Stochastic approximation algorithms with constant step size whose average is cooperaitve

Annals of Applied Probability

Volumn 9, Issue 1, 1999, Pages 216-241

  Benaïm, Michel ^a   Hirsch, Morris W ^b  

^a INSTITUT DE MATHÉMATIQUES DE TOULOUSE   (France)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Cooperative vector fields; Large deviations; Ordinary differential equation method; Stochastic approximation; Theory of learning in games; Weak convergence

Indexed keywords

EID: 0039521654     PISSN: 10505164     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (33)

1. AZENCOTT, R. and RUGET, G. (1977). Mélanges d'équations differentielles et grands écarts à la loi des grands nombres. Z. Wahrsch. Verw. Gebiete 38 1-54.
2. BENAÏM, M. (1996a). A dynamical system approach to stochastic approximations. Siam J. Control Optim. 34 437-472.
3. BENAÏM, M. (1998). Recursive algorithms, urn processes and chaining number of chain recurrent sets. Ergodic Theory Dynam. Systems 18 53-87.
4. BENAÏM, M. and HIRSCH, M. W. (1995). Dynamics of Morse Smale urn processes. Ergodic Theory Dynam. Systems 15 1005-1030.
5. BENAÏM, M. and HIRSCH, M. W. (1996). Asymptotic pseudo-trajectories and chain-recurrent flows, with applications. J. Dynam. Differential Equations 8 141-174.
6. BENAÏM, M. and HIRSCH, M. W. (1997). Information games, Dynamical systems and stochastic approximation. Discussion paper for IIASA workshop Oct 17-20.
7. BENVENISTE, A. MÉTIVIER, M. and PRIOURET, P. (1990). Stochastic Approximations and Adaptive Algorithms. Springer, New York.
8. CHOW, S. N. and HALE, J. (1982). Methods of Bifurcation Theory. Springer, New York.
9. CONLEY, C. C. (1978). Isolated Invariant Sets and the Morse Index. Amer. Math. Soc., Providence, RI.
10. DUFLO, M. (1996). Algorithmes Stochastiques. Springer, Berlin.
11. DUFLO, M. (1997). Random Iterative Models. Springer, New York.
12. DUPUIS, P. (1988). Large deviations analysis of some recursive algorithms with state dependent noise. Ann. Probab. 6 1509-1536.
13. DUPUIS, P. and ELLIS R. S. (1997). A Weak Convergence Approach to the Theory of Large Deviations. Wiley, New York.
14. FORT, J. C. and PAGES, G. (1996). Asymptotics behaviour of a Markovian constant step size algorithm. Siam J. Control Optim. To appear.
15. FREIDLIN, M. I. (1978). The averaging principle and theorems on large deviations. Russian Math. Surveys 33 117-176.
16. FREIDLIN, M. I. and WENTZELL, A. D. (1984). Random Perturbation of Dynamical Systems. Springer, New York.
17. FUDENBERG, D. and LEVINE, D. (1998). Theory of Learning in Games. MIT Press.
18. HIRSCH, M. W. (1985). Systems of differential equations that are competitive or cooperative II: convergence almost everywhere. SIAM J. Math. Anal. 16 423-439.
19. HIRSCH, M. W. (1988a). Systems of differential equations which are competitive or cooperative III: competing species. Nonlinearity 1 51-71.
20. HIRSCH, M. W. (1988b). Stability and convergence in strongly monotone dynamical systems. J. Reine Angew. Math. 383 1-53.
21. HIRSCH, M. W. (1996). Chain transitive sets for smooth strongly monotone maps. Preprint, Univ. California, Berkeley.
22. JIANG, J.-F. (1991). Attractors for strongly monotone flows. J. Math. Anal. Appl. 162 210-222.
23. JIANG, J.-F. and YU, S.-X. (1995). Stable cycles for attractors of strongly monotone discrete-time dynamical systems. Comm. Appl. Nonlinear Anal. 2 455-458.
24. KIFER, Y. (1988). Random Perturbation of Dynamical Systems. Birkhäuser, Boston.
25. KUNZE, H. and SIEGEL, D. (1994). A graph theoretic approach to monotonicity with respect to initial conditions. In Comparison Methods and Stability Theory (X. Liu and D. Siegel, eds.) Dekker, New York.
26. KUSHNER, H. J. and CLARK, D. S. (1978). Stochastic Approximation Methods for Constrained and Unconstrained Systems. Springer, New York.
27. LJUNG, L. (1977). Analysis of recursive stochastic algorithms. IEEE Trans. Automat. Control AC-22 551-575.
28. MIERCZYŃSKI, J. (1994). P-arcs in strongly monotone discrete-time dynamical systems. Differential Integral Equations 7 1473-1494.
29. SMITH, H. (1995). Monotone Dynamical Systems. Amer. Math. Soc., Providence, RI.
30. TAKÁČ, P. (1992). Domains of attraction of generic ω-limit sets for strongly monotone discrete-time semigroups. J. Reine Angew. Math. 423 101-173.
31. 1 smooth strongly monotone discrete-time dynamical systems. Preprint.
32. VARADHAN, (1984). Large Deviations and Applications. SIAM, Philadelphia.
33. WEIBULL, J. W. (1995). Evolutionary Game Theory. MIT Press.