Minimizing Dynamic Regret on Geodesic Metric Spaces

TL;DR

This paper develops adaptive online learning algorithms with optimistic regret bounds for minimizing dynamic regret on Riemannian manifolds, addressing challenges posed by curvature and extending Euclidean results to non-Euclidean spaces.

Contribution

It introduces the first algorithms for general dynamic regret on geodesic metric spaces, particularly manifolds with non-positive curvature, allowing improper learning.

Findings

Achieved optimistic regret bounds on non-positively curved manifolds

Developed adaptive no-regret algorithms for Riemannian settings

Extended online learning theory from Euclidean to geodesic metric spaces

Abstract

In this paper, we consider the sequential decision problem where the goal is to minimize the general dynamic regret on a complete Riemannian manifold. The task of offline optimization on such a domain, also known as a geodesic metric space, has recently received significant attention. The online setting has received significantly less attention, and it has remained an open question whether the body of results that hold in the Euclidean setting can be transplanted into the land of Riemannian manifolds where new challenges (e.g., curvature) come into play. In this paper, we show how to get optimistic regret bound on manifolds with non-positive curvature whenever improper learning is allowed and propose an array of adaptive no-regret algorithms. To the best of our knowledge, this is the first work that considers general dynamic regret and develops "optimistic" online learning algorithms which can be employed on geodesic metric spaces.