Online Optimization on Hadamard Manifolds: Curvature Independent Regret Bounds on Horospherically Convex Objectives
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911153621303296 |
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| author | Sahinoglu, Emre Shahrampour, Shahin |
| author_facet | Sahinoglu, Emre Shahrampour, Shahin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_11236 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Online Optimization on Hadamard Manifolds: Curvature Independent Regret Bounds on Horospherically Convex Objectives Sahinoglu, Emre Shahrampour, Shahin Machine Learning Optimization and Control 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. |
| title | Online Optimization on Hadamard Manifolds: Curvature Independent Regret Bounds on Horospherically Convex Objectives |
| topic | Machine Learning Optimization and Control |
| url | https://arxiv.org/abs/2509.11236 |