Query-Efficient Fixpoints of $ell_p$-Contractions

2025-03-20

AI Summary

This paper studies the efficient query problem for ε-approximate fixed points of λ-contractive mappings (f) on the high-dimensional unit cube ([0,1]^d) under the (ell_p) metric ((p in [1,infty])). Prior to this work, query complexity was exponential in (d), (log(1/varepsilon)), or (log(1/(1-lambda))) for (p otin {2,infty}). Methodologically, we unify and extend the (ell_infty) fixed-point query framework to all (ell_p) metrics; introduce the notion of (ell_p)-halfspaces and establish a generalized centerpoint theorem; and design a discrete grid-based query scheme for (ell_1), placing it in the FP(^{ ext{d}t}) complexity class. Our main contribution is reducing the query complexity to (mathcal{O}(d^2(log 1/varepsilon + log 1/(1-lambda)))), achieving the first polynomial-time algorithm—breaking the prior exponential barrier—and providing the first finite-granularity implementable algorithm for (ell_1).

Technology Category

• Search and Optimization: Learning to Search
• Machine Learning: Learning with Manifolds
• Data Mining & Knowledge Management: Intelligent Query Processing

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

Abstract

We prove that an $epsilon$-approximate fixpoint of a map $f:[0,1]^d ightarrow [0,1]^d$ can be found with $mathcal{O}(d^2(logfrac{1}{epsilon} + logfrac{1}{1-lambda}))$ queries to $f$ if $f$ is $lambda$-contracting with respect to an $ell_p$-metric for some $pin [1,infty)cup{infty}$. 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 $pin [1,infty) setminus {2}$ were either exponential in $d$, $logfrac{1}{epsilon}$, or $logfrac{1}{1-lambda}$. 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 $ extsf{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).

Problem

• Efficiently find ε-approximate fixpoints for ℓp-contractions
• Generalize query upper bounds from ℓ∞ to all ℓp-metrics
• Extend discrete grid rounding techniques to ℓ1-case

Innovation

• Efficient query algorithm for $ell_p$-contractions
• Generalized centerpoint theorem for $ell_p$-halfspaces
• Discrete grid rounding for $ell_1$-case