1. Atwo-pole 60 Hz induction motor has a slip of2%on full load. The full-load speed is (a) 58.8 r/s (b) 3528 r/min (c) 60 r/s (d) 2880 r/min 2.The….

## Argue that the probability that for all of these chunks we have LD(A(S))> minh∈H LD(h)+_ is at most δk0 ≤ δ/2.

1. Consider the mapping *φ *: R*d *→R*d*+1 defined by *φ*(x) = (x*,* x 2). Show that if x1*, . . .,*x*m *are shattered by *Bd *then *φ*(x1)*, . . .,φ*(x*m*) are shattered by the class of halfspaces in R*d*+1 (in this question we assume that sign(0) = 1). What does this tell us about VCdim(*Bd*)?

2. (*) Find a set of *d *+1 points in R*d *that is shattered by *Bd *. Conclude that *d *+1 ≤ VCdim(*Bd *) ≤ *d *+2.

3. Boosting the Confidence: Let *A *be an algorithm that guarantees the following: There exist some constant *δ*0 ∈ (0*,*1) and a function *mH *: (0*, *1) → N such that for every *_ *∈ (0*, *1), if *m *≥ *mH*(*_*) then for every distribution *D *it holds that with probability of at least 1−*δ*0, *LD*(*A*(*S*))≤ min*h*∈*H LD*(*h*)+*_*. Suggest a procedure that relies on *A *and learns *H *in the usual agnostic PAC learning model and has a sample complexity of *mH*(*_, δ*) ≤ *kmH*(*_*)+

2log(4*k**/δ*) *_*2 *,* where *k *=_log(*δ*)*/*log(*δ*0)_.

*Hint: *Divide the data into *k *+1 chunks, where each of the first *k *chunks is of size *mH*(*_*) examples. Train the first *k *chunks using *A*. Argue that the probability that for all of these chunks we have *LD*(*A*(*S*))*> *min*h*∈*H LD*(*h*)+*_ *is at most *δ**k*0 ≤ *δ/*2.

Finally, use the last chunk to choose from the *k *hypotheses that *A *generated from the *k *chunks (by relying on Corollary 4.6).