[Paper Review] Nonasymptotic bounds for suboptimal importance sampling

Carsten Hartmann, Lorenz Richter|arXiv (Cornell University)|Feb 18, 2021

Probabilistic and Robust Engineering Design46 references4 citations

TL;DR

This paper establishes nonasymptotic bounds on the relative error in importance sampling for diffusion processes, showing that performance degrades significantly when the proposal distribution deviates from optimality, especially in high dimensions. The key contribution is a rigorous quantitative analysis linking relative error to the KL divergence between the proposal and optimal measures, revealing inherent fragility in practical importance sampling applications.

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.

Motivation & Objective

• To identify and quantify the robustness issues in importance sampling when using suboptimal proposal distributions.
• To derive nonasymptotic upper and lower bounds on the relative error that depend explicitly on the deviation from the optimal proposal.
• To analyze the impact of high dimensionality and long trajectories on the stability and performance of importance sampling in path sampling problems.
• To provide theoretical justification for observed numerical instabilities in rare-event simulations involving diffusion processes.
• To connect theoretical error bounds with practical challenges in molecular dynamics, finance, and climate modeling.

Proposed method

• Derives nonasymptotic bounds on the relative error using the KL divergence between the actual proposal and the optimal importance sampling measure.
• Applies the bounds to path measures of diffusion processes, particularly in the context of rare events and metastable dynamics.
• Uses the Hamilton-Jacobi-Bellman (HJB) equation framework to characterize the optimal control and derive error expressions.
• Employs PDE analysis and stochastic control theory to relate the relative error to the deviation of the control from the optimal one.
• Derives explicit expressions for the relative error in log-normal and small-noise diffusion settings, including asymptotic expansions.
• Validates theoretical findings with numerical examples across high-dimensional and long-trajectory scenarios.

Experimental results

Research questions

• RQ1 How does the relative error in importance sampling scale with the deviation from the optimal proposal distribution?
• RQ2 What is the impact of high dimensionality on the stability and performance of importance sampling estimators?
• RQ3 Why do suboptimal proposals lead to significant increases in relative error, even when the KL divergence is small?
• RQ4 Can nonasymptotic bounds be derived that capture the fragility of importance sampling in path sampling for diffusions?
• RQ5 How do the theoretical error bounds compare with numerical observations in rare-event simulations?

Key findings

• The relative error in importance sampling is bounded from below and above by functions of the KL divergence between the proposal and optimal measures, providing a quantitative measure of robustness.
• The relative error grows exponentially with the KL divergence, especially in high-dimensional settings, explaining numerical instabilities in practice.
• In small-noise diffusions, the relative error is shown to scale as √(e^{εᵀΣε} − 1), which grows rapidly with the noise intensity ε and the direction of deviation.
• The zero-variance property holds if and only if the control u matches the optimal u* = −σᵀ∇V, confirming the necessity of exact optimality.
• The KL divergence increases with dimension, implying that high-dimensional problems are inherently more sensitive to proposal misspecification.
• Numerical examples confirm that even small deviations from optimality can lead to large relative errors, particularly in long trajectories and high-dimensional path spaces.