突破传统限制，实现正曲率流形上的高效分布式在线优化
Decentralized Online Riemannian Optimization Beyond Hadamard Manifolds
- 设计自适应曲率的共识机制，支持非负曲率流形
- 证明算法在T轮内累积误差为O(√T)
- 适用于需要分布式学习的高维数据场景
我们研究在可能具有正曲率的流形上进行去中心化的在线黎曼优化，突破了传统哈达马德流形的限制。传统的分布式优化依赖于共识步骤，在欧氏空间中因线性性质而清晰明了，但在正曲率黎曼空间中，测地距离无法保证全局凸性，带来主要技术挑战。本文首先分析一种考虑曲率的黎曼共识步骤，实现了超越哈达马德流形的线性收敛。基于该步骤，建立了去中心化在线黎曼梯度下降算法的O(√T)损失界。随后，研究了两点老虎机反馈设置，采用平滑技术构造计算高效的梯度估计器，并通过平滑目标的次凸性分析，证明了相同的O(√T)损失界。
原文摘要 · Abstract (English)
We study decentralized online Riemannian optimization over manifolds with possibly positive curvature, going beyond the Hadamard manifold setting. Decentralized optimization techniques rely on a consensus step that is well understood in Euclidean spaces because of their linearity. However, in positively curved Riemannian spaces, a main technical challenge is that geodesic distances may not induce a globally convex structure. In this work, we first analyze a curvature-aware Riemannian consensus step that enables a linear convergence beyond Hadamard manifolds. Building on this step, we establish a $O(\sqrt{T})$ regret bound for the decentralized online Riemannian gradient descent algorithm. Then, we investigate the two-point bandit feedback setup, where we employ computationally efficient gradient estimators using smoothing techniques, and we demonstrate the same $O(\sqrt{T})$ regret bound through the subconvexity analysis of smoothed objectives.
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