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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