Grokking as a First-Order Phase Transition in Two-Layer Neural Networks

Core Concepts

Grokking, a sudden increase in test accuracy, is shown to be a result of a first-order phase transition in the internal representations of deep neural networks. This transition leads to the emergence of new features that significantly improve the sample complexity compared to Gaussian Process limits.

Abstract

The paper investigates the phenomenon of "Grokking" in deep neural networks, where a sudden increase in test accuracy is observed after a period of constant and poor performance. The authors apply a recent theoretical framework, the adaptive kernel approach, to two teacher-student models with cubic-polynomial and modular addition teachers.
The key findings are:

Grokking can be mapped to the theory of first-order phase transitions in physics. Before the transition, the network's internal representations are well described by Gaussian Feature Learning (GFL), where the covariance matrices of neuron pre-activations gradually adapt to the task.

At the phase transition, a new set of features emerge, leading to a mixed phase (GMFL-I) where some neurons fluctuate around the teacher-aware minima while others remain in the GFL phase. Further increasing the dataset or network size leads to a pure GMFL-II phase where all neurons are specialized to the teacher.

The authors show that the complex prior induced by a finite DNN can learn the features from the readily accessible linear components of the target and apply them to the non-linear ones, leading to a significant reduction in sample complexity compared to the Gaussian Process limit.

For the modular addition teacher, the authors provide an analytical prediction for the location of the phase transition, demonstrating the power of their approach.

Overall, the paper provides a unifying framework to classify and predict feature learning effects in deep neural networks, with potential implications for network pruning, interpretability, and sample complexity.

Stats

The paper does not contain explicit numerical data or statistics. The key results are analytical predictions and their validation through numerical simulations.

Quotes

"Grokking, a sudden increase in test accuracy, is shown to be a result of a first-order phase transition in the internal representations of deep neural networks."
"The complex prior induced by a finite DNN can learn the features from the readily accessible linear components of the target and apply them to the non-linear ones, leading to a significant reduction in sample complexity compared to the Gaussian Process limit."

Key Insights Distilled From

Grokking as a First Order Phase Transition in Two Layer Networks

by Noa Rubin,In... at arxiv.org 05-07-2024

https://arxiv.org/pdf/2310.03789.pdf

Deeper Inquiries

How can the insights from this work be applied to improve the interpretability of deep neural networks

The insights from this work can significantly enhance the interpretability of deep neural networks by providing a formal framework to understand feature learning and Grokking phenomena. By mapping Grokking to a first-order phase transition in two-layer networks, we gain a deeper understanding of how neural networks learn new features during training. This understanding can be leveraged to develop better visualization techniques that can help researchers and practitioners interpret the internal representations learned by deep neural networks.
Specifically, the phase transition framework can be used to identify critical points in the training process where sudden improvements in test accuracy occur, indicating a phase transition akin to Grokking. By analyzing the network's behavior before and after these transitions, we can gain insights into the internal representations that the network has developed. This can lead to the identification of key features or patterns in the data that the network has learned to capture, thereby improving interpretability.
Furthermore, the framework can also be used to track the sample complexity of deep neural networks, providing a quantitative measure of how much data is required for the network to learn specific features. This information can be invaluable for understanding the generalization capabilities of the network and optimizing its performance on real-world tasks.

Can the phase transition framework be extended to other types of neural network architectures beyond the two-layer models considered here

The phase transition framework developed in this work for two-layer networks can potentially be extended to other types of neural network architectures beyond the models considered. While the specific details of the phase transitions may vary based on the network architecture and complexity, the underlying principles of feature learning and Grokking can still be applicable.
For example, the concept of phase transitions in neural networks can be explored in deeper architectures such as convolutional neural networks (CNNs) or recurrent neural networks (RNNs). By analyzing the behavior of these networks in terms of phase transitions, researchers can gain insights into how different layers interact during training and how features are learned and refined across the network.
Additionally, the phase transition framework can be adapted to unsupervised learning tasks or generative models, where understanding the learning dynamics and feature representations is crucial. By extending the framework to these domains, researchers can uncover new insights into the learning processes of complex neural network architectures.

What are the implications of the discovered sample complexity improvements for real-world applications of deep learning

The discovered sample complexity improvements in deep learning have significant implications for real-world applications of neural networks. By reducing the sample complexity required for learning new features, deep learning models can become more efficient and effective in a variety of tasks.

Faster Training: With reduced sample complexity, deep learning models can potentially be trained faster, leading to quicker deployment of models in production environments. This can be particularly beneficial in time-sensitive applications such as real-time data analysis or decision-making systems.

Improved Generalization: Lower sample complexity implies that deep learning models can generalize better from limited data, making them more robust in scenarios with sparse or noisy datasets. This can enhance the performance of neural networks in practical applications where data availability is a challenge.

Reduced Data Requirements: The ability to learn complex features with fewer data points can lower the data acquisition costs for training deep learning models. This can make deep learning more accessible to organizations with limited resources or in domains where data collection is expensive or time-consuming.

Overall, the discovered sample complexity improvements open up new possibilities for the widespread adoption of deep learning in various domains, leading to more efficient and effective applications of neural networks in real-world settings.