Nonlinear Hamiltonian Monte Carlo and its Particle Approximation

Abstract

In the talk, a nonlinear (in the sense of McKean) generalization of Hamiltonian Monte Carlo (HMC) termed nonlinear HMC (nHMC) is presented to sample from nonlinear probability measures of meanfield type. Provided the underlying confinement potential is L-gradient Lipschitz and asymptotically K-strongly convex, we prove contraction in L 1 -Wasserstein distance. Further, we show that uniformly in time nHMC can be approximated by unadjusted HMC for its corresponding mean-field particle system in L 1 -Wasserstein distance. In particular, we analyse the number of gradient evaluations needed for unadjusted HMC with randomized time integration to achieve an ϵ-accuracy of a d-dimensional nonlinear probability measure in L 1 -Wasserstein distance. The talk is based on joint work with Nawaf Bou-Rabee (arXiv:2308.11491).