# Convergence of Unadjusted Langevin in High Dimensions: Delocalization of Bias

arXiv: https://arxiv.org/abs/2408.13115 · HTML: https://mlsanity.com/paper/2408.13115/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 44 references that are in the map.
- First posted on arXiv 20 Aug 2024.

## Builds on (most cited first, 25 of 25)

- [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019)
- [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)
- [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)
- [Convergence of Langevin MCMC in KL-divergence](https://mlsanity.com/paper/1705.09048/) (2017)
- [Structured Logconcave Sampling with a Restricted Gaussian Oracle](https://mlsanity.com/paper/2010.03106/) (2020)
- [The Randomized Midpoint Method for Log-Concave Sampling](https://mlsanity.com/paper/1909.05503/) (2019)
- [Improved analysis for a proximal algorithm for sampling](https://mlsanity.com/paper/2202.06386/) (2022)
- [Optimal dimension dependence of the Metropolis-Adjusted Langevin Algorithm](https://mlsanity.com/paper/2012.12810/) (2020)
- [On the Convergence of Langevin Monte Carlo: The Interplay between Tail Growth and Smoothness](https://mlsanity.com/paper/2005.13097/) (2020)
- [Fast mixing of Metropolized Hamiltonian Monte Carlo: Benefits of multi-step gradients](https://mlsanity.com/paper/1905.12247/) (2019)
- [Minimax Mixing Time of the Metropolis-Adjusted Langevin Algorithm for Log-Concave Sampling](https://mlsanity.com/paper/2109.13055/) (2021)
- [Convergence of Langevin Monte Carlo in Chi-Squared and Renyi Divergence](https://mlsanity.com/paper/2007.11612/) (2020)
- [Improved Bounds for Discretization of Langevin Diffusions: Near-Optimal Rates without Convexity](https://mlsanity.com/paper/1907.11331/) (2019)
- [Stochastic Runge-Kutta Accelerates Langevin Monte Carlo and Beyond](https://mlsanity.com/paper/1906.07868/) (2019)
- [Improved dimension dependence of a proximal algorithm for sampling](https://mlsanity.com/paper/2302.10081/) (2023)
- [Langevin Monte Carlo without smoothness](https://mlsanity.com/paper/1905.13285/) (2019)
- [The Langevin Monte Carlo algorithm in the non-smooth log-concave case](https://mlsanity.com/paper/2101.10695/) (2021)
- [Sqrt(d) Dimension Dependence of Langevin Monte Carlo](https://mlsanity.com/paper/2109.03839/) (2021)
- [On the Ergodicity, Bias and Asymptotic Normality of Randomized Midpoint Sampling Method](https://mlsanity.com/paper/2011.03176/) (2020)
- [Lower Bounds on Metropolized Sampling Methods for Well-Conditioned Distributions](https://mlsanity.com/paper/2106.05480/) (2021)
- [Asymptotic bias of inexact Markov Chain Monte Carlo methods in high dimension](https://mlsanity.com/paper/2108.00682/) (2021)
- [A Simple Proof of the Mixing of Metropolis-Adjusted Langevin Algorithm under Smoothness and Isoperimetry](https://mlsanity.com/paper/2304.04095/) (2023)

## Built on by (most cited first, 4 of 4)

- [Sampling from Bayesian Neural Network Posteriors with Symmetric Minibatch Splitting Langevin Dynamics](https://mlsanity.com/paper/2410.19780/) (2024)
- [Stein's method for marginals on large graphical models](https://mlsanity.com/paper/2410.11771/) (2024)
- [Wasserstein mixing time of the unadjusted Langevin algorithm](https://mlsanity.com/paper/2608.02430/) (2026)
- [A hierarchical entropy method for the delocalization of bias in high-dimensional Langevin Monte Carlo](https://mlsanity.com/paper/2509.08619/) (2025)

## Abstract (arXiv)

The unadjusted Langevin algorithm is commonly used to sample probability distributions in extremely high-dimensional settings. However, existing analyses of the algorithm for strongly log-concave distributions suggest that, as the dimension d of the problem increases, the number of iterations required to ensure convergence within a desired error in the W2 metric scales in proportion to d or √d. In this paper, we argue that, despite this poor scaling of the W2 error for the full set of variables, the behavior for a small number of variables can be significantly better: a number of iterations proportional to K, up to logarithmic terms in d, often suffices for the algorithm to converge to within a desired W2 error for all K-marginals. We refer to this effect as delocalization of bias. We show that the delocalization effect does not hold universally and prove its validity for Gaussian distributions and strongly log-concave distributions with certain sparse interactions. Our analysis relies on a novel W2,ℓ^∞ metric to measure convergence. A key technical challenge we address is the lack of a one-step contraction property in this metric. Finally, we use asymptotic arguments to explore potential generalizations of the delocalization effect beyond the Gaussian and sparse interactions setting.