An Optimal Algorithm for Stochastic Three-Composite Optimization

Abstract

We develop an optimal primal-dual first-order algorithm for a class of stochastic three-composite convex minimization problems. The convergence rate of our method not only improves upon the existing methods, but also matches a lower bound derived for all first-order methods that solve this problem. We extend our proposed algorithm to solve a composite stochastic program with any finite number of nonsmooth functions. In addition, we generalize an optimal stochastic alternating direction method of multipliers (SADMM) algorithm proposed for the two-composite case to solve this problem, and establish its connection to our optimal primal-dual algorithm. We perform extensive numerical experiments on a variety of machine learning applications to demonstrate the superiority of our method via-a-vis the state-of-the-art.