标题： 高阶朗之万蒙特卡罗算法 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 Monte Carlo（LMC）算法在总变差和Wasserstein距离中具有改进的收敛速率。所有这些结果都是在从一个目标分布$\pi$进行采样的背景下获得的，该分布在一个$\mathbb{R}^d$上具有密度，但仅知道归一化常数以外的部分。关键的是，与目标分布$\pi$相关的Langevin SDE被假设具有局部Lipschitz的漂移系数，其二阶导数在指数$\beta \in (0,1]$下是局部Hölder连续的。对于新采样方法的平稳性收敛，获得了非渐近界，其在Wasserstein距离中的收敛速率为$1+ \beta/2$，同时表明即使在没有凸性的情况下，速率在总变差中仍为1。最后，在梯度为Lipschitz的情况下，提供了显式常数。 显示英文摘要

摘要： A new (unadjusted) Langevin Monte Carlo (LMC) algorithm with improved rates in total variation and in Wasserstein distance is presented. All these are obtained in the context of sampling from a target distribution $\pi$ that has a density on $\mathbb{R}^d$ 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 $\beta \in (0,1]$. 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 even in the absence of convexity. Finally, in the case of a Lipschitz gradient, explicit constants are provided.