  We analyze the properties of adversarial training for learning adversarially
robust halfspaces in the presence of agnostic label noise. Denoting
$\mathsf{OPT}_{p,r}$ as the best robust classification error achieved by a
halfspace that is robust to perturbations of $\ell_{p}$ balls of radius $r$, we
show that adversarial training on the standard binary cross-entropy loss yields
adversarially robust halfspaces up to (robust) classification error $\tilde
O(\sqrt{\mathsf{OPT}_{2,r}})$ for $p=2$, and $\tilde O(d^{1/4}
\sqrt{\mathsf{OPT}_{\infty, r}} + d^{1/2} \mathsf{OPT}_{\infty,r})$ when
$p=\infty$. Our results hold for distributions satisfying anti-concentration
properties enjoyed by log-concave isotropic distributions among others. We
additionally show that if one instead uses a nonconvex sigmoidal loss,
adversarial training yields halfspaces with an improved robust classification
error of $O(\mathsf{OPT}_{2,r})$ for $p=2$, and $O(d^{1/4}\mathsf{OPT}_{\infty,
r})$ when $p=\infty$. To the best of our knowledge, this is the first work to
show that adversarial training provably yields robust classifiers in the
presence of noise.
