A self consistent theory of Gaussian Processes captures feature learning effects in finite CNNs

Deep neural networks (DNNs) in the infinite width/channel limit have received much attention recently, as they provide a clear analytical window to deep learning via mappings to Gaussian Processes (GPs). Despite its theoretical appeal, this viewpoint lacks a crucial ingredient of deep learning in finite DNNs, laying at the heart of their success --- \textit{feature learning}. Here we consider DNNs trained with noisy gradient descent on a large training set and derive a self-consistent Gaussian Process theory accounting for \textit{strong} finite-DNN and feature learning effects. Applying this to a toy model of a two-layer linear convolutional neural network (CNN) shows good agreement with experiments. We further identify, both analytically and numerically, a sharp transition between a feature learning regime and a lazy learning regime in this model.