Abstract

While convex losses for binary classification are attractive due to the existence of numerous (provably) efficient methods for finding their global optima, they are sensitive to outliers. On the other hand, while the nonconvex 0–1 loss is robust to outliers, it is NP-hard to optimize and thus rarely directly optimized in practice. In this paper, however, we do just that: we explore a variety of practical methods for direct (approximate) optimization of the 0–1 loss based on branch and bound search, combinatorial search, and coordinate descent on smooth, differentiable relaxations of 0–1 loss. Empirically, we compare our proposed algorithms to logistic regression, SVM, and the Bayes point machine showing that the proposed 0–1 loss optimization algorithms perform at least as well and offer a clear advantage in the presence of outliers. To this end, we believe this work reiterates the importance of 0–1 loss and its robustness properties while challenging the notion that it is difficult to directly optimize.