By Johan A. K. Suykens
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Additional info for Advances in learning theory: methods, models, and applications
The result now follows since -^ therefore || (id + ^) || > 1. is positive definite and D 38 F. Cucker, S. 4 Estimating the Sample Error The expression |£(/7) — £(/7,z)| is called the sample error (of /7,z). In the previous section we estimated the confidence of obtaining a small sample error when the sample size ra and an error bound e are given. 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*.
The capacity factor (to choose the element Sn with the appropriate value of VC dimension). We confine ourselves now to the pattern recognition case and consider two type of learning machines: 1. Neural Networks (NN) that were inspired on the biological analogy with the brain 2. The support vector machines that were inspired on statistical learning theory. We discuss how each corresponding machine can control these factors. , i. ,wn) (weights) which minimize the empirical risk functional There are several methods for minimizing this functional.
5 Problem of constructing rigorous (distribution dependent) bounds To construct rigorous bounds for the rate of convergence one has to take into account information about the probability measure. Let PQ be a set of all probability measures and let P C PQ be a subset of the set PQ. We say that one has prior information about an unknown probability measure P(z) if one knows the set of measures P that contains P(z). , ze). A(£). For another extreme case where P contains only one function P(z) the Generalized growth function coincides with the annealed VC-entropy.
Advances in learning theory: methods, models, and applications by Johan A. K. Suykens