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: The paper studies real-valued input matrices to the minimum feedback arc set problem on tournaments (MFAST). The matrices are assumed to have low rank and originate from pairwise preference matrices under certain link functions. The authors present a geometrical condition that enables a polynomial-time algorithm to solve MFAST for rank-2 matrices. They apply their algorithm in experiments on learning to rank from pairwise comparisons.