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

• URL: http://arxiv.org/abs/2510.21468v1
• Date: Fri, 24 Oct 2025 13:55:43 GMT
• Title: Finite-Time Analysis of Stochastic Nonconvex Nonsmooth Optimization on the Riemannian Manifolds
• Authors: Emre Sahinoglu, Youbang Sun, Shahin Shahrampour,
• Abstract summary: This work addresses the finitetime analysis of nonsmooth nonsmooth optimization under Riemannian manifold constraints.<n>We adapt the notion of nonsmooth optimization as a performance metric for nonsmooth optimization on a sample of $delta,epsilon$ .<n>We develop a zeroth version of RO2NC algorithm (ZORO2NC) which matches the optimal complexity in the gradient.<n>The numerical results support the practical effectiveness algorithms.
• Score: 10.615029276620511
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $O(\epsilon^{-3}\delta^{-1})$ 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.

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