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

2026-09-11

AI Summary

本文解决了Hadamard流形上分布式在线优化对曲率的依赖问题，通过使用h-凸函数和D-ROGD算法，实现了与欧氏空间相匹配的无曲率依赖的遗憾界。

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.

Problem

• Distributed Online Optimization
• Hadamard Manifolds
• Curvature-Independent
• h-convex Functions
• Regret Bounds

Innovation

• Curvature-Independent
• D-ROGD
• Hadamard Manifolds
• h-convex functions
• Regret Bounds