Define such a function w when for all n Hn is countable (possibly infinite).

1. Let H={hn ∈N} be an infinite countable hypothesis class for binary classification. Show that it is impossible to assign weights to the hypotheses in such that _ could be learned nonuniformly using these weights. That is, the weighting function :H→[0,1] should satisfy the condition _ hHw(h) ≤ 1. _ The weights would be monotonically nondecreasing. That is, if <>j, then w(hi ) ≤ w(h j ).

2.  _ Consider a hypothesis class = _∞ n=1 Hn, where for every ∈ N, Hn is finite. Find a weighting function :H [01] such that _ hHw(h)≤1 and so that for all ∈ Hw(h) is determined by n(h) = min{∈ Hn} and by |Hn(h)|. _ (*) Define such a function when for all n Hn is countable (possibly infinite).


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