Revisiting Frank-Wolfe for Polytopes: Strict Complementarity and Sparsity
This work addresses convergence guarantees for optimization algorithms in machine learning and operations research, offering improved theoretical understanding for large-scale problems, though it is incremental as it builds on existing assumptions and methods.
The paper tackles the issue of dimension-dependent convergence rates in Frank-Wolfe algorithms for convex minimization over polytopes, showing that under a strict complementarity assumption, the method with away-steps and line-search achieves linear convergence with rates depending only on the dimension of the optimal face, and proves that this condition ensures sparsity-robustness of solutions to noise.
In recent years it was proved that simple modifications of the classical Frank-Wolfe algorithm (aka conditional gradient algorithm) for smooth convex minimization over convex and compact polytopes, converge with linear rate, assuming the objective function has the quadratic growth property. However, the rate of these methods depends explicitly on the dimension of the problem which cannot explain their empirical success for large scale problems. In this paper we first demonstrate that already for very simple problems and even when the optimal solution lies on a low-dimensional face of the polytope, such dependence on the dimension cannot be avoided in worst case. We then revisit the addition of a strict complementarity assumption already considered in Wolfe's classical book \cite{Wolfe1970}, and prove that under this condition, the Frank-Wolfe method with away-steps and line-search converges linearly with rate that depends explicitly only on the dimension of the optimal face. We motivate strict complementarity by proving that it implies sparsity-robustness of optimal solutions to noise.