# Hausdorff Dimension, Heavy Tails, and Generalization in Neural Networks

arXiv: https://arxiv.org/abs/2006.09313 · HTML: https://mlsanity.com/paper/2006.09313/

## Key facts

- Cited by 62 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Jun 2020.

## Built on by (most cited first, 30 of 62)

- [Intrinsic Dimension, Persistent Homology and Generalization in Neural Networks](https://mlsanity.com/paper/2111.13171/) (2021)
- [Neural Networks Efficiently Learn Low-Dimensional Representations with SGD](https://mlsanity.com/paper/2209.14863/) (2022)
- [Heavy Tails in SGD and Compressibility of Overparametrized Neural Networks](https://mlsanity.com/paper/2106.03795/) (2021)
- [Strength of Minibatch Noise in SGD](https://mlsanity.com/paper/2102.05375/) (2021)
- [A Fully Spiking Hybrid Neural Network for Energy-Efficient Object Detection](https://mlsanity.com/paper/2104.10719/) (2021)
- [Fractal Structure and Generalization Properties of Stochastic Optimization Algorithms](https://mlsanity.com/paper/2106.04881/) (2021)
- [Rate-Distortion Theoretic Generalization Bounds for Stochastic Learning Algorithms](https://mlsanity.com/paper/2203.02474/) (2022)
- [Temperature Balancing, Layer-wise Weight Analysis, and Neural Network Training](https://mlsanity.com/paper/2312.00359/) (2023)
- [Algorithmic Stability of Heavy-Tailed SGD with General Loss Functions](https://mlsanity.com/paper/2301.11885/) (2023)
- [Generalization Bounds with Data-dependent Fractal Dimensions](https://mlsanity.com/paper/2302.02766/) (2023)
- [Metric Space Magnitude and Generalisation in Neural Networks](https://mlsanity.com/paper/2305.05611/) (2023)
- [Rate-Distortion Theoretic Bounds on Generalization Error for Distributed Learning](https://mlsanity.com/paper/2206.02604/) (2022)
- [On the Sample Complexity and Metastability of Heavy-tailed Policy Search in Continuous Control](https://mlsanity.com/paper/2106.08414/) (2021)
- [Topology and geometry of data manifold in deep learning](https://mlsanity.com/paper/2204.08624/) (2022)
- [Algorithmic Stability of Heavy-Tailed Stochastic Gradient Descent on Least Squares](https://mlsanity.com/paper/2206.01274/) (2022)
- [HTMuon: Improving Muon via Heavy-Tailed Spectral Correction](https://mlsanity.com/paper/2603.10067/) (2026)
- [On the Hidden Biases of Policy Mirror Ascent in Continuous Action Spaces](https://mlsanity.com/paper/2201.12332/) (2022)
- [Generalization Bounds for Stochastic Gradient Descent via Localized varepsilon-Covers](https://mlsanity.com/paper/2209.08951/) (2022)
- [Two Facets of SDE Under an Information-Theoretic Lens: Generalization of SGD via Training Trajectories and via Terminal States](https://mlsanity.com/paper/2211.10691/) (2022)
- [Minimum Description Length and Generalization Guarantees for Representation Learning](https://mlsanity.com/paper/2402.03254/) (2024)
- [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)
- [Predicting the generalization gap in neural networks using topological data analysis](https://mlsanity.com/paper/2203.12330/) (2022)
- [Chaotic Regularization and Heavy-Tailed Limits for Deterministic Gradient Descent](https://mlsanity.com/paper/2205.11361/) (2022)
- [Data-dependent Generalization Bounds via Variable-Size Compressibility](https://mlsanity.com/paper/2303.05369/) (2023)
- [Uniform Generalization Bounds on Data-Dependent Hypothesis Sets via PAC-Bayesian Theory on Random Sets](https://mlsanity.com/paper/2404.17442/) (2024)
- [From Spikes to Heavy Tails: Unveiling the Spectral Evolution of Neural Networks](https://mlsanity.com/paper/2406.04657/) (2024)
- [Characterization of Generalizability of Spike Timing Dependent Plasticity trained Spiking Neural Networks](https://mlsanity.com/paper/2105.14677/) (2021)
- [Heterogeneous Neuronal and Synaptic Dynamics for Spike-Efficient Unsupervised Learning: Theory and Design Principles](https://mlsanity.com/paper/2302.11618/) (2023)
- [Heavy-Tailed Regularization of Weight Matrices in Deep Neural Networks](https://mlsanity.com/paper/2304.02911/) (2023)

## Abstract (arXiv)

Despite its success in a wide range of applications, characterizing the generalization properties of stochastic gradient descent (SGD) in non-convex deep learning problems is still an important challenge. While modeling the trajectories of SGD via stochastic differential equations (SDE) under heavy-tailed gradient noise has recently shed light over several peculiar characteristics of SGD, a rigorous treatment of the generalization properties of such SDEs in a learning theoretical framework is still missing. Aiming to bridge this gap, in this paper, we prove generalization bounds for SGD under the assumption that its trajectories can be well-approximated by a Feller process, which defines a rich class of Markov processes that include several recent SDE representations (both Brownian or heavy-tailed) as its special case. We show that the generalization error can be controlled by the Hausdorff dimension of the trajectories, which is intimately linked to the tail behavior of the driving process. Our results imply that heavier-tailed processes should achieve better generalization; hence, the tail-index of the process can be used as a notion of "capacity metric". We support our theory with experiments on deep neural networks illustrating that the proposed capacity metric accurately estimates the generalization error, and it does not necessarily grow with the number of parameters unlike the existing capacity metrics in the literature.