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.

Optimization and Control (math.OC); Machine Learning (cs.LG); Multiagent Systems (cs.MA)

arXiv:2509.07779 [math.OC]

(or arXiv:2509.07779v2 [math.OC] for this version)

https://doi.org/10.48550/arXiv.2509.07779 Focus to learn more arXiv-issued DOI via DataCite

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