# Analysing heavy-tail properties of Stochastic Gradient Descent by means of Stochastic Recurrence Equations

arXiv: https://arxiv.org/abs/2403.13868 · HTML: https://mlsanity.com/paper/2403.13868/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 25 references that are in the map.
- First posted on arXiv 20 Mar 2024.

## Builds on (most cited first, 5 of 5)

- [Cyclical Learning Rates for Training Neural Networks](https://mlsanity.com/paper/1506.01186/) (2015)
- [A disciplined approach to neural network hyper-parameters: Part 1 -- learning rate, batch size, momentum, and weight decay](https://mlsanity.com/paper/1803.09820/) (2018)
- [The Heavy-Tail Phenomenon in SGD](https://mlsanity.com/paper/2006.04740/) (2020)
- [A Variational Analysis of Stochastic Gradient Algorithms](https://mlsanity.com/paper/1602.02666/) (2016)
- [Cyclic and Randomized Stepsizes Invoke Heavier Tails in SGD than Constant Stepsize](https://mlsanity.com/paper/2302.05516/) (2023)

## Built on by (most cited first, 2 of 2)

- [Statistical Inference for Stochastic Gradient Descent: Beyond Finite Variance](https://mlsanity.com/paper/2605.26000/) (2026)
- [Gaussian Limits for SGD Without Stationary Moments](https://mlsanity.com/paper/2610.05599/) (2026)

## Abstract (arXiv)

In recent works on the theory of machine learning, it has been observed that heavy tail properties of Stochastic Gradient Descent (SGD) can be studied in the probabilistic framework of stochastic recursions. In particular, G"urb"uzbalaban et al. (arXiv:2006.04740) considered a setup corresponding to linear regression for which iterations of SGD can be modelled by a multivariate affine stochastic recursion Xk=Ak Xk-1+Bk, for independent and identically distributed pairs (Ak, Bk), where Ak is a random symmetric matrix and Bk is a random vector. In this work, we will answer several open questions of the quoted paper and extend their results by applying the theory of irreducible-proximal (i-p) matrices.