Purdue University（普渡大学）；The Ohio State University（俄亥俄州立大学）
Recently, there have been significant interests in studying the so-called "double-descent" of the generalization error of linear regression models under the overparameterized and overfitting regime, with the hope that such analysis may provide the first step towards understanding why overparameterized deep neural networks (DNN) still generalize well. However, to date most of these studies focused on the min $\ell_2$-norm solution that overfits the data. In contrast, in this paper we study the overfitting solution that minimizes the $\ell_1$-norm, which is known as Basis Pursuit (BP) in the com…
MEAT: Median-Ensemble Adversarial Training for Improving Robustness and Generalization
Fujian Province Key Laboratory of Information Security（福建省信息安全重点实验室）；College of Computer Science and Big Data, Fuzhou University（福州大学计算机与大数据学院）；Yuan Ze University（元智大学）
Huazhong University of Science and Technology（华中科技大学）；Peking University（北京大学）
In the field of adversarial robustness, there is a common practice that adopts the single-step adversarial training for quickly developing adversarially robust models. However, the single-step adversarial training is most likely to cause catastrophic overfitting, as after a few training epochs it will be hard to generate strong adversarial examples to continuously boost the adversarial robustness. In this work, we aim to avoid the catastrophic overfitting by introducing multi-step adversarial examples during the single-step adversarial training. Then, to balance the large training overhead of …
The Hong Kong University of Science and Technology（香港科技大学）；Tencent AI Lab（腾讯人工智能实验室）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
While adversarial training and its variants have shown to be the most effective algorithms to defend against adversarial attacks, their extremely slow training process makes it hard to scale to large datasets like ImageNet. The key idea of recent works to accelerate adversarial training is to substitute multi-step attacks (e.g., PGD) with single-step attacks (e.g., FGSM). However, these single-step methods suffer from catastrophic overfitting, where the accuracy against PGD attack suddenly drops to nearly 0% during training, destroying the robustness of the networks. In this work, we study the…
Improving Fast Adversarial Training with Prior-Guided Knowledge
Institute of Information Engineering, Chinese Academy of Sciences（中国科学院信息工程研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Tencent Inc.（腾讯公司）；Beihang University（北京航空航天大学）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；Shenzhen Research Institute of Big Data（深圳大数据研究院）；Sun Yat-sen University（中山大学）
University of California, Los Angeles（加州大学洛杉矶分校）；Johns Hopkins University（约翰斯·霍普金斯大学）；University of Washington（华盛顿大学）；Microsoft Research（微软研究院）
There is an increasing realization that algorithmic inductive biases are central in preventing overfitting; empirically, we often see a benign overfitting phenomenon in overparameterized settings for natural learning algorithms, such as stochastic gradient descent (SGD), where little to no explicit regularization has been employed. This work considers this issue in arguably the most basic setting: constant-stepsize SGD (with iterate averaging or tail averaging) for linear regression in the overparameterized regime. Our main result provides a sharp excess risk bound, stated in terms of the full…
Balance, Imbalance, and Rebalance: Understanding Robust Overfitting from a Minimax Game Perspective
Department of Engineering, University of Cambridge（剑桥大学工程系）；School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Institute for Artificial Intelligence, Peking University（北京大学人工智能研究院）；Peng Cheng Laboratory（鹏城实验室）
EPFL（洛桑联邦理工学院）；Theory of Machine Learning Lab（机器学习理论实验室）
A recent line of work focused on making adversarial training computationally efficient for deep learning models. In particular, Wong et al. (2020) showed that $\ell_\infty$-adversarial training with fast gradient sign method (FGSM) can fail due to a phenomenon called "catastrophic overfitting", when the model quickly loses its robustness over a single epoch of training. We show that adding a random step to FGSM, as proposed in Wong et al. (2020), does not prevent catastrophic overfitting, and that randomness is not important per se -- its main role being simply to reduce the magnitude of the p…
There has been a recent surge in single-step adversarial training as it shows robustness and efficiency. However, a phenomenon referred to as ``catastrophic overfitting" has been observed, which is prevalent in single-step defenses and may frustrate attempts to use FGSM adversarial training. To address this issue, we propose a novel method, Stable and Efficient Adversarial Training (SEAT), which mitigates catastrophic overfitting by harnessing on local properties that distinguish a robust model from that of a catastrophic overfitted model. The proposed SEAT has strong theoretical justification…
Improving Fast Adversarial Training Paradigm: An Example Taxonomy Perspective
School of Cyber Science and Engineering, Southeast University（东南大学网络空间科学与工程学院）；Purple Mountain Laboratories（紫金山实验室）；Department of Computer Science, City University of Hong Kong（香港城市大学计算机科学系）；Department of Computer and Information Science, University of Macau（澳门大学计算机与信息科学系）
The University of Sydney（悉尼大学）；Hong Kong Baptist University（香港浸会大学）；The University of Melbourne（墨尔本大学）；JD Explore Academy（京东探索研究院）；Institute of Information Engineering, Chinese Academy of Sciences（中国科学院信息工程研究所）；Wuhan University（武汉大学）
Overfitting widely exists in adversarial robust training of deep networks. An effective remedy is adversarial weight perturbation, which injects the worst-case weight perturbation during network training by maximizing the classification loss on adversarial examples. Adversarial weight perturbation helps reduce the robust generalization gap; however, it also undermines the robustness improvement. A criterion that regulates the weight perturbation is therefore crucial for adversarial training. In this paper, we propose such a criterion, namely Loss Stationary Condition (LSC) for constrained pert…
Single-step adversarial training (AT) has received wide attention as it proved to be both efficient and robust. However, a serious problem of catastrophic overfitting exists, i.e., the robust accuracy against projected gradient descent (PGD) attack suddenly drops to 0% during the training. In this paper, we approach this problem from a novel perspective of optimization and firstly reveal the close link between the fast-growing gradient of each sample and overfitting, which can also be applied to understand robust overfitting in multi-step AT. To control the growth of the gradient, we propose a…
Unveiling the Backdoor Mechanism Hidden Behind Catastrophic Overfitting in Fast Adversarial Training
School of Computer Science and Technology, Anhui University（安徽大学计算机科学与技术学院）；School of Information and Communication Engineering, Dalian University of Technology（大连理工大学信息与通信工程学院）；School of Computer Science and Technology, Zhejiang University（浙江大学计算机科学与技术学院）
State-of-the-art machine learning models can be vulnerable to very small input perturbations that are adversarially constructed. Adversarial training is an effective approach to defend against such examples. It is formulated as a min-max problem, searching for the best solution when the training data was corrupted by the worst-case attacks. For linear regression problems, adversarial training can be formulated as a convex problem. We use this reformulation to make two technical contributions: First, we formulate the training problem as an instance of robust regression to reveal its connection …
ETH Zurich（苏黎世联邦理工学院）；Institute for Machine Learning（机器学习研究所）
Recently, FGSM adversarial training is found to be able to train a robust model which is comparable to the one trained by PGD but an order of magnitude faster. However, there is a failure mode called catastrophic overfitting (CO) that the classifier loses its robustness suddenly during the training and hardly recovers by itself. In this paper, we find CO is not only limited to FGSM, but also happens in $\mbox{DF}^{\infty}$-1 adversarial training. Then, we analyze the geometric properties for both FGSM and $\mbox{DF}^{\infty}$-1 and find they have totally different decision boundaries after CO.…
The Ohio State University（俄亥俄州立大学）；Columbia University（哥伦比亚大学）；Cold Spring Harbor Laboratory（冷泉港实验室）
Many modern machine learning models are trained to achieve zero or near-zero training error in order to obtain near-optimal (but non-zero) test error. This phenomenon of strong generalization performance for "overfitted" / interpolated classifiers appears to be ubiquitous in high-dimensional data, having been observed in deep networks, kernel machines, boosting and random forests. Their performance is consistently robust even when the data contain large amounts of label noise. Very little theory is available to explain these observations. The vast majority of theoretical analyses of generaliza…
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
Catastrophic overfitting is a phenomenon observed during Adversarial Training (AT) with the Fast Gradient Sign Method (FGSM) where the test robustness steeply declines over just one epoch in the training stage. Prior work has attributed this loss in robustness to a sharp decrease in $\textit{local linearity}$ of the neural network with respect to the input space, and has demonstrated that introducing a local linearity measure as a regularization term prevents catastrophic overfitting. Using a simple neural network architecture, we experimentally demonstrate that maintaining high local linearit…
High cost of training time caused by multi-step adversarial example generation is a major challenge in adversarial training. Previous methods try to reduce the computational burden of adversarial training using single-step adversarial example generation schemes, which can effectively improve the efficiency but also introduce the problem of catastrophic overfitting, where the robust accuracy against Fast Gradient Sign Method (FGSM) can achieve nearby 100\% whereas the robust accuracy against Projected Gradient Descent (PGD) suddenly drops to 0\% over a single epoch. To address this problem, we …
Adversarial attacks against neural networks in a regression setting are a critical yet understudied problem. In this work, we advance the state of the art by investigating adversarial attacks against regression networks and by formulating a more effective defense against these attacks. In particular, we take the perspective that adversarial attacks are likely caused by numerical instability in learned functions. We introduce a stability inducing, regularization based defense against adversarial attacks in the regression setting. Our new and easy to implement defense is shown to outperform prio…
While deep learning in the form of recurrent neural networks (RNNs) has caused a significant improvement in neural language modeling, the fact that they are extremely prone to overfitting is still a mainly unresolved issue. In this paper we propose a regularization method based on generative adversarial networks (GANs) and adversarial training (AT), that can prevent overfitting in neural language models. Unlike common adversarial training methods such as the fast gradient sign method (FGSM) that require a second back-propagation through time, and therefore effectively require at least twice th…
We study the error of linear regression in the face of adversarial attacks. In this framework, an adversary changes the input to the regression model in order to maximize the prediction error. We provide bounds on the prediction error in the presence of an adversary as a function of the parameter norm and the error in the absence of such an adversary. We show how these bounds make it possible to study the adversarial error using analysis from non-adversarial setups. The obtained results shed light on the robustness of overparameterized linear models to adversarial attacks. Adding features migh…
