# Quasi-potential as an implicit regularizer for the loss function in the stochastic gradient descent

arXiv: https://arxiv.org/abs/1901.06054 · HTML: https://mlsanity.com/paper/1901.06054/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Jan 2019.

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

- [Exponential escape efficiency of SGD from sharp minima in non-stationary regime](https://mlsanity.com/paper/2111.04004/) (2021)
- [Doubly Stochastic Models: Learning with Unbiased Label Noises and Inference Stability](https://mlsanity.com/paper/2304.00320/) (2023)
- [Convergence, Sticking and Escape: Stochastic Dynamics Near Critical Points in SGD](https://mlsanity.com/paper/2505.18535/) (2025)

## Abstract (arXiv)

We interpret the variational inference of the Stochastic Gradient Descent (SGD) as minimizing a new potential function named the textitquasi-potential. We analytically construct the quasi-potential function in the case when the loss function is convex and admits only one global minimum point. We show in this case that the quasi-potential function is related to the noise covariance structure of SGD via a partial differential equation of Hamilton-Jacobi type. This relation helps us to show that anisotropic noise leads to faster escape than isotropic noise. We then consider the dynamics of SGD in the case when the loss function is non-convex and admits several different local minima. In this case, we demonstrate an example that shows how the noise covariance structure plays a role in "implicit regularization", a phenomenon in which SGD favors some particular local minimum points. This is done through the relation between the noise covariance structure and the quasi-potential function. Our analysis is based on Large Deviations Theory (LDT), and they are validated by numerical experiments.