Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices [article]

Santosh S. Vempala, Andre Wibisono
2022 arXiv   pre-print
We study the Unadjusted Langevin Algorithm (ULA) for sampling from a probability distribution ν = e^-f on ℝ^n. We prove a convergence guarantee in Kullback-Leibler (KL) divergence assuming ν 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 Rényi divergence of order q > 1 assuming the limit of ULA satisfies either the log-Sobolev or Poincaré inequality. We also prove a
more » ... ound on the bias of the limiting distribution of ULA assuming third-order smoothness of f, without requiring isoperimetry.
arXiv:1903.08568v4 fatcat:v2v5fnz4jvfufpywyut5zf2aom