The effect of smooth parametrizations on nonconvex optimization landscapes

TL;DR

This paper introduces a framework to analyze how smooth parametrizations affect the landscapes of nonconvex optimization problems, revealing that landscape relations are often independent of the cost function and enabling new guarantees across various applications.

Contribution

The paper develops a general framework to study the impact of smooth parametrizations on optimization landscapes, providing new theoretical guarantees for multiple nonconvex problems.

Findings

Landscape relations are often determined by parametrization, not the cost function.

The framework applies to low-rank matrix and tensor optimization, semidefinite programming, neural network training, and symmetry quotienting.

New guarantees are derived for these applications, improving understanding of nonconvex landscapes.

Abstract

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.