(1) Sometimes machine learning is used on imperfect training data - for example, data collected via...

 

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(1) Sometimes machine learning is used on imperfect training data - for example, data collected via noisy sensors. In these cases, we might try to correct for noise while training the classifier. Consider the following formulation for training a logistic regression classifier w Rd on a noisy training data set (x(), y()),..., (x(n), y(n)), where for each i, y(i) {1, +1}. For simplicity, we ignore the bias term b. Suppose we know that the noise magnitude is at most r. Then, instead of the standard logistic regression loss, we might want to minimize the following loss: L(w) = -1 Li(w), where, L (w) = log(1 + exp(-y) w z(i))), max z(i):||z(i)-x(i)||r where |v|| means the L2-norm of vector v. (a) (5 points) Prove that for all i, L (w) Mi (w), where M (w) log(1 + exp(r||w|| - yw x))). For full credit, show all the steps in your proof. = (1) Sometimes machine learning is used on imperfect training data - for example, data collected via noisy sensors. In these cases, we might try to correct for noise while training the classifier. Consider the following formulation for training a logistic regression classifier w Rd on a noisy training data set (x(), y()),..., (x(n), y(n)), where for each i, y(i) {1, +1}. For simplicity, we ignore the bias term b. Suppose we know that the noise magnitude is at most r. Then, instead of the standard logistic regression loss, we might want to minimize the following loss: L(w) = -1 Li(w), where, L (w) = log(1 + exp(-y) w z(i))), max z(i):||z(i)-x(i)||r where |v|| means the L2-norm of vector v. (a) (5 points) Prove that for all i, L (w) Mi (w), where M (w) log(1 + exp(r||w|| - yw x))). For full credit, show all the steps in your proof. =