Stochastic Frank-Wolfe for Constrained Finite-Sum Minimization

We propose a novel Stochastic Frank-Wolfe (a.k.a.\ Conditional Gradient) algorithm with a fixed batch size tailored to the constrained optimization of a finite sum of smooth objectives. The design of our method hinges on a primal-dual interpretation of the Frank-Wolfe algorithm. Recent work to design stochastic variants of the Frank-Wolfe algorithm falls into two categories: algorithms with increasing batch size, and algorithms with a given, constant, batch size. The former have faster convergence rates but are impractical; the latter are practical but slower. The proposed method combines the advantages of both: it converges for unit batch size, and has faster theoretical worst-case rates than previous unit batch size algorithms. Our experiments also show faster empirical convergence than previous unit batch size methods for several tasks. Finally, we construct a stochastic estimator of the Frank-Wolfe gap. It allows us to bound the true Frank-Wolfe gap, which in the convex setting bounds the primal-dual gap in the convex case while in general is a measure of stationarity. Our gap estimator can therefore be used as a practical stopping criterion in all cases.