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The analysis of projection-free first order methods is often complicated bythe presence of different kinds of "good" and "bad" steps. In this article, wepropose a unifying framework for projection-free methods, aiming to simplifythe converge analysis by getting rid of such a distinction between steps. Themain tool employed in our framework is the Short Step Chain (SSC) procedure,which skips gradient computations in consecutive short steps until properstopping conditions are satisfied. This technique allows us to give a unifiedanalysis and converge rates in the general smooth non convex setting, as wellas convergence rates under a Kurdyka-Lojasiewicz (KL) property, a setting that,to our knowledge, has not been analyzed before for the projection-free methodsunder study. In this context, we prove local convergence rates comparable tothose of projected gradient methods under the same conditions. Our analysisrelies on a sufficient slope condition, ensuring that the directions selectedby the methods have the steepest slope possible up to a constant among feasibledirections. This condition is satisfied, among others, by several Frank-Wolfe(FW) variants on polytopes, and by some projection-free methods on convex setswith smooth boundary.

A unifying framework for the analysis of projection-free first-order methods under a sufficient slope condition