Part of Advances in Neural Information Processing Systems 37 (NeurIPS 2024) Main Conference Track
Joel Daniel Andersson, Monika Henzinger, Rasmus Pagh, Teresa Steiner, Jalaj Upadhyay
Differential privacy with gradual expiration models the setting where data items arrive in a stream and at a given time $t$ the privacy loss guaranteed for a data item seen at time $(t-d)$ is $\epsilon g(d)$, where $g$ is a monotonically non-decreasing function. We study the fundamental *continual (binary) counting* problem where each data item consists of a bit and the algorithm needs to output at each time step the sum of all the bits streamed so far. For a stream of length $T$ and privacy *without* expiration continual counting is possible with maximum (over all time steps) additive error $O(\log^2(T)/\varepsilon)$ and the best known lower bound is $\Omega(\log(T)/\varepsilon)$; closing this gap is a challenging open problem. We show that the situation is very different for privacy with gradual expiration by giving upper and lower bounds for a large set of expiration functions $g$. Specifically, our algorithm achieves an additive error of $O(\log(T)/\epsilon)$ for a large set of privacy expiration functions. We also give a lower bound that shows that if $C$ is the additive error of any $\epsilon$-DP algorithm for this problem, then the product of $C$ and the privacy expiration function after $2C$ steps must be $\Omega(\log(T)/\epsilon)$. Our algorithm matches this lower bound as its additive error is $O(\log(T)/\epsilon)$, even when $g(2C) = O(1)$.Our empirical evaluation shows that we achieve a slowly growing privacy loss that has significantly smaller empirical privacy loss for large values of $d$ than a natural baseline algorithm.