Private Counting of Distinct Elements in the Turnstile Model and Extensions

TL;DR

This paper advances private algorithms for counting distinct elements in data streams, introducing a simple method with tight error bounds and analyzing privacy guarantees across different models, addressing open questions in the field.

Contribution

It presents a simple, tight-error private algorithm based on the sparse vector technique and extends lower bounds from item-level to event-level differential privacy.

Findings

A simple sparse vector-based algorithm achieves tight error bounds.

Lower bounds for item-level privacy extend to event-level privacy.

Addresses open questions from prior work on privacy models.

Abstract

Privately counting distinct elements in a stream is a fundamental data analysis problem with many applications in machine learning. In the turnstile model, Jain et al. [NeurIPS2023] initiated the study of this problem parameterized by the maximum flippancy of any element, i.e., the number of times that the count of an element changes from 0 to above 0 or vice versa. They give an item-level $(\epsilon,\delta)$-differentially private algorithm whose additive error is tight with respect to that parameterization. In this work, we show that a very simple algorithm based on the sparse vector technique achieves a tight additive error for item-level $(\epsilon,\delta)$-differential privacy and item-level $\epsilon$-differential privacy with regards to a different parameterization, namely the sum of all flippancies. Our second result is a bound which shows that for a large class of algorithms, including all existing differentially private algorithms for this problem, the lower bound from item-level differential privacy extends to event-level differential privacy. This partially answers an open question by Jain et al. [NeurIPS2023].