We introduce efficient differentially private (DP) algorithms for several linear algebraic tasks, including solving linear equalities over arbitrary fields, linear inequalities over the reals, and computing affine spans and convex hulls. As an application, we obtain efficient DP algorithms for learning halfspaces and affine subspaces. Our algorithms addressing equalities are strongly polynomial, whereas those addressing inequalities are weakly polynomial. Furthermore, this distinction is inevitable: no DP algorithm for linear programming can be strongly polynomial-time efficient.

翻译：我们针对若干线性代数任务提出了高效的差分隐私（DP）算法，包括在任意域上求解线性等式、在实数域上求解线性不等式，以及计算仿射张成空间与凸包。作为应用，我们获得了学习半空间与仿射子空间的高效DP算法。处理等式问题的算法具有强多项式时间复杂度，而处理不等式问题的算法仅具有弱多项式时间复杂度。此外，这一区分具有必然性：任何用于线性规划的DP算法均无法实现强多项式时间效率。