Stochastic Compositional Optimization with Compositional Constraints
This work addresses a limitation in stochastic optimization for data-driven and risk management applications, such as risk-averse optimization, by enabling handling of complex constraints, though it is incremental in extending the SCO framework.
The paper tackles stochastic compositional optimization with compositional constraints, which existing methods cannot handle, and proposes a primal-dual algorithm achieving an O(1/√N) convergence rate for both single-level and two-level constraints.
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.