The Annals of Applied Probability

Multi-level stochastic approximation algorithms

Noufel Frikha

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This paper studies multi-level stochastic approximation algorithms. Our aim is to extend the scope of the multi-level Monte Carlo method recently introduced by Giles [Oper. Res. 56 (2008) 607–617] to the framework of stochastic optimization by means of stochastic approximation algorithm. We first introduce and study a two-level method, also referred as statistical Romberg stochastic approximation algorithm. Then its extension to a multi-level method is proposed. We prove a central limit theorem for both methods and give optimal parameters. Numerical results confirm the theoretical analysis and show a significant reduction in the initial computational cost.

Article information

Ann. Appl. Probab., Volume 26, Number 2 (2016), 933-985.

Received: October 2013
Revised: February 2015
First available in Project Euclid: 22 March 2016

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 62K12 65C05: Monte Carlo methods 60H35: Computational methods for stochastic equations [See also 65C30]

Multi-level Monte Carlo methods stochastic approximation Ruppert–Polyak averaging principle Euler scheme


Frikha, Noufel. Multi-level stochastic approximation algorithms. Ann. Appl. Probab. 26 (2016), no. 2, 933--985. doi:10.1214/15-AAP1109.

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