A mean-field theory of lazy training in two-layer neural nets: entropic regularization and controlled McKean-Vlasov dynamics

We consider the problem of universal approximation of functions by two-layer neural nets with random weights that are "nearly Gaussian" in the sense of Kullback-Leibler divergence. This problem is motivated by recent works on lazy training, where the weight updates generated by stochastic gradient descent do not move appreciably from the i.i.d. Gaussian initialization. We first consider the mean-field limit, where the finite population of neurons in the hidden layer is replaced by a continual ensemble, and show that our problem can be phrased as global minimization of a free-energy functional on the space of probability measures over the weights. This functional trades off the $L^2$ approximation risk against the KL divergence with respect to a centered Gaussian prior. We characterize the unique global minimizer and then construct a controlled nonlinear dynamics in the space of probability measures over weights that solves a McKean--Vlasov optimal control problem. This control problem is closely related to the Schr\"odinger bridge (or entropic optimal transport) problem, and its value is proportional to the minimum of the free energy. Finally, we show that SGD in the lazy training regime (which can be ensured by jointly tuning the variance of the Gaussian prior and the entropic regularization parameter) serves as a greedy approximation to the optimal McKean--Vlasov distributional dynamics and provide quantitative guarantees on the $L^2$ approximation error.