Memory-limited non-U-shaped learning with solved open problems

Humans appear to be able to learn new concepts without needing to be programmed explicitly in any conventional sense. In this paper we regard learning as the phenomenon of knowledge acquisition in the absence of explicit programming. We give a precise methodology for studying this phenomenon from a computational viewpoint. It consists of choosing an appropriate information gathering mechanism, the learning protocol, and exploring the class of concepts that can be learned using it in a reasonable (polynomial) number of steps. Although inherent algorithmic complexity appears to set serious limits to the range of concepts that can be learned, we show that there are some important nontrivial classes of propositional concepts that can be learned in a realistic sense.

Journal of Computer and System Sciences, 2014

Automatic classes are classes of languages for which a finite automaton can decide whether a given element is in a set given by its index. The present work studies the learnability of automatic families by automatic learners which, in each round, output a hypothesis and update a long term memory, depending on the input datum, via an automatic function, that is, via a function whose graph is recognised by a finite automaton. Many variants of automatic learners are investigated: where the long term memory is restricted to be the just prior hypothesis whenever this exists, cannot be of size larger than the size of the longest example or has to consist of a constant number of examples seen so far. Furthermore, learnability is also studied with respect to queries which reveal information about past data or past computation history; the number of queries per round is bounded by a constant. These models are generalisations of the model of feedback queries, given by Lange, Wiehagen and Zeugmann.

Machine Learning, 2008

This paper solves an important problem left open in the literature by showing that U-shapes are unnecessary in iterative learning from positive data. A U-shape occurs when a learner first learns, then unlearns, and, finally, relearns, some target concept. Iterative learning is a Gold-style learning model in which each of a learner's output conjectures depends only upon the learner's most recent conjecture and input element. Previous results had shown, for example, that U-shapes are unnecessary for explanatory learning, but are necessary for behaviorally correct learning. Work on the aforementioned problem led to the consideration of an iterative-like learning model, in which each of a learner's conjectures may, in addition, depend upon the number of elements so far presented to the learner. Learners in this new model are strictly more powerful than traditional iterative learners, yet not as powerful as full explanatory learners. Can any class of languages learnable in this new model be learned without U-shapes? For now, this problem is left open.

Cybernetics, 1974

In this article we present an example of a growing structure that consists at each instant of time of a specified automaton, i.e., a head and finitely many finite automata (elements) of the same type. This structure interacts with a pair of Bernoullian random generators that yield ones and zeros with probabilities unknown to the structure (which interrogates one of the generators during each interval). The principal result is as follows. For any initial state of the structure and any generators, the limiting average frequency of interrogation of the generator that yields a "one" with a higher probability is equal to 1. In this sense the structure realizes an optimal training algorithm. The algorithm consists of alternating (socalled) learning and operating cycles, in Sec. i we shall formulate a sufficient condition of optimality of such algorithms. In Sec. 2 we shall describe the automata forming the structure, and in Sec. 3 we shall show that the algorithm realized by it satisfies this sufficient condition. Each element of the structure has three states, and the minimum possible number of states is not smaller than two in any case. In Sec. 4 we present bounds related to a probability-theoretical scheme whose simulation underlies the operation of the structure; these bounds are perhaps of an intrinsic interest.

Nº.: UC3M Working papers. Economics 2000-72-26, 2000

The paper studies infinite repetition of finite strategic form games. Players use a learning behavior and face bounds on their cognitive capacities. We show that for any given beliefprobability over the set of possible outcomes where players have no experience. games can be payoff classified and there always exists a stationary state in the space of action profiles. In particular, if the belief-probability assumes all possible outcomes without experience to be equally likely, in one class of Prisoners' Dilemmas where the average defecting payoff is higher than the cooperative payoff and the average cooperative payoff is lower than the defecting payoff. play converges in the long run to the static Nash equilibrium while in the other class of Prisoners' Dilemmas where the reserve holds, play converges to cooperation. Results are applied to a largc class of 2 x 2 games.

Annals of Mathematics and Artificial Intelligence, 1995

To implement a program that somehow "learns", it is necessary to fix a set of concepts to be learned and to develop a representation for the concepts and examples of the concepts. In order to investigate general properties of machine learning, it is necessary to work in as representation-independent fashion as possible. In this work, we consider machines that learn programs for recursive (total effectively computable) functions. Several authors have argued that such studies are general enough to include a wide array of learning situations .

2006

A model for learning in the limit is defined where a (so-called iterative) learner gets all positive examples from the target language, tests every new conjecture with a teacher (oracle) if it is a subset of the target language (and if it is not, then it receives a negative counterexample), and uses only limited long-term memory (incorporated in conjectures). Three variants of this model are compared: when a learner receives least negative counterexamples, the ones whose size is bounded by the maximum size of input seen so far, and arbitrary ones. We also compare our learnability model with other relevant models of learnability in the limit, study how our model works for indexed classes of recursive languages, and show that learners in our model can work in non-U-shaped way — never abandoning the first right conjecture.

Information and Computation, 2003

The present work is dedicated to the study of modes of data-presentation in the range between text and informant within the framework of inductive inference. In this study, the learner alternatingly requests sequences of positive and negative data. We define various formalizations of valid data presentations in such a scenario. We resolve the relationships between these different formalizations, and show that one of these is equivalent to learning from informant. We also show a hierarchy formed (for each of the formalizations studied) by considering the number of switches between requests for positive and negative data.

2020

We initiate a study of learning with computable learners and computable output predictors. Recent results in statistical learning theory have shown that there are basic learning problems whose learnability can not be determined within ZFC (Ben-David et al. (2017, 2019)). This motivates us to consider learnability by algorithms with computable output predictors (both learners and predictors are then representable as finite objects). We thus propose the notion of CPAC learnability, by adding some basic computability requirements into a PAC learning framework. As a first step towards a characterization, we show that in this framework learnability of a binary hypothesis class is not implied by finiteness of its VC-dimension anymore. We also present some situations where we are guaranteed to have a computable learner.

Journal of Computer and System Sciences, 2008

U-shaped learning behaviour in cognitive development involves learning, unlearning and relearning. It occurs, for example, in learning irregular verbs. The prior cognitive science literature is occupied with how humans do it, for example, general rules versus tables of exceptions. This paper is mostly concerned with whether U-shaped learning behaviour may be necessary in the abstract mathematical setting of inductive inference, that is, in the computational learning theory following the framework of Gold. All notions considered are learning from text, that is, from positive data. Previous work showed that U-shaped learning behaviour is necessary for behaviourally correct learning but not for syntactically convergent, learning in the limit (= explanatory learning). The present paper establishes the necessity for the hierarchy of classes of vacillatory learning where a behaviourally correct learner has to satisfy the additional constraint that it vacillates in the limit between at most b grammars, where b∈{2,3,…,∗}b∈{2,3,…,∗}. Non-U-shaped vacillatory learning is shown to be restrictive: every non-U-shaped vacillatorily learnable class is already learnable in the limit. Furthermore, if vacillatory learning with the parameter b=2b=2 is possible then non-U-shaped behaviourally correct learning is also possible. But for b=3b=3, surprisingly, there is a class witnessing that this implication fails.