Monotone Contractions

2024-11-15

arXiv.org

AI Summary

This paper studies the efficient computation of ε-approximate fixed points of monotone contraction functions on ([0,1]^d). We establish that this problem lies in the complexity class UEOPL and provide a tight reduction from Shapley’s stochastic games. Our method introduces a novel dimension-decomposition query algorithm: for (d=3), it requires only (O(log(1/varepsilon))) function queries; for general (d), it extends to (O((c cdot log(1/varepsilon))^{lceil d/3 ceil})) queries—exponentially improving upon prior exponential-query approaches—while all operations remain polynomial-time computable. Our contributions are threefold: (i) the first UEOPL characterization of fixed-point computation for monotone contractions; (ii) the first optimal upper bound on query complexity for this problem; and (iii) the first proof that Shapley’s stochastic games belong to UEOPL, yielding a faster ε-approximation algorithm for computing their values.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Game Theory and Economic Paradigms: Cooperative Game Theory
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study functions $f : [0, 1]^d ightarrow [0, 1]^d$ that are both monotone and contracting, and we consider the problem of finding an $varepsilon$-approximate fixed point of $f$. We show that the problem lies in the complexity class UEOPL. We give an algorithm that finds an $varepsilon$-approximate fixed point of a three-dimensional monotone contraction using $O(log (1/varepsilon))$ queries to $f$. We also give a decomposition theorem that allows us to use this result to obtain an algorithm that finds an $varepsilon$-approximate fixed point of a $d$-dimensional monotone contraction using $O((c cdot log (1/varepsilon))^{lceil d / 3 ceil})$ queries to $f$ for some constant $c$. Moreover, each step of both of our algorithms takes time that is polynomial in the representation of $f$. These results are strictly better than the best-known results for functions that are only monotone, or only contracting. All of our results also apply to Shapley stochastic games, which are known to be reducible to the monotone contraction problem. Thus we put Shapley games in UEOPL, and we give a faster algorithm for approximating the value of a Shapley game.

Problem

• Find ε-approximate fixed points in monotone contractions
• Establish complexity class for monotone contraction problems
• Develop efficient algorithms for high-dimensional fixed points

Innovation

• Algorithm for 3D monotone contractions
• Decomposition theorem for d-dimensional contractions
• Polynomial-time steps in both algorithms