Year: 2026
Type: article
Abstract: ABSTRACT 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 of the problem increases, the number of iterations required to ensure convergence within a desired error in the metric scales in proportion to or . In this paper, we argue that, despite this poor scaling of the 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 , up to logarithmic terms in , often suffices for the algorithm to converge to within a desired error for all ‐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 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.
DOI: 10.1002/cpa.70032
Authors: Yifan Chen, Xiaoou Cheng, Jonathan Niles‐Weed, Jonathan Weare
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
Awards: DMS/NIGMS 1: Design and Analysis of Machine Learning Approaches for Long Timescale Prediction from Short Trajectory Data, Collaborative Research: Statistical Optimal Transport in High Dimensional Mixtures, G332372917, Collaborative Research: CAIG: Developing AI Emulator Tools for Extreme Events with Application to Heat Waves and Cold Snaps, G4720003262, CAREER: Statistical foundations of particle tracking and trajectory inference, G8876996369
OpenAlex ID: W7126273032