A Dynamical Central Limit Theorem for Shallow Neural Networks
This work provides theoretical insights into the training dynamics of neural networks, which is incremental for researchers in machine learning theory.
The authors tackled the problem of understanding fluctuations in wide shallow neural networks during gradient descent training, proving that asymptotic deviations from the mean-field limit remain bounded in mean square and can vanish under certain conditions, with the bound depending on a 2-norm that also controls generalization error.
Recent theoretical works have characterized the dynamics of wide shallow neural networks trained via gradient descent in an asymptotic mean-field limit when the width tends towards infinity. At initialization, the random sampling of the parameters leads to deviations from the mean-field limit dictated by the classical Central Limit Theorem (CLT). However, since gradient descent induces correlations among the parameters, it is of interest to analyze how these fluctuations evolve. Here, we use a dynamical CLT to prove that the asymptotic fluctuations around the mean limit remain bounded in mean square throughout training. The upper bound is given by a Monte-Carlo resampling error, with a variance that that depends on the 2-norm of the underlying measure, which also controls the generalization error. This motivates the use of this 2-norm as a regularization term during training. Furthermore, if the mean-field dynamics converges to a measure that interpolates the training data, we prove that the asymptotic deviation eventually vanishes in the CLT scaling. We also complement these results with numerical experiments.