Higher Order Langevin Monte Carlo Algorithm
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A new (unadjusted) Langevin Monte Carlo (LMC) algorithm with improved rates in total variation and Wasserstein distance is presented in the context of sampling from a target distribution π under the assumption that its density on R^d is known up to a normalizing constant. Crucially, the Langevin SDE associated with the target distribution π is assumed to have a locally Lipschitz drift coefficient such that its second derivative is locally Hölder continuous with exponent β∈ (0,1]. In addition, non-asymptotic bounds are obtained for the convergence to stationarity of the new sampling method with convergence rate 1+ β/2 in Wasserstein distance, while it is shown that the rate is 1 in total variation. The bounds between the invariant measure of the LMC algorithm and π are also given.
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