Calibrated Surrogate Losses for Adversarially Robust Classification
This addresses the theoretical consistency of optimization methods for adversarial robustness, which is crucial for reliable machine learning in security-sensitive domains, but it is incremental as it builds on prior work on convex surrogates and calibration.
The paper tackles the problem of finding consistent surrogate losses for adversarially robust classification, showing that no convex surrogate is calibrated for linear models, but nonconvex losses can be calibrated under certain conditions, and convex losses may work under Massart's noise condition.
Adversarially robust classification seeks a classifier that is insensitive to adversarial perturbations of test patterns. This problem is often formulated via a minimax objective, where the target loss is the worst-case value of the 0-1 loss subject to a bound on the size of perturbation. Recent work has proposed convex surrogates for the adversarial 0-1 loss, in an effort to make optimization more tractable. A primary question is that of consistency, that is, whether minimization of the surrogate risk implies minimization of the adversarial 0-1 risk. In this work, we analyze this question through the lens of calibration, which is a pointwise notion of consistency. We show that no convex surrogate loss is calibrated with respect to the adversarial 0-1 loss when restricted to the class of linear models. We further introduce a class of nonconvex losses and offer necessary and sufficient conditions for losses in this class to be calibrated. We also show that if the underlying distribution satisfies Massart's noise condition, convex losses can also be calibrated in the adversarial setting.