  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.
