Adaptive Smoothness-weighted Adversarial Training for Multiple Perturbations with Its Stability Analysis

Abstract: Adversarial Training (AT) has been demonstrated as one of the most effective methods against adversarial examples. While most existing works focus on AT with a single type of perturbation e.g., the $\ell_\infty$ attacks), DNNs are facing threats from different types of adversarial examples. Therefore, adversarial training for multiple perturbations (ATMP) is proposed to generalize the adversarial robustness over different perturbation types (in $\ell_1$, $\ell_2$, and $\ell_\infty$ norm-bounded perturbations). However, the resulting model exhibits trade-off between different attacks. Meanwhile, there is no theoretical analysis of ATMP, limiting its further development. In this paper, we first provide the smoothness analysis of ATMP and show that $\ell_1$, $\ell_2$, and $\ell_\infty$ adversaries give different contributions to the smoothness of the loss function of ATMP. Based on this, we develop the stability-based excess risk bounds and propose adaptive smoothness-weighted adversarial training for multiple perturbations. Theoretically, our algorithm yields better bounds. Empirically, our experiments on CIFAR10 and CIFAR100 achieve the state-of-the-art performance against the mixture of multiple perturbations attacks.