  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.
