# On the Ergodicity, Bias and Asymptotic Normality of Randomized Midpoint Sampling Method

arXiv: https://arxiv.org/abs/2011.03176 · HTML: https://mlsanity.com/paper/2011.03176/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Nov 2020.

## Built on by (most cited first, 23 of 23)

- [Faster high-accuracy log-concave sampling via algorithmic warm starts](https://mlsanity.com/paper/2302.10249/) (2023)
- [Towards a Complete Analysis of Langevin Monte Carlo: Beyond Poincar'e Inequality](https://mlsanity.com/paper/2303.03589/) (2023)
- [Improved dimension dependence of a proximal algorithm for sampling](https://mlsanity.com/paper/2302.10081/) (2023)
- [Faster Diffusion Sampling with Randomized Midpoints: Sequential and Parallel](https://mlsanity.com/paper/2406.00924/) (2024)
- [Improved Discretization Analysis for Underdamped Langevin Monte Carlo](https://mlsanity.com/paper/2302.08049/) (2023)
- [Zeroth-Order Sampling Methods for Non-Log-Concave Distributions: Alleviating Metastability by Denoising Diffusion](https://mlsanity.com/paper/2402.17886/) (2024)
- [Error bounds for particle gradient descent, and extensions of the log-Sobolev and Talagrand inequalities](https://mlsanity.com/paper/2403.02004/) (2024)
- [Langevin Monte Carlo for strongly log-concave distributions: Randomized midpoint revisited](https://mlsanity.com/paper/2306.08494/) (2023)
- [Parallelized Midpoint Randomization for Langevin Monte Carlo](https://mlsanity.com/paper/2402.14434/) (2024)
- [GIST: Gibbs self-tuning for locally adaptive Hamiltonian Monte Carlo](https://mlsanity.com/paper/2404.15253/) (2024)
- [Assessing the Quality of Denoising Diffusion Models in Wasserstein Distance: Noisy Score and Optimal Bounds](https://mlsanity.com/paper/2506.09681/) (2025)
- [Unadjusted Hamiltonian MCMC with Stratified Monte Carlo Time Integration](https://mlsanity.com/paper/2211.11003/) (2022)
- [Sampling from the Mean-Field Stationary Distribution](https://mlsanity.com/paper/2402.07355/) (2024)
- [Heavy-tailed Sampling via Transformed Unadjusted Langevin Algorithm](https://mlsanity.com/paper/2201.08349/) (2022)
- [A Separation in Heavy-Tailed Sampling: Gaussian vs. Stable Oracles for Proximal Samplers](https://mlsanity.com/paper/2405.16736/) (2024)
- [The Poisson Midpoint Method for Langevin Dynamics: Provably Efficient Discretization for Diffusion Models](https://mlsanity.com/paper/2405.17068/) (2024)
- [Mean-Square Analysis of Discretized Ito Diffusions for Heavy-tailed Sampling](https://mlsanity.com/paper/2303.00570/) (2023)
- [Convergence of Unadjusted Langevin in High Dimensions: Delocalization of Bias](https://mlsanity.com/paper/2408.13115/) (2024)
- [A Dynamical System View of Langevin-Based Non-Convex Sampling](https://mlsanity.com/paper/2210.13867/) (2022)
- [Mean-field underdamped Langevin dynamics and its spacetime discretization](https://mlsanity.com/paper/2312.16360/) (2023)
- [Randomized Midpoint Method for Log-Concave Sampling under Constraints](https://mlsanity.com/paper/2405.15379/) (2024)
- [Randomized Runge-Kutta-Nystr"om Methods for Unadjusted Hamiltonian and Kinetic Langevin Monte Carlo](https://mlsanity.com/paper/2310.07399/) (2023)
- [Poisson Midpoint Method for Log Concave Sampling: Beyond the Strong Error Lower Bounds](https://mlsanity.com/paper/2506.07614/) (2025)

## Abstract (arXiv)

The randomized midpoint method, proposed by [SL19], has emerged as an optimal discretization procedure for simulating the continuous time Langevin diffusions. Focusing on the case of strong-convex and smooth potentials, in this paper, we analyze several probabilistic properties of the randomized midpoint discretization method for both overdamped and underdamped Langevin diffusions. We first characterize the stationary distribution of the discrete chain obtained with constant step-size discretization and show that it is biased away from the target distribution. Notably, the step-size needs to go to zero to obtain asymptotic unbiasedness. Next, we establish the asymptotic normality for numerical integration using the randomized midpoint method and highlight the relative advantages and disadvantages over other discretizations. Our results collectively provide several insights into the behavior of the randomized midpoint discretization method, including obtaining confidence intervals for numerical integrations.