Differential Privacy (DP) is a rigorous framework for protecting individual privacy in data analysis and machine learning. Since its introduction by Dwork et al. in 2006, DP has undergone continuous development and has become a central theoretical tool in privacy-preserving research. Notably, differential privacy has advanced not only in academia but also in industry, where leading technology companies such as Apple and Microsoft have adopted DP techniques in their products and services to balance privacy protection with the utility of large-scale data analysis.

In recent years, studies have investigated estimation error under Local Differential Privacy (LDP). Duchi, Jordan, and Wainwright (2018) analyzed the estimation error under ε-LDP for certain statistical tasks, while Duchi and Ruan (2024) extended the results to a more general framework, obtaining the same convergence rate. These works demonstrate that estimation error induced by LDP is non-negligible in statistical inference. However, most existing methods rely on task-specific mechanisms, which restrict their versatility. In particular, the commonly used Noisy-SGD algorithm, while broadly applicable to M-estimation tasks, often requires reallocation of the privacy budget when applied to new analytical tasks, limiting its general applicability.

To address these limitations, this paper proposes a unified and versatile privacy-preserving mechanism that enables consistent parameter estimation and inference under the M-estimation framework, while accommodating non-smooth loss functions. Specifically, the authors design the Zero-Inflated Symmetric Multivariate Laplace (ZIL) distribution as the noise distribution to simplify estimation and inference, and extend ε-LDP to a more general f-LDP framework to characterize the privacy level. Building on this, the paper further introduces a Doubly Random (DR) procedure, which adds symmetric multivariate Laplace noise to the output of the ZIL mechanism to yield an unbiased corrected loss estimator. This method requires no tuning and is applicable to a broad range of non-smooth loss functions, including quantile regression, support vector machine classification, and ReLU neural network models.

The paper establishes the consistency and asymptotic normality of the DR corrected loss estimator, and derives its variance for inference. Compared with the classical Gaussian and Laplace mechanisms, the proposed method avoids integration or differentiation, making it simpler to implement and applicable to a wider range of problems. Furthermore, when the loss functions are restricted to be second-order smooth, the paper proposes a Smoothed Doubly Random (SDR) procedure, whose estimation error is, in a certain sense, comparable to the results of Duchi and Ruan (2024). Finally, the article provides a systematic discussion of the trade-off between privacy protection levels and estimation efficiency for the proposed estimators.