# On the diffusion approximation of nonconvex stochastic gradient descent

arXiv: https://arxiv.org/abs/1705.07562 · HTML: https://mlsanity.com/paper/1705.07562/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 May 2017.

## Built on by (most cited first, 5 of 5)

- [Why (and When) does Local SGD Generalize Better than SGD?](https://mlsanity.com/paper/2303.01215/) (2023)
- [Weak Convergence of Approximate reflection coupling and its Application to Non-convex Optimization](https://mlsanity.com/paper/2205.11970/) (2022)
- [Uniform Generalization Bound on Time and Inverse Temperature for Gradient Descent Algorithm and its Application to Analysis of Simulated Annealing](https://mlsanity.com/paper/2205.12959/) (2022)
- [Adam-like Algorithm with Smooth Clipping Attains Global Minima: Analysis Based on Ergodicity of Functional SDEs](https://mlsanity.com/paper/2312.02182/) (2023)
- [A General Continuous-Time Formulation of Stochastic ADMM and Its Variants](https://mlsanity.com/paper/2404.14358/) (2024)

## Abstract (arXiv)

We study the Stochastic Gradient Descent (SGD) method in nonconvex optimization problems from the point of view of approximating diffusion processes. We prove rigorously that the diffusion process can approximate the SGD algorithm weakly using the weak form of master equation for probability evolution. In the small step size regime and the presence of omnidirectional noise, our weak approximating diffusion process suggests the following dynamics for the SGD iteration starting from a local minimizer (resp.~saddle point): it escapes in a number of iterations exponentially (resp.~almost linearly) dependent on the inverse stepsize. The results are obtained using the theory for random perturbations of dynamical systems (theory of large deviations for local minimizers and theory of exiting for unstable stationary points). In addition, we discuss the effects of batch size for the deep neural networks, and we find that small batch size is helpful for SGD algorithms to escape unstable stationary points and sharp minimizers. Our theory indicates that one should increase the batch size at later stage for the SGD to be trapped in flat minimizers for better generalization.