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We consider global efficiency of algorithms for minimizing a sum of a convexfunction and a composition of a Lipschitz convex function with a smooth map.The basic algorithm we rely on is the prox-linear method, which in eachiteration solves a regularized subproblem formed by linearizing the smooth map.When the subproblems are solved exactly, the method has efficiency$\mathcal{O}(\varepsilon^{-2})$, akin to gradient descent for smoothminimization. We show that when the subproblems can only be solved byfirst-order methods, a simple combination of smoothing, the prox-linear method,and a fast-gradient scheme yields an algorithm with complexity$\widetilde{\mathcal{O}}(\varepsilon^{-3})$. The technique readily extends tominimizing an average of $m$ composite functions, with complexity$\widetilde{\mathcal{O}}(m/\varepsilon^{2}+\sqrt{m}/\varepsilon^{3})$ inexpectation. We round off the paper with an inertial prox-linear method thatautomatically accelerates in presence of convexity.

Efficiency of minimizing compositions of convex functions and smooth maps