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Optimality of Staircase Mechanisms for Vector Queries under Differential Privacy

James Melbourne, Mario Diaz, Shahab Asoodeh

TL;DR

The paper resolves the vector-query optimal noise design problem under ε-DP by connecting differential privacy to convex rearrangement theory. It shows that, among all additive ε-DP mechanisms, there exists an optimal staircase noise distribution for any dimension, norm, and norm-monotone cost, by reducing to a one-dimensional radial profile and characterizing extreme points. The main contributions are a general optimality proof for staircase mechanisms, a geometric explanation for their extremal role, and an efficient sampling method that enables practical performance comparisons against Laplace noise. This work provides a unifying principle for noise design in vector DP and informs practical mechanism construction with provable optimality guarantees across broad settings.

Abstract

We study the optimal design of additive mechanisms for vector-valued queries under $ε$-differential privacy (DP). Given only the sensitivity of a query and a norm-monotone cost function measuring utility loss, we ask which noise distribution minimizes expected cost among all additive $ε$-DP mechanisms. Using convex rearrangement theory, we show that this infinite-dimensional optimization problem admits a reduction to a one-dimensional compact and convex family of radially symmetric distributions whose extreme points are the staircase distributions. As a consequence, we prove that for any dimension, any norm, and any norm-monotone cost function, there exists an $ε$-DP staircase mechanism that is optimal among all additive mechanisms. This result resolves a conjecture of Geng, Kairouz, Oh, and Viswanath, and provides a geometric explanation for the emergence of staircase mechanisms as extremal solutions in differential privacy.

Optimality of Staircase Mechanisms for Vector Queries under Differential Privacy

TL;DR

The paper resolves the vector-query optimal noise design problem under ε-DP by connecting differential privacy to convex rearrangement theory. It shows that, among all additive ε-DP mechanisms, there exists an optimal staircase noise distribution for any dimension, norm, and norm-monotone cost, by reducing to a one-dimensional radial profile and characterizing extreme points. The main contributions are a general optimality proof for staircase mechanisms, a geometric explanation for their extremal role, and an efficient sampling method that enables practical performance comparisons against Laplace noise. This work provides a unifying principle for noise design in vector DP and informs practical mechanism construction with provable optimality guarantees across broad settings.

Abstract

We study the optimal design of additive mechanisms for vector-valued queries under -differential privacy (DP). Given only the sensitivity of a query and a norm-monotone cost function measuring utility loss, we ask which noise distribution minimizes expected cost among all additive -DP mechanisms. Using convex rearrangement theory, we show that this infinite-dimensional optimization problem admits a reduction to a one-dimensional compact and convex family of radially symmetric distributions whose extreme points are the staircase distributions. As a consequence, we prove that for any dimension, any norm, and any norm-monotone cost function, there exists an -DP staircase mechanism that is optimal among all additive mechanisms. This result resolves a conjecture of Geng, Kairouz, Oh, and Viswanath, and provides a geometric explanation for the emergence of staircase mechanisms as extremal solutions in differential privacy.
Paper Structure (29 sections, 25 theorems, 164 equations, 2 figures, 1 algorithm)

This paper contains 29 sections, 25 theorems, 164 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. More Related Work
  3. Paper Organization
  4. Notation
  5. Basic Definitions and Preliminaries
  6. Differential Privacy
  7. Radially Symmetric Decreasing Rearrangements
  8. Stochastic Domination
  9. From Rearrengement to Differential Privacy
  10. Optimality of the Staircase Mechanism
  11. Efficient Sampling from Staircase Mechanism
  12. Analytical Properties of the Staircase Density
  13. Two-Stage Sampling Procedure
  14. Numerical Evaluation and Comparison
  15. Proof of Theorem \ref{['Thm:DPequivalentEnlarged']}
  16. ...and 14 more sections

Key Result

Theorem 1

Given a norm $\|\cdot\|$ and a norm-monotone cost function $\Phi:{\mathbb R}^n\to{\mathbb R}$, there exists $\gamma^*\in[0,1]$ such that the staircase mechanism $M_{\gamma^*}$ minimizes expected cost among all $\varepsilon$-DP additive mechanisms.

Figures (2)

  • Figure 1: The utility-privacy tradeoff for staircase mechanism and Laplace mechanism for $n = 3$ (left) and $n= 15$ (right) with $\|\cdot\|_1$-sensitivity one, where the utility is measure via the cost function $\Phi(x) = \|x\|_1$. The gray dotted curve shows the optimal $\gamma^*$, the solution of the optimization problem \ref{['eq:Gamma^*']}.
  • Figure 2: The utility-privacy tradeoff for staircase mechanism and Laplace mechanism for $\varepsilon = 8$ (left) and $\varepsilon = 15$ (right) with $\|\cdot\|_1$-sensitivity one, where the utility is measure via the cost function $\Phi(x) = \|x\|_1$. The gray dotted curve shows the optimal $\gamma^*$, the solution of the optimization problem \ref{['eq:Gamma^*']}.

Theorems & Definitions (52)

  • Theorem 1: Informal
  • Definition 1: Differential privacy Dwork_Calibration
  • Definition 2: Staircase mechanism geng2015optimalgeng2015staircase
  • Theorem 2: Geng et al. geng2015staircase
  • Definition 3
  • Definition 4
  • Definition 5
  • Proposition 1
  • Definition 6
  • Theorem 3
  • ...and 42 more