Zeroth-Order Distributed Stochastic Optimization Over Riemannian Manifolds

Due to its ability to handle strict constraints on feasible domains, distributed optimization over a Riemannian manifold offers an attractive solution for many practical applications. To develop such an algorithm for scenarios where the explicit expression of the cost function is unavailable, we introduce the zeroth-order (ZO) Riemannian stochastic gradient into distributed optimization on a Riemannian manifold. Specifically, an intermediate estimate is first obtained through a local update step using the ZO Riemannian stochastic gradient, which is approximated based on two function evaluations. Subsequently, an improved estimate is derived by minimizing the weighted Fréchet mean over the manifold using information from neighboring nodes. To further enhance performance, a mini-batch strategy is incorporated into the gradient estimation process. Finally, simulation results are presented to validate the effectiveness of the proposed algorithm.