The objectives of this technical report is to provide additional results on the generalized conditional gradient methods introduced by [BLM05]. Indeed , when the objective function is smooth, we provide a novel certificate of optimality and we show that the algorithm has a linear convergence rate. Applications of this algorithm are also discussed.
Here, we propose to take advantage of the composite structure of the objective function and of an efficient solver of that partially linearized problem. Note that a result similar to Lemma , denoted as the surrogate duality gap in @cite_7 , exists for the conditional gradient: @math Using the convexity of @math one can see that @math This means that the bound expressed in Lemma is at least as good that the one provided by the classical CG. In addition when @math is strictly convex, our bond is strictly better which suggests that our approach provides a better control of the convergence along the iterations.
@cite_7
905619
In this paper, we present a new conditional gradient type method for convex optimization by calling a linear optimization ( @math ) 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 @math oracle but also the number of gradient evaluations. More specifically, we show that the CGS method requires @math and @math gradient evaluations, respectively, for solving smooth and strongly convex problems, while still maintaining the optimal @math bound on the number of calls to the @math 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...