EE769 - Data Matrices and Target Vectors for Linear Regression assignment

Assignment Task

Instructions

• You can discuss the concepts with the other students, but each student should write their own code and
• Write copious Each line should ideally have a trailing comment to explain what it does. Before each code block a text block should explain the intent of the trailing code block. Include a trailing text block to note your observations.
• Include some kind of unit testing in code blocks that do not have an output, such as those defining functions
• Max points 55, which will be scaled to approximately 8% towards course

Tasks

1. Write a function to generate a data matrix Inputs: Number of samples, feature dimension. Output: Data matrix X.
2. Write a function to generated dependent variable column
   1. Inputs: Data matrix X, weight vector for each column, bias w0, noise variance
   2. Output: Target vector t
3. Write a function to compute a linear regression
   1. Input: data matrix X and weight vector w
   2. Output: y
4. Write a function to compute the mean square error of two vectors y and
5. Write a function to estimate the weights of linear regression using pseudo-inverse, assuming L2 regularization
   1. Input: X, t, and lambda
   2. Output: w, MSE, y
6. Write a function to compute the gradient of MSE with respect to its weight
   1. Input: X matrix, t vector, and w vector
   2. Output: gradient vector
7. Write a function to compute L2 norm of a vector w passed as a numpy Exclude bias w0.
8. Write a function to compute the gradient of L2 norm with respect to the weight
   1. Input: X matrix and w vector
   2. Output: gradient vector, where gradient with respect to w0 is
9. Write a function to compute L1 norm of a vector w passed as a numpy Exclude bias w0.
10. Write a function to compute the gradient of L1 norm with respect to the weight
    1. Input: X matrix and w vector
    2. Output: gradient vector, where gradient with respect to w0 is
11. Write a function for a single update of weights of linear regression using gradient
    1. Input: X, t, w, eta, lambda 2, lambda Note that the weight of MSE will be 1
    2. Output: updated weight and updated MSE
12. Write a function to estimate the weights of linear regression using gradient
    1. Inputs: X, t, lambda2 (default 0), lambda1 (default 0), eta, max_iter, min_change_NRMSE
    2. Output: Final w, final RMSE normalized with respect to variance of
    3. Stopping criteria: Either max_iter has been reached, or the normalized RMSE does not change by more than min_change_NRMSE
13. Run multiple experiments (with different random seeds) for, plot the results of (box plots), and comment on the trends and potential reasons for the following relations:
    • Training and validation NRMSE obtained using pseudo inverse with number of training samples
    • Training and validation NRMSE obtained using pseudo inverse with number of variables
    • Training and validation NRMSE obtained using pseudo inverse with noise variance
    • Training and validation NRMSE obtained using pseudo inverse with w0
    • Training and validation NRMSE obtained using pseudo inverse with lambda2
    • Time taken to solve pseudo inverse with number of samples and number of variables and its breaking points
    • Training and validation NRMSE obtained using gradient descent with max_iter
    • Training and validation NRMSE obtained using gradient descent with eta
    • Time taken to solve gradient descent with number of samples and number of variables and its breaking points
    • Time taken to solve gradient descent with number of variables and its breaking point
    • Training and validation NRMSE and number of nearly zero weights obtained using gradient descent with lambda2
    • Training and validation NRMSE and number of nearly zero weights obtained using gradient descent with lambda1
    • Training and validation NRMSE for optimal lambda2 with noise variance
    • Training and validation NRMSE for optimal lambda1 with noise variance
    • Experiment (f) but, this time with number of training samples and number of variables
14. Write your overall learning points by doing entire [4 + 2 bonus for diligent work and critical thinking]
15. Quote your references, including roll numbers of fellow students with whom you Be specific about which part was inspired by what source or which friend.

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