  We aim to understand grokking, a phenomenon where models generalize long
after overfitting their training set. We present both a microscopic analysis
anchored by an effective theory and a macroscopic analysis of phase diagrams
describing learning performance across hyperparameters. We find that
generalization originates from structured representations whose training
dynamics and dependence on training set size can be predicted by our effective
theory in a toy setting. We observe empirically the presence of four learning
phases: comprehension, grokking, memorization, and confusion. We find
representation learning to occur only in a "Goldilocks zone" (including
comprehension and grokking) between memorization and confusion. We find on
transformers the grokking phase stays closer to the memorization phase
(compared to the comprehension phase), leading to delayed generalization. The
Goldilocks phase is reminiscent of "intelligence from starvation" in Darwinian
evolution, where resource limitations drive discovery of more efficient
solutions. This study not only provides intuitive explanations of the origin of
grokking, but also highlights the usefulness of physics-inspired tools, e.g.,
effective theories and phase diagrams, for understanding deep learning.

Machine Learning;Disordered Systems and Neural Networks;Statistical Mechanics;Artificial Intelligence;Classical Physics