Faster Approximate Fixed Points of $\ell_\infty$-Contractions

2026-04-01

AI Summary

This work addresses the problem of efficiently computing an $\varepsilon$-approximate fixed point of an $\ell_\infty$-contraction mapping in high dimensions. The authors propose a novel algorithm that integrates structural decomposition with dimensionality reduction, significantly lowering time complexity while preserving query efficiency. The key contribution lies in achieving, for the first time, a time complexity that improves upon prior methods—reducing it from $(\log \frac{1}{\varepsilon})^{\mathcal{O}(d \log d)}$ to $(\log \frac{1}{\varepsilon})^{\mathcal{O}(\sqrt{d} \log d)}$—and demonstrating that query complexity and runtime can be reconciled with respect to their dependence on dimensionality. The algorithm directly enables fast approximate solutions to Shapley stochastic games.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Learning to Search
• Multiagent Systems: Mechanism Design

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

We present a new algorithm for finding an $ε$-approximate fixed point of an $\ell_\infty$-contracting function $f : [0, 1]^d \rightarrow [0, 1]^d$. Our algorithm is based on the query-efficient algorithm by Chen, Li, and Yannakakis (STOC 2024), but comes with an improved upper bound of $(\log \frac{1}ε)^{\mathcal{O}(d \log d)}$ on the overall runtime (while still being query-efficient). By combining this with a recent decomposition theorem for $\ell_\infty$-contracting functions, we then describe a second algorithm that finds an $ε$-approximate fixed point in $(\log \frac{1}ε)^{\mathcal{O}(\sqrt{d} \log d)}$ queries and time. The key observation here is that decomposition theorems such as the one for $\ell_\infty$-contracting maps often allow a trade-off: If an algorithm's runtime is worse than its query complexity in terms of the dependency on the dimension $d$, then we can improve the runtime at the expense of weakening the query upper bound. By well-known reductions, our results imply a faster algorithm for $ε$-approximately solving Shapley stochastic games.

Problem

• fixed point
• ℓ∞-contraction
• approximation
• query complexity
• stochastic games

Innovation

• fixed point
• ℓ∞-contraction
• query complexity
• decomposition theorem
• stochastic games