Conditional gradient type methods for composite nonlinear and stochastic optimization

Abstract

In this paper, we present a conditional gradient type (CGT) method for solving a class of composite optimization problems where the objective function consists of a (weakly) smooth term and a (strongly) convex regularization term. While including a strongly convex term in the subproblems of the classical conditional gradient (CG) method improves its rate of convergence, it does not cost per iteration as much as general proximal type algorithms. More specifically, we present a unified analysis for the CGT method in the sense that it achieves the best-known rate of convergence when the weakly smooth term is nonconvex and possesses (nearly) optimal complexity if it turns out to be convex. While implementation of the CGT method requires explicitly estimating problem parameters like the level of smoothness of the first term in the objective function, we also present a few variants of this method which relax such estimation. Unlike general proximal type parameter free methods, these variants of the CGT method do not require any additional effort for computing (sub)gradients of the objective function and/or solving extra subproblems at each iteration. We then generalize these methods under stochastic setting and present a few new complexity results. To the best of our knowledge, this is the first time that such complexity results are presented for solving stochastic weakly smooth nonconvex and (strongly) convex optimization problems.