  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 for
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. Recently, a first
step towards this direction was made by Gao et al. using tools from online
learning, but they require the width of the net to be \emph{exponential} in
input dimension $d$, and with an unnatural activation function. Our work proves
convergence to low robust training loss for \emph{polynomial} width instead of
exponential, under natural assumptions and with the ReLU activation. Key
element of our proof is showing that ReLU networks near initialization can
approximate the step function, which may be of independent interest.
