Grokking Beyond Neural Networks: An Empirical Exploration with Model Complexity (2310.17247v2)

Abstract: In some settings neural networks exhibit a phenomenon known as \textit{grokking}, where they achieve perfect or near-perfect accuracy on the validation set long after the same performance has been achieved on the training set. In this paper, we discover that grokking is not limited to neural networks but occurs in other settings such as Gaussian process (GP) classification, GP regression, linear regression and Bayesian neural networks. We also uncover a mechanism by which to induce grokking on algorithmic datasets via the addition of dimensions containing spurious information. The presence of the phenomenon in non-neural architectures shows that grokking is not restricted to settings considered in current theoretical and empirical studies. Instead, grokking may be possible in any model where solution search is guided by complexity and error.

• The paper shows that grokking, characterized by delayed generalization, occurs in non-neural models such as Gaussian Processes and linear regression.
• The authors analyze how model complexity, initialization, and regularization guide the trajectory from overfitting to a more generalizable solution.
• The study introduces data augmentation with spurious features to extend the grokking gap, offering fresh insights into model selection and learning dynamics.

Grokking Beyond Neural Networks: An Empirical Exploration with Model Complexity

The paper "Grokking Beyond Neural Networks: An Empirical Exploration with Model Complexity" extends the paper of grokking, a phenomenon typically associated with neural networks, to encompass other predictive models like Gaussian Processes (GPs) and linear regression. Grokking is identified by the delayed generalization of a model, where high validation performance is achieved significantly after the training set performance stabilizes. This research presents empirical evidence that grokking can be observed in non-neural settings, challenging existing assumptions about the phenomenon's exclusivity to neural network architectures.

Key Experiments and Findings

Grokking in Non-Neural Models

The researchers demonstrate grokking in GP classification and linear regression models. Notably, for linear regression, grokking manifests under specific conditions involving sparse training data, highlighting the nuanced conditions under which similar phenomena can emerge in different models. Their GP experiments utilized variations in kernel hyperparameters to achieve grokking-like effects. This broadens the scope of grokking beyond neural network idiosyncrasies, indicating that various models, when optimized under complexity considerations, can exhibit similar delayed generalization characteristics.

Complexity Considerations in Model Trajectories

A significant part of the paper focused on analyzing the trajectory of model parameters across complexity and error landscapes. By adjusting the initialization of model parameters and employing different complexity penalties, the paper illustrated how certain initialization regions (characterized by high complexity and low error) could prolong the period before a model transitions to a lower complexity solution that generalizes well. This analysis underpins a proposed mechanism where regularization plays a critical role in guiding models from overfitting solutions towards general solutions, aligning with previously established theories emphasizing model parsimony.

Novel Contributions

1. Generalizability of Grokking: The research adds to the understanding of grokking by establishing its occurrence across different types of models, suggesting that it is not inherently tied to specific neural network properties.
2. Data Augmentation via Concealment: Introducing the concept of augmenting data with spurious features, the paper shows how increased dimensionality can artificially induce grokking. This technique was shown to extend the grokking gap in algorithmic tasks, offering a tool to further probe the dynamics of the phenomenon.
3. Hypothesis on Model Complexity: The paper posits that the accessibility of regions characterized by minimal-complexity solutions is central to understanding grokking. When optimization paths in high-complexity areas are more accessible, models demonstrate a delayed shift to generalization. This notion aligns grokking with broader principles of model selection and generalization theory, unifying it under a complexity-guided learning framework.

Implications and Future Directions

This paper suggests several critical implications for the design and understanding of learning systems:

• Model Selection: By highlighting the role of complexity in grokking, the paper emphasizes the significance of model initialization and regularization strategies that explicitly account for complexity in enhancing generalization.
• Algorithmic Data Learning: The findings encourage the exploration of complexity and learning dynamics in algorithmic data tasks, offering new methodologies to understand the grokking behavior in practical settings.
• Broader Model Exploration: The work prompts further exploration in non-neural models, stressing the need for comprehensive theories that transcend model-specific dynamics and focus on universal principles governing learning and generalization.

Future research could expand on this by exploring how these dynamics play out in real-world datasets and across different categories of machine learning models. Additionally, understanding the precise mathematical characterizations of accessibility in complexity landscapes could lead to more formal theories that predict grokking behavior under various conditions.

In conclusion, this essay provides a structured and detailed summary of the research paper while offering insights into the paper's contributions to the broader understanding of the grokking phenomenon across different model architectures.