Linearly Convergent Frank-Wolfe with Backtracking Line-Search

Structured constraints in Machine Learning have recently brought the Frank-Wolfe (FW) family of algorithms back into the spotlight. While the classical FW algorithm has poor local convergence properties, Away-steps FW and Pairwise FW have emerged as improved variants with faster convergence. However, these improved variants suffer from two practical limitations: they require at each iteration to solve a 1-dimensional minimization problem to set the step-size and also require the Frank-Wolfe linear subproblems to be solved exactly. In this paper, we propose variants of Away-steps and Pairwise FW that lift both restrictions simultaneously. The proposed methods set the step-size based on a sufficient decrease condition, and do not require prior knowledge of the objective. Furthermore, they inherit all the favorable convergence properties of the exact line-search version, including linear convergence for strongly convex functions over polytopes. Benchmarks on different machine learning problems illustrate large performance gains of the proposed variants.