  Projection-free optimization via different variants of the Frank-Wolfe (FW),
a.k.a. Conditional Gradient 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 guarantees for 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)
after k iterations. If the problem admits a stronger local linear minimization
oracle, we construct a novel FW method with linear convergence rate for SC
functions.
