By the bias-variance decomposition at some ; we have E [(fm(1.) - f(2.)] = (Elfm(2.)] - (2.)2 + E[(fm(Z.) - Elfm(z.)])2] Bias? (1.) Variance(ri) a. (5 points) Intuitively, how do you expect the bias and variance to behave for small values of m? What about large values of m? b. [5 points) If we define f) =m E: (-1)m+1 f(x;) and the average bias-squared as , ET, (Elfm(x.)]- f(x;)), show that j=1i=(-1 )m+1 C. (5 points) If we define the average variance as E : Et(fm(x.) -Elfm(x.)])2|, show (both equali- ties) m d. [15 points) Let n = 256, o2 = 1, and f(x) = 4sin(x) cos(67x2). For values of m = 1, 2, 4, 8, 16, 32 plot the average empirical error " Et (fm(.) - f(x:)) using randomly drawn data as a function of m on the r-axis. On the same plot, using parts b and c of above, plot the average bias-squared, the average variance, and their sum (the average error). Thus, there should be 4 lines on your plot, each described in a legend. e. (5 points) By the Mean-Value theorem we have that min i (j-1 )m+1,....j f( a. ) 0. Show that the average bias-squared is O( ,"). Using the expression for average variance above, the total error behaves like O(LT + "). Minimize this expression with respect to m. Does this value of m, and the total error when you plug this value of m back in, behave in an intuitive way with respect to n, L, 2? That is, how does m scale with each of these parameters? It turns out that this simple estimator (with the optimized choice of m) obtains the best achievable error rate up to a universal constant in this setup for this class of L-Lipschitz functions (see Tsybakov's Introduction to Nonparametric Estimation for details). This setup of each r; deterministically placed at i is a good approximation for the more natural setting where each z; is drawn uniformly at random from [0, 1]. In fact, one can redo this problem and obtain nearly identical conclusions, but the calculations are messier