Close the efficiency–utility gap for the explicit mechanism

Close the \(O(\log d)\) multiplicative gap between the error of the computationally efficient \(\ell_\infty\)-mechanism, which is \(O(d\log d/\varepsilon)\), and the optimal error bound \(O(d/\varepsilon)\) achieved by the non-efficient mechanism.

Background

The paper gives an information-theoretic mechanism based on a probabilistically constructed multi-scale secluded partition that achieves $O(d/\varepsilon)$ error using $O(\log d)$ random bits. For the $\ell_\infty$-norm, it also constructs an explicit multi-scale secluded partition whose rounding function is efficiently computable. The explicit construction has weaker parameters, causing the resulting efficient mechanism to incur an additional $O(\log d)$ factor in its error. The authors explicitly leave removal of this factor as an open problem.