Revisiting Frank-Wolfe for Polytopes: Strict Complementarity and Sparsity
arXiv:2006.00558
Abstract
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.
arXiv commentAccepted to Conference on Neural Information Processing Systems (NeurIPS) 2020, spotlight presentation. This version corrects a mistake in the last part of the proof of Theorem 5
References in corpus (8)
- Faster Rates for the Frank-Wolfe Method over Strongly-Convex Sets
- Projection-free Online Learning
- Minding the Gaps for Block Frank-Wolfe Optimization of Structured SVMs
- Linear-memory and Decomposition-invariant Linearly Convergent Conditional Gradient Algorithm for Structured Polytopes
- A Unified Optimization View on Generalized Matching Pursuit and Frank-Wolfe
- Frank-Wolfe Splitting via Augmented Lagrangian Method
- Linear Convergence of Stochastic Frank Wolfe Variants
- Logarithmic Regret for Online Gradient Descent Beyond Strong Convexity