# On the Geometric Ergodicity of Hamiltonian Monte Carlo

arXiv: https://arxiv.org/abs/1601.08057 · HTML: https://mlsanity.com/paper/1601.08057/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Jan 2016.

## Built on by (most cited first, 19 of 19)

- [An Introduction to Hamiltonian Monte Carlo Method for Sampling](https://mlsanity.com/paper/2108.12107/) (2021)
- [Second Order Ensemble Langevin Method for Sampling and Inverse Problems](https://mlsanity.com/paper/2208.04506/) (2022)
- [Entropy-based adaptive Hamiltonian Monte Carlo](https://mlsanity.com/paper/2110.14625/) (2021)
- [Geometric Methods for Sampling, Optimisation, Inference and Adaptive Agents](https://mlsanity.com/paper/2203.10592/) (2022)
- [Listening to the Noise: Blind Denoising with Gibbs Diffusion](https://mlsanity.com/paper/2402.19455/) (2024)
- [A Unifying and Canonical Description of Measure-Preserving Diffusions](https://mlsanity.com/paper/2105.02845/) (2021)
- [On the convergence of dynamic implementations of Hamiltonian Monte Carlo and No U-Turn Samplers](https://mlsanity.com/paper/2307.03460/) (2023)
- [Ensemble-Based Annealed Importance Sampling](https://mlsanity.com/paper/2401.15645/) (2024)
- [Reflection coupling for unadjusted generalized Hamiltonian Monte Carlo in the nonconvex stochastic gradient case](https://mlsanity.com/paper/2310.18774/) (2023)
- [On Lq Convergence of the Hamiltonian Monte Carlo](https://mlsanity.com/paper/2101.08688/) (2021)
- [Hamiltonian Monte Carlo with Asymmetrical Momentum Distributions](https://mlsanity.com/paper/2110.12907/) (2021)
- [Differentially Private Statistical Inference through β-Divergence One Posterior Sampling](https://mlsanity.com/paper/2307.05194/) (2023)
- [Non-asymptotic bounds for forward processes in denoising diffusions: Ornstein-Uhlenbeck is hard to beat](https://mlsanity.com/paper/2408.13799/) (2024)
- [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)
- [An MCMC Method to Sample from Lattice Distributions](https://mlsanity.com/paper/2101.06453/) (2021)
- [Geometric Ergodicity in Modified Variations of Riemannian Manifold and Lagrangian Monte Carlo](https://mlsanity.com/paper/2301.01409/) (2023)
- [Learning variational autoencoders via MCMC speed measures](https://mlsanity.com/paper/2308.13731/) (2023)
- [Repelling-Attracting Hamiltonian Monte Carlo](https://mlsanity.com/paper/2403.04607/) (2024)
- [LazyHMC: Hamiltonian Monte Carlo Simulation for Lazy, Infinite Dimensional Probabilistic Programs](https://mlsanity.com/paper/2608.08588/) (2026)

## Abstract (arXiv)

We establish general conditions under which Markov chains produced by the Hamiltonian Monte Carlo method will and will not be geometrically ergodic. We consider implementations with both position-independent and position-dependent integration times. In the former case we find that the conditions for geometric ergodicity are essentially a gradient of the log-density which asymptotically points towards the centre of the space and grows no faster than linearly. In an idealised scenario in which the integration time is allowed to change in different regions of the space, we show that geometric ergodicity can be recovered for a much broader class of tail behaviours, leading to some guidelines for the choice of this free parameter in practice.