Tracking and regret bounds for online zeroth-order Euclidean and Riemannian optimisation

TL;DR

This paper develops zeroth-order online optimization algorithms for geodesically-convex functions on Riemannian manifolds, providing regret bounds and analyzing the influence of manifold curvature, with applications demonstrated through numerical simulations.

Contribution

It extends zeroth-order online optimization methods to Hadamard manifolds and derives the first regret bounds even in Euclidean settings, considering curvature effects.

Findings

Derived bounds on expected instantaneous tracking error.

Provided algorithm parameters minimizing performance.

Demonstrated applicability via numerical simulations on Karcher mean problem.

Abstract

We study numerical optimisation algorithms that use zeroth-order information to minimise time-varying geodesically-convex cost functions on Riemannian manifolds. In the Euclidean setting, zeroth-order algorithms have received a lot of attention in both the time-varying and time-invariant cases. However, the extension to Riemannian manifolds is much less developed. We focus on Hadamard manifolds, which are a special class of Riemannian manifolds with global nonpositive curvature that offer convenient grounds for the generalisation of convexity notions. Specifically, we derive bounds on the expected instantaneous tracking error, and we provide algorithm parameter values that minimise the algorithm's performance. Our results illustrate how the manifold geometry in terms of the sectional curvature affects these bounds. Additionally, we provide dynamic regret bounds for this online optimisation setting. To the best of our knowledge, these are the first regret bounds even for the Euclidean version of the problem. Lastly, via numerical simulations, we demonstrate the applicability of our algorithm on an online Karcher mean problem.