The recursive teaching dimension (RTD) of a concept class C {0,1}n, introduced by Zilles et al. [ZLHZ11], is a complexity parameter measured by the worst-case number of labeled examples needed to learn any target concept of C in the recursive teaching model. In this paper, we study the quantitative relation between RTD and the well-known learning complexity measure VC dimension (VCD), and improve the best known upper and (worst-case) lower bounds on the recursive teaching dimension with respect to the VC dimension. Given a concept class C {0,1}n with VCD(C) = d, we first show that RTD(C) is at most d 2d+1. This is the first upper bound for RTD(C)that depends only on VCD(C), independent of the size of the concept class |C| and its domain size n. Before our work, the best known upper bound for RTD(C) is O(d2d loglog|C|), obtained by Moran et al. [MSWY15].

Algorithmic machine teaching studies the interaction between a teacher and a learner where the teacher selects labeled examples aiming at teaching a target hypothesis.

In this paper, we consider the problem of iterative machine teaching, where a teacher provides examples sequentially based on the current iterative learner.

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 $\operatorname{TS}_{\min} = O(d)$.

We study the empirical strategies that humans follow as they teach a target concept with a simple 1D threshold to a robot.

In this paper, we study the quantitative relation between RTD and the well-known learning complexity measure VC dimension (VCD), and improve the best known upper and (worst-case) lower bounds on the recursive teaching dimension with respect to the VC dimension.

We study a model of machine teaching where the teacher mapping is constructed from a size function on both concepts and examples. The main question in machine teaching is the minimum number of examples needed for any concept, the so-called teaching dimension. A recent paper [7] conjectured that the worst case for this model, as a function of the size of the concept class, occurs when the consistency matrix contains the binary representations of numbers from zero and up. In this paper we prove their conjecture. The result can be seen as a generalization of a theorem resolving the edge isoperimetry problem for hypercubes [12], and our proof is based on a lemma of [10].

We study optimal teaching of Linear Behavior Cloning (LBC) learners. In this setup, the teacher can select which states to demonstrate to an LBC learner. The learner maintains a version space of infinite linear hypotheses consistent with the demonstration. The goal of the teacher is to teach a realizable target policy to the learner using minimum number of state demonstrations. This number is known as the Teaching Dimension(TD). We present a teaching algorithm called ``Teach using Iterative Elimination(TIE)" that achieves instance optimal TD. However, we also show that finding optimal teaching set computationally is NP-hard. We further provide an approximation algorithm that guarantees an approximation ratio of $\log(|A|-1)$ on the teaching dimension. Finally, we provide experimental results to validate the efficiency and effectiveness of our algorithm.

Machine learning researchers and practitioners steadily enlarge the multitude of successful learning models. They achieve this through in-depth theoretical analyses and experiential heuristics. However, there is no known general-purpose procedure for rigorously evaluating whether newly proposed models indeed successfully learn from data. We show that such a procedure cannot exist. For PAC binary classification, uniform and universal online learning, and exact learning through teacher-learner interactions, learnability is in general undecidable, both in the sense of independence of the axioms in a formal system and in the sense of uncomputability. Our proofs proceed via computable constructions that encode the consistency problem for formal systems and the halting problem for Turing machines into whether certain function classes are trivial/finite or highly complex, which we then relate to whether these classes are learnable via established characterizations of learnability through complexity measures. Our work shows that undecidability appears in the theoretical foundations of artificial intelligence: There is no one-size-fits-all algorithm for deciding whether a machine learning model can be successful. We cannot in general automatize the process of assessing new learning models.