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 O\big(d^{1{3}}/\varepsilon^{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 O\big(d^{2{5}}/\varepsilon^{2{5}}\big) and O\big(d/\varepsilon^{1{3}}\big) steps when Lipschitz continuity is assumed for the Hessian and third derivative of f. By discretizing this ODE using a fourth order splitting method, we  obtain a practical MCMC method that requires just three additional gradient evaluations in each step. In our experiment, where the target comes from a logistic regression, this method shows faster convergence compared to other unadjusted Langevin MCMC algorithms.
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 O\big(d^{1{3}}/\varepsilon^{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 O\big(d^{2{5}}/\varepsilon^{2{5}}\big) and O\big(d/\varepsilon^{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.
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fourth order splitting
third order Runge-Kutta
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can
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requires just three
uses just two
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in each
per