Underdamped Langevin Monte Carlo (ULMC) is an algorithm used to sample from unnormalized densities by leveraging the momentum of a particle moving in a potential well. We provide a novel analysis of ULMC, motivated by two central questions: (1) Can we obtain improved sampling guarantees beyond strong log-concavity? (2) Can we achieve acceleration for sampling? For (1), prior results for ULMC only hold under a log-Sobolev inequality together with a restrictive Hessian smoothness condition. Here, we relax these assumptions by removing the Hessian smoothness condition and by considering distributions satisfying a Poincar\'e inequality. Our analysis achieves the state of art dimension dependence, and is also flexible enough to handle weakly smooth potentials. As a byproduct, we also obtain the first KL divergence guarantees for ULMC without Hessian smoothness under strong log-concavity, which is based on a new result on the log-Sobolev constant along the underdamped Langevin diffusion. For (2), the recent breakthrough of Cao, Lu, and Wang (2020) established the first accelerated result for sampling in continuous time via PDE methods. Our discretization analysis translates their result into an algorithmic guarantee, which indeed enjoys better condition number dependence than prior works on ULMC, although we leave open the question of full acceleration in discrete time. Both (1) and (2) necessitate R\'enyi discretization bounds, which are more challenging than the typically used Wasserstein coupling arguments. We address this using a flexible discretization analysis based on Girsanov's theorem that easily extends to more general settings.

翻译：欠阻尼朗之万蒙特卡洛（ULMC）是一种通过利用粒子在势阱中运动的动量从非归一化密度采样的算法。我们受两个核心问题驱动，对ULMC进行了新的分析：（1）能否在强对数凹性之外获得改进的采样保证？（2）能否实现采样的加速？针对问题（1），先前ULMC的结果仅在对数索博列夫不等式及严格的Hessian光滑性条件下成立。我们通过移除Hessian光滑性条件并考虑满足庞加莱不等式的分布来放宽这些假设。我们的分析实现了最优的维度依赖，且具有足够灵活性以处理弱光滑势函数。作为副产品，我们还在强对数凹性条件下首次获得了无需Hessian光滑性的ULMC的KL散度保证，这基于欠阻尼朗之万扩散过程中对数索博列夫常数的新结果。针对问题（2），Cao、Lu和Wang（2020）的最新突破通过偏微分方程方法建立了连续时间采样的首个加速结果。我们的离散化分析将其转化为算法保证，确实比先前ULMC研究具有更好的条件数依赖，尽管我们未完全解决离散时间加速问题。问题（1）和（2）均需要Rényi离散化界，这比常用的Wasserstein耦合论证更具挑战性。我们通过基于Girsanov定理的灵活离散化分析解决了这一问题，该方法可轻松推广至更一般的设定。