Accelerating Langevin Sampling with Birth-death

TL;DR

This paper introduces a birth-death mechanism to accelerate Langevin sampling, effectively overcoming metastability in multimodal distributions and improving convergence rates in Bayesian inference.

Contribution

It proposes a novel birth-death based Langevin sampling algorithm with a PDE framework, achieving convergence independent of potential barriers.

Findings

Convergence rate is independent of potential barriers.

Algorithm outperforms standard Langevin methods in multimodal sampling.

Numerical experiments validate theoretical advantages.

Abstract

A fundamental problem in Bayesian inference and statistical machine learning is to efficiently sample from multimodal distributions. Due to metastability, multimodal distributions are difficult to sample using standard Markov chain Monte Carlo methods. We propose a new sampling algorithm based on a birth-death mechanism to accelerate the mixing of Langevin diffusion. Our algorithm is motivated by its mean field partial differential equation (PDE), which is a Fokker-Planck equation supplemented by a nonlocal birth-death term. This PDE can be viewed as a gradient flow of the Kullback-Leibler divergence with respect to the Wasserstein-Fisher-Rao metric. We prove that under some assumptions the asymptotic convergence rate of the nonlocal PDE is independent of the potential barrier, in contrast to the exponential dependence in the case of the Langevin diffusion. We illustrate the efficiency of the birth-death accelerated Langevin method through several analytical examples and numerical experiments.