mlsanityData to 14 Oct 2026

Differentially private Riemannian optimization

arXiv 2205.09494 ↗First posted 19 May 2022math.OC cs.CR cs.LGcited by 12

Not on the map. Why?

Abstract (arXiv)

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-Lojasiewicz condition. We show the efficacy of the proposed framework in several applications.

Key facts. Cited by 12 papers in the map (data through 14 Oct 2026). Builds on 34 of its 88 references that are in the map. First posted on arXiv 19 May 2022.

Builds on

34 of 88 references.

References · 34

  1. 2016Deep Learning with Differential Privacycited by 2,728
  2. 2017Poincar'e Embeddings for Learning Hierarchical Representationscited by 684
  3. 2017Renyi Differential Privacycited by 650
  4. 2016Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Conditioncited by 539
  5. 2019Hyperbolic Graph Convolutional Neural Networkscited by 402
  6. 2018Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometrycited by 265
  7. 2018Subsampled R'enyi Differential Privacy and Analytical Moments Accountantcited by 236
  8. 2019Hyperbolic Image Embeddingscited by 221
  9. 2019Private Stochastic Convex Optimization with Optimal Ratescited by 186
  10. 2018Poincar'e GloVe: Hyperbolic Word Embeddingscited by 156
  11. 2020Private Stochastic Convex Optimization: Optimal Rates in Linear Timecited by 153
  12. 2018Differentially Private Empirical Risk Minimization Revisited: Faster and More Generalcited by 128
22 more
  1. 2013Manopt, a Matlab toolbox for optimization on manifoldscited by 113
  2. 2018Privately Learning High-Dimensional Distributionscited by 104
  3. 2016First-order Methods for Geodesically Convex Optimizationcited by 93
  4. 2020CoinPress: Practical Private Mean and Covariance Estimationcited by 74
  5. 2016Riemannian SVRG: Fast Stochastic Optimization on Riemannian Manifoldscited by 64
  6. 2016Global rates of convergence for nonconvex optimization on manifoldscited by 62
  7. 2017Efficient Private ERM for Smooth Objectivescited by 61
  8. 2021Private Adaptive Gradient Methods for Convex Optimizationcited by 46
  9. 2021Differentially Private Stochastic Optimization: New Results in Convex and Non-Convex Settingscited by 45
  10. 2021On Riemannian Optimization over Positive Definite Matrices with the Bures-Wasserstein Geometrycited by 32
  11. 2019Efficient Privacy-Preserving Stochastic Nonconvex Optimizationcited by 27
  12. 2019A Continuous-time Perspective for Modeling Acceleration in Riemannian Optimizationcited by 25
  13. 2020Dual Policy Distillationcited by 25
  14. 2014A Stochastic PCA and SVD Algorithm with an Exponential Convergence Ratecited by 18
  15. 2021Differential Privacy Over Riemannian Manifoldscited by 16
  16. 2020Differentially Private Accelerated Optimization Algorithmscited by 13
  17. 2015Wishart Mechanism for Differentially Private Principal Components Analysiscited by 9
  18. 2019KNG: The K-Norm Gradient Mechanismcited by 8
  19. 2020Riemannian stochastic recursive momentum method for non-convex optimizationcited by 7
  20. 2018Manifold Structured Predictioncited by 5
  21. 2020Hyperbolic Manifold Regressioncited by 5
  22. 2022Stochastic and Private Nonconvex Outlier-Robust PCAcited by 4

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 3; 2023: 3; 2024: 3; 2025: 1; 2026: 2.

12 papers.

  1. 2023DPZero: Private Fine-Tuning of Language Models without Backpropagationcited by 26
  2. 2023DIFF2: Differential Private Optimization via Gradient Differences for Nonconvex Distributed Learningcited by 10
  3. 2022Shape And Structure Preserving Differential Privacycited by 6
  4. 2023Gaussian Differential Privacy on Riemannian Manifoldscited by 6
  5. 2024Riemannian coordinate descent algorithms on matrix manifoldscited by 4
  6. 2024Riemannian Federated Learning via Averaging Gradient Streamscited by 4
  7. 2022Rieoptax: Riemannian Optimization in JAXcited by 1
  8. 2026Geometric Renyi Differential Privacy: Ricci Curvature Characterized by Heat Diffusion Mechanismscited by 1
  9. 2022Changes from Classical Statistics to Modern Statistics and Data Sciencecited by 0
  10. 2024Gaussian Differentially Private Human Faces Under a Face Radial Curve Representationcited by 0
  11. 2025Differentially Private Geodesic Regressioncited by 0
  12. 2026Locally Private Inference for Riemannian Stochastic Optimizationcited by 0