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