Finite-Time Analysis of Stochastic Nonconvex Nonsmooth Optimization on the Riemannian Manifolds

Emre Sahinoglu, Youbang Sun, Shahin Shahrampour
10/24/2025
math.OCcs.LG

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Abstract

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 $O(ε^{-3}δ^{-1})$ in finding $(δ,ε)$-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.

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Cite this paper

@article{sahinoglu2025finitetime,
  title  = {Finite-Time Analysis of Stochastic Nonconvex Nonsmooth Optimization on the Riemannian Manifolds},
  author = {Emre Sahinoglu and Youbang Sun and Shahin Shahrampour},
  year   = {2025},
  eprint = {2510.21468},
  archivePrefix = {arXiv},
  url    = {http://arxiv.org/abs/2510.21468v1}
}

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