Electronic Journal of Probability

Transport-Entropy inequalities and deviation estimates for stochastic approximation schemes

Max Fathi and Noufel Frikha

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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.

Article information

Source
Electron. J. Probab. Volume 18 (2013), paper no. 67, 36 pp.

Dates
Accepted: 6 July 2013
First available in Project Euclid: 4 June 2016

Permanent link to this document
http://projecteuclid.org/euclid.ejp/1465064292

Digital Object Identifier
doi:10.1214/EJP.v18-2586

Mathematical Reviews number (MathSciNet)
MR3084653

Zentralblatt MATH identifier
1284.60137

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

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

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

Citation

Fathi, Max; Frikha, Noufel. Transport-Entropy inequalities and deviation estimates for stochastic approximation schemes. Electron. J. Probab. 18 (2013), paper no. 67, 36 pp. doi:10.1214/EJP.v18-2586. http://projecteuclid.org/euclid.ejp/1465064292.


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References

  • A. Alfonsi and A. Kohatsu-Higa and B. Jourdain: Pathwise optimal transport bounds between a one-dimensional diffusion and its Euler scheme. arXiv.org:1209.0576
    Mathematical Reviews (MathSciNet): MR3199980
    Digital Object Identifier: doi:10.1214/13-AAP941
    Project Euclid: euclid.aoap/1398258095
  • Bally, Vlad; Talay, Denis. The law of the Euler scheme for stochastic differential equations. II. Convergence rate of the density. Monte Carlo Methods Appl. 2 (1996), no. 2, 93–128.
    Mathematical Reviews (MathSciNet): MR1401964
    Digital Object Identifier: doi:10.1515/mcma.1996.2.2.93
  • F. Barthe and C. Bordenave: Combinatorial optimization over two random point sets. To appear in Séminaire de Probabilités, arXiv.org:1103.2734
    Mathematical Reviews (MathSciNet): MR3185927
    Digital Object Identifier: doi:10.1007/978-3-319-00321-4_19
  • Blower, Gordon; Bolley, François. Concentration of measure on product spaces with applications to Markov processes. Studia Math. 175 (2006), no. 1, 47–72.
    Mathematical Reviews (MathSciNet): MR2261699
    Digital Object Identifier: doi:10.4064/sm175-1-3
  • Boissard, Emmanuel. Simple bounds for convergence of empirical and occupation measures in 1-Wasserstein distance. Electron. J. Probab. 16 (2011), no. 83, 2296–2333.
    Mathematical Reviews (MathSciNet): MR2861675
    Digital Object Identifier: doi:10.1214/EJP.v16-958
  • 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.
    Mathematical Reviews (MathSciNet): MR2172583
    Digital Object Identifier: doi:10.5802/afst.1095
  • 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.
    Mathematical Reviews (MathSciNet): MR2280433
    Digital Object Identifier: doi:10.1007/s00440-006-0004-7
  • Djellout, H.; Guillin, A.; Wu, L. Transportation cost-information inequalities and applications to random dynamical systems and diffusions. Ann. Probab. 32 (2004), no. 3B, 2702–2732.
    Mathematical Reviews (MathSciNet): MR2078555
    Digital Object Identifier: doi:10.1214/009117904000000531
    Project Euclid: euclid.aop/1091813628
  • 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
    Mathematical Reviews (MathSciNet): MR1612815
  • Frikha, Noufel; Menozzi, Stéphane. Concentration bounds for stochastic approximations. Electron. Commun. Probab. 17 (2012), no. 47, 15 pp.
    Mathematical Reviews (MathSciNet): MR2988393
  • Gozlan, Nathael; Léonard, Christian. A large deviation approach to some transportation cost inequalities. Probab. Theory Related Fields 139 (2007), no. 1-2, 235–283.
    Mathematical Reviews (MathSciNet): MR2322697
    Digital Object Identifier: doi:10.1007/s00440-006-0045-y
  • Gozlan, N.; Léonard, C. Transport inequalities. A survey. Markov Process. Related Fields 16 (2010), no. 4, 635–736.
    Mathematical Reviews (MathSciNet): MR2895086
  • Johnson, W. B.; Schechtman, G.; Zinn, J. Best constants in moment inequalities for linear combinations of independent and exchangeable random variables. Ann. Probab. 13 (1985), no. 1, 234–253.
    Mathematical Reviews (MathSciNet): MR770640
    Digital Object Identifier: doi:10.1214/aop/1176993078
    Project Euclid: euclid.aop/1176993078
  • 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
    Mathematical Reviews (MathSciNet): MR1993642
  • Laruelle, Sophie; Pagès, Gilles. Stochastic approximation with averaging innovation applied to finance. Monte Carlo Methods Appl. 18 (2012), no. 1, 1–51.
    Mathematical Reviews (MathSciNet): MR2908605
    Digital Object Identifier: doi:10.1515/mcma-2011-0018
  • Lemaire, V.; Menozzi, S. On some non asymptotic bounds for the Euler scheme. Electron. J. Probab. 15 (2010), no. 53, 1645–1681.
    Mathematical Reviews (MathSciNet): MR2735377
  • Malrieu, Florent; Talay, Denis. Concentration inequalities for Euler schemes. Monte Carlo and quasi-Monte Carlo methods 2004, 355–371, Springer, Berlin, 2006.
    Mathematical Reviews (MathSciNet): MR2208718
    Digital Object Identifier: doi:10.1007/3-540-31186-6_21
  • Polyak, B. T.; Juditsky, A. B. Acceleration of stochastic approximation by averaging. SIAM J. Control Optim. 30 (1992), no. 4, 838–855.
    Mathematical Reviews (MathSciNet): MR1167814
    Digital Object Identifier: doi:10.1137/0330046
  • Robbins, Herbert; Monro, Sutton. A stochastic approximation method. Ann. Math. Statistics 22, (1951). 400–407.
    Mathematical Reviews (MathSciNet): MR42668
    Digital Object Identifier: doi:10.1214/aoms/1177729586
    Project Euclid: euclid.aoms/1177729586
  • Rachev, Svetlozar T.; Rüschendorf, Ludger. Mass transportation problems. Vol. II. Applications. Probability and its Applications (New York). Springer-Verlag, New York, 1998. xxvi+430 pp. ISBN: 0-387-98352-X
    Mathematical Reviews (MathSciNet): MR1619171
  • Ruppert, David. Stochastic approximation. Handbook of sequential analysis, 503–529, Statist. Textbooks Monogr., 118, Dekker, New York, 1991.
    Mathematical Reviews (MathSciNet): MR1174318
  • 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).
    Mathematical Reviews (MathSciNet): MR1091544
    Digital Object Identifier: doi:10.1080/07362999008809220

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