  Sampling from log-concave distributions is a well researched problem that has
many applications in statistics and machine learning. We study the
distributions of the form $p^{*}\propto\exp(-f(x))$, where
$f:\mathbb{R}^{d}\rightarrow\mathbb{R}$ has an $L$-Lipschitz gradient and is
$m$-strongly convex. In our paper, we propose a Markov chain Monte Carlo (MCMC)
algorithm based on the underdamped Langevin diffusion (ULD). It can achieve
$\epsilon\cdot D$ error (in 2-Wasserstein distance) in
$\tilde{O}\left(\kappa^{7/6}/\epsilon^{1/3}+\kappa/\epsilon^{2/3}\right)$
steps, where $D\overset{\mathrm{def}}{=}\sqrt{\frac{d}{m}}$ is the effective
diameter of the problem and $\kappa\overset{\mathrm{def}}{=}\frac{L}{m}$ is the
condition number. Our algorithm performs significantly faster than the
previously best known algorithm for solving this problem, which requires
$\tilde{O}\left(\kappa^{1.5}/\epsilon\right)$ steps. Moreover, our algorithm
can be easily parallelized to require only $O(\kappa\log\frac{1}{\epsilon})$
parallel steps.
  To solve the sampling problem, we propose a new framework to discretize
stochastic differential equations. We apply this framework to discretize and
simulate ULD, which converges to the target distribution $p^{*}$. The framework
can be used to solve not only the log-concave sampling problem, but any problem
that involves simulating (stochastic) differential equations.
