  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.

Optimization and Control