Fixed Point Computation: Beating Brute Force with Smoothed Analysis

TL;DR

This paper introduces a novel algorithm for finding approximate fixed points of smooth functions in high-dimensional spaces, achieving faster runtimes than brute-force methods under smoothed analysis, and establishes lower bounds on query complexity.

Contribution

It presents the first algorithm with sub-exponential runtime for fixed point computation in the smoothed analysis framework, surpassing exhaustive search complexity.

Findings

Algorithm runs in $e^{O(n)}/\varepsilon$ time, faster than $(1/\varepsilon)^{O(n)}$

Provides a lower bound of $e^{\Omega(n)}$ on query complexity for fixed points

First to analyze fixed point computation on the unit ball in smoothed analysis

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.