Banach's fixed point theorem for contraction maps has been widely used to analyze the convergence of iterative methods in nonconvex problems. It is a common experience, however, that iterative maps fail to be globally contracting under the natural metric in their domain, making the applicability of Banach's theorem limited. We explore how generally we can apply Banach's fixed point theorem to establish the convergence of iterative methods when pairing it with carefully designed metrics.
Our first result is a strong converse of Banach's theorem, showing that it is a universal analysis tool for establishing global convergence of iterative methods to unique fixed points, and for bounding their convergence rate. In other words, we show that, whenever an iterative map globally converges to a unique fixed point, there exists a metric under which the iterative map is contracting and which can be used to bound the number of iterations until convergence. We illustrate our approach in the widely used power method, providing a new way of bounding its convergence rate through contraction arguments.
We next consider the computational complexity of Banach's fixed point theorem. Making the proof of our converse theorem constructive, we show that computing a fixed point whose existence is guaranteed by Banach's fixed point theorem is CLScomplete. We thus provide the first natural complete problem for the class CLS, which was defined in [Daskalakis, Papadimitriou 2011] to capture the complexity of problems such as Pmatrix LCP, computing KKTpoints, and finding mixed Nash equilibria in congestion and network coordination games.
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A Converse to Banach's Fixed Point Theorem and its CLS Completeness
Banach's fixed point theorem for contraction maps has been widely used to analyze the convergence of iterative methods in nonconvex problems. It is a common experience, however, that iterative maps fail to be globally contracting under the natural metric in their domain, making the applicability of Banach's theorem limited. We explore how generally we can apply Banach's fixed point theorem to establish the convergence of iterative methods when pairing it with carefully designed metrics. Our first result is a strong converse of Banach's theorem, showing that it is a universal analysis tool for establishing global convergence of iterative methods to unique fixed points, and for bounding their convergence rate. In other words, we show that, whenever an iterative map globally converges to a unique fixed point, there exists a metric under which the iterative map is contracting and which can be used to bound the number of iterations until convergence. We illustrate our approach in the widely used power method, providing a new way of bounding its convergence rate through contraction arguments.
We next consider the computational complexity of Banach's fixed point theorem. Making the proof of our converse theorem constructive, we show that computing a fixed point whose existence is guaranteed by Banach's fixed point theorem is CLScomplete. We thus provide the first natural complete problem for the class CLS, which was defined in [DP11] to capture the complexity of problems such as Pmatrix LCP, computing KKTpoints, and finding mixed Nash equilibria in congestion and network coordination games.
more »
« less
 NSFPAR ID:
 10079724
 Date Published:
 Journal Name:
 Proceedings of the annual ACM Symposium on Theory of Computing
 ISSN:
 07378017
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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