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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arXiv preprint arXiv:2301.02679 , year=
18 Pith papers cite this work, alongside 3 external citations. Polarity classification is still indexing.
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abstract
We present a simple neural network that can learn modular arithmetic tasks and exhibits a sudden jump in generalization known as ``grokking''. Concretely, we present (i) fully-connected two-layer networks that exhibit grokking on various modular arithmetic tasks under vanilla gradient descent with the MSE loss function in the absence of any regularization; (ii) evidence that grokking modular arithmetic corresponds to learning specific feature maps whose structure is determined by the task; (iii) analytic expressions for the weights -- and thus for the feature maps -- that solve a large class of modular arithmetic tasks; and (iv) evidence that these feature maps are also found by vanilla gradient descent as well as AdamW, thereby establishing complete interpretability of the representations learnt by the network.
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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.
Slingshot loss spikes are produced by low-precision arithmetic that breaks the zero-sum gradient constraint and drives exponential growth via Numerical Feature Inflation.
EGD equalizes gradient speeds across singular directions, eliminating or shortening grokking plateaus on modular addition and sparse parity problems.
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
Derives an interaction measure between crosscoder features from reconstruction error in compact proofs and applies it to produce computationally sparse crosscoders retaining 60% MLP performance with single-feature selection versus 10% for standard crosscoders.
Deep linear network theory derives logarithmic decay for cross-entropy loss under gap-growth conditions versus polynomial closure for Schatten-regularized structural energy under late-time KL tails, separating fitting from simplification; conditional reductions extend this to ReLU MLPs with fixed ac
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.
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.
Deep sequence models develop geometric memory in embeddings that encodes novel global relationships, transforming l-fold composition tasks into 1-step navigation via a natural spectral bias connected to Node2Vec.
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.
SingGuard introduces a policy-adaptive multimodal LLM guardrail with dynamic reasoning regimes and SingGuard-Bench, reporting SOTA F1 scores across 35 datasets and improved policy-following accuracy under runtime shifts.
Modular arithmetic induces cyclic rank-2 geometries via layerwise subspace locking and entropy-regularized phase alignment on S^1, prevailing over neural collapse simplices due to a Theta(K) advantage under weight-decay surrogates.
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.
A compact parameterized quantum circuit with geometric phase embedding achieves exact, zero-variance generalization on Z11 arithmetic and S4 group multiplication, a regime the authors call crystallization.
The paper introduces a taxonomy of AI safety for LLMs organized into Trustworthy AI, Responsible AI, and Safe AI perspectives, accompanied by a review of state-of-the-art methods, challenges, and future directions.
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citing papers explorer
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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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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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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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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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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
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Interactions Between Crosscoder Features: A Compact Proofs Perspective
Derives an interaction measure between crosscoder features from reconstruction error in compact proofs and applies it to produce computationally sparse crosscoders retaining 60% MLP performance with single-feature selection versus 10% for standard crosscoders.
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Deciphering Two Training Clocks in Grokking via Deep Linear Network Theory with Conditional ReLU Reduction
Deep linear network theory derives logarithmic decay for cross-entropy loss under gap-growth conditions versus polynomial closure for Schatten-regularized structural energy under late-time KL tails, separating fitting from simplification; conditional reductions extend this to ReLU MLPs with fixed ac
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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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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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Deep sequence models tend to memorize geometrically; it is unclear why
Deep sequence models develop geometric memory in embeddings that encodes novel global relationships, transforming l-fold composition tasks into 1-step navigation via a natural spectral bias connected to Node2Vec.
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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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SingGuard: A Policy-Adaptive Multimodal LLM Guardrail with Dynamic Reasoning
SingGuard introduces a policy-adaptive multimodal LLM guardrail with dynamic reasoning regimes and SingGuard-Bench, reporting SOTA F1 scores across 35 datasets and improved policy-following accuracy under runtime shifts.
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Beyond Neural Collapse: Task-Intrinsic Geometry Governs Neural Representations in Modular Arithmetic
Modular arithmetic induces cyclic rank-2 geometries via layerwise subspace locking and entropy-regularized phase alignment on S^1, prevailing over neural collapse simplices due to a Theta(K) advantage under weight-decay surrogates.
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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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Universal Quantum Transformer
A compact parameterized quantum circuit with geometric phase embedding achieves exact, zero-variance generalization on Z11 arithmetic and S4 group multiplication, a regime the authors call crystallization.
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AI Safety Landscape for Large Language Models: Taxonomy, State-of-the-art, and Future Directions
The paper introduces a taxonomy of AI safety for LLMs organized into Trustworthy AI, Responsible AI, and Safe AI perspectives, accompanied by a review of state-of-the-art methods, challenges, and future directions.
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There Will Be a Scientific Theory of Deep Learning
A mechanics of the learning process is emerging in deep learning theory, characterized by dynamics, coarse statistics, and falsifiable predictions across idealized settings, limits, laws, hyperparameters, and universal behaviors.
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