When Are Nonconvex Problems Not Scary?
Abstract
In this note, we focus on smooth nonconvex optimization problems that obey: (1) all local minimizers are also global; and (2) around any saddle point or local maximizer, the objective has a negative directional curvature. Concrete applications such as dictionary learning, generalized phase retrieval, and orthogonal tensor decomposition are known to induce such structures. We describe a second-order trust-region algorithm that provably converges to a global minimizer efficiently, without special initializations. Finally we highlight alternatives, and open problems in this direction.
Cite
@article{arxiv.1510.06096,
title = {When Are Nonconvex Problems Not Scary?},
author = {Ju Sun and Qing Qu and John Wright},
journal= {arXiv preprint arXiv:1510.06096},
year = {2016}
}
Comments
6 pages, 3 figures. New examples on phase synchronization and community detection added; emphasis on all local minimizers being global added; exposition is polished. This is a concise expository article that avoids much technical rigor. We will make a separate submission with full technical details in future