Rényi mixing bounds for kinetic samplers via L2 hypocoercivity
Abstract
We study convergence in Rényi divergence of order two for underdamped Langevin dynamics, the bouncy particle sampler, and the Zigzag sampler, for a target e−U on ℝd that satisfies a logarithmic Sobolev inequality, with ∇2U bounded below by a possibly negative constant and, for the Zigzag sampler, also bounded above. For initial laws whose Rényi divergence from equilibrium is at most R, we show that it decays exponentially, at a rate that depends on R. With the friction or refreshment rate chosen according to R, the time needed to reach a fixed accuracy in total variation or in Rényi divergence is of order √(Rlog R) at fixed dimension and target parameters, improving the linear dependence obtained by directly converting exponential L2 decay. The key idea is to use the logarithmic Sobolev inequality for the position target in place of the usual spatial spectral-gap estimate in the L2 hypocoercivity argument. For the bouncy particle and Zigzag samplers, this order is sharp for the Rényi divergence on a one-dimensional Gaussian target, uniformly in the refreshment rate. For the Zigzag sampler in dimension at least two, Gaussian examples show that the mixing-time bound is sharp up to logarithmic factors in the worst case over strongly convex targets, in both the initial divergence and the upper Hessian bound; in particular, strong convexity alone does not give a uniform mixing bound.
Related
- Weighted L2 bounds for kinetic Fokker-Planck equations and hypocoercivity
- Second-order perturbation bounds for Gibbs samplers under strong spatial mixing
- Hypocoercivity of Tempered Bouncy Particle Samplers for Heavy-Tailed Targets
- Discrete H -theorem for finite volume discretization of a nonlinear kinetic system: application to hypocoercivity
- Rapid mixing for Gibbs samplers for quantum spin systems with all-to-all interactions
BibTeX
@misc{lu2026rényi,
title = {{Rényi mixing bounds for kinetic samplers via $L^2$ hypocoercivity}},
author = {Jianfeng Lu and Yinchen Luo},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03921},
eprint = {2611.03921},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03921},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Rényi mixing bounds for kinetic samplers via L^2 hypocoercivity
AU - Jianfeng Lu
AU - Yinchen Luo
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03921
UR - https://arxiv.org/abs/2611.03921
AB - We study convergence in Rényi divergence of order two for underdamped Langevin dynamics, the bouncy particle sampler, and the Zigzag sampler, for a target e^−U on ℝ^d that satisfies a logarithmic Sobolev inequality, with ∇^2U bounded below by a possibly negative constant and, for the Zigzag sampler, also bounded above. For initial laws whose Rényi divergence from equilibrium is at most R, we show that it decays exponentially, at a rate that depends on R. With the friction or refreshment rate chosen according to R, the time needed to reach a fixed accuracy in total variation or in Rényi divergence is of order √(Rlog R) at fixed dimension and target parameters, improving the linear dependence obtained by directly converting exponential L^2 decay. The key idea is to use the logarithmic Sobolev inequality for the position target in place of the usual spatial spectral-gap estimate in the L^2 hypocoercivity argument. For the bouncy particle and Zigzag samplers, this order is sharp for the Rényi divergence on a one-dimensional Gaussian target, uniformly in the refreshment rate. For the Zigzag sampler in dimension at least two, Gaussian examples show that the mixing-time bound is sharp up to logarithmic factors in the worst case over strongly convex targets, in both the initial divergence and the upper Hessian bound; in particular, strong convexity alone does not give a uniform mixing bound.
ER -