🤖 AI Summary
Sampling from high-dimensional target distributions with superlinearly growing potentials—such as nonconvex functions that become convex at infinity—remains challenging.
Method: We propose two accelerated higher-order Langevin algorithms, aHOLA and aHOLLA, which integrate higher-order discretizations of Langevin dynamics with momentum-based acceleration. Our theoretical analysis leverages local Hölder continuity, convexity at infinity, and dissipativity conditions.
Contribution/Results: We establish the first non-asymptotic convergence bound in Wasserstein-1/2 distance for nonconvex settings, achieving a Wasserstein-1 convergence rate of order $1 + q/2$, where $q$ is the Hölder exponent—surpassing existing rates and attaining the best-known rate for nonconvex sampling. Numerical experiments across diverse nonconvex distributions validate the algorithms’ accelerated convergence, stability, and practical efficiency.
📝 Abstract
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 $1+q/2$ and $1/2+q/4$, respectively, under a local H""{o}lder condition with exponent $qin(0,1]$ 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. Numerical experiments are conducted to sample from several distributions and the results support our main findings.