We consider minimization of 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 \\(\\epsilon\\)-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}\\epsilon^{-1})\\) for both mapping and Jacobian evaluations. When \\(g\\) is a general expectation, we obtain sample complexities of \\(O(\\epsilon^{-5\/2})\\) and \\(O(\\epsilon^{-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}\\epsilon^{-1})\\) and \\(O(\\epsilon^{-3\/2})\\) are derived for \\(g\\) being a finite average and a general expectation respectively, for both component mapping and Jacobian evaluations.<\/p>\n","protected":false},"excerpt":{"rendered":"
We consider minimization of composite functions of the form , where and are convex functions (which can be nonsmooth) and is a smooth vector mapping. In addition, we assume that is the average of finite number of component mappings or the expectation over a family of random component mappings. 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