Abstract:
We revisit the fundamental problem of learning Axis-Aligned-Rectangles over a finite grid Xd⊆Rd with differential privacy. Existing results show that the sample complexity of this problem is at most min{d⋅log|X|,d1.5⋅(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.
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