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Transport-entropy inequalities and deviation estimates for stochastic approximation schemes. (English) Zbl 1284.60137

Summary: 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 by N. Frikha and S. Menozzi [Electron. Commun. Probab. 17, Paper No. 47, 15 p. (2012; Zbl 1252.60065)]. 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 bounds for stochastic approximations with averaging of trajectories, in particular, we prove that averaging a stochastic approximation algorithm with a slow decreasing step sequence gives rise to an optimal concentration rate.

MSC:

60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
65C30 Numerical solutions to stochastic differential and integral equations
65C05 Monte Carlo methods

Citations:

Zbl 1252.60065
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