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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arXiv preprint arXiv:2309.02390 , year=
20 Pith papers cite this work, alongside 2 external citations. Polarity classification is still indexing.
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abstract
One of the most surprising puzzles in neural network generalisation is grokking: a network with perfect training accuracy but poor generalisation will, upon further training, transition to perfect generalisation. We propose that grokking occurs when the task admits a generalising solution and a memorising solution, where the generalising solution is slower to learn but more efficient, producing larger logits with the same parameter norm. We hypothesise that memorising circuits become more inefficient with larger training datasets while generalising circuits do not, suggesting there is a critical dataset size at which memorisation and generalisation are equally efficient. We make and confirm four novel predictions about grokking, providing significant evidence in favour of our explanation. Most strikingly, we demonstrate two novel and surprising behaviours: ungrokking, in which a network regresses from perfect to low test accuracy, and semi-grokking, in which a network shows delayed generalisation to partial rather than perfect test accuracy.
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In grokking modular arithmetic, weight direction portably carries circuit identity across independent runs while weight norm only sets susceptibility to overwrite and a weak delay effect.
During pretraining, language models exhibit natural ungrokking where learned rules are forgotten based on their support frequency in the corpus, with asymmetric editability of rule survival.
FSD, a permutation-tested metric of Fourier circuit synchronization, precedes grokking by a mean of 1722 steps across nine modular addition setups and causally controls grokking timing when weight decay is varied at the FSD ceiling.
Two steps of gradient descent on first-layer weights in linear-width two-layer networks produce a spiked random matrix with floor(alpha2/(1/2-alpha1)) outliers, each a learned direction, and batch reuse allows capturing directions with information exponent exceeding one.
Grokking delay follows T_grok - T_mem = Θ(γ_eff^{-1} log(‖θ_mem‖² / ‖θ_post‖²)), derived from norm separation in regularized optimization and validated with high correlations across 293 runs.
Multi-task grokking in Transformers produces staggered generalization, low-dimensional manifolds, weight-decay phase structure, holographic solutions, and transverse redundancy.
EGD equalizes gradient speeds across singular directions, eliminating or shortening grokking plateaus on modular addition and sparse parity problems.
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.
A 2x2 ablation shows repeated shared access enables grokking while addressable memory (not recurrence) enables edit propagation in transformer variants on synthetic KG QA.
Larger models succeed on rare and complex tasks by reducing gradient interference from common tasks, allowing rare-task features to accumulate, as shown via synthetic task mixtures and OLMo pretraining from 4M to 4B parameters.
In a structured-output NW matrix task, Transformers generalize fastest at intermediate dataset sizes while larger sets can accelerate memorization in partial-competence regimes.
A latent mediation framework with sparse autoencoders enables non-additive token-level influence attribution in LLMs by learning orthogonal features and back-propagating attributions.
Grokking occurs as the effective dimensionality of the gradient field transitions from sub-diffusive to super-diffusive at the onset of generalization, exhibiting self-organized criticality.
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.
SingGuard introduces a policy-adaptive multimodal LLM guardrail with dynamic reasoning regimes and SingGuard-Bench, reporting SOTA F1 scores across 35 datasets and improved policy-following accuracy under runtime shifts.
Grokking emerges near the model size where memorization timescale T_mem(P) intersects generalization timescale T_gen(P) on modular arithmetic.
Preconditioned gradient descent mitigates spectral bias and reduces grokking delays by enabling uniform parameter space exploration in the NTK regime, confirming grokking as a transition to the rich regime.
Varying evaluation metrics and corruption methods in activation patching produces different localization and circuit discovery outcomes in language models, leading to recommendations for preferred practices.
A dynamical systems analysis of constant-step SGD explains memorization in generative models by combining two-time-scale dynamics with a collapse model.
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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Cross-Trajectory Chimera Interventions Reveal Dissociable Roles of Weight Magnitude and Direction in Grokking
In grokking modular arithmetic, weight direction portably carries circuit identity across independent runs while weight norm only sets susceptibility to overwrite and a weak delay effect.
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Natural Ungrokking: Asymmetric Control of Which Rules Survive Pretraining
During pretraining, language models exhibit natural ungrokking where learned rules are forgotten based on their support frequency in the corpus, with asymmetric editability of rule survival.
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Circuit Synchronization Precedes Generalization: A Causal Precursor to Grokking
FSD, a permutation-tested metric of Fourier circuit synchronization, precedes grokking by a mean of 1722 steps across nine modular addition setups and causally controls grokking timing when weight decay is varied at the FSD ceiling.
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Feature Learning in Linear-Width Two-Layer Networks: Two vs. One Step of Gradient Descent
Two steps of gradient descent on first-layer weights in linear-width two-layer networks produce a spiked random matrix with floor(alpha2/(1/2-alpha1)) outliers, each a learned direction, and batch reuse allows capturing directions with information exponent exceeding one.
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The Norm-Separation Delay Law of Grokking: A First-Principles Theory of Delayed Generalization
Grokking delay follows T_grok - T_mem = Θ(γ_eff^{-1} log(‖θ_mem‖² / ‖θ_post‖²)), derived from norm separation in regularized optimization and validated with high correlations across 293 runs.
-
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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Egalitarian Gradient Descent: A Simple Approach to Accelerated Grokking
EGD equalizes gradient speeds across singular directions, eliminating or shortening grokking plateaus on modular addition and sparse parity problems.
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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.
-
Repeated Shared Access Enables Grokking, but Edit Propagation Depends on an Addressable Memory
A 2x2 ablation shows repeated shared access enables grokking while addressable memory (not recurrence) enables edit propagation in transformer variants on synthetic KG QA.
-
Why Larger Models Learn More: Effects of Capacity, Interference, and Rare-Task Retention
Larger models succeed on rare and complex tasks by reducing gradient interference from common tasks, allowing rare-task features to accumulate, as shown via synthetic task mixtures and OLMo pretraining from 4M to 4B parameters.
-
Slower Generalization, Faster Memorization: A Sweet Spot in Algorithmic Learning
In a structured-output NW matrix task, Transformers generalize fastest at intermediate dataset sizes while larger sets can accelerate memorization in partial-competence regimes.
-
Correcting Influence: Unboxing LLM Outputs with Orthogonal Latent Spaces
A latent mediation framework with sparse autoencoders enables non-additive token-level influence attribution in LLMs by learning orthogonal features and back-propagating attributions.
-
Grokking as Dimensional Phase Transition in Neural Networks
Grokking occurs as the effective dimensionality of the gradient field transitions from sub-diffusive to super-diffusive at the onset of generalization, exhibiting self-organized criticality.
-
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.
-
SingGuard: A Policy-Adaptive Multimodal LLM Guardrail with Dynamic Reasoning
SingGuard introduces a policy-adaptive multimodal LLM guardrail with dynamic reasoning regimes and SingGuard-Bench, reporting SOTA F1 scores across 35 datasets and improved policy-following accuracy under runtime shifts.
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Model Capacity Determines Grokking through Competing Memorisation and Generalisation Speeds
Grokking emerges near the model size where memorization timescale T_mem(P) intersects generalization timescale T_gen(P) on modular arithmetic.
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On the Convergence Behavior of Preconditioned Gradient Descent Toward the Rich Learning Regime
Preconditioned gradient descent mitigates spectral bias and reduces grokking delays by enabling uniform parameter space exploration in the NTK regime, confirming grokking as a transition to the rich regime.
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Towards Best Practices of Activation Patching in Language Models: Metrics and Methods
Varying evaluation metrics and corruption methods in activation patching produces different localization and circuit discovery outcomes in language models, leading to recommendations for preferred practices.
-
Adynamical systems view of training generativemodels and the memorization phenomenon
A dynamical systems analysis of constant-step SGD explains memorization in generative models by combining two-time-scale dynamics with a collapse model.