Position: Solve Layerwise Linear Models First to Understand Neural Dynamical Phenomena (Neural Collapse, Emergence, Lazy/Rich Regime, and Grokking) (2502.21009v2)

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.

Understanding Neural Dynamical Phenomena through Layerwise Linear Models

The paper "Position: Solve Layerwise Linear Models First to Understand Neural Dynamical Phenomena" provides a structured perspective on studying deep neural networks (DNNs) by emphasizing the utility of layerwise linear models. The authors argue that these models, despite their simplicity, offer insights into complex dynamical phenomena observed in neural networks, such as neural collapse, emergence, lazy/rich regime dynamics, and grokking.

Key Insights and Contributions

The central thesis of the paper is that layerwise linear models can serve as valuable tools for understanding the dynamics of neural networks. The authors advocate for solving these models thoroughly to gain insights into various neural phenomena. Here are the main contributions and insights from the paper:

1. Dynamical Feedback Principle: The paper introduces the dynamical feedback principle, which highlights how the interactions between consecutive layers in a network can influence the evolution of their parameters. This principle is pivotal for explaining several dynamic behaviors in neural networks.
2. Emergence and Sigmoidal Dynamics: The authors show how layerwise linear models can be used to model emergent dynamics observed in LLMs. Emergence, characterized by a sudden enhancement in capability with increasing model capacity or data, can be formalized using sigmoidal dynamics typical in layerwise linear models.
3. Neural Collapse Analysis: The work explores neural collapse, where class features in the penultimate layer of neural networks collapse to form equiangular tight frames. This phenomenon is explained through the low-rank dynamics inherent in layerwise linear models, which naturally prioritize significant features under specific initializations and training dynamics.
4. Lazy and Rich Regimes: Through layerwise linear models, the authors identify the conditions under which neural networks exhibit lazy or rich training dynamics. The paper discusses how layer imbalance plays a critical role in determining these regimes, where layer imbalance (i.e., differences in weight magnitudes between layers) dictates the extent of non-linear feature learning.
5. Grokking Dynamics: Grokking, or the delayed generalization in neural networks, is interpreted as a transition from lazy to rich dynamics. The paper provides insights into controlling grokking by adjusting the weight-to-target ratio, thereby promoting richer dynamics from the onset of training.

Practical and Theoretical Implications

The implications of this research are multifaceted:

• Accelerated Exploration of Neural Dynamics: By focusing on layerwise linear models, researchers can more easily identify the effects of various factors such as initialization and layer imbalance on DNN dynamics. This approach reduces complexity while retaining core dynamic properties, enabling more rapid exploration and hypothesis testing.
• Foundations for Nonlinear Extensions: Understanding the dynamics in linearized models sets a foundation for extending results to more complex, non-linear networks. While the primary focus is on linear dynamics, insights gained can be informative when studying networks with non-linear activations.
• Design of More Efficient Models: Insights into dynamics such as emergence and neural collapse could be used to design networks that capitalize on these phenomena, leading to efficient architectures with better generalization.

Future Directions

The paper calls for future research to further investigate the implications of layerwise dynamics, particularly in exploring the role of network depth and how non-linearities might be integrated as perturbations to the linear model. Additionally, there is a potential to extend these findings to scenarios involving different data distributions and tasks.

Overall, this paper provides a compelling argument for the utility of layerwise linear models in understanding complex neural phenomena and sets the stage for further theoretical advancements in the field of deep learning dynamics.