Lecture Notes on Linear Neural Networks: A Tale of Optimization and Generalization in Deep Learning

TL;DR

This paper presents a mathematical theory of linear neural networks focusing on optimization and generalization, using dynamical tools to deepen understanding and explore practical implications in deep learning.

Contribution

It introduces a novel dynamical systems approach to analyze linear neural networks, advancing theoretical understanding of their optimization and generalization properties.

Findings

The theory provides insights into the training dynamics of linear neural networks.

Practical applications derived from the theory demonstrate its relevance.

The approach enhances the mathematical understanding of deep learning models.

Abstract

These notes are based on a lecture delivered by NC on March 2021, as part of an advanced course in Princeton University on the mathematical understanding of deep learning. They present a theory (developed by NC, NR and collaborators) of linear neural networks -- a fundamental model in the study of optimization and generalization in deep learning. Practical applications born from the presented theory are also discussed. The theory is based on mathematical tools that are dynamical in nature. It showcases the potential of such tools to push the envelope of our understanding of optimization and generalization in deep learning. The text assumes familiarity with the basics of statistical learning theory. Exercises (without solutions) are included.