Private learning of halfspaces: Simplifying the construction & reducing the sample complexity

We present a differentially private learner for halfspaces over a finite grid G in R^d with sample complexity ˜ d^2.5 · 2^{log* |}G^|, which improves the state-of-the-art result of [Beimel et al., COLT 2019] by a d² 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 = 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.