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.
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