Rank Aggregation

1. Rank Aggregation

2. Rank Aggregation: Settings • Multiple items • Web-pages, cars, apartments,…. • Multiple scores for each item • By different reviewers, users, according to different features… • Some aggregation function on the scores • Sum, Average, Max… • Goal: compute the top-k items

3. Rank Aggregation Example

4. Naïve Algorithm • Compute the aggregated rank for all items • Find the best one, then the second best one… the k best one • Good for small-scale problems • Still not feasible for web scales…

5. Can we do any better? • An assumption to help us: each individual list comes sorted • Reasonable for search engines, user rankings… • Another assumption: monotonicity of the aggregation function • Now can we do any better?

6. Fagin's algorithm (FA) • Do sorted access on all lists in parallel • For every item do random access to the other lists to fetch all of its values • Stop when at least k items were seen (in the sorted access) in all lists • Sort the list • Why is this enough?

7. Example Beauty Comfort Average

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12. Example (top-3) Beauty Comfort Average How do we know not to look further?

13. Complexity • Probabilistic analysis on the order of items can be used to show better bounds (with good probability) • Can we do even better?

14. Cost model • This is a very simple settings so we can define a finer cost model than worst case complexity • In a web context it is important to do so • Since the scale is huge • We associate some cost Cs with every sorted access , and some cost Cr with every random access • Denote the cost for algorithm A on input instance I by cost(A,I)

15. Instance-optimality • An algorithm A is instance-optimal if for every input instance I, cost(A,I) = O(cost(A',I)) for every algorithm A' • A very strong notion • But we can realize it here!

16. Threshold Algorithm (TA) • Idea: sometimes we can stop before seeing k objects in every list • Use a threshold on how good can a score of an unseen object be. • Based on aggregating the minimal score seen so far in all lists

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18. Example Beauty Comfort Average T=9.5

19. Example Beauty Comfort Average T=9.5

20. Example Beauty Comfort Average T=7

21. Example Beauty Comfort Average T=4 One step less!

22. Instance-optimality • Theorem: If the aggregation function is strictly monotone and every two items in a list have distinct grades, then TA is instance-optimal • Intuition: If an algorithm stops on input I before reaching the threshold, then we can design an input I' on which it is wrong, by changing values it did not see • TA sees at most K items more than any algorithm on any input • Strict monotonicity is needed to avoid "lucky guesses" in breaking ties • Thm. In general no instance-optimal algorithm exists • Theorem: TA is instance-optimal against all algorithms that do not "guess" • i.e. do not do random access to an item they did not see in sorted access

23. Restricted Sorted Access • Some rankings are not available as sorted • E.g. distances from a map site • Then we can revise TA to do sorted access only on the list where it is possible • And still instance-optimal! (Against algorithms that work under the same restrictions, of course)

24. No Random Access • Maintain bottom and upper bounds for every item (worst and best grades) • Best is the aggregation of what we have seen and the worst we have seen in every list, Worst is the aggregation with what we have seen and zeros • Keep in the list those with top-K "worst" grades • Break ties by "best" grades • Halt if we have k items in the list, and the best grade for every item out of the list is less than the k'th in the list

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31. Example Beauty Comfort Average Score(D)<3