1. An example of the neuron model is the sigmoid function which is defined by:

φ(v) = 1/(1+exp(-v))

For this problem, you need to prove that

(∂φ(v))/∂v = φ(v)(1-φ(v))

2. Consider the following cost function:

E(w)=1/2 σ² - r_xd^T w+1/2 w^T R_x w

where σ² is some constant, and

(a) Find the optimal value w^* for which E(w) reaches its minimum value, and give this minimum cost.

(b) Use the method of gradient descent to compute w^* for the following two values of learning-rate parameter:

η = 0.3

η = 1.0

For each case, plot the trajectories of two weights and its corresponding cost with the number of iterations. These two weights should converge to the optimal values as you will give in (a).

Notice: for this problem, you need to prepare one additional document to attach your code and figures. Print out the document and staple them to the back of this sheet.