Model Function Based Conditional Gradient Method with Armijo-like Line Search

The Conditional Gradient Method is generalized to a class of non-smooth non-convex optimization problems with many applications in machine learning. The proposed algorithm iterates by minimizing so-called model functions over the constraint set. Complemented with an Amijo line search procedure, we prove that subsequences converge to a stationary point. The abstract framework of model functions provides great flexibility for the design of concrete algorithms. As special cases, for example, we develop an algorithm for additive composite problems and an algorithm for non-linear composite problems which leads to a Gauss--Newton-type algorithm. Both instances are novel in non-smooth non-convex optimization and come with numerous applications in machine learning. Moreover, we obtain a hybrid version of Conditional Gradient and Proximal Minimization schemes for free, which combines advantages of both. Our algorithm is shown to perform favorably on a sparse non-linear robust regression problem and we discuss the flexibility of the proposed framework in several matrix factorization formulations.