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Grokking modular arithmetic
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Grokking modular arithmetic
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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.
Forward citations
Cited by 24 Pith papers
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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 t...
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Feature Learning in Linear-Width Two-Layer Networks: Two vs. One Step of Gradient Descent
In the linear-width regime, the second GD step yields a spiked random matrix whose number of outliers is floor(alpha2 / (1/2 - alpha1)), and batch reuse enables learning directions with information exponent greater th...
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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 capturi...
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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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Grokking or Glitching? How Low-Precision Drives Slingshot Loss Spikes
Slingshot loss spikes result from floating-point precision limits that round correct-class gradients to zero, triggering Numerical Feature Inflation and breaking gradient zero-sum constraints.
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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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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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Structure-Specific Representational Priors Causally Control the Grokking Delay
The grokking delay is causally the time to form the right feature-level representational structure, not a fixed optimization constant or a pure weight-norm effect.
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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 interpretabil...
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Grokking Is Conditional and Fragile: A Fully-Tractable, Multi-Seed Study at 12K Parameters
In a fully tractable 12K Llama-style model, grokking is a conditional fragile phase transition gated by coverage (tracking modulus more than structure), weight decay, and floating-point reduction order, so evidence mu...
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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 sel...
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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...
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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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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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Deep sequence models tend to memorize geometrically; it is unclear why
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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 presents a policy-adaptive multimodal LLM guardrail family with hybrid reasoning regimes and a new benchmark of 56,340 examples, claiming SOTA F1 across 35 datasets and improved policy adherence under runtim...
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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-dec...
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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 ...
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Universal Quantum Transformer
UQT on 5 qubits achieves exact deterministic learning of Z_11 modular arithmetic and S_4 non-Abelian algebra via quantum-native mechanisms, claiming to bypass classical attention limits and run on NISQ hardware.
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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 universa...
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