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

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-parametrized 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, Feynmann 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 problems. It also explains in a concrete manner why DNNs generalize well despite being highly over-parametrized, this due to an entropic bias to simple functions -- low order polynomials in the input vector, for the case of fully-connected DNNs. 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.