Algebraic methods proving Sauer's bound for teaching complexity

Samei, Rahim; Semukhin, Pavel; Yang, Boting; Zilles, Sandra

doi:10.1016/j.tcs.2014.09.024

Cited by 6 publications

(6 citation statements)

References 13 publications

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“…Consider, for instance, the model of recursive teaching (introduced by [18]). As shown by [13], Φ d (n) also upper-bounds the size of any concept class which has recursive teaching dimension d and is defined over a domain of size n. As shown by [10], 2 d n d upper-bounds the size of any concept class of NC-teaching dimension d that is defined over a domain of size n. While the upper bound Φ d (n) is tight if d equals the VC-dimension or the recursive teaching dimension, the corresponding bound 2 d n d in case of d = NCTD(C) is tight only for d = 1 (as we will show in this paper).…”

Section: Bounds Of the Sauer-shelah Typementioning

confidence: 84%

Tournaments, Johnson Graphs, and NC-Teaching

Simon¹

2022

Preprint

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Quite recently a teaching model, called "No-Clash Teaching" or simply "NC-Teaching", had been suggested that is provably optimal in the following strong sense. First, it satisfies Goldman and Matthias' collusion-freeness condition. Second, the NC-teaching dimension (= NCTD) is smaller than or equal to the teaching dimension with respect to any other collusion-free teaching model. It has also been shown that any concept class which has NC-teaching dimension d and is defined over a domain of size n can have at most 2 d n d concepts. The main results in this paper are as follows. First, we characterize the maximum concept classes of NC-teaching dimension 1 as classes which are induced by tournaments (= complete oriented graphs) in a very natural way. Second, we show that there exists a family (Cn) n≥1 of concept classes such that the well known recursive teaching dimension (= RTD) of Cn grows logarithmically in n = |Cn| while, for every n ≥ 1, the NC-teaching dimension of Cn equals 1. Since the recursive teaching dimension of a finite concept class C is generally bounded log |C|, the family (Cn) n≥1 separates RTD from NCTD in the most striking way. The proof of existence of the family (Cn) n≥1 makes use of the probabilistic method and random tournaments. Third, we improve the afore-mentioned upper bound 2 d n d by a factor of order √ d. The verification of the superior bound makes use of Johnson graphs and maximum subgraphs not containing large narrow cliques.

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Section: Bounds Of the Sauer-shelah Typementioning

confidence: 84%

Tournaments, Johnson Graphs, and NC-Teaching

Simon¹

2022

Preprint

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“…Machine teaching is shown useful in reinforcement learning [24,32,63,76], humanin-the-loop learning [9,31,55], crowd sourcing [72,99,100] and cyber security [2,57,96,97,98]. [9,13,21,35,65,107] study machine teaching from a more theoretical point of view. Cooperative communication.…”

Section: Related Workmentioning

confidence: 99%

Iterative Teaching by Label Synthesis

Liu

Wang

et al. 2021

Preprint

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In this paper, we consider the problem of iterative machine teaching, where a teacher provides examples sequentially based on the current iterative learner. In contrast to previous methods that have to scan over the entire pool and select teaching examples from it in each iteration, we propose a label synthesis teaching framework where the teacher randomly selects input teaching examples (e.g., images) and then synthesizes suitable outputs (e.g., labels) for them. We show that this framework can avoid costly example selection while still provably achieving exponential teachability. We propose multiple novel teaching algorithms in this framework. Finally, we empirically demonstrate the value of our framework.35th Conference on Neural Information Processing Systems (NeurIPS 2021).

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“…[48], Doliwa et al [12], and Samei et al [40], uses a natural hierarchy on the concept class C which is defined as follows. The first layer in the hierarchy consists of all concepts whose teaching set has minimal size.…”

Section: Teachingmentioning

confidence: 99%

“…Goldman and Mathias [20] and Angluin and Krikis [2] therefore suggested less restrictive teaching models, and more efficient teaching schemes were indeed discovered in these models. One approach, studied by Zilles et al [48], Doliwa et al [12], and Samei et al [40], uses a natural hierarchy on the concept class C which is defined as follows. The first layer in the hierarchy consists of all concepts whose teaching set has minimal size.…”

Section: Teachingmentioning

confidence: 99%

“…The study of mathematical foundations of learning and teaching has been very fruitful, revealing fundamental connections to various other areas of mathematics, such as geometry, topology, and combinatorics. Many key ideas and notions emerged from this study: Vapnik and Chervonenkis's VC-dimension [45], Valiant's seminal definition of PAC learning [44], Littlestone and Warmuth's sample compression schemes [32], Goldman and Kearns's teaching dimension [19], recursive teaching dimension (RT-dimension, for short) [48,12,40] and more.…”

Section: Introductionmentioning

confidence: 99%

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Teaching and Compressing for Low VC-Dimension

Moran

Shpilka

Wigderson³

et al. 2017

A Journey Through Discrete Mathematics

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In this work we study the quantitative relation between VC-dimension and two other basic parameters related to learning and teaching. Namely, the quality of sample compression schemes and of teaching sets for classes of low VC-dimension. Let C be a binary concept class of size m and VC-dimension d. Prior to this work, the best known upper bounds for both parameters were log(m), while the best lower bounds are linear in d. We present significantly better upper bounds on both as follows. Set k = O(d2 d log log |C|).We show that there always exists a concept c in C with a teaching set (i.e. a list of c-labeled examples uniquely identifying c in C) of size k. This problem was studied by Kuhlmann (1999). Our construction implies that the recursive teaching (RT) dimension of C is at most k as well. The RT-dimension was suggested by Zilles et al. and Doliwa et al. (2010). The same notion (under the name partial-ID width) was independently studied by Wigderson and Yehudayoff (2013). An upper bound on this parameter that depends only on d is known just for the very simple case d = 1, and is open even for d = 2. We also make small progress towards this seemingly modest goal.We further construct sample compression schemes of size k for C, with additional information of k log(k) bits. Roughly speaking, given any list of C-labelled examples of arbitrary length, we can retain only k labeled examples in a way that allows to recover the labels of all others examples in the list, using additional k log(k) information bits. This problem was first suggested by Littlestone and Warmuth (1986). * A preliminary version of this work, combined with the paper "Sample compression schemes for VC classes" by the first and the last authors, was published in the proceeding of FOCS'15.

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Algebraic methods proving Sauer's bound for teaching complexity

Cited by 6 publications

References 13 publications

Tournaments, Johnson Graphs, and NC-Teaching

Tournaments, Johnson Graphs, and NC-Teaching

Iterative Teaching by Label Synthesis

Teaching and Compressing for Low VC-Dimension

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