Differential Privacy for Euclidean Jordan Algebra with Applications to Private Symmetric Cone Programming
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908549751242752 |
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| author | Song, Zhao Xue, Jianfei Zhang, Lichen |
| author_facet | Song, Zhao Xue, Jianfei Zhang, Lichen |
| contents | In this paper, we study differentially private mechanisms for functions whose outputs lie in a Euclidean Jordan algebra. Euclidean Jordan algebras capture many important mathematical structures and form the foundation of linear programming, second-order cone programming, and semidefinite programming. Our main contribution is a generic Gaussian mechanism for such functions, with sensitivity measured in $\ell_2$, $\ell_1$, and $\ell_\infty$ norms. Notably, this framework includes the important case where the function outputs are symmetric matrices, and sensitivity is measured in the Frobenius, nuclear, or spectral norm. We further derive private algorithms for solving symmetric cone programs under various settings, using a combination of the multiplicative weights update method and our generic Gaussian mechanism. As an application, we present differentially private algorithms for semidefinite programming, resolving a major open question posed by [Hsu, Roth, Roughgarden, and Ullman, ICALP 2014]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_16915 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Differential Privacy for Euclidean Jordan Algebra with Applications to Private Symmetric Cone Programming Song, Zhao Xue, Jianfei Zhang, Lichen Optimization and Control Cryptography and Security Data Structures and Algorithms Machine Learning In this paper, we study differentially private mechanisms for functions whose outputs lie in a Euclidean Jordan algebra. Euclidean Jordan algebras capture many important mathematical structures and form the foundation of linear programming, second-order cone programming, and semidefinite programming. Our main contribution is a generic Gaussian mechanism for such functions, with sensitivity measured in $\ell_2$, $\ell_1$, and $\ell_\infty$ norms. Notably, this framework includes the important case where the function outputs are symmetric matrices, and sensitivity is measured in the Frobenius, nuclear, or spectral norm. We further derive private algorithms for solving symmetric cone programs under various settings, using a combination of the multiplicative weights update method and our generic Gaussian mechanism. As an application, we present differentially private algorithms for semidefinite programming, resolving a major open question posed by [Hsu, Roth, Roughgarden, and Ullman, ICALP 2014]. |
| title | Differential Privacy for Euclidean Jordan Algebra with Applications to Private Symmetric Cone Programming |
| topic | Optimization and Control Cryptography and Security Data Structures and Algorithms Machine Learning |
| url | https://arxiv.org/abs/2509.16915 |