In previous work, we introduced a method for determining convergence rates for integration methods for the kinetic Langevin equation for $M$-$\nabla$Lipschitz $m$-log-concave densities [arXiv:2302.10684, 2023]. In this article, we exploit this method to treat several additional schemes including the method of Brunger, Brooks and Karplus (BBK) and stochastic position/velocity Verlet. We introduce a randomized midpoint scheme for kinetic Langevin dynamics, inspired by the recent scheme of Bou-Rabee and Marsden [arXiv:2211.11003, 2022]. We also extend our approach to stochastic gradient variants of these schemes under minimal extra assumptions. We provide convergence rates of $\mathcal{O}(m/M)$, with explicit stepsize restriction, which are of the same order as the stability thresholds for Gaussian targets and are valid for a large interval of the friction parameter. We compare the contraction rate estimates of many kinetic Langevin integrators from molecular dynamics and machine learning. Finally we present numerical experiments for a Bayesian logistic regression example.

翻译：在先前工作中，我们提出了一种确定$M$-$\nabla$Lipschitz $m$-对数凹密度[arXiv:2302.10684, 2023]的动郎文方程积分方法收敛率的方法。本文利用该方法处理了多种其他方案，包括Brunger-Brooks-Karplus（BBK）方法与随机位置/速度Verlet方法。受Bou-Rabee和Marsden近期方案[arXiv:2211.11003, 2022]启发，我们引入了一种用于动郎文动力学的随机中点方案。此外，在最小额外假设条件下，我们将方法扩展至这些方案的随机梯度变体。我们提供了$\mathcal{O}(m/M)$的收敛率（含显式步长限制），该收敛率与高斯目标的稳定性阈值同阶，且适用于摩擦参数的大区间。我们对来自分子动力学与机器学习的多种动郎文积分器的收缩率估计进行了比较。最后，我们针对贝叶斯逻辑回归示例展示了数值实验。