On the Global Linear Convergence of Frank-Wolfe Optimization Variants

TL;DR

This paper proves that several variants of the Frank-Wolfe algorithm, including away-steps and pairwise methods, achieve global linear convergence under weaker conditions than strong convexity, with implications for structured optimization problems.

Contribution

It establishes the first proof of global linear convergence for multiple Frank-Wolfe variants under weaker assumptions, introducing a geometric condition number of the constraint set.

Findings

All variants enjoy global linear convergence.

The convergence rate constant involves a new geometric condition number.

Applications include flow polytope, marginal polytope, and submodular optimization.

Abstract

The Frank-Wolfe (FW) optimization algorithm has lately re-gained popularity thanks in particular to its ability to nicely handle the structured constraints appearing in machine learning applications. However, its convergence rate is known to be slow (sublinear) when the solution lies at the boundary. A simple less-known fix is to add the possibility to take 'away steps' during optimization, an operation that importantly does not require a feasibility oracle. In this paper, we highlight and clarify several variants of the Frank-Wolfe optimization algorithm that have been successfully applied in practice: away-steps FW, pairwise FW, fully-corrective FW and Wolfe's minimum norm point algorithm, and prove for the first time that they all enjoy global linear convergence, under a weaker condition than strong convexity of the objective. The constant in the convergence rate has an elegant interpretation as the product of the (classical) condition number of the function with a novel geometric quantity that plays the role of a 'condition number' of the constraint set. We provide pointers to where these algorithms have made a difference in practice, in particular with the flow polytope, the marginal polytope and the base polytope for submodular optimization.