# Conditional Gradient Sliding for Convex Optimization

In this paper, we present a new conditional gradient type method for convex optimization by utilizing a linear optimization (LO) oracle to minimize a series of linear functions over the feasible set. Different from the classic conditional gradient method, the conditional gradient sliding (CGS) algorithm developed herein can skip the computation of gradients from time to time, and as a result, can achieve the optimal complexity bounds in terms of not only the number of calls to the LO oracle, but also the number of gradient evaluations. More specifically, we show that the CGS method requires ${\cal O}(1/\sqrt{\epsilon})$ and ${\cal O}(\log (1/\epsilon))$ gradient evaluations, respectively, for solving smooth and strongly convex problems, while still maintaining the optimal ${\cal O}(1/\epsilon)$ bound on the number of calls to the LO oracle. We also develop variants of the CGS method which can achieve the optimal complexity bounds for solving stochastic optimization problems and an important class of saddle point optimization problems. To the best of our knowledge, this is the first time that these types of projection-free optimal first-order methods have been developed in the literature. Some preliminary numerical results have also been provided to demonstrate the advantages of the CGS method.

## Citation

Technical Report, Department of Industrial and Systems Engineering, University of Florida.