Blended Conditional Gradients: the unconditioning of conditional gradients
This work addresses optimization challenges in machine learning and operations research by providing an efficient, projection-free algorithm with sparse solutions, though it is incremental as it builds on existing conditional gradient methods.
The paper tackles the problem of minimizing a smooth convex function over a polytope by introducing a blended conditional gradient approach that combines Frank-Wolfe with gradient-based steps, achieving linear convergence for strongly convex functions and outperforming lazy conditional gradient algorithms in terms of primal and dual gap reductions in iterations and wall-clock time.
We present a blended conditional gradient approach for minimizing a smooth convex function over a polytope P, combining the Frank--Wolfe algorithm (also called conditional gradient) with gradient-based steps, different from away steps and pairwise steps, but still achieving linear convergence for strongly convex functions, along with good practical performance. Our approach retains all favorable properties of conditional gradient algorithms, notably avoidance of projections onto P and maintenance of iterates as sparse convex combinations of a limited number of extreme points of P. The algorithm is lazy, making use of inexpensive inexact solutions of the linear programming subproblem that characterizes the conditional gradient approach. It decreases measures of optimality (primal and dual gaps) rapidly, both in the number of iterations and in wall-clock time, outperforming even the lazy conditional gradient algorithms of [arXiv:1410.8816]. We also present a streamlined version of the algorithm for the probability simplex.