  We consider a singularly perturbed system of stochastic differential
equations proposed by Chaudhari et al. (Res. Math. Sci. 2018) to approximate
the Entropic Gradient Descent in the optimization of deep neural networks, via
homogenisation. We embed it in a much larger class of two-scale stochastic
control problems and rely on convergence results for Hamilton-Jacobi-Bellman
equations with unbounded data proved recently by ourselves (ESAIM Control
Optim. Calc. Var. 2023). We show that the limit of the value functions is
itself the value function of an effective control problem with extended
controls, and that the trajectories of the perturbed system converge in a
suitable sense to the trajectories of the limiting effective control system.
These rigorous results improve the understanding of the convergence of the
algorithms used by Chaudhari et al., as well as of their possible extensions
where some tuning parameters are modelled as dynamic controls.
