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Calibrated Surrogate Losses for Classification with Label-Dependent Costs

We present surrogate regret bounds for arbitrary surrogate losses in the context of binary classification with label-dependent costs. Such bounds relate a classifier's risk, assessed with respect to a surrogate loss, to its cost-sensitive classification risk. Two approaches to surrogate regret bounds are developed. The first is a direct generalization of Bartlett et al. [2006], who focus on margin-based losses and cost-insensitive classification, while the second adopts the framework of Steinwart [2007] based on calibration functions. Nontrivial surrogate regret bounds are shown to exist precisely when the surrogate loss satisfies a "calibration" condition that is easily verified for many common losses. We apply this theory to the class of uneven margin losses, and characterize when these losses are properly calibrated. The uneven hinge, squared error, exponential, and sigmoid losses are then treated in detail.