This paper studies exact learning of unions of non-discretized geometric concepts in the model of preference-based teaching. In particular, it focuses on upper and lower bounds of the corresponding sample complexity parameter, the preference-based teaching dimension (PBTD), when learning disjoint unions of a bounded number of geometric concepts of various types -- for instance balls, axis-aligned cubes, or axis-aligned boxes -- in arbitrary dimensions. It is shown that the PBTD of disjoint unions of some such types of concepts grows linearly with the number of concepts in the union, independent of the dimensionality. Teaching the union of potentially overlapping objects turns out to be more involved and is hence considered here only for unions of up to two objects.

We introduce a new model of teaching named preference-based teaching and a corresponding complexity parameter---the preference-based teaching dimension (PBTD)---representing the worst-case number of examples needed to teach any concept in a given concept class. Although the PBTD coincides with the well- known recursive teaching dimension (RTD) on finite classes, it is radically different on infinite ones: the RTD becomes infinite already for trivial infinite classes (such as half- intervals) whereas the PBTD evaluates to reasonably small values for a wide collection of infinite classes including classes consisting of so-called closed sets w.r.t. a given closure operator, including various classes related to linear sets over $\mathbb{N}_0$ (whose RTD had been studied quite recently) and including the class of Euclidean half-spaces. On top of presenting these concrete results, we provide the reader with a theoretical framework (of a combinatorial flavor) which helps to derive bounds on the PBTD.

We introduce a new model of teaching named “preference-based teaching” and a corresponding complexity parameter—the preference-based teaching dimension (PBTD)—representing the worst-case number of examples needed to teach any concept in a given concept class. Although the PBTD coincides with the well-known recursive teaching dimension (RTD) on finite classes, it is radically different on infinite ones: the RTD becomes infinite already for trivial infinite classes (such as half-intervals) whereas the PBTD evaluates to reasonably small values for a wide collection of infinite classes including classes consisting of so-called closed sets w.r.t. a given closure operator, including various classes related to linear sets over \mathbbN_0 (whose RTD had been studied quite recently) and including the class of Euclidean half-spaces (and some other geometric classes). On top of presenting these concrete results, we provide the reader with a theoretical framework (of a combinatorial flavor) which helps to derive bounds on the PBTD.