Description
I. Pen-and-paper [12v]
Consider the problem of learning a regression model from 5 univariate observations ((0.8), (1), (1.2), (1.4), (1.6)) with targets .
- [5v] Consider the basis function, 𝜙𝑗 𝑥𝑗, for performing a 3-order polynomial regression,
𝑧̂𝑤𝑗𝜙𝑗.
𝑗
Learn the Ridge regression ( regularization) on the transformed data space using the closed form solution with 𝜆 .
Hint: use numpy matrix operations (e.g., linalg.pinv for inverse) to validate your calculus.
- [1v] Compute the training RMSE for the learnt regression model.
- [6v] Consider a multi-layer perceptron characterized by one hidden layer with 2 nodes. Using the activation function 𝑓(𝑥) = 𝑒1𝑥 on all units, all weights initialized as 1 (including biases), and the half squared error loss, perform one batch gradient descent update (with learning rate 𝜂 = 0.1) for the first three observations (0.8), (1) and (1.2).
II. Programming and critical analysis [8v]
Consider the following three regressors applied on kin8nm.arff data (available at the webpage):
| − | linear regression with Ridge regularization term of 0.1 |
| − | two MLPs – 𝑀𝐿𝑃1 and 𝑀𝐿𝑃2 – each with two hidden layers of size 10, hyperbolic tangent function as the activation function of all nodes, a maximum of 500 iterations, and a fixed seed (random_state=0). 𝑀𝐿𝑃1 should be parameterized with early stopping while 𝑀𝐿𝑃2 should not consider early stopping. Remaining parameters (e.g., loss function, batch size, regularization term, solver) should be set as default. |
Using a 70-30 training-test split with a fixed seed (random_state=0):
- [4v] Compute the MAE of the three regressors: linear regression, 𝑀𝐿𝑃1 and 𝑀𝐿𝑃2.
- [1.5v] Plot the residues (in absolute value) using two visualizations: boxplots and histograms. Hint: consider using boxplot and hist functions from matplotlib.pyplot to this end 6) [1v] How many iterations were required for 𝑀𝐿𝑃1 and 𝑀𝐿𝑃2 to converge?
7) [1.5v] What can be motivating the unexpected differences on the number of iterations?
Hypothesize one reason underlying the observed performance differences between the MLPs.
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