梯度下降的复杂性：CLS = PPAD $\cap$ PLS
The Complexity of Gradient Descent: CLS = PPAD $\cap$ PLS
摘要：
我们研究搜索问题，可以通过执行梯度下降有界凸多面体域上解决，并表明这一类是等于两个著名的类：PPAD和PLS的交集。作为我们的主要基础技术的贡献，我们表明，计算Karush-Kuhn-Tucker（KKT）点的连续可微功能的域$[0,1]^2$是PPAD $\cap$ PLS完成。这是这门课上第一个完整的非人工问题。我们的研究结果还意味着，类CLS（连续局部搜索）-这是由Daskalakis和Papadimitriou定义为一个更“自然”的对应PPAD $\cap$ PLS，并包含许多有趣的问题-本身等于PPAD $\cap$ PLS。
英文摘要：
We study search problems that can be solved by performing Gradient Descent on a bounded convex polytopal domain and show that this class is equal to the intersection of two well-known classes: PPAD and PLS. As our main underlying technical contribution, we show that computing a Karush-Kuhn-Tucker (KKT) point of a continuously differentiable function over the domain $[0,1]^2$ is PPAD $\cap$ PLS-complete. This is the first non-artificial problem to be shown complete for this class. Our results also imply that the class CLS (Continuous Local Search) - which was defined by Daskalakis and Papadimitriou as a more "natural" counterpart to PPAD $\cap$ PLS and contains many interesting problems - is itself equal to PPAD $\cap$ PLS.
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