Stochastic approximations of set-valued dynamical systems: Convergence with positive probability to an attractor

TL;DR

This paper extends the analysis of stochastic approximation algorithms to set-valued dynamical systems, proving convergence with positive probability to attractors under certain conditions, generalizing previous results for differential equations and inclusions.

Contribution

It introduces a convergence result for stochastic approximations of set-valued dynamical systems, broadening the scope of prior work on differential equations and inclusions.

Findings

Proves convergence with positive probability to attractors in set-valued systems.

Extends previous results from differential equations to differential inclusions.

Provides a theoretical foundation for analyzing stochastic algorithms with set-valued dynamics.

Abstract

A succesful method to describe the asymptotic behavior of a discrete time stochastic process governed by some recursive formula is to relate it to the limit sets of a well chosen mean differential equation. Under an attainability condition, convergence to a given attractor of the flow induced by this dynamical system was proved to occur with positive probability (Bena\"im, 1999) for a class of Robbins Monro algorithms. Bena\"im et al. (2005) generalised this approach for stochastic approximation algorithms whose average behavior is related to a differential inclusion instead. We pursue the analogy by extending to this setting the result of convergence with positive probability to an attractor.