Doubly Robust Stein-Kernelized Monte Carlo Estimator: Simultaneous Bias-Variance Reduction and Supercanonical Convergence

TL;DR

This paper introduces a new doubly robust Stein-kernelized Monte Carlo estimator that achieves supercanonical convergence, reducing bias and variance simultaneously, especially effective with biased and noisy samples.

Contribution

The paper proposes a general framework that unifies control variate and importance sampling techniques, enhancing Monte Carlo error reduction under biased and noisy conditions.

Findings

Outperforms existing methods in mean squared error rates

Achieves supercanonical convergence in Monte Carlo estimation

Demonstrates superior numerical performance in experiments

Abstract

Standard Monte Carlo computation is widely known to exhibit a canonical square-root convergence speed in terms of sample size. Two recent techniques, one based on control variate and one on importance sampling, both derived from an integration of reproducing kernels and Stein's identity, have been proposed to reduce the error in Monte Carlo computation to supercanonical convergence. This paper presents a more general framework to encompass both techniques that is especially beneficial when the sample generator is biased and noise-corrupted. We show our general estimator, which we call the doubly robust Stein-kernelized estimator, outperforms both existing methods in terms of mean squared error rates across different scenarios. We also demonstrate the superior performance of our method via numerical examples.