Title: On Differential Private $\ell_1$, $\ell_2$ and $\ell_p^p$ Distance Queries
Primary area: alignment, fairness, safety, privacy, and societal considerations
Keywords: Differential Privacy, Kernel Density Estimation, Distance Query, Data Structure, Balanced Binary Tree

Abstract: We introduce a refined differentially private (DP) data structure for kernel density estimation (KDE) with $\ell_1$, $\ell_2$ and $\ell_p^p$ kernels. 
This new DP data structure offers not only improved privacy-utility tradeoff but also better query 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 $\alpha^{-1}d \log^2 n$, 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 use the same space and pre-processing time, improve the best previous result [Backurs, Lin, Mahabadi, Silwal, and Tarnawski, ICLR 2024] in three aspects
- We reduce query time by $\alpha^{-1} \log n$ factor
- 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.