On Classification-Calibration of Gamma-Phi Losses

TL;DR

This paper establishes a general sufficient condition for the classification-calibration of Gamma-Phi losses, a family of multiclass loss functions, providing the first fully justified nonconvex surrogate losses for multiclass classification.

Contribution

It introduces the first general sufficient condition for classification-calibration of Gamma-Phi losses and corrects a misconception about a previously proposed condition.

Findings

First general sufficient condition for CC of Gamma-Phi losses

Identification of a previously proposed insufficient condition

Highlights a technical issue in multiclass CC analysis

Abstract

Gamma-Phi losses constitute a family of multiclass classification loss functions that generalize the logistic and other common losses, and have found application in the boosting literature. We establish the first general sufficient condition for the classification-calibration (CC) of such losses. To our knowledge, this sufficient condition gives the first family of nonconvex multiclass surrogate losses for which CC has been fully justified. In addition, we show that a previously proposed sufficient condition is in fact not sufficient. This contribution highlights a technical issue that is important in the study of multiclass CC but has been neglected in prior work.