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1. (500) Classifying Handwritten Digits: 1 vs. 5
Pick one of the following 3 classification algorithms for non-separable data:
(i) Linear Regression for classification followed by pocket for improvement.
(ii) Linear Programming for classification.
(iii) Logistic regression for classification using gradient descent.
Use your chosen algorithm to find the best separator you can using the training data only (use
your 2 features from a previous assignment as the inputs). The output is +1 if the example is a
1 and -1 for a 5.
(a) Give separate plots of the training and test data, together with the separators.
(b) Compute Ein on your training data and Etest, the test error on the test data.
(c) Obtain a bound on the true out-of-sample error. You should get two bounds, one based on
Ein and one based on Etest. Use a tolerance δ = 0.05.
Which is the better bound?
(d) Now repeat using a 3rd order polynomial transform.
(e) As your final deliverable to a customer, would you use the linear model with or without the
3rd order polynomial transform? Explain.
2. (200) Gradient Descent on a “Simple” Function
Consider the function f(x, y) = x
2 + 2y
2 + 2 sin(2πx) sin(2πy).
(a) Implement gradient descent to minimize this function. Let the initial values be x0 = 0.1; y0 =
0.1, let the learning rate be η = 0.01 and let the number of iterations be 50; Give a plot of
the how the function value drops with the number of iterations performed.
Repeat this problem for a learning rate of η = 0.1. What happened?
(b) Obtain the “minimum” value and the location of the minimum you get for gradient descent
using the same η and number of iterations as in part (a), starting from the following initial
points: (0.1, 0.1),(1, 1),(−0.5, −0.5),(−1, −1). A table with the location of the minimum
and the minimum values will suffice. You should now appreciate why finding the “true”
global minimum of an arbitrary function is a hard problem.
3. (300) Problem 3.16 in LFD
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