Grokking at the Edge of Linear Separability (2410.04489v1)

Abstract: We study the generalization properties of binary logistic classification in a simplified setting, for which a "memorizing" and "generalizing" solution can always be strictly defined, and elucidate empirically and analytically the mechanism underlying Grokking in its dynamics. We analyze the asymptotic long-time dynamics of logistic classification on a random feature model with a constant label and show that it exhibits Grokking, in the sense of delayed generalization and non-monotonic test loss. We find that Grokking is amplified when classification is applied to training sets which are on the verge of linear separability. Even though a perfect generalizing solution always exists, we prove the implicit bias of the logisitc loss will cause the model to overfit if the training data is linearly separable from the origin. For training sets that are not separable from the origin, the model will always generalize perfectly asymptotically, but overfitting may occur at early stages of training. Importantly, in the vicinity of the transition, that is, for training sets that are almost separable from the origin, the model may overfit for arbitrarily long times before generalizing. We gain more insights by examining a tractable one-dimensional toy model that quantitatively captures the key features of the full model. Finally, we highlight intriguing common properties of our findings with recent literature, suggesting that grokking generally occurs in proximity to the interpolation threshold, reminiscent of critical phenomena often observed in physical systems.

• The paper demonstrates that grokking emerges when data nears linear separability, leading to delayed generalization and eventual overfitting.
• It employs both analytical and numerical methods, including logistic dynamics in random feature models, to uncover critical training behaviors.
• The study connects grokking with phenomena like double descent and critical slowing down, offering insights into neural network generalization limits.

An Essay on "Grokking at the Edge of Linear Separability"

The paper "Grokking at the Edge of Linear Separability" examines the phenomenon of grokking within the framework of binary logistic classification, focusing on training dynamics and generalization. The authors investigate this behavior in a synthetic, analytically tractable setup where "memorizing" and "generalizing" solutions can be sharply defined. This offers insights into the mechanisms underlying grokking, characterized by delayed generalization and non-monotonic test loss.

Key Findings

A primary observation is that grokking is pronounced when the training data is near the threshold of linear separability from the origin. In these situations, although a solution that perfectly generalizes exists, logistic loss bias can lead to overfitting. The research importantly shows that for non-separable data, the model asymptotically generalizes perfectly, while for separable data, it overfits, resulting in suboptimal generalization.

The paper's implications are significant both practically and theoretically, particularly regarding the model's behavior close to the interpolation threshold, where grokking predominantly occurs.

Analytical and Numerical Insights

The authors approach the problem both numerically and analytically by examining logistic classification's long-time dynamics within a random feature model and a simplified one-dimensional toy model. They show that grokking manifests near critical points in training dynamics. This is akin to critical phenomena observed in physical systems, where a pronounced change in behavior occurs near specific thresholds.

Theoretical Implications

The setup reveals that grokking may be related to phenomena such as double descent and critical slowing down in long-term dynamics. The paper posits that such critical behaviors reflect deep connections between intrinsic data structures and neural network generalization capabilities.

Future Directions

Several avenues for future exploration arise from this work. The relationship between grokking and other emergent neural phenomena, such as phase transitions in generalization, is of particular interest. Additionally, extending the analysis to non-linear regimes and different model architectures may shed light on how these findings apply to broader machine learning contexts.

Conclusion

By placing grokking within the broader context of separability and interpolation thresholds, this paper deepens the understanding of model dynamics and potential mechanisms for delayed generalization. This paper's insights, especially in how they relate to critical transitions in neural networks, can inspire further research on both theoretical predictions and practical applications of machine learning models.