Decentralized Online Riemannian Optimization Beyond Hadamard Manifolds

• URL: http://arxiv.org/abs/2509.07779v1
• Date: Tue, 09 Sep 2025 14:14:46 GMT
• Title: Decentralized Online Riemannian Optimization Beyond Hadamard Manifolds
• Authors: Emre Sahinoglu, Shahin Shahrampour,
• Abstract summary: We analyze a curvature-aware geodesic step that enables a convergence beyond Hadamard linear distances.<n>We employ gradient, smoothing techniques, and we demonstrate $O(sqrtT)$ regret bound through the same subity analysis.
• Score: 9.940555460165545
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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