Dynamics of on-line learning in radial basis function networks

On-line learning is examined for the radial basis function network, an important and practical type of neural network. The evolution of generalization error is calculated within a framework which allows the phenomena of the learning process, such as the specialization of the hidden units, to be analyzed. The distinct stages of training are elucidated, and the role of the learning rate described. The three most important stages of training, the symmetric phase, the symmetry-breaking phase, and the convergence phase, are analyzed in detail; the convergence phase analysis allows derivation of maximal and optimal learning rates. As well as finding the evolution of the mean system parameters, the variances of these parameters are derived and shown to be typically small. Finally, the analytic results are strongly confirmed by simulations.

Curvature-driven smoothing: a learning algorithm for feed-forward networks

The performance of feed-forward neural networks in real applications can be often be improved significantly if use is made of a-priori information. For interpolation problems this prior knowledge frequently includes smoothness requirements on the network mapping, and can be imposed by the addition to the error function of suitable regularization terms. The new error function, however, now depends on the derivatives of the network mapping, and so the standard back-propagation algorithm cannot be applied. In this paper, we derive a computationally efficient learning algorithm, for a feed-forward network of arbitrary topology, which can be used to minimize the new error function. Networks having a single hidden layer, for which the learning algorithm simplifies, are treated as a special case.

Measuring performance of virtual learning environment system in higher education

Purpose – The purpose of this paper is to measure the performance of commercial virtual learning environment (VLE) systems, which helps the decision makers to select the appropriate system for their institutions. Design/methodology/approach – This paper develops an integrated multiple criteria decision making approach, which combines the analytic hierarchy process (AHP) and quality function deployment (QFD), to evaluate and select the best system. The evaluating criteria are derived from the requirements of those who use the system. A case study is provided to demonstrate how the integrated approach works. Findings – The major advantage of the integrated approach is that the evaluating criteria are of interest to the stakeholders. This ensures that the selected system will achieve the requirements and satisfy the stakeholders most. Another advantage is that the approach can guarantee the benchmarking to be consistent and reliable. From the case study, it is proved that the performance of a VLE system being used at the university is the best. Therefore, the university should continue to run the system in order to support and facilitate both teaching and learning. Originality/value – It is believed that there is no study that measures the performance of VLE systems, and thus decision makers may have difficulties in system evaluation and selection for their institutions.

The exact sample complexity of PAC-learning problems with unit VC dimension

The Vapnik-Chervonenkis (VC) dimension is a combinatorial measure of a certain class of machine learning problems, which may be used to obtain upper and lower bounds on the number of training examples needed to learn to prescribed levels of accuracy. Most of the known bounds apply to the Probably Approximately Correct (PAC) framework, which is the framework within which we work in this paper. For a learning problem with some known VC dimension, much is known about the order of growth of the sample-size requirement of the problem, as a function of the PAC parameters. The exact value of sample-size requirement is however less well-known, and depends heavily on the particular learning algorithm being used. This is a major obstacle to the practical application of the VC dimension. Hence it is important to know exactly how the sample-size requirement depends on VC dimension, and with that in mind, we describe a general algorithm for learning problems having VC dimension 1. Its sample-size requirement is minimal (as a function of the PAC parameters), and turns out to be the same for all non-trivial learning problems having VC dimension 1. While the method used cannot be naively generalised to higher VC dimension, it suggests that optimal algorithm-dependent bounds may improve substantially on current upper bounds.

On-line learning in radial basis functions networks

An analytic investigation of the average case learning and generalization properties of Radial Basis Function Networks (RBFs) is presented, utilising on-line gradient descent as the learning rule. The analytic method employed allows both the calculation of generalization error and the examination of the internal dynamics of the network. The generalization error and internal dynamics are then used to examine the role of the learning rate and the specialization of the hidden units, which gives insight into decreasing the time required for training. The realizable and over-realizable cases are studied in detail; the phase of learning in which the hidden units are unspecialized (symmetric phase) and the phase in which asymptotic convergence occurs are analyzed, and their typical properties found. Finally, simulations are performed which strongly confirm the analytic results.

Recursive self-organizing map as a contractive iterative function system

Recently, there has been a considerable research activity in extending topographic maps of vectorial data to more general data structures, such as sequences or trees. However, the representational capabilities and internal representations of the models are not well understood. We rigorously analyze a generalization of the Self-Organizing Map (SOM) for processing sequential data, Recursive SOM (RecSOM [1]), as a non-autonomous dynamical system consisting off a set of fixed input maps. We show that contractive fixed input maps are likely to produce Markovian organizations of receptive fields o the RecSOM map. We derive bounds on parameter $\beta$ (weighting the importance of importing past information when processing sequences) under which contractiveness of the fixed input maps is guaranteed.

A randomized controlled trial of the effect of thyroxine replacement on cognitive function in community-living elderly subjects with subclinical hypothyroidism: the Birmingham elderly thyroid study

Context: Subclinical hypothyroidism (SCH) and cognitive dysfunction are both common in the elderly and have been linked. It is important to determine whether T4 replacement therapy in SCH confers cognitive benefit. Objective: Our objective was to determine whether administration of T4 replacement to achieve biochemical euthyroidism in subjects with SCH improves cognitive function. Design and Setting: We conducted a double-blind placebo-controlled randomized controlled trial in the context of United Kingdom primary care. Patients: Ninety-four subjects aged 65 yr and over (57 females, 37 males) with SCH were recruited from a population of 147 identified by screening. Intervention: T4 or placebo was given at an initial dosage of one tablet of either placebo or 25 µg T4 per day for 12 months. Thyroid function tests were performed at 8-weekly intervals with dosage adjusted in one-tablet increments to achieve TSH within the reference range for subjects in treatment arm. Fifty-two subjects received T4 (31 females, 21 males; mean age 73.5 yr, range 65–94 yr); 42 subjects received placebo (26 females, 16 males; mean age 74.2 yr, 66–84 yr). Main Outcome Measures: Mini-Mental State Examination, Middlesex Elderly Assessment of Mental State (covering orientation, learning, memory, numeracy, perception, attention, and language skills), and Trail-Making A and B were administered. Results: Eighty-two percent and 84% in the T4 group achieved euthyroidism at 6- and 12-month intervals, respectively. Cognitive function scores at baseline and 6 and 12 months were as follows: Mini-Mental State Examination T4 group, 28.26, 28.9, and 28.28, and placebo group, 28.17, 27.82, and 28.25 [not significant (NS)]; Middlesex Elderly Assessment of Mental State T4 group, 11.72, 11.67, and 11.78, and placebo group, 11.21, 11.47, and 11.44 (NS); Trail-Making A T4 group, 45.72, 47.65, and 44.52, and placebo group, 50.29, 49.00, and 46.97 (NS); and Trail-Making B T4 group, 110.57, 106.61, and 96.67, and placebo group, 131.46, 119.13, and 108.38 (NS). Linear mixed-model analysis demonstrated no significant changes in any of the measures of cognitive function over time and no between-group difference in cognitive scores at 6 and 12 months. Conclusions: This RCT provides no evidence for treating elderly subjects with SCH with T4 replacement therapy to improve cognitive function.

The development of a multi-function computer-assisted instruction system using a hierarchical database structure as the lesson compendium.

Computer-Based Learning systems of one sort or another have been in existence for almost 20 years, but they have yet to achieve real credibility within Commerce, Industry or Education. A variety of reasons could be postulated for this, typically: - cost - complexity - inefficiency - inflexibility - tedium Obviously different systems deserve different levels and types of criticism, but it still remains true that Computer-Based Learning (CBL) is falling significantly short of its potential. Experience of a small, but highly successful CBL system within a large, geographically distributed industry (the National Coal Board) prompted an investigation into currently available packages, the original intention being to purchase the most suitable software and run it on existing computer hardware, alongside existing software systems. It became apparent that none of the available CBL packages were suitable, and a decision was taken to develop an in-house Computer-Assisted Instruction system according to the following criteria: - cheap to run; - easy to author course material; - easy to use; - requires no computing knowledge to use (as either an author or student) ; - efficient in the use of computer resources; - has a comprehensive range of facilities at all levels. This thesis describes the initial investigation, resultant observations and the design, development and implementation of the SCHOOL system. One of the principal characteristics c£ SCHOOL is that it uses a hierarchical database structure for the storage of course material - thereby providing inherently a great deal of the power, flexibility and efficiency originally required. Trials using the SCHOOL system on IBM 303X series equipment are also detailed, along with proposed and current development work on what is essentially an operational CBL system within a large-scale Industrial environment.

Content-based VLE designs improve learning efficiency in constructivist statistics education

Background: We introduced a series of computer-supported workshops in our undergraduate statistics courses, in the hope that it would help students to gain a deeper understanding of statistical concepts. This raised questions about the appropriate design of the Virtual Learning Environment (VLE) in which such an approach had to be implemented. Therefore, we investigated two competing software design models for VLEs. In the first system, all learning features were a function of the classical VLE. The second system was designed from the perspective that learning features should be a function of the course's core content (statistical analyses), which required us to develop a specific-purpose Statistical Learning Environment (SLE) based on Reproducible Computing and newly developed Peer Review (PR) technology. Objectives: The main research question is whether the second VLE design improved learning efficiency as compared to the standard type of VLE design that is commonly used in education. As a secondary objective we provide empirical evidence about the usefulness of PR as a constructivist learning activity which supports non-rote learning. Finally, this paper illustrates that it is possible to introduce a constructivist learning approach in large student populations, based on adequately designed educational technology, without subsuming educational content to technological convenience. Methods: Both VLE systems were tested within a two-year quasi-experiment based on a Reliable Nonequivalent Group Design. This approach allowed us to draw valid conclusions about the treatment effect of the changed VLE design, even though the systems were implemented in successive years. The methodological aspects about the experiment's internal validity are explained extensively. Results: The effect of the design change is shown to have substantially increased the efficiency of constructivist, computer-assisted learning activities for all cohorts of the student population under investigation. The findings demonstrate that a content-based design outperforms the traditional VLE-based design. © 2011 Wessa et al.

Learning in ultrametric committee machines

The problem of learning by examples in ultrametric committee machines (UCMs) is studied within the framework of statistical mechanics. Using the replica formalism we calculate the average generalization error in UCMs with L hidden layers and for a large enough number of units. In most of the regimes studied we find that the generalization error, as a function of the number of examples presented, develops a discontinuous drop at a critical value of the load parameter. We also find that when L>1 a number of teacher networks with the same number of hidden layers and different overlaps induce learning processes with the same critical points.

A novel adaptive weight selection algorithm for multi-objective multi-agent reinforcement learning

To solve multi-objective problems, multiple reward signals are often scalarized into a single value and further processed using established single-objective problem solving techniques. While the field of multi-objective optimization has made many advances in applying scalarization techniques to obtain good solution trade-offs, the utility of applying these techniques in the multi-objective multi-agent learning domain has not yet been thoroughly investigated. Agents learn the value of their decisions by linearly scalarizing their reward signals at the local level, while acceptable system wide behaviour results. However, the non-linear relationship between weighting parameters of the scalarization function and the learned policy makes the discovery of system wide trade-offs time consuming. Our first contribution is a thorough analysis of well known scalarization schemes within the multi-objective multi-agent reinforcement learning setup. The analysed approaches intelligently explore the weight-space in order to find a wider range of system trade-offs. In our second contribution, we propose a novel adaptive weight algorithm which interacts with the underlying local multi-objective solvers and allows for a better coverage of the Pareto front. Our third contribution is the experimental validation of our approach by learning bi-objective policies in self-organising smart camera networks. We note that our algorithm (i) explores the objective space faster on many problem instances, (ii) obtained solutions that exhibit a larger hypervolume, while (iii) acquiring a greater spread in the objective space.

Learning in friendship groups:developing students’ conceptual understanding through social interaction

The role that student friendship groups play in learning was investigated here. Employing a critical realist design, two focus groups on undergraduates were conducted to explore their experience of studying. Data from the "case-by-case" analysis suggested student-to-student friendships produced social contexts which facilitated conceptual understanding through discussion, explanation, and application to "real life" contemporary issues. However, the students did not conceive this as a learning experience or suggest the function of their friendships involved learning. These data therefore challenge the perspective that student groups in higher education are formed and regulated for the primary function of learning. Given these findings, further research is needed to assess the role student friendships play in developing disciplinary conceptual understanding.