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 $\mu(dx) \propto e^{-U(x)} dx$ where $U: \mathbb{R}^d \to \mathbb{R}_{\ge 0}$ is $K$-strongly convex but only $L$-gradient Lipschitz, and initial distributions $\nu$ 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 $O\left((d/K)^{1/3} (L/K)^{5/3} \varepsilon^{-2/3} \log( \boldsymbol{\mathcal{W}}^2(\mu, \nu) / \varepsilon)^+\right)$ gradient evaluations; whereas for such rough target densities the corresponding complexity of the uHMC algorithm with Verlet time integration is in general $O\left((d/K)^{1/2} (L/K)^2 \varepsilon^{-1} \log( \boldsymbol{\mathcal{W}}^2(\mu, \nu) / \varepsilon)^+ \right)$. Metropolis-adjustable randomized time integrators are also provided.

翻译：本文提出了一种用于未调整哈密顿蒙特卡洛（uHMC）的随机化时间积分器，该积分器仅需对标准Verlet时间积分器进行极微小的修改，因而易于实现。对于形式为 $\mu(dx) \propto e^{-U(x)} dx$ 的目标分布，其中 $U: \mathbb{R}^d \to \mathbb{R}_{\ge 0}$ 是 $K$-强凸但仅 $L$-梯度Lipschitz的，以及具有有限二阶矩的初始分布 $\nu$，耦合证明表明：采用随机化时间积分的uHMC算法能够以 $O\left((d/K)^{1/3} (L/K)^{5/3} \varepsilon^{-2/3} \log( \boldsymbol{\mathcal{W}}^2(\mu, \nu) / \varepsilon)^+\right)$ 次梯度求值，在 $L^2$-Wasserstein距离 $\boldsymbol{\mathcal{W}}^2$ 下获得目标分布的 $\varepsilon$-精度近似；而对于此类粗糙的目标密度，采用Verlet时间积分的uHMC算法的相应复杂度通常为 $O\left((d/K)^{1/2} (L/K)^2 \varepsilon^{-1} \log( \boldsymbol{\mathcal{W}}^2(\mu, \nu) / \varepsilon)^+ \right)$。本文还提供了可进行Metropolis调整的随机化时间积分器。