In this paper, we revisit the recently established theoretical guarantees for the convergence of the Langevin Monte Carlo algorithm of sampling from a smooth and (strongly) log-concave density. We improve, in terms of constants, the existing results when the accuracy of sampling is measured in the Wasserstein distance and provide further insights on relations between, on the one hand, the Langevin Monte Carlo for sampling and, on the other hand, the gradient descent for optimization. More importantly, we establish non-asymptotic guarantees for the accuracy of a version of the Langevin Monte Carlo algorithm that is based on inaccurate evaluations of the gradient. Finally, we propose a variable-step version of the Langevin Monte Carlo algorithm that has two advantages. First, its step-sizes are independent of the target accuracy and, second, its rate provides a logarithmic improvement over the constant-step Langevin Monte Carlo algorithm.

本文研究了从已知平滑和强对数凹概率密度函数中采样的方法， 分析了基于过渡态随机游走的近似采样方法，并提出了几种保证误差的方法， 包括第一阶Langevin Monte Carlo算法的误差上界、误差上界和梯度评估不准确的情况， 以及二阶Langevin Monte Carlo算法利用log密度的海森矩阵的保证。

Langevin Monte Carlo 与不准确梯度的用户友好保证