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Modern machine learning approaches to classification, including AdaBoost,support vector machines, and deep neural networks, utilize surrogate losstechniques to circumvent the computational complexity of minimizing empiricalclassification risk. These techniques are also useful for causal policylearning problems, since estimation of individualized treatment rules can becast as a weighted (cost-sensitive) classification problem. Consistency of thesurrogate loss approaches studied in Zhang (2004) and Bartlett et al. (2006)crucially relies on the assumption of correct specification, meaning that thespecified set of classifiers is rich enough to contain a first-best classifier.This assumption is, however, less credible when the set of classifiers isconstrained by interpretability or fairness, leaving the applicability ofsurrogate loss based algorithms unknown in such second-best scenarios. Thispaper studies consistency of surrogate loss procedures under a constrained setof classifiers without assuming correct specification. We show that in thesetting where the constraint restricts the classifier's prediction set only,hinge losses (i.e., $\ell_1$-support vector machines) are the only surrogatelosses that preserve consistency in second-best scenarios. If the constraintadditionally restricts the functional form of the classifier, consistency of asurrogate loss approach is not guaranteed even with hinge loss. We thereforecharacterize conditions for the constrained set of classifiers that canguarantee consistency of hinge risk minimizing classifiers. Exploiting ourtheoretical results, we develop robust and computationally attractive hingeloss based procedures for a monotone classification problem.

Constrained Classification and Policy Learning