Self-concordant analysis of Frank-Wolfe algorithm

Part of Proceedings of the International Conference on Machine Learning 1 pre-proceedings (ICML 2020)

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Pavel Dvurechenskii, Petr Ostroukhov, Kamil Safin, Shimrit Shtern, Mathias Staudigl


<p>Projection-free optimization via different variants of the Frank-Wolfe (FW) method has become one of the cornerstones in optimization for machine learning since in many cases the linear minimization oracle is much cheaper to implement than projections and some sparsity needs to be preserved. In a number of applications, e.g. Poisson inverse problems or quantum state tomography, the loss is given by a self-concordant (SC) function having unbounded curvature, implying absence of theoretical guaranteesfor the existing FW methods. We use the theory of SC functions to provide a new adaptive step size for FW methods and prove global convergence rate O(1/k), k being the iteration counter. If the problem can be represented by a local linear minimization oracle, we are the first to propose a FW method with linear convergence rate without assuming neither strong convexity nor a Lipschitz continuous gradient.</p>