Machine Learning  Written Problems Week 10 and 11

EECS 440: Machine Learning  Written Problems Week 10 and 11
General Instructions: Write or type your answers neatly and remember to show all relevant work. All
questions are worth 10 points. Each answer should be a separate pdf, and you can turn in the pdfs on
canvas in the appropriate assignment. Some questions may be very challenging; significant partial credit
is available for reasonable attempts at solutions. Since each question is worth the same number of points,
do not waste too much time on any one. Ask me or the TAs for help if stuck.
Some of the questions require you to write short programs to simulate things. You can use any
language/software to do this, and you do not need to turn in your code.
Upload your answers to Canvas as a pdf file by 11:59pm on the due date specified after the question. You
will receive a 10% bonus for a solution turned in a week or more in advance of the due date. You can use
one late day each week (up to Saturday 11:59pm) with a penalty of 20%. Submissions after Saturday
11:59pm for any week will not be graded.
Each group must do their own work. Only one submission is needed from each group. Do not use any
source other than the lecture notes, textbook(s) and readings on the class website to answer these
questions. Only those who contributed equally to a submission should have their names and Case IDs on
the submission. Those not listed as contributing will not receive points.
37. Consider a regression problem with examples described by 2 continuous attributes, x and y.
Each example is sampled according to the uniform distribution on (−1,1)2 and labeled with
f(x,y)=1−x2
−y
2
. A learner’s hypothesis class is h(x,y)=ax+by+c. Calculate its bias and
variance as a function of x and y if the learner sees an arbitrarily large training sample.
(11/13)
38. Using MATLAB/python/R etc, find the (x,y) with the largest bias and the (x,y) with the
largest variance for samples of size 10 in the previous setup. Can you intuitively justify your
findings? (11/13)
39. Show that the VC dimension of the concept class of m-dimensional hyperplanes is at least m.
A concept in this class is a hyperplane f(x)=w∙x+b. It classifies an example as positive if
f(x)≥0. (11/20)
40. Show that the concept class of 3CNF Boolean formulae is efficiently PAC learnable. A
3CNF formula is a conjunction where each conjunct is a disjunction of at most 3 literals.
(11/20)
All done! 