Quadratic Speedup for Computing Contraction Fixed Points

TL;DR

This work studies the problem of finding $\varepsilon$-fixed points of $(1-\gamma)$-contraction maps on $[0,1]^k$ under the $\ell_\infty$ and $\ell_1$ norms in a black-box setting. It introduces the generalized nonexpansive problem $\textsc{NonExp}_\infty^\dagger(\varepsilon,k)$ and proves a decomposition theorem that enables a quadratic speedup, achieving $O(\log^{\lceil k/2\rceil}(1/\varepsilon))$ time for $\textsc{NonExp}_\infty(\varepsilon,k)$ and $\textsc{NonExp}_1(\varepsilon,k)$ for constant $k$, improving the prior $O(\log^k(1/\varepsilon))$ bounds. For the $\ell_1$-norm, it reduces to a contraction problem via $g(x)=(1-\varepsilon/(2k))f(x)$ and employs a recursive, dominating-coordinate strategy to obtain $O(k\log^{\lceil k/2\rceil}(1/(\varepsilon\gamma)))$ time. The results advance the algorithmic boundary for contraction fixed points in fixed dimensions and raise natural open questions about the $k=3$ case and time–query trade-offs in this landscape.

Abstract

We study the problem of finding an $ε$-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 $O(\log^{\lceil k/2\rceil}(1/ε))$, for any constant $k$. These improve upon the previous best $O(\log^k(1/ε))$-time algorithm for the $\ell_{\infty}$-norm by Shellman and Sikorski [SS03], and the previous best $O(\log^k (1/ε))$-time algorithm for the $\ell_{1}$-norm by Fearnley, Gordon, Mehta and Savani [FGMS20].

Theorems & Definitions (18)

• Theorem 1
• Theorem 2
• Definition 1: Contraction
• Definition 2: $\textsc{Contraction}_p(\varepsilon,\gamma,k)$
• Definition 3: $\textsc{NonExp}_p(\varepsilon,k)$
• Definition 4: $\textsc{NonExp}_\infty^\dagger(\varepsilon,k)$
• Lemma 1
• proof
• Theorem 3
• proof : Proof of \ref{['thm: linfty decomposition']}