Q2) Gradient Descent Algorithm (2 marks) i) Consider the last 4 digits of your mobile number (Note : In case there is a

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Q2) Gradient Descent Algorithm (2 marks)
i) Consider the last 4 digits of your mobile number (Note : In case
there is a 0 in one of the digits replace it by 3). Let it be
n1n2n3n4. Generate a random matrix A of size n1n2 ×n3n4. For
example, if the last four digits are 2311, generate a random
matrix of size 23 ×11. Write a code to calculate the l∞ norm
of this matrix.
Deliverable(s) : The code that generates the results. (0.5)
ii) Generate a random vector b of size n1n2 ×1 and consider the
function
f(x) = 1/2∥Ax −b∥22 where ∥ · ∥2 is the vector ℓ2 norm. Its
gradient is given to be ∇f(x) = A⊤Ax −A⊤b. Write a code to
find the local minima of this function by using the gradient
descent algorithm (by using the gradient expression given
to you). The step size τ in the iteration xk+1 = xk −τ∇f(xk)
should be chosen by the formula
τ = gTk gk/gTk
ATAgk where gk = ∇f(xk) = A⊤Axk −A⊤b. The algorithm
should execute until ∥xk −xk−1∥2 < 10−4.
Deliverable(s) : The code that finds the minimum of the given
function 2 and the expression for τ. The values of xk and
f(xk) should be stored in a file. (1)
iii) Generate the graph of f(xk) vs k where k is the iteration
number and xk is the current estimate of x at iteration k.
This graph should convey the decreasing nature of function
values.
Deliverable(s) : The graph that is generated.