# Mean-field Langevin System, Optimal Control and Deep Neural Networks

arXiv: https://arxiv.org/abs/1909.07278 · HTML: https://mlsanity.com/paper/1909.07278/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Sep 2019.

## Built on by (most cited first, 4 of 4)

- [Deep Generalized Schr"odinger Bridge](https://mlsanity.com/paper/2209.09893/) (2022)
- [Convergence of Policy Gradient for Entropy Regularized MDPs with Neural Network Approximation in the Mean-Field Regime](https://mlsanity.com/paper/2201.07296/) (2022)
- [Second-Order Neural ODE Optimizer](https://mlsanity.com/paper/2109.14158/) (2021)
- [Dynamic Game Theoretic Neural Optimizer](https://mlsanity.com/paper/2105.03788/) (2021)

## Abstract (arXiv)

In this paper, we study a regularised relaxed optimal control problem and, in particular, we are concerned with the case where the control variable is of large dimension. We introduce a system of mean-field Langevin equations, the invariant measure of which is shown to be the optimal control of the initial problem under mild conditions. Therefore, this system of processes can be viewed as a continuous-time numerical algorithm for computing the optimal control. As an application, this result endorses the solvability of the stochastic gradient descent algorithm for a wide class of deep neural networks.