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Energy-based models (EBMs) are a simple yet powerful framework for generative modeling. They are based on a trainable energy function which defines an associated Gibbs measure, and they can be trained and sampled from via well-established statistical tools, such as MCMC. Neural networks may be used as energy function approximators, providing both a rich class of expressive models as well as a flexible device to incorporate data structure. In this work we focus on shallow neural networks. Building from the incipient theory of overparametrized neural networks, we show that models trained in the so-called “active” regime provide a statistical advantage over their associated “lazy” or kernel regime, leading to improved adaptivity to hidden low-dimensional structure in the data distribution, as already observed in supervised learning. Our study covers both maximum likelihood and Stein Discrepancy estimators, and we validate our theoretical results with numerical experiments on synthetic data. 

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.