Extreme Points of the $(0,δ)$-LDP Polytope with Small Input Size and Arbitrary Output Sizes

2026-06-08

AI Summary

This work addresses the incomplete understanding of the polyhedral structure of $(0,\delta)$-local differential privacy (LDP) mechanisms for small input alphabets. By integrating tools from convex geometry and linear programming, the authors conduct a geometric analysis of mechanism matrices to fully characterize all extreme points of the $(0,\delta)$-LDP polytope for input sizes $k=2$ and $k=3$. Furthermore, they identify a novel class of star-shaped extreme mechanisms that generalize to larger input domains. The study reveals fundamental geometric distinctions between $(0,\delta)$-LDP and $(\varepsilon,0)$-LDP, thereby deepening the theoretical understanding of the mechanism space under $(0,\delta)$-LDP and laying a foundation for the design of optimal privacy-preserving mechanisms.

Abstract

The structure of locally differentially private (LDP) mechanisms can be understood through the geometry of the corresponding privacy polytope. While the extreme points of the \( (ε,0)\)-LDP polytope are well characterized (Kairouz \emph{et al.}, 2014; Holohan \emph{et al.}, 2017; Pensia \emph{et al.}, 2017), comparatively little is known for the \((ε,δ)\)-LDP polytope with \(δ>0\). Recent work (Elangovan and Jog, 2024) has shown that even in the special case \(ε=0\), the \( (0,δ) \)-LDP privacy polytope exhibits fundamentally different behaviour. In this work, we provide complete characterizations of the extreme points for the low-input-alphabet regime \(k=2\) and \(k=3\) and with arbitrary output alphabet size \(m \). We also identify new extreme mechanisms for larger input alphabet sizes $k$, of the star configuration type, as introduced by Elangovan and Jog (2024).

Problem

• local differential privacy
• privacy polytope
• extreme points
• (0,δ)-LDP
• input alphabet size

Innovation

• local differential privacy
• (0,δ)-LDP
• privacy polytope
• extreme points
• star configuration