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

arXiv:2606.09161v1 Announce Type: new 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).