Is There a Curse of Dimensionality for Contraction Fixed Points in the Worst Case?

Abstract

This paper analyzes the complexity of the contraction fixed point problem: compute an ε-approximation to the fixed point V * = Γ(V * ) of a contraction mapping Γ that maps a Banach space B d of continuous functions of d variables into itself. We focus on quasi linear contractions where Γ 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 Γ, 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 ε-approximation to a fixed point V * ∈ B d increases exponentially in d. We show that the curse of dimensionality disappears if the domain of Γ 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 ε-approximation of V * is bounded by Cε −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.

Swapping the Nested Fixed Point Algorithm: A Class of Estimators for Discrete Markov Decision Models

“…We first note that the smoothed Bellman operator has a unique fixed point. Rust, Traub, and Wozniakowski (2002) show that the additivity of the social surplus makes the smooth Bellman operator "quasilinear" and this is used to prove that is a contraction. Second, we show that if P is a fixed point of , then P is a fixed point of .…”

Advances in Numerical Dynamic Programming and New Applications

“…In this section we describe recent theoretical work on these issues. Rust (1997) and Rust et al (2002) are two recent papers that prove that the curse of dimensionality is a problem for large classes of dynamic programming problems. However, before one becomes too pessimistic about solving high-dimensional dynamic programming problems, he should remember how the curse of dimensionality is defined.…”

Comment

“…It is then a quite natural consequence to use iterations to arrive at equilibrium, because fixed-point problems are usually solved using a contraction argument and iterating the fixedpoint equation an infinite number of times (Rust, Traub, and Wozniakowski 2002). This iteration technique sets two kinds of questions, questions of the interpretation of the iterations and also of the uniqueness of solutions.…”