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.
翻译:我们研究了一系列广泛的随机近似算法,以解决里曼尼方块上的非线性问题。这种算法自然地出现在里曼尼优化、游戏理论和最佳运输的研究中,但与欧几里德案相比,它们的行为远不那么为人所理解。我们通过引入一个合适的Fermi协调框架克服了这一困难,这个框架使我们能够将里曼尼罗·罗宾斯-蒙罗(RRM)的无症状算法与一个相关的确定性动态系统(RRM)的无症状算法(RRM)绘制成图。在这样做时,我们提供了一个几乎可以肯定的趋同结果的一般模板,它反映并扩展了欧几里登-罗宾斯-蒙特罗计划的现有理论,尽管由于根本方块的曲解和地貌学而出现重大复杂因素。我们展示了拟议框架的灵活性,将它应用于一系列基于回溯性、乐观/超等级方法的、解决最小化问题和游戏的折叠变法。我们提供了一种统一的处理办法。