Assignment-2 Solution

Description

Please use Google Classroom to upload your submission by the deadline mentioned above. Your submission should comprise of a single le (PDF/ZIP), named <Your Roll No> Assign2, with all your solutions.

For late submissions, 10% is deducted for each day (including weekend) late after an assign-ment is due. Note that each student begins the course with 7 grace days for late submission of assignments. Late submissions will automatically use your grace days balance, if you have any left. You can see your balance on the CS6510 Marks and Grace Days document.

You have to use Python for the programming questions.

Please read the department plagiarism policy. Do not engage in any form of cheating – strict penalties will be imposed for both givers and takers. Please talk to instructor or TA if you have concerns.

Questions: Theory

(No programming required)

1. Kernels: (5 marks) Let k₁ and k₂ be valid kernel functions. Comment about the validity of the following kernel functions, and justify your answer with proof or counter-examples as required:

1. 1. k(x; z) = k₁(x; z) + k₂(x; z)

1. 2. k(x; z) = k₁(x; z)k₂(x; z)

1. 3. k(x; z) = h(k₁(x; z)) where h is a polynomial function with positive co-e cients

(d) k(x; z) = exp(k₁(x; z))

(e) k(x; z) = exp	x zk22
	2

2. Perceptron Classi er: (3 marks) Solve the three simple classi cation problems shown in the gure below by drawing a decision boundary. Find weight and bias values that result in single-neuron perceptrons with the chosen decision boundaries:

3. Neural Networks: (2+2=4 marks) Given the following training data/label inputs:

fx₁ = [11]; y₁ = 1g; fx₂ = [1 1]; y₂ = 1g

and that these inputs are used to train a perceptron with no bias using the LMS (Widrow-

Ho /Delta learning) rule, answer the following questions:

1. What does the mean error surface look like? In particular, comment on its curvature along the w₁ and w₂ axes, as well as where the surface is centered on (the minimum).

2. What can you say about the eigenvalues of the Hessian of the error function?

Questions: Programming

4. SVMs: (3 + 2 + 4 + 3 = 12 marks) In this question, you will be working on a soft-margin SVM. You may nd it helpful to review the Scikit Learn’s SVM documentation: http://scikit-learn.org/stable/modules/svm.html.

We will apply soft-margin SVM to handwritten digits from the processed US Postal Service Zip Code data set. The data (extracted features of intensity and symmetry) for training and testing are available at:

http://www.amlbook.com/data/zip/features.train http://www.amlbook.com/data/zip/features.test

We will train a one-versus-one (one digit is class +1 and another digit is class -1) classi er for the digits ‘1’ (+1) and ‘5’ (-1).

1. Consider the linear kernel K(x_n; x_m) = x^T_n x_m. Train using the provided training data and test using the provided test data, and report your accuracy over the entire test set, and the number of support vectors.

2. In continuation, train only using the rst f50; 100; 200; 800g points with the linear kernel. Report the accuracy over the entire test set, and the number of support vectors in each of these cases.

2. Consider the polynomial kernel K(x_n; x_m) = (1 + x^T_n x_m)^Q, where Q is the degree of the polynomial. Comparing Q = 2 with Q = 5, comment whether each of the following statements is TRUE or FALSE.

2. 1. When C = 0:0001, training error is higher at Q = 5.

2. 2. When C = 0:001, the number of support vectors is lower at Q = 5.

2. 3. When C = 0:01, training error is higher at Q = 5.

2. 4. When C = 1, test error is lower at Q = 5.

(d) Consider the radial basis function (RBF) kernel K(x_n; x_m) = e( x_n x_mjj²) in the soft-margin SVM approach. Which value of C 2 f0:01; 1; 100; 10⁴; 10⁶g results in the

lowest training error? The lowest test error? Show the error values for all the C values.

Deliverables:

Code

Brief report (PDF) with your solutions for the above questions

5. SVMs (contd): (3 + 4 = 7 marks) (5 marks) GISETTE (https://archive.ics. uci.edu/ml/datasets/Gisette) is a handwritten digit recognition problem. The problem is to separate the highly confusible digits ‘4’ and ‘9’. This dataset is one of ve datasets of the NIPS 2003 feature selection challenge. The dataset for this problem is large, so please budget time accordingly for this problem.

1. Standard run: Use all the 6000 training samples from the training set to train the model, and test over all test instances, using the linear kernel. Report the train error, test error, and number of support vectors.

2. Kernel variations: In addition to the basic linear kernel, investigate two other standard

kernels: RBF (a.k.a. Gaussian kernel; set = 0:001), Polynomial kernel (set degree = 2, coef0 = 1; e.g, (1 + x^T x)²). Which kernel yields the lowest training error? Report the train error, test error, and number of support vectors for both these kernels.

Deliverables:

Code

Brief report (PDF) with your solutions for the above questions

6. Random Forests: (5 + 2 + 2 = 9 marks)

1. Write your own random forest classi er (this should be relatively easy, given you have written your own decision tree code) to apply to the Spam dataset [data, information]. Use 30% of the provided data as test data and the remaining for training. Compare your results in terms of accuracy and time taken with Scikitlearn’s built-in random forest classi er.

2. Explore the sensitivity of Random Forests to the parameter m (the number of features used for best split).

3. Plot the OOB (out-of-bag) error (you have to nd what this is, and read about it!) and the test error against a suitably chosen range of values for m.

Deliverables:

Code

Brief report (PDF) with your solutions for the above questions

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