In this paper, we study the problem of sampling from a given probability density function that is known to be smooth and strongly log-concave. We analyze several methods of approximate sampling based on discretizations of the (highly overdamped) Langevin diffusion and establish guarantees on its error measured in the Wasserstein-2 distance. Our guarantees improve or extend the state-of-the-art results in three directions. First, we provide an upper bound on the error of the first-order Langevin Monte Carlo (LMC) algorithm with optimized varying step-size. This result has the advantage of being horizon free (we do not need to know in advance the target precision) and to improve by a logarithmic factor the corresponding result for the constant step-size. Second, we study the case where accurate evaluations of the gradient of the log-density are unavailable, but one can have access to approximations of the aforementioned gradient. In such a situation, we consider both deterministic and stochastic approximations of the gradient and provide an upper bound on the sampling error of the first-order LMC that quantifies the impact of the gradient evaluation inaccuracies. Third, we establish upper bounds for two versions of the second-order LMC, which leverage the Hessian of the log-density. We provide nonasymptotic guarantees on the sampling error of these second-order LMCs. These guarantees reveal that the second-order LMC algorithms improve on the first-order LMC in ill-conditioned settings.

翻译：本文研究从已知光滑且强对数凹的概率密度函数中采样的问题。我们分析了基于（高度过阻尼）朗之万扩散离散化的几种近似采样方法，并建立了以Wasserstein-2距离度量的误差保证。我们的保证在以下三个方面改进或拓展了现有最优结果。首先，我们为采用优化变步长的一阶朗之万蒙特卡洛（LMC）算法提供了误差上界。该结果的优点是无需预先设定目标精度（即无界性），且相较于常步长对应的结果在对数因子方面有所改进。其次，我们研究了无法精确评估对数密度梯度但可获取该梯度近似值的情形。在此情况下，我们同时考虑了梯度的确定性与随机性近似，并给出了量化梯度评估不精确性影响的一阶LMC采样误差上界。第三，我们建立了利用对数密度海森矩阵的两种二阶LMC算法的上界，并提供了这些二阶LMC采样误差的非渐近保证。这些保证表明，在病态条件下二阶LMC算法较一阶LMC具有显著改进。