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 J Observe 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) 6

Select Informative Matches Assume matches are expensive but computation is cheap Previous Work (Finkelstein) Use Ranking Algorithm to make better use of information Select 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 innovation Select matches based on Shannon’s mutual information 7

Related Work Sensor 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 8

Ranking Algorithms – Linear Model Each player has a skill level µ The probability that Player I beats Player J is a function of the difference µ i - µ j Transitive Use Maximum Likelihood Thurstone-Mosteller Model Q function Performance has Gaussian distribution about the mean µ Bradley-Terry Model Logistic function 9

Examples Elo’s chess ranking system Based on Bradley-Terry model Sagarin’s sports rankings 10

Mutual Information 11 Mutual Information: Conditional Mutual information

Entropy 12 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 O k be the outcome of the k th match At time k, scheduler chooses the pair (i k+1, j k+1 ): 13

Why use Mutual Information? Additive Property Fano’s Inequality Related entropy to probability of error For small error: Continuous distributions: MSE bounds differential entropy 14

Greedy is Not Optimal 15 Consider Huffman codes---Greedy is not optimal

Performance (MSE) 16

Performance (Gambling Penalty) 17

Identify correct ranking 18

Find strongest player 19

Find strongest player 20

Evaluating Goodness-of-Fit 21 Ranking: Inversions Skill Level Estimates: Mean squared error (MSE) Kullback-Leibler (KL) divergence (relative entropy) Others Betting risk Sampling inconsistency

Numerical Techniques Calculate mutual information Importance sampling Convex 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 Comparisons Bradley Terry Model: Examples: A hockey team scores Poisson- goals in a game Two cities compete to have the tallest person is the population

Computing Mutual Information 25 Importance Sampling: Multidimensional integral Probability distributions Skill level estimates Why is it good for estimating skill levels? –Faster than convex optimization –Efficient memory use Skill level of player 1 Skill level of player 2

Results 26 (for a 10 player tournament and100 experiments)

Visualizing the Algorithm 27 PlayerABCD A0233 B0072 C0205 D1220 ABCD A B C D AB C D ? Outcomes Scheduling