Problem 2. Single predictor soft-thresholding derivation (25 points) Consider a single predictor setting, based on samples {(1}, y,) 3:1. Assume that the data 2 has been standardized (Le. Hmlb = 1). The problem is to minimize with respect to the 71 function: 1 T2; 2n _ 1,: 1 (92' @532 + AW (4) where A 2 0. The standard approach to this univariate minimization problem would be to take the gradient with respect to ,8, and set it to zero. There is a complication, however, because the absolute value of the function | | does not have a derivative at B = 0. However, we can proceed by direct inspection of the function in equation (4) and nd , n zz; ya: A if i zyzlyix, > ,\\ 5 = SA (71 Z%%) = 0 if $21.11 y,:c,-| g A . (5) i=1 % 2:121 yzliz' 'i' A if $22121 yiffz' 0. Under this scenario which one of the following options must be true: . 5:0 wage - {320 . 60 Explain your reasoning. (a) Case 1a: 227; 34.33.; g A. Under this scenario, nd the optimal value for ,6. Explain your reasoning. (b) Case 1b: 3 23:1 yizri > A. Under this scenario, nd the optimal value for 5 (Hint: 71, think of the solution for the problem minxeR (1332 + bx + c where (1,5,0 6 R are constants). o. (5 points) Case 2: 22; yimi = 0. Under this scenario which one of the following options must be true: . :0 43:0 4320 . B0 Explain your reasoning. d. (5 points) Case 3: $22; yimi 0 Explain your reasoning. (a) Case 3a: 2221 yimi > )\\. Under this scenario, nd the optimal value for 5. Explain your reasoning. (1)) Case 3b: $22121 yiccz-