Skip to Main content Skip to Navigation
Journal articles

Constant step stochastic approximations involving differential inclusions: stability, long-run convergence and applications

Pascal Bianchi 1 Walid Hachem 2, 3 Adil Salim 2
1 S2A - Signal, Statistique et Apprentissage
LTCI - Laboratoire Traitement et Communication de l'Information
2 COMNUM - Communications Numériques
LTCI - Laboratoire Traitement et Communication de l'Information
Abstract : We consider a Markov chain (xn) whose kernel is indexed by a scaling parameter γ > 0, referred to as the step size. The aim is to analyze the behavior of the Markov chain in the doubly asymptotic regime where n → ∞ then γ → 0. First, under mild assumptions on the so-called drift of the Markov chain, we show that the interpolated process converges narrowly to the solutions of a Differential Inclusion (DI) involving an upper semicontinuous set-valued map with closed and convex values. Second, we provide verifiable conditions which ensure the stability of the iterates. Third, by putting the above results together, we establish the long run convergence of the iterates as γ → 0, to the Birkhoff center of the DI. The ergodic behavior of the iterates is also provided. Application examples are investigated. We apply our findings to 1) the problem of nonconvex proximal stochastic optimization and 2) a fluid model of parallel queues.
Document type :
Journal articles
Complete list of metadata

Cited literature [33 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-02369439
Contributor : Adil Salim Connect in order to contact the contributor
Submitted on : Tuesday, November 19, 2019 - 7:50:50 AM
Last modification on : Tuesday, October 19, 2021 - 11:26:22 AM
Long-term archiving on: : Thursday, February 20, 2020 - 2:10:47 PM

File

revised_dicst.pdf
Files produced by the author(s)

Identifiers

Citation

Pascal Bianchi, Walid Hachem, Adil Salim. Constant step stochastic approximations involving differential inclusions: stability, long-run convergence and applications. Stochastics: An International Journal of Probability and Stochastic Processes, Taylor & Francis: STM, Behavioural Science and Public Health Titles, 2019, 91 (2), pp.288-320. ⟨10.1080/17442508.2018.1539086⟩. ⟨hal-02369439⟩

Share

Metrics

Record views

119

Files downloads

206