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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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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A Survey of Large Language Models
This survey reviews the background, key techniques, and evaluation methods for large language models, emphasizing emergent abilities that appear at large scales.