# A Dynamical Central Limit Theorem for Shallow Neural Networks

Recent theoretical work has characterized the dynamics of wide shallow neural networks trained via gradient descent in an asymptotic regime called the mean-field limit as the number of parameters tends towards infinity. At initialization, the randomly sampled parameters lead to a deviation from the mean-field limit that is dictated by the classical Central Limit Theorem (CLT). However, the dynamics of training introduces correlations among the parameters, raising the question of how the fluctuations evolve during training. Here, we analyze the mean-field dynamics as a Wasserstein gradient flow and prove that the deviations from the mean-field limit scaled by the width, in the width-asymptotic limit, remain bounded throughout training. In particular, they eventually vanish in the CLT scaling if the mean-field dynamics converges to a measure that interpolates the training data. This observation has implications for both the approximation rate and the generalization: the upper bound we obtain is given by a Monte-Carlo type resampling error, which does not depend explicitly on the dimension. This bound motivates a regularizaton term on the 2-norm of the underlying measure, which is also connected to generalization via the variation-norm function spaces.

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