Symmetric Rank-k Methods

This paper proposes a novel class of block quasi-Newton methods for convex optimization which we call symmetric rank-k (SR-k) methods. Each iteration of SR-k incorporates the curvature information with~k Hessian-vector products achieved from the greedy or random strategy. We prove that SR-k methods have the local superlinear convergence rate of Obig((1-k/d)t(t-1)/2big) for minimizing smooth and strongly convex function, where d is the problem dimension and t is the iteration counter. This is the first explicit superlinear convergence rate for block quasi-Newton methods, and it successfully explains why block quasi-Newton methods converge faster than ordinary quasi-Newton methods in practice. We also leverage the idea of SR-k methods to study the block BFGS and block DFP methods, showing their superior convergence rates.