Electronic Communications in Probability

Concentration bounds for stochastic approximations

Noufel Frikha and Stéphane Menozzi

Full-text: Open access

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 an Euler like discretization 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. Also, no specific non-degeneracy conditions are assumed.

An Erratum is available in ECP volume 17 paper number 60.

Article information

Source
Electron. Commun. Probab., Volume 17 (2012), paper no. 47, 15 pp.

Dates
Accepted: 7 October 2012
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465263180

Digital Object Identifier
doi:10.1214/ECP.v17-1952

Mathematical Reviews number (MathSciNet)
MR2988393

Zentralblatt MATH identifier
1252.60065

Subjects
Primary: 60H35: Computational methods for stochastic equations [See also 65C30]
Secondary: 65C30: Stochastic differential and integral equations 65C05: Monte Carlo methods

Keywords
Non asymptotic bounds Euler scheme Stochastic approximation algorithms Gaussian concentration

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Frikha, Noufel; Menozzi, Stéphane. Concentration bounds for stochastic approximations. Electron. Commun. Probab. 17 (2012), paper no. 47, 15 pp. doi:10.1214/ECP.v17-1952. https://projecteuclid.org/euclid.ecp/1465263180


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References

  • Bally, V.; Talay, D. The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function. Probab. Theory Related Fields 104 (1996), no. 1, 43–60.
  • Bhattacharya, R. N.; Ranga Rao, R. Normal approximation and asymptotic expansions. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, New York-London-Sydney, 1976. xiv+274 pp.
  • Blower, Gordon; Bolley, François. Concentration of measure on product spaces with applications to Markov processes. Studia Math. 175 (2006), no. 1, 47–72.
  • Boissard, Emmanuel. Simple bounds for convergence of empirical and occupation measures in 1-Wasserstein distance. Electron. J. Probab. 16 (2011), no. 83, 2296–2333.
  • Bolley, François; Guillin, Arnaud; Villani, Cédric. Quantitative concentration inequalities for empirical measures on non-compact spaces. Probab. Theory Related Fields 137 (2007), no. 3-4, 541–593.
  • Bolley, François; Villani, Cédric. Weighted Csiszár-Kullback-Pinsker inequalities and applications to transportation inequalities. Ann. Fac. Sci. Toulouse Math. (6) 14 (2005), no. 3, 331–352.
  • Duflo, Marie. Algorithmes stochastiques. (French) [Stochastic algorithms] Mathématiques & Applications (Berlin) [Mathematics & Applications], 23. Springer-Verlag, Berlin, 1996. xiv+319 pp. ISBN: 3-540-60699-8
  • Konakov, Valentin; Mammen, Enno. Local limit theorems for transition densities of Markov chains converging to diffusions. Probab. Theory Related Fields 117 (2000), no. 4, 551–587.
  • Konakov, Valentin; Mammen, Enno. Local approximations of Markov random walks by diffusions. Stochastic Process. Appl. 96 (2001), no. 1, 73–98.
  • Konakov, Valentin; Mammen, Enno. Edgeworth type expansions for Euler schemes for stochastic differential equations. Monte Carlo Methods Appl. 8 (2002), no. 3, 271–285.
  • Kushner, Harold J.; Yin, G. George. Stochastic approximation and recursive algorithms and applications. Second edition. Applications of Mathematics (New York), 35. Stochastic Modelling and Applied Probability. Springer-Verlag, New York, 2003. xxii+474 pp. ISBN: 0-387-00894-2
  • Laruelle, Sophie; Pagès, Gilles. Stochastic approximation with averaging innovation applied to finance. Monte Carlo Methods Appl. 18 (2012), no. 1, 1–51.
  • Lemaire, V.; Menozzi, S. On some non asymptotic bounds for the Euler scheme. Electron. J. Probab. 15 (2010), no. 53, 1645–1681.
  • Malrieu, Florent; Talay, Denis. Concentration inequalities for Euler schemes. Monte Carlo and quasi-Monte Carlo methods 2004, 355–371, Springer, Berlin, 2006.
  • Shiryaev, A. N. Probability. Translated from the first (1980) Russian edition by R. P. Boas. Second edition. Graduate Texts in Mathematics, 95. Springer-Verlag, New York, 1996. xvi+623 pp. ISBN: 0-387-94549-0
  • Talay, Denis; Tubaro, Luciano. Expansion of the global error for numerical schemes solving stochastic differential equations. Stochastic Anal. Appl. 8 (1990), no. 4, 483–509 (1991).