Description
1
Problem 1: Partial Derivatives
Consider the following functions of the variables u, v, and w. Assume the
variables x, y, x
(i) and y
(i) are constants: they represent numbers that will not
change during the execution of a machine learning algorithm (e.g., the training
data). (2 points for each problem)
f(u, v) = 8u
2
v
4 + 4v
3 + 6u
g(u, v, w) = x log(u) + yuvw3 + 13x
3
h(u, v) = Xm
i=1
1
2
(x
(i)u + y
(i)
v)
2
Write the following partial derivatives:
1. ∂
∂u f(u, v)
2. ∂
∂v f(u, v)
3. ∂
∂u g(u, v, w)
4. ∂
∂v g(u, v, w)
5. ∂
∂w g(u, v, w)
6. ∂
∂u h(u, v)
7. ∂
∂v h(u, v)
Problem 2: Partial Derivative Intuition
Consider the following contour plot of a function f(u, v):
2
20
20
20
20
20
20
40
40
40
40
40
60
u
v
−5 −4 −3 −2 −1 0 1 2 3 4 5
−5
−4
−3
−2
−1
0
1
2
3
4
5
For each of the following partial derivatives, state whether it is positive, negative, or equal to zero. Briefly
explain. These questions can be answered from the contour plot without knowing the formula for the function.
(Note: for two numbers a and b we will use the notation ∂
∂u f(a, b) to mean “the partial derivative of f(u, v)
with respect to u at the point where u = a and v = b”. This notation is succinct but obfuscates the original
variable names. A more explicit way to write the same thing is ∂
∂u f(u, v)|u=a,v=b.)
1.
∂
∂uf(−2, −2)
2.
∂
∂v f(−2, −2)
3.
∂
∂uf(3, −3)
4. ∂
∂v f(3, −3)
5. To the nearest integer, estimate the values of u and v that minimize f(u, v).
Problems (10 points each)
The first few problems will consider the “slope-only” (i.e., θ0 = 0) linear regression model from class, which
has the following hypothesis and cost function:
Hypothesis : hθ(x) = θ1x
Cost function : J(θ1) = 1
2
Xm
i=1
(hθ(x
(i)
) − y
(i)
)
2
3
For each of the following partial derivatives, state whether it is positive, negative, or equal to zero. Briefly explain. These questions can be answered from
the contour plot without knowing the formula for the function. (2 points for
each problem)
(Note: for two numbers a and b we will use the notation ∂
∂u f(a, b) to mean
”the partial derivative of f(u, v) with respect to u at the point where u = a and
v = b”. This notation is succinct but obfuscates the original variable names. A
more explicit way to write the same thing is ∂
∂u f(a, b)|u=a,v=b )
1. ∂
∂u f(−2, −2)
2. ∂
∂v f(−2, −2)
3. ∂
∂u f(3, −3)
4. ∂
∂v f(3, −3)
5. To the nearest integer, estimate the values of u and v that minimize f(u, v).
3
Problem 3: Matrix Manipulation I
Matrix multiplication practice (2 points each for the first 5 problems and 5
points for the last problem).
1. Write the result of the following matrix-matrix multiplication. Your answer should be written in terms of u, v, a and b.
3 −1
2 5
−2 2
·
u a
v b
2. Suppose A ∈ R
2×2
, B ∈ R
2×4
. Does the product AB exist? If so, what
size is it?
3. Suppose A ∈ R
3×5
, B ∈ R
4×1
. Does the product AB exist? If so, what
size is it?
4. Suppose A ∈ R
3×2
, y ∈ R
3
. Is y
T A a row vector or a column vector?
5. Suppose A ∈ R
3×2
, x ∈ R
2
. Is Ax a row vector or a column vector?
6. Suppose (Bx + y)
T AT = 0, where A and B are both invertible n × n
matrices, x and y are vectors in R
n, and 0 is a vector of all zeros. Use
the properties of multiplication, transpose, and inverse to show that x =
−B−1y. Show your work.
Problem 4: Matrix Manipulation II
(10 points) Create a jupyter notebook called exercise 1.ipynb and write code
to do the following.
1. Enter the following matrices and vectors
A =
−2 −3
1 0
, B =
1 1
1 0
, x =
−1
1
2. Compute C = A−1
3. Check that AC = I and CA = I
4. Compute Ax
5. Compute AT A
6. Compute Ax − Bx
7. Compute ||x|| (use the dot product)
4
8. Compute ||Ax − Bx||
9. Print the first column of A (do not use a loop – use array ”sliding” instead)
10. Assign the vector x to the first column of B (do not use a loop – use ”array
slicing” instead)
11. Compute the element-wise product between the first column of A and the
second column of A
Problem 5: Linear Regression
The problems consider linear regression based on the following hypothesis and
function. (10 points each)
hθ(x) = θ0 + θ1x, J(θ0, θ1) = 1
2
Xm
i=1
(hθ(x
(i)
) − y
(i)
)
2
Consider the following small data set:
x y
2 5
-1 -1
1 3
1. Solve for the values of θ0 and θ1 that minimize the cost function by substituting the value from the training set into the cost function, setting the
derivatives (with respect to both θ0 and θ1) equal to zero, and solving the
system of two equations for θ0 and θ1. Show your work.
2. Now do the same thing, but do not substitute the values of the training
set into the cost function. Instead. leave the x
(i) and y
(i) variables, take
the derivatives with respect to both θ0 and θ1, set them equal to zero, and
solve for θ0 and θ1. This will give you a general expression for θ0 and θ1
in terms of the training data.
Check your answer by plugging in the training data from the previous
problem into your expression for θ0 and θ1. You should get the same
values for θ0 and θ1 that you got in that problem.
3. In this problem you will implement gradient descent for linear regression.
Open the notebook gradient-descent.ipynb in Jupyter and follow the
instructions to complete the problem.
Problem 6: Polynomial Regression
(30 points) In this problem you will implement methods for multivariate linear
regression and use them to solve a polynomial regression problem. The purpose
of this problem is:
5
1. To practice writing “vectorized” versions of algorithms in Python
2. To understand how feature expansion can be used to fit non-linear hypotheses using linear methods
3. To understand feature normalization and its impact on numerical optimization for machine learning. Open the notebook multivariate-linearregression.ipynb and follow the instructions to complete the problem.
6
