Structures of Neural Network Effective Theories

We develop a diagrammatic approach to effective field theories (EFTs) corresponding to deep neural networks at initialization, which dramatically simplifies computations of finite-width corrections to neuron statistics. The structures of EFT calculations make it transparent that a single condition governs criticality of all connected correlators of neuron preactivations. Understanding of such EFTs may facilitate progress in both deep learning and field theory simulations.