Fast John Ellipsoid Computation with Differential Privacy Optimization

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 $\Theta(\xi^{-2}(\log(n/\delta_0) + (L\epsilon_0)^{-2}))$ 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.