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：我们提出了一个框架，用于证明 $M$-$\nabla$Lipschitz $m$-log-concave 密度下动力学 Langevin 动力学离散化的收敛速率。我们的方法给出了 $\mathcal{O}(m/M)$ 的收敛速率，具有明确的步长限制，该速率与高斯目标的稳定性阈值同阶，并且适用于一个较大的摩擦参数区间。我们将此方法应用于分子动力学和机器学习领域中流行的多种积分方法。最后，我们引入了“$\gamma$-极限收敛”(GLC) 性质来描述在高摩擦极限下收敛到过阻尼动力学、且步长限制与摩擦参数无关的欠阻尼 Langevin 格式；我们通过展示该类及其补集中的方法，表明该性质并非普遍存在。