Randomness lower bound beyond the high-privacy regime

Extend the known lower bound of \(\Omega(\log d)\) on the randomness complexity of differentially private mechanisms answering \(d\) linear queries from the regime \(\varepsilon\leq 1/d\) to larger values of the privacy parameter \(\varepsilon\).

Background

The paper establishes an $O(\log d)$-randomness mechanism for answering $d$ linear queries with near-optimal error. Prior work shows that $\Omega(\log d)$ random bits are necessary only in the high-privacy regime where $\varepsilon\leq 1/d$. The authors explicitly identify extending this lower bound to larger $\varepsilon$ as unresolved, leaving open whether the logarithmic randomness requirement remains necessary outside that regime.