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：我们研究针对吉布斯分布的Langevin型算法，其中势函数是耗散的，且其弱梯度具有有限连续性模。基于Liptser--Shiryaev理论和泛函不等式，我们的主要结果是吉布斯分布与基于一般Langevin型算法的分布之间的2- Wasserstein距离的非渐近上界。应用该上界，我们证明当势函数具有耗散性且梯度具有α>1/3的α- Hölder连续性时，通过适当控制参数，Langevin蒙特卡洛算法能够收敛。此外，针对非凸或非连续可微的势函数，我们提出了带有球面平滑的Langevin型算法。