# Algorithmic warm starts for Hamiltonian Monte Carlo

arXiv: https://arxiv.org/abs/2603.22741 · HTML: https://mlsanity.com/paper/2603.22741/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 160 references that are in the map.
- First posted on arXiv 24 Mar 2026.

## Builds on (most cited first, 7 of 7)

- [Composable Effects for Flexible and Accelerated Probabilistic Programming in NumPyro](https://mlsanity.com/paper/1912.11554/) (2019)
- [TensorFlow Distributions](https://mlsanity.com/paper/1711.10604/) (2017)
- [When does Metropolized Hamiltonian Monte Carlo provably outperform Metropolis-adjusted Langevin algorithm?](https://mlsanity.com/paper/2304.04724/) (2023)
- [Algorithmic Theory of ODEs and Sampling from Well-conditioned Logconcave Densities](https://mlsanity.com/paper/1812.06243/) (2018)
- [Mixing of the No-U-Turn Sampler and the Geometry of Gaussian Concentration](https://mlsanity.com/paper/2410.06978/) (2024)
- [Tail-Sensitive KL and R'enyi Convergence of Unadjusted Hamiltonian Monte Carlo via One-Shot Couplings](https://mlsanity.com/paper/2601.09019/) (2026)
- [Analysis of Langevin midpoint methods using an anticipative Girsanov theorem](https://mlsanity.com/paper/2507.12791/) (2025)

## Built on by (most cited first, 2 of 2)

- [Delocalization of bias in unadjusted Hamiltonian Monte Carlo and underdamped Langevin](https://mlsanity.com/paper/2607.15208/) (2026)
- [Efficient Mass Matrix Estimation with Gaussian Cooling](https://mlsanity.com/paper/2610.04806/) (2026)

## Abstract (arXiv)

Generating samples from a continuous probability density is a central algorithmic problem across statistics, engineering, and the sciences. For high-dimensional settings, Hamiltonian Monte Carlo (HMC) is the default algorithm across mainstream software packages. However, despite the extensive line of work on HMC and its widespread empirical success, it remains unclear how many iterations of HMC are required as a function of the dimension d. On one hand, a variety of results show that Metropolized HMC converges in O(d1/4) iterations from a warm start close to stationarity. On the other hand, Metropolized HMC is significantly slower without a warm start, e.g., requiring Omega(d1/2) iterations even for simple target distributions such as isotropic Gaussians. Finding a warm start is therefore the computational bottleneck for HMC. We resolve this issue for the well-studied setting of sampling from a probability distribution satisfying strong log-concavity (or isoperimetry) and third-order derivative bounds. We prove that non-Metropolized HMC generates a warm start in tildeO(d1/4) iterations, after which we can exploit the warm start using Metropolized HMC. Our final complexity of tildeO(d1/4) is the fastest algorithm for high-accuracy sampling under these assumptions, improving over the prior best of tildeO(d1/2). This closes the long line of work on the dimensional complexity of MHMC for such settings, and also provides a simple warm-start prescription for practical implementations.