Stochastic variance-reduced prox-linear algorithms for nonconvex composite optimization.

We consider the problem of minimizing composite functions of the form f (g (x)) + h (x) , where f and h are convex functions (which can be nonsmooth) and g is a smooth vector mapping. In addition, we assume that g is the average of finite number of component mappings or the expectation over a family of random component mappings. We propose a class of stochastic variance-reduced prox-linear algorithms for solving such problems and bound their sample complexities for finding an ϵ -stationary point in terms of the total number of evaluations of the component mappings and their Jacobians. When g is a finite average of N components, we obtain sample complexity O (N + N 4 / 5 ϵ - 1) for both mapping and Jacobian evaluations. When g is a general expectation, we obtain sample complexities of O (ϵ - 5 / 2) and O (ϵ - 3 / 2) for component mappings and their Jacobians respectively. If in addition f is smooth, then improved sample complexities of O (N + N 1 / 2 ϵ - 1) and O (ϵ - 3 / 2) are derived for g being a finite average and a general expectation respectively, for both component mapping and Jacobian evaluations. [ABSTRACT FROM AUTHOR]