Hans U. Simon

Farnam Mansouri, Hans U. Simon, Adish Singla et al.

Machine learning can greatly benefit from providing learning algorithms with pairs of contrastive training examples -- typically pairs of instances that differ only slightly, yet have different class labels. Intuitively, the difference in the instances helps explain the difference in the class labels. This paper proposes a theoretical framework in which the effect of various types of contrastive examples on active learners is studied formally. The focus is on the sample complexity of learning concept classes and how it is influenced by the choice of contrastive examples. We illustrate our results with geometric concept classes and classes of Boolean functions. Interestingly, we reveal a connection between learning from contrastive examples and the classical model of self-directed learning.

David Kirkpatrick, Hans U. Simon, Sandra Zilles

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 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 any concept class $\mathcal{C}$ and 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.