Convexity of Proper Composite Binary Losses

Mark Reid, Robert Williamson ; JMLR W&CP 9:637-644, 2010.

Abstract

A composite loss assigns a penalty to a real-valued prediction by associating the prediction with a probability via a link function then applying a class probability estimation (CPE) loss. If the risk for a composite loss is always minimised by predicting the value associated with the true class probability the composite loss is proper. We provide a novel, explicit and complete characterisation of the convexity of any proper composite loss in terms of its link and its ``weight function'' associated with its proper CPE loss.