Differential privacy and Sublinear time are incompatible sometimes
Jeremiah Blocki, Hendrik Fichtenberger, Elena Grigorescu, Tamalika Mukherjee
TL;DR
This paper investigates the compatibility of differential privacy with sublinear-time computation and shows, via fingerprinting-code-based lower bounds, that the two properties can be inherently incompatible in general. By constructing a problem based on one-way marginals and employing a PR-PPC fingerprinting-code framework augmented with a secret-sharing encoding, the authors establish a separation: there exist DP algorithms running in near-linear time, while sublinear-time DP algorithms cannot achieve nontrivial accuracy without violating DP, up to logarithmic factors. The main technical contribution is a two-pronged lower-bound approach (random-oracle warm-up and secret-sharing encoding) that demonstrates any $(1/3,1/3)$-accurate DP algorithm must read essentially all data, yielding a near-linear time requirement. Overall, the work clarifies fundamental limits on achieving DP under sublinear access models and provides a principled methodology for deriving such lower bounds in the presence of complex access patterns.
Abstract
Differential privacy and sublinear algorithms are both rapidly emerging algorithmic themes in times of big data analysis. Although recent works have shown the existence of differentially private sublinear algorithms for many problems including graph parameter estimation and clustering, little is known regarding hardness results on these algorithms. In this paper, we initiate the study of lower bounds for problems that aim for both differentially-private and sublinear-time algorithms. Our main result is the incompatibility of both the desiderata in the general case. In particular, we prove that a simple problem based on one-way marginals yields both a differentially-private algorithm, as well as a sublinear-time algorithm, but does not admit a ``strictly'' sublinear-time algorithm that is also differentially private.