Over-parameterized Adversarial Training: An Analysis Overcoming the Curse of Dimensionality

Adversarial training is a popular method to give neural nets robustness against adversarial perturbations. In practice adversarial training leads to low robust training loss. However, a rigorous explanation for why this happens under natural conditions is still missing. Recently a convergence theory of standard (non-adversarial) supervised training was developed by various groups for {\em very overparametrized} nets. It is unclear how to extend these results to adversarial training because of the min-max objective.