Sept 22, 2021

Ensemble Kalman Inversion, Ensemble Kalman Sampling, and mean-field limit

Qin Li, University of Wisconsin–Madison

Abstract:

How to sample from a target distribution function is one of the core problems in Bayesian inference. It is used widely in machine learning, data assimilation and inverse problems. During the past decade, ensemble type algorithms have been very popular, among which, Ensemble Kalman Inversion (EKI) and Ensemble Kalman Sampling (EKS) may have garnered the most attention. While numerical experiments suggest consistency and fast convergence, rigorous mathematical justification has been in lack.

To prove the validity of the two methods, we utilize the mean-field limit argument. In the continuum limit, the algorithms are translated into a set of coupled stochastic differential equations, whose mean-field limits (as the number of particles $N$ goes to infinity) are Fokker-Planck equations that reconstruct the target distribution exponentially fast under some conditions. We prove, when the algorithms converge, the convergence rate is optimal in the Wasserstein sense, meaning the ensemble distribution converges to the target distribution in $N^{-1/2}$.

It is a joint work with Zhiyan Ding.