Quantifying the Memorization-to-Generalization Transition: Scaling Laws and Phase Structure in Grokking

A study posted to arXiv (2609.10657v1) mapped the memorization-to-generalization boundary across 384 configurations of two-hidden-layer MLPs on modular arithmetic, fitting a power-law scaling relation for grokking onset time of T_grok ∝ H^-0.27 D^-2.04 η^-0.50 λ^-0.64 (R² = 0.732; 0.821 with interactions). The exponent hierarchy shows data complexity (D^-2.04) dominates the regime transition over model capacity (H^-0.27): doubling data accelerates generalization by roughly 4x, while doubling width yields only about 1.2x. A sharp phase boundary at weight decay λ ≳ 1.0 separates grokking from non-grokking configurations, and weight norm trajectories show monotonic compression during the transition, consistent with implicit regularization selecting low-complexity solutions.

arXiv:2609.10657v1 Announce Type: new Abstract: Neural networks trained past memorization frequently undergo a delayed transition to generalization, a phenomenon known as grokking. Despite theoretical progress on \emph{why} this transition occurs, the quantitative structure of \emph{when} it occurs in hyperparameter space remains uncharacterized. We map the memorization-to-generalization boundary across 384 configurations of two-hidden-layer MLPs on modular arithmetic, fitting a power-law scaling relation for generalization onset time: $T {\mathrm{grok}} \propto H^{-0.27}\, D^{-2.04}\, \eta^{-0.50}\, \lambda^{-0.64}$ $R^2 = 0.732$; $0.821$ with interactions . The exponent hierarchy reveals that data complexity $D^{-2.04}$ is the dominant driver of regime transition, not model capacity $H^{-0.27}$ : doubling data accelerates generalization by ${\sim}4\times$, while doubling width yields only ${\sim}1.2\times$. A sharp phase boundary at weight decay $\lambda \gtrsim 1.0$ separates grokking from non-grokking configurations, and weight norm trajectories show monotonic compression during the transition, consistent with implicit regularization selecting low-complexity solutions. These results provide a quantitative foundation for predicting and controlling regime transitions in overparameterized networks.