Voici les éléments 1 - 10 sur 49
  • Publication
    Métadonnées seulement
    A dynamical system approach to stochastic approximations
    It is known that some problems of almost sure convergence for stochastic approximation processes can be analyzed via an ordinary differential equation (ODE) obtained by suitable averaging. The goal of this paper is to show that the asymptotic behavior of such a process can be related to the asymptotic behavior of the ODE without any particular assumption concerning the dynamics of this ODE. The main results are as follows: a) The limit sets of trajectory solutions to the stochastic approximation recursion are, under classical assumptions, almost surely nonempty compact connected sets invariant under the how of the ODE and contained in its set of chain-recurrence. b) If the gain parameter goes to zero at a suitable rate depending on the expansion rate of the ODE, any trajectory solution to the recursion is almost surely asymptotic to a forward trajectory solution to the ODE.
  • Publication
    Métadonnées seulement
    A Bakry-Emery criterion for self-interacting diffusions
    (: Birkhauser Boston, 2005) ;
    Raimond, Olivier
    ;
    Dalang, Marco
    ;
    Dozzi, Marco
    ;
    Russo, Francesco
    We give a Bakry-Emery type criterion for self-interacting diffusions on a compact manifold.
  • Publication
    Métadonnées seulement
    Dynamics of Morse-Smale urn processes
    (1995) ;
    Hirsch, Morris W
    We consider stochastic processes {x(n)}(n greater than or equal to 0) of the form x(n+1)-x(n)=gamma(n+1)(F(x(n))+U-n+1) where F : R(m) --> R(m) is C-2, {gamma(i)}(i greater than or equal to 1) is a Sequence of positive numbers decreasing to 0 and {U-i}(i greater than or equal to 1) is a sequence of uniformly bounded R(m)-valued random variables forming suitable martingale differences. We show that when the vector field F is Morse-Smale, almost surely every sample path approaches an asymptotically stable periodic orbit of the deterministic dynamical system dy/dt = F(y). In the case of certain generalized urn processes we show that for each such orbit Gamma, the probability of sample paths approaching Gamma is positive. This gives the generic behavior of three-color urn models.
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    Métadonnées seulement
  • Publication
    Métadonnées seulement
    On invariant hypersurfaces of strongly monotone maps
    For C-k, k>1, strongly monotone discrete-time dynamical systems, we present simple criteria which ensure that nonmonotone invariant manifolds are C-k. (C) 1997 Academic Press.
  • Publication
    Métadonnées seulement
    Stochastic approximations and differential inclusions, part II: Applications
    (2006) ;
    Hofbauer, Josef
    ;
    Sorin, Sylvain
    We apply the theoretical results on "stochastic approximations and differential inclusions" developed in Benaim et al. [M. Benaim, J. Hofbauer, S. Sorin. 2005. Stochastic approximations and differential inclusions. SIAM J. Control Optim. 44 328-348] to several adaptive processes used in game theory, including classical and generalized approachability, no-regret potential procedures (Hart and Mas-Colell [S. Hart, A. Mas-Colell. 2003. Regret-based continuous time dynamics. Games Econom. Behav. 45 375-394]), and smooth fictitious play [D. Fudenberg, D. K. Levine. 1995. Consistency and cautious fictitious play. J. Econom. Dynam. Control 19 1065-1089].
  • Publication
    Métadonnées seulement
    Recursive algorithms, urn processes and chaining number of chain recurrent sets
    This paper investigates the dynamical properties of a class of urn processes and recursive stochastic algorithms with constant gain which arise frequently in control, pattern recognition, learning theory, and elsewhere. It is shown that, under suitable conditions, invariant measures of the process tend to concentrate on the Birkhoff center of irreducible (i.e. chain transitive) attractors of some vector field F : R-d --> R-d obtained by averaging. Applications are given to simple situations including the cases where F is Axiom A or Morse-Smale, F is gradient-like, F is a planar vector field, F has finitely many alpha and omega limit sets.
  • Publication
    Métadonnées seulement
    Quantitative ergodicity for some switched dynamical systems
    (2012) ;
    Le Borgne, Stéphane
    ;
    Malrieu, Florent
    ;
    Zitt, Pierre-André
  • Publication
    Métadonnées seulement
    Almost sure convergence of the one-dimensional Kohonen algorithm
    (1998) ;
    Fort, Jean-Claude
    ;
    Pages, Gilles
  • Publication
    Métadonnées seulement
    Stochastic approximation algorithms with constant step size whose average is cooperative
    (1999) ;
    Hirsch, Morris W
    We consider stochastic approximation algorithms with constant step size whose average ordinary differential equation (ODE) is cooperative and irreducible. We show that, under mild conditions on the noise process, invariant measures and empirical occupations measures of the process weakly converge (as the time goes to infinity and the step size goes to zero) toward measures which are supported by stable equilibria of the ODE. These results are applied to analyzing the long-term behavior of a class of learning processes arising in game theory.