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→Rd+1 defined by φ(x) = (x,       x              2). Show that if x1, . . .,xare shattered by Bd then φ(x1), . . .,φ(xm) are shattered by the class of halfspaces in Rd+1 (in this question we assume that sign(0) = 1). What does this tell us about VCdim(Bd)?

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

3. Boosting the Confidence: Let be an algorithm that guarantees the following: There exist some constant δ0 ∈ (0,1) and a function mH : (01) → N such that for every ∈ (01), if ≥ mH(_) then for every distribution it holds that with probability of at least 1−δ0, LD(A(S))≤ minhH LD(h)+_. Suggest a procedure that relies on and learns in the usual agnostic PAC learning model and has a sample complexity of mH(_, δ) ≤ kmH(_)+

2log(4k_, where =_log(δ)/log(δ0)_.

Hint: Divide the data into +1 chunks, where each of the first chunks is of size mH(_) examples. Train the first chunks using A. Argue that the probability that for all of these chunks we have LD(A(S))minhH LD(h)+is at most δk0 ≤ δ/2.

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

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