Stochastic Approximation for Smooth Functions

Abstract

The problem of approximating a point $\theta$ of minimum of a function $f \varepsilon \mathscr{C}$ (see 2.1) is considered. An approximation procedure of the type described in Fabian (1967) using the design described in Fabian (1968), but with the size of design increasing, achieves the speed \begin{equation*}\tag{1}E|X_n - \theta|^2 = o(t^{-1}_n \log ^3 t_n);\end{equation*} here $X_n$ is the $n$th approximation and $t_n$ the number of observations necessary to construct $X_1, X_2, \cdots, X_n$.