发表机构
Northeastern University（东北大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出分布式黎曼在线梯度下降（D-ROGD）算法，在Hadamard流形上针对h-凸函数实现曲率无关的遗憾界，分别达到O(√T)和O(log T)的静态遗憾，并通过实验验证。
AI 中文摘要
本文研究了Hadamard流形上的分布式在线黎曼优化问题。以往在测地凸性（g-凸性）假设下的工作，在优化分析中可能需要曲率信息，通常通过截面曲率的有限下界来体现。曲率也可能进入切空间黎曼共识方案的步长或收缩因子。本文针对一类更窄的球面凸（h-凸）函数，放宽了对曲率的依赖。我们研究了分布式黎曼在线梯度下降（D-ROGD），该方法将局部黎曼h-次梯度更新与隐式Fréchet均值共识相结合。对于h-凸和强h-凸的局部目标函数，我们分别建立了$O(\sqrt{T})$和$O(\log T)$的静态遗憾界，与相应的欧几里得速率在$T$上匹配，网络依赖性仅由谱间隙决定。据我们所知，这是首次在Hadamard流形上为分布式在线优化提供与曲率无关的遗憾保证。在双曲嵌入上的实验验证了预测的速率，且未观察到因曲率导致的性能退化。
英文摘要
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.
Comments6 pages, 2 figures
