Preference Elicitation for Single Crossing Domain

Transcription

1 Preference Elicitation for Single Crossing Domain joint work with Neeldhara Misra (IIT Gandhinagar) March 6, 2017 Appeared in IJCAI 2016

2 Motivation for Preference Elicitation One often wants to learn how people like different items Netflix would like to know which movies their users like more Zomato is interested in knowing which are popular restaurants Wikipedia would like to know quality of their articles using user feedback In any voting applications, one first have to know voters preferences

3 Motivation for Preference Elicitation One often wants to learn how people like different items Netflix would like to know which movies their users like more Zomato is interested in knowing which are popular restaurants Wikipedia would like to know quality of their articles using user feedback In any voting applications, one first have to know voters preferences Preference elicitation is fundamental in many applications

4 General Setting: Preferences Set C of alternatives. C = m Set V of agents. V = n Every agent has preference over C Preference single alternatives subset of C ranking over C... In this talk: Preferences are rankings over C

5 The Comparison Query Model We often have a huge number of alternatives in applications Movies in Netflix: tens of thousands Restaurants listed in Zomato: thousands Wikipedia pages: millions

6 The Comparison Query Model We often have a huge number of alternatives in applications Movies in Netflix: tens of thousands Restaurants listed in Zomato: thousands Wikipedia pages: millions Users cannot provide their preferences over such a large set

7 The Comparison Query Model We often have a huge number of alternatives in applications Movies in Netflix: tens of thousands Restaurants listed in Zomato: thousands Wikipedia pages: millions Users cannot provide their preferences over such a large set Users can easily compare two given alternatives! Goal: Learn preferences using comparisons only

8 Single crossing domain: Opinions about subsidy Low income Wants subsidy Voters High income Does not want subsidy

9 Single crossing domain: Opinions about subsidy Voters Low income High income Wants subsidy Does not want subsidy (x, y) C C voters with x y are contiguous

10 Single crossing domain: Opinions about subsidy Voters Low income High income Wants subsidy Does not want subsidy (x, y) C C voters with x y are contiguous Properties Always exists a Condorcet winner for odd number of voters Many computational problems, Kemeny aggregation for example, are tractable

11 Single crossing domain: Opinions about subsidy Voters Low income High income Wants subsidy Does not want subsidy (x, y) C C voters with x y are contiguous Properties Always exists a Condorcet winner for odd number of voters Many computational problems, Kemeny aggregation for example, are tractable Question: Does preference elicitation get easier? Yes!

12 Summary of results Ordering Access model Upper Bound Query Complexity Lower Bound Random O(m 2 log n) Ω(m log m + m log n) Known Sequential single crossing order O(mn + m 2 ) Sequential any order O(mn + m 2 log n) Ω(m log m + mn) Unknown Sequential any order O(mn + m 3 log m) Random O(mn + m 3 log m) Ω(m log m + mn)

13 Random access + Known single crossing order

14 Algorithm v 1, v 2,..., v n be a single crossing order For every x, y C, find i such that x i y and y i+1 x Query complexity: O(m 2 log n)

15 Algorithm v 1, v 2,..., v n be a single crossing order For every x, y C, find i such that x i y and y i+1 x Query complexity: O(m 2 log n) Lower Bound Let C = {c 1, c 2,..., c m }, m even Fix {c 1, c 2 } {c 3, c 4 } {c m 1, c m } for everyone

16 Algorithm v 1, v 2,..., v n be a single crossing order For every x, y C, find i such that x i y and y i+1 x Query complexity: O(m 2 log n) Lower Bound Let C = {c 1, c 2,..., c m }, m even Fix {c 1, c 2 } {c 3, c 4 } {c m 1, c m } for everyone Each {c 2i, c 2i+1 } needs to be queried log n times Query complexity: Ω(m log n)

17 Sequential access in single crossing order

18 Algorithm Use insertion sort Elicit the first preference: O(m log m) comparisons Elicit i th preference using insertion sort in (i 1) th preference order

19 Algorithm Use insertion sort Elicit the first preference: O(m log m) comparisons Elicit i th preference using insertion sort in (i 1) th preference order Total number of comparisons Good cost (x, y) good if i th preference orders x, y same as (i 1) th Bad cost (x, y) bad if (x, y) not good

20 Algorithm Use insertion sort Elicit the first preference: O(m log m) comparisons Elicit i th preference using insertion sort in (i 1) th preference order Total number of comparisons Good cost (x, y) good if i th preference orders x, y same as (i 1) th Bad cost (x, y) bad if (x, y) not good Single crossing 1 bad cost per pair of candidates

21 Algorithm Use insertion sort Elicit the first preference: O(m log m) comparisons Elicit i th preference using insertion sort in (i 1) th preference order Total number of comparisons Good cost (x, y) good if i th preference orders x, y same as (i 1) th Bad cost (x, y) bad if (x, y) not good Single crossing 1 bad cost per pair of candidates Query Complexity: O ( mn + ( )) m 2

22 Lower Bound Let C = {c 1, c 2,..., c m }, m even Fix c 1 c 2 c m for everyone Each {c 2i, c 2i+1 } needs to be queried for every agent times Query complexity: Ω(mn + m log m)

23 Sequential access in any order + Known single crossing order

24 Algorithm Again use insertion sort Elicit the first preference: O(m log m) comparisons Maintain a set S of already elicited preferences Elicit current preference using insertion sort in closest known order

25 Algorithm Again use insertion sort Elicit the first preference: O(m log m) comparisons Maintain a set S of already elicited preferences Elicit current preference using insertion sort in closest known order Total number of comparisons Good cost (x, y) good if current preference orders x, y same as baseline Bad cost (x, y) bad if (x, y) not good

26 Algorithm Again use insertion sort Elicit the first preference: O(m log m) comparisons Maintain a set S of already elicited preferences Elicit current preference using insertion sort in closest known order Good cost (x, y) good if current preference orders x, y same as baseline Total number of comparisons Bad cost (x, y) bad if (x, y) not good Single crossing log n bad cost per pair of candidates

27 Algorithm Again use insertion sort Elicit the first preference: O(m log m) comparisons Maintain a set S of already elicited preferences Elicit current preference using insertion sort in closest known order Good cost (x, y) good if current preference orders x, y same as baseline Total number of comparisons Bad cost (x, y) bad if (x, y) not good Single crossing log n bad cost per pair of candidates Query Complexity: O ( mn + ( ) ) m 2 log n

28 Sequential access in any order + Unknown single crossing order

29 Algorithm: any sequential + unknown single crossing order There can be at most ( m 2 ) + 1 distinct preference

30 Algorithm: any sequential + unknown single crossing order There can be at most ( m 2 ) + 1 distinct preference Let S be single crossing of distinct preferences Then there exists two alternatives x, y such that exactly S /2 prefers x over y

31 Algorithm: any sequential + unknown single crossing order There can be at most ( m 2 ) + 1 distinct preference Let S be single crossing of distinct preferences Then there exists two alternatives x, y such that exactly S /2 prefers x over y Let S be single crossing of distinct preferences It can be checked in O(log m + m) comparisons whether a preference is in S

32 Algorithm: any sequential + unknown single crossing order... Elicit the first preference: O(m log m) comparisons

33 Algorithm: any sequential + unknown single crossing order... Elicit the first preference: O(m log m) comparisons Maintain the set S of preferences found so far

34 Algorithm: any sequential + unknown single crossing order... Elicit the first preference: O(m log m) comparisons Maintain the set S of preferences found so far Check if i th preference is in S: O(m)

35 Algorithm: any sequential + unknown single crossing order... Elicit the first preference: O(m log m) comparisons Maintain the set S of preferences found so far Check if i th preference is in S: O(m) If not then elicit: O(m log m)

36 Algorithm: any sequential + unknown single crossing order... Elicit the first preference: O(m log m) comparisons Maintain the set S of preferences found so far Check if i th preference is in S: O(m) If not then elicit: O(m log m) This can happen O (( m 2 )) times

37 Algorithm: any sequential + unknown single crossing order... Elicit the first preference: O(m log m) comparisons Maintain the set S of preferences found so far Check if i th preference is in S: O(m) If not then elicit: O(m log m) This can happen O (( m 2 )) times Total number of comparisons: O(mn + m 3 log m)

38 Conclusions Summary Optimal algorithm for preference elicitation for various realistic scenarios

39 Conclusions Summary Optimal algorithm for preference elicitation for various realistic scenarios Future work Find exact query complexity for preference elicitation when random access is given and a single crossing order is known How query complexity gets affected in the presence of k outliers

40 Thank You!