# Radial Suppression Accelerates Algorithmic Generalization: A Geometric Analysis of Delayed Generalization

arXiv: https://arxiv.org/abs/2606.32000 · HTML: https://mlsanity.com/paper/2606.32000/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 20 references that are in the map.
- First posted on arXiv 30 Jun 2026.

## Builds on (most cited first, 19 of 19)

- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Progress measures for grokking via mechanistic interpretability](https://mlsanity.com/paper/2301.05217/) (2023)
- [Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets](https://mlsanity.com/paper/2201.02177/) (2022)
- [Weight Normalization: A Simple Reparameterization to Accelerate Training of Deep Neural Networks](https://mlsanity.com/paper/1602.07868/) (2016)
- [Gradient Descent on Neural Networks Typically Occurs at the Edge of Stability](https://mlsanity.com/paper/2103.00065/) (2021)
- [Towards Understanding Grokking: An Effective Theory of Representation Learning](https://mlsanity.com/paper/2205.10343/) (2022)
- [Regularizing and Optimizing LSTM Language Models](https://mlsanity.com/paper/1708.02182/) (2017)
- [Hidden Progress in Deep Learning: SGD Learns Parities Near the Computational Limit](https://mlsanity.com/paper/2207.08799/) (2022)
- [The Clock and the Pizza: Two Stories in Mechanistic Explanation of Neural Networks](https://mlsanity.com/paper/2306.17844/) (2023)
- [Towards Resolving the Implicit Bias of Gradient Descent for Matrix Factorization: Greedy Low-Rank Learning](https://mlsanity.com/paper/2012.09839/) (2020)
- [Teaching Arithmetic to Small Transformers](https://mlsanity.com/paper/2307.03381/) (2023)
- [A Tale of Two Circuits: Grokking as Competition of Sparse and Dense Subnetworks](https://mlsanity.com/paper/2303.11873/) (2023)
- [Grokking at the Edge of Numerical Stability](https://mlsanity.com/paper/2501.04697/) (2025)
- [Grokfast: Accelerated Grokking by Amplifying Slow Gradients](https://mlsanity.com/paper/2405.20233/) (2024)
- [Let Me Grok for You: Accelerating Grokking via Embedding Transfer from a Weaker Model](https://mlsanity.com/paper/2504.13292/) (2025)
- [Dimensional Collapse in Transformer Attention Outputs: A Challenge for Sparse Dictionary Learning](https://mlsanity.com/paper/2508.16929/) (2025)
- [The Geometric Inductive Bias of Grokking: Bypassing Phase Transitions via Architectural Topology](https://mlsanity.com/paper/2603.05228/) (2026)
- [Minimizing Layerwise Activation Norm Improves Generalization in Federated Learning](https://mlsanity.com/paper/2512.08314/) (2025)

## Built on by (most cited first, 1 of 1)

- [Structure-Specific Representational Priors Causally Control the Grokking Delay](https://mlsanity.com/paper/2607.04333/) (2026)

## Abstract (arXiv)

Why do neural networks memorize algorithmic training data long before they generalize? We present a geometric case study demonstrating that, on tasks where generalization requires discovering structured low-dimensional circuits, the memorization-generalization delay is driven by radial inflation of hidden representations under cross-entropy optimization. We formalize a radial-angular decomposition of activation-space dynamics and derive three testable propositions: (i) that penalizing radial inflation induces anisotropic, data-dependent weight regularization; (ii) that it suppresses radial gradient energy below the isotropic random baseline, forcing predominantly angular updates; and (iii) that it biases convergence toward flatter minima. To empirically validate these propositions, we study a single-hyperparameter norm penalty that softly constrains activations to a sqrt(d)-radius hypersphere. On modular arithmetic, this penalty accelerates grokking up to 6x across MLPs and Transformers, and halves training steps for a 10M-parameter nanoGPT on 3-digit addition.