mlsanityData to 14 Oct 2026

Feature emergence via margin maximization: case studies in algebraic tasks

arXiv 2311.07568 ↗First posted 13 Nov 2023cs.LGcited by 33

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Abstract (arXiv)

Understanding the internal representations learned by neural networks is a cornerstone challenge in the science of machine learning. While there have been significant recent strides in some cases towards understanding how neural networks implement specific target functions, this paper explores a complementary question -- why do networks arrive at particular computational strategies? Our inquiry focuses on the algebraic learning tasks of modular addition, sparse parities, and finite group operations. Our primary theoretical findings analytically characterize the features learned by stylized neural networks for these algebraic tasks. Notably, our main technique demonstrates how the principle of margin maximization alone can be used to fully specify the features learned by the network. Specifically, we prove that the trained networks utilize Fourier features to perform modular addition and employ features corresponding to irreducible group-theoretic representations to perform compositions in general groups, aligning closely with the empirical observations of Nanda et al. and Chughtai et al. More generally, we hope our techniques can help to foster a deeper understanding of why neural networks adopt specific computational strategies.

Key facts. Cited by 33 papers in the map (data through 14 Oct 2026). Builds on 29 of its 37 references that are in the map. First posted on arXiv 13 Nov 2023.

Builds on

29 of 37 references.

References · 29

  1. 2021Measuring Mathematical Problem Solving With the MATH Datasetcited by 5,371
  2. 2021Convex optimizationcited by 2,515
  3. 2022Solving Quantitative Reasoning Problems with Language Modelscited by 1,392
  4. 2022Toy Models of Superpositioncited by 971
  5. 2023Progress measures for grokking via mechanistic interpretabilitycited by 839
  6. 2017The Implicit Bias of Gradient Descent on Separable Datacited by 655
  7. 2022Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasetscited by 629
  8. 2019Analysing Mathematical Reasoning Abilities of Neural Modelscited by 335
  9. 2019Gradient Descent Maximizes the Margin of Homogeneous Neural Networkscited by 320
  10. 2020Implicit Bias of Gradient Descent for Wide Two-layer Neural Networks Trained with the Logistic Losscited by 268
  11. 2018Implicit Bias of Gradient Descent on Linear Convolutional Networkscited by 251
  12. 2022Towards Understanding Grokking: An Effective Theory of Representation Learningcited by 230
17 more
  1. 2022Hidden Progress in Deep Learning: SGD Learns Parities Near the Computational Limitcited by 187
  2. 2020Directional convergence and alignment in deep learningcited by 178
  3. 2023The Clock and the Pizza: Two Stories in Mechanistic Explanation of Neural Networkscited by 156
  4. 2022Omnigrok: Grokking Beyond Algorithmic Datacited by 146
  5. 2023A Toy Model of Universality: Reverse Engineering How Networks Learn Group Operationscited by 144
  6. 2018Regularization Matters: Generalization and Optimization of Neural Nets v.s. their Induced Kernelcited by 131
  7. 2022On the Implicit Bias in Deep-Learning Algorithmscited by 86
  8. 2021Gradient Descent on Two-layer Nets: Margin Maximization and Simplicity Biascited by 83
  9. 2023Grokking modular arithmeticcited by 83
  10. 2020Learning Parities with Neural Networkscited by 78
  11. 2022A Theoretical Analysis on Feature Learning in Neural Networks: Emergence from Inputs and Advantage over Fixed Featurescited by 49
  12. 2022Implicit Bias in Leaky ReLU Networks Trained on High-Dimensional Datacited by 46
  13. 2022Random Feature Amplification: Feature Learning and Generalization in Neural Networkscited by 35
  14. 2021On Margin Maximization in Linear and ReLU Networkscited by 34
  15. 2023Benign Overfitting in Linear Classifiers and Leaky ReLU Networks from KKT Conditions for Margin Maximizationcited by 29
  16. 2023Simplicity Bias in 1-Hidden Layer Neural Networkscited by 26
  17. 2023Pareto Frontiers in Neural Feature Learning: Data, Compute, Width, and Luckcited by 13

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Citing papers per year: 2023: 2; 2024: 9; 2025: 9; 2026: 13.

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  1. 2024Not All Language Model Features Are One-Dimensionally Linearcited by 212
  2. 2023Dichotomy of Early and Late Phase Implicit Biases Can Provably Induce Grokkingcited by 80
  3. 2026When Models Manipulate Manifolds: The Geometry of a Counting Taskcited by 46
  4. 2024Pre-trained Large Language Models Use Fourier Features to Compute Additioncited by 39
  5. 2024Why Do You Grok? A Theoretical Analysis of Grokking Modular Additioncited by 36
  6. 2026There Will Be a Scientific Theory of Deep Learningcited by 29
  7. 2024Emergence in non-neural models: grokking modular arithmetic via average gradient outer productcited by 28
  8. 2026Manifold Steering Reveals the Shared Geometry of Neural Network Representation and Behaviorcited by 26
  9. 2023Grokking Group Multiplication with Cosetscited by 24
  10. 2024Progressive distillation induces an implicit curriculumcited by 15
  11. 2025Provable Scaling Laws of Feature Emergence from Learning Dynamics of Grokkingcited by 15
  12. 2024Towards Empirical Interpretation of Internal Circuits and Properties in Grokked Transformers on Modular Polynomialscited by 14
21 more
  1. 2025Alternating Gradient Flows: A Theory of Feature Learning in Two-layer Neural Networkscited by 14
  2. 2025Let Me Grok for You: Accelerating Grokking via Embedding Transfer from a Weaker Modelcited by 13
  3. 2024Towards a unified and verified understanding of group-operation networkscited by 10
  4. 2026Sequential Group Composition: A Window into the Mechanics of Deep Learningcited by 10
  5. 2026Why Larger Models Learn More: Effects of Capacity, Interference, and Rare-Task Retentioncited by 10
  6. 2024Composing Global Solutions to Reasoning Tasks via Algebraic Objects in Neural Netscited by 6
  7. 2025In-Context Algebracited by 5
  8. 2026The Geometric Inductive Bias of Grokking: Bypassing Phase Transitions via Architectural Topologycited by 5
  9. 2025On the geometry and topology of representations: the manifolds of modular additioncited by 3
  10. 2025Learning richness modulates equality reasoning in neural networkscited by 2
  11. 2025The Features at Convergence Theorem: a first-principles alternative to the Neural Feature Ansatz for how networks learn representationscited by 2
  12. 2025Adam or Gauss-Newton? A Comparative Study In Terms of Basis Alignment and SGD Noisecited by 2
  13. 2026Gradient Flow Through Diagram Expansions: Learning Regimes and Explicit Solutionscited by 1
  14. 2026Boule or Baguette? A Study on Task Topology, Length Generalization, and the Benefit of Reasoning Tracescited by 1
  15. 2026Breaking Data Symmetry is Needed For Generalization in Feature Learning Kernelscited by 1
  16. 2026Unveiling Memorization-Generalization Coexistence: A Case Study on Arithmetic Tasks with Label Noisecited by 1
  17. 2024Do Mice Grok? Glimpses of Hidden Progress During Overtraining in Sensory Cortexcited by 0
  18. 2025Low Rank and Sparse Fourier Structure in Recurrent Networks Trained on Modular Additioncited by 0
  19. 2026Assign and Add: A Mechanistic Study of Compositional Arithmeticcited by 0
  20. 2026A Counterexample to Fourier Alignment in Single-Neuron Modular Additioncited by 0
  21. 2026Symmetry without a manifold: intrinsic dimension on orbitscited by 0