Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices
This work addresses the challenge of efficient sampling in machine learning and statistics, particularly for non-convex settings, by providing theoretical guarantees that relax traditional assumptions, making it incremental but practically relevant.
The paper tackles the problem of analyzing the convergence of the Unadjusted Langevin Algorithm (ULA) for sampling from non-log-concave distributions, proving convergence guarantees in Kullback-Leibler and Rényi divergences under weaker assumptions like log-Sobolev inequalities, without requiring convexity or higher derivative bounds.
We study the Unadjusted Langevin Algorithm (ULA) for sampling from a probability distribution $ν= e^{-f}$ on $\mathbb{R}^n$. 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 bound on the bias of the limiting distribution of ULA assuming third-order smoothness of $f$, without requiring isoperimetry.