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.
تحميل البحث