Differentially private Riemannian optimization

TL;DR

This paper introduces a novel framework for differentially private Riemannian optimization, adding Gaussian noise to the tangent space gradients, with theoretical privacy and utility guarantees for various objective functions.

Contribution

It extends differential privacy to Riemannian optimization by adapting Gaussian mechanisms to the tangent space, providing a simple analysis and privacy guarantees.

Findings

Effective privacy guarantees using tangent space noise addition.

Utility guarantees for convex, nonconvex, and Polyak-Łojasiewicz conditions.

Demonstrated applicability in multiple Riemannian optimization tasks.

Abstract

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.