{"ID":2887642,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.01392","arxiv_id":"2508.01392","title":"Quenched large deviations for Monte Carlo integration with Coulomb gases","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.","short_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 re...","url_abs":"https://arxiv.org/abs/2508.01392","url_pdf":"https://arxiv.org/pdf/2508.01392v1","authors":"[\"Rémi Bardenet\",\"Mylène Maïda\",\"Martin Rouault\"]","published":"2025-08-02T14:52:06Z","proceeding":"cs.LG","tasks":"[\"cs.LG\",\"math.PR\",\"stat.ML\"]","methods":"[]","has_code":false}