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Monotone Contractions

Eleni Batziou, John Fearnley, Spencer Gordon, Ruta Mehta, Rahul Savani

TL;DR

The paper analyzes monotone contractions f: [0,1]^d -> [0,1]^d and focuses on efficiently finding ε-approximate fixed points leveraging monotone structure plus contraction. It introduces DMAC, a discrete surrogate on a grid, and proves DMAC lies in UEOPL by reducing to OPDC, with a verifiable least-fixed-point test to ensure a unique, well-defined solution. A decomposition theorem shows that solving in lower dimensions yields a product-like bound, enabling a 3D algorithm with O(log(1/ε)) queries and polynomial-time per step; this extends to d-dimensional monotone contractions with O((c log(1/ε))^{⌈d/3⌉}) queries. The approach improves upon prior results for functions that are solely monotone or solely contracting, and directly applies to Shapley stochastic games, placing them in UEOPL and offering faster ε-approximation of game values. Overall, the work reveals that monotone contractions are computationally easier (in UEOPL) than either monotone alone or contracting alone, via a discrete-to-continuous pipeline and a robust decomposition framework.

Abstract

We study functions $f : [0, 1]^d \rightarrow [0, 1]^d$ that are both monotone and contracting, and we consider the problem of finding an $\varepsilon$-approximate fixed point of $f$. We show that the problem lies in the complexity class UEOPL. We give an algorithm that finds an $\varepsilon$-approximate fixed point of a three-dimensional monotone contraction using $O(\log (1/\varepsilon))$ queries to $f$. We also give a decomposition theorem that allows us to use this result to obtain an algorithm that finds an $\varepsilon$-approximate fixed point of a $d$-dimensional monotone contraction using $O((c \cdot \log (1/\varepsilon))^{\lceil d / 3 \rceil})$ queries to $f$ for some constant $c$. Moreover, each step of both of our algorithms takes time that is polynomial in the representation of $f$. These results are strictly better than the best-known results for functions that are only monotone, or only contracting. All of our results also apply to Shapley stochastic games, which are known to be reducible to the monotone contraction problem. Thus we put Shapley games in UEOPL, and we give a faster algorithm for approximating the value of a Shapley game.

Monotone Contractions

TL;DR

The paper analyzes monotone contractions f: [0,1]^d -> [0,1]^d and focuses on efficiently finding ε-approximate fixed points leveraging monotone structure plus contraction. It introduces DMAC, a discrete surrogate on a grid, and proves DMAC lies in UEOPL by reducing to OPDC, with a verifiable least-fixed-point test to ensure a unique, well-defined solution. A decomposition theorem shows that solving in lower dimensions yields a product-like bound, enabling a 3D algorithm with O(log(1/ε)) queries and polynomial-time per step; this extends to d-dimensional monotone contractions with O((c log(1/ε))^{⌈d/3⌉}) queries. The approach improves upon prior results for functions that are solely monotone or solely contracting, and directly applies to Shapley stochastic games, placing them in UEOPL and offering faster ε-approximation of game values. Overall, the work reveals that monotone contractions are computationally easier (in UEOPL) than either monotone alone or contracting alone, via a discrete-to-continuous pipeline and a robust decomposition framework.

Abstract

We study functions that are both monotone and contracting, and we consider the problem of finding an -approximate fixed point of . We show that the problem lies in the complexity class UEOPL. We give an algorithm that finds an -approximate fixed point of a three-dimensional monotone contraction using queries to . We also give a decomposition theorem that allows us to use this result to obtain an algorithm that finds an -approximate fixed point of a -dimensional monotone contraction using queries to for some constant . Moreover, each step of both of our algorithms takes time that is polynomial in the representation of . These results are strictly better than the best-known results for functions that are only monotone, or only contracting. All of our results also apply to Shapley stochastic games, which are known to be reducible to the monotone contraction problem. Thus we put Shapley games in UEOPL, and we give a faster algorithm for approximating the value of a Shapley game.

Paper Structure

This paper contains 109 sections, 86 theorems, 66 equations, 114 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Shapley games.
  3. Monotone functions.
  4. Contracting functions.
  5. Our contribution.
  6. Techniques.
  7. MonotoneContraction and DMAC: Problem Definitions
  8. Technical Overview
  9. UEOPL containment of DMAC
  10. Verifiable least fixed points.
  11. Handling violations.
  12. Paths of fixed points.
  13. Algorithmic Results for DMAC
  14. 1DUniqueDMAC and Surfaces
  15. 1DUniqueDMAC.
  16. ...and 94 more sections

Key Result

Lemma 6

A point $y \in G$ is a D1 solution of the $\mathsf{DMAC}$ instance arising from our reduction only if $\hat{y} = \hat{\varepsilon} \cdot y \in \hat{G}$ is an M1 solution of the original $\mathsf{MonotoneContraction}$ instance.

Figures (114)

  • Figure 1: Surfaces in a two-dimensional $\mathsf{DMAC}$ instance.
  • Figure 4: The structure of $\mathop{\mathrm{Up}}\nolimits(f)$ in a two-dimensional slice.
  • Figure 5: A CB-config, and the corresponding region that is ruled out.
  • Figure 6: An example in which $\mathop{\mathrm{Up}}\nolimits(f) \cap s_i$ lies in a lobe.
  • Figure 7: Handling left lobes.
  • ...and 109 more figures

Theorems & Definitions (170)

  • Definition 1: $\mathsf{MonotoneContraction}\xspace$
  • Definition 2: Discrete Monotone Approximate Contraction ($\mathsf{DMAC}\xspace$)
  • Definition 3
  • Definition 4
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Lemma 8
  • proof
  • ...and 160 more