# Dynamical behavior of a stochastic forward-backward algorithm using random monotone operators

Abstract : The purpose of this paper is to study the dynamical behavior of the sequence (x n) produced by the forward-backward algorithm $y_{n+1} \in B(u_{n+1}, x_n)$, $x_{n+1} = ( I + \gamma_{n+1} A(u_{n+1}, \cdot))^{-1} ( x_n - \gamma_{n+1} y_{n+1} )$, where $A(\xi) = A(\xi, \cdot)$ and $B(\xi) = B(\xi, \cdot)$ are two functions valued in the set of maximal monotone operators on $\mathbb{R}^N$, $(u_n)$ is a sequence of independent and identically distributed random variables, and $(y_n)$ is a sequence of vanishing step sizes. Following the approach of the recent paper [16], we define the operators ${\mathcal A}(x) = {\mathbb E}[A(u_1 , x)]$ and ${\mathcal B}(x) = {\mathbb E}[B(u_1 , x)]$ where the expectations are the set-valued Aumann integrals with respect to the law of $u_1$ , and assume that the monotone operator ${\mathcal A} + {\mathcal B}$ is maximal (sufficient conditions for maximality are provided). It is shown that with probability one, the interpolated process obtained from the iterates x n is an asymptotic pseudo trajectory in the sense of Benaim and Hirsch of the differential inclusion $\dot z(t) \in - ( {\mathcal A} + {\mathcal B} )(z(t))$. The convergence of the empirical means of the $x_n$'s towards a zero of ${\mathcal A} + {\mathcal B}$ follows, as well as the convergence of the sequence $(x_n)$ itself to such a zero under a demipositivity assumption. These results find applications in a wide range of optimization or variational inequality problems in random environments.
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https://hal.archives-ouvertes.fr/hal-01183959
Contributor : Walid Hachem <>
Submitted on : Wednesday, August 12, 2015 - 2:56:36 PM
Last modification on : Monday, October 19, 2020 - 12:07:09 PM
Long-term archiving on: : Friday, November 13, 2015 - 11:38:44 AM

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• HAL Id : hal-01183959, version 1

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Pascal Bianchi, Walid Hachem. Dynamical behavior of a stochastic forward-backward algorithm using random monotone operators. Journal of Optimization Theory and Applications, Springer Verlag, 2016, 171. ⟨hal-01183959⟩

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