Homework 1 Solution

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  1. (20 points) LFD Exercise 1.10. Note that you should be able to run a single simulation (flip-ping 1000 fair coins independently 10 times each, and finding the frequency of heads for each of them) in one line of Matlab code. For part (a), in addition to answering the question, also write the single line of Matlab code you used for this. There is no need to submit any additional code.

  1. (20 points) Consider the following experiment on perceptron learning for random training sets of dimension 10:

Generate an 11-dimensional weight vector w , where the first dimension is 0 and the other 10 dimensions are sampled independently at random from the uniform (0; 1) dis-tribution (the first dimension will serve as the threshold, and we’ll just set it to 0 for convenience).

Generate a random training set with 100 examples, where each dimension of each train-ing example is sampled independently at random from the uniform ( 1; 1) distribu-tion, and the examples are all classified in accordance with w .

Run the perceptron learning algorithm, starting with the zero weight vector, on the training set you just generated, and keep track of the number of iterations it takes to learn a hypothesis that correctly separates the training data.

Write code in Matlab to perform the above experiment and then repeat it 1000 times (note that you’re generating a new w and a new training set each time). Plot a histogram of the

number of iterations the algorithm takes to learn a linear separator. How does the number of iterations compare with the bound on the number of errors we derived in class? Note that this bound will be different for each instantiation of w and the training set, so you should either analyze the distribution of bounds across training sets and make a general statement, or directly analyze the distribution of differences.

There is no need to submit your code for this problem, but we may ask you for it if we have any questions about your results when grading the homework.

For up to 10 points of extra credit, can you characterize the situations in which the algorithm takes more iterations to correctly learn a hypothesis that separates the training data? Back up your answer with evidence from your experiments. Hint: You may want to try and visualize a lower dimensional version, and/or hold w fixed and vary the training set as you try to figure this out.

  1. (15 points) LFD Problem 1.1

  1. (15 points) LFD Problem 1.6

  1. (15 points) LFD Problem 1.8

  1. (15 points) LFD Problem 1.12