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Zbl 0955.62085
Bena\" im, Michel
Dynamics of stochastic approximation algorithms. (English)
[A] Azéma, Jacques (ed.) et al., Séminaire de probabilités XXXIII. Berlin: Springer. Lect. Notes Math. 1709, 1-68 (1999). ISBN 3-540-66342-8

This is the author's DEA course given at ENS Cachan during 1996-98. From the author's abstract: The aim of this course is to introduce the reader to the dynamical system aspects of the theory of stochastic approximation.\par The general form of stochastic approximation processes is $x_{n+ 1}-x_n= \gamma_n(F(x_n) +U_n)$, $n\in\bbfN$. It is difficult to summarize in few lines all results given in this dense course; every section should be reviewed separately. The basic concept is ``asymptotic pseudotrajectories'', whose definition is given in section 3. All other sections give results about the link between asymptotic pseudotrajectories and stochastic approximation.\par From the author's introduction: In section 4 classical results on stochastic approximations are (re)formulated in the language of asymptotic pseudotrajectories. It is shown that, under suitable conditions, the continuous time process obtained by a convenient interpolation of $\{x_n\}$ is almost surely an asymptotic pseudotrajectory of the semiflow induced by the associated ODE $dx/dt=F(x)\dots$\par Section 5 characterizes the limit sets of asymptotic pseudotrajectories. The main result of the section establishes that limit sets of precompact asymptotic pseudotrajectories are internally chain-transitive\dots \par Section 6 applies the abstract results of section 5 in various situations. It is shown how assumptions on the deterministic dynamics can help to identify the possible limit sets of stochastic approximation processes with a great deal of generality.\dots\par Section 7 establishes simple sufficient conditions ensuring that a given attractor of an ODE has a positive probability to host the limit set of the stochastic approximation process\dots\par Section 8 considers the question of shadowing\dots Section 9 focusses on the behavior of stochastic approximation processes near `unstable' sets\dots\par Section 10 introduces the notion of a stochastic process being a weak asymptotic pseudodotrajectory for a semiflow and analyzes properties of its empirical occupation measure.
[A.Grorud (Marseille)]

MSC 2000:: *62L20 Stochastic approximation
37L30 Attractors and their dimensions

Keywords: dynamical systems; Lyapunov functions; asymptotic pseudotrajectories; limit sets; attractor; shadowing; empirical occupation measure

Cited in: Zbl 1068.60072 Zbl 1068.65104

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