It is well known that adversarial attacks can fool deep neural networks with imperceptible perturbations. Although adversarial training significantly improves model robustness, failure cases of defense still broadly exist. In this work, we find that the adversarial attacks can also be vulnerable to small perturbations. Namely, on adversarially-trained models, perturbing adversarial examples with a small random noise may invalidate their misled predictions. After carefully examining state-of-the-art attacks of various kinds, we find that all these attacks have this deficiency to different extents. Enlightened by this finding, we propose to counter attacks by crafting more effective defensive perturbations. Our defensive perturbations leverage the advantage that adversarial training endows the ground-truth class with smaller local Lipschitzness. By simultaneously attacking all the classes, the misled predictions with larger Lipschitzness can be flipped into correct ones. We verify our defensive perturbation with both empirical experiments and theoretical analyses on a linear model. On CIFAR10, it boosts the state-of-the-art model from 66.16% to 72.66% against the four attacks of AutoAttack, including 71.76% to 83.30% against the Square attack. On ImageNet, the top-1 robust accuracy of FastAT is improved from 33.18% to 38.54% under the 100-step PGD attack.
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