# A hierarchical entropy method for the delocalization of bias in high-dimensional Langevin Monte Carlo

arXiv: https://arxiv.org/abs/2509.08619 · HTML: https://mlsanity.com/paper/2509.08619/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 16 references that are in the map.
- First posted on arXiv 10 Sep 2025.

## Builds on (most cited first, 4 of 4)

- [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019)
- [Asymptotic bias of inexact Markov Chain Monte Carlo methods in high dimension](https://mlsanity.com/paper/2108.00682/) (2021)
- [Convergence of Unadjusted Langevin in High Dimensions: Delocalization of Bias](https://mlsanity.com/paper/2408.13115/) (2024)
- [Stein's method for marginals on large graphical models](https://mlsanity.com/paper/2410.11771/) (2024)

## Built on by (most cited first, 1 of 1)

- [Delocalization of bias in unadjusted Hamiltonian Monte Carlo and underdamped Langevin](https://mlsanity.com/paper/2607.15208/) (2026)

## Abstract (arXiv)

The unadjusted Langevin algorithm is widely used for sampling from complex high-dimensional distributions. It is well known to be biased, with the bias typically scaling linearly with the dimension when measured in squared Wasserstein distance. However, the recent paper of Chen et al. (2024) identifies an intriguing new delocalization effect: For a class of distributions with sparse interactions, the bias between low-dimensional marginals scales only with the lower dimension, not the full dimension. In this work, we strengthen the results of Chen et al. (2024) in the sparse interaction regime by removing a logarithmic factor, measuring distance in relative entropy (a.k.a. KL-divergence), and relaxing the strong log-concavity assumption. In addition, we expand the scope of the delocalization phenomenon by showing that it holds for a class of distributions with weak interactions. Our proofs are based on a hierarchical analysis of the marginal relative entropies, inspired by the authors' recent work on propagation of chaos.