{"ID":2873062,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.07779","arxiv_id":"2509.07779","title":"Decentralized Online Riemannian Optimization Beyond Hadamard Manifolds","abstract":"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.","short_abstract":"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 Riemanni...","url_abs":"https://arxiv.org/abs/2509.07779","url_pdf":"https://arxiv.org/pdf/2509.07779v1","authors":"[\"Emre Sahinoglu\",\"Shahin Shahrampour\"]","published":"2025-09-09T14:14:46Z","proceeding":"math.OC","tasks":"[\"math.OC\",\"cs.LG\",\"cs.MA\"]","methods":"[]","has_code":false}