This paper considers Robbins-Monro stochastic approximation when the regression function changes with time. At time $n$, one can select $X_n$ and observe an unbiased estimator of the regression function evaluated at $X_n$. Let $\theta_n$ be the root of the regression function at time $n$. Our goal is to select the sequence $X_n$ so that $X_n - \theta_n$ converges to 0. It is assumed that $\theta_n = f(s_n)$ for $s_n$ known at time $n$ and $f$ an unknown element of a class of functions. Under certain conditions on this class and on the sequence of regression functions, we obtain a random sequence $X_n$ such that $|X_n - \theta_n|$ converges to 0 in Cesaro mean with probability 1. Under more stringent conditions, $X_n - \theta_n$ converges to 0 with probability 1.
Publié le : 1979-11-14
Classification:
Stochastic approximation,
Robbins and Monro procedure,
changing root,
Cesaro mean convergence with probability one,
inner product space,
62L20,
62J99
@article{1176344839,
author = {Ruppert, David},
title = {A New Dynamic Stochastic Approximation Procedure},
journal = {Ann. Statist.},
volume = {7},
number = {1},
year = {1979},
pages = { 1179-1195},
language = {en},
url = {http://dml.mathdoc.fr/item/1176344839}
}
Ruppert, David. A New Dynamic Stochastic Approximation Procedure. Ann. Statist., Tome 7 (1979) no. 1, pp. 1179-1195. http://gdmltest.u-ga.fr/item/1176344839/