Step-Size Adaptivity in Projection-Free Optimization

We propose and analyze adaptive step-size (also known as inexact line search) variants of different projection-free algorithms such as Frank-Wolfe, its Away-Steps and Pairwise variants as well as Matching Pursuit. The proposed methods leverage local information of the objective through a backtracking line search strategy. This has two key advantages: First, it does rely on constants of the objective that might be costly to compute, such as the Lipschitz or the curvature constant. Second, the proposed criterion is adaptive to local information of the objective, allowing for larger step-sizes. For all proposed methods, we derive convergence rates on convex and non-convex problems that asymptotically match the strongest known bounds for non-adaptive variants. As a side-product of our analysis we obtain the first known bounds for matching pursuit on non-convex objectives. Benchmarks on three different datasets illustrate the practical advantages of the method.