A novel randomized time integrator is suggested for unadjusted Hamiltonian Monte Carlo (uHMC) in place of the usual Verlet integrator; namely, a stratified Monte Carlo (sMC) integrator which involves a minor modification to Verlet, 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 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 additional assumptions the corresponding complexity of the uHMC algorithm with Verlet time integration is in general $O\left((d/K)^{1/2} (L/K)^2 \varepsilon^{-1} \log( \boldsymbol{\mathcal{W}}^2(\mu, \nu) / \varepsilon)^+ \right)$. Duration randomization, which has a similar effect as partial momentum refreshment, is also treated. In this case, without additional assumptions on the target distribution, the complexity of duration-randomized uHMC with sMC time integration improves to $O\left(\max\left((d/K)^{1/4} (L/K)^{3/2} \varepsilon^{-1/2},(d/K)^{1/3} (L/K)^{4/3} \varepsilon^{-2/3} \right) \right)$ up to logarithmic factors. The improvement due to duration randomization turns out to be analogous to that of time integrator randomization.
翻译:为未调整的 汉密尔顿 Monte Carlo (uHMC) 提供新的随机时间整合器, 取代通常的 Verlet 3/3 集成器; 即 一個分解的 Monte Carlo (sMC) 集成器, 包括微小修改 Verlet, 因此, 容易执行。 对于表格 $mu( dx)\ proto e ⁇ e ⁇ - U(x) dx 的分布器, 其中 $:\ mathb{R ⁇ d\ to\ mathbrob{( w ⁇ d) =2} Rge 0} 美元, 包括 3K+K 和 legient Lipsitz, 最初分配 $\nu$, 其中, $\ valepliplall $2$(dxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx