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Adversarial training is a popular method to give neural nets robustnessagainst adversarial perturbations. In practice adversarial training leads tolow robust training loss. However, a rigorous explanation for why this happensunder natural conditions is still missing. Recently a convergence theory forstandard (non-adversarial) supervised training was developed by various groupsfor {\em very overparametrized} nets. It is unclear how to extend these resultsto adversarial training because of the min-max objective. Recently, a firststep towards this direction was made by Gao et al. using tools from onlinelearning, but they require the width of the net to be \emph{exponential} ininput dimension $d$, and with an unnatural activation function. Our work provesconvergence to low robust training loss for \emph{polynomial} width instead ofexponential, under natural assumptions and with the ReLU activation. Keyelement of our proof is showing that ReLU networks near initialization canapproximate the step function, which may be of independent interest.

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