On the Sample Complexity of Privately Learning Axis-Aligned Rectangles

TL;DR

This paper introduces a new differentially private learning algorithm for axis-aligned rectangles that significantly reduces sample complexity, especially removing dependence on the size of the data domain, and employs a novel data deletion technique.

Contribution

It presents a novel algorithm that achieves near-optimal sample complexity for private learning of axis-aligned rectangles, improving over previous methods by eliminating the dependence on domain size.

Findings

Achieves sample complexity of  ext{d}( ext{log}^*|X|)^{1.5}

Reduces dependence on |X| in sample complexity

Introduces a new data deletion technique for private algorithms

Abstract

We revisit the fundamental problem of learning Axis-Aligned-Rectangles over a finite grid $X^d\subseteq{\mathbb{R}}^d$ with differential privacy. Existing results show that the sample complexity of this problem is at most $\min\left\{ d{\cdot}\log|X| \;,\; d^{1.5}{\cdot}\left(\log^*|X| \right)^{1.5}\right\}$. 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 $\tilde{O}\left\{d{\cdot}\left(\log^*|X|\right)^{1.5}\right\}$, 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.