Non-asymptotic estimates for accelerated high order Langevin Monte Carlo algorithms

In this paper, we propose two new algorithms, namely, aHOLA and aHOLLA, to sample from high-dimensional target distributions with possibly super-linearly growing potentials. We establish non-asymptotic convergence bounds for aHOLA in Wasserstein-1 and Wasserstein-2 distances with rates of convergence equal to [Formula: see text] and [Formula: see text], respectively, under a local Hölder condition with exponent [Formula: see text] and a convexity at infinity condition on the potential of the target distribution. Similar results are obtained for aHOLLA under certain global continuity conditions and a dissipativity condition. Crucially, we achieve state-of-the-art rates of convergence of the proposed algorithms in the non-convex setting which are higher than those of the existing algorithms. Examples from high-dimensional sampling and logistic regression are presented, and numerical results support our main findings.