为欧几里得乔丹代数上的函数设计差分隐私机制，可私有化半定规划等优化问题。
Differential Privacy for Euclidean Jordan Algebra with Applications to Private Symmetric Cone Programming
- 基于高斯机制，在ℓ₂、ℓ₁、ℓ∞范数下测量敏感度
- 首次实现对称锥规划的差分隐私算法，解决2014年遗留难题
- 适用于矩阵输出场景，支持弗罗贝尼乌斯、核范数等敏感度度量
本文研究输出位于欧几里得乔丹代数上的函数的差分隐私机制。该代数涵盖线性规划、二阶锥规划和半定规划等重要数学结构。核心贡献是一个通用的高斯机制，敏感度在ℓ₂、ℓ₁、ℓ∞范数下定义。特别地，该框架包含输出为对称矩阵的情形，敏感度可采用弗罗贝尼乌斯、核范数或谱范数衡量。结合乘法权重更新法与该高斯机制，我们推导出在多种设置下求解对称锥规划的私有算法。作为应用，给出了半定规划的差分隐私算法，解决了Hsu等人（ICALP 2014）提出的一个重大开放问题。
原文摘要 · Abstract (English)
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].
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