Greedy Quasi-Newton Methods with Explicit Superlinear Convergence

In this paper, we study greedy variants of quasi-Newton methods. They are based on the updating formulas from a certain subclass Broyden family. particular, includes well-known DFP, BFGS, and SR1 updates. However, in contrast to classical methods, which use difference successive iterates for Hessian approximations, our methods apply basis vectors, greedily selected so as maximize measure progress. For establish an explicit nonasymptotic bound their rate local superlinear convergence, applied minimizing strongly convex self-concordant functions (and, with Lipschitz continuous Hessian). The established convergence contains contraction factor, depends square iteration counter. We also show that produce approximations whose deviation exact Hessians linearly converges zero.