Relating Regularization and Generalization through the Intrinsic Dimension of Activations
This work provides empirical insights into generalization mechanisms in machine learning, though it is incremental as it builds on existing regularization and intrinsic dimension concepts.
The paper investigates the relationship between regularization and generalization by analyzing the intrinsic dimension of model activations, showing that regularization reduces last-layer intrinsic dimension and affects generalization, and that a sudden drop in intrinsic dimension coincides with grokking in models.
Given a pair of models with similar training set performance, it is natural to assume that the model that possesses simpler internal representations would exhibit better generalization. In this work, we provide empirical evidence for this intuition through an analysis of the intrinsic dimension (ID) of model activations, which can be thought of as the minimal number of factors of variation in the model's representation of the data. First, we show that common regularization techniques uniformly decrease the last-layer ID (LLID) of validation set activations for image classification models and show how this strongly affects generalization performance. We also investigate how excessive regularization decreases a model's ability to extract features from data in earlier layers, leading to a negative effect on validation accuracy even while LLID continues to decrease and training accuracy remains near-perfect. Finally, we examine the LLID over the course of training of models that exhibit grokking. We observe that well after training accuracy saturates, when models ``grok'' and validation accuracy suddenly improves from random to perfect, there is a co-occurent sudden drop in LLID, thus providing more insight into the dynamics of sudden generalization.