BKI

212:
Artif
icial Intelligence
: from search to planning
E
xam part 2
,
April 19
th
, 2007, 8
.45

1
0
.30
Write your name,
student number and “BKI 212 exam part
2
06/07”
at the top of the answer
sheet.
There are 4 questions.
Answer the questions in a full, clear, but also relevant way. Omissions,
unclarities as well as irrelevant elaborations will lower your g
rade.
The maximum attainable score is 100 points.
Question 1
.
MDPs and Reinforcement Learning
(max.
5+5+5+5+5
= 25 points)
Consider the following world:
a
b
+1
c
d

1
In this world there are 4 possible
(deterministic)
actions: left (
), up (
), righ
t (
), and down (
)
resulting
in moving to the give
n
neighbour state when possible or staying in the same state w
hen the
action is not possible.
Choose
the discount factor
equal to 0.1
.
The
reward
of the states
a
,
b
,
c
, and
d
is
0;
the reward of
the stat
e marked by +1 is
+1
;
the reward of the state marked by

1
is

1
.
The states with reward +1
and

1 are terminal states.
a.
Compute the utilities for the states:
U
(
a
),
U
(
b
),
U
(
c
), and
U
(
d
).
b.
Compute the Q

function
for the state

action pairs
(
up
,
b
)
, (
left
,
b
)
, (
right
,
b
),
and
(
down
,
b
)
in this world.
c.
W
h
at is/
are the optimal polic(y)(ies) f
or a
n agent in
this wor
ld?
Answer the following questions in a general world with states
s
,
s
, actions
a
,
a
, utility
estimates
U
(
s
), and Q

function estimates
Q
(
a
,
s
).
d.
Ex
plain how TD learning
is applied when learning a Q

function.
e.
Why does an optimistic utility
estimate
cause exploration?
Question 2
.
Machine Learning
(max.
8+8+9 =
25 points)
Given are the following training points (
x
,
y
):
the points (0,0) and (2,0) as n
egative instances, and
the points (0,1), and (0,

1) as positive instances.
Construct a support vector machine which classifies these examples correctly. Take the values
–
1 and +1 (instead of 0 and 1) for the input and output values.
a.
Draw the input v
ectors in the
u

v
plane defined by
u
=
x
and
v
=
y
2

1
.
b.
Draw the “maximal margin separator” in the
u

v
plane.
c.
Also draw the corresponding decision line/curve in the original Euclidian plane defined by
x
and
y
.
BKI

212:
Artif
icial Intelligence
: from search to planning
E
xam part 2
,
April 19
th
, 2007, 8
.45

1
0
.30
Question 3
.
Ensemble Learning
(max.
7+18
= 25 points)
a.
Suppose you have a classifier that is an ensemble of decision trees
generated
using bagging.
Denote the type of this classifier with
T
.
Do you expect to gain in accuracy if you produce an ensemble of classifiers of type
T
(
an
ense
mble of ensembles
)
? Explain your answer.
b.
Consider a two

class classification task. Assume we have three classifiers A, B, and C, with
errors
e
(0 <
e
< 1
) each.
Consider
the ensemble of
classifier
s
consisting of
the
3 cl
assifiers
A, B, and C
,
and usin
g
majority voting for classification
.
Estimate the
error
of
this ensemble
when
e
= 1/3 and when
e
= 2/3
for all three classifiers A, B,
and C
:
i.
In the worst case;
ii.
In the best case;
iii.
In the case that the errors of the classifi
ers are completely independent
.
Question
4
.
PAC Learning
(max.
13+12
= 25 points)
a.
Consider the space of instances
X
corresponding to all points in the
x

y
plane. Give the VC

dimension of the following hypothesis spaces:
i.
H
r
= the set of all rectangles in the
x

y
plane. Points in
side the rectangle are classified as
positive examples.
ii.
H
c
= the set of all
triangles
in the
x

y
plane. Points inside the
triangle
are classified as positive
examples.
Motivate your answers.
b.
For PAC Learning
with a finite hypothesis space
H
, which cont
ains the concept to be learned,
Haussler
derived
the following formula for the number of training examples needed:
i.
Explain how to apply this formula (give the meaning of
m
,
ε
,

H

,
δ
, and the conditions for
both training instan
ces and test instances
).
ii.
This formula follows from the inequality:
Explain this
inequality
and also explain why this
inequality
results in an overestimation of the
number of training examples needed.
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