Õptimal Differentially Private Learning of Thresholds and Quasi-Concave Optimization
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The problem of learning threshold functions is a fundamental one in machine learning. Classical learning theory implies sample complexity of O(ξ^-1log(1/β)) (for generalization error ξ with confidence 1-β). The private version of the problem, however, is more challenging and in particular, the sample complexity must depend on the size |X| of the domain. Progress on quantifying this dependence, via lower and upper bounds, was made in a line of works over the past decade. In this paper, we finally close the gap for approximate-DP and provide a nearly tight upper bound of Õ(log^* |X|), which matches a lower bound by Alon et al (that applies even with improper learning) and improves over a prior upper bound of Õ((log^* |X|)^1.5) by Kaplan et al. We also provide matching upper and lower bounds of Θ̃(2^log^*|X|) for the additive error of private quasi-concave optimization (a related and more general problem). Our improvement is achieved via the novel Reorder-Slice-Compute paradigm for private data analysis which we believe will have further applications.
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