Position: Solve Layerwise Linear Models First to Understand Neural Dynamical Phenomena (Neural Collapse, Emergence, Lazy/Rich Regime, and Grokking)

TL;DR

This paper advocates for using layerwise linear models to better understand complex neural phenomena like neural collapse, emergence, and grokking, aiming to accelerate deep learning research.

Contribution

It introduces the dynamical feedback principle in layerwise linear models as a unifying framework for neural phenomena, emphasizing their role in understanding deep neural network dynamics.

Findings

Layerwise linear models explain neural collapse and emergence.

Dynamical feedback principle unifies various neural phenomena.

Using simplified models accelerates deep learning understanding.

Abstract

In physics, complex systems are often simplified into minimal, solvable models that retain only the core principles. In machine learning, layerwise linear models (e.g., linear neural networks) act as simplified representations of neural network dynamics. These models follow the dynamical feedback principle, which describes how layers mutually govern and amplify each other's evolution. This principle extends beyond the simplified models, successfully explaining a wide range of dynamical phenomena in deep neural networks, including neural collapse, emergence, lazy and rich regimes, and grokking. In this position paper, we call for the use of layerwise linear models retaining the core principles of neural dynamical phenomena to accelerate the science of deep learning.