In real-world applications of education, an effective teacher adaptively chooses the next example to teach based on the learner's current state. However, most existing work in algorithmic machine teachingfocuses on the batch setting, where adaptivity plays no role. In this paper, we study the case of teaching consistent, version space learners in an interactive setting. At any time step, the teacher provides an example, the learner performs an update, and the teacher observes the learner'snew state.

More concretely, we consider a learner who asks equivalence queries (i.e., is the queried hypothesis the target hypothesis?), and a teacher responds either yes or no along with a counterexample to the queried hypothesis. This learning paradigm has been extensively studied when the learner receives worst-case or random counterexamples; in this paper, we consider the optimal teacher who picks best-case counterexamples to teach the target hypothesis within a hypothesis class. For this optimal teacher, we introduce LwEQ-TD, a notion of TD capturing the teaching complexity (i.e., the number of queries made) in this paradigm. We show that a significant reduction in queries can be achieved with best-case counterexamples, in contrast to worst-case or random counterexamples, for different hypothesis classes. Furthermore, we establish new connections of LwEQ-TD to the well-studied notions of TD in the learning-from-samples paradigm.

In real-world applications of education, an effective teacher adaptively chooses the next example to teach based on the learner's current state. However, most existing work in algorithmic machine teaching focuses on the batch setting, where adaptivity plays no role. In this paper, we study the case of teaching consistent, version space learners in an interactive setting. At any time step, the teacher provides an example, the learner performs an update, and the teacher observes the learner's new state.

More concretely, we consider a learner who asks equivalence queries (i.e., "is the queried hypothesis the target hypothesis?"), and a teacher responds either "yes" or "no" along with a counterexample to the queried hypothesis. This learning paradigm has been extensively studied when the learner receives worst-case or random counterexamples; in this paper, we consider the optimal teacher who picks best-case counterexamples to teach the target hypothesis within a hypothesis class. For this optimal teacher, we introduce LwEQ-TD, a notion of TD capturing the teaching complexity (i.e., the number of queries made) in this paradigm. We show that a significant reduction in queries can be achieved with best-case counterexamples, in contrast to worst-case or random counterexamples, for different hypothesis classes. Furthermore, we establish new connections of LwEQ-TD to the well-studied notions of TD in the learning-from-samples paradigm.

We study the problem of active learning with the added twist that the learner is assisted by a helpful teacher. We consider the following natural interaction protocol: At each round, the learner proposes a query asking for the label of an instance $x^q$, the teacher provides the requested label $\{x^q, y^q\}$ along with explanatory information to guide the learning process. In this paper, we view this information in the form of an additional contrastive example ($\{x^c, y^c\}$) where $x^c$ is picked from a set constrained by $x^q$ (e.g., dissimilar instances with the same label). Our focus is to design a teaching algorithm that can provide an informative sequence of contrastive examples to the learner to speed up the learning process. We show that this leads to a challenging sequence optimization problem where the algorithm's choices at a given round depend on the history of interactions. We investigate an efficient teaching algorithm that adaptively picks these contrastive examples. We derive strong performance guarantees for our algorithm based on two problem-dependent parameters and further show that for specific types of active learners (e.g., a generalized binary search learner), the proposed teaching algorithm exhibits strong approximation guarantees. Finally, we illustrate our bounds and demonstrate the effectiveness of our teaching framework via two numerical case studies.

This paper proposes a data-driven systematic, consistent and non-exhaustive approach to Model Selection, that is an extension of the classical agnostic PAC learning model. In this approach, learning problems are modeled not only by a hypothesis space $\mathcal{H}$, but also by a Learning Space $\mathbb{L}(\mathcal{H})$, a poset of subspaces of $\mathcal{H}$, which covers $\mathcal{H}$ and satisfies a property regarding the VC dimension of related subspaces, that is a suitable algebraic search space for Model Selection algorithms. Our main contributions are a data-driven general learning algorithm to perform regularized Model Selection on $\mathbb{L}(\mathcal{H})$ and a framework under which one can, theoretically, better estimate a target hypothesis with a given sample size by properly modeling $\mathbb{L}(\mathcal{H})$ and employing high computational power. A remarkable consequence of this approach are conditions under which a non-exhaustive search of $\mathbb{L}(\mathcal{H})$ can return an optimal solution. The results of this paper lead to a practical property of Machine Learning, that the lack of experimental data may be mitigated by a high computational capacity. In a context of continuous popularization of computational power, this property may help understand why Machine Learning has become so important, even where data is expensive and hard to get.

In part \textit{I} we proposed a structure for a general Hypotheses Space $\mathcal{H}$, the Learning Space $\mathbb{L}(\mathcal{H})$, which can be employed to avoid \textit{overfitting} when estimating in a complex space with relative shortage of examples. Also, we presented the U-curve property, which can be taken advantage of in order to select a Hypotheses Space without exhaustively searching $\mathbb{L}(\mathcal{H})$. In this paper, we carry further our agenda, by showing the consistency of a model selection framework based on Learning Spaces, in which one selects from data the Hypotheses Space on which to learn. The method developed in this paper adds to the state-of-the-art in model selection, by extending Vapnik-Chervonenkis Theory to \textit{random} Hypotheses Spaces, i.e., Hypotheses Spaces learned from data. In this framework, one estimates a random subspace $\hat{\mathcal{M}} \in \mathbb{L}(\mathcal{H})$ which converges with probability one to a target Hypotheses Space $\mathcal{M}^{\star} \in \mathbb{L}(\mathcal{H})$ with desired properties. As the convergence implies asymptotic unbiased estimators, we have a consistent framework for model selection, showing that it is feasible to learn the Hypotheses Space from data. Furthermore, we show that the generalization errors of learning on $\hat{\mathcal{M}}$ are lesser than those we commit when learning on $\mathcal{H}$, so it is more efficient to learn on a subspace learned from data.