Implicit Regularization of Large Neural Networks via Mean-Field Formulation
Abstract
We propose a mathematical framework to explain implicit regularization from early stopping during the training of overparametrized neural networks. In the mean-field limit, the parameter distribution evolves according to a gradient flow on the space of probability measures. We show that these dynamics admit an equivalent McKean-Vlasov stochastic control formulation through the corresponding Hamilton-Jacobi-Bellman (HJB) equation. The control viewpoint yields a Dynamic Programming Principle (DPP), which we use to define a new metric on probability measures. This metric can be viewed as a mean-field generalization of the control representation of the Wasserstein-2 distance, and it naturally appears as a regularization term selected by early stopping. We further obtain non-asymptotic bounds describing how the induced regularization depends on the stopping time.
Cite
@article{arxiv.2603.20892,
title = {Implicit Regularization of Large Neural Networks via Mean-Field Formulation},
author = {Beatrice Acciaio and Jakob Heiss and Gudmund Pammer and Qinxin Yan},
journal= {arXiv preprint arXiv:2603.20892},
year = {2026}
}