Concentration Bounds for Stochastic Approximations

TL;DR

This paper derives non-asymptotic Gaussian concentration bounds for stochastic approximations, including Euler schemes of diffusion processes and stochastic algorithms, without bias or non-degeneracy assumptions.

Contribution

It provides new Gaussian concentration bounds for stochastic approximations, improving upon previous work by removing bias and non-degeneracy conditions.

Findings

Concentration bounds are Gaussian under suitable assumptions.

No systematic bias in the estimates for Euler schemes.

Applicable to stochastic approximation algorithms without degeneracy conditions.

Abstract

We obtain non asymptotic concentration bounds for two kinds of stochastic approximations. We first consider the deviations between the expectation of a given function of the Euler scheme of some diffusion process at a fixed deterministic time and its empirical mean obtained by the Monte-Carlo procedure. We then give some estimates concerning the deviation between the value at a given time-step of a stochastic approximation algorithm and its target. Under suitable assumptions both concentration bounds turn out to be Gaussian. The key tool consists in exploiting accurately the concentration properties of the increments of the schemes. For the first case, as opposed to the previous work of Lemaire and Menozzi (EJP, 2010), we do not have any systematic bias in our estimates. Also, no specific non-degeneracy conditions are assumed.