The Complexity of Gradient Descent: CLS = PPAD boolean AND PLS
STOC '21: PROCEEDINGS OF THE 53RD ANNUAL ACM SIGACT SYMPOSIUM ON THEORY OF COMPUTING(2023)
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 boolean AND 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 boolean AND PLS and contains many interesting problems - is itself equal to PPAD boolean AND PLS.
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关键词
TFNP, computational complexity, continuous optimization
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