# Large Sample Mean-Field Stochastic Optimization

## TL;DR

This paper analyzes mean-field stochastic optimization problems with finite samples, proving convergence of solutions to a limiting problem characterized by a nonlinear Fokker-Planck-Kolmogorov equation, and relates it to neural network training.

## Contribution

It establishes the existence of optimal relaxed controls for finite samples and proves their convergence to the mean-field limit using $ extGamma$-convergence, linking to neural network training.

## Key findings

- Convergence of finite sample solutions to the mean-field limit.
- Connection between optimization minimizers and neural network weights.
- Existence of optimal relaxed controls in finite sample scenarios.

## Abstract

We study a class of sampled stochastic optimization problems, where the underlying state process has diffusive dynamics of the mean-field type. We establish the existence of optimal relaxed controls when the sample set has finite size. The core of our paper is to prove, via $\Gamma$-convergence, that the minimizer of the finite sample relaxed problem converges to that of the limiting optimization problem. We connect the limit of the sampled objective functional to the unique solution, in the trajectory sense, of a nonlinear Fokker-Planck-Kolmogorov (FPK) equation in a random environment. We highlight the connection between the minimizers of our optimization problems and the optimal training weights of a deep residual neural network.