On Accelerated Mixing of the No-U-turn Sampler

TL;DR

This paper provides a rigorous analysis of the No-U-turn Sampler's ability to accelerate convergence in sampling, combining theoretical tools to understand its automatic tuning and mixing properties.

Contribution

It offers the first theoretical investigation into the accelerated convergence of the No-U-turn Sampler, linking automatic tuning to mixing performance.

Findings

Rigorous mixing guarantees for Gaussian targets.

Characterization of the sampler's acceleration capabilities.

Insights into limitations of accelerated convergence.

Abstract

Recent progress on the theory of variational hypocoercivity established that Randomized Hamiltonian Monte Carlo -- at criticality -- can achieve pronounced acceleration in its convergence and hence sampling performance over diffusive dynamics. Manual critical tuning being unfeasible in practice has motivated automated algorithmic solutions, notably the No-U-turn Sampler. Beyond its empirical success, a rigorous study of this method's ability to achieve accelerated convergence has been missing. We initiate this investigation combining a concentration of measure approach to examine the automatic tuning mechanism with a coupling based mixing analysis for Hamiltonian Monte Carlo. In certain Gaussian target distributions, this yields a precise characterization of the sampler's behavior resulting, in particular, in rigorous mixing guarantees describing the algorithm's ability and limitations in achieving accelerated convergence.