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Pré-Publication, Document De Travail Année : 2016

Convergence of Markovian Stochastic Approximation with discontinuous dynamics

Résumé

This paper is devoted to the convergence analysis of stochastic approximation algorithms of the form $\theta_{n+1} = \theta_n + \gamma_{n+1} H_{\theta_n}(X_{n+1})$ where $\{\theta_nn, n \geq 0\}$ is a $R^d$-valued sequence, $\{\gamma, n \geq 0\}$ is a deterministic step-size sequence and $\{X_n, n \geq 0\}$ is a controlled Markov chain. We study the convergence under weak assumptions on smoothness-in-$\theta$ of the function $\theta \mapsto H_{\theta}(x)$. It is usually assumed that this function is continuous for any $x$; in this work, we relax this condition. Our results are illustrated by considering stochastic approximation algorithms for (adaptive) quantile estimation and a penalized version of the vector quantization.
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Dates et versions

hal-00966187 , version 1 (26-03-2014)
hal-00966187 , version 2 (22-01-2016)

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Citer

Gersende Fort, Eric Moulines, Amandine Schreck, Matti Vihola. Convergence of Markovian Stochastic Approximation with discontinuous dynamics. 2016. ⟨hal-00966187v2⟩
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