We propose a general optimization-based framework for computing differentially private M-estimators and a new method for constructing differentially private confidence regions. Firstly, we show that robust statistics can be used in conjunction with noisy gradient descent or noisy Newton methods in order to obtain optimal private estimators with global linear or quadratic convergence, respectively. We establish local and global convergence guarantees, under both local strong convexity and self-concordance, showing that our private estimators converge with high probability to a small neighborhood of the non-private M-estimators. Secondly, we tackle the problem of parametric inference by constructing differentially private estimators of the asymptotic variance of our private M-estimators. This naturally leads to approximate pivotal statistics for constructing confidence regions and conducting hypothesis testing. We demonstrate the effectiveness of a bias correction that leads to enhanced small-sample empirical performance in simulations. We illustrate the benefits of our methods in several numerical examples.

翻译：我们提出了一种通用的基于优化的框架，用于计算差分隐私M估计量，并给出了一种构建差分隐私置信区域的新方法。首先，我们证明，结合稳健统计量与带噪声梯度下降或带噪声牛顿方法，可以分别获得具有全局线性或二次收敛性的最优隐私估计量。我们在局部强凸性和自和谐性条件下建立了局部与全局收敛性保证，表明我们的隐私估计量以高概率收敛到非隐私M估计量的小邻域内。其次，我们通过构建隐私M估计量渐近方差的差分隐私估计量，解决了参数推断问题。这自然引出了用于构建置信区域和进行假设检验的近似枢轴统计量。我们展示了偏差校正的有效性，该校正增强了模拟中的小样本实证性能。最后，通过若干数值示例说明我们方法的优势。