Research

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 e-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 (e^-5/2 ) and O (e^-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.