Certifying the Absence of Spurious Local Minima at Infinity
Certifying the Absence of Spurious Local Minima at Infinity
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DOI：
10.1137/22m1479531
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发表时间：
2023-03
期刊：
影响因子：
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通讯作者：
C. Josz;Xiaopeng Li
中科院分区：
文献类型：
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作者：
C. Josz;Xiaopeng Li
在求解非凸无约束优化问题的全局最优解时，要求每个局部最优解都是全局最优解。这种没有虚假局部极小值的特性在当今的各种感兴趣的问题中都是真实的，包括主成分分析，矩阵传感和线性神经网络。然而，由于这些问题是非强制性的，它们可能在无穷远处有虚假的局部最小值。用于分析优化景观的经典工具，即梯度和Hessian，无法检测到无穷远处的虚假局部极小值。在本文中，我们确定的条件，证明虚假的局部极小值在无穷远，其中之一是有界次梯度轨迹的情况下。我们检查它们在几个感兴趣的应用程序中是否有效。
When searching for global optima of nonconvex unconstrained optimization problems, it is desirable that every local minimum be a global minimum. This property of having no spurious local minima is true in various problems of interest nowadays, including principal component analysis, matrix sensing, and linear neural networks. However, since these problems are non-coercive, they may yet have spurious local minima at infinity. The classical tools used to analyze the optimization landscape, namely the gradient and the Hessian, are incapable of detecting spurious local minima at infinity. In this paper, we identify conditions that certify the absence of spurious local minima at infinity, one of which is having bounded subgradient trajectories. We check that they hold in several applications of interest.