2022/05/19 by Andi Han, Han, Andi, Bamdev Mishra +5
Computer Science · Mathematics · #Cryptography and Security (cs.CR) #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Optimization and Control (math.OC) #Point processes and geometric inequalities #Statistical Methods and Inference #Stochastic Gradient Optimization Techniques
paper · pdf · doi:10.48550/arxiv.2205.09494
openalex publication_date 2022/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.