Convergence in total variation for the kinetic Langevin algorithm

TL;DR

This work analyzes sampling from a log-density $\mu(dx)=e^{-V(x)}dx$ on $\mathbb{R}^n$ using the kinetic (underdamped) Langevin algorithm and provides non-asymptotic total-variation guarantees in high dimension under Poincaré and smoothness assumptions on $V$. It deploys a hypocoercivity framework, leveraging a space-time Poincaré inequality and a divergence equation to obtain explicit decay bounds for the chi-square divergence along the kinetic diffusion, despite degeneracy. The discretization error is controlled via a Girsanov-based relative-entropy argument, leading to concrete step-size and iteration-count prescriptions that achieve $TV(x_k,\mu)\le\varepsilon$, with dimension dependence improved from $O(n)$ to $O(\sqrt{n})$ relative to the overdamped case; log-concavity yields further improvements. The results highlight the practical and theoretical advantages of the kinetic approach, offering polynomial warm-start dependence and potential extensions to related sampling schemes such as Hamiltonian Monte Carlo.

Abstract

We prove non asymptotic total variation estimates for the kinetic Langevin algorithm in high dimension when the target measure satisfies a Poincaré 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 $O(\sqrt n)$.

Theorems & Definitions (20)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4
• Lemma 5: Bochner formula
• Corollary 6
• proof
• Proposition 7
• Remark
• Lemma 8