  The Frank-Wolfe method (a.k.a. conditional gradient algorithm) for smooth
optimization has regained much interest in recent years in the context of large
scale optimization and machine learning. A key advantage of the method is that
it avoids projections - the computational bottleneck in many applications -
replacing it by a linear optimization step. Despite this advantage, the known
convergence rates of the FW method fall behind standard first order methods for
most settings of interest. It is an active line of research to derive faster
linear optimization-based algorithms for various settings of convex
optimization.
  In this paper we consider the special case of optimization over strongly
convex sets, for which we prove that the vanila FW method converges at a rate
of $\frac{1}{t^2}$. This gives a quadratic improvement in convergence rate
compared to the general case, in which convergence is of the order
$\frac{1}{t}$, and known to be tight. We show that various balls induced by
$\ell_p$ norms, Schatten norms and group norms are strongly convex on one hand
and on the other hand, linear optimization over these sets is straightforward
and admits a closed-form solution. We further show how several previous
fast-rate results for the FW method follow easily from our analysis.
