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) maps the memorization-to-generalization boundary in grokking across 384 configurations of two-hidden-layer MLPs trained on modular arithmetic. The authors fit a power-law scaling relation for the 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 is the dominant driver of the regime transition rather than model capacity: doubling the dataset accelerates generalization roughly 4x, while doubling width only yields about a 1.2x speedup. A sharp phase boundary exists near weight decay λ ≳ 1.0, separating grokking from non-grokking configurations. Weight-norm trajectories show monotone compression during the transition, consistent with implicit regularization selecting low-complexity solutions. The results provide a quantitative basis for predicting and controlling regime transitions in over-parameterized networks.

Paper Overview

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

Summary

Neural networks trained past memorization frequently undergo a delayed transition to generalization, a phenomenon known as grokking. While there has been theoretical progress on *why* this transition occurs, the quantitative structure of *when* it occurs in hyperparameter space has remained uncharacterized.

The author maps the memorization-to-generalization boundary across 384 configurations of two-hidden-layer MLPs on modular arithmetic, fitting a power-law scaling relation for the generalization onset time:

T_{\mathrm{grok}} \propto H^{-0.27}\, D^{-2.04}\, \eta^{-0.50}\, \lambda^{-0.64} \quad (R^2 = 0.732;\ 0.821 \text{ with interactions})

Key Findings

Exponent hierarchy: Data complexity ( $D^{-2.04}$ ) is the dominant driver of the regime transition, not model capacity ( $H^{-0.27}$ ). Doubling the data accelerates generalization by roughly 4x, whereas doubling the width only speeds it up by about 1.2x.
Sharp phase boundary: There is a distinct boundary near weight decay $\lambda \gtrsim 1.0$ separating grokking from non-grokking configurations.
Weight norm dynamics: Weight-norm trajectories exhibit monotone 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 over-parameterized networks.

--- *Auto-collected on 2026-09-12*

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

Summary

Paper Overview

Summary

Key Findings

Significance

Tags