Influence of sampling on the convergence rates of greedy algorithms for parameter-dependent random variables

TL;DR

This paper analyzes how Monte-Carlo sampling affects the convergence of greedy algorithms for parameter-dependent expectations, providing theoretical guarantees and a practical heuristic for sample size selection.

Contribution

It offers a theoretical analysis of sampling effects on greedy algorithms' convergence and proposes a heuristic for practical sample size determination.

Findings

Theoretical conditions for sample sizes ensuring convergence with high probability.

A heuristic procedure for choosing Monte-Carlo samples at each iteration.

Numerical tests demonstrating the effectiveness of the heuristic.

Abstract

The main focus of this article is to provide a mathematical study of the algorithm proposed in \cite{boyaval2010variance} where the authors proposed a variance reduction technique for the computation of parameter-dependent expectations using a reduced basis paradigm. We study the effect of Monte-Carlo sampling on the theoretical properties of greedy algorithms. In particular, using concentration inequalities for the empirical measure in Wasserstein distance proved in \cite{fournier2015rate}, we provide sufficient conditions on the number of samples used for the computation of empirical variances at each iteration of the greedy procedure to guarantee that the resulting method algorithm is a weak greedy algorithm with high probability. These theoretical results are not fully practical and we therefore propose a heuristic procedure to choose the number of Monte-Carlo samples at each iteration, inspired from this theoretical study, which provides satisfactory results on several numerical test cases.