CS221 Midterm Solutions
CS221
November 18, 2014 Name:
| {z }
by writing my name I agree to abide by the honor code
SUNet ID:
Read all of the following information before starting the exam:
• This test has 3 proble
...
CS221 Midterm Solutions
CS221
November 18, 2014 Name:
| {z }
by writing my name I agree to abide by the honor code
SUNet ID:
Read all of the following information before starting the exam:
• This test has 3 problems and is worth 150 points total. It is your responsibility to
make sure that you have all of the pages.
• Keep your answers precise and concise. Show all work, clearly and in order, or else
points will be deducted, even if your final answer is correct.
• Don’t spend too much time on one problem. Read through all the problems carefully
and do the easy ones first. Try to understand the problems intuitively; it really helps
to draw a picture.
• Good luck!
Problem Part Max Score Score
1
a 10
b 10
c 10
d 10
e 10
2
a 10
b 10
c 10
d 10
e 10
3
a 10
b 10
c 10
d 10
e 10
Total Score: + + =
1
1. Enchaining Realm (50 points)
This problem is about machine learning.
a. (10 points)
Suppose we want to predict a real-valued output y ∈ R given an input x = (x1, x2) ∈ R
2
,
which is represented by a feature vector φ(x) = (x1, |x1 − x2|).
Consider the following training set of (x, y) pairs:
Dtrain = {((1, 2), 2),((1, 1), 1),((2, 1), 3)}. (1)
We use a modified squared loss function, which penalizes overshooting twice as much as
undershooting:
Loss(x, y, w) = (
1
2
(w · φ(x) − y)
2
if w · φ(x) < y
(w · φ(x) − y)
2 otherwise
(2)
Using a fixed learning rate of η = 1, apply the stochastic gradient descent algorithm on
this training set starting from w = [0, 0] after looping through each example (x, y) in order
and performing the following update:
w ← w − η ∇wLoss(x, y, w). (3)
For each example in the training set, calculate the loss on that example and update the
weight vector w to fill in the table below:
x φ(x) Loss(x, y, w) ∇wLoss(x, y, w) weights w
Initialization n/a n/a n/a n/a [0, 0]
After example 1 (1,2)
After example 2 (1,1)
After example 3 (2,1)
Solution
x φ(x) Loss(x, y, w) ∇wLoss(x, y, w) weights w
Initialization n/a n/a n/a n/a [0, 0]
After example 1 (1,2) (1,1) 2 [−2, −2] [2, 2]
After example 2 (1,1) (1,0) 1 [2, 0] [0, 2]
After example 3 (2,1) (2,1) 0.5 [−2, −1] [2, 3]
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