论文标题
差异私人利曼尼亚优化
Differentially private Riemannian optimization
论文作者
论文摘要
在本文中，我们研究了差异性的经验风险最小化问题，其中参数被限制为riemannian歧管。我们通过向切线空间的Riemannian梯度添加噪声来介绍一个差异私有Riemannian优化的框架。噪声遵循相对于riemannian度量的固有定义的高斯分布。我们将高斯机制从欧几里得空间调整到与这种广义高斯分布兼容的切线空间。我们表明，与直接在歧管上添加噪声相比，该策略提出了一个简单的分析。我们进一步显示了使用时刻会计技术扩展的拟议差异私人riemannian（随机）梯度下降的隐私保证。此外，我们证明了在地球（强）凸，一般非凸目标以及Riemannian Polyak-olojasiewicz条件下的效用保证。我们显示了所提出的框架在几种应用中的功效。
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-Łojasiewicz condition. We show the efficacy of the proposed framework in several applications.
