Exploring Local Norms in Exp-concave Statistical Learning
This addresses a theoretical gap in statistical learning for exp-concave losses, offering improved risk bounds that could benefit researchers in optimization and machine learning, though it appears incremental as it builds on prior works.
The paper tackles the problem of stochastic convex optimization with exp-concave losses using Empirical Risk Minimization, providing an excess risk bound of O(d/n + log(1/δ)/n) for a wide class of bounded exp-concave losses, where d is dimension, n is sample size, and δ is confidence level.
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.