Fixed Point Computation: Beating Brute Force with Smoothed Analysis

• URL: http://arxiv.org/abs/2501.10884v1
• Date: Sat, 18 Jan 2025 21:32:26 GMT
• Title: Fixed Point Computation: Beating Brute Force with Smoothed Analysis
• Authors: Idan Attias, Yuval Dagan, Constantinos Daskalakis, Rui Yao, Manolis Zampetakis,
• Abstract summary: We propose a new algorithm that finds an $varepsilon$-approximate fixed point of a smooth function from the $n$-dimensional $ell$ unit ball to itself. The algorithm's runtime is bounded by $eO(n)/varepsilon$, under the smoothed-analysis framework.
• Score: 28.978340288565118
• License:
• 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.

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