Learning to Partition using Score Based Compatibilities

TL;DR

This paper introduces algorithms for learning optimal user groupings based on compatibilities, analyzing computational complexity, and proposing a score-based structure that enables polynomial solutions and an online PAC learning algorithm, validated on datasets.

Contribution

It proposes a novel score-based structure for user compatibilities, enabling polynomial-time solutions and an online PAC algorithm for optimal partitioning.

Findings

Score structure makes many problems polynomial-time solvable.

Optimal groupings relate to homophilous and heterophilous partitions.

The online PAC algorithm performs well on datasets.

Abstract

We study the problem of learning to partition users into groups, where one must learn the compatibilities between the users to achieve optimal groupings. We define four natural objectives that optimize for average and worst case compatibilities and propose new algorithms for adaptively learning optimal groupings. When we do not impose any structure on the compatibilities, we show that the group formation objectives considered are $NP$ hard to solve and we either give approximation guarantees or prove inapproximability results. We then introduce an elegant structure, namely that of \textit{intrinsic scores}, that makes many of these problems polynomial time solvable. We explicitly characterize the optimal groupings under this structure and show that the optimal solutions are related to \emph{homophilous} and \emph{heterophilous} partitions, well-studied in the psychology literature. For one of the four objectives, we show $NP$ hardness under the score structure and give a $\frac{1}{2}$ approximation algorithm for which no constant approximation was known thus far. Finally, under the score structure, we propose an online low sample complexity PAC algorithm for learning the optimal partition. We demonstrate the efficacy of the proposed algorithm on synthetic and real world datasets.