On Stochastic Approximations Driven by Sample Averages: Convergence Results via the ODE Method

Abstract

We consider a class of projected stochastic approximation algorithms drive by sample averages. These algorithms arise naturally in problems of on-line parametric optimization for discrete event dynamical systems., e.g., queueing systems and Petri net models. We develop a general framework for investigating the a.s. convergence of the iterate sequence, and show how such convergence results can be obtained by means of the ordinary differential equation (ODE) method under a condition of exponential convergence. We relate this condition of exponential convergence to certain Large Deviations upper bounds which are uniform in both the parameter q and the initial condition x. To demonstrate the applicability of the results, we specialize them to two specific classes of state processes, namely sequences of i.i.d. random variables and finite state time-homogeneous Markov chains. In both cases, we identify simple (and checkable) conditions that ensure the validity of a uniform Large Deviations upper bound.