  We consider variants of the classical Frank-Wolfe algorithm for constrained
smooth convex minimization, that instead of access to the standard oracle for
minimizing a linear function over the feasible set, have access to an oracle
that can find an extreme point of the feasible set that is closest in Euclidean
distance to a given vector. We first show that for many feasible sets of
interest, such an oracle can be implemented with the same complexity as the
standard linear optimization oracle. We then show that with such an oracle we
can design new Frank-Wolfe variants which enjoy significantly improved
complexity bounds in case the set of optimal solutions lies in the convex hull
of a subset of extreme points with small diameter (e.g., a low-dimensional face
of a polytope). In particular, for many $0\text{--}1$ polytopes, under
quadratic growth and strict complementarity conditions, we obtain the first
linearly convergent variant with rate that depends only on the dimension of the
optimal face and not on the ambient dimension.
