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Recall that we defined a class C to be (improperly) PAC-learnable by a hypothesis class H if our PAC-learning algorithm runs in time polynomial in

Recall that we defined a class C to be (improperly) PAC-learnable by a hypothesis class H if our PAC-learning algorithm runs in time polynomial in n (the number of attributes), the size of the target c C, 1/, and 1/, and with probability 1 produces a representation h H as output that agrees with the labels given by c with probability 1 . We restricted our attention to representations that are efficiently evaluatable, i.e., to classes H such that there is a polynomial time algorithm that, given as input a representation h and example x, computes h(x). Show that if a class C is improperly PAC-learnable by any efficiently evaluatable hypothesis class, then it is PAC-learnable by Boolean circuits. (Hint. You may find it useful that, as stated in lecture, circuits can be efficiently generated for any efficient algorithm.)

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