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.