Rate of Convergence of Polynomial Networks to Gaussian Processes

We examine one-hidden-layer neural networks with random weights. It is well-known that in the limit of infinitely many neurons they simplify to Gaussian processes. For networks with a polynomial activation, we demonstrate that the rate of this convergence in 2-Wasserstein metric is O(n 1 2 ), where n is the number of hidden neurons. We suspect this rate is asymptotically sharp. We improve the known convergence rate for other activations, to power-law in n for ReLU and inverse-square-root up to logarithmic factors for erf. We explore the interplay between spherical harmonics, Stein kernels and optimal transport in the non-isotropic setting.