Quadratic Upper Bound for Recursive Teaching Dimension of Finite VC Classes

TL;DR

This paper establishes a quadratic upper bound on the recursive teaching dimension for finite VC classes, significantly improving previous exponential bounds and advancing understanding of the relationship between RTD and VC dimension.

Contribution

The paper proves a quadratic upper bound on RTD in terms of VC dimension, moving closer to resolving an open problem in concept class complexity.

Findings

RTD is bounded by O(d^2) for classes with VC dimension d

Previous bounds were exponential, now improved to quadratic

Discussion of challenges in fully resolving the open problem

Abstract

In this work we study the quantitative relation between the recursive teaching dimension (RTD) and the VC dimension (VCD) of concept classes of finite sizes. The RTD of a concept class $\mathcal C \subseteq \{0, 1\}^n$, introduced by Zilles et al. (2011), is a combinatorial complexity measure characterized by the worst-case number of examples necessary to identify a concept in $\mathcal C$ according to the recursive teaching model. For any finite concept class $\mathcal C \subseteq \{0,1\}^n$ with $\mathrm{VCD}(\mathcal C)=d$, Simon & Zilles (2015) posed an open problem $\mathrm{RTD}(\mathcal C) = O(d)$, i.e., is RTD linearly upper bounded by VCD? Previously, the best known result is an exponential upper bound $\mathrm{RTD}(\mathcal C) = O(d \cdot 2^d)$, due to Chen et al. (2016). In this paper, we show a quadratic upper bound: $\mathrm{RTD}(\mathcal C) = O(d^2)$, much closer to an answer to the open problem. We also discuss the challenges in fully solving the problem.