  Stochastic compositional optimization (SCO) has attracted considerable
attention because of its broad applicability to important real-world problems.
However, existing works on SCO assume that the projection within a solution
update is simple, which fails to hold for problem instances where the
constraints are in the form of expectations, such as empirical conditional
value-at-risk constraints. We study a novel model that incorporates
single-level expected value and two-level compositional constraints into the
current SCO framework. Our model can be applied widely to data-driven
optimization and risk management, including risk-averse optimization and
high-moment portfolio selection, and can handle multiple constraints. We
further propose a class of primal-dual algorithms that generates sequences
converging to the optimal solution at the rate of
$\cO(\frac{1}{\sqrt{N}})$under both single-level expected value and two-level
compositional constraints, where $N$ is the iteration counter, establishing the
benchmarks in expected value constrained SCO.
