NeurIPS Poster Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices

Poster

Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices

Santosh Vempala · Andre Wibisono

East Exhibition Hall B, C #181

Keywords: [ Information Theory ] [ Theory ] [ Algorithms ] [ Stochastic Methods ]

[ Abstract ]

Abstract: We study the Unadjusted Langevin Algorithm (ULA) for sampling from a probability distribution

ν = e^{- f}

$\nu = e^{-f}$ on

\R^{n}

$\R^n$ . We prove a convergence guarantee in Kullback-Leibler (KL) divergence assuming

ν

$\nu$ satisfies log-Sobolev inequality and

f

$f$ has bounded Hessian. Notably, we do not assume convexity or bounds on higher derivatives. We also prove convergence guarantees in R\'enyi divergence of order

q > 1

$q > 1$ assuming the limit of ULA satisfies either log-Sobolev or Poincar\'e inequality.

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