Unadjusted Hamiltonian MCMC with Stratified Monte Carlo Time Integration

TL;DR

This work develops a simple yet effective randomized time integration scheme for unadjusted Hamiltonian Monte Carlo (uHMC) by introducing a stratified Monte Carlo (sMC) time integrator. Under $K$-strong convexity and $L$-gradient Lipschitz conditions, the authors prove $L^2$-Wasserstein contractivity and quantify the asymptotic bias between the invariant measure of the discretized chain and the target distribution, establishing a $3/2$-order $L^2$-accuracy for the sMC integrator. The resulting complexity bound shows that, for rough potentials, uHMC with sMC achieves $\varepsilon$-approximation in $\mathcal{W}^2$ with significantly better gradient-evaluation scaling than Verlet-based uHMC, namely $O\left((d/K)^{1/3} (L/K)^{5/3} \varepsilon^{-2/3} \log(\cdot)^+\right)$ versus $O\left((d/K)^{1/2} (L/K)^2 \varepsilon^{-1} \log(\cdot)^+\right)$ in the rough target setting. The paper also extends the framework to adjustable randomized time integrators and to duration-randomized variants, showing potential further gains in contraction and bias, with detailed proofs and an appendix on related implementations.

Abstract

A randomized time integrator is suggested for unadjusted Hamiltonian Monte Carlo (uHMC) which involves a very minor modification to the usual Verlet time integrator, and hence, is easy to implement. For target distributions of the form $μ(dx) \propto e^{-U(x)} dx$ where $U: \mathbb{R}^d \to \mathbb{R}_{\ge 0}$ is $K$-strongly convex but only $L$-gradient Lipschitz, and initial distributions $ν$ with finite second moment, coupling proofs reveal that an $\varepsilon$-accurate approximation of the target distribution in $L^2$-Wasserstein distance $\boldsymbol{\mathcal{W}}^2$ can be achieved by the uHMC algorithm with randomized time integration using $O\left((d/K)^{1/3} (L/K)^{5/3} \varepsilon^{-2/3} \log( \boldsymbol{\mathcal{W}}^2(μ, ν) / \varepsilon)^+\right)$ gradient evaluations; whereas for such rough target densities the corresponding complexity of the uHMC algorithm with Verlet time integration is in general $O\left((d/K)^{1/2} (L/K)^2 \varepsilon^{-1} \log( \boldsymbol{\mathcal{W}}^2(μ, ν) / \varepsilon)^+ \right)$. Metropolis-adjustable randomized time integrators are also provided.

Figures (1)

• Figure 1: $L^2$-Accuracy Verification. Left Image: A plot of the $L^2$-error in $(x,v)$-space of the sMC time integrator for the linear oscillator with Hamiltonian $H(x,v) = (1/2) (v^2 + x^2)$. Right Image: A plot of the $L^2$-error in $(x,v)$-space of the sMC time integrator for a double-well system with Hamiltonian $H(x,v) = (1/2) (v^2 + (1-x^2)^2)$. Both simulations have initial condition $(2,1)$ and unit duration. The time step sizes tested are $2^{-n}$ where $n$ is given on the horizontal axis. The dashed curve is $2^{-3 n/2} = h^{3/2}$ versus $n$.

Theorems & Definitions (34)

• Definition 1: uHMC with sMC time integration
• Remark 3
• Remark 4
• Lemma 5: Almost Sure Contractivity of sMC Time Integrator
• Theorem 6: $\boldsymbol{\mathcal{W}}^2$-Contractivity of uHMC
• proof
• Lemma 7: $L^2$-accuracy of sMC Time Integrator
• proof : Proof of Lemma \ref{['lem:smc_strong_accuracy']}
• Remark 8
• Remark 9