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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Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets
45 Pith papers cite this work. Polarity classification is still indexing.
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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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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.
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.
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.
In sparse regimes, models exploit normal alignment of Jacobians to minimize loss and maximize robustness; GrokAlign induces this alignment to accelerate training and RFAMs improve adversarial robustness.
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.
Slingshot loss spikes arise from floating-point precision limits that round correct-class gradients to zero, breaking zero-sum constraints and driving exponential parameter growth through numerical feature inflation.
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.
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.
The grokking delay in encoder-decoder models on one-step Collatz prediction stems from decoder inability to use early-learned encoder representations of parity and residue structure, with numeral base acting as a strong inductive bias that can raise accuracy from failure to 99.8%.
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.
A Random Matrix Theory method identifies growing Correlation Traps in neural network weight spectra during an 'anti-grokking' overfitting phase, and applies the same diagnostic to some foundation LLMs.
Emergent misalignment arises from overtraining after primary task convergence and is preventable by early stopping, which retains 93% of task performance on average.
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.
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.
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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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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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
In sparse regimes, models exploit normal alignment of Jacobians to minimize loss and maximize robustness; GrokAlign induces this alignment to accelerate training and RFAMs improve adversarial robustness.
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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 arise from floating-point precision limits that round correct-class gradients to zero, breaking zero-sum constraints and driving exponential parameter growth through 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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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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The Long Delay to Arithmetic Generalization: When Learned Representations Outrun Behavior
The grokking delay in encoder-decoder models on one-step Collatz prediction stems from decoder inability to use early-learned encoder representations of parity and residue structure, with numeral base acting as a strong inductive bias that can raise accuracy from failure to 99.8%.
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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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Detecting overfitting in Neural Networks during long-horizon grokking using Random Matrix Theory
A Random Matrix Theory method identifies growing Correlation Traps in neural network weight spectra during an 'anti-grokking' overfitting phase, and applies the same diagnostic to some foundation LLMs.
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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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Spectral Lens: Activation and Gradient Spectra as Diagnostics of LLM Optimization
Spectral analysis of activations and gradients provides new diagnostics that link batch size to representation geometry, early covariance tails to token efficiency, and spectral shifts to learning dynamics in decoder-only LLMs, backed by a mechanistic model.
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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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Can Transformers predict system collapse in dynamical systems?
Transformers fail to predict catastrophic collapse in unseen parameter regimes of nonlinear dynamical systems, while reservoir computing reliably succeeds.
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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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Convergent Evolution: How Different Language Models Learn Similar Number Representations
Diverse language models converge on similar periodic number features with a two-tier hierarchy of Fourier sparsity and geometric separability, acquired via language co-occurrences or multi-token arithmetic.
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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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Nexus: Same Pretraining Loss, Better Downstream Generalization via Common Minima
Nexus optimizer improves LLM downstream performance by converging to common minima across data sources despite identical pretraining loss.
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Training Deep Visual Networks Beyond Loss and Accuracy Through a Dynamical Systems Approach
Introduces integration, metastability, and dynamical stability index measures from layer activations and reports patterns distinguishing CIFAR-10 from CIFAR-100 difficulty plus early convergence signals across ResNet variants, DenseNet, MobileNetV2, VGG-16, and a Vision Transformer.
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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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Language Models (Mostly) Know What They Know
Language models show good calibration when asked to estimate the probability that their own answers are correct, with performance improving as models get larger.
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Constrained Stochastic Spectral Preconditioning Converges for Nonconvex Objectives
Proximal stochastic spectral preconditioning converges for nonconvex constrained objectives under heavy-tailed noise, with a variance-reduced version achieving faster rates and a refined analysis of Muon iterations.
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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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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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Artificial Jagged Intelligence as Uneven Optimization Energy Allocation Capability Concentration, Redistribution, and Optimization Governance
AJI frames jagged AI capabilities as lower bounds on performance dispersion arising from concentrated optimization energy allocation under anisotropic objectives, with theorems on tradeoffs and redistribution interventions.
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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.