Grokking in Linear Estimators -- A Solvable Model that Groks without Understanding
This work provides a solvable model for grokking, offering insights into generalization dynamics, but it is incremental as it extends known concepts to linear settings.
The paper tackles the phenomenon of grokking in machine learning by showing it can occur in linear networks on linear tasks, deriving exact predictions for grokking time based on factors like dimensionality and regularization, and arguing that the sharp generalization increase may not indicate a transition from memorization to understanding.
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 with Gaussian inputs. In this setting, the full training dynamics is derived in terms of the 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 initialization. We demonstrate that the sharp increase in generalization accuracy may not imply a transition from "memorization" to "understanding", but can simply be an artifact of the accuracy measure. We provide empirical verification for our calculations, along with preliminary results indicating that some predictions also hold for deeper networks, with non-linear activations.