Limit theorems on the Robbins-Monro process for different variance behaviors of the stochastic gradient

Abstract

For finding a solution x* ∈ ℝ^r of the system of nonlinear equations G(x) = 0 in case that only estimations $\hat{G}\left( x \right) - G\left( x \right) + Z\left( x \right)$ at each x ∈ ℝ^r are available, we consider the stochastic approximation procedure

$$ {{X}_{{n + 1}}}: - {{X}_{n}} - {{r}_{n}}{{\hat{G}}_{n}}\left( {{{X}_{n}}} \right), where {{\hat{G}}_{n}}({{X}_{n}}): - G\left( {{{X}_{n}}} \right) + {{Z}_{n}}, n - 1,2, \ldots , $$

((1))

with the estimation error Z_n: -Z(X_n). In this paper the limiting distribution of the random sequence ${{\left( {{{\tau }_{n}}\left( {{{X}_{n}} - {{x}^{*}}} \right)} \right)}_{n}}$ is considered for different sequences (r_n)_n of deterministic step sizes, where (τ _n)_n is a sequence of positive numbers such that

$$ E{{\left\| {{{X}_{n}} - {{x}^{*}}} \right\|}^{2}} - 0\left( {{{1} \left/ {{\tau _{n}^{2}}} \right.}} \right). $$

Approximations for the limit covariance matrices of (τ _n(X_n-x*))_n are given for the case where the estimation error Z_n in (1) has different variances for indices n from different subsets ℕ^(k) of ℕ. Particular attention is paid to the semi-stochastic case where Z -0 for all n contained in an infinite subset N⁽¹⁾ of ℕ.