# The Anisotropic Noise in Stochastic Gradient Descent: Its Behavior of Escaping from Sharp Minima and Regularization Effects

arXiv: https://arxiv.org/abs/1803.00195 · HTML: https://mlsanity.com/paper/1803.00195/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Mar 2018.

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

- [Homogenization of SGD in high-dimensions: Exact dynamics and generalization properties](https://mlsanity.com/paper/2205.07069/) (2022)
- [Why (and When) does Local SGD Generalize Better than SGD?](https://mlsanity.com/paper/2303.01215/) (2023)
- [Clip Body and Tail Separately: High Probability Guarantees for DPSGD with Heavy Tails](https://mlsanity.com/paper/2405.17529/) (2024)
- [Vanishing Curvature and the Power of Adaptive Methods in Randomly Initialized Deep Networks](https://mlsanity.com/paper/2106.03763/) (2021)
- [Enhance Diffusion to Improve Robust Generalization](https://mlsanity.com/paper/2306.02618/) (2023)
- [Shift-Curvature, SGD, and Generalization](https://mlsanity.com/paper/2108.09507/) (2021)
- [Combining resampling and reweighting for faithful stochastic optimization](https://mlsanity.com/paper/2105.14694/) (2021)
- [Why SGD is not Brownian Motion: A New Perspective on Stochastic Dynamics](https://mlsanity.com/paper/2605.22644/) (2026)
- [Stochastic Gradient Optimization with Model-Assisted Sampling](https://mlsanity.com/paper/2606.27171/) (2026)

## Abstract (arXiv)

Understanding the behavior of stochastic gradient descent (SGD) in the context of deep neural networks has raised lots of concerns recently. Along this line, we study a general form of gradient based optimization dynamics with unbiased noise, which unifies SGD and standard Langevin dynamics. Through investigating this general optimization dynamics, we analyze the behavior of SGD on escaping from minima and its regularization effects. A novel indicator is derived to characterize the efficiency of escaping from minima through measuring the alignment of noise covariance and the curvature of loss function. Based on this indicator, two conditions are established to show which type of noise structure is superior to isotropic noise in term of escaping efficiency. We further show that the anisotropic noise in SGD satisfies the two conditions, and thus helps to escape from sharp and poor minima effectively, towards more stable and flat minima that typically generalize well. We systematically design various experiments to verify the benefits of the anisotropic noise, compared with full gradient descent plus isotropic diffusion (i.e. Langevin dynamics).