  Algorithmic machine teaching studies the interaction between a teacher and a
learner where the teacher selects labeled examples aiming at teaching a target
hypothesis. In a quest to lower teaching complexity and to achieve more natural
teacher-learner interactions, several teaching models and complexity measures
have been proposed for both the batch settings (e.g., worst-case, recursive,
preference-based, and non-clashing models) as well as the sequential settings
(e.g., local preference-based model). To better understand the connections
between these different batch and sequential models, we develop a novel
framework which captures the teaching process via preference functions
$\Sigma$. In our framework, each function $\sigma \in \Sigma$ induces a
teacher-learner pair with teaching complexity as $\TD(\sigma)$. We show that
the above-mentioned teaching models are equivalent to specific types/families
of preference functions in our framework. This equivalence, in turn, allows us
to study the differences between two important teaching models, namely $\sigma$
functions inducing the strongest batch (i.e., non-clashing) model and $\sigma$
functions inducing a weak sequential (i.e., local preference-based) model.
Finally, we identify preference functions inducing a novel family of sequential
models with teaching complexity linear in the VC dimension of the hypothesis
class: this is in contrast to the best known complexity result for the batch
models which is quadratic in the VC dimension.
