Lower Bounds for Greedy Teaching Set Constructions

• URL: http://arxiv.org/abs/2505.03223v1
• Date: Tue, 06 May 2025 06:30:01 GMT
• Title: Lower Bounds for Greedy Teaching Set Constructions
• Authors: Spencer Compton, Chirag Pabbaraju, Nikita Zhivotovskiy,
• Abstract summary: We prove lower bounds on the performance of a greedy algorithm for small $k$.<n>Most consequentially, our lower bound extends up to $k le lceil c d rceil$ for small constant $c>0$.
• Score: 12.186950360560145
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: A fundamental open problem in learning theory is to characterize the best-case teaching dimension $\operatorname{TS}_{\min}$ of a concept class $\mathcal{C}$ with finite VC dimension $d$. Resolving this problem will, in particular, settle the conjectured upper bound on Recursive Teaching Dimension posed by [Simon and Zilles; COLT 2015]. Prior work used a natural greedy algorithm to construct teaching sets recursively, thereby proving upper bounds on $\operatorname{TS}_{\min}$, with the best known bound being $O(d^2)$ [Hu, Wu, Li, and Wang; COLT 2017]. In each iteration, this greedy algorithm chooses to add to the teaching set the $k$ labeled points that restrict the concept class the most. In this work, we prove lower bounds on the performance of this greedy approach for small $k$. Specifically, we show that for $k = 1$, the algorithm does not improve upon the halving-based bound of $O(\log(|\mathcal{C}|))$. Furthermore, for $k = 2$, we complement the upper bound of $O\left(\log(\log(|\mathcal{C}|))\right)$ from [Moran, Shpilka, Wigderson, and Yuhudayoff; FOCS 2015] with a matching lower bound. Most consequentially, our lower bound extends up to $k \le \lceil c d \rceil$ for small constant $c>0$: suggesting that studying higher-order interactions may be necessary to resolve the conjecture that $\operatorname{TS}_{\min} = O(d)$.

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