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 natural 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.

中文翻译：

---

梯度下降的复杂度：CLS = PPAD $\cap$ PLS

我们研究了可以通过在有界凸多边形域上执行梯度下降来解决的搜索问题，并表明该类等于两个众所周知的类的交集：PPAD 和 PLS。作为我们的主要基础技术贡献，我们表明计算域 $[0,1]^2$ 上连续可微函数的 Karush-Kuhn-Tucker (KKT) 点是 PPAD $\cap$ PLS-complete。这是本课程第一个完整显示的自然问题。我们的结果还暗示类 CLS（连续本地搜索）——由 Daskalakis 和 Papadimitriou 定义为 PPAD $\cap$ PLS 的更“自然”对应物并包含许多有趣的问题——本身等于 PPAD $\cap$ PLS。