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

Stochastic Compositional Optimization with Compositional Constraints