Fast John Ellipsoid Computation with Differential Privacy Optimization

TL;DR

This work introduces the first differentially private algorithm for fast John Ellipsoid computation by integrating truncated Gaussian noise with sketching and weighted leverage-score sampling. The method achieves $(\epsilon,\delta)$-DP under an $\epsilon_0$-close neighborhood definition for neighboring polytopes and converges to a $(1+\xi)$-approximation of the optimal JE within $T=\Theta(\xi^{-2}(\log(n/\delta_0)+(L\epsilon_0)^{-2}))$ iterations, while running in $O((\mathrm{nnz}(A)+d^\omega)T)$ time. A Lipschitz analysis of Lewis weights under $\epsilon_0$ perturbations underpins the privacy guarantees, and the framework offers a robust approach for balancing utility and privacy in privacy-preserving JE computations. Overall, the paper bridges fast JE optimization with formal differential privacy, enabling privacy-aware geometric computations in ML and optimization contexts.

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 $(ε,δ)$-differential privacy and the privacy guarantee holds for neighboring datasets that are $ε_0$-close, allowing flexibility in the privacy definition; (2) our algorithm still converges to a $(1+ξ)$-approximation of the optimal John ellipsoid in $Θ(ξ^{-2}(\log(n/δ_0) + (Lε_0)^{-2}))$ iterations where $n$ is the number of data point, $L$ is the Lipschitz constant, $δ_0$ is the failure probability, and $ε_0$ 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.

Theorems & Definitions (86)

• Definition 1.1: Symmetric convex polytope, Definition 4.1 in syyz22
• Definition 1.2: Neighboring polytopes
• Definition 1.3: Differential privacy
• Theorem 1.4: Main Results, informal version of Theorem \ref{['thm:main_informal']}
• Definition 3.1: Primal program of JE computation
• Definition 3.2: Dual program of JE computation
• Lemma 3.3: Optimal solution, Proposition 2.5 in todd16
• Definition 3.4: $(1+\xi)$-approximate John Ellipsoid, Definition 4.3 in syyz22
• Lemma 3.5: $(1+\xi)$-approximation is good rounding, Lemma 3.5 in syyz22
• Remark 3.6