Differentially private Riemannian optimization
It addresses privacy-preserving optimization for machine learning problems constrained to manifolds, such as in robotics or computer vision, representing an incremental extension of differential privacy to non-Euclidean settings.
The paper tackles differentially private optimization on Riemannian manifolds by adding noise to Riemannian gradients, achieving privacy guarantees and utility bounds for convex, nonconvex, and Polyak-Łojasiewicz objectives.
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.