Nested Expectations with Kernel Quadrature

TL;DR

This paper introduces a novel nested kernel quadrature estimator for nested expectations, achieving faster convergence and requiring fewer samples than traditional Monte Carlo methods in applications like Bayesian optimization and finance.

Contribution

The paper proposes a new nested kernel quadrature estimator with proven faster convergence for smooth integrands, outperforming existing Monte Carlo-based methods.

Findings

Faster convergence rate than baseline methods for smooth functions

Requires fewer samples in real-world applications

Effective in Bayesian optimization, option pricing, and health economics

Abstract

This paper considers the challenging computational task of estimating nested expectations. Existing algorithms, such as nested Monte Carlo or multilevel Monte Carlo, are known to be consistent but require a large number of samples at both inner and outer levels to converge. Instead, we propose a novel estimator consisting of nested kernel quadrature estimators and we prove that it has a faster convergence rate than all baseline methods when the integrands have sufficient smoothness. We then demonstrate empirically that our proposed method does indeed require fewer samples to estimate nested expectations on real-world applications including Bayesian optimisation, option pricing, and health economics.