Quadratic Upper Bound for Recursive Teaching Dimension of Finite VC Classes
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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 C ⊆{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 C according to the recursive teaching model. For any finite concept class C ⊆{0,1}^n with VCD( C)=d, Simon & Zilles (2015) posed an open problem RTD( C) = O(d), i.e., is RTD linearly upper bounded by VCD? Previously, the best known result is an exponential upper bound RTD( C) = O(d · 2^d), due to Chen et al. (2016). In this paper, we show a quadratic upper bound: RTD( C) = O(d^2), much closer to an answer to the open problem. We also discuss the challenges in fully solving the problem.
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