Nonasymptotic bounds for suboptimal importance sampling

DOI10.1137/21M1427760arXiv2102.09606OpenAlexW4394821450MaRDI QIDQ6360995

Publication date: 18 February 2021

Abstract: Importance sampling is a popular variance reduction method for Monte Carlo estimation, where a notorious question is how to design good proposal distributions. While in most cases optimal (zero-variance) estimators are theoretically possible, in practice only suboptimal proposal distributions are available and it can often be observed numerically that those can reduce statistical performance significantly, leading to large relative errors and therefore counteracting the original intention. In this article, we provide nonasymptotic lower and upper bounds on the relative error in importance sampling that depend on the deviation of the actual proposal from optimality, and we thus identify potential robustness issues that importance sampling may have, especially in high dimensions. We focus on path sampling problems for diffusion processes, for which generating good proposals comes with additional technical challenges, and we provide numerous numerical examples that support our findings.

Full work available at URL: https://doi.org/10.1137/21m1427760

Mathematics Subject Classification ID

Probabilistic models, generic numerical methods in probability and statistics (65C20) Applications of optimal control and differential games (49N90) Approximations to statistical distributions (nonasymptotic) (62E17) Monte Carlo methods applied to problems in statistical mechanics (82M31)