On the Sample Complexity of Privately Learning Axis-Aligned Rectangles

We revisit the fundamental problem of learning Axis-Aligned-Rectangles over a finite grid Xdsubseteqℝd with differential privacy. Existing results show that the sample complexity of this problem is at most minleft d·log|X| ;,; d1.5·left(log^*|X| right)1.5right. That is, existing constructions either require sample complexity that grows linearly with log|X|, or else it grows super linearly with the dimension d. We present a novel algorithm that reduces the sample complexity to only tildeOleftd·left(log^*|X|right)1.5right, attaining a dimensionality optimal dependency without requiring the sample complexity to grow with log|X|.The technique used in order to attain this improvement involves the deletion of "exposed" data-points on the go, in a fashion designed to avoid the cost of the adaptive composition theorems. The core of this technique may be of individual interest, introducing a new method for constructing statistically-efficient private algorithms.