Optimal Collusion-Free Teaching

David Kirkpatrick, Hans U. Simon, Sandra Zilles
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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v98-kirkpatrick19a, title = {Optimal Collusion-Free Teaching}, author = {Kirkpatrick, David and Simon, Hans U. and Zilles, Sandra}, booktitle = {Proceedings of the 30th International Conference on Algorithmic Learning Theory}, pages = {506--528}, year = {2019}, editor = {Garivier, Aurélien and Kale, Satyen}, volume = {98}, series = {Proceedings of Machine Learning Research}, month = {22--24 Mar}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v98/kirkpatrick19a/kirkpatrick19a.pdf}, url = {https://proceedings.mlr.press/v98/kirkpatrick19a.html}, 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.} }
Endnote
%0 Conference Paper %T Optimal Collusion-Free Teaching %A David Kirkpatrick %A Hans U. Simon %A Sandra Zilles %B Proceedings of the 30th International Conference on Algorithmic Learning Theory %C Proceedings of Machine Learning Research %D 2019 %E Aurélien Garivier %E Satyen Kale %F pmlr-v98-kirkpatrick19a %I PMLR %P 506--528 %U https://proceedings.mlr.press/v98/kirkpatrick19a.html %V 98 %X 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.
APA
Kirkpatrick, D., Simon, H.U. & Zilles, S.. (2019). Optimal Collusion-Free Teaching. Proceedings of the 30th International Conference on Algorithmic Learning Theory, in Proceedings of Machine Learning Research 98:506-528 Available from https://proceedings.mlr.press/v98/kirkpatrick19a.html.

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