  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.
