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Benaim, Michel
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Benaim, Michel
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michel.benaim@unine.ch
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Voici les éléments 1 - 10 sur 49
- PublicationMétadonnées seulementA dynamical system approach to stochastic approximations(1996)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.
- PublicationMétadonnées seulementA Bakry-Emery criterion for self-interacting diffusions(: Birkhauser Boston, 2005)
; - PublicationMétadonnées seulementDynamics of Morse-Smale urn processes(1995)
; Hirsch, Morris WWe 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. - PublicationMétadonnées seulementOn invariant hypersurfaces of strongly monotone maps(1997)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.
- PublicationMétadonnées seulementStochastic approximations and differential inclusions, part II: Applications(2006)
; - PublicationMétadonnées seulementRecursive algorithms, urn processes and chaining number of chain recurrent sets(1998)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.
- PublicationMétadonnées seulementQuantitative ergodicity for some switched dynamical systems(2012)
; - PublicationMétadonnées seulementAlmost sure convergence of the one-dimensional Kohonen algorithm(1998)
; - PublicationMétadonnées seulementStochastic approximation algorithms with constant step size whose average is cooperative(1999)
; Hirsch, Morris WWe 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.