Improving Adversarial Training for Multiple Perturbations through the Lens of Uniform Stability

In adversarial training (AT), most existing works focus on AT with a single type of perturbation, such as the $\ell_\infty$ attacks. However, deep neural networks (DNNs) are vulnerable to different types of adversarial examples, necessitating the development of adversarial training for multiple perturbations (ATMP). Despite the benefits of ATMP, there exists a trade-off between different types of attacks. Furthermore, there is a lack of theoretical analyses of ATMP, which hinders its further development. To address these issues, we conduct a smoothness analysis of ATMP. Our analysis reveals that $\ell_1$, $\ell_2$, and $\ell_\infty$ adversaries contribute differently to the smoothness of the loss function in ATMP. Leveraging these smoothness properties, we investigate the improvement of ATMP through the lens of uniform stability. Through our research, we demonstrate that employing an adaptive smoothness-weighted learning rate leads to enhanced uniform stability bounds, thus improving adversarial training for multiple perturbations. We validate our findings through experiments on CIFAR-10 and CIFAR-100 datasets, where our approach achieves competitive performance against various mixtures of multiple perturbation attacks. This work contributes to a deeper understanding of ATMP and provides practical insights for improving the robustness of DNNs against diverse adversarial examples.