The exponentially weighted average forecaster in geodesic spaces of non-positive curvature

TL;DR

This paper extends the exponentially weighted average forecaster to geodesic spaces with non-positive curvature, leveraging barycenters and geometric insights for improved prediction and aggregation.

Contribution

It introduces a novel extension of the classical forecaster to non-Euclidean spaces with non-positive curvature, using barycenters and geometric methods.

Findings

Extended the forecaster to Alexandrov spaces with non-positive curvature

Adapted online to batch conversion in this geometric setting

Discussed applications in aggregation and barycenter estimation

Abstract

This paper addresses the problem of prediction with expert advice for outcomes in a geodesic space with non-positive curvature in the sense of Alexandrov. Via geometric considerations, and in particular the notion of barycenters, we extend to this setting the definition and analysis of the classical exponentially weighted average forecaster. We also adapt the principle of online to batch conversion to this setting. We shortly discuss the application of these results in the context of aggregation and for the problem of barycenter estimation.