The Norm-Separation Delay Law of Grokking: A First-Principles Theory of Delayed Generalization

TL;DR

This paper presents a quantitative theory explaining the delay in grokking, showing it as a norm-driven phase transition in regularised training dynamics, with predictions validated across multiple tasks.

Contribution

It introduces the Norm-Separation Delay Law, linking grokking delay to effective contraction rate and norm ratios, supported by extensive empirical validation.

Findings

Grokking delay inversely scales with weight decay and learning rate.

Logarithmic dependence of delay on norm ratio confirmed.

AdamW optimizer enables grokking where SGD fails.

Abstract

Grokking -- the sudden generalisation that appears long after a model has perfectly memorised its training data -- has been widely observed but lacks a quantitative theory explaining the length of the delay. We show that grokking is a norm-driven representational phase transition in regularised training dynamics, and establish the Norm-Separation Delay Law: $T_{\mathrm{grok}} - T_{\mathrm{mem}} = \Theta(\gamma_{\mathrm{eff}}^{-1} \log(\|\theta_{\mathrm{mem}}\|^2 / \|\theta_{\mathrm{post}}\|^2))$, where $\gamma_{\mathrm{eff}}$ is the optimiser's effective contraction rate ($\gamma_{\mathrm{eff}} = \eta\lambda$ for SGD, $\gamma_{\mathrm{eff}} \ge \eta\lambda$ for AdamW). The upper bound follows from a discrete Lyapunov contraction argument; the matching lower bound from dynamical constraints of regularised first-order optimisation. Across 293 training runs spanning modular addition, modular multiplication, and sparse parity, we confirm three falsifiable predictions: inverse scaling with weight decay ($R^2 = 0.97$), inverse scaling with learning rate ($R^2 = 0.92$), and logarithmic dependence on the norm ratio (Pearson $r = 0.91$). A fourth finding reveals that grokking requires an optimiser capable of decoupling memorisation from contraction: SGD fails entirely at the same hyperparameters where AdamW reliably groks. These results reframe grokking not as a mysterious optimisation artefact but as a predictable consequence of norm separation between competing interpolating representations. We further derive a practical three-input algorithm that predicts grokking delay at memorisation time with 34.6% mean absolute error (bootstrap 95% CI [30.0%, 39.4%], $N=60$ seeds), enabling principled early stopping.