This work addresses the finite-time analysis of nonsmooth nonconvex stochastic optimization under Riemannian manifold constraints. We adapt the notion of Goldstein stationarity to the Riemannian setting as a performance metric for nonsmooth optimization on manifolds. We then propose a Riemannian Online to NonConvex (RO2NC) algorithm, for which we establish the sample complexity of $O(\epsilon^{-3}\delta^{-1})$ in finding $(\delta,\epsilon)$-stationary points. This result is the first-ever finite-time guarantee for fully nonsmooth, nonconvex optimization on manifolds and matches the optimal complexity in the Euclidean setting. When gradient information is unavailable, we develop a zeroth order version of RO2NC algorithm (ZO-RO2NC), for which we establish the same sample complexity. The numerical results support the theory and demonstrate the practical effectiveness of the algorithms.

翻译：本研究针对黎曼流形约束下的非光滑非凸随机优化问题，提出了有限时间分析框架。我们将Goldstein平稳性概念推广至黎曼几何场景，作为流形上非光滑优化的性能度量指标。随后，我们提出一种黎曼在线转非凸（RO2NC）算法，并证明该算法在寻找$(\delta,\epsilon)$-平稳点时具有$O(\epsilon^{-3}\delta^{-1})$的样本复杂度。该结果是流形上完全非光滑、非凸优化问题的首个有限时间收敛保证，且与欧几里得空间中的最优复杂度相匹配。在梯度信息不可得的情况下，我们进一步开发了RO2NC算法的零阶版本（ZO-RO2NC），并证明了其具有相同的样本复杂度。数值实验结果验证了理论分析，并证明了算法的实际有效性。