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By Johan A. K. Suykens

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45) (the number of errors on the training set). If it happens that at the stage of designing the network one constructs a network that is too complex (for the given amount of training data), the confidence interval ^(p) will be large. In this case, even if one could minimize the empirical risk as small as zero, the amount of errors on the test set can become big. This case is called overfitting. To avoid overfitting (and get a small confidence interval) one has to construct networks with small VC-dimension.

In this section we will fix a confidence 1 — 6 and a sample size m and obtain bounds for the sample error. Lemma 7 Let ci, GI > 0 and s > q > 0. Then the equation xs - cixq - c2 = 0 has a unique positive zero x*. In addition PROOF. Let

C. Burges, Geometry and invariance in kernel based methods, In B. Scholkopf, C. Burges, and A. Smola (ed) Advances in Kernel Methods. Support Vector Learning, MIT Press (1999). [8] C. N. Vapnik, Support Vector Networks, Machine Learning 20 (1995) 273-297. [9] L. Devroye, L. Gyorfi and G. Y. (1996). [10] F. Girosi, An equivalence between sparse approximation and support vector machines, Neural Computation 10(6) (1998) 1455-1480. [11] F. Girosi, M. Jones, and T. Poggio, Regularization theory and neural networks architectures, Neural Computation 7(2) (1995) 219-269.

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