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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 Nesterovs 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.
We consider the use of a curvature-adaptive step size in gradient-based iterative methods, including quasi-Newton methods, for minimizing self-concordant functions, extending an approach first proposed for Newtons method by Nesterov. This step size h
In this paper, we follow the recent works about the explicit superlinear convergence rate of quasi-Newton methods. We focus on classical Broydens methods for solving nonlinear equations and establish explicit (local) superlinear convergence if the in
In this paper, we propose some new proximal quasi-Newton methods with line search or without line search for a special class of nonsmooth multiobjective optimization problems, where each objective function is the sum of a twice continuously different
We present two sampled quasi-Newton methods (sampled LBFGS and sampled LSR1) for solving empirical risk minimization problems that arise in machine learning. Contrary to the classical variants of these methods that sequentially build Hessian or inver
In this paper, we consider stochastic second-order methods for minimizing a finite summation of nonconvex functions. One important key is to find an ingenious but cheap scheme to incorporate local curvature information. Since the true Hessian matrix