Prove that if a class H is nonuniformly learnable then there are classes Hn so that H= _ n∈N Hn and, for every n ∈ N, VCdim(Hn) is finite.

1. Let be a nonuniform learner for a class H. For each ∈ N define HA n = {H: mNUL(0.10.1,h) ≤ n}. Prove that each such class Hn has a finite VC-dimension.

2. Prove that if a class is nonuniformly learnable then there are classes Hn so that H= _ n∈N Hn and, for every ∈ N, VCdim(Hn) is finite.

3. Let be a class that shatters an infinite set. Then, for every sequence of classes (Hn ∈ N) such that = _ n∈N

Hn, there exists some for which VCdim(Hn)=∞. Hint: Given a class H that shatters some infinite set K, and a sequence of classes (Hn ∈ N), each having a finite VC-dimension, start by defining subsets Kn ⊆ K such that, for all n, |Kn|>VCdim(Hnand for any n _=m, Kn ∩ Km =∅. Now, pick for each such Kn a function fn Kn →{0,1} so that no h ∈ Hn agrees with fn on the domain Kn. Finally, define f →{0,1} by combining these fn’s and prove that f ∈ _ H\_ n∈N Hn _

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