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Assignment
Answer all parts of the four questions below. The number of marks available for each question (part) is indicated in the Criteria for Assessment section below. There are 100 marks
available in total for this assignment.
Question 1.
a) Consider the class of hypotheses consisting of all Boolean functions that can be represented as a list of if-then rules of the form:
If l1 then b1, else if l2 then b2, . . . , else bk
where li (for all 1 ≤ i ≤ k) is a literal, that is, a variable xj or its negation xj and
bi ∈ {1, 0} is the assigned label. For example, “If x2 then 1, else if x1 then 0, else 1”
is a decision list.
Assuming that only n variables can be used. Is the class of decision lists learnable in the PAC model? Justify your answer with a formal argument (in maximum
2 pages .
b) For the class H of axis-aligned n-dimensional rectangles in R
n
, that is
H = {[a1, b1] × · · · × [an, bn] : ai
, bi ∈ R}
(1) Give a PAC-learning algorithm.
(2) Prove that your algorithm satisfies the conditions for PAC learnability.
(3) How does the sample complexity vary as a function of n?
(4) Your answer should not exceed 2 pages.