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Summary: STABILITY OF STOCHASTIC APPROXIMATION UNDER VERIFIABLE
CONDITIONS
CHRISTOPHE ANDRIEU , ´ERIC MOULINES , AND PIERRE PRIOURET
Abstract. In this paper we address the problem of the stability and convergence of the stochastic approximation
procedure
n+1 = n + n+1[h(n) + n+1].
The stability of such sequences {n} is known to heavily rely on the behaviour of the mean field h at the boundary of the
parameter set and the magnitude of the stepsizes used. The conditions typically required to ensure convergence, and in
particular the boundedness or stability of {n}, are either too difficult to check in practice or not satisfied at all. This is the
case even for very simple models. The most popular technique to circumvent the stability problem consists of constraining
{n} to a compact subset K in the parameter space. This is obviously not a satisfactory solution as the choice of K is a
delicate one. In the present contribution we first prove a "deterministic" stability result which relies on simple conditions
on the sequences {n} and {n}. We then propose and analyze an algorithm based on projections on adaptive truncation
sets which ensures that the aforementioned conditions required for stability are satisfied. We focus in particular on the
case where {n} is a so-called Markov state-dependent noise. We establish both the stability and convergence w.p. 1 of the
algorithm under a set of simple and verifiable assumptions. We illustrate our results with an example related to adaptive
Markov chain Monte Carlo algorithms.
Key words. Stochastic approximation, state-dependent noise, randomly varying truncation, Adaptive Markov Chain
Monte Carlo.
AMS subject classifications. 62L20,90C15
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