Hostname: page-component-669899f699-vbsjw Total loading time: 0 Render date: 2025-04-26T18:51:06.821Z Has data issue: false hasContentIssue false
  • English
  • Français

The dynamic system method and the traps

Part of: Sequential methods

Published online by Cambridge University Press:  01 July 2016

Odile Brandière
Odile Brandière*
Affiliation:
Université de Marne-la-Vallée
*
∗ Postal address: Université de Marne-la-Vallée, Equipe d'Analyse et de Mathématiques Appliquées, 2, rue de la Butte Verte, 93166 Noisy-le-Grand, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

We transpose the ordinary differential equation method (used for decreasing stepsize stochastic algorithms) to a dynamical system method to study dynamical systems disturbed by a noise decreasing to zero. We prove that such an algorithm does not fall into a regular trap if the noise is exciting in an unstable direction.

Keywords

Stochastic approximationdynamical systemstraps

MSC classification

Primary: 62L20: Stochastic approximation
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 1 , March 1998 , pp. 137 - 151
Copyright
Copyright © Applied Probability Trust 1998 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1] Benaïm, M., (1996). A dynamical system approach to stochastic approximations. SIAM J. Control Optim. 34, 437–472.CrossRefGoogle Scholar
[2] Benaïm, M. and Hirsch, M.W. (1995). Chain recurrence in surface flows. Discrete Cont. Dynam. Syst. 1, 1–17.Google Scholar
[3] Benveniste, A., Métivier, M. and Priouret, P. (1990). Adaptive Algorithms and Stochastic Approximation. Springer, Berlin.CrossRefGoogle Scholar
[4] Biscarat, J.C. (1994). Almost sure convergence of a class of stochastic algorithms. Stoch. Proc. Appl. 50, 83–99.Google Scholar
[5] Brandière, O., (1996). Autour des pièges des algorithmes stochastiques. Thesis. Université de Marne-la-Vallée, France.Google Scholar
[6] Brandière, O. and Duflo, M. (1996). Les algorithmes stochastiques contournent-ils les pièges? Ann. Inst. Henri Poincaré 32, 395–427.Google Scholar
[7] Celeux, G. and Diebolt, J. (1993). A stochastic approximation type EM algorithm for mixture problem. Stoch. Stoch. Rep. 41, 119–134.CrossRefGoogle Scholar
[8] Chen, H.-F., Guo, L. and Gao, A.J. (1988). Convergence and robustness of the Robbins–Monro algorithm truncated at randomly varying bounds. Stoch. Proc. Appl. 27, 217–231.CrossRefGoogle Scholar
[9] Delyon, B. (1996). General convergence results on stochastic approximation IEEE Trans. Automatic Control 41, 9.CrossRefGoogle Scholar
[10] Duflo, M. (1996). Algorithmes stochastiques. In Collection Math. et Appl. Vol. 23. Springer, Berlin.Google Scholar
[11] Fort, J.C. and Pagès, G. (1996). Convergence of stochastic algorithms: from Kushner–Clark theorem to the Lyapounov functional method. Adv. Appl. Prob. 28, 1072–1094.CrossRefGoogle Scholar
[12] Hale, J.K. (1988). Asymptotic behavior of dissipative systems. In Mathematical Surveys and Monographs. Vol. 25. American Mathematical Society, Providence, RI.Google Scholar
[13] Haraux, A. (1991). Systèmes dynamiques dissipatifs et applications. Masson, Paris.Google Scholar
[14] Hartman, P. (1982). Ordinary Differential Equations. 2nd edn, Wiley, New York.Google Scholar
[15] Hirsch, M.W. (1993). Asymptotic phase, shadowing and reaction diffusion, control theory, dynamical systems and geometry of dynamics. ed Elworthy, K.D. and Everitts, W.N.. Dekker, New York. pp. 87–99.Google Scholar
[16] Kesten, H. (1972). Limit theorems for stochastic growth models I and II. Adv. Appl. Prob. 4, 193–232, 393–428.CrossRefGoogle Scholar
[17] Kushner, H.J. and Clark, D.S. (1978). Stochastic approximation for constrained and unconstrained systems. Applied Math. Science Series. Vol. 26, Springer, Berlin.Google Scholar
[18] Lai, T.Z. and Wei, C.Z. (1983). A note on martingale difference sequences satisfying the local Marcinkie-wicz–Zygmund condition. Bull. Inst. Math. Acad. Sinica 11, 1–13.Google Scholar
[19] Lazarev, V.A. (1992). Convergence of stochastic approximation procedures in case of regression equation with several roots. Prob. Peredachi Inform. 28, 75–88.Google Scholar
[20] Ljung, L. (1977). Analysis of recursive stochastic algorithms. IEEE Trans. Automatic Control 22, 551–575.Google Scholar
[21] Pierre Loti Viaud, D. (1995). Random perturbations of recursive sequences with an application to an epidemic model. J. Appl. Prob. 32, 559–578.Google Scholar