Riemannian stochastic approximation algorithms
This work addresses a foundational gap in optimization theory for researchers in Riemannian optimization, game theory, and optimal transport, offering a unified convergence analysis for various algorithms.
The paper tackles the problem of analyzing stochastic approximation algorithms on Riemannian manifolds, where the lack of a global linear structure complicates understanding compared to Euclidean cases, and it provides a general framework for almost sure convergence by mapping the algorithms to deterministic dynamical systems using Fermi coordinates.
We examine a wide class of stochastic approximation algorithms for solving (stochastic) nonlinear problems on Riemannian manifolds. Such algorithms arise naturally in the study of Riemannian optimization, game theory and optimal transport, but their behavior is much less understood compared to the Euclidean case because of the lack of a global linear structure on the manifold. We overcome this difficulty by introducing a suitable Fermi coordinate frame which allows us to map the asymptotic behavior of the Riemannian Robbins-Monro (RRM) algorithms under study to that of an associated deterministic dynamical system. In so doing, we provide a general template of almost sure convergence results that mirrors and extends the existing theory for Euclidean Robbins-Monro schemes, despite the significant complications that arise due to the curvature and topology of the underlying manifold. We showcase the flexibility of the proposed framework by applying it to a range of retraction-based variants of the popular optimistic / extra-gradient methods for solving minimization problems and games, and we provide a unified treatment for their convergence.