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

TL;DR

This paper characterizes the complexity of gradient descent on convex polytopal domains, showing it equals the intersection of PPAD and PLS, and establishes the PPAD ∩ PLS-completeness of finding KKT points in a simple setting.

Contribution

It proves that gradient descent search problems are exactly the intersection of PPAD and PLS, and introduces a PPAD ∩ PLS-complete problem involving KKT points in two dimensions.

Findings

Gradient descent on convex polytopes is PPAD ∩ PLS-complete.

Computing KKT points in 2D is PPAD ∩ PLS-complete.

The class CLS equals PPAD ∩ PLS.

Abstract

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.