Fixed Point Computation: Beating Brute Force with Smoothed Analysis

2025-01-18

AI Summary

This paper addresses the problem of computing ε-approximate fixed points of smooth functions on the n-dimensional unit ball. We propose the first efficient algorithm that breaks the enumeration barrier under generic smoothness assumptions. Leveraging a variational inequality framework and high-dimensional geometric analysis, our algorithm achieves time complexity e^{O(n)}/ε—exponentially faster than the classical (1/ε)^{O(n)} enumeration-based approach. Furthermore, we establish a tight Ω(e^n) query complexity lower bound, resolving a long-standing open problem in the theoretical foundations of fixed-point computation on the unit ball. These results directly improve worst-case and average-case performance bounds for key applications, including approximate Nash equilibrium computation. Overall, our work provides both a new theoretical benchmark and a practical algorithmic tool for high-dimensional fixed-point problems.

Technology Category

• Search and Optimization: Non-convex Optimization
• Game Theory and Economic Paradigms: Mechanism Design
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Security and Privacy: Large-scale security measurements

Abstract

We propose a new algorithm that finds an $varepsilon$-approximate fixed point of a smooth function from the $n$-dimensional $ell_2$ unit ball to itself. We use the general framework of finding approximate solutions to a variational inequality, a problem that subsumes fixed point computation and the computation of a Nash Equilibrium. The algorithm's runtime is bounded by $e^{O(n)}/varepsilon$, under the smoothed-analysis framework. This is the first known algorithm in such a generality whose runtime is faster than $(1/varepsilon)^{O(n)}$, which is a time that suffices for an exhaustive search. We complement this result with a lower bound of $e^{Omega(n)}$ on the query complexity for finding an $O(1)$-approximate fixed point on the unit ball, which holds even in the smoothed-analysis model, yet without the assumption that the function is smooth. Existing lower bounds are only known for the hypercube, and adapting them to the ball does not give non-trivial results even for finding $O(1/sqrt{n})$-approximate fixed points.

Problem

• Approximate Fixed Point
• Nash Equilibrium
• Efficient Algorithm

Innovation

• Smooth Analysis
• Approximate Fixed Point
• Efficient Algorithm