Feature Repulsion and Spectral Lock-in: An Empirical Study of Two-Layer Network Grokking

grokking relu-activation parameter-updates feature-repulsion spectral-lock-in matrix-inversion activation-derivative eigengap-detection

Abstract

Tian (2025) proves a repulsion theorem (Theorem 6) for the matrix $ B = (\widetilde{F}^\top \widetilde{F} + \eta I)^{-1} $ during the interactive feature-learning stage of grokking: similar features have negative off-diagonal entries $ B_{j\ell} $, producing an effective repulsive force that drives them apart. However, the theorem does not specify when this mechanism becomes empirically observable, nor whether it leaves a measurable spectral signature in the parameter updates. We test this directly on Tian's modular addition setup ($ M = 71 $, $ K = 2048 $, MSE loss) and observe a clear structure-mechanism dissociation. The predicted sign rule holds robustly on the top-200 most-similar feature pairs across activations (empirical sign-match rising from 0.865 to 0.985 on $ \sigma = x^2 $ across 5 seeds, and saturating at 1.000 on $ \sigma = \operatorname{ReLU} $). However, the spectral signature in the parameter updates is strongly activation-dependent. With $ \sigma = x^2 $, a simple slope detector on the rolling eigengap $ \sigma_2 / \sigma_3 $ of $ \Delta W $ fires in 15/15 grokking seeds at epoch 174 (IQR [173,174]) and in 0/15 non-grokking controls, with 229$ \times $ late-stage magnitude separation; the spectrum is rank-2. In contrast, with $ \sigma = \operatorname{ReLU} $, the detector never fires and the spectrum remains effectively rank-1. This dissociation aligns with Tian's Theorem 5 distinction between focused (power-law) and spreading (ReLU) memorization: while the sign structure of $ B $ depends only on $ \widetilde{F}^\top \widetilde{F} $, how feature repulsion translates into weight updates critically depends on the activation derivative $ \sigma' $.

한국어 요약