Mutual Information Scheduling for Ranking

Hamza Aftab

Nevin Raj

Paul Cuff

Sanjeev Kulkarni

Adam Finkelstein

1

Applications of Ranking

2

Pair-wise Comparisons

3

Query:A > B ?Ask a voter whether candidate I is better

than candidate JObserve the outcome of a match

Scheduling

4

Design queries dynamically, based on past observations.

Example: Kitten Wars

5

Example: All Our Ideas(Matthew Salganik – Princeton)

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Select Informative MatchesAssume matches are expensive but computation is

cheap

Previous Work (Finkelstein)Use Ranking Algorithm to make better use of informationSelect matches by giving priority based on two criterion

Lack of information: Has a team been in a lot of matches already?Comparability of the match: Are the two teams roughly equal in

strength?

Our innovationSelect matches based on Shannon’s mutual information

7

Related WorkSensor Management (tracking)

Information-Driven [Manyika, Durrant-Whyte 1994][Zhao et. al. 2002] – Bayesian filtering[Aoki et. al. 2011] – This session

Learning Network Topology[Hayek, Spuckler 2010]

Noisy Sort

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Ranking Algorithms – Linear ModelEach player has a skill level µThe probability that Player I beats Player J is a

function of the difference µi - µj

TransitiveUse Maximum Likelihood

Thurstone-Mosteller ModelQ function

Performance has Gaussian distribution about the mean µ

Bradley-Terry ModelLogistic function

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ExamplesElo’s chess ranking system

Based on Bradley-Terry modelSagarin’s sports rankings

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Mutual Information

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Mutual Information:

Conditional Mutual information

Entropy

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Entropy:

Conditional Entropy

High entropy Low entropy

Mutual Information Scheduling

Let R be the information we wish to learn(i.e. ranking or skill levels)

Let Ok be the outcome of the kth match

At time k, scheduler chooses the pair (ik+1, jk+1):

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Why use Mutual Information?Additive Property

Fano’s InequalityRelated entropy to probability of errorFor small error:

Continuous distributions: MSE bounds differential entropy

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Greedy is Not Optimal

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Consider Huffman codes---Greedy is not optimal

Performance (MSE)

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Performance (Gambling Penalty)

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Identify correct ranking

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Find strongest player

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Find strongest player

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Evaluating Goodness-of-Fit

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Ranking: Inversions

Skill Level Estimates:Mean squared error (MSE)Kullback-Leibler (KL) divergence

(relative entropy)Others

Betting riskSampling inconsistency

1234

1

4

Xx xq

xpxpqpD

)(

)(log)()||(

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Numerical Techniques

Calculate mutual informationImportance samplingConvex Optimization (tracking of ML

estimate)

Summary of Main Idea

Get the most out of measurements for estimating a ranking

Schedule each match to maximize(Greedy, to make the computation tractable)Flexible

S is any parameter of interest, discrete or continuous(skill levels; best candidate; etc.)

Simple design---competes well with other heuristics

Ranking Based on Pair-wise ComparisonsBradley Terry Model:

Examples:A hockey team scores Poisson- goals in a

gameTwo cities compete to have the tallest person

is the population

Computing Mutual Information

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Importance Sampling:Multidimensional integralProbability distributionsSkill level estimates

• Why is it good for estimating skill levels?– Faster than convex optimization– Efficient memory use

Skill level of player 1

Ski

ll le

vel o

f pl

ayer

2

0 100 200 300 400 5000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Number of games

Ave

rage

num

ber

of in

vers

ions

ELOTrueSkillRandom SchedulingMinGames/ClosestSkillMutual InformationGraph Based

Results

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(for a 10 player tournament and100 experiments)

220 230 240 250 260 270 2800

0.1

0.2

0.3

0.4

0.5

Number of games

Aver

age

num

ber o

f inv

ersi

ons

20 30 40 50 60 70

0.3

0.4

0.5

0.6

0.7

Number of games

Ave

rage

num

ber

of in

vers

ions

Visualizing the Algorithm

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Player A B C D

A 0 2 3 3

B 0 0 7 2

C 0 2 0 5

D 1 2 2 0

Player A B C D

A 0 0.031 0.025 0.024

B 0.031 0 0.023 0.033

C 0.025 0.023 0 0.030

D 0.024 0.033 0.030 0

A B

C D

?

Outcomes

Scheduling