Control variates for variance-reduced ratio of means estimators

TL;DR

This work addresses variance reduction for Monte Carlo estimators of ratios of expectations by introducing a variance-reduced CV/CV estimator that applies control variates to both the numerator and denominator and optimizes the pair $(\alpha,\beta)$ jointly. It extends the method to approximate control variates and provides a theoretical guarantee of variance reduction under the mild condition $|\text{Corr}(B,D)|<1$, with closed-form solutions for the optimal coefficients. The authors demonstrate substantial variance reductions in simulations and two multi-fidelity applications (aircraft design and electromagnetic cVar), highlighting the practical impact when HF/LF variables are highly correlated. They also compare against prior coefficient choices (e.g., Gordon 1982) and discuss when those may fail, outlining directions for extending the approach to other ratio-based estimators and tail statistics.

Abstract

The control variates method is a classical variance reduction technique for Monte Carlo estimators that exploits correlated auxiliary variables without introducing bias. In many applications, the quantity of interest can be expressed as a ratio of expectations. We propose a variance-reduced estimator for such ratios, which applies control variates to both the numerator and the denominator. The control variate coefficients are optimized jointly to minimize the variance of the resulting estimator. This approach theoretically guarantees variance reduction and naturally extends to approximate control variates. Simulation studies show significant variance reduction, particularly when correlations between variables and control variates are strong. The practical value of the method is illustrated on multi-fidelity applications: estimating a proportion in an aircraft design use case and a conditional value-at-risk in an electromagnetic dataset.

Figures (7)

• Figure 1: Boxplots of ratio of means estimations, comparing the MC/MC, CV/MC, and CV/CV estimators - Strong variance reduction with all estimators
• Figure 2: Boxplots of ratio of means estimations, comparing the MC/MC and CV/CV estimators - Variance increase with non-optimal coefficients
• Figure 3: Boxplots of ratio of means estimations, comparing the MC/MC and CV/CV estimator - Best variance reduction with the coefficients from gordon_efficient_1982 with $n=10$
• Figure 4: Boxplots of ratio of means estimations, comparing the MC/MC and CV/CV estimator - Equivalent variance reduction performances with $n=100$
• Figure 5: Illustrations of the $1,252$ coupled HF and LF samples of the aircraft design dataset

Theorems & Definitions (25)

• Definition 1: Monte Carlo estimator
• Definition 2: Control variates estimator asmussen_variance-reduction_2007
• Definition 3: Approximate control variates gorodetsky_generalized_2020
• Definition 4: CV/CV estimator
• Proposition 1: Coefficients for the CV/CV estimator
• Proposition 2: Optimal coefficients for the CV/CV estimator
• Proposition 3
• Proposition 4
• Definition 5: CV/MC estimator
• Proposition 5