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

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.

💡 Research Summary

The paper studies the classic problem of prediction with expert advice in the setting where outcomes and predictions lie in a geodesic metric space of non‑positive curvature (NPC), also known as a Hadamard or CAT(0) space. In Euclidean spaces the exponentially weighted average (EWA) forecaster combines expert predictions by a linear weighted average; this construction relies on the vector space structure and convexity of the loss. The authors replace the linear average with the barycenter (intrinsic mean) of a probability distribution over the expert predictions, thereby obtaining a natural generalization of EWA to arbitrary NPC spaces.

The main technical contributions are as follows:

1. Geometric Foundations – Section 2 reviews NPC spaces, geodesic convexity, and the notion of barycenters. Key facts include the convexity of the squared distance function (2‑convexity) and the existence‑uniqueness of barycenters for probability measures with finite second moment (Theorem 2.9). A generalized Jensen inequality for geodesically convex functions (Theorem 2.10) is also recalled.

2. Generalized EWA Forecaster – In Section 3.1 the forecaster is defined by first forming the exponential weights
   \