Large language models have become multimodal, and many of them are said to integrate their modalities using common representations. If this were true, a drawing of a car as an image, for instance, should map to a similar area in the latent space as a textual description of the strokes that form the drawing. To explore this in a black-box access regime to these models, we propose the use of machine teaching, a theory that studies the minimal set of examples a teacher needs to choose so that the learner captures the concept. In this paper, we evaluate the complexity of teaching vision-language models a subset of objects in the Quick, Draw! dataset using two presentations: raw images as bitmaps and trace coordinates in TikZ format. The results indicate that image-based representations generally require fewer segments and achieve higher accuracy than coordinate-based representations. But, surprisingly, the teaching size usually ranks concepts similarly across both modalities, even when controlling for (a human proxy of) concept priors, suggesting that the simplicity of concepts may be an inherent property that transcends modality representations.

Recent research in machine teaching has explored the instruction of any concept expressed in a universal language. In this compositional context, new experimental results have shown that there exist data teaching sets surprisingly shorter than the concept description itself. However, there exists a bound for those remarkable experimental findings through teaching size and concept complexity that we further explore here. As concepts are rarely taught in isolation we investigate the best configuration of concepts to teach a given set of concepts, where those that have been acquired first can be reused for the description of new ones. This new notion of conditional teaching size uncovers new insights, such as the interposition phenomenon: certain prior knowledge generates simpler compatible concepts that increase the teaching size of the concept that we want to teach. This does not happen for conditional Kolmogorov complexity. Furthermore, we provide an algorithm that constructs optimal curricula based on interposition avoidance. This paper presents a series of theoretical results, including their proofs, and some directions for future work. New research possibilities in curriculum teaching in compositional scenarios are now wide open to exploration.