The effect of smooth parametrizations on nonconvex optimization landscapes
研究了光滑参数化如何改变非凸优化问题的景观，发现参数化本身而非代价函数决定了局部极小点和临界点之间的关系，为低秩矩阵分解、半定规划、神经网络训练等问题提供了新保证。
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.