# On the convergence of dynamic implementations of Hamiltonian Monte Carlo and No U-Turn Samplers

arXiv: https://arxiv.org/abs/2307.03460 · HTML: https://mlsanity.com/paper/2307.03460/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 46 references that are in the map.
- First posted on arXiv 7 Jul 2023.

## Builds on (most cited first, 9 of 9)

- [A Conceptual Introduction to Hamiltonian Monte Carlo](https://mlsanity.com/paper/1701.02434/) (2017)
- [Coupling and Convergence for Hamiltonian Monte Carlo](https://mlsanity.com/paper/1805.00452/) (2018)
- [Geometric integrators and the Hamiltonian Monte Carlo method](https://mlsanity.com/paper/1711.05337/) (2017)
- [The Geometric Foundations of Hamiltonian Monte Carlo](https://mlsanity.com/paper/1410.5110/) (2014)
- [On the Geometric Ergodicity of Hamiltonian Monte Carlo](https://mlsanity.com/paper/1601.08057/) (2016)
- [Mixing Time Guarantees for Unadjusted Hamiltonian Monte Carlo](https://mlsanity.com/paper/2105.00887/) (2021)
- [A general perspective on the Metropolis-Hastings kernel](https://mlsanity.com/paper/2012.14881/) (2020)
- [Learning Generative Models with Visual Attention](https://mlsanity.com/paper/1312.6110/) (2013)
- [The Apogee to Apogee Path Sampler](https://mlsanity.com/paper/2112.08187/) (2021)

## Built on by (most cited first, 4 of 4)

- [GIST: Gibbs self-tuning for locally adaptive Hamiltonian Monte Carlo](https://mlsanity.com/paper/2404.15253/) (2024)
- [Mixing of the No-U-Turn Sampler and the Geometry of Gaussian Concentration](https://mlsanity.com/paper/2410.06978/) (2024)
- [On Accelerated Mixing of the No-U-turn Sampler](https://mlsanity.com/paper/2507.13259/) (2025)
- [A Theoretical Comparison of No-U-Turn Sampler Variants: Necessary and Sufficient Convergence Conditions and Mixing Time Analysis under Gaussian Targets](https://mlsanity.com/paper/2603.18640/) (2026)

## Abstract (arXiv)

There is substantial empirical evidence about the success of dynamic implementations of Hamiltonian Monte Carlo (HMC), such as the No U-Turn Sampler (NUTS), in many challenging inference problems but theoretical results about their behavior are scarce. The aim of this paper is to fill this gap. More precisely, we consider a general class of MCMC algorithms we call dynamic HMC. We show that this general framework encompasses NUTS as a particular case, implying the invariance of the target distribution as a by-product. Second, we establish conditions under which NUTS is irreducible and aperiodic and as a corrolary ergodic. Under conditions similar to the ones existing for HMC, we also show that NUTS is geometrically ergodic. Finally, we improve existing convergence results for HMC showing that this method is ergodic without any boundedness condition on the stepsize and the number of leapfrog steps, in the case where the target is a perturbation of a Gaussian distribution.