Online learning with exponential weights in metric spaces

TL;DR

This paper extends online learning algorithms based on exponential weights to general metric spaces, utilizing barycenters and curvature bounds to analyze performance.

Contribution

It introduces a framework for online learning with exponential weights in metric spaces, generalizing Euclidean-based methods using barycenters and curvature concepts.

Findings

Extended exponential weights analysis to metric spaces

Developed barycenter-based online learning algorithms

Applied results to statistical learning via online-to-batch conversion

Abstract

This paper addresses the problem of online learning in metric spaces using exponential weights. We extend the analysis of the exponentially weighted average forecaster, traditionally studied in a Euclidean settings, to a more abstract framework. Our results rely on the notion of barycenters, a suitable version of Jensen's inequality and a synthetic notion of lower curvature bound in metric spaces known as the measure contraction property. We also adapt the online-to-batch conversion principle to apply our results to a statistical learning framework.