An Introduction to Hamiltonian Monte Carlo Method for Sampling

2021/08/27 by Nisheeth K. Vishnoi, Vishnoi, Nisheeth K.
Computer Science · Mathematics · Physics and Astronomy · #Computation (stat.CO) #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cs.DS #cs.LG #math.PR #stat.CO #stat.ML

paper · pdf · doi:10.48550/arxiv.2108.12107

This exposition is to supplement the talk by the author at the Bootcamp in the semester on Geometric Methods for Optimization and Sampling at the Simons Institute for the Theory of Computing

arxiv created 2021/08/27 · openalex publication_date 2021/08/27 · arxiv updated 2021/08/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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) ∝ 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.

Citations

• Constrained reaction coordinate dynamics for the simulation of rare events
• Stochastic Gradient Hamiltonian Monte Carlo
• On the convergence of Hamiltonian Monte Carlo
• Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm

Related

• Hamiltonian Monte Carlo for efficient Gaussian sampling: long and random steps
• Understanding the Hamiltonian Monte Carlo through its Physics Fundamentals and Examples
• Unbiased constrained sampling with Self-Concordant Barrier Hamiltonian Monte Carlo
• On the convergence of Hamiltonian Monte Carlo
• Stochastic dynamics without detailed balance condition connecting simple gradient method and Hamiltonian Monte Carlo