Optimality of Matrix Mechanism on $\ell_p^p$-metric
Jingcheng Liu, Jalaj Upadhyay, Zongrui Zou
TL;DR
This paper introduces the $\,\ell_p^p$-error metric for answering linear queries under differential privacy, providing tight bounds and characterizations that generalize previous results for specific error metrics.
Contribution
It characterizes the optimality of the matrix mechanism under the $\,\ell_p^p$-error metric for all constant $p$, extending prior work on $\,\ell_2^2$ and $\,\ell_p^2$ metrics.
Findings
Provides tight bounds for prefix sum and parity queries under differential privacy.
Generalizes previous bounds from $p=2$ to all constant $p$.
Characterizes the error of the matrix mechanism under the $\,\ell_p^p$-metric.
Abstract
In this paper, we introduce the -error metric (for ) when answering linear queries under the constraint of differential privacy. We characterize such an error under -differential privacy. Before this paper, tight characterization in the hardness of privately answering linear queries was known under -error metric (Edmonds et al., STOC 2020) and -error metric for unbiased mechanisms (Nikolov and Tang, ITCS 2024). As a direct consequence of our results, we give tight bounds on answering prefix sum and parity queries under differential privacy for all constant in terms of the error, generalizing the bounds in Henzinger et al. (SODA 2023) for .
Peer Reviews
Decision·ICLR 2025 Poster
* I like the motivation of interpolating between $\ell_2$ and $\ell_\infty$ error via $\ell_p^p$. * The introduction of the paper is largely clear and easy to follow (perhaps with some background knowledge).
* Although the gentle introduction is clear, once it goes into a more technical part, it becomes significantly harder for me to appreciate the discussion. I take Lines 254-255 "The main technical obstacle of Theorem 2.1 lies in ......" as the punchline of this paper. However, I have little idea what this means in terms of technical novelty. Given my limited time reading, I hope the authors can clarify some of my questions below. * Some discussions in the introduction may be improved (see some o
1. The problem addressed is of fundamental importance. 2. The paper successfully derives asymptotic lower bounds for matrix mechanisms under $\ell_p$ error. 3. It demonstrates sophisticated applications of existing techniques.
1. The paper demonstrates that the matrix mechanism is asymptotically optimal, rather than instance optimal, for the error metric considered. 2. The writing is generally good, but there are parts that lack consistency.
1. Linear query release is a fundamental problem in private data analysis. This article characterizes the error of this task with respect to $\ell_p^p$-metric, contributing to a deeper understanding of the role of differential privacy. 2. The writing is clear. Also, the technical details are well-organized and easy to follow.
1. It appears that the upper bound and lower bound only match in the high privacy regime. In the low privacy regime, where $\varepsilon = \Omega(1)$, the presented upper bound cannot match the lower bound. 2. When $p$ is not a constant, the proposed bounds may not be tight for certain regimes of the parameters. For example, when $n = p$, the upper bound in Theorem 1.5 is $O(\log^{1.5} (p))$ while the lower bound is $\Omega(\log (p))$. However, I think both of the above are not serious issues.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Advanced Differential Geometry Research · Mathematics and Applications