By Johan A. K. Suykens

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We say that the VC dimension of the set of indicator functions Q(z, a), a € A is infinite if the Growth function for this set of functions is linear. We say that the VC dimension of the set of indicator functions Q(z, a), a e A is finite and equals h if the Growth function is bounded by a logarithmic function with coefficient h. The finiteness of the VC-dimension of the set of indicator functions implemented by the learning machine forms the necessary and sufficient condition for consistency of the ERM method independent of the probability measure.

In what follows, for / : X —* B, and x € Xm, we denote by /[x] the point (/(zi), • • •, /(zm)) € Hm. , |um|}. Proposition 2 For all 7, e > 0, Prob PROOF OF THEOREM 2. - /7)Z[x]||max < 26} > 1 - 4me 2C M2 - <^c-)2. Recall, The first and last terms are each bounded by e with probabilities at least 67 by Proposition 1 and the fact that r7 > PL/. For the middle term note that j_ m 1=1 < I Now apply Proposition 2 to bound this term by 2e with probability at least mcV l-4me ac'jfM'dr+cjc)' and the conclusion follows by noting that 2C2YM2(7 + C# )2 < 8M4(7 + CK}4 and by replacing e by 6/4.

12 V. 18). 3 Two important examples Example 1 1. ,zn] is equal to h = n + 1, since using functions of this set one can shatter at most n+ 1 vectors. Here #{•} is the step function, which takes value 1 if the expression between brackets is positive and takes value 0 otherwise. 2. ,zn} is also equal to h = n + 1 because the VC-dimension of the corresponding linear indicator functions is equal to n+1 (using aQ—(3 instead of &Q does not change the set of indicator functions). Example 2 We call a hyperplane (w*>x)-b = Q, K| = l the A-margin separating hyperplane if it classifies vectors x as follows _ f 1 if (w* • x) - b > A y ~ { -1 if (w* • x) - b < -A.