Stochastic Approximation on Riemannian manifolds

TL;DR

This paper extends stochastic approximation theory to Riemannian manifolds, introducing new analysis tools for constrained stochastic algorithms on manifolds, including non-differentiable cases.

Contribution

It develops a framework for stochastic approximation on Riemannian manifolds using retraction mappings, including for non-differentiable constraints, expanding the applicability of SA methods.

Findings

Extended ODE method for Riemannian constraints

Framework for projected SA with approximate retractions

Applicable to non-differentiable constraint sets

Abstract

The standard theory of stochastic approximation (SA) is extended to the case when the constraint set is a Riemannian manifold. Specifically, the standard ODE method for analyzing SA schemes is extended to iterations constrained to stay on a manifold using a retraction mapping. In addition, for submanifolds of a Euclidean space, a framework is developed for a projected SA scheme with approximate retractions. The framework is also extended to non-differentiable constraint sets.