On the asymptotic convergence and acceleration of gradient methods

TL;DR

This paper analyzes the asymptotic behavior of a family of gradient methods, revealing their zigzagging pattern and proposing spectral-based acceleration techniques, including a new stepsize and a periodic method inspired by the Barzilai-Borwein approach.

Contribution

It introduces a spectral property-based acceleration for gradient methods, including a new stepsize and a periodic method, improving convergence on quadratic functions.

Findings

Gradient methods asymptotically zigzag between two directions.

Proposed stepsize converges to the reciprocal of the largest eigenvalue.

Numerical results show the new method's promising performance.

Abstract

We consider the asymptotic behavior of a family of gradient methods, which include the steepest descent and minimal gradient methods as special instances. It is proved that each method in the family will asymptotically zigzag between two directions. Asymptotic convergence results of the objective value, gradient norm, and stepsize are presented as well. To accelerate the family of gradient methods, we further exploit spectral properties of stepsizes to break the zigzagging pattern. In particular, a new stepsize is derived by imposing finite termination on minimizing two-dimensional strictly convex quadratic function. It is shown that, for the general quadratic function, the proposed stepsize asymptotically converges to the reciprocal of the largest eigenvalue of the Hessian. Furthermore, based on this spectral property, we propose a periodic gradient method by incorporating the Barzilai-Borwein method. Numerical comparisons with some recent successful gradient methods show that our new method is very promising.