arXiv2026
We study a class of nonsmooth composite stochastic optimization on Riemannian manifolds, where the objective is the sum of an expectation function and a nonsmooth regularizer. These types of problems appear widely in various application fields, such as machine learning. Although operator-splitting methods naturally exploit this separable structure, existing Riemannian variants primarily target deterministic problems, and stochastic extensions are limited to more restrictive settings. In this work, we propose a momentum-based adaptive Riemannian stochastic alternating direction method of multipliers (MARS-ADMM), which combines a recursive variance-reduced estimator for the expectation component with a prediction-correction update for the nonsmooth block. This yields a single-loop algorithm requiring only two proximal updates, one Riemannian gradient evaluation, and $\mathcal{O}(1)$ stochastic gradient samples per iteration. Under standard assumptions, we prove that MARS-ADMM attains an $ε$-KKT point with an oracle complexity of $\mathcal{O}(ε^{-3})$, improving upon the previously best-known rate of $\widetilde{\mathcal{O}}(ε^{-3.5})$ for stochastic Riemannian primal-dual methods. This complexity also matches the best-known bounds in deterministic nonsmooth Riemannian optimization, demonstrating that deterministic-level accuracy can be achieved using only constant-size stochastic samples. Numerical experiments on two types of test problems reveal promising performances of the proposed algorithm. To the best of our knowledge, MARS-ADMM is the first stochastic Riemannian ADMM with provable optimal complexity guarantees.