Breaking Data Symmetry is Needed For Generalization in Feature Learning Kernels
This addresses generalization challenges in feature learning for algebraic problems, but it is incremental as it builds on prior work on grokking and RFM.
The study investigates grokking in algebraic tasks using the Recursive Feature Machine (RFM) algorithm, finding that generalization occurs only when training data symmetry is broken, and RFM recovers the underlying invariance group action.
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.