Calibration and Consistency of Adversarial Surrogate Losses

Adversarial robustness is an increasingly critical property of classifiers in applications. The design of robust algorithms relies on surrogate losses since the optimization of the adversarial loss with most hypothesis sets is NP-hard. But which surrogate losses should be used and when do they benefit from theoretical guarantees? We present an extensive study of this question, including a detailed analysis of the H-calibration and H-consistency of adversarial surrogate losses. We show that, under some general assumptions, convex loss functions, or the supremum-based convex losses often used in applications, are not H-calibrated for important functions classes such as generalized linear models or one-layer neural networks. We then give a characterization of H-calibration and prove that some surrogate losses are indeed H-calibrated for the adversarial loss, with these function classes. Next, we show that H-calibration is not sufficient to guarantee consistency and prove that, in the absence of any distributional assumption, no continuous surrogate loss is consistent in the adversarial setting. This, in particular, falsifies a claim made in a COLT 2020 publication. Next, we identify natural conditions under which some surrogate losses that we describe in detail are H-consistent for function classes such as generalized linear models and one-layer neural networks. We also report a series of empirical results with simulated data, which show that many H-calibrated surrogate losses are indeed not H-consistent, and validate our theoretical assumptions.