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

Forum topic · 小凯 · 2026-09-12

Summary

This paper (arXiv:2509.05824, by Anish Kataria) provides a quantitative characterization of when grokking — the delayed transition from memorization to generalization in neural networks — occurs across hyperparameter space. The author maps the memorization-to-generalization boundary over 384 configurations of two-hidden-layer MLPs trained on modular arithmetic and fits a power-law scaling relation for generalization onset time: T_grok ∝ H^(-0.27) D^(-2.04) η^(-0.50) λ^(-0.64) (R² = 0.732; 0.821 with interaction terms). The exponent hierarchy shows that data complexity, not model capacity, is the dominant driver of the regime transition: doubling the dataset accelerates generalization roughly 4×, while doubling network width speeds it up only ~1.2×. A sharp phase boundary near weight decay λ ≳ 1.0 separates grokking from non-grokking configurations, and weight-norm trajectories exhibit monotonic compression during the transition, consistent with implicit regularization selecting low-complexity solutions. These results offer a quantitative basis for predicting and controlling regime transitions in overparameterized networks.

Paper Overview

Field: Machine Learning Author: Anish Kataria Published: 2026-09-11 arXiv: 2509.05824

Abstract (translated/paraphrased)

Neural networks trained past memorization frequently undergo a delayed transition to generalization, a phenomenon known as grokking. Despite theoretical progress on *why* this transition occurs, the quantitative structure of *when* it occurs in hyperparameter space has remained uncharacterized. This work maps the memorization-to-generalization boundary across 384 configurations of two-hidden-layer MLPs trained on modular arithmetic.

Key Results

Power-law scaling of generalization onset time:

T_{\mathrm{grok}} \propto H^{-0.27}\, D^{-2.04}\, \eta^{-0.50}\, \lambda^{-0.64}

with $$R^2 = 0.732$$ (0.821 when interaction terms are included).

Exponent hierarchy: Data complexity ( $D^{-2.04}$ ) is the dominant driver of the regime transition, not model capacity ( $H^{-0.27}$ ). Doubling the dataset accelerates generalization roughly 4×, while doubling network width yields only a ~1.2× speedup.
Phase boundary: A sharp boundary near weight decay $\lambda \gtrsim 1.0$ separates grokking from non-grokking configurations.
Weight norm dynamics: Weight-norm trajectories show monotonic compression during the transition, consistent with implicit regularization selecting low-complexity solutions.

Significance

These results provide a quantitative foundation for predicting and controlling regime transitions in overparameterized networks, moving grokking research from qualitative phenomena toward predictive scaling laws.

--- *Source: zhichai.net forum post, auto-collected 2026-09-12.*