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.
Our model is closely related to that validated the advantages of adopting the mixed membership perspective. models each latent ranking factor as a single permutation and is a degenerate special case of Mallows distribution over the permutations in M4. Therefore, while both @cite_5 and M4 can capture the inconsistent behavior semming from the influence of multiple latent factors, M4 can further account for the inconsistency as the consequence of the randomness within each Mallows components. Our approach has similar polynomial time and sample guarantees as in @cite_5 . We note that motivated by social choice application, proposed another mixed membership ranking model where the latent topics'' are PL models. An MCMC based approach is used for estimation without theoretical guarantees. Table. summarizes all the closely related works.
@cite_5
1574648663 1827387789 2115822807 2394967154
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. The Markov Chain Monte Carlo (MCMC) family of methods form a valuable part of the toolbox of social modeling and prediction techniques, enabling modelers to generate samples and summary statistics of a population of interest with minimal information. It has been used successfully to model changes over time in many types of social systems, including patterns of disease spread, adolescent smoking, and geopolitical conflicts. In MCMC an initial proposal distribution is iteratively refined until it approximates the posterior distribution. However, the selection of the proposal distribution can have a significant impact on model convergence. In this paper, we propose a new hybrid modeling technique in which an agent-based model is used to initialize the proposal distribution of the MCMC simulation. We demonstrate the use of our modeling technique in an urban transportation prediction scenario and show that the hybrid combined model produces more accurate predictions than either of the parent models. 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. Recently, Stochastic Gradient Markov Chain Monte Carlo (SG-MCMC) methods have been proposed for scaling up Monte Carlo computations to large data problems. Whilst these approaches have proven useful in many applications, vanilla SG-MCMC might suffer from poor mixing rates when random variables exhibit strong couplings under the target densities or big scale differences. In this study, we propose a novel SG-MCMC method that takes the local geometry into account by using ideas from Quasi-Newton optimization methods. These second order methods directly approximate the inverse Hessian by using a limited history of samples and their gradients. Our method uses dense approximations of the inverse Hessian while keeping the time and memory complexities linear with the dimension of the problem. We provide a formal theoretical analysis where we show that the proposed method is asymptotically unbiased and consistent with the posterior expectations. We illustrate the effectiveness of the approach on both synthetic and real datasets. Our experiments on two challenging applications show that our method achieves fast convergence rates similar to Riemannian approaches while at the same time having low computational requirements similar to diagonal preconditioning approaches.