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# From Ordinal Ranking to Binary Classiﬁcation

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## From Ordinal Ranking to Binary Classification

Hsuan-Tien Lin

Learning Systems Group, California Institute of Technology

Talk at Dept. of CSIE, National Taiwan University March 21, 2008

Benefited from joint work with Dr. Ling Li (ALT’06, NIPS’06)

& discussions with Prof. Yaser Abu-Mostafa and Dr. Amrit Pratap

(2)

## Outline

1 Introduction to Machine Learning

2 The Ordinal Ranking Setup

3 Reduction from Ordinal Ranking to Binary Classification Algorithmic Usefulness of Reduction

Theoretical Usefulness of Reduction Experimental Performance of Reduction

4 Conclusion

(3)

## Apple, Orange, or Strawberry?

?

apple orange strawberry

how can machine learn to classify?

(4)

## Supervised Machine Learning

Parent

?

(picture, category) pairs

?

Kid’s good

decision function brain

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&

\$

% -

6 possibilities

Truth f (x ) + noise e(x )

?

examples (picture xn, category yn)

?

learning good

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

'

&

\$

% -

6

learning model {hα(x )}

challenge:

see only {(xn,yn)}without knowing f (x ) or e(x )

=⇒? generalize to unseen (x , y ) w.r.t. f (x )

(5)

## Machine Learning Research

What can the machines learn?

concrete applications:

computer vision, multimedia analysis, architecture optimization, information retrieval, bio-informatics, computational finance, · · · abstract setups:

classification, regression, · · · How can the machines learn?

faster algorithms

algorithms with bettergeneralization performance Why can the machines learn?

statistical learning, reinforcement learning, interactive learning, · · · generalization guarantees

new opportunities of machine learning keep coming from new applications/setups

(6)

## Outline

1 Introduction to Machine Learning

2 The Ordinal Ranking Setup

3 Reduction from Ordinal Ranking to Binary Classification Algorithmic Usefulness of Reduction

Theoretical Usefulness of Reduction Experimental Performance of Reduction

4 Conclusion

(7)

## Which Age-Group?

2

rank: a finite ordered set of labels Y = {1, 2, · · · , K }

(8)

## Properties of Ordinal Ranking (1/2)

ranks representorder information

infant (1)

child (2)

teen (3)

## <

adult (4) general multiclass classification cannot

properly use order information

(9)

## Hot or Not?

http://www.hotornot.com

rank: natural representation of human preferences

(10)

## Properties of Ordinal Ranking (2/2)

ranks donot carry numerical information rating 9 not 2.25 times “hotter” than rating 4

actual metric hidden

infant (ages 1–3)

child (ages 4–12)

teen (ages 13–19)

adult (ages 20–) general metric regression deteriorates

without correct numerical information

(11)

## How Much Did You Like These Movies?

http://www.netflix.com

goal: use “movies you’ve rated” to automatically predict your preferences (ranks) on future movies

(12)

## Ordinal Ranking Setup

Given

N examples (input xn,rank yn) ∈ X × Y

age-group: X = encoding(human pictures), Y = {1, · · · , 4}

hotornot: X = encoding(human pictures), Y = {1, · · · , 10}

netflix: X = encoding(movies), Y = {1, · · · , 5}

Goal

an ordinal ranker (decision function) r (x ) that “closely predicts”

the ranks y associated with someunseen inputs x

ordinal ranking: a hot and important research problem

(13)

## Importance of Ordinal Ranking

relatively new for machine learning connecting classification and regression

matching human preferences—many applications in social science, information retrieval, psychology, and recommendation systems

Ongoing Heat: Netflix Million Dollar Prize

(14)

## Ongoing Heat: Netflix Million Dollar Prize

(since 10/2006)

Given

each user u (480,189 users) rates Nu (from tens to thousands) movies x —a total ofP

uNu=100,480,507 examples Goal

personalized ordinal rankers ru(x ) evaluated on 2,817,131

“unseen” queries (u, x )

the first team being 10% better than original Netflix system getsa million USD

(15)

## Cost of Wrong Prediction

ranks carry no numerical information: how to say “better”?

artificially quantify thecost of being wrong

e.g. loss of customer royalty when the system

says but you feel

cost vectorc of example (x , y , c):

c[k ] = cost when predicting (x , y ) as rank k

e.g. for ( Sweet Home Alabama , ), a proper cost isc = (1, 0, 2, 10, 15)

closely predict: small testing cost

(16)

## Ordinal Cost Vectors

For an ordinal example (x , y ,c), the cost vector c should follow the rank y :c[y ] = 0; c[k ] ≥ 0

respect the ordinal information: V-shaped (ordinal) or even convex (strongly ordinal)

1: infant 2: child 3: teenager 4: adult Cy, k

V-shaped: pay more when predicting further away

1: infant 2: child 3: teenager 4: adult Cy, k

convex: payincreasingly more when further away c[k ] =Jy 6= k K c[k ] =

y − k

c[k ] = (y − k )2 classification: absolute: squared (Netflix):

ordinal strongly strongly

ordinal ordinal

(1, 0, 1, 1, 1) (1, 0, 1, 2, 3) (1, 0, 1, 4, 9)

(17)

## Our Contributions

a theoretical and algorithmic foundation of ordinal ranking, which ...

provides a methodology for designing new ordinal ranking algorithms withany ordinal cost effortlessly takes many existing ordinal ranking algorithms as special cases

introducesnew theoretical guarantee on the generalization performance of ordinal rankers leads tosuperior experimental results

0 0.2 0.4 0.6 0.8 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.2 0.4 0.6 0.8 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.2 0.4 0.6 0.8 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

(18)

## Central Idea: Reduction

(iPod)

complex ordinal ranking problems

(cassette player)

simpler binary classification problems with well-known results on models, algorithms, and theories

If I have seen further it is by

standing on the shoulders of Giants—I. Newton

(19)

## Outline

1 Introduction to Machine Learning

2 The Ordinal Ranking Setup

3 Reduction from Ordinal Ranking to Binary Classification Algorithmic Usefulness of Reduction

Theoretical Usefulness of Reduction Experimental Performance of Reduction

4 Conclusion

(20)

## Threshold Model

If we can first get an ideal score s(x ) of a movie x , how can we construct the discrete r (x ) from an analog s(x )?

x x - θ1

d d d

θ2

t tt t θ3

??

1 2 3 4 ordinal rankerr (x )

score function s(x )

1 2 3 4 target rank y

quantize s(x ) by someordered threshold θ commonly used in previous work:

threshold perceptrons (PRank, Crammer and Singer, 2002)

threshold hyperplanes (SVOR, Chu and Keerthi, 2005)

threshold ensembles (ORBoost, Lin and Li, 2006)

threshold model: r (x ) = min {k : s(x ) < θk}

(21)

## Key of Reduction: Associated Binary Queries

getting the rank using a threshold model

1 is s(x ) > θ1? Yes

2 is s(x ) > θ2? No

3 is s(x ) > θ3? No

4 is s(x ) > θ4? No

generally, how do we query the rank of a movie x ?

1 is movie x better than rank 1?Yes

2 is movie x better than rank 2?No

3 is movie x better than rank 3?No

4 is movie x better than rank 4?No associated binary queries:

is movie x better than rank k ?

(22)

Reduction from Ordinal Ranking to Binary Classification

## More on Associated Binary Queries

say, the machine uses g(x , k ) to answer the query

“is movie x better than rank k ?”

e.g. threshold model g(x , k ) = sign(s(x ) − θk) K − 1 binary classification problems w.r.t. each k

x x d d d t tt t ?? -

1 2 3 4 rg(x )

s(x )

1 2 3 4 y

N N θ1 Y Y Y Y YY Y YY

(z)1

θ1 g(x , 1)

N N N N N Y YY Y YY

(z)2

θ2 g(x , 2)

N N N N N N NNN YY

(z)3

θ3 g(x , 3)

let (x , k ), (z)k be binary examples (x , k ): extended input w.r.t. k -th query (z)k: desired binary answerY/N If g(x , k ) = (z)k for all k ,

we can compute rg(x )from g(x , k ) s.t. rg(x ) = y.

(23)

## Computing Ranks from Associated Binary Queries

when g(x , k ) answers “is movie x better than rank k ?”

Consider g(x , 1), g(x , 2), · · · , g(x , K −1), consistent predictions: (Y,Y,N,N,N,N,N) extracting the rank from consistent predictions:

minimum index searching: rg(x ) = min {k : g(x , k ) =N}

counting: rg(x ) = 1 +P

kJg (x , k ) =YK

two approaches equivalent for consistent predictions noisy/inconsistent predictions? e.g. (Y,N,Y,Y,N,N,Y)

counting: simpler to analyze and robust to noise

(24)

## The Counting Approach

Say y = 5, i.e., (z)1, (z)2, · · · , (z)7 = (Y,Y,Y,Y,N,N,N) if g1(x , k ) reports consistent predictions (Y,Y,N,N,N,N,N)

g1(x , k ) made 2 binary classification errors rg1(x ) = 3 by counting: the absolute cost is 2

absolute cost = # of binary classification errors

if g2(x , k ) reports inconsistent predictions (Y,N,Y,Y,N,N,Y) g2(x , k ) made 2 binary classification errors

rg2(x ) = 5 by counting: the absolute cost is 0

absolute cost ≤ # of binary classification errors If (z)k = desired answer & rg computed by counting,

y − rg(x ) ≤

K−1

P

k =1

q(z)k 6= g(x, k )y .

(25)

## Binary Classification Error v.s. Ordinal Ranking Cost

Say y = 5, i.e., (z)1, (z)2, · · · , (z)7 = (Y,Y,Y,Y,N,N,N) if g1(x , k ) reports consistent predictions (Y,Y,N,N,N,N,N)

g1(x , k ) made 2 binary classification errors rg1(x ) = 3 by counting: thesquared cost is 4

if g3(x , k ) reports consistent predictions (Y,N,N,N,N,N,N) g3(x , k ) made 3 binary classification errors

rg3(x ) = 2 by counting: thesquared cost is 9

now 1 binary classification error can introduce up to 5 more ordinal ranking cost—how to take this into account?

(26)

## Importance of Associated Binary Queries

(z)k Y Y Y Y N N N

g1(x , k ) Y Y N N N N N crg1(x ) = c[3] = 4 g3(x , k ) Y N N N N N N crg3(x ) = c[2] = 9

(w )k 7 5 3 1 1 3 5

(w )k

c[k + 1] − c[k ]

: the importance of (x , k ), (z)k per-example cost bound(Li and Lin, 2007; Lin, 2008):

forconsistent predictions or strongly ordinal costs

crg(x ) ≤

K−1

X

k =1

(w )kq(z)k 6= g(x, k )y

accurate binary predictions =⇒ correct ranks

(27)

## The Reduction Framework (1/2)

1 transform ordinal examples (xn,yn,cn)to weighted binary examples (xn,k ), (zn)k, (wn)k

2 use your favorite algorithm on the weighted binary examples and get K −1 binary classifiers (i.e., one big joint binary classifier) g(x , k )

3 for each new input x , predict its rank using rg(x ) = 1 +P

kJg (x , k ) =YK the reduction framework:

systematic & easy to implement







 ordinal examples (xn, yn, cn)





@ AA

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weighted binary examples

(xn, k), (zn)k,(wn)k

 k= 1, · · · , K −1

core

binary classification

algorithm

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associated binary classifiers

g(x, k) k= 1, · · · , K −1

AA

@









 ordinal

ranker rg(x)

(28)

## The Reduction Framework (2/2)

performance guarantee:

accurate binary predictions =⇒ correct ranks wide applicability:

works with any ordinalc & any binary classification algorithm simplicity:

mild computation overheads with O(NK ) binary examples up-to-date:

allows new improvements in binary classification to be immediately inherited by ordinal ranking







 ordinal examples (xn, yn, cn)





@ AA

%

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weighted binary examples

(xn, k), (zn)k,(wn)k

 k= 1, · · · , K −1

core

binary classification

algorithm

%

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associated binary classifiers

g(x, k) k= 1, · · · , K −1

AA

@









 ordinal

ranker rg(x)

(29)

## Theoretical Guarantees of Reduction (1/3)

is reduction a practical approach? YES!

error transformation theorem(Li and Lin, 2007)

Forconsistent predictions or strongly ordinal costs, if g makes test error ∆ in the induced binary problem, then rgpays test cost at most ∆ in ordinal ranking.

a one-step extension of the per-example cost bound conditions: general and minor

performance guarantee in the absolute sense:

accuracy in binary classification =⇒ correctness in ordinal ranking Is reduction reallyoptimal?

—what if the induced binary problem is “too hard”?

(30)

## Theoretical Guarantees of Reduction (2/3)

is reduction an optimal approach?YES!

regret transformation theorem(Lin, 2008)

For a general class ofordinal costs,

if g is -close to the optimal binary classifier g, then rgis -close to the optimal ordinal ranker r. error guarantee in the relative setting:

regardless of the absolute hardness of the induced binary prob., optimality in binary classification =⇒ optimality in ordinal ranking reduction does not introduce additional hardness

“reduction to binary” sufficient, but necessary?

i.e., is reduction aprincipled approach?

(31)

## Theoretical Guarantees of Reduction (3/3)

is reduction a principled approach? YES!

equivalence theorem(Lin, 2008)

For a general class ofordinal costs,

ordinal ranking is learnable by a learning model if and only if binary classification is learnable by the associated learning model.

a surprising equivalence:

ordinal ranking isas easy as binary classification reduction to binary classification:

practical, optimal, and principled

(32)

## Outline

1 Introduction to Machine Learning

2 The Ordinal Ranking Setup

3 Reduction from Ordinal Ranking to Binary Classification Algorithmic Usefulness of Reduction

Theoretical Usefulness of Reduction Experimental Performance of Reduction

4 Conclusion

(33)

## Unifying Existing Algorithms

ordinal ranking = reduction + cost + binary classification

ordinal ranking cost binary classification algorithm PRank absolute modified perceptron rule

(Crammer and Singer, 2002)

kernel ranking classification modified hard-margin SVM

(Rajaram et al., 2003)

SVOR-EXP classification modified soft-margin SVM SVOR-IMC absolute modified soft-margin SVM

(Chu and Keerthi, 2005)

(Lin and Li, 2006)

development and implementation time could have been saved e.g. correctness proof significantly simplified (PRank)

algorithmic structure revealed (SVOR, ORBoost) variants of existing algorithms can be designed quickly by tweaking reduction

(34)

## Designing New Algorithms Effortlessly

ordinal ranking = reduction + cost + binary classification ordinal ranking cost binary classification algorithm Reduction-C4.5 absolute standard C4.5 decision tree Reduction-SVM absolute standard soft-margin SVM SVOR (modified SVM) v.s. Reduction-SVM (standard SVM):

ban com cal cen

0 1 2 3 4 5 6

avg. training time (hour)

SVOR RED−SVM

advantages of core binary classification algorithm inherited in the new ordinal ranking one

(35)

## Designing New Algorithms Easily (1/2)

say, we have some ordinal rankers that predict your preference on movies:

r1(x ) = an ordinal ranker based on actor performance r2(x ) = an ordinal ranker based on actress performance r3(x ) = an ordinal ranker based on an expert opinion r4(x ) = an ordinal ranker based on box reports

no single ordinal ranker can explain your preference well, but a combination of them possibly can

ensemble learning:

how can machines combine simple functions to make complicated decisions?

previously: no good ensemble algorithm for ordinal ranking

(36)

## Designing New Algorithms Easily (2/2)

good ensemble alg. for bin. class.:

for t = 1, 2, · · · , T ,

1 find a simple gt that matches best with the current “view” of {(xn,yn)}

2 give a larger weight vt to gt if the match is stronger

3 update “view” by emphasizing the weights of those (xn,yn) that gt doesn’t predict well prediction:

majority vote of

vt,gt(x )

good ensemble alg. for ord. rank.:

for t = 1, 2, · · · , T ,

1 find a simplert that matches best with the current “view” of {(xn,yn)}

2 give a larger weight vt tort if the match is stronger

3 update “view” by emphasizing the costscnof those (xn,yn) that rt doesn’t predict well prediction:

weighted median of

= reduction + any cost + AdaBoost + math derivation

(37)

## Outline

1 Introduction to Machine Learning

2 The Ordinal Ranking Setup

3 Reduction from Ordinal Ranking to Binary Classification Algorithmic Usefulness of Reduction

Theoretical Usefulness of Reduction Experimental Performance of Reduction

4 Conclusion

(38)

## Proving New Generalization Theorems

Ordinal Ranking(Lin, 2008)

For AdaBoost.OR, with prob. > 1 − δ, expected test abs. cost of r

N1

N

X

n=1 K−1

X

k =1

q ¯ρ r (xn),yn,k ≤ Φy

| {z }

ambiguous training predictions w.r.t.

criteria Φ

+ O

 poly

 K ,log N

N,Φ1, q

log1δ



| {z }

deviation that decreases with stronger criteria or

more examples

Bin. Class. (Schapire et al., 1998)

For AdaBoost, with prob. > 1 − δ, expected test err. of g

N1

N

X

n=1

q ¯ρ g(xn),yn ≤ Φy

| {z }

ambiguous training predictions w.r.t.

criteria Φ

+ O

 poly



log N N,Φ1,

q log1δ



| {z }

deviation that decreases with stronger criteria or

more examples

new ordinal ranking theorem

= reduction + any cost + bin. thm. + math derivation

(39)

## Outline

1 Introduction to Machine Learning

2 The Ordinal Ranking Setup

3 Reduction from Ordinal Ranking to Binary Classification Algorithmic Usefulness of Reduction

Theoretical Usefulness of Reduction Experimental Performance of Reduction

4 Conclusion

(40)

## Reduction-C4.5 v.s. SVOR

pyr mac bos aba ban com cal cen

0 0.5 1 1.5 2 2.5

avg. test absolute cost

SVOR (Gauss)

RED−C4.5 C4.5: a (too) simple

binary classifier

—decision trees SVOR:

state-of-the-art ordinal ranking algorithm

even simple Reduction-C4.5 sometimes beats SVOR

(41)

## Reduction-SVM v.s. SVOR

pyr mac bos aba ban com cal cen

0 0.5 1 1.5 2 2.5

avg. test absolute cost

SVOR (Gauss)

RED−SVM (Perc.) SVM: one of the most

powerful binary classification algorithm SVOR:

state-of-the-art ordinal ranking algorithm extended from modified SVM

Reduction-SVM without modification often better than SVOR and faster

(42)

## Outline

1 Introduction to Machine Learning

2 The Ordinal Ranking Setup

3 Reduction from Ordinal Ranking to Binary Classification Algorithmic Usefulness of Reduction

Theoretical Usefulness of Reduction Experimental Performance of Reduction

4 Conclusion

(43)

## Conclusion

reduction framework:

not only simple, intuitive, and useful

but alsopractical, optimal, and principled algorithmic reduction:

take existing ordinal ranking algorithms asspecial cases design new and better ordinal ranking algorithmseasily theoretic reduction:

derivenew generalization guarantee of ordinal rankers superior experimental results:

better performance and faster training time reduction keeps ordinal ranking up-to-date with binary classification

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our reduction to boosting approaches results in significantly better ensemble ranking

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For consistent predictions or strongly ordinal costs, if g makes test error ∆ in the induced binary problem, then r g pays test cost at most ∆ in ordinal ranking. a one-step