Suppose that we wish to estimate a user's preference vector @math from paired comparisons of the form "does user @math prefer item @math or item @math ?," where both the user and items are embedded in a low-dimensional Euclidean space with distances that reflect user and item similarities. Such observations arise in numerous settings, including psychometrics and psychology experiments, search tasks, advertising, and recommender systems. In such tasks, queries can be extremely costly and subject to varying levels of response noise; thus, we aim to actively choose pairs that are most informative given the results of previous comparisons. We provide new theoretical insights into the benefits and challenges of greedy information maximization in this setting, and develop two novel strategies that maximize lower bounds on information gain and are simpler to analyze and compute respectively. We use simulated responses from a real-world dataset to validate our strategies through their similar performance to greedy information maximization, and their superior preference estimation over state-of-the-art selection methods as well as random queries.
There is a rich literature investigating statistical inference from paired comparisons and related ordinal query types. However, many of these works target a different problem than considered here, such as constructing item embeddings @cite_21 , training classifiers @cite_28 , selecting arms of bandits @cite_5 , and learning rankings @cite_16 @cite_13 @cite_36 @cite_8 @cite_24 or scores @cite_27 @cite_26 over items.
@cite_26 @cite_8 @cite_28 @cite_36 @cite_21 @cite_24 @cite_27 @cite_5 @cite_16 @cite_13
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The question of aggregating pairwise 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 ranking, finding 'scores' for each object (e.g. player's rating) is of interest to understanding the intensity of the preferences. In this paper, we propose a novel iterative rank aggregation algorithm for discovering scores for objects from pairwise comparisons. The algorithm has a natural random walk interpretation over the graph of objects with edges present between two objects if they are compared; the scores turn out to be the stationary probability of this random walk. The algorithm is model independent. To establish the efficacy of our method, however, we consider the popular Bradley-Terry-Luce (BTL) model in which each object has an associated score which determines the probabilistic outcomes of pairwise comparisons between objects. We bound the finite sample error rates between the scores assumed by the BTL model and those estimated by our algorithm. This, in essence, leads to order-optimal dependence on the number of samples required to learn the scores well by our algorithm. Indeed, the experimental evaluation shows that our (model independent) algorithm performs as well as the Maximum Likelihood Estimator of the BTL model and outperforms a recently proposed algorithm by Ammar and Shah [1]. Given a collection of N items with some unknown underlying ranking, we examine how to use pairwise comparisons to determine the top ranked items in the set. Resolving the top items from pairwise comparisons has application in diverse elds ranging from recommender systems to image-based search to protein structure analysis. In this paper we introduce techniques to resolve the top ranked items using signicantly fewer than all the possible pairwise comparisons using both random and adaptive sampling methodologies. Using randomly-chosen comparisons, a graph-based technique is shown to eciently resolve the top O(logN) items when there are no comparison errors. In terms of adaptively-chosen comparisons, we show how the top O(logN) items can be found, even in the presence of corrupted observations, using a voting methodology that only requires O N log 2 N pairwise comparisons. This paper explores the preference-based top-K 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-K ranked items, based on partially revealed preferences. We focus on the Bradley-Terry-Luce 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-K 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 Kth and (K + 1)th ranked items. The minimum sample complexity required for reliable top-K ranking scales inversely with the separation measure. To approach this minimax limit, we propose a nearly linear-time ranking scheme, called Spectral MLE, that returns the indices of the top-K 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-K item identification under minimal sample complexity. The practical applicability of Spectral MLE is further corroborated by numerical experiments. We introduce an algorithm that, given n objects, learns a similarity matrix over all n2 pairs, from crowdsourced data alone. The algorithm samples responses to adaptively chosen triplet-based relative-similarity queries. Each query has the form "is object a more similar to b or to c?" and is chosen to be maximally informative given the preceding responses. The output is an embedding of the objects into Euclidean space (like MDS); we refer to this as the "crowd kernel." SVMs reveal that the crowd kernel captures prominent and subtle features across a number of domains, such as "is striped" among neckties and "vowel vs. consonant" among letters. Data in the form of pairwise comparisons arises in many domains, including preference elicitation, sporting competitions, and peer grading among others. We consider parametric ordinal models for such pairwise comparison data involving a latent vector w* e Rd that represents the "qualities" of the d items being compared; this class of models includes the two most widely used parametric models|the Bradley-Terry-Luce (BTL) and the Thurstone models. Working within a standard minimax framework, we provide tight upper and lower bounds on the optimal error in estimating the quality score vector w* under this class of models. The bounds depend on the topology of the comparison graph induced by the subset of pairs being compared, via the spectrum of the Laplacian of the comparison graph. Thus, in settings where the subset of pairs may be chosen, our results provide principled guidelines for making this choice. Finally, we compare these error rates to those under cardinal measurement models and show that the error rates in the ordinal and cardinal settings have identical scalings apart from constant pre-factors. The ranking of n objects based on pairwise comparisons is a core machine learning problem, arising in recommender systems, ad placement, player ranking, biological applications and others. In many practical situations the true pairwise comparisons cannot be actively measured, but a subset of all n(n-1) 2 comparisons is passively and noisily observed. Optimization algorithms (e.g., the SVM) could be used to predict a ranking with fixed expected Kendall tau distance, while achieving an Ω (n) lower bound on the corresponding sample complexity. However, due to their centralized structure they are difficult to extend to online or distributed settings. In this paper we show that much simpler algorithms can match the same Ω (n) lower bound in expectation. Furthermore, if an average of O(n log(n)) binary comparisons are measured, then one algorithm recovers the true ranking in a uniform sense, while the other predicts the ranking more accurately near the top than the bottom. We discuss extensions to online and distributed ranking, with benefits over traditional alternatives. The paper is concerned with learning to rank, which is to construct a model or a function for ranking objects. Learning to rank is useful for document retrieval, collaborative filtering, and many other applications. Several methods for learning to rank have been proposed, which take object pairs as 'instances' in learning. We refer to them as the pairwise approach in this paper. Although the pairwise approach offers advantages, it ignores the fact that ranking is a prediction task on list of objects. The paper postulates that learning to rank should adopt the listwise approach in which lists of objects are used as 'instances' in learning. The paper proposes a new probabilistic method for the approach. Specifically it introduces two probability models, respectively referred to as permutation probability and top k probability, to define a listwise loss function for learning. Neural Network and Gradient Descent are then employed as model and algorithm in the learning method. Experimental results on information retrieval show that the proposed listwise approach performs better than the pairwise approach.