# Riemannian stochastic approximation algorithms

arXiv: https://arxiv.org/abs/2206.06795 · HTML: https://mlsanity.com/paper/2206.06795/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 70 references that are in the map.
- First posted on arXiv 14 Jun 2022.

## Builds on (most cited first, 12 of 12)

- [First-order Methods for Geodesically Convex Optimization](https://mlsanity.com/paper/1602.06053/) (2016)
- [Global rates of convergence for nonconvex optimization on manifolds](https://mlsanity.com/paper/1605.08101/) (2016)
- [The limits of min-max optimization algorithms: convergence to spurious non-critical sets](https://mlsanity.com/paper/2006.09065/) (2020)
- [Averaging Stochastic Gradient Descent on Riemannian Manifolds](https://mlsanity.com/paper/1802.09128/) (2018)
- [Escaping from saddle points on Riemannian manifolds](https://mlsanity.com/paper/1906.07355/) (2019)
- [Efficiently escaping saddle points on manifolds](https://mlsanity.com/paper/1906.04321/) (2019)
- [On the Characterization of Local Nash Equilibria in Continuous Games](https://mlsanity.com/paper/1411.2168/) (2014)
- [An accelerated first-order method for non-convex optimization on manifolds](https://mlsanity.com/paper/2008.02252/) (2020)
- [On Riemannian Stochastic Approximation Schemes with Fixed Step-Size](https://mlsanity.com/paper/2102.07586/) (2021)
- [Stochastic Approximation on Riemannian manifolds](https://mlsanity.com/paper/1711.10754/) (2017)
- [Convergence Analysis of Riemannian Stochastic Approximation Schemes](https://mlsanity.com/paper/2005.13284/) (2020)
- [Curvature-Dependant Global Convergence Rates for Optimization on Manifolds of Bounded Geometry](https://mlsanity.com/paper/2008.02517/) (2020)

## Built on by (most cited first, 2 of 2)

- [Minimizing Dynamic Regret on Geodesic Metric Spaces](https://mlsanity.com/paper/2302.08652/) (2023)
- [Stochastic Modified Flows for Riemannian Stochastic Gradient Descent](https://mlsanity.com/paper/2402.03467/) (2024)

## Abstract (arXiv)

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.