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Kalman-Langevin dynamics : exponential convergence, particle approximation and numerical approximation

Axel Ringh, Akash Sharma

TL;DR

This work develops and analyzes Kalman-Langevin dynamics, a covariance-preconditioned mean-field Langevin framework for sampling and optimization under non-Gaussian potentials. The authors establish exponential convergence to the Gibbs measure in relative entropy, along with uniform moment bounds and Wasserstein convergence, extending prior Gaussian results to broader energy landscapes. They also introduce a weak, derivative-free particle approximation and prove propagation of chaos, linking the particle system to the mean-field limit. Finally, they provide a robust time discretization analysis with taming that yields uniform-in-N convergence of the numerical scheme to the continuous dynamics, enabling practical, stable simulations for high-dimensional problems.

Abstract

Langevin dynamics has found a large number of applications in sampling, optimization and estimation. Preconditioning the gradient in the dynamics with the covariance - an idea that originated in literature related to solving estimation and inverse problems using Kalman techniques - results in a mean-field (McKean-Vlasov) SDE. We demonstrate exponential convergence of the time marginal law of the mean-field SDE to the Gibbs measure with non-Gaussian potentials. This extends previous results, obtained in the Gaussian setting, to a broader class of potential functions. We also establish uniform in time bounds on all moments and convergence in $p$-Wasserstein distance. Furthermore, we show convergence of a weak particle approximation, that avoids computing the square root of the empirical covariance matrix, to the mean-field limit. Finally, we prove that an explicit numerical scheme for approximating the particle dynamics converges, uniformly in number of particles, to its continuous-time limit, addressing non-global Lipschitzness in the measure.

Kalman-Langevin dynamics : exponential convergence, particle approximation and numerical approximation

TL;DR

This work develops and analyzes Kalman-Langevin dynamics, a covariance-preconditioned mean-field Langevin framework for sampling and optimization under non-Gaussian potentials. The authors establish exponential convergence to the Gibbs measure in relative entropy, along with uniform moment bounds and Wasserstein convergence, extending prior Gaussian results to broader energy landscapes. They also introduce a weak, derivative-free particle approximation and prove propagation of chaos, linking the particle system to the mean-field limit. Finally, they provide a robust time discretization analysis with taming that yields uniform-in-N convergence of the numerical scheme to the continuous dynamics, enabling practical, stable simulations for high-dimensional problems.

Abstract

Langevin dynamics has found a large number of applications in sampling, optimization and estimation. Preconditioning the gradient in the dynamics with the covariance - an idea that originated in literature related to solving estimation and inverse problems using Kalman techniques - results in a mean-field (McKean-Vlasov) SDE. We demonstrate exponential convergence of the time marginal law of the mean-field SDE to the Gibbs measure with non-Gaussian potentials. This extends previous results, obtained in the Gaussian setting, to a broader class of potential functions. We also establish uniform in time bounds on all moments and convergence in -Wasserstein distance. Furthermore, we show convergence of a weak particle approximation, that avoids computing the square root of the empirical covariance matrix, to the mean-field limit. Finally, we prove that an explicit numerical scheme for approximating the particle dynamics converges, uniformly in number of particles, to its continuous-time limit, addressing non-global Lipschitzness in the measure.

Paper Structure

This paper contains 14 sections, 12 theorems, 191 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Exponential convergence of non-linear dynamics to equilibrium
  4. Proof of Lemma \ref{['min_eigen_bound_lemma']}
  5. Proof of Theorem \ref{['main_thrm_exp_conv']}
  6. Proof of Lemma \ref{['max_eigen_bound_lemma']}
  7. Proof of Theorem \ref{['lem:X_bounded_moments']}
  8. Particle approximation and propagation of chaos
  9. Proof of Theorem \ref{['weak_prop_chaos']}
  10. Tightness of random measure
  11. Identification of limit
  12. Uniqueness of limit
  13. Time discretization analysis
  14. Proof of Theorem \ref{['thm:convergence_time_disc']}

Key Result

Theorem 2.1

Let Assumption first_assum_on_U hold. Then the non-linear Langevin SDEs (eq:ek_sdes) converges to its invariant measure exponentially in relative entropy, i.e., there exist positive constants $c_1$ and $c_2$ independent of $t$ such that

Figures (1)

  • Figure 1: The difference in behavior of overdamped Langevin dynamics and Kalman-Langevin dynamics for potential function $U = 0.26 (x^2 + y^2) - 0.48 x y$ at time $T = 1$ with $2000$ particles uniformly initialized in $[-15, 15]^2$ with $\beta = 1$.

Theorems & Definitions (27)

  • Remark 2.1
  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.2
  • Theorem 2.3
  • Corollary 2.1
  • proof
  • Corollary 2.2
  • proof
  • Lemma 2.4
  • ...and 17 more