The Complexity of Gradient Descent: CLS = PPAD $\cap$ PLS
This work resolves a foundational complexity theory problem by establishing the equality of CLS and PPAD ∩ PLS, impacting theoretical computer science and optimization.
The paper tackles the problem of classifying the complexity of gradient descent on bounded convex polytopal domains, showing that this class equals PPAD ∩ PLS, and proves that computing a KKT point over [0,1]^2 is PPAD ∩ PLS-complete, the first non-artificial problem with this completeness result.
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.