The complexity of unsupervised learning of lexicographic preferences

TL;DR

This paper investigates the computational complexity of learning lexicographic preferences trees (LP-trees) from limited data, providing bounds on sample complexity and polynomial-time algorithms for certain subclasses.

Contribution

It offers new theoretical insights into the sample complexity and computational feasibility of learning LP-trees for user preferences.

Findings

Sample complexity is logarithmic in the number of attributes.

Empirical risk minimization for linear LP-trees is polynomial-time solvable.

Provides bounds and algorithms for preference learning in combinatorial settings.

Abstract

This paper considers the task of learning users' preferences on a combinatorial set of alternatives, as generally used by online configurators, for example. In many settings, only a set of selected alternatives during past interactions is available to the learner. Fargier et al. [2018] propose an approach to learn, in such a setting, a model of the users' preferences that ranks previously chosen alternatives as high as possible; and an algorithm to learn, in this setting, a particular model of preferences: lexicographic preferences trees (LP-trees). In this paper, we study complexity-theoretical problems related to this approach. We give an upper bound on the sample complexity of learning an LP-tree, which is logarithmic in the number of attributes. We also prove that computing the LP tree that minimises the empirical risk can be done in polynomial time when restricted to the class of linear LP-trees.