Dynamics of stochastic approximation algorithms
Benaïm, Michel
Séminaire de probabilités de Strasbourg, Tome 33 (1999), p. 1-68 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU) 
@article{SPS_1999__33__1_0,
     author = {Bena\"\i m, Michel},
     title = {Dynamics of stochastic approximation algorithms},
     journal = {S\'eminaire de probabilit\'es de Strasbourg},
     volume = {33},
     year = {1999},
     pages = {1-68},
     mrnumber = {1767993},
     zbl = {0955.62085},
     language = {en},
     url = {http://dml.mathdoc.fr/item/SPS_1999__33__1_0}
}

Benaïm, Michel. Dynamics of stochastic approximation algorithms. Séminaire de probabilités de Strasbourg, Tome 33 (1999) pp. 1-68. http://gdmltest.u-ga.fr/item/SPS_1999__33__1_0/

Akin, E. (1993). The General Topology of Dynamical Systems. American Mathematical Society, Providence. | MR 1219737 | Zbl 0781.54025

Arthur, B., Ermol'Ev, Y., and Kaniovskii, Y. (1983). A generalized urn problem and its applications. Cybernetics, 19:61-71. | Zbl 0534.90049

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. | MR 1120101

Benaïm, M. (1996). A dynamical systems approach to stochastic approximations. SIAM Journal on Control and Optimization, 34:141-176. | MR 1377706 | Zbl 0841.62072

Benaïm, M. (1997). Vertex reinforced random walks and a conjecture of Pemantle. The Annals of Probability, 25:361-392. | MR 1428513 | Zbl 0873.60044

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

Benaïm, M. and Hirsch, M.W. (1995a). Chain recurrence in surface flows. Discrete and Continuous Dynamical Systems, 1(1):1-16. | MR 1355862 | Zbl 0871.58062

Benaïm, M. and Hirsch, M.W. (1995b). Dynamics of morse-smale urn processes. Ergodic Theory and Dynamical Systems, 15:1005-1030. | MR 1366305 | Zbl 0846.60054

Benaïm, M. and Hirsch, M.W. (1996). Asymptotic pseudotrajectories and chain recurrent flows, with applications. J. Dynam. Differential Equations, 8:141-176. | MR 1388167 | Zbl 0878.58053

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

Benveniste, A., Métivier, M., and Priouret, P. (1990). Stochastic Approximation and Adaptive Algorithms. Springer-Verlag, Berlin and New York. | MR 1082341 | Zbl 0752.93073

Bowen, R. (1975). Omega limit sets of Axiom A diffeomorphisms. J. Diff. Eq, 18:333-339. | MR 413181 | Zbl 0315.58019

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

Brandière., O. (1997). Some pathological traps for stochastic approximation. SIAM Journal on Control and Optimization. To Appear. | MR 1618037 | Zbl 0980.62068

Brandière, O. and Duflo., M. (1996). Les algorithmes stochastique contournent ils les pièges. Annales de l'IHP, 32:395-427. | Numdam | MR 1387397 | Zbl 0849.62043

Conley, C.C. (1978). Isolated invariant sets and the Morse index. CBMS Regional conference series in mathematics. American Mathematical Society, Providence. | MR 511133 | Zbl 0397.34056

Delyon, B. (1996), General convergence results on stochastic approximation. IEEE trans. on automatic control, 41:1245-1255. | MR 1409470 | Zbl 0867.93075

Duflo, M. (1990). Méthodes Récursives Aléatoires. Masson. English Translation: Random Iterative Models, Springer Verlag 1997. | Zbl 0703.62084

Duflo, M. (1996). Algorithmes Stochastiques. Mathématiques et Applications. Springer-Verlag. | MR 1612815 | Zbl 0882.60001

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

Ethier, S.N. and Kurtz, T.G. (1986). Markov Processes, Characterization and Convergence. John Wiley and Sons, Inc. | MR 838085 | Zbl 0592.60049

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

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. | MR 1418247 | Zbl 0881.62085

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

Fudenberg, D. and Kreps, K. (1993). Learning mixed equilibria. Games and Econom. Behav., 5:320-367. | MR 1227915 | Zbl 0790.90092

Fudenberg, F. and Levine, D. (1998). Theory of Learning in Games. MIT Press, Cambridge, MA. In Press. | MR 1629477 | Zbl 0939.91004

Hartman, P. (1964). Ordinary Differential Equationq. Wiley, New York. | MR 171038 | Zbl 0125.32102

Hill, B.M., Lane, D., and Sudderth, W. (1980). A strong law for some generalized urn processes. Annals of Probability, 8:214-226. | MR 566589 | Zbl 0429.60021

Hirsch, M.W. (1976). Differential Topology. Springer-Verlag, Berlin, New York, Heidelberg. | MR 448362 | Zbl 0356.57001

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. | MR 1243195 | Zbl 0795.93055

Hirsch, M.W. and Pugh, C.C. (1988). Cohomology of chain recurrent sets. Ergodic Theory and Dynamical Systems, 8:73-80. | MR 939061 | Zbl 0643.54039

Kaniovski, Y. and Young, H. (1995). Learning dynamics in games with stochastic perturbations. Games and Econom. Behav., 11:330-363. | MR 1360043 | Zbl 0841.90124

Kiefer, J. and Wolfowitz, J. (1952). Stochastic estimation of the maximum of a regression function. Ann. Math. Statis, 23:462-466. | MR 50243 | Zbl 0049.36601

Kushner, H.J. and Clarck, C.C. (1978). Stochastic Approximation for Constrained and Unconstrained Systems. Springer-Verlag, Berlin and New York. | MR 499560 | Zbl 0381.60004

Kushner, H.J. and Yin, G.G. (1997). Stochastic Approximation Algorithms and Applications. Springer-Verlag, New York. | MR 1453116 | Zbl 0914.60006

Ljung, L. (1977). Analysis of recursive stochastic algorithms. IEEE Trans. Automat. Control., AC-22:551-575. | MR 465458 | Zbl 0362.93031

Ljung, L. (1986). System Identification Theory for the User. Prentice Hall, Englewood Cliffs, NJ. | Zbl 0615.93004

Ljung, L. and Söderström, T. (1983). Theory and Practice of Recursive Identification. MIT Press, Cambridge, MA. | MR 719192 | Zbl 0548.93075

Mañé, R. (1987). Ergodic Theory and Differentiable Dynamics. Springer-Verlag, New York. | MR 889254 | Zbl 0616.28007

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. | MR 873887 | Zbl 0588.62153

Munkres, J.R. (1975). Topology a first course. Prentice Hall. | MR 464128 | Zbl 0306.54001

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

Pemantle, R. (1990). Nonconvergence to unstable points in urn models and stochastic approximations. Annals of Probability, 18:698-712. | MR 1055428 | Zbl 0709.60054

Pemantle, R. (1992). Vertex reinforced random walk. Probability Theory and Related Fields, 92:117-136. | MR 1156453 | Zbl 0741.60029

Robbins, H. and Monro, S. (1951). A stochastic approximation method. Ann. Math. Statis, 22:400-407. | MR 42668 | Zbl 0054.05901

Robinson, C.. (1977). Stability theorems and hyperbolicity in dynamical systems. Rocky Journal of Mathematics, 7:425-434. | MR 494300 | Zbl 0375.58016

Robinson, C. (1995). Introduction to the Theory of Dynamical Systems. Studies in Advances Mathematics. CRC Press, Boca Raton. | MR 1396532

Schreiber, S.J. (1997). Expansion rates and Lyapunov exponents. Discrete and Conts. Dynam. Sys., 3:433-438. | MR 1444204 | Zbl 0948.37019

Shub, M. (1987). Global Stability of Dynamical Systems. Springer-Verlag, Berlin, New York, Heidelberg. | MR 869255 | Zbl 0606.58003

Stroock, D.W. (1993). Probability Theory. An analytic view. Cambridge University Press. | MR 1267569 | Zbl 0925.60004

White, H. (1992). Artificial Neural Networks: Approximation and Learning Theory. Blackwell, Cambridge, Massachussets. | MR 1203316