Université Paris 6
Pierre et Marie Curie
Université Paris 7
Denis Diderot

CNRS U.M.R. 7599
``Probabilités et Modèles Aléatoires''

Stability of stochastic approximation under verifiable conditions


Code(s) de Classification MSC:

Résumé: In this paper we address the problem of the stability of the stochastic approximation procedure. The stability of such algorithms is known to rely heavily on the growth of the mean field at the boundary of the parameter set and the magnitude of the sizesteps used in the procedure. The conditions typically required to ensure convergence are either too difficult to check in practice or not satisfied at all, even for simple models. The most popular technique to circumvent this problem consists of constraining the parameter to a compact subset in the parameter space. We propose and analyze here an alternative, based on projection on adaptive truncation sets, extending previous works in this direction. This procedure allows for the adaptive tuning of the magnitude of the stepsizes, which is key to ensuring stability. The stability - with probability one - of the scheme is proved under a set of verifiable assumptions. We illustrate these claims in the so-called controlled Markovian setting and present two substantial examples. The first example is related to the minimum prediction error estimation of the parameters of stable and invertible ARMA processes and the second example is related to controlled Markov chain Monte Carlo algorithms.

Mots Clés: Stochastic approximation ; state-dependent noise ; randomly varying truncation ; Adaptive Markov Chain Monte Carlo

Date: 2003-02-10

Prépublication numéro: PMA-791

Postscript file: PMA-791.ps