CSC 411 Lecture 5: Ensembles IIrgrosse/courses/csc411_f18/... · CSC 411 Lecture 5: Ensembles II...

CSC 411 Lecture 5: Ensembles II

Roger Grosse, Amir-massoud Farahmand, and Juan Carrasquilla

University of Toronto

UofT CSC 411: 05-Ensembles II 1 / 22

Boosting

Recall that an ensemble is a set of predictors whose individual decisions arecombined in some way to classify new examples.

(Previous lecture) Bagging: Train classifiers independently on randomsubsets of the training data.

(This lecture) Boosting: Train classifiers sequentially, each time focusing ontraining data points that were previously misclassified.

Let us start with the concept of weak learner/classifier (or base classifiers).


Weak Learner/Classifier

(Informal) Weak learner is a learning algorithm that outputs a hypothesis(e.g., a classifier) that performs slightly better than chance, e.g., it predictsthe correct label with probability 0.6.

We are interested in weak learners that are computationally efficient.

I Decision treesI Even simpler: Decision Stump: A decision tree with only a single split

[Formal definition of weak learnability has quantifies such as “for any distribution over data” and the requirement that itsguarantee holds only probabilistically.]


Weak Classifiers

These weak classifiers, which are decision stumps, consist of the set of horizontaland vertical half spaces.

Vertical half spaces Horizontal half spaces


Weak Classifiers

Vertical half spaces Horizontal half spaces

A single weak classifier is not capable of making the training error verysmall. It only perform slightly better than chance, i.e., the error of classifierh according to the given weights w = (w1, . . . ,wN) (with

∑Ni=1 wi = 1 and

wi ≥ 0)

err =N∑

i=1

wi I{h(xi ) 6= yi}

is at most 12 − γ for some γ > 0.

Can we combine a set of weak classifiers in order to make a better ensembleof classifiers?

Boosting: Train classifiers sequentially, each time focusing on training datapoints that were previously misclassified.


AdaBoost (Adaptive Boosting)

Key steps of AdaBoost:

1. At each iteration we re-weight the training samples by assigning largerweights to samples (i.e., data points) that were classified incorrectly.

2. We train a new weak classifier based on the re-weighted samples.3. We add this weak classifier to the ensemble of classifiers. This is our

new classifier.4. We repeat the process many times.

The weak learner needs to minimize weighted error.

AdaBoost reduces bias by making each classifier focus on previous mistakes.


AdaBoost Example

Training data

[Slide credit: Verma & Thrun]


AdaBoost Example

Round 1

w =

(1

10, . . . ,

1

10

)⇒ Train a classifier (using w)⇒ err1 =

∑10i=1 wi I{h1(x(i)) 6= t(i)}∑N

i=1 wi

=3

10

⇒α1 =1

2log

1− err1

err1=

1

2log(

1

0.3− 1) ≈ 0.42⇒ H(x) = sign (α1h1(x))



AdaBoost Example

Round 2

w = updated weights⇒ Train a classifier (using w)⇒ err2 =

∑10i=1 wi I{h1(x(i)) 6= t(i)}∑N

i=1 wi

= 0.21

⇒α2 =1

2log

1− err3

err3=

1

2log(

1

0.21− 1) ≈ 0.66⇒ H(x) = sign (α1h1(x) + α2h2(x))

[Slide credit: Verma & Thrun]UofT CSC 411: 05-Ensembles II 9 / 22

AdaBoost Example

Round 3

w = updated weights⇒ Train a classifier (using w)⇒ err3 =

∑10i=1 wi I{h1(x(i)) 6= t(i)}∑N

i=1 wi

= 0.14

⇒α3 =1

2log

1− err2

err2=

1

2log(

1

0.14− 1) ≈ 0.91⇒ H(x) = sign (α1h1(x) + α2h2(x) + α3h3(x))



AdaBoost Example

Final classifier



AdaBoost Algorithm

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Re-weighted Samples

Re-weighted Samples

Re-weighted Samples

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H(x) = sign

TX

t=1

↵tht(x)

!

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errt =

PNi=1 wiI{ht(x

(i) 6= t(i)}PN

i=1 wi<latexit sha1_base64="5U5SI+PZ7uJNFYG/ReSMtHFZ5Po=">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</latexit><latexit sha1_base64="5U5SI+PZ7uJNFYG/ReSMtHFZ5Po=">AAACqXicdVFNaxsxENVuvxL3y2mPvYiaFpeC2Q2F5hIIzaWnNIE4Mc06i1aetYX1sZFm2xixP7LH/pJeI9t7SJzmCcHTm3lieFNUUjhMkr9R/Ojxk6fPtrY7z1+8fPW6u/PmzJnachhyI40dFcyBFBqGKFDCqLLAVCHhvJgfLuvnv8A6YfQpLioYKzbVohScYZDy7jxDuEYP1jY50n2alZZxTzNXq9yL/bS5PKK/c0GzHxr8LLT0s6L0182l74tPDc00XFFsH+F0tteg//miybu9ZJCsQO+TtCU90uI434k+ZhPDawUauWTOXaRJhWPPLAouoelktYOK8TmbwkWgmilwY79KpaEfgjKhpbHhaqQr9bbDM+XcQhWhUzGcuc3aUnyohjPV3NXk1FgRZMEfKGxMi+Xe2Atd1Qiar4cta0nR0OWa6ERY4CgXgTAe/IJTPmNhNxiW2clWRn9olGJ64pbJpps53idnu4M0GaQnX3oH39qMt8g78p70SUq+kgPynRyTIeHkD/kXkSiKP8cn8Sj+uW6No9bzltxBzG8AkrLQdg==</latexit><latexit sha1_base64="5U5SI+PZ7uJNFYG/ReSMtHFZ5Po=">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</latexit><latexit sha1_base64="5U5SI+PZ7uJNFYG/ReSMtHFZ5Po=">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</latexit>

↵t =1

2log

✓1 � errt

errt

◆

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wi wi exp⇣2↵tI{ht(x

(i)) 6= t(i)}⌘

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AdaBoost Algorithm

Input: Data DN = {x(i), t(i)}Ni=1, weak classifier WeakLearn (a classificationprocedure that return a classifier from base hypothesis space H withh : x→ {−1,+1} for h ∈ H), number of iterations T

Output: Classifier H(x)

Initialize sample weights: wi = 1N

for i = 1, . . . ,N

For t = 1, . . . ,T

I Fit a classifier to data using weighted samples (ht ←WeakLearn(DN ,w)),e.g.,

ht ← argminh∈H

N∑i=1

wi I{h(x(i)) 6= t(i)}

I Compute weighted error errt =∑N

i=1 wi I{ht (x(i)) 6=t(i)}∑Ni=1 wi

I Compute classifier coefficient αt = 12

log 1−errterrt

I Update data weights

wi ← wi exp(−αtt

(i)ht(x(i))) [≡ wi exp

(2αtI{ht(x(i)) 6= t(i)}

)]Return H(x) = sign

(∑Tt=1 αtht(x)

)UofT CSC 411: 05-Ensembles II 13 / 22

AdaBoost Example

Each figure shows the number m of base learners trained so far, the decisionof the most recent learner (dashed black), and the boundary of the ensemble(green)


AdaBoost Minimizes the Training Error

TheoremAssume that at each iteration of AdaBoost the WeakLearn returns a hypothesiswith error errt ≤ 1

2 − γ for all t = 1, . . . ,T with γ > 0. The training error of the

output hypothesis H(x) = sign(∑T

t=1 αtht(x))

is at most

LN(H) =1

N

N∑

i=1

I{H(x(i)) 6= t(i))} ≤ exp(−2γ2T

).

This is under the simplifying assumption that each weak learner is γ-betterthan a random predictor.

Analyzing the convergence of AdaBoost is generally difficult.


Generalization Error of AdaBoost

AdaBoost’s training error (loss) converges to zero. What about the testerror of H?

As we add more weak classifiers, the overall classifier H becomes more“complex”.

We expect more complex classifiers overfit.

If one runs AdaBoost long enough, it can in fact overfit.

Overfitting Can HappenOverfitting Can HappenOverfitting Can HappenOverfitting Can HappenOverfitting Can Happen

0

5

10

15

20

25

30

1 10 100 1000

test

train

erro

r

# rounds

(boosting “stumps” onheart-disease dataset)

• but often doesn’t...


Generalization Error of AdaBoost

But often it does not!

Sometimes the test error decreases even after the training error is zero!Actual Typical RunActual Typical RunActual Typical RunActual Typical RunActual Typical Run

10 100 10000

5

10

15

20er

ror

test

train

)T# of rounds (

(boosting C4.5 on“letter” dataset)

• test error does not increase, even after 1000 rounds• (total size > 2,000,000 nodes)

• test error continues to drop even after training error is zero!

# rounds5 100 1000

train error 0.0 0.0 0.0

test error 8.4 3.3 3.1

• Occam’s razor wrongly predicts “simpler” rule is better

How does that happen?

We will provide an alternative viewpoint on AdaBoost later in the course.

[Slide credit: Robert Shapire’s Slides, http://www.cs.princeton.edu/courses/archive/spring12/cos598A/schedule.html ]


http://www.cs.princeton.edu/courses/archive/spring12/cos598A/schedule.html

AdaBoost for Face Recognition

Viola and Jones created a very fast face detector that can be scanned acrossa large image to find the faces.

The base classifier/weak learner just compares the total intensity in tworectangular pieces of the image.

I There is a neat trick for computing the total intensity in a rectangle ina few operations.

I So it is easy to evaluate a huge number of base classifiers and they arevery fast at runtime.

I The algorithm adds classifiers greedily based on their quality on theweighted training cases.


AdaBoost for Face Detection

Famous application of boosting: detecting faces in images

A few twists on standard algorithm

I Pre-define weak classifiers, so optimization=selectionI Change loss function for weak learners: false positives less costly than

missesI Smart way to do inference in real-time (in 2001 hardware)


AdaBoost Face Detection Results


Summary

Boosting reduces bias by generating an ensemble of weak classifiers.

Each classifier is trained to reduce errors of previous ensemble.

It is quite resilient to overfitting, though it can overfit.

We will later provide a loss minimization viewpoint to AdaBoost. It allowsus to derive other boosting algorithms for regression, ranking, etc.


Ensembles Recap

Ensembles combine classifiers to improve performance

Boosting

I Reduces biasI Increases variance (large ensemble can cause overfitting)I SequentialI High dependency between ensemble elements

Bagging

I Reduces variance (large ensemble can’t cause overfitting)I Bias is not changed (much)I ParallelI Want to minimize correlation between ensemble elements.

Next Lecture: Linear Regression


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