A single-layer compartmental SNN with apical recurrence matching leaky online Widrow-Hoff LMS achieves seed-stable ICL on high-dimensional Garg-2022 tasks where Transformers fail, with a linear probe recovering the LMS trajectory at R²=0.93.
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Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets
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In this paper we propose to study generalization of neural networks on small algorithmically generated datasets. In this setting, questions about data efficiency, memorization, generalization, and speed of learning can be studied in great detail. In some situations we show that neural networks learn through a process of "grokking" a pattern in the data, improving generalization performance from random chance level to perfect generalization, and that this improvement in generalization can happen well past the point of overfitting. We also study generalization as a function of dataset size and find that smaller datasets require increasing amounts of optimization for generalization. We argue that these datasets provide a fertile ground for studying a poorly understood aspect of deep learning: generalization of overparametrized neural networks beyond memorization of the finite training dataset.
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- abstract In this paper we propose to study generalization of neural networks on small algorithmically generated datasets. In this setting, questions about data efficiency, memorization, generalization, and speed of learning can be studied in great detail. In some situations we show that neural networks learn through a process of "grokking" a pattern in the data, improving generalization performance from random chance level to perfect generalization, and that this improvement in generalization can happen well past the point of overfitting. We also study generalization as a function of dataset size and f
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cs.LG 72 cs.AI 12 cs.CL 11 cs.CV 5 stat.ML 4 quant-ph 3 cs.NE 2 cond-mat.dis-nn 1 cs.HC 1 math.OC 1roles
background 16representative citing papers
Two-layer neural networks provably converge almost surely to irreducible representations of finite groups when trained on the group composition task, with the dynamics governed by Riemannian gradient ascent on a representation-theoretic energy functional.
Derives geodesic ridge regularization and Riemannian Gibbs Process prior for feature-learning wide neural networks, generalizing kernel-regime results via function-space axiomatization.
In the high-dimensional limit the spherical Boltzmann machine admits exact equations for training dynamics, Bayesian evidence, and cascades of phase transitions tied to mode alignment with data, which connect to generative phenomena including double descent and out-of-equilibrium biases.
Transformer weight spectra exhibit transient compression waves that propagate layer-wise, persistent non-monotonic depth gradients in power-law exponents, and Q/K-V asymmetry, with the spectral exponent alpha predicting layer importance and enabling pruning gains of 1.1x-3.6x over Last-N baselines.
Content-based routing succeeds only when models provide bidirectional context and perform pairwise comparisons, with bidirectional Mamba plus rank-1 projection reaching 99.7% precision at linear inference cost.
Infinite-width transformers exhibit an inductive bias against high-complexity polynomial-time algorithms, with derived upper bounds on capturable tasks like sorting and string matching.
Grokking reflects escape from a metastable low-dimensional regime where transverse curvature accumulates before generalization, with subspace motion necessary but curvature boost insufficient.
The AI Scientist framework enables LLMs to independently conduct the full scientific process from idea generation to paper writing and review, demonstrated across three ML subfields with papers costing under $15 each.
Grokking arises from gradual amplification of a Fourier-based circuit in the weights followed by removal of memorizing components.
Toy models demonstrate that polysemanticity arises when neural networks store more sparse features than neurons via superposition, producing a phase transition tied to polytope geometry and increased adversarial vulnerability.
A descent-free method recovers the singularity order k of dead directions in neural networks from the directional-Fisher rate, classifies them, and assembles global learning coefficients matching closed forms.
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.
Dead-Direction Signatures provide closed-form spectral readings of dead directions in network activations and gradients that track rank deficits at singular minima, offering a cheap directional alternative to SGLD-based LLC.
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.
Fragility, the activation noise level causing probe accuracy collapse, reveals evolving lexical-to-compositional moral encoding, layer robustness gradients, and fine-tuning differences invisible to saturated probing accuracy.
Dead directions recover Watanabe's RLCT contribution and triple (λ, m, ν) from directional Fisher curvature decay rates in original parameter space for singular models, extended via K-FAC to networks and gauge-equivariant optimizers.
RFLO learning restricts solutions to low-rank perturbations of initial parameters in linear RNNs and produces qualitatively different stability and convergence behavior than BPTT.
Self-evolving rubric with anti-gaming fitness reveals that objective capability scaling fails to transfer to subjective LLM behaviors, with advice-restraint as the universal lowest dimension that can regress.
Apparent phase transitions during fine-tuning on near-synonym tasks are phantoms originating in the softmax readout; an order parameter isolates kinematic and structural failure modes and a few dimensionless quantities predict critical learning rates across architectures via blind test.
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.
Generalization is a testable hedging property of the learner's response law, recovered via f-divergence regularizers that induce information-geometric curves between training loss and sample dependence.
Temporal correlations from lazy random walks enable efficient SGD learning of k-juntas via temporal-difference loss on ReLU networks, achieving linear sample complexity in d.
Normal alignment is the rank-one Jacobian structure that lets classifiers minimize loss and maximize local robustness in sparse regimes; the paper proves its optimality and uses it to create GrokAlign and RFAMs.
citing papers explorer
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Dendritic In-Context Learning in a Single-Layer Spiking Neural Network
A single-layer compartmental SNN with apical recurrence matching leaky online Widrow-Hoff LMS achieves seed-stable ICL on high-dimensional Garg-2022 tasks where Transformers fail, with a linear probe recovering the LMS trajectory at R²=0.93.
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Neural Networks Provably Learn Spectral Representations for Group Composition
Two-layer neural networks provably converge almost surely to irreducible representations of finite groups when trained on the group composition task, with the dynamics governed by Riemannian gradient ascent on a representation-theoretic energy functional.
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Canonical Regularisation of Wide Feature-Learning Neural Networks
Derives geodesic ridge regularization and Riemannian Gibbs Process prior for feature-learning wide neural networks, generalizing kernel-regime results via function-space axiomatization.
-
Spherical Boltzmann machines: a solvable theory of learning and generation in energy-based models
In the high-dimensional limit the spherical Boltzmann machine admits exact equations for training dynamics, Bayesian evidence, and cascades of phase transitions tied to mode alignment with data, which connect to generative phenomena including double descent and out-of-equilibrium biases.
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The Spectral Lifecycle of Transformer Training: Transient Compression Waves, Persistent Spectral Gradients, and the Q/K--V Asymmetry
Transformer weight spectra exhibit transient compression waves that propagate layer-wise, persistent non-monotonic depth gradients in power-law exponents, and Q/K-V asymmetry, with the spectral exponent alpha predicting layer importance and enabling pruning gains of 1.1x-3.6x over Last-N baselines.
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When Does Content-Based Routing Work? Representation Requirements for Selective Attention in Hybrid Sequence Models
Content-based routing succeeds only when models provide bidirectional context and perform pairwise comparisons, with bidirectional Mamba plus rank-1 projection reaching 99.7% precision at linear inference cost.
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Algorithmic Task Capture, Computational Complexity, and Inductive Bias of Infinite Transformers
Infinite-width transformers exhibit an inductive bias against high-complexity polynomial-time algorithms, with derived upper bounds on capturable tasks like sorting and string matching.
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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.
-
The AI Scientist: Towards Fully Automated Open-Ended Scientific Discovery
The AI Scientist framework enables LLMs to independently conduct the full scientific process from idea generation to paper writing and review, demonstrated across three ML subfields with papers costing under $15 each.
-
Progress measures for grokking via mechanistic interpretability
Grokking arises from gradual amplification of a Fourier-based circuit in the weights followed by removal of memorizing components.
-
Toy Models of Superposition
Toy models demonstrate that polysemanticity arises when neural networks store more sparse features than neurons via superposition, producing a phase transition tied to polytope geometry and increased adversarial vulnerability.
-
Measuring Dead Directions: Decomposing and Classifying Singular Structure off Canonical Alignment
A descent-free method recovers the singularity order k of dead directions in neural networks from the directional-Fisher rate, classifies them, and assembles global learning coefficients matching closed forms.
-
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.
-
Dead-Direction Signatures: A Cheap Spectral Reading of Singular Complexity
Dead-Direction Signatures provide closed-form spectral readings of dead directions in network activations and gradients that track rank deficits at singular minima, offering a cheap directional alternative to SGLD-based LLC.
-
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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When Probing Accuracy Saturates, Fragility Resolves: A Complementary Metric for LLM Pre-Training Analysis
Fragility, the activation noise level causing probe accuracy collapse, reveals evolving lexical-to-compositional moral encoding, layer robustness gradients, and fine-tuning differences invisible to saturated probing accuracy.
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Dead Directions: Geometric Singular Learning
Dead directions recover Watanabe's RLCT contribution and triple (λ, m, ν) from directional Fisher curvature decay rates in original parameter space for singular models, extended via K-FAC to networks and gauge-equivariant optimizers.
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Dynamics and Representation Structure of Local Approximations to Gradient-Based Learning in Linear Recurrent Neural Networks
RFLO learning restricts solutions to low-rank perturbations of initial parameters in linear RNNs and produces qualitatively different stability and convergence behavior than BPTT.
-
Does Capability Transfer to Subjective Behavior -- and Would Our Instruments Tell Us? A Self-Evolving, Trust-by-Construction Evaluation Paradigm
Self-evolving rubric with anti-gaming fitness reveals that objective capability scaling fails to transfer to subjective LLM behaviors, with advice-restraint as the universal lowest dimension that can regress.
-
Phantom transitions in language model fine-tuning
Apparent phase transitions during fine-tuning on near-synonym tasks are phantoms originating in the softmax readout; an order parameter isolates kinematic and structural failure modes and a few dimensionless quantities predict critical learning rates across architectures via blind test.
-
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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Bounded-Rationality, Hedging, and Generalization
Generalization is a testable hedging property of the learner's response law, recovered via f-divergence regularizers that induce information-geometric curves between training loss and sample dependence.
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The Benefits of Temporal Correlations: SGD Learns k-Juntas from Random Walks Efficiently
Temporal correlations from lazy random walks enable efficient SGD learning of k-juntas via temporal-difference loss on ReLU networks, achieving linear sample complexity in d.
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The Geometric Structure of Models Learning Sparse Data
Normal alignment is the rank-one Jacobian structure that lets classifiers minimize loss and maximize local robustness in sparse regimes; the paper proves its optimality and uses it to create GrokAlign and RFAMs.
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Topological Signatures of Grokking
Persistent homology detects a sharp increase in maximum and total H1 persistence during grokking on modular arithmetic, offering a topological diagnostic that links representation geometry to generalization.
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Grokking or Glitching? How Low-Precision Drives Slingshot Loss Spikes
Slingshot loss spikes are produced by low-precision arithmetic that breaks the zero-sum gradient constraint and drives exponential growth via Numerical Feature Inflation.
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Estimating Implicit Regularization in Deep Learning
Gradient matching empirically recovers implicit regularization effects such as l2 penalties from early stopping and dropout in neural networks.
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Layerwise LQR for Geometry-Aware Optimization of Deep Networks
Steepest descent under divergence-induced quadratic models equals an LQR problem, enabling learning of diagonal or Kronecker-factored inverse preconditioners via a global layerwise objective for scalable geometry-aware training.
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A Theory of Generalization in Deep Learning
A theory shows SGD accumulates coherent signal via linear drift in NTK signal directions while trapping noise in orthogonal low-eigenvalue dimensions, enabling generalization even under O(1) kernel evolution and yielding an exact population-risk objective from one run that acts as an Adam SNR boost.
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ILDR: Geometric Early Detection of Grokking
ILDR detects the geometric reorganization preceding grokking by measuring when inter-class centroid separation exceeds intra-class scatter by 2.5 times its baseline in penultimate-layer representations.
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Grokking of Diffusion Models: Case Study on Modular Addition
Diffusion models show grokking on modular addition by composing periodic operand representations in simple data regimes or by separating arithmetic computation from visual denoising across timesteps in varied regimes.
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Scalable Neural Decoders for Practical Fault-Tolerant Quantum Computation
Neural decoder for quantum LDPC codes achieves ~10^{-10} logical error at 0.1% physical error with 17x improvement and high throughput, enabling practical fault tolerance at modest code sizes.
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Is your algorithm unlearning or untraining?
Machine unlearning conflates reversing the influence of specific training examples (untraining) with removing the full underlying distribution or behavior (unlearning).
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Spectral Edge Dynamics Reveal Functional Modes of Learning
Spectral edge dynamics during grokking reveal task-dependent low-dimensional functional modes over inputs, such as Fourier modes for modular addition and cross-term decompositions for x squared plus y squared.
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Dimensional Criticality at Grokking Across MLPs and Transformers
Effective cascade dimension D(t) crosses D=1 at the grokking transition in MLPs and Transformers, with opposite directions for modular addition versus XOR, consistent with attraction to a shared critical manifold.
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Rethinking the Harmonic Loss via Non-Euclidean Distance Layers
Non-Euclidean distance variants of harmonic loss improve accuracy, gradient stability, and energy efficiency over cross-entropy and Euclidean harmonic loss in vision backbones and large language models.
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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.
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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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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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Reinforcement Learning for Reasoning in Large Language Models with One Training Example
One training example via RLVR boosts LLM math reasoning from 17.6% to 35.7% average across six benchmarks.
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In-context Learning and Induction Heads
Induction heads, which implement pattern completion in attention, develop at the same training stage as a sudden rise in in-context learning, providing evidence they are the primary mechanism for in-context learning in transformers.
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Radial Suppression Accelerates Algorithmic Generalization: A Geometric Analysis of Delayed Generalization
A norm penalty constraining activations to a sqrt(d)-radius hypersphere accelerates grokking by up to 6x on modular arithmetic via radial suppression in activation dynamics.
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Patnaik-Pearson intrinsic dimension for internal representations of neural networks
Introduces the Patnaik-Pearson intrinsic dimension estimator, proves some of its properties, relates it to HTSR/SETOL for Pareto spectra, and applies it to track embedding dimension evolution in BERT-base and DeepSeek-R1-Distill-Qwen-1.
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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.
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Learning Dynamics of Chain-of-Thought State Tracking in a Solvable Transformer Model
Mean-field equations for attention retrieval, teacher alignment, and logic overlap quantitatively match simulations and predict a sharp accuracy transition in a solvable transformer for permutation state tracking.
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XtrAIn: Training-Guided Occlusion for Feature Attribution
XtrAIn shifts occlusion from input space to parameter space along the training trajectory to produce cleaner feature attributions than standard methods.
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Deciphering Two Training Clocks in Grokking via Deep Linear Network Theory with Conditional ReLU Reduction
Deep linear network theory derives logarithmic decay for cross-entropy loss under gap-growth conditions versus polynomial closure for Schatten-regularized structural energy under late-time KL tails, separating fitting from simplification; conditional reductions extend this to ReLU MLPs with fixed ac
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Arithmetic Pedagogy for Language Models
A small GPT-2 model trained from scratch on GASING-derived CoT supervision for arithmetic reaches over 80% held-out accuracy, exhibits three learning phases, and develops both procedural and associative reasoning.
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Consistency Training while Mitigating Obfuscation via Rate Matching
RMCT matches the rate of target behaviors like bias-following across input perturbations to reduce sycophancy in LLMs while preserving verbalization of bias cues.
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Emergent Ordinal Geometry in Transformers Trained on Local Comparisons
Transformers trained on local comparisons spontaneously form a rank-aligned one-dimensional embedding manifold that reproduces the symbolic distance effect.