# The Impact of Local Geometry and Batch Size on Stochastic Gradient Descent for Nonconvex Problems

arXiv: https://arxiv.org/abs/1709.04718 · HTML: https://mlsanity.com/paper/1709.04718/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Sep 2017.

## Built on by (most cited first, 2 of 2)

- [Stochastic gradient descent with noise of machine learning type. Part II: Continuous time analysis](https://mlsanity.com/paper/2106.02588/) (2021)
- [Stochastic Ratios Tracking Algorithm for Large Scale Machine Learning Problems](https://mlsanity.com/paper/2305.09978/) (2023)

## Abstract (arXiv)

In several experimental reports on nonconvex optimization problems in machine learning, stochastic gradient descent (SGD) was observed to prefer minimizers with flat basins in comparison to more deterministic methods, yet there is very little rigorous understanding of this phenomenon. In fact, the lack of such work has led to an unverified, but widely-accepted stochastic mechanism describing why SGD prefers flatter minimizers to sharper minimizers. However, as we demonstrate, the stochastic mechanism fails to explain this phenomenon. Here, we propose an alternative deterministic mechanism that can accurately explain why SGD prefers flatter minimizers to sharper minimizers. We derive this mechanism based on a detailed analysis of a generic stochastic quadratic problem, which generalizes known results for classical gradient descent. Finally, we verify the predictions of our deterministic mechanism on two nonconvex problems.