标题： 高阶郎之万蒙特卡罗算法 显示英文标题

标题： Higher Order Langevin Monte Carlo Algorithm

摘要： 一种新的（未调整的）Langevin蒙特卡罗（LMC）算法在从目标分布$\pi$采样的背景下被提出，在假设其密度在$\mathbb{R}^d$上已知到归一化常数的情况下。关键的是，与目标分布$\pi$相关的Langevin SDE被假定具有局部Lipschitz的漂移系数，使其二阶导数在指数$\beta \in (0,1]$下局部Hölder连续。此外，还获得了新采样方法收敛到平稳性的非渐近界，在Wasserstein距离下的收敛率为$1+ \beta/2$，同时表明该速率在总变差中为1。还给出了LMC算法的不变测度与$\pi$之间的界限。 显示英文摘要

摘要： A new (unadjusted) Langevin Monte Carlo (LMC) algorithm with improved rates in total variation and Wasserstein distance is presented in the context of sampling from a target distribution $\pi$ under the assumption that its density on $\mathbb{R}^d$ is known up to a normalizing constant. Crucially, the Langevin SDE associated with the target distribution $\pi$ is assumed to have a locally Lipschitz drift coefficient such that its second derivative is locally H\"{o}lder continuous with exponent $\beta \in (0,1]$. In addition, non-asymptotic bounds are obtained for the convergence to stationarity of the new sampling method with convergence rate $1+ \beta/2$ in Wasserstein distance, while it is shown that the rate is 1 in total variation. The bounds between the invariant measure of the LMC algorithm and $\pi$ are also given.