Subset-Based Instance Optimality in Private Estimation

TL;DR

This paper introduces a new, stronger notion of instance optimality for differentially private estimation algorithms, and constructs algorithms that achieve this optimality for various dataset properties, including means and quantiles.

Contribution

It defines a novel, stronger benchmark for private algorithms and demonstrates how to achieve this optimality for multiple estimation tasks.

Findings

Constructed private algorithms that meet the new instance optimality criteria.

Algorithms match or outperform existing methods for mean estimation under various distributions.

The new benchmark is significantly stronger than previous privacy benchmarks.

Abstract

We propose a new definition of instance optimality for differentially private estimation algorithms. Our definition requires an optimal algorithm to compete, simultaneously for every dataset $D$, with the best private benchmark algorithm that (a) knows $D$ in advance and (b) is evaluated by its worst-case performance on large subsets of $D$. That is, the benchmark algorithm need not perform well when potentially extreme points are added to $D$; it only has to handle the removal of a small number of real data points that already exist. This makes our benchmark significantly stronger than those proposed in prior work. We nevertheless show, for real-valued datasets, how to construct private algorithms that achieve our notion of instance optimality when estimating a broad class of dataset properties, including means, quantiles, and $\ell_p$-norm minimizers. For means in particular, we provide a detailed analysis and show that our algorithm simultaneously matches or exceeds the asymptotic performance of existing algorithms under a range of distributional assumptions.