2024/12/04 by Pierre Monmarché, Monmarché, Pierre, Katharina Schuh +1 · 1 citation
Mathematics · Physics and Astronomy · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Statistical Mechanics and Entropy #Statistical Methods and Inference
paper · pdf · doi:10.48550/arxiv.2412.03560
openalex publication_date 2024/12/04 · openalex created_date 2024/12/06 · openalex updated_date 2026/07/28
The problem of sampling according to the probability distribution minimizing a given free energy, using interacting particles unadjusted kinetic Langevin Monte Carlo, is addressed. In this setting, three sources of error arise, related to three parameters: the number of particles N, the discretization step size h, and the length of the trajectory n. The main result of the present work is a quantitative estimate of strong convergence in relative entropy, implying non-asymptotic bounds for the quadratic risk of Monte Carlo estimators for bounded observables. The numerical discretization scheme considered here is a second-order splitting method, as commonly used in practice. In addition to N,h,n, the dependency in the ambient dimension d of the problem is also made explicit, under suitable conditions. The main results are proven under general conditions (regularity, moments, log-Sobolev inequality), for which tractable conditions are then provided. In particular, a Lyapunov analysis is conducted under more general conditions than previous works; the nonlinearity may not be small and it may not be convex along linear interpolations between measures.