Grokking through the Lens of Minimum-Norm Interpolation

2026-09-29

AI Summary

This study addresses the unclear intrinsic mechanisms of delayed generalization (grokking) by quantifying the influence of inductive biases and signal structure. It constructs a statistical theoretical framework to analyze how regularization geometry and signal sparsity govern generalization during near-interpolation. Specifically, this work proposes a “0-1 generalization law,” revealing the statistical instability of minimum-norm interpolation and the boundaries of sparsity-induced gains. These theoretical findings are systematically validated through high-dimensional regression, convex norm analysis, and experiments on linear networks and Transformers. By precisely characterizing the evolution trajectories of training and generalization errors, this research demonstrates the significant advantages of sparse regularization under noiseless data conditions, thereby providing a rigorous theoretical foundation for understanding the grokking phenomenon.

Abstract

Grokking shows that fitting the training data and learning the underlying signal can occur at very different stages. However, existing theories offer limited quantitative insight into how this delayed generalization depends on inductive bias and signal structure. Our work addresses the gap by developing a statistical theory that characterizes how regularization geometry and signal sparsity govern generalization near interpolation. In particular, we focus on the prototypical setting of high-dimensional regression and identify regimes in which sparsity-promoting regularization makes exact interpolation much more accurate than approximate fitting. In strongly overparameterized noiseless problems, we prove a zero--one generalization law and construct a family of convex norms whose interpolators transition from the trivial risk of the all-zero predictor to exact recovery, while keeping the training error equal to $0$. Furthermore, when feature dimension and sample size are proportional, we provide a precise characterization of training and generalization errors along $\ell_r$-regularization paths. This in turn allows us to quantify the generalization gain that remains near interpolation: we show that this gain increases as the norm becomes more sparsity-promoting and as the target becomes sparser, with a sharp drop in generalization reached for noiseless data and $\ell_1$ regularization. Experiments on diagonal linear networks and transformers trained on modular arithmetic demonstrate the generality of our theoretical predictions. Finally, beyond grokking, our work reveals a statistical instability in minimum-norm interpolation: small perturbations in the regularization strength can lead to drastically different generalization, while preserving small training error.

Innovation

• Grokking
• Minimum-Norm Interpolation
• Sparsity-Promoting Regularization
• High-Dimensional Regression
• Statistical Instability