Interactive Fingerprinting Codes and the Hardness of Preventing False Discovery

TL;DR

This paper establishes a tight computational hardness bound on answering adaptively chosen statistical queries with limited samples, linking the problem to interactive fingerprinting codes and advancing understanding of privacy and query-answering limitations.

Contribution

It introduces a new Fourier-analytic method for analyzing fingerprinting codes and proves a tight hardness bound for answering statistical queries efficiently, connecting privacy, complexity, and coding theory.

Findings

No efficient algorithm can answer more than O(n^2) adaptive queries with n samples under standard assumptions.

A new Fourier-analytic approach simplifies and improves fingerprinting code analysis.

The results imply that answering arbitrary adaptive queries efficiently is as hard as ensuring differential privacy.

Abstract

We show an essentially tight bound on the number of adaptively chosen statistical queries that a computationally efficient algorithm can answer accurately given $n$ samples from an unknown distribution. A statistical query asks for the expectation of a predicate over the underlying distribution, and an answer to a statistical query is accurate if it is "close" to the correct expectation over the distribution. This question was recently studied by Dwork et al., who showed how to answer $\tilde{\Omega}(n^2)$ queries efficiently, and also by Hardt and Ullman, who showed that answering $\tilde{O}(n^3)$ queries is hard. We close the gap between the two bounds and show that, under a standard hardness assumption, there is no computationally efficient algorithm that, given $n$ samples from an unknown distribution, can give valid answers to $O(n^2)$ adaptively chosen statistical queries. An implication of our results is that computationally efficient algorithms for answering arbitrary, adaptively chosen statistical queries may as well be differentially private. We obtain our results using a new connection between the problem of answering adaptively chosen statistical queries and a combinatorial object called an interactive fingerprinting code. In order to optimize our hardness result, we give a new Fourier-analytic approach to analyzing fingerprinting codes that is simpler, more flexible, and yields better parameters than previous constructions.