Primal-Dual Block Frank-Wolfe

Particularly, we are interested in problems whose solution has special "simple" structure like low-rank or sparsity. The sparsity constraint applies to large-scale multiclass/multi-label classification, low-degree polynomial data mapping [5], random feature kernel machines [32], and Elastic Net [39]. Motivated by recent applications in low-rank multi-class SVM, phase retrieval, matrix completion, affine rank minimization and other problems (e.g., [9, 31, 2, 3]), we also consider settings where the constraint x C (e.g., trace norm ball) while convex, may be difficult to project onto. A wish-list for this class of problems would include an algorithm that (1) exploits the function finite-sum form and the simple structure of the solution, (2) achieves linear convergence for smooth and strongly convex problems, (3) does not pay a heavy price for the projection step. We propose a Frank-Wolfe (FW) type method that attains these three goals. This does not come without challenges: Although it is currently well-appreciated that FW type algorithms avoid the cost of projection [14, 1], the benefits are limited to constraints that are hard to project onto, like the trace norm ball. For problems like phase retrieval and ERM for multi-label multi-class classification, the gradient computation requires large matrix multiplications. This dominates the per-iteration cost, and the existing FW type methods do not asymptotically reduce time complexity per iteration, even without paying the expensive projection step.