mlsanityData to 14 Oct 2026

Outliers with Opposing Signals Have an Outsized Effect on Neural Network Optimization

arXiv 2311.04163 ↗First posted 7 Nov 2023cs.LG cs.AI cs.CVcited by 13

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Abstract (arXiv)

We identify a new phenomenon in neural network optimization which arises from the interaction of depth and a particular heavy-tailed structure in natural data. Our result offers intuitive explanations for several previously reported observations about network training dynamics. In particular, it implies a conceptually new cause for progressive sharpening and the edge of stability; we also highlight connections to other concepts in optimization and generalization including grokking, simplicity bias, and Sharpness-Aware Minimization. Experimentally, we demonstrate the significant influence of paired groups of outliers in the training data with strong opposing signals: consistent, large magnitude features which dominate the network output throughout training and provide gradients which point in opposite directions. Due to these outliers, early optimization enters a narrow valley which carefully balances the opposing groups; subsequent sharpening causes their loss to rise rapidly, oscillating between high on one group and then the other, until the overall loss spikes. We describe how to identify these groups, explore what sets them apart, and carefully study their effect on the network's optimization and behavior. We complement these experiments with a mechanistic explanation on a toy example of opposing signals and a theoretical analysis of a two-layer linear network on a simple model. Our finding enables new qualitative predictions of training behavior which we confirm experimentally. It also provides a new lens through which to study and improve modern training practices for stochastic optimization, which we highlight via a case study of Adam versus SGD.

Key facts. Cited by 13 papers in the map (data through 14 Oct 2026). Builds on 54 of its 64 references that are in the map. First posted on arXiv 7 Nov 2023.

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54 of 64 references.

References · 54

  1. 2020An Image is Worth 16x16 Words: Transformers for Image Recognition at Scalecited by 30,186
  2. 2014Very Deep Convolutional Networks for Large-Scale Image Recognitioncited by 15,364
  3. 2015Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shiftcited by 7,005
  4. 2018The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networkscited by 2,262
  5. 2017Understanding Black-box Predictions via Influence Functionscited by 1,741
  6. 2020Sharpness-Aware Minimization for Efficiently Improving Generalizationcited by 1,417
  7. 2018Reconciling modern machine learning practice and the bias-variance trade-offcited by 927
  8. 2023Progress measures for grokking via mechanistic interpretabilitycited by 839
  9. 2013Exact solutions to the nonlinear dynamics of learning in deep linear neural networkscited by 763
  10. 2019Deep Double Descent: Where Bigger Models and More Data Hurtcited by 672
  11. 2022Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasetscited by 629
  12. 2023Symbolic Discovery of Optimization Algorithmscited by 434
38 more
  1. 2021Gradient Descent on Neural Networks Typically Occurs at the Edge of Stabilitycited by 427
  2. 2020Dataset Cartography: Mapping and Diagnosing Datasets with Training Dynamicscited by 425
  3. 2018How Does Batch Normalization Help Optimization?cited by 357
  4. 2020The Pitfalls of Simplicity Bias in Neural Networkscited by 351
  5. 2019An Investigation into Neural Net Optimization via Hessian Eigenvalue Densitycited by 292
  6. 2017Empirical Analysis of the Hessian of Over-Parametrized Neural Networkscited by 290
  7. 2017Three Factors Influencing Minima in SGDcited by 273
  8. 2020The large learning rate phase of deep learning: the catapult mechanismcited by 216
  9. 2019SGD on Neural Networks Learns Functions of Increasing Complexitycited by 204
  10. 2022Hidden Progress in Deep Learning: SGD Learns Parities Near the Computational Limitcited by 187
  11. 2018Deep learning generalizes because the parameter-function map is biased towards simple functionscited by 183
  12. 2018Algorithmic Regularization in Learning Deep Homogeneous Models: Layers are Automatically Balancedcited by 177
  13. 2019Towards Explaining the Regularization Effect of Initial Large Learning Rate in Training Neural Networkscited by 162
  14. 2016Eigenvalues of the Hessian in Deep Learning: Singularity and Beyondcited by 159
  15. 2022Self-Stabilization: The Implicit Bias of Gradient Descent at the Edge of Stabilitycited by 155
  16. 2021The Low-Rank Simplicity Bias in Deep Networkscited by 154
  17. 2020The Break-Even Point on Optimization Trajectories of Deep Neural Networkscited by 150
  18. 2022Understanding Gradient Descent on Edge of Stability in Deep Learningcited by 128
  19. 2023Noise Is Not the Main Factor Behind the Gap Between SGD and Adam on Transformers, but Sign Descent Might Becited by 114
  20. 2022Adaptive Gradient Methods at the Edge of Stabilitycited by 101
  21. 2020Traces of Class/Cross-Class Structure Pervade Deep Learning Spectracited by 94
  22. 2022The Slingshot Mechanism: An Empirical Study of Adaptive Optimizers and the Grokking Phenomenoncited by 94
  23. 2018On the Relation Between the Sharpest Directions of DNN Loss and the SGD Step Lengthcited by 88
  24. 2023A Tale of Two Circuits: Grokking as Competition of Sparse and Dense Subnetworkscited by 83
  25. 2018A Walk with SGDcited by 78
  26. 2022Domain-Adjusted Regression or: ERM May Already Learn Features Sufficient for Out-of-Distribution Generalizationcited by 69
  27. 2019Measurements of Three-Level Hierarchical Structure in the Outliers in the Spectrum of Deepnet Hessianscited by 68
  28. 2019Generalization Guarantees for Neural Networks via Harnessing the Low-rank Structure of the Jacobiancited by 67
  29. 2022The alignment property of SGD noise and how it helps select flat minima: A stability analysiscited by 64
  30. 2021On Linear Stability of SGD and Input-Smoothness of Neural Networkscited by 63
  31. 2022Analyzing Sharpness along GD Trajectory: Progressive Sharpening and Edge of Stabilitycited by 63
  32. 2020Catastrophic Fisher Explosion: Early Phase Fisher Matrix Impacts Generalizationcited by 62
  33. 2022Understanding Edge-of-Stability Training Dynamics with a Minimalist Examplecited by 56
  34. 2023Simfluence: Modeling the Influence of Individual Training Examples by Simulating Training Runscited by 53
  35. 2023Toward Understanding Why Adam Converges Faster Than SGD for Transformerscited by 53
  36. 2019Emergent properties of the local geometry of neural loss landscapescited by 46
  37. 2022Beyond the Quadratic Approximation: the Multiscale Structure of Neural Network Loss Landscapescited by 43
  38. 2019Hessian based analysis of SGD for Deep Nets: Dynamics and Generalizationcited by 37

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Citing papers per year: 2024: 7; 2025: 4; 2026: 2.

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  1. 2024Heavy-Tailed Class Imbalance and Why Adam Outperforms Gradient Descent on Language Modelscited by 80
  2. 2024Iteration Head: A Mechanistic Study of Chain-of-Thoughtcited by 31
  3. 2024Decomposing and Editing Predictions by Modeling Model Computationcited by 19
  4. 2024Learning Associative Memories with Gradient Descentcited by 16
  5. 2025Hidden Breakthroughs in Language Model Trainingcited by 15
  6. 2024Where Do Large Learning Rates Lead Us?cited by 6
  7. 2025Understanding Sharpness Dynamics in NN Training with a Minimalist Example: The Effects of Dataset Difficulty, Depth, Stochasticity, and Morecited by 5
  8. 2025A Minimalist Example of Edge-of-Stability and Progressive Sharpeningcited by 4
  9. 2024Reevaluating Theoretical Analysis Methods for Optimization in Deep Learningcited by 3
  10. 2026Too Sharp, Too Sure: When Calibration Follows Curvaturecited by 3
  11. 2025Large Learning Rates Simultaneously Achieve Robustness to Spurious Correlations and Compressibilitycited by 2
  12. 2024Training on the Edge of Stability Is Caused by Layerwise Jacobian Alignmentcited by 1
1 more
  1. 2026Edge of Stability Selectively Shapes Learning Across the Data Distributioncited by 0