We study instrumental variable regression (IVaR) under differential privacy constraints. Classical IVaR methods (like two-stage least squares regression) rely on solving moment equations that directly use sensitive covariates and instruments, creating significant risks of privacy leakage and posing challenges in designing algorithms that are both statistically efficient and differentially private. We propose a noisy two-state gradient descent algorithm that ensures $ρ$-zero-concentrated differential privacy by injecting carefully calibrated noise into the gradient updates. Our analysis establishes finite-sample convergence rates for the proposed method, showing that the algorithm achieves consistency while preserving privacy. In particular, we derive precise bounds quantifying the trade-off among optimization, privacy, and sampling error. To the best of our knowledge, this is the first work to provide both privacy guarantees and provable convergence rates for instrumental variable regression in linear models. We further validate our theoretical findings with experiments on both synthetic and real datasets, demonstrating that our method offers practical accuracy-privacy trade-offs.

翻译：本研究探讨差分隐私约束下的工具变量回归问题。经典工具变量回归方法（如两阶段最小二乘回归）依赖于求解直接使用敏感协变量和工具变量的矩方程，这存在显著的隐私泄露风险，并为设计兼具统计效率与差分隐私特性的算法带来挑战。我们提出一种噪声双阶段梯度下降算法，通过在梯度更新中注入经精确校准的噪声，确保满足$ρ$-零集中差分隐私。理论分析建立了该方法的有限样本收敛速率，证明算法在保护隐私的同时具有一致性。特别地，我们推导出量化优化误差、隐私损失与抽样误差之间权衡关系的精确界。据我们所知，这是首个为线性模型中的工具变量回归同时提供隐私保证与可证明收敛速率的研究工作。我们进一步通过合成数据与真实数据集的实验验证理论结果，证明该方法能提供实用的精度-隐私权衡。