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

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.

arXiv comment41 pages, 1 figure; error in Theorem 1 corrected; Theorem 3 extended to the concave energy case, with application presented in the new Section 4

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