Abstract
Surrogate losses underlie numerous state-of-the-art binary classification algorithms, such as support vector machines and boosting. The impact of a surrogate loss on the statistical performance of an algorithm is well-understood in symmetric classification settings, where the misclassification costs are equal and the loss is a margin loss. In particular, classification-calibrated losses are known to imply desirable properties such as consistency. While numerous efforts have been made to extend surrogate loss-based algorithms to asymmetric settings, to deal with unequal misclassification costs or training data imbalance, considerably less attention has been paid to whether the modified loss is still calibrated in some sense. This article extends the theory of classification-calibrated losses to asymmetric problems. As in the symmetric case, it is shown that calibrated asymmetric surrogate losses give rise to excess risk bounds, which control the expected misclassification cost in terms of the excess surrogate risk. This theory is illustrated on the class of uneven margin losses, and the uneven hinge, squared error, exponential, and sigmoid losses are treated in detail.
Highlights
Surrogate losses are key ingredients in many of the most successful modern classification algorithms, including support vector machines and boosting
We develop the notion of α-classification calibrated losses, and show that non-trivial excess risk bounds exist when L is α-classification calibrated, where α ∈ (0, 1) represents the misclassification cost asymmetry
To illustrate the theory of calibrated asymmetric surrogate losses, we study in some detail the class of uneven margin losses, which have the form
Summary
Surrogate losses are key ingredients in many of the most successful modern classification algorithms, including support vector machines and boosting. These losses are valued for their computational qualities, such as convexity, and facilitate the development of efficient algorithms for large-scale data sets. Given the considerable interest in asymmetric binary classification problems, and given the proliferation of heuristic asymmetric surrogate losses, there is a need for a theory to guide practitioners in the design of such losses, and to enable performance analysis. An elegant theory for calibrated surrogate losses was developed by Bartlett, Jordan and McAuliffe [2] and extended by Steinwart [31] These works do not consider the asymmetric classification problem considered here. Appendices A, B, and C, respectively, contain additional connections to Steinwart [31] and calibration functions, proofs of supporting lemmas, and uneven margin loss details
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