Show that the two approaches are equivalent, i.e. they will produce the same solution for. 1.2 Linearly Separable Data In this problem we will investigate the behavior linearly separable. " 1. Show that the decision boundary of logistic regression is given by x: xTu w 0. Note that the set will not change if we multiply the weights by some constantc. 1.3 2. Suppose the data is linearly separable and by gradient descent/ascent we have reached a decision boundary defined by w where all examples are classified correctly. Show that we can increase the likelihood of the data by increasing a scaler c on w unboundedly, which means that MLE is not well-defined in this case. [Hint: You can show this by taking the derivative of L(cw) with respect to c, where L is the likelihood function] 2 Regularized Logistic Regression As we've shown in Section 1.2, when the data is linearly separable MLE for logistic regression may end up with very large weights, which is a sign of overfitting. In this part, we will apply regularization to fix the problem The regularized logistic regression objective function J logistic (w) = n (w) + X || w || 1 n of MLE for logistic regression when the data is i=1 log 1. Prove that the objective function J. the convex optimization notes. 1 + exp can be defined as (i) w Tx(i) 3. Complete the fit logistic_regression_function function from scipy.optimize " logistic (w) is convex. You may use any facts mentioned in + X || w || . code, 2. Complete the f_objective function in the skeleton which computes the objective function for J. logistic (w). (Hint: you may get numerical overflow when computing the expo- nential literally, e.g., try e 1000 in Numpy. Make sure to read about the log-sum-exp trick and use the numpy function logaddexp to get accurate calculations and to prevent overflow. " in the skeleton code using the minimize Use this function to train a model on the provided data. Make sure to take the appropriate preprocessing steps such as standardizing the data and adding a column for the bias term. 2 4. Find the ' regularization parameter that minimizes the log -likelihood on the validation set. Plot the log - likelihood for different values of the regularization parameter -w Tr 5. [Optional] It seems reasonable to interpret the prediction f(x) (w Tx) 1/ 1+e-u as the probability that y = 1, for a randomly drawn pair (x, y). Since we only have a finite sample (and we are regularizing, which will bias things a bit) there is a question of how well = =