A major challenge in defending against adversarial attacks is the enormous space of possible attacks that even a simple adversary might perform. To address this, prior work has proposed a variety of defenses that effectively reduce the size of this space. These include randomized smoothing methods that add noise to the input to take away some of the adversary's impact. Another approach is input discretization which limits the adversary's possible number of actions. Motivated by these two approaches, we introduce a new notion of adversarial loss which we call distributional adversarial loss, to unify these two forms of effectively weakening an adversary. In this notion, we assume for each original example, the allowed adversarial perturbation set is a family of distributions (e.g., induced by a smoothing procedure), and the adversarial loss over each example is the maximum loss over all the associated distributions. The goal is to minimize the overall adversarial loss. We show generalization guarantees for our notion of adversarial loss in terms of the VC-dimension of the hypothesis class and the size of the set of allowed adversarial distributions associated with each input. We also investigate the role of randomness in achieving robustness against adversarial attacks in the methods described above. We show a general derandomization technique that preserves the extent of a randomized classifier's robustness against adversarial attacks. We corroborate the procedure experimentally via derandomizing the Random Projection Filters framework of \cite{dong2023adversarial}. Our procedure also improves the robustness of the model against various adversarial attacks.
View on arXiv