  We study singular perturbations of a class of two-scale stochastic control
systems with unbounded data. The assumptions are designed to cover some
relaxation problems for deep neural networks. We construct effective
Hamiltonian and initial data and prove the convergence of the value function to
the solution of a limit (effective) Cauchy problem for a parabolic equation of
HJB type. We use methods of probability, viscosity solutions and
homogenization.
