Subset-Based Instance Optimality in Private Estimation

• Main Authors: Dick, Travis, Kulesza, Alex, Sun, Ziteng, Suresh, Ananda Theertha
• Format: Journal Article
• Language: English
• Published: 01.03.2023
• Subjects: Computer Science - Cryptography and Security; Computer Science - Information Theory; Computer Science - Learning; Mathematics - Information Theory
• DOI: 10.48550/arxiv.2303.01262

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.