Exploring Local Norms in Exp-concave Statistical Learning

02/21/2023

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by   Nikita Puchkin, et al.

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We consider the problem of stochastic convex optimization with exp-concave losses using Empirical Risk Minimization in a convex class. Answering a question raised in several prior works, we provide a O( d / n + log( 1 / δ) / n ) excess risk bound valid for a wide class of bounded exp-concave losses, where d is the dimension of the convex reference set, n is the sample size, and δ is the confidence level. Our result is based on a unified geometric assumption on the gradient of losses and the notion of local norms.