winnow vs perceptron

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SIMS 290-2: Applied Natural Language Processing

Barbara RosarioOctober 4, 2004

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Today

Algorithms for Classification Binary classification

PerceptronWinnowSupport Vector Machines (SVM)Kernel Methods

Multi-Class classificationDecision TreesNaïve BayesK nearest neighbor

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Binary Classification: examples

Spam filtering (spam, not spam)Customer service message classification (urgent vs. not urgent)Information retrieval (relevant, not relevant)Sentiment classification (positive, negative)Sometime it can be convenient to treat a multi-way problem like a binary one: one class versus all the others, for all classes

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Binary Classification

Given: some data items that belong to a positive (+1 ) or a negative (-1 ) classTask: Train the classifier and predict the class for a new data itemGeometrically: find a separator

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Linear versus Non Linear algorithms

Linearly separable data: if all the data points can be correctly classified by a linear (hyperplanar) decision boundary

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Linearly separable data

Class1Class2Linear Decision boundary

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Non linearly separable data

Class1Class2

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Non linearly separable data

Non Linear Classifier Class1Class2

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Linear or Non linear separable data?We can find out only empirically

Linear algorithms (algorithms that find a linear decision boundary)

When we think the data is linearly separableAdvantages

– Simpler, less parameters

Disadvantages– High dimensional data (like for NLT) is usually not linearly

separable

Examples: Perceptron, Winnow, SVMNote: we can use linear algorithms also for non linear problems (see Kernel methods)

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Non LinearWhen the data is non linearly separableAdvantages

– More accurate

Disadvantages– More complicated, more parameters

Example: Kernel methods

Note: the distinction between linear and non linear applies also for multi-class classification (we’ll see this later)

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Simple linear algorithms

Perceptron and Winnow algorithmLinearBinary classificationOnline (process data sequentially, one data point at the time)Mistake drivenSimple single layer Neural Networks

12From Gert Lanckriet, Statistical Learning Theory Tutorial

Linear binary classification

Data: {(xi,yi)}i=1...n

x in Rd (x is a vector in d-dimensional space) feature vector

y in {-1,+1} label (class, category)

Question: Design a linear decision boundary: wx + b (equation of hyperplane) such that the classification rule associated with it has minimal probability of error classification rule:

– y = sign(w x + b) which means:– if wx + b > 0 then y = +1– if wx + b < 0 then y = -1


Linear binary classification

Find a good hyperplane (w,b) in Rd+1

that correctly classifies data points as much as possible

In online fashion: one data point at the time, update weights as necessary

wx + b = 0

Classification Rule: y = sign(wx + b)


Perceptron algorithmInitialize: w1 = 0

Updating rule For each data point x

If class(x) != decision(x,w)then

wk+1 wk + yixi

k k + 1 else

wk+1 wk

Function decision(x, w)If wx + b > 0 return +1

Else return -1

wk

0

+1

-1wk x + b = 0

wk+1

Wk+1 x + b = 0


Perceptron algorithm

Online: can adjust to changing target, over timeAdvantages

Simple and computationally efficientGuaranteed to learn a linearly separable problem (convergence, global optimum)

LimitationsOnly linear separationsOnly converges for linearly separable dataNot really “efficient with many features”


Winnow algorithm

Another online algorithm for learning perceptron weights:

f(x) = sign(wx + b)Linear, binary classification Update-rule: again error-driven, but multiplicative (instead of additive)


Winnow algorithm

wk

0

+1

-1wk x + b= 0

wk+1

Wk+1 x + b = 0

Initialize: w1 = 0Updating rule For each data point x

If class(x) != decision(x,w)then

wk+1 wk + yixi Perceptron

wk+1 wk *exp(yixi) Winnow

k k + 1 else

wk+1 wk

Function decision(x, w)If wx + b > 0 return +1Else return -1


Perceptron vs. Winnow

AssumeN available featuresonly K relevant items, with K<<N

Perceptron: number of mistakes: O( K N)Winnow: number of mistakes: O(K log N)

Winnow is more robust to high-dimensional feature

spaces


Perceptron vs. WinnowPerceptronOnline: can adjust to changing target, over timeAdvantages

Simple and computationally efficientGuaranteed to learn a linearly separable problem

Limitationsonly linear separationsonly converges for linearly separable datanot really “efficient with many features”

WinnowOnline: can adjust to changing target, over timeAdvantages

Simple and computationally efficientGuaranteed to learn a linearly separable problem Suitable for problems with many irrelevant attributes

Limitationsonly linear separationsonly converges for linearly separable datanot really “efficient with many features”

Used in NLP

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Weka

Winnow in Weka


Another family of linear algorithmsIntuition (Vapnik, 1965) If the classes are linearly separable:

Separate the dataPlace hyper-plane “far” from the data: large marginStatistical results guarantee good generalization

Large margin classifier

BAD


GOOD

Maximal Margin Classifier

Intuition (Vapnik, 1965) if linearly separable:

Separate the dataPlace hyperplane “far” from the data: large marginStatistical results guarantee good generalization



If not linearly separableAllow some errorsStill, try to place hyperplane “far” from each class


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Large Margin Classifiers

AdvantagesTheoretically better (better error bounds)

LimitationsComputationally more expensive, large quadratic programming


Support Vector Machine (SVM)

Large Margin Classifier

Linearly separable case

Goal: find the hyperplane that maximizes the margin

wT x + b = 0

M wTxa + b = 1

wTxb + b = -1

Support vectors


Support Vector Machine (SVM)

Text classificationHand-writing recognitionComputational biology (e.g., micro-array data)Face detection Face expression recognition Time series prediction

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Non Linear problem

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Non Linear problem


Non Linear problem

Kernel methodsA family of non-linear algorithmsTransform the non linear problem in a linear one (in a different feature space)Use linear algorithms to solve the linear problem in the new space

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Main intuition of Kernel methods

(Copy here from black board)


X=[x z]

Basic principle kernel methods : Rd RD (D >> d)

(X)=[x2 z2 xz]

f(x) = sign(w1x2+w2z2+w3xz +b)

wT(x)+b=0


Basic principle kernel methods

Linear separability: more likely in high dimensionsMapping: maps input into high-dimensional feature spaceClassifier: construct linear classifier in high-dimensional feature spaceMotivation: appropriate choice of leads to linear separabilityWe can do this efficiently!

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Basic principle kernel methods

We can use the linear algorithms seen before (Perceptron, SVM) for classification in the higher dimensional space

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Multi-class classification

Given: some data items that belong to one of M possible classes Task: Train the classifier and predict the class for a new data itemGeometrically: harder problem, no more simple geometry

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Multi-class classification

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Multi-class classification: Examples

Author identificationLanguage identificationText categorization (topics)

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(Some) Algorithms for Multi-class classification

LinearParallel class separators: Decision TreesNon parallel class separators: Naïve Bayes

Non LinearK-nearest neighbors

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Linear, parallel class separators (ex: Decision Trees)

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Linear, NON parallel class separators (ex: Naïve Bayes)

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Non Linear (ex: k Nearest Neighbor)

41http://dms.irb.hr/tutorial/tut_dtrees.php

Decision Trees

Decision tree is a classifier in the form of a tree structure, where each node is either:

Leaf node - indicates the value of the target attribute (class) of examples, orDecision node - specifies some test to be carried out on a single attribute-value, with one branch and sub-tree for each possible outcome of the test.

A decision tree can be used to classify an example by starting at the root of the tree and moving through it until a leaf node, which provides the classification of the instance.

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Training Examples

NoStrongHighMildRainD14

YesWeakNormalHotOvercastD13

YesStrongHighMildOvercastD12

YesStrongNormalMildSunnyD11

YesStrongNormalMildRainD10

YesWeakNormalColdSunnyD9

NoWeakHighMildSunnyD8

YesWeakNormalCoolOvercastD7

NoStrongNormalCoolRainD6

YesWeakNormalCoolRainD5

YesWeakHighMildRain D4

YesWeakHighHotOvercastD3

NoStrongHighHotSunnyD2

NoWeakHighHotSunnyD1

Play TennisWindHumidityTemp.OutlookDay

Goal: learn when we can play Tennis and when we cannot

43www.math.tau.ac.il/~nin/ Courses/ML04/DecisionTreesCLS.pp

Decision Tree for PlayTennis

Outlook

Sunny Overcast Rain

Humidity

High Normal

Wind

Strong Weak

No Yes

Yes

YesNo



Outlook

Sunny Overcast Rain

Humidity

High Normal

No Yes

Each internal node tests an attribute

Each branch corresponds to anattribute value node

Each leaf node assigns a classification


No


Outlook

Sunny Overcast Rain

Humidity

High Normal

Wind

Strong Weak

No Yes

Yes

YesNo

Outlook Temperature Humidity Wind PlayTennis Sunny Hot High Weak ?

46Foundations of Statistical Natural Language Processing, Manning and Schuetze

Decision Tree for Reuter classification


Decision Tree for Reuter classification

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Building Decision Trees

Given training data, how do we construct them?The central focus of the decision tree growing algorithm is selecting which attribute to test at each node in the tree. The goal is to select the attribute that is most useful for classifying examples. Top-down, greedy search through the space of possible decision trees.

That is, it picks the best attribute and never looks back to reconsider earlier choices.

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Building Decision Trees

Splitting criterionFinding the features and the values to split on

– for example, why test first “cts” and not “vs”? – Why test on “cts < 2” and not “cts < 5” ?

Split that gives us the maximum information gain (or the maximum reduction of uncertainty)

Stopping criterionWhen all the elements at one node have the same class, no need to split further

In practice, one first builds a large tree and then one prunes it back (to avoid overfitting)

See Foundations of Statistical Natural Language Processing, Manning and Schuetze for a good introduction


Decision Trees: Strengths

Decision trees are able to generate understandable rules. Decision trees perform classification without requiring much computation. Decision trees are able to handle both continuous and categorical variables. Decision trees provide a clear indication of which features are most important for prediction or classification.


Decision Trees: weaknesses

Decision trees are prone to errors in classification problems with many classes and relatively small number of training examples. Decision tree can be computationally expensive to train.

Need to compare all possible splitsPruning is also expensive

Most decision-tree algorithms only examine a single field at a time. This leads to rectangular classification boxes that may not correspond well with the actual distribution of records in the decision space.

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Decision Trees

Decision Trees in Weka

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Naïve Bayes

More powerful that Decision Trees

Decision Trees Naïve Bayes

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Naïve Bayes Models

Graphical Models: graph theory plus probability theoryNodes are variablesEdges are conditional probabilities

A

B C

P(A) P(B|A)P(C|A)

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Naïve Bayes Models

Graphical Models: graph theory plus probability theoryNodes are variablesEdges are conditional probabilitiesAbsence of an edge between nodes implies independence between the variables of the nodes

A

B C

P(A) P(B|A)P(C|A) P(C|A,B)


Naïve Bayes for text classification

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earn

Shr 34 cts vs shrper

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The words depend on the topic: P(wi| Topic)

P(cts|earn) > P(tennis| earn)

Naïve Bayes assumption: all words are independent given the topicFrom training set we learn the probabilities P(wi| Topic) for each word and for each topic in the training set

Topic

w1 w2 w3 w4 wnwn-1

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To: Classify new exampleCalculate P(Topic | w1, w2, … wn) for each topic

Bayes decision rule:Choose the topic T’ for which P(T’ | w1, w2, … wn) > P(T | w1, w2, … wn) for each T T’

Topic

w1 w2 w3 w4 wnwn-1

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Naïve Bayes: Math

Naïve Bayes define a joint probability distribution: P(Topic , w1, w2, … wn) = P(Topic) P(wi| Topic)

We learn P(Topic) and P(wi| Topic) in training

Test: we need P(Topic | w1, w2, … wn)

P(Topic | w1, w2, … wn) = P(Topic , w1, w2, … wn) / P(w1, w2, … wn)

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Naïve Bayes: Strengths

Very simple modelEasy to understandVery easy to implement

Very efficient, fast training and classificationModest space storageWidely used because it works really well for text categorizationLinear, but non parallel decision boundaries

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Naïve Bayes: weaknesses

Naïve Bayes independence assumption has two consequences:

The linear ordering of words is ignored (bag of words model)The words are independent of each other given the class: False

– President is more likely to occur in a context that contains election than in a context that contains poet

Naïve Bayes assumption is inappropriate if there are strong conditional dependencies between the variables(But even if the model is not “right”, Naïve Bayes models do well in a surprisingly large number of cases because often we are interested in classification accuracy and not in accurate probability estimations)

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Naïve Bayes

Naïve Bayes in Weka

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k Nearest Neighbor Classification

Nearest Neighbor classification rule: to classify a new object, find the object in the training set that is most similar. Then assign the category of this nearest neighborK Nearest Neighbor (KNN): consult k nearest neighbors. Decision based on the majority category of these neighbors. More robust than k = 1

Example of similarity measure often used in NLP is cosine similarity

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1-Nearest Neighbor

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1-Nearest Neighbor

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3-Nearest Neighbor

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3-Nearest Neighbor

Assign the category of the majority of the neighbors

But this is closer..We can weight neighbors according to their similarity

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k Nearest Neighbor Classification

StrengthsRobustConceptually simpleOften works wellPowerful (arbitrary decision boundaries)

WeaknessesPerformance is very dependent on the similarity measure used (and to a lesser extent on the number of neighbors k used)Finding a good similarity measure can be difficultComputationally expensive

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Summary

Algorithms for Classification Linear versus non linear classificationBinary classification

Perceptron WinnowSupport Vector Machines (SVM)Kernel Methods

Multi-Class classificationDecision TreesNaïve BayesK nearest neighbor

On Wednesday: Weka

winnow vs perceptron

Technology

linear algorithms linear

b linear

linear separable data

linear separations

linear problems

data items

data points

new data item