On The Structure of Parametric Tournaments with Application to Ranking from Pairwise Comparisons

1 Introduction . A tournament is a complete directed graph . Given a tournament T , the classical feedback arc set on tournament ( MFAST ) problem asks for the minimum number of edges that must be removed ( or whose orientation reversed ) in T to make it acyclic [ 3 ] . The problem is known to be NP-hard for general tournaments [ 6 ] . We investigate the MFAST problem for several classes of tournaments which naturally occur in learning to rank from pairwise comparisons . In particular , we wish to study the MFAST problem on tournament classes which arise out of parametric pairwise preference classes . Popular parametric preference models such as the Bradley-Terry-Luce ( BTL ) [ 4 ; 17 ] and Thurstone models [ 25 ] give rise to acyclic tournament matrices for which the MFAST problem is trivial . Our ∗Cohesity , Inc , has neither endorsed nor sponsored this research or its publication in any way ; the author ’ s research opinions and findings are his own . 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . goal is to identify and study non-trivial classes of parametric preference matrices which give rise to potentially cyclic tournaments . Cyclic tournaments arise naturally in several real world preference datasets . Figure 1 shows a few real world cyclic tournament structures and Figure 2 shows the fraction of tournaments out of 290 real world tournaments that are acyclic and those that are cyclic and can arise out of the models proposed in this paper . Clearly , simple models such as BTL/Thurstone are insufficient to model these tournaments . Every pairwise preference matrix P ∈ ( 0 , 1 ) n×n can be associated to a tournament via a skew symmetric link function φ i.e , φ ( P ) is a skew symmetric matrix and the tournament T associated to P has an edge from i to j if and only if φ ( Pij ) > 0 . While preference matrices themselves are always of high rank [ 11 ] , several well studied pairwise preference classes give rise to skew symmetric matrices of low rank under a suitable link function . For example , the BTL model gives rise to a rank 2 skew symmetric matrix under the logit link whereas the Thurstone Model does so under a probit link [ 22 ] . It is important to understand the possible tournaments any parametric preference class can give rise to . Indeed , it might be natural and much easier for a modeller/domain expert to make a structural statement such as The class of tournaments of interest should contain cycles of length at most 4 as opposed to an algebraic statement like The class of tournaments should be associated with a preference matrix which on a skew symmetric transformation leads to a rank 8 matrix . We study the structure of rank r tournament class i.e , tournaments which arise out of preference matrices that lead to some rank r matrix under a skew-symmetric transformation . This class can be interpreted as each node in the tournament having an embedding in Rr and the preference relation between nodes depends on a suitable notion of similarity between the embeddings ( see Section 4 for details ) . It is known that such classes can model cyclic relations [ 22 ] . To investigate the MFAST problem for the rank r tournament classes , we first structurally understand these tournament via the notion of forbidden configurations [ 8 ] . Forbidden configurations in tournaments have been studied in other non-parametric contexts . The simplest example is the class of acyclic tournaments which has the 3 cycle as a forbidden configuration . A complicated forbidden configuration class using Möbius ladders was studied in [ 8 ; 9 ] . In both these cases , the MFAST problem is poly-time solvable . Our Contributions . Our first contribution is to derive upper and lower bounds for the size of a forbidden configuration for any rank r tournament class . We do this by introducing a novel combinatorial parameter that we call the Tournament Dimension . When rank r = 2 , we completely characterize the associated tournament class by identifying the exact forbidden configuration . We use this forbidden configuration to understand the geometry of this tournament class and propose an algorithm ( Rank2Rank ) to solve the MFAST problem in polynomial time for this class of tournaments . Our next contribution is to propose a flexible class of tournaments called the Block Rank 2 class which builds on the rank 2 class . Tournaments in this class can arise from the local-global model - a general pairwise preference model that we propose . The model can capture locally cyclic relations as well as globally acyclic relations and is well suited for several practical applications including those in Figure 1 . The model subsumes several popular pairwise ranking models including the BTL and Thurstone models . We show that the MFAST problem on the associated Block Rank 2 tournament class is polynomial time solvable and propose an algorithm ( BlockRank2Rank ) for the problem . Our third contribution is to study an application of our work to the problem of learning to rank from pairwise comparisons . We propose a matrix completion based algorithm called PairwiseBlockRank for this problem and derive sample complexity bounds for the same under the proposed local-global preference model . Finally , we evaluate our algorithms on real world and synthetic data-sets and show improved performance over several existing algorithms . 2 Related Work . Several works have considered learning from transitive pairwise models especially focusing on the Bradley-Terry-Luce ( BTL ) model [ 12 ; 20 ; 21 ; 24 ; 14 ] . As we focus on cyclic relations , we discuss only the models which can lead to cyclic preferences . While the BTL model can be seen as using a 1 dimensional embedding of the item using a score vector , studies have considered higher dimensional embeddings . A 2 dimensional embedding was considered in [ 5 ] . While their model can give rise to cyclic tournaments , it is not clear what type of cycles are possible . [ 18 ; 19 ] propose the Majority vote model which is a random utility model ( RUM ) with a d dimensional feature embedding for each item . The Majority vote model is powerful enough to produce arbitrarily long cycles and can express any probability sub-matrix over a fixed triplet . The Rank 2 model we study is as powerful as the Majority vote model . Furthermore , we give a complete structural characterization of the tournaments of our model which is not known for the Majority vote model . The Blade-Chest inner ( BCI ) model [ 7 ] embeds each item into two d dimensional vectors ( blade vector and a chest vector ) and a score vector s where the probability of i being preferred over j depends on < ichest , jblade > − < iblade , jchest > +si − sj . The rank-2 model can be seen as a special case of the BCI model where d = 1 and s is a constant vector . When d is O ( n ) , the BCI model can give rise to any tournament as there are O ( n2 ) parameters and so the MFAST problem becomes intractable . We focus on the special case of d = 1 where there are only 2n parameters and propose algorithms to obtain optimal rankings . Finally , we discuss the low rank pairwise rank ( LRPR ) class of models which result in a low rank matrix under a transformation using a suitable link function [ 22 ] . Previous work [ 12,22 ] have proposed matrix completion based algorithms to obtain optimal ranking for the LRPR type models assuming transitivity of preferences . In this work , we make no such assumptions . 3 Preliminaries . A Tournament is a complete directed graph . We use i T j to denote that there is a directed edge from node i to j in the tournament T. We call a set of edges Fσ in a tournament T as the feedback arc set of permutation σ w.r.t T if σ ( i ) > σ ( j ) ( j ranked ahead of i in σ ) but i T j . Indeed , if one reverses the orientation of the Fσ edges in T , one gets an acyclic tournament whose topological sort would yield σ . We call a permutation σ∗ as MFAST-Optimal for T if σ∗ ∈ arg minσ |Fσ| . We call P ∈ ( 0 , 1 ) n×n a pairwise preference matrix if Pij + Pji = 1 ∀i 6= j . We assume Pij 6= 0.5 ∀i , j . We will say TP is a tournament associated with P if a directed edge from node i to j is present in TP if and only if Pij > 0.5 . We call P a rank r preference matrix w.r.t a link function φ if rank ( φ ( P ) ) = r where the function φ is applied element-wise to P. We will call φ a skew symmetric link function if φ ( P ) is a skew symmetric matrix for any pairwise preference matrix P. Examples of such φ include the logit and probit links where logit ( x ) = log ( x/ ( 1− x ) ) and probit ( x ) = Φ−1 ( p ) where Φ is the standard normal cdf . We use T n , φr to denote the class of all tournaments on n nodes which are associated with a rank r preference matrix w.r.t φ . Our goal is the study the tournament class T n , φr where φ is a skew symmetric link function . Skewsymmetric matrices are naturally associated to tournaments where the edge directions in the tournament are determined by the sign of the corresponding matrix entry . Our results would apply for any skew symmetric φ and so we will drop the φ in T n , φr when it is clear from the context . As skew symmetric matrices have even rank , T n , φr is non-empty only for even r. We wish to investigate the structural constraints on these tournament classes that arise out of the algebraic ( rank ) restriction that define them . We note that it was known earlier that for the special case when r = 2 , T n , φ2 contains both cyclic and acyclic tournaments [ 22 ] . However , nothing further was known about this class . Algorithm 1 Tournament Game 1 : Input : Integers k , d 2 : Player-1 chooses a labelled dataset { ( x1 , y1 ) , . . . ( xk , yk ) } where D = { xi } ki=1 ∈ Rd induces a Tournament , yi ∈ { +1 , −1 } 3 : Player-2 chooses a Tournament preserving mapping f w.r.t D 4 : if ∃w ∈ Rd such that sign ( wT f ( xi ) ) = yi ∀i ∈ [ k ] then 5 : Player-2 wins the game 6 : else 7 : Player-1 wins the game 8 : end if Definition 1 . A class of tournaments T is said to have a forbidden configuration of size ` if ` is the smallest integer such that there exists a tournament T ` on ` nodes which does not appear as an induced sub-tournament for any T ∈ T In other words , no induced sub-tournament on l nodes of T ∈ T is isomorphic to T ` . For example , the class of all acyclic tournaments T acyclic has C3 - the 3-cycle as a forbidden configuration . Taking advantage of this forbidden configuration , one can immediately come up with an algorithm to solve the MFAST problem on T acyclic - indeed , a topological sort solves this problem . Our interest is to understand if there are similar forbidden configurations for the class T n , φr . Towards this , we start the next section by defining a novel combinatorial parameter called the Tournament dimension .

Summary: This paper focus on minimum feedback arc set problem in tournaments (MFAST) and characterizes the relation between the preference matrix rank with forbidden configuration. The authors propose a novel polynomial-time algorithm on MFAST with preference matrix rank 2. The authors further extend the setting to a block rank 2 tournament, where nodes can be clustered into several blocks and in each block, the preference matrix has rank 2. The proposed algorithm can also work on the non-tournament cases with a matrix-completion step. The experiments shows that the proposed algorithm reaches better performance than other works, in both synthetic (non-tournament) and real-world datasets.