Anderson Y. Zhang

Duc Nguyen, Anderson Y. Zhang

Mixture models of Plackett-Luce (PL) -- one of the most fundamental ranking models -- are an active research area of both theoretical and practical significance. Most previously proposed parameter estimation algorithms instantiate the EM algorithm, often with random initialization. However, such an initialization scheme may not yield a good initial estimate and the algorithms require multiple restarts, incurring a large time complexity. As for the EM procedure, while the E-step can be performed efficiently, maximizing the log-likelihood in the M-step is difficult due to the combinatorial nature of the PL likelihood function (Gormley and Murphy 2008). Therefore, previous authors favor algorithms that maximize surrogate likelihood functions (Zhao et al. 2018, 2020). However, the final estimate may deviate from the true maximum likelihood estimate as a consequence. In this paper, we address these known limitations. We propose an initialization algorithm that can provide a provably accurate initial estimate and an EM algorithm that maximizes the true log-likelihood function efficiently. Experiments on both synthetic and real datasets show that our algorithm is competitive in terms of accuracy and speed to baseline algorithms, especially on datasets with a large number of items.

Duc Nguyen, Anderson Y. Zhang

Item response theory (IRT) is the study of how people make probabilistic decisions, with diverse applications in education testing, recommendation systems, among others. The Rasch model of binary response data, one of the most fundamental models in IRT, remains an active area of research with important practical significance. Recently, Nguyen and Zhang (2022) proposed a new spectral estimation algorithm that is efficient and accurate. In this work, we extend their results in two important ways. Firstly, we obtain a refined entrywise error bound for the spectral algorithm, complementing the `average error' $\ell_2$ bound in their work. Notably, under mild sampling conditions, the spectral algorithm achieves the minimax optimal error bound (modulo a log factor). Building on the refined analysis, we also show that the spectral algorithm enjoys optimal sample complexity for top-$K$ recovery (e.g., identifying the best $K$ items from approval/disapproval response data), explaining the empirical findings in the previous work. Our second contribution addresses an important but understudied topic in IRT: privacy. Despite the human-centric applications of IRT, there has not been any proposed privacy-preserving mechanism in the literature. We develop a private extension of the spectral algorithm, leveraging its unique Markov chain formulation and the discrete Gaussian mechanism (Canonne et al., 2020). Experiments show that our approach is significantly more accurate than the baselines in the low-to-moderate privacy regime.