The Adversarial Consistency of Surrogate Risks for Binary Classification
This work addresses a foundational issue in adversarial machine learning for researchers, offering theoretical insights into robust classifier design.
The paper tackles the problem of identifying which surrogate loss functions remain consistent for robust binary classification under adversarial perturbations, and provides a complete characterization of such surrogates, showing they are fewer than in standard settings.
We study the consistency of surrogate risks for robust binary classification. It is common to learn robust classifiers by adversarial training, which seeks to minimize the expected $0$-$1$ loss when each example can be maliciously corrupted within a small ball. We give a simple and complete characterization of the set of surrogate loss functions that are \emph{consistent}, i.e., that can replace the $0$-$1$ loss without affecting the minimizing sequences of the original adversarial risk, for any data distribution. We also prove a quantitative version of adversarial consistency for the $ρ$-margin loss. Our results reveal that the class of adversarially consistent surrogates is substantially smaller than in the standard setting, where many common surrogates are known to be consistent.