The Mallows model, introduced in the seminal paper of Mallows 1957, is one of the most fundamental ranking distribution over the symmetric group @math . To analyze more complex ranking data, several studies considered the Generalized Mallows model defined by Fligner and Verducci 1986. Despite the significant research interest of ranking distributions, the exact sample complexity of estimating the parameters of a Mallows and a Generalized Mallows Model is not well-understood. The main result of the paper is a tight sample complexity bound for learning Mallows and Generalized Mallows Model. We approach the learning problem by analyzing a more general model which interpolates between the single parameter Mallows Model and the @math parameter Mallows model. We call our model Mallows Block Model -- referring to the Block Models that are a popular model in theoretical statistics. Our sample complexity analysis gives tight bound for learning the Mallows Block Model for any number of blocks. We provide essentially matching lower bounds for our sample complexity results. As a corollary of our analysis, it turns out that, if the central ranking is known, one single sample from the Mallows Block Model is sufficient to estimate the spread parameters with error that goes to zero as the size of the permutations goes to infinity. In addition, we calculate the exact rate of the parameter estimation error.
The parameter estimation of the Generalized Mallows Model has been examined from a practical point of view by @cite_7 but no theoretical guarantees for the sample complexity have been provided. Several ranking models are routinely used in analyzing ranking data , such as Plackett-Luce model , Babington-Smith model and spectral analysis based methods and non-parametric methods . However, to our best knowledge, none of these ranking methods have been analyzed from point of distribution learning which comes with guarantee on some information theoretic distance. considered the problem of learning parameters of Plackett-Luce model and they came up with high probability bounds for their estimator that is tight in a sense that there is no algorithm which can achieve lower estimation error with fewer examples.
@cite_7
2113815377 1526445072 1574648663 2115822807
We analyze the generalized Mallows model, a popular exponential model over rankings. Estimating the central (or consensus) ranking from data is NP-hard. We obtain the following new results: (1) We show that search methods can estimate both the central ranking pi0 and the model parameters theta exactly. The search is n! in the worst case, but is tractable when the true distribution is concentrated around its mode; (2) We show that the generalized Mallows model is jointly exponential in (pi0; theta), and introduce the conjugate prior for this model class; (3) The sufficient statistics are the pairwise marginal probabilities that item i is preferred to item j. Preliminary experiments confirm the theoretical predictions and compare the new algorithm and existing heuristics. This paper introduces two new methods for label ranking based on a probabilistic model of ranking data, called the Plackett-Luce model. The idea of the first method is to use the PL model to fit locally constant probability models in the context of instance-based learning. As opposed to this, the second method estimates a global model in which the PL parameters are represented as functions of the instance. Comparing our methods with previous approaches to label ranking, we find that they offer a number of advantages. Experimentally, we moreover show that they are highly competitive to start-of-the-art methods in terms of predictive accuracy, especially in the case of training data with incomplete ranking information. We propose a novel parameterized family of Mixed Membership Mallows Models (M4) to account for variability in pairwise comparisons generated by a heterogeneous population of noisy and inconsistent users. M4 models individual preferences as a user-specific probabilistic mixture of shared latent Mallows components. Our key algorithmic insight for estimation is to establish a statistical connection between M4 and topic models by viewing pairwise comparisons as words, and users as documents. This key insight leads us to explore Mallows components with a separable structure and leverage recent advances in separable topic discovery. While separability appears to be overly restrictive, we nevertheless show that it is an inevitable outcome of a relatively small number of latent Mallows components in a world of large number of items. We then develop an algorithm based on robust extreme-point identification of convex polygons to learn the reference rankings, and is provably consistent with polynomial sample complexity guarantees. We demonstrate that our new model is empirically competitive with the current state-of-the-art approaches in predicting real-world preferences. This paper is concerned with rank aggregation, which aims to combine multiple input rankings to get a better ranking. A popular approach to rank aggregation is based on probabilistic models on permutations, e.g., the Luce model and the Mallows model. However, these models have their limitations in either poor expressiveness or high computational complexity. To avoid these limitations, in this paper, we propose a new model, which is defined with a coset-permutation distance, and models the generation of a permutation as a stagewise process. We refer to the new model as coset-permutation distance based stagewise (CPS) model. The CPS model has rich expressiveness and can therefore be used in versatile applications, because many different permutation distances can be used to induce the coset-permutation distance. The complexity of the CPS model is low because of the stagewise decomposition of the permutation probability and the efficient computation of most coset-permutation distances. We apply the CPS model to supervised rank aggregation, derive the learning and inference algorithms, and empirically study their effectiveness and efficiency. Experiments on public datasets show that the derived algorithms based on the CPS model can achieve state-of-the-art ranking accuracy, and are much more efficient than previous algorithms.