Privately Learning Thresholds: Closing the Exponential Gap

Abstract

We study the sample complexity of learning threshold functions under the constraint of differential privacy. It is assumed that each labeled example in the training data is the information of one individual and we would like to come up with a generalizing hypothesis h while guaranteeing differential privacy for the individuals. Intuitively, this means that any single labeled example in the training data should not have a significant effect on the choice of the hypothesis. This problem has received much attention recently; unlike the non-private case, where the sample complexity is independent of the domain size and just depends on the desired accuracy and confidence, for private learning the sample complexity must depend on the domain size X (even for approximate differential privacy). Alon et al. (STOC 2019) showed a lower bound of Ω(log * |X|) on the sample complexity and Bun et al. (FOCS 2015) presented an approximate-private learner with sample complexity Õ 2 log * |X| . In this work we reduce this gap significantly, almost settling the sample complexity. We first present a new upper bound (algorithm) of Õ (log * |X|) 2 on the sample complexity and then present an improved version with sample complexity Õ (log * |X|) 1.5 .