\section{Conclusion}
\label{sec-conclusion}

In this paper, we address the challenge of answering KDE queries under LDP, where users privatize their data locally before sending them to an untrusted server. 
To ensure local privacy while preserving high-utility KDE results, we adopt a relaxed definition of LDP based on metrics called mLDP. 
Then, we propose a novel \textsc{mLDP-KDE} framework, which augments an LSH-based sketch method to provide unbiased estimations to KDE queries within a bounded additive error in sublinear time and space with high probability subject to mLDP.
Experimental results on five data sets demonstrate the superiority of \textsc{mLDP-KDE} over existing KDE methods under both LDP and mLDP, especially showing significant advantages on large, high-dimensional data.

A current limitation of \textsc{mLDP-KDE} is that it tends to yield less promising KDE results in the high privacy regime, as indicated in Theorem~\ref{thm-circdep} that \textsc{mLDP-KDE} might not achieve any approximation bound when $\varepsilon = o(\frac{\log(1/\eta)}{\alpha^{2}})$ and evidenced in Section~\ref{sec-exp} that \textsc{mLDP-KDE} provides KDEs with high errors when $\varepsilon \leq 1$.
For future work, we intend to find better privacy and approximation bounds to improve its performance under tighter privacy budgets.
