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Item response theory (IRT) is the study of how people make probabilisticdecisions, with diverse applications in education testing, recommendationsystems, among others. The Rasch model of binary response data, one of the mostfundamental models in IRT, remains an active area of research with importantpractical significance. Recently, Nguyen and Zhang (2022) proposed a newspectral estimation algorithm that is efficient and accurate. In this work, weextend their results in two important ways. Firstly, we obtain a refinedentrywise error bound for the spectral algorithm, complementing the `averageerror' $\ell_2$ bound in their work. Notably, under mild sampling conditions,the spectral algorithm achieves the minimax optimal error bound (modulo a logfactor). Building on the refined analysis, we also show that the spectralalgorithm 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 contributionaddresses an important but understudied topic in IRT: privacy. Despite thehuman-centric applications of IRT, there has not been any proposedprivacy-preserving mechanism in the literature. We develop a private extensionof the spectral algorithm, leveraging its unique Markov chain formulation andthe discrete Gaussian mechanism (Canonne et al., 2020). Experiments show thatour approach is significantly more accurate than the baselines in thelow-to-moderate privacy regime.

Optimal and Private Learning from Human Response Data