Provable Robustness of Adversarial Training for Learning Halfspaces with Noise

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.