# Homework 8 DSCI 552 Supervised, Semi-Supervised, and Unsupervised Learning solution

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## 1. Supervised, Semi-Supervised, and Unsupervised Learning

https://archive.ics.uci.edu/ml/datasets/Breast+Cancer+Wisconsin+
machine-learning-databases/breast-cancer-wisconsin/wdbc.data, which
has IDs, classes (Benign=B, Malignant=M), and 30 attributes. This data has
two output classes.

(b) Monte-Carlo Simulation: Repeat the following procedures for supervised, unsupervised, and semi-supervised learning M = 30 times, and use randomly selected train and test data (make sure you use 20% of both the positve and negative classes as the test set). Then compare the average scores (accuracy, precision,
recall, F1-score, and AUC) that you obtain from each algorithm.

i. Supervised Learning: Train an L1-penalized SVM to classify the data.
Use 5 fold cross validation to choose the penalty parameter. Use normalized
data. Report the average accuracy, precision, recall, F1-score, and AUC, for
both training and test sets over your M runs. Plot the ROC and report the
confusion matrix for training and testing in one of the runs.

ii. Semi-Supervised Learning/ Self-training: select 50% of the positive
class along with 50% of the negative class in the training set as labeled data
and the rest as unlabelled data. You can select them randomly.

A. Train an L1-penalized SVM to classify the labeled data Use normalized
data. Choose the penalty parameter using 5 fold cross validation.

B. Find the unlabeled data point that is the farthest to the decision boundary
of the SVM. Let the SVM label it (ignore its true label), and add it to
the labeled data, and retrain the SVM. Continue this process until all
unlabeled data are used. Test the final SVM on the test data andthe
average accuracy, precision, recall, F1-score, and AUC, for both training
and test sets over your M runs. Plot the ROC and report the confusion
matrix for training and testing in one of the runs.

iii. Unsupervised Learning: Run k-means algorithm on the whole training
set. Ignore the labels of the data, and assume k = 2.

A. Run the k-means algorithm multiple times. Make sure that you initialize
the algoritm randomly. How do you make sure that the algorithm was
not trapped in a local minimum?

B. Compute the centers of the two clusters and find the closest 30 data
points to each center. Read the true labels of those 30 data points and
take a majority poll within them. The majority poll becomes the label
predicted by k-means for the members of each cluster. Then compare the
labels provided by k-means with the true labels of the training data and
report the average accuracy, precision, recall, F1-score, and AUC over M
runs, and ROC and the confusion matrix for one of the runs.1

1Here we are using k-means as a classifier. The closest 30 data points to each center are labeled by

C. Classify test data based on their proximity to the centers of the clusters.
Report the average accuracy, precision, recall, F1-score, and AUC over
M runs, and ROC and the confusion matrix for one of the runs for the
test data.2

iv. Spectral Clustering: Repeat 1(b)iii using spectral clustering, which is clustering based on kernels.3 Research what spectral clustering is. Use RBF
kernel with gamma=1 or find a gamma for which the two clutsres have the
same balance as the one in original data set (if the positive class has p and the
negative class has n samples, the two clusters must have p and n members).

Do not label data based on their proximity to cluster center, because spectral
clustering may give you non-convex clusters . Instead, use fit − predict
method.

v. One can expect that supervised learning on the full data set works better than
semi-supervised learning with half of the data set labeled.One can expect that
unsupervised learning underperforms in such situations. Compare the results
you obtained by those methods.

## 2. Active Learning Using Support Vector Machines

uci.edu/ml/datasets/banknote+authentication. Choose 472 data points randomly as the test set, and the remaining 900 points as the training set. This is a
binary classification problem.

(b) Repeat each of the following two procedures 50 times. You will have 50 errors for
90 SVMs per each procedure.

i. Train a SVM with a pool of 10 randomly selected data points from the training
set using linear kernel and L1 penalty. Select the penalty parameter using
5-fold cross validation.4 Repeat this process by adding 10 other randomly
experts, so as to use k-means for classification. Obviously, this is a na¨ıve approach.

2K-means algorithm does not provide probabilities, so one can use the distances from cluster center and
pass them through a softmax to calculate probabilities. Alternatively, one can calculate the ROC curve by
varying the threshold for majority polling. Usually, a majority is achieved when t = 50% of the data are in
a class. one can vary t and obtain an ROC curve.

3Because Spectral Clustering will not give you cluster centers, instead of considering 30 closest data points
to the center, consider labeling based on either 30 randomly selected data points or the entire points in each
cluster. Also, for ROC curves, you can vary the threshold of majority polling to obtain an ROC.

4How to choose parameter ranges for SVMs? One can use wide ranges for the parameters and a fine
grid (e.g. 1000 points) for cross validation; however,this method may be computationally expensive. An
alternative way is to train the SVM with very large and very small parameters on the whole training data
and find very large and very small parameters for which the training accuracy is not below a threshold (e.g.,
70%). Then one can select a fixed number of parameters (e.g., 20) between those points for cross validation.

For the penalty parameter, usually one has to consider increments in log(λ). For example, if one found that
the accuracy of a support vector machine will not be below 70% for λ = 10−3 and λ = 106
, one has to choose
log(λ) ∈ {−3, −2, . . . , 4, 5, 6}. For the Gaussian Kernel parameter, one usually chooses linear increments,e.g.
σ ∈ {.1, .2, . . . , 2}.

When both σ and λ are to be chosen using cross-validation, combinations of very small
and very large λ’s and σ’s that keep the accuracy above a threshold (e.g.70%) can be used to determine the
ranges for σ and λ. Please note that these are very rough rules of thumb, not general procedures.

selected data points to the pool, until you use all the 900 points. Do NOT
replace the samples back into the training set at each step. Calculate the
test error for each SVM. You will have 90 SVMs that were trained using 10,
20, 30, … , 900 data points and their 90 test errors. You have implemented
passive learning.

ii. Train a SVM with a pool of 10 randomly selected data points from the training
set5 using linear kernel and L1 penalty. Select the parameters of the SVM
with 5-fold cross validation. Choose the 10 closest data points in the training
set to the hyperplane of the SVM6 and add them to the pool. Do not replace
the samples back into the training set. Train a new SVM using the pool.

Repeat this process until all training data is used. You will have 90 SVMs
that were trained using 10, 20, 30,…, 900 data points and their 90 test errors.
You have implemented active learning.

(c) Average the 50 test errors for each of the incrementally trained 90 SVMs in 2(b)i
and 2(b)ii. By doing so, you are performing a Monte Carlo simulation. Plot
average test error versus number of training instances for both active and passive
learners on the same figure and report your conclusions. Here, you are actually
obtaining a learning curve by Monte-Carlo simulation.

If all selected data points are from one class, select another set of 10 data points randomly.
6You may use the result from linear algebra about the distance of a point from a hyperplane.