Learning Classifiers with Fenchel-Young Losses: Generalized Entropies, Margins, and Algorithms

We study in this paper Fenchel-Young losses, a generic way to construct convex loss functions from a convex regularizer. We provide an in-depth study of their properties in a broad setting and show that they unify many well-known loss functions. When constructed from a generalized entropy, which includes well-known entropies such as Shannon and Tsallis entropies, we show that Fenchel-Young losses induce a predictive probability distribution and develop an efficient algorithm to compute that distribution for separable entropies. We derive conditions for generalized entropies to yield a distribution with sparse support and losses with a separation margin. Finally, we present both primal and dual algorithms to learn predictive models with generic Fenchel-Young losses.