Differentially Private Quasi-Concave Optimization: Bypassing the Lower Bound and Application to Geometric Problems

We study the sample complexity of differentially private optimization of quasi-concave functions. For a fixed input domain X, Cohen et al. (STOC 2023) proved that any generic private optimizer for low sensitive quasi-concave functions must have sample complexity Omega(2log^*|X|). We show that the lower bound can be bypassed for a series of ``natural'' problems. We define a new class of approximated quasi-concave functions, and present a generic differentially private optimizer for approximated quasi-concave functions with sample complexity tildeO(log^*|X|). As applications, we use our optimizer to privately select a center point of points in d dimensions and probably approximately correct (PAC) learn d-dimensional halfspaces. In previous works, Bun et al. (FOCS 2015) proved a lower bound of Omega(log^*|X|) for both problems. Beimel et al. (COLT 2019) and Kaplan et al. (NeurIPS 2020) gave an upper bound of tildeO(d2.5· 2log^*|X|) for the two problems, respectively. We improve the dependency of the upper bounds on the cardinality of the domain by presenting a new upper bound of tildeO(d5.5·log^*|X|) for both problems. To the best of our understanding, this is the first work to reduce the sample complexity dependency on |X| for these two problems from exponential in log^* |X| to log^* |X|.