Decentralized Online Riemannian Optimization Beyond Hadamard Manifolds
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918138204913664 |
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| author | Sahinoglu, Emre Shahrampour, Shahin |
| author_facet | Sahinoglu, Emre Shahrampour, Shahin |
| contents | We study decentralized online Riemannian optimization over manifolds with possibly positive curvature, going beyond the Hadamard manifold setting. Decentralized optimization techniques rely on a consensus step that is well understood in Euclidean spaces because of their linearity. However, in positively curved Riemannian spaces, a main technical challenge is that geodesic distances may not induce a globally convex structure. In this work, we first analyze a curvature-aware Riemannian consensus step that enables a linear convergence beyond Hadamard manifolds. Building on this step, we establish a $O(\sqrt{T})$ regret bound for the decentralized online Riemannian gradient descent algorithm. Then, we investigate the two-point bandit feedback setup, where we employ computationally efficient gradient estimators using smoothing techniques, and we demonstrate the same $O(\sqrt{T})$ regret bound through the subconvexity analysis of smoothed objectives. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_07779 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Decentralized Online Riemannian Optimization Beyond Hadamard Manifolds Sahinoglu, Emre Shahrampour, Shahin Optimization and Control Machine Learning Multiagent Systems We study decentralized online Riemannian optimization over manifolds with possibly positive curvature, going beyond the Hadamard manifold setting. Decentralized optimization techniques rely on a consensus step that is well understood in Euclidean spaces because of their linearity. However, in positively curved Riemannian spaces, a main technical challenge is that geodesic distances may not induce a globally convex structure. In this work, we first analyze a curvature-aware Riemannian consensus step that enables a linear convergence beyond Hadamard manifolds. Building on this step, we establish a $O(\sqrt{T})$ regret bound for the decentralized online Riemannian gradient descent algorithm. Then, we investigate the two-point bandit feedback setup, where we employ computationally efficient gradient estimators using smoothing techniques, and we demonstrate the same $O(\sqrt{T})$ regret bound through the subconvexity analysis of smoothed objectives. |
| title | Decentralized Online Riemannian Optimization Beyond Hadamard Manifolds |
| topic | Optimization and Control Machine Learning Multiagent Systems |
| url | https://arxiv.org/abs/2509.07779 |