# Complexity analysis of second-order line-search algorithms for smooth nonconvex optimization

## TL;DR

This paper introduces a line-search-based second-order optimization algorithm for smooth nonconvex functions, offering simple analysis and practical inexact computation methods with favorable complexity guarantees.

## Contribution

It proposes a novel line-search-only second-order method with straightforward analysis and extends results to inexact linear algebra-based search directions.

## Key findings

- Algorithm achieves optimal iteration complexity.
- Simple backtracking line search ensures sufficient decrease.
- Inexact methods maintain favorable convergence properties.

## Abstract

There has been much recent interest in finding unconstrained local minima of smooth functions, due in part of the prevalence of such problems in machine learning and robust statistics. A particular focus is algorithms with good complexity guarantees. Second-order Newton-type methods that make use of regularization and trust regions have been analyzed from such a perspective. More recent proposals, based chiefly on first-order methodology, have also been shown to enjoy optimal iteration complexity rates, while providing additional guarantees on computational cost. In this paper, we present an algorithm with favorable complexity properties that differs in two significant ways from other recently proposed methods. First, it is based on line searches only: Each step involves computation of a search direction, followed by a backtracking line search along that direction. Second, its analysis is rather straightforward, relying for the most part on the standard technique for demonstrating sufficient decrease in the objective from backtracking. In the latter part of the paper, we consider inexact computation of the search directions, using iterative methods in linear algebra: the conjugate gradient and Lanczos methods. We derive modified convergence and complexity results for these more practical methods.