On conditions for convergence rates of stochastic approximation algorithms

Abstract

We develop deterministic necessary and sufficient conditions on individual noise sequences of a stochastic approximation algorithm for the error of the iterates to converge at a given rate. Specifically, suppose {p/sub n/} is a given positive sequence converging monotonically to 0. Consider a stochastic approximation algorithm x/sub n+1/=x/sub n/-a/sub n/(A/sub n/x/sub n/-b/sub n/)+a/sub n/e/sub n/, where {x/sub n/} is the iterate sequence, {a/sub n/} is the step size sequence, {e/sub n/} is the noise sequence, and x* is the desired zero of the function f(x)=Ax-b. We show that x/sub n/-x*=o(/spl rho//sub n/) if and only if the sequence {e/sub n/} satisfies one of five equivalent conditions. These conditions are based on well known formulas for noise sequences found in the literature.