For massive and heterogeneous modern datasets, it is of fundamental interest to provide guarantees on the accuracy of estimation when computational resources are limited. In the application of learning to rank, we provide a hierarchy of rank-breaking mechanisms ordered by the complexity in thus generated sketch of the data. This allows the number of data points collected to be gracefully traded off against computational resources available, while guaranteeing the desired level of accuracy. Theoretical guarantees on the proposed generalized rank-breaking implicitly provide such trade-offs, which can be explicitly characterized under certain canonical scenarios on the structure of the data.
On the theoretical side, when samples consist of pairwise comparisons, @cite_3 first established consistency and asymptotic normality of the maximum likelihood estimate when all teams play against each other. For a broader class of scenarios where we allow for sparse observations, where the number of total comparisons grow linearly in the number of teams, @cite_15 show that Rank Centrality achieves optimal sample complexity by comparing it to a lower bound on the minimax rate. For a more general class of traditional observations, including pairwise comparisons, @cite_17 provide similar optimal guarantee for the maximum likelihood estimator. @cite_7 introduced Spectral MLE that applies Rank Centrality followed by MLE, and showed that the resulting estimate is optimal in @math error as well as the previously analyzed @math error. @cite_18 study a new measure of the error induced by the Laplacian of the comparisons graph and prove a sharper upper and lower bounds that match up to a constant factor.
@cite_18 @cite_7 @cite_3 @cite_15 @cite_17
2951817470 2132572278 1757513105 2952852844
The question of aggregating pair-wise comparisons to obtain a global ranking over a collection of objects has been of interest for a very long time: be it ranking of online gamers (e.g. MSR's TrueSkill system) and chess players, aggregating social opinions, or deciding which product to sell based on transactions. In most settings, in addition to obtaining a ranking, finding scores' for each object (e.g. player's rating) is of interest for understanding the intensity of the preferences. In this paper, we propose Rank Centrality, an iterative rank aggregation algorithm for discovering scores for objects (or items) from pair-wise comparisons. The algorithm has a natural random walk interpretation over the graph of objects with an edge present between a pair of objects if they are compared; the score, which we call Rank Centrality, of an object turns out to be its stationary probability under this random walk. To study the efficacy of the algorithm, we consider the popular Bradley-Terry-Luce (BTL) model (equivalent to the Multinomial Logit (MNL) for pair-wise comparisons) in which each object has an associated score which determines the probabilistic outcomes of pair-wise comparisons between objects. In terms of the pair-wise marginal probabilities, which is the main subject of this paper, the MNL model and the BTL model are identical. We bound the finite sample error rates between the scores assumed by the BTL model and those estimated by our algorithm. In particular, the number of samples required to learn the score well with high probability depends on the structure of the comparison graph. When the Laplacian of the comparison graph has a strictly positive spectral gap, e.g. each item is compared to a subset of randomly chosen items, this leads to dependence on the number of samples that is nearly order-optimal. This paper explores the preference-based top- @math rank aggregation problem. Suppose that a collection of items is repeatedly compared in pairs, and one wishes to recover a consistent ordering that emphasizes the top- @math ranked items, based on partially revealed preferences. We focus on the Bradley-Terry-Luce (BTL) model that postulates a set of latent preference scores underlying all items, where the odds of paired comparisons depend only on the relative scores of the items involved. We characterize the minimax limits on identifiability of top- @math ranked items, in the presence of random and non-adaptive sampling. Our results highlight a separation measure that quantifies the gap of preference scores between the @math and @math ranked items. The minimum sample complexity required for reliable top- @math ranking scales inversely with the separation measure irrespective of other preference distribution metrics. To approach this minimax limit, we propose a nearly linear-time ranking scheme, called , that returns the indices of the top- @math items in accordance to a careful score estimate. In a nutshell, Spectral MLE starts with an initial score estimate with minimal squared loss (obtained via a spectral method), and then successively refines each component with the assistance of coordinate-wise MLEs. Encouragingly, Spectral MLE allows perfect top- @math item identification under minimal sample complexity. The practical applicability of Spectral MLE is further corroborated by numerical experiments. This paper is motivated by two applications, namely i) generalizations of cuckoo hashing, a computationally simple approach to assigning keys to objects, and ii) load balancing in content distribution networks, where one is interested in determining the impact of content replication on performance. These two problems admit a common abstraction: in both scenarios, performance is characterized by the maximum weight of a generalization of a matching in a bipartite graph, featuring node and edge capacities. Our main result is a law of large numbers characterizing the asymptotic maximum weight matching in the limit of large bipartite random graphs, when the graphs admit a local weak limit that is a tree. This result specializes to the two application scenarios, yielding new results in both contexts. In contrast with previous results, the key novelty is the ability to handle edge capacities with arbitrary integer values. An analysis of belief propagation algorithms (BP) with multivariate belief vectors underlies the proof. In particular, we show convergence of the corresponding BP by exploiting monotonicity of the belief vectors with respect to the so-called upshifted likelihood ratio stochastic order. This auxiliary result can be of independent interest, providing a new set of structural conditions which ensure convergence of BP. The probability that a user will click a search result depends both on its relevance and its position on the results page. The position based model explains this behavior by ascribing to every item an attraction probability, and to every position an examination probability. To be clicked, a result must be both attractive and examined. The probabilities of an item-position pair being clicked thus form the entries of a rank- @math matrix. We propose the learning problem of a Bernoulli rank- @math bandit where at each step, the learning agent chooses a pair of row and column arms, and receives the product of their Bernoulli-distributed values as a reward. This is a special case of the stochastic rank- @math bandit problem considered in recent work that proposed an elimination based algorithm Rank1Elim, and showed that Rank1Elim's regret scales linearly with the number of rows and columns on "benign" instances. These are the instances where the minimum of the average row and column rewards @math is bounded away from zero. The issue with Rank1Elim is that it fails to be competitive with straightforward bandit strategies as @math . In this paper we propose Rank1ElimKL which simply replaces the (crude) confidence intervals of Rank1Elim with confidence intervals based on Kullback-Leibler (KL) divergences, and with the help of a novel result concerning the scaling of KL divergences we prove that with this change, our algorithm will be competitive no matter the value of @math . Experiments with synthetic data confirm that on benign instances the performance of Rank1ElimKL is significantly better than that of even Rank1Elim, while experiments with models derived from real data confirm that the improvements are significant across the board, regardless of whether the data is benign or not.