Prove that the hypothesis class of all conjunctions over d variables is PAC learnable and bound its sample complexity.

1.  In this question, we study the hypothesis class of Boolean conjunctions defined as follows. The instance space is ={0,1}and the label set is ={0,1}. A literal over the variables x1, . . ., xd is a simple Boolean function that takes the form (x)= xi, for some ∈ [d], or (x) = 1−xi for some ∈ [d]. We use the notation .xi as a shorthand

for 1− xi . A conjunction is any product of literals. In Boolean logic, the product is denoted using the ∧ sign. For example, the function h(x) = x1 · (1− x2) is written as x1 ∧ .x2. We consider the hypothesis class of all conjunctions of literals over the variables. The empty conjunction is interpreted as the all-positive hypothesis (namely, the function that returns h(x) = 1 for all x). The conjunction x1 ∧ .x1 (and similarly any conjunction involving a literal and its negation) is allowed and interpreted as the all-negative hypothesis (namely, the conjunction that returns h(x) = 0 for all x). We assume realizability: Namely, we assume that there exists a Boolean conjunction that generates the labels. Thus, each example (xy) ∈ × consists of an assignment to the Boolean variables x1, . . ., xd , and its truth value (0 for false and 1 for true). For instance, let = 3 and suppose that the true conjunction is x1 ∧ .x2. Then, the training set might contain the following instances: ((1,11)0)((1,01)1)((0,10),0)((1,00)1).

Prove that the hypothesis class of all conjunctions over variables is PAC learnable and bound its sample complexity. Propose an algorithm that implements the ERM rule, whose runtime is polynomial in ·m.

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