# Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices

arXiv: https://arxiv.org/abs/1903.08568 · HTML: https://mlsanity.com/paper/1903.08568/

## Key facts

- Cited by 227 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Mar 2019.

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## Abstract (arXiv)

We study the Unadjusted Langevin Algorithm (ULA) for sampling from a probability distribution nu = e-f on ℝn. We prove a convergence guarantee in Kullback-Leibler (KL) divergence assuming nu satisfies a log-Sobolev inequality and the Hessian of f is bounded. Notably, we do not assume convexity or bounds on higher derivatives. We also prove convergence guarantees in R'enyi divergence of order q > 1 assuming the limit of ULA satisfies either the log-Sobolev or Poincar'e inequality. We also prove a bound on the bias of the limiting distribution of ULA assuming third-order smoothness of f, without requiring isoperimetry.