Multi-cut stochastic approximation methods for solving stochastic convex composite optimization

This paper considers the stochastic convex composite optimization problem and presents multi-cut stochastic approximation (SA) methods for solving it, whose models in expectation overestimate its objective function. The multi-cut model obtained by taking the maximum of a finite number of linearizations of the stochastic objective function provides a biased estimate of the objective function, with the error being uncontrollable. Instead, our proposed SA method uses models obtained by taking the maximum of a finite number of one-cut models, i.e., suitable convex combinations of linearizations of the stochastic objective function. It is shown that the proposed methods achieve nearly optimal convergence rate and have computational performance comparable, and sometimes superior, to other SA-type methods.