# On the Noisy Gradient Descent that Generalizes as SGD

arXiv: https://arxiv.org/abs/1906.07405 · HTML: https://mlsanity.com/paper/1906.07405/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Jun 2019.

## Built on by (most cited first, 4 of 4)

- [Revisiting the Noise Model of Stochastic Gradient Descent](https://mlsanity.com/paper/2303.02749/) (2023)
- [Revisiting the Characteristics of Stochastic Gradient Noise and Dynamics](https://mlsanity.com/paper/2109.09833/) (2021)
- [Shift-Curvature, SGD, and Generalization](https://mlsanity.com/paper/2108.09507/) (2021)
- [Disentangling the Mechanisms Behind Implicit Regularization in SGD](https://mlsanity.com/paper/2211.15853/) (2022)

## Abstract (arXiv)

The gradient noise of SGD is considered to play a central role in the observed strong generalization abilities of deep learning. While past studies confirm that the magnitude and the covariance structure of gradient noise are critical for regularization, it remains unclear whether or not the class of noise distributions is important. In this work we provide negative results by showing that noises in classes different from the SGD noise can also effectively regularize gradient descent. Our finding is based on a novel observation on the structure of the SGD noise: it is the multiplication of the gradient matrix and a sampling noise that arises from the mini-batch sampling procedure. Moreover, the sampling noises unify two kinds of gradient regularizing noises that belong to the Gaussian class: the one using (scaled) Fisher as covariance and the one using the gradient covariance of SGD as covariance. Finally, thanks to the flexibility of choosing noise class, an algorithm is proposed to perform noisy gradient descent that generalizes well, the variant of which even benefits large batch SGD training without hurting generalization.