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(1)-joint differential privacy with high probability; (2) o([Formula presented])-approximate Bayes Nash equilibrium for (1−o(1))-fraction of agents; (3) o(1) error in parameter estimation; (4) individual rationality for (1−o(1)) of agents; (5) o(1) payment budget. We then extend our approach to linear regression with heavy-tailed data, using an ℓ₄-norm shrinkage operator to propose a similar estimator and payment scheme.