A Multi-Step Richardson-Romberg Extrapolation Method For Stochastic Approximation

TL;DR

This paper extends the Richardson-Romberg extrapolation method to stochastic approximation algorithms, improving estimation accuracy and reducing computational costs in stochastic optimization tasks like quantile estimation of diffusion processes.

Contribution

It develops a multi-step Richardson-Romberg extrapolation approach for stochastic approximation, enhancing efficiency in stochastic optimization problems.

Findings

Significant reduction in computational cost demonstrated.

Theoretical analysis confirms improved accuracy.

Effective application to quantile estimation of diffusion processes.

Abstract

We obtain an expansion of the implicit weak discretization error for the target of stochastic approximation algorithms introduced and studied in [Frikha2013]. This allows us to extend and develop the Richardson-Romberg extrapolation method for Monte Carlo linear estimator (introduced in [Talay & Tubaro 1990] and deeply studied in [Pag{\`e}s 2007]) to the framework of stochastic optimization by means of stochastic approximation algorithm. We notably apply the method to the estimation of the quantile of diffusion processes. Numerical results confirm the theoretical analysis and show a significant reduction in the initial computational cost.