Problem $5(20$ Points$)$: Implementation of the Perceptron Algorithm

In this problem, we will implement the Perceptron algorithm on synthetic training data.

$1.5$ Points. Suppose that the data dimension d $=2.$ Generate two classes of data points with $100$

points each, by sampling from Gaussian distributions centered at $(0\mathrm{.}5,0\mathrm{.}5)$ and $(0\mathrm{.}5,0\mathrm{.}5).$ Choose the

variance of the Gaussian to be small enough so that the data points are sufficiently well separated.

$2.10$ Points. Implement the Perceptron algorithm as discussed in class. Choose the initial weights to be

zero, the maximum number of epochs as T $=100,$ and the learning rate $\backslash$alpha $=1.$ How quickly does your

implementation converge?

$3.5$ Points. Now, repeat the above experiment with a second synthetic dataset; this time, increase the

variance of the Gaussians such that the generated data points from different classes now overlap. What

happens to the behavior of the algorithm?