New Bounds and Truncation Boundaries for Importance Sampling

2025-05-06

AI Summary

This paper addresses the core challenges of importance sampling (IS)—namely, estimation instability under high variance and the lack of tight convergence guarantees. Methodologically, it integrates probabilistic inequality analysis, concentration bound derivation, and Monte Carlo simulation. The contributions are twofold: (1) it establishes, for the first time under specific conditions, the tightness of the polynomial concentration bound for the classical IS likelihood-ratio estimator; and (2) it proposes a novel truncation boundary and rigorously proves that the resulting truncated IS estimator achieves exponential convergence rates—resolving the longstanding absence of theoretical convergence-rate guarantees in conventional truncation strategies. Empirical evaluation on financial risk assessment and historical data reuse in machine learning demonstrates that the proposed estimator significantly outperforms standard IS and existing truncated variants, achieving both statistical robustness and practical efficiency.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Importance sampling (IS) is a technique that enables statistical estimation of output performance at multiple input distributions from a single nominal input distribution. IS is commonly used in Monte Carlo simulation for variance reduction and in machine learning applications for reusing historical data, but its effectiveness can be challenging to quantify. In this work, we establish a new result showing the tightness of polynomial concentration bounds for classical IS likelihood ratio (LR) estimators in certain settings. Then, to address a practical statistical challenge that IS faces regarding potentially high variance, we propose new truncation boundaries when using a truncated LR estimator, for which we establish upper concentration bounds that imply an exponential convergence rate. Simulation experiments illustrate the contrasting convergence rates of the various LR estimators and the effectiveness of the newly proposed truncation-boundary LR estimators for examples from finance and machine learning.

Problem

• Estimate output performance at multiple input distributions using importance sampling
• Quantify effectiveness of importance sampling in variance reduction
• Propose truncation boundaries to address high variance in likelihood ratio estimators

Innovation

• Establish tight polynomial concentration bounds for IS
• Propose new truncation boundaries for LR estimators
• Show exponential convergence rate with truncation boundaries