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.

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