{"ID":2871341,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.11236","arxiv_id":"2509.11236","title":"Online Optimization on Hadamard Manifolds: Curvature Independent Regret Bounds on Horospherically Convex Objectives","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 Fréchet mean computation, showing the application of h-convexity in practice.","short_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 grad...","url_abs":"https://arxiv.org/abs/2509.11236","url_pdf":"https://arxiv.org/pdf/2509.11236v1","authors":"[\"Emre Sahinoglu\",\"Shahin Shahrampour\"]","published":"2025-09-14T12:27:31Z","proceeding":"cs.LG","tasks":"[\"cs.LG\",\"math.OC\",\"stat.ML\"]","methods":"[]","has_code":false}