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).

翻译：局部差分隐私(LDP)机制的结构可通过对应隐私多面体的几何性质来理解。尽管(ε,0)-LDP多面体的极值点已得到充分刻画(Kairouz等人，2014；Holohan等人，2017；Pensia等人，2017)，但对于δ>0的(ε,δ)-LDP多面体的研究相对较少。近期工作(Elangovan和Jog，2024)表明，即使在ε=0的特殊情形下，(0,δ)-LDP隐私多面体也表现出根本不同的行为。本文针对低输入字母表情形k=2与k=3且输出字母表规模m任意的情况，给出了极值点的完整刻画。同时，我们识别出更大输入字母表规模k下新型星型配置结构的极值机制，该结构由Elangovan和Jog(2024)首次提出。