Self-interacting approximation to McKean-Vlasov long-time limit: a Markov chain Monte Carlo method

TL;DR

This paper introduces a self-interacting approximation method for McKean-Vlasov processes, demonstrating ergodicity and measure approximation, with applications to neural network training.

Contribution

It presents a novel self-interacting process approach that approximates McKean-Vlasov invariant measures and proves ergodicity under broad conditions.

Findings

Proves ergodicity of the self-interacting dynamics.

Shows approximation of McKean-Vlasov invariant measures.

Applies method to neural network weight learning.

Abstract

For a certain class of McKean-Vlasov processes, we introduce proxy processes that substitute the mean-field interaction with self-interaction, employing a weighted occupation measure. Our study encompasses two key achievements. First, we demonstrate the ergodicity of the self-interacting dynamics, under broad conditions, by applying the reflection coupling method. Second, in scenarios where the drifts are negative intrinsic gradients of convex mean-field potential functionals, we use entropy and functional inequalities to demonstrate that the stationary measures of the self-interacting processes approximate the invariant measures of the corresponding McKean-Vlasov processes. As an application, we show how to learn the optimal weights of a two-layer neural network by training a single neuron.