Higher Order Langevin Monte Carlo Algorithm

Abstract: A new (unadjusted) Langevin Monte Carlo (LMC) algorithm with improved rates in total variation and in Wasserstein distance is presented. All these are obtained in the context of sampling from a target distribution $\pi$ that has a density $\hat{\pi}$ on $\mathbb{R}^d$ known up to a normalizing constant. Moreover, $-\log \hat{\pi}$ is assumed to have a locally Lipschitz gradient and its third derivative is locally H\"{o}lder continuous with exponent $\beta \in (0,1]$. Non-asymptotic bounds are obtained for the convergence to stationarity of the new sampling method with convergence rate $1+ \beta/2$ in Wasserstein distance, while it is shown that the rate is 1 in total variation even in the absence of convexity. Finally, in the case where $-\log \hat{\pi}$ is strongly convex and its gradient is Lipschitz continuous, explicit constants are provided.