Influence and interaction indexes for pseudo-Boolean functions: a unified least squares approach

TL;DR

This paper introduces a unified least squares approach to analyze influence and interaction indexes for pseudo-Boolean functions, revealing new properties and relationships, especially in weighted scenarios, with implications for function reconstruction.

Contribution

It extends the least squares approximation framework to include influence and interaction indexes, introduces weighted versions, and compares their properties and reconstruction behaviors.

Findings

Banzhaf influence index derived from approximation problems

Weighted influence indexes exhibit distinct properties from non-weighted cases

Reconstruction of functions varies significantly between weighted and non-weighted scenarios

Abstract

The Banzhaf power and interaction indexes for a pseudo-Boolean function (or a cooperative game) appear naturally as leading coefficients in the standard least squares approximation of the function by a pseudo-Boolean function of a specified degree. We first observe that this property still holds if we consider approximations by pseudo-Boolean functions depending only on specified variables. We then show that the Banzhaf influence index can also be obtained from the latter approximation problem. Considering certain weighted versions of this approximation problem, we introduce a class of weighted Banzhaf influence indexes, analyze their most important properties, and point out similarities between the weighted Banzhaf influence index and the corresponding weighted Banzhaf interaction index. We also discuss the issue of reconstructing a pseudo-Boolean function from prescribed influences and point out very different behaviors in the weighted and non-weighted cases.