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

Part of Advances in Neural Information Processing Systems 33 pre-proceedings (NeurIPS 2020)

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Authors

Haim Kaplan, Yishay Mansour, Uri Stemmer, Eliad Tsfadia

Abstract

We present a differentially private learner for halfspaces over a finite grid $G$ in $\R^d$ with sample complexity $\approx d^{2.5}\cdot 2^{\log^*|G|}$, which improves the state-of-the-art result of [Beimel et al., COLT 2019] by a $d^2$ 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 {\em privately} identify a solution $x$ that satisfies {\em most} of the constraints. Our algorithm is iterative, where each iteration determines the next coordinate of the constructed solution $x$.