LAB 01.01 - WARM UP

!wget -nc --no-cache -O -q
import init; init.init(force_download=False); init.get_weblink()
from local.lib.rlxmoocapi import submit, session
import inspect
session.LoginSequence(endpoint=init.endpoint, course_id=init.course_id, lab_id="L01.01", varname="student");

Task 1: An operation with matrices

Given the following matrices and vectors:

  • \(X \in \mathbb{R}^{m\times n}\)

  • \(y \in \mathbb{R}^m\)

  • \(W \in \mathbb{R}^{n\times 1}\)

  • \(b \in \mathbb{R}\)

Complete the following function so that it computes the following value:

\[\text{mean}\bigg(\big(\text{relu}(X \times W + b) - y\big)^2\bigg)\]

observe that:

  • \(X\times W \in \mathbb{R}^m\) and \(b \in \mathbb{R}\), so \(b\) gets added (broadcasted) to all elements of \(X \times W\)

  • \(\text{mean}\) is the mean of the elements of an vector with \(m\) elements.

  • The result is a number \(\in \mathbb{R}\)

  • \(\text{relu}(z)=\text{max}(0,z)\) is a function \(\mathbb{R}\rightarrow\mathbb{R}\) that when applied to a vector is also broadcasted (applied individually to each element of the vector)

CHALLENGE: Solve it with a single line of Python code (not counting the relu function definition).

import numpy as np

def operation(X,y,W,b):
    relu = lambda x: x*(x>0)
    return ... # YOUR CODE HERE

test your code with the following case, which should result in 0.15848

X = np.array([[-0.09348275, -0.17182042, -0.29143506],
              [ 0.34581753,  0.37816707,  0.39850916],
              [ 0.23478876, -0.07832256,  0.10793716],
              [-0.1746856 , -0.10240038, -0.27959607]])

y = np.array([[-0.47312685],
              [ 0.42086142],
              [ 0.44194868],
              [ 0.46536898]])
W = np.array([[0.12650597],

b = -0.02

test your code with random input values. This is actually what the automatic grader does

m,n = np.random.randint(5, size=2)+2
X = np.random.random(size=(m,n))-0.5
W = np.random.random(size=(n,1))-0.5
b = np.random.random()-0.5
y = np.random.random(size=n)-0.5
print ("X=\n", X)
print ("y=\n", y)
print ("W=\n", W)
print ("b=\n", b)
print ("an_operation=", operation(X,y,W,b))

Submit your solution

student.submit_task(namespace=globals(), task_id='T1');

Task 2: Function argmax

Complete the following function such that when, given as argument a function f(x) with \(x\in\mathbb{R}\), returns the value of \(x\) which maximizes f(x). If there is more than one value that maximizes the function, just return any one of those.

Your return value must be exact up to 1 decimal position, and must be a number of type float (not a numpy array or any other type of object)

HINT: Use scipy.optimize.minimize with the BFGS method, with a lambda function.

CHALLENGE: Solve it with one single line of code (not counting the import)

def argmax(f):
    from scipy.optimize import minimize
    return ...

Test your code, the following two functions must have their max on x=1 and x=-2 approx.

def A(x):
    return -(x-1)**2

B = lambda x: -(x+2)**4
import matplotlib.pyplot as plt
%matplotlib inline

x = np.linspace(-6,4, 100)
plt.subplot(121); plt.plot(x, A(x)); plt.title("A(X)")
plt.subplot(122); plt.plot(x, B(x)); plt.title("B(X)")
argmax(A), argmax(B)

Submit your solution

student.submit_task(namespace=globals(), task_id='T2');