  In the past decade, deep neural networks (DNNs) came to the fore as the
leading machine learning algorithms for a variety of tasks. Their raise was
founded on market needs and engineering craftsmanship, the latter based more on
trial and error than on theory. While still far behind the application
forefront, the theoretical study of DNNs has recently made important
advancements in analyzing the highly over-parameterized regime where some exact
results have been obtained. Leveraging these ideas and adopting a more
physics-like approach, here we construct a versatile field-theory formalism for
supervised deep learning, involving renormalization group, Feynman diagrams and
replicas. In particular we show that our approach leads to highly accurate
predictions of learning curves of truly deep DNNs trained on polynomial
regression tasks and that these predictions can be used for efficient
hyper-parameter optimization. In addition, they explain how DNNs generalize
well despite being highly over-parameterized, this due to an entropic bias to
simple functions which, for the case of fully-connected DNNs with data sampled
on the hypersphere, are low order polynomials in the input vector. Being a
complex interacting system of artificial neurons, we believe that such tools
and methodologies borrowed from condensed matter physics would prove essential
for obtaining an accurate quantitative understanding of deep learning.
