Convergence of Noise-Free Sampling Algorithms with Regularized Wasserstein Proximals

TL;DR

This paper analyzes the convergence of the backward regularized Wasserstein proximal method for sampling, showing it outperforms classical Langevin algorithms in speed and bias reduction, with theoretical guarantees and numerical validation.

Contribution

The paper provides a convergence analysis of the BRWP method, establishing conditions and rates, and demonstrates its advantages over classical Langevin Monte Carlo methods.

Findings

Guaranteed convergence in KL divergence for strongly log-concave targets.

Identification of optimal step sizes for convergence.

Numerical experiments confirm theoretical results and improved performance.

Abstract

In this work, we investigate the convergence properties of the backward regularized Wasserstein proximal (BRWP) method for sampling a target distribution. The BRWP approach can be shown as a semi-implicit time discretization for a probability flow ODE with the score function whose density satisfies the Fokker-Planck equation of the overdamped Langevin dynamics. Specifically, the evolution of the density, hence the score function, is approximated via a kernel representation derived from the regularized Wasserstein proximal operator. By applying the dual formulation and a localized Taylor series to obtain the asymptotic expansion of this kernel formula, we establish guaranteed convergence in terms of the Kullback-Leibler divergence for the BRWP method towards a strongly log-concave target distribution. Our analysis also identifies the optimal and maximum step sizes for convergence. Furthermore, we demonstrate that the deterministic and semi-implicit BRWP scheme outperforms many classical Langevin Monte Carlo methods, such as the Unadjusted Langevin Algorithm (ULA), by offering faster convergence and reduced bias. Numerical experiments further validate the convergence analysis of the BRWP method.