  The Frank-Wolfe algorithm has become a popular first-order optimization
algorithm for it is simple and projection-free, and it has been successfully
applied to a variety of real-world problems. Its main drawback however lies in
its convergence rate, which can be excessively slow due to naive descent
directions. We propose to speed up the Frank-Wolfe algorithm by better aligning
the descent direction with that of the negative gradient via a subroutine. This
subroutine chases the negative gradient direction in a matching pursuit-style
while still preserving the projection-free property. Although the approach is
reasonably natural, it produces very significant results. We derive convergence
rates $\mathcal{O}(1/t)$ to $\mathcal{O}(e^{-\omega t})$ of our method and we
demonstrate its competitive advantage both per iteration and in CPU time over
the state-of-the-art in a series of computational experiments.

Optimization and Control;Machine Learning