A Momentum-Guided Frank-Wolfe Algorithm
Bingcong Li, Mario Coutiño, Georgios B. Giannakis, Geert Leus
Abstract
With the well-documented popularity of Frank Wolfe (FW) algorithms in machine learning tasks, the present paper establishes links between FW subproblems and the notion of momentum emerging in accelerated gradient methods (AGMs). On the one hand, these links reveal why momentum is unlikely to be effective for FW-type algorithms on general problems. On the other hand, it is established that momentum accelerates FW on a class of signal processing and machine learning applications. Specifically, it is proved that a momentum variant of FW, here termed accelerated Frank Wolfe (AFW), converges with a faster rate <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> (\frac1k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) on such a family of problems, despite the same <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> (\frac1k) rate of FW on general cases. Distinct from existing fast convergent FW variants, the faster rates here rely on parameter-free step sizes. Numerical experiments on benchmarked machine learning tasks corroborate the theoretical findings.