Sample Complexity Bounds for Active Ranking from Multi-wise Comparisons

1 Introduction . 1.1 Background and motivation . Ranking from comparisons is a class of fundamental problems that underpin many areas in machine learning , and has found various applications in problems involving crowd-sourcing , social choices , recommendation , and searching . In such ranking problems , there is a hidden ranking among multiple items to be recovered , where items may refer to candidates , products , movies , advertisements , etc . In this paper , we study ranking from multi-wise comparisons . A multi-wise comparison refers to a query on m items about the most preferred one ( the winner feedback ) or the full ranking ( the full-ranking feedback ) of these items . These comparisons may be deterministic or non-deterministic ( i.e. , noisy or they may return incorrect results ) . The noise comes from the uncertain nature of humans , the lack of information , or the underlying physics . In this paper , we focus on two goals . One is to find the top-k items ( ranking or ordering these items are not necessary ) , and the other is to find the full ranking . Our focus is on active ranking ( e.g. , [ 4–7 , 10 , 17–19 ] ) , where “ active ” means that after each comparison , the learner can adaptively choose the next items to be compared according to past observations 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . and comparison results . Active ranking can be viewed as active learning for ranking problems and the comparisons refer to the samples . The opposite of active ranking is passive ranking , where the learner first obtains a set of comparison results and then recovers a ranking from there . Active ranking can greatly reduce the sample complexity in many scenarios , e.g. , if all comparisons return correct results with probability 23 , passively ranking n items needs Ω ( n 2 ) comparisons [ 3 ] , while active ranking only needs O ( n log n ) 1 comparisons [ 9 , 18 ] . Most existing works have focused on ranking from pairwise comparisons . In contrast , we focus on ranking from multi-wise ( or m-wise ) comparisons . The pairwise comparisons can be viewed as multi-wise comparisons with m = 2 . One motivation is that in many scenarios , multi-wise comparisons are more common . For instance , in video streaming websites or e-shopping apps , customers or users are normally presented more than two options , and the choices made by users can be viewed as multi-wise comparisons that suggest their preferences over these options . Studying ranking from multi-wise comparisons is useful for these types of applications . Besides , ranking from multi-wise comparisons may also reduce the cost of the learning process . In some applications , conducting the comparisons may be expensive . For instance , to find the candidates that are most preferred by the voters , people may need to do a series of surveys . Each survey contains a query about the preference order of a voter . In this application , the cost of finding the voter and asking this voter to fill the survey could far outweigh the cost of filling the survey itself . This means that the cost of conducting a multi-wise comparison is almost the same as that for a pairwise comparison , and how to reduce the number of samples by multi-wise comparisons becomes more interesting . Thus , it is not only interesting but also significant to study whether and to what degree we can reduce the sample complexity for ranking by using multi-wise comparisons . We focus on a non-parametric model , where each comparison returns the correct result with a certain probability q > 12 and an arbitrary incorrect result otherwise . When q = 1 , the comparisons are deterministic and always return correct results , and when q < 1 , we say the comparisons are nondeterministic or noisy . This differs from parametric models2 that assume that each item holds a value representing the users ’ preference on this item . In parametric models , the comparisons may provide more information than the setting in this paper3 , and thus , the conclusion drawn under parametric models can not be directly applied to this paper . 1.2 Problem formulation . Assume that there are n items , indexed by 1 , 2 , 3 , ... n , that form the item set [ n ] 4 . We further assume that these items have a unique unknown true ranking r1 ≻ r2 ≻ · · · ≻ rn , where for any items i and j , notation i ≻ j means that item i ranks higher ( or is more preferred ) than item j . Assume that we can compare at most m items at a time . The comparisons can be either deterministic or non-deterministic . We also assume that the comparisons are independent across time , items , and sets , which is standard in the literature ( e.g. , [ 5–7 , 10 , 11 , 13 , 17–22 ] ) . Here , we note that the independence is based on the assumption that the hidden parameters and ranking are some fixed values . When the comparisons are deterministic , the comparisons always return the correct results , i.e. , the best item under the winner feedback model or the true ranking of the compared items under the full-ranking feedback model . In the deterministic case , our goal is to find the exact top-k items or the true ranking . When the comparisons are non-deterministic , we assume that they return the correct results with a certain fixed probability q ∈ ( 12 , 1 ) . In this case , it is infeasible to rank the items with 100 % confidence , and thus , our goal is to find the ranking with confidence 1− δ for some δ ∈ ( 0 , 12 ) . We focus on the case where q = 23 . We note that this does not lose much generality . When q > 2 3 , the 1All log in this paper are natural log unless explicitly noted . 2For instance , parametric models can be the Bradley-Terry-Luce ( BTL ) model [ 2 ] , the Plackett-Luce ( PL ) model [ 15 ] , or the multinomial logit ( MNL ) model [ 14 ] . 3For instance , given three items i , j , k with i being the best and k being the worst , in the setting of this paper with q = 0.7 , the multi-wise comparison returns the best item i with probability 0.7 , and return j or k with some unknown probabilities , which does not provides information about the orders of j and k. In contrast , in a parametric model , the comparison over { i , j , k } would return i , j , or k with probability 0.7 , 0.2 , and 0.1 , respectively , which not only provides information about the best one but also information for ordering j and k. 4For any positive integer l , we define [ l ] : = { 1 , 2 , 3 , ... , l } . algorithms and sample complexity bounds in this paper can be directly applied . When q < 23 , we can use repeated comparisons to simulate one comparison with correct probability at least 23 . To see this , consider a set over which a comparison returns the correct result with probability 12 +∆ > 1 2 . By comparing this set for ⌈ 12∆2 log 3⌉ times , the item or permutation that occurs most often is the correct result with probability at least 23 ( by Hoeffding Inequality ) . Thus , we can use the above method to substitute the comparisons in the algorithms designed for the case where q = 23 while only introducing an additional 1∆2 factor on the sample complexity . This ( 12 +∆ ) m-wise comparison model can also be justified in many scenarios that use an iterative subroutine to conduct m-wise comparisons over a set in a smaller time-scale . Here , we give a simple example . We use pi , S to denote the probability that item i wins the comparison performed on set S ( assuming |S| ≤ m ) and we use i∗ to denote the best item of S. If pi∗ , S ≥ pi , S +∆ for any item i ̸= i∗ and any set S , then by repeatedly comparing S for Θ ( 1∆2 log |S| ) times , one can find the best item of S with confidence 23 . We can use the above procedure as a subroutine in each iteration of our algorithms , and get the algorithms for this case while only introducing an additional logm factor to the sample complexities . In more general cases , we can use similar tricks but the additional factors may vary . 1.3 Main results . The main results of this paper are summarized in Table 1 . We note that due to space limitation , all proofs in this paper are left to the supplementary material . 2 Related works . When m = 2 and the comparisons are deterministic , the top-k selection problem becomes the classical pairwise k-selection problem , which requires Θ ( n ) comparisons [ 1 ] , and the full ranking problem becomes the classical pairwise sorting problem , which requires Θ ( n log n ) comparisons . Thus , ranking from multi-wise comparisons can also be viewed as extensions of these foundational problems . Surprisingly , these extensions have not been well understood . For top-k selection from non-deterministic pairwise comparisons , the authors of [ 9 ] showed that Θ ( n log kδ ) comparisons are necessary and sufficient to reach confidence level 1− δ . For probably approximately correct ( PAC ) 5 top-k selection with error tolerance ϵ > 0 , the authors of [ 5 , 7 ] proved a Θ ( nϵ2 log 1 δ ) bound for k = 1 , and the authors of [ 16 , 18 ] proved a Θ ( n ϵ2 log k δ ) bound for k ≤ n 2 . For full ranking from non-deterministic pairwise comparisons , an early work was [ 9 ] , whose authors proved that when the comparisons of all pairs have the same noise level , then to get the full ranking with confidence 1 − δ , Θ ( n log nδ ) comparisons are necessary and sufficient . The authors of [ 17 ] 5The PAC setting means that we want to find the ranking approximately with an error tolerance ϵ and a confidence 1− δ , where the ϵ tolerance means that for any two items i and j , item i is viewed to rank higher than item j if item i wins the comparisons over item j with probability 1 2 − ϵ or higher . extend the above results to the case where the comparisons on different pairs have different noise levels , and also obtained the Θ ( n log nδ ) bounds . The authors of [ 5–7 ] showed the Θ ( n ϵ2 log n δ ) order-wise tight upper and lower bounds for PAC full ranking with error tolerance ϵ > 0 . Thus , we can see that for m = 2 ( i.e. , ranking from pairwise comparisons ) , people have already established optimal sample complexity bounds for top-k selection and full ranking . However , the corresponding multi-wise ranking problems have been relatively under-explored . Most of these works have focused on parametric models such as the PL model and the MNL model . Under the MNL model , the work in [ 4 ] showed that under certain cases , one can get an m-reduction from m-wise comparisons over ranking from pairwise comparisons . However , for most cases , using m-wise comparisons does not reduce the sample complexity ( ignoring constant factors ) . For multi-wise full ranking under the MNL model , the works in [ 16 , 17 ] showed that the lower bound is the same as that using pairwise comparisons , i.e. , there is no reduction by using m-wise comparisons . Under the list-wise PL model with certain types of feedback , we may achieve up to m-reduction in the sample complexity according to [ 19 ] . In contrast to the above works , we do not assume any parametric model , and instead use a nonparametric model where each comparison returns a correct result with a certain fixed probability q . As noted in Section 1.1 , the results drawn from the parametric models can not be directly applied to this non-parametric model . Multi-wise ranking under this non-parametric setting is even less under-explored in the literature . The most related work to this paper is [ 17 ] , where the authors showed a Θ ( n lognlogm ) bound for full ranking from the winner feedback , which solves one of the eight cases ( top-k selection or full ranking , winner feedback or full-ranking feedback , deterministic feedback or non-deterministic feedback ) that is studied in this paper and shown in Table 1 . We will study the rest seven cases in this paper . 3 Top-k selection from winner feedback

Summary: This paper studies the problem of active ranking from ($m$) multi-wise comparisons. The focus on past literature was on pairwise comparisons $m=2$. For this problem the authors considered both  top-$k$ items selection and full ranking. For both problems the authors provide theoretical guarantees (lower and upper bounds) on the sample complexity, i.e., the number of comparison. Furthermore, contrary to previous related works, the current model is not parametric and quite general.