Convergence of Stochastic Approximation via Martingale and Converse Lyapunov Methods

TL;DR

This paper offers a new, simpler proof for the convergence of stochastic approximation algorithms using martingale methods, extending stability conditions beyond previous ODE-based approaches.

Contribution

It introduces an alternative martingale-based proof technique and new stability theorems, broadening the understanding of SA convergence and boundedness.

Findings

Provides a new sufficient condition for global asymptotic stability.

Establishes a converse Lyapunov theorem with a bounded Hessian.

Demonstrates the theory's applicability to cases beyond existing results.

Abstract

In this paper, we study the almost sure boundedness and the convergence of the stochastic approximation (SA) algorithm. At present, most available convergence proofs are based on the ODE method, and the almost sure boundedness of the iterations is an assumption and not a conclusion. In Borkar-Meyn (2000), it is shown that if the ODE has only one globally attractive equilibrium, then under additional assumptions, the iterations are bounded almost surely, and the SA algorithm converges to the desired solution. Our objective in the present paper is to provide an alternate proof of the above, based on martingale methods, which are simpler and less technical than those based on the ODE method. As a prelude, we prove a new sufficient condition for the global asymptotic stability of an ODE. Next we prove a "converse" Lyapunov theorem on the existence of a suitable Lyapunov function with a globally bounded Hessian, for a globally exponentially stable system. Both theorems are of independent interest to researchers in stability theory. Then, using these results, we provide sufficient conditions for the almost sure boundedness and the convergence of the SA algorithm. We show through examples that our theory covers some situations that are not covered by currently known results, specifically Borkar-Meyn (2000).