Frank-Wolfe Algorithms for Saddle Point Problems

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Gauthier Gidel, Tony Jebara, Simon Lacoste-Julien ;
Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, PMLR 54:362-371, 2017.

Abstract

We extend the Frank-Wolfe (FW) optimization algorithm to solve constrained smooth convex-concave saddle point (SP) problems. Remarkably, the method only requires access to linear minimization oracles. Leveraging recent advances in FW optimization, we provide the first proof of convergence of a FW-type saddle point solver over polytopes, thereby partially answering a 30 year-old conjecture. We also survey other convergence results and highlight gaps in the theoretical underpinnings of FW-style algorithms. Motivating applications without known efficient alternatives are explored through structured prediction with combinatorial penalties as well as games over matching polytopes involving an exponential number of constraints.

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