Quadratic Speedup for Computing Contraction Fixed Points

TL;DR

This paper presents a new algorithm that computes contraction fixed points more efficiently, achieving a quadratic speedup over previous methods for fixed dimensions under both the $\,\ell_$-norm and the $\,\ell_1$-norm.

Contribution

The authors introduce an algorithm with $O(\log^{\lceil k/2\rceil}(1/\epsilon))$ runtime for fixed points, improving upon prior $O(\log^k(1/\epsilon))$ algorithms for both norms.

Findings

Achieves quadratic speedup for fixed point computation.

Improves runtime from $O(\log^k(1/\epsilon))$ to $O(\log^{\lceil k/2\rceil}(1/\epsilon))$.

Applicable for fixed dimensions under both $\,\ell_$-norm and $\,\ell_1$-norm.

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].