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1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

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.

Keywords : deviation bounds transportation-entropy inequalities Euler scheme stochastic approximation algorithms stochastic approximation with averaging





Author: Max Fathi - Noufel Frikha -

Source: https://hal.archives-ouvertes.fr/



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