Nonlinear Hamiltonian Monte Carlo & its Particle Approximation

TL;DR

This paper introduces a nonlinear generalization of Hamiltonian Monte Carlo (HMC) called nonlinear HMC (nHMC) for sampling from mean-field type nonlinear probability measures, providing complexity bounds and particle approximation methods.

Contribution

The paper develops nonlinear HMC (nHMC) for mean-field measures and analyzes its efficiency and particle approximation, extending HMC to nonlinear probability measures.

Findings

nHMC approximates nonlinear measures with $O((L/K) \, \log(1/\varepsilon))$ steps.

Particle approximation with randomized time integration achieves $\varepsilon$-accuracy in $L^1$-Wasserstein distance.

Complexity bounds are extended to non-logconcave, nonlinear measures of mean-field type.

Abstract

We present a nonlinear (in the sense of McKean) generalization of Hamiltonian Monte Carlo (HMC) termed nonlinear HMC (nHMC) capable of sampling from nonlinear probability measures of mean-field type. When the underlying confinement potential is $K$-strongly convex and $L$-gradient Lipschitz, and the underlying interaction potential is gradient Lipschitz, nHMC can produce an $\varepsilon$-accurate approximation of a $d$-dimensional nonlinear probability measure in $L^1$-Wasserstein distance using $O((L/K) \log(1/\varepsilon))$ steps. Owing to a uniform-in-steps propagation of chaos phenomenon, and without further regularity assumptions, unadjusted HMC with randomized time integration for the corresponding particle approximation can achieve $\varepsilon$-accuracy in $L^1$-Wasserstein distance using $O( (L/K)^{5/3} (d/K)^{4/3} (1/\varepsilon)^{8/3} \log(1/\varepsilon) )$ gradient evaluations. These mixing/complexity upper bounds are a specific case of more general results developed in the paper for a larger class of non-logconcave, nonlinear probability measures of mean-field type.