In this paper, we study the differentially private empirical risk minimization problem where the parameter is constrained to a Riemannian manifold. We introduce a framework of differentially private Riemannian optimization by adding noise to the Riemannian gradient on the tangent space. The noise follows a Gaussian distribution intrinsically defined with respect to the Riemannian metric. We adapt the Gaussian mechanism from the Euclidean space to the tangent space compatible to such generalized Gaussian distribution. We show that this strategy presents a simple analysis as compared to directly adding noise on the manifold. We further show privacy guarantees of the proposed differentially private Riemannian (stochastic) gradient descent using an extension of the moments accountant technique. Additionally, we prove utility guarantees under geodesic (strongly) convex, general nonconvex objectives as well as under the Riemannian Polyak-{\L}ojasiewicz condition. We show the efficacy of the proposed framework in several applications.
翻译:在本文中,我们研究了该参数受里曼尼方块限制的、有差别的私人实验风险最小化问题。我们引入了一个有差别的里曼尼私人优化框架,在正切空间增加里曼尼梯度的噪音。噪音沿着与里曼尼度有关固有的高斯分布法。我们把高西安机制从欧几里德空间调整到符合这种普遍高斯分布法的正切空间。我们表明,这一战略提供了简单的分析,与直接在多管上添加噪音相比。我们进一步展示了使用时钟会计技术对拟议的里曼尼个人差异性梯度下降的隐私保障。此外,我们还证明了在地质学(强力)二次曲线下、一般非convex目标以及里曼尼加聚亚克-L} ojasiewicz条件下的效用保障。我们在若干应用中展示了拟议框架的功效。