# Three Factors Influencing Minima in SGD

arXiv: https://arxiv.org/abs/1711.04623 · HTML: https://mlsanity.com/paper/1711.04623/

## Key facts

- Cited by 273 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Nov 2017.

## Built on by (most cited first, 30 of 273)

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- [What Happens after SGD Reaches Zero Loss? --A Mathematical Framework](https://mlsanity.com/paper/2110.06914/) (2021)
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- [MMPareto: Boosting Multimodal Learning with Innocent Unimodal Assistance](https://mlsanity.com/paper/2405.17730/) (2024)
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- [How Does Sharpness-Aware Minimization Minimize Sharpness?](https://mlsanity.com/paper/2211.05729/) (2022)
- [The alignment property of SGD noise and how it helps select flat minima: A stability analysis](https://mlsanity.com/paper/2207.02628/) (2022)
- [On Linear Stability of SGD and Input-Smoothness of Neural Networks](https://mlsanity.com/paper/2105.13462/) (2021)
- [On the Maximum Hessian Eigenvalue and Generalization](https://mlsanity.com/paper/2206.10654/) (2022)
- [Gradient Norm Aware Minimization Seeks First-Order Flatness and Improves Generalization](https://mlsanity.com/paper/2303.03108/) (2023)
- [Large Learning Rate Tames Homogeneity: Convergence and Balancing Effect](https://mlsanity.com/paper/2110.03677/) (2021)
- [The Implicit Regularization of Dynamical Stability in Stochastic Gradient Descent](https://mlsanity.com/paper/2305.17490/) (2023)
- [A Loss Curvature Perspective on Training Instability in Deep Learning](https://mlsanity.com/paper/2110.04369/) (2021)
- [How Much Data Are Augmentations Worth? An Investigation into Scaling Laws, Invariance, and Implicit Regularization](https://mlsanity.com/paper/2210.06441/) (2022)
- [Strength of Minibatch Noise in SGD](https://mlsanity.com/paper/2102.05375/) (2021)
- [On the Effectiveness of Lipschitz-Driven Rehearsal in Continual Learning](https://mlsanity.com/paper/2210.06443/) (2022)

## Abstract (arXiv)

We investigate the dynamical and convergent properties of stochastic gradient descent (SGD) applied to Deep Neural Networks (DNNs). Characterizing the relation between learning rate, batch size and the properties of the final minima, such as width or generalization, remains an open question. In order to tackle this problem we investigate the previously proposed approximation of SGD by a stochastic differential equation (SDE). We theoretically argue that three factors - learning rate, batch size and gradient covariance - influence the minima found by SGD. In particular we find that the ratio of learning rate to batch size is a key determinant of SGD dynamics and of the width of the final minima, and that higher values of the ratio lead to wider minima and often better generalization. We confirm these findings experimentally. Further, we include experiments which show that learning rate schedules can be replaced with batch size schedules and that the ratio of learning rate to batch size is an important factor influencing the memorization process.