# Directional Analysis of Stochastic Gradient Descent via von Mises-Fisher Distributions in Deep learning

arXiv: https://arxiv.org/abs/1810.00150 · HTML: https://mlsanity.com/paper/1810.00150/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Sep 2018.

## Built on by (most cited first, 3 of 3)

- [HCFL: A High Compression Approach for Communication-Efficient Federated Learning in Very Large Scale IoT Networks](https://mlsanity.com/paper/2204.06760/) (2022)
- [AdaTerm: Adaptive T-Distribution Estimated Robust Moments for Noise-Robust Stochastic Gradient Optimization](https://mlsanity.com/paper/2201.06714/) (2022)
- [No Shifted Augmentations (NSA): compact distributions for robust self-supervised Anomaly Detection](https://mlsanity.com/paper/2203.10344/) (2022)

## Abstract (arXiv)

Although stochastic gradient descent (SGD) is a driving force behind the recent success of deep learning, our understanding of its dynamics in a high-dimensional parameter space is limited. In recent years, some researchers have used the stochasticity of minibatch gradients, or the signal-to-noise ratio, to better characterize the learning dynamics of SGD. Inspired from these work, we here analyze SGD from a geometrical perspective by inspecting the stochasticity of the norms and directions of minibatch gradients. We propose a model of the directional concentration for minibatch gradients through von Mises-Fisher (VMF) distribution, and show that the directional uniformity of minibatch gradients increases over the course of SGD. We empirically verify our result using deep convolutional networks and observe a higher correlation between the gradient stochasticity and the proposed directional uniformity than that against the gradient norm stochasticity, suggesting that the directional statistics of minibatch gradients is a major factor behind SGD.