Decentralized Online Riemannian Optimization Extended to Positively Curved Manifolds
Researchers have developed a decentralized online Riemannian optimization framework that extends beyond Hadamard manifolds to handle spaces with positive curvature. The key innovation is a curvature-aware consensus step that achieves linear convergence even when geodesic distances do not induce a globally convex structure. The work establishes O(√T) regret bounds for both gradient descent and bandit feedback settings, broadening the applicability of decentralized optimization to a wider class of geometric spaces.
A preprint posted to arXiv presents a theoretical framework for decentralized online optimization over Riemannian manifolds that may have positive curvature, a setting previously difficult to handle because standard consensus techniques rely on Euclidean linearity or the globally convex structure of Hadamard (non-positively curved) manifolds. The authors introduce a curvature-aware Riemannian consensus step and prove it achieves linear convergence in this more general setting. Building on this, they establish an O(√T) regret bound for a decentralized online Riemannian gradient descent algorithm. The paper also addresses the two-point bandit feedback scenario, where only function evaluations rather than gradients are available, using smoothing-based gradient estimators and a subconvexity analysis of smoothed objectives to recover the same O(√T) regret guarantee. The results are significant for multi-agent machine learning and optimization problems where data or computation is distributed across nodes and the underlying parameter space has non-trivial geometric structure, such as covariance matrices or rotation groups.
What's missing
The paper is a preprint and has not yet undergone formal peer review. It is unclear how the curvature-aware consensus step scales computationally with network size or manifold dimension. The tightness of the O(√T) bound (i.e., whether a matching lower bound exists in this setting) is not addressed.
What different sources said
- arXiv cs.LGCenter
Decentralized Online Riemannian Optimization Beyond Hadamard Manifolds
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