We study the Langevin-type algorithms for Gibbs distributions such that the potentials are dissipative and their weak gradients have the finite moduli of continuity. Our main result is a non-asymptotic upper bound of the 2-Wasserstein distance between the Gibbs distribution and the law of general Langevin-type algorithms based on the Liptser--Shiryaev theory and functional inequalities. We apply this bound to show that the dissipativity of the potential and the $\alpha$-H\"{o}lder continuity of the gradient with $\alpha>1/3$ are sufficient for the convergence of the Langevin Monte Carlo algorithm with appropriate control of the parameters. We also propose Langevin-type algorithms with spherical smoothing for potentials without convexity or continuous differentiability.

翻译：我们研究了Gibbs分布的Langevin型算法，其中势函数是耗散的，其弱梯度具有有限的连续性模。我们的主要结果是基于Liptser-Shiryaev理论和函数不等式的一种2- Wasserstein距离的非渐进上界，用于描述Gibbs分布和一般Langevin型算法的距离。我们将这个上界应用到实际问题中，例如：当势函数的耗散性和其梯度保持$\alpha$-Holder连续性且$\alpha>1/3$时，可以对参数进行适当控制，使Langevin Monte Carlo算法收敛。此外，我们还提出了具有球形平滑特性的Langevin型算法，用于处理没有凸性或连续可微性的势函数。