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

## TL;DR

This paper introduces a physics-inspired field-theory approach to analyze deep neural networks, accurately predicting learning curves and explaining their generalization despite over-parameterization.

## Contribution

It develops a novel field-theory formalism for supervised deep learning, providing precise learning curve predictions and insights into generalization mechanisms.

## Key findings

- Accurate predictions of deep DNN learning curves on polynomial regression tasks
- Efficient hyper-parameter optimization based on the theoretical model
- Explanation of generalization in over-parameterized DNNs via entropic bias

## Abstract

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.