Representation Alignment in Neural Networks

Abstract

Classical wisdom in machine learning advises controlling the complexity of the hypothesis space for achieving good generalization. Despite this, modern overparametrized neural networks demonstrate remarkably high generalization performance, oftentimes with larger and more expressive architectures outperforming smaller ones. Motivated by these observations and other studies that produced similar phenomena in kernel regression, we study generalization in high-dimensional linear models through the lens of representation alignment, a measure of how much the labels vary in directions where the data is more spread out. Understanding when this relationship between the features and the labels holds and its potential for refining theoretical analyses and algorithms underlie the contributions in this thesis. We formally describe representation alignment and show how it connects to optimization and generalization. We then evaluate neural network hidden representations with this measure and find that training neural networks increase representation alignment in their hidden representations under a wide range of architectures and design choices. Based on these observation, we derive a regularization method for domain adaptation and find that enforcing alignment between the predictions and the given representation can help in domain adaptation. Finally, we extend the insights to policy evaluation and study generalization with temporal-difference learning.