Ensemble Kalman Sampler: mean-field limit and convergence analysis

Abstract

Ensemble Kalman Sampler (EKS) is a method to find approximately $i.i.d.$ samples from a target distribution. As of today, why the algorithm works and how it converges is mostly unknown. The continuous version of the algorithm is a set of coupled stochastic differential equations (SDEs). In this paper, we prove the wellposedness of the SDE system, justify its mean-field limit is a Fokker-Planck equation, whose long time equilibrium is the target distribution. We further demonstrate that the convergence rate is near-optimal ($J^{-1/2}$, with $J$ being the number of particles). These results, combined with the in-time convergence of the Fokker-Planck equation to its equilibrium, justify the validity of EKS, and provide the convergence rate as a sampling method.

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• Gradient flow structure and convergence analysis of the ensemble Kalman inversion for nonlinear forward models
• Covariance-modulated optimal transport and gradient flows
• Particle Dual Averaging: Optimization of Mean Field Neural Networks with Global Convergence Rate Analysis
• Constrained Ensemble Langevin Monte Carlo
• Uniform-in-time propagation of chaos for the Cucker--Smale model
• Variance reduction for Random Coordinate Descent-Langevin Monte Carlo
• EnKSGD: A Class Of Preconditioned Black Box Optimization And Inversion Algorithms