Enhancing Adversarial Robustness through Stable Adversarial Training

Deep neural network models are vulnerable to attacks from adversarial methods, such as gradient attacks. Evening small perturbations can cause significant differences in their predictions. Adversarial training (AT) aims to improve the model’s adversarial robustness against gradient attacks by generating adversarial samples and optimizing the adversarial training objective function of the model. Existing methods mainly focus on improving robust accuracy, balancing natural and robust accuracy and suppressing robust overfitting. They rarely consider the AT problem from the characteristics of deep neural networks themselves, such as the stability properties under certain conditions. From a mathematical perspective, deep neural networks with stable training processes may have a better ability to suppress overfitting, as their training process is smoother and avoids sudden drops in performance. We provide a proof of the existence of Ulam stability for deep neural networks. Ulam stability not only determines the existence of the solution for an operator inequality, but it also provides an error bound between the exact and approximate solutions. The feature subspace of a deep neural network with Ulam stability can be accurately characterized and constrained by a function with special properties and a controlled error boundary constant. This restricted feature subspace leads to a more stable training process. Based on these properties, we propose an adversarial training framework called Ulam stability adversarial training (US-AT). This framework can incorporate different Ulam stability conditions and benchmark AT models, optimize the construction of the optimal feature subspace, and consistently improve the model’s robustness and training stability. US-AT is simple and easy to use, and it can be easily integrated with existing multi-class AT models, such as GradAlign and TRADES. Experimental results show that US-AT methods can consistently improve the robust accuracy and training stability of benchmark models.