Curvature-Independent Regret Bounds for Distributed Online Optimization on Hadamard Manifolds

Abstract:This work addresses decentralized online Riemannian optimization on Hadamard manifolds. Prior work under geodesic convexity (g-convexity) may require curvature information in the optimization analysis, typically through a finite lower bound on the sectional curvature. Curvature may also enter the step size or contraction factor of tangent-space Riemannian consensus schemes. In this work, we relax the curvature dependence for a narrower class of horospherical convex (h-convex) functions. We study Distributed Riemannian Online Gradient Descent (D-ROGD), which combines local Riemannian h-subgradient updates with an implicit Fréchet-mean consensus. For h-convex and strongly h-convex local objectives, we establish $O(\sqrt{T})$ and $O(\log T)$ static regret, respectively, matching the corresponding Euclidean rates with respect to $T$, with network dependence governed solely by the spectral gap. To our knowledge, these are the first curvature-independent regret guarantees for decentralized online optimization on Hadamard manifolds. Experiments on hyperbolic embeddings corroborate the predicted rates, with no observable degradation due to curvature.

Machine Learning (cs.LG); Optimization and Control (math.OC); Machine Learning (stat.ML)

arXiv:2609.13646 [cs.LG]

(or arXiv:2609.13646v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2609.13646 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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