A Quasi-Newton Primal-Dual Algorithm with Line Search

TL;DR

This paper introduces a line search variant of a quasi-Newton primal-dual algorithm that enhances efficiency and flexibility for non-smooth large-scale problems, with proven convergence and superior performance in image deblurring.

Contribution

It develops a new line search quasi-Newton primal-dual method that improves step size flexibility and convergence analysis for non-smooth optimization.

Findings

Proven convergence and convergence rates for the proposed method.

Outperforms related algorithms in large-scale image deblurring.

Adds flexibility and larger steps per iteration in the algorithm.

Abstract

Quasi-Newton methods refer to a class of algorithms at the interface between first and second order methods. They aim to progress as substantially as second order methods per iteration, while maintaining the computational complexity of first order methods. The approximation of second order information by first order derivatives can be expressed as adopting a variable metric, which for (limited memory) quasi-Newton methods is of type ``identity $\pm$ low rank''. This paper continues the effort to make these powerful methods available for non-smooth systems occurring, for example, in large scale Machine Learning applications by exploiting this special structure. We develop a line search variant of a recently introduced quasi-Newton primal-dual algorithm, which adds significant flexibility, admits larger steps per iteration, and circumvents the complicated precalculation of a certain operator norm. We prove convergence, including convergence rates, for our proposed method and outperform related algorithms in a large scale image deblurring application.