Homework 4 (Week 5)

    1. Consider the email spam classification problem of Murphy Problem 8.1. Suppose you intend to use a linear perceptron classifier on that data (not logistic regression as directed in Problem 8.1). In the parts below, unless stated otherwise, assume the
dataset of
N = 4601 samples is split into
NTr = 3000  for training and
NTest = 1601
for testing.
Also, for the tolerance δ  in the VC generalization bound, use 0.1 (for a
certainty of 0.9).  The parts below have short answers.



Hint:  You may use the relation that if
H  is a linear perceptron classifier in D
dimensions (D features),
dVC (H ) = D + 1 . ( This will be proved in Problem 2.)

a)
What is the VC dimension of the hypothesis set?



b)
Expressing
the
upper
bound
on
the
out-of-sample
error
as

Eout (hg ) ≤ Ein (hg )+ εvc







For  Ein (hg )
measured on the training data, use  dvc
from part (a) to get a value

for εvc  .








    c) To get a lower εvc , suppose you reduce the number of features to D = 10 , and also increase the training set size to 10,000. Now what is εvc ?

    d) Suppose  that  you  had  control  over  the  number  of  training  samples  NTr  (by

collecting more email data). How many training samples would ensure a generalization error of εvc = 0.1 again with probability 0.9 (the same tolerance

        ◦ = 0.1), and using the reduced feature set (10 features)?

    e) Instead suppose you use the test set to measure Ein (hg ) , so let’s call it Etest (hg ) . What is the hypothesis set now? What is its cardinality?

    f) Continuing from part (e), use the bound:

Eout (hg ) ≤ Etest (hg )+ ε
Use the original feature set and the original test set, so that NTest = 1601 . Give an appropriate expression for ε and calculate it numerically.


    2. AML Exercise 2.4 (page 52). In addition to the hints given in the book, you can solve the problem by following the steps outlined below.




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For part (a):

i.
Write a point x
as a
d +1 dimensional vector;



i






    ii. Construct the (d + 1) × (d + 1) matrix suggested by the book;
    iii. Write h( X ) , the output of the perceptron, as function of X and the weights w (note that h( X ) is a d +1 dimensional vector with elements +1 and -1);
    iv. Using the nonsingularity of X , justify how any h( X ) can be obtained.
For part (b):

        i. Write a point xk  as a linear combination of the other d +1 points;

        ii. Write h( xk ) (output for the chosen point) and substitute the value of xk by the expression just found on the previous item (Hint: use the sgn{i} function);

        iii. What part of your expression in (ii) determines the class assignment of each point

                x i , for i ≠ k  ?
        iv. You have just proven (part (a)) that h( X ) with X (d +1)×(d +1) can be shattered. When we add a (d + 2)th line to X can it still be shattered? In other words, can you choose the value of h( xk ) ? Justify your answer. Hint: you can choose the class label of the other (d + 1) points.

    3. AML Problem 2.24 (page 75), except

            ▪ Replace part (a) with:

(a.1) For a single given dataset, give an expression for g(D) (x) . (AML notation)

(a.2)  Find g (x) analytically; express your answer in simplest form.

            ▪ For parts (b) and (c), obtain ΕD {Eout } by direct numerical computation, not by adding bias and var.

            ▪ For part (d), obtain bias(x), var(x), bias, var, and ΕD {Eout } , all by analytical (pencil and paper) techniques.

    4. AML Problem 2.13 (a), (b).








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    5. AML Problem 4.4 (a)-(c), plus additional parts (i)-(iii) below.

>>    For part (c), assume both g10 (x) and    f (x ) are given as functions of x, and you

can
express
your
answer
}
in
terms
of
them;
and
define




2

.





Eout (g10 ) = Εx, y { g10
(x) − y(x)







    (i) In Fig. 4.3(a), set σ! = 0.5, and traverse the horizontal line from N ≈ 60 to N ≈ 130. Explain why ℋ"# transitions from overfit to good fit (relative to ℋ!).

    (ii) Also in Fig. 4.3(a), set N = 100, and traverse the vertical line from σ! = 0 to σ! = 2. Explain why ℋ"# transitions from good fit to overfit (relative to ℋ!).

    (iii) In Fig. 4.3(b), set N ≈ 75, and traverse the vertical line from Q$ = 0 to Q$ = 100. Explain the behavior.












































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