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Ensemble Kalman Sampler: mean-field limit and convergence analysis

Zhiyan Ding, Qin Li

TL;DR

This paper analyzes the continuous version of EKS, a coupled SDE system, and justifies its mean-filed limit is a Fokker-Planck equation, whose equilibrium state is the target distribution, proving that in long time, the samples generated by EKS indeed are approximately i.i.d. samples from a target distribution.

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.

Ensemble Kalman Sampler: mean-field limit and convergence analysis

TL;DR

This paper analyzes the continuous version of EKS, a coupled SDE system, and justifies its mean-filed limit is a Fokker-Planck equation, whose equilibrium state is the target distribution, proving that in long time, the samples generated by EKS indeed are approximately i.i.d. samples from a target distribution.

Abstract

Ensemble Kalman Sampler (EKS) is a method to find approximately 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 (, with 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.

Paper Structure

This paper contains 13 sections, 18 theorems, 129 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Ensemble Kalman Sampler and the continuum limit
  3. Algorithm description
  4. Continuum limit of Ensemble Kalman sampler
  5. Main results and strategy of our proof
  6. Wellposedness of Noisy Ensemble Kalman Flow
  7. Mean-field limit of \ref{['CAmunew']}
  8. Comparing $\rho(t,u)$ and $M_{v_t}$
  9. Comparing $\{v^j_t\}$ and $\{u^j_t\}$ systems
  10. Conclusion
  11. Proof of Proposition \ref{['Bforv']}
  12. Two Lemmas
  13. EKS with nonlinear $\mathcal{G}$

Key Result

Theorem 3.1

Suppose $\mathcal{G}$ satisfies linear, let $\rho(t,u)$ solve FKPK with initial data $\rho_0$ and $\{u^j_t\}$ solve CAmunew with $u^j_0$ i.i.d. drawn from the distribution induced by $\rho_0$. Define $M_{u_T}(u)$ to be the ensemble distribution of $\{u^j_T\}$ as in eqn:empirical. Assume then for any $0<\delta\ll 1$, there exists $T_\delta>0$ and $J_{T_\delta}>0$ so that

Theorems & Definitions (34)

  • Remark 2.1
  • Definition 1
  • Theorem 3.1: Main result
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • proof : Proof of Theorem \ref{['thm:main']}
  • proof : Proof of Theorem \ref{['ThforM']}
  • Proposition 4.1
  • Lemma 4.1
  • ...and 24 more