  Training generative adversarial networks (GANs) is known to be difficult,
especially for financial time series. This paper first analyzes the
well-posedness problem in GANs minimax games and the convexity issue in GANs
objective functions. It then proposes a stochastic control framework for
hyper-parameters tuning in GANs training. The weak form of dynamic programming
principle and the uniqueness and the existence of the value function in the
viscosity sense for the corresponding minimax game are established. In
particular, explicit forms for the optimal adaptive learning rate and batch
size are derived and are shown to depend on the convexity of the objective
function, revealing a relation between improper choices of learning rate and
explosion in GANs training. Finally, empirical studies demonstrate that
training algorithms incorporating this adaptive control approach outperform the
standard ADAM method in terms of convergence and robustness.
  From GANs training perspective, the analysis in this paper provides
analytical support for the popular practice of ``clipping'', and suggests that
the convexity and well-posedness issues in GANs may be tackled through
appropriate choices of hyper-parameters.
