Avoiding bad steps in Frank-Wolfe variants

The study of Frank-Wolfe (FW) variants is often complicated by the presence of different kinds of "good" and "bad" steps. In this article, we aim to simplify the convergence analysis of specific variants by getting rid of such a distinction between steps, and to improve existing rates by ensuring a non-trivial bound at each iteration. In order to do this, we define the Short Step Chain (SSC) procedure, which skips gradient computations in consecutive short steps until proper conditions are satisfied. This algorithmic tool allows us to give a unified analysis and converge rates in the general smooth non convex setting, as well as a linear convergence rate under a Kurdyka-Lojasiewicz (KL) property. While the KL setting has been widely studied for proximal gradient type methods, to our knowledge, it has never been analyzed before for the Frank-Wolfe variants considered in the paper. An angle condition, ensuring that the directions selected by the methods have the steepest slope possible up to a constant, is used to carry out our analysis. We prove that such a condition is satisfied, when considering minimization problems over a polytope, by the away step Frank-Wolfe (AFW), the pairwise Frank-Wolfe (PFW), and the Frank-Wolfe method with in face directions (FDFW).