On algorithmically boosting fixed-point computations

TL;DR

This paper introduces a general algorithmic boosting method for fixed-point computations, demonstrating exponential speedups and convergence improvements for Nash equilibria and eigenvector problems.

Contribution

It presents a novel boosting framework that accelerates fixed-point algorithms, including for Nash equilibria and eigenvector calculations, with theoretical and practical insights.

Findings

Exponential convergence speedups for linear maps.

Conversion of non-convergent iterators into convergent ones.

Constructive proof of Nash equilibrium existence via averaging.

Abstract

The main topic of this paper are algorithms for computing Nash equilibria. We cast our particular methods as instances of a general algorithmic abstraction, namely, a method we call {\em algorithmic boosting}, which is also relevant to other fixed-point computation problems. Algorithmic boosting is the principle of computing fixed points by taking (long-run) averages of iterated maps and it is a generalization of exponentiation. We first define our method in the setting of nonlinear maps. Secondly, we restrict attention to convergent linear maps (for computing dominant eigenvectors, for example, in the PageRank algorithm) and show that our algorithmic boosting method can set in motion {\em exponential speedups in the convergence rate}. Thirdly, we show that algorithmic boosting can convert a (weak) non-convergent iterator to a (strong) convergent one. We also consider a {\em variational approach} to algorithmic boosting providing tools to convert a non-convergent continuous flow to a convergent one. Then, by embedding the construction of averages in the design of the iterated map, we constructively prove the existence of Nash equilibria (and, therefore, Brouwer fixed points). We then discuss implementations of averaging and exponentiation, an important matter even for the scalar case. We finally discuss a relationship between dominant (PageRank) eigenvectors and Nash equilibria.