We provide a framework to prove convergence rates for discretizations of kinetic Langevin dynamics for $M$-$\nabla$Lipschitz $m$-log-concave densities. Our approach provides convergence rates of $\mathcal{O}(m/M)$, with explicit stepsize restrictions, which are of the same order as the stability threshold for Gaussian targets and are valid for a large interval of the friction parameter. We apply this methodology to various integration methods which are popular in the molecular dynamics and machine learning communities. Finally we introduce the property ``$\gamma$-limit convergent" (GLC) to characterise underdamped Langevin schemes that converge to overdamped dynamics in the high friction limit and which have stepsize restrictions that are independent of the friction parameter; we show that this property is not generic by exhibiting methods from both the class and its complement.

翻译：我们建立了一个框架，用于证明针对 $M$-$\nabla$利普希茨 $m$-对数凹密度的动能朗之万动力学离散化的收敛率。我们的方法提供了 $\mathcal{O}(m/M)$ 阶的收敛率，带有显式的步长限制，这些步长限制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。我们将此方法论应用于分子动力学和机器学习社区中流行的多种积分方法。最后，我们引入了“$\gamma$-极限收敛”性质，以刻画在高摩擦极限下收敛到过阻尼动力学、且步长限制独立于摩擦参数的欠阻尼朗之万方案；我们通过展示来自该性质类及其补类的方法，证明了该性质并非普适。