# Differential Privacy for Euclidean Jordan Algebra with Applications to Private Symmetric Cone Programming

arXiv: https://arxiv.org/abs/2509.16915 · HTML: https://mlsanity.com/paper/2509.16915/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 34 of its 84 references that are in the map.
- First posted on arXiv 21 Sep 2025.

## Builds on (most cited first, 30 of 34)

- [Deep Learning with Differential Privacy](https://mlsanity.com/paper/1607.00133/) (2016)
- [Renyi Differential Privacy](https://mlsanity.com/paper/1702.07476/) (2017)
- [Differentially Private Federated Learning: A Client Level Perspective](https://mlsanity.com/paper/1712.07557/) (2017)
- [Differential Privacy Has Disparate Impact on Model Accuracy](https://mlsanity.com/paper/1905.12101/) (2019)
- [Subsampled R'enyi Differential Privacy and Analytical Moments Accountant](https://mlsanity.com/paper/1808.00087/) (2018)
- [Differentially Private Learning with Adaptive Clipping](https://mlsanity.com/paper/1905.03871/) (2019)
- [Improving the Gaussian Mechanism for Differential Privacy: Analytical Calibration and Optimal Denoising](https://mlsanity.com/paper/1805.06530/) (2018)
- [LDP-Fed: Federated Learning with Local Differential Privacy](https://mlsanity.com/paper/2006.03637/) (2020)
- [Differential Privacy for Text Analytics via Natural Text Sanitization](https://mlsanity.com/paper/2106.01221/) (2021)
- [Differentially Private Federated Learning on Heterogeneous Data](https://mlsanity.com/paper/2111.09278/) (2021)
- [High-Dimensional Robust Mean Estimation in Nearly-Linear Time](https://mlsanity.com/paper/1811.09380/) (2018)
- [Low Rank Matrix Completion via Robust Alternating Minimization in Nearly Linear Time](https://mlsanity.com/paper/2302.11068/) (2023)
- [Faster Algorithms for High-Dimensional Robust Covariance Estimation](https://mlsanity.com/paper/1906.04661/) (2019)
- [Sketching for First Order Method: Efficient Algorithm for Low-Bandwidth Channel and Vulnerability](https://mlsanity.com/paper/2210.08371/) (2022)
- [Sketching Meets Differential Privacy: Fast Algorithm for Dynamic Kronecker Projection Maintenance](https://mlsanity.com/paper/2210.11542/) (2022)
- [SynTF: Synthetic and Differentially Private Term Frequency Vectors for Privacy-Preserving Text Mining](https://mlsanity.com/paper/1805.00904/) (2018)
- [Adversarially Robust Streaming Algorithms via Differential Privacy](https://mlsanity.com/paper/2004.05975/) (2020)
- [Near-Optimal Discrete Optimization for Experimental Design: A Regret Minimization Approach](https://mlsanity.com/paper/1711.05174/) (2017)
- [Dynamic Algorithms Against an Adaptive Adversary: Generic Constructions and Lower Bounds](https://mlsanity.com/paper/2111.03980/) (2021)
- [Faster Algorithms for Structured Linear and Kernel Support Vector Machines](https://mlsanity.com/paper/2307.07735/) (2023)
- [A Framework for Adversarial Streaming via Differential Privacy and Difference Estimators](https://mlsanity.com/paper/2107.14527/) (2021)
- [Efficient Alternating Minimization with Applications to Weighted Low Rank Approximation](https://mlsanity.com/paper/2306.04169/) (2023)
- [Privately Solving Linear Programs](https://mlsanity.com/paper/1402.3631/) (2014)
- [DPAUC: Differentially Private AUC Computation in Federated Learning](https://mlsanity.com/paper/2208.12294/) (2022)
- [Robust Algorithms on Adaptive Inputs from Bounded Adversaries](https://mlsanity.com/paper/2304.07413/) (2023)
- [Matrix Completion in Almost-Verification Time](https://mlsanity.com/paper/2308.03661/) (2023)
- [Matrix Completion from O(n) Samples in Linear Time](https://mlsanity.com/paper/1702.02267/) (2017)
- [Private Optimization Without Constraint Violations](https://mlsanity.com/paper/2007.01181/) (2020)
- [Differentially Private Convex Optimization with Feasibility Guarantees](https://mlsanity.com/paper/2006.12338/) (2020)
- [A Primal-Dual Framework for Symmetric Cone Programming](https://mlsanity.com/paper/2405.09157/) (2024)

## Abstract (arXiv)

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 ℓ2, ℓ1, and ℓ_∞ 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].