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 81 cs.AI 13 cs.CL 11 cs.CV 5 quant-ph 4 stat.ML 4 cs.NE 2 cond-mat.dis-nn 1 cs.HC 1 math.OC 1roles
background 17representative citing papers
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.
Memorized facts in fine-tuned LLMs often sit off the mid-layer reasoning path; relocating those representations recovers most multi-hop generalization failures.
Fixed-architecture networks of width O(D) and depth O(r) approximate Hölder functions with parameter magnitude log P = O(ε^{-2D/(r+γ)} log(1/ε)) via CRT encoding.
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.
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.
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.
Introduces the Generalization Spectrum evaluation framework to track per-example generalization across transfer distances in competitive programming tasks.
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.
Gradient flow on a two-layer network trained to compose finite-group elements provably pushes each neuron to a single irreducible representation with rank-one cross-layer alignment; for Abelian groups it yields a uniformly diversified, Haar-phase majority-vote predictor.
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.
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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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.
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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.
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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.
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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.
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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.
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Towards Mechanistically Understanding Why Memorized Knowledge Fails to Generalize in Large Language Model Finetuning
Memorized facts in fine-tuned LLMs often sit off the mid-layer reasoning path; relocating those representations recovers most multi-hop generalization failures.
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On Explicit Super-Expressive Approximation for Neural Networks
Fixed-architecture networks of width O(D) and depth O(r) approximate Hölder functions with parameter magnitude log P = O(ε^{-2D/(r+γ)} log(1/ε)) via CRT encoding.
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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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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.
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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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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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The Generalization Spectrum: A Chromatographic Approach to Evaluating Learning Algorithms
Introduces the Generalization Spectrum evaluation framework to track per-example generalization across transfer distances in competitive programming tasks.
-
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.
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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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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.
-
Neural Networks Provably Learn Spectral Representations for Group Composition
Gradient flow on a two-layer network trained to compose finite-group elements provably pushes each neuron to a single irreducible representation with rank-one cross-layer alignment; for Abelian groups it yields a uniformly diversified, Haar-phase majority-vote predictor.
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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.
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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.
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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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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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Final Checkpoints Are Not Enough: Analyzing Latent Reasoning Faithfulness Along Training Trajectories
Latent reasoning faithfulness is a property of training stage and answer format, not of architecture or the final checkpoint alone.
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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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World-Model Collapse as a Phase Transition
Long-horizon LLM agents exhibit a finite-grid phase transition in world-model fidelity under state cardinality and dependency density, with world-state collapse preceding action failure.
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Grokking and epoch-wise double descent in quantum neural networks
Overparameterized two-qubit SU(4) QNNs exhibit grokking and epoch-wise double descent; depth raises generalization success, and weak L2 regularization anchors the post-grokking state against weight-norm drift.
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Multiplication Beyond Groups: Stratified Fourier Mechanisms in Transformer Circuits
Small transformers learning composite-modular multiplication partition inputs into algebraic J-classes and apply local Fourier-based group mechanisms within each class, extending representation-theoretic interpretability beyond globally invertible group operations