具有有界截面曲率的黎曼流形上强测地凸函数的分散式在线黎曼优化

中文摘要 AI 辅助

我们研究了具有有界截面曲率（包括正曲率流形）的黎曼流形上强测地凸（强g-凸）损失的分散式在线优化。在集中式黎曼优化中，强g-凸性将最优遗憾从$O(\sqrt{T})$收紧到$O(\log T)$。然而，在分散式黎曼设置中，现有方法仅处理g-凸损失，强g-凸情况未被探索。挑战在于集中式所需的递减步长与现有网络误差分析不兼容。首先，我们为时变调度提供了一般网络误差分析。其次，基于此分析为分散式在线黎曼梯度下降建立了首个$O(\log T)$静态遗憾界，与强凸欧几里得在线优化的极小极大最优速率相匹配。最后，利用损失函数平滑版本的新颖强次凸性论证为两点策略反馈设置证明了相同的$O(\log T)$遗憾界。

英文摘要

We study decentralized online optimization for strongly geodesically convex (strongly g-convex) losses on Riemannian manifolds with bounded sectional curvature, including positively curved manifolds. In centralized Riemannian optimization, strong g-convexity tightens the optimal regret from $O(\sqrt{T})$ to $O(\log T)$, where $T$ is the time horizon; in the decentralized Riemannian setting, however, existing methods address only g-convex losses, leaving the strongly g-convex regime unexplored. One challenge is that the required decaying step size in the centralized regime is incompatible with existing network-error analyses, which typically assume a fixed step size. First, we provide a general network-error analysis for time-varying schedules. Next, we build on this analysis to establish the first $O(\log T)$ static regret bound for decentralized online Riemannian gradient descent, matching the minimax-optimal rate for strongly-convex Euclidean online optimization. Finally, we prove the same $O(\log T)$ regret bound for the two-point bandit feedback setting using novel strong subconvexity arguments for the smoothed versions of the loss functions.

发表机构

• Department of Mechanical & Industrial Engineering at Northeastern University（东北大学机械与工业工程系）

机构由 AI 辅助整理，请以论文原文为准。