We investigate the phenomenon of grokking -- delayed generalization accompanied by non-monotonic test loss behavior -- in a simple binary logistic classification task, for which "memorizing" and "generalizing" solutions can be strictly defined. Surprisingly, we find that grokking arises naturally even in this minimal model when the parameters of the problem are close to a critical point, and provide both empirical and analytical insights into its mechanism. Concretely, by appealing to the implicit bias of gradient descent, we show that logistic regression can exhibit grokking when the training dataset is nearly linearly separable from the origin and there is strong noise in the perpendicular directions. The underlying reason is that near the critical point, "flat" directions in the loss landscape with nearly zero gradient cause training dynamics to linger for arbitrarily long times near quasi-stable solutions before eventually reaching the global minimum. Finally, we highlight similarities between our findings and the recent literature, strengthening the conjecture that grokking generally occurs in proximity to the interpolation threshold, reminiscent of critical phenomena often observed in physical systems.

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.