Accelerating Langevin Sampling with Birth-death

TL;DR

The paper tackles the challenge of sampling from multimodal distributions by introducing a birth-death mechanism that accelerates Langevin diffusion. It formulates a nonlocal birth-death term in the Fokker-Planck equation and reveals a KL-divergence gradient-flow structure under the Wasserstein-Fisher-Rao metric. The authors prove that, under mild assumptions, the asymptotic convergence rate becomes independent of potential barriers, and they validate the approach with analytical examples and a practical interacting-particle algorithm (BDLS) alongside numerical experiments on torus, Gaussian mixtures, and Bayesian GMMs. The work offers a scalable, globally moving mass mechanism that enhances mixing across modes and provides a foundation for combining birth-death dynamics with other sampling schemes.

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.

Figures (9)

• Figure 1: Convergence of continuous dynamics and particles systems in Example 1. The left figure shows decay of the KL divergence in semilogy scale along the evolution of three continuous dynamics. The middle (or the right) figure shows the decay in loglog scale of mean square errors of estimating mean (or variance) using varying number of particles.
• Figure 2: Scatter plots of particles and their marginal distributions (computed by kernel density estimators) in Example 2. Top left figure displays the target density and the bottom left shows initial locations of particles. Each column in the rest shows the scatter plots and the marginal distributions of particles computed using parallel ULA (top, blue) and BDLS (bottom, red) at different iterations.
• Figure 3: The absolute errors of estimating $\mathbb{E}[f(x,y)]$ with various observables $f$ in Example 2. In the third figure $\chi(x,y) = \mathbf{1}_{|x|\leq 5, |y-2|\leq 0.8}$. The blue dash-dot and red-dot lines are estimation errors along iterations using ULA and BDLS respectively. The total number of iterations is $2\times 10^{5}$. For the purpose of resolution, we plot the error for every 400 iterations.
• Figure 4: Evolution of particles in $(\mu_1,\mu_2)$-coordinate for Example 3. The first column shows the histogram (top) of the synthetic data and the initial locations (bottom) of particles in $(\mu_1,\mu_2)$-coordinate. The rest columns compare the scatter plots of particles in $(\mu_1,\mu_2)$ and their marginals computed using parallel ULA (top, blue) and BDLS (bottom, red) at different iterations.
• Figure C.5: Solutions of continuous dynamics (top row) at varying times and distributions (kernel density estimators) of the corresponding particle algorithms (bottom row) at different iterations in Example 1. The initial distribution is $\mathcal{N}(0,0.2)$. The solid black lines are the target density and the blue (resp., blue dash-dot) lines are solutions of the FPE (resp., iterates of parallel ULA). The green and green dotted lines are solutions of BDE and the distributions of particles computed using BDS respectively. The red lines and red dashed lines are solutions of BDL-FPE and the distributions of particles computed using BDLS respectively.

Theorems & Definitions (12)

• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4
• Proposition 5.1
• Proposition A.1
• proof
• proof : Proof of Theorem \ref{['thm:gf']}
• proof : Proof of Theorem \ref{['thm:conv2']}
• proof : Proof of Theorem \ref{['thm:converge2']}