1. Consider the problem of learning halfspaces with the hinge loss. We limit our domain to the Euclidean ball with radius R. That is, X ={x :    x2 ≤ R}. The label set is Y = {±1} and the loss function        is defined by     (w, (x, y)) = max{0,1− y_w,x_}. We already know that the loss function is convex. Show that it is R-Lipschitz.

2.  (*) Convex-Lipschitz-Boundedness Is Not Sufficient for Computational Efficiency: In the next chapter we show that from the statistical perspective, all convex- Lipschitz-bounded problems are learnable (in the agnostic PAC model). However, our main motivation to learn such problems resulted from the computational perspective – convex optimization is often efficiently solvable. Yet the goal of this exercise is to show that convexity alone is not sufficient for efficiency. We show that even for the case d = 1, there is a convex-Lipschitz-bounded problem which cannot be learned by any computable learner. Let the hypothesis class be H=[0, 1] and let the example domain, Z, be the set of all Turing machines. Define the loss function as follows. For every Turing machine

T ∈ Z, let               (0,T )=1 if T halts on the input 0 and         (0,T )=0 if T doesn’t halt on the input 0. Similarly, let        (1,T )=0 if T halts on the input 0 and                (1,T )=1 if T doesn’t halt on the input 0. Finally, for h ∈ (0, 1), let                 (h,T ) = h              (0,T )+(1−h) (1,T ).