# Birth-death dynamics for sampling: Global convergence, approximations and their asymptotics arXiv: https://arxiv.org/abs/2211.00450 ยท HTML: https://mlsanity.com/paper/2211.00450/ ## Key facts - Cited by 30 papers in the map (data through 14 Oct 2026). - Builds on 23 of its 84 references that are in the map. - First posted on arXiv 1 Nov 2022. ## Builds on (most cited first, 23 of 23) - [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016) - [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019) - [Theoretical guarantees for approximate sampling from smooth and log-concave densities](https://mlsanity.com/paper/1412.7392/) (2014) - [Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1507.05021/) (2015) - [Stein Variational Gradient Descent as Gradient Flow](https://mlsanity.com/paper/1704.07520/) (2017) - [Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures](https://mlsanity.com/paper/1508.07941/) (2015) - [Regularization Matters: Generalization and Optimization of Neural Nets v.s. their Induced Kernel](https://mlsanity.com/paper/1810.05369/) (2018) - [Maximum Mean Discrepancy Gradient Flow](https://mlsanity.com/paper/1906.04370/) (2019) - [Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem](https://mlsanity.com/paper/1802.08089/) (2018) - [Variational inference via Wasserstein gradient flows](https://mlsanity.com/paper/2205.15902/) (2022) - [On the geometry of Stein variational gradient descent](https://mlsanity.com/paper/1912.00894/) (2019) - [SVGD as a kernelized Wasserstein gradient flow of the chi-squared divergence](https://mlsanity.com/paper/2006.02509/) (2020) - [Mean-Field Langevin Dynamics: Exponential Convergence and Annealing](https://mlsanity.com/paper/2202.01009/) (2022) - [Kernel Stein Discrepancy Descent](https://mlsanity.com/paper/2105.09994/) (2021) - [Accelerating Langevin Sampling with Birth-death](https://mlsanity.com/paper/1905.09863/) (2019) - [Scaling limit of the Stein variational gradient descent: the mean field regime](https://mlsanity.com/paper/1805.04035/) (2018) - [The Bouncy Particle Sampler: A Non-Reversible Rejection-Free Markov Chain Monte Carlo Method](https://mlsanity.com/paper/1510.02451/) (2015) - [Analysis of a Two-Layer Neural Network via Displacement Convexity](https://mlsanity.com/paper/1901.01375/) (2019) - [Global convergence of neuron birth-death dynamics](https://mlsanity.com/paper/1902.01843/) (2019) - [A consensus-based model for global optimization and its mean-field limit](https://mlsanity.com/paper/1604.05648/) (2016) - [Polyak-L ojasiewicz inequality on the space of measures and convergence of mean-field birth-death processes](https://mlsanity.com/paper/2206.02774/) (2022) - [Second Order Ensemble Langevin Method for Sampling and Inverse Problems](https://mlsanity.com/paper/2208.04506/) (2022) - [On optimal transport of matrix-valued measures](https://mlsanity.com/paper/1808.05064/) (2018) ## Built on by (most cited first, 30 of 30) - [Learning Gaussian Mixtures Using the Wasserstein-Fisher-Rao Gradient Flow](https://mlsanity.com/paper/2301.01766/) (2023) - [Sampling via Gradient Flows in the Space of Probability Measures](https://mlsanity.com/paper/2310.03597/) (2023) - [Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approach](https://mlsanity.com/paper/2506.03979/) (2025) - [An Explicit Expansion of the Kullback-Leibler Divergence along its Fisher-Rao Gradient Flow](https://mlsanity.com/paper/2302.12229/) (2023) - [A connection between Tempering and Entropic Mirror Descent](https://mlsanity.com/paper/2310.11914/) (2023) - [Two-Scale Gradient Descent Ascent Dynamics Finds Mixed Nash Equilibria of Continuous Games: A Mean-Field Perspective](https://mlsanity.com/paper/2212.08791/) (2022) - [Fisher-Rao Gradient Flow: Geodesic Convexity and Functional Inequalities](https://mlsanity.com/paper/2407.15693/) (2024) - [Weighted quantization using MMD: From mean field to mean shift via gradient flows](https://mlsanity.com/paper/2502.10600/) (2025) - [Stein transport for Bayesian inference](https://mlsanity.com/paper/2409.01464/) (2024) - [Interaction-Force Transport Gradient Flows](https://mlsanity.com/paper/2405.17075/) (2024) - [Efficient, Multimodal, and Derivative-Free Bayesian Inference With Fisher-Rao Gradient Flows](https://mlsanity.com/paper/2406.17263/) (2024) - [Kernel Approximation of Fisher-Rao Gradient Flows](https://mlsanity.com/paper/2410.20622/) (2024) - [Hellinger-Kantorovich Gradient Flows: Global Exponential Decay of Entropy Functionals](https://mlsanity.com/paper/2501.17049/) (2025) - [Sequential Monte Carlo approximations of Wasserstein--Fisher--Rao gradient flows](https://mlsanity.com/paper/2506.05905/) (2025) - [Accelerate Langevin Sampling with Birth-Death Process and Exploration Component](https://mlsanity.com/paper/2305.05529/) (2023) - [Ensemble-Based Annealed Importance Sampling](https://mlsanity.com/paper/2401.15645/) (2024) - [Inclusive KL Gradient Flows: Otto-Wasserstein, Fisher-Rao-Gaussian, and Local-Estimator Dynamics](https://mlsanity.com/paper/2411.00214/) (2024) - [Connections between sequential Bayesian inference and evolutionary dynamics](https://mlsanity.com/paper/2411.16366/) (2024) - [New affine invariant ensemble samplers and their dimensional scaling](https://mlsanity.com/paper/2505.02987/) (2025) - [A note on the unique properties of the Kullback--Leibler divergence for sampling via gradient flows](https://mlsanity.com/paper/2507.04330/) (2025) - [Multi-Objective Optimization via Wasserstein-Fisher-Rao Gradient Flow](https://mlsanity.com/paper/2311.13159/) (2023) - [A Fisher-Rao gradient flow for entropic mean-field min-max games](https://mlsanity.com/paper/2405.15834/) (2024) - [A Particle Algorithm for Mean-Field Variational Inference](https://mlsanity.com/paper/2412.20385/) (2024) - [Stable Derivative Free Gaussian Mixture Variational Inference for Bayesian Inverse Problems](https://mlsanity.com/paper/2501.04259/) (2025) - [Gradient Flow Sampler-based Distributionally Robust Optimization](https://mlsanity.com/paper/2510.25956/) (2025) - [An operator splitting analysis of Wasserstein--Fisher--Rao gradient flows](https://mlsanity.com/paper/2511.18060/) (2025) - [On propagation of chaos for the Fisher-Rao gradient flow in entropic mean-field optimization](https://mlsanity.com/paper/2602.15094/) (2026) - [Properties and limitations of geometric tempering for gradient flow dynamics](https://mlsanity.com/paper/2604.20301/) (2026) - [Nesterov acceleration in optimizing over probability measures](https://mlsanity.com/paper/2607.23008/) (2026) - [Preservation of Log-Concavity and Convergence of Wasserstein-Fisher-Rao Gradient Flows](https://mlsanity.com/paper/2609.18118/) (2026) ## Abstract (arXiv) Motivated by the challenge of sampling Gibbs measures with nonconvex potentials, we study a continuum birth-death dynamics. We improve results in previous works [51,57] and provide weaker hypotheses under which the probability density of the birth-death governed by Kullback-Leibler divergence or by chi2 divergence converge exponentially fast to the Gibbs equilibrium measure, with a universal rate that is independent of the potential barrier. To build a practical numerical sampler based on the pure birth-death dynamics, we consider an interacting particle system, which is inspired by the gradient flow structure and the classical Fokker-Planck equation and relies on kernel-based approximations of the measure. Using the technique of Gamma-convergence of gradient flows, we show that on the torus, smooth and bounded positive solutions of the kernelized dynamics converge on finite time intervals, to the pure birth-death dynamics as the kernel bandwidth shrinks to zero. Moreover we provide quantitative estimates on the bias of minimizers of the energy corresponding to the kernelized dynamics. Finally we prove the long-time asymptotic results on the convergence of the asymptotic states of the kernelized dynamics towards the Gibbs measure.