"Delle Rose et al. (COLT’23) introduced an effective version of the Vapnik-Chervonenkis dimension, and showed that it characterizes improper PAC learning with total computable learners. In this paper, we introduce and study a similar effectivization of the notion of Littlestone dimension. Finite effective Littlestone dimension is a necessary condition for computable online learning but is not a sufficient one—which we already establish for classes of the effective Littlestone dimension 2. However, the effective Littlestone dimension equals the optimal mistake bound for computable learners in two special cases: a) for classes of Littlestone dimension 1 and b) when the learner receives as additional information a bound on the numbers to be guessed. Interestingly, finite effective Littlestone dimension also guarantees that the class consists only of computable functions. Keywords: Online learning, Littlestone dimension, computable machine learning"

Despite the wide use of k-Nearest Neighbors as classification models, their explainability properties remain poorly understood from a theoretical perspective. While nearest neighbors classifiers offer interpretability from a ''data perspective'', in which the classification of an input vector x is explained by identifying the vectors v1, ..., vk in the training set that determine the classification of x, we argue that such explanations can be impractical in high-dimensional applications, where each vector has hundreds or thousands of features and it is not clear what their relative importance is. Hence, we focus on understanding nearest neighbor classifications through a ''feature perspective'', in which the goal is to identify how the values of the features in x affect its classification. Concretely, we study abductive explanations such as ''minimum sufficient reasons'', which correspond to sets of features in x that are enough to guarantee its classification, and counterfactual explanations based on the minimum distance feature changes one would have to perform in x to change its classification. We present a detailed landscape of positive and negative complexity results for counterfactual and abductive explanations, distinguishing between discrete and continuous feature spaces, and considering the impact of the choice of distance function involved. Finally, we show that despite some negative complexity results, Integer Quadratic Programming and SAT solving allow for computing explanations in practice.

Publisher:  LATIN 2024: Theoretical Informatics Link>

ABSTRACT

In this paper, we construct a winning condition W over a finite set of colors such that, first, every finite arena has a strategy with 2 states of general memory which is optimal w.r.t. W, and second, there exists no k such that every finite arena has a strategy with k states of chromatic memory which is optimal w.r.t. W.