  This book develops an effective theory approach to understanding deep neural
networks of practical relevance. Beginning from a first-principles
component-level picture of networks, we explain how to determine an accurate
description of the output of trained networks by solving layer-to-layer
iteration equations and nonlinear learning dynamics. A main result is that the
predictions of networks are described by nearly-Gaussian distributions, with
the depth-to-width aspect ratio of the network controlling the deviations from
the infinite-width Gaussian description. We explain how these effectively-deep
networks learn nontrivial representations from training and more broadly
analyze the mechanism of representation learning for nonlinear models. From a
nearly-kernel-methods perspective, we find that the dependence of such models'
predictions on the underlying learning algorithm can be expressed in a simple
and universal way. To obtain these results, we develop the notion of
representation group flow (RG flow) to characterize the propagation of signals
through the network. By tuning networks to criticality, we give a practical
solution to the exploding and vanishing gradient problem. We further explain
how RG flow leads to near-universal behavior and lets us categorize networks
built from different activation functions into universality classes.
Altogether, we show that the depth-to-width ratio governs the effective model
complexity of the ensemble of trained networks. By using information-theoretic
techniques, we estimate the optimal aspect ratio at which we expect the network
to be practically most useful and show how residual connections can be used to
push this scale to arbitrary depths. With these tools, we can learn in detail
about the inductive bias of architectures, hyperparameters, and optimizers.
