Recall that ridge regression refers to loss min wRd,bR Xw+b1-y+|||| error where X Rx and...
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Recall that ridge regression refers to loss min wЄRd,bЄR Xw+b1-y+|||| error where X € Rx and y € R" are the given dataset and A > 0 is the regularization hyperparameter. 1. (1 pt) Show that the derivatives are a (1) X(Xw+b1-y)+2\w Ow = 11 (Xw+b1-y). მს (2) (3) 2. (2 pts) Implement the gradient descent algorithm for solving ridge regression. The following incomplete pseudo-code may of help. Test your implementation on the Boston housing dataset (to predict the median house price, i.e., y). Train and test splits are provided on course website. Try A e {0, 10} and report your training error, training loss and test error. [Your training loss should monotonically decrease during iteration; if not try to tune your step size n, e.g. make it smaller.] Algorithm 1: Gradient descent for ridge regression. Input: XRxd, y € R", wo=0d, b = 0, max-pass € N, 77 > 0, tol > 0) Output: w, b 1 for t= 1,2,..., max.pass do 2 W₁ <- 3 bet 4 if w-w-tol then 5 break 6 ww₁, b<br // can use other stopping criteria 3. (1 pt) We note that given w, we can actually solve b by setting the derivative (3) to 0. Re-implement Line 3 in Algorithm 1 with this closed-form solution. Does the modification converge to the same solution? 4. (1 pt) If we center our data beforehand, i.e., by subtracting their mean we get XT1 = 0 and 1 y = 0. What is the optimal value of b in this case? [You may verify your result by running your code above.] Recall that ridge regression refers to loss min wЄRd,bЄR Xw+b1-y+|||| error where X € Rx and y € R" are the given dataset and A > 0 is the regularization hyperparameter. 1. (1 pt) Show that the derivatives are a (1) X(Xw+b1-y)+2\w Ow = 11 (Xw+b1-y). მს (2) (3) 2. (2 pts) Implement the gradient descent algorithm for solving ridge regression. The following incomplete pseudo-code may of help. Test your implementation on the Boston housing dataset (to predict the median house price, i.e., y). Train and test splits are provided on course website. Try A e {0, 10} and report your training error, training loss and test error. [Your training loss should monotonically decrease during iteration; if not try to tune your step size n, e.g. make it smaller.] Algorithm 1: Gradient descent for ridge regression. Input: XRxd, y € R", wo=0d, b = 0, max-pass € N, 77 > 0, tol > 0) Output: w, b 1 for t= 1,2,..., max.pass do 2 W₁ <- 3 bet 4 if w-w-tol then 5 break 6 ww₁, b<br // can use other stopping criteria 3. (1 pt) We note that given w, we can actually solve b by setting the derivative (3) to 0. Re-implement Line 3 in Algorithm 1 with this closed-form solution. Does the modification converge to the same solution? 4. (1 pt) If we center our data beforehand, i.e., by subtracting their mean we get XT1 = 0 and 1 y = 0. What is the optimal value of b in this case? [You may verify your result by running your code above.]
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Accounting for Governmental and Nonprofit Entities
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17th edition
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