Cost-Sensitive Classiﬁcation: Algorithms and Advances

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Cost-Sensitive Classification:

Algorithms and Advances

Hsuan-Tien Lin htlin@csie.ntu.edu.tw

Department of Computer Science

& Information Engineering

National Taiwan University

Tutorial for ACML @ Canberra, Australia November 13, 2013

Hsuan-Tien Lin (NTU CSIE) Cost-Sensitive Classification: Algorithms and Advances 0/99

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More about Me

Associate Professor Dept. CSIE National Taiwan

University

• co-leader of KDDCup world champion teams at NTU: 2010–2013

• research on multi-label classification, ranking, active learning, etc.

• research on cost-sensitive classification:

2007–Present

• Secretary General, Taiwanese Association for Artificial Intelligence

• instructor of Mandarin-teaching MOOC of Machine Learning on NTU-Coursera:

2013.11–

https://www.coursera.org/course/ntumlone

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Cost-Sensitive Binary Classification

Outline

Bayesian Perspective of Cost-Sensitive Binary Classification Non-Bayesian Perspective of Cost-Sensitive Binary Classification Cost-Sensitive Multiclass Classification

Bayesian Perspective of Cost-Sensitive Multiclass Classification Cost-Sensitive Classification by Reweighting and Relabeling Cost-Sensitive Classification by Binary Classification

Cost-Sensitive Classification by Regression

Cost-and-Error-Sensitive Classification with Bioinformatics Application Cost-Sensitive Ordinal Ranking with Information Retrieval Application Summary

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Is This Your Fingerprint?

f





 +1 you

−1 intruder

?

you intruder

• abinary classification problem

—grouping “fingerprint pictures” intotwo different “categories”

C’mon, we know about binary classification all too well!:-)

(5)

Supervised Machine Learning

parent

?

(picture, category) pairs

?

kid’s good

decision function brain

'

&

$

% -

6 possibilities

truth f (x) + noise e(x)

?

examples (picturexn, category yn)

?

learning good

decision function g(x)≈ f (x) algorithm

'

&

$

% -

6

learning model{h(x)}

how toevaluatewhetherg(x)≈ f (x)?

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Performance Evaluation

Fingerprint Verification

f





 +1 you

−1 intruder

example/figure borrowed from Amazon ML best-seller textbook

“Learning from Data” (Abu-Mostafa, Magdon-Ismail, , 2013)

two types of error: false acceptandfalse reject g

+1 -1

f +1 no error false reject -1 false accept no error

g +1 -1

f +1 0 1

-1 1 0

simplest choice:

penalizes both typesequallyand calculateaveragepenalties

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Fingerprint Verification for Supermarket

f





 +1 you

−1 intruder

+1 -1

g +1 -1

f +1 0 10

-1 1 0

• supermarket: fingerprint for discount

• false reject:very unhappy customer, lose future business

• false accept:give a minor discount, intruder left fingerprint:-)

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Fingerprint Verification for CIA

f





 +1 you

−1 intruder

+1 -1

g

+1 -1

f +1 0 1

-1 1000 0

• CIA: fingerprint for entrance

• false accept:very serious consequences!

• false reject:unhappy employee, but so what? :-)

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Regular Binary Classification

penalizes both typesequally

h(x) +1 -1

y +1 0 1

-1 1 0

in-sample error for any hypothesis h E_in(h) = 1

N u w v y_n

|{z}

f (xn)+noise

6= h(xn) }

~

out-of-sample error for any hypothesis h E_out(h) = E

(x,y )

u w

v y

|{z}

f (x)+noise

6= h(x) }

~

regular binary classification:

well-studied in machine learning

—ya, we know! :-)

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Class-Weighted Cost-Sensitive Binary Classification

Supermarket Cost (Error, Loss, . . .) Matrix

h(x) +1 -1

y +1 0 10

-1 1 0

in-sample

E_in(h) = 1 N

XN n=1

10 if yn = +1 1 if y_n =−1

·Jyn6= h(xn)K out-of-sample

E_out(h) = E

(x,y )

10 if y = +1 1 if y =−1

·Jy 6= h(x)K class-weighted cost-sensitive binary classification:different

‘weight’ for different y

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Setup: Class-Weighted Cost-Sensitive Binary Classification Given

N examples, each

(inputx_n, label yn)∈ X × {−1, +1}

and weightsw₊,w₋

representing the two entries of the cost matrix

h(x) +1 -1 y +1 0 w+

-1 w− 0

Goal

a classifier g(x) that

pays a small cost wyJy 6= g (x)K

on futureunseen example (x, y ), i.e., achieves low E_out(g)

regular classification: w₊ =w−(= 1)

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Supermarket Revisited

f





 +1 you

−1 intruder

+1 -1

g +1 -1 f big customer 0 100

usual customer 0 10

intruder 1 0

• supermarket: fingerprint for discount

• big customer: really don’t want to lose her/his business

• usual customer: don’t want to lose business, but not so serious

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Example-Weighted Cost-Sensitive Binary Classification

Supermarket Cost Vectors(Rows)

h(x) +1 -1

y

big 0 100

usual 0 10

intruder 1 0

in-sample

E_in(h) = 1 N

XN n=1

wn

|{z}

importance

·Jyⁿ6= h(xn)K

out-of-sample E_out(h) = E

(x,y ,w)w·Jy 6= h(x)K example-weighted cost-sensitive binary classification:

different w for different (x, y )

—seen this in AdaBoost? :-)

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Setup: Example-Weighted Cost-Sensitive Binary Classification

Given

N examples, each (inputxn, label yn)∈ X × {−1, +1}

and weight wn∈ R⁺

Goal

pays a small cost wJy 6= g (x)K

on futureunseen example (x, y , w), i.e., achieves low Eout(g)

regular⊂ class-weighted ⊂ example-weighted

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Bayesian Perspective of Cost-Sensitive Binary Classification

Outline

Non-Bayesian Perspective of Cost-Sensitive Binary Classification Cost-Sensitive Multiclass Classification

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Key Idea: Conditional Probability Estimator

Goal (Class-Weighted Setup)

a classifier g(x) that pays a small cost wyJy 6= g (x)K on future unseen example (x, y )

• expected error for predicting +1 onx: w−P(−1|x)

• expected error for predicting -1 onx: w₊P(+1|x)

if P(y|x) known

Bayes optimal g^∗(x) = sign

w₊P(+1|x) − w⁻P(−1|x)

if p(x)≈ P(+1|x) well

approximately good g_p(x) = sign

w₊p(x)− w⁻(1− p(x)) how to get conditional probability estimator p?

logistic regression, Naïve Bayes,. . .

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Approximate Bayes-Optimal Decision

if p(x)≈ P(+1|x) well

approximately good g_p(x) = sign

w₊p(x)− w⁻(1− p(x)) that is(Elkan, 2001),

g_p(x) = +1 iff w₊p(x)− w−(1− p(x)) > 0 iff p(x)> w−

w₊+w₋ : ₁₁¹ for supermarket; ¹⁰⁰₁₀₁ for CIA

Approximate Bayes-Optimal Decision (ABOD) Approach

1 use your favorite algorithm on{(xn, yn)} to getp(x)≈ P(+1|x)

2 for each new inputx, predict its class using gp(x) = sign(p(x)−w+^w+w⁻−)

‘simplest’ approach:

probability estimate+threshold changing

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ABOD on Artificial Data

1 use your favorite algorithm on{(xⁿ, yn)} to getp(x)≈ P(+1|x)

2 for each new inputx, predict its class using gp(x) = sign(p(x)−_w₊^w_+w⁻₋)

LogReg

−→

g +1 -1

y +1 0 10

-1 1 0

regular supermarket

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Pros and Cons of ABOD

Pros

• optimal: if goodprobability estimate: p(x) really close to P(+1|x)

• simple: training (probability estimate)unchanged, andprediction (threshold)changed only a little Cons

• ‘difficult’: goodprobability estimateoften more difficult than goodbinary classification

• ‘restricted’: only applicable toclass-weighted setup

—need ‘full picture’ of cost matrix

approach for theexample-weighted setup?

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Non-Bayesian Perspective of Cost-Sensitive Binary Classification

Outline

Bayesian Perspective of Cost-Sensitive Binary Classification Non-Bayesian Perspective of Cost-Sensitive Binary Classification

Cost-Sensitive Multiclass Classification

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Key Idea: Example Weight = Copying

Goal

pays a small cost wJy 6= g (x)K on futureunseen example (x, y , w)

on one (x, y )

wrong prediction charged by w

—regular classification

onw copiesof (x, y )

wrong prediction charged by 1

—regular classification

how to copy?over-sampling

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Example-Weighted Classification by Over-Sampling

copy each (x_n, yn)for w_ntimes original problem

evaluate with

h(x) +1 -1

y

big 0 100

usual 0 10

intruder 1 0

(x₁,-1,1) (x₂,+1,10) (x₃,+1,100)

(x₄,+1,10) (x₅,-1,1)

equivalent problem evaluate with

h(x) +1 -1

y

big 0 1

usual 0 1

intruder 1 0

(x₁,-1) (x₂,+1),. . ., (x2,+1) (x₃,+1),. . ., (x₃,+1),. . ., (x₃,+1)

(x₄,+1),. . ., (x₄,+1) (x₅,-1)

how to learn a good g for RHS?

SVM, NNet,. . .

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Cost-Proportionate Example Weighting

Cost-Proportionate Example Weighting (CPEW) Approach

1 effectively transform{(xⁿ, yn, wn)} to{(x^m, ym)}such that the

‘copies’ of (xn, yn)in{(x^m, ym)} is proportional to wⁿ

• over/under-sampling with normalized wn(Elkan, 2001)

• under-sampling by rejection(Zadrozny, 2003)

• modify existing algorithms equivalently(Zadrozny, 2003) 2 use your favorite algorithm on{(xm, ym)}to get binary

classifierg(x)

3 for each new inputx, predict its class usingg(x)

simple and general:

very popular for cost-sensitive binary classification

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CPEW by Modification

‘copies’ of (x_n, yn)in{(x^m, ym)} is proportional to wⁿ

• modify existing algorithms equivalently(Zadrozny, 2003) 2 use your favorite algorithm on{(xm, ym)} to get binary

classifierg(x)

Regular Linear SVM

minw,b

1 2hw, wi +

N

X

n=1

Cξn

ξn=max (1 − yn(hw, xni + b), 0)

Modified Linear SVM

minw,b

1 2hw, wi +

N

X

n=1

C · wn· ξn

ξn=max (1 − yn(hw, xni + b), 0)

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CPEW by Modification on Artificial Data

1 effectively transform{(xⁿ, yn, wn)} to{(x^m, ym)}by modifying existing algorithms equivalently(Zadrozny, 2003)

2 use your favorite algorithm on{(x^m, ym)} to getg(x)

modify

−→

g +1 -1

y +1 0 10

-1 1 0

regular supermarket

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CPEW by Rejection Sampling

COSTING Algorithm(Zadrozny, 2003)

‘copies’ of (xn, yn)in{(x^m, ym)} is proportional to wⁿ

• under-sampling by rejection(Zadrozny, 2003)

2 use your favorite algorithm on{(xm, ym)} to get binary classifierg(x)

3 repeat 1 and 2 to get multiplegandaggregate them

4 for each new inputx, predict its class usingaggregatedg(x)

commonly used when your favorite algorithm is ablack box rather than awhite box

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Biased Personal Favorites

• CPEW by Modification if possible

• COSTING: fast training and stable performance

• ABOD if in the mood for Bayesian :-)

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Outline

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Which Digit Did You Write?

?

one (1) two (2) three (3) four (4)

• amulticlass classification problem

—grouping “pictures” into different “categories”

C’mon, we know about

multiclass classification all too well!:-)

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Performance Evaluation

^(g(x)≈ f (x)?)

?

• ZIP code recognition:

1:wrong; 2:right; 3: wrong; 4: wrong

• check value recognition:

1:one-dollar mistake; 2:no mistake;

3:one-dollar mistake; 4:two-dollar mistake

• evaluation by formation similarity:

1:not very similar; 2: very similar;

3:somewhat similar; 4: asilly prediction different applications:

evaluate mis-predictions differently

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ZIP Code Recognition

?

1:wrong; 2:right; 3: wrong; 4: wrong

• regular multiclass classification: only right or wrong

• wrong cost: 1;right cost: 0

• prediction error of h on some (x, y ):

classification cost =Jy 6= h(x)K

—as discussed in regular binary classification

regular multiclass classification:

well-studied, many good algorithms

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Check Value Recognition

?

1:one-dollar mistake; 2:no mistake;

3:one-dollar mistake; 4: two-dollar mistake

• cost-sensitive multiclass classification:

different costs for different mis-predictions

• e.g. prediction error of h on some (x, y ):

absolute cost =|y − h(x)|

next: cost-sensitivemulticlass classification

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What is the Status of the Patient?

?

H1N1-infected cold-infected healthy

• anotherclassification problem

—grouping “patients” into different “status”

are all mis-prediction costs equal?

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Patient Status Prediction

error measure = society cost XXXXactual XXXXXX

predicted

H7N9 cold healthy

H7N9 0 1000 100000

cold 100 0 3000

healthy 100 30 0

• H7N9 mis-predicted as healthy:very high cost

• cold mis-predicted as healthy: high cost

• cold correctly predicted as cold: no cost

human doctors consider costs of decision;

can computer-aided diagnosis do the same?

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What is the Type of the Movie?

? romance fiction terror

customer 1 who hates romance but likes terror error measure = non-satisfaction

XXXXactual XXXXXX predicted

romance fiction terror

romance 0 5 100

customer 2 who likes terror and romance

romance 0 5 3

different customers:

evaluate mis-predictions differently

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Cost-Sensitive Multiclass Classification Tasks

movie classification with non-satisfaction

customer 1, romance 0 5 100

customer 2, romance 0 5 3

patient diagnosis with society cost

H7N9 cold healthy

H7N9 0 1000 100000

cold 100 0 3000

healthy 100 30 0

check digit recognition with absolute cost C(y, h(x)) = |y − h(x)|

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Cost Vector

cost vectorc: a row of cost components

• customer 1 on a romance movie: c = (0,5,100)

• an H7N9 patient: c = (0,1000,100000)

• absolute cost for digit 2: c = (1,0,1,2)

• “regular” classification cost for label 2: c⁽²⁾_c = (1,0,1,1)

regular classification:

special case of cost-sensitive classification

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Setup: Matrix-Based Cost-Sensitive Binary Classification Given

N examples, each (inputxn, label yn)∈ X × {1, 2, . . . , K } and cost matrixC ∈ R^{K ×K}

—will assumeC(y, y) =0=min_{1≤k ≤K}C(y, k) Goal

pays a small costC(y, g(x)) on futureunseen example (x, y )

extension of ‘class-weighted’ cost-sensitive binary classification

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Setup: Vector-Based Cost-Sensitive Binary Classification Given

N examples, each (inputxn, label yn)∈ X × {1, 2, . . . , K } and cost vectorc_n∈ R^K

—will assumec_n[y_n] =0=min_{1≤k ≤K}c_n[k ] Goal

a classifier g(x) that pays a small cost c[g(x)] on future unseen example (x, y , c)

• will assumec[y ] =0=c_min =min_{1≤k ≤K}c[k ]

• note: y not really needed in evaluation

extension of ‘example-weighted’

cost-sensitive binary classification

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Which Age-Group?

?

infant (1) child (2) teen (3) adult (4)

• small mistake—classify a child as a teen;

big mistake—classify an infant as an adult

• cost matrixC(y, g(x)) for embedding ‘order’: C =







0 1 4 5 1 0 1 3 3 1 0 2 5 4 1 0







cost-sensitive classification can help solve many other problems, such asordinal ranking

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Bayesian Perspective of Cost-Sensitive Multiclass Classification

Outline

Cost-Sensitive Classification by Reweighting and Relabeling Cost-Sensitive Classification by Binary Classification

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Key Idea: Conditional Probability Estimator

Goal (Matrix Setup)

a classifier g(x) that pays a small costC(y, g(x)) on future unseen example (x, y )

if P(y|x) known

Bayes optimal g^∗(x) =

argmin

1≤k ≤K

XK y =1

P(y|x)C(y, k)

if p(y, x)≈ P(y|x) well

approximately good g_p(x) =

argmin

1≤k ≤K

XK y =1

p(y, x)C(y, k)

how to get conditional probability estimator p?

logistic regression, Naïve Bayes,. . .

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Approximate Bayes-Optimal Decision

if p(y, x)≈ P(+1|x) well

(Domingos, 1999)

approximately good gp(x) = argmink ∈{1,2,...,K }

PK

y =1p(y, x)C(y, k) Approximate Bayes-Optimal Decision (ABOD) Approach

1 use your favorite algorithm on{(xⁿ, yn)} to getp(y, x)≈ P(y|x)

2 for each new inputx, predict its class using g_p(x) above a simple extension from binary classification:

probability estimate+Bayes-optimal decision

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ABOD on Artificial Data

1 use your favorite algorithm on{(xn, yn)} to getp(y, x)≈ P(y|x)

2 for each new inputx, predict its class using g_p(x)

LogReg

−→

g

1 2 3 4

y

1 0 1 2 4

2 4 0 1 2

3 2 4 0 1

4 1 2 4 0

regular rotate

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Pros and Cons of ABOD

Pros

• optimal: if goodprobability estimate: p(y, x) really close to P(y|x)

• simple: withtraining (probability estimate)unchanged, andprediction (threshold)changed only a little

Cons

• ‘difficult’: goodprobability estimateoften more difficult than goodmulticlass classification

• ‘restricted’: only applicable toclass-weighted setup

—need ‘full picture’ of cost matrix

• ‘slow prediction’: need sophisticated calculation at prediction stage

can we use anymulticlass classification algorithmfor ABOD?

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MetaCost Approach

Approximate Bayes-Optimal Decision (ABOD) Approach

1 use your favorite algorithm on{(xn, y_n)} to getp(y, x)≈ P(y|x)

2 for each new inputx, predict its class using gp(x) MetaCost Approach (Domingos, 1999)

1 use your favoritemulticlass classificationalgorithm on bootstrapped{(xn, yn)} and aggregate the classifiers to get p(y, x)≈ P(y|x)

2 for each given inputx_n,relabel it to y_n⁰ using gp(x)

3 run your favoritemulticlass classificationalgorithm on relabeled {(xⁿ, y_n⁰)} to get final classifier g

4 for each new inputx, predict its class using g(x)

pros: anymulticlass classificationalgorithm can be used

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MetaCost on Semi-Real Data

0 500 1000 1500 2000 2500 3000 3500

0 300 600 900 1200 1500 1800

C4.5R

MetaCost Multiclass Two-class y=x

0 500 1000 1500 2000 2500

0 300 600 900 1200 1500 1800

Undersampling

0 500 1000 1500 2000 2500

0 300 600 900 1200 1500 1800

Oversampling

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(Domingos, 1999)

• some “random” cost with UCI data

• MetaCost+C4.5:

cost-sensitive

• C4.5: regular

not surprisingly,

considering the cost properly does help

(48)

Cost-Sensitive Classification by Reweighting and Relabeling

Outline

Bayesian Perspective of Cost-Sensitive Multiclass Classification Cost-Sensitive Classification by Reweighting and Relabeling

Cost-Sensitive Classification by Binary Classification Cost-Sensitive Classification by Regression

(49)

Recall: Example-Weighting Useful for Binary

can example weighting be used for multiclass?

Yes! an elegant solution if using costmatrixwith special properties

(Zhou, 2010)

C(i, j) C(j, i) = w_i

w_j

what if using costvectorswithout special properties?

(50)

Key Idea: Cost Transformation

0 1000

| {z }

c

= 1000 0

| {z }

#of copies

·

0 1 1 0

| {z } classification costs

3 2 3 4

| {z }

costc

= 1 2 1 0

| {z }

mixture weightsq`

·







0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0







| {z }

classification costs

• split the cost-sensitive example:

(x, 2) with c = (3, 2, 3, 4) equivalent to

a weighted mixture{(x, 1, 1), (x, 2, 2), (x, 3, 1)}

cost equivalence: c[h(x)] = PK

`=1

q_`J` 6= h(x)K for any h

(51)

Meaning of Cost Equivalence

c[h(x)] = XK

`=1

q_`J` 6= h(x)K

on one (x, y , c)

wrong prediction charged by c[h(x)]

—weighted classification

onall(x, `, q`)

wrong prediction charged by total weighted classification error

—weighted classification weighted classification =⇒ regular classification?

same as binary(with CPEW) when q_` ≥ 0

min_g expected LHS (original cost-sensitive problem)

= min_g expected RHS (a regular problem when q_`≥ 0)

(52)

Cost Transformation Methodology: Preliminary

1 split each training example (x_n, y_n, c_n)to a weighted mixture {(xn, `, qn,`)}^K_`=1

2 apply regular/weighted classification algorithm on the weighted mixtures

SN

n=1{(xn, `, q_n,`)}^K_`=1

• byc[g(x)] =PK

`=1q_`J` 6= g (x)K (cost equivalence), good g for new regular classification problem

=

good g for original cost-sensitive classification problem

• regular classification: needs q_n,`≥ 0

but what if q_n,`negative?

(53)

Similar Cost Vectors

1 0 1 2 3 2 3 4

| {z }

costs

=

1/3 4/3 1/3 −2/3

1 2 1 0

| {z }

mixture weights q`

·







0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0







| {z }

• negative q_`: cannot split

• but ˆc = (1, 0, 1, 2) is similar to c = (3, 2, 3, 4):

for any classifier g,

c[g(x)] + constant = c[g(x)] =ˆ XK

`=1q_`J` 6= g (x)K

• constant can be dropped during minimization

ming expected ˆc[g(x)] (original cost-sensitive problem)

= min_g expected LHS (a regular problem when q_`≥ 0)

(54)

Cost Transformation Methodology: Revised

1 shift each training cost ˆc_n to a similar and “splittable”c_n

2 split (x_n, yn, cn)to a weighted mixture{(xn, `, qn,`)}^K_`=1

3 apply regular classification algorithm on the weighted mixtures SN

n=1{(xⁿ, `, q_n,`)}^K_`=1

• splittable: q_n,`≥ 0

• by cost equivalence after shifting:

good g for new regular classification problem

=

but infinitely many similar and splittable c_n!

(55)

Uncertainty in Mixture

• a single example{(x, 2)}

—certain that the desired label is 2

• a mixture{(x, 1, 1), (x, 2, 2), (x, 3, 1)} sharing the same x

—uncertainty in the desired label (25% : 1, 50% : 2, 25% : 3)

• over-shifting adds unnecessary mixture uncertainty:

3 2 3 4

33 32 33 34

| {z }

costs

=

1 2 1 0

11 12 11 10

| {z }

mixture weights

·







0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0







| {z }

should choose a similar and splittablec withminimum mixture uncertainty

(56)

Cost Transformation Methodology: Final

Cost Transformation Methodology(Lin, 2010)

1 shift each training cost ˆc_n to a similar and splittablec_nwith minimum “mixture uncertainty”

2 split (xn, yn, cn)to a weighted mixture{(xⁿ, `, qn,`)}^K_`=1

3 apply regular classification algorithm on the weighted mixtures SN

n=1

{(xn, `, qn,`)}^K_`=1

• mixture uncertainty: entropy of normalized (q₁, q₂,· · · , qK)

• a simple and unique optimal shifting exists for every ˆc

good g for new regular classification problem

=

(57)

Data Space Expansion Approach

Data Space Expansion (DSE) Approach(Abe, 2004) 1 for each (x_n, yn, cn)and`, letqn,`= max

1≤k ≤Kc_n[k ]− cⁿ[`]

2 apply your favoritemulticlass classification algorithmon the weighted mixtures

SN

n=1{(xn, `, q_n,`)}^K_`=1to get g(x)

3 for each new inputx, predict its class using g(x)

• detailed explanation provided by the cost transformation methodology discussed above(Lin, 2010)

• extension of Cost-Proportionate Example Weighting, but now with relabeling!

pros: anymulticlass classificationalgorithm can be used

(58)

DSE versus MetaCost on Semi-Real Data

(Abe, 2004)

MetaCost DSE annealing 206.8 127.1

solar 5317 110.9

kdd99 49.39 46.68

letter 129.6 114.0 splice 49.95 135.5 satellite 104.4 116.8

• C4.5 with COSTING for weighted

classification

DSE comparable to MetaCost

(59)

Cons of DSE: Unavoidable (Minimum) Uncertainty

Original Cost-Sensitive Classification Problem

1 2 3 4

individual examples with certainty

+absolute cost =

New Regular Classification Problem

mixtures with unavoidable uncertainty

• cost embedded as weight + label

• new problem usuallyharder than original one

needrobustmulticlass classification algorithm to deal with uncertainty

(60)

Cost-Sensitive Classification by Binary Classification

Outline

(61)

Key Idea: Design Robust Multiclass Algorithm

One-Versus-One: A Popular Classification Meta-Method

1 for a pair (i, j), take all examples (xn, yn)that yn=i or j (original one-versus-one)

2 for a pair (i, j), from each weighted mixture{(xn, `, q_n,`)} with q_n,i > qn,j, take (x_n, i) with weight qn,i− qn,j ; vice versa(robust one-versus-one)