Versatile differentially private learning for general loss functions

This paper aims to provide a versatile privacy-preserving release mechanism along with a unified approach for subsequent parameter estimation and statistical inference. We propose a privacy mechanism based on zero-inflated symmetric multivariate Laplace (ZIL) noise, which requires no prior specification of subsequent analysis tasks, allows for general loss functions under minimal conditions, imposes no limit on the number of analyses, and is adaptable to increasing data volume in online scenarios. We derive the trade-off function for the proposed ZIL mechanism, which characterizes its privacy protection level. Furthermore, to formalize the local differential privacy (LDP) property of the ZIL mechanism, we extend the classical $\varepsilon$-LDP to a more general $f$-LDP framework. To address scenarios where only individual attribute values require protection, we propose attribute-level differential privacy (ADP) and its local version. Within the M-estimation framework, we introduce a novel doubly random (DR) corrected loss for the ZIL mechanism, which yields consistent and asymptotically normal M-estimates under differential privacy constraints. The proposed approach is computationally efficient and does not require numerical integration or differentiation for noisy data. It applies to a broad class of loss functions, including non-smooth ones. Two alternative estimators for smooth loss are also proposed with asymptotic properties. The cost of privacy in terms of estimation efficiency for these three estimators is evaluated both theoretically and numerically.