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# Convexity of Proper Composite Binary Losses

*Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics*, PMLR 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.