On a Combinatorial Problem Arising in Machine Teaching

TL;DR

This paper proves a conjecture in machine teaching regarding the worst-case scenario for the minimum number of examples needed, linking it to a combinatorial problem involving binary representations and hypercube isoperimetry.

Contribution

It proves a conjecture about the worst-case teaching dimension in a model of machine teaching, generalizing a hypercube isoperimetry theorem.

Findings

Confirmed the conjecture that the worst case occurs with binary representations from zero upwards.

Connected the problem to hypercube edge isoperimetry.

Provided a proof based on a lemma from prior work.

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].