Uniform-in-time concentration in two-layer neural networks via transportation inequalities

We quantify, uniformly over time and with high probability, the discrepancy between the predictions of a two-layer neural network trained by stochastic gradient descent (SGD) and their mean-field limit, for quadratic loss and ridge regularization. As a key ingredient, we establish T p transportation inequalities (p ∈ 1, 2) for the law of the SGD parameters, with explicit constants independent of the iteration index. We then prove uniform-in-time concentration of the empirical parameter measure around its mean-field limit in the Wasserstein distance W 1 , and we translate these bounds into prediction-error estimates against a fixed test function Phi. We also derive analogous concentration bounds in the sliced-Wasserstein distance SW 1 , leading to dimension-free rates.