[Paper Review] Optimal Collusion-Free Teaching

David Kirkpatrick, Hans Ulrich Simon+1 more|arXiv (Cornell University)|Mar 10, 2019

Machine Learning and Algorithms4 citations

TL;DR

This paper introduces no-clash teaching, a novel collusion-free teaching model that achieves the minimal possible teaching dimension across all such models under Goldman and Mathias’s criterion. It proves that no-clash teaching is optimal in that its teaching dimension, NCTD(C), is always less than or equal to that of any other collusion-free model, and establishes connections to constrained matchings in bipartite graphs and sample compression.

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 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.

Motivation & Objective

• To identify the minimal teaching complexity achievable under Goldman and Mathias’s collusion-freeness criterion.
• To define a new teaching model—no-clash teaching—that provably achieves this minimal complexity.
• To establish theoretical connections between no-clash teaching, VC-dimension, and sample compression.
• To analyze the computational complexity of deciding whether NCTD(C) = k or NCTD⁺(C) = k for a given concept class C and integer k.
• To explore the relationship between no-clash teaching and constrained matchings in bipartite graphs

Proposed method

• Proposes no-clash teaching as a new model of collusion-free teaching, defined by the condition that no two distinct concepts are consistent with the union of their teaching sets.
• Defines the no-clash teaching dimension, NCTD(C), as the minimal number of examples needed to teach any concept in class C under this model.
• Establishes a one-to-one correspondence between no-clash teaching and uniquely restricted matchings in a bipartite graph constructed from the concept class.
• Reduces the decision problem NCTD(C) = k (or NCTD⁺(C) = k) to the existence of a constrained matching of size k in a bipartite graph.
• Develops a linear-time algorithm for checking whether a uniquely restricted matching of size |V₀| exists in a bipartite graph, based on iterative removal of degree-1 vertices.
• Uses this algorithm to decide NCTD(C) ≤ k in time linear in the size of the associated bipartite graph

Experimental results

Research questions

• RQ1 What is the smallest possible teaching dimension achievable among all collusion-free teaching models under Goldman and Mathias’s criterion?
• RQ2 Can the no-clash teaching model be characterized via combinatorial structures such as constrained matchings in bipartite graphs?
• RQ3 Is the decision problem NCTD(C) = k computationally tractable, and how does it relate to known problems in graph theory?
• RQ4 Does the no-clash teaching dimension NCTD(C) grow linearly with the VC-dimension of C, or is it bounded by a function of it?
• RQ5 Can no-clash teaching provide insights into the size of minimal sample compression schemes?

Key findings

• No-clash teaching is provably optimal: for any concept class C and any collusion-free teaching model M, NCTD(C) ≤ M-TD(C).
• The decision problem NCTD(C) = k is equivalent to the existence of a uniquely restricted matching of size k in a bipartite graph derived from C.
• The problem of deciding whether a uniquely restricted matching of size k exists is NP-complete, even when k is part of the input.
• A linear-time algorithm exists for deciding whether a uniquely restricted matching of size |V₀| exists in a bipartite graph, based on iterative removal of degree-1 vertices.
• The same algorithmic approach applies to the positive-data-only variant, NCTD⁺(C), by modifying the bipartite graph to include only positive samples of size at most k.
• The results establish a strong connection between no-clash teaching, constrained matchings, and sample compression, suggesting that no-clash teaching may resolve open problems related to linear bounds on VC-dimension and compression schemes.