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.

翻译：Gamma-Phi损失函数是一类多类别分类损失函数族，它推广了逻辑损失及其他常见损失函数，并在提升方法文献中得到应用。我们首次建立了此类损失函数分类校准(CC)的通用充分条件。据我们所知，该充分条件给出了首个完全证明具有CC特性的非凸多类别替代损失函数族。此外，我们证明先前提出的一个充分条件实际上并不充分。这一贡献揭示了多类别分类校准研究中一个重要的技术问题，该问题在现有工作中长期被忽视。