  Despite the fundamental distinction between adversarial and natural training
(AT and NT), AT methods generally adopt momentum SGD (MSGD) for the outer
optimization. This paper aims to analyze this choice by investigating the
overlooked role of outer optimization in AT. Our exploratory evaluations reveal
that AT induces higher gradient norm and variance compared to NT. This
phenomenon hinders the outer optimization in AT since the convergence rate of
MSGD is highly dependent on the variance of the gradients. To this end, we
propose an optimization method called ENGM which regularizes the contribution
of each input example to the average mini-batch gradients. We prove that the
convergence rate of ENGM is independent of the variance of the gradients, and
thus, it is suitable for AT. We introduce a trick to reduce the computational
cost of ENGM using empirical observations on the correlation between the norm
of gradients w.r.t. the network parameters and input examples. Our extensive
evaluations and ablation studies on CIFAR-10, CIFAR-100, and TinyImageNet
demonstrate that ENGM and its variants consistently improve the performance of
a wide range of AT methods. Furthermore, ENGM alleviates major shortcomings of
AT including robust overfitting and high sensitivity to hyperparameter
settings.
