ISYE 6740 Homework 3 Total 100 points. 1. Basic optimization. (30 points.) Consider a simplied logistic regression problem. Given m training samples (xi; yi), i

ISYE 6740 Homework 3 Total 100 points. 1. Basic optimization. (30 points.) Consider a simplied logistic regression problem. Given m training samples (xi; yi), i = 1; : : : ;m. The data xi 2 R (note that we only have one feature for each sample), and yi 2 f0; 1g. To t a logistic regression model for classication, we solve the following optimization problem, where 2 R is a parameter we aim to nd: max `(); (1) where the log-likelhood function `() = Xm i=1 f???? log(1 + expf????xig) + (yi ???? 1)xig : (a) (10 points) Show step-by-step mathematical derivation for the gradient of the cost function `() in (1) and write a pseudo-code for performing gradient descent to nd the optimizer . This is essentially what the training procedure does. (pseudo-code means you will write down the steps of the algorithm, not necessarily any specic programming language.) (b) (10 points) Present a stochastic gradient descent algorithm to solve the training of logistic regression problem (1). (c) (10 points) We will show that the training problem in basic logistic regression problem is concave. Derive the Hessian matrix of `() and based on this, show the training problem (1) is concave (note that in this case, since we only have one feature, the Hessian matrix is just a scalar). Explain why the problem can be solved eciently and gradient descent will achieve a unique global optimizer, as we discussed in class. 2. Comparing Bayes, logistic, and KNN classiers. (30 points) In lectures we learn three dierent classiers. This question is to implement and compare them. We are suggest use Scikit-learn, which is a commonly-used and powerful Python library with various machine learning tools. But you can also use other similar library in other languages of your choice to perform the tasks. Part One (Divorce classication/prediction). (20 points) This dataset is about participants who completed the personal information form and a divorce predic- tors scale. The data is a modied version of the publicly available at https://archive.ics.uci.edu/ml/datasets/ Divorce+Predictors+data+set (by injecting noise so you will not replicate the results on uci web- site). There are 170 participants and 54 attributes (or predictor variables) that are all real-valued. The dataset marriage.csv. The last column of the CSV le is label y (1 means \divorce”, 0 means \no divorce”). Each column is for one feature (predictor variable), and each row is a sample (participant). A detailed explanation for each feature (predictor variable) can be found at the website link above. Our goal is to build a classier using training data, such that given a test sample, we can classify (or essentially predict) whether its label is 0 (\no divorce”) or 1 (\divorce”). 1 Build three classiers using (Naive Bayes, Logistic Regression, KNN). Use the rst 80% data for training and the remaining 20% for testing. If you use scikit-learn you can use train test split to split the dataset. Remark: note that, here, for Naive Bayes, this means that we have to estimate the variance for each individual feature from training data. When estimating the variance, if the variance is zero to close to zero (meaning that there is very little variability in the feature), you can set the variance to be a small number, e.g., = 10????3. We do not want to have include zero or nearly variance in Naive Bayes. This tip holds for both Part One and Part Two of this question. (a) (10 points) Report testing accuracy for each of the three classiers. Comment on their perfor- mance: which performs the best and make a guess why they perform the best in this setting. (b) (10 points) Use the rst two features to train three new classiers. Plot the data points and decision boundary of each classier. Comment on the dierence between the decision boundary for the three classiers. clearly represent the data points with dierent labels using dierent colors. Part Two (Handwritten digits classication). (10 points) Repeat the above using the MNIST Data in our Homework 2. Here, give \digit” 6 label y = 1, and give \digit” 2 label y = 0. All the pixels in each image will be the feature (predictor variables) for that sample (i.e., image). Our goal is to build classier to such that given a new test sample, we can tell is it a 2 or a 6. Using the rst 80% of the samples for training and remaining 20% for testing. Report the classication accuracy on testing data, for each of the three classiers. Comment on their performance: which performs the best and make a guess why they perform the best in this setting. 3. Naive Bayes for spam ltering. (40 points) In this problem we will use the Naive Bayes algorithm to t a spam lter by hand. This will en- hance your understanding to Bayes classier and build intuition. This question does not involve any programming but only derivation and hand calculation. Spam lters are used in all email services to classify received emails as \Spam” or \Not Spam”. A simple approach involves maintaining a vocabulary of words that commonly occur in \Spam” emails and classifying an email as \Spam” if the number of words from the dictionary that are present in the email is over a certain threshold. We are given the vocabulary consists of 15 words V = fsecret, oer, low, price, valued, customer, today, dollar, million, sports, is, for, play, healthy, pizzag: We will use Vi to represent the ith word in V . As our training dataset, we are also given 3 example spam messages, • million dollar oer • secret oer today • secret is secret and 4 example non-spam messages • low price for valued customer • play secret sports today • sports is healthy • low price pizza 2 Recall that the Naive Bayes classier assumes the probability of an input depends on its input feature. The feature for each sample is dened as x(i) = [x(i) 1 ; x(i) 2 ; : : : ; x(i) d ]T , i = 1; : : : ;m and the class of the ith sample is y(i). In our case the length of the input vector is d = 15, which is equal to the number of words in the vocabulary V . Each entry x(i) j is equal to the number of times word Vj occurs in the i-th message. (a) (5 points) Calculate class prior P(y = 0) and P(y = 1) from the training data, where y = 0 corresponds to spam messages, and y = 1 corresponds to non-spam messages. Note that these class prior essentially corresponds to the frequency of each class in the training sample. (b) (10 points) Write down the feature vectors for each spam and non-spam messages. (c) (15 points) In the Naive Bayes model, assuming the keywords are independent of each other (this is a simplication), the likelihood of a sentence with its feature vector x given a class c is given by P(xjy = c) = Yd k=1 xk c;k; c = f0; 1g where 0 c;k 1 is the probability of word k appearing in class c, which satises Xd k=1 c;k = 1; 8c: Given this, the complete log-likelihood function for our training data is given by `(1;1; : : : ; 1;d; 2;1; : : : ; 2;d) = Xm i=1 Xd k=1 x(i) k log y(i);k (In this example, m = 7.) Calculate the maximum likelihood estimates of 0;1, 0;7, 1;1, 1;15 by maximizing the log-likelihood function above. (Hint: We are solving a constrained maximization problem. To do this, remember, you need to introduce two Lagrangian multiplier because you have two constraints.) (d) (10 points) Given a test message \today is secret”, using the Naive Bayes classier that you have trained in Part (a)-(c), to calculate the posterior and decide whether it is spam or not spam. 3

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