Continual counting with gradual privacy expiration

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 εg(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(log2 (T)/ε) and the best known lower bound is Ω(log(T)/ε); 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)/ε) 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 ε-DP algorithm for this problem, then the product of C and the privacy expiration function after 2C steps must be Ω(log(T)/ε). Our algorithm matches this lower bound as its additive error is O(log(T)/ε), even when g(2C) = O(1). Our empirical evaluation shows that we achieve a slowly growing privacy loss with significantly smaller empirical privacy loss for large values of d than a natural baseline algorithm.