A Line-Search Algorithm Inspired by the Adaptive Cubic Regularization Framework and Complexity Analysis

TL;DR

This paper introduces a novel line-search algorithm inspired by adaptive cubic regularization, employing a scaled norm to unify and improve convergence and complexity analysis for nonconvex optimization.

Contribution

It proposes a new scaled norm approach that makes adaptive cubic regularization behave as a line-search method, enhancing theoretical properties and practical performance.

Findings

Algorithm achieves optimal complexity for first-order stationary points.

Method performs well on large-scale optimization problems.

Unified approach improves convergence analysis for trust-region and line-search methods.

Abstract

Adaptive regularized framework using cubics has emerged as an alternative to line-search and trust-region algorithms for smooth nonconvex optimization, with an optimal complexity amongst second-order methods. In this paper, we propose and analyze the use of an iteration dependent scaled norm in the adaptive regularized framework using cubics. Within such scaled norm, the obtained method behaves as a line-search algorithm along the quasi-Newton direction with a special backtracking strategy. Under appropriate assumptions, the new algorithm enjoys the same convergence and complexity properties as adaptive regularized algorithm using cubics. The complexity for finding an approximate first-order stationary point can be improved to be optimal whenever a second order version of the proposed algorithm is regarded. In a similar way, using the same scaled norm to define the trust-region neighborhood, we show that the trust-region algorithm behaves as a line-search algorithm. The good potential of the obtained algorithms is shown on a set of large scale optimization problems.