A D
ECISION

T
HEORETIC
G
ENERALIZATION
OF
O
N

L
INE
L
EARNING
AND
AN
A
PPLICATION
TO
B
OOSTING
By
Yoav
Freund
and Robert E.
Schapire
Presented by David Leach
Original
Slides by Glenn
Rachlin
1
O
UTLINE
:
Background
On

line allocation of resources
Introduction
The Problem
The Hedge Algorithm
Analysis
Boosting
Introduction
The Problem
The
AdaBoost
Algorithm
Analysis
Applications
Extensions
Conclusions
Questions for Final exam
2
U
SEFUL
D
EFINITIONS
:
On

Line Learning
–
Information comes in one
step at a time, learner must apply model, make
prediction, observe true value, then adjust model
accordingly.
Weak Learner
–
Algorithm has higher accuracy
than random guessing, but is impractical by itself
for most real

world applications.
PAC
–
Probably Approximately Correct; most of
the time the prediction returned will be close to
the actual result.
3
E
NSEMBLE
L
EARNING
:
A machine learning paradigm where multiple
learners are used to solve the problem
Problem
… ...
… ...
Problem
Learner
Learner
Learner
Learner
Previously:
Ensemble:
•
The
generalization ability of the ensemble is usually
significantly better than that of an individual learner
•
Boosting
is one of the most important families of ensemble
methods
4
B
OOSTING
:
A
BACKGROUND
Significant advantages:
Solid theoretical foundation
High level of accuracy
Simple to implement
Wide range of applications
R.
Schapire
and Y. Freund won the
2003
Godel
Prize
(one of the most prestigious awards in
theoretical computer science)
Prize winning paper (which introduced
AdaBoost
):
"A decision theoretic generalization of on

line
learning and an application to Boosting,“ Journal of
Computer and System Sciences, 1997, 55: 119

139.
5
H
OW
WAS
A
DABOOST
BORN
?
In 1988, M. Kearns and L.G. Valiant
posed an interesting question:
Can a “weak” learning algorithm that performs just
slightly better than random guess can be “boosted”
into an arbitrarily accurate “strong” learning
algorithm?
More simply, can we transform one or more weak
learners into a single strong learner?
6
H
OW
WAS
A
DABOOST
BORN
?
In R.
Schapire’s
MLJ90 paper, Rob said “yes” and
gave a proof to the question. The proof is a
construction, which is the first Boosting algorithm
(“Recursive Majority Gate Formulation”)
Then, in Y. Freund’s
Phd
thesis (1993),
Yoav
gave a
scheme of combining weak learners by “Majority Vote”
Though theoretically strong, both algorithms relied on
knowledge of each weak learner’s accuracy
Later, at AT&T Bell Labs, they published the 1997
paper
(in fact the work was done in 1995)
, which proposed
the
AdaBoost
algorithm, a practical, “adaptable”
algorithm
7
B
OOSTING
T
IMELINE
1990
–
Boost

by

majority algorithm (Freund)
1995
–
AdaBoost
(Freund &
Schapire
)
1997
–
Generalized version of
AdaBoost
(
Schapire
& Singer)
2001
–
AdaBoost
in Face Detection (Viola &
Jones)
8
O
N

LINE
A
LLOCATION
OF
R
ESOURCES
:
I
NTRODUCTION
Problem: “... dynamically apportioning resources
among a set of options...”
In other words, “Given a set of individual
predictions, how much should we value each
one?”
The Gambler Example (A recurring theme)
9
T
HE
G
AMBLER
:
A Gambler wants to make money on horse

racing
by consulting a group of experts.
He discovers that experts tend to use certain “rules of
thumb” for races that dictate results to some degree
(“Horse with the best odds”, etc.).
Hard to find one particular rule that works for
multiple circumstances.
How can he use the network of various predictions
(each of which tends to use a given rule of thumb) to
win money?
More specifically, how should he split his money
among the experts?
10
O
N

LINE
A
LLOCATION
OF
R
ESOURCES
:
P
ROBLEM
F
ORMULATION
The on

line allocation model:
Allocation agent:
A

the gambler
A certain strategy:
i
–
one expert’s behavior
# of options/strategies:
{1,2,3, ... ,N}

the # of
experts to choose from
# of time steps:
{1,2,3, ... ,T}

the # of races
distribution over strategies:
p
t

how much money
he spends on each expert
l
oss:
l
–
money lost (or not gained)
11
O
N

LINE
A
LLOCATION
OF
R
ESOURCES
:
H
EDGE
(
Β
)
Basis: “The algorithm and its analysis are direct
generalizations of
Littlestone
and
Warmuth’s
weighted majority algorithm”
Assumptions:
The loss suffered by any strategy be bounded
All weights be nonnegative
Initial weights sum up to 1 (optional)
12
Algorithm Hedge (β)
Parameters: β
∈
[0,1]
initial weight vector: ω
1
∈
[0,1]
N
with
number of trials T
Do for t = 1,2, ..., T
1.
Choose allocation from environment
2.
Receive loss vector
3.
Suffer loss
4.
Set new weights vector to be
Goal:
minimize difference between expected total loss and
minimal total loss of repeating one action
13
T
HE
G
AMBLER
R
EVISITED
o
The gambler uses his fancy new algorithm as
follows:
o
1. The gambler splits his money evenly between 3
experts, giving $5 to each
o
p
1
= <.33,.33,.33>
o
2. The gambler records the loss to each expert
o
Expert 1 loses $2
o
Expert 2 loses $1
o
Expert 3 loses $4
o
loss vector
l
t
= <2,1,4>
o
total loss = .33x2 + .33x1 + .33x4 = 2.33
14
T
HE
G
AMBLER
R
EVISITED
3. The gambler sets new weights using this data
and a beta of .5
Expert 1 is weighted .33
x
.5
2
= .083
Expert 2 is now weighted .33 x .5
1
= .167
Expert 3 is now weighted .33 x .5
4
= .063
Total weight = .083 + .167 + .063 = .313
4. The gambler repeats the process, now
“hedging” his bets as follows:
p
2
= <.083/.313, .167/.313, .063/.313>
= <.265, .533, .202>
15
O
N

LINE
A
LLOCATION
OF
R
ESOURCES
:
B
OUNDS
OF
H
EDGE
(
Β
)
16
B
OUNDS
,
C
ON
’
T
β
= given parameter
T = total number of time steps or trials
N = number of options
𝑐
≥
ln
1
𝛽
1
−
𝛽
𝑜𝑟
𝑎
≥
1
1
−
𝛽
17
C
HOOSING
BETA
Set
Where L~ is the bound on the best strategy, and
Then:
And if we know T:
18
O
N

LINE
A
LLOCATION
OF
R
ESOURCES
:
E
VALUATION
The authors show that the
Hedge(β
) algorithm
“
yield[s
] bounds that are slightly weaker in some
cases, [than those produced by the algorithm
proposed by
Littlestone
and
Warmuth
, 1994] but
applicable to a considerably more general class of
learning problems.”
Not only binary decisions
Not only discrete loss
19
M
ORE
G
AMBLING
The gambler now wants to avoid the experts, and
opts to write a program that predicts the winner.
He must take input data
Odds
Previous results
Track conditions
And predict the outcome
Win or loss
He notices that “rules of thumb” once again
emerge, where simple heuristics can provide
some predictive accuracy, but not enough
How can he use this information to make money?
20
B
OOSTING
: I
NTRODUCTION
Aim: “.. converting a weak learning algorithm
that performs just slightly better than random
guessing into one with arbitrarily high accuracy.”
Example: Constructing an expert computer
program
Two problems:
Choosing data
Combining rules
“Boosting refers to this general problem of
producing a very accurate prediction rule by
combining rough and moderately inaccurate
rules

of

thumb.”
21
T
RADITIONAL
B
OOSTING
1.
Split a training data set into multiple
overlapping subsets.
2.
Train a weak learner on one equally
weighted example set, until accuracy
is > 50%.
3.
Train a weak learner on a new
example set, now weighted to focus on
errors.
4.
Repeat until all example sets are
exhausted.
5.
Apply all learners to test set to
determine final hypothesis.
22
T
HE
P
ROBLEM
Previous algorithms by the same authors “work
by calling a given weak learning algorithm
WeakLearn
multiple times, each time presenting
it with a different distribution [of examples], and
finally combining all the generated hypotheses
into a single hypothesis.”
Problems
Too much has to be known in advance
Improvement of the overall performance depends on
the weakest rules
23
A
DABOOST
: A
DAPTIVE
B
OOSTING
Instead of sampling, re

weight
Can be used to train weak classifiers
Final classification based on weighted
vote of weak classifiers
24
A
DA
B
OOST
:
If the underlying classifiers are linear networks, then
AdaBoost
builds multilayer
perceptrons
one node at a
time.
However, the underlying classifier can be anything,
decision trees, neural networks, etc…
25
T
HE
A
DA
B
OOST
A
LGORITHM
:
26
I
NITIAL
A
NALYSIS
:
The weight of each example is adjusted so that
the multiplier will be beta if correct (<1), or 1 if
incorrect (beta^0). Remember that the weight
will be normalized, so no decrease is effectively
an increase.
Each learner gets a vote inversely proportional to
the logarithm of its beta, which in turn was
proportional to its error.
27
B
ETA
What implications does this have?
1. If the error is .5, or equivalent to random guessing,
no information is gained and the time step isn’t used.
2. For a common error <.5, we can weight examples
proportionally to error, and weight votes inversely
proportional to error.
3. In the final hypothesis generation, if error was >
.5, the time step will actually have an inverse vote.
28
S
TILL
M
ORE
G
AMBLING
The gambler now has a pretty good scheme to make
money, and downloads the entire race history from
the track’s database
1. He finds that odds are a PAC predictor, and comes
up with hypotheses accordingly.
2. He calculates the error of using this predictor.
3. He looks at the data in a different way, focusing on
examples that odds could not easily predict, and
comes up with a new heuristic (when the track is
muddy, the horse with the most experienced jockey
wins)
4. He repeats this process until no more viable
heuristics can be determined.
5. When enough of the heuristics indicate a win for a
given horse, he places a bet.
29
T
HEORETICAL
P
ROPERTIES
:
Y. Freund and R.
Schapire
[JCSS97]
have proved that
the training error of
AdaBoost
is bounded by:
where
Thus, if each base classifier is slightly better than
random so that for some ,
then
the
training error drops exponentially fast
in
T
since
the above bound is at
most
30
T
HEORETICAL
P
ROPERTIES
C
ON
’
T
Y. Freund and R.
Schapire
[JCSS97]
have tried to bound
the generalization error as:
where denotes empirical probability
on training sample,
s
is the sample size,
d is the VC

dim of base learner
The above bounds suggest that Boosting will
overfit
if
T
is large.
However, empirical studies show that
Boosting
often does
not
overfit
R.
Schapire
et al.
[AnnStat98]
gave a margin

based bound:
for any
> 0 with high probability
where
31
T
OY
E
XAMPLE
–
TAKEN
FROM
A
NTONIO
T
ORRALBA
@MIT
Weak learners from
the family of lines
h => p(error) = 0.5 it is at chance
Each data point
has
a class label:
w
t
=1
and a weight:
+1 ( )

1 ( )
y
t
=
32
T
OY
EXAMPLE
This one seems to be the best
Each data point
has
a class label:
w
t
=1
and a weight:
+1 ( )

1 ( )
y
t
=
This is a ‘
weak classifier
’: It performs slightly better than chance.
33
T
OY
EXAMPLE
We set a new problem for which the previous weak
classifier
performs
better than chance again
Each data point
has
a class label:
w
t
w
t
exp
{

y
t
H
t
}
We update the weights:
+1 ( )

1 ( )
y
t
=
34
T
OY
EXAMPLE
We set a new problem for which the previous weak classifier
performs
better than
chance again
Each data point
has
a class label:
w
t
w
t
exp
{

y
t
H
t
}
We update the weights:
+1 ( )

1 ( )
y
t
=
35
T
OY
EXAMPLE
We set a new problem for which the previous weak classifier performs at chance again
Each data point
has
a class label:
w
t
w
t
exp{

y
t
H
t
}
We update the weights:
+1 ( )

1 ( )
y
t
=
36
T
OY
EXAMPLE
We set a new problem for which the previous weak classifier
performs
better than
chance again
Each data point
has
a class label:
w
t
w
t
exp{

y
t
H
t
}
We update the weights:
+1 ( )

1 ( )
y
t
=
37
T
OY
EXAMPLE
The strong (non

linear) classifier is built as the
combination of all the weak (linear) classifiers.
f
1
f
2
f
3
f
4
38
F
ORMAL
P
ROCEDURE
OF
A
DA
B
OOST
39
P
ROCEDURE
OF
A
DABOOST
:
40
E
RROR
ON
T
RAINING
S
ET
:
41
O
VERFITTING
Will
Adaboost
screw up with a fat complex
classifier finally?
Occam’s razor
–
simple is the best
Over fitting
Shall we stop before over fitting? If only over fitting happens.
42
A
CTUAL
T
YPICAL
R
UN
43
A
N
EXPLANATION
BY
MARGIN
This margin is not the margin in SVM
44
M
ARGIN
D
ISTRIBUTION
Although final classifier is getting
larger, margins are still increasing
Final classifier is actually getting to
simpler classifer
45
P
RACTICAL
A
DVANTAGES
OF
A
DA
B
OOST
:
Simple and easy to program.
No parameters to tune (except T).
Effective, provided it can consistently find rough
rules of thumb.
Goal is to find hypothesis barely better than
guessing.
Can combine with any (or many) classifiers to
find weak hypotheses: neural networks, decision
trees, simple rules of thumb, nearest

neighbor
classifiers, etc.
46
E
XTENSIONS
AdaBoost.M1: First Multiclass
AdaBoost.M2: Second Multiclass
AdaBoost.R
: Weak Regression
47
A
DA
B
OOST
.M1
We modify error calculation as follows:
With the caveat:
And come up with a final hypothesis by:
48
C
ONCLUDING
R
EMARKS
This paper was the introduction of
AdaBoost
, an
award

winning, widely used algorithm featured
in the top 10 algorithms.
The paper also included the Hedge algorithm, a
much less widely known algorithm.
It should be noted that Hedge was a solution to a
problem that prevented adaptable boosting for a
long time, and has therefore had a significant
impact on data mining since.
49
E
XAM
Q
UESTIONS
:
1.
What are we seeking
to minimize in resource
allocation?
2.
What is the goal of boosting?
3.
What makes
Adaboost
adaptable?
50
E
XAM
Q
UESTION
I: W
HAT
ARE
WE
SEEKING
TO
MINIMIZE
IN
RESOURCE
ALLOCATION
?
More simply,
we seek to minimize the total loss of the
allocator with respect to the loss of the best learner.
This gives us a consistent “worst case scenario”,
effectively hedging our bets.
51
E
XAM
Q
UESTION
II: W
HAT
IS
THE
GOAL
OF
BOOSTING
?
The goal is to use one or more weak learners as an
arbitrarily accurate strong learner
In other words, to use better

than

chance heuristics in
ensemble for high predictive accuracy
52
E
XAM
Q
UESTION
III: W
HAT
MAKES
A
DA
B
OOST
ADAPTABLE
?
The classifiers used in the final decision function have
all been modified to account for the weaknesses in the
preceding classifiers.
As
long as we at least one initial learner that tells
us
something
about the data, the algorithm will infer
everything else it needs to.
53
Σχόλια 0
Συνδεθείτε για να κοινοποιήσετε σχόλιο