On Loss Functions and Regret Bounds for Multi-Category Classification

We develop new approaches in multi-class settings for constructing loss functions and establishing corresponding regret bounds with respect to the zero-one or cost-weighted classification loss. provide general representations of losses by deriving inverse mappings from a concave generalized entropy through convex dissimilarity function related multi-distribution $f$ -divergence. This approach is then applied study both hinge-like proper scoring rules. In first case, we derive losses, which are tighter extensions outside probability simplex than geometrically simpler fewer non-differentiable edges. also establish bound all same as loss, thereby substantially extending improving existing results. second identify sets rules different types reveal interesting relationships between various composite currently use. and, applications, simple meaningful two specific These results generalize, time, previous two-class settings.