# Mean Field Optimization Problem Regularized by Fisher Information

arXiv: https://arxiv.org/abs/2302.05938 · HTML: https://mlsanity.com/paper/2302.05938/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 25 references that are in the map.
- First posted on arXiv 12 Feb 2023.

## Builds on (most cited first, 8 of 8)

- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Mean-field theory of two-layers neural networks: dimension-free bounds and kernel limit](https://mlsanity.com/paper/1902.06015/) (2019)
- [Mean-Field Langevin Dynamics: Exponential Convergence and Annealing](https://mlsanity.com/paper/2202.01009/) (2022)
- [Convex Analysis of the Mean Field Langevin Dynamics](https://mlsanity.com/paper/2201.10469/) (2022)
- [Mean-Field Langevin Dynamics and Energy Landscape of Neural Networks](https://mlsanity.com/paper/1905.07769/) (2019)
- [Uniform-in-time propagation of chaos for mean field Langevin dynamics](https://mlsanity.com/paper/2212.03050/) (2022)
- [Global convergence of neuron birth-death dynamics](https://mlsanity.com/paper/1902.01843/) (2019)
- [Ergodicity of the underdamped mean-field Langevin dynamics](https://mlsanity.com/paper/2007.14660/) (2020)

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

- [Sampling from the Mean-Field Stationary Distribution](https://mlsanity.com/paper/2402.07355/) (2024)
- [Mean-field underdamped Langevin dynamics and its spacetime discretization](https://mlsanity.com/paper/2312.16360/) (2023)

## Abstract (arXiv)

Recently there is a rising interest in the research of mean field optimization, in particular because of its role in analyzing the training of neural networks. In this paper by adding the Fisher Information as the regularizer, we relate the regularized mean field optimization problem to a so-called mean field Schrodinger dynamics. We develop an energy-dissipation method to show that the marginal distributions of the mean field Schrodinger dynamics converge exponentially quickly towards the unique minimizer of the regularized optimization problem. Remarkably, the mean field Schrodinger dynamics is proved to be a gradient flow on the probability measure space with respect to the relative entropy. Finally we propose a Monte Carlo method to sample the marginal distributions of the mean field Schrodinger dynamics.