Boosting

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# Boosting - PowerPoint PPT Presentation

Boosting. LING 572 Fei Xia 02/01/06. Outline. Basic concepts Theoretical validity Case study: POS tagging Summary. Basic concepts. Overview of boosting. Introduced by Schapire and Freund in 1990s. “Boosting”: convert a weak learning algorithm into a strong one.

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Presentation Transcript

### Boosting

LING 572

Fei Xia

02/01/06

Outline
• Basic concepts
• Theoretical validity
• Case study:
• POS tagging
• Summary

### Basic concepts

Overview of boosting
• Introduced by Schapire and Freund in 1990s.
• “Boosting”: convert a weak learning algorithm into a strong one.
• Main idea: Combine many weak classifiers to produce a powerful committee.
• Algorithms:
• BrownBoost
Bagging

ML

Random sample

with replacement

f1

ML

f2

f

ML

fT

Random sample

with replacement

Boosting

Weighted Sample

ML

f1

Training Sample

ML

Weighted Sample

f2

f

ML

fT

Main ideas
• Train a set of weak hypotheses: h1, …., hT.
• The combined hypothesis H is a weighted majority vote of the T weak hypotheses.
• Each hypothesis ht has a weight αt.
• During the training, focus on the examples that are misclassified.

 At round t, example xi has the weight Dt(i).

Algorithm highlight
• Training time: (h1, 1), …., (ht, t), …
• Test time: for x,
• Call each classifier ht, and calculate ht(x)
• Calculate the sum: tt * ht(x)
Basic Setting
• Binary classification problem
• Training data:
• Dt(i): the weight of xi at round t. D1(i)=1/m.
• A learner L that finds a weak hypothesis ht: X  Y given the training set and Dt
• The error of a weak hypothesis ht:
• For t=1, …, T
• Train weak learner ht : X  {-1, 1}using training data and Dt
• Get the error rate:
• Choose classifier weight:
• Update the instance weights:
The new weights

When

When

Two iterations

Initial weights:

1st iteration:

2nd iteration:

The basic and general algorithms
• In the basic algorithm, it can be proven that
• The hypothesis weight αt is decided at round t
• Di (The weight distribution of training examples) is updated at every round t.
• Choice of weak learner:
• its error should be less than 0.5:
• Ex: DT (C4.5), decision stump
Experiment results(Freund and Schapire, 1996)

Error rate on a set of 27 benchmark problems

### Theoretical validity

Training error of H(x)

Final hypothesis:

Training error is defined to be

It can be proved that training error

Training error for basic algorithm

Let

Training error

 Training error drops exponentially fast.

Generalization error (expected test error)
• Generalization error, with high probability, is at most

T: the number of rounds of boosting

m: the size of the sample

d: VC-dimension of the base classifier space

Selecting weak hypotheses
• Training error
• Choose ht that minimize Zt.
• See “case study” for details.

### Multiclass boosting

Two ways
• Converting a multiclass problem to binary problem first:
• One-vs-all
• All-pairs
• ECOC
• Extending boosting directly
• AdaBoost.M2  Prob 2 in Hw5

### Case study

Overview(Abney, Schapire and Singer, 1999)
• Boosting applied to Tagging and PP attachment
• Issues:
• How to learn weak hypotheses?
• How to deal with multi-class problems?
• Local decision vs. globally best sequence
Weak hypotheses
• In this paper, a weak hypothesis h simply tests a predicate (a.k.a. feature), Φ:

h(x) = p1 if Φ(x) is true, h(x)=p0 o.w.

 h(x)=pΦ(x)

• Examples:
• POS tagging: Φ is “PreviousWord=the”
• PP attachment: Φ is “V=accused, N1=president, P=of”
• Choosing a list of hypotheses  choosing a list of features.
Finding weak hypotheses
• The training error of the combined hypothesis is at most

where

 choose ht that minimizes Zt.

• ht corresponds to a (Φt, p0, p1) tuple.
Finding weak hypotheses (cont)
• For each Φ, calculate Zt

Choose the one with min Zt.

Sequential model
• Sequential model: a Viterbi-style optimization to choose a globally best sequence of labels.

### Summary

Main ideas
• Boosting combines many weak classifiers to produce a powerful committee.
• Base learning algorithms that only need to be better than random.
• The instance weights are updated during training to put more emphasis on hard examples.
• Theoretical validity: it comes with a set of theoretical guarantee (e.g., training error, test error)
• It performs well on many tasks.
• It can identify outliners: i.e. examples that are either mislabeled or that are inherently ambiguous and hard to categorize.
• The actual performance of boosting depends on the data and the base learner.
• Boosting seems to be especially susceptible to noise.
• When the number of outliners is very large, the emphasis placed on the hard examples can hurt the performance.

Other properties
• Simplicity (conceptual)
• Efficiency at training
• Efficiency at testing time
• Handling multi-class
• Interpretability
Bagging vs. Boosting (Freund and Schapire 1996)
• Bagging always uses resampling rather than reweighting.
• Bagging does not modify the weight distribution over examples or mislabels, but instead always uses the uniform distribution
• In forming the final hypothesis, bagging gives equal weight to each of the weak hypotheses
Relation to other topics
• Game theory
• Linear programming
• Bregman distances
• Support-vector machines
• Brownian motion
• Logistic regression
• Maximum-entropy methods such as iterative scaling.

Sources of Bias and Variance
• Bias arises when the classifier cannot represent the true function – that is, the classifier underfits the data
• Variance arises when the classifier overfits the data
• There is often a tradeoff between bias and variance
Effect of Bagging
• If the bootstrap replicate approximation were correct, then bagging would reduce variance without changing bias.
• In practice, bagging can reduce both bias and variance
• For high-bias classifiers, it can reduce bias
• For high-variance classifiers, it can reduce variance
Effect of Boosting
• In the early iterations, boosting is primary a bias-reducing method
• In later iterations, it appears to be primarily a variance-reducing method
How to choose αt for ht with range [-1,1]?
• Training error
• Choose αt that minimize Zt.

Issues
• Given ht, how to choose αt?
• How to select ht?