Non-asymptotic entropic bounds for non-linear kinetic Langevin sampler with second-order splitting scheme

Abstract

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.