1. Suppose that a list contains the values 20 44 48 55 62 66 74 88 93 99 at index positions 0 through 9. Trace the values of the variables….

## Prove that for every learning algorithm A there exist a probability distribution, D, and a learning algorithm B such that A is not better than B w.r.t. D.

1. Let *X *be a domain and {0*,*1} be a set of labels. Prove that for every distribution *D *over *X *× {0*,*1}, there exist a learning algorithm *AD *that is better than any other learning algorithm with respect to *D*.

2. Prove that for every learning algorithm *A *there exist a probability distribution, *D*, and a learning algorithm *B *such that *A *is not better than *B *w.r.t. *D*.

3. Consider a variant of the PAC model in which there are two example oracles: one that generates positive examples and one that generates negative examples, both according to the underlying distribution *D *on *X*. Formally, given a target function *f *: *X *→ {0*,*1}, let *D*+ be the distribution over *X*+ = {*x *∈ *X *: *f *(*x*) = 1} defined by *D*+(*A*)= *D*(*A*)*/**D*(*X*+), for every *A *⊂ *X*+. Similarly, *D*− is the distribution over *X*− induced by *D*. The definition of PAC learnability in the two-oracle model is the same as the standard definition of PAC learnability except that here the learner has access to *m*+ *H*(*_, δ*) i.i.d. examples from*D*+ and *m*−(*_, δ*) i.i.d. examples from*D*−. The learner’s goal is to output *h *s.t. with probability at least 1 − *δ *(over the choice of the two training sets, and possibly over the nondeterministic decisions made by the learning algorithm), both *L*(*D*+ *, **f *)(*h*) ≤ *_ *and *L*(*D*−*, **f *)(*h*) ≤ *_*.