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 $\mathcal{ O}(N+ N^{ 4/5}\epsilon^{ -1})$ for both mapping and Jacobian evaluations. When $g$ is a general expectation, we obtain sample complexities of $\mathcal{ O}(\epsilon^{ -5/2})$ and $\mathcal{ O}(\epsilon^{ -3/2})$ for component mappings and their Jacobians respectively. If in addition $f$ is smooth, then improved sample complexities of $\mathcal{ O}(N+N^{ 1/2}\epsilon^{ -1})$ and $\mathcal{ O}(\epsilon^{ -3/2})$ are derived for $g$ being a finite average and a general expectation respectively, for both component mapping and Jacobian evaluations.
Categories: math.OC