论文标题
未经调整的哈密顿MCMC与分层的蒙特卡洛时间整合
Unadjusted Hamiltonian MCMC with Stratified Monte Carlo Time Integration
论文作者
论文摘要
建议对未经调整的汉密尔顿蒙特卡洛（UHMC）提出一个随机时间集成器，涉及对通常的Verlet时间集成器的非常小的修改，因此很容易实现。对于形式的目标分布，$μ（dx）\ propto e^{ - u（x）} dx $其中$ u：\ mathbb {r}^d \ to \ mathbb {r} _ {\ ge 0} $ $ \ varepsilon $ -Carcurate $ l^2 $ -WassErstein距离$ \ boldsymbol {\ Mathcal {w}}^2 $可以通过UHMC算法可以实现，并使用$ o \ weft（d/k）^{1/3}（l/k）^{1/1/3}（1/1/3}（{1/1/3}） \ varepsilon^{ - 2/3} \ log（\ boldsymbol {\ mathcal {w}}}^2（μ，μ，ν） / \ varepsilon）^+\ right）$渐变评估；尽管对于这种粗糙的目标密度，verlet时间积分的UHMC算法的相应复杂性在一般而言$ o \ left（（（d/k）^{1/2} {1/2}（l/k）^2 \ varepsilon^{ - 1} \ log log（\ boldsymbol \ varepsilon）^+ \ right）$。还提供了大都市调整的随机时间集成符。
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.
