Multi-level stochastic approximation algorithms

TL;DR

This paper extends multilevel Monte Carlo methods to stochastic approximation algorithms, introducing two-level and multi-level algorithms with proven convergence properties and demonstrating significant computational cost reductions through numerical experiments.

Contribution

It introduces multi-level stochastic approximation algorithms, including a two-level method and its extension, with theoretical analysis and practical efficiency improvements.

Findings

Proven central limit theorems for the proposed methods

Significant reduction in computational cost demonstrated

Optimal step size sequences identified

Abstract

This paper studies multi-level stochastic approximation algorithms. Our aim is to extend the scope of the multilevel Monte Carlo method recently introduced by Giles (Giles 2008) 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 multi-level is proposed. We prove a central limit theorem for both methods and describe the possible optimal choices of step size sequence. Numerical results confirm the theoretical analysis and show a significant reduction in the initial computational cost.