Proximal Langevin Algorithm: Rapid Convergence Under Isoperimetry
This work addresses the challenge of efficient sampling in high-dimensional spaces for machine learning and statistics, representing an incremental improvement over existing methods like ULA.
The paper tackles the problem of sampling from probability distributions under isoperimetric conditions, proving that the Proximal Langevin Algorithm achieves rapid convergence in KL divergence with improved dependence on the LSI constant, matching the fastest known rate for sampling under LSI without a Metropolis filter.
We study the Proximal Langevin Algorithm (PLA) for sampling from a probability distribution $ν= e^{-f}$ on $\mathbb{R}^n$ under isoperimetry. We prove a convergence guarantee for PLA in Kullback-Leibler (KL) divergence when $ν$ satisfies log-Sobolev inequality (LSI) and $f$ has bounded second and third derivatives. This improves on the result for the Unadjusted Langevin Algorithm (ULA), and matches the fastest known rate for sampling under LSI (without Metropolis filter) with a better dependence on the LSI constant. We also prove convergence guarantees for PLA in Rényi divergence of order $q > 1$ when the biased limit satisfies either LSI or Poincaré inequality.