On loss functions and regret bounds for multi-category classification
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We develop new approaches in multi-class settings for constructing proper scoring rules and hinge-like losses and establishing corresponding regret bounds with respect to the zero-one or cost-weighted classification loss. Our construction of losses involves deriving new inverse mappings from a concave generalized entropy to a loss through the use of a convex dissimilarity function related to the multi-distribution f-divergence. We identify new classes of multi-class proper scoring rules, which recover and reveal interesting relationships between various composite losses currently in use. We also derive new hinge-like convex losses, which are tighter convex extensions than related hinge-like losses and geometrically simpler with fewer non-differentiable edges, while achieving similar regret bounds. Finally, we establish new classification regret bounds in general for multi-class proper scoring rules by exploiting the Bregman divergences of the associated generalized entropies, and, as applications, provide simple meaningful regret bounds for two specific classes of proper scoring rules.
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