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.

翻译：我们研究在可能具有正曲率的流形上的去中心化在线黎曼优化，突破了哈达玛流形的限制。去中心化优化技术依赖于共识步骤，该步骤在欧几里得空间中因线性性质而易于理解。然而，在正曲率黎曼空间中，一个主要技术挑战是测地线距离可能无法诱导全局凸结构。本文首先分析了一种曲率感知的黎曼共识步骤，该步骤能够实现超越哈达玛流形的线性收敛。基于此步骤，我们为去中心化在线黎曼梯度下降算法建立了$O(\sqrt{T})$遗憾界。随后，我们研究了双点赌博反馈设置，采用基于平滑技术的计算高效梯度估计器，并通过平滑目标的次凸性分析证明了相同的$O(\sqrt{T})$遗憾界。