Article Dans Une Revue Journal de Mathématiques Pures et Appliquées Année : 2026

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

Résumé

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.

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hal-05309416 , version 1 (11-10-2025)

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Kai Du, Zhenjie Ren, Florin Suciu, Songbo Wang. Self-interacting approximation to McKean-Vlasov long-time limit: a Markov chain Monte Carlo method. Journal de Mathématiques Pures et Appliquées, 2026, 205, pp.103782. ⟨10.1016/j.matpur.2025.103782⟩. ⟨hal-05309416⟩
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