# An Introduction to Hamiltonian Monte Carlo Method for Sampling

arXiv: https://arxiv.org/abs/2108.12107 · HTML: https://mlsanity.com/paper/2108.12107/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 58 references that are in the map.
- First posted on arXiv 27 Aug 2021.

## Builds on (most cited first, 19 of 19)

- [Stochastic Gradient Hamiltonian Monte Carlo](https://mlsanity.com/paper/1402.4102/) (2014)
- [Identifying and attacking the saddle point problem in high-dimensional non-convex optimization](https://mlsanity.com/paper/1406.2572/) (2014)
- [Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysis](https://mlsanity.com/paper/1702.03849/) (2017)
- [Theoretical guarantees for approximate sampling from smooth and log-concave densities](https://mlsanity.com/paper/1412.7392/) (2014)
- [Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1507.05021/) (2015)
- [High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1605.01559/) (2016)
- [Underdamped Langevin MCMC: A non-asymptotic analysis](https://mlsanity.com/paper/1707.03663/) (2017)
- [Log-concave sampling: Metropolis-Hastings algorithms are fast](https://mlsanity.com/paper/1801.02309/) (2018)
- [Structured Logconcave Sampling with a Restricted Gaussian Oracle](https://mlsanity.com/paper/2010.03106/) (2020)
- [Hamiltonian Monte Carlo for Hierarchical Models](https://mlsanity.com/paper/1312.0906/) (2013)
- [Fast mixing of Metropolized Hamiltonian Monte Carlo: Benefits of multi-step gradients](https://mlsanity.com/paper/1905.12247/) (2019)
- [Coupling and Convergence for Hamiltonian Monte Carlo](https://mlsanity.com/paper/1805.00452/) (2018)
- [Convergence Rate of Riemannian Hamiltonian Monte Carlo and Faster Polytope Volume Computation](https://mlsanity.com/paper/1710.06261/) (2017)
- [Optimal Convergence Rate of Hamiltonian Monte Carlo for Strongly Logconcave Distributions](https://mlsanity.com/paper/1905.02313/) (2019)
- [On the Geometric Ergodicity of Hamiltonian Monte Carlo](https://mlsanity.com/paper/1601.08057/) (2016)
- [Geodesic Walks in Polytopes](https://mlsanity.com/paper/1606.04696/) (2016)
- [Rapid Mixing of Hamiltonian Monte Carlo on Strongly Log-Concave Distributions](https://mlsanity.com/paper/1708.07114/) (2017)
- [Optimizing The Integrator Step Size for Hamiltonian Monte Carlo](https://mlsanity.com/paper/1411.6669/) (2014)
- [Rapid Mixing of Geodesic Walks on Manifolds with Positive Curvature](https://mlsanity.com/paper/1609.02901/) (2016)

## Built on by (most cited first, 8 of 8)

- [Improved Discretization Analysis for Underdamped Langevin Monte Carlo](https://mlsanity.com/paper/2302.08049/) (2023)
- [Langevin Soft Actor-Critic: Efficient Exploration through Uncertainty-Driven Critic Learning](https://mlsanity.com/paper/2501.17827/) (2025)
- [Sampling from Log-Concave Distributions with Infinity-Distance Guarantees](https://mlsanity.com/paper/2111.04089/) (2021)
- [Hamiltonian Monte Carlo for efficient Gaussian sampling: long and random steps](https://mlsanity.com/paper/2209.12771/) (2022)
- [Accelerating Hamiltonian Monte Carlo via Chebyshev Integration Time](https://mlsanity.com/paper/2207.02189/) (2022)
- [A blob method for inhomogeneous diffusion with applications to multi-agent control and sampling](https://mlsanity.com/paper/2202.12927/) (2022)
- [Bayesian Inference with Latent Hamiltonian Neural Networks](https://mlsanity.com/paper/2208.06120/) (2022)
- [On the Dissipation of Ideal Hamiltonian Monte Carlo Sampler](https://mlsanity.com/paper/2209.07438/) (2022)

## Abstract (arXiv)

The goal of this article is to introduce the Hamiltonian Monte Carlo (HMC) method -- a Hamiltonian dynamics-inspired algorithm for sampling from a Gibbs density π(x) propto e-f(x). We focus on the "idealized" case, where one can compute continuous trajectories exactly. We show that idealized HMC preserves π and we establish its convergence when f is strongly convex and smooth.