Grokking as Slow Transport: A Quantitative Framework of Delayed Generalization

Delayed generalization, or grokking, has been explained by many different mechanisms in the literature, with conclusions that sometimes appear to conflict. We show that these mechanisms can be understood as different kinetic regimes of a common process: the motion of a slow coordinate out of a metastable state. Starting from the minibatch update rule, we reduce plateau training dynamics to a one-dimensional first-passage problem and derive a governing equation for the grokking delay. This yields a quantitative regime classification, where each regime has a distinct delay law and seed-to-seed variability law. We further derive explicit delay laws for SGD and Adam-family optimizers by mapping their hyperparameters to the effective drift and diffusion of the slow coordinate. Our framework reconciles existing explanations as regime-specific limits and accounts for diverse grokking phenomena observed in prior work. Code will be publicly available at https://github.com.