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**Example text**

Thus, for a ﬁnite hypothesis set H, R(h) ≤ R(h) + O log2 |H| m . As already pointed out, log2 |H| can be interpreted as the number of bits needed to represent H. Several other remarks similar to those made on the generalization bound in the consistent case can be made here: a larger sample size m guarantees better generalization, and the bound increases with |H|, but only logarithmically. 2 |H| ; it varies as the square But, here, the bound is a less favorable function of logm root of this term.

No, since hS is not a ﬁxed hypothesis, but a random variable depending on the training sample S drawn. 3), the generalization error R(hS ) is a random variable and in general distinct from the expectation E[R(hS )], which is a constant. Thus, as in the proof for the consistent case, we need to derive a uniform convergence bound, that is a bound that holds with high probability for all hypotheses h ∈ H. 2 Learning bound — ﬁnite H, inconsistent case Let H be a ﬁnite hypothesis set. Then, for any δ > 0, with probability at least 1 − δ, the following inequality holds: ∀h ∈ H, R(h) ≤ R(h) + log |H| + log 2δ .

2 Growth function Here we will show how the Rademacher complexity can be bounded in terms of the growth function. ,xm }⊆X h(x1 ), . . , h(xm ) : h ∈ H . 19) Thus, ΠH (m) is the maximum number of distinct ways in which m points can be classiﬁed using hypotheses in H. This provides another measure of the richness of the hypothesis set H. However, unlike the Rademacher complexity, this measure does not depend on the distribution, it is purely combinatorial. 2 Growth function 39 To relate the Rademacher complexity to the growth function, we will use Massart’s lemma.