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

Andrei Feodorov, Sebastian Haslebacher

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.

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

Abstract

We present a new algorithm for finding an -approximate fixed point of an -contracting function . Our algorithm is based on the query-efficient algorithm by Chen, Li, and Yannakakis (STOC 2024), but comes with an improved upper bound of on the overall runtime (while still being query-efficient). By combining this with a recent decomposition theorem for -contracting functions, we then describe a second algorithm that finds an -approximate fixed point in queries and time. The key observation here is that decomposition theorems such as the one for -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 , 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 -approximately solving Shapley stochastic games.

Paper Structure

This paper contains 22 sections, 26 theorems, 39 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Technical Overview
  3. Preliminaries
  4. The Proof of \ref{['thm:main_result_1']}
  5. $\ell_\infty$-Halfspaces and $\ell_\infty$-Centerpoints
  6. Query-Efficient Algorithm
  7. Correctness and Query Complexity
  8. Valid Search Space
  9. Finding Centerpoints and \ref{['thm:main_result_1']}
  10. Bit Complexity and Precision
  11. Proof of \ref{['thm:main_result_2']}
  12. Finding Approximate Centerpoints
  13. Overview of Ideas
  14. The Pulling Operation
  15. Proof of \ref{['thm:approximate_centerpoint_theorem']}
  16. ...and 7 more sections

Key Result

Theorem 1.1

Given $\varepsilon > 0$ and query-access to a $\lambda$-contracting (in the $\ell_\infty$-metric) function $f : [0, 1]^d \rightarrow [0, 1]^d$, there is an algorithm that finds an $\varepsilon$-approximate fixed point $x \in [0, 1]^d$ (i.e. with $\|x - f(x)\|_\infty \leq \varepsilon$) in $\mathcal{O

Figures (8)

  • Figure 1: An $\ell_\infty$-halfspace $H_{v}(0)$ around the origin with $v = (-1, -1, -1)$ in $\mathbb{R}^3$ (coordinate axes drawn in gray). In particular, observe that $\ell_\infty$-halfspaces are in general not convex. Figure created with Desmos 3D calculator (https://www.desmos.com/3d).
  • Figure 2: A pyramid around the origin in $\mathbb{R}^3$. Observe that all pyramids are convex. Figure created with Desmos 3D calculator (https://www.desmos.com/3d).
  • Figure 3: A simple example of \ref{['lemma:pulling']} for $d = 2$. Observe how pulling $c$ in the direction $u = (0, 1)$ (indicated by the arrow) implies that $P^+_2(c)$ may lose points $x$ and $y$ to $P^-_1(c)$ and $P^+_1(c)$ respectively (or it could even lose them to $P^-_2(c)$ if we pull far enough). Similarly, $P^-_1(c)$ might lose point $z$ to $P^-_2$. However, $P^-_2$ will not lose any points.
  • Figure 4: \ref{['lemma:pulling_upper_bound']} gives a sufficient bound for how far we have to pull $c$ in the direction $(1, 0)$ until $P^-_1(c)$ contains $[0, 1]^2$.
  • Figure 5: A two-dimensional sketch of one iteration in the proof of \ref{['lemma:balancing_pyramids']}. Observe that $\mathop{\mathrm{vol}}\nolimits(X \cap P^-_2(c))$ is much larger than $\mathop{\mathrm{vol}}\nolimits(X \cap P^-_1(c))$, and thus we want to pull $c$ in the direction $(1, 0)$ (indicated by the arrow) to balance this out. It is not hard to see that in two dimensions, one iteration of the algorithm in \ref{['lemma:balancing_pyramids']} actually suffices to balance the two negative pyramids.
  • ...and 3 more figures

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1: $\ell_\infty$-Centerpoint Theorem chenComputingFixedPoint2025
  • proof
  • Lemma 2.1: haslebacherQueryEfficientFixpointsLpContractions2025
  • proof
  • Definition 2.3: Valid Search Space
  • Lemma 2.4: Volume Computation
  • Theorem 2.5: Finding Approximate Centerpoints
  • Theorem 2.5
  • ...and 37 more