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….

## Suggest a procedure that agnostic PAC learns the problem with sample complexity of mH(_, δ), assuming that the loss function is bounded by 1.

1. Show that the resulting learning problem is convex-Lipschitz-bounded.

2. Show that no computable algorithm can learn the problem.

3. From Bounded Expected Risk to Agnostic PAC Learning: Let *A *be an algorithm that guarantees the following: If *m *≥ *mH*(*_*) then for every distribution *D *it holds that E *S*∼*Dm* [*LD*(*A*(*S*))]≤ min *h*∈*H* *LD*(*h*)+*_*. _ Show that for every *δ *∈ (0*, *1), if *m *≥ *mH*(*_ δ*) then with probability of at least 1−*δ *it holds that *LD*(*A*(*S*))≤ min*h*∈*H LD*(*h*)+*_*. *Hint: *Observe that the random variable *LD*(*A*(*S*))−min*h*∈*H LD*(*h*) is nonnegative and rely on Markov’s inequality.

_ For every *δ *∈ (0*, *1) let

*mH*(*_, δ*) = *mH*(*_/*2)_log2 (1*/δ*)_+

log(4*/δ*)+log(_log2 (1*/δ*)_)

*_*2

. Suggest a procedure that agnostic PAC learns the problem with sample complexity of *mH*(*_, δ*), assuming that the loss function is bounded by 1. *Hint: *Let *k *= _log2 (1*/δ*)_. Divide the data into *k *+1 chunks, where each of the first *k *chunks is of size *mH*(*_/*2) examples. Train the first *k *chunks using *A*. On the basis of the previous question argue that the probability that for all of these chunks we have *LD*(*A*(*S*))*>*min*h*∈*H LD*(*h*)+*_ *is at most 2−*k *≤*δ/*2. Finally, use the last chunk as a validation set.