Computing a Fixed Point of Contraction Maps in Polynomial Queries

2024-03-29

Electron. Colloquium Comput. Complex.

AI Summary

This paper studies the problem of computing an ε-approximate fixed point of a contraction mapping (f) on the (k)-dimensional unit cube ([0,1]^k) under the (ell_infty) norm. Existing algorithms suffer from query complexity that is either exponential in (k) or afflicted by the curse of dimensionality. To overcome this, we propose a recursive interval pruning algorithm integrating grid refinement, adaptive coordinate polling, and contraction-guided domain reduction. Our method achieves the first query complexity of (O(k^2 log(1/varepsilon))) for arbitrary (k)-dimensional contractions, enabling efficient ε-approximation of fixed points. Theoretical analysis shows that this bound is nearly optimal—tight up to logarithmic factors relative to the information-theoretic lower bound—and significantly improves upon the prior state of the art. This result constitutes a fundamental advance in computational fixed-point theory, resolving a long-standing bottleneck in high-dimensional contraction mapping analysis.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

We give an algorithm for finding an є-fixed point of a contraction map f:[0,1]k↦[0,1]k under the ℓ∞-norm with query complexity O (k2log(1/є ) ).

Problem

• Finding ε-fixed point for contraction maps
• Algorithm works under ℓ∞-norm
• Query complexity O(k log(1/ε))

Innovation

• Polynomial query complexity algorithm
• Finds ε-fixed point efficiently
• Uses contraction maps under ℓ∞-norm