The Complexity Dynamics of Grokking (2412.09810v1)

Abstract: We investigate the phenomenon of generalization through the lens of compression. In particular, we study the complexity dynamics of neural networks to explain grokking, where networks suddenly transition from memorizing to generalizing solutions long after over-fitting the training data. To this end we introduce a new measure of intrinsic complexity for neural networks based on the theory of Kolmogorov complexity. Tracking this metric throughout network training, we find a consistent pattern in training dynamics, consisting of a rise and fall in complexity. We demonstrate that this corresponds to memorization followed by generalization. Based on insights from rate--distortion theory and the minimum description length principle, we lay out a principled approach to lossy compression of neural networks, and connect our complexity measure to explicit generalization bounds. Based on a careful analysis of information capacity in neural networks, we propose a new regularization method which encourages networks towards low-rank representations by penalizing their spectral entropy, and find that our regularizer outperforms baselines in total compression of the dataset.

• The paper introduces a complexity measure based on Kolmogorov complexity and lossy compression, achieving 30-40x higher compression rates.
• The paper examines grokking dynamics by tracking the network's complexity evolution, revealing a sudden transition from memorization to generalization.
• The paper proposes a regularization strategy that penalizes spectral entropy to lower network complexity and enhance model generalization.

The Complexity Dynamics of Grokking: A Comprehensive Analysis

This study undertakes an exploration into the intricate phenomenon of "grokking" in neural networks. Grokking refers to an observed behavior where networks make a sudden transition from memorizing training data to generalizing well to unseen data, occurring after the network has presumably overfit the training set. The authors of the paper explore the underlying complexity dynamics of neural networks through the use of information theory, particularly the concepts of Kolmogorov complexity and rate-distortion theory, to provide an explanation for this phenomenon.

To ground their investigation, the authors introduce a novel measure for assessing the intrinsic complexity of neural network models. This measure leverages the principles of Kolmogorov complexity, which seeks the length of the shortest possible description of an object or dataset. Since Kolmogorov complexity is theoretically incomputable, practical approximations involve compression techniques. In this research, compression operates within a framework akin to lossy data compression, controlled by a distortion bound reflecting the permissible loss level while maintaining acceptable performance on the model.

Key Contributions

1. Introduction of Complexity Measure: The research introduces a complexity measure based on the Kolmogorov complexity and rate-distortion theory. It acts as a form of lossy compression for neural networks, surpassing typical naïve compression methods by achieving much higher compression rates, between 30 and 40 times greater.
2. Investigation of Grokking Dynamics: By following the time-evolving complexity of neural networks throughout their training cycles, the paper illustrates the dynamics of complexity as networks transition from memorization to generalization. The characterization of this transition provides insights into the grokking phenomenon.
3. Regularization Strategy: A proposed regularization technique discourages high complexity in networks by penalizing their spectral entropy. Spectral entropy serves as a proxy for the effective dimensionality of the network—a lower value indicates a simpler, potentially more generalizable model.

Implications and Future Directions

The insights derived from this study establish a deeper understanding of the network generalization process and the factors that influence it. The relationship between complexity and generalization underscores the potential for utilizing compression as an analytical tool in model evaluation beyond mere performance metrics.

In practical terms, the application of this regularization approach suggests avenues for enhancing model robustness and efficiency, particularly in scenarios requiring minimized data storage without compromising model efficacy. It emphasizes low-rank representations, which are now becoming increasingly relevant in large-scale models and in fields like efficient neural network deployment on mobile devices.

Theoretically, extending the methods outlined in this paper could provide more predictive measures of a model's generalization capacity before deployment. As models continue to scale in size, developing effective bounds for generalization based on intrinsic complexity could become increasingly significant.

Conclusion

The paper provides an innovative take on understanding and quantifying network complexity dynamics as a pivotal factor in the transition from memorization to generalization, contributing to the broader field of study on model interpretability and optimization. The results emphasize the need for smarter regularization techniques that balance model performance with low complexity, which is crucial for deploying efficient machine learning systems in practical applications. Future research could extend these ideas to different network architectures or investigate the effects of various forms of regularization on model complexity and generalization trade-offs.