Accelerating Langevin Sampling with Birth-death
This addresses a fundamental bottleneck in sampling for multimodal distributions, which is critical for applications in Bayesian inference and machine learning, representing a novel method rather than an incremental improvement.
The paper tackles the problem of efficiently sampling from multimodal distributions in Bayesian inference and statistical machine learning by proposing a new sampling algorithm that accelerates Langevin diffusion with a birth-death mechanism, achieving an asymptotic convergence rate independent of potential barriers, unlike the exponential dependence in standard methods.
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.