We study Langevin-type algorithms for sampling from Gibbs distributions such that the potentials are dissipative and their weak gradients have finite moduli of continuity not necessarily convergent to zero. Our main result is a non-asymptotic upper bound of the 2-Wasserstein distance between a Gibbs distribution and the law of general Langevin-type algorithms based on the Liptser--Shiryaev theory and Poincar\'{e} inequalities. We apply this bound to show that the Langevin Monte Carlo algorithm can approximate Gibbs distributions with arbitrary accuracy if the potentials are dissipative and their gradients are uniformly continuous. We also propose Langevin-type algorithms with spherical smoothing for distributions whose potentials are not convex or continuously differentiable.

翻译：我们研究用于从吉布斯分布采样的Langevin型算法，其中势能是耗散性的，且其弱梯度具有不一定收敛到零的有限连续性模量。基于Liptser-Shiryaev理论和庞加莱不等式，我们的主要结果是吉布斯分布与一般Langevin型算法分布之间的2-瓦瑟斯坦距离的非渐近上界。应用该上界，我们证明了当势能具有耗散性且其梯度一致连续时，Langevin蒙特卡洛算法能以任意精度逼近吉布斯分布。针对势能非凸或非连续可微的分布，我们还提出了带有球形平滑处理的Langevin型算法。