Quadratic Upper Bound for Recursive Teaching Dimension of Finite VC Classes

Lunjia Hu, Ruihan Wu, Tianhong Li, Liwei Wang
Proceedings of the 2017 Conference on Learning Theory, PMLR 65:1147-1156, 2017.

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 $\mathcal C⊆{0,1}^n$ , 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 $\mathcal C⊆{0,1}^n$ with $\mathrm{VCD}(\mathcal C) = d$, Simon and 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 $\mathrm{RTD}(\mathcal C) = O(d\cdot2^d)$, 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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v65-hu17a, title = {Quadratic Upper Bound for Recursive Teaching Dimension of Finite VC Classes}, author = {Hu, Lunjia and Wu, Ruihan and Li, Tianhong and Wang, Liwei}, booktitle = {Proceedings of the 2017 Conference on Learning Theory}, pages = {1147--1156}, year = {2017}, editor = {Kale, Satyen and Shamir, Ohad}, volume = {65}, series = {Proceedings of Machine Learning Research}, month = {07--10 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v65/hu17a/hu17a.pdf}, url = {https://proceedings.mlr.press/v65/hu17a.html}, 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 $\mathcal C⊆{0,1}^n$ , 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 $\mathcal C⊆{0,1}^n$ with $\mathrm{VCD}(\mathcal C) = d$, Simon and 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 $\mathrm{RTD}(\mathcal C) = O(d\cdot2^d)$, 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.} }
Endnote
%0 Conference Paper %T Quadratic Upper Bound for Recursive Teaching Dimension of Finite VC Classes %A Lunjia Hu %A Ruihan Wu %A Tianhong Li %A Liwei Wang %B Proceedings of the 2017 Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2017 %E Satyen Kale %E Ohad Shamir %F pmlr-v65-hu17a %I PMLR %P 1147--1156 %U https://proceedings.mlr.press/v65/hu17a.html %V 65 %X 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 $\mathcal C⊆{0,1}^n$ , 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 $\mathcal C⊆{0,1}^n$ with $\mathrm{VCD}(\mathcal C) = d$, Simon and 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 $\mathrm{RTD}(\mathcal C) = O(d\cdot2^d)$, 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.
APA
Hu, L., Wu, R., Li, T. & Wang, L.. (2017). Quadratic Upper Bound for Recursive Teaching Dimension of Finite VC Classes. Proceedings of the 2017 Conference on Learning Theory, in Proceedings of Machine Learning Research 65:1147-1156 Available from https://proceedings.mlr.press/v65/hu17a.html.

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