Formal Learning Theory Research Articles

Editors' Introduction: Why Formal Learning Theory Matters for Cognitive Science

This article reviews a number of different areas in the foundations of formal learning theory. After outlining the general framework for formal models of learning, the Bayesian approach to learning is summarized. This leads to a discussion of Solomonoff's Universal Prior Distribution for Bayesian learning. Gold's model of identification in the limit is also outlined. We next discuss a number of aspects of learning theory raised in contributed papers, related to both computational and representational complexity. The article concludes with a description of how semi-supervised learning can be applied to the study of cognitive learning models. Throughout this overview, the specific points raised by our contributing authors are connected to the models and methods under review.

The neurobiology of syntax: beyond string sets.

The human capacity to acquire language is an outstanding scientific challenge to understand. Somehow our language capacities arise from the way the human brain processes, develops and learns in interaction with its environment. To set the stage, we begin with a summary of what is known about the neural organization of language and what our artificial grammar learning (AGL) studies have revealed. We then review the Chomsky hierarchy in the context of the theory of computation and formal learning theory. Finally, we outline a neurobiological model of language acquisition and processing based on an adaptive, recurrent, spiking network architecture. This architecture implements an asynchronous, event-driven, parallel system for recursive processing. We conclude that the brain represents grammars (or more precisely, the parser/generator) in its connectivity, and its ability for syntax is based on neurobiological infrastructure for structured sequence processing. The acquisition of this ability is accounted for in an adaptive dynamical systems framework. Artificial language learning (ALL) paradigms might be used to study the acquisition process within such a framework, as well as the processing properties of the underlying neurobiological infrastructure. However, it is necessary to combine and constrain the interpretation of ALL results by theoretical models and empirical studies on natural language processing. Given that the faculty of language is captured by classical computational models to a significant extent, and that these can be embedded in dynamic network architectures, there is hope that significant progress can be made in understanding the neurobiology of the language faculty.

No Answer to Hume

In a recent article in this journal, Daniel Steel charges me with committing a fallacy in my discussion of inductive rules. I show that the charge is false, and that Steel's own attempt to validate enumerative induction in terms of formal learning theory is itself fallacious. I go on to argue that, contra Steel, formal learning theory is in principle incapable of answering Hume's famous claim that any attempt to justify induction will beg the question.

On Not Changing the Problem: A Reply to Howson

Howson's critique of my essay on Hume's problem of induction levels two main charges. First, Howson claims that I have attributed to him an error that he never made, and in fact which he warned against in the very text that I cite. Secondly, Howson argues that my proposed solution to Hume's problem is flawed on technical and philosophical grounds. In response to the first charge, I explain how Howson's text justifies attributing to him the claim that the principle of induction is shown to be inconsistent by Goodman's riddle. In regards to the second, I show that Howson's objections rest on misunderstandings of formal learning theory and on conflating the problem of induction with the problem of unconceived alternatives.

Finite identification from the viewpoint of epistemic update

Formal learning theory constitutes an attempt to describe and explain the phenomenon of learning, in particular of language acquisition. The considerations in this domain are also applicable in philosophy of science, where it can be interpreted as a description of the process of scientific inquiry. The theory focuses on various properties of the process of hypothesis change over time. Treating conjectures as informational states, we link the process of conjecture-change to epistemic update. We reconstruct and analyze the temporal aspect of learning in the context of dynamic and temporal logics of epistemic change. We first introduce the basic formal notions of learning theory and basic epistemic logic. We provide a translation of the components of learning scenarios into the domain of epistemic logic. Then, we propose a characterization of finite identifiability in an epistemic temporal language. In the end we discuss consequences and possible extensions of our work.

FEW 2009 Special Issue: Preface

The Annual Formal Epistemology Workshop (FEW) is now entering its seventh year. I am very happy to report that the first six FEWs have been a great success (see http://fitelson.org/few/ for detailed information about past and current FEWs). We have managed to foster a thriving community of students, professors, and other researchers with a wide range of overlapping interests. And, this year, FEW is going international—with our first meeting in Europe (FEW 2010 is being held in Konstanz). This trend will continue, with future FEWs taking place in various hotbeds of formal epistemology activity around the world. Stay tuned! This special issue contains a selection of five of the papers presented at FEW 2009, which was held at CMU in June 2009 〈http://fitelson.org/few/few_09/〉. Three of these papers (by Maher, Norton, and Novack) are concerned (broadly) with inductive logic and inductive inference. One of the papers (by Kelly & Mayo-Wilson) is about formal learning theory. And, the fifth paper (by Poproski) is about rational choice. This is a nice sample of the excellent papers (and topics) that were presented (and discussed) at FEW 2009. It is our intention to continue to publish selected papers from future FEWs in subsequent special issues of this journal.

Adaptive partitioning schemes for bipartite ranking

Recursive partitioning methods are among the most popular techniques in machine learning. The purpose of this paper is to investigate how to adapt this methodology to the bipartite ranking problem. Following in the footsteps of the TreeRank approach developed in Clémençon and Vayatis (Proceedings of the 2008 Conference on Algorithmic Learning Theory, 2008 and IEEE Trans. Inf. Theory 55(9):4316–4336, 2009), we present tree-structured algorithms designed for learning to rank instances based on classification data. The main contributions of the present work are the following: the practical implementation of the TreeRank algorithm, well-founded solutions to the crucial issues related to the splitting rule and the choice of the “right” size for the ranking tree. From the angle embraced in this paper, splitting is viewed as a cost-sensitive classification task with data-dependent cost. Hence, up to straightforward modifications, any classification algorithm may serve as a splitting rule. Also, we propose to implement a cost-complexity pruning method after the growing stage in order to produce a “right-sized” ranking sub-tree with large AUC. In particular, performance bounds are established for pruning schemes inspired by recent work on nonparametric model selection. Eventually, we propose indicators for variable importance and variable dependence, plus various simulation studies illustrating the potential of our method.

What If the Principle of Induction Is Normative? Formal Learning Theory and Hume’s Problem

This article argues that a successful answer to Hume’s problem of induction can be developed from a sub‐genre of philosophy of science known as formal learning theory. One of the central concepts of formal learning theory is logical reliability: roughly, a method is logically reliable when it is assured of eventually settling on the truth for every sequence of data that is possible given what we know. I show that the principle of induction (PI) is necessary and sufficient for logical reliability in what I call simple enumerative induction. This answer to Hume’s problem rests on interpreting PI as a normative claim justified by a non‐empirical epistemic means–ends argument. In such an argument, a rule of inference is shown by mathematical or logical proof to promote a specified epistemic end. Since the proof concerning PI and logical reliability is not based on inductive reasoning, this argument avoids the circularity that Hume argued was inherent in any attempt to justify PI.

Striatal Prediction Error Modulates Cortical Coupling

Both perceptual inference and motor responses are shaped by learned probabilities. For example, stimulus-induced responses in sensory cortices and preparatory activity in premotor cortex reflect how (un)expected a stimulus is. This is in accordance with predictive coding accounts of brain function, which posit a fundamental role of prediction errors for learning and adaptive behavior. We used functional magnetic resonance imaging and recent advances in computational modeling to investigate how (failures of) learned predictions about visual stimuli influence subsequent motor responses. Healthy volunteers discriminated visual stimuli that were differentially predicted by auditory cues. Critically, the predictive strengths of cues varied over time, requiring subjects to continuously update estimates of stimulus probabilities. This online inference, modeled using a hierarchical Bayesian learner, was reflected behaviorally: speed and accuracy of motor responses increased significantly with predictability of the stimuli. We used nonlinear dynamic causal modeling to demonstrate that striatal prediction errors are used to tune functional coupling in cortical networks during learning. Specifically, the degree of striatal trial-by-trial prediction error activity controls the efficacy of visuomotor connections and thus the influence of surprising stimuli on premotor activity. This finding substantially advances our understanding of striatal function and provides direct empirical evidence for formal learning theories that posit a central role for prediction error-dependent plasticity.

Laboratory phonology 8

Reviewed by: Laboratory phonology 8 Jaye Padgett Laboratory phonology 8. Ed. by Louis Goldstein, D. H. Whalen, and Catherine T. Best. (Phonology and phonetics 4-2.) Berlin: Mouton de Gruyter, 2006. Pp. xvi, 675. ISBN 9783110176780. $192 (Hb). This collection of papers is the end-product of the eighth Conference on Laboratory Phonology (LabPhon), held in New Haven, Connecticut, in June 2002, and hosted by Yale University and Haskins Laboratories. The volume is dedicated to the memory of Catherine P. Browman. If the LabPhon conferences and volumes were a bit renegade when they began, they are now more of an institution. It was still unusual in the 1980s to combine phonological theorizing with experimental methods and with theories drawn from phonetics and psycholinguistics, but to do so now seems more the norm. Out of thirteen papers published in the 1989 issue of Phonology, three incorporate experimental methodologies. (I construe ‘experimental’ broadly to include, for example, gestural or neural network modeling and formal learning theory.) In 2009 it was nine out of thirteen. (Six of these were from a special issue called ‘Phonological models and experimental data’; the reader can decide whether this strengthens or weakens the point.) The early ‘labphon’ movement can take credit for much of this change. This year we should see the inaugural publication of a laboratory phonology journal to replace the published volumes. In my view, this shift to a regular, peer-reviewed, and more accessible forum is very welcome. The book contains twenty-six contributions (including four commentary pieces) and an introduction. It is divided into three sections (two of them further subdivided): ‘Qualitative and variable faces of phonological competence’, ‘Sources of variation and their role in the acquisition of phonological competence’, and ‘Knowledge of language-specific organization of speech gestures’. I found these groupings to be nebulous; what comes through much more clearly is a second theme, on sign languages and comparisons between spoken and sign language. The deployment of laboratory methods and ‘philosophy’ in exploring sign languages is an exciting development. Other leitmotifs in the book draw on gestural phonology, exemplar modeling, acquisition, and the roles of abstract and categorical vs. concrete and gradient notions in representation and usage. Some of the papers are probably longer and less clearly written than they might be, but this is a minor complaint about a very interesting collection of works. Given space limitations here, I could not do justice to all twenty-six contributions; instead I focus on highlighting a few of them. Mirjam Ernestus and Harald Baayen, in ‘The functionality of incomplete neutralization in Dutch: The case of past-tense formation’ (27–49), replicate, for one speaker, the finding in Warner et al. 2004, 2006 of incomplete neutralization (IN) of final devoicing in Dutch, based on a reading task involving nonce verb forms. (Unlike in Warner et al., the forms were not presented as minimal pairs.) Particularly interesting are the results of their perception experiments using the speaker’s productions as stimuli. Ernestus and Baayen show that subjects not only detected IN but also used it to choose the appropriate past-tense ending (-te or -de) for the nonce verb stimulus forms, a task that requires the listener to infer the underlying voicing of the stem-final obstruent. The authors argue that IN, as well as their perception results, are due to the storage of lexical paradigms, among other things. Consider for example the form [vεrvεit] ‘widen’ and its infinitival form [vεrvεid n]. Even if the former is stored in its surface form (contrary to the assumption of [End Page 957] most generative phonologists), both the production and the perception of its final consonant will be influenced by activation of the associated form [vεrvεid n] (see also Bybee 2001). IN has posed a serious problem for the traditional understanding of the phonology-phonetics relation, in which discrete phonology is transduced into continuous phonetics, because if /vεrvεid/ is categorically devoiced to [vεrvεit] by phonology, then phonetic implementation has no means of recovering underlying voicing in order to produce IN. The storage and use of entire paradigms circumvents this problem. Since not all...

Computational intelligence in early diabetes diagnosis: a review.

The development of an effective diabetes diagnosis system by taking advantage of computational intelligence is regarded as a primary goal nowadays. Many approaches based on artificial network and machine learning algorithms have been developed and tested against diabetes datasets, which were mostly related to individuals of Pima Indian origin. Yet, despite high accuracies of up to 99% in predicting the correct diabetes diagnosis, none of these approaches have reached clinical application so far. One reason for this failure may be that diabetologists or clinical investigators are sparsely informed about, or trained in the use of, computational diagnosis tools. Therefore, this article aims at sketching out an outline of the wide range of options, recent developments, and potentials in machine learning algorithms as diabetes diagnosis tools. One focus is on supervised and unsupervised methods, which have made significant impacts in the detection and diagnosis of diabetes at primary and advanced stages. Particular attention is paid to algorithms that show promise in improving diabetes diagnosis. A key advance has been the development of a more in-depth understanding and theoretical analysis of critical issues related to algorithmic construction and learning theory. These include trade-offs for maximizing generalization performance, use of physically realistic constraints, and incorporation of prior knowledge and uncertainty. The review presents and explains the most accurate algorithms, and discusses advantages and pitfalls of methodologies. This should provide a good resource for researchers from all backgrounds interested in computational intelligence-based diabetes diagnosis methods, and allows them to extend their knowledge into this kind of research.

Grammar Induction by Unification of Type-logical Lexicons

A method is described for inducing a type-logical grammar from a sample of bare sentence trees which are annotated by lambda terms, called term-labelled trees. Any type logic from a permitted class of multimodal logics may be specified for use with the procedure, which induces the lexicon of the grammar including the grammatical categories. A first stage of semantic bootstrapping is performed, which induces a general form lexicon from the sample of term-labelled trees using Fulop's (J Log Lang Inf 14(1):49---86, 2005) procedure. Next we present a two-stage procedure for performing distributional learning by unifying the lexical types that are initially discovered. The first structural unification algorithm in essence unifies the initial family of sets of types so that the resulting grammar will generate all term-labelled trees that follow the usage patterns evident from the learning sample. Further altering the lexical categories to generate a recursively extended language can be accomplished by a second unification. The combined unification algorithm is shown to yield a new type-logical lexicon that extends the learning sample to a possibly infinite (and possibly context-sensitive) language in a principled fashion. Finally, the complete learning strategy is analyzed from the perspective of algorithmic learning theory; the range of the procedure is shown to be a class of term-labelled tree languages which is finitely learnable from good examples (Lange et al in Algorithmic learning theory, Vol 872 of lecture notes in artificial intelligence, Springer, Berlin, pp 423---437), and so is identifiable in the limit as a corollary.

Testability and Ockham’s Razor: How Formal and Statistical Learning Theory Converge in the New Riddle of Induction

Nelson Goodman’s new riddle of induction forcefully illustrates a challenge that must be confronted by any adequate theory of inductive inference: provide some basis for choosing among alternative hypotheses that fit past data but make divergent predictions. One response to this challenge is to distinguish among alternatives by means of some epistemically significant characteristic beyond fit with the data. Statistical learning theory takes this approach by showing how a concept similar to Popper’s notion of degrees of testability is linked to minimizing expected predictive error. In contrast, formal learning theory appeals to Ockham’s razor, which it justifies by reference to the goal of enhancing efficient convergence to the truth. In this essay, I show that, despite their differences, statistical and formal learning theory yield precisely the same result for a class of inductive problems that I call strongly VC ordered, of which Goodman’s riddle is just one example.

Pre-Service Teachers’ Manipulations of Formal Learning Theories

Pre-Service Teachers’ Manipulations of Formal Learning Theories

A sequent calculus for limit computable mathematics

We introduce an implication-free fragment PA inf + - Exch of ω -arithmetic, having Exchange rule for sequents dropped. Exchange rule for formulas is, instead, an admissible rule in PA inf + - Exch . Our main result is that cut-free proofs of PA inf + - Exch are isomorphic with recursive winning strategies of a set of games called “1-backtracking games” in [S. Berardi, Th. Coquand, S. Hayashi, Games with 1-backtracking, Games for Logic and Programming Languages, Edinburgh, April 2005]. We also show that PA inf + - Exch is a sound and complete formal system for the implication-free fragment of LCM (Limit Computable Mathematics, see [S. Hayashi, Mathematics based on Learning, Algorithmic Learning Theory, in: LNAI, vol. 2533, Springer, pp. 7–21; S. Hayashi, Can proofs be animated by games? in: P. Urzyczyn (Ed.), Seventh International Conference on Typed Lambda Calculi and Applications, in: Lecture Notes in Computer Science, vol. 3461, 2005, pp. 11–22. Invited paper]). A simple syntactical description of this fragment was still missing. No sound and complete formal system is known for LCM itself.

Provably Fast Training Algorithms for Support Vector Machines

Support Vector Machines are a family of algorithms for the analysis of data based on convex Quadratic Programming. We derive randomized algorithms for training SVMs, based on a variation of Random Sampling Techniques; these have been successfully used for similar problems. We formally prove an upper bound on the expected running time which is quasilinear with respect to the number of data points and polynomial with respect to the other parameters, i.e., the number of attributes and the inverse of a chosen soft margin parameter. [This is the combined journal version of the conference papers (Balcazar, J.L. et al. in Proceedings of 12th International Conference on Algorithmic Learning Theory (ALT’01), pp. 119–134, [2001]; Balcazar, J.L. et al. in Proceedings of First IEEE International Conference on Data Mining (ICDM’01), pp. 43–50, [2001]; and Balcazar, J.L. et al. in Proceedings of SIAM Workshop in Discrete Mathematics and Data Mining, pp. 19–29, [2002]).]

Machine Learning for Detection and Diagnosis of Disease

Machine learning offers a principled approach for developing sophisticated, automatic, and objective algorithms for analysis of high-dimensional and multimodal biomedical data. This review focuses on several advances in the state of the art that have shown promise in improving detection, diagnosis, and therapeutic monitoring of disease. Key in the advancement has been the development of a more in-depth understanding and theoretical analysis of critical issues related to algorithmic construction and learning theory. These include trade-offs for maximizing generalization performance, use of physically realistic constraints, and incorporation of prior knowledge and uncertainty. The review describes recent developments in machine learning, focusing on supervised and unsupervised linear methods and Bayesian inference, which have made significant impacts in the detection and diagnosis of disease in biomedicine. We describe the different methodologies and, for each, provide examples of their application to specific domains in biomedical diagnostics.

From learning in the limit to stochastic finite learning

Inductive inference can be considered as one of the fundamental paradigms of algorithmic learning theory. We survey results recently obtained and show their impact to potential applications. Since the main focus is put on the efficiency of learning, we also deal with postulates of naturalness and their impact to the efficiency of limit learners. In particular, we look at the learnability of the class of all pattern languages and ask whether or not one can design a learner within the paradigm of learning in the limit that is nevertheless efficient. For achieving this goal, we deal with iterative learning and its interplay with the hypothesis spaces allowed. This interplay has also a severe impact to postulates of naturalness satisfiable by any learner. Furthermore, since a limit learner is only supposed to converge, one never knows at any particular learning stage whether or not the learner did already succeed. The resulting uncertainty may be prohibitive in many applications. We survey results to resolve this problem by outlining a new learning model, called stochastic finite learning. Though pattern languages can neither be finitely inferred from positive data nor PAC-learned, our approach can be extended to a stochastic finite learner that exactly infers all pattern languages from positive data with high confidence. Finally, we apply the techniques developed to the problem of learning conjunctive concepts.

Sequential sampling techniques for algorithmic learning theory

A sequential sampling algorithm or adaptive sampling algorithm is a sampling algorithm that obtains instances sequentially one by one and determines from these instances whether it has already seen enough number of instances for achieving a given task. In this paper, we present two typical sequential sampling algorithms. By using simple estimation problems for our example, we explain when and how to use such sampling algorithms for designing adaptive learning algorithms.

Human Neural Learning Depends on Reward Prediction Errors in the Blocking Paradigm

Learning occurs when an outcome deviates from expectation (prediction error). According to formal learning theory, the defining paradigm demonstrating the role of prediction errors in learning is the blocking test. Here, a novel stimulus is blocked from learning when it is associated with a fully predicted outcome, presumably because the occurrence of the outcome fails to produce a prediction error. We investigated the role of prediction errors in human reward-directed learning using a blocking paradigm and measured brain activation with functional magnetic resonance imaging. Participants showed blocking of behavioral learning with juice rewards as predicted by learning theory. The medial orbitofrontal cortex and the ventral putamen showed significantly lower responses to blocked, compared with nonblocked, reward-predicting stimuli. In reward-predicting control situations, deactivations in orbitofrontal cortex and ventral putamen occurred at the time of unpredicted reward omissions. Responses in discrete parts of orbitofrontal cortex correlated with the degree of behavioral learning during, and after, the learning phase. These data suggest that learning in primary reward structures in the human brain correlates with prediction errors in a manner that complies with principles of formal learning theory.