Convex Analysis of the Mean Field Langevin Dynamics

TL;DR

This paper provides a concise convergence rate analysis of the mean field Langevin dynamics, linking it to convex optimization and empirical risk minimization, with implications for neural network training.

Contribution

It introduces a novel convergence analysis framework for mean field Langevin dynamics using a proximal Gibbs distribution and connects it to duality gaps in empirical risk minimization.

Findings

Established convergence rates for continuous and discrete time dynamics.

Linked the proximal Gibbs distribution to the duality gap.

Enabled empirical evaluation of convergence through this connection.

Abstract

As an example of the nonlinear Fokker-Planck equation, the mean field Langevin dynamics recently attracts attention due to its connection to (noisy) gradient descent on infinitely wide neural networks in the mean field regime, and hence the convergence property of the dynamics is of great theoretical interest. In this work, we give a concise and self-contained convergence rate analysis of the mean field Langevin dynamics with respect to the (regularized) objective function in both continuous and discrete time settings. The key ingredient of our proof is a proximal Gibbs distribution $p_q$ associated with the dynamics, which, in combination with techniques in [Vempala and Wibisono (2019)], allows us to develop a simple convergence theory parallel to classical results in convex optimization. Furthermore, we reveal that $p_q$ connects to the duality gap in the empirical risk minimization setting, which enables efficient empirical evaluation of the algorithm convergence.