A DECISION-THEORETIC
GENERALIZATION OF ON-LINE
LEARNING AND AN APPLICATION TO
BOOSTING
1
By Yoav Freund and Robert E. Schapire
Presented by David Leach
Original Slides by Glenn Rachlin
OUTLINE:


Background
On-line allocation of resources





Boosting








Introduction
The Problem
The Hedge Algorithm
Analysis
Introduction
The Problem
The AdaBoost Algorithm
Analysis
Applications
Extensions
Conclusions
Questions for Final exam
2
USEFUL DEFINITIONS:
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
ENSEMBLE LEARNING:
A machine learning paradigm where multiple
learners are used to solve the problem
Previously:
Ensemble:
Problem
Problem
Learner
Learner
… ...
… ...
Learner
Learner
• 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
BOOSTING: 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
HOW WAS ADABOOST 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
HOW WAS ADABOOST 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
BOOSTING TIMELINE
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
ON-LINE ALLOCATION OF RESOURCES:
INTRODUCTION
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
THE GAMBLER:
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
ON-LINE ALLOCATION OF RESOURCES:
PROBLEM FORMULATION
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: pt -how much money
he spends on each expert
loss: l –money lost (or not gained)
11
ON-LINE ALLOCATION OF RESOURCES:
HEDGE(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: ω1∈ [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
THE GAMBLER REVISITED
o
o
The gambler uses his fancy new algorithm as
follows:
1. The gambler splits his money evenly between 3
experts, giving $5 to each
o
o
p1 = <.33,.33,.33>
2. The gambler records the loss to each expert
o
o
o
o
o
Expert 1 loses $2
Expert 2 loses $1
Expert 3 loses $4
loss vector lt = <2,1,4>
total loss = .33x2 + .33x1 + .33x4 = 2.33
14
THE GAMBLER REVISITED

3. The gambler sets new weights using this data
and a beta of .5
Expert 1 is weighted .33 x .52 = .083
 Expert 2 is now weighted .33 x .51 = .167
 Expert 3 is now weighted .33 x .54 = .063
 Total weight = .083 + .167 + .063 = .313


4. The gambler repeats the process, now
“hedging” his bets as follows:

p2 = <.083/.313, .167/.313, .063/.313>
= <.265, .533, .202>
15
ON-LINE ALLOCATION OF RESOURCES:
BOUNDS OF HEDGE(Β)
16
BOUNDS, CON’T
β = given parameter
 T = total number of time steps or trials
 N = number of options

1

𝑐≥
ln 𝛽
1−𝛽
𝑜𝑟 𝑎 ≥
1
1−𝛽
17
CHOOSING BETA
Set
Where L~ is the bound on the best strategy, and
Then:
And if we know T:
18
ON-LINE ALLOCATION OF RESOURCES:
EVALUATION

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
MORE GAMBLING
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
BOOSTING: INTRODUCTION
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
TRADITIONAL BOOSTING
1.
2.
3.
4.
5.
Split a training data set into multiple
overlapping subsets.
Train a weak learner on one equally
weighted example set, until accuracy
is > 50%.
Train a weak learner on a new
example set, now weighted to focus on
errors.
Repeat until all example sets are
exhausted.
Apply all learners to test set to
determine final hypothesis.
22
THE PROBLEM
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
ADABOOST: ADAPTIVE BOOSTING
 Instead
 Can
of sampling, re-weight
be used to train weak classifiers
 Final
classification based on weighted
vote of weak classifiers
24
ADABOOST:
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
THE ADABOOST ALGORITHM:
26
INITIAL ANALYSIS:


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
BETA

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
STILL MORE GAMBLING






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
THEORETICAL PROPERTIES:

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
THEORETICAL PROPERTIES CON’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
TOY EXAMPLE – TAKEN FROM ANTONIO TORRALBA
@MIT
Each data point
has
a class label:
+1 ( )
yt =
-1 ( )
and a weight:
wt =1
Weak learners from
the family of lines
h => p(error) = 0.5 it is at chance
32
TOY EXAMPLE
Each data point
has
a class label:
+1 ( )
yt =
-1 ( )
and a weight:
wt =1
This one seems to be the best
33
This is a ‘weak classifier’: It performs slightly better than chance.
TOY EXAMPLE
Each data point
has
a class label:
+1 ( )
yt =
-1 ( )
We update the weights:
wt
wt exp{-yt Ht}
We set a new problem for which the previous weak
classifier performs better than chance again
34
TOY EXAMPLE
Each data point
has
a class label:
+1 ( )
yt =
-1 ( )
We update the weights:
wt
wt exp{-yt Ht}
35
We set a new problem for which the previous weak classifier
performs better than chance again
TOY EXAMPLE
Each data point
has
a class label:
+1 ( )
yt =
-1 ( )
We update the weights:
wt
wt exp{-yt Ht}
36
We set a new problem for which the previous weak classifier performs at chance again
TOY EXAMPLE
Each data point
has
a class label:
+1 ( )
yt =
-1 ( )
We update the weights:
wt
wt exp{-yt Ht}
37
We set a new problem for which the previous weak classifier
performs better than chance again
TOY EXAMPLE
f1
f2
f4
f3
The strong (non- linear) classifier is built as the
combination of all the weak (linear) classifiers.
38
FORMAL PROCEDURE OF ADABOOST
39
PROCEDURE OF ADABOOST:
40
ERROR ON TRAINING SET:
41
OVERFITTING

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
ACTUAL TYPICAL RUN
43
AN EXPLANATION BY MARGIN

This margin is not the margin in SVM
44
MARGIN DISTRIBUTION
Although final classifier is getting
larger, margins are still increasing
Final classifier is actually getting to
simpler classifer
45
PRACTICAL ADVANTAGES OF ADABOOST:
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
EXTENSIONS

AdaBoost.M1: First Multiclass

AdaBoost.M2: Second Multiclass

AdaBoost.R: Weak Regression
47
ADABOOST.M1

We modify error calculation as follows:

With the caveat:

And come up with a final hypothesis by:
48
CONCLUDING REMARKS
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
EXAM QUESTIONS:
1.
What are we seeking to minimize in resource
allocation?
2.
What is the goal of boosting?
3.
What makes Adaboost adaptable?
50
EXAM QUESTION I: WHAT 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
EXAM QUESTION II: WHAT 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
EXAM QUESTION III: WHAT MAKES
ADABOOST 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