Ranking from Pairwise Comparisons

Sewoong Oh

Department of ISEUniversity of Illinois at Urbana-Champaign

Joint work with Sahand Negahban(MIT) and Devavrat Shah(MIT)

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Rank Aggregation

AlgorithmData Decision

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Score Ranking

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Example

AlgorithmData Decision

kittenwar.com3 / 19

Example

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AlgorithmData Decision

kittenwar.com3 / 19

Example

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AlgorithmData Decision

kittenwar.com3 / 19

Example

AlgorithmData Decision

www.allourideas.org4 / 19

Example

AlgorithmData Decision

www.allourideas.org4 / 19

Comparisons data

Group decisions, recommendation and advertisement, sports and game

Universal (ratings can be converted into comparisons)

Consistent and reliable

Natural (e.g. Sports and games, MSR’s TrueSkill)

Aggregation is challenging

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NP-hard approach: Kemeny optimal algorithm

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3 2

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a12a21

Find the most consistent ranking

arg minσ

∑i 6=j

aijI(σ(j) > σ(i))

NP-hard combinatorial optimization

Optimal for a specific model

Strong Weak

1 2 3 4 5 76

4 5

54

w.p. 1− p

w.p. p

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Traditional approach: `1 Ranking

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3 2

4

56

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a12a21

Compute score:

s(i) =1

|∂i|∑j∈∂i

ajiaij + aji

n: number of items

m: number of samples

Compute in O(m) time

Works when everyone plays everyone else:m = Ω(n2) [Ammar, Shah ’12]

Strong Weak

1 .9 .8 .7 .6 .4 .31 .2 .1 0

All wins are equally weighted

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Traditional approach: `1 Ranking

1

3 2

4

56

7

a12a21

Compute score:

s(i) =1

|∂i|∑j∈∂i

ajiaij + aji

n: number of items

m: number of samples

Compute in O(m) time

Works when everyone plays everyone else:m = Ω(n2) [Ammar, Shah ’12]

Strong Weak

1 .9 .8 .7 .6 .4 .31 .2 .1 0

All wins are equally weighted

7 / 19

Traditional approach: `1 Ranking

1

3 2

4

56

7

a12a21

Compute score:

s(i) =1

|∂i|∑j∈∂i

ajiaij + aji

n: number of items

m: number of samples

Compute in O(m) time

Works when everyone plays everyone else:m = Ω(n2) [Ammar, Shah ’12]

Strong Weak

1 .9 .8 .7 .6 .4 .31 .2 .1 0

All wins are equally weighted

7 / 19

Traditional approach: `1 Ranking

1

3 2

4

56

7

a12a21

Compute score:

s(i) =1

|∂i|∑j∈∂i

ajiaij + aji

s(j)

n: number of items

m: number of samples

Compute in O(m) time

Works when everyone plays everyone else:m = Ω(n2) [Ammar, Shah ’12]

Strong Weak

1 .9 .8 .7 .6 .4 .31 .2 .1 0

All wins are equally weighted

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Prior work

AlgorithmData Decision

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6

5

Score Ranking

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/`1 ranking`p ranking

Kemeny optimal

MNL

Mixed MNL

etc.

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Challenges

High-dimensional regime

# of samples ∝ n(log n)

Low computational complexity

# of operations ∝ # of samples

Model independent

Makes sense

Under BTL model, Rank Centrality achieves optimal performance

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Challenges

High-dimensional regime

# of samples ∝ n(log n)

Low computational complexity

# of operations ∝ # of samples

Model independent

Makes sense

Under BTL model, Rank Centrality achieves optimal performance

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Rank Centrality

1

3 2

4

56

7

a12a21

Define a random walk on G

Pij =1

dmax

aijaij + aji

Pii = 1−1

dmax

∑j 6=i

aijaij + aji

(unique) stationary distribution

sTP = sT

Random walk spends more time on ‘stronger’ nodes

Higher score for beating a ‘stronger’ node

s(i) =(

1− 1dmax

∑j 6=i

aijaij + aji

)s(i) +

∑j 6=i

Pjis(j)

=1

Zi

∑j 6=i

ajiaij + aji

s(j)

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Rank Centrality

1

3 2

4

56

7

a12a21

Define a random walk on G

Pij =1

dmax

aijaij + aji

Pii = 1−1

dmax

∑j 6=i

aijaij + aji

(unique) stationary distribution

sTP = sT

Random walk spends more time on ‘stronger’ nodes

Higher score for beating a ‘stronger’ node

s(i) =(

1− 1dmax

∑j 6=i

aijaij + aji

)s(i) +

∑j 6=i

Pjis(j)

=1

Zi

∑j 6=i

ajiaij + aji

s(j)

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Rank Centrality

1

3 2

4

56

7

a12a21

Define a random walk on G

Pij =1

dmax

aijaij + aji

Pii = 1−1

dmax

∑j 6=i

aijaij + aji

(unique) stationary distribution

sTP = sT

Random walk spends more time on ‘stronger’ nodes

Higher score for beating a ‘stronger’ node

s(i) =(

1− 1dmax

∑j 6=i

aijaij + aji

)s(i) +

∑j 6=i

Pjis(j)

=1

Zi

∑j 6=i

ajiaij + aji

s(j)

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Rank Centrality

1

3 2

4

56

7

a12a21

Define a random walk on G

Pij =1

dmax

aijaij + aji

Pii = 1−1

dmax

∑j 6=i

aijaij + aji

(unique) stationary distribution

sTP = sT

Random walk spends more time on ‘stronger’ nodes

Higher score for beating a ‘stronger’ node

s(i) =(

1− 1dmax

∑j 6=i

aijaij + aji

)s(i) +

∑j 6=i

Pjis(j)

=1

Zi

∑j 6=i

ajiaij + aji

s(j)

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Rank Centrality

1

3 2

4

56

7

a12a21

Define a random walk on G

Pij =1

dmax

aijaij + aji

Pii = 1−1

dmax

∑j 6=i

aijaij + aji

(unique) stationary distribution

sTP = sT

Random walk spends more time on ‘stronger’ nodes

Higher score for beating a ‘stronger’ node

s(i) =(

1− 1dmax

∑j 6=i

aijaij + aji

)s(i) +

∑j 6=i

Pjis(j)

=1

Zi

∑j 6=i

ajiaij + aji

s(j)

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Rank Centrality

1

3 2

4

56

7

a12a21

Define a random walk on G

Pij =1

dmax

aijaij + aji

Pii = 1−1

dmax

∑j 6=i

aijaij + aji

(unique) stationary distribution

sTP = sT

Random walk spends more time on ‘stronger’ nodesHigher score for beating a ‘stronger’ node

s(i) =(

1− 1dmax

∑j 6=i

aijaij + aji

)s(i) +

∑j 6=i

Pjis(j)

=1

Zi

∑j 6=i

ajiaij + aji

s(j)

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Experiment: Polling public opinionsWashington Post - allourideas

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Experiment: PollingGround truth: what the algorithm produces with complete dataError = 1n

∑i(σi − σ̂i)

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L1 rankingRank Centrality

Error

Sampling rate12 / 19

ModelAlgorithm is model independentFor experiments and analysis, need model to generate dataComparisons model

I Bradley-Terry-Luce (BTL) modelthere is a true ranking {wi}when a pair is compared, the noise is modelled by

P (i < j) =wj

wi + wj

Sampling modelI Sample each pair with probability d/nI k comparisons for each pair

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k1

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k − 3

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ModelAlgorithm is model independentFor experiments and analysis, need model to generate dataComparisons model

I Bradley-Terry-Luce (BTL) modelthere is a true ranking {wi}when a pair is compared, the noise is modelled by

P (i < j) =wj

wi + wj

Sampling modelI Sample each pair with probability d/nI k comparisons for each pair

1

3 2

4

56

7

d

k1

3 2

4

56

7

k − 3

3

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Experiment: BTL model

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d

k

Error= 1‖w‖∑

i>j(wi − wj)2I((σ̂i−σ̂j)(wi−wj)>0)

0.0001

0.001

0.01

0.1

1 10 100

Ratio MatrixL1 ranking

Rank CentralityML estimateError

k

0.001

0.01

0.1

0.1 1

Ratio MatrixL1 ranking

Rank CentralityML estimateError

d/n

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Performance guarantee

For d = Ω(log n)

BTL model {wi} with wmin = Θ(wmax)

Theorem (Negahban, O., Shah, ’12)

Rank Centrality achieves

‖w − s‖‖w‖

≤ C√

log n

k d

Information-theoretic lower bound:

infs

supw∈W

‖w − s‖‖w‖

≥ C ′√

1

k d

# of samples = O(n log n) sufficies to achieve arbitrary small error

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Performance guarantee

For d = Ω(log n)

BTL model {wi} with wmin = Θ(wmax)

Theorem (Negahban, O., Shah, ’12)

Rank Centrality achieves

‖w − s‖‖w‖

≤ C√

log n

k d

Information-theoretic lower bound:

infs

supw∈W

‖w − s‖‖w‖

≥ C ′√

1

k d

# of samples = O(n log n) sufficies to achieve arbitrary small error

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Performance guarantee for general graphs

Oftentimes we do not control how data is collected

Let G denote the (undirected) graph of given data

Compute spectral gap of G (cf. mixing time of natural random walk)

ξ ≡ 1− λ1(G)λ2(G)

Theorem (Negahban, O., Shah, ’12)

Rank Centrality achieves

‖w − s‖‖w‖

≤ C dmaxξ dmin

√log n

k dmax

# of samples = O(n log n) sufficies to achieve arbitrary small error

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Performance guarantee for general graphs

Oftentimes we do not control how data is collected

Let G denote the (undirected) graph of given data

Compute spectral gap of G (cf. mixing time of natural random walk)

ξ ≡ 1− λ1(G)λ2(G)

Theorem (Negahban, O., Shah, ’12)

Rank Centrality achieves

‖w − s‖‖w‖

≤ C dmaxξ dmin

√log n

k dmax

# of samples = O(n log n) sufficies to achieve arbitrary small error

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Proof technique

1. Spectral analysis of reversible Markov chainsMarkov chain (P, π) is revisible iff π(i)Pij = π(j)Pji

Pij =1

dmax

aijaij + aji

is not reversible, but the expectation π̃(i)P̃ij = π̃(j)P̃ji

P̃ij =1

dmax

wjwi + wj

π̃(i) ∝ wi

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Proof technique

1. Spectral analysis of reversible Markov chainsFor any Markov chain (P, π) and any reversible MC (P̃ , π̃)

‖π − π̃‖ = ‖P Tπ − P̃ T π̃‖≤ ‖P Tπ − P T π̃‖+ ‖P T π̃ − P̃ T π̃‖≤ ‖P̃ T (π − π̃)‖︸ ︷︷ ︸

Reversible

+‖(P − P̃ )T (π − π̃)‖+ ‖P − P̃‖2 ‖π̃‖

≤(λ2(P̃ ) + ‖P − P̃‖2

)‖π − π̃‖+ ‖P − P̃‖2 ‖π̃‖

‖π − π̃‖‖π̃‖

≤ ‖P − P̃‖21− λ2(P̃ )− ‖P − P̃‖2

2. Bound 1− λ2(P̃ ) by comparisons theorem3. Bound ‖P − P̃‖2 by concentration of measure inequality for matrices

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Proof technique

1. Spectral analysis of reversible Markov chainsFor any Markov chain (P, π) and any reversible MC (P̃ , π̃)

‖π − π̃‖ = ‖P Tπ − P̃ T π̃‖≤ ‖P Tπ − P T π̃‖+ ‖P T π̃ − P̃ T π̃‖≤ ‖P̃ T (π − π̃)‖︸ ︷︷ ︸

Reversible

+‖(P − P̃ )T (π − π̃)‖+ ‖P − P̃‖2 ‖π̃‖

≤(λ2(P̃ ) + ‖P − P̃‖2

)‖π − π̃‖+ ‖P − P̃‖2 ‖π̃‖

‖π − π̃‖‖π̃‖

≤ ‖P − P̃‖21− λ2(P̃ )− ‖P − P̃‖2

2. Bound 1− λ2(P̃ ) by comparisons theorem3. Bound ‖P − P̃‖2 by concentration of measure inequality for matrices

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Conclusion

Rank aggregation from comparisons

High-dimensional regime nkd ∼ n log nLow complexity O(nkd log n)

Model independent

Optimal under BTL up to log n

Other Markov chain approaches e.g.”Rank Aggregation Revisited”C. Dwork, R. Kumar, M. Naor, and D. Sivakumar

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