The Power of Factorization Mechanisms in Local and Central Differential Privacy

TL;DR

This paper provides new characterizations of the sample complexity for answering linear queries under differential privacy in both local and central models, highlighting the role of factorization mechanisms and norms.

Contribution

It offers the first approximate characterization of sample complexity in the non-interactive local model and tighter, simpler bounds in the central model, advancing understanding of privacy mechanisms.

Findings

Sample complexity bounds are tight up to polylogarithmic factors.

Factorization norms of query matrices determine optimal sample complexity.

Results apply to both empirical and population estimation.

Abstract

We give new characterizations of the sample complexity of answering linear queries (statistical queries) in the local and central models of differential privacy: *In the non-interactive local model, we give the first approximate characterization of the sample complexity. Informally our bounds are tight to within polylogarithmic factors in the number of queries and desired accuracy. Our characterization extends to agnostic learning in the local model. *In the central model, we give a characterization of the sample complexity in the high-accuracy regime that is analogous to that of Nikolov, Talwar, and Zhang (STOC 2013), but is both quantitatively tighter and has a dramatically simpler proof. Our lower bounds apply equally to the empirical and population estimation problems. In both cases, our characterizations show that a particular factorization mechanism is approximately optimal, and the optimal sample complexity is bounded from above and below by well studied factorization norms of a matrix associated with the queries.