Spectral Method and Regularized MLE Are Both Optimal for Top-K Ranking

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1 Spectral Method and Regularized MLE Are Both Optimal for Top-K Ranking Yuxin Chen Electrical Engineering, Princeton University Joint work with Jianqing Fan, Cong Ma and Kaizheng Wang

2 Ranking A fundamental problem in a wide range of contexts web search, recommendation systems, admissions, sports competitions, voting,... PageRank figure credit: Dzenan Hamzic Top-K ranking 2/ 21

3 Rank aggregation from pairwise comparisons pairwise comparisons for ranking top tennis players figure credit: Bozóki, Csató, Temesi Top-K ranking 3/ 21

4 Parametric models Assign latent score to each of n items w = [w 1,, w n] w i :preferencescore i: rank Top-K ranking 4/ 21

5 Parametric models Assign latent score to each of n items w = [w 1,, w n] w i :preferencescore i: rank This work: Bradley-Terry-Luce (logistic) model P {item j beats item i} = wi + w j Other models: Thurstone model, low-rank model,... Top-K ranking 4/ 21 w j

6 Typical ranking procedures Estimate latent scores rank items based on score estimates Top-K ranking 5/ 21

7 Top-K ranking Estimate latent scores rank items based on score estimates Goal: identify the set of top-k items under minimal sample size Top-K ranking 5/ 21

8 Model: random sampling Comparison graph: Erdős Rényi graph G G(n, p) For each (i, j) G, obtain L paired comparisons y (l) wj ind. 1, with prob. w i,j = i +w j 1 l L 0, else Top-K ranking 6/ 21

9 Model: random sampling Comparison graph: Erdős Rényi graph G G(n, p) For each (i, j) G, obtain L paired comparisons y i,j = 1 L L l=1 y (l) i,j (sufficient statistic) Top-K ranking 6/ 21

10 Prior art mean square error for estimating scores top-k ranking accuracy Spectral method MLE Spectral MLE Negahban et al. 12 Negahban et al. 12 Hajek et al. 14 Chen & Suh. 15 Top-K ranking 7/ 21

11 Prior art a meta metric mean square error for estimating scores top-k ranking accuracy Spectral method MLE Spectral MLE Negahban et al. 12 Negahban et al. 12 Hajek et al. 14 Chen & Suh. 15 Top-K ranking 7/ 21

12 Small l 2 loss high ranking accuracy Top-K ranking 8/ 21

13 Small l 2 loss high ranking accuracy Top-K ranking 8/ 21

14 Small l 2 loss high ranking accuracy These two estimates have same l 2 loss, but output different rankings Top-K ranking 8/ 21

15 Small l 2 loss high ranking accuracy These two estimates have same l 2 loss, but output different rankings Need to control entrywise error! Top-K ranking 8/ 21

16 Optimality? Is spectral method or MLE alone optimal for top-k ranking? Top-K ranking 9/ 21

17 Optimality? Is spectral method or MLE alone optimal for top-k ranking? Partial answer (Jang et al 16): spectral method works if comparison graph is sufficiently dense Top-K ranking 9/ 21

18 Optimality? Is spectral method or MLE alone optimal for top-k ranking? Partial answer (Jang et al 16): spectral method works if comparison graph is sufficiently dense This work: affirmative answer for both methods + entire regime }{{} inc. sparse graphs Top-K ranking 9/ 21

19 Spectral method (Rank Centrality) Negahban, Oh, Shah 12 Construct a probability transition matrix P, whose off-diagonal entries obey { yi,j, if (i, j) G P i,j 0, if (i, j) / G Return score estimate as leading left eigenvector of P Top-K ranking 10/ 21

20 Rationale behind spectral method In large-sample case, P P, whose off-diagonal entries obey wj Pi,j w, if (i, j) G i +w j 0, if (i, j) / G Stationary distribution of } reversible {{} P check detailed balance π [w 1, w 2,..., w n] }{{} true score Top-K ranking 11/ 21

21 (i,j) G Regularized MLE Negative log-likelihood L (w) := { } w i w j y j,i log + (1 y j,i ) log w i + w j w i + w j L (w) becomes convex after reparametrization: w θ = [θ 1,, θ n ], θ i = log w i Top-K ranking 12/ 21

22 (i,j) G Regularized MLE Negative log-likelihood L (w) := { } w i w j y j,i log + (1 y j,i ) log w i + w j w i + w j L (w) becomes convex after reparametrization: w θ = [θ 1,, θ n ], θ i = log w i (Regularized MLE) minimize θ L λ (θ) := L (θ) λ θ 2 2 choose λ np log n L Top-K ranking 12/ 21

23 Main result comparison graph G(n, p); sample size pn2 L Theorem 1 (Chen, Fan, Ma, Wang 17) When p & logn n, both spectral method and regularized MLE achieve optimal sample complexity for top-k ranking! Top-K ranking 13/ 21

separation achievable by both methods infeasible feasible Jang et al 16 sample size Our work / optimal sample size Jan

24 mple size Jang et al 16 K 1/4 log n Main result relatively dense Our work / optimal sample size Jang 1/4 log n Our work / optimal sample size Jang et al 16 K n easible sample size n Our work / optimal sample size separation: K : score separation achievable by both methods infeasible feasible Jang et al 16 sample size Our work / optimal sample size Jan 1/4 log n separation: K : score separation w g et al 16 K w(k) n (K+1) K := : score separation kw k nking 14/ 21 Top-K ranking 13/ 21

25 Comparison with Jang et al 16 log n Jang et al 16: spectral method controls entrywise error if p n }{{} relatively dense Top-K ranking 14/ 21

26 Comparison with Jang et al 16 log n Jang et al 16: spectral method controls entrywise error if p n }{{} relatively dense sample size Our work / optimal sample size Jang et al 16 1/4 log n n K : score separation Top-K ranking 14/ 21

27 top-k ranking accuracy Empirical top-k ranking accuracy Spectral Method Regularized MLE " K : score separation n = 200, p = 0.25, L = 20 Top-K ranking 15/ 21

28 Optimal control of entrywise error w 1 w 2 w 3 w K w K+1 w 1 w 2 w 3 w K K z } { {z} w K+1 {z} < 1 2 K < 1 2 K true score score estimates z Theorem 2 Suppose p log n n and sample size n log n. Then with high prob., 2 K the estimates w returned by both methods obey (up to global scaling) w w w < 1 2 K Top-K ranking 16/ 21

29 Key ingredient: leave-one-out analysis For each 1 m n, introduce leave-one-out estimate w (m) m n m =) w (m) n y =[y i,j ] 1applei,japplen Top-K ranking 17/ 21

30 Key ingredient: leave-one-out analysis For each 1 m n, introduce leave-one-out estimate w (m) statistical independence stability Top-K ranking 17/ 21

31 Exploit statistical independence m n m =) w (m) n y =[y i,j ] 1applei,japplen leave-one-out estimate w (m) = all data related to mth item Top-K ranking 18/ 21

32 Leave-one-out stability leave-one-out estimate w (m) true estimate w Top-K ranking 19/ 21

33 Leave-one-out stability leave-one-out estimate w (m) true estimate w Spectral method: eigenvector perturbation bound π π π π(p P ) π spectral-gap new Davis-Kahan bound for probability transition matrices }{{} asymmetric Top-K ranking 19/ 21

34 Leave-one-out stability leave-one-out estimate w (m) true estimate w Spectral method: eigenvector perturbation bound π π π π(p P ) π spectral-gap new Davis-Kahan bound for probability transition matrices }{{} asymmetric MLE: local strong convexity θ θ 2 L λ (θ; ŷ) 2 strong convexity parameter Top-K ranking 19/ 21

35 A small sample of related works Parametric models Ford 57 Hunter 04 Negahban, Oh, Shah 12 Rajkumar, Agarwal 14 Hajek, Oh, Xu 14 Chen, Suh 15 Rajkumar, Agarwal 16 Jang, Kim, Suh, Oh 16 Suh, Tan, Zhao 17 Non-parametric models Shah, Wainwright 15 Shah, Balakrishnan, Guntuboyina, Wainwright 16 Chen, Gopi, Mao, Schneider 17 Leave-one-out analysis El Karoui, Bean, Bickel, Lim, Yu 13 Zhong, Boumal 17 Abbe, Fan, Wang, Zhong 17 Top-K ranking 20/ 21

36 Summary Spectral method Regularized MLE Optimal sample complexity Linear-time computational complexity Novel entrywise perturbation analysis for spectral method and convex optimization Paper: Spectral method and regularized MLE are both optimal for top-k ranking, Y. Chen, J. Fan, C. Ma, K. Wang, arxiv: , 2017 Top-K ranking 21/ 21

ELE 538B: Mathematics of High-Dimensional Data. Spectral methods. Yuxin Chen Princeton University, Fall 2018

ELE 538B: Mathematics of High-Dimensional Data Spectral methods Yuxin Chen Princeton University, Fall 2018 Outline A motivating application: graph clustering Distance and angles between two subspaces Eigen-space