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The Frank-Wolfe (FW) optimization algorithm has lately re-gained popularitythanks in particular to its ability to nicely handle the structured constraintsappearing in machine learning applications. However, its convergence rate isknown to be slow (sublinear) when the solution lies at the boundary. A simpleless-known fix is to add the possibility to take 'away steps' duringoptimization, an operation that importantly does not require a feasibilityoracle. In this paper, we highlight and clarify several variants of theFrank-Wolfe optimization algorithm that have been successfully applied inpractice: away-steps FW, pairwise FW, fully-corrective FW and Wolfe's minimumnorm point algorithm, and prove for the first time that they all enjoy globallinear convergence, under a weaker condition than strong convexity of theobjective. The constant in the convergence rate has an elegant interpretationas the product of the (classical) condition number of the function with a novelgeometric quantity that plays the role of a 'condition number' of theconstraint set. We provide pointers to where these algorithms have made adifference in practice, in particular with the flow polytope, the marginalpolytope and the base polytope for submodular optimization.

On the Global Linear Convergence of Frank-Wolfe Optimization Variants