The Complexity of Gradient Descent: CLS = PPAD $cap$ PLS
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