Wasserstein mixing time of the unadjusted Langevin algorithm
2026-08-03 • Machine Learning
Machine Learning
AI summaryⓘ
The authors study the unadjusted Langevin algorithm, which is used to sample from certain probability distributions. They focus on cases where the probabilities have nice properties called log-smooth and strongly log-concave. The authors provide new estimates on how close the algorithm's output is to the true distribution, measured by Wasserstein distance. Their results show that the algorithm mixes faster than previously known, with a time that depends on the condition number, the dimension, and the desired accuracy. This improves earlier estimates by a significant factor.
Unadjusted Langevin algorithmWasserstein distanceMixing timeLog-smoothStrongly log-concaveCondition numberDimensionAsymptotic biasSampling algorithmsMarkov chain Monte Carlo
Authors
Francesco Pedrotti, Peter A. Whalley
Abstract
We provide new estimates in Wasserstein distance for the asymptotic bias of the unadjusted Langevin algorithm, in the classical setting of log-smooth strongly log-concave measures. Our bound implies a Wasserstein mixing time of order $κ\sqrt{d}/\varepsilon$, where $κ$ is the condition number, $d$ is the dimension, and $\varepsilon$ is the target precision: this improves by a factor of $\sqrt{d}/\varepsilon$ over the previous state-of-the-art results.