# Quenched large deviations for Monte Carlo integration with Coulomb gases

arXiv: https://arxiv.org/abs/2508.01392 · HTML: https://mlsanity.com/paper/2508.01392/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 45 references that are in the map.
- First posted on arXiv 2 Aug 2025.

## Builds on (most cited first, 17 of 17)

- [Measuring Sample Quality with Kernels](https://mlsanity.com/paper/1703.01717/) (2017)
- [Log-concave sampling: Metropolis-Hastings algorithms are fast](https://mlsanity.com/paper/1801.02309/) (2018)
- [Kernel Stein Discrepancy Descent](https://mlsanity.com/paper/2105.09994/) (2021)
- [Kernel Thinning](https://mlsanity.com/paper/2105.05842/) (2021)
- [Stein Points](https://mlsanity.com/paper/1803.10161/) (2018)
- [Support points](https://mlsanity.com/paper/1609.01811/) (2016)
- [Stein's Method Meets Computational Statistics: A Review of Some Recent Developments](https://mlsanity.com/paper/2105.03481/) (2021)
- [Stein Point Markov Chain Monte Carlo](https://mlsanity.com/paper/1905.03673/) (2019)
- [Generalized Kernel Thinning](https://mlsanity.com/paper/2110.01593/) (2021)
- [Kernel quadrature with DPPs](https://mlsanity.com/paper/1906.07832/) (2019)
- [Monte Carlo with Determinantal Point Processes](https://mlsanity.com/paper/1605.00361/) (2016)
- [Kernel interpolation with continuous volume sampling](https://mlsanity.com/paper/2002.09677/) (2020)
- [Distribution Compression in Near-linear Time](https://mlsanity.com/paper/2111.07941/) (2021)
- [Controlling Moments with Kernel Stein Discrepancies](https://mlsanity.com/paper/2211.05408/) (2022)
- [Efficient Numerical Integration in Reproducing Kernel Hilbert Spaces via Leverage Scores Sampling](https://mlsanity.com/paper/2311.13548/) (2023)
- [Kernel Stein Discrepancy thinning: a theoretical perspective of pathologies and a practical fix with regularization](https://mlsanity.com/paper/2301.13528/) (2023)
- [Monte Carlo with kernel-based Gibbs measures: Guarantees for probabilistic herding](https://mlsanity.com/paper/2402.11736/) (2024)

## Abstract (arXiv)

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 Rd, 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 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.