Calm Composite Losses: Being Improper Yet Proper Composite

Strict proper losses are fundamental loss functions inducing classifiers capable of estimating class probabilities.While practitioners have devised many loss functions, their properness is often unverified.In this paper, we identify several losses as improper, calling into question the validity of class probability estimates derived from their simplex-projected outputs.Nevertheless, we show that these losses are strictly proper composite with appropriate link functions, allowing predictions to be mapped into true class probabilities.We invent the calmness condition, which we prove suffices to identify that a loss has a strictly proper composite representation, and provide the general form of the inverse link.To further understand proper composite losses, we explore proper composite losses through the framework of property elicitation, revealing a connection between inverse link functions and Bregman projections.Numerical simulations are provided to demonstrate the behavior of proper composite losses and the effectiveness of the inverse link function.