Asynchronous Stochastic Approximation with Differential Inclusions

TL;DR

This paper extends the asymptotic pseudo-trajectory approach to asynchronous stochastic approximation with set-valued mean fields, enabling convergence analysis without many previous restrictions and demonstrating applicability in Markov decision processes.

Contribution

It introduces a new framework for asynchronous stochastic approximation with set-valued mean fields, removing many restrictive assumptions and extending to coupled two-timescale processes.

Findings

Convergence results similar to synchronous processes are established.

Framework applicability is demonstrated in Markov decision process learning.

Many restrictive assumptions of prior asynchronous methods are removed.

Abstract

The asymptotic pseudo-trajectory approach to stochastic approximation of Benaim, Hofbauer and Sorin is extended for asynchronous stochastic approximations with a set-valued mean field. The asynchronicity of the process is incorporated into the mean field to produce convergence results which remain similar to those of an equivalent synchronous process. In addition, this allows many of the restrictive assumptions previously associated with asynchronous stochastic approximation to be removed. The framework is extended for a coupled asynchronous stochastic approximation process with set-valued mean fields. Two-timescales arguments are used here in a similar manner to the original work in this area by Borkar. The applicability of this approach is demonstrated through learning in a Markov decision process.