  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 norm-based complexity, based on a PAC-Bayes approach
(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. Compared
to existing bounds, our bound offers two significant advantages: Firstly, it
does not depend on additional assumptions. Secondly, it is considerably
tighter, aligning with the bounds of standard generalization. Therefore, our
result provides a different perspective on understanding robust generalization:
The mismatch terms between standard and robust generalization bounds shown in
previous studies do not contribute to the poor robust generalization. Instead,
these disparities solely due to mathematical issues. Finally, we extend the
main result to adversarial robustness against general non-$\ell_p$ attacks and
other neural network architectures.
