标题： 高阶Langevin蒙特卡洛算法 显示英文标题

标题： Higher Order Langevin Monte Carlo Algorithm

摘要： 提出了一种新的（未经调整的）Langevin Monte Carlo (LMC) 算法，在总变差距离和Wasserstein距离下均具有改进的收敛速率。 所有这些结果均在从目标分布 $\pi$ 抽样时获得，该分布的密度为 $\hat{\pi}$，定义在已知归一化常数的 $\mathbb{R}^d$ 上。 此外，假设 $-\log \hat{\pi}$ 的梯度局部 Lipschitz 连续，且其三阶导数局部 Hölder 连续，指数为 $\beta \in (0,1]$。 对于新抽样方法在Wasserstein距离下的收敛到平稳态，得到了非渐近界，收敛速率为 $1+ \beta/2$，而在总变差距离下即使在非凸情形下也证明了收敛速率为1。 最后，在 $-\log \hat{\pi}$ 强凸且其梯度 Lipschitz 连续的情况下，给出了显式常数。 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 $\hat{\pi}$ on $\mathbb{R}^d$ known up to a normalizing constant. Moreover, $-\log \hat{\pi}$ is assumed to have a locally Lipschitz gradient and its third 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 where $-\log \hat{\pi}$ is strongly convex and its gradient is Lipschitz continuous, explicit constants are provided.