Stochastic approximation in infinite dimensions

Abstract

Stochastic Approximation (SA) was introduced in the early 1950's and has been an active area of research for several decades. While the initial focus was on statistical questions, it was seen to have applications to signal processing, convex optimisation. %Over the last decade, there has been a revival of interest in SA as In later years SA has found application in Reinforced Learning (RL) and led to revival of interest. While bulk of the literature is on SA for the case when the observations are from a finite dimensional Euclidian space, there has been interest in extending the same to infinite dimension. Extension to Hilbert spaces is relatively easier to do, but this is not so when we come to a Banach space - since in the case of a Banach space, even {\em law of large numbers} is not true in general. We consider some cases where approximation works in a Banach space. Our framework includes case when the Banach space $\Bb$ is $\Cb([0,1],\R^d)$, as well as $\L^1([0,1],\R^d)$, the two cases which do not even have the Radon-Nikodym property.