EECS 440: Machine Learning Written Problems Week 4

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.

15. Show that the entropy of a Bernoulli random variable is a concave function (i.e. the

negative entropy is a convex function).

16. Show that the set C={x|Ax ≥ b} , A in Rmxn

, x in Rn

, b in Rm, is a convex set. Note that

this describes the constraint set of a linear program.

17. A function f is said to have a global minimum at x if for all y, f(y) ≥ f(x). It is said to have

a local minimum at x if there exists a neighborhood H around x so that for all y in H, f(y)≥

f(x). Show that, if f is convex, every local minimum is a global minimum. [Hint: Prove by

contradiction using Jensen’s inequality.]

18. Consider the LP: min c

T

x s.t. Ax ≥ b, x ≥ 0, where A is the 4×2 matrix: [ 0 −1; −1 −1; −1

2; 1 −1], b is a 4×1 vector [−5; −9;0; −3] and c is a 2×1 vector [−1; −2]. (a) Draw the

feasible region in R2

. (b) Draw the contours of c

T

x =−12, c

T

x =−14 and c

T

x =−16 and

determine the solution graphically. Does the minimum go through a vertex? Can you

intuitively explain why?

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