We call a learning environment partially blind when there is an admixture o f supervised and unsupervised (or blind) learning. Such situations typically arise in practice when supervised training data labelled by a teacher are scarce or expensive and are supplemented by inexpensive unlabelled (or blind) data available in relative profusion. Vapnik-eervonenkis theory can be deployed in such sett...

Abstract. The Vapnik-Chervonenkis (VC) dimension and the Sauer-Shelah lemma have found applications in numerous areas including set theory, combinatorial geometry, graph theory and statistical learning theory. Estimation of the complexity of discrete structures associated with the search space of algorithms often amounts to estimating the cardinality of a simpler class which is effectively indu...

One fundamental goal in any learning algorithm is to mitigate its risk for overfitting. Mathematically, this requires that the learning algorithm enjoys a small generalization risk, which is defined either in expectation or in probability. Both types of generalization are commonly used in the literature. For instance, generalization in expectation has been used to analyze algorithms, such as ri...

Very often symbolic regression, as addressed in Genetic Programming (GP), is equivalent to approximate interpolation. This means that, in general, GP algorithms try to fit the sample as better as possible but no notion of generalization error is considered. As a consequence, overfitting, code-bloat and noisy data are problems which are not satisfactorily solved under this approach. Motivated by...

The Vapnik-Chervonenkis (VC) dimension (also known as the trace number) and the Sauer-Shelah lemma have found applications in numerous areas including set theory, combinatorial geometry, graph theory and statistical learning theory. Estimation of the complexity of discrete structures associated with the search space of algorithms often amounts to estimating the cardinality of a simpler class wh...

In this paper we present a case study of a 1-dimensional higher order neuron using a statistical approach to learning theory which incorporates some information on the distribution on the sample space and can be viewed as a modification of the Vapnik-Chervonenkis formalism (VC-formalism). We concentrate on learning curves defined as averages of the worst generalization error of binary hypothesi...

The Vapnik-Chervonenkis (VC) dimension and the Sauer-Shelah lemma have found applications in numerous areas including set theory, combinatorial geometry, graph theory and statistical learning theory. Estimation of the complexity of discrete structures associated with the search space of algorithms often amounts to estimating the cardinality of a simpler class which is effectively induced by som...