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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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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background 16representative citing papers
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.
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.
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.
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.
Gradient matching empirically recovers implicit regularization effects such as l2 penalties from early stopping and dropout in neural 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.
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.
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.
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.
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.
Machine unlearning conflates reversing the influence of specific training examples (untraining) with removing the full underlying distribution or behavior (unlearning).
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.
citing papers explorer
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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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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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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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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.
-
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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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 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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A Pre-Training Analogue of Grokking in Language Models: Tracing Delayed Grammatical Generalization
An exposure-based split on BLiMP data reveals delayed generalization in five grammatical phenomena during LLM pre-training, with post-generalization shifts in concept vector predictiveness and attention patterns.
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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.
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Understanding Goal Generalisation in Sequential Reinforcement Learning
Empirical analysis of over 100 sequential RL training pipelines across 250+ OOD environments finds salient features drive generalization and early goals persist, with latent policy gradients simulating latent variable evolution to predict OOD behavior from training history.
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Weight Decay Regimes in Grokking Transformers: Cheap Online Diagnostics
Weight decay controls distinct learning regimes in grokking transformers on modular arithmetic, tracked by new cheap attention-based diagnostics with empirical critical value and exponent fits.
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Transformers Linearly Represent Highly Structured World Models
Transformers trained on Sudoku traces develop constraint-structured internal world models and a monosemantic naked-single circuit.
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Detecting overfitting in Neural Networks during long-horizon grokking using Random Matrix Theory
Random Matrix Theory detects overfitting via growing Correlation Traps in weight spectra during the anti-grokking phase of neural network training.
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Overtrained, Not Misaligned
Emergent misalignment arises from overtraining after primary task convergence and is preventable by early stopping, which retains 93% of task performance on average.
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The two clocks and the innovation window: When and how generative models learn rules
Generative models learn rules before memorizing data, creating an innovation window whose width depends on dataset size and rule complexity, observed in both diffusion and autoregressive architectures.
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Selection Plateau and a Sparsity-Dependent Hierarchy of Pruning Features
All rank-monotone pruning scorers converge to identical accuracy at fixed sparsity, but non-monotone features with sparsity-dependent complexity can escape this plateau, as shown by the SICS hypothesis on ViT-Small/CIFAR-10.
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Learning Large-Scale Modular Addition with an Auxiliary Modulus
An auxiliary modulus during training reduces wrap-around issues and preserves train-test input distributions, enabling better accuracy and sample efficiency for large N and q in modular addition learning.
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The Weight Gram Matrix Captures Sequential Feature Linearization in Deep Networks
Gradient descent in deep networks implicitly drives features toward target-linear structure as captured by the weight Gram matrix and a derived virtual covariance.
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Critical Windows of Complexity Control: When Transformers Decide to Reason or Memorize
Transformers show a sharp, task-specific critical window for weight decay application that determines reasoning versus memorization, with middle placement optimal and boundaries as narrow as 100 steps.
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Finite-Size Gradient Transport in Large Language Model Pretraining: From Cascade Size to Intensive Transport Efficiency
A gradient-transport framework with observables D, z, β, δ, v_rel applied to Pico-LM and Pythia datasets shows distinct scaling regimes in duration and efficiency while sharing a near-unity cascade-size backbone.
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Generalization at the Edge of Stability
Training at the edge of stability causes neural network optimizers to converge on fractal attractors whose effective dimension, measured via a new sharpness dimension from the Hessian spectrum, bounds generalization error in a way not captured by prior trace or norm measures.
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Spectral Entropy Collapse as a Phase Transition in Delayed Generalisation: An Interventional and Predictive Framework for Grokkin
Spectral entropy collapse in learned representations precedes and predicts grokking, with interventions showing it is not explained by parameter norm alone.
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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.
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Autolearn: Learn by Surprise, Commit by Proof
Autolearn uses high-loss passages and self-generated Q&A training to drive the perturbation gap below baseline, improving novel fact acquisition while suppressing memorization in language models.
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Spectral Edge Dynamics: An Analytical-Empirical Study of Phase Transitions in Neural Network Training
Spectral gaps in the Gram matrix of parameter updates control phase transitions such as grokking in neural network training.
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Feature Identification via the Empirical NTK
Eigenanalysis of the empirical NTK surfaces feature directions that align with Fourier features in modular addition networks and grammatical features in Gemma-3-270M, outperforming PCA baselines on activations.
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Axiomatizing Neural Networks via Pursuit of Subspaces
Authors introduce the Pursuit of Subspaces (PoS) hypothesis, an axiomatic geometric framework that unifies explanations for representation, computation, and generalization in shallow and deep neural networks.
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Unveiling Memorization-Generalization Coexistence: A Case Study on Arithmetic Tasks with Label Noise
Experiments on modular arithmetic with heavy label noise show that over-parameterized networks form a distributed internal generalization structure that can be extracted via frequency methods to achieve high accuracy despite 80% noise.
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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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Phase Transitions in Driven Informational Systems: A Two-Field Perspective on Learning Theory and Non-Equilibrium Chemistry
Proposes a two-gradient-field model with candidate order parameters alpha_dagger and kappa_c to unify phase transitions across learning theory and non-equilibrium chemistry.
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Gradient-Direction Sensitivity Reveals Linear-Centroid Coupling Hidden by Optimizer Trajectories
Gradient-based SVD diagnostic uncovers hidden SED-LCH coupling in single and multitask settings and shows rank-3 subspace constraints speed up grokking by 2.3x.
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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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Learning the symmetric group: large from small
Transformer trained on S10 permutation prediction from transpositions generalizes to S25 with near 100% accuracy using identity augmentation and partitioned windows.
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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.