Learning Curves for Deep Neural Networks: A Gaussian Field Theory Perspective

A series of recent works suggest that deep neural networks (DNNs), of fixed depth, are equivalent to certain Gaussian Processes (NNGP/NTK) in the highly over-parameterized regime (width or number-of-channels going to infinity). Other works suggest that this limit is relevant for real-world DNNs. These results invite further study into the generalization properties of Gaussian Processes of the NNGP and NTK type. Here we make several contributions along this line. First, we develop a formalism, based on field theory tools, for calculating learning curves perturbatively in one over the dataset size. For the case of NNGPs, this formalism naturally extends to finite width corrections. Second, in cases where one can diagonalize the covariance-function of the NNGP/NTK, we provide analytic expressions for the asymptotic learning curves of any given target function. These go beyond the standard equivalence kernel results. Last, we provide closed analytic expressions for the eigenvalues of NNGP/NTK kernels of depth 2 fully-connected ReLU networks. For datasets on the hypersphere, the eigenfunctions of such kernels, at any depth, are hyperspherical harmonics. A simple coherent picture emerges wherein fully-connected DNNs have a strong entropic bias towards functions which are low order polynomials of the input.