Efficient Algorithm for Computing Approximate Fixed Points of Contraction Maps

Core Concepts

We give an O(k^2 log(1/ε))-query algorithm for finding an ε-fixed point of a (1-γ)-contraction map over the k-cube [0,1]^k under the ℓ∞-norm.

Abstract

The paper presents an efficient algorithm for computing approximate fixed points of contraction maps over the k-cube [0,1]^k under the ℓ∞-norm.
Key highlights:

The authors give an O(k^2 log(1/ε))-query algorithm for finding an ε-fixed point of a (1-γ)-contraction map. This improves upon previous exponential upper bounds.
The algorithm works by discretizing the search space and efficiently cutting down the set of candidate solutions at each iteration.
The authors prove the existence of a "balanced point" in the discretized grid that allows for effective pruning of the candidate set.
The algorithm also extends to finding strong ε-fixed points and computing fixed points of non-expansive maps.
In contrast to many other fixed point problems, the authors show that the query complexity of the total search version is the same as the promise version.
The paper provides a significant advance in the understanding of the query complexity of computing fixed points of contraction maps, with potential applications in areas like optimization, game theory, and verification.

Stats

n := ⌈16/(γε)⌉
g(x) := n · f(x/n)

Quotes

"We give an algorithm for finding an ε-fixed point of a contraction map f : [0, 1]^k ↦→[0, 1]^k under the ℓ∞-norm with query complexity O(k^2 log(1/ε))."
"Crucially, in all these reductions, both the approximation parameter ε and the contraction parameter γ are inversely exponential in the input size. Therefore, efficient algorithms in this context are those with a complexity upper bound that is polynomial in k, log(1/ε) and log(1/γ)."

Key Insights Distilled From

Computing a Fixed Point of Contraction Maps in Polynomial Queries

by Xi Chen,Yuha... at arxiv.org 04-01-2024

https://arxiv.org/pdf/2403.19911.pdf

Deeper Inquiries

How can the time complexity of the algorithm be improved to match the polynomial query complexity

To improve the time complexity of the algorithm to match the polynomial query complexity, we can explore more efficient search strategies and optimization techniques. One approach could be to incorporate heuristics or machine learning algorithms to guide the search process towards potential balanced points more effectively. Additionally, refining the criteria for selecting query points and optimizing the search space could help reduce the number of iterations required for convergence. By fine-tuning the algorithm's decision-making process and leveraging advanced search algorithms, we can aim to achieve a balance between query complexity and time complexity, ultimately improving the overall efficiency of the algorithm.

Can the techniques developed in this paper be extended to other fixed point problems beyond contraction maps

The techniques developed in this paper for finding fixed points of contraction maps can potentially be extended to other fixed point problems in various domains. By adapting the algorithm to different types of maps and metric spaces, we can explore applications in optimization, machine learning, and mathematical modeling. For example, the approach could be applied to non-expansive maps, convex optimization problems, or even nonlinear systems of equations. By generalizing the methodology and adapting it to different problem settings, we can address a broader range of fixed point problems beyond contraction maps.

What are the implications of these results for the computational complexity of related problems in areas like game theory and verification

The results presented in this paper have significant implications for the computational complexity of related problems in areas such as game theory and verification. By demonstrating a polynomial query complexity for finding fixed points of contraction maps, the study opens up possibilities for more efficient algorithms in dynamic programming, reinforcement learning, and optimization. These results could lead to advancements in solving complex decision-making problems, analyzing game strategies, and verifying system behaviors. The computational tractability of fixed point problems in these domains could pave the way for developing faster algorithms and enhancing the scalability of existing models and frameworks.