Private Learning of Halfspaces: Simplifying the Construction and Reducing the Sample Complexity

Abstract: We present a differentially private learner for halfspaces over a finite grid $G$ in $\mathbb{R}<sup>d$ with sample complexity $$\approx d^{2.5}\cdot 2<sup>{\log<sup>*|G|}$$, which improves the state-of-the-art result of [Beimel et al., COLT 2019] by a $d<sup>2$ factor. The building block for our learner is a new differentially private algorithm for approximately solving the linear feasibility problem: Given a feasible collection of $m$ linear constraints of the form $Ax\geq b$, the task is to privately identify a solution $x$ that satisfies most of the constraints. Our algorithm is iterative, where each iteration determines the next coordinate of the constructed solution $x$.