Why Grokking Takes So Long: A First-Principles Theory of Representational Phase Transitions

• URL: http://arxiv.org/abs/2603.13331v1
• Date: Thu, 05 Mar 2026 17:28:39 GMT
• Title: Why Grokking Takes So Long: A First-Principles Theory of Representational Phase Transitions
• Abstract summary: Grokking is the sudden memorization that appears long after a model has memorized its training data.<n>We present a first-principles theory showing that grokking arises from a norm-driven representational phase transition in regularized training dynamics.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Grokking is the sudden generalization that appears long after a model has perfectly memorized its training data. Although this phenomenon has been widely observed, there is still no quantitative theory explaining the length of the delay between memorization and generalization. Prior work has noted that weight decay plays an important role, but no result derives tight bounds for the delay or explains its scaling behavior. We present a first-principles theory showing that grokking arises from a norm-driven representational phase transition in regularized training dynamics. Training first converges to a high-norm memorization solution and only later contracts toward a lower-norm structured representation that generalizes. Our main result establishes a scaling law for the delay: T_grok - T_mem = Theta((1 / gamma_eff) * log(||theta_mem||^2 / ||theta_post||^2)), where gamma_eff is the effective contraction rate of the optimizer (gamma_eff = eta * lambda for SGD and gamma_eff >= eta * lambda for AdamW). The upper bound follows from a discrete Lyapunov contraction argument, and the matching lower bound arises from dynamical constraints of regularized first-order optimization. Across 293 training runs spanning modular addition, modular multiplication, and sparse parity tasks, we confirm three predictions: inverse scaling with weight decay, inverse scaling with learning rate, and logarithmic dependence on the norm ratio (R^2 > 0.97). We further find that grokking requires an optimizer that can decouple memorization from contraction: SGD fails under hyperparameters where AdamW reliably groks. These results show that grokking is a predictable consequence of norm separation between competing interpolating representations and provide the first quantitative scaling law for the delay of grokking.

Related papers

• Quantifying the Memorization-to-Generalization Transition: Scaling Laws and Phase Structure in Grokking [0.0]
  We map the $0.8-to-generalization boundary across 384 configurations of two-hidden generalizations on modular arithmetic.<n>Data complexity is the dominant driver of regime transition, not model capacity.
  arXiv  Detail & Related papers  (2026-09-09T16:47:03Z)
• Sharp Capacity Scaling of Spectral Optimizers in Learning Associative Memory [77.27772368491698]
  Spectrals such as Muon have recently shown strong empirical performance in large-scale language model training.<n>We study this question through the linear associative memory problem.<n>We show that the storage capacity of Muon significantly exceeds that of SGD.
  arXiv  Detail & Related papers  (2026-03-27T16:13:18Z)
• Grokking as a Variance-Limited Phase Transition: Spectral Gating and the Epsilon-Stability Threshold [0.0]
  Generalization requires the textitanisotropic rectification unique to adaptives, which directs noise into the tangent space of the solution.<n>This work analyzes AdamW dynamics on modular arithmetic tasks, revealing a Spectral Gating'' mechanism that regulates the transition from generalization to memorization.
  arXiv  Detail & Related papers  (2026-03-16T16:18:18Z)
• Early-Warning Signals of Grokking via Loss-Landscape Geometry [0.0]
  We study two sequence-learning benchmarks: SCAN compositional generalization and Dyck-1 depth prediction.<n>Across both tasks and a wide range of learning rates, the commutator defect rises well before generalization.<n>These results identify the commutator defect as a robust, architecture-agnostic, causally implicated early-warning signal for delayed generalization in transformers.
  arXiv  Detail & Related papers  (2026-02-19T00:14:36Z)
• To Grok Grokking: Provable Grokking in Ridge Regression [24.785366757570202]
  We study grokking, the onset of generalization long after overfitting, in a classical ridge regression setting.<n>We show, both theoretically and empirically, that grokking can be amplified or eliminated in a principled manner.<n>Our results suggest grokking is not an inherent failure mode of deep learning, but rather a consequence of specific training conditions.
  arXiv  Detail & Related papers  (2026-01-27T16:52:04Z)
• The Geometry of Grokking: Norm Minimization on the Zero-Loss Manifold [5.076419064097734]
  We argue that post-memorization learning can be understood through the lens of constrained optimization.<n>We show that gradient descent effectively minimizes the weight norm on the zero-loss manifold.<n>Experiments confirm that simulating the training process using our predicted gradients reproduces both the delayed generalization and representation learning characteristic of grokking.
  arXiv  Detail & Related papers  (2025-11-02T18:44:42Z)
• Compute-Optimal LLMs Provably Generalize Better With Scale [102.29926217670926]
  We develop generalization bounds on the pretraining objective of large language models (LLMs) in the compute-optimal regime.<n>We introduce a novel, fully empirical Freedman-type martingale concentration that tightens existing bounds by accounting for the variance of the loss function.<n>We produce a scaling law for the generalization gap, with bounds that become predictably stronger with scale.
  arXiv  Detail & Related papers  (2025-04-21T16:26:56Z)
• Beyond Progress Measures: Theoretical Insights into the Mechanism of Grokking [50.465604300990904]
  Grokking refers to the abrupt improvement in test accuracy after extended overfitting.<n>In this work, we investigate the grokking mechanism underlying the Transformer in the task of prime number operations.
  arXiv  Detail & Related papers  (2025-04-04T04:42:38Z)
• Towards Understanding the Generalizability of Delayed Stochastic Gradient Descent [63.43247232708004]
  A gradient descent performed in an asynchronous manner plays a crucial role in training large-scale machine learning models.<n>Existing generalization error bounds are rather pessimistic and cannot reveal the correlation between asynchronous delays and generalization.<n>Our theoretical results indicate that asynchronous delays reduce the generalization error of the delayed SGD algorithm.
  arXiv  Detail & Related papers  (2023-08-18T10:00:27Z)
• Lassoed Tree Boosting [53.56229983630983]
  We prove that a gradient boosted tree algorithm with early stopping faster than $n-1/4$ L2 convergence in the large nonparametric space of cadlag functions of bounded sectional variation. Our convergence proofs are based on a novel, general theorem on early stopping with empirical loss minimizers of nested Donsker classes.
  arXiv  Detail & Related papers  (2022-05-22T00:34:41Z)
• Direction Matters: On the Implicit Bias of Stochastic Gradient Descent with Moderate Learning Rate [105.62979485062756]
  This paper attempts to characterize the particular regularization effect of SGD in the moderate learning rate regime. We show that SGD converges along the large eigenvalue directions of the data matrix, while GD goes after the small eigenvalue directions.
  arXiv  Detail & Related papers  (2020-11-04T21:07:52Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.