# Feature emergence via margin maximization: case studies in algebraic tasks

arXiv: https://arxiv.org/abs/2311.07568 · HTML: https://mlsanity.com/paper/2311.07568/

## 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 (most cited first, 29 of 29)

- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Solving Quantitative Reasoning Problems with Language Models](https://mlsanity.com/paper/2206.14858/) (2022)
- [Toy Models of Superposition](https://mlsanity.com/paper/2209.10652/) (2022)
- [Progress measures for grokking via mechanistic interpretability](https://mlsanity.com/paper/2301.05217/) (2023)
- [The Implicit Bias of Gradient Descent on Separable Data](https://mlsanity.com/paper/1710.10345/) (2017)
- [Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets](https://mlsanity.com/paper/2201.02177/) (2022)
- [Analysing Mathematical Reasoning Abilities of Neural Models](https://mlsanity.com/paper/1904.01557/) (2019)
- [Gradient Descent Maximizes the Margin of Homogeneous Neural Networks](https://mlsanity.com/paper/1906.05890/) (2019)
- [Implicit Bias of Gradient Descent for Wide Two-layer Neural Networks Trained with the Logistic Loss](https://mlsanity.com/paper/2002.04486/) (2020)
- [Implicit Bias of Gradient Descent on Linear Convolutional Networks](https://mlsanity.com/paper/1806.00468/) (2018)
- [Towards Understanding Grokking: An Effective Theory of Representation Learning](https://mlsanity.com/paper/2205.10343/) (2022)
- [Hidden Progress in Deep Learning: SGD Learns Parities Near the Computational Limit](https://mlsanity.com/paper/2207.08799/) (2022)
- [Directional convergence and alignment in deep learning](https://mlsanity.com/paper/2006.06657/) (2020)
- [The Clock and the Pizza: Two Stories in Mechanistic Explanation of Neural Networks](https://mlsanity.com/paper/2306.17844/) (2023)
- [Omnigrok: Grokking Beyond Algorithmic Data](https://mlsanity.com/paper/2210.01117/) (2022)
- [A Toy Model of Universality: Reverse Engineering How Networks Learn Group Operations](https://mlsanity.com/paper/2302.03025/) (2023)
- [Regularization Matters: Generalization and Optimization of Neural Nets v.s. their Induced Kernel](https://mlsanity.com/paper/1810.05369/) (2018)
- [On the Implicit Bias in Deep-Learning Algorithms](https://mlsanity.com/paper/2208.12591/) (2022)
- [Gradient Descent on Two-layer Nets: Margin Maximization and Simplicity Bias](https://mlsanity.com/paper/2110.13905/) (2021)
- [Grokking modular arithmetic](https://mlsanity.com/paper/2301.02679/) (2023)
- [Learning Parities with Neural Networks](https://mlsanity.com/paper/2002.07400/) (2020)
- [A Theoretical Analysis on Feature Learning in Neural Networks: Emergence from Inputs and Advantage over Fixed Features](https://mlsanity.com/paper/2206.01717/) (2022)
- [Implicit Bias in Leaky ReLU Networks Trained on High-Dimensional Data](https://mlsanity.com/paper/2210.07082/) (2022)
- [Random Feature Amplification: Feature Learning and Generalization in Neural Networks](https://mlsanity.com/paper/2202.07626/) (2022)
- [On Margin Maximization in Linear and ReLU Networks](https://mlsanity.com/paper/2110.02732/) (2021)
- [Benign Overfitting in Linear Classifiers and Leaky ReLU Networks from KKT Conditions for Margin Maximization](https://mlsanity.com/paper/2303.01462/) (2023)
- [Simplicity Bias in 1-Hidden Layer Neural Networks](https://mlsanity.com/paper/2302.00457/) (2023)
- [Pareto Frontiers in Neural Feature Learning: Data, Compute, Width, and Luck](https://mlsanity.com/paper/2309.03800/) (2023)

## Built on by (most cited first, 30 of 33)

- [Not All Language Model Features Are One-Dimensionally Linear](https://mlsanity.com/paper/2405.14860/) (2024)
- [Dichotomy of Early and Late Phase Implicit Biases Can Provably Induce Grokking](https://mlsanity.com/paper/2311.18817/) (2023)
- [When Models Manipulate Manifolds: The Geometry of a Counting Task](https://mlsanity.com/paper/2601.04480/) (2026)
- [Pre-trained Large Language Models Use Fourier Features to Compute Addition](https://mlsanity.com/paper/2406.03445/) (2024)
- [Why Do You Grok? A Theoretical Analysis of Grokking Modular Addition](https://mlsanity.com/paper/2407.12332/) (2024)
- [There Will Be a Scientific Theory of Deep Learning](https://mlsanity.com/paper/2604.21691/) (2026)
- [Emergence in non-neural models: grokking modular arithmetic via average gradient outer product](https://mlsanity.com/paper/2407.20199/) (2024)
- [Manifold Steering Reveals the Shared Geometry of Neural Network Representation and Behavior](https://mlsanity.com/paper/2605.05115/) (2026)
- [Grokking Group Multiplication with Cosets](https://mlsanity.com/paper/2312.06581/) (2023)
- [Progressive distillation induces an implicit curriculum](https://mlsanity.com/paper/2410.05464/) (2024)
- [Provable Scaling Laws of Feature Emergence from Learning Dynamics of Grokking](https://mlsanity.com/paper/2509.21519/) (2025)
- [Towards Empirical Interpretation of Internal Circuits and Properties in Grokked Transformers on Modular Polynomials](https://mlsanity.com/paper/2402.16726/) (2024)
- [Alternating Gradient Flows: A Theory of Feature Learning in Two-layer Neural Networks](https://mlsanity.com/paper/2506.06489/) (2025)
- [Let Me Grok for You: Accelerating Grokking via Embedding Transfer from a Weaker Model](https://mlsanity.com/paper/2504.13292/) (2025)
- [Towards a unified and verified understanding of group-operation networks](https://mlsanity.com/paper/2410.07476/) (2024)
- [Sequential Group Composition: A Window into the Mechanics of Deep Learning](https://mlsanity.com/paper/2602.03655/) (2026)
- [Why Larger Models Learn More: Effects of Capacity, Interference, and Rare-Task Retention](https://mlsanity.com/paper/2605.29548/) (2026)
- [Composing Global Solutions to Reasoning Tasks via Algebraic Objects in Neural Nets](https://mlsanity.com/paper/2410.01779/) (2024)
- [In-Context Algebra](https://mlsanity.com/paper/2512.16902/) (2025)
- [The Geometric Inductive Bias of Grokking: Bypassing Phase Transitions via Architectural Topology](https://mlsanity.com/paper/2603.05228/) (2026)
- [On the geometry and topology of representations: the manifolds of modular addition](https://mlsanity.com/paper/2512.25060/) (2025)
- [Learning richness modulates equality reasoning in neural networks](https://mlsanity.com/paper/2503.09781/) (2025)
- [The Features at Convergence Theorem: a first-principles alternative to the Neural Feature Ansatz for how networks learn representations](https://mlsanity.com/paper/2507.05644/) (2025)
- [Adam or Gauss-Newton? A Comparative Study In Terms of Basis Alignment and SGD Noise](https://mlsanity.com/paper/2510.13680/) (2025)
- [Gradient Flow Through Diagram Expansions: Learning Regimes and Explicit Solutions](https://mlsanity.com/paper/2602.04548/) (2026)
- [Boule or Baguette? A Study on Task Topology, Length Generalization, and the Benefit of Reasoning Traces](https://mlsanity.com/paper/2602.14404/) (2026)
- [Breaking Data Symmetry is Needed For Generalization in Feature Learning Kernels](https://mlsanity.com/paper/2604.00316/) (2026)
- [Unveiling Memorization-Generalization Coexistence: A Case Study on Arithmetic Tasks with Label Noise](https://mlsanity.com/paper/2605.18022/) (2026)
- [Do Mice Grok? Glimpses of Hidden Progress During Overtraining in Sensory Cortex](https://mlsanity.com/paper/2411.03541/) (2024)
- [Low Rank and Sparse Fourier Structure in Recurrent Networks Trained on Modular Addition](https://mlsanity.com/paper/2503.22059/) (2025)

## 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.