Is There a Curse of Dimensionality for Contraction Fixed Points in the Worst Case?
AbstractThis paper analyzes the complexity of the "contraction fixed point problem": compute an epsilon-approximation to the fixed point "V"*Gamma("V"*) of a contraction mapping Gamma that maps a Banach space "B-sub-d" of continuous functions of "d" variables into itself. We focus on "quasi linear contractions" where Gamma is a nonlinear functional of a finite number of conditional expectation operators. This class includes contractive Fredholm integral equations that arise in asset pricing applications and the contractive Bellman equation from dynamic programming. In the absence of further restrictions on the domain of Gamma, the quasi linear fixed point problem is subject to the curse of dimensionality, i.e., in the worst case the minimal number of function evaluations and arithmetic operations required to compute an epsilon-approximation to a fixed point "V"* is an element of "B-sub-d" increases exponentially in "d". We show that the curse of dimensionality disappears if the domain of Gamma has additional special structure. We identify a particular type of special structure for which the problem is "strongly tractable" even in the worst case, i.e., the number of function evaluations and arithmetic operations needed to compute an epsilon-approximation of "V"* is bounded by "C"epsilon-super- - "p" where "C" and "p" are constants independent of "d". We present examples of economic problems that have this type of special structure including a class of rational expectations asset pricing problems for which the optimal exponent "p"1 is nearly achieved. Copyright The Econometric Society 2002.
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Bibliographic InfoArticle provided by Econometric Society in its journal Econometrica.
Volume (Year): 70 (2002)
Issue (Month): 1 (January)
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- Manuel Santos & John Rust, . "Convergence Properties of Policy Iteration," Working Papers 2133377, Department of Economics, W. P. Carey School of Business, Arizona State University.
- Magnac, Thierry, 2008. "Comment on the Identification Power in Games by Andres Aradilla-Lopez and Elie Tamer," Open Access publications from University of Toulouse 1 Capitole http://neeo.univ-tlse1.fr, University of Toulouse 1 Capitole.
- Aguirregabiria, Victor, 2005.
"Nonparametric identification of behavioral responses to counterfactual policy interventions in dynamic discrete decision processes,"
Elsevier, vol. 87(3), pages 393-398, June.
- Victor Aguirregabiria, 2004. "Nonparametric Identification of Behavioral Responses to Counterfactual Policy Interventions in Dynamic Discrete Decision Processes," Econometrics 0408004, EconWPA.
- Joao Macieira, 2010. "Oblivious Equilibrium in Dynamic Discrete Games," 2010 Meeting Papers 680, Society for Economic Dynamics.
- George Hall and John Rust, Yale University, 2001.
"Econometric Methods for Endogenously Sampled Time Series: The Case of Commodity Price Speculation in the Steel Market,"
Computing in Economics and Finance 2001
274, Society for Computational Economics.
- George Hall & John Rust, 2002. "Econometric Methods for Endogenously Sampled Time Series: The Case of Commodity Price Speculation in the Steel Market," NBER Technical Working Papers 0278, National Bureau of Economic Research, Inc.
- George Hall & John Rust, 2002. "Econometric Methods for Endogenously Sampled Time Series: The Case of Commodity Price Speculation in the Steel Market," Cowles Foundation Discussion Papers 1376, Cowles Foundation for Research in Economics, Yale University.
- Alexei Onatski & Noah Williams, 2003.
"Modeling Model Uncertainty,"
NBER Working Papers
9566, National Bureau of Economic Research, Inc.
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