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In this paper, we consider the underdamped Langevin diffusion (ULD) andpropose a numerical approximation using its associated ordinary differentialequation (ODE). When used as a Markov Chain Monte Carlo (MCMC) algorithm, weshow 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 underthe standard smoothness and strong convexity assumptions on the targetdistribution. This matches the complexity of the randomized midpoint methodproposed by Shen and Lee [NeurIPS 2019] which was shown to be order optimal byCao, Lu and Wang. However, the main feature of the proposed numerical method isthat it can utilize additional smoothness of the target log-density $f$. Moreconcretely, we show that the ODE approximation achieves a $2$-Wasserstein errorof $\varepsilon$ 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 Lipschitzcontinuity is assumed for the Hessian and third derivative of $f$. Bydiscretizing this ODE using a third order Runge-Kutta method, we can obtain apractical MCMC method that uses just two additional gradient evaluations perstep. In our experiment, where the target comes from a logistic regression,this method shows faster convergence compared to other unadjusted Langevin MCMCalgorithms.

The shifted ODE method for underdamped Langevin MCMC