Query-Efficient Fixpoints of $\ell_p$-Contractions

TL;DR

This work studies the problem of finding $\varepsilon$-approximate fixed points of $\lambda$-contracting maps on $([0,1]^d, ||\cdot||_p)$ for $p\in[1,\infty]\cup\{\infty\}$, extending prior $\ell_\infty$-norm results to all $p$. It introduces $\ell_p$-halfspaces and proves a generalized $\ell_p$-centerpoint theorem via a Brouwer fixed-point construction, enabling a centerpoint-based, query-efficient reduction that discards a constant fraction of the search space per query. The authors achieve a unified query upper bound of $\mathcal{O}(d^2(\log\frac{1}{\varepsilon}+\log\frac{1}{1-\lambda}))$ for all $p$, and provide a grid-rounding scheme for the $\ell_1$-case, placing the discrete problem in $\mathsf{FP}^{\text{dt}}$. They also relate their results to existing $p=2$ and $p=\infty$ work, discuss CLS-completeness in general Banach spaces, and outline open problems on efficient centerpoint computation and potential time-efficient algorithms. Overall, the paper advances the understanding of contraction-fixpoint computation across $\ell_p$-geometries by combining geometric centerpoint techniques with fixed-point arguments, offering a new pathway toward polynomial-query, and potentially polynomial-time, algorithms in structured normed spaces.

Abstract

We prove that an $ε$-approximate fixpoint of a map $f:[0,1]^d\rightarrow [0,1]^d$ can be found with $\mathcal{O}(d^2(\log\frac{1}ε + \log\frac{1}{1-λ}))$ queries to $f$ if $f$ is $λ$-contracting with respect to an $\ell_p$-metric for some $p\in [1,\infty)\cup\{\infty\}$. This generalizes a recent result of Chen, Li, and Yannakakis [STOC'24] from the $\ell_\infty$-case to all $\ell_p$-metrics. Previously, all query upper bounds for $p\in [1,\infty) \setminus \{2\}$ were either exponential in $d$, $\log\frac{1}ε$, or $\log\frac{1}{1-λ}$. Chen, Li, and Yannakakis also show how to ensure that all queries to $f$ lie on a discrete grid of limited granularity in the $\ell_\infty$-case. We provide such a rounding for the $\ell_1$-case, placing an appropriately defined version of the $\ell_1$-case in $\textsf{FP}^{dt}$. To prove our results, we introduce the notion of $\ell_p$-halfspaces and generalize the classical centerpoint theorem from discrete geometry: for any $p \in [1, \infty) \cup \{\infty\}$ and any mass distribution (or point set), we prove that there exists a centerpoint $c$ such that every $\ell_p$-halfspace defined by $c$ and a normal vector contains at least a $\frac{1}{d+1}$-fraction of the mass (or points).

Figures (4)

• Figure 1: The $\ell_5$-halfspace $\mathcal{H}^{5}_{\mathbf{0},v}$ for $v=(-2,-0.5,-0.6)$ is drawn in red. The vector shown in the image is $-v$. Image created with the Desmos 3D calculator.
• Figure 2: Examples of \ref{['lem:structureoflimithalfspace']}. Two $\ell_1$-halfspaces, both are unions of rays starting at $x$, but only the left one (with non-degenerate $v$) has a boundary consisting of a union of lines through $x$.
• Figure 3: Sketch of \ref{['lem:anglesonlimithalfspaces']}. No point in the red cone is contained in the $\ell_p$-halfspace $\mathcal{H}^{p}_{x,v}$, but all points in the green cone are.
• Figure 4: Sketch of \ref{['lemma:pull_towards_zero']}. The cube $[0,1]^d$ is contained in all $\mathcal{H}^p_{x,-x}$ for $||x||_2$ large enough.

Theorems & Definitions (68)

• Definition 2.1: Contraction Map
• Theorem 2.2: Banach Fixpoint Theorem banach1922operations
• Definition 2.3: $\ell_p$-Norm
• Definition 2.4: $\varepsilon$-Approximate Fixpoint
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Definition 2.8
• Lemma 2.9
• proof