Greedy and Random Quasi-Newton Methods with Faster Explicit Superlinear Convergence

Optimization is important in machine learning problems, and quasi-Newton methods have a reputation as the most efficient numerical methods for smooth unconstrained optimization. In this paper, we study the explicit superlinear convergence rates of quasi-Newton methods and address two open problems mentioned by Rodomanov and Nesterov (2021b). First, we extend Rodomanov and Nesterov (2021b)'s results to random quasi-Newton methods, which include common DFP, BFGS, SR1 methods. Such random methods employ a random direction for updating the approximate Hessian matrix in each iteration. Second, we focus on the specific quasi-Newton methods: SR1 and BFGS methods. We provide improved versions of greedy and random methods with provable better explicit (local) superlinear convergence rates. 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.