# Mapping the Emergence of Regularization-Driven Dynamics in Grokking

arXiv: https://arxiv.org/abs/2608.25813 · HTML: https://mlsanity.com/paper/2608.25813/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 32 references that are in the map.
- First posted on arXiv 26 Aug 2026.

## Builds on (most cited first, 25 of 25)

- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [Progress measures for grokking via mechanistic interpretability](https://mlsanity.com/paper/2301.05217/) (2023)
- [The Implicit Bias of Gradient Descent on Separable Data](https://mlsanity.com/paper/1710.10345/) (2017)
- [Linear Mode Connectivity and the Lottery Ticket Hypothesis](https://mlsanity.com/paper/1912.05671/) (2019)
- [Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets](https://mlsanity.com/paper/2201.02177/) (2022)
- [The Role of Permutation Invariance in Linear Mode Connectivity of Neural Networks](https://mlsanity.com/paper/2110.06296/) (2021)
- [Characterizing Implicit Bias in Terms of Optimization Geometry](https://mlsanity.com/paper/1802.08246/) (2018)
- [Towards Understanding Grokking: An Effective Theory of Representation Learning](https://mlsanity.com/paper/2205.10343/) (2022)
- [Deep learning versus kernel learning: an empirical study of loss landscape geometry and the time evolution of the Neural Tangent Kernel](https://mlsanity.com/paper/2010.15110/) (2020)
- [Hidden Progress in Deep Learning: SGD Learns Parities Near the Computational Limit](https://mlsanity.com/paper/2207.08799/) (2022)
- [Omnigrok: Grokking Beyond Algorithmic Data](https://mlsanity.com/paper/2210.01117/) (2022)
- [Predicting the Generalization Gap in Deep Networks with Margin Distributions](https://mlsanity.com/paper/1810.00113/) (2018)
- [Grokking as the Transition from Lazy to Rich Training Dynamics](https://mlsanity.com/paper/2310.06110/) (2023)
- [A Tale of Two Circuits: Grokking as Competition of Sparse and Dense Subnetworks](https://mlsanity.com/paper/2303.11873/) (2023)
- [Dichotomy of Early and Late Phase Implicit Biases Can Provably Induce Grokking](https://mlsanity.com/paper/2311.18817/) (2023)
- [Time Matters in Regularizing Deep Networks: Weight Decay and Data Augmentation Affect Early Learning Dynamics, Matter Little Near Convergence](https://mlsanity.com/paper/1905.13277/) (2019)
- [Grokking Beyond the Euclidean Norm of Model Parameters](https://mlsanity.com/paper/2506.05718/) (2025)
- [A Theoretical Framework for Grokking: Interpolation followed by Riemannian Norm Minimisation](https://mlsanity.com/paper/2505.20172/) (2025)
- [The Butterfly Effect: Neural Network Training Trajectories Are Highly Sensitive to Initial Conditions](https://mlsanity.com/paper/2506.13234/) (2025)
- [What Can Grokking Teach Us About Learning Under Nonstationarity?](https://mlsanity.com/paper/2507.20057/) (2025)
- [To Grok Grokking: Provable Grokking in Ridge Regression](https://mlsanity.com/paper/2601.19791/) (2026)
- [The Weight Norm Sets the Grokking Timescale: A Causal Delay Law](https://mlsanity.com/paper/2606.13753/) (2026)
- [New Evidence of the Two-Phase Learning Dynamics of Neural Networks](https://mlsanity.com/paper/2505.13900/) (2025)
- [On the Cone Effect in the Learning Dynamics](https://mlsanity.com/paper/2503.16316/) (2025)
- [Thermodynamic Weight Decay: Exploring Grokking Acceleration via Attention Specific Heat](https://mlsanity.com/paper/2607.20552/) (2026)

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

- [An Exploratory Replica-Overlap Probe of the Grokking Transition](https://mlsanity.com/paper/2609.25634/) (2026)

## Abstract (arXiv)

For overparameterized neural networks, many solutions can fit the training data equally well while behaving very differently on unseen samples. Grokking separates training fit from visible generalization, providing a window for studying how this selection develops during training. We sweep short, fixed-duration weight decay (WD) perturbations across the pre-generalization plateau and measure how they shift later generalization time. Across three grokking tasks, these shifts are unordered early in the plateau but later form a stable dose ordering before visible generalization, with stronger WD increases leading to earlier generalization and stronger WD decreases leading to later generalization. Test-loss barriers between perturbed and baseline generalization checkpoints collapse toward zero while the ordered timing effects persist. A similar response reorganization is observed under ℓ1 regularization in the grokking setting of Junior et al. (2025). Drawing on Waddington's developmental landscape as an analogy, we call this combination of increasingly constrained solution selection and persistent dose-ordered timing shifts the canalization of grokking solution selection. Together, our response maps and loss-barrier measurements reveal a dynamical reorganization before visible generalization that is consistent with the theoretical picture of regularization-driven motion along a stable slow manifold (Boursier et al., 2025).