Grokking is the intriguing phenomenon where a model learns to generalize long after it has fit the training data.     We show both analytically and numerically that grokking can surprisingly occur in linear networks performing linear tasks in a simple teacher-student setup. In this setting, the full training dynamics is derived in terms of the expected training and generalization data covariance matrix. We present exact predictions on how the grokking time depends on input and output dimensionality, train sample size, regularization, and network parameters initialization.    The key findings are that late generalization increase may not imply a transition from "memorization" to "understanding", but can simply be an artifact of the accuracy measure.    We provide empirical verification for these propositions, along with preliminary results indicating that some predictions also hold for deeper networks, with non-linear activations.
Grokking in Linear Estimators -- A Solvable Model that Groks without Understanding
Noam Itzhak Levi Alon Beck Yohai Bar-Sinai