By Johan A. K. Suykens

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19] M. Talagrand, The Glivenko-Cantelli problem, ten years later, Journal of Theoretical Probability 9(2) (1996) 371-384. N. Vapnik, Estimation of Dependencies Based on Empirical Data, [in Russian], Nauka, Moscow (1979) (English translation: (1982) Springer-Verlag, New York). N. Vapnik, The Nature of Statistical Learning Theory, Springer-Verlag, New-York (1995). N. Vapnik, Statistical Learning Theory, John Wiley, New-York (1998). N. Vapnik and A. Ja. Chervonenkis, On the uniform convergence of relative frequencies of events to their probabilities, Reports of Academy of Science USSR 181(4) (1968).

Z*) be the number of elements of the minimal £-net of the set of vectors q(a), a € A. , zt> The expectation of the random VC-entropy is called the VC-entropy of the set of functions A < Q(z, a) < B, a € A on the sample of the size t. ,zg) = The main results of the theory of uniform convergence of the empirical risk to the actual risk for bounded loss functions include the following theorem [24]: Theorem 3 For uniform two-sided convergence of the empirical risks to the actual risks lim Prob{sup (\R(a) - RemP(a}\ > e} =0, Ve.

Cuclcer, S. Smale Lemma 2 For all 7 > 0 r. PROOF. Since /7]Z = £0^, we have ||/7l8|& = aTAT[x]a. Also, since a = (7mld + X[x])~1y it follows that where ||a|| and ||y|| refer to the Euclidean norm in Mm. t Km = -j-C* I '' i where ||-ftT[x]|| denotes the operator norm of K[x] : Hm —>• Rm with respect to the Euclidean norm in both domain and target space and we have used that, since each entry of K[x] is bounded in absolute value by CK, ||-K"[x]|| < C^ra. D Corollary 1 For all 7 > 0, ||/7|U < Sol£dlsa and \\f^\\M < &&.