Fast John Ellipsoid Computation with Differential Privacy Optimization

2024-08-12

arXiv.org

AI Summary

This work addresses the private computation of the John ellipsoid—the maximum-volume inscribed ellipsoid—in a convex polytope. We propose the first efficient (ε,δ)-differentially private algorithm for this problem. To balance privacy flexibility and geometric approximation accuracy, we introduce the notion of ε₀-proximal datasets and integrate noise injection, randomized sketching, and leverage-score sampling within a differentially private iterative optimization framework. Theoretically, our algorithm converges to a (1+ξ)-approximate optimal solution in Θ(ξ⁻²(log(n/δ₀)+(Lε₀)⁻²)) iterations, with rigorous guarantees on both differential privacy and utility lower bounds. This is the first systematic incorporation of differential privacy into John ellipsoid computation, offering significant implications for privacy-preserving machine learning and optimization over sensitive geometric data.

Technology Category

Machine Learning: PrivacySearch and Optimization: Non-convex OptimizationReasoning under Uncertainty: Stochastic Optimization

Application Category

Security and Privacy: Data transparency and provenanceGraph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsUser Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

Determining the John ellipsoid - the largest volume ellipsoid contained within a convex polytope - is a fundamental problem with applications in machine learning, optimization, and data analytics. Recent work has developed fast algorithms for approximating the John ellipsoid using sketching and leverage score sampling techniques. However, these algorithms do not provide privacy guarantees for sensitive input data. In this paper, we present the first differentially private algorithm for fast John ellipsoid computation. Our method integrates noise perturbation with sketching and leverages score sampling to achieve both efficiency and privacy. We prove that (1) our algorithm provides $(epsilon,delta)$-differential privacy and the privacy guarantee holds for neighboring datasets that are $epsilon_0$-close, allowing flexibility in the privacy definition; (2) our algorithm still converges to a $(1+xi)$-approximation of the optimal John ellipsoid in $Theta(xi^{-2}(log(n/delta_0) + (Lepsilon_0)^{-2}))$ iterations where $n$ is the number of data point, $L$ is the Lipschitz constant, $delta_0$ is the failure probability, and $epsilon_0$ is the closeness of neighboring input datasets. Our theoretical analysis demonstrates the algorithm's convergence and privacy properties, providing a robust approach for balancing utility and privacy in John ellipsoid computation. This is the first differentially private algorithm for fast John ellipsoid computation, opening avenues for future research in privacy-preserving optimization techniques.

Problem

• Differentially private John ellipsoid computation
• Fast approximation with privacy guarantees
• Balancing efficiency and data privacy

Innovation

• Differentially private algorithm integration
• Noise perturbation with sketching
• Leverage score sampling for privacy