Convergence and Cycling in Walker-type Saddle Search Algorithms

TL;DR

This paper analyzes idealized saddle search algorithms, revealing their limitations by demonstrating quasi-periodic solutions and providing improved estimates on saddle attraction regions, highlighting the challenges in achieving global convergence.

Contribution

The paper offers new theoretical insights into the limitations of dimer and GAD saddle search algorithms, including improved attraction estimates and the construction of quasi-periodic solutions.

Findings

Improved estimate on the attraction region of saddles

Construction of quasi-periodic solutions showing convergence limitations

Evidence that globally convergent variants of these algorithms are impossible

Abstract

Algorithms for computing local minima of smooth objective functions enjoy a mature theory as well as robust and efficient implementations. By comparison, the theory and practice of saddle search is destitute. In this paper we present results for idealized versions of the dimer and gentlest ascent (GAD) saddle search algorithms that show-case the limitations of what is theoretically achievable within the current class of saddle search algorithms: (1) we present an improved estimate on the region of attraction of saddles; and (2) we construct quasi-periodic solutions which indicate that it is impossible to obtain globally convergent variants of dimer and GAD type algorithms.