  Stochastic convex optimization problems with expectation constraints (SOECs)
are encountered in statistics and machine learning, business, and engineering.
In data-rich environments, the SOEC objective and constraints contain
expectations defined with respect to large datasets. Therefore, efficient
algorithms for solving such SOECs need to limit the fraction of data points
that they use, which we refer to as algorithmic data complexity. Recent
stochastic first order methods exhibit low data complexity when handling SOECs
but guarantee near-feasibility and near-optimality only at convergence. These
methods may thus return highly infeasible solutions when heuristically
terminated, as is often the case, due to theoretical convergence criteria being
highly conservative. This issue limits the use of first order methods in
several applications where the SOEC constraints encode implementation
requirements. We design a stochastic feasible level set method (SFLS) for SOECs
that has low data complexity and emphasizes feasibility before convergence.
Specifically, our level-set method solves a root-finding problem by calling a
novel first order oracle that computes a stochastic upper bound on the
level-set function by extending mirror descent and online validation
techniques. We establish that SFLS maintains a high-probability feasible
solution at each root-finding iteration and exhibits favorable iteration
complexity compared to state-of-the-art deterministic feasible level set and
stochastic subgradient methods. Numerical experiments on three diverse
applications validate the low data complexity of SFLS relative to the former
approach and highlight how SFLS finds feasible solutions with small optimality
gaps significantly faster than the latter method.
