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Optimality of Matrix Mechanism on $\ell_p^p$-metric

Jingcheng Liu, Jalaj Upadhyay, Zongrui Zou

TL;DR

This work defines the $\ell_p^p$-error metric forprivately answering linear queries and proves that the matrix mechanism, via a factorization $A=LR$, achieves near-optimal accuracy up to polylog factors for all $p\ge 2$. It derives a general lower bound on $\mathsf{err}_{\ell_p^p}$ in terms of the generalized factorization norm $\gamma_{(p)}(A)$, and connects this to a geometric quantity $\Lambda_p(A)$ and width parameter $\kappa(A)$ to extend to arbitrary DP settings. The paper also provides tight bounds for prefix-sum and parity queries, showing optimality (up to log factors) of the matrix mechanism in these canonical workloads. The results generalize prior $p=2$ instances and offer a principled, norm-based framework to understand private query workloads with higher-moment error metrics, informing both theory and mechanism design in differential privacy. Overall, it establishes fundamental limits and near-optimal constructions for private linear queries under the $\ell_p^p$-error metric, with implications for private continual counting and parity-based hardness results.

Abstract

In this paper, we introduce the $\ell_p^p$-error metric (for $p \geq 2$) when answering linear queries under the constraint of differential privacy. We characterize such an error under $(ε,δ)$-differential privacy. Before this paper, tight characterization in the hardness of privately answering linear queries was known under $\ell_2^2$-error metric (Edmonds et al., STOC 2020) and $\ell_p^2$-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 $p$ in terms of the $\ell_p^p$ error, generalizing the bounds in Henzinger et al. (SODA 2023) for $p=2$.

Optimality of Matrix Mechanism on $\ell_p^p$-metric

TL;DR

This work defines the -error metric forprivately answering linear queries and proves that the matrix mechanism, via a factorization , achieves near-optimal accuracy up to polylog factors for all . It derives a general lower bound on in terms of the generalized factorization norm , and connects this to a geometric quantity and width parameter to extend to arbitrary DP settings. The paper also provides tight bounds for prefix-sum and parity queries, showing optimality (up to log factors) of the matrix mechanism in these canonical workloads. The results generalize prior instances and offer a principled, norm-based framework to understand private query workloads with higher-moment error metrics, informing both theory and mechanism design in differential privacy. Overall, it establishes fundamental limits and near-optimal constructions for private linear queries under the -error metric, with implications for private continual counting and parity-based hardness results.

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 .
Paper Structure (31 sections, 36 theorems, 104 equations, 1 figure)

This paper contains 31 sections, 36 theorems, 104 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Organization of the proof.
  4. Lower Bound for $(\varepsilon,\delta)$-DP and its Application
  5. Lower bound on additive noise mechanisms
  6. Proof of Lg
  7. Lg
  8. Lg
  9. Lg
  10. Discussion and Limitations
  11. Basic Definitions and Preliminaries
  12. Matrix theory and Convex Geometry
  13. Differential privacy.
  14. Error Metric.
  15. High-Level Overview of Our Techniques
  16. ...and 16 more sections

Key Result

Theorem 2

Fix $A \in \mathbb R^{m \times n}$ be a matrix representing $m$ linear queries, and let $\mathcal{M}:\mathbb{R}^n\rightarrow \mathbb{R}^m$ be any $(\varepsilon,\delta)$-DP algorithm. Then, there exists a factorization of $A = LR$ such that $\mathcal{M}_{\mathsf{matrix}}(x) = L(Rx + z)$ with $z\sim \

Figures (1)

  • Figure 1: A geometric intuition of Lemma \ref{['lem:width_of_count']}.

Theorems & Definitions (57)

  • Definition 1
  • Theorem 2: Informal statement of \ref{['thm:lower_bound_DP_intro']} and \ref{['thm:upper_bound']}
  • Theorem 3
  • Theorem 4: Lower bound for $(\varepsilon,\delta)$-DP
  • Theorem 5
  • Theorem 6
  • Theorem 7: Lower bound for additive noise mechanisms
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • ...and 47 more