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# Optimal Collusion-Free Teaching

*Proceedings of the 30th International Conference on Algorithmic Learning Theory*, PMLR 98:506-528, 2019.

#### Abstract

Formal models of learning from teachers need to respect certain criteria to avoid collusion. The most commonly accepted notion of collusion-freeness was proposed by Goldman and Mathias (1996), and various teaching models obeying their criterion have been studied. For each model $M$ and each concept class $\mathcal{C}$, a parameter $M$-$\mathrm{TD}(\mathcal{C})$ refers to the \emph{teaching dimension} of concept class $\mathcal{C}$ in model $M$—defined to be the number of examples required for teaching a concept, in the worst case over all concepts in $\mathcal{C}$.
This paper introduces a new model of teaching, called no-clash teaching, together with the corresponding parameter $\mathrm{NCTD}(\mathcal{C})$. No-clash teaching is provably optimal in the strong sense that, given \emph{any}\/{concept} class $\mathcal{C}$ and \emph{any}\/{model} $M$ obeying Goldman and Mathias’s collusion-freeness criterion, one obtains $\mathrm{NCTD}(\mathcal{C})\le M$-$\mathrm{TD}(\mathcal{C})$. We also study a corresponding notion $\mathrm{NCTD}^+$ for the case of learning from positive data only, establish useful bounds on $\mathrm{NCTD}$ and $\mathrm{NCTD}^+$, and discuss relations of these parameters to the VC-dimension and to sample compression.
In addition to formulating an optimal model of collusion-free teaching,
our main results are on the computational complexity of deciding whether $\mathrm{NCTD}^+(\mathcal{C})=k$ (or $\mathrm{NCTD}(\mathcal{C})=k$) for given $\mathcal{C}$ and $k$. We show some such decision problems to be equivalent to
the existence question for certain constrained matchings in bipartite
graphs. Our NP-hardness results for the latter are of independent interest in the study of constrained graph matchings.