This paper investigates the theory of robustness against adversarial attacks. It focuses on the family of randomization techniques that consist in injecting noise in the network at inference time. These techniques have proven effective in many contexts, but lack theoretical arguments. We close this gap by presenting a theoretical analysis of these approaches, hence explaining why they perform well in practice. More precisely, we provide the first result relating the randomization rate to robustness to adversarial attacks. This result applies for the general family of exponential distributions, and thus extends and unifies the previous approaches. We support our theoretical claims with a set of experiments.
Even though several explicit or implicit definitions can be found in the literature, e.g. @cite_17 @cite_23 @cite_4 , there is no broadly accepted definition of robustness to adversarial examples attacks. Recently @cite_1 proposed general definitions and a taxonomy of these. The authors divide the definitions from the literature into three categories: error-region, prediction-change and corrupted instance. In this paper we introduce a definition of robustness that generalizes the one of prediction-change, in the sense that it relies on probabilistic mappings in arbitrary metric spaces, and is not restricted to classification tasks, as discussed in the sequel.
@cite_1 @cite_4 @cite_23 @cite_17
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The prediction accuracy has been the long-lasting and sole standard for comparing the performance of different image classification models, including the ImageNet competition. However, recent studies have highlighted the lack of robustness in well-trained deep neural networks to adversarial examples. Visually imperceptible perturbations to natural images can easily be crafted and mislead the image classifiers towards misclassification. To demystify the trade-offs between robustness and accuracy, in this paper we thoroughly benchmark 18 ImageNet models using multiple robustness metrics, including the distortion, success rate and transferability of adversarial examples between 306 pairs of models. Our extensive experimental results reveal several new insights: (1) linear scaling law - the empirical ( _2 ) and ( _ ) distortion metrics scale linearly with the logarithm of classification error; (2) model architecture is a more critical factor to robustness than model size, and the disclosed accuracy-robustness Pareto frontier can be used as an evaluation criterion for ImageNet model designers; (3) for a similar network architecture, increasing network depth slightly improves robustness in ( _ ) distortion; (4) there exist models (in VGG family) that exhibit high adversarial transferability, while most adversarial examples crafted from one model can only be transferred within the same family. Experiment code is publicly available at https: github.com huanzhang12 Adversarial_Survey. We identify a trade-off between robustness and accuracy that serves as a guiding principle in the design of defenses against adversarial examples. Although this problem has been widely studied empirically, much remains unknown concerning the theory underlying this trade-off. In this work, we decompose the prediction error for adversarial examples (robust error) as the sum of the natural (classification) error and boundary error, and provide a differentiable upper bound using the theory of classification-calibrated loss, which is shown to be the tightest possible upper bound uniform over all probability distributions and measurable predictors. Inspired by our theoretical analysis, we also design a new defense method, TRADES, to trade adversarial robustness off against accuracy. Our proposed algorithm performs well experimentally in real-world datasets. The methodology is the foundation of our entry to the NeurIPS 2018 Adversarial Vision Challenge in which we won the 1st place out of 2,000 submissions, surpassing the runner-up approach by @math in terms of mean @math perturbation distance. The goal of this paper is to analyze an intriguing phenomenon recently discovered in deep networks, namely their instability to adversarial perturbations (Szegedy et. al., 2014). We provide a theoretical framework for analyzing the robustness of classifiers to adversarial perturbations, and show fundamental upper bounds on the robustness of classifiers. Specifically, we establish a general upper bound on the robustness of classifiers to adversarial perturbations, and then illustrate the obtained upper bound on the families of linear and quadratic classifiers. In both cases, our upper bound depends on a distinguishability measure that captures the notion of difficulty of the classification task. Our results for both classes imply that in tasks involving small distinguishability, no classifier in the considered set will be robust to adversarial perturbations, even if a good accuracy is achieved. Our theoretical framework moreover suggests that the phenomenon of adversarial instability is due to the low flexibility of classifiers, compared to the difficulty of the classification task (captured by the distinguishability). Moreover, we show the existence of a clear distinction between the robustness of a classifier to random noise and its robustness to adversarial perturbations. Specifically, the former is shown to be larger than the latter by a factor that is proportional to d (with d being the signal dimension) for linear classifiers. This result gives a theoretical explanation for the discrepancy between the two robustness properties in high dimensional problems, which was empirically observed in the context of neural networks. To the best of our knowledge, our results provide the first theoretical work that addresses the phenomenon of adversarial instability recently observed for deep networks. Our analysis is complemented by experimental results on controlled and real-world data. 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 part which measures the prediction stability in the presence of perturbations, and the accuracy part which evaluates the standard classification accuracy. As the stability part does not depend on any label information, we can optimize this part using unlabeled data. We further prove that for a specific Gaussian mixture problem illustrated by schmidt2018adversarially , adversarially robust generalization can be almost as easy as the standard generalization in supervised learning if a sufficiently large amount of unlabeled data is provided. Inspired by the theoretical findings, we propose a new algorithm called PASS by leveraging unlabeled data during adversarial training. We show that in the transductive and semi-supervised settings, PASS achieves higher robust accuracy and defense success rate on the Cifar-10 task.