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Convergence of stochastic algorithms: from the Kushner–Clark theorem to the Lyapounov functional method

Part of: Limit theorems Sequential methods

Published online by Cambridge University Press:  01 July 2016

Jean-Claude Fort  and
Gilles Pagès
Jean-Claude Fort*
Affiliation:
Université de Nancy 1 and Université de Paris 1
Gilles Pagès*
Affiliation:
Université Pierre et Marie Curie and Université de Paris 12
*
∗ Postal address: Université de Nancy I, Fac. Sciences, Dpt de Math., B.P. 239, F-54506 Vandoeuvre-Lès-Nancy Cedex, France and SAMOS, Université de Paris 1, UFR 27, 90, rue de Tolbiac F-75634 Paris Cedex 13, France.
∗∗ Postal address: Laboratoire de Probabilités, URA 224, Université P. et M. Curie, Pl. Jussieu, F-75252 Paris Cedex 05, France and Université de Paris XII, UFR, Sciences, 61 av. du Gal de Gaulle, 94010 Créteil Cedex, France.
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Abstract

In the first part of this paper a global Kushner–Clark theorem about the convergence of stochastic algorithms is proved: we show that, under some natural assumptions, one can ‘read' from the trajectories of its ODE whether or not an algorithm converges. The classical stochastic optimization results are included in this theorem. In the second part, the above smoothness assumption on the mean vector field of the algorithm is relaxed using a new approach based on a path-dependent Lyapounov functional. Several applications, for non-smooth mean vector fields and/or bounded Lyapounov function settings, are derived. Examples and simulations are provided that illustrate and enlighten the field of application of the theoretical results.

Keywords

STOCHASTIC ALGORITHMORDINARY DIFFERENTIAL EQUATIONLYAPOUNOV FUNCTIONAL

MSC classification

Primary: 62L20: Stochastic approximation
Secondary: 60F05: Central limit and other weak theorems
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 28 , Issue 4 , December 1996 , pp. 1072 - 1094
Copyright
Copyright © Applied Probability Trust 1996 

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