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# Lower Bounds for Differential Privacy Under Continual Observation and Online Threshold Queries

*Proceedings of Thirty Seventh Conference on Learning Theory*, PMLR 247:1200-1222, 2024.

#### Abstract

One of the most basic problems for studying the “price of privacy over time” is the so called {\em private counter problem}, introduced by Dwork et al. (2010) and Chan et al. (2011). In this problem, we aim to track the number of {\em events} that occur over time, while hiding the existence of every single event. More specifically, in every time step $t\in[T]$ we learn (in an online fashion) that $\Delta_t\geq 0$ new events have occurred, and must respond with an estimate $n_t\approx\sum_{j=1}^t \Delta_j$. The privacy requirement is that {\em all of the outputs together}, across all time steps, satisfy {\em event level} differential privacy. The main question here is how our error needs to depend on the total number of time steps $T$ and the total number of events $n$. Dwork et al. (2015) showed an upper bound of $O\left(\log(T)+\log^2(n)\right)$, and Henzinger et al. (2023) showed a lower bound of $\Omega\left( \min\{\log n, \log T\} \right)$. We show a new lower bound of $\Omega\left(\min\{n,\log T\}\right)$, which is tight w.r.t. the dependence on $T$, and is tight in the sparse case where $\log^2 n=O(\log T)$. Our lower bound has the following implications: \begin{itemize} \item We show that our lower bound extends to the {\em online thresholds} problem, where the goal is to privately answer many “quantile queries” when these queries are presented one-by-one. This resolves an open question of Bun et al. (2017). \item Our lower bound implies, for the first time, a separation between the number of mistakes obtainable by a private online learner and a non-private online learner. This partially resolves a COLT’22 open question published by Sanyal and Ramponi. \item Our lower bound also yields the first separation between the standard model of private online learning and a recently proposed relaxed variant of it, called {\em private online prediction}. \end{itemize}