CS60050MachineLearning

Classification:LogisticRegression

Some slides taken from course materials of Andrew Ng

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Classification

Email:Spam/NotSpam?OnlineTransactions:Fraudulent /Genuine?Tumor:Malignant/Benign?

0:“NegativeClass”(e.g.,benigntumor)1:“PositiveClass”(e.g.,malignanttumor)

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TumorSize

Malignant?

(Yes)1

(No)0

Canwesolvetheproblemusinglinearregression?

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TumorSize

Thresholdclassifieroutput at0.5:

If,predict“y=1”

If,predict“y=0”

Malignant?

(Yes)1

(No)0

Canwesolvetheproblemusinglinearregression?E.g., fitastraightlineanddefineathresholdat0.5

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TumorSize

Thresholdclassifieroutput at0.5:

If,predict“y=1”

If,predict“y=0”

Malignant?

(Yes)1

(No)0

Canwesolvetheproblemusinglinearregression?E.g., fitastraightlineanddefineathresholdat0.5

Failureduetoaddinganewpoint

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Classification:y=0or1

canbe>1or<0

LogisticRegression:

Anotherdrawbackofusinglinearregressionforthisproblem

Whatweneed:

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SigmoidfunctionLogisticfunction

LogisticRegressionModelWant

0

Ausefulproperty:easytocomputedifferentialatanypoint

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InterpretationofHypothesisOutput

=estimatedprobabilitythaty=1oninputx

Tellpatientthat70%chanceoftumorbeingmalignant

Example:If

“probabilitythaty=1,givenx,parameterized by”

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Logisticregression

Supposepredict““if

predict““if

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Logisticregression

Supposepredict““if

predict““if

When𝛩Tx ≥0

When𝛩Tx <0

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Separatingtwoclassesofpoints• Weareattemptingtoseparatetwogivensets/classesofpoints

• Separatetworegionsofthefeaturespace• ConceptofDecisionBoundary• Findingagooddecisionboundary=>learnappropriatevaluesfortheparameters𝛩

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x1

x2

DecisionBoundary

1 2 3

1

2

3

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x1

x2

DecisionBoundary

1 2 3

1

2

3

Predictif

Howtogettheparametervalues– willbediscussedsoon

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Non-linear decisionboundaries

x1

x2

1-1

-1

1

Wecanlearnmorecomplexdecisionboundarieswhere thehypothesisfunctioncontainshigherorderterms.

(rememberpolynomialregression)

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Non-linear decisionboundaries

x1

x2

Predictif

1-1

-1

1

Howtogettheparametervalues– willbediscussedsoon

CostfunctionforLogisticRegression

Howtogettheparameter values?

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Trainingset:

Howtochooseparameters?

mexamples

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Costfunction

Linearregression:

Howeverthiscostfunction isnon-convexforthehypothesisoflogisticregression.

Squarederrorcostfunction:

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Logisticregressioncostfunction

Cost

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Logisticregressioncostfunction

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Logisticregressioncostfunction

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Logisticregressioncostfunction

Tofitparameters:Thiscostfunctionisconvex

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GradientDescent

Want:Repeat

(simultaneouslyupdateall)

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GradientDescent

Want:

(simultaneouslyupdateall)

Repeat

Algorithmlooksidenticaltolinearregression,butthehypothesisfunctionisdifferentforlogisticregression.

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Thuswecangradientdescenttolearnparametervalues,andhencecomputeforanewinput:

Output

Tomakeapredictiongivennew :

=estimatedprobabilitythaty=1oninputx

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Howtousetheestimatedprobability?• Refrainingfromclassifyingunlessconfident• Rankingitems• Multi-classclassification

Multi-classclassification:onevs.all

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Multiclassclassification

Newsarticletagging:Politics,Sports,Movies,Religion,…

Medicaldiagnosis:Notill,Cold,Flu,Fever

Weather:Sunny,Cloudy,Rain,Snow

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x1

x2

x1

x2

Binaryclassification: Multi-classclassification:

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x1

x2

One-vs-all(one-vs-rest):

Class1:Class2:Class3:

x1

x2

x1

x2

x1

x2

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One-vs-all

Trainalogisticregressionclassifierforeachclasstopredicttheprobabilitythat.

Onanewinput,tomakeaprediction,picktheclassthatmaximizes

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AdvancedOptimization algorithms (notpartofthiscourse)

Optimizationalgorithms:- Gradientdescent- Conjugategradient- BFGS- L-BFGS

Advantagesoftheotheralgorithms:- Noneedtomanuallypicklearningrate- Oftenconvergesfasterthangradientdescent

Disadvantages:- Morecomplex