1. Strong Convexity with Respect to General Norms: Throughout the section we used the            2 norm. In this exercise we generalize some of the results to general norms. Let       ·               be some arbitrary norm, and let f be a strongly convex function with respect to this norm (see Definition 13.4).

2. Show that items 2–3 of Lemma 13.5 hold for every norm.

3. (*) Give an example of a norm for which item 1 of Lemma 13.5 does not hold.

4. Let R(w) be a function that is (2λ)-strongly convex with respect to some nor. Let A be an RLM rule with respect to R, namely, A(S)= argmin W _ LS(w)+ R(w) _ .

Assume that for every z, the loss function            (·, z) is ρ-Lipschitz with respect to the same norm, namely, ∀z, ∀w, v,     (w, z)−          (v, z) ≤ ρ               w−v       .

Prove that A is on-average-replace-one-stable with rate 2ρ2 λm .