# The Role of Interactivity in Local Differential Privacy

## TL;DR

This paper investigates the power of interactivity in local differential privacy, demonstrating that fully interactive protocols can be more sample-efficient than sequentially interactive ones, but noninteractive tests are often optimal for hypothesis testing.

## Contribution

It classifies locally private protocols by compositionality, provides a transformation from fully to sequentially interactive protocols, and establishes tight bounds on their relative power.

## Key findings

- Fully interactive protocols can be exponentially more sample-efficient than sequential ones.
- Noninteractive hypothesis tests are optimal for a broad class of problems.
- The paper provides tight bounds and transformations between different protocol types.

## Abstract

We study the power of interactivity in local differential privacy. First, we focus on the difference between fully interactive and sequentially interactive protocols. Sequentially interactive protocols may query users adaptively in sequence, but they cannot return to previously queried users. The vast majority of existing lower bounds for local differential privacy apply only to sequentially interactive protocols, and before this paper it was not known whether fully interactive protocols were more powerful. We resolve this question. First, we classify locally private protocols by their compositionality, the multiplicative factor $k \geq 1$ by which the sum of a protocol's single-round privacy parameters exceeds its overall privacy guarantee. We then show how to efficiently transform any fully interactive $k$-compositional protocol into an equivalent sequentially interactive protocol with an $O(k)$ blowup in sample complexity. Next, we show that our reduction is tight by exhibiting a family of problems such that for any $k$, there is a fully interactive $k$-compositional protocol which solves the problem, while no sequentially interactive protocol can solve the problem without at least an $\tilde \Omega(k)$ factor more examples. We then turn our attention to hypothesis testing problems. We show that for a large class of compound hypothesis testing problems --- which include all simple hypothesis testing problems as a special case --- a simple noninteractive test is optimal among the class of all (possibly fully interactive) tests.