When Are Nonconvex Problems Not Scary?

Ju Sun; Qing Qu; John Wright

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arXiv · 1510.06096

When Are Nonconvex Problems Not Scary?

Ju Sun·Qing Qu·John Wright

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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.

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Ju Sun, Qing Qu, John Wright. 2016-04-23. When Are Nonconvex Problems Not Scary?. https://arxiv.org/abs/1510.06096

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