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Motivated principally by the low-rank matrix completion problem, we present an extension of the Frank-Wolfe method that is designed to induce near-optimal solutions on low-dimensional faces of the feasible region. This is accomplished by a new approach to generating ``in-face" directions at each iteration, as well as through new choice rules for selecting between in-face and ``regular" Frank-Wolfe steps. Our framework for generating in-face directions generalizes the notion of away-steps introduced by Wolfe. In particular, the in-face directions always keep the next iterate within the minimal face containing the current iterate. We present computational guarantees for the new method that trade off efficiency in computing near-optimal solutions with upper bounds on the dimension of minimal faces of iterates. We apply the new method to the matrix completion problem, where low-dimensional faces correspond to low-rank matrices. We present computational results that demonstrate the effectiveness of our methodological approach at producing nearly-optimal solutions of very low rank. On both artificial and real datasets, we demonstrate significant speed-ups in computing very low-rank nearly-optimal solutions as compared to either the Frank-Wolfe method or its traditional away-step variant., Comment: 25 pages, 3 tables and 2 figues

We present new results for the Frank-Wolfe method (also known as the conditional gradient method). We derive computational guarantees for arbitrary step-size sequences, which are then applied to various step-size rules, including simple averaging and constant step-sizes. We also develop step-size rules and computational guarantees that depend naturally on the warm-start quality of the initial (and subsequent) iterates. Our results include computational guarantees for both duality/bound gaps and the so-called FW gaps. Lastly, we present complexity bounds in the presence of approximate computation of gradients and/or linear optimization subproblem solutions., Comment: Changed the name of the method from "conditional gradient" to "Frank-Wolfe"