# Understanding Short-Range Memory Effects in Deep Neural Networks

arXiv: https://arxiv.org/abs/2105.02062 · HTML: https://mlsanity.com/paper/2105.02062/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 73 references that are in the map.
- First posted on arXiv 5 May 2021.

## Builds on (most cited first, 27 of 27)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima](https://mlsanity.com/paper/1609.04836/) (2016)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [Fantastic Generalization Measures and Where to Find Them](https://mlsanity.com/paper/1912.02178/) (2019)
- [Opening the Black Box of Deep Neural Networks via Information](https://mlsanity.com/paper/1703.00810/) (2017)
- [Sharp Minima Can Generalize For Deep Nets](https://mlsanity.com/paper/1703.04933/) (2017)
- [Entropy-SGD: Biasing Gradient Descent Into Wide Valleys](https://mlsanity.com/paper/1611.01838/) (2016)
- [Don't Decay the Learning Rate, Increase the Batch Size](https://mlsanity.com/paper/1711.00489/) (2017)
- [Stochastic Gradient Descent as Approximate Bayesian Inference](https://mlsanity.com/paper/1704.04289/) (2017)
- [Train longer, generalize better: closing the generalization gap in large batch training of neural networks](https://mlsanity.com/paper/1705.08741/) (2017)
- [Three Factors Influencing Minima in SGD](https://mlsanity.com/paper/1711.04623/) (2017)
- [A Tail-Index Analysis of Stochastic Gradient Noise in Deep Neural Networks](https://mlsanity.com/paper/1901.06053/) (2019)
- [Stochastic modified equations and adaptive stochastic gradient algorithms](https://mlsanity.com/paper/1511.06251/) (2015)
- [Stochastic gradient descent performs variational inference, converges to limit cycles for deep networks](https://mlsanity.com/paper/1710.11029/) (2017)
- [Hessian-based Analysis of Large Batch Training and Robustness to Adversaries](https://mlsanity.com/paper/1802.08241/) (2018)
- [In Search of Robust Measures of Generalization](https://mlsanity.com/paper/2010.11924/) (2020)
- [Preconditioned Stochastic Gradient Descent](https://mlsanity.com/paper/1512.04202/) (2015)
- [First Exit Time Analysis of Stochastic Gradient Descent Under Heavy-Tailed Gradient Noise](https://mlsanity.com/paper/1906.09069/) (2019)
- [diffGrad: An Optimization Method for Convolutional Neural Networks](https://mlsanity.com/paper/1909.11015/) (2019)
- [Non-Gaussianity of Stochastic Gradient Noise](https://mlsanity.com/paper/1910.09626/) (2019)
- [Accelerating Minibatch Stochastic Gradient Descent using Typicality Sampling](https://mlsanity.com/paper/1903.04192/) (2019)

## Built on by (most cited first, 1 of 1)

- [Learning Non-Vacuous Generalization Bounds from Optimization](https://mlsanity.com/paper/2206.04359/) (2022)

## Abstract (arXiv)

Stochastic gradient descent (SGD) is of fundamental importance in deep learning. Despite its simplicity, elucidating its efficacy remains challenging. Conventionally, the success of SGD is ascribed to the stochastic gradient noise (SGN) incurred in the training process. Based on this consensus, SGD is frequently treated and analyzed as the Euler-Maruyama discretization of stochastic differential equations (SDEs) driven by either Brownian or Levy stable motion. In this study, we argue that SGN is neither Gaussian nor Levy stable. Instead, inspired by the short-range correlation emerging in the SGN series, we propose that SGD can be viewed as a discretization of an SDE driven by fractional Brownian motion (FBM). Accordingly, the different convergence behavior of SGD dynamics is well-grounded. Moreover, the first passage time of an SDE driven by FBM is approximately derived. The result suggests a lower escaping rate for a larger Hurst parameter, and thus SGD stays longer in flat minima. This happens to coincide with the well-known phenomenon that SGD favors flat minima that generalize well. Extensive experiments are conducted to validate our conjecture, and it is demonstrated that short-range memory effects persist across various model architectures, datasets, and training strategies. Our study opens up a new perspective and may contribute to a better understanding of SGD.