By Edwin F. Beckenbach (Editor)

ISBN-10: 1842022032

ISBN-13: 9781842022030

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ISBN-13: 9782582812943

ISBN-10: 2592612742

ISBN-13: 9782592612748

ISBN-10: 3443453473

ISBN-13: 9783443453473

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University of the Basque Country, Dept. of Computer Science and AI, PO Box 649 E-20080 San Sebastian, Spain. , Royal Holloway, University of London, Egham, Surrey TW20 0EX UK. , Derbyshire Constabulary Headquarters, Butterley Hall, Ripley, Derbyshire DE5 3RS UK. , University of the Basque Country, Dept. of Computer Science and AI, PO Box 649 E-20080 San Sebastian, Spain. , Royal Institute of Technology, Studies of Artifical Neural Systems, Department of Numerical Analysis and Computing Science, S-100 44 Stockholm, Sweden.

13) over the set of real bounded functions as well [16]. 3 Three Milestones of Learning Theory In this section we consider the set of indicator functions Q(z, a), aÎL (we consider the problem of pattern recognition). Analysis of this set of functions plays crucial part in learning theory. For the set of indicator functions (footnote continued from previous page) vector (in a given metric). For a C metric that means 2. N is minimal number of vectors which possess this property. 5 Note that VC-entropy is different from classical metrical e-entropy where NL(e) is cardinality of the minimal e-net of the set of functions Q(z, a), aÎL.

5 Four Parts of Learning Theory Learning theory has to address the following four questions [11]: 1. What are (necessary and sufficient) conditions for consistency of the learning process based on the ERM principle? This means to specify the necessary and sufficient conditions for convergence in probability2 both the values of risks for the functions minimizing the empirical risk Remp(a)) to the minimal possible value of the risk R(a0) and value of obtained empirical risks 2Convergence in probability of values means that for any e > 0 and for any h > 0 there exists a number such, that for any with probability at least 1 - h the inequality holds true.

### Applied Combinatorial Mathematics by Edwin F. Beckenbach (Editor)

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