Online Optimization on Hadamard Manifolds: Curvature Independent Regret Bounds on Horospherically Convex Objectives

• URL: http://arxiv.org/abs/2509.11236v1
• Date: Sun, 14 Sep 2025 12:27:31 GMT
• Title: Online Optimization on Hadamard Manifolds: Curvature Independent Regret Bounds on Horospherically Convex Objectives
• Authors: Emre Sahinoglu, Shahin Shahrampour,
• Abstract summary: We analyze the geodesicity framework (g-ity) for h-in-variants of the curvature manifold.<n>In particular, we investigate the application of a Fr-in-variant with the definite (SPD) with a gradient.
• Score: 9.940555460165545
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study online Riemannian optimization on Hadamard manifolds under the framework of horospherical convexity (h-convexity). Prior work mostly relies on the geodesic convexity (g-convexity), leading to regret bounds scaling poorly with the manifold curvature. To address this limitation, we analyze Riemannian online gradient descent for h-convex and strongly h-convex functions and establish $O(\sqrt{T})$ and $O(\log(T))$ regret guarantees, respectively. These bounds are curvature-independent and match the results in the Euclidean setting. We validate our approach with experiments on the manifold of symmetric positive definite (SPD) matrices equipped with the affine-invariant metric. In particular, we investigate online Tyler's $M$-estimation and online Fr\'echet mean computation, showing the application of h-convexity in practice.

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