Fairness in Ranking under Disparate Uncertainty

Transcript

1. October 29, 2024 Fairness in Ranking under Disparate Uncertainty Richa Rastogi, Thorsten Joachims
2. Motivation College Admissions Housing Loans Hiring Clinical Trials
3. A First Example ri ∼ Bernoulli(0.5) ri ∼ Bernoulli(0.9) ri ∼ Bernoulli(0.1) 1000 candidates Expected relevant candidates 0.5*2000 = 1000 Expected relevant candidates = 0.1*1000 + 0.9*1000 = 1000 1000 candidates 2000 candidates Select 100 candidates Only candidates selected from Group A Group B
4. Research Question How does disparate uncertainty between groups of options induce un-fairness in selection processes ? How can we mitigate this unfairness ?
5. Model • Ranking policy — computes a ranking of candidates • Binary Relevance of each candidate • Relevance can only be revealed through a human decision maker (Principal) • Policy has access to a predictive model of relevance , trained on prior human decisions and candidate features • The principal goes through the ranking from the top to some position • Goal of the principal — fi nd as many relevant candidates as possible π σ n ri ∈ {0,1} i ri π pi = ℙ(ri | 𝒟 ) 𝒟 σ k
6. Probability Ranking Principle 0.7 0.9 0.9 0.1 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.5 0.5 0.5 0.5 0.5 0.5 Group A Group B Select top 3 candidates
7. Probability Ranking Principle 0.9 σPRP How can we quantify unfairness due to disparate uncertainty ? 0.5 0.5 0.5 0.5 0.5 0.5 0.1 0.05 0.05 0.05 0.05 0.05 0.9 0.7 0.7 0.9 0.9 0.05 0.05 0.05 Group A Group B Select top 3 candidates
8. Relevant Candidates nRel(g|πk ) = ∑ i∈g ℙ(i ∈ πk )ℙ(ri = 1| 𝒟 ) nRel(A|σPRP 3 ) = 2.5 nRel(B|σPRP 3 ) = 0.0 σPRP 0.7 0.9 0.9 nRel(g) = ∑ i∈g ℙ(ri = 1| 𝒟 ) Expected number of relevant candidates nRel(B) = 3.0 nRel(A) = 3.0
9. Cost of Opportunity Uncertainty of the predictive model imposes costs on the groups and the principal Cost Burden on candidate groups Cost of missing out on the opportunity to be selected when truly relevant Cost Burden on the principal Cost of missing a relevant candidate Cost: Expected fraction of relevant candidates that don’t get selected (overall for the principal, or for each group)
10. Probability Ranking Principle Quali fi ed, Unquali fi ed 0.17 Cost Burden Cost Burden 1.0 Principal Cost Burden 0.58 low cost burden on the principal but unfair nRel(A|σPRP 3 ) nRel(A) = 2.5 3.0 = 0.83 nRel(B|σPRP 3 ) nRel(B) = 0.0 0.9 0.5 0.5 0.5 0.5 0.5 0.5 0.1 0.05 0.05 0.05 0.05 0.05 0.9 0.7 0.05 0.05 0.05 σPRP 0.7 0.9 0.9 Group A Group B nRel(B) = 3.0 nRel(A) = 3.0 nRel(A|σPRP 3 ) + nRel(B|σPRP 3 ) nRel(A) + nRel(B) = 2.5 6.0 = 0.42
11. Demographic Parity in Rankings 0.7 0.9 0.9 0.1 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.5 0.5 0.5 0.5 0.5 0.5 Group A Group B
12. Demographic Parity in Rankings σDP 0.9 0.5 0.5 0.5 0.5 0.5 0.5 0.1 0.05 0.05 0.05 0.05 0.05 0.9 0.7 0.5 0.9 0.9 0.05 0.05 0.05 Group A Group B
13. Demographic Parity in Rankings σDP 0.9 0.5 0.5 0.5 0.5 0.5 0.5 0.1 0.05 0.05 0.05 0.05 0.05 0.9 0.7 0.5 0.9 0.9 0.05 0.05 0.05 0.4 Cost Burden Cost Burden 0.83 Principal Cost Burden 0.62 Still unfair nRel(A|σDP 3 ) nRel(A) = 1.8 3.0 = 0.6 nRel(B|σunif 3 ) nRel(B) = 0.5 3.0 = 0.17 Quali fi ed, Unquali fi ed Group A Group B nRel(B) = 3.0 nRel(A) = 3.0 nRel(A|σPRP 3 ) + nRel(B|σPRP 3 ) nRel(A) + nRel(B) = 2.3 6.0 = 0.38
14. A possible Solution- Uniform Policy ? 0.7 0.9 0.9 0.1 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.5 0.5 0.5 0.5 0.5 0.5 Group A Group B
15. Uniform Policy 0.83 Cost Burden Cost Burden 0.83 Principal Cost Burden 0.83 Fair and still high cost πunif 0.7 0.9 0.9 0.1 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.5 0.5 0.5 0.5 0.5 0.5 Group A Group B nRel(B) = 3.0 nRel(A) = 3.0 Equality of lottery, Ben Saunders, 2008
16. Research Question How does disparate uncertainty between groups of options induce un-fairness in rankings ? Can we learn policy that is as fair as random lottery but more e ff ective ?
17. Fair Lottery Axiom of Fair Lottery: In expectation an equal fraction of the relevant candidates from each group are selected in the top-k subsets fo r every pre fi x of the ranking. πk k
18. Fair Lottery ∀k nRel(A|σk ) nRel(A) = nRel(B|σk ) nRel(B) Selected relevant candidates in expectation from group A Selected relevant candidates in expectation from group B Total relevant candidates in expectation from group A Total relevant candidates in expectation from group B Equivalent to group-wise fair lottery among relevant candidates nRel(B) = 3.0 nRel(A) = 3.0 Pick relevant candidates from both groups in expectation l nRel(B) = 3.0 nRel(A) = 6.0 Pick relevant candidates from group A and B in expectation 2l : l 0.7 0.9 0.9 0.1 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.5 0.5 0.5 0.5 0.5 0.5
19. Equality of Opportunity (EOR) Fairness Selected relevant candidates in expectation from group A Selected relevant candidates in expectation from group B Total relevant candidates in expectation from group A Total relevant candidates in expectation from group B nRel(B) = 3.0 nRel(A) = 3.0 0.7 0.9 0.9 0.1 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.5 0.5 0.5 0.5 0.5 0.5 -EOR-Fair Ranking Policy δ ∀k nRel(A|σk ) nRel(A) − nRel(B|σk ) nRel(B) ≤ δ
20. EOR Criterion Selected relevant candidates in expectation from group A Selected relevant candidates in expectation from group B Total relevant candidates in expectation from group A Total relevant candidates in expectation from group B nRel(B) = 3.0 nRel(A) = 3.0 0.7 0.9 0.9 0.1 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.5 0.5 0.5 0.5 0.5 0.5 δ(σk ) = ∑ i∈A∩σk ℙ(ri | 𝒟 ) ∑ i∈A ℙ(ri | 𝒟 ) − ∑ i∈B∩σk ℙ(ri | 𝒟 ) ∑ i∈B ℙ(ri | 𝒟 ) 0.5 δ(σ1 ) = − 0.5 3.0 = − 0.17 0.9 δ(σ2 ) = 0.9 − 0.5 3.0 = 0.13 0.5 δ(σ3 ) = 0.9 − 1.0 3.0 = − 0.03
21. EOR Policy 0.7 0.9 0.9 0.1 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.5 0.5 0.5 0.5 0.5 0.5 Group A Group B
22. EOR Policy σEOR 0.7 0.9 0.9 0.1 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.5 0.5 0.5 0.5 0.5 0.5 Group A Group B 0.5 0.5 0.9
23. EOR Policy σEOR 0.7 0.9 0.9 0.1 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.5 0.5 0.5 0.5 0.5 0.5 Group A Group B 0.7 Cost Burden Cost Burden 0.67 Principal Cost Burden 0.68 Fair and low cost nRel(A|σEOR 3 ) nRel(A) = 0.9 3.0 = 0.3 nRel(B|σEOR 3 ) nRel(B) = 1.0 3.0 = 0.33 Quali fi ed, Unquali fi ed 0.5 0.5 0.9 nRel(B) = 3.0 nRel(A) = 3.0 nRel(A|σPRP 3 ) + nRel(B|σPRP 3 ) nRel(A) + nRel(B) = 1.9 6.0 = 0.32
24. Cost Distribution Ranking Top-3 EOR criterion Cost Burden on Cost Burden on Cost Burden on Principal 0.03 0.7 0.67 0.68 0.83 0.17 1.0 0.57 0.43 0.4 0.83 0.62 0.13 0.95 0.83 0.89 0.37 0.47 0.83 0.65 0.5 0.99 0.5 0.75 σPRP σunif σEOR σDP σTS σExposure |δ(σ3 )| ≈ ≈
25. EOR Algorithm Algorithm 1: EOR Algorithm Input: Groups g 2 {A, B}; Rankings P RP,g per group in the sorted (decreasing) order of relevance probabilities P(r i |D) Initialize: j 0; empty ranking EOR 1: while j < k do 2: l g P RP,g[1] 8g 2 {A, B} 3: g⇤ arg min g2{A,B} ( EOR [ {l g }) , where (.) is computed according to (7) 4: l g⇤ P RP,g ⇤ [1]; P RP,g ⇤ P RP,g ⇤ n{l g⇤ } 5: EOR EOR [ {l g⇤ }; j j + 1 6: end while 7: return EOR • Goal: • Minimize unfairness • Select relevant candidates • Input — PRP ranking for both groups • Select candidate that minimizes the EOR criterion • Greedy Algorithm, w. runtime complexity δ(σk ) σPRP,A, σPRP,B O(n log n)
26. Cost Optimality Cost Approximation Guarantee at : The EOR-fair ranking produced by Algorithm 1 is at least cost optimal for any pre fi x of the ranking. (Details of in the paper) k σEOR ϕδ(σEOR k ) k ϕ Proof Sketch: Formulate the problem as Integer Linear Program. Relax the problem to Linear Program. Construct feasible dual solutions. Use Linear Duality to bound gap between cost-optimal and EOR solution.
27. Empirical Results Un-fairness # E↵ectiveness " ⇡ \Disp. Unc. High Medium Low High Medium Low ⇡EOR 1.07 ±0.01 1.02 ±0.00 1.02 ±0.00 10.44 ±0.15 11.89 ±0.04 14.58 ±0.10 ⇡DP 11.09 ±0.38 6.02 ±0.07 2.42 ±0.20 10.07 ±0.20 11.33 ±0.04 14.49 ±0.11 ⇡P RP 15.41 ±0.69 7.68 ±0.13 2.63 ±0.17 12.11 ±0.20 12.00 ±0.02 14.62 ±0.09 ⇡T S 11.77 ±0.57 4.96 ±0.07 4.49 ±0.45 7.66 ±0.04 9.62 ±0.06 12.81 ±0.69 ⇡unif 5.96 ±0.13 5.80 ±0.00 6.49 ±0.09 0.00 ±0.00 0.00 ±0.00 0.00 ±0.00 ⇡EXP 9.23 ±0.77 5.62 ±0.01 3.26 ±0.62 11.59 ±0.23 11.97 ±0.03 14.62 ±0.09 ⇡RA 13.97 ±0.71 6.57 ±0.16 2.40 ±0.00 12.02 ±0.19 12.00 ±0.02 14.60 ±0.00 ⇡F S 13.33 ±0.70 7.04 ±0.16 2.95 ±0.17 11.98 ±0.20 12.00 ±0.02 14.62 ±0.09 Synthetic Data
28. Empirical Results More experiments in the paper w. Normalized DCG metric Synthetic Data
29. Empirical Results 2 groups 4 groups US Census 2018, State of NY Evaluation w. true relevance label Binary classi fi cation task of predicting whether the income of an individual is >$50K
30. Empirical Results Amazon Logging Dataset Audit a production system
31. Summary • Disparate Uncertainty induces un-fairness in selection processes • EOR is equivalent to group-fair lottery among the relevant candidates • Distributes more even cost burden on all stakeholders and provides effective rankings, even when true relevance labels are unknown • An approximation guarantee for cost optimality of EOR rankings to the decision maker • Checkout our paper for more !
32. Thanks!