  In this paper, we consider the underdamped Langevin diffusion (ULD) and
propose a numerical approximation using its associated ordinary differential
equation (ODE). When used as a Markov Chain Monte Carlo (MCMC) algorithm, we
show that the ODE approximation achieves a $2$-Wasserstein error of
$\varepsilon$ in
$\mathcal{O}\big(d^{\frac{1}{3}}/\varepsilon^{\frac{2}{3}}\big)$ steps under
the standard smoothness and strong convexity assumptions on the target
distribution. This matches the complexity of the randomized midpoint method
proposed by Shen and Lee [NeurIPS 2019] which was shown to be order optimal by
Cao, Lu and Wang. However, the main feature of the proposed numerical method is
that it can utilize additional smoothness of the target log-density $f$. More
concretely, we show that the ODE approximation achieves a $2$-Wasserstein error
of $\varepsilon$ in
$\mathcal{O}\big(d^{\frac{2}{5}}/\varepsilon^{\frac{2}{5}}\big)$ and
$\mathcal{O}\big(\sqrt{d}/\varepsilon^{\frac{1}{3}}\big)$ steps when Lipschitz
continuity is assumed for the Hessian and third derivative of $f$. By
discretizing this ODE using a third order Runge-Kutta method, we can obtain a
practical MCMC method that uses just two additional gradient evaluations per
step. In our experiment, where the target comes from a logistic regression,
this method shows faster convergence compared to other unadjusted Langevin MCMC
algorithms.
