  The analysis 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 some of these variants by getting rid
of such a distinction between steps, and to improve existing rates by ensuring
a sizable decrease of the objective 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 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 a linear convergence rate under a
Kurdyka-Lojasiewicz (KL) property. While this setting has been widely studied
for proximal gradient type methods, to our knowledge, it has not been analyzed
before for the Frank Wolfe variants under study. 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 this condition
is satisfied on polytopes by the away step Frank-Wolfe (AFW), the pairwise
Frank-Wolfe (PFW), and the Frank-Wolfe method with in face directions (FDFW).
