On the Sample Complexity of Privately Learning Axis-Aligned Rectangles

Abstract

We revisit the fundamental problem of learning Axis-Aligned-Rectangles over a finite grid X d ⊆ R d with differential privacy. Existing results show that the sample complexity of this problem is at most min d• log |X| , d 1.5 • (log * |X|) 1.5 . 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 O d• (log * |X|) 1.5 , 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.