Variable Smoothing for Weakly Convex Composite Functions

Abstract: We study minimization of a structured objective function, being the sum of a smooth function and a composition of a weakly convex function with a linear operator. Applications include image reconstruction problems with regularizers that introduce less bias than the standard convex regularizers. We develop a variable smoothing algorithm, based on the Moreau envelope with a decreasing sequence of smoothing parameters, and prove a complexity of $\mathcal{O}(\epsilon^{-3})$ to achieve an $\epsilon$-approximate solution. This bound interpolates between the $\mathcal{O}(\epsilon^{-2})$ bound for the smooth case and the $\mathcal{O}(\epsilon^{-4})$ bound for the subgradient method. Our complexity bound is in line with other works that deal with structured nonsmoothness of weakly convex functions.