  Adversarial training has gained great popularity as one of the most effective
defenses for deep neural network and more generally for gradient-based machine
learning models against adversarial perturbations on data points. This paper
establishes a continuous-time approximation for the mini-max game of
adversarial training. This approximation approach allows for precise and
analytical comparisons between stochastic gradient descent and its adversarial
training counterpart; and confirms theoretically the robustness of adversarial
training from a new gradient-flow viewpoint. The analysis is then corroborated
through various analytical and numerical examples.
