PAC-Bayesian Adversarially Robust Generalization Bounds for Deep Neural Networks

Deep neural networks (DNNs) are vulnerable to adversarial attacks. It is found empirically that adversarially robust generalization is crucial in establishing defense algorithms against adversarial attacks. Therefore, it is interesting to study the theoretical guarantee of robust generalization. This paper focuses on PAC-Bayes analysis (Neyshabur et al., 2017). The main challenge lies in extending the key ingredient, which is a weight perturbation bound in standard settings, to the robust settings. Existing attempts heavily rely on additional strong assumptions, leading to loose bounds. In this paper, we address this issue and provide a spectrally-normalized robust generalization bound for DNNs. Our bound is at least as tight as the standard generalization bound, differing only by a factor of the perturbation strength $\epsilon$. In comparison to existing robust generalization bounds, our bound offers two significant advantages: 1) it does not depend on additional assumptions, and 2) it is considerably tighter. We present a framework that enables us to derive more general results. Specifically, we extend the main result to 1) adversarial robustness against general non-$\ell_p$ attacks, and 2) other neural network architectures, such as ResNet.