Transport-entropy inequalities and deviation estimates for stochastic approximation schemes

TL;DR

This paper develops new transport-entropy inequalities and deviation bounds for discrete stochastic approximation schemes, improving existing results and providing optimal concentration rates for averaged algorithms.

Contribution

It introduces novel transport-entropy inequalities for Euler schemes and stochastic approximation, enhancing deviation estimates and concentration bounds.

Findings

Improved deviation estimates for Euler discretizations of diffusion processes.

Non-asymptotic bounds for stochastic approximation with averaging.

Optimal concentration rates for averaged stochastic approximation algorithms.

Abstract

We obtain new transport-entropy inequalities and, as a by-product, new deviation estimates for the laws of two kinds of discrete stochastic approximation schemes. The first one refers to the law of an Euler like discretization scheme of a diffusion process at a fixed deterministic date and the second one concerns the law of a stochastic approximation algorithm at a given time-step. Our results notably improve and complete those obtained in [Frikha, Menozzi,2012]. The key point is to properly quantify the contribution of the diffusion term to the concentration regime. We also derive a general non-asymptotic deviation bound for the difference between a function of the trajectory of a continuous Euler scheme associated to a diffusion process and its mean. Finally, we obtain non-asymptotic bound for stochastic approximation with averaging of trajectories, in particular we prove that averaging a stochastic approximation algorithm with a slow decreasing step sequence gives rise to optimal concentration rate.