Poster
Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices
Santosh Vempala · Andre Wibisono
East Exhibition Hall B + C #181
Keywords: [ Stochastic Methods ] [ Algorithms ] [ Theory ] [ Information Theory ]
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Abstract
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Abstract:
We study the Unadjusted Langevin Algorithm (ULA) for sampling from a probability distribution $\nu = e^{-f}$ on $\R^n$. We prove a convergence guarantee in Kullback-Leibler (KL) divergence assuming $\nu$ satisfies log-Sobolev inequality and $f$ 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 $q > 1$ assuming the limit of ULA satisfies either log-Sobolev or Poincar\'e inequality.
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