For the first part of this assignment we will consider a synthetic prediction problem to develop our intuition about the error decomposition. Consider the random variables xX=[0,1] distributed uniformely (xUnif([0,1])) and yY=R defined as a polynomial of degree 2 of x : there exists (a0,a1,a2)R3 such that the values of x and y are linked as y=g(x)= a0+a1x+a2x2. Note that this relation fixes the joint distribution PXY. From the knowledge of a sample {xi,yi}i=1N, we would like to predict the relation between x and y, that is find a function f to make predictions y^=f(x). We note Hd, the set of polynomial functions on R of degree d:Hd={f:xb0+bx+bdxd;bkRk{0,d}}. We will consider the hypothesis classes Hd varying d. We will minimize the squared loss (y^,y)= 21(y^y)2 to solve the regression problem. 1. (2 Points) Recall the definition of the expected risk R(f) of a predictor f. While this cannot be computed in general note that here we defined PXY. Which function f is an obvious Bayes predictor? Make sure to explain why the risk R(f) is minimum at f. 2. (2 Points) Using H2 as your hypothesis class, which function fH2 is a risk minimizer in H2 ? Recall the definition of the approximation error. What is the approximation error achieved by fH2 ? 3. (2 Points) Considering now Hd, with d>2. Justify an inequality between R(fH2) and R(fHd). Which function fHd is a risk minimizer in Hd ? What is the approximation error achieved by fHd ? 4. (4 Points) For this question we assume a0=0. Considering H={f:xb1x;b1R}, which function fH is a risk minimizer in H ? What is the approximation error achieved by fH ? In particular what is the approximation error achieved if furthermore a2=0 in the definition of true underlying relation g(x) above