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

ArXiv ID: 2407.18504 “View on arXiv”

Authors: Unknown

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 Cramér’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.

Keywords: Sample Average Approximation (SAA), Multilevel Monte Carlo (MLMC), Optimization, Conditional Value-at-Risk (CVaR), Cramer-Lundberg, General / Derivatives

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 4.0/10
• Quadrant: Lab Rats
• Why: The paper is densely mathematical, featuring advanced theoretical analysis using Cramér’s large deviation theory and empirical process theory to derive convergence rates and sample complexity, placing it in the ‘Lab Rats’ quadrant. While it includes numerical examples, the empirical component is primarily illustrative of the theoretical claims rather than a heavy backtesting/implementation-focused study.

  flowchart TD
    A["Research Goal: Enhance SAA for Optimization<br>using Biased MLMC Estimators"] --> B["Key Methodology"]
    B --> C["Theoretical Analysis<br>Cramer's Large Deviation & Empirical Process"]
    B --> D["Algorithm Development<br>MLMC SAA Framework"]
    
    C & D --> E["Computational Processes"]
    E --> F["Numerical Experiments<br>CVaR Estimation: GBM & Nested Expectation"]
    
    F --> G["Key Findings & Outcomes"]
    G --> H["Uniform Convergence & Sample Complexity<br>MLMC Superiority in Computational Efficiency"]