Rényi mixing bounds for kinetic samplers via $L^2$ 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 $\mathbb R^d$ that satisfies a logarithmic Sobolev inequality, with $\nabla^2U$ 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 $\sqrt{R\log R}$ 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.