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

NeurIPS 2025 · Federated learning and privacy › Differential privacy · approach: Privacy accounting and mechanisms · 0 citations (OpenAlex)

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Gives a generic Gaussian mechanism for Euclidean Jordan algebras (with l1/l2/linf sensitivity) and uses it with multiplicative weights to build differentially private solvers for symmetric cone programs, including SDPs.

Problem

Differentially private algorithms for symmetric cone programs (LP, SOCP, SDP) whose outputs lie in Euclidean Jordan algebras, an area with little prior DP work beyond linear programs.

Approach

Define a Gaussian mechanism on an EJA by sampling a Gaussian vector in R^k and mapping it through an isometry, with sensitivity measured in l2 (and l1/linf via norm inequalities); combine with dense multiplicative weights over constraints (high sensitivity) and an MWU over variables with a noisy/exponential-mechanism dual oracle (low sensitivity).

Key results

• Generic Gaussian mechanism over EJAs gives (epsilon, delta)-DP under l2 sensitivity; l1 and linf cases follow via norm inequalities (linf sensitivity scaled by sqrt(r)).
• High-sensitivity covering SDP solver is epsilon-DP with at most s = O~(r/epsilon) violated constraints and alpha = O(OPT).
• Low-sensitivity constraint SCP solver is (epsilon, delta)-DP with alpha = O~(Delta^{1/2} r^{1/4} k^{1/4} / epsilon^{1/2}), a k^{1/4} factor worse than the LP analogue.

Surprising findings

• Naive perturbation of only eigenvalues with scalar noise does not give differential privacy; the Jordan frame must also be randomized.

Limitations

• Private solvers satisfy constraints only approximately; high-sensitivity setting permits a small number of violated constraints.
• Uses basic Gaussian mechanism composition rather than moment accounting or Renyi DP, leaving tighter trade-offs open.
• Low-sensitivity bound loses a k^{1/4} factor relative to LP due to bounding the Gaussian element by sqrt(k).

Experimental setup

Metrics: privacy parameters (epsilon, delta) constraint violation alpha number of violated constraints s

Baselines: Laplace mechanism private linear programming algorithms (Hsu et al. 2014)

Builds on: Hsu et al. private linear programming (dense MWU) symmetric cone multiplicative weights update Gaussian mechanism exponential mechanism

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