Optimality of Staircase Mechanisms for Vector Queries under Differential Privacy

TL;DR

This paper proves that staircase mechanisms are optimal for vector queries under differential privacy, regardless of dimension, norm, or utility cost, by reducing the problem to a one-dimensional convex family.

Contribution

It establishes that staircase mechanisms are universally optimal for additive vector queries under differential privacy, resolving a longstanding conjecture.

Findings

Staircase mechanisms are optimal for all vector queries under DP.

The optimization reduces to a one-dimensional convex problem.

The result applies to any dimension, norm, and monotone cost function.

Abstract

We study the optimal design of additive mechanisms for vector-valued queries under $\epsilon$-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 $\epsilon$-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 $\epsilon$-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.