The Convexity and Design of Composite Multiclass Losses

Mark D. Reid, Robert C. Williamson, Peng Sun

Abstract: We consider composite loss functions for multiclass prediction comprising a proper (i.e., Fisher-consistent) loss over probability distributions and an inverse link function. We establish conditions for their (strong) convexity and explore the implications. We also show how the separation of concerns afforded by using this composite representation allows for the design of families of losses with the same Bayes risk.