Langevin diffusion processes and their discretizations are often used for sampling from a target density. The most convenient framework for assessing the quality of such a sampling scheme corresponds to smooth and strongly log-concave densities defined on $\mathbb R^p$. The present work focuses on this framework and studies the behavior of the Monte Carlo algorithm based on discretizations of the kinetic Langevin diffusion. We first prove the geometric mixing property of the kinetic Langevin diffusion with a mixing rate that is optimal in terms of its dependence on the condition number. We then use this result for obtaining improved guarantees of sampling using the kinetic Langevin Monte Carlo method, when the quality of sampling is measured by the Wasserstein distance. We also consider the situation where the Hessian of the log-density of the target distribution is Lipschitz-continuous. In this case, we introduce a new discretization of the kinetic Langevin diffusion and prove that this leads to a substantial improvement of the upper bound on the sampling error measured in Wasserstein distance.

使用Langevin扩散过程进行离散化的蒙特卡洛算法可用于对光滑且强对数凹密度进行采样, 本文主要研究了这个框架, 并证明了基于kinetic Langevin扩散的Monte Carlo算法的混合性质和采样质量, 进一步证明了Hessian矩阵Lipschitz连续的情况下, 使用新的离散化方法可以显著提高采样误差的上界。

使用运动朗之万扩散从对数凹密度中采样