Ranking from Pairwise Comparisons : The Role of the Pairwise Preference Matrix
Ranking a set of candidates or items from pair-wise comparisons is a fundamental problem that arises in many settings such as elections, recommendation systems, sports team rankings, document rankings and so on. Indeed it is well known in the psychology literature that when a large number of items are to be ranked, it is easier for humans to give pair-wise comparisons as opposed to complete rankings. The problem of ranking from pair-wise comparisons has been studied in multiple communities such as machine learning, operations research, linear algebra, statistics etc., and several algorithms (both classic and recent) have been proposed. However, it is not well under-stood under what conditions these different algorithms perform well. In this thesis, we aim to fill this fundamental gap, by elucidating precise conditions under which different algorithms perform well, as well as giving new algorithms that provably perform well under broader conditions. In particular, we consider a natural statistical model wherein for every pair of items (i; j), there is a probability Pij such that each time items i and j are compared, item j beats item i with probability Pij . Such models, which we summarize through a matrix containing all these pair-wise probabilities, have been used explicitly or implicitly in much previous work in the area; we refer to the resulting matrix as the pair-wise preference matrix, and elucidate clearly the crucial role it plays in determining the performance of various algorithms. In the first part of the thesis, we consider a natural generative model where all pairs of items can be sampled and where the underlying preferences are assumed to be acyclic. Under this setting, we elucidate the conditions on the pair-wise preference matrix under which popular algorithms such as matrix Borda, spectral ranking, least squares and maximum likelihood under a Bradley-Terry-Luce (BTL) model produce optimal rankings that minimize the pair-wise disagreement error. Specifically, we derive explicit sample complexity bounds for each of these algorithms to output an optimal ranking under interesting subclasses of the class of all acyclic pair-wise preference matrices. We show that none of these popular algorithms is guaranteed to produce optimal rankings for all acyclic preference matrices. We then pro-pose a novel support vector machine based rank aggregation algorithm that provably does so. In the second part of the thesis, we consider the setting where preferences may contain cycles. Here, finding a ranking that minimizes the pairwise disagreement error is in general NP-hard. However, even in the presence of cycles, one may wish to rank 'good' items ahead of the rest. We develop a framework for this setting using notions of winners based on tournament solution concepts from social choice theory. We first show that none of the existing algorithms are guaranteed to rank winners ahead of the rest for popular tournament solution based winners such as top cycle, Copeland set, Markov set etc. We propose three algorithms - matrix Copeland, unweighted Markov and parametric Markov - which provably rank winners at the top for these popular tournament solutions. In addition to ranking winners at the top, we show that the rankings output by the matrix Copeland and the parametric Markov algorithms also minimize the pair-wise disagreement error for certain classes of acyclic preference matrices. Finally, in the third part of the thesis, we consider the setting where the number of items to be ranked is large and it is impractical to obtain comparisons among all pairs. Here, one samples a small set of pairs uniformly at random and compares each pair a fixed number of times; in particular, the goal is to come up with good algorithms that sample comparisons among only O(nlog(n)) item pairs (where n is the number of items). Unlike existing results for such settings, where one either assumes a noisy permutation model (under which there is a true underlying ranking and the outcome of every comparison differs from the true ranking with some fixed probability) or assumes a BTL or Thurstone model, we develop a general algorithmic framework based on ideas from matrix completion, termed low-rank pair-wise ranking, which provably produces an good ranking by comparing only O(nlog(n)) pairs, O(log(n)) times each, not only for popular classes of models such as BTL and Thurstone, but also for much more general classes of models wherein a suitable transform of the pair-wise probabilities leads to a low-rank matrix; this subsumes the guarantees of many previous algorithms in this setting. Overall, our results help to understand at a fundamental level the statistical properties of various algorithms for the problem of ranking from pair-wise comparisons, and under various natural settings, lead to novel algorithms with improved statistical guarantees compared to existing algorithms for this problem.