  We study Frank-Wolfe methods for nonconvex stochastic and finite-sum
optimization problems. Frank-Wolfe methods (in the convex case) have gained
tremendous recent interest in machine learning and optimization communities due
to their projection-free property and their ability to exploit structured
constraints. However, our understanding of these algorithms in the nonconvex
setting is fairly limited. In this paper, we propose nonconvex stochastic
Frank-Wolfe methods and analyze their convergence properties. For objective
functions that decompose into a finite-sum, we leverage ideas from variance
reduction techniques for convex optimization to obtain new variance reduced
nonconvex Frank-Wolfe methods that have provably faster convergence than the
classical Frank-Wolfe method. Finally, we show that the faster convergence
rates of our variance reduced methods also translate into improved convergence
rates for the stochastic setting.
