The effect of smooth parametrizations on nonconvex optimization landscapes

• 成果类型: Article
• 署名作者: Levin, Eitan; Kileel, Joe; Boumal, Nicolas
• 署名单位: California Institute of Technology; University of Texas System; University of Texas Austin; University of Texas System; University of Texas Austin; Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-024-02058-3
• 发表日期: 2025
• 页码: 63-111
• 关键词: Rank; CONVERGENCE; matrices

摘要

We develop new tools to study landscapes in nonconvex optimization. Given one optimization problem, we pair it with another by smoothly parametrizing the domain. This is either for practical purposes (e.g., to use smooth optimization algorithms with good guarantees) or for theoretical purposes (e.g., to reveal that the landscape satisfies a strict saddle property). In both cases, the central question is: how do the landscapes of the two problems relate? More precisely: how do desirable points such as local minima and critical points in one problem relate to those in the other problem? A key finding in this paper is that these relations are often determined by the parametrization itself, and are almost entirely independent of the cost function. Accordingly, we introduce a general framework to study parametrizations by their effect on landscapes. The framework enables us to obtain new guarantees for an array of problems, some of which were previously treated on a case-by-case basis in the literature. Applications include: optimizing low-rank matrices and tensors through factorizations; solving semidefinite programs via the Burer-Monteiro approach; training neural networks by optimizing their weights and biases; and quotienting out symmetries.