  In this paper, we follow the work (Rodomanov and Nesterov 2021) to study
quasi-Newton methods, which is based on the updating formulas from a certain
subclass of the Broyden family. We focus on the common SR1 and BFGS
quasi-Newton methods to establish better explicit superlinear convergence.
First, based on greedy quasi-Newton update in Rodomanov and Nesterov's work,
which greedily selected the direction so as to maximize a certain measure of
progress, we improve the linear convergence rate to a condition-number-free
superlinear convergence rate, when applied with the well-known SR1 update, and
BFGS update. Moreover, our results can also be applied to the inverse
approximation of the SR1 update. Second, based on random update, that selects
the direction randomly from any spherical symmetry distribution we show the
same superlinear convergence rate established as above. Our analysis is closely
related to the approximation of a given Hessian matrix, unconstrained quadratic
objective, as well as the general strongly convex, smooth and strongly
self-concordant functions.
