测地优化中近期方法的障碍
Barriers for recent methods in geodesic optimization
AI总结：
本文研究了多维数组缩放和张量缩放等测地优化问题，通过构造具有指数级直径界限和双指数条件数的实例，证明了信赖域等方法无法保证多项式时间求解，并指出 margin 和 gap 呈指数级小，从而激发了对不依赖直径界限的内点法等高级算法的寻找。
摘要：
我们研究了一类优化问题，包括矩阵缩放、矩阵平衡、多维数组缩放、算子缩放和张量缩放，这些问题在理论和实践中频繁出现。其中一些问题，如矩阵和数组缩放，在欧氏意义上是凸的，但其他问题如算子缩放和张量缩放在不同的黎曼流形上是测地凸的。已知信赖域方法（包括盒约束牛顿法）能够非常快速地为矩阵缩放和矩阵平衡生成高精度解（Cohen 等人，FOCS 2017；Allen-Zhu 等人，FOCS 2017），并为算子缩放等测地凸问题导出多项式时间算法（Garg 等人，STOC 2018；Bürgisser 等人，FOCS 2019）。人们不禁要问，这些保证是否也适用于多维数组缩放和张量缩放。我们通过展示具有指数级直径界限的实例证明情况并非如此：我们构造了 3 维数组缩放和 3 张量缩放的多项式规模实例，其所有近似解都具有双指数级的条件数。此外，我们研究了被称为 margin 和 gap 的凸几何复杂度概念，这些概念用于界定此类问题所有现有优化算法的运行时间。我们表明，对于包括数组缩放、张量缩放和多项式缩放在内的几个问题，margin 和 gap 呈指数级小。我们的结果表明，仅基于直径界限不可能为张量缩放证明多项式运行时间界限。因此，我们的工作激发了对测地凸优化中不依赖多项式直径界限的更复杂算法（如内点法）类似物的寻找。
英文摘要：
We study a class of optimization problems including matrix scaling, matrix balancing, multidimensional array scaling, operator scaling, and tensor scaling that arise frequently in theory and in practice. Some of these problems, such as matrix and array scaling, are convex in the Euclidean sense, but others such as operator scaling and tensor scaling are geodesically convex on a different Riemannian manifold. Trust region methods, which include box-constrained Newton's method, are known to produce high precision solutions very quickly for matrix scaling and matrix balancing (Cohen et. al., FOCS 2017, Allen-Zhu et. al. FOCS 2017), and result in polynomial time algorithms for some geodesically convex problems like operator scaling (Garg et. al. STOC 2018, Bürgisser et. al. FOCS 2019). One is led to ask whether these guarantees also hold for multidimensional array scaling and tensor scaling. We show that this is not the case by exhibiting instances with exponential diameter bound: we construct polynomial-size instances of 3-dimensional array scaling and 3-tensor scaling whose approximate solutions all have doubly exponential condition number. Moreover, we study convex-geometric notions of complexity known as margin and gap, which are used to bound the running times of all existing optimization algorithms for such problems. We show that margin and gap are exponentially small for several problems including array scaling, tensor scaling and polynomial scaling. Our results suggest that it is impossible to prove polynomial running time bounds for tensor scaling based on diameter bounds alone. Therefore, our work motivates the search for analogues of more sophisticated algorithms, such as interior point methods, for geodesically convex optimization that do not rely on polynomial diameter bounds.
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