# From Spikes to Heavy Tails: Unveiling the Spectral Evolution of Neural Networks

arXiv: https://arxiv.org/abs/2406.04657 · HTML: https://mlsanity.com/paper/2406.04657/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 43 of its 70 references that are in the map.
- First posted on arXiv 7 Jun 2024.

## Builds on (most cited first, 30 of 43)

- [Deep Neural Networks as Gaussian Processes](https://mlsanity.com/paper/1711.00165/) (2017)
- [signSGD: Compressed Optimisation for Non-Convex Problems](https://mlsanity.com/paper/1802.04434/) (2018)
- [The generalization error of random features regression: Precise asymptotics and double descent curve](https://mlsanity.com/paper/1908.05355/) (2019)
- [Gaussian Process Behaviour in Wide Deep Neural Networks](https://mlsanity.com/paper/1804.11271/) (2018)
- [Implicit Self-Regularization in Deep Neural Networks: Evidence from Random Matrix Theory and Implications for Learning](https://mlsanity.com/paper/1810.01075/) (2018)
- [A Tail-Index Analysis of Stochastic Gradient Noise in Deep Neural Networks](https://mlsanity.com/paper/1901.06053/) (2019)
- [Neural Networks can Learn Representations with Gradient Descent](https://mlsanity.com/paper/2206.15144/) (2022)
- [High-dimensional Asymptotics of Feature Learning: How One Gradient Step Improves the Representation](https://mlsanity.com/paper/2205.01445/) (2022)
- [SGD learning on neural networks: leap complexity and saddle-to-saddle dynamics](https://mlsanity.com/paper/2302.11055/) (2023)
- [Generalisation error in learning with random features and the hidden manifold model](https://mlsanity.com/paper/2002.09339/) (2020)
- [The Heavy-Tail Phenomenon in SGD](https://mlsanity.com/paper/2006.04740/) (2020)
- [Online stochastic gradient descent on non-convex losses from high-dimensional inference](https://mlsanity.com/paper/2003.10409/) (2020)
- [Predicting trends in the quality of state-of-the-art neural networks without access to training or testing data](https://mlsanity.com/paper/2002.06716/) (2020)
- [Noise Is Not the Main Factor Behind the Gap Between SGD and Adam on Transformers, but Sign Descent Might Be](https://mlsanity.com/paper/2304.13960/) (2023)
- [The Gaussian equivalence of generative models for learning with shallow neural networks](https://mlsanity.com/paper/2006.14709/) (2020)
- [Random Features for Kernel Approximation: A Survey on Algorithms, Theory, and Beyond](https://mlsanity.com/paper/2004.11154/) (2020)
- [Learning Single-Index Models with Shallow Neural Networks](https://mlsanity.com/paper/2210.15651/) (2022)
- [Normalization Techniques in Training DNNs: Methodology, Analysis and Application](https://mlsanity.com/paper/2009.12836/) (2020)
- [A Random Matrix Approach to Neural Networks](https://mlsanity.com/paper/1702.05419/) (2017)
- [Spectra of the Conjugate Kernel and Neural Tangent Kernel for linear-width neural networks](https://mlsanity.com/paper/2005.11879/) (2020)
- [The Step Decay Schedule: A Near Optimal, Geometrically Decaying Learning Rate Procedure For Least Squares](https://mlsanity.com/paper/1904.12838/) (2019)
- [How Two-Layer Neural Networks Learn, One (Giant) Step at a Time](https://mlsanity.com/paper/2305.18270/) (2023)
- [Neural Networks Efficiently Learn Low-Dimensional Representations with SGD](https://mlsanity.com/paper/2209.14863/) (2022)
- [Hausdorff Dimension, Heavy Tails, and Generalization in Neural Networks](https://mlsanity.com/paper/2006.09313/) (2020)
- [Smoothing the Landscape Boosts the Signal for SGD: Optimal Sample Complexity for Learning Single Index Models](https://mlsanity.com/paper/2305.10633/) (2023)
- [The Benefits of Reusing Batches for Gradient Descent in Two-Layer Networks: Breaking the Curse of Information and Leap Exponents](https://mlsanity.com/paper/2402.03220/) (2024)
- [A Theory of Non-Linear Feature Learning with One Gradient Step in Two-Layer Neural Networks](https://mlsanity.com/paper/2310.07891/) (2023)
- [Neural network learns low-dimensional polynomials with SGD near the information-theoretic limit](https://mlsanity.com/paper/2406.01581/) (2024)
- [Heavy-Tailed Universality Predicts Trends in Test Accuracies for Very Large Pre-Trained Deep Neural Networks](https://mlsanity.com/paper/1901.08278/) (2019)
- [First Exit Time Analysis of Stochastic Gradient Descent Under Heavy-Tailed Gradient Noise](https://mlsanity.com/paper/1906.09069/) (2019)

## Built on by (most cited first, 9 of 9)

- [AlphaPruning: Using Heavy-Tailed Self Regularization Theory for Improved Layer-wise Pruning of Large Language Models](https://mlsanity.com/paper/2410.10912/) (2024)
- [AlphaLoRA: Assigning LoRA Experts Based on Layer Training Quality](https://mlsanity.com/paper/2410.10054/) (2024)
- [HTMuon: Improving Muon via Heavy-Tailed Spectral Correction](https://mlsanity.com/paper/2603.10067/) (2026)
- [Model Balancing Helps Low-data Training and Fine-tuning](https://mlsanity.com/paper/2410.12178/) (2024)
- [Eigenspectrum Analysis of Neural Networks without Aspect Ratio Bias](https://mlsanity.com/paper/2506.06280/) (2025)
- [One LR Doesn't Fit All: Heavy-Tail Guided Layerwise Learning Rates for LLMs](https://mlsanity.com/paper/2605.22297/) (2026)
- [Diffract: Spectral View of LLM Domain Adaptation](https://mlsanity.com/paper/2608.10850/) (2026)
- [Dimension-Corrected Hitting Times for Heavy-Tailed Spectral Emergence in Neural Optimizer Dynamics](https://mlsanity.com/paper/2609.12994/) (2026)
- [A Full Adam Theorem for Spectral Heavy-Tail Onset](https://mlsanity.com/paper/2609.12996/) (2026)

## Abstract (arXiv)

Training strategies for modern deep neural networks (NNs) tend to induce a heavy-tailed (HT) empirical spectral density (ESD) in the layer weights. While previous efforts have shown that the HT phenomenon correlates with good generalization in large NNs, a theoretical explanation of its occurrence is still lacking. Especially, understanding the conditions which lead to this phenomenon can shed light on the interplay between generalization and weight spectra. Our work aims to bridge this gap by presenting a simple, rich setting to model the emergence of HT ESD. In particular, we present a theory-informed setup for 'crafting' heavy tails in the ESD of two-layer NNs and present a systematic analysis of the HT ESD emergence without any gradient noise. This is the first work to analyze a noise-free setting, and we also incorporate optimizer (GD/Adam) dependent (large) learning rates into the HT ESD analysis. Our results highlight the role of learning rates on the Bulk+Spike and HT shape of the ESDs in the early phase of training, which can facilitate generalization in the two-layer NN. These observations shed light on the behavior of large-scale NNs, albeit in a much simpler setting.