2. (14.0 points) Suppose we want to fit a constant model, y = 0, to a dataset {y1, yz, ..., yn} (a) (3.0 pt) For the first two parts of this question, suppose we use the following loss function: We choose to use the following loss function: L(yi, 0) = - In ( exp ( - kil)) What is the derivative of the average loss R (i.e. empirical risk) for this choice of model and loss function with respect to 0? . Hint: exp(x) = ex, and In(x) = log.(x). . Hint: In(a exp(b)) = In(a) + b. O dR = 0 O do = A Elyilexp( -luz!) O dR do = ly exp( - lul) O dR O do (b) (3.0 pt) Determine the value of 0 that minimizes average loss for the above choice of model and loss function. O 0 = 1 En elyil 0 0= 1Elyil n E- 1 yi(c) (2.0 pt) For the remaining parts of this question, consider the following two loss functions. LAWi) = 2(yz' 9J2 LB(yia6) = 3ly 9| Let 6:4 and 6}, be the values of 6 that minimize average L A loss and average L B loss, respectively, for the constant model and a given dataset. Which of the following is true regarding the relationship between 6:4 and 6}; for the dataset y = [1, 2, 3, 4, 5]? O a; = 633 O a; > a}; O 634 00, what happens to 6A and 6B? O 634 and 619 both increase 0 Both 6;; and 63 remain constant 0 6A increases, but 63 remains constant 0 6A remains constant, but 63 increases (f) (2.0 pt) Now consider another new loss function, Lc(yi, 9) = (in 26)? True or False: For every dataset consisting of positive real numbers 3; = [y1, y2, ...,yn], the value of 6 that minimizes average L A loss is equal to the value of 6 that minimizes average LC loss (in other words, 9,, = 90). 0 True 0 False