Synthetic Data Generation for Differential Privacy Using Maximum Weight Matching

Abstract

Differential privacy synthetic data is one of the most effective methods for privacy preserving data release. However, the existing schemes still suffer from high computational complexity and inability to directly handle values of large domain size when synthesizing high-dimensional data. To mitigate this gap, we propose synthetic data generation for differential privacy using maximum weight matching (DPMWM), a method for automatically synthesizing tabular data in high-dimensional large domain size via differential privacy. Specifically, DPMWM uses differential privacy maximum weight matching for low-dimensional marginal selection and then automatically synthesizes multiple records based on the filtered marginals. The experimental results show that DPMWM outperforms the state-of-the-art in terms of accuracy for counting queries and classification tasks on datasets with larger domain size.

A Proof of Lemma

Proof

We assume a dataset D contains n records, and consider two attributes a and b. Denote the frequency of different values of attribute a is $\left\{ a_1,a_2,\ldots \right\}$ and the frequency of b is $\left\{ b_1,b_2,\ldots \right\}$. For 2-way marginal of (a, b), denote its frequency of joint distribution is $\left\{ c_{11},c_{12},\ldots \right\}$.

The metric w(a, b) is

$$\begin{aligned} w(a, b)=\frac{1}{2} \sum _{i j}\left| \frac{a_i b_j}{n}-c_{i j}\right| \end{aligned}$$

If we add a user with value $u$ for $a$ and $v$ for $b$, then

$$ \begin{aligned} w^{\prime }(a, b) & =\frac{1}{2} \sum _{i \ne u, j \ne v}\left| \frac{a_{i} b_{j}}{n+1}-c_{i j}\right| \\ & +\frac{1}{2} \sum _{i \ne u}\left| \frac{a_{i}\left( b_{v}+1\right) }{n+1}-c_{i v}\right| \\ & +\frac{1}{2} \sum _{j \ne v}\left| \frac{\left( a_{u}+1\right) b_{j}}{n+1}-c_{u j}\right| \\ & +\frac{1}{2}\left| \frac{\left( a_{u}+1\right) \left( b_{v}+1\right) }{n+1}-\left( c_{u v}+1\right) \right| \end{aligned} $$

The sensitivity is given by

$$\begin{aligned} \begin{aligned} \varDelta _{w} & =\left| w(a, b)-w^{\prime }(a, b)\right| \\ & \le \frac{1}{2} \sum _{i \ne u, j \ne v}\left| \frac{a_{i} b_{j}}{n(n+1)}\right| +\frac{1}{2} \sum _{i \ne u}\left| \frac{a_{i} b_{v}}{n(n+1)}-\frac{a_{i}}{n+1}\right| \\ & +\frac{1}{2} \sum _{j \ne v}\left| \frac{a_{u}b_j}{n(n+1)}-\frac{b_{j}}{n+1}\right| +\frac{1}{2}\left| \frac{(n+1)a_{u} b_{v}-n\left( a_{u}+1\right) (b_{v}+1)+n(n+1)}{n(n+1)}\right| \\ & =\frac{\frac{1}{2} \sum _{i \ne u,j \ne v} a_{i} b_{j}-\frac{1}{2} \sum _{i \ne u}\left( a_{i} b_{v}-na_{i}\right) -\frac{1}{2} \sum _{j \ne v}\left( a_{u} b_{j}-n b_{j}\right) }{n(n+1)} \\ & + \frac{(n+1) a_{u} b_{v}-n\left( a_{u}+1\right) \left( b_{v}+1\right) +n(n+1)}{2 n(n+1)} \\ & =\frac{(n-a_u)(n-b_v)-(n-a_u)(b_v-n)-(a_u-n)(n-b_v)+(n-a_u)(n-b_v)}{2 n(n+1)} \\ & =\frac{4(n-a_u) \cdot (n-b_v)}{2 n(n+1)} \\ & =\frac{2\left( n-a_{u}\right) \left( n-b_{v}\right) }{n(n+1)} \le 2 \\ \end{aligned} \end{aligned}$$

(8)

For the above formula, some details are $\sum _{i \ne u} a_i=n-a_u$, $\sum _{j \ne v}b_j=n-b_v$ and $\sum _{i \ne u,j \ne v} a_{i} b_{j}=(n-a_u)(n-b_v)$.