Smoothness Analysis of Loss Functions of Adversarial Training

Deep neural networks are vulnerable to adversarial attacks. Recent studies of adversarial robustness focus on the loss landscape in the parameter space since it is related to optimization performance. These studies conclude that it is hard to optimize the loss function for adversarial training with respect to parameters because the loss function is not smooth: i.e., its gradient is not Lipschitz continuous. However, this analysis ignores the dependence of adversarial attacks on parameters. Since adversarial attacks are the worst noise for the models, they should depend on the parameters of the models. In this study, we analyze the smoothness of the loss function of adversarial training for binary linear classification considering the dependence. We reveal that the Lipschitz continuity depends on the types of constraints of adversarial attacks in this case. Specifically, under the L2 constraints, the adversarial loss is smooth except at zero.