Deep Linear Network Training Dynamics from Random Initialization: Data, Width, Depth, and Hyperparameter Transfer

TL;DR

This paper provides a theoretical analysis of gradient descent dynamics in deep linear networks, revealing how data, width, depth, and hyperparameters influence training behavior and transfer effects.

Contribution

It offers a comprehensive theoretical framework for understanding training dynamics in deep linear networks, including effects of width, depth, data, and hyperparameters, with asymptotic descriptions and transfer insights.

Findings

Wider networks exhibit better training dynamics.

Hyperparameter transfer effects depend on parameterization.

Power law structured data shows accelerated training dynamics.

Abstract

We theoretically characterize gradient descent dynamics in deep linear networks trained at large width from random initialization and on large quantities of random data. Our theory captures the ``wider is better" effect of mean-field/maximum-update parameterized networks as well as hyperparameter transfer effects, which can be contrasted with the neural-tangent parameterization where optimal learning rates shift with model width. We provide asymptotic descriptions of both non-residual and residual neural networks, the latter of which enables an infinite depth limit when branches are scaled as $1/\sqrt{\text{depth}}$. We also compare training with one-pass stochastic gradient descent to the dynamics when training data are repeated at each iteration. Lastly, we show that this model recovers the accelerated power law training dynamics for power law structured data in the rich regime observed in recent works.