Adversarially Robust Learning of Real-Valued Functions

We study robustness to test-time adversarial attacks in the regression setting with $\ell_p$ losses and arbitrary perturbation sets. We address the question of which function classes are PAC learnable in this setting. We show that classes of finite fat-shattering dimension are learnable. Moreover, for convex function classes, they are even properly learnable. In contrast, some non-convex function classes provably require improper learning algorithms. We also discuss extensions to agnostic learning. Our main technique is based on a construction of an adversarially robust sample compression scheme of a size determined by the fat-shattering dimension.