An Overview of Schema Theory

The purpose of this paper is to give an introduction to the field of Schema Theory written by a mathematician and for mathematicians. In particular, we endeavor to to highlight areas of the field which might be of interest to a mathematician, to point out some related open problems, and to suggest some large-scale projects. Schema theory seeks to give a theoretical justification for the efficacy of the field of genetic algorithms, so readers who have studied genetic algorithms stand to gain the most from this paper. However, nothing beyond basic probability theory is assumed of the reader, and for this reason we write in a fairly informal style. Because the mathematics behind the theorems in schema theory is relatively elementary, we focus more on the motivation and philosophy. Many of these results have been proven elsewhere, so this paper is designed to serve a primarily expository role. We attempt to cast known results in a new light, which makes the suggested future directions natural. This involves devoting a substantial amount of time to the history of the field. We hope that this exposition will entice some mathematicians to do research in this area, that it will serve as a road map for researchers new to the field, and that it will help explain how schema theory developed. Furthermore, we hope that the results collected in this document will serve as a useful reference. Finally, as far as the author knows, the questions raised in the final section are new.Comment: 27 pages. Originally written in 2009 and hosted on my website, I've decided to put it on the arXiv as a more permanent home. The paper is primarily expository, so I don't really know where to submit it, but perhaps one day I will find an appropriate journa

Probabilistic and fuzzy reasoning in simple learning classifier systems

This paper is concerned with the general stimulus-response problem as addressed by a variety of simple learning c1assifier systems (CSs). We suggest a theoretical model from which the assessment of uncertainty emerges as primary concern. A number of representation schemes borrowing from fuzzy logic theory are reviewed, and sorne connections with a well-known neural architecture revisited. In pursuit of the uncertainty measuring goal, usage of explicit probability distributions in the action part of c1assifiers is advocated. Sorne ideas supporting the design of a hybrid system incorpo'rating bayesian learning on top of the CS basic algorithm are sketched

Learning Computer Programs with the Bayesian Optimization Algorithm

The hierarchical Bayesian Optimization Algorithm (hBOA) [24, 25] learns bit-strings by constructing explicit centralized models of a population and using them to generate new instances. This thesis is concerned with extending hBOA to learning open-ended program trees. The new system, BOA programming (BOAP), improves on previous probabilistic model building GP systems (PMBGPs) in terms of the expressiveness and open-ended flexibility of the models learned, and hence control over the distribution of individuals generated. BOAP is studied empirically on a toy problem (learning linear functions) in various configurations, and further experimental results are presented for two real-world problems: prediction of sunspot time series, and human gene function inference

The effect of multiple knowledge sources on learning and teaching

Current paradigms for machine-based learning and teaching tend to perform their task in isolation from a rich context of existing knowledge. In contrast, the research project presented here takes the view that bringing multiple sources of knowledge to bear is of central importance to learning in complex domains. As a consequence teaching must both take advantage of and beware of interactions between new and existing knowledge. The central process which connects learning to its context is reasoning by analogy, a primary concern of this research. In teaching, the connection is provided by the explicit use of a learning model to reason about the choice of teaching actions. In this learning paradigm, new concepts are incrementally refined and integrated into a body of expertise, rather than being evaluated against a static notion of correctness. The domain chosen for this experimentation is that of learning to solve "algebra story problems." A model of acquiring problem solving skills in this domain is described, including: representational structures for background knowledge, a problem solving architecture, learning mechanisms, and the role of analogies in applying existing problem solving abilities to novel problems. Examples of learning are given for representative instances of algebra story problems. After relating our views to the psychological literature, we outline the design of a teaching system. Finally, we insist on the interdependence of learning and teaching and on the synergistic effects of conducting both research efforts in parallel

Runtime analysis of convex evolutionary search algorithm with standard crossover

This is the final version. Available on open access from Elsevier via the DOI in this recordEvolutionary Algorithms (EAs) with no mutation can be generalized across representations as Convex Evolu- tionary Search algorithms (CSs). However, the crossover operator used by CSs does not faithfully generalize the standard two-parents crossover: it samples a convex hull instead of a segment. Segmentwise Evolutionary Search algorithms (SESs) are defined as a more faithful generalization, equipped with a crossover operator that samples the metric segment of two parents. In metric spaces where the union of all possible segments of a given set is always a convex set, a SES is a particular CS. Consequently, the representation-free analysis of the CS on quasi- concave landscapes can be extended to the SES in these particular metric spaces. When instantiated to binary strings of the Hamming space (resp. -ary strings of the Manhattan space), a polynomial expected runtime upper bound is obtained for quasi-concave landscapes with at most polynomially many level sets for well-chosen popu- lation sizes. In particular, the SES solves Leading Ones in at most 288 ln [4 (2 + 1)] expected fitness evaluations when the population size is equal to 144 ln [4 (2 + 1)]