Learning conditionally lexicographic preference relations Richard Booth 1 , Yann Chevaleyre 2 , J´ erˆ ome Lang 2 J´ erˆ ome Mengin 3 and Chattrakul Sombattheera 4 Abstract. We consider the problem of learning a user’s ordinal pref- erences on a multiattribute domain, assuming that her preferences are lexicographic. We introduce a general graphical representation called LP-trees which captures various natural classes of such preference relations, depending on whether the importance order between at- tributes and/or the local preferences on the domain of each attribute is conditional on the values of other attributes. For each class we determine the Vapnik-Chernovenkis dimension, the communication complexity of preference elicitation, and the complexity of identify- ing a model in the class consistent with a set of user-provided exam- ples. 1 Introduction In many applications, especially electronic commerce, it is impor- tant to be able to learn the preferences of a user on a set of alterna- tives that has a combinatorial (or multiattribute) structure: each al- ternative is a tuple of values for each of a given number of variables (or attributes). Whereas learning numerical preferences (i.e., utility functions) on multiattribute domains has been considered in various places, learning ordinal preferences (i.e., order relations) on multiat- tribute domains has been given less attention. Two streams of work are worth mentioning. First, a series of very recent works focus on the learning of prefer- ence relations enjoying some preferential independencies conditions. Passive learning of separable preferences is considered by Lang & Mengin (2009), whereas passive (resp. active) learning of acyclic CP-nets is considered by Dimopoulos et al. (2009) (resp. Koriche & Zanuttini, 2009). The second stream of work, on which we focus in this paper, is the class of lexicographic preferences, considered in Schmitt & Mar- tignon (2006); Dombi et al. (2007); Yaman et al. (2008). These works only consider very simple classes of lexicographic prefer- ences, in which both the importance order of attributes and the local preference relations on the attributes are unconditional. In this pa- per we build on these papers, and go considerably beyond, since we consider conditionally lexicographic preference relations. Consider a user who wants to buy a computer and who has a lim- ited amount of money, and a web site whose objective is to find the best computer she can afford. She prefers a laptop to a desktop, and this preference overrides everything else. There are two other at- tributes: colour, and type of optical drive (whether it has a simple DVD-reader or a powerful DVD-writer). Now, for a laptop, the next most important attribute is colour (because she does not want to be 1 University of Luxembourg. Also affiliated to Mahasarakham University, Thailand as Adjunct Lecturer. richard.booth@uni.lu 2 LAMSADE, Universit´ e Paris-Dauphine, France. {yann.chevaleyre, lang}@lamsade.dauphine.fr 3 IRIT, Universit´ e de Toulouse, France. mengin@irit.fr 4 Mahasarakham University, Thailand. chattrakul.s@msu.ac.th seen at a meeting with the usual bland, black laptop) and she prefers a flashy yellow laptop to a black one; whereas for a desktop, she prefers black to flashy yellow, and also, colour is less important than the type of optical drive. In this example, both the importance of the attributes and the local preference on the values of some attributes may be conditioned by the values of some other attributes: the rel- ative importance of colour and type of optical drive depends on the type of computer, and the preferred colour depends on the type of computer as well. In this paper we consider various classes of lexicographic prefer- ence models, where the importance relation between attributes and/or the local preference on an attribute may depend on the values of some more important attributes. In Section 2 we give a general model for lexicographic preference relations, and define six classes of lexico- graphic preference relations, only two of which have already been considered from a learning perspective. Then each of the following sections focuses on a specific kind of learning problem: in Section 3 we consider preference elicitation, a.k.a. active learning, in Sec- tion 4 we address the sample complexity of learning lexicographic preferences, and in Section 5 we consider passive learning, and more specifically model identification and approximation. 2 Lexicographic preference relations: a general model 2.1 Lexicographic preferences trees We consider a set A of n attributes. For the sake of simplicity, all attributes we consider are binary (however, our notions and results would still hold in the more general case of attributes with finite value domains). The domain of attribute X ∈A is X = {x, x}. If U ⊆A, then U is the Cartesian product of the domains of the attributes in U . Attributes and sets of attributes are denoted by upper-case Roman letters (X, Xi , A etc.). An outcome is an element of A ; outcomes are denoted by lower case Greek letters (α, β, etc.). Given a (partial) assignment u ∈ U for some U ⊆A, and V ⊆A, we denote by u(V ) the assignment made by u to the attributes in U ∩ V . In our learning setting, we assume that, when asked to compare two outcomes, the user, whose preferences we wish to learn, is al- ways able to choose one of them. Formally, we assume that the (un- known) user’s preference relation on A is a linear order, which is a rather classical assumption in learning preferences on multi-attribute domains (see, e.g., Koriche & Zanuttini (2009); Lang & Mengin (2009); Dimopoulos et al. (2009); Dombi et al. (2007)). Allowing for indifference in our model would not be difficult, and most results would extend, but would require heavier notations and more details. Lexicographic comparisons order pairs of outcomes (α, β) by looking at the attributes in sequence, according to their importance, until we reach an attribute X such that α(X) = β(X); α and β are