Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userConcentric mixtures of Mallows models for top-$k$ rankings: sampling and identifiability
52
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper, we consider mixtures of two Mallows models for top-$k$ rankings, both with the same location parameter but with different scale parameters, i.e., a mixture of concentric Mallows models. This situation arises when we have a heterogeneous population of voters formed by two homogeneous populations, one of which is a subpopulation of expert voters while the other includes the non-expert voters. We propose efficient sampling algorithms for Mallows top-$k$ rankings. We show the identifiability of both components, and the learnability of their respective parameters in this setting by, first, bounding the sample complexity for the Borda algorithm with top-$k$ rankings and second, proposing polynomial time algorithm for the separation of the rankings in each component. Finally, since the rank aggregation will suffer from a large amount of noise introduced by the non-expert voters, we adapt the Borda algorithm to be able to recover the ground truth consensus ranking which is especially consistent with the expert rankings.
rate research
Read More
Word embeddings are commonly obtained as optimizers of a criterion function $f$ of a text corpus, but assessed on word-task performance using a different evaluation function $g$ of the test data. We contend that a possible source of disparity in performance on tasks is the incompatibility between classes of transformations that leave $f$ and $g$ invariant. In particular, word embeddings defined by $f$ are not unique; they are defined only up to a class of transformations to which $f$ is invariant, and this class is larger than the class to which $g$ is invariant. One implication of this is that the apparent superiority of one word embedding over another, as measured by word task performance, may largely be a consequence of the arbitrary elements selected from the respective solution sets. We provide a formal treatment of the above identifiability issue, present some numerical examples, and discuss possible resolutions.
This paper is concerned with the problem of top-$K$ ranking from pairwise comparisons. Given a collection of $n$ items and a few pairwise comparisons across them, one wishes to identify the set of $K$ items that receive the highest ranks. To tackle this problem, we adopt the logistic parametric model --- the Bradley-Terry-Luce model, where each item is assigned a latent preference score, and where the outcome of each pairwise comparison depends solely on the relative scores of the two items involved. Recent works have made significant progress towards characterizing the performance (e.g. the mean square error for estimating the scores) of several classical methods, including the spectral method and the maximum likelihood estimator (MLE). However, where they stand regarding top-$K$ ranking remains unsettled. We demonstrate that under a natural random sampling model, the spectral method alone, or the regularized MLE alone, is minimax optimal in terms of the sample complexity --- the number of paired comparisons needed to ensure exact top-$K$ identification, for the fixed dynamic range regime. This is accomplished via optimal control of the entrywise error of the score estimates. We complement our theoretical studies by numerical experiments, confirming that both methods yield low entrywise errors for estimating the underlying scores. Our theory is established via a novel leave-one-out trick, which proves effective for analyzing both iterative and non-iterative procedures. Along the way, we derive an elementary eigenvector perturbation bound for probability transition matrices, which parallels the Davis-Kahan $sinTheta$ theorem for symmetric matrices. This also allows us to close the gap between the $ell_2$ error upper bound for the spectral method and the minimax lower limit.
Learning latent variable models with stochastic variational inference is challenging when the approximate posterior is far from the true posterior, due to high variance in the gradient estimates. We propose a novel rejection sampling step that discards samples from the variational posterior which are assigned low likelihoods by the model. Our approach provides an arbitrarily accurate approximation of the true posterior at the expense of extra computation. Using a new gradient estimator for the resulting unnormalized proposal distribution, we achieve average improvements of 3.71 nats and 0.21 nats over state-of-the-art single-sample and multi-sample alternatives respectively for estimating marginal log-likelihoods using sigmoid belief networks on the MNIST dataset.
The Quadratic Assignment Problem (QAP) is a well-known permutation-based combinatorial optimization problem with real applications in industrial and logistics environments. Motivated by the challenge that this NP-hard problem represents, it has captured the attention of the optimization community for decades. As a result, a large number of algorithms have been proposed to tackle this problem. Among these, exact methods are only able to solve instances of size $n<40$. To overcome this limitation, many metaheuristic methods have been applied to the QAP. In this work, we follow this direction by approaching the QAP through Estimation of Distribution Algorithms (EDAs). Particularly, a non-parametric distance-based exponential probabilistic model is used. Based on the analysis of the characteristics of the QAP, and previous work in the area, we introduce Kernels of Mallows Model under the Hamming distance to the context of EDAs. Conducted experiments point out that the performance of the proposed algorithm in the QAP is superior to (i) the classical EDAs adapted to deal with the QAP, and also (ii) to the specific EDAs proposed in the literature to deal with permutation problems.
We study the problem of sampling from the power posterior distribution in Bayesian Gaussian mixture models, a robust version of the classical posterior. This power posterior is known to be non-log-concave and multi-modal, which leads to exponential mixing times for some standard MCMC algorithms. We introduce and study the Reflected Metropolis-Hastings Random Walk (RMRW) algorithm for sampling. For symmetric two-component Gaussian mixtures, we prove that its mixing time is bounded as $d^{1.5}(d + Vert theta_{0} Vert^2)^{4.5}$ as long as the sample size $n$ is of the order $d (d + Vert theta_{0} Vert^2)$. Notably, this result requires no conditions on the separation of the two means. En route to proving this bound, we establish some new results of possible independent interest that allow for combining Poincar{e} inequalities for conditional and marginal densities.
suggested questions
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
