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

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 \( (\epsilon,0)\)-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 \((\epsilon,\delta)\)-LDP polytope with \(\delta>0\). Recent work (Elangovan and Jog, 2024) has shown that even in the special case \(\epsilon=0\), the \( (0,\delta) \)-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).