  We consider stochastic optimization problems involving an expected value of a
nonlinear function of a base random vector and a conditional expectation of
another function depending on the base random vector, a dependent random
vector, and the decision variables. We call such problems conditional
stochastic optimization problems. They arise in many applications, such as
uplift modeling, reinforcement learning, and contextual optimization. We
propose a specialized single time-scale stochastic method for nonconvex
constrained conditional stochastic optimization problems with a Lipschitz
smooth outer function and a generalized differentiable inner function. In the
method, we approximate the inner conditional expectation with a rich parametric
model whose mean squared error satisfies a stochastic version of a
{\L}ojasiewicz condition. The model is used by an inner learning algorithm. The
main feature of our approach is that unbiased stochastic estimates of the
directions used by the method can be generated with one observation from the
joint distribution per iteration, which makes it applicable to real-time
learning. The directions, however, are not gradients or subgradients of any
overall objective function. We prove the convergence of the method with
probability one, using the method of differential inclusions and a specially
designed Lyapunov function, involving a stochastic generalization of the
Bregman distance. Finally, a numerical illustration demonstrates the viability
of our approach.
