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Dynamics of stochastic approximation algorithms

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Séminaire de Probabilités XXXIII

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 1709))

Abstract

These notes were written for a D.E.A. course given at Ecole Normale Supérieure de Cachan during the 1996–97 and 1997–98 academic years and at University Toulouse III during the 1997–98 academic year. Their aim is to introduce the reader to the dynamical system aspects of the theory of stochastic approximations.

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References

  • Akin, E. (1993). The General Topology of Dynamical Systems. American Mathematical Society, Providence.

    MATH  Google Scholar 

  • Arthur, B., Ermol'ev, Y., and Kaniovskii, Y. (1983). A generalized urn problem and its applications. Cybernetics, 19: 61–71.

    Article  MathSciNet  Google Scholar 

  • Arthur, B. M. (1988). Self-reinforcing mechanisms in economics. In W. A. P., Arrow, K. J., and Pines, D., editors, The Economy as an Evolving Complex System, SFI Studies in the Sciences of Complexity. Addison-Wesley.

    Google Scholar 

  • Benaïm, M. (1996). A dynamical systems approach to stochastic approximations. SIAM Journal on Control and Optimization, 34: 141–176.

    Article  MATH  Google Scholar 

  • Benaïm, M. (1997). Vertex reinforced random walks and a conjecture of Pemantle. The Annals of Probability, 25: 361–392.

    Article  MathSciNet  MATH  Google Scholar 

  • Benaïm, M. and Hirsch, M. W. (1994). Learning processes, mixed equilibria and dynamical systems arising from repeated games. Submitted.

    Google Scholar 

  • Benaïm, M. and Hirsch, M. W. (1995a). Chain recurrence in surface flows. Discrete and Continuous Dynamical Systems, 1(1): 1–16.

    MathSciNet  MATH  Google Scholar 

  • Benaïm, M. and Hirsch, M. W. (1995b). Dynamics of morse-smale urn processes. Ergodic Theory and Dynamical Systems, 15: 1005–1030.

    Article  MathSciNet  MATH  Google Scholar 

  • Benaïm, M. and Hirsch, M. W. (1996). Asymptotic pseudotrajectories and chain recurrent flows, with applications. J. Dynam. Differential Equations, 8: 141–176.

    Article  MathSciNet  MATH  Google Scholar 

  • Benaïm, M. and Schreiber, S. J. (1997). Weak asymptotic pseudotrajectories for semiflows: Ergodic properties. Preprint.

    Google Scholar 

  • Benveniste, A., Métivier, M., and Priouret, P. (1990). Stochastic Approximation and Adaptive Algorithms. Springer-Verlag, Berlin and New York.

    Book  MATH  Google Scholar 

  • Bowen, R. (1975). Omega limit sets of Axiom A diffeomorphisms. J. Diff. Eq, 18: 333–339.

    Article  MathSciNet  MATH  Google Scholar 

  • Brandière, O. (1996). Autour des pièges des algorithmes stochastiques. Thèse de Doctorat, Université de Marne-la-Vallée.

    Google Scholar 

  • Brandière, O. (1997). Some pathological traps for stochastic approximation. SIAM Journal on Control and Optimization. To Appear.

    Google Scholar 

  • Brandière, O. and Duflo, M. (1996). Les algorithmes stochastique contournent ils les pièges. Annales de l'IHP, 32: 395–427.

    Google Scholar 

  • Conley, C. C. (1978). Isolated invariant sets and the Morse index. CBMS Regional conference series in mathematics. American Mathematical Society, Providence.

    Book  MATH  Google Scholar 

  • Delyon, B. (1996). General convergence results on stochastic approximation. IEEE trans. on automatic control, 41: 1245–1255.

    Article  MathSciNet  MATH  Google Scholar 

  • Duflo, M. (1990). Méthodes Récursives Aléatoires. Masson. English Translation: Random Iterative Models, Springer Verlag 1997.

    Google Scholar 

  • Duflo, M. (1996). Algorithmes Stochastiques. Mathématiques et Applications. Springer-Verlag.

    Google Scholar 

  • Duflo, M. (1997). Cibles atteignables avec une probabilité positive d'après m. benaim. Unpublished manuscript.

    Google Scholar 

  • Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes, Characterization and Convergence. John Wiley and Sons, Inc.

    Google Scholar 

  • Fort, J. C., and Pages, G. (1994). Résaux de neurones: des méthodes connexionnistes d'apprentissage. Matapli, 37: 31–48.

    Google Scholar 

  • Fort, J. C. and Pages, G. (1996). Convergence of stochastic algorithms: From Kushner-Clark theorem to the lyapounov functional method. Adv. Appl. Prob, 28: 1072–1094.

    MathSciNet  MATH  Google Scholar 

  • Fort, J. C. and Pages, G. (1997). Stochastic algorithm with non constant step: a.s. weak convergence of empirical measures. Preprint.

    Google Scholar 

  • Fudenberg, D., and Kreps, K. (1993). Learning mixed equilibria. Games and Econom. Behav., 5: 320–367.

    Article  MathSciNet  MATH  Google Scholar 

  • Fudenberg, F., and Levine, D. (1998). Theory of Learning in Games. MIT Press, Cambridge, MA. In Press.

    MATH  Google Scholar 

  • Hartman, P. (1964). Ordinary Differential Equationq. Wiley, New York.

    Google Scholar 

  • Hill, B. M., Lane, D., and Sudderth, W. (1980). A strong law for some generalized urn processes. Annals of Probability, 8: 214–226.

    Article  MathSciNet  MATH  Google Scholar 

  • Hirsch, M. W. (1976). Differential Topology. Springer-Verlag, Berlin, New York, Heidelberg.

    Book  MATH  Google Scholar 

  • Hirsch, M. W. (1994). Asymptotic phase, shadowing and reaction-diffusion systems. In Differential equations, dynamical systems and control science, volume 152 of Lectures notes in pure and applied mathematics, pages 87–99. Marcel Dekker, New-York.

    Google Scholar 

  • Hirsch, M. W., and Pugh, C. C. (1988). Cohomology of chain recurrent sets. Ergodic Theory and Dynamical Systems, 8: 73–80.

    Article  MathSciNet  MATH  Google Scholar 

  • Kaniovski, Y., and Young, H. (1995). Learning dynamics in games with stochastic perturbations. Games and Econom. Behav., 11: 330–363.

    Article  MathSciNet  MATH  Google Scholar 

  • Kiefer, J., and Wolfowitz, J. (1952). Stochastic estimation of the maximum of a regression function. Ann. Math. Statis, 23: 462–466.

    Article  MathSciNet  MATH  Google Scholar 

  • Kushner, H. J., and Clarck, C. C. (1978). Stochastic Approximation for Constrained and Unconstrained Systems. Springer-Verlag, Berlin and New York.

    Book  Google Scholar 

  • Kushner, H. J., and Yin, G. G. (1997). Stochastic Approximation Algorithms and Applications. Springer-Verlag, New York.

    Book  MATH  Google Scholar 

  • Ljung, L. (1977). Analysis of recursive stochastic algorithms. IEEE Trans. Automat. Control., AC-22: 551–575.

    Article  MathSciNet  MATH  Google Scholar 

  • Ljung, L. (1986). System Identification Theory for the User. Prentice Hall, Englewood Cliffs, NJ.

    MATH  Google Scholar 

  • Ljung, L., and Sőderstrőm, T. (1983). Theory and Practice of Recursive Identification. MIT Press, Cambridge, MA.

    MATH  Google Scholar 

  • Mañé, R. (1987). Ergodic Theory and Differentiable Dynamics. Springer-Verlag, New York.

    Book  MATH  Google Scholar 

  • Métivier, M., and Priouret, P. (1987). Théorèmes de convergence presque sure pour une classe d'algorithmes stochastiques à pas décroissant. Probability Theory and Related Fields, 74: 403–428.

    Article  MathSciNet  MATH  Google Scholar 

  • Munkres, J. R. (1975). Topology a first course. Prentice Hall.

    Google Scholar 

  • Nevelson, M. B., and Khasminskii, R. Z. (1976). Stochastic Approximation and Recursive Estimation. Translation of Math. Monographs. American Mathematical Society, Providence.

    Google Scholar 

  • Pemantle, R. (1990). Nonconvergence to unstable points in urn models and stochastic approximations. Annals of Probability, 18: 698–712.

    Article  MathSciNet  MATH  Google Scholar 

  • Pemantle, R. (1992). Vertex reinforced random walk. Probability Theory and Related Fields, 92: 117–136.

    Article  MathSciNet  MATH  Google Scholar 

  • Robbins, H., and Monro, S. (1951). A stochastic approximation method. Ann. Math. Statis, 22: 400–407.

    Article  MathSciNet  MATH  Google Scholar 

  • Robinson, C. (1977). Stability theorems and hyperbolicity in dynamical systems. Rocky Journal of Mathematics, 7: 425–434.

    Article  MathSciNet  MATH  Google Scholar 

  • Robinson, C. (1995). Introduction to the Theory of Dynamical Systems. Studies in Advances Mathematics. CRC Press, Boca Raton.

    Google Scholar 

  • Schreiber, S. J. (1997). Expansion rates and Lyapunov exponents. Discrete and Conts. Dynam. Sys., 3: 433–438.

    Article  MathSciNet  MATH  Google Scholar 

  • Shub, M. (1987). Global Stability of Dynamical Systems. Springer-Verlag, Berlin, New York, Heidelberg.

    Book  MATH  Google Scholar 

  • Stroock, D. W. (1993). Probability Theory. An analytic view. Cambridge University Press.

    Google Scholar 

  • White, H. (1992). Artificial Neural Networks: Approximation and Learning Theory. Blackwell, Cambridge, Massachussets. *** DIRECT SUPPORT *** A00I6C60 00002

    Google Scholar 

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Authors

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Jacques Azéma Michel Émery Michel Ledoux Marc Yor

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Benaïm, M. (1999). Dynamics of stochastic approximation algorithms. In: Azéma, J., Émery, M., Ledoux, M., Yor, M. (eds) Séminaire de Probabilités XXXIII. Lecture Notes in Mathematics, vol 1709. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0096509

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  • DOI: https://doi.org/10.1007/BFb0096509

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  • Print ISBN: 978-3-540-66342-3

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