On a Combinatorial Problem Arising in Machine Teaching

TL;DR

This work resolves a conjecture on the worst-case teaching dimension in a formal machine teaching framework by showing that the maximal teaching demand occurs when the consistency matrix $M$ has rows given by the binary representations of $0$ through $k-1$ (the matrix $H_{n,k}$). The authors formalize the target quantity $m_q(n,k)$ as the minimum over all $k\times n$ binary matrices of the sum of distinct-row counts across all $q$-column projections, and prove $m_q(n,k)=h_q(n,k)$, where $h_q(n,k)=m_q(H_{n,k})$. The proof hinges on a recursive construction of $H_{n,k}$, a recurrence for $h_q(n,k)$, and a generalized Graham lemma about binary weights to establish that the minimum over the split parameter $x$ occurs at $x=\lceil k/2 \rceil$, thereby generalizing the edge isoperimetry results on hypercubes. This yields a precise, computable characterization of the minimal teaching sets and provides insights into the efficiency of Greedy-based teaching strategies in general machine teaching models.

Abstract

We study a model of machine teaching where the teacher mapping is constructed from a size function on both concepts and examples. The main question in machine teaching is the minimum number of examples needed for any concept, the so-called teaching dimension. A recent paper [7] conjectured that the worst case for this model, as a function of the size of the concept class, occurs when the consistency matrix contains the binary representations of numbers from zero and up. In this paper we prove their conjecture. The result can be seen as a generalization of a theorem resolving the edge isoperimetry problem for hypercubes [12], and our proof is based on a lemma of [10].

Theorems & Definitions (20)

• Theorem 2.1
• Theorem 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Lemma 4.1
• proof