A Converse to Banach's Fixed Point Theorem and its CLS Completeness
Name
1702.07339.pdf
Description
Submitted version
Size
364.5 KB
Format
Unknown
Checksum (MD5)
2d73f0ee206d97a33d4d0558288c8520
Author(s)
Daskalakis, Konstantinos
Tzamos, Christos
Zampetakis, Manolis
Date Issued
June 2018
Publisher
Association for Computing Machinery (ACM)
Citation
Daskalakis, Constantinos, Tzamos, Christos and Zampetakis, Manolis. 2018. "A Converse to Banach's Fixed Point Theorem and its CLS Completeness."
Version
Original manuscript
Abstract
Banach's fixed point theorem for contraction maps has been widely used to analyze the convergence of iterative methods in non-convex 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 CLS-complete. 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 P-matrix LCP, computing KKT-points, and finding mixed Nash equilibria in congestion and network coordination games.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1145/3188745.3188968
