Truthful and privacy-preserving generalized linear models

This paper explores estimating Generalized Linear Models (GLMs) when agents are strategic and privacy-conscious. We aim to design mechanisms that encourage truthful reporting, protect privacy, and ensure outputs are close to the true parameters. Initially, we address models with sub-Gaussian covariates and heavy-tailed responses with finite fourth moments, proposing a novel private, closed-form estimator. Our mechanism features: (1) $o\left(1\right)$-joint differential privacy with high probability; (2) $o\left(\frac{1}{n}\right)$-approximate Bayes Nash equilibrium for $(1-o(1\left)\right)$-fraction of agents; (3) $o\left(1\right)$ error in parameter estimation; (4) individual rationality for $(1-o(1\left)\right)$ of agents; (5) $o\left(1\right)$ payment budget. We then extend our approach to linear regression with heavy-tailed data, using an ${\ell }_4$-norm shrinkage operator to propose a similar estimator and payment scheme.