Noise conditions for prespecified convergence rates of stochastic approximation algorithms

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 {ρ_n} is a given positive sequence converging monotonically to zero. Consider a stochastic approximation algorithm x _n+1=x_n-a_n(A_nx_n-b _n)+a_ne_n, where {x_n} is the iterate sequence, {a_n} is the step size sequence, {e_n } is the noise sequence, and x* is the desired zero of the function f(x)=Ax-b. Then, under appropriate assumptions, we show that x _n-x*=o(ρ_n) if and only if the sequence {e_n} satisfies one of five equivalent conditions. These conditions are based on well-known formulas for noise sequences: Kushner and Clark´s (1978) condition, Chen´s (see Proc. IFAC World Congr., p.375-80, 1996) condition, Kulkarni and Horn´s (see IEEE Trails Automat. Contr., vol.41, p.419-24, 1996) condition, a decomposition condition, and a weighted averaging condition. Our necessary and sufficient condition on {e_n} to achieve a convergence rate of {ρ_n} is basically that the sequence {e_n/ρ _n} satisfies any one of the above five well-known conditions. We provide examples to illustrate our result. In particular, we easily recover the familiar result that if a_n=a/n and {e_n} is a martingale difference process with bounded variance, then x_n-x*=o(n^-1/2(log(n))^β ) for any β>1/2