Quenched large deviations for Monte Carlo integration with Coulomb gases

2025-08-02

AI Summary

This paper addresses the slow convergence of Monte Carlo numerical integration caused by the mismatch between the equilibrium measure and the target distribution π. We propose a randomized particle integration method grounded in Gibbs measures—such as the Coulomb gas—designed to circumvent precise hyperparameter tuning. Leveraging the quenched large deviation principle (within the Garcia–Zelada framework), we rigorously analyze integration error, enabling the use of stochastic approximate potentials in place of exact interaction kernels and confinement potentials. This preserves fast large-deviation performance while substantially reducing preprocessing computational cost. Theoretical analysis establishes uniform convergence under the target distribution π for both nonsingular and Coulomb-type kernels. Empirical evaluation and theoretical analysis jointly demonstrate that the proposed method achieves a faster integration error decay rate than both independent sampling and conventional Markov chain quadrature.

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: User modeling and simulation for interactive and conversational systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Gibbs measures, such as Coulomb gases, are popular in modelling systems of interacting particles. Recently, we proposed to use Gibbs measures as randomized numerical integration algorithms with respect to a target measure $π$ on $mathbb R^d$, following the heuristics that repulsiveness between particles should help reduce integration errors. A major issue in this approach is to tune the interaction kernel and confining potential of the Gibbs measure, so that the equilibrium measure of the system is the target distribution $π$. Doing so usually requires another Monte Carlo approximation of the emph{potential}, i.e. the integral of the interaction kernel with respect to $π$. Using the methodology of large deviations from Garcia--Zelada (2019), we show that a random approximation of the potential preserves the fast large deviation principle that guarantees the proposed integration algorithm to outperform independent or Markov quadratures. For non-singular interaction kernels, we make minimal assumptions on this random approximation, which can be the result of a computationally cheap Monte Carlo preprocessing. For the Coulomb interaction kernel, we need the approximation to be based on another Gibbs measure, and we prove in passing a control on the uniform convergence of the approximation of the potential.

Problem

• Optimizing interaction kernel and confining potential for Gibbs measures
• Ensuring equilibrium measure matches target distribution in Monte Carlo integration
• Validating random potential approximations for fast large deviation principles

Innovation

• Uses Gibbs measures for Monte Carlo integration
• Tunes interaction kernel via random approximation
• Ensures fast large deviation for accuracy