On Differentially Private Linear Algebra

TL;DR

This paper presents new efficient differentially private algorithms for linear algebra tasks like solving equalities, inequalities, and computing affine spans, with applications to learning halfspaces and affine subspaces.

Contribution

It introduces the first strongly polynomial DP algorithms for equalities and discusses the limitations for inequalities and linear programming.

Findings

Strongly polynomial DP algorithms for equalities

Weakly polynomial DP algorithms for inequalities

No strongly polynomial DP algorithm exists for linear programming

Abstract

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.