The lazy and rich regimes: a gentle tutorial

A central theme of the modern machine learning paradigm is that larger neural networks achieve better performance on a variety of metrics. Theoretical analyses of these overparameterized models have recently centered around studying very wide neural networks. In this tutorial, we provide a nonrigorous but illustrative derivation of the following fact: in order to train wide networks effectively, there is only one degree of freedom in choosing hyperparameters such as the learning rate and the size of the initial weights. This degree of freedom controls the richness of training behavior: at minimum, the wide network trains lazily like a kernel machine, and at maximum, it exhibits feature learning in the so-called $\mu$P regime. In this paper, we explain this richness scale, synthesize recent research results into a coherent whole, offer new perspectives and intuitions, and provide empirical evidence supporting our claims. In doing so, we hope to encourage further study of the richness scale, as it may be key to developing a scientific theory of feature learning in practical deep neural networks.

More is Better in Modern Machine Learning: when Infinite Overparameterization is Optimal and Overfitting is Obligatory

In our era of enormous neural networks, empirical progress has been driven by the philosophy that more is better. Recent deep learning practice has found repeatedly that larger model size, more data, and more computation (resulting in lower training loss) improves performance. In this paper, we give theoretical backing to these empirical observations by showing that these three properties hold in random feature (RF) regression, a class of models equivalent to shallow networks with only the last layer trained. Concretely, we first show that the test risk of RF regression decreases monotonically with both the number of features and the number of samples, provided the ridge penalty is tuned optimally. In particular, this implies that infinite width RF architectures are preferable to those of any finite width. We then proceed to demonstrate that, for a large class of tasks characterized by powerlaw eigenstructure, training to near-zero training loss is obligatory: near-optimal performance can only be achieved when the training error is much smaller than the test error. Grounding our theory in real-world data, we find empirically that standard computer vision tasks with convolutional neural tangent kernels clearly fall into this class. Taken together, our results tell a simple, testable story of the benefits of overparameterization, overfitting, and more data in random feature models.