Continual Counting with Gradual Privacy Expiration
CoRR(2024)
Abstract
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 ϵ 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)/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.
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