Multilevel Monte Carlo in Sample Average Approximation: Convergence, Complexity and Application

TL;DR

This paper analyzes the convergence and complexity of Sample Average Approximation methods enhanced with Multilevel Monte Carlo, providing theoretical insights and numerical validation for efficient risk estimation in stochastic optimization.

Contribution

It introduces MLMC into the SAA framework, offering a comprehensive convergence analysis and demonstrating improved computational efficiency over standard methods.

Findings

MLMC improves sample complexity in SAA for biased Monte Carlo estimators.

Uniform convergence and convergence rates are established using large deviation theory.

Numerical experiments show MLMC's advantages in CVaR estimation under stochastic models.

Abstract

In this paper, we examine the Sample Average Approximation (SAA) procedure within a framework where the Monte Carlo estimator of the expectation is biased. We also introduce Multilevel Monte Carlo (MLMC) in the SAA setup to enhance the computational efficiency of solving optimization problems. In this context, we conduct a thorough analysis, exploiting Cram\'er's large deviation theory, to establish uniform convergence, quantify the convergence rate, and determine the sample complexity for both standard Monte Carlo and MLMC paradigms. Additionally, we perform a root-mean-squared error analysis utilizing tools from empirical process theory to derive sample complexity without relying on the finite moment condition typically required for uniform convergence results. Finally, we validate our findings and demonstrate the advantages of the MLMC estimator through numerical examples, estimating Conditional Value-at-Risk (CVaR) in the Geometric Brownian Motion and nested expectation framework.