# Condition-number-independent convergence rate of Riemannian Hamiltonian Monte Carlo with numerical integrators

arXiv: https://arxiv.org/abs/2210.07219 · HTML: https://mlsanity.com/paper/2210.07219/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 53 references that are in the map.
- First posted on arXiv 13 Oct 2022.

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

- [Pyro: Deep Universal Probabilistic Programming](https://mlsanity.com/paper/1810.09538/) (2018)
- [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019)
- [Underdamped Langevin MCMC: A non-asymptotic analysis](https://mlsanity.com/paper/1707.03663/) (2017)
- [Analysis of Langevin Monte Carlo via convex optimization](https://mlsanity.com/paper/1802.09188/) (2018)
- [Log-concave sampling: Metropolis-Hastings algorithms are fast](https://mlsanity.com/paper/1801.02309/) (2018)
- [The Randomized Midpoint Method for Log-Concave Sampling](https://mlsanity.com/paper/1909.05503/) (2019)
- [Optimal dimension dependence of the Metropolis-Adjusted Langevin Algorithm](https://mlsanity.com/paper/2012.12810/) (2020)
- [Wasserstein Control of Mirror Langevin Monte Carlo](https://mlsanity.com/paper/2002.04363/) (2020)
- [Efficient constrained sampling via the mirror-Langevin algorithm](https://mlsanity.com/paper/2010.16212/) (2020)
- [Fast mixing of Metropolized Hamiltonian Monte Carlo: Benefits of multi-step gradients](https://mlsanity.com/paper/1905.12247/) (2019)
- [The Mirror Langevin Algorithm Converges with Vanishing Bias](https://mlsanity.com/paper/2109.12077/) (2021)
- [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)
- [Sampling with Riemannian Hamiltonian Monte Carlo in a Constrained Space](https://mlsanity.com/paper/2202.01908/) (2022)
- [Convergence of the Riemannian Langevin Algorithm](https://mlsanity.com/paper/2204.10818/) (2022)
- [Logsmooth Gradient Concentration and Tighter Runtimes for Metropolized Hamiltonian Monte Carlo](https://mlsanity.com/paper/2002.04121/) (2020)
- [Geodesic Walks in Polytopes](https://mlsanity.com/paper/1606.04696/) (2016)
- [Strong Self-Concordance and Sampling](https://mlsanity.com/paper/1911.05656/) (2019)
- [Introducing an Explicit Symplectic Integration Scheme for Riemannian Manifold Hamiltonian Monte Carlo](https://mlsanity.com/paper/1910.06243/) (2019)

## Built on by (most cited first, 10 of 10)

- [In-and-Out: Algorithmic Diffusion for Sampling Convex Bodies](https://mlsanity.com/paper/2405.01425/) (2024)
- [Fast sampling from constrained spaces using the Metropolis-adjusted Mirror Langevin algorithm](https://mlsanity.com/paper/2312.08823/) (2023)
- [Algorithmic Aspects of the Log-Laplace Transform and a Non-Euclidean Proximal Sampler](https://mlsanity.com/paper/2302.06085/) (2023)
- [Sampling with Barriers: Faster Mixing via Lewis Weights](https://mlsanity.com/paper/2303.00480/) (2023)
- [Unbiased constrained sampling with Self-Concordant Barrier Hamiltonian Monte Carlo](https://mlsanity.com/paper/2210.11925/) (2022)
- [Non-Log-Concave and Nonsmooth Sampling via Langevin Monte Carlo Algorithms](https://mlsanity.com/paper/2305.15988/) (2023)
- [R'enyi-infinity constrained sampling with d3 membership queries](https://mlsanity.com/paper/2407.12967/) (2024)
- [Oracle-based Uniform Sampling from Convex Bodies](https://mlsanity.com/paper/2510.02983/) (2025)
- [Tukey Depth Mechanisms for Practical Private Mean Estimation](https://mlsanity.com/paper/2502.18698/) (2025)
- [Beyond the d2.5-mixing bound for Dikin walks on polytopes](https://mlsanity.com/paper/2607.13943/) (2026)

## Abstract (arXiv)

We study the convergence rate of discretized Riemannian Hamiltonian Monte Carlo on sampling from distributions in the form of e-f(x) on a convex body Msubsetℝn. We show that for distributions in the form of e^-αtopx on a polytope with m constraints, the convergence rate of a family of commonly-used integrators is independent of leftVert αrightVert 2 and the geometry of the polytope. In particular, the implicit midpoint method (IMM) and the generalized Leapfrog method (LM) have a mixing time of widetildeOleft(mn3right) to achieve ε total variation distance to the target distribution. These guarantees are based on a general bound on the convergence rate for densities of the form e-f(x) in terms of parameters of the manifold and the integrator. Our theoretical guarantee complements the empirical results of [KLSV22], which shows that RHMC with IMM can sample ill-conditioned, non-smooth and constrained distributions in very high dimension efficiently in practice.