Distributed Stochastic Proximal Algorithm on Riemannian Submanifolds for Weakly-convex Functions

This paper aims to investigate the distributed stochastic optimization problems on compact embedded submanifolds (in the Euclidean space) for multi-agent network systems. To address the manifold structure, we propose a distributed Riemannian stochastic proximal algorithm framework by utilizing the retraction and Riemannian consensus protocol, and analyze three specific algorithms: the distributed Riemannian stochastic subgradient, proximal point, and prox-linear algorithms. When the local costs are weakly-convex and the initial points satisfy certain conditions, we show that the iterates generated by this framework converge to a nearly stationary point in expectation while achieving consensus. We further establish the convergence rate of the algorithm framework as $\mathcal{O}(\frac{1+\kappa_g}{\sqrt{k}})$ where $k$ denotes the number of iterations and $\kappa_g$ shows the impact of manifold geometry on the algorithm performance. Finally, numerical experiments are implemented to demonstrate the theoretical results and show the empirical performance.