  The conditional gradient algorithm (also known as the Frank-Wolfe algorithm)
has recently regained popularity in the machine learning community due to its
projection-free property to solve constrained problems. Although many variants
of the conditional gradient algorithm have been proposed to improve
performance, they depend on first-order information (gradient) to optimize.
Naturally, these algorithms are unable to function properly in the field of
increasingly popular zeroth-order optimization, where only zeroth-order
information (function value) is available. To fill in this gap, we propose a
novel Accelerated variance-Reduced Conditional gradient Sliding (ARCS)
algorithm for finite-sum problems, which can use either first-order or
zeroth-order information to optimize. To the best of our knowledge, ARCS is the
first zeroth-order conditional gradient sliding type algorithms solving convex
problems in zeroth-order optimization. In first-order optimization, the
convergence results of ARCS substantially outperform previous algorithms in
terms of the number of gradient query oracle. Finally we validated the
superiority of ARCS by experiments on real-world datasets.
