Stochastic Approximation for Smooth Functions
Václav Fabian · The Annals of Mathematical Statistics · 1969
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 $X_1, X_2, \cdots, X_n$.