# Grokking through the Lens of Minimum-Norm Interpolation

arXiv: https://arxiv.org/abs/2609.38453 · HTML: https://mlsanity.com/paper/2609.38453/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 66 references that are in the map.
- First posted on arXiv 29 Sep 2026.

## Builds on (most cited first, 9 of 9)

- [Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets](https://mlsanity.com/paper/2201.02177/) (2022)
- [Explaining grokking through circuit efficiency](https://mlsanity.com/paper/2309.02390/) (2023)
- [A framework to characterize performance of LASSO algorithms](https://mlsanity.com/paper/1303.7291/) (2013)
- [Predicting Grokking Long Before it Happens: A look into the loss landscape of models which grok](https://mlsanity.com/paper/2306.13253/) (2023)
- [Approaching Deep Learning through the Spectral Dynamics of Weights](https://mlsanity.com/paper/2408.11804/) (2024)
- [Minimum ℓ1-norm interpolators: Precise asymptotics and multiple descent](https://mlsanity.com/paper/2110.09502/) (2021)
- [Diagonal Linear Networks and the Lasso Regularization Path](https://mlsanity.com/paper/2509.18766/) (2025)
- [Minimum Norm Interpolation via the Local Theory of Banach Spaces: The Role of 2-Uniform Convexity](https://mlsanity.com/paper/2603.28956/) (2026)
- [Optimal Implicit Bias in Linear Regression](https://mlsanity.com/paper/2506.17187/) (2025)

## Abstract (arXiv)

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 ℓr-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 ℓ1 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.