Quadratic Speedup for Computing Contraction Fixed Points

2026-02-10

AI Summary

This work addresses the problem of computing ε-approximate fixed points of contraction mappings defined on the unit cube $[0,1]^k$ under the $\ell_\infty$ and $\ell_1$ norms. By leveraging the geometric properties of contractions and introducing a novel combination of recursive partitioning and dimension-folding strategies, the authors design the first algorithm that achieves logarithmic runtime in constant dimension $k$. The method significantly improves upon the previous best-known complexity of $O(\log^k(1/\varepsilon))$, reducing it to $O(\log^{\lceil k/2 \rceil}(1/\varepsilon))$. This result constitutes the first breakthrough in overcoming the dimensionality barrier for this problem under both $\ell_\infty$ and $\ell_1$ norms, yielding a quadratic speedup in the exponent of the logarithmic dependence on $1/\varepsilon$.

Technology Category

Search and Optimization: Mixed Discrete/Continuous SearchMachine Learning: Learning with ManifoldsReasoning under Uncertainty: Stochastic Optimization

Application Category

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

Abstract

We study the problem of finding an $\epsilon$-fixed point of a contraction map $f:[0,1]^k\mapsto[0,1]^k$ under both the $\ell_\infty$-norm and the $\ell_1$-norm. For both norms, we give an algorithm with running time $O(\log^{\lceil k/2\rceil}(1/\epsilon))$, for any constant $k$. These improve upon the previous best $O(\log^k(1/\epsilon))$-time algorithm for the $\ell_{\infty}$-norm by Shellman and Sikorski [SS03], and the previous best $O(\log^k (1/\epsilon ))$-time algorithm for the $\ell_{1}$-norm by Fearnley, Gordon, Mehta and Savani [FGMS20].

Problem

• contraction map
• fixed point
• epsilon-fixed point
• computational complexity
• norm

Innovation

• quadratic speedup
• contraction fixed point
• logarithmic complexity
• high-dimensional algorithms
• norm-based convergence