Grokking reflects escape from a metastable low-dimensional regime where transverse curvature accumulates before generalization, with subspace motion necessary but curvature boost insufficient.
Grokking as the transition from lazy to rich training dynamics.arXiv preprint arXiv:2310.06110
8 Pith papers cite this work. Polarity classification is still indexing.
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abstract
We propose that the grokking phenomenon, where the train loss of a neural network decreases much earlier than its test loss, can arise due to a neural network transitioning from lazy training dynamics to a rich, feature learning regime. To illustrate this mechanism, we study the simple setting of vanilla gradient descent on a polynomial regression problem with a two layer neural network which exhibits grokking without regularization in a way that cannot be explained by existing theories. We identify sufficient statistics for the test loss of such a network, and tracking these over training reveals that grokking arises in this setting when the network first attempts to fit a kernel regression solution with its initial features, followed by late-time feature learning where a generalizing solution is identified after train loss is already low. We find that the key determinants of grokking are the rate of feature learning -- which can be controlled precisely by parameters that scale the network output -- and the alignment of the initial features with the target function $y(x)$. We argue this delayed generalization arises when (1) the top eigenvectors of the initial neural tangent kernel and the task labels $y(x)$ are misaligned, but (2) the dataset size is large enough so that it is possible for the network to generalize eventually, but not so large that train loss perfectly tracks test loss at all epochs, and (3) the network begins training in the lazy regime so does not learn features immediately. We conclude with evidence that this transition from lazy (linear model) to rich training (feature learning) can control grokking in more general settings, like on MNIST, one-layer Transformers, and student-teacher networks.
years
2026 8representative citing papers
Observable Matrix Dynamics (OMD) is a new diagnostic framework that uses random matrix theory on distance matrices to distinguish diffusive relaxations from phase-transition-like reorganizations during neural network training.
Grokking delay under cross-entropy is mediated primarily by logit scale and resulting softmax saturation, with weight norm acting only as an upstream handle that adds 1-2% beyond the scale.
Multi-task grokking in Transformers produces staggered generalization, low-dimensional manifolds, weight-decay phase structure, holographic solutions, and transverse redundancy.
Embedding effective rank at grokking is a transient that overstates the converged floor by 3–5× (MLP) / 1.3–1.5× (transformer), and compression lags generalization by order T_grok, modulated by LayerNorm.
Introduces 9 synthetic annotation tasks and benchmarks for behavioral cloning, finding hierarchical skill learning, scaling benefits, effective multi-task pretraining, and shared internal representations of task phases and mistakes.
Grokking delay on Collatz prediction is a decoder access bottleneck after early encoder structure learning, with numeral base as a strong inductive bias on learnability.
Empirical tests confirm robust feature repulsion signs but reveal activation-dependent spectral lock-in in grokking, with x^2 yielding rank-2 updates at epoch ~174 and ReLU remaining rank-1.
citing papers explorer
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Low-Dimensional and Transversely Curved Optimization Dynamics in Grokking
Grokking reflects escape from a metastable low-dimensional regime where transverse curvature accumulates before generalization, with subspace motion necessary but curvature boost insufficient.
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Learning as Observable Matrix Dynamics: Diffusive Relaxations versus Phase Transitions
Observable Matrix Dynamics (OMD) is a new diagnostic framework that uses random matrix theory on distance matrices to distinguish diffusive relaxations from phase-transition-like reorganizations during neural network training.
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What Does the Weight Norm Control in Grokking? Logit-Scale Mediation under Cross-Entropy
Grokking delay under cross-entropy is mediated primarily by logit scale and resulting softmax saturation, with weight norm acting only as an upstream handle that adds 1-2% beyond the scale.
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The Geometry of Multi-Task Grokking: Transverse Instability, Superposition, and Weight Decay Phase Structure
Multi-task grokking in Transformers produces staggered generalization, low-dimensional manifolds, weight-decay phase structure, holographic solutions, and transverse redundancy.
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At-Grok Is Not Converged:A Measurement-Validity Audit for Grokking Representation Metrics
Embedding effective rank at grokking is a transient that overstates the converged floor by 3–5× (MLP) / 1.3–1.5× (transformer), and compression lags generalization by order T_grok, modulated by LayerNorm.
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A Systematic Study of Behavioral Cloning for Scientific Data Annotation
Introduces 9 synthetic annotation tasks and benchmarks for behavioral cloning, finding hierarchical skill learning, scaling benefits, effective multi-task pretraining, and shared internal representations of task phases and mistakes.
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The Long Delay to Arithmetic Generalization: When Learned Representations Outrun Behavior
Grokking delay on Collatz prediction is a decoder access bottleneck after early encoder structure learning, with numeral base as a strong inductive bias on learnability.
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Feature Repulsion and Spectral Lock-in: An Empirical Study of Two-Layer Network Grokking
Empirical tests confirm robust feature repulsion signs but reveal activation-dependent spectral lock-in in grokking, with x^2 yielding rank-2 updates at epoch ~174 and ReLU remaining rank-1.