An Adaptive High Order Method for Finding Third-Order Critical Points of Nonconvex Optimization

TL;DR

This paper introduces AHOM, an adaptive higher-order method designed to efficiently find third-order critical points in nonconvex optimization, with proven iteration complexity and promising numerical results.

Contribution

It presents a practical, implementable third-order method with adaptive parameter tuning, filling the gap between theoretical development and numerical experiments.

Findings

AHOM can escape degenerate saddle points.

The method has established iteration complexity.

Preliminary results show promising performance.

Abstract

It is well known that finding a global optimum is extremely challenging for nonconvex optimization. There are some recent efforts \cite{anandkumar2016efficient, cartis2018second, cartis2020sharp, chen2019high} regarding the optimization methods for computing higher-order critical points, which can exclude the so-called degenerate saddle points and reach a solution with better quality. Desipte theoretical development in \cite{anandkumar2016efficient, cartis2018second, cartis2020sharp, chen2019high}, the corresponding numerical experiments are missing. In this paper, we propose an implementable higher-order method, named adaptive high order method (AHOM), that aims to find the third-order critical points. This is achieved by solving an ``easier'' subproblem and incorporating the adaptive strategy of parameter-tuning in each iteration of the algorithm. The iteration complexity of the proposed method is established. Some preliminary numerical results are provided to show AHOM is able to escape the degenerate saddle points, where the second-order method could possibly get stuck.