  In this work, we consider constrained stochastic optimization problems under
hidden convexity, i.e., those that admit a convex reformulation via non-linear
(but invertible) map $c(\cdot)$. A number of non-convex problems ranging from
optimal control, revenue and inventory management, to convex reinforcement
learning all admit such a hidden convex structure. Unfortunately, in the
majority of applications considered, the map $c(\cdot)$ is unavailable or
implicit; therefore, directly solving the convex reformulation is not possible.
On the other hand, the stochastic gradients with respect to the original
variable are often easy to obtain. Motivated by these observations, we examine
the basic projected stochastic (sub-) gradient methods for solving such
problems under hidden convexity. We provide the first sample complexity
guarantees for global convergence in smooth and non-smooth settings.
Additionally, in the smooth setting, we improve our results to the last iterate
convergence in terms of function value gap using the momentum variant of
projected stochastic gradient descent.
