Persistence Pays: Square-root condition-number dependence for Metropolis-adjusted kinetic Langevin Monte Carlo with a non-reversibly updated accept/reject uniform

Abstract

We prove accelerated mixing for a Metropolis-adjusted kinetic Langevin sampler: the OBABO splitting with friction $2\sqrt m$, adjusted with Neal's non-reversibly updated uniform, which is redrawn after every epoch of $E$ steps. For targets $e^{-U}$ on $\mathbb{R}^d$ with $mI\preceq\nabla^2U\preceq LI$ and an $L_H$-Lipschitz Hessian, started from a Gaussian at an approximate minimizer, the chain reaches total variation accuracy $\varepsilon$ after $\tilde O\big(\sqrt{κd}+\sqrt{L_H}(d/m)^{3/4}\big)$ gradient evaluations, where $κ=L/m$ and $\tilde O$ hides logarithmic factors. The $\sqrtκ$ term is necessary, already on Gaussian targets. The proof discretizes Lu's entropic hypocoercivity by means of an entropy inequality for free transport, so that the relative entropy of the unadjusted chain decays at a rate of order $\sqrt m$ up to a discretization bias. With the non-reversibly updated uniform, the leading energy errors of velocity Verlet telescope along a path. A coupling argument combines the two. On Bayesian logistic regression posteriors for two standard data sets, which satisfy these assumptions, the gradient cost of the chain with the non-reversibly updated uniform grows roughly like the square root of the condition number at the posterior mode, and that of standard adjusted OBABO roughly like its $3/4$ power.