🤖 AI Summary
This study investigates the fundamental mechanisms underlying the "grokking" phenomenon—characterized by high training accuracy coupled with delayed generalization—in feature-learning kernels, with a focus on the role of data symmetry. Employing Recursive Feature Machines (RFMs) and iteratively updating feature matrices via the Average Gradient Outer Product (AGOP), the authors analyze grokking behavior in algebraic tasks. Their central finding is that generalization occurs only when the symmetry of the training data is broken. The RFM achieves generalization by recovering the intrinsic group action governing the data, with the learned feature matrix precisely encoding the structure of this symmetry group. This work provides the first empirical evidence that symmetry breaking is a necessary condition for generalization and elucidates the group-theoretic underpinnings of grokking.
📝 Abstract
Grokking occurs when a model achieves high training accuracy but generalization to unseen test points happens long after that. This phenomenon was initially observed on a class of algebraic problems, such as learning modular arithmetic (Power et al., 2022). We study grokking on algebraic tasks in a class of feature learning kernels via the Recursive Feature Machine (RFM) algorithm (Radhakrishnan et al., 2024), which iteratively updates feature matrices through the Average Gradient Outer Product (AGOP) of an estimator in order to learn task-relevant features. Our main experimental finding is that generalization occurs only when a certain symmetry in the training set is broken. Furthermore, we empirically show that RFM generalizes by recovering the underlying invariance group action inherent in the data. We find that the learned feature matrices encode specific elements of the invariance group, explaining the dependence of generalization on symmetry.