On Classification-Calibration of Gamma-Phi Losses
This addresses a technical gap in multiclass classification theory, particularly for boosting applications, but is incremental as it builds on existing loss function families.
The paper tackles the problem of establishing classification-calibration for Gamma-Phi losses in multiclass classification, showing that a previously proposed sufficient condition is insufficient and providing the first general sufficient condition for such nonconvex losses.
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.