We study the consistency of surrogate risks for robust binary classification.

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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 $\rho$-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.

We extend the recent framework of Osokin et al.

We study the consistency of surrogate risks for robust binary classification.

We study the consistency of surrogate risks for robust binary classification.

From now on, we can therefore focus on proving (24).

A central concern in classification is the vulnerability of machine learning models to adversarial attacks. Adversarial training is one of the most popular techniques for training robust classifiers, which involves minimizing an adversarial surrogate risk. Recent work characterized when a minimizing sequence of an adversarial surrogate risk is also a minimizing sequence of the adversarial classification risk for binary classification-- a property known as adversarial consistency . However, these results do not address the rate at which the adversarial classification risk converges to its optimal value for such a sequence of functions that minimize the adversarial surrogate. This paper provides surrogate risk bounds that quantify that convergence rate. Additionally, we derive distribution-dependent surrogate risk bounds in the standard (non-adversarial) learning setting, that may be of independent interest.