Convergence in total variation for the kinetic Langevin algorithm

TL;DR

This paper provides improved non-asymptotic total variation bounds for the kinetic Langevin algorithm in high dimensions, showing a significant reduction in dimension dependence compared to the non-kinetic version, under certain regularity conditions.

Contribution

The paper establishes sharper total variation convergence bounds for the kinetic Langevin algorithm, reducing the dimension dependence from linear to square root order.

Findings

Dimension dependence drops from O(n) to O(√n)

Improved convergence bounds under Poincaré inequality

Applicable to high-dimensional target measures with Lipschitz gradients

Abstract

We prove non asymptotic total variation estimates for the kinetic Langevin algorithm in high dimension when the target measure satisfies a Poincar\'e inequality and has gradient Lipschitz potential. The main point is that the estimate improves significantly upon the corresponding bound for the non kinetic version of the algorithm, due to Dalalyan. In particular the dimension dependence drops from $O(n)$ to $O(\sqrt n)$.