Lower Bounds for Differential Privacy Under Continual Observation and Online Threshold Queries
IACR Cryptol. ePrint Arch.(2024)
摘要
One of the most basic problems for studying the "price of privacy over time"
is the so called private counter problem, introduced by Dwork et al. (2010) and
Chan et al. (2010). In this problem, we aim to track the number of events that
occur over time, while hiding the existence of every single event. More
specifically, in every time step t∈[T] we learn (in an online fashion) that
Δ_t≥ 0 new events have occurred, and must respond with an estimate
n_t≈∑_j=1^t Δ_j. The privacy requirement is that all of the
outputs together, across all time steps, satisfy 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(log(T)+log^2(n)), and
Henzinger et al. (2023) showed a lower bound of Ω(min{log n, log
T}). We show a new lower bound of Ω(min{n,log
T}), 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:
∙ We show that our lower bound extends to the "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).
∙ 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.
∙ 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 private online prediction.
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