Differentially Private Kernel Density Estimation
<|reference_start|>Differentially Private Kernel Density Estimation: We introduce a refined differentially private (DP) data structure for kernel density estimation (KDE), offering not only improved privacy-utility tradeoff but also better efficiency over prior results. Specifically, we study the mathematical problem: given a similarity function $f$ (or DP KDE) and a private dataset $X \subset \mathbb{R}^d$, our goal is to preprocess $X$ so that for any query $y\in\mathbb{R}^d$, we approximate $\sum_{x \in X} f(x, y)$ in a differentially private fashion. The best previous algorithm for $f(x,y) =\| x - y \|_1$ is the node-contaminated balanced binary tree by [Backurs, Lin, Mahabadi, Silwal, and Tarnawski, ICLR 2024]. Their algorithm requires $O(nd)$ space and time for preprocessing with $n=|X|$. For any query point, the query time is $d \log n$, with an error guarantee of $(1+\alpha)$-approximation and $\epsilon^{-1} \alpha^{-0.5} d^{1.5} R \log^{1.5} n$.   In this paper, we improve the best previous result [Backurs, Lin, Mahabadi, Silwal, and Tarnawski, ICLR 2024] in three aspects:   - We reduce query time by a factor of $\alpha^{-1} \log n$.   - We improve the approximation ratio from $\alpha$ to 1.   - We reduce the error dependence by a factor of $\alpha^{-0.5}$.   From a technical perspective, our method of constructing the search tree differs from previous work [Backurs, Lin, Mahabadi, Silwal, and Tarnawski, ICLR 2024]. In prior work, for each query, the answer is split into $\alpha^{-1} \log n$ numbers, each derived from the summation of $\log n$ values in interval tree countings. In contrast, we construct the tree differently, splitting the answer into $\log n$ numbers, where each is a smart combination of two distance values, two counting values, and $y$ itself. We believe our tree structure may be of independent interest.<|reference_end|>
@article{liu2024differentially,
    title={Differentially Private Kernel Density Estimation},
    author={Erzhi Liu, Jerry Yao-Chieh Hu, Alex Reneau, Zhao Song, Han Liu},
    journal={arXiv preprint arXiv:2409.01688},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.01688},
    primaryClass={cs.DS cs.AI cs.LG stat.ML}
}