Learning versus Refutation in Noninteractive Local Differential Privacy

TL;DR

This paper characterizes the sample complexity for agnostic PAC learning in non-interactive local differential privacy, revealing an equivalence between learning and refutation tasks based on the approximate gamma_2 norm.

Contribution

It provides a complete characterization of sample complexity for agnostic PAC learning in non-interactive LDP using the approximate gamma_2 norm, establishing an equivalence with refutation.

Findings

Sample complexity is captured by the approximate gamma_2 norm.

Establishes an equivalence between learning and refutation in the agnostic setting.

Provides a complete characterization for non-interactive LDP protocols.

Abstract

We study two basic statistical tasks in non-interactive local differential privacy (LDP): learning and refutation. Learning requires finding a concept that best fits an unknown target function (from labelled samples drawn from a distribution), whereas refutation requires distinguishing between data distributions that are well-correlated with some concept in the class, versus distributions where the labels are random. Our main result is a complete characterization of the sample complexity of agnostic PAC learning for non-interactive LDP protocols. We show that the optimal sample complexity for any concept class is captured by the approximate $\gamma_2$~norm of a natural matrix associated with the class. Combined with previous work [Edmonds, Nikolov and Ullman, 2019] this gives an equivalence between learning and refutation in the agnostic setting.