  Conditional gradients constitute a class of projection-free first-order
algorithms for smooth convex optimization. As such, they are frequently used in
solving smooth convex optimization problems over polytopes, for which the
computational cost of orthogonal projections would be prohibitive. However,
they do not enjoy the optimal convergence rates achieved by projection-based
accelerated methods; moreover, achieving such globally-accelerated rates is
information-theoretically impossible for these methods. To address this issue,
we present Locally Accelerated Conditional Gradients -- an algorithmic
framework that couples accelerated steps with conditional gradient steps to
achieve local acceleration on smooth strongly convex problems. Our approach
does not require projections onto the feasible set, but only on (typically
low-dimensional) simplices, thus keeping the computational cost of projections
at bay. Further, it achieves the optimal accelerated local convergence. Our
theoretical results are supported by numerical experiments, which demonstrate
significant speedups of our framework over state of the art methods in both
per-iteration progress and wall-clock time.
