Unadjusted Hamiltonian MCMC with Stratified Monte Carlo Time Integration

A novel unadjusted Hamiltonian Monte Carlo (uHMC) algorithm is suggested that uses a stratified Monte Carlo (SMC) time integrator for the underlying Hamiltonian dynamics in place of the usual Verlet time integrator. For target distributions of the form $\mu(dx) \propto e^{-U(x)} dx$ where $U: \mathbb{R}^d \to \mathbb{R}_{\ge 0}$ is both $K$-strongly convex and $L$-gradient Lipschitz, and initial distributions $\nu$ with finite second moment, coupling proofs reveal that an $\varepsilon$-accurate approximation of the target distribution $\mu$ in $L^2$-Wasserstein distance $\boldsymbol{\mathcal{W}}^2$ can be achieved by the uHMC algorithm with SMC time integration using $O\left((d/K)^{1/3} (L/K)^{5/3} \varepsilon^{-2/3} \log( \boldsymbol{\mathcal{W}}^2(\mu, \nu) / \varepsilon)^+\right)$ gradient evaluations; whereas without any additional assumptions the corresponding complexity of the uHMC algorithm with Verlet time integration is in general $O\left((d/K)^{1/2} (L/K)^2 \varepsilon^{-1} \log( \boldsymbol{\mathcal{W}}^2(\mu, \nu) / \varepsilon)^+ \right)$. The SMC time integrator involves a minor modification to Verlet, and hence, is easy to implement.