Solving Linear Programs with Differential Privacy

TL;DR

This paper develops differentially private algorithms for solving linear programs, providing improved bounds on constraint violations for both homogeneous and general LPs, advancing privacy-preserving optimization methods.

Contribution

It introduces new efficient differentially private algorithms for linear programming that improve upon previous bounds for constraint violations, applicable to both homogeneous and general LPs.

Findings

For homogeneous LPs, finds solutions violating fewer constraints than prior bounds.

For general LPs, drops significantly more constraints while maintaining privacy.

Builds upon and refines existing private algorithms for LPs with novel techniques.

Abstract

We study the problem of solving linear programs of the form $Ax\le b$, $x\ge0$ with differential privacy. For homogeneous LPs $Ax\ge0$, we give an efficient $(\epsilon,\delta)$-differentially private algorithm which with probability at least $1-\beta$ finds in polynomial time a solution that satisfies all but $O(\frac{d^{2}}{\epsilon}\log^{2}\frac{d}{\delta\beta}\sqrt{\log\frac{1}{\rho_{0}}})$ constraints, for problems with margin $\rho_{0}>0$. This improves the bound of $O(\frac{d^{5}}{\epsilon}\log^{1.5}\frac{1}{\rho_{0}}\mathrm{poly}\log(d,\frac{1}{\delta},\frac{1}{\beta}))$ by [Kaplan-Mansour-Moran-Stemmer-Tur, STOC '25]. For general LPs $Ax\le b$, $x\ge0$ with potentially zero margin, we give an efficient $(\epsilon,\delta)$-differentially private algorithm that w.h.p drops $O(\frac{d^{4}}{\epsilon}\log^{2.5}\frac{d}{\delta}\sqrt{\log dU})$ constraints, where $U$ is an upper bound for the entries of $A$ and $b$ in absolute value. This improves the result by Kaplan et al. by at least a factor of $d^{5}$. Our techniques build upon privatizing a rescaling perceptron algorithm by [Hoberg-Rothvoss, IPCO '17] and a more refined iterative procedure for identifying equality constraints by Kaplan et al.